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A Multi-Agent System for Building Dynamic Ontologies

Ontologies building from text is still a time-consuming task which justifies the growth of Ontology Learning. Our system named Dynamo is designed along this domain but following an original approach based on an adaptive multi-agent architecture. In this paper we present a distributed hierarchical clustering algorithm, core of our approach. It is evaluated and compared to a more conventional centralized algorithm. We also present how it has been improved using a multi-criteria approach. With those results in mind, we discuss the limits of our system and add as perspectives the modifications required to reach a complete ontology building solution.

1. INTRODUCTION Nowadays, it is well established that ontologies are needed for semantic web, knowledge management, B2B... For knowledge management, ontologies are used to annotate documents and to enhance the information retrieval. But building an ontology manually is a slow, tedious, costly, complex and time consuming process. Currently, a real challenge lies in building them automatically or semi-automatically and keeping them up to date. It would mean creating dynamic ontologies [10] and it justifies the emergence of ontology learning techniques [14] [13]. Our research focuses on Dynamo (an acronym of DYNAMic Ontologies), a tool based on an adaptive multi-agent system to construct and maintain an ontology from a domain specific set of texts. Our aim is not to build an exhaustive, general hierarchical ontology but a domain specific one. We propose a semi-automated tool since an external resource is required: the "ontologist". An ontologist is a kind of cognitive engineer, or analyst, who is using information from texts and expert interviews to design ontologies. In the multi-agent field, ontologies generally enable agents to understand each other [12]. They"re sometimes used to ease the ontology building process, in particular for collaborative contexts [3], but they rarely represent the ontology itself [16]. Most works interested in the construction of ontologies [7] propose the refinement of ontologies. This process consists in using an existing ontology and building a new one from it. This approach is different from our approach because Dynamo starts from scratch. Researchers, working on the construction of ontologies from texts, claim that the work to be automated requires external resources such as a dictionary [14], or web access [5]. In our work, we propose an interaction between the ontologist and the system, our external resource lies both in the texts and the ontologist. This paper first presents, in section 2, the big picture of the Dynamo system. In particular the motives that led to its creation and its general architecture. Then, in section 3 we discuss the distributed clustering algorithm used in Dynamo and compare it to a more classic centralized approach. Section 4 is dedicated to some enhancement of the agents behavior that got designed by taking into account criteria ignored by clustering. And finally, in section 5, we discuss the limitations of our approach and explain how it will be addressed in further work. 2. DYNAMO OVERVIEW 2.1 Ontology as a Multi-Agent System Dynamo aims at reducing the need for manual actions in processing the text analysis results and at suggesting a concept network kick-off in order to build ontologies more efficiently. The chosen approach is completely original to our knowledge and uses an adaptive multi-agent system. This choice comes from the qualities offered by multi-agent system: they can ease the interactive design of a system [8] (in our case, a conceptual network), they allow its incremental building by progressively taking into account new data (coming from text analysis and user interaction), and last but not least they can be easily distributed across a computer network. Dynamo takes a syntactical and terminological analysis of texts as input. It uses several criteria based on statistics computed from the linguistic contexts of terms to create and position the concepts. As output, Dynamo provides to the analyst a hierarchical organization of concepts (the multi-agent system itself) that can be validated, refined of modified, until he/she obtains a satisfying state of 1286 978-81-904262-7-5 (RPS) c 2007 IFAAMAS the semantic network. An ontology can be seen as a stable map constituted of conceptual entities, represented here by agents, linked by labelled relations. Thus, our approach considers an ontology as a type of equilibrium between its concept-agents where their forces are defined by their potential relationships. The ontology modification is a perturbation of the previous equilibrium by the appearance or disappearance of agents or relationships. In this way, a dynamic ontology is a self-organizing process occurring when new texts are included into the corpus, or when the ontologist interacts with it. To support the needed flexibility of such a system we use a selforganizing multi-agent system based on a cooperative approach [9]. We followed the ADELFE method [4] proposed to drive the design of this kind of multi-agent system. It justifies how we designed some of the rules used by our agents in order to maximize the cooperation degree within Dynamo"s multi-agent system. 2.2 Proposed Architecture In this section, we present our system architecture. It addresses the needs of Knowledge Engineering in the context of dynamic ontology management and maintenance when the ontology is linked to a document collection. The Dynamo system consists of three parts (cf. figure 1): • a term network, obtained thanks to a term extraction tool used to preprocess the textual corpus, • a multi-agent system which uses the term network to make a hierarchical clustering in order to obtain a taxonomy of concepts, • an interface allowing the ontologist to visualize and control the clustering process. ?? Ontologist Interface System Concept Agent Term Term network Terms Extraction Tool Figure 1: System architecture The term extractor we use is Syntex, a software that has efficiently been used for ontology building tasks [11]. We mainly selected it because of its robustness and the great amount of information extracted. In particular, it creates a "Head-Expansion" network which has already proven to be interesting for a clustering system [1]. In such a network, each term is linked to its head term1 and 1 i.e. the maximum sub-phrase located as head of the term its expansion term2 , and also to all the terms for which it is a head or an expansion term. For example, "knowledge engineering from text" has "knowledge engineering" as head term and "text" as expansion term. Moreover, "knowledge engineering" is composed of "knowledge" as head term and "engineering" as expansion term. With Dynamo, the term network obtained as the output of the extractor is stored in a database. For each term pair, we assume that it is possible to compute a similarity value in order to make a clustering [6] [1]. Because of the nature of the data, we are only focusing on similarity computation between objects described thanks to binary variables, that means that each item is described by the presence or absence of a characteristic set [15]. In the case of terms we are generally dealing with their usage contexts. With Syntex, those contexts are identified by terms and characterized by some syntactic relations. The Dynamo multi-agent system implements the distributed clustering algorithm described in detail in section 3 and the rules described in section 4. It is designed to be both the system producing the resulting structure and the structure itself. It means that each agent represent a class in the taxonomy. Then, the system output is the organization obtained from the interaction between agents, while taking into account feedback coming from the ontologist when he/she modifies the taxonomy given his needs or expertise. 3. DISTRIBUTED CLUSTERING This section presents the distributed clustering algorithm used in Dynamo. For the sake of understanding, and because of its evaluation in section 3.1, we recall the basic centralized algorithm used for a hierarchical ascending clustering in a non metric space, when a symmetrical similarity measure is available [15] (which is the case of the measures used in our system). Algorithm 1: Centralized hierarchical ascending clustering algorithm Data: List L of items to organize as a hierarchy Result: Root R of the hierarchy while length(L) > 1 do max ← 0; A ← nil; B ← nil; for i ← 1 to length(L) do I ← L[i]; for j ← i + 1 to length(L) do J ← L[j]; sim ← similarity(I, J); if sim > max then max ← sim; A ← I; B ← J; end end end remove(A, L); remove(B, L); append((A, B), L); end R ← L[1]; In algorithm 1, for each clustering step, the pair of the most similar elements is determined. Those two elements are grouped in a cluster, and the resulting class is appended to the list of remaining elements. This algorithm stops when the list has only one element left. 2 i.e. the maximum sub-phrase located as tail of the term The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1287 The hierarchy resulting from algorithm 1 is always a binary tree because of the way grouping is done. Moreover grouping the most similar elements is equivalent to moving them away from the least similar ones. Our distributed algorithm is designed relying on those two facts. It is executed concurrently in each of the agents of the system. Note that, in the following of this paper, we used for both algorithms an Anderberg similarity (with α = 0.75) and an average link clustering strategy [15]. Those choices have an impact on the resulting tree, but they impact neither the global execution of the algorithm nor its complexity. We now present the distributed algorithm used in our system. It is bootstrapped in the following way: • a TOP agent having no parent is created, it will be the root of the resulting taxonomy, • an agent is created for each term to be positioned in the taxonomy, they all have TOP as parent. Once this basic structure is set, the algorithm runs until it reaches equilibrium and then provides the resulting taxonomy. Ak−1 Ak AnA2A1 P ...... ...... A1 Figure 2: Distributed classification: Step 1 The process first step (figure 2) is triggered when an agent (here Ak) has more than one brother (since we want to obtain a binary tree). Then it sends a message to its parent P indicating its most dissimilar brother (here A1). Then P receives the same kind of message from each of its children. In the following, this kind of message will be called a "vote". Ak−1 Ak AnA2A1 P P" ...... ...... P" P" Figure 3: Distributed clustering: Step 2 Next, when P has got messages from all its children, it starts the second step (figure 3). Thanks to the received messages indicating the preferences of its children, P can determine three sub-groups among its children: • the child which got the most "votes" by its brothers, that is the child being the most dissimilar from the greatest number of its brothers. In case of a draw, one of the winners is chosen randomly (here A1), • the children that allowed the "election" of the first group, that is the agents which chose their brother of the first group as being the most dissimilar one (here Ak to An), • the remaining children (here A2 to Ak−1). Then P creates a new agent P (having P as parent) and asks agents from the second group (here agents Ak to An) to make it their new parent. Ak−1 Ak AnA2A1 P P" ...... ...... Figure 4: Distributed clustering: Step 3 Finally, step 3 (figure 4) is trivial. The children rejected by P (here agent A2 to An) take its message into account and choose P as their new parent. The hierarchy just created a new intermediate level. Note that this algorithm generally converges, since the number of brothers of an agent drops. When an agent has only one remaining brother, its activity stops (although it keeps processing messages coming from its children). However in a few cases we can reach a "circular conflict" in the voting procedure when for example A votes against B, B against C and C against A. With the current system no decision can be taken. The current procedure should be improved to address this, probably using a ranked voting method. 3.1 Quantitative Evaluation Now, we evaluate the properties of our distributed algorithm. It requires to begin with a quantitative evaluation, based on its complexity, while comparing it with the algorithm 1 from the previous section. Its theoretical complexity is calculated for the worst case, by considering the similarity computation operation as elementary. For the distributed algorithm, the worst case means that for each run, only a two-item group can be created. Under those conditions, for a given dataset of n items, we can determine the amount of similarity computations. For algorithm 1, we note l = length(L), then the most enclosed "for" loop is run l − i times. And its body has the only similarity computation, so its cost is l−i. The second "for" loop is ran l times for i ranging from 1 to l. Then its cost is Pl i=1(l − i) which can be simplified in l×(l−1) 2 . Finally for each run of the "while" loop, l is decreased from n to 1 which gives us t1(n) as the amount of similarity computations for algorithm 1: t1(n) = nX l=1 l × (l − 1) 2 (1) For the distributed algorithm, at a given step, each one of the l agents evaluates the similarity with its l −1 brothers. So each steps has a l × (l − 1) cost. Then, groups are created and another vote occurs with l decreased by one (since we assume worst case, only groups of size 2 or l −1 are built). Since l is equal to n on first run, we obtain tdist(n) as the amount of similarity computations for the distributed algorithm: tdist(n) = nX l=1 l × (l − 1) (2) Both algorithms then have an O(n3 ) complexity. But in the worst case, the distributed algorithm does twice the number of el1288 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) ementary operations done by the centralized algorithm. This gap comes from the local decision making in each agent. Because of this, the similarity computations are done twice for each agent pair. We could conceive that an agent sends its computation result to its peer. But, it would simply move the problem by generating more communication in the system. 0 20000 40000 60000 80000 100000 120000 140000 160000 180000 10 20 30 40 50 60 70 80 90 100 Amountofcomparisons Amount of input terms 1. Distributed algorithm (on average, with min and max) 2. Logarithmic polynomial 3. Centralized algorithm Figure 5: Experimental results In a second step, the average complexity of the algorithm has been determined by experiments. The multi-agent system has been executed with randomly generated input data sets ranging from ten to one hundred terms. The given value is the average of comparisons made for one hundred of runs without any user interaction. It results in the plots of figure 5. The algorithm is then more efficient on average than the centralized algorithm, and its average complexity is below the worst case. It can be explained by the low probability that a data set forces the system to create only minimal groups (two items) or maximal (n − 1 elements) for each step of reasoning. Curve number 2 represents the logarithmic polynomial minimizing the error with curve number 1. The highest degree term of this polynomial is in n2 log(n), then our distributed algorithm has a O(n2 log(n)) complexity on average. Finally, let"s note the reduced variation of the average performances with the maximum and the minimum. In the worst case for 100 terms, the variation is of 1,960.75 for an average of 40,550.10 (around 5%) which shows the good stability of the system. 3.2 Qualitative Evaluation Although the quantitative results are interesting, the real advantage of this approach comes from more qualitative characteristics that we will present in this section. All are advantages obtained thanks to the use of an adaptive multi-agent system. The main advantage to the use of a multi-agent system for a clustering task is to introduce dynamic in such a system. The ontologist can make modifications and the hierarchy adapts depending on the request. It is particularly interesting in a knowledge engineering context. Indeed, the hierarchy created by the system is meant to be modified by the ontologist since it is the result of a statistic computation. During the necessary look at the texts to examine the usage contexts of terms [2], the ontologist will be able to interpret the real content and to revise the system proposal. It is extremely difficult to realize this with a centralized "black-box" approach. In most cases, one has to find which reasoning step generated the error and to manually modify the resulting class. Unfortunately, in this case, all the reasoning steps that occurred after the creation of the modified class are lost and must be recalculated by taking the modification into account. That is why a system like ASIUM [6] tries to soften the problem with a system-user collaboration by showing to the ontologist the created classes after each step of reasoning. But, the ontologist can make a mistake, and become aware of it too late. Figure 6: Concept agent tree after autonomous stabilization of the system In order to illustrate our claims, we present an example thanks to a few screenshots from the working prototype tested on a medical related corpus. By using test data and letting the system work by itself, we obtain the hierarchy from figure 6 after stabilization. It is clear that the concept described by the term "lésion" (lesion) is misplaced. It happens that the similarity computations place it closer to "femme" (woman) and "chirurgien" (surgeon) than to "infection", "gastro-entérite" (gastro-enteritis) and "hépatite" (hepatitis). This wrong position for "lesion" is explained by the fact that without ontologist input the reasoning is only done on statistics criteria. Figure 7: Concept agent tree after ontologist modification Then, the ontologist replaces the concept in the right branch, by affecting "ConceptAgent:8" as its new parent. The name "ConceptAgent:X" is automatically given to a concept agent that is not described by a term. The system reacts by itself and refines the clustering hierarchy to obtain a binary tree by creating "ConceptAgent:11". The new stable state if the one of figure 7. This system-user coupling is necessary to build an ontology, but no particular adjustment to the distributed algorithm principle is needed since each agent does an autonomous local processing and communicates with its neighborhood by messages. Moreover, this algorithm can de facto be distributed on a computer network. The communication between agents is then done by sending messages and each one keeps its decision autonomy. Then, a system modification to make it run networked would not require to adjust the algorithm. On the contrary, it would only require to rework the communication layer and the agent creation process since in our current implementation those are not networked. 4. MULTI-CRITERIA HIERARCHY In the previous sections, we assumed that similarity can be computed for any term pair. But, as soon as one uses real data this property is not verified anymore. Some terms do not have any similarity value with any extracted term. Moreover for leaf nodes it is sometimes interesting to use other means to position them in the hierarchy. For this low level structuring, ontologists generally base The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1289 their choices on simple heuristics. Using this observation, we built a new set of rules, which are not based on similarity to support low level structuring. 4.1 Adding Head Coverage Rules In this case, agents can act with a very local point of view simply by looking at the parent/child relation. Each agent can try to determine if its parent is adequate. It is possible to guess this because each concept agent is described by a set of terms and thanks to the "Head-Expansion" term network. In the following TX will be the set of terms describing concept agent X and head(TX ) the set of all the terms that are head of at least one element of TX . Thanks to those two notations we can describe the parent adequacy function a(P, C) between a parent P and a child C: a(P, C) = |TP ∩ head(TC )| |TP ∪ head(TC )| (3) Then, the best parent for C is the P agent that maximizes a(P, C). An agent unsatisfied by its parent can then try to find a better one by evaluating adequacy with candidates. We designed a complementary algorithm to drive this search: When an agent C is unsatisfied by its parent P, it evaluates a(Bi, C) with all its brothers (noted Bi) the one maximizing a(Bi, C) is then chosen as the new parent. Figure 8: Concept agent tree after autonomous stabilization of the system without head coverage rule We now illustrate this rule behavior with an example. Figure 8 shows the state of the system after stabilization on test data. We can notice that "hépatite viral" (viral hepatitis) is still linked to the taxonomy root. It is caused by the fact that there is no similarity value between the "viral hepatitis" term and any of the term of the other concept agents. Figure 9: Concept agent tree after activation of the head coverage rule After activating the head coverage rule and letting the system stabilize again we obtain figure 9. We can see that "viral hepatitis" slipped through the branch leading to "hepatitis" and chose it as its new parent. It is a sensible default choice since "viral hepatitis" is a more specific term than "hepatitis". This rule tends to push agents described by a set of term to become leafs of the concept tree. It addresses our concern to improve the low level structuring of our taxonomy. But obviously our agents lack a way to backtrack in case of modifications in the taxonomy which would make them be located in the wrong branch. That is one of the point where our system still has to be improved by adding another set of rules. 4.2 On Using Several Criteria In the previous sections and examples, we only used one algorithm at a time. The distributed clustering algorithm tends to introduce new layers in the taxonomy, while the head coverage algorithm tends to push some of the agents toward the leafs of the taxonomy. It obviously raises the question on how to deal with multiple criteria in our taxonomy building, and how agents determine their priorities at a given time. The solution we chose came from the search for minimizing non cooperation within the system in accordance with the ADELFE method. Each agent computes three non cooperation degrees and chooses its current priority depending on which degree is the highest. For a given agent A having a parent P, a set of brothers Bi and which received a set of messages Mk having the priority pk the three non cooperation degrees are: • μH (A) = 1 − a(P, A), is the "head coverage" non cooperation degree, determined by the head coverage of the parent, • μB(A) = max(1 − similarity(A, Bi)), is the "brotherhood" non cooperation degree, determined by the worst brother of A regarding similarities, • μM (A) = max(pk), is the "message" non cooperation degree, determined by the most urgent message received. Then, the non cooperation degree μ(A) of agent A is: μ(A) = max(μH (A), μB(A), μM (A)) (4) Then, we have three cases determining which kind of action A will choose: • if μ(A) = μH (A) then A will use the head coverage algorithm we detailed in the previous subsection • if μ(A) = μB(A) then A will use the distributed clustering algorithm (see section 3) • if μ(A) = μM (A) then A will process Mk immediately in order to help its sender Those three cases summarize the current activities of our agents: they have to find the best parent for them (μ(A) = μH (A)), improve the structuring through clustering (μ(A) = μB(A)) and process other agent messages (μ(A) = μM (A)) in order to help them fulfill their own goals. 4.3 Experimental Complexity Revisited We evaluated the experimental complexity of the whole multiagent system when all the rules are activated. In this case, the metric used is the number of messages exchanged in the system. Once again the system has been executed with input data sets ranging from ten to one hundred terms. The given value is the average of message amount sent in the system as a whole for one hundred runs without user interaction. It results in the plots of figure 10. Curve number 1 represents the average of the value obtained. Curve number 2 represents the average of the value obtained when only the distributed clustering algorithm is activated, not the full rule set. Curve number 3 represents the polynomial minimizing the error with curve number 1. The highest degree term of this polynomial is in n3 , then our multi-agent system has a O(n3 ) complexity 1290 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 0 5000 10000 15000 20000 25000 10 20 30 40 50 60 70 80 90 100 Amountofmessages Amount of input terms 1. Dynamo, all rules (on average, with min and max) 2. Distributed clustering only (on average) 2. Cubic polynomial Figure 10: Experimental results on average. Moreover, let"s note the very small variation of the average performances with the maximum and the minimum. In the worst case for 100 terms, the variation is of 126.73 for an average of 20,737.03 (around 0.6%) which proves the excellent stability of the system. Finally the extra head coverage rules are a real improvement on the distributed algorithm alone. They introduce more constraints and stability point is reached with less interactions and decision making by the agents. It means that less messages are exchanged in the system while obtaining a tree of higher quality for the ontologist. 5. DISCUSSION & PERSPECTIVES 5.1 Current Limitation of our Approach The most important limitation of our current algorithm is that the result depends on the order the data gets added. When the system works by itself on a fixed data set given during initialization, the final result is equivalent to what we could obtain with a centralized algorithm. On the contrary, adding a new item after a first stabilization has an impact on the final result. Figure 11: Concept agent tree after autonomous stabilization of the system To illustrate our claims, we present another example of the working system. By using test data and letting the system work by itself, we obtain the hierarchy of figure 11 after stabilization. Figure 12: Concept agent tree after taking in account "hepatitis" Then, the ontologist interacts with the system and adds a new concept described by the term "hepatitis" and linked to the root. The system reacts and stabilizes, we then obtain figure 12 as a result. "hepatitis" is located in the right branch, but we have not obtained the same organization as the figure 6 of the previous example. We need to improve our distributed algorithm to allow a concept to move along a branch. We are currently working on the required rules, but the comparison with centralized algorithm will become very difficult. In particular since they will take into account criteria ignored by the centralized algorithm. 5.2 Pruning for Ontologies Building In section 3, we presented the distributed clustering algorithm used in the Dynamo system. Since this work was first based on this algorithm, it introduced a clear bias toward binary trees as a result. But we have to keep in mind that we are trying to obtain taxonomies which are more refined and concise. Although the head coverage rule is an improvement because it is based on how the ontologists generally work, it only addresses low level structuring but not the intermediate levels of the tree. By looking at figure 7, it is clear that some pruning could be done in the taxonomy. In particular, since "lésion" moved, "ConceptAgent:9" could be removed, it is not needed anymore. Moreover the branch starting with "ConceptAgent:8" clearly respects the constraint to make a binary tree, but it would be more useful to the user in a more compact and meaningful form. In this case "ConceptAgent:10" and "ConceptAgent:11" could probably be merged. Currently, our system has the necessary rules to create intermediate levels in the taxonomy, or to have concepts shifting towards the leaf. As we pointed, it is not enough, so new rules are needed to allow removing nodes from the tree, or move them toward the root. Most of the work needed to develop those rules consists in finding the relevant statistic information that will support the ontologist. 6. CONCLUSION After being presented as a promising solution, ensuring model quality and their terminological richness, ontology building from textual corpus analysis is difficult and costly. It requires analyst supervising and taking in account the ontology aim. Using natural languages processing tools ease the knowledge localization in texts through language uses. That said, those tools produce a huge amount of lexical or grammatical data which is not trivial to examine in order to define conceptual elements. Our contribution lies in this step of the modeling process from texts, before any attempts to normalize or formalize the result. We proposed an approach based on an adaptive multi-agent system to provide the ontologist with a first taxonomic structure of concepts. Our system makes use of a terminological network resulting from an analysis made by Syntex. The current state of our software allows to produce simple structures, to propose them to the ontologist and to make them evolve depending on the modifications he made. Performances of the system are interesting and some aspects are even comparable to their centralized counterpart. Its strengths are mostly qualitative since it allows more subtle user interactions and a progressive adaptation to new linguistic based information. From the point of view of ontology building, this work is a first step showing the relevance of our approach. It must continue, both to ensure a better robustness during classification, and to obtain richer structures semantic wise than simple trees. From this improvements we are mostly focusing on the pruning to obtain better taxonomies. We"re currently working on the criterion to trigger the complementary actions of the structure changes applied by our clustering algorithm. In other words this algorithm introduces inThe Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1291 termediate levels, and we need to be able to remove them if necessary, in order to reach a dynamic equilibrium. Also from the multi-agent engineering point of view, their use in a dynamic ontology context has shown its relevance. This dynamic ontologies can be seen as complex problem solving, in such a case self-organization through cooperation has been an efficient solution. And, more generally it"s likely to be interesting for other design related tasks, even if we"re focusing only on knowledge engineering in this paper. Of course, our system still requires more evaluation and validation work to accurately determine the advantages and flaws of this approach. We"re planning to work on such benchmarking in the near future. 7. REFERENCES [1] H. Assadi. Construction of a regional ontology from text and its use within a documentary system. Proceedings of the International Conference on Formal Ontology and Information Systems - FOIS"98, pages 236-249, 1998. [2] N. Aussenac-Gilles and D. Sörgel. Text analysis for ontology and terminology engineering. Journal of Applied Ontology, 2005. [3] J. Bao and V. Honavar. Collaborative ontology building with wiki@nt. Proceedings of the Workshop on Evaluation of Ontology-Based Tools (EON2004), 2004. [4] C. Bernon, V. Camps, M.-P. Gleizes, and G. Picard. Agent-Oriented Methodologies, chapter 7. Engineering Self-Adaptive Multi-Agent Systems : the ADELFE Methodology, pages 172-202. Idea Group Publishing, 2005. [5] C. Brewster, F. Ciravegna, and Y. Wilks. Background and foreground knowledge in dynamic ontology construction. Semantic Web Workshop, SIGIR"03, August 2003. [6] D. Faure and C. Nedellec. A corpus-based conceptual clustering method for verb frames and ontology acquisition. LREC workshop on adapting lexical and corpus resources to sublanguages and applications, 1998. [7] F. Gandon. Ontology Engineering: a Survey and a Return on Experience. INRIA, 2002. [8] J.-P. Georgé, G. Picard, M.-P. Gleizes, and P. Glize. Living Design for Open Computational Systems. 12th IEEE International Workshops on Enabling Technologies, Infrastructure for Collaborative Enterprises, pages 389-394, June 2003. [9] M.-P. Gleizes, V. Camps, and P. Glize. A Theory of emergent computation based on cooperative self-organization for adaptive artificial systems. Fourth European Congress of Systems Science, September 1999. [10] J. Heflin and J. Hendler. Dynamic ontologies on the web. American Association for Artificial Intelligence Conference, 2000. [11] S. Le Moigno, J. Charlet, D. Bourigault, and M.-C. Jaulent. Terminology extraction from text to build an ontology in surgical intensive care. Proceedings of the AMIA 2002 annual symposium, 2002. [12] K. Lister, L. Sterling, and K. Taveter. Reconciling Ontological Differences by Assistant Agents. AAMAS"06, May 2006. [13] A. Maedche. Ontology learning for the Semantic Web. Kluwer Academic Publisher, 2002. [14] A. Maedche and S. Staab. Mining Ontologies from Text. EKAW 2000, pages 189-202, 2000. [15] C. D. Manning and H. Schütze. Foundations of Statistical Natural Language Processing. The MIT Press, Cambridge, Massachusetts, 1999. [16] H. V. D. Parunak, R. Rohwer, T. C. Belding, and S. Brueckner. Dynamic decentralized any-time hierarchical clustering. 29th Annual International ACM SIGIR Conference on Research & Development on Information Retrieval, August 2006. 1292 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07)

cooperation;parent adequacy function;ontology;dynamic equilibrium;hepatitis;emergent behavior;quantitative evaluation;black-box;model quality;multi-agent field;dynamo;terminological richness

train_I-71

A Formal Model for Situated Semantic Alignment

Ontology matching is currently a key technology to achieve the semantic alignment of ontological entities used by knowledge-based applications, and therefore to enable their interoperability in distributed environments such as multiagent systems. Most ontology matching mechanisms, however, assume matching prior integration and rely on semantics that has been coded a priori in concept hierarchies or external sources. In this paper, we present a formal model for a semantic alignment procedure that incrementally aligns differing conceptualisations of two or more agents relative to their respective perception of the environment or domain they are acting in. It hence makes the situation in which the alignment occurs explicit in the model. We resort to Channel Theory to carry out the formalisation.

1. INTRODUCTION An ontology is commonly defined as a specification of the conceptualisation of a particular domain. It fixes the vocabulary used by knowledge engineers to denote concepts and their relations, and it constrains the interpretation of this vocabulary to the meaning originally intended by knowledge engineers. As such, ontologies have been widely adopted as a key technology that may favour knowledge sharing in distributed environments, such as multi-agent systems, federated databases, or the Semantic Web. But the proliferation of many diverse ontologies caused by different conceptualisations of even the same domain -and their subsequent specification using varying terminology- has highlighted the need of ontology matching techniques that are capable of computing semantic relationships between entities of separately engineered ontologies. [5, 11] Until recently, most ontology matching mechanisms developed so far have taken a classical functional approach to the semantic heterogeneity problem, in which ontology matching is seen as a process taking two or more ontologies as input and producing a semantic alignment of ontological entities as output [3]. Furthermore, matching often has been carried out at design-time, before integrating knowledge-based systems or making them interoperate. This might have been successful for clearly delimited and stable domains and for closed distributed systems, but it is untenable and even undesirable for the kind of applications that are currently deployed in open systems. Multi-agent communication, peer-to-peer information sharing, and webservice composition are all of a decentralised, dynamic, and open-ended nature, and they require ontology matching to be locally performed during run-time. In addition, in many situations peer ontologies are not even open for inspection (e.g., when they are based on commercially confidential information). Certainly, there exist efforts to efficiently match ontological entities at run-time, taking only those ontology fragment that are necessary for the task at hand [10, 13, 9, 8]. Nevertheless, the techniques used by these systems to establish the semantic relationships between ontological entities -even though applied at run-time- still exploit a priori defined concept taxonomies as they are represented in the graph-based structures of the ontologies to be matched, use previously existing external sources such as thesauri (e.g., WordNet) and upper-level ontologies (e.g., CyC or SUMO), or resort to additional background knowledge repositories or shared instances. We claim that semantic alignment of ontological terminology is ultimately relative to the particular situation in which the alignment is carried out, and that this situation should be made explicit and brought into the alignment mechanism. Even two agents with identical conceptualisation capabilities, and using exactly the same vocabulary to specify their respective conceptualisations may fail to interoperate 1278 978-81-904262-7-5 (RPS) c 2007 IFAAMAS in a concrete situation because of their differing perception of the domain. Imagine a situation in which two agents are facing each other in front of a checker board. Agent A1 may conceptualise a figure on the board as situated on the left margin of the board, while agent A2 may conceptualise the same figure as situated on the right. Although the conceptualisation of ‘left" and ‘right" is done in exactly the same manner by both agents, and even if both use the terms left and right in their communication, they still will need to align their respective vocabularies if they want to successfully communicate to each other actions that change the position of figures on the checker board. Their semantic alignment, however, will only be valid in the scope of their interaction within this particular situation or environment. The same agents situated differently may produce a different alignment. This scenario is reminiscent to those in which a group of distributed agents adapt to form an ontology and a shared lexicon in an emergent, bottom-up manner, with only local interactions and no central control authority [12]. This sort of self-organised emergence of shared meaning is namely ultimately grounded on the physical interaction of agents with the environment. In this paper, however, we address the case in which agents are already endowed with a top-down engineered ontology (it can even be the same one), which they do not adapt or refine, but for which they want to find the semantic relationships with separate ontologies of other agents on the grounds of their communication within a specific situation. In particular, we provide a formal model that formalises situated semantic alignment as a sequence of information-channel refinements in the sense of Barwise and Seligman"s theory of information flow [1]. This theory is particularly useful for our endeavour because it models the flow of information occurring in distributed systems due to the particular situations -or tokens- that carry information. Analogously, the semantic alignment that will allow information to flow ultimately will be carried by the particular situation agents are acting in. We shall therefore consider a scenario with two or more agents situated in an environment. Each agent will have its own viewpoint of the environment so that, if the environment is in a concrete state, both agents may have different perceptions of this state. Because of these differences there may be a mismatch in the meaning of the syntactic entities by which agents describe their perceptions (and which constitute the agents" respective ontologies). We state that these syntactic entities can be related according to the intrinsic semantics provided by the existing relationship between the agents" viewpoint of the environment. The existence of this relationship is precisely justified by the fact that the agents are situated and observe the same environment. In Section 2 we describe our formal model for Situated Semantic Alignment (SSA). First, in Section 2.1 we associate a channel to the scenario under consideration and show how the distributed logic generated by this channel provides the logical relationships between the agents" viewpoints of the environment. Second, in Section 2.2 we present a method by which agents obtain approximations of this distributed logic. These approximations gradually become more reliable as the method is applied. In Section 3 we report on an application of our method. Conclusions and further work are analyzed in Section 4. Finally, an appendix summarizes the terms and theorems of Channel theory used along the paper. We do not assume any knowledge of Channel Theory; we restate basic definitions and theorems in the appendix, but any detailed exposition of the theory is outside the scope of this paper. 2. A FORMAL MODEL FOR SSA 2.1 The Logic of SSA Consider a scenario with two agents A1 and A2 situated in an environment E (the generalization to any numerable set of agents is straightforward). We associate a numerable set S of states to E and, at any given instant, we suppose E to be in one of these states. We further assume that each agent is able to observe the environment and has its own perception of it. This ability is faithfully captured by a surjective function seei : S → Pi, where i ∈ {1, 2}, and typically see1 and see2 are different. According to Channel Theory, information is only viable where there is a systematic way of classifying some range of things as being this way or that, in other words, where there is a classification (see appendix A). So in order to be within the framework of Channel Theory, we must associate classifications to the components of our system. For each i ∈ {1, 2}, we consider a classification Ai that models Ai"s viewpoint of E. First, tok(Ai) is composed of Ai"s perceptions of E states, that is, tok(Ai) = Pi. Second, typ(Ai) contains the syntactic entities by which Ai describes its perceptions, the ones constituting the ontology of Ai. Finally, |=Ai synthesizes how Ai relates its perceptions with these syntactic entities. Now, with the aim of associating environment E with a classification E we choose the power classification of S as E, which is the classification whose set of types is equal to 2S , whose tokens are the elements of S, and for which a token e is of type ε if e ∈ ε. The reason for taking the power classification is because there are no syntactic entities that may play the role of types for E since, in general, there is no global conceptualisation of the environment. However, the set of types of the power classification includes all possible token configurations potentially described by types. Thus tok(E) = S, typ(E) = 2S and e |=E ε if and only if e ∈ ε. The notion of channel (see appendix A) is fundamental in Barwise and Seligman"s theory. The information flow among the components of a distributed system is modelled in terms of a channel and the relationships among these components are expressed via infomorphisms (see appendix A) which provide a way of moving information between them. The information flow of the scenario under consideration is accurately described by channel E = {fi : Ai → E}i∈{1,2} defined as follows: • ˆfi(α) = {e ∈ tok(E) | seei(e) |=Ai α} for each α ∈ typ(Ai) • ˇfi(e) = seei(e) for each e ∈ tok(E) where i ∈ {1, 2}. Definition of ˇfi seems natural while ˆfi is defined in such a way that the fundamental property of the infomorphisms is fulfilled: ˇfi(e) |=Ai α iff seei(e) |=Ai α (by definition of ˇfi) iff e ∈ ˆfi(α) (by definition of ˆfi) iff e |=E ˆfi(α) (by definition of |=E) The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1279 Consequently, E is the core of channel E and a state e ∈ tok(E) connects agents" perceptions ˇf1(e) and ˇf2(e) (see Figure 1). typ(E) typ(A1) ˆf1 99ttttttttt typ(A2) ˆf2 eeJJJJJJJJJ tok(E) |=E ˇf1yyttttttttt ˇf2 %%JJJJJJJJJ tok(A1) |=A1 tok(A2) |=A2 Figure 1: Channel E E explains the information flow of our scenario by virtue of agents A1 and A2 being situated and perceiving the same environment E. We want to obtain meaningful relations among agents" syntactic entities, that is, agents" types. We state that meaningfulness must be in accord with E. The sum operation (see appendix A) gives us a way of putting the two agents" classifications of channel E together into a single classification, namely A1 +A2, and also the two infomorphisms together into a single infomorphism, f1 +f2 : A1 + A2 → E. A1 + A2 assembles agents" classifications in a very coarse way. tok(A1 + A2) is the cartesian product of tok(A1) and tok(A2), that is, tok(A1 + A2) = { p1, p2 | pi ∈ Pi}, so a token of A1 + A2 is a pair of agents" perceptions with no restrictions. typ(A1 + A2) is the disjoint union of typ(A1) and typ(A2), and p1, p2 is of type i, α if pi is of type α. We attach importance to take the disjoint union because A1 and A2 could use identical types with the purpose of describing their respective perceptions of E. Classification A1 + A2 seems to be the natural place in which to search for relations among agents" types. Now, Channel Theory provides a way to make all these relations explicit in a logical fashion by means of theories and local logics (see appendix A). The theory generated by the sum classification, Th(A1 + A2), and hence its logic generated, Log(A1 + A2), involve all those constraints among agents" types valid according to A1 +A2. Notice however that these constraints are obvious. As we stated above, meaningfulness must be in accord with channel E. Classifications A1 + A2 and E are connected via the sum infomorphism, f = f1 + f2, where: • ˆf( i, α ) = ˆfi(α) = {e ∈ tok(E) | seei(e) |=Ai α} for each i, α ∈ typ(A1 + A2) • ˇf(e) = ˇf1(e), ˇf2(e) = see1(e), see2(e) for each e ∈ tok(E) Meaningful constraints among agents" types are in accord with channel E because they are computed making use of f as we expound below. As important as the notion of channel is the concept of distributed logic (see appendix A). Given a channel C and a logic L on its core, DLogC(L) represents the reasoning about relations among the components of C justified by L. If L = Log(C), the distributed logic, we denoted by Log(C), captures in a logical fashion the information flow inherent in the channel. In our case, Log(E) explains the relationship between the agents" viewpoints of the environment in a logical fashion. On the one hand, constraints of Th(Log(E)) are defined by: Γ Log(E) Δ if ˆf[Γ] Log(E) ˆf[Δ] (1) where Γ, Δ ⊆ typ(A1 + A2). On the other hand, the set of normal tokens, NLog(E), is equal to the range of function ˇf: NLog(E) = ˇf[tok(E)] = { see1(e), see2(e) | e ∈ tok(E)} Therefore, a normal token is a pair of agents" perceptions that are restricted by coming from the same environment state (unlike A1 + A2 tokens). All constraints of Th(Log(E)) are satisfied by all normal tokens (because of being a logic). In this particular case, this condition is also sufficient (the proof is straightforward); as alternative to (1) we have: Γ Log(E) Δ iff for all e ∈ tok(E), if (∀ i, γ ∈ Γ)[seei(e) |=Ai γ] then (∃ j, δ ∈ Δ)[seej(e) |=Aj δ] (2) where Γ, Δ ⊆ typ(A1 + A2). Log(E) is the logic of SSA. Th(Log(E)) comprises the most meaningful constraints among agents" types in accord with channel E. In other words, the logic of SSA contains and also justifies the most meaningful relations among those syntactic entities that agents use in order to describe their own environment perceptions. Log(E) is complete since Log(E) is complete but it is not necessarily sound because although Log(E) is sound, ˇf is not surjective in general (see appendix B). If Log(E) is also sound then Log(E) = Log(A1 +A2) (see appendix B). That means there is no significant relation between agents" points of view of the environment according to E. It is just the fact that Log(E) is unsound what allows a significant relation between the agents" viewpoints. This relation is expressed at the type level in terms of constraints by Th(Log(E)) and at the token level by NLog(E). 2.2 Approaching the logic of SSA through communication We have dubbed Log(E) the logic of SSA. Th(Log(E)) comprehends the most meaningful constraints among agents" types according to E. The problem is that neither agent can make use of this theory because they do not know E completely. In this section, we present a method by which agents obtain approximations to Th(Log(E)). We also prove these approximations gradually become more reliable as the method is applied. Agents can obtain approximations to Th(Log(E)) through communication. A1 and A2 communicate by exchanging information about their perceptions of environment states. This information is expressed in terms of their own classification relations. Specifically, if E is in a concrete state e, we assume that agents can convey to each other which types are satisfied by their respective perceptions of e and which are not. This exchange generates a channel C = {fi : Ai → 1280 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) C}i∈{1,2} and Th(Log(C)) contains the constraints among agents" types justified by the fact that agents have observed e. Now, if E turns to another state e and agents proceed as before, another channel C = {fi : Ai → C }i∈{1,2} gives account of the new situation considering also the previous information. Th(Log(C )) comprises the constraints among agents" types justified by the fact that agents have observed e and e . The significant point is that C is a refinement of C (see appendix A). Theorem 2.1 below ensures that the refined channel involves more reliable information. The communication supposedly ends when agents have observed all the environment states. Again this situation can be modeled by a channel, call it C∗ = {f∗ i : Ai → C∗ }i∈{1,2}. Theorem 2.2 states that Th(Log(C∗ )) = Th(Log(E)). Theorem 2.1 and Theorem 2.2 assure that applying the method agents can obtain approximations to Th(Log(E)) gradually more reliable. Theorem 2.1. Let C = {fi : Ai → C}i∈{1,2} and C = {fi : Ai → C }i∈{1,2} be two channels. If C is a refinement of C then: 1. Th(Log(C )) ⊆ Th(Log(C)) 2. NLog(C ) ⊇ NLog(C) Proof. Since C is a refinement of C then there exists a refinement infomorphism r from C to C; so fi = r ◦ fi . Let A =def A1 + A2, f =def f1 + f2 and f =def f1 + f2. 1. Let Γ and Δ be subsets of typ(A) and assume that Γ Log(C ) Δ, which means ˆf [Γ] C ˆf [Δ]. We have to prove Γ Log(C) Δ, or equivalently, ˆf[Γ] C ˆf[Δ]. We proceed by reductio ad absurdum. Suppose c ∈ tok(C) does not satisfy the sequent ˆf[Γ], ˆf[Δ] . Then c |=C ˆf(γ) for all γ ∈ Γ and c |=C ˆf(δ) for all δ ∈ Δ. Let us choose an arbitrary γ ∈ Γ. We have that γ = i, α for some α ∈ typ(Ai) and i ∈ {1, 2}. Thus ˆf(γ) = ˆf( i, α ) = ˆfi(α) = ˆr ◦ ˆfi (α) = ˆr( ˆfi (α)). Therefore: c |=C ˆf(γ) iff c |=C ˆr( ˆfi (α)) iff ˇr(c) |=C ˆfi (α) iff ˇr(c) |=C ˆf ( i, α ) iff ˇr(c) |=C ˆf (γ) Consequently, ˇr(c) |=C ˆf (γ) for all γ ∈ Γ. Since ˆf [Γ] C ˆf [Δ] then there exists δ∗ ∈ Δ such that ˇr(c) |=C ˆf (δ∗ ). A sequence of equivalences similar to the above one justifies c |=C ˆf(δ∗ ), contradicting that c is a counterexample to ˆf[Γ], ˆf[Δ] . Hence Γ Log(C) Δ as we wanted to prove. 2. Let a1, a2 ∈ tok(A) and assume a1, a2 ∈ NLog(C). Therefore, there exists c token in C such that a1, a2 = ˇf(c). Then we have ai = ˇfi(c) = ˇfi ◦ ˇr(c) = ˇfi (ˇr(c)), for i ∈ {1, 2}. Hence a1, a2 = ˇf (ˇr(c)) and a1, a2 ∈ NLog(C ). Consequently, NLog(C ) ⊇ NLog(C) which concludes the proof. Remark 2.1. Theorem 2.1 asserts that the more refined channel gives more reliable information. Even though its theory has less constraints, it has more normal tokens to which they apply. In the remainder of the section, we explicitly describe the process of communication and we conclude with the proof of Theorem 2.2. Let us assume that typ(Ai) is finite for i ∈ {1, 2} and S is infinite numerable, though the finite case can be treated in a similar form. We also choose an infinite numerable set of symbols {cn | n ∈ N}1 . We omit informorphisms superscripts when no confusion arises. Types are usually denoted by greek letters and tokens by latin letters so if f is an infomorphism, f(α) ≡ ˆf(α) and f(a) ≡ ˇf(a). Agents communication starts from the observation of E. Let us suppose that E is in state e1 ∈ S = tok(E). A1"s perception of e1 is f1(e1 ) and A2"s perception of e1 is f2(e1 ). We take for granted that A1 can communicate A2 those types that are and are not satisfied by f1(e1 ) according to its classification A1. So can A2 do. Since both typ(A1) and typ(A2) are finite, this process eventually finishes. After this communication a channel C1 = {f1 i : Ai → C1 }i=1,2 arises (see Figure 2). C1 A1 f1 1 ==|||||||| A2 f1 2 aaCCCCCCCC Figure 2: The first communication stage On the one hand, C1 is defined by: • tok(C1 ) = {c1 } • typ(C1 ) = typ(A1 + A2) • c1 |=C1 i, α if fi(e1 ) |=Ai α (for every i, α ∈ typ(A1 + A2)) On the other hand, f1 i , with i ∈ {1, 2}, is defined by: • f1 i (α) = i, α (for every α ∈ typ(Ai)) • f1 i (c1 ) = fi(e1 ) Log(C1 ) represents the reasoning about the first stage of communication. It is easy to prove that Th(Log(C1 )) = Th(C1 ). The significant point is that both agents know C1 as the result of the communication. Hence they can compute separately theory Th(C1 ) = typ(C1 ), C1 which contains the constraints among agents" types justified by the fact that agents have observed e1 . Now, let us assume that E turns to a new state e2 . Agents can proceed as before, exchanging this time information about their perceptions of e2 . Another channel C2 = {f2 i : Ai → C2 }i∈{1,2} comes up. We define C2 so as to take also into account the information provided by the previous stage of communication. On the one hand, C2 is defined by: • tok(C2 ) = {c1 , c2 } 1 We write these symbols with superindices because we limit the use of subindices for what concerns to agents. Note this set is chosen with the same cardinality of S. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1281 • typ(C2 ) = typ(A1 + A2) • ck |=C2 i, α if fi(ek ) |=Ai α (for every k ∈ {1, 2} and i, α ∈ typ(A1 + A2)) On the other hand, f2 i , with i ∈ {1, 2}, is defined by: • f2 i (α) = i, α (for every α ∈ typ(Ai)) • f2 i (ck ) = fi(ek ) (for every k ∈ {1, 2}) Log(C2 ) represents the reasoning about the former and the later communication stages. Th(Log(C2 )) is equal to Th(C2 ) = typ(C2 ), C2 , then it contains the constraints among agents" types justified by the fact that agents have observed e1 and e2 . A1 and A2 knows C2 so they can use these constraints. The key point is that channel C2 is a refinement of C1 . It is easy to check that f1 defined as the identity function on types and the inclusion function on tokens is a refinement infomorphism (see at the bottom of Figure 3). By Theorem 2.1, C2 constraints are more reliable than C1 constraints. In the general situation, once the states e1 , e2 , . . . , en−1 (n ≥ 2) have been observed and a new state en appears, channel Cn = {fn i : Ai → Cn }i∈{1,2} informs about agents communication up to that moment. Cn definition is similar to the previous ones and analogous remarks can be made (see at the top of Figure 3). Theory Th(Log(Cn )) = Th(Cn ) = typ(Cn ), Cn contains the constraints among agents" types justified by the fact that agents have observed e1 , e2 , . . . , en . Cn fn−1  A1 fn−1 1 99PPPPPPPPPPPPP fn 1 UUnnnnnnnnnnnnn f2 1 %%44444444444444444444444444 f1 1 "",,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, A2 fn 2 ggPPPPPPPPPPPPP fn−1 2 wwnnnnnnnnnnnnn f2 2 ÕÕ f1 2 ØØ Cn−1  . . .  C2 f1  C1 Figure 3: Agents communication Remember we have assumed that S is infinite numerable. It is therefore unpractical to let communication finish when all environment states have been observed by A1 and A2. At that point, the family of channels {Cn }n∈N would inform of all the communication stages. It is therefore up to the agents to decide when to stop communicating should a good enough approximation have been reached for the purposes of their respective tasks. But the study of possible termination criteria is outside the scope of this paper and left for future work. From a theoretical point of view, however, we can consider the channel C∗ = {f∗ i : Ai → C∗ }i∈{1,2} which informs of the end of the communication after observing all environment states. On the one hand, C∗ is defined by: • tok(C∗ ) = {cn | n ∈ N} • typ(C∗ ) = typ(A1 + A2) • cn |=C∗ i, α if fi(en ) |=Ai α (for n ∈ N and i, α ∈ typ(A1 + A2)) On the other hand, f∗ i , with i ∈ {1, 2}, is defined by: • f∗ i (α) = i, α (for α ∈ typ(Ai)) • f∗ i (cn ) = fi(en ) (for n ∈ N) Theorem below constitutes the cornerstone of the model exposed in this paper. It ensures, together with Theorem 2.1, that at each communication stage agents obtain a theory that approximates more closely to the theory generated by the logic of SSA. Theorem 2.2. The following statements hold: 1. For all n ∈ N, C∗ is a refinement of Cn . 2. Th(Log(E)) = Th(C∗ ) = Th(Log(C∗ )). Proof. 1. It is easy to prove that for each n ∈ N, gn defined as the identity function on types and the inclusion function on tokens is a refinement infomorphism from C∗ to Cn . 2. The second equality is straightforward; the first one follows directly from: cn |=C∗ i, α iff ˇfi(en ) |=Ai α (by definition of |=C∗ ) iff en |=E ˆfi(α) (because fi is infomorphim) iff en |=E ˆf( i, α ) (by definition of ˆf) E C∗ gn  A1 fn 1 99OOOOOOOOOOOOO f∗ 1 UUooooooooooooo f1 cc A2 f∗ 2 ggOOOOOOOOOOOOO fn 2 wwooooooooooooo f2 ••????????????????? Cn 1282 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 3. AN EXAMPLE In the previous section we have described in great detail our formal model for SSA. However, we have not tackled the practical aspect of the model yet. In this section, we give a brushstroke of the pragmatic view of our approach. We study a very simple example and explain how agents can use those approximations of the logic of SSA they can obtain through communication. Let us reflect on a system consisting of robots located in a two-dimensional grid looking for packages with the aim of moving them to a certain destination (Figure 4). Robots can carry only one package at a time and they can not move through a package. Figure 4: The scenario Robots have a partial view of the domain and there exist two kinds of robots according to the visual field they have. Some robots are capable of observing the eight adjoining squares but others just observe the three squares they have in front (see Figure 5). We call them URDL (shortened form of Up-Right-Down-Left) and LCR (abbreviation for Left-Center-Right) robots respectively. Describing the environment states as well as the robots" perception functions is rather tedious and even unnecessary. We assume the reader has all those descriptions in mind. All robots in the system must be able to solve package distribution problems cooperatively by communicating their intentions to each other. In order to communicate, agents send messages using some ontology. In our scenario, there coexist two ontologies, the UDRL and LCR ontologies. Both of them are very simple and are just confined to describe what robots observe. Figure 5: Robots field of vision When a robot carrying a package finds another package obstructing its way, it can either go around it or, if there is another robot in its visual field, ask it for assistance. Let us suppose two URDL robots are in a situation like the one depicted in Figure 6. Robot1 (the one carrying a package) decides to ask Robot2 for assistance and sends a request. This request is written below as a KQML message and it should be interpreted intuitively as: Robot2, pick up the package located in my Up square, knowing that you are located in my Up-Right square. ` request :sender Robot1 :receiver Robot2 :language Packages distribution-language :ontology URDL-ontology :content (pick up U(Package) because UR(Robot2) ´ Figure 6: Robot assistance Robot2 understands the content of the request and it can use a rule represented by the following constraint: 1, UR(Robot2) , 2, UL(Robot1) , 1, U(Package) 2, U(Package) The above constraint should be interpreted intuitively as: if Robot2 is situated in Robot1"s Up-Right square, Robot1 is situated in Robot2"s Up-Left square and a package is located in Robot1"s Up square, then a package is located in Robot2"s Up square. Now, problems arise when a LCR robot and a URDL robot try to interoperate. See Figure 7. Robot1 sends a request of the form: ` request :sender Robot1 :receiver Robot2 :language Packages distribution-language :ontology LCR-ontology :content (pick up R(Robot2) because C(Package) ´ Robot2 does not understand the content of the request but they decide to begin a process of alignment -corresponding with a channel C1 . Once finished, Robot2 searches in Th(C1 ) for constraints similar to the expected one, that is, those of the form: 1, R(Robot2) , 2, UL(Robot1) , 1, C(Package) C1 2, λ(Package) where λ ∈ {U, R, D, L, UR, DR, DL, UL}. From these, only the following constraints are plausible according to C1 : The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1283 Figure 7: Ontology mismatch 1, R(Robot2) , 2, UL(Robot1) , 1, C(Package) C1 2, U(Package) 1, R(Robot2) , 2, UL(Robot1) , 1, C(Package) C1 2, L(Package) 1, R(Robot2) , 2, UL(Robot1) , 1, C(Package) C1 2, DR(Package) If subsequently both robots adopting the same roles take part in a situation like the one depicted in Figure 8, a new process of alignment -corresponding with a channel C2 - takes place. C2 also considers the previous information and hence refines C1 . The only constraint from the above ones that remains plausible according to C2 is : 1, R(Robot2) , 2, UL(Robot1) , 1, C(Package) C2 2, U(Package) Notice that this constraint is an element of the theory of the distributed logic. Agents communicate in order to cooperate successfully and success is guaranteed using constrains of the distributed logic. Figure 8: Refinement 4. CONCLUSIONS AND FURTHER WORK In this paper we have exposed a formal model of semantic alignment as a sequence of information-channel refinements that are relative to the particular states of the environment in which two agents communicate and align their respective conceptualisations of these states. Before us, Kent [6] and Kalfoglou and Schorlemmer [4, 10] have applied Channel Theory to formalise semantic alignment using also Barwise and Seligman"s insight to focus on tokens as the enablers of information flow. Their approach to semantic alignment, however, like most ontology matching mechanisms developed to date (regardless of whether they follow a functional, design-time-based approach, or an interaction-based, runtime-based approach), still defines semantic alignment in terms of a priori design decisions such as the concept taxonomy of the ontologies or the external sources brought into the alignment process. Instead the model we have presented in this paper makes explicit the particular states of the environment in which agents are situated and are attempting to gradually align their ontological entities. In the future, our effort will focus on the practical side of the situated semantic alignment problem. We plan to further refine the model presented here (e.g., to include pragmatic issues such as termination criteria for the alignment process) and to devise concrete ontology negotiation protocols based on this model that agents may be able to enact. The formal model exposed in this paper will constitute a solid base of future practical results. Acknowledgements This work is supported under the UPIC project, sponsored by Spain"s Ministry of Education and Science under grant number TIN2004-07461-C02- 02 and also under the OpenKnowledge Specific Targeted Research Project (STREP), sponsored by the European Commission under contract number FP6-027253. Marco Schorlemmer is supported by a Ram´on y Cajal Research Fellowship from Spain"s Ministry of Education and Science, partially funded by the European Social Fund. 5. REFERENCES [1] J. Barwise and J. Seligman. Information Flow: The Logic of Distributed Systems. Cambridge University Press, 1997. [2] C. Ghidini and F. Giunchiglia. Local models semantics, or contextual reasoning = locality + compatibility. Artificial Intelligence, 127(2):221-259, 2001. [3] F. Giunchiglia and P. Shvaiko. Semantic matching. The Knowledge Engineering Review, 18(3):265-280, 2004. [4] Y. Kalfoglou and M. Schorlemmer. IF-Map: An ontology-mapping method based on information-flow theory. In Journal on Data Semantics I, LNCS 2800, 2003. [5] Y. Kalfoglou and M. Schorlemmer. Ontology mapping: The sate of the art. The Knowledge Engineering Review, 18(1):1-31, 2003. [6] R. E. Kent. Semantic integration in the Information Flow Framework. In Semantic Interoperability and Integration, Dagstuhl Seminar Proceedings 04391, 2005. [7] D. Lenat. CyC: A large-scale investment in knowledge infrastructure. Communications of the ACM, 38(11), 1995. [8] V. L´opez, M. Sabou, and E. Motta. PowerMap: Mapping the real Semantic Web on the fly. Proceedings of the ISWC"06, 2006. [9] F. McNeill. Dynamic Ontology Refinement. PhD 1284 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) thesis, School of Informatics, The University of Edinburgh, 2006. [10] M. Schorlemmer and Y. Kalfoglou. Progressive ontology alignment for meaning coordination: An information-theoretic foundation. In 4th Int. Joint Conf. on Autonomous Agents and Multiagent Systems, 2005. [11] P. Shvaiko and J. Euzenat. A survey of schema-based matching approaches. In Journal on Data Semantics IV, LNCS 3730, 2005. [12] L. Steels. The Origins of Ontologies and Communication Conventions in Multi-Agent Systems. In Journal of Autonomous Agents and Multi-Agent Systems, 1(2), 169-194, 1998. [13] J. van Diggelen et al. ANEMONE: An Effective Minimal Ontology Negotiation Environment In 5th Int. Joint Conf. on Autonomous Agents and Multiagent Systems, 2006 APPENDIX A. CHANNEL THEORY TERMS Classification: is a tuple A = tok(A), typ(A), |=A where tok(A) is a set of tokens, typ(A) is a set of types and |=A is a binary relation between tok(A) and typ(A). If a |=A α then a is said to be of type α. Infomorphism: f : A → B from classifications A to B is a contravariant pair of functions f = ˆf, ˇf , where ˆf : typ(A) → typ(B) and ˇf : tok(B) → tok(A), satisfying the following fundamental property: ˇf(b) |=A α iff b |=B ˆf(α) for each token b ∈ tok(B) and each type α ∈ typ(A). Channel: consists of two infomorphisms C = {fi : Ai → C}i∈{1,2} with a common codomain C, called the core of C. C tokens are called connections and a connection c is said to connect tokens ˇf1(c) and ˇf2(c).2 Sum: given classifications A and B, the sum of A and B, denoted by A + B, is the classification with tok(A + B) = tok(A) × tok(B) = { a, b | a ∈ tok(A) and b ∈ tok(B)}, typ(A + B) = typ(A) typ(B) = { i, γ | i = 1 and γ ∈ typ(A) or i = 2 and γ ∈ typ(B)} and relation |=A+B defined by: a, b |=A+B 1, α if a |=A α a, b |=A+B 2, β if b |=B β Given infomorphisms f : A → C and g : B → C, the sum f + g : A + B → C is defined on types by ˆ(f + g)( 1, α ) = ˆf(α) and ˆ(f + g)( 2, β ) = ˆg(β), and on tokens by ˇ(f + g)(c) = ˇf(c), ˇg(c) . Theory: given a set Σ, a sequent of Σ is a pair Γ, Δ of subsets of Σ. A binary relation between subsets of Σ is called a consequence relation on Σ. A theory is a pair T = Σ, where is a consequence relation on Σ. A sequent Γ, Δ of Σ for which Γ Δ is called a constraint of the theory T. T is regular if it satisfies: 1. Identity: α α 2. Weakening: if Γ Δ, then Γ, Γ Δ, Δ 2 In fact, this is the definition of a binary channel. A channel can be defined with an arbitrary index set. 3. Global Cut: if Γ, Π0 Δ, Π1 for each partition Π0, Π1 of Π (i.e., Π0 ∪ Π1 = Π and Π0 ∩ Π1 = ∅), then Γ Δ for all α ∈ Σ and all Γ, Γ , Δ, Δ , Π ⊆ Σ.3 Theory generated by a classification: let A be a classification. A token a ∈ tok(A) satisfies a sequent Γ, Δ of typ(A) provided that if a is of every type in Γ then it is of some type in Δ. The theory generated by A, denoted by Th(A), is the theory typ(A), A where Γ A Δ if every token in A satisfies Γ, Δ . Local logic: is a tuple L = tok(L), typ(L), |=L , L , NL where: 1. tok(L), typ(L), |=L is a classification denoted by Cla(L), 2. typ(L), L is a regular theory denoted by Th(L), 3. NL is a subset of tok(L), called the normal tokens of L, which satisfy all constraints of Th(L). A local logic L is sound if every token in Cla(L) is normal, that is, NL = tok(L). L is complete if every sequent of typ(L) satisfied by every normal token is a constraint of Th(L). Local logic generated by a classification: given a classification A, the local logic generated by A, written Log(A), is the local logic on A (i.e., Cla(Log(A)) = A), with Th(Log(A)) = Th(A) and such that all its tokens are normal, i.e., NLog(A) = tok(A). Inverse image: given an infomorphism f : A → B and a local logic L on B, the inverse image of L under f, denoted f−1 [L], is the local logic on A such that Γ f−1[L] Δ if ˆf[Γ] L ˆf[Δ] and Nf−1[L] = ˇf[NL ] = {a ∈ tok(A) | a = ˇf(b) for some b ∈ NL }. Distributed logic: let C = {fi : Ai → C}i∈{1,2} be a channel and L a local logic on its core C, the distributed logic of C generated by L, written DLogC(L), is the inverse image of L under the sum f1 + f2. Refinement: let C = {fi : Ai → C}i∈{1,2} and C = {fi : Ai → C }i∈{1,2} be two channels with the same component classifications A1 and A2. A refinement infomorphism from C to C is an infomorphism r : C → C such that for each i ∈ {1, 2}, fi = r ◦fi (i.e., ˆfi = ˆr ◦ ˆfi and ˇfi = ˇfi ◦ˇr). Channel C is a refinement of C if there exists a refinement infomorphism r from C to C. B. CHANNEL THEORY THEOREMS Theorem B.1. The logic generated by a classification is sound and complete. Furthermore, given a classification A and a logic L on A, L is sound and complete if and only if L = Log(A). Theorem B.2. Let L be a logic on a classification B and f : A → B an infomorphism. 1. If L is complete then f−1 [L] is complete. 2. If L is sound and ˇf is surjective then f−1 [L] is sound. 3 All theories considered in this paper are regular. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1285

constraint;information-channel refinement;ontology;distributed logic;semantic alignment;distribute logic;knowledge-based system;channel refinement;sum infomorphism;multi-agent system;semantic web;disjoint union;federated database

train_I-72

Learning Consumer Preferences Using Semantic Similarity

In online, dynamic environments, the services requested by consumers may not be readily served by the providers. This requires the service consumers and providers to negotiate their service needs and offers. Multiagent negotiation approaches typically assume that the parties agree on service content and focus on finding a consensus on service price. In contrast, this work develops an approach through which the parties can negotiate the content of a service. This calls for a negotiation approach in which the parties can understand the semantics of their requests and offers and learn each other"s preferences incrementally over time. Accordingly, we propose an architecture in which both consumers and producers use a shared ontology to negotiate a service. Through repetitive interactions, the provider learns consumers" needs accurately and can make better targeted offers. To enable fast and accurate learning of preferences, we develop an extension to Version Space and compare it with existing learning techniques. We further develop a metric for measuring semantic similarity between services and compare the performance of our approach using different similarity metrics.

1. INTRODUCTION Current approaches to e-commerce treat service price as the primary construct for negotiation by assuming that the service content is fixed [9]. However, negotiation on price presupposes that other properties of the service have already been agreed upon. Nevertheless, many times the service provider may not be offering the exact requested service due to lack of resources, constraints in its business policy, and so on [3]. When this is the case, the producer and the consumer need to negotiate the content of the requested service [15]. However, most existing negotiation approaches assume that all features of a service are equally important and concentrate on the price [5, 2]. However, in reality not all features may be relevant and the relevance of a feature may vary from consumer to consumer. For instance, completion time of a service may be important for one consumer whereas the quality of the service may be more important for a second consumer. Without doubt, considering the preferences of the consumer has a positive impact on the negotiation process. For this purpose, evaluation of the service components with different weights can be useful. Some studies take these weights as a priori and uses the fixed weights [4]. On the other hand, mostly the producer does not know the consumer"s preferences before the negotiation. Hence, it is more appropriate for the producer to learn these preferences for each consumer. Preference Learning: As an alternative, we propose an architecture in which the service providers learn the relevant features of a service for a particular customer over time. We represent service requests as a vector of service features. We use an ontology in order to capture the relations between services and to construct the features for a given service. By using a common ontology, we enable the consumers and producers to share a common vocabulary for negotiation. The particular service we have used is a wine selling service. The wine seller learns the wine preferences of the customer to sell better targeted wines. The producer models the requests of the consumer and its counter offers to learn which features are more important for the consumer. Since no information is present before the interactions start, the learning algorithm has to be incremental so that it can be trained at run time and can revise itself with each new interaction. Service Generation: Even after the producer learns the important features for a consumer, it needs a method to generate offers that are the most relevant for the consumer among its set of possible services. In other words, the question is how the producer uses the information that was learned from the dialogues to make the best offer to the consumer. For instance, assume that the producer has learned that the consumer wants to buy a red wine but the producer can only offer rose or white wine. What should the producer"s offer 1301 978-81-904262-7-5 (RPS) c 2007 IFAAMAS contain; white wine or rose wine? If the producer has some domain knowledge about semantic similarity (e.g., knows that the red and rose wines are taste-wise more similar than white wine), then it can generate better offers. However, in addition to domain knowledge, this derivation requires appropriate metrics to measure similarity between available services and learned preferences. The rest of this paper is organized as follows: Section 2 explains our proposed architecture. Section 3 explains the learning algorithms that were studied to learn consumer preferences. Section 4 studies the different service offering mechanisms. Section 5 contains the similarity metrics used in the experiments. The details of the developed system is analyzed in Section 6. Section 7 provides our experimental setup, test cases, and results. Finally, Section 8 discusses and compares our work with other related work. 2. ARCHITECTURE Our main components are consumer and producer agents, which communicate with each other to perform content-oriented negotiation. Figure 1 depicts our architecture. The consumer agent represents the customer and hence has access to the preferences of the customer. The consumer agent generates requests in accordance with these preferences and negotiates with the producer based on these preferences. Similarly, the producer agent has access to the producer"s inventory and knows which wines are available or not. A shared ontology provides the necessary vocabulary and hence enables a common language for agents. This ontology describes the content of the service. Further, since an ontology can represent concepts, their properties and their relationships semantically, the agents can reason the details of the service that is being negotiated. Since a service can be anything such as selling a car, reserving a hotel room, and so on, the architecture is independent of the ontology used. However, to make our discussion concrete, we use the well-known Wine ontology [19] with some modification to illustrate our ideas and to test our system. The wine ontology describes different types of wine and includes features such as color, body, winery of the wine and so on. With this ontology, the service that is being negotiated between the consumer and the producer is that of selling wine. The data repository in Figure 1 is used solely by the producer agent and holds the inventory information of the producer. The data repository includes information on the products the producer owns, the number of the products and ratings of those products. Ratings indicate the popularity of the products among customers. Those are used to decide which product will be offered when there exists more than one product having same similarity to the request of the consumer agent. The negotiation takes place in a turn-taking fashion, where the consumer agent starts the negotiation with a particular service request. The request is composed of significant features of the service. In the wine example, these features include color, winery and so on. This is the particular wine that the customer is interested in purchasing. If the producer has the requested wine in its inventory, the producer offers the wine and the negotiation ends. Otherwise, the producer offers an alternative wine from the inventory. When the consumer receives a counter offer from the producer, it will evaluate it. If it is acceptable, then the negotiation will end. Otherwise, the customer will generate a new request or stick to the previous request. This process will continue until some service is accepted by the consumer agent or all possible offers are put forward to the consumer by the producer. One of the crucial challenges of the content-oriented negotiation is the automatic generation of counter offers by the service producer. When the producer constructs its offer, it should consider Figure 1: Proposed Negotiation Architecture three important things: the current request, consumer preferences and the producer"s available services. Both the consumer"s current request and the producer"s own available services are accessible by the producer. However, the consumer"s preferences in most cases will not be available. Hence, the producer will have to understand the needs of the consumer from their interactions and generate a counter offer that is likely to be accepted by the consumer. This challenge can be studied in three stages: • Preference Learning: How can the producers learn about each customer"s preferences based on requests and counter offers? (Section 3) • Service Offering: How can the producers revise their offers based on the consumer"s preferences that they have learned so far? (Section 4) • Similarity Estimation: How can the producer agent estimate similarity between the request and available services? (Section 5) 3. PREFERENCE LEARNING The requests of the consumer and the counter offers of the producer are represented as vectors, where each element in the vector corresponds to the value of a feature. The requests of the consumers represent individual wine products whereas their preferences are constraints over service features. For example, a consumer may have preference for red wine. This means that the consumer is willing to accept any wine offered by the producers as long as the color is red. Accordingly, the consumer generates a request where the color feature is set to red and other features are set to arbitrary values, e.g. (Medium, Strong, Red). At the beginning of negotiation, the producer agent does not know the consumer"s preferences but will need to learn them using information obtained from the dialogues between the producer and the consumer. The preferences denote the relative importance of the features of the services demanded by the consumer agents. For instance, the color of the wine may be important so the consumer insists on buying the wine whose color is red and rejects all 1302 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) Table 1: How DCEA works Type Sample The most The most general set specific set + (Full,Strong,White) {(?, ?, ?)} {(Full,Strong,White)} {{(?-Full), ?, ? }, - (Full,Delicate,Rose) {?, (?-Delicate), ?}, {(Full,Strong,White)} {?, ?, (?-Rose)}} {{(?-Full), ?, ?}, {{(Full,Strong,White)}, + (Medium,Moderate,Red) {?,(?-Delicate), ?}, {(Medium,Moderate,Red)}} {?, ?, (?-Rose)}} the offers involving the wine whose color is white or rose. On the contrary, the winery may not be as important as the color for this customer, so the consumer may have a tendency to accept wines from any winery as long as the color is red. To tackle this problem, we propose to use incremental learning algorithms [6]. This is necessary since no training data is available before the interactions start. We particularly investigate two approaches. The first one is inductive learning. This technique is applied to learn the preferences as concepts. We elaborate on Candidate Elimination Algorithm (CEA) for Version Space [10]. CEA is known to perform poorly if the information to be learned is disjunctive. Interestingly, most of the time consumer preferences are disjunctive. Say, we are considering an agent that is buying wine. The consumer may prefer red wine or rose wine but not white wine. To use CEA with such preferences, a solid modification is necessary. The second approach is decision trees. Decision trees can learn from examples easily and classify new instances as positive or negative. A well-known incremental decision tree is ID5R [18]. However, ID5R is known to suffer from high computational complexity. For this reason, we instead use the ID3 algorithm [13] and iteratively build decision trees to simulate incremental learning. 3.1 CEA CEA [10] is one of the inductive learning algorithms that learns concepts from observed examples. The algorithm maintains two sets to model the concept to be learned. The first set is the most general set G. G contains hypotheses about all the possible values that the concept may obtain. As the name suggests, it is a generalization and contains all possible values unless the values have been identified not to represent the concept. The second set is the most specific set S. S contains only hypotheses that are known to identify the concept that is being learned. At the beginning of the algorithm, G is initialized to cover all possible concepts while S is initialized to be empty. During the interactions, each request of the consumer can be considered as a positive example and each counter offer generated by the producer and rejected by the consumer agent can be thought of as a negative example. At each interaction between the producer and the consumer, both G and S are modified. The negative samples enforce the specialization of some hypotheses so that G does not cover any hypothesis accepting the negative samples as positive. When a positive sample comes, the most specific set S should be generalized in order to cover the new training instance. As a result, the most general hypotheses and the most special hypotheses cover all positive training samples but do not cover any negative ones. Incrementally, G specializes and S generalizes until G and S are equal to each other. When these sets are equal, the algorithm converges by means of reaching the target concept. 3.2 Disjunctive CEA Unfortunately, CEA is primarily targeted for conjunctive concepts. On the other hand, we need to learn disjunctive concepts in the negotiation of a service since consumer may have several alternative wishes. There are several studies on learning disjunctive concepts via Version Space. Some of these approaches use multiple version space. For instance, Hong et al. maintain several version spaces by split and merge operation [7]. To be able to learn disjunctive concepts, they create new version spaces by examining the consistency between G and S. We deal with the problem of not supporting disjunctive concepts of CEA by extending our hypothesis language to include disjunctive hypothesis in addition to the conjunctives and negation. Each attribute of the hypothesis has two parts: inclusive list, which holds the list of valid values for that attribute and exclusive list, which is the list of values which cannot be taken for that feature. EXAMPLE 1. Assume that the most specific set is {(Light, Delicate, Red)} and a positive example, (Light, Delicate, White) comes. The original CEA will generalize this as (Light, Delicate, ?), meaning the color can take any value. However, in fact, we only know that the color can be red or white. In the DCEA, we generalize it as {(Light, Delicate, [White, Red] )}. Only when all the values exist in the list, they will be replaced by ?. In other words, we let the algorithm generalize more slowly than before. We modify the CEA algorithm to deal with this change. The modified algorithm, DCEA, is given as Algorithm 1. Note that compared to the previous studies of disjunctive versions, our approach uses only a single version space rather than multiple version space. The initialization phase is the same as the original algorithm (lines 1, 2). If any positive sample comes, we add the sample to the special set as before (line 4). However, we do not eliminate the hypotheses in G that do not cover this sample since G now contains a disjunction of many hypotheses, some of which will be conflicting with each other. Removing a specific hypothesis from G will result in loss of information, since other hypotheses are not guaranteed to cover it. After some time, some hypotheses in S can be merged and can construct one hypothesis (lines 6, 7). When a negative sample comes, we do not change S as before. We only modify the most general hypotheses not to cover this negative sample (lines 11-15). Different from the original CEA, we try to specialize the G minimally. The algorithm removes the hypothesis covering the negative sample (line 13). Then, we generate new hypotheses as the number of all possible attributes by using the removed hypothesis. For each attribute in the negative sample, we add one of them at each time to the exclusive list of the removed hypothesis. Thus, all possible hypotheses that do not cover the negative sample are generated (line 14). Note that, exclusive list contains the values that the attribute cannot take. For example, consider the color attribute. If a hypothesis includes red in its exclusive list and ? in its inclusive list, this means that color may take any value except red. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1303 Algorithm 1 Disjunctive Candidate Elimination Algorithm 1: G ←the set of maximally general hypotheses in H 2: S ←the set of maximally specific hypotheses in H 3: For each training example, d 4: if d is a positive example then 5: Add d to S 6: if s in S can be combined with d to make one element then 7: Combine s and d into sd {sd is the rule covers s and d} 8: end if 9: end if 10: if d is a negative example then 11: For each hypothesis g in G does cover d 12: * Assume : g = (x1, x2, ..., xn) and d = (d1, d2, ..., dn) 13: - Remove g from G 14: - Add hypotheses g1, g2, gn where g1= (x1-d1, x2,..., xn), g2= (x1, x2-d2,..., xn),..., and gn= (x1, x2,..., xn-dn) 15: - Remove from G any hypothesis that is less general than another hypothesis in G 16: end if EXAMPLE 2. Table 1 illustrates the first three interactions and the workings of DCEA. The most general set and the most specific set show the contents of G and S after the sample comes in. After the first positive sample, S is generalized to also cover the instance. The second sample is negative. Thus, we replace (?, ?, ?) by three disjunctive hypotheses; each hypothesis being minimally specialized. In this process, at each time one attribute value of negative sample is applied to the hypothesis in the general set. The third sample is positive and generalizes S even more. Note that in Table 1, we do not eliminate {(?-Full), ?, ?} from the general set while having a positive sample such as (Full, Strong, White). This stems from the possibility of using this rule in the generation of other hypotheses. For instance, if the example continues with a negative sample (Full, Strong, Red), we can specialize the previous rule such as {(?-Full), ?, (?-Red)}. By Algorithm 1, we do not miss any information. 3.3 ID3 ID3 [13] is an algorithm that constructs decision trees in a topdown fashion from the observed examples represented in a vector with attribute-value pairs. Applying this algorithm to our system with the intention of learning the consumer"s preferences is appropriate since this algorithm also supports learning disjunctive concepts in addition to conjunctive concepts. The ID3 algorithm is used in the learning process with the purpose of classification of offers. There are two classes: positive and negative. Positive means that the service description will possibly be accepted by the consumer agent whereas the negative implies that it will potentially be rejected by the consumer. Consumer"s requests are considered as positive training examples and all rejected counter-offers are thought as negative ones. The decision tree has two types of nodes: leaf node in which the class labels of the instances are held and non-leaf nodes in which test attributes are held. The test attribute in a non-leaf node is one of the attributes making up the service description. For instance, body, flavor, color and so on are potential test attributes for wine service. When we want to find whether the given service description is acceptable, we start searching from the root node by examining the value of test attributes until reaching a leaf node. The problem with this algorithm is that it is not an incremental algorithm, which means all the training examples should exist before learning. To overcome this problem, the system keeps consumer"s requests throughout the negotiation interaction as positive examples and all counter-offers rejected by the consumer as negative examples. After each coming request, the decision tree is rebuilt. Without doubt, there is a drawback of reconstruction such as additional process load. However, in practice we have evaluated ID3 to be fast and the reconstruction cost to be negligible. 4. SERVICE OFFERING After learning the consumer"s preferences, the producer needs to make a counter offer that is compatible with the consumer"s preferences. 4.1 Service Offering via CEA and DCEA To generate the best offer, the producer agent uses its service ontology and the CEA algorithm. The service offering mechanism is the same for both the original CEA and DCEA, but as explained before their methods for updating G and S are different. When producer receives a request from the consumer, the learning set of the producer is trained with this request as a positive sample. The learning components, the most specific set S and the most general set G are actively used in offering service. The most general set, G is used by the producer in order to avoid offering the services, which will be rejected by the consumer agent. In other words, it filters the service set from the undesired services, since G contains hypotheses that are consistent with the requests of the consumer. The most specific set, S is used in order to find best offer, which is similar to the consumer"s preferences. Since the most specific set S holds the previous requests and the current request, estimating similarity between this set and every service in the service list is very convenient to find the best offer from the service list. When the consumer starts the interaction with the producer agent, producer agent loads all related services to the service list object. This list constitutes the provider"s inventory of services. Upon receiving a request, if the producer can offer an exactly matching service, then it does so. For example, for a wine this corresponds to selling a wine that matches the specified features of the consumer"s request identically. When the producer cannot offer the service as requested, it tries to find the service that is most similar to the services that have been requested by the consumer during the negotiation. To do this, the producer has to compute the similarity between the services it can offer and the services that have been requested (in S). We compute the similarities in various ways as will be explained in Section 5. After the similarity of the available services with the current S is calculated, there may be more than one service with the maximum similarity. The producer agent can break the tie in a number of ways. Here, we have associated a rating value with each service and the producer prefers the higher rated service to others. 4.2 Service Offering via ID3 If the producer learns the consumer"s preferences with ID3, a similar mechanism is applied with two differences. First, since ID3 does not maintain G, the list of unaccepted services that are classified as negative are removed from the service list. Second, the similarities of possible services are not measured with respect to S, but instead to all previously made requests. 4.3 Alternative Service Offering Mechanisms In addition to these three service offering mechanisms (Service Offering with CEA, Service Offering with DCEA, and Service Offering with ID3), we include two other mechanisms.. 1304 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) • Random Service Offering (RO): The producer generates a counter offer randomly from the available service list, without considering the consumer"s preferences. • Service Offering considering only the current request (SCR): The producer selects a counter offer according to the similarity of the consumer"s current request but does not consider previous requests. 5. SIMILARITY ESTIMATION Similarity can be estimated with a similarity metric that takes two entries and returns how similar they are. There are several similarity metrics used in case based reasoning system such as weighted sum of Euclidean distance, Hamming distance and so on [12]. The similarity metric affects the performance of the system while deciding which service is the closest to the consumer"s request. We first analyze some existing metrics and then propose a new semantic similarity metric named RP Similarity. 5.1 Tversky"s Similarity Metric Tversky"s similarity metric compares two vectors in terms of the number of exactly matching features [17]. In Equation (1), common represents the number of matched attributes whereas different represents the number of the different attributes. Our current assumption is that α and β is equal to each other. SMpq = α(common) α(common) + β(different) (1) Here, when two features are compared, we assign zero for dissimilarity and one for similarity by omitting the semantic closeness among the feature values. Tversky"s similarity metric is designed to compare two feature vectors. In our system, whereas the list of services that can be offered by the producer are each a feature vector, the most specific set S is not a feature vector. S consists of hypotheses of feature vectors. Therefore, we estimate the similarity of each hypothesis inside the most specific set S and then take the average of the similarities. EXAMPLE 3. Assume that S contains the following two hypothesis: { {Light, Moderate, (Red, White)} , {Full, Strong, Rose}}. Take service s as (Light, Strong, Rose). Then the similarity of the first one is equal to 1/3 and the second one is equal to 2/3 in accordance with Equation (1). Normally, we take the average of it and obtain (1/3 + 2/3)/2, equally 1/2. However, the first hypothesis involves the effect of two requests and the second hypothesis involves only one request. As a result, we expect the effect of the first hypothesis to be greater than that of the second. Therefore, we calculate the average similarity by considering the number of samples that hypotheses cover. Let ch denote the number of samples that hypothesis h covers and (SM(h,service)) denote the similarity of hypothesis h with the given service. We compute the similarity of each hypothesis with the given service and weight them with the number of samples they cover. We find the similarity by dividing the weighted sum of the similarities of all hypotheses in S with the service by the number of all samples that are covered in S. AV G−SM(service,S) = |S| |h| (ch ∗ SM(h,service)) |S| |h| ch (2) Figure 2: Sample taxonomy for similarity estimation EXAMPLE 4. For the above example, the similarity of (Light, Strong, Rose) with the specific set is (2 ∗ 1/3 + 2/3)/3, equally 4/9. The possible number of samples that a hypothesis covers can be estimated with multiplying cardinalities of each attribute. For example, the cardinality of the first attribute is two and the others is equal to one for the given hypothesis such as {Light, Moderate, (Red, White)}. When we multiply them, we obtain two (2 ∗ 1 ∗ 1 = 2). 5.2 Lin"s Similarity Metric A taxonomy can be used while estimating semantic similarity between two concepts. Estimating semantic similarity in a Is-A taxonomy can be done by calculating the distance between the nodes related to the compared concepts. The links among the nodes can be considered as distances. Then, the length of the path between the nodes indicates how closely similar the concepts are. An alternative estimation to use information content in estimation of semantic similarity rather than edge counting method, was proposed by Lin [8]. The equation (3) [8] shows Lin"s similarity where c1 and c2 are the compared concepts and c0 is the most specific concept that subsumes both of them. Besides, P(C) represents the probability of an arbitrary selected object belongs to concept C. Similarity(c1, c2) = 2 × log P(c0) log P(c1) + log P(c2) (3) 5.3 Wu & Palmer"s Similarity Metric Different from Lin, Wu and Palmer use the distance between the nodes in IS-A taxonomy [20]. The semantic similarity is represented with Equation (4) [20]. Here, the similarity between c1 and c2 is estimated and c0 is the most specific concept subsuming these classes. N1 is the number of edges between c1 and c0. N2 is the number of edges between c2 and c0. N0 is the number of IS-A links of c0 from the root of the taxonomy. SimW u&P almer(c1, c2) = 2 × N0 N1 + N2 + 2 × N0 (4) 5.4 RP Semantic Metric We propose to estimate the relative distance in a taxonomy between two concepts using the following intuitions. We use Figure 2 to illustrate these intuitions. • Parent versus grandparent: Parent of a node is more similar to the node than grandparents of that. Generalization of The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1305 a concept reasonably results in going further away that concept. The more general concepts are, the less similar they are. For example, AnyWineColor is parent of ReddishColor and ReddishColor is parent of Red. Then, we expect the similarity between ReddishColor and Red to be higher than that of the similarity between AnyWineColor and Red. • Parent versus sibling: A node would have higher similarity to its parent than to its sibling. For instance, Red and Rose are children of ReddishColor. In this case, we expect the similarity between Red and ReddishColor to be higher than that of Red and Rose. • Sibling versus grandparent: A node is more similar to it"s sibling then to its grandparent. To illustrate, AnyWineColor is grandparent of Red, and Red and Rose are siblings. Therefore, we possibly anticipate that Red and Rose are more similar than AnyWineColor and Red. As a taxonomy is represented in a tree, that tree can be traversed from the first concept being compared through the second concept. At starting node related to the first concept, the similarity value is constant and equal to one. This value is diminished by a constant at each node being visited over the path that will reach to the node including the second concept. The shorter the path between the concepts, the higher the similarity between nodes. Algorithm 2 Estimate-RP-Similarity(c1,c2) Require: The constants should be m > n > m2 where m, n ∈ R[0, 1] 1: Similarity ← 1 2: if c1 is equal to c2 then 3: Return Similarity 4: end if 5: commonParent ← findCommonParent(c1, c2) {commonParent is the most specific concept that covers both c1 and c2} 6: N1 ← findDistance(commonParent, c1) 7: N2 ← findDistance(commonParent, c2) {N1 & N2 are the number of links between the concept and parent concept} 8: if (commonParent == c1) or (commonParent == c2) then 9: Similarity ← Similarity ∗ m(N1+N2) 10: else 11: Similarity ← Similarity ∗ n ∗ m(N1+N2−2) 12: end if 13: Return Similarity Relative distance between nodes c1 and c2 is estimated in the following way. Starting from c1, the tree is traversed to reach c2. At each hop, the similarity decreases since the concepts are getting farther away from each other. However, based on our intuitions, not all hops decrease the similarity equally. Let m represent the factor for hopping from a child to a parent and n represent the factor for hopping from a sibling to another sibling. Since hopping from a node to its grandparent counts as two parent hops, the discount factor of moving from a node to its grandparent is m2 . According to the above intuitions, our constants should be in the form m > n > m2 where the value of m and n should be between zero and one. Algorithm 2 shows the distance calculation. According to the algorithm, firstly the similarity is initialized with the value of one (line 1). If the concepts are equal to each other then, similarity will be one (lines 2-4). Otherwise, we compute the common parent of the two nodes and the distance of each concept to the common parent without considering the sibling (lines 5-7). If one of the concepts is equal to the common parent, then there is no sibling relation between the concepts. For each level, we multiply the similarity by m and do not consider the sibling factor in the similarity estimation. As a result, we decrease the similarity at each level with the rate of m (line9). Otherwise, there has to be a sibling relation. This means that we have to consider the effect of n when measuring similarity. Recall that we have counted N1+N2 edges between the concepts. Since there is a sibling relation, two of these edges constitute the sibling relation. Hence, when calculating the effect of the parent relation, we use N1+N2 −2 edges (line 11). Some similarity estimations related to the taxonomy in Figure 2 are given in Table 2. In this example, m is taken as 2/3 and n is taken as 4/7. Table 2: Sample similarity estimation over sample taxonomy Similarity(ReddishColor, Rose) = 1 ∗ (2/3) = 0.6666667 Similarity(Red, Rose) = 1 ∗ (4/7) = 0.5714286 Similarity(AnyW ineColor,Rose) = 1 ∗ (2/3)2 = 0.44444445 Similarity(W hite,Rose) = 1 ∗ (2/3) ∗ (4/7) = 0.3809524 For all semantic similarity metrics in our architecture, the taxonomy for features is held in the shared ontology. In order to evaluate the similarity of feature vector, we firstly estimate the similarity for feature one by one and take the average sum of these similarities. Then the result is equal to the average semantic similarity of the entire feature vector. 6. DEVELOPED SYSTEM We have implemented our architecture in Java. To ease testing of the system, the consumer agent has a user interface that allows us to enter various requests. The producer agent is fully automated and the learning and service offering operations work as explained before. In this section, we explain the implementation details of the developed system. We use OWL [11] as our ontology language and JENA as our ontology reasoner. The shared ontology is the modified version of the Wine Ontology [19]. It includes the description of wine as a concept and different types of wine. All participants of the negotiation use this ontology for understanding each other. According to the ontology, seven properties make up the wine concept. The consumer agent and the producer agent obtain the possible values for the these properties by querying the ontology. Thus, all possible values for the components of the wine concept such as color, body, sugar and so on can be reached by both agents. Also a variety of wine types are described in this ontology such as Burgundy, Chardonnay, CheninBlanc and so on. Intuitively, any wine type described in the ontology also represents a wine concept. This allows us to consider instances of Chardonnay wine as instances of Wine class. In addition to wine description, the hierarchical information of some features can be inferred from the ontology. For instance, we can represent the information Europe Continent covers Western Country. Western Country covers French Region, which covers some territories such as Loire, Bordeaux and so on. This hierarchical information is used in estimation of semantic similarity. In this part, some reasoning can be made such as if a concept X covers Y and Y covers Z, then concept X covers Z. For example, Europe Continent covers Bordeaux. 1306 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) For some features such as body, flavor and sugar, there is no hierarchical information, but their values are semantically leveled. When that is the case, we give the reasonable similarity values for these features. For example, the body can be light, medium, or strong. In this case, we assume that light is 0.66 similar to medium but only 0.33 to strong. WineStock Ontology is the producer"s inventory and describes a product class as WineProduct. This class is necessary for the producer to record the wines that it sells. Ontology involves the individuals of this class. The individuals represent available services that the producer owns. We have prepared two separate WineStock ontologies for testing. In the first ontology, there are 19 available wine products and in the second ontology, there are 50 products. 7. PERFORMANCE EVALUATION We evaluate the performance of the proposed systems in respect to learning technique they used, DCEA and ID3, by comparing them with the CEA, RO (for random offering), and SCR (offering based on current request only). We apply a variety of scenarios on this dataset in order to see the performance differences. Each test scenario contains a list of preferences for the user and number of matches from the product list. Table 3 shows these preferences and availability of those products in the inventory for first five scenarios. Note that these preferences are internal to the consumer and the producer tries to learn these during negotiation. Table 3: Availability of wines in different test scenarios ID Preference of consumer Availability (out of 19) 1 Dry wine 15 2 Red and dry wine 8 3 Red, dry and moderate wine 4 4 Red and strong wine 2 5 Red or rose, and strong 3 7.1 Comparison of Learning Algorithms In comparison of learning algorithms, we use the five scenarios in Table 3. Here, first we use Tversky"s similarity measure. With these test cases, we are interested in finding the number of iterations that are required for the producer to generate an acceptable offer for the consumer. Since the performance also depends on the initial request, we repeat our experiments with different initial requests. Consequently, for each case, we run the algorithms five times with several variations of the initial requests. In each experiment, we count the number of iterations that were needed to reach an agreement. We take the average of these numbers in order to evaluate these systems fairly. As is customary, we test each algorithm with the same initial requests. Table 4 compares the approaches using different learning algorithm. When the large parts of inventory is compatible with the customer"s preferences as in the first test case, the performance of all techniques are nearly same (e.g., Scenario 1). As the number of compatible services drops, RO performs poorly as expected. The second worst method is SCR since it only considers the customer"s most recent request and does not learn from previous requests. CEA gives the best results when it can generate an answer but cannot handle the cases containing disjunctive preferences, such as the one in Scenario 5. ID3 and DCEA achieve the best results. Their performance is comparable and they can handle all cases including Scenario 5. Table 4: Comparison of learning algorithms in terms of average number of interactions Run DCEA SCR RO CEA ID3 Scenario 1: 1.2 1.4 1.2 1.2 1.2 Scenario 2: 1.4 1.4 2.6 1.4 1.4 Scenario 3: 1.4 1.8 4.4 1.4 1.4 Scenario 4: 2.2 2.8 9.6 1.8 2 Scenario 5: 2 2.6 7.6 1.75+ No offer 1.8 Avg. of all cases: 1.64 2 5.08 1.51+No offer 1.56 7.2 Comparison of Similarity Metrics To compare the similarity metrics that were explained in Section 5, we fix the learning algorithm to DCEA. In addition to the scenarios shown in Table 3, we add following five new scenarios considering the hierarchical information. • The customer wants to buy wine whose winery is located in California and whose grape is a type of white grape. Moreover, the winery of the wine should not be expensive. There are only four products meeting these conditions. • The customer wants to buy wine whose color is red or rose and grape type is red grape. In addition, the location of wine should be in Europe. The sweetness degree is wished to be dry or off dry. The flavor should be delicate or moderate where the body should be medium or light. Furthermore, the winery of the wine should be an expensive winery. There are two products meeting all these requirements. • The customer wants to buy moderate rose wine, which is located around French Region. The category of winery should be Moderate Winery. There is only one product meeting these requirements. • The customer wants to buy expensive red wine, which is located around California Region or cheap white wine, which is located in around Texas Region. There are five available products. • The customer wants to buy delicate white wine whose producer in the category of Expensive Winery. There are two available products. The first seven scenarios are tested with the first dataset that contains a total of 19 services and the last three scenarios are tested with the second dataset that contains 50 services. Table 5 gives the performance evaluation in terms of the number of interactions needed to reach a consensus. Tversky"s metric gives the worst results since it does not consider the semantic similarity. Lin"s performance are better than Tversky but worse than others. Wu Palmer"s metric and RP similarity measure nearly give the same performance and better than others. When the results are examined, considering semantic closeness increases the performance. 8. DISCUSSION We review the recent literature in comparison to our work. Tama et al. [16] propose a new approach based on ontology for negotiation. According to their approach, the negotiation protocols used in e-commerce can be modeled as ontologies. Thus, the agents can perform negotiation protocol by using this shared ontology without the need of being hard coded of negotiation protocol details. While The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1307 Table 5: Comparison of similarity metrics in terms of number of interactions Run Tversky Lin Wu Palmer RP Scenario 1: 1.2 1.2 1 1 Scenario 2: 1.4 1.4 1.6 1.6 Scenario 3: 1.4 1.8 2 2 Scenario 4: 2.2 1 1.2 1.2 Scenario 5: 2 1.6 1.6 1.6 Scenario 6: 5 3.8 2.4 2.6 Scenario 7: 3.2 1.2 1 1 Scenario 8: 5.6 2 2 2.2 Scenario 9: 2.6 2.2 2.2 2.6 Scenario 10: 4.4 2 2 1.8 Average of all cases: 2.9 1.82 1.7 1.76 Tama et al. model the negotiation protocol using ontologies, we have instead modeled the service to be negotiated. Further, we have built a system with which negotiation preferences can be learned. Sadri et al. study negotiation in the context of resource allocation [14]. Agents have limited resources and need to require missing resources from other agents. A mechanism which is based on dialogue sequences among agents is proposed as a solution. The mechanism relies on observe-think-action agent cycle. These dialogues include offering resources, resource exchanges and offering alternative resource. Each agent in the system plans its actions to reach a goal state. Contrary to our approach, Sadri et al."s study is not concerned with learning preferences of each other. Brzostowski and Kowalczyk propose an approach to select an appropriate negotiation partner by investigating previous multi-attribute negotiations [1]. For achieving this, they use case-based reasoning. Their approach is probabilistic since the behavior of the partners can change at each iteration. In our approach, we are interested in negotiation the content of the service. After the consumer and producer agree on the service, price-oriented negotiation mechanisms can be used to agree on the price. Fatima et al. study the factors that affect the negotiation such as preferences, deadline, price and so on, since the agent who develops a strategy against its opponent should consider all of them [5]. In their approach, the goal of the seller agent is to sell the service for the highest possible price whereas the goal of the buyer agent is to buy the good with the lowest possible price. Time interval affects these agents differently. Compared to Fatima et al. our focus is different. While they study the effect of time on negotiation, our focus is on learning preferences for a successful negotiation. Faratin et al. propose a multi-issue negotiation mechanism, where the service variables for the negotiation such as price, quality of the service, and so on are considered traded-offs against each other (i.e., higher price for earlier delivery) [4]. They generate a heuristic model for trade-offs including fuzzy similarity estimation and a hill-climbing exploration for possibly acceptable offers. Although we address a similar problem, we learn the preferences of the customer by the help of inductive learning and generate counter-offers in accordance with these learned preferences. Faratin et al. only use the last offer made by the consumer in calculating the similarity for choosing counter offer. Unlike them, we also take into account the previous requests of the consumer. In their experiments, Faratin et al. assume that the weights for service variables are fixed a priori. On the contrary, we learn these preferences over time. In our future work, we plan to integrate ontology reasoning into the learning algorithm so that hierarchical information can be learned from subsumption hierarchy of relations. Further, by using relationships among features, the producer can discover new knowledge from the existing knowledge. These are interesting directions that we will pursue in our future work. 9. REFERENCES [1] J. Brzostowski and R. Kowalczyk. On possibilistic case-based reasoning for selecting partners for multi-attribute agent negotiation. In Proceedings of the 4th Intl. Joint Conference on Autonomous Agents and MultiAgent Systems (AAMAS), pages 273-278, 2005. [2] L. Busch and I. Horstman. A comment on issue-by-issue negotiations. Games and Economic Behavior, 19:144-148, 1997. [3] J. K. Debenham. Managing e-market negotiation in context with a multiagent system. In Proceedings 21st International Conference on Knowledge Based Systems and Applied Artificial Intelligence, ES"2002:, 2002. [4] P. Faratin, C. Sierra, and N. R. Jennings. Using similarity criteria to make issue trade-offs in automated negotiations. Artificial Intelligence, 142:205-237, 2002. [5] S. Fatima, M. Wooldridge, and N. Jennings. Optimal agents for multi-issue negotiation. In Proceeding of the 2nd Intl. Joint Conference on Autonomous Agents and MultiAgent Systems (AAMAS), pages 129-136, 2003. [6] C. Giraud-Carrier. A note on the utility of incremental learning. AI Communications, 13(4):215-223, 2000. [7] T.-P. Hong and S.-S. Tseng. Splitting and merging version spaces to learn disjunctive concepts. IEEE Transactions on Knowledge and Data Engineering, 11(5):813-815, 1999. [8] D. Lin. An information-theoretic definition of similarity. In Proc. 15th International Conf. on Machine Learning, pages 296-304. Morgan Kaufmann, San Francisco, CA, 1998. [9] P. Maes, R. H. Guttman, and A. G. Moukas. Agents that buy and sell. Communications of the ACM, 42(3):81-91, 1999. [10] T. M. Mitchell. Machine Learning. McGraw Hill, NY, 1997. [11] OWL. OWL: Web ontology language guide, 2003. http://www.w3.org/TR/2003/CR-owl-guide-20030818/. [12] S. K. Pal and S. C. K. Shiu. Foundations of Soft Case-Based Reasoning. John Wiley & Sons, New Jersey, 2004. [13] J. R. Quinlan. Induction of decision trees. Machine Learning, 1(1):81-106, 1986. [14] F. Sadri, F. Toni, and P. Torroni. Dialogues for negotiation: Agent varieties and dialogue sequences. In ATAL 2001, Revised Papers, volume 2333 of LNAI, pages 405-421. Springer-Verlag, 2002. [15] M. P. Singh. Value-oriented electronic commerce. IEEE Internet Computing, 3(3):6-7, 1999. [16] V. Tamma, S. Phelps, I. Dickinson, and M. Wooldridge. Ontologies for supporting negotiation in e-commerce. Engineering Applications of Artificial Intelligence, 18:223-236, 2005. [17] A. Tversky. Features of similarity. Psychological Review, 84(4):327-352, 1977. [18] P. E. Utgoff. Incremental induction of decision trees. Machine Learning, 4:161-186, 1989. [19] Wine, 2003. http://www.w3.org/TR/2003/CR-owl-guide20030818/wine.rdf. [20] Z. Wu and M. Palmer. Verb semantics and lexical selection. In 32nd. Annual Meeting of the Association for Computational Linguistics, pages 133 -138, 1994. 1308 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07)

incremental decision tree;candidate elimination algorithm;rp similarity;disjunctive cea;consumer preference;disjunctive hypothesis;service;ontology;decision tree;semantic similarity;price;datum repository;negotiation;preference learning;inductive learn;multiple version space;learning set;similarity metric;consumer agent;id3

train_I-73

Exchanging Reputation Values among Heterogeneous Agent Reputation Models: An Experience on ART Testbed

In open MAS it is often a problem to achieve agents' interoperability. The heterogeneity of its components turns the establishment of interaction or cooperation among them into a non trivial task, since agents may use different internal models and the decision about trust other agents is a crucial condition to the formation of agents' cooperation. In this paper we propose the use of an ontology to deal with this issue. We experiment this idea by enhancing the ART reputation model with semantic data obtained from this ontology. This data is used during interaction among heterogeneous agents when exchanging reputation values and may be used for agents that use different reputation models.

1. INTRODUCTION Open multiagent systems (MAS) are composed of autonomous distributed agents that may enter and leave the agent society at their will because open systems have no centralized control over the development of its parts [1]. Since agents are considered as autonomous entities, we cannot assume that there is a way to control their internal behavior. These features are interesting to obtain flexible and adaptive systems but they also create new risks about the reliability and the robustness of the system. Solutions to this problem have been proposed by the way of trust models where agents are endowed with a model of other agents that allows them to decide if they can or cannot trust another agent. Such trust decision is very important because it is an essential condition to the formation of agents' cooperation. The trust decision processes use the concept of reputation as the basis of a decision. Reputation is a subject that has been studied in several works [4][5][8][9] with different approaches, but also with different semantics attached to the reputation concept. Casare and Sichman [2][3] proposed a Functional Ontology of Reputation (FORe) and some directions about how it could be used to allow the interoperability among different agent reputation models. This paper describes how the FORe can be applied to allow interoperability among agents that have different reputation models. An outline of this approach is sketched in the context of a testbed for the experimentation and comparison of trust models, the ART testbed [6]. 2. THE FUNCTIONAL ONTOLOGY OF REPUTATION (FORe) In the last years several computational models of reputation have been proposed [7][10][13][14]. As an example of research produced in the MAS field we refer to three of them: a cognitive reputation model [5], a typology of reputation [7] and the reputation model used in the ReGret system [9][10]. Each model includes its own specific concepts that may not exist in other models, or exist with a different name. For instance, Image and Reputation are two central concepts in the cognitive reputation model. These concepts do not exist in the typology of reputation or in the ReGret model. In the typology of reputation, we can find some similar concepts such as direct reputation and indirect reputation but there are some slight semantic differences. In the same way, the ReGret model includes four kinds of reputation (direct, witness, neighborhood and system) that overlap with the concepts of other models but that are not exactly the same. The Functional Ontology of Reputation (FORe) was defined as a common semantic basis that subsumes the concepts of the main reputation models. The FORe includes, as its kernel, the following concepts: reputation nature, roles involved in reputation formation and propagation, information sources for reputation, evaluation of reputation, and reputation maintenance. The ontology concept ReputationNature is composed of concepts such as IndividualReputation, GroupReputation and ProductReputation. Reputation formation and propagation involves several roles, played by the entities or agents that participate in those processes. The ontology defines the concepts ReputationProcess and ReputationRole. Moreover, reputation can be classified according to the origin of beliefs and opinions that can derive from several sources. The ontology defines the concept ReputationType which can be PrimaryReputation or SecondaryReputation. PrimaryReputation is composed of concepts ObservedReputation and DirectReputation and the concept SecondaryReputation is composed of concepts such as PropagatedReputation and CollectiveReputation. More details about the FORe can be found on [2][3]. 3. MAPPING THE AGENT REPUTATION MODELS TO THE FORe Visser et al [12] suggest three different ways to support semantic integration of different sources of information: a centralized approach, where each source of information is related to one common domain ontology; a decentralized approach, where every source of information is related to its own ontology; and a hybrid approach, where every source of information has its own ontology and the vocabulary of these ontologies are related to a common ontology. This latter organizes the common global vocabulary in order to support the source ontologies comparison. Casare and Sichman [3] used the hybrid approach to show that the FORe serves as a common ontology for several reputation models. Therefore, considering the ontologies which describe the agent reputation models we can define a mapping between these ontologies and the FORe whenever the ontologies use a common vocabulary. Also, the information concerning the mappings between the agent reputation models and the FORe can be directly inferred by simply classifying the resulting ontology from the integration of a given reputation model ontology and the FORe in an ontology tool with reasoning engine. For instance, a mapping between the Cognitive Reputation Model ontology and the FORe relates the concepts Image and Reputation to PrimaryReputation and SecondaryReputation from FORe, respectively. Also, a mapping between the Typology of Reputation and the FORe relates the concepts Direct Reputation and Indirect Reputation to PrimaryReputation and SecondaryReputation from FORe, respectively. Nevertheless, the concepts Direct Trust and Witness Reputation from the Regret System Reputation Model are mapped to PrimaryReputation and PropagatedReputation from FORe. Since PropagatedReputation is a sub-concept of SecondaryReputation, it can be inferred that Witness Reputation is also mapped to SecondaryReputation. 4. EXPERIMENTAL SCENARIOS USING THE ART TESTBED To exemplify the use of mappings from last section, we define a scenario where several agents are implemented using different agent reputation models. This scenario includes the agents" interaction during the simulation of the game defined by ART [6] in order to describe the ways interoperability is possible between different trust models using the FORe. 4.1 The ART testbed The ART testbed provides a simulation engine on which several agents, using different trust models, may run. The simulation consists in a game where the agents have to decide to trust or not other agents. The game"s domain is art appraisal, in which agents are required to evaluate the value of paintings based on information exchanged among other agents during agents" interaction. The information can be an opinion transaction, when an agent asks other agents to help it in its evaluation of a painting; or a reputation transaction, when the information required is about the reputation of another agent (a target) for a given era. More details about the ART testbed can be found in [6]. The ART common reputation model was enhanced with semantic data obtained from FORe. A general agent architecture for interoperability was defined [11] to allow agents to reason about the information received from reputation interactions. This architecture contains two main modules: the Reputation Mapping Module (RMM) which is responsible for mapping concepts between an agent reputation model and FORe; and the Reputation Reasoning Module (RRM) which is responsible for deal with information about reputation according to the agent reputation model. 4.2 Reputation transaction scenarios While including the FORe to the ART common reputation model, we have incremented it to allow richer interactions that involve reputation transaction. In this section we describe scenarios concerning reputation transactions in the context of ART testbed, but the first is valid for any kind of reputation transaction and the second is specific for the ART domain. 4.2.1 General scenario Suppose that agents A, B and C are implemented according to the aforementioned general agent architecture with the enhanced ART common reputation model, using different reputation models. Agent A uses the Typology of Reputation model, agent B uses the Cognitive Reputation Model and agent C uses the ReGret System model. Consider the interaction about reputation where agents A and B receive from agent C information about the reputation of agent Y. A big picture of this interaction is showed in Figure 2. ReGret Ontology (Y, value=0.8, witnessreputation) C Typol. Ontology (Y, value=0.8, propagatedreputation) A CogMod. Ontology (Y, value=0.8, reputation) B (Y, value=0.8, PropagatedReputation) (Y, value=0.8, PropagatedReputation) ReGret Ontology (Y, value=0.8, witnessreputation) C ReGret Ontology (Y, value=0.8, witnessreputation) ReGret Ontology (Y, value=0.8, witnessreputation) (Y, value=0.8, witnessreputation) C Typol. Ontology (Y, value=0.8, propagatedreputation) A Typol. Ontology (Y, value=0.8, propagatedreputation) Typol. Ontology (Y, value=0.8, propagatedreputation) (Y, value=0.8, propagatedreputation) A CogMod. Ontology (Y, value=0.8, reputation) B CogMod. Ontology (Y, value=0.8, reputation) CogMod. Ontology (Y, value=0.8, reputation) (Y, value=0.8, reputation) B (Y, value=0.8, PropagatedReputation) (Y, value=0.8, PropagatedReputation) (Y, value=0.8, PropagatedReputation) (Y, value=0.8, PropagatedReputation) Figure 1. Interaction about reputation The information witness reputation from agent C is treated by its RMM and is sent as PropagatedReputation to both agents. The corresponding information in agent A reputation model is propagated reputation and in agent B reputation model is reputation. The way agents A and B make use of the information depends on their internal reputation model and their RRM implementation. 1048 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 4.2.2 ART scenario Considering the same agents A and B and the art appraisal domain of ART, another interesting scenario describes the following situation: agent A asks to agent B information about agents it knows that have skill on some specific painting era. In this case agent A wants information concerning the direct reputation agent B has about agents that have skill on an specific era, such as cubism. Following the same steps of the previous scenario, agent A message is prepared in its RRM using information from its internal model. A big picture of this interaction is in Figure 2. Typol. Ontology (agent = ?, value = ?, skill = cubism, reputation = directreputation) A (agent = ?, value = ?, skill = cubism, reputation = PrimaryReputation) CogMod. Ontology (agent = ?, value = ?, skill = cubism, reputation = image) B Typol. Ontology (agent = ?, value = ?, skill = cubism, reputation = directreputation) A (agent = ?, value = ?, skill = cubism, reputation = PrimaryReputation) CogMod. Ontology (agent = ?, value = ?, skill = cubism, reputation = image) B Figure 2. Interaction about specific types of reputation values Agent B response to agent A is processed in its RRM and it is composed of tuples (agent, value, cubism, image) , where the pair (agent, value) is composed of all agents and associated reputation values whose agent B knows their expertise about cubism by its own opinion. This response is forwarded to the RMM in order to be translated to the enriched common model and to be sent to agent A. After receiving the information sent by agent B, agent A processes it in its RMM and translates it to its own reputation model to be analyzed by its RRM. 5. CONCLUSION In this paper we present a proposal for reducing the incompatibility between reputation models by using a general agent architecture for reputation interaction which relies on a functional ontology of reputation (FORe), used as a globally shared reputation model. A reputation mapping module allows agents to translate information from their internal reputation model into the shared model and vice versa. The ART testbed has been enriched to use the ontology during agent transactions. Some scenarios were described to illustrate our proposal and they seem to be a promising way to improve the process of building reputation just using existing technologies. 6. ACKNOWLEDGMENTS Anarosa A. F. Brandão is supported by CNPq/Brazil grant 310087/2006-6 and Jaime Sichman is partially supported by CNPq/Brazil grants 304605/2004-2, 482019/2004-2 and 506881/2004-1. Laurent Vercouter was partially supported by FAPESP grant 2005/02902-5. 7. REFERENCES [1] Agha, G. A. Abstracting Interaction Patterns: A Programming Paradigm for Open Distributed Systems, In (Eds) E. Najm and J.-B. Stefani, Formal Methods for Open Object-based Distributed Systems IFIP Transactions, 1997, Chapman Hall. [2] Casare,S. and Sichman, J.S. Towards a Functional Ontology of Reputation, In Proc of the 4th Intl Joint Conference on Autonomous Agents and Multi Agent Systems (AAMAS"05), Utrecht, The Netherlands, 2005, v.2, pp. 505-511. [3] Casare, S. and Sichman, J.S. Using a Functional Ontology of Reputation to Interoperate Different Agent Reputation Models, Journal of the Brazilian Computer Society, (2005), 11(2), pp. 79-94. [4] Castelfranchi, C. and Falcone, R. Principles of trust in MAS: cognitive anatomy, social importance and quantification. In Proceedings of ICMAS"98, Paris, 1998, pp. 72-79. [5] Conte, R. and Paolucci, M. Reputation in Artificial Societies: Social Beliefs for Social Order, Kluwer Publ., 2002. [6] Fullam, K.; Klos, T.; Muller, G.; Sabater, J.; Topol, Z.; Barber, S.;Rosenchein, J.; Vercouter, L. and Voss, M. A specification of the agent reputation and trust (art) testbed: experimentation and competition for trust in agent societies. In Proc. of the 4th Intl. Joint Conf on Autonomous Agents and Multiagent Systems (AAMAS"05), ACM, 2005, 512-158. [7] Mui, L.; Halberstadt, A.; Mohtashemi, M. Notions of Reputation in Multi-Agents Systems: A Review. In: Proc of 1st Intl. Joint Conf. on Autonomous Agents and Multi-agent Systems (AAMAS 2002), Bologna, Italy, 2002, 1, 280-287. [8] Muller, G. and Vercouter, L. Decentralized monitoring of agent communication with a reputation model. In Trusting Agents for Trusting Electronic Societies, LNCS 3577, 2005, pp. 144-161. [9] Sabater, J. and Sierra, C. ReGret: Reputation in gregarious societies. In Müller, J. et al (Eds) Proc. of the 5th Intl. Conf. on Autonomous Agents, Canada, 2001, ACM, 194-195. [10] Sabater, J. and Sierra, C. Review on Computational Trust and Reputation Models. In: Artificial Intelligence Review, Kluwer Acad. Publ., (2005), v. 24, n. 1, pp. 33 - 60. [11] Vercouter,L, Casare, S., Sichman, J. and Brandão, A. An experience on reputation models interoperability based on a functional ontology In Proc. of the 20th IJCAI, Hyderabad, India, 2007, pp.617-622. [12] Visser, U.; Stuckenschmidt, H.; Wache, H. and Vogele, T. Enabling technologies for inter-operability. In: In U. Visser and H. Pundt, Eds, Workshop on the 14th Intl Symp. of Computer Science for Environmental Protection, Bonn, Germany, 2000, pp. 35-46. [13] Yu, B. and Singh, M.P. An Evidential Model of Distributed Reputation Management. In: Proc. of the 1st Intl Joint Conf. on Autonomous Agents and Multi-agent Systems (AAMAS 2002), Bologna, Italy, 2002, part 1, pp. 294 - 301. [14] Zacharia, G. and Maes, P. Trust Management Through Reputation Mechanisms. In: Applied Artificial Intelligence, 14(9), 2000, pp. 881-907. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1049

interoperability;reputation model;agent architecture;functional ontology of reputation;ontology;heterogeneous agent;reputation value;autonomous distributed agent;reputation formation;reputation;art testbed;trust;art testb;multiagent system

train_I-74

On the relevance of utterances in formal inter-agent dialogues

Work on argumentation-based dialogue has defined frameworks within which dialogues can be carried out, established protocols that govern dialogues, and studied different properties of dialogues. This work has established the space in which agents are permitted to interact through dialogues. Recently, there has been increasing interest in the mechanisms agents might use to choose how to act - the rhetorical manoeuvring that they use to navigate through the space defined by the rules of the dialogue. Key in such considerations is the idea of relevance, since a usual requirement is that agents stay focussed on the subject of the dialogue and only make relevant remarks. Here we study several notions of relevance, showing how they can be related to both the rules for carrying out dialogues and to rhetorical manoeuvring.

1. INTRODUCTION Finding ways for agents to reach agreements in multiagent systems is an area of active research. One mechanism for achieving agreement is through the use of argumentation - where one agent tries to convince another agent of something during the course of some dialogue. Early examples of argumentation-based approaches to multiagent agreement include the work of Dignum et al. [7], Kraus [14], Parsons and Jennings [16], Reed [23], Schroeder et al. [25] and Sycara [26]. The work of Walton and Krabbe [27], popularised in the multiagent systems community by Reed [23], has been particularly influential in the field of argumentation-based dialogue. This work influenced the field in a number of ways, perhaps most deeply in framing multi-agent interactions as dialogue games in the tradition of Hamblin [13]. Viewing dialogues in this way, as in [2, 21], provides a powerful framework for analysing the formal properties of dialogues, and for identifying suitable protocols under which dialogues can be conducted [18, 20]. The dialogue game view overlaps with work on conversation policies (see, for example, [6, 10]), but differs in considering the entire dialogue rather than dialogue segments. In this paper, we extend the work of [18] by considering the role of relevance - the relationship between utterances in a dialogue. Relevance is a topic of increasing interest in argumentation-based dialogue because it relates to the scope that an agent has for applying strategic manoeuvering to obtain the outcomes that it requires [19, 22, 24]. Our work identifes the limits on such rhetorical manoeuvering, showing when it can and cannot have an effect. 2. BACKGROUND We begin by introducing the formal system of argumentation that underpins our approach, as well as the corresponding terminology and notation, all taken from [2, 8, 17]. A dialogue is a sequence of messages passed between two or more members of a set of agents A. An agent α maintains a knowledge base, Σα, containing formulas of a propositional language L and having no deductive closure. Agent α also maintains the set of its past utterances, called the commitment store, CSα. We refer to this as an agent"s public knowledge, since it contains information that is shared with other agents. In contrast, the contents of Σα are private to α. Note that in the description that follows, we assume that is the classical inference relation, that ≡ stands for logical equivalence, and we use Δ to denote all the information available to an agent. Thus in a dialogue between two agents α and β, Δα = Σα ∪ CSα ∪ CSβ, so the commitment store CSα can be loosely thought of as a subset of Δα consisting of the assertions that have been made public. In some dialogue games, such as those in [18] anything in CSα is either in Σα or can be derived from it. In other dialogue games, such as 1006 978-81-904262-7-5 (RPS) c 2007 IFAAMAS those in [2], CSα may contain things that cannot be derived from Σα. Definition 2.1. An argument A is a pair (S, p) where p is a formula of L and S a subset of Δ such that (i) S is consistent; (ii) S p; and (iii) S is minimal, so no proper subset of S satisfying both (1) and (2) exists. S is called the support of A, written S = Support(A) and p is the conclusion of A, written p = Conclusion(A). Thus we talk of p being supported by the argument (S, p). In general, since Δ may be inconsistent, arguments in A(Δ), the set of all arguments which can be made from Δ, may conflict, and we make this idea precise with the notion of undercutting: Definition 2.2. Let A1 and A2 be arguments in A(Δ). A1 undercuts A2 iff ∃¬p ∈ Support(A2) such that p ≡ Conclusion(A1). In other words, an argument is undercut if and only if there is another argument which has as its conclusion the negation of an element of the support for the first argument. To capture the fact that some beliefs are more strongly held than others, we assume that any set of beliefs has a preference order over it. We consider all information available to an agent, Δ, to be stratified into non-overlapping subsets Δ1, . . . , Δn such that beliefs in Δi are all equally preferred and are preferred over elements in Δj where i > j. The preference level of a nonempty subset S ⊂ Δ, where different elements s ∈ S may belong to different layers Δi, is valued at the highest numbered layer which has a member in S and is referred to as level(S). In other words, S is only as strong as its weakest member. Note that the strength of a belief as used in this context is a separate concept from the notion of support discussed earlier. Definition 2.3. Let A1 and A2 be arguments in A(Δ). A1 is preferred to A2 according to Pref , A1 Pref A2, iff level(Support(A1)) > level(Support(A2)). If A1 is preferred to A2, we say that A1 is stronger than A2. We can now define the argumentation system we will use: Definition 2.4. An argumentation system is a triple: A(Δ), Undercut, Pref such that: • A(Δ) is a set of the arguments built from Δ, • Undercut is a binary relation representing the defeat relationship between arguments, Undercut ⊆ A(Δ) × A(Δ), and • Pref is a pre-ordering on A(Δ) × A(Δ). The preference order makes it possible to distinguish different types of relations between arguments: Definition 2.5. Let A1, A2 be two arguments of A(Δ). • If A2 undercuts A1 then A1 defends itself against A2 iff A1 Pref A2. Otherwise, A1 does not defend itself. • A set of arguments A defends A1 iff for every A2 that undercuts A1, where A1 does not defend itself against A2, then there is some A3 ∈ A such that A3 undercuts A2 and A2 does not defend itself against A3. We write AUndercut,Pref to denote the set of all non-undercut arguments and arguments defending themselves against all their undercutting arguments. The set A(Δ) of acceptable arguments of the argumentation system A(Δ), Undercut, Pref is [1] the least fixpoint of a function F: A ⊆ A(Δ) F(A) = {(S, p) ∈ A(Δ) | (S, p) is defended by A} Definition 2.6. The set of acceptable arguments for an argumentation system A(Δ), Undercut, Pref is recursively defined as: A(Δ) = [ Fi≥0(∅) = AUndercut,Pref ∪ h[ Fi≥1(AUndercut,Pref ) i An argument is acceptable if it is a member of the acceptable set, and a proposition is acceptable if it is the conclusion of an acceptable argument. An acceptable argument is one which is, in some sense, proven since all the arguments which might undermine it are themselves undermined. Definition 2.7. If there is an acceptable argument for a proposition p, then the status of p is accepted, while if there is not an acceptable argument for p, the status of p is not accepted. Argument A is said to affect the status of another argument A if changing the status of A will change the status of A . 3. DIALOGUES Systems like those described in [2, 18], lay down sets of locutions that agents can make to put forward propositions and the arguments that support them, and protocols that define precisely which locutions can be made at which points in the dialogue. We are not concerned with such a level of detail here. Instead we are interested in the interplay between arguments that agents put forth. As a result, we will consider only that agents are allowed to put forward arguments. We do not discuss the detail of the mechanism that is used to put these arguments forward - we just assume that arguments of the form (S, p) are inserted into an agent"s commitment store where they are then visible to other agents. We then have a typical definition of a dialogue: Definition 3.1. A dialogue D is a sequence of moves: m1, m2, . . . , mn. A given move mi is a pair α, Ai where Ai is an argument that α places into its commitment store CSα. Moves in an argumentation-based dialogue typically attack moves that have been made previously. While, in general, a dialogue can include moves that undercut several arguments, in the remainder of this paper, we will only consider dialogues that put forward moves that undercut at most one argument. For now we place no additional constraints on the moves that make up a dialogue. Later we will see how different restrictions on moves lead to different kinds of dialogue. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1007 The sequence of arguments put forward in the dialogue is determined by the agents who are taking part in the dialogue, but they are usually not completely free to choose what arguments they make. As indicated earlier, their choice is typically limited by a protocol. If we write the sequence of n moves m1, m2, . . . , mn as mn, and denote the empty sequence as m0, then we can define a profocol in the following way: Definition 3.2. A protocol P is a function on a sequence of moves mi in a dialogue D that, for all i ≥ 0, identifies a set of possible moves Mi+1 from which the mi+1th move may be drawn: P : mi → Mi+1 In other words, for our purposes here, at every point in a dialogue, a protocol determines a set of possible moves that agents may make as part of the dialogue. If a dialogue D always picks its moves m from the set M identified by protocol P, then D is said to conform to P. Even if a dialogue conforms to a protocol, it is typically the case that the agent engaging in the dialogue has to make a choice of move - it has to choose which of the moves in M to make. This excercise of choice is what we refer to as an agent"s use of rhetoric (in its oratorical sense of influencing the thought and conduct of an audience). Some of our results will give a sense of how much scope an agent has to exercise rhetoric under different protocols. As arguments are placed into commitment stores, and hence become public, agents can determine the relationships between them. In general, after several moves in a dialogue, some arguments will undercut others. We will denote the set of arguments {A1, A2, . . . , Aj} asserted after moves m1, m2, . . . , mj of a dialogue to be Aj - the relationship of the arguments in Aj can be described as an argumentation graph, similar to those described in, for example, [3, 4, 9]: Definition 3.3. An argumentation graph AG over a set of arguments A is a directed graph (V, E) such that every vertex v, v ∈ V denotes one argument A ∈ A, every argument A is denoted by one vertex v, and every directed edge e ∈ E from v to v denotes that v undercuts v . We will use the term argument graph as a synonym for argumentation graph. Note that we do not require that the argumentation graph is connected. In other words the notion of an argumentation graph allows for the representation of arguments that do not relate, by undercutting or being undercut, to any other arguments (we will come back to this point very shortly). We adapt some standard graph theoretic notions in order to describe various aspects of the argumentation graph. If there is an edge e from vertex v to vertex v , then v is said to be the parent of v and v is said to be the child of v. In a reversal of the usual notion, we define a root of an argumentation graph1 as follows: Definition 3.4. A root of an argumentation graph AG = (V, E) is a node v ∈ V that has no children. Thus a root of a graph is a node to which directed edges may be connected, but from which no directed edges connect to other nodes. Thus a root is a node representing an 1 Note that we talk of a root rather than the root - as defined, an argumentation graph need not be a tree. v v" Figure 1: An example argument graph argument that is undercut, but which itself does no undercutting. Similarly: Definition 3.5. A leaf of an argumentation graph AG = (V, E) is a node v ∈ V that has no parents. Thus a leaf in an argumentation graph represents an argument that undercuts another argument, but does no undercutting. Thus in Figure 1, v is a root, and v is a leaf. The reason for the reversal of the usual notions of root and leaf is that, as we shall see, we will consider dialogues to construct argumentation graphs from the roots (in our sense) to the leaves. The reversal of the terminology means that it matches the natural process of tree construction. Since, as described above, argumentation graphs are allowed to be not connected (in the usual graph theory sense), it is helpful to distinguish nodes that are connected to other nodes, in particular to the root of the tree. We say that node v is connected to node v if and only if there is a path from v to v . Since edges represent undercut relations, the notion of connectedness between nodes captures the influence that one argument may have on another: Proposition 3.1. Given an argumentation graph AG, if there is any argument A, denoted by node v that affects the status of another argument A , denoted by v , then v is connected to v . The converse does not hold. Proof. Given Definitions 2.5 and 2.6, the only ways in which A can affect the status of A is if A either undercuts A , or if A undercuts some argument A that undercuts A , or if A undercuts some A that undercuts some A that undercuts A , and so on. In all such cases, a sequence of undercut relations relates the two arguments, and if they are both in an argumentation graph, this means that they are connected. Since the notion of path ignores the direction of the directed arcs, nodes v and v are connected whether the edge between them runs from v to v or vice versa. Since A only undercuts A if the edge runs from v to v , we cannot infer that A will affect the status of A from information about whether or not they are connected. The reason that we need the concept of the argumentation graph is that the properties of the argumentation graph tell us something about the set of arguments A the graph represents. When that set of arguments is constructed through a dialogue, there is a relationship between the structure of the argumentation graph and the protocol that governs the dialogue. It is the extent of the relationship between structure and protocol that is the main subject of this paper. To study this relationship, we need to establish a correspondence between a dialogue and an argumentation graph. Given the definitions we have so far, this is simple: Definition 3.6. A dialogue D, consisting of a sequence of moves mn, and an argument graph AG = (V, E) correspond to one another iff ∀m ∈ mn, the argument Ai that 1008 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) is advanced at move mi is represented by exactly one node v ∈ V , and ∀v ∈ V , v represents exactly one argument Ai that has been advanced by a move m ∈ mn. Thus a dialogue corresponds to an argumentation graph if and only if every argument made in the dialogue corresponds to a node in the graph, and every node in the graph corresponds to an argument made in the dialogue. This one-toone correspondence allows us to consider each node v in the graph to have an index i which is the index of the move in the dialogue that put forward the argument which that node represents. Thus we can, for example, refer to the third node in the argumentation graph, meaning the node that represents the argument put forward in the third move of the dialogue. 4. RELEVANCE Most work on dialogues is concerned with what we might call coherent dialogues, that is dialogues in which the participants are, as in the work of Walton and Krabbe [27], focused on resolving some question through the dialogue2 To capture this coherence, it seems we need a notion of relevance to constrain the statements made by agents. Here we study three notions of relevance: Definition 4.1. Consider a dialogue D, consisting of a sequence of moves mi, with a corresponding argument graph AG. The move mi+1, i > 1, is said to be relevant if one or more of the following hold: R1 Making mi+1 will change the status of the argument denoted by the first node of AG. R2 Making mi+1 will add a node vi+1 that is connected to the first node of AG. R3 Making mi+1 will add a node vi+1 that is connected to the last node to be added to AG. R2-relevance is the form of relevance defined by [3] in their study of strategic and tactical reasoning3 . R1-relevance was suggested by the notion used in [15], and though it differs somewhat from that suggested there, we believe it captures the essence of its predecessor. Note that we only define relevance for the second move of the dialogue onwards because the first move is taken to identify the subject of the dialogue, that is, the central question that the dialogue is intended to answer, and hence it must be relevant to the dialogue, no matter what it is. In assuming this, we focus our attention on the same kind of dialogues as [18]. We can think of relevance as enforcing a form of parsimony on a dialogue - it prevents agents from making statements that do not bear on the current state of the dialogue. This promotes efficiency, in the sense of limiting the number of moves in the dialogue, and, as in [15], prevents agents revealing information that they might better keep hidden. Another form of parsimony is to insist that agents are not allowed to put forward arguments that will be undercut by arguments that have already been made during the dialogue. We therefore distinguish such arguments. 2 See [11, 12] for examples of dialogues where this is not the case. 3 We consider such reasoning sub-types of rhetoric. Definition 4.2. Consider a dialogue D, consisting of a sequence of moves mi, with a corresponding argument graph AG. The move mi+1 and the argument it puts forward, Ai+1, are both said to be pre-empted, if Ai+1 is undercut by some A ∈ Ai. We use the term pre-empted because if such an argument is put forward, it can seem as though another agent anticipated the argument being made, and already made an argument that would render it useless. In the rest of this paper, we will only deal with protocols that permit moves that are relevant, in any of the senses introduced above, and are not allowed to be pre-empted. We call such protocols basic protocols, and dialogues carried out under such protocols basic dialogues. The argument graph of a basic dialogue is somewhat restricted. Proposition 4.1. Consider a basic dialogue D. The argumentation graph AG that corresponds to D is a tree with a single root. Proof. Recall that Definition 3.3 requires only that AG be a directed graph. To show that it is a tree, we have to show that it is acyclic and connected. That the graph is connected follows from the construction of the graph under a protocol that enforces relevance. If the notion of relevance is R3, each move adds a node that is connected to the previous node. If the notion of relevance is R2, then every move adds a node that is connected to the root, and thus is connected to some node in the graph. If the notion of relevance is R1, then every move has to change the status of the argument denoted by the root. Proposition 3.1 tells us that to affect the status of an argument A , the node v representing the argument A that is effecting the change has to be connected to v , the node representing A , and so it follows that every new node added as a result of an R1relevant move will be connected to the argumentation graph. Thus AG is connected. Since a basic dialogue does not allow moves that are preempted, every edge that is added during construction is directed from the node that is added to one already in the graph (thus denoting that the argument A denoted by the added node, v, undercuts the argument A denoted by the node to which the connection is made, v , rather than the other way around). Since every edge that is added is directed from the new node to the rest of the graph, there can be no cycles. Thus AG is a tree. To show that AG has a single root, consider its construction from the initial node. After m1 the graph has one node, v1 that is both a root and a leaf. After m2, the graph is two nodes connected by an edge, and v1 is now a root and not a leaf. v2 is a leaf and not a root. However the third node is added, the argument earlier in this proof demonstrates that there will be a directed edge from it to some other node, making it a leaf. Thus v1 will always be the only root. The ruling out of pre-empted moves means that v1 will never cease to be a root, and so the argumentation graph will always have one root. Since every argumentation graph constructed by a basic dialogue is a tree with a single root, this means that the first node of every argumentation graph is the root. Although these results are straightforward to obtain, they allow us to show how the notions of relevance are related. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1009 Proposition 4.2. Consider a basic dialogue D, consisting of a sequence of moves mi, with a corresponding argument graph AG. 1. Every move mi+1 that is R1-relevant is R2-relevant. The converse does not hold. 2. Every move mi+1 that is R3-relevant is R2-relevant. The converse does not hold. 3. Not every move mi+1 that is R1-relevant is R3-relevant, and not every move mi+1 that is R3-relevant is R1relevant Proof. For 1, consider how move mi+1 can satisfy R1. Proposition 3.1 tells us that if Ai+1 can change the status of the argument denoted by the root v1 (which, as observed above, is the first node) of AG, then vi+1 must be connected to the root. This is precisely what is required to satisfy R2, and the relatiosnhip is proved to hold. To see that the converse does not hold, we have to consider what it takes to change the status of r (since Proposition 3.1 tells us that connectedness is not enough to ensure a change of status - if it did, R1 and R2 relevance would coincide). For mi+1 to change the status of the root, it will have to (1) make the argument A represented by r either unacceptable, if it were acceptable before the move, or (2) acceptable if it were unacceptable before the move. Given the definition of acceptability, it can achieve (1) either by directly undercutting the argument represented by r, in which case vi+1 will be directly connected to r by some edge, or by undercutting some argument A that is part of the set of non-undercut arguments defending A. In the latter case, vi+1 will be directly connected to the node representing A and by Proposition 4.1 to r. To achieve (2), vi+1 will have to undercut an argument A that is either currently undercutting A, or is undercutting an argument that would otherwise defend A. Now, further consider that mi+1 puts forward an argument Ai+1 that undercuts the argument denoted by some node v , but this latter argument defends itself against Ai+1. In such a case, the set of acceptable arguments will not change, and so the status of Ar will not change. Thus a move that is R2-relevant need not be R1-relevant. For 2, consider that mi+1 can satisfy R3 simply by adding a node that is connected to vi, the last node to be added to AG. By Proposition 4.1, it is connected to r and so is R2-relevant. To see that the converse does not hold, consider that an R2-relevant move can connect to any node in AG. The first part of 3 follows by a similar argument to that we just used - an R1-relevant move does not have to connect to vi, just to some v that is part of the graph - and the second part follows since a move that is R3-relevant may introduce an argument Ai+1 that undercuts the argument Ai put forward by the previous move (and so vi+1 is connected to vi), but finds that Ai defends itself against Ai+1, preventing a change of status at the root. What is most interesting is not so much the results but why they hold, since this reveals some aspects of the interplay between relevance and the structure of argument graphs. For example, to restate a case from the proof of Proposition 4.2, a move that is R3-relevant by definition has to add a node to the argument graph that is connected to the last node that was added. Since a move that is R2-relevant can add a node that connects anywhere on an argument graph, any move that is R3-relevant will be R2-relevant, but the converse does not hold. It turns out that we can exploit the interplay between structure and relevance that Propositions 4.1 and 4.2 have started to illuminate to establish relationships between the protocols that govern dialogues and the argument graphs constructed during such dialogues. To do this we need to define protocols in such a way that they refer to the structure of the graph. We have: Definition 4.3. A protocol is single-path if all dialogues that conform to it construct argument graphs that have only one branch. Proposition 4.3. A basic protocol P is single-path if, for all i, the set of permitted moves Mi at move i are all R3relevant. The converse does not hold. Proof. R3-relevance requires that every node added to the argument graph be connected to the previous node. Starting from the first node this recursively constructs a tree with just one branch, and the relationship holds. The converse does not hold because even if one or more moves in the protocol are R1- or R2-relevant, it may be the case that, because of an agent"s rhetorical choice or because of its knowledge, every argument that is chosen to be put forward will undercut the previous argument and so the argument graph is a one-branch tree. Looking for more complex kinds of protocol that construct more complex kinds of argument graph, it is an obvious move to turn to: Definition 4.4. A basic protocol is multi-path if all dialogues that conform to it can construct argument graphs that are trees. But, on reflection, since any graph with only one branch is also a tree: Proposition 4.4. Any single-path protocol is an instance of a multi-path protocol. and, furthermore: Proposition 4.5. Any basic protocol P is multi-path. Proof. Immediate from Proposition 4.1 So the notion of a multi-path protocol does not have much traction. As a result we distinguish multi-path protocols that permit dialogues that can construct trees that have more than one branch as bushy protocols. We then have: Proposition 4.6. A basic protocol P is bushy if, for some i, the set of permitted moves Mi at move i are all R1- or R2-relevant. Proof. From Proposition 4.3 we know that if all moves are R3-relevant then we"ll get a tree with one branch, and from Proposition 4.1 we know that all basic protocols will build an argument graph that is a tree, so providing we exclude R3-relevant moves, we will get protocols that can build multi-branch trees. 1010 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) Of course, since, by Proposition 4.2, any move that is R3relevant is R2-relevant and can quite possibly be R1-relevant (all that Proposition 4.2 tells us is that there is no guarantee that it will be), all that Proposition 4.6 tells us is that dialogues that conform to bushy protocols may have more than one branch. All we can do is to identify a bound on the number of branches: Proposition 4.7. Consider a basic dialogue D that includes m moves that are not R3-relevant, and has a corresponding argumentation graph AG. The number of branches in AG is less than or equal to m + 1. Proof. Since it must connect a node to the last node added to AG, an R3-relevant move can only extend an existing branch. Since they do not have the same restriction, R1 and R2-relevant moves may create a new branch by connecting to a node that is not the last node added. Every such move could create a new branch, and if they do, we will have m branches. If there were R3-relevant moves before any of these new-branch-creating moves, then these m branches are in addition to the initial branch created by the R3-relevant moves, and we have a maximum of m + 1 possible branches. We distinguish bushy protocols from multi-path protocols, and hence R1- and R2-relevance from R3-relevance, because of the kinds of dialogue that R3-relevance enforces. In a dialogue in which all moves must be R3-relevant, the argumentation graph has a single branch - the dialogue consists of a sequence of arguments each of which undercuts the previous one and the last move to be made is the one that settles the dialogue. This, as we will see next, means that such a dialogue only allows a subset of all the moves that would otherwise be possible. 5. COMPLETENESS The above discussion of the difference between dialogues carried out under single-path and bushy protocols brings us to the consideration of what [18] called predeterminism, but we now prefer to describe using the term completeness. The idea of predeterminism, as described in [18], captures the notion that, under some circumstances, the result of a dialogue can be established without actually having the dialogue - the agents have sufficiently little room for rhetorical manoeuver that were one able to see the contents of all the Σi of all the αi ∈ A, one would be able to identify the outcome of any dialogue on a given subject4 . We develop this idea by considering how the argument graphs constructed by dialogues under different protocols compare to benchmark complete dialogues. We start by developing ideas of what complete might mean. One reasonable definition is that: Definition 5.1. A basic dialogue D between the set of agents A with a corresponding argumentation graph AG is topic-complete if no agent can construct an argument A that undercuts any argument A represented by a node in AG. The argumentation graph constructed by a topic-complete dialogue is called a topic-complete argumentation graph and is denoted AG(D)T . 4 Assuming that the Σi do not change during the dialogue, which is the usual assumption in this kind of dialogue. A dialogue is topic-complete when no agent can add anything that is directly connected to the subject of the dialogue. Some protocols will prevent agents from making moves even though the dialogue is not topic-complete. To distinguish such cases we have: Definition 5.2. A basic dialogue D between the set of agents A with a corresponding argumentation graph AG is protocol-complete under a protocol P if no agent can make a move that adds a node to the argumentation graph that is permitted by P. The argumentation graph constructed by a protocol-complete dialogue is called a protocol-complete argumentation graph and is denoted AG(D)P . Clearly: Proposition 5.1. Any dialogue D under a basic protocol P is protocol-complete if it is topic-complete. The converse does not hold in general. Proof. If D is topic-complete, no agent can make a move that will extend the argumentation graph. This means that no agent can make a move that is permitted by a basic protocol, and so D is also protocol complete. The converse does not hold since some basic dialogues (under a protocol that only permits R3-relevant moves, for example) will not permit certain moves (like the addition of a node that connects to the root of the argumentation graph after more than two moves) that would be allowed in a topiccomplete dialogue. Corollary 5.1. For a basic dialogue D, AG(D)P is a sub-graph of AG(D)T . Obviously, from the definition of a sub-graph, the converse of Corollary 5.1 does not hold in general. The important distinction between topic- and protocolcompleteness is that the former is determined purely by the state of the dialogue - as captured by the argumentation graph - and is thus independent of the protocol, while the latter is determined entirely by the protocol. Any time that a dialogue ends in a state of protocol-completeness rather than topic completeness, it is ending when agents still have things to say but can"t because the protocol won"t allow them to. With these definitions of completeness, our task is to relate topic-completeness - the property that ensures that agents can say everything that they have to say in a dialogue that is, in some sense, important - to the notions of relevance we have developed - which determine what agents are allowed to say. When we need very specific conditions to make protocol-complete dialogues topic-complete, it means that agents have lots of room for rhetorical maneouver when those conditions are not in force. That is there are many ways they can bring dialogues to a close before everything that can be said has been said. Where few conditions are required, or conditions are absent, then dialogues between agents with the same knowledge will always play out the same way, and rhetoric has no place. We have: Proposition 5.2. A protocol-complete basic dialogue D under a protocol which only allows R3-relevant moves will be topic-complete only when AG(D)T has a single branch in which the nodes are labelled in increasing order from the root. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1011 Proof. Given what we know about R3-relevance, the condition on AG(D)P having a single branch is obvious. This is not a sufficient condition on its own because certain protocols may prevent - through additional restrictions, like strict turn-taking in a multi-party dialogue - all the nodes in AG(D)T , which is not subject to such restrictions, being added to the graph. Only when AG(D)T includes the nodes in the exact order that the corresponding arguments are put forward is it necessary that a topic-complete argumentation graph be constructed. Given Proposition 5.1, these are the conditions under which dialogues conducted under the notion of R3-relevance will always be predetermined, and given how restrictive the conditions are, such dialogues seem to have plenty of room for rhetoric to play a part. To find similar conditions for dialogues composed of R1and R2-relevant moves, we first need to distinguish between them. We can do this in terms of the structure of the argumentation graph: Proposition 5.3. Consider a basic dialogue D, with argumentation graph AG which has root r denoting an argument A. If argument A , denoted by node v is an an R2relevant move m, m is not R1-relevant if and only if: 1. there are two nodes v and v on the path between v and r, and the argument denoted by v defends itself against the argument denoted by v ; or 2. there is an argument A , denoted by node v , that affects the status of A, and the path from v to r has one or more nodes in common with the path from v to r. Proof. For the first condition, consider that since AG is a tree, v is connected to r. Thus there is a series of undercut relations between A and A , and this corrresponds to a path through AG. If this path is the only branch in the tree, then A will affect the status of A unless the chain of affect is broken by an undercut that can"t change the status of the undercut argument because the latter defends itself. For the second condition, as for the first, the only way that A cannot affect the status of A is if something is blocking its influence. If this is not due to defending against, it must be because there is some node u on the path that represents an argument whose status is fixed somehow, and that must mean that there is another chain of undercut relations, another branch of the tree, that is incident at u. Since this second branch denotes another chain of arguments, and these affect the status of the argument denoted by u, they must also affect the status of A. Any of these are the A in the condition. So an R2-relevant move m is not R1-relevant if either its effect is blocked because an argument upstream is not strong enough, or because there is another line of argument that is currently determining the status of the argument at the root. This, in turn, means that if the effect is not due to defending against, then there is an alternative move that is R1-relevant - a move that undercuts A in the second condition above5 . We can now show 5 Though whether the agent in question can make such a move is another question. Proposition 5.4. A protocol-complete basic dialogue D will always be topic-complete under a protocol which only includes R2-relevant moves and allows every R2-relevant move to be made. The restriction on R2-relevant rules is exactly that for topiccompleteness, so a dialogue that has only R2-relevant moves will continue until every argument that any agent can make has been put forward. Given this, and what we revealed about R1-relevance in Proposition 5.3, we can see that: Proposition 5.5. A protocol-complete basic dialogue D under a protocol which only includes R1-relevant moves will be topic-complete if AG(D)T : 1. includes no path with adjacent nodes v, denoting A, and v , denoting A , such that A undercuts A and A is stronger that A; and 2. is such that the nodes in every branch have consecutive indices and no node with degree greater than two is an odd number of arcs from a leaf node. Proof. The first condition rules out the first condition in Proposition 5.3, and the second deals with the situation that leads to the second condition in Proposition 5.3. The second condition ensures that each branch is constructed in full before any new branch is added, and when a new branch is added, the argument that is undercut as part of the addition will be acceptable, and so the addition will change the status of the argument denoted by that node, and hence the root. With these conditions, every move required to construct AG(D)T will be permitted and so the dialogue will be topic-complete when every move has been completed. The second part of this result only identifies one possible way to ensure that the second condition in Proposition 5.3 is met, so the converse of this result does not hold. However, what we have is sufficient to answer the question about predetermination that we started with. For dialogues to be predetermined, every move that is R2-relevant must be made. In such cases every dialogue is topic complete. If we do not require that all R2-relevant moves are made, then there is some room for rhetoric - the way in which alternative lines of argument are presented becomes an issue. If moves are forced to be R3-relevant, then there is considerable room for rhetorical play. 6. SUMMARY This paper has studied the different ideas of relevance in argumentation-based dialogue, identifying the relationship between these ideas, and showing how they can impact the extent to which the way that agents choose moves in a dialogue - what some authors have called the strategy and tactics of a dialogue. This extends existing work on relvance, such as [3, 15] by showing how different notions of relevance can have an effect on the outcome of a dialogue, in particular when they render the outcome predetermined. This connection extends the work of [18] which considered dialogue outcome, but stopped short of identifying the conditions under which it is predetermined. There are two ways we are currently trying to extend this work, both of which will generalise the results and extend its applicability. First, we want to relax the restrictions that 1012 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) we have imposed, the exclusion of moves that attack several arguments (without which the argument graph can be mulitply-connected) and the exclusion of pre-empted moves, without which the argument graph can have cycles. Second, we want to extend the ideas of relevance to cope with moves that do not only add undercutting arguments, but also supporting arguments, thus taking account of bipolar argumentation frameworks [5]. Acknowledgments The authors are grateful for financial support received from the EC, through project IST-FP6-002307, and from the NSF under grants REC-02-19347 and NSF IIS-0329037. They are also grateful to Peter Stone for a question, now several years old, which this paper has finally answered. 7. REFERENCES [1] L. Amgoud and C. Cayrol. On the acceptability of arguments in preference-based argumentation framework. In Proceedings of the 14th Conference on Uncertainty in Artificial Intelligence, pages 1-7, 1998. [2] L. Amgoud, S. Parsons, and N. Maudet. Arguments, dialogue, and negotiation. In W. Horn, editor, Proceedings of the Fourteenth European Conference on Artificial Intelligence, pages 338-342, Berlin, Germany, 2000. IOS Press. [3] J. Bentahar, M. Mbarki, and B. Moulin. Strategic and tactic reasoning for communicating agents. In N. Maudet, I. Rahwan, and S. Parsons, editors, Proceedings of the Third Workshop on Argumentation in Muliagent Systems, Hakodate, Japan, 2006. [4] P. Besnard and A. Hunter. A logic-based theory of deductive arguments. Artificial Intelligence, 128:203-235, 2001. [5] C. Cayrol, C. Devred, and M.-C. Lagasquie-Schiex. Handling controversial arguments in bipolar argumentation frameworks. In P. E. Dunne and T. J. M. Bench-Capon, editors, Computational Models of Argument: Proceedings of COMMA 2006, pages 261-272. IOS Press, 2006. [6] B. Chaib-Draa and F. Dignum. Trends in agent communication language. Computational Intelligence, 18(2):89-101, 2002. [7] F. Dignum, B. Dunin-K¸eplicz, and R. Verbrugge. Agent theory for team formation by dialogue. In C. Castelfranchi and Y. Lesp´erance, editors, Seventh Workshop on Agent Theories, Architectures, and Languages, pages 141-156, Boston, USA, 2000. [8] P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77:321-357, 1995. [9] P. M. Dung, R. A. Kowalski, and F. Toni. Dialectic proof procedures for assumption-based, admissable argumentation. Artificial Intelligence, 170(2):114-159, 2006. [10] R. A. Flores and R. C. Kremer. To commit or not to commit. Computational Intelligence, 18(2):120-173, 2002. [11] D. M. Gabbay and J. Woods. More on non-cooperation in Dialogue Logic. Logic Journal of the IGPL, 9(2):321-339, 2001. [12] D. M. Gabbay and J. Woods. Non-cooperation in Dialogue Logic. Synthese, 127(1-2):161-186, 2001. [13] C. L. Hamblin. Mathematical models of dialogue. Theoria, 37:130-155, 1971. [14] S. Kraus, K. Sycara, and A. Evenchik. Reaching agreements through argumentation: a logical model and implementation. Artificial Intelligence, 104(1-2):1-69, 1998. [15] N. Oren, T. J. Norman, and A. Preece. Loose lips sink ships: A heuristic for argumentation. In N. Maudet, I. Rahwan, and S. Parsons, editors, Proceedings of the Third Workshop on Argumentation in Muliagent Systems, Hakodate, Japan, 2006. [16] S. Parsons and N. R. Jennings. Negotiation through argumentation - a preliminary report. In Proceedings of Second International Conference on Multi-Agent Systems, pages 267-274, 1996. [17] S. Parsons, M. Wooldridge, and L. Amgoud. An analysis of formal inter-agent dialogues. In 1st International Conference on Autonomous Agents and Multi-Agent Systems. ACM Press, 2002. [18] S. Parsons, M. Wooldridge, and L. Amgoud. On the outcomes of formal inter-agent dialogues. In 2nd International Conference on Autonomous Agents and Multi-Agent Systems. ACM Press, 2003. [19] H. Prakken. On dialogue systems with speech acts, arguments, and counterarguments. In Proceedings of the Seventh European Workshop on Logic in Artificial Intelligence, Berlin, Germany, 2000. Springer Verlag. [20] H. Prakken. Relating protocols for dynamic dispute with logics for defeasible argumentation. Synthese, 127:187-219, 2001. [21] H. Prakken and G. Sartor. Modelling reasoning with precedents in a formal dialogue game. Artificial Intelligence and Law, 6:231-287, 1998. [22] I. Rahwan, P. McBurney, and E. Sonenberg. Towards a theory of negotiation strategy. In I. Rahwan, P. Moraitis, and C. Reed, editors, Proceedings of the 1st International Workshop on Argumentation in Multiagent Systems, New York, NY, 2004. [23] C. Reed. Dialogue frames in agent communications. In Y. Demazeau, editor, Proceedings of the Third International Conference on Multi-Agent Systems, pages 246-253. IEEE Press, 1998. [24] M. Rovatsos, I. Rahwan, F. Fisher, and G. Weiss. Adaptive strategies for practical argument-based negotiation. In I. Rahwan, P. Moraitis, and C. Reed, editors, Proceedings of the 1st International Workshop on Argumentation in Multiagent Systems, New York, NY, 2004. [25] M. Schroeder, D. A. Plewe, and A. Raab. Ultima ratio: should Hamlet kill Claudius. In Proceedings of the 2nd International Conference on Autonomous Agents, pages 467-468, 1998. [26] K. Sycara. Argumentation: Planning other agents" plans. In Proceedings of the Eleventh Joint Conference on Artificial Intelligence, pages 517-523, 1989. [27] D. N. Walton and E. C. W. Krabbe. Commitment in Dialogue: Basic Concepts of Interpersonal Reasoning. State University of New York Press, Albany, NY, USA, 1995. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1013

node;status;relevance;graph;tree;leaf;dialogue;argument;argumentation;multiagent system

train_I-75

Hypotheses Refinement under Topological Communication Constraints

We investigate the properties of a multiagent system where each (distributed) agent locally perceives its environment. Upon perception of an unexpected event, each agent locally computes its favoured hypothesis and tries to propagate it to other agents, by exchanging hypotheses and supporting arguments (observations). However, we further assume that communication opportunities are severely constrained and change dynamically. In this paper, we mostly investigate the convergence of such systems towards global consistency. We first show that (for a wide class of protocols that we shall define), the communication constraints induced by the topology will not prevent the convergence of the system, at the condition that the system dynamics guarantees that no agent will ever be isolated forever, and that agents have unlimited time for computation and arguments exchange. As this assumption cannot be made in most situations though, we then set up an experimental framework aiming at comparing the relative efficiency and effectiveness of different interaction protocols for hypotheses exchange. We study a critical situation involving a number of agents aiming at escaping from a burning building. The results reported here provide some insights regarding the design of optimal protocol for hypotheses refinement in this context.

1. INTRODUCTION We consider a multiagent system where each (distributed) agent locally perceives its environment, and we assume that some unexpected event occurs in that system. If each agent computes only locally its favoured hypothesis, it is only natural to assume that agents will seek to coordinate and refine their hypotheses by confronting their observations with other agents. If, in addition, the communication opportunities are severely constrained (for instance, agents can only communicate when they are close enough to some other agent), and dynamically changing (for instance, agents may change their locations), it becomes crucial to carefully design protocols that will allow agents to converge to some desired state of global consistency. In this paper we exhibit some sufficient conditions on the system dynamics and on the protocol/strategy structures that allow to guarantee that property, and we experimentally study some contexts where (some of) these assumptions are relaxed. While problems of diagnosis are among the venerable classics in the AI tradition, their multiagent counterparts have much more recently attracted some attention. Roos and colleagues [8, 9] in particular study a situation where a number of distributed entities try to come up with a satisfying global diagnosis of the whole system. They show in particular that the number of messages required to establish this global diagnosis is bound to be prohibitive, unless the communication is enhanced with some suitable protocol. However, they do not put any restrictions on agents" communication options, and do not assume either that the system is dynamic. The benefits of enhancing communication with supporting information to make convergence to a desired global state of a system more efficient has often been put forward in the literature. This is for instance one of the main idea underlying the argumentation-based negotiation approach [7], where the desired state is a compromise between agents with conflicting preferences. Many of these works however make the assumption that this approach is beneficial to start with, and study the technical facets of the problem (or instead emphasize other advantages of using argumentation). Notable exceptions are the works of [3, 4, 2, 5], which studied in contexts different from ours the efficiency of argumentation. The rest of the paper is as follows. Section 2 specifies the basic elements of our model, and Section 3 goes on to presenting the different protocols and strategies used by the agents to exchange hypotheses and observations. We put special attention at clearly emphasizing the conditions on the system dynamics and protocols/strategies that will be exploited in the rest of the paper. Section 4 details one of 998 978-81-904262-7-5 (RPS) c 2007 IFAAMAS the main results of the paper, namely the fact that under the aforementioned conditions, the constraints that we put on the topology will not prevent the convergence of the system towards global consistency, at the condition that no agent ever gets completely lost forever in the system, and that unlimited time is allowed for computation and argument exchange. While the conditions on protocols and strategies are fairly mild, it is also clear that these system requirements look much more problematic, even frankly unrealistic in critical situations where distributed approaches are precisely advocated. To get a clearer picture of the situation induced when time is a critical factor, we have set up an experimental framework that we introduce and discuss in Section 5. The critical situation involves a number of agents aiming at escaping from a burning building. The results reported here show that the effectiveness of argument exchange crucially depends upon the nature of the building, and provide some insights regarding the design of optimal protocol for hypotheses refinement in this context. 2. BASIC NOTIONS We start by defining the basic elements of our system. Environment Let O be the (potentially infinite) set of possible observations. We assume the sensors of our agents to be perfect, hence the observations to be certain. Let H be the set of hypotheses, uncertain and revisable. Let Cons(h, O) be the consistency relation, a binary relation between a hypothesis h ∈ H and a set of observations O ⊆ O. In most cases, Cons will refer to classical consistency relation, however, we may overload its meaning and add some additional properties to that relation (in which case we will mention it). The environment may include some dynamics, and change over the course of time. We define below sequences of time points to deal with it: Definition 1 (Sequence of time points). A sequence of time points t1, t2, . . . , tn from t is an ordered set of time points t1, t2, . . . , tn such that t1 ≥ t and ∀i ∈ [1, n − 1], ti+1 ≥ ti. Agent We take a system populated by n agents a1, . . . , an. Each agent is defined as a tuple F, Oi, hi , where: • F, the set of facts, common knowledge to all agents. • Oi ∈ 2O , the set of observations made by the agent so far. We assume a perfect memory, hence this set grows monotonically. • hi ∈ H, the favourite hypothesis of the agent. A key notion governing the formation of hypotheses is that of consistency, defined below: Definition 2 (Consistency). We say that: • An agent is consistent (Cons(ai)) iff Cons(hi, Oi) (that is, its hypothesis is consistent with its observation set). • An agent ai consistent with a partner agent aj iff Cons(ai) and Cons(hi, Oj) (that is, this agent is consistent and its hypothesis can explain the observation set of the other agent). • Two agents ai and aj are mutually consistent (MCons(ai, aj)) iff Cons(ai, aj) and Cons(aj, ai). • A system is consistent iff ∀(i, j)∈[1, n]2 it is the case that MCons(ai, aj). To ensure its consistency, each agent is equipped with an abstract reasoning machinery that we shall call the explanation function Eh. This (deterministic) function takes a set of observation and returns a single prefered hypothesis (2O → H). We assume h = Eh(O) to be consistent with O by definition of Eh, so using this function on its observation set to determine its favourite hypothesis is a sure way for the agent to achieve consistency. Note however that an hypothesis does not need to be generated by Eh to be consistent with an observation set. As a concrete example of such a function, and one of the main inspiration of this work, one can cite the Theorist reasoning system [6] -as long as it is coupled with a filter selecting a single prefered theory among the ones initially selected by Theorist. Note also that hi may only be modified as a consequence of the application Eh. We refer to this as the autonomy of the agent: no other agent can directly impose a given hypothesis to an agent. As a consequence, only a new observation (being it a new perception, or an observation communicated by a fellow agent) can result in a modification of its prefered hypothesis hi (but not necessarily of course). We finally define a property of the system that we shall use in the rest of the paper: Definition 3 (Bounded Perceptions). A system involves a bounded perception for agents iff ∃n0 s.t. ∀t| ∪N i=1 Oi| ≤ n0. (That is, the number of observations to be made by the agents in the system is not infinite.) Agent Cycle Now we need to see how these agents will evolve and interact in their environment. In our context, agents evolve in a dynamic environment, and we classicaly assume the following system cycle: 1. Environment dynamics: the environment evolves according to the defined rules of the system dynamics. 2. Perception step : agents get perceptions from the environment. These perceptions are typically partial (e.g. the agent can only see a portion of a map). 3. Reasoning step: agents compare perception with predictions, seek explanations for (potential) difference(s), refine their hypothesis, draw new conclusions. 4. Communication step: agents can communicate hypotheses and observations with other agents through a defined protocol. Any agent can only be involved in one communication with another agent by step. 5. Action step: agents do some practical reasoning using the models obtained from the previous steps and select an action. They can then modify the environment by executing it. The communication of the agents will be further constrained by topological consideration. At a given time, an agent will only be able to communicate with a number of neighbours. Its connexions with these others agents may The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 999 evolve with its situation in the environment. Typically, an agent can only communicate with agents that it can sense, but one could imagine evolving topological constraints on communication based on a network of communications between agents where the links are not always active. Communication In our system, agents will be able to communicate with each other. However, due to the aforementionned topological constraints, they will not be able to communicate with any agents at anytime. Who an agent can communicate with will be defined dynamically (for instance, this can be a consequence of the agents being close enough to get in touch). We will abstractly denote by C(ai, aj, t) the communication property, in other words, the fact that agents ai and aj can communicate at time t (note that this relation is assumed to be symetric, but of course not transitive). We are now in a position to define two essential properties of our system. Definition 4 (Temporal Path). There exists a temporal communication path at horizon tf (noted Ltf (aI , aJ )) between ai and aj iff there exists a sequence of time points t1, t2, . . . , tn from tf and a sequence of agents k1, k2, . . . , kn s.t. (i) C(aI , ak1 , t1), (ii) C(akn , aJ , tn+1), (iii) ∀i ∈ [1, n], C(aki , aki+1 , ti) Intuitively, what this property says is that it is possible to find a temporal path in the future that would allow to link agent ai and aj via a sequence of intermediary agents. Note that the time points are not necessarily successive, and that the sequence of agents may involve the same agents several times. Definition 5 (Temporal Connexity). A system is temporaly connex iff ∀t ∀(i, j)∈[1, n]2 Lt(ai, aj) In short, a temporaly connex system guarantees that any agent will be able to communicate with any other agents, no matter how long it might take to do so, at any time. To put it another way, it is never the case that an agent will be isolated for ever from another agent of the system. We will next discuss the detail of how communication concretely takes place in our system. Remember that in this paper, we only consider the case of bilateral exchanges (an agent can only speak to a single other agent), and that we also assume that any agent can only engage in a single exchange in a given round. 3. PROTOCOLS AND STRATEGIES In this section, we discuss the requirements of the interaction protocols that govern the exchange of messages between agents, and provide some example instantiation of such protocols. To clarify the presentation, we distinguish two levels: the local level, which is concerned with the regulation of bilateral exchanges; and the global level,which essentially regulates the way agents can actually engage into a conversation. At each level, we separate what is specified by the protocol, and what is left to agents" strategies. Local Protocol and Strategies We start by inspecting local protocols and strategies that will regulate the communication between the agents of the system. As we limit ourselves to bilateral communication, these protocols will simply involve two agents. Such protocol will have to meet one basic requirement to be satisfying. • consistency (CONS)- a local protocol has to guarantee the mutual consistency of agents upon termination (which implies termination of course). Figure 1: A Hypotheses Exchange Protocol [1] One example such protocol is the protocol described in [1] that is pictured in Fig. 1. To further illustrate how such protocol can be used by agents, we give some details on a possible strategy: upon receiving a hypothesis h1 (propose(h1) or counterpropose(h1)) from a1, agent a2 is in state 2 and has the following possible replies: counterexample (if the agent knows an example contradicting the hypothesis, or not explained by this hypothesis), challenge (if the agents lacks evidence to accept this hypothesis), counterpropose (if the agent agrees with the hypothesis but prefers another one), or accept (if it is indeed as good as its favourite hypothesis). This strategy guarantees, among other properties, the eventual mutual logical consistency of the involved agents [1]. Global Protocol The global protocol regulates the way bilateral exchanges will be initiated between agents. At each turn, agents will concurrently send one weighted request to communicate to other agents. This weight is a value measuring the agent"s willingness to converse with the targeted agent (in practice, this can be based on different heuristics, but we shall make some assumptions on agents" strategies, see below). Sending such a request is a kind of conditional commitment for the agent. An agent sending a weighted request commits to engage in conversation with the target if he does not receive and accept himself another request. Once all request have been received, each agent replies with either an acccept or a reject. By answering with an accept, an agent makes a full commitment to engage in conversation with the sender. Therefore, it can only send one accept in a given round, as an agent can only participate in one conversation per time step. When all response have been received, each agent receiving an accept can either initiate a conversation using the local protocol or send a cancel if it has accepted another request. At the end of all the bilateral exchanges, the agents engaged in conversation are discarded from the protocol. Then each of the remaining agents resends a request and the process iterates until no more requests are sent. Global Strategy We now define four requirements for the strategies used by agents, depending on their role in the protocol: two are concerned with the requestee role (how to decide who the 1000 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) agent wishes to communicate with?), the other two with the responder role (how to decide which communication request to accept or not?). • Willingness to solve inconsistancies (SOLVE)-agents want to communicate with any other agents unless they know they are mutually consistent. • Focus on solving inconsistencies (FOCUS)-agents do not request communication with an agent with whom they know they are mutually consistent. • Willingness to communicate (COMM)-agents cannot refuse a weighted communication request, unless they have just received or send a request with a greater weight. • Commitment to communication request (REQU)agents cannot accept a weighted communication request if they have themselves sent a communication request with a greater weight. Therefore, they will not cancel their request unless they have received a communicational request with greater weight. Now the protocol structure, together with the properties COMM+REQU, ensure that a request can only be rejected if its target agent engages in communication with another agent. Suppose indeed that agent ai wants to communicate with aj by sending a request with weight w. COMM guarantees that an agent receiving a weighted request will either accept this communication, accept a communication with a greater weight or wait for the answer to a request with a greater weight. This ensures that the request with maximal weight will be accepted and not cancelled (as REQU ensures that an agent sending a request can only cancel it if he accepts another request with greater weight). Therefore at least two agents will engage in conversation per round of the global protocol. As the protocol ensures that ai can resend its request while aj is not engaged in a conversation, there will be a turn in which aj must engage in a conversation, either with ai or another agent. These requirements concern request sending and acceptation, but agents also need some strategy of weight attribution. We describe below an altruist strategy, used in our experiments. Being cooperative, an agent may want to know more of the communication wishes of other agents in order to improve the overall allocation of exchanges to agents. A context request step is then added to the global protocol. Before sending their chosen weighted request, agents attribute a weight to all agents they are prepared to communicate with, according to some internal factors. In the simplest case, this weight will be 1 for all agent with whom the agent is not sure of being mutually consistent (ensuring SOLVE), other agent being not considered for communication (ensuring FOCUS). The agent then sends a context request to all agents with whom communication is considered. This request also provides information about the sender (list of considered communications along with their weight). After reception of all the context requests, agents will either reply with a deny, iff they are already engaged in a conversation (in which case, the requesting agent will not consider communication with them anymore in this turn), or an inform giving the requester information about the requests it has sent and received. When all replies have been received, each agent can calculate the weight of all requests concerning it. It does so by substracting from the weight of its request the weight of all requests concerning either it or its target (that is, the final weight of the request from ai to aj is Wi,j = wi,j +wj,i − ( P k∈R(i)−{j} wi,k + P k∈S(i)−{j} wk,i + P k∈R(j)−{i} wj,k + P k∈S(j)−{i} wk,j) where wi,j is the weight of the request of ai to aj, R(i) is the set of indice of agents having received a request from ai and S(i) is the set of indice of agents having send a request to ai). It then finally sends a weighted request to the agents who maximise this weight (or wait for a request) as described in the global protocol. 4. (CONDITIONAL) CONVERGENCE TO GLOBAL CONSISTENCY In this section we will show that the requirements regarding protocols and strategies just discussed will be sufficient to ensure that the system will eventually converge towards global consistency, under some conditions. We first show that, if two agents are not mutually consistent at some time, then there will be necessarily a time in the future such that an agent will learn a new observation, being it because it is new for the system, or by learning it from another agent. Lemma 1. Let S be a system populated by n agents a1, a2, ..., an, temporaly connex, and involving bounded perceptions for these agents. Let n1 be the sum of cardinalities of the intersection of pairwise observation sets. (n1 = P (i,j)∈[1,n]2 |Oi ∩ Oj|) Let n2 be the cardinality of the union of all agents" observations sets. (n2 = | ∪N i=1 Oi|). If ¬MCons(ai, aj) at time t0, there is necessarily a time t > t0 s.t. either n1 or n2 will increase. Proof. Suppose that there exist a time t0 and indices (i, j) s.t. ¬MCons(ai, aj). We will use mt0 = P (k,l)∈[1,n]2 εComm(ak, al, t0) where εComm(ak, al, t0) = 1 if ak and al have communicated at least once since t0, and 0 otherwise. Temporal connexity guarantees that there exist t1, ..., tm+1 and k1, ..., km s.t. C(ai, ak1 , t1), C(akm , aj, tm+1), and ∀p ∈ [1, m], C(akp , akp+1 , tp). Clearly, if MCons(ai, ak1 ), MCons(akm , aj) and ∀p, MCons(akp , akp+1 ), we have MCons(ai, aj) which contradicts our hypothesis (MCons being transitive, MCons(ai, ak1 )∧MCons(ak1 , ak2 ) implies that MCons(ai, ak2 ) and so on till MCons(ai, akm )∧ MCons(akm , aj) which implies MCons(ai, aj) ). At least two agents are then necessarily inconsistent (¬MCons(ai, ak1 ), or ¬MCons(akm , aj), or ∃p0 t.q. ¬MCons(akp0 , akp0+1 )). Let ak and al be these two neighbours at a time t > t0 1 . The SOLVE property ensures that either ak or al will send a communication request to the other agent at time t . As shown before, this in turn ensures that at least one of these agents will be involved in a communication. Then there are two possibilities: (case i) ak and al communicate at time t . In this case, we know that ¬MCons(ak, al). This and the CONS property ensures that at least one of the agents must change its 1 Strictly speaking, the transitivity of MCons only ensure that ak and al are inconsistent at a time t ≥ t0 that can be different from the time t at which they can communicate. But if they become consistent between t and t (or inconsistent between t and t ), it means that at least one of them have changed its hypothesis between t and t , that is, after t0. We can then apply the reasoning of case iib. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1001 hypothesis, which in turn, since agents are autonomous, implies at least one exchange of observation. But then |Ok ∩Ol| is bound to increase: n1(t ) > n1(t0). (case ii) ak communicates with ap at time t . We then have again two possibilities: (case iia) ak and ap did not communicate since t0. But then εComm(ak, ap, t0) had value 0 and takes value 1. Hence mt0 increases. (case iib) ak and ap did communicate at some time t0 > t0. The CONS property of the protocol ensures that MCons(ak, ap) at that time. Now the fact that they communicate and FOCUS implies that at least one of them did change its hypothesis in the meantime. The fact that agents are autonomous implies in turn that a new observation (perceived or received from another agent) necessarily provoked this change. The latter case would ensure the existence of a time t > t0 and an agent aq s.t. either |Op ∩Oq| or |Ok ∩Oq| increases of 1 at that time (implying n1(t ) > n1(t0)). The former case means that the agent gets a new perception o at time t . If that observation was unknown in the system before, then n2(t ) > n2(t0). If some agent aq already knew this observation before, then either Op ∩ Oq or Ok ∩ Oq increases of 1 at time t (which implies that n1(t ) > n1(t0)). Hence, ¬MCons(ai, aj) at time t0 guarantees that, either: −∃t > t0 t.q. n1(t ) > n1(t0); or −∃t > t0 t.q. n2(t ) > n2(t0); or −∃t > t0 t.q. mt0 increases of 1 at time t . By iterating the reasoning with t (but keeping t0 as the time reference for mt0 ), we can eliminate the third case (mt0 is integer and bounded by n2 , which means that after a maximum of n2 iterations, we necessarily will be in one of two other cases.) As a result, we have proven that if ¬MCons(ai, aj) at time t0, there is necessarily a time t s.t. either n1 or n2 will increase. Theorem 1 (Global consistency). Let S be a system populated by n agents a1, a2, ..., an, temporaly connex, and involving bounded perceptions for these agents. Let Cons(ai, aj) be a transitive consistency property. Then any protocol and strategies satisfying properties CONS, SOLVE, FOCUS, COMM and REQU guarantees that the system will converge towards global consistency. Proof. For the sake of contradiction, let us assume ∃I, J ∈ [1, N] s.t. ∀t, ∃t0 > t, t.q. ¬Cons(aI , aJ , t0). Using the lemma, this implies that ∃t > t0 s.t. either n1(t ) > n1(t0) or n2(t ) > n2(t0). But we can apply the same reasoning taking t = t , which would give us t1 > t > t0 s.t. ¬Cons(aI , aJ , t1), which gives us t > t1 s.t. either n1(t ) > n1(t1) or n2(t ) > n2(t1). By successive iterations we can then construct a sequence t0, t1, ..., tn, which can be divided in two sub-sequences t0, t1, ...tn and t0 , t1 , ..., tn s.t. n1(t0) < n1(t1) < ... < n1(tn) and n2(t0 ) < n2(t1 ) < ... < n2(tn). One of these sub-sequences has to be infinite. However, n1(ti) and n2(ti ) are strictly growing, integer, and bounded, which implies that both are finite. Contradiction. What the previous result essentially shows is that, in a system where no agent will be isolated from the rest of the agents for ever, only very mild assumptions on the protocols and strategies used by agents suffice to guarantee convergence towards system consistency in a finite amount of time (although it might take very long). Unfortunately, in many critical situations, it will not be possible to assume this temporal connexity. As distributed approaches as the one advocated in this paper are precisely often presented as a good way to tackle problems of reliability or problems of dependence to a center that are of utmost importance in these critical applications, it is certainly interesting to further explore how such a system would behave when we relax this assumption. 5. EXPERIMENTAL STUDY This experiment involves agents trying to escape from a burning building. The environment is described as a spatial grid with a set of walls and (thankfully) some exits. Time and space are considered discrete. Time is divided in rounds. Agents are localised by their position on the spatial grid. These agents can move and communicate with other agents. In a round, an agent can move of one cell in any of the four cardinal directions, provided it is not blocked by a wall. In this application, agents communicate with any other agent (but, recall, a single one) given that this agent is in view, and that they have not yet exchanged their current favoured hypothesis. Suddenly, a fire erupts in these premises. From this moment, the fire propagates. Each round, for each cases where there is fire, the fire propagates in the four directions. However, the fire cannot propagate through a wall. If the fire propagates in a case where an agent is positioned, that agent burns and is considered dead. It can of course no longer move nor communicate. If an agent gets to an exit, it is considered saved, and can no longer be burned. Agents know the environment and the rules governing the dynamics of this environment, that is, they know the map as well as the rules of fire propagation previously described. They also locally perceive this environment, but cannot see further than 3 cases away, in any direction. Walls also block the line of view, preventing agents from seeing behind them. Within their sight, they can see other agents and whether or not the cases they see are on fire. All these perceptions are memorised. We now show how this instantiates the abstract framework presented the paper. • O = {Fire(x, y, t), NoFire(x, y, t), Agent(ai, x, y, t)} Observations can then be positive (o ∈ P(O) iff ∃h ∈ H s.t. h |= o) or negative (o ∈ N(O) iff ∃h ∈ H s.t. h |= ¬o). • H={FireOrigin(x1, y1, t1)∧...∧FireOrigin(xl, yl, tl)} Hypotheses are conjunctions of FireOrigins. • Cons(h, O) consistency relation satisfies: - coherence : ∀o ∈ N(O), h |= ¬o. - completeness : ∀o ∈ P(O), h |= o. - minimality : For all h ∈ H, if h is coherent and complete for O, then h is prefered to h according to the preference relation (h ≤p h ).2 2 Selects first the minimal number of origins, then the most recent (least preemptive strategy [6]), then uses some arbitrary fixed ranking to discriminate ex-aequo. The resulting relation is a total order, hence minimality implies that there will be a single h s.t.Cons(O, h) for a given O. This in turn means that MCons(ai, aj) iff Cons(ai), Cons(aj), and hi = hj. This relation is then transitive and symmetric. 1002 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) • Eh takes O as argument and returns min≤p of the coherent and complete hypothesis for O 5.1 Experimental Evaluation We will classically (see e.g. [3, 4]) assess the effectiveness and efficiency of different interaction protocols. Effectiveness of a protocol The proportion of agents surviving the fire over the initial number of agents involved in the experiment will determine the effectiveness of a given protocol. If this value is high, the protocol has been effective to propagate the information and/or for the agents to refine their hypotheses and determine the best way to the exit. Efficiency of a protocol Typically, the use of supporting information will involve a communication overhead. We will assume here that the efficiency of a given protocol is characterised by the data flow induced by this protocol. In this paper we will only discuss this aspect wrt. local protocols. The main measure that we shall then use here is the mean total size of messages that are exchanged by agents per exchange (hence taking into account both the number of messages and the actual size of the messages, because it could be that messages happen to be very big, containing e.g. a large number of observations, which could counter-balance a low number of messages). 5.2 Experimental Settings The chosen experimental settings are the following: • Environmental topology- Performances of information propagation are highly constrained by the environment topology. The perception skills of the agents depend on the openness of the environment. With a large number of walls the perceptions of agents are limited, and also the number of possible inter-agent communications, whereas an open environment will provide optimal possibilities of perception and information propagation. Thus, we propose a topological index (see below) as a common basis to charaterize the environments (maps) used during experimentations. The topological index (TI) is the ratio of the number of cells that can be perceived by agents summed up from all possible positions, divided by the number of cells that would be perceived from the same positions but without any walls. (The closer to 1, the more open the environment). We shall also use two additional, more classical [10], measures: the characteristic path length3 (CPL) and the clustering coefficient4 (CC). • Number of agents- The propagation of information also depends on the initial number of agents involved during an experimentation. For instance, the more agents, the more potential communications there is. This means that there will be more potential for propagation, but also that the bilateral exchange restriction will be more crucial. 3 The CPL is the median of the means of the shortest path lengths connecting each node to all other nodes. 4 characterising the isolation degree of a region of an environment in terms of acessibility (number of roads still usable to reach this region). Map T.I. (%) C.P.L. C.C. 69-1 69,23 4,5 0,69 69-2 68,88 4,38 0,65 69-3 69,80 4,25 0,67 53-1 53,19 5,6 0,59 53-2 53,53 6,38 0,54 53-3 53,92 6,08 0,61 38-1 38,56 8,19 0,50 38-2 38,56 7,3 0,50 38-3 38,23 8,13 0,50 Table 1: Topological Characteristics of the Maps • Initial positions of the agents- Initial positions of the agents have a significant influence on the overall behavior of an instance of our system: being close from an exit will (in general) ease the escape. 5.3 Experimental environments We choose to realize experiments on three very different topological indexes (69% for open environments, 53% for mixed environments, and 38% for labyrinth-like environments). Figure 2: Two maps (left: TI=69%, right TI=38%) We designed three different maps for each index (Fig. 2 shows two of them), containing the same maximum number of agents (36 agents max.) with a maximum density of one agent per cell, the same number of exits and a similar fire origin (e.g. starting time and position). The three differents maps of a given index are designed as follows. The first map is a model of an existing building floor. The second map has the same enclosure, exits and fire origin as the first one, but the number and location of walls are different (wall locations are designed by an heuristic which randomly creates walls on the spatial grid such that no fully closed rooms are created and that no exit is closed). The third map is characterised by geometrical enclosure in wich walls location is also designed with the aforementioned heuristic. Table 1 summarizes the different topological measures characterizing these different maps. It is worth pointing out that the values confirm the relevance of TI (maps with a high TI have a low CPL and a high CC. However the CPL and CC allows to further refine the difference between the maps, e.g. between 53-1 and 53-2). 5.4 Experimental Results For each triple of maps defined as above we conduct the same experiments. In each experiment, the society differs in terms of its initial proportion of involved agents, from 1% to 100%. This initial proportion represents the percentage of involved agents with regards to the possible maximum number of agents. For each map and each initial proportion, The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1003 we select randomly 100 different initial agents" locations. For each of those different locations we execute the system one time for each different interaction protocol. Effectiveness of Communication and Argumentation The first experiment that we set up aims at testing how effective is hypotheses exchange (HE), and in particular how the topological aspects will affect this effectiveness. In order to do so, we have computed the ratio of improvement offered by that protocol over a situation where agents could simply not communicate (no comm). To get further insights as to what extent the hypotheses exchange was really crucial, we also tested a much less elaborated protocol consisting of mere observation exchanges (OE). More precisely, this protocol requires that each agent stores any unexpected observation that it perceives, and agents simply exchange their respective lists of observations when they discuss. In this case, the local protocol is different (note in particular that it does not guarantee mutual consistency), but the global protocol remains the same (at the only exception that agents" motivation to communicate is to synchronise their list of observations, not their hypothesis). If this protocol is at best as effective as HE, it has the advantage of being more efficient (this is obvious wrt the number of messages which will be limited to 2, less straightforward as far as the size of messages is concerned, but the rough observation that the exchange of observations can be viewed as a flat version of the challenge is helpful to see this). The results of these experiments are reported in Fig. 3. Figure 3: Comparative effectiveness ratio gain of protocols when the proportion of agents augments The first observation that needs to be made is that communication improves the effectiveness of the process, and this ratio increases as the number of agents grows in the system. The second lesson that we learn here is that closeness relatively makes communication more effective over non communication. Maps exhibiting a T.I. of 38% are constantly above the two others, and 53% are still slightly but significantly better than 69%. However, these curves also suggest, perhaps surprisingly, that HE outperforms OE in precisely those situations where the ratio gain is less important (the only noticeable difference occurs for rather open maps where T.I. is 69%). This may be explained as follows: when a map is open, agents have many potential explanation candidates, and argumentation becomes useful to discriminate between those. When a map is labyrinth-like, there are fewer possible explanations to an unexpected event. Importance of the Global Protocol The second set of experiments seeks to evaluate the importance of the design of the global protocol. We tested our protocol against a local broadcast (LB) protocol. Local broadcast means that all the neighbours agents perceived by an agent will be involved in a communication with that agent in a given round -we alleviate the constraint of a single communication by agent. This gives us a rough upper bound upon the possible ratio gain in the system (for a given local protocol). Again, we evaluated the ratio gain induced by that LB over our classical HE, for the three different classes of maps. The results are reported in Fig. 4. Figure 4: Ratio gain of local broadcast over hypotheses exchange Note to begin with that the ratio gain is 0 when the proportion of agents is 5%, which is easily explained by the fact that it corresponds to situations involving only two agents. We first observe that all classes of maps witness a ratio gain increasing when the proportion of agents augments: the gain reaches 10 to 20%, depending on the class of maps considered. If one compares this with the improvement reported in the previous experiment, it appears to be of the same magnitude. This illustrates that the design of the global protocol cannot be ignored, especially when the proportion of agents is high. However, we also note that the effectiveness ratio gain curves have very different shapes in both cases: the gain induced by the accuracy of the local protocol increases very quickly with the proportion of agents, while the curve is really smooth for the global one. Now let us observe more carefully the results reported here: the curve corresponding to a TI of 53% is above that corresponding to 38%. This is so because the more open a map, the more opportunities to communicate with more than one agent (and hence benefits from broadcast). However, we also observe that curve for 69% is below that for 53%. This is explained as follows: in the case of 69%, the potential gain to be made in terms of surviving agents is much lower, because our protocols already give rather efficient outcomes anyway (quickly reaching 90%, see Fig. 3). A simple rule of thumb could be that when the number of agents is small, special attention should be put on the local 1004 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) protocol, whereas when that number is large, one should carefully design the global one (unless the map is so open that the protocol is already almost optimally efficient). Efficiency of the Protocols The final experiment reported here is concerned with the analysis of the efficiency of the protocols. We analysis here the mean size of the totality of the messages that are exchanged by agents (mean size of exchanges, for short) using the following protocols: HE, OE, and two variant protocols. The first one is an intermediary restricted hypotheses exchange protocol (RHE). RHE is as follows: it does not involve any challenge nor counter-propose, which means that agents cannot switch their role during the protocol (this differs from RE in that respect). In short, RHE allows an agent to exhaust its partner"s criticism, and eventually this partner will come to adopt the agent"s hypothesis. Note that this means that the autonomy of the agent is not preserved here (as an agent will essentially accept any hypothesis it cannot undermine), with the hope that the gain in efficiency will be significant enough to compensate a loss in effectiveness. The second variant protocol is a complete observation exchange protocol (COE). COE uses the same principles as OE, but includes in addition all critical negative examples (nofire) in the exchange (thus giving all examples used as arguments by the hypotheses exchanges protocol), hence improving effectiveness. Results for map 69-1 are shown on Fig. 5. Figure 5: Mean size of exchanges First we can observe the fact that the ordering of the protocols, from the least efficient to the most efficient, is COE, HE, RHE and then OE. HE being more efficient than COE proves that the argumentation process gains efficiency by selecting when it is needed to provide negative example, which have less impact that positive ones in our specific testbed. However, by communicating hypotheses before eventually giving observation to support it (HE) instead of directly giving the most crucial observations (OE), the argumentation process doubles the size of data exchanges. It is the cost for ensuring consistency at the end of the exchange (a property that OE does not support). Also significant is the fact the the mean size of exchanges is slightly higher when the number of agents is small. This is explained by the fact that in these cases only a very few agents have relevant informations in their possession, and that they will need to communicate a lot in order to come up with a common view of the situation. When the number of agents increases, this knowledge is distributed over more agents which need shorter discussions to get to mutual consistency. As a consequence, the relative gain in efficiency of using RHE appears to be better when the number of agents is small: when it is high, they will hardly argue anyway. Finally, it is worth noticing that the standard deviation for these experiments is rather high, which means that the conversation do not converge to any stereotypic pattern. 6. CONCLUSION This paper has investigated the properties of a multiagent system where each (distributed) agent locally perceives its environment, and tries to reach consistency with other agents despite severe communication restrictions. In particular we have exhibited conditions allowing convergence, and experimentally investigated a typical situation where those conditions cannot hold. There are many possible extensions to this work, the first being to further investigate the properties of different global protocols belonging to the class we identified, and their influence on the outcome. There are in particular many heuristics, highly dependent on the context of the study, that could intuitively yield interesting results (in our study, selecting the recipient on the basis of what can be inferred from his observed actions could be such a heuristic). One obvious candidate for longer term issues concern the relaxation of the assumption of perfect sensing. 7. REFERENCES [1] G. Bourgne, N. Maudet, and S. Pinson. When agents communicate hypotheses in critical situations. In Proceedings of DALT-2006, May 2006. [2] P. Harvey, C. F. Chang, and A. Ghose. Support-based distributed search: a new approach for multiagent constraint processing. In Proceedings of AAMAS06, 2006. [3] H. Jung and M. Tambe. Argumentation as distributed constraint satisfaction: Applications and results. In Proceedings of AGENTS01, 2001. [4] N. C. Karunatillake and N. R. Jennings. Is it worth arguing? In Proceedings of ArgMAS 2004, 2004. [5] S. Onta˜n´on and E. Plaza. Arguments and counterexamples in case-based joint deliberation. In Proceedings of ArgMAS-2006, May 2006. [6] D. Poole. Explanation and prediction: An architecture for default and abductive reasoning. Computational Intelligence, 5(2):97-110, 1989. [7] I. Rahwan, S. D. Ramchurn, N. R. Jennings, P. McBurney, S. Parsons, and L. Sonenberg. Argumention-based negotiation. The Knowledge Engineering Review, 4(18):345-375, 2003. [8] N. Roos, A. ten Tije, and C. Witteveen. A protocol for multi-agent diagnosis with spatially distributed knowledge. In Proceedings of AAMAS03, 2003. [9] N. Roos, A. ten Tije, and C. Witteveen. Reaching diagnostic agreement in multiagent diagnosis. In Proceedings of AAMAS04, 2004. [10] T. Takahashi, Y. Kaneda, and N. Ito. Preliminary study - using robocuprescue simulations for disasters prevention. In Proceedings of SRMED2004, 2004. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1005

global consistency;hypothesis exchange protocol;inter-agent communication;favoured hypothesis;bounded perception;context request step;negotiation and argumentation;bilateral exchange;temporal path;sequence of time point;agent communication language and protocol;time point sequence;topological constraint;mutual consistency;multiagent system;consistency;observation set

train_I-76

Negotiation by Abduction and Relaxation

This paper studies a logical framework for automated negotiation between two agents. We suppose an agent who has a knowledge base represented by a logic program. Then, we introduce methods of constructing counter-proposals in response to proposals made by an agent. To this end, we combine the techniques of extended abduction in artificial intelligence and relaxation in cooperative query answering for databases. These techniques are respectively used for producing conditional proposals and neighborhood proposals in the process of negotiation. We provide a negotiation protocol based on the exchange of these proposals and develop procedures for computing new proposals.

1. INTRODUCTION Automated negotiation has been received increasing attention in multi-agent systems, and a number of frameworks have been proposed in different contexts ([1, 2, 3, 5, 10, 11, 13, 14], for instance). Negotiation usually proceeds in a series of rounds and each agent makes a proposal at every round. An agent that received a proposal responds in two ways. One is a critique which is a remark as to whether or not (parts of) the proposal is accepted. The other is a counter-proposal which is an alternative proposal made in response to a previous proposal [13]. To see these proposals in one-to-one negotiation, suppose the following negotiation dialogue between a buyer agent B and a seller agent S. (Bi (or Si) represents an utterance of B (or S) in the i-th round.) B1: I want to buy a personal computer of the brand b1, with the specification of CPU:1GHz, Memory:512MB, HDD: 80GB, and a DVD-RW driver. I want to get it at the price under 1200 USD. S1: We can provide a PC with the requested specification if you pay for it by cash. In this case, however, service points are not added for this special discount. B2: I cannot pay it by cash. S2: In a normal price, the requested PC costs 1300 USD. B3: I cannot accept the price. My budget is under 1200 USD. S3: We can provide another computer with the requested specification, except that it is made by the brand b2. The price is exactly 1200 USD. B4: I do not want a PC of the brand b2. Instead, I can downgrade a driver from DVD-RW to CD-RW in my initial proposal. S4: Ok, I accept your offer. In this dialogue, in response to the opening proposal B1, the counter-proposal S1 is returned. In the rest of the dialogue, B2, B3, S4 are critiques, while S2, S3, B4 are counterproposals. Critiques are produced by evaluating a proposal in a knowledge base of an agent. In contrast, making counter-proposals involves generating an alternative proposal which is more favorable to the responding agent than the original one. It is known that there are two ways of producing counterproposals: extending the initial proposal or amending part of the initial proposal. According to [13], the first type appears in the dialogue: A: I propose that you provide me with service X. B: I propose that I provide you with service X if you provide me with service Z. The second type is in the dialogue: A: I propose that I provide you with service Y if you provide me with service X. B: I propose that I provide you with service X if you provide me with service Z. A negotiation proceeds by iterating such give-andtake dialogues until it reaches an agreement/disagreement. In those dialogues, agents generate (counter-)proposals by reasoning on their own goals or objectives. The objective of the agent A in the above dialogues is to obtain service X. The agent B proposes conditions to provide the service. In the process of negotiation, however, it may happen that agents are obliged to weaken or change their initial goals to reach a negotiated compromise. In the dialogue of 1022 978-81-904262-7-5 (RPS) c 2007 IFAAMAS a buyer agent and a seller agent presented above, a buyer agent changes its initial goal by downgrading a driver from DVD-RW to CD-RW. Such behavior is usually represented as specific meta-knowledge of an agent or specified as negotiation protocols in particular problems. Currently, there is no computational logic for automated negotiation which has general inference rules for producing (counter-)proposals. The purpose of this paper is to mechanize a process of building (counter-)proposals in one-to-one negotiation dialogues. We suppose an agent who has a knowledge base represented by a logic program. We then introduce methods for generating three different types of proposals. First, we use the technique of extended abduction in artificial intelligence [8, 15] to construct a conditional proposal as an extension of the original one. Second, we use the technique of relaxation in cooperative query answering for databases [4, 6] to construct a neighborhood proposal as an amendment of the original one. Third, combining extended abduction and relaxation, conditional neighborhood proposals are constructed as amended extensions of the original proposal. We develop a negotiation protocol between two agents based on the exchange of these counter-proposals and critiques. We also provide procedures for computing proposals in logic programming. This paper is organized as follows. Section 2 introduces a logical framework used in this paper. Section 3 presents methods for constructing proposals, and provides a negotiation protocol. Section 4 provides methods for computing proposals in logic programming. Section 5 discusses related works, and Section 6 concludes the paper. 2. PRELIMINARIES Logic programs considered in this paper are extended disjunctive programs (EDP) [7]. An EDP (or simply a program) is a set of rules of the form: L1 ; · · · ; Ll ← Ll+1 , . . . , Lm, not Lm+1 , . . . , not Ln (n ≥ m ≥ l ≥ 0) where each Li is a positive/negative literal, i.e., A or ¬A for an atom A, and not is negation as failure (NAF). not L is called an NAF-literal. The symbol ; represents disjunction. The left-hand side of the rule is the head, and the right-hand side is the body. For each rule r of the above form, head(r), body+ (r) and body− (r) denote the sets of literals {L1, . . . , Ll}, {Ll+1, . . . , Lm}, and {Lm+1, . . . , Ln}, respectively. Also, not body− (r) denotes the set of NAF-literals {not Lm+1, . . . , not Ln}. A disjunction of literals and a conjunction of (NAF-)literals in a rule are identified with its corresponding sets of literals. A rule r is often written as head(r) ← body+ (r), not body− (r) or head(r) ← body(r) where body(r) = body+ (r)∪not body− (r). A rule r is disjunctive if head(r) contains more than one literal. A rule r is an integrity constraint if head(r) = ∅; and r is a fact if body(r) = ∅. A program is NAF-free if no rule contains NAF-literals. Two rules/literals are identified with respect to variable renaming. A substitution is a mapping from variables to terms θ = {x1/t1, . . . , xn/tn}, where x1, . . . , xn are distinct variables and each ti is a term distinct from xi. Given a conjunction G of (NAF-)literals, Gθ denotes the conjunction obtained by applying θ to G. A program, rule, or literal is ground if it contains no variable. A program P with variables is a shorthand of its ground instantiation Ground(P), the set of ground rules obtained from P by substituting variables in P by elements of its Herbrand universe in every possible way. The semantics of an EDP is defined by the answer set semantics [7]. Let Lit be the set of all ground literals in the language of a program. Suppose a program P and a set of literals S(⊆ Lit). Then, the reduct P S is the program which contains the ground rule head(r) ← body+ (r) iff there is a rule r in Ground(P) such that body− (r)∩S = ∅. Given an NAF-free EDP P, Cn(P) denotes the smallest set of ground literals which is (i) closed under P, i.e., for every ground rule r in Ground(P), body(r) ⊆ Cn(P) implies head(r) ∩ Cn(P) = ∅; and (ii) logically closed, i.e., it is either consistent or equal to Lit. Given an EDP P and a set S of literals, S is an answer set of P if S = Cn(P S ). A program has none, one, or multiple answer sets in general. An answer set is consistent if it is not Lit. A program P is consistent if it has a consistent answer set; otherwise, P is inconsistent. Abductive logic programming [9] introduces a mechanism of hypothetical reasoning to logic programming. An abductive framework used in this paper is the extended abduction introduced by Inoue and Sakama [8, 15]. An abductive program is a pair P, H where P is an EDP and H is a set of literals called abducibles. When a literal L ∈ H contains variables, any instance of L is also an abducible. An abductive program P, H is consistent if P is consistent. Throughout the paper, abductive programs are assumed to be consistent unless stated otherwise. Let G = L1, . . . , Lm, not Lm+1, . . . , not Ln be a conjunction, where all variables in G are existentially quantified at the front and range-restricted, i.e., every variable in Lm+1, . . . , Ln appears in L1, . . . , Lm. A set S of ground literals satisfies the conjunction G if { L1θ, . . . , Lmθ } ⊆ S and { Lm+1θ, . . . , Lnθ }∩ S = ∅ for some ground instance Gθ with a substitution θ. Let P, H be an abductive program and G a conjunction as above. A pair (E, F) is an explanation of an observation G in P, H if1 1. (P \ F) ∪ E has an answer set which satisfies G, 2. (P \ F) ∪ E is consistent, 3. E and F are sets of ground literals such that E ⊆ H\P and F ⊆ H ∩ P. When (P \ F) ∪ E has an answer set S satisfying the above three conditions, S is called a belief set of an abductive program P, H satisfying G (with respect to (E, F)). Note that if P has a consistent answer set S satisfying G, S is also a belief set of P, H satisfying G with respect to (E, F) = (∅, ∅). Extended abduction introduces/removes hypotheses to/from a program to explain an observation. Note that normal abduction (as in [9]) considers only introducing hypotheses to explain an observation. An explanation (E, F) of an observation G is called minimal if for any explanation (E , F ) of G, E ⊆ E and F ⊆ F imply E = E and F = F. Example 2.1. Consider the abductive program P, H : P : flies(x) ← bird(x), not ab(x) , ab(x) ← broken-wing(x) , bird(tweety) ← , bird(opus) ← , broken-wing(tweety) ← . H : broken-wing(x) . The observation G = flies(tweety) has the minimal explanation (E, F) = (∅, {broken-wing(tweety)}). 1 This defines credulous explanations [15]. Skeptical explanations are used in [8]. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1023 3. NEGOTIATION 3.1 Conditional Proposals by Abduction We suppose an agent who has a knowledge base represented by an abductive program P, H . A program P consists of two types of knowledge, belief B and desire D, where B represents objective knowledge of an agent, while D represents subjective knowledge in general. We define P = B ∪ D, but do not distinguish B and D if such distinction is not important in the context. In contrast, abducibles H are used for representing permissible conditions to make a compromise in the process of negotiation. Definition 3.1. A proposal G is a conjunction of literals and NAF-literals: L1, . . . , Lm, not Lm+1, . . . , not Ln where every variable in G is existentially quantified at the front and range-restricted. In particular, G is called a critique if G = accept or G = reject where accept and reject are the reserved propositions. A counter-proposal is a proposal made in response to a proposal. Definition 3.2. A proposal G is accepted in an abductive program P, H if P has an answer set satisfying G. When a proposal is not accepted, abduction is used for seeking conditions to make it acceptable. Definition 3.3. Let P, H be an abductive program and G a proposal. If (E, F) is a minimal explanation of Gθ for some substitution θ in P, H , the conjunction G : Gθ, E, not F is called a conditional proposal (for G), where E, not F represents the conjunction: A1, . . . , Ak, not Ak+1, . . . , not Al for E = {A1, . . . , Ak} and F = { Ak+1, . . . , Al }. Proposition 3.1. Let P, H be an abductive program and G a proposal. If G is a conditional proposal, there is a belief set S of P, H satisfying G . Proof. When G = Gθ, E, not F, (P \ F) ∪ E has a consistent answer set S satisfying Gθ and E ∩ F = ∅. In this case, S satisfies Gθ, E, not F. A conditional proposal G provides a minimal requirement for accepting the proposal G. If Gθ has multiple minimal explanations, several conditional proposals exist accordingly. When (E, F) = (∅, ∅), a conditional proposal is used as a new proposal made in response to the proposal G. Example 3.1. An agent seeks a position of a research assistant at the computer department of a university with the condition that the salary is at least 50,000 USD per year. The agent makes his/her request as the proposal:2 G = assist(compt dept), salary(x), x ≥ 50, 000. The university has the abductive program P, H : P : salary(40, 000) ← assist(compt dept), not has PhD, salary(60, 000) ← assist(compt dept), has PhD, salary(50, 000) ← assist(math dept), salary(55, 000) ← system admin(compt dept), 2 For notational convenience, we often include mathematical (in)equations in proposals/programs. They are written by literals, for instance, x ≥ y by geq(x, y) with a suitable definition of the predicate geq. employee(x) ← assist(x), employee(x) ← system admin(x), assist(compt dept); assist(math dept) ; system admin(compt dept) ←, H : has PhD, where available positions are represented by disjunction. According to P, the base salary of a research assistant at the computer department is 40,000 USD, but if he/she has PhD, it is 60,000 USD. In this case, (E, F) = ({has PhD}, ∅) becomes the minimal explanation of Gθ = assist(compt dept), salary(60, 000) with θ = { x/60, 000 }. Then, the conditional proposal made by the university becomes assist(compt dept), salary(60, 000), has PhD . 3.2 Neighborhood Proposals by Relaxation When a proposal is unacceptable, an agent tries to construct a new counter-proposal by weakening constraints in the initial proposal. We use techniques of relaxation for this purpose. Relaxation is used as a technique of cooperative query answering in databases [4, 6]. When an original query fails in a database, relaxation expands the scope of the query by relaxing the constraints in the query. This allows the database to return neighborhood answers which are related to the original query. We use the technique for producing proposals in the process of negotiation. Definition 3.4. Let P, H be an abductive program and G a proposal. Then, G is relaxed to G in the following three ways: Anti-instantiation: Construct G such that G θ = G for some substitution θ. Dropping conditions: Construct G such that G ⊂ G. Goal replacement: If G is a conjunction G1, G2, where G1 and G2 are conjunctions, and there is a rule L ← G1 in P such that G1θ = G1 for some substitution θ, then build G as Lθ, G2. Here, Lθ is called a replaced literal. In each case, every variable in G is existentially quantified at the front and range-restricted. Anti-instantiation replaces constants (or terms) with fresh variables. Dropping conditions eliminates some conditions in a proposal. Goal replacement replaces the condition G1 in G with a literal Lθ in the presence of a rule L ← G1 in P under the condition G1θ = G1. All these operations generalize proposals in different ways. Each G obtained by these operations is called a relaxation of G. It is worth noting that these operations are also used in the context of inductive generalization [12]. The relaxed proposal can produce new offers which are neighbor to the original proposal. Definition 3.5. Let P, H be an abductive program and G a proposal. 1. Let G be a proposal obtained by anti-instantiation. If P has an answer set S which satisfies G θ for some substitution θ and G θ = G, G θ is called a neighborhood proposal by anti-instantiation. 2. Let G be a proposal obtained by dropping conditions. If P has an answer set S which satisfies G θ for some substitution θ, G θ is called a neighborhood proposal by dropping conditions. 1024 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 3. Let G be a proposal obtained by goal replacement. For a replaced literal L ∈ G and a rule H ← B in P such that L = Hσ and (G \ {L}) ∪ Bσ = G for some substitution σ, put G = (G \ {L}) ∪ Bσ. If P has an answer set S which satisfies G θ for some substitution θ, G θ is called a neighborhood proposal by goal replacement. Example 3.2. (cont. Example 3.1) Given the proposal G = assist(compt dept), salary(x), x ≥ 50, 000, • G1 = assist(w), salary(x), x ≥ 50, 000 is produced by substituting compt dept with a variable w. As G1θ1 = assist(math dept), salary(50, 000) with θ1 = { w/math dept } is satisfied by an answer set of P, G1θ1 becomes a neighborhood proposal by anti-instantiation. • G2 = assist(compt dept), salary(x) is produced by dropping the salary condition x ≥ 50, 000. As G2θ2 = assist(compt dept), salary(40, 000) with θ2 = { x/40, 000 } is satisfied by an answer set of P, G2θ2 becomes a neighborhood proposal by dropping conditions. • G3 = employee(compt dept), salary(x), x ≥ 50, 000 is produced by replacing assist(compt dept) with employee(compt dept) using the rule employee(x) ← assist(x) in P. By G3 and the rule employee(x) ← system admin(x) in P, G3 = sys admin(compt dept), salary(x), x ≥ 50, 000 is produced. As G3 θ3 = sys admin(compt dept), salary(55, 000) with θ3 = { x/55, 000 } is satisfied by an answer set of P, G3 θ3 becomes a neighborhood proposal by goal replacement. Finally, extended abduction and relaxation are combined to produce conditional neighborhood proposals. Definition 3.6. Let P, H be an abductive program and G a proposal. 1. Let G be a proposal obtained by either anti-instantiation or dropping conditions. If (E, F) is a minimal explanation of G θ(= G) for some substitution θ, the conjunction G θ, E, not F is called a conditional neighborhood proposal by anti-instantiation/dropping conditions. 2. Let G be a proposal obtained by goal replacement. Suppose G as in Definition 3.5(3). If (E, F) is a minimal explanation of G θ for some substitution θ, the conjunction G θ, E, not F is called a conditional neighborhood proposal by goal replacement. A conditional neighborhood proposal reduces to a neighborhood proposal when (E, F) = (∅, ∅). 3.3 Negotiation Protocol A negotiation protocol defines how to exchange proposals in the process of negotiation. This section presents a negotiation protocol in our framework. We suppose one-to-one negotiation between two agents who have a common ontology and the same language for successful communication. Definition 3.7. A proposal L1, ..., Lm, not Lm+1, ..., not Ln violates an integrity constraint ← body+ (r), not body− (r) if for any substitution θ, there is a substitution σ such that body+ (r)σ ⊆ { L1θ, . . . , Lmθ }, body− (r)σ∩{ L1θ, . . . , Lmθ } = ∅, and body− (r)σ ⊆ { Lm+1θ, . . . , Lnθ }. Integrity constraints are conditions which an agent should satisfy, so that they are used to explain why an agent does not accept a proposal. A negotiation proceeds in a series of rounds. Each i-th round (i ≥ 1) consists of a proposal Gi 1 made by one agent Ag1 and another proposal Gi 2 made by the other agent Ag2. Definition 3.8. Let P1, H1 be an abductive program of an agent Ag1 and Gi 2 a proposal made by Ag2 at the i-th round. A critique set of Ag1 (at the i-th round) is a set CSi 1(P1, Gj 2) = CSi−1 1 (P1, Gj−1 2 ) ∪ { r | r is an integrity constraint in P1 and Gj 2 violates r } where j = i − 1 or i, and CS0 1 (P1, G0 2) = CS1 1 (P1, G0 2) = ∅. A critique set of an agent Ag1 accumulates integrity constraints which are violated by proposals made by another agent Ag2. CSi 2(P2, Gj 1) is defined in the same manner. Definition 3.9. Let Pk, Hk be an abductive program of an agent Agk and Gj a proposal, which is not a critique, made by any agent at the j(≤ i)-th round. A negotiation set of Agk (at the i-th round) is a triple NSi k = (Si c, Si n, Si cn), where Si c is the set of conditional proposals, Si n is the set of neighborhood proposals, and Si cn is the set of conditional neighborhood proposals, produced by Gj and Pk, Hk . A negotiation set represents the space of possible proposals made by an agent. Si x (x ∈ {c, n, cn}) accumulates proposals produced by Gj (1 ≤ j ≤ i) according to Definitions 3.3, 3.5, and 3.6. Note that an agent can construct counter-proposals by modifying its own previous proposals or another agent"s proposals. An agent Agk accumulates proposals that are made by Agk but are rejected by another agent, in the failed proposal set FP i k (at the i-th round), where FP 0 k = ∅. Suppose two agents Ag1 and Ag2 who have abductive programs P1, H1 and P2, H2 , respectively. Given a proposal G1 1 which is satisfied by an answer set of P1, a negotiation starts. In response to the proposal Gi 1 made by Ag1 at the i-th round, Ag2 behaves as follows. 1. If Gi 1 = accept, an agreement is reached and negotiation ends in success. 2. Else if Gi 1 = reject, put FP i 2 = FPi−1 2 ∪{Gi−1 2 } where {G0 2} = ∅. Proceed to the step 4(b). 3. Else if P2 has an answer set satisfying Gi 1, Ag2 returns Gi 2 = accept to Ag1. Negotiation ends in success. 4. Otherwise, Ag2 behaves as follows. Put FP i 2 = FPi−1 2 . (a) If Gi 1 violates an integrity constraint in P2, return the critique Gi 2 = reject to Ag1, together with the critique set CSi 2(P2, Gi 1). (b) Otherwise, construct NSi 2 as follows. (i) Produce Si c. Let μ(Si c) = { p | p ∈ Si c \ FPi 2 and p satisfies the constraints in CSi 1(P1, Gi−1 2 )}. If μ(Si c) = ∅, select one from μ(Si c) and propose it as Gi 2 to Ag1; otherwise, go to (ii). (ii) Produce Si n. If μ(Si n) = ∅, select one from μ(Si n) and propose it as Gi 2 to Ag1; otherwise, go to (iii). (iii) Produce Si cn. If μ(Si cn) = ∅, select one from μ(Si cn) and propose it as Gi 2 to Ag1; otherwise, negotiation ends in failure. This means that Ag2 can make no counter-proposal or every counterproposal made by Ag2 is rejected by Ag1. In the step 4(a), Ag2 rejects the proposal Gi 1 and returns the reason of rejection as a critique set. This helps for Ag1 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1025 in preparing a next counter-proposal. In the step 4(b), Ag2 constructs a new proposal. In its construction, Ag2 should take care of the critique set CSi 1(P1, Gi−1 2 ), which represents integrity constraints, if any, accumulated in previous rounds, that Ag1 must satisfy. Also, FP i 2 is used for removing proposals which have been rejected. Construction of Si x (x ∈ {c, n, cn}) in NSi 2 is incrementally done by adding new counter-proposals produced by Gi 1 or Gi−1 2 to Si−1 x . For instance, Si n in NSi 2 is computed as Si n = Si−1 n ∪{ p | p is a neighborhood proposal made by Gi 1 } ∪ { p | p is a neighborhood proposal made by Gi−1 2 }, where S0 n = ∅. That is, Si n is constructed from Si−1 n by adding new proposals which are obtained by modifying the proposal Gi 1 made by Ag1 at the i-th round or modifying the proposal Gi−1 2 made by Ag2 at the (i − 1)-th round. Si c and Si cn are obtained as well. In the above protocol, an agent produces Si c at first, secondly Si n, and finally Si cn. This strategy seeks conditions which satisfy the given proposal, prior to neighborhood proposals which change the original one. Another strategy, which prefers neighborhood proposals to conditional ones, is also considered. Conditional neighborhood proposals are to be considered in the last place, since they differ from the original one to the maximal extent. The above protocol produces the candidate proposals in Si x for each x ∈ {c, n, cn} at once. We can consider a variant of the protocol in which each proposal in Si x is constructed one by one (see Example 3.3). The above protocol is repeatedly applied to each one of the two negotiating agents until a negotiation ends in success/failure. Formally, the above negotiation protocol has the following properties. Theorem 3.2. Let Ag1 and Ag2 be two agents having abductive programs P1, H1 and P2, H2 , respectively. 1. If P1, H1 and P2, H2 are function-free (i.e., both Pi and Hi contain no function symbol), any negotiation will terminate. 2. If a negotiation terminates with agreement on a proposal G, both P1, H1 and P2, H2 have belief sets satisfying G. Proof. 1. When an abductive program is function-free, abducibles and negotiation sets are both finite. Moreover, if a proposal is once rejected, it is not proposed again by the function μ. Thus, negotiation will terminate in finite steps. 2. When a proposal G is made by Ag1, P1, H1 has a belief set satisfying G. If the agent Ag2 accepts the proposal G, it is satisfied by an answer set of P2 which is also a belief set of P2, H2 . Example 3.3. Suppose a buying-selling situation in the introduction. A seller agent has the abductive program Ps, Hs in which Ps consists of belief Bs and desire Ds: Bs : pc(b1, 1G, 512M, 80G) ; pc(b2, 1G, 512M, 80G) ←,(1) dvd-rw ; cd-rw ←, (2) Ds : normal price(1300) ← pc(b1, 1G, 512M, 80G), dvd-rw, (3) normal price(1200) ← pc(b1, 1G, 512M, 80G), cd-rw, (4) normal price(1200) ← pc(b2, 1G, 512M, 80G), dvd-rw, (5) price(x) ← normal price(x), add point, (6) price(x ∗ 0.9) ← normal price(x), pay cash, not add point,(7) add point ←, (8) Hs : add point, pay cash. Here, (1) and (2) represent selection of products. The atom pc(b1, 1G, 512M, 80G) represents that the seller agent has a PC of the brand b1 such that CPU is 1GHz, memory is 512MB, and HDD is 80GB. Prices of products are represented as desire of the seller. The rules (3) - (5) are normal prices of products. A normal price is a selling price on the condition that service points are added (6). On the other hand, a discount price is applied if the paying method is cash and no service point is added (7). The fact (8) represents the addition of service points. This service would be withdrawn in case of discount prices, so add point is specified as an abducible. A buyer agent has the abductive program Pb, Hb in which Pb consists of belief Bb and desire Db: Bb : drive ← dvd-rw, (9) drive ← cd-rw, (10) price(x) ←, (11) Db : pc(b1, 1G, 512M, 80G) ←, (12) dvd-rw ←, (13) cd-rw ← not dvd-rw, (14) ← pay cash, (15) ← price(x), x > 1200, (16) Hb : dvd-rw. Rules (12) - (16) are the buyer"s desire. Among them, (15) and (16) impose constraints for buying a PC. A DVD-RW is specified as an abducible which is subject to concession. (1st round) First, the following proposal is given by the buyer agent: G1 b : pc(b1, 1G, 512M, 80G), dvd-rw, price(x), x ≤ 1200. As Ps has no answer set which satisfies G1 b , the seller agent cannot accept the proposal. The seller takes an action of making a counter-proposal and performs abduction. As a result, the seller finds the minimal explanation (E, F) = ({ pay cash }, { add point }) which explains G1 b θ1 with θ1 = { x/1170 }. The seller constructs the conditional proposal: G1 s : pc(b1, 1G, 512M, 80G), dvd-rw, price(1170), pay cash, not add point and offers it to the buyer. (2nd round) The buyer does not accept G1 s because he/she cannot pay it by cash (15). The buyer then returns the critique G2 b = reject to the seller, together with the critique set CS2 b (Pb, G1 s) = {(15)}. In response to this, the seller tries to make another proposal which satisfies the constraint in this critique set. As G1 s is stored in FP 2 s and no other conditional proposal satisfying the buyer"s requirement exists, the seller produces neighborhood proposals. He/she relaxes G1 b by dropping x ≤ 1200 in the condition, and produces pc(b1, 1G, 512M, 80G), dvd-rw, price(x). As Ps has an answer set which satisfies G2 s : pc(b1, 1G, 512M, 80G), dvd-rw, price(1300), 1026 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) the seller offers G2 s as a new counter-proposal. (3rd round) The buyer does not accept G2 s because he/she cannot pay more than 1200USD (16). The buyer again returns the critique G3 b = reject to the seller, together with the critique set CS3 b (Pb, G2 s) = CS2 b (Pb, G1 s) ∪ {(16)}. The seller then considers another proposal by replacing b1 with a variable w, G1 b now becomes pc(w, 1G, 512M, 80G), dvd-rw, price(x), x ≤ 1200. As Ps has an answer set which satisfies G3 s : pc(b2, 1G, 512M, 80G), dvd-rw, price(1200), the seller offers G3 s as a new counter-proposal. (4th round) The buyer does not accept G3 s because a PC of the brand b2 is out of his/her interest and Pb has no answer set satisfying G3 s. Then, the buyer makes a concession by changing his/her original goal. The buyer relaxes G1 b by goal replacement using the rule (9) in Pb, and produces pc(b1, 1G, 512M, 80G), drive, price(x), x ≤ 1200. Using (10), the following proposal is produced: pc(b1, 1G, 512M, 80G), cd-rw, price(x), x ≤ 1200. As Pb \ { dvd-rw } has a consistent answer set satisfying the above proposal, the buyer proposes the conditional neighborhood proposal G4 b : pc(b1, 1G, 512M, 80G), cd-rw, not dvd-rw, price(x), x ≤ 1200 to the seller agent. Since Ps also has an answer set satisfying G4 b , the seller accepts it and sends the message G4 s = accept to the buyer. Thus, the negotiation ends in success. 4. COMPUTATION In this section, we provide methods of computing proposals in terms of answer sets of programs. We first introduce some definitions from [15]. Definition 4.1. Given an abductive program P, H , the set UR of update rules is defined as: UR = { L ← not L, L ← not L | L ∈ H } ∪ { +L ← L | L ∈ H \ P } ∪ { −L ← not L | L ∈ H ∩ P } , where L, +L, and −L are new atoms uniquely associated with every L ∈ H. The atoms +L and −L are called update atoms. By the definition, the atom L becomes true iff L is not true. The pair of rules L ← not L and L ← not L specify the situation that an abducible L is true or not. When p(x) ∈ H and p(a) ∈ P but p(t) ∈ P for t = a, the rule +L ← L precisely becomes +p(t) ← p(t) for any t = a. In this case, the rule is shortly written as +p(x) ← p(x), x = a. Generally, the rule becomes +p(x) ← p(x), x = t1, . . . , x = tn for n such instances. The rule +L ← L derives the atom +L if an abducible L which is not in P is to be true. In contrast, the rule −L ← not L derives the atom −L if an abducible L which is in P is not to be true. Thus, update atoms represent the change of truth values of abducibles in a program. That is, +L means the introduction of L, while −L means the deletion of L. When an abducible L contains variables, the associated update atom +L or −L is supposed to have exactly the same variables. In this case, an update atom is semantically identified with its ground instances. The set of all update atoms associated with the abducibles in H is denoted by UH, and UH = UH+ ∪ UH− where UH+ (resp. UH− ) is the set of update atoms of the form +L (resp. −L). Definition 4.2. Given an abductive program P, H , its update program UP is defined as the program UP = (P \ H) ∪ UR . An answer set S of UP is called U-minimal if there is no answer set T of UP such that T ∩ UH ⊂ S ∩ UH. By the definition, U-minimal answer sets exist whenever UP has answer sets. Update programs are used for computing (minimal) explanations of an observation. Given an observation G as a conjunction of literals and NAF-literals possibly containing variables, we introduce a new ground literal O together with the rule O ← G. In this case, O has an explanation (E, F) iff G has the same explanation. With this replacement, an observation is assumed to be a ground literal without loss of generality. In what follows, E+ = { +L | L ∈ E } and F − = { −L | L ∈ F } for E ⊆ H and F ⊆ H. Proposition 4.1. ([15]) Let P, H be an abductive program, UP its update program, and G a ground literal representing an observation. Then, a pair (E, F) is an explanation of G iff UP ∪ { ← not G } has a consistent answer set S such that E+ = S ∩ UH+ and F− = S ∩ UH− . In particular, (E, F) is a minimal explanation iff S is a U-minimal answer set. Example 4.1. To explain the observation G = flies(t) in the program P of Example 2.1, first construct the update program UP of P:3 UP : flies(x) ← bird(x), not ab(x), ab(x) ← broken-wing(x) , bird(t) ← , bird(o) ← , broken-wing(x) ← not broken-wing(x), broken-wing(x) ← not broken-wing(x), +broken-wing(x) ← broken-wing(x), x = t , −broken-wing(t) ← not broken-wing(t) . Next, consider the program UP ∪ { ← not flies(t) }. It has the single U-minimal answer set: S = { bird(t), bird(o), flies(t), flies(o), broken-wing(t), broken-wing(o), −broken-wing(t) }. The unique minimal explanation (E, F) = (∅, {broken-wing(t)}) of G is expressed by the update atom −broken-wing(t) in S ∩ UH− . Proposition 4.2. Let P, H be an abductive program and G a ground literal representing an observation. If P ∪ { ← not G } has a consistent answer set S, G has the minimal explanation (E, F) = (∅, ∅) and S satisfies G. Now we provide methods for computing (counter-)proposals. First, conditional proposals are computed as follows. input : an abductive program P, H , a proposal G; output : a set Sc of proposals. If G is a ground literal, compute its minimal explanation (E, F) in P, H using the update program. Put G, E, not F in Sc. Else if G is a conjunction possibly containing variables, consider the abductive program 3 t represents tweety and o represents opus. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1027 P ∪{ O ← G }, H with a ground literal O. Compute a minimal explanation of O in P ∪ { O ← G }, H using its update program. If O has a minimal explanation (E, F) with a substitution θ for variables in G, put Gθ, E, not F in Sc. Next, neighborhood proposals are computed as follows. input : an abductive program P, H , a proposal G; output : a set Sn of proposals. % neighborhood proposals by anti-instantiation; Construct G by anti-instantiation. For a ground literal O, if P ∪ { O ← G } ∪ { ← not O } has a consistent answer set satisfying G θ with a substitution θ and G θ = G, put G θ in Sn. % neighborhood proposals by dropping conditions; Construct G by dropping conditions. If G is a ground literal and the program P ∪ { ← not G } has a consistent answer set, put G in Sn. Else if G is a conjunction possibly containing variables, do the following. For a ground literal O, if P ∪{ O ← G }∪{ ← not O } has a consistent answer set satisfying G θ with a substitution θ, put G θ in Sn. % neighborhood proposals by goal replacement; Construct G by goal replacement. If G is a ground literal and there is a rule H ← B in P such that G = Hσ and Bσ = G for some substitution σ, put G = Bσ. If P ∪ { ← not G } has a consistent answer set satisfying G θ with a substitution θ, put G θ in Sn. Else if G is a conjunction possibly containing variables, do the following. For a replaced literal L ∈ G , if there is a rule H ← B in P such that L = Hσ and (G \ {L}) ∪ Bσ = G for some substitution σ, put G = (G \ {L}) ∪ Bσ. For a ground literal O, if P ∪ { O ← G } ∪ { ← not O } has a consistent answer set satisfying G θ with a substitution θ, put G θ in Sn. Theorem 4.3. The set Sc (resp. Sn) computed above coincides with the set of conditional proposals (resp. neighborhood proposals). Proof. The result for Sc follows from Definition 3.3 and Proposition 4.1. The result for Sn follows from Definition 3.5 and Proposition 4.2. Conditional neighborhood proposals are computed by combining the above two procedures. Those proposals are computed at each round. Note that the procedure for computing Sn contains some nondeterministic choices. For instance, there are generally several candidates of literals to relax in a proposal. Also, there might be several rules in a program for the usage of goal replacement. In practice, an agent can prespecify literals in a proposal for possible relaxation or rules in a program for the usage of goal replacement. 5. RELATED WORK As there are a number of literature on automated negotiation, this section focuses on comparison with negotiation frameworks based on logic and argumentation. Sadri et al. [14] use abductive logic programming as a representation language of negotiating agents. Agents negotiate using common dialogue primitives, called dialogue moves. Each agent has an abductive logic program in which a sequence of dialogues are specified by a program, a dialogue protocol is specified as constraints, and dialogue moves are specified as abducibles. The behavior of agents is regulated by an observe-think-act cycle. Once a dialogue move is uttered by an agent, another agent that observed the utterance thinks and acts using a proof procedure. Their approach and ours both employ abductive logic programming as a platform of agent reasoning, but the use of it is quite different. First, they use abducibles to specify dialogue primitives of the form tell(utterer, receiver, subject, identifier, time), while we use abducibles to specify arbitrary permissible hypotheses to construct conditional proposals. Second, a program pre-specifies a plan to carry out in order to achieve a goal, together with available/missing resources in the context of resource-exchanging problems. This is in contrast with our method in which possible counter-proposals are newly constructed in response to a proposal made by an agent. Third, they specify a negotiation policy inside a program (as integrity constraints), while we give a protocol independent of individual agents. They provide an operational model that completely specifies the behavior of agents in terms of agent cycle. We do not provide such a complete specification of the behavior of agents. Our primary interest is to mechanize construction of proposals. Bracciali and Torroni [2] formulate abductive agents that have knowledge in abductive logic programs. To explain an observation, two agents communicate by exchanging integrity constraints. In the process of communication, an agent can revise its own integrity constraints according to the information provided by the other agent. A set IC of integrity constraints relaxes a set IC (or IC tightens IC ) if any observation that can be proved with respect to IC can also be proved with respect to IC . For instance, IC : ← a, b, c relaxes IC : ← a, b. Thus, they use relaxation for weakening the constraints in an abductive logic program. In contrast, we use relaxation for weakening proposals and three different relaxation methods, anti-instantiation, dropping conditions, and goal replacement, are considered. Their goal is to explain an observation by revising integrity constraints of an agent through communication, while we use integrity constraints for communication to explain critiques and help other agents in making counter-proposals. Meyer et al. [11] introduce a logical framework for negotiating agents. They introduce two different modes of negotiation: concession and adaptation. They provide rational postulates to characterize negotiated outcomes between two agents, and describe methods for constructing outcomes. They provide logical conditions for negotiated outcomes to satisfy, but they do not describe a process of negotiation nor negotiation protocols. Moreover, they represent agents by classical propositional theories, which is different from our abductive logic programming framework. Foo et al. [5] model one-to-one negotiation as a one-time encounter between two extended logic programs. An agent offers an answer set of its program, and their mutual deal is regarded as a trade on their answer sets. Starting from the initial agreement set S∩T for an answer set S of an agent and an answer set T of another agent, each agent extends this set to reflect its own demand while keeping consistency with demand of the other agent. Their algorithm returns new programs having answer sets which are consistent with each other and keep the agreement set. The work is extended to repeated encounters in [3]. In their framework, two agents exchange answer sets to produce a common belief set, which is different from our framework of exchanging proposals. There are a number of proposals for negotiation based 1028 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) on argumentation. An advantage of argumentation-based negotiation is that it constructs a proposal with arguments supporting the proposal [1]. The existence of arguments is useful to convince other agents of reasons why an agent offers (counter-)proposals or returns critiques. Parsons et al. [13] develop a logic of argumentation-based negotiation among BDI agents. In one-to-one negotiation, an agent A generates a proposal together with its arguments, and passes it to another agent B. The proposal is evaluated by B which attempts to build arguments against it. If it conflicts with B"s interest, B informs A of its objection by sending back its attacking argument. In response to this, A tries to find an alternative way of achieving its original objective, or a way of persuading B to drop its objection. If either type of argument can be found, A will submit it to B. If B finds no reason to reject the new proposal, it will be accepted and the negotiation ends in success. Otherwise, the process is iterated. In this negotiation processes, the agent A never changes its original objective, so that negotiation ends in failure if A fails to find an alternative way of achieving the original objective. In our framework, when a proposal is rejected by another agent, an agent can weaken or change its objective by abduction and relaxation. Our framework does not have a mechanism of argumentation, but reasons for critiques can be informed by responding critique sets. Kakas and Moraitis [10] propose a negotiation protocol which integrates abduction within an argumentation framework. A proposal contains an offer corresponding to the negotiation object, together with supporting information representing conditions under which this offer is made. Supporting information is computed by abduction and is used for constructing conditional arguments during the process of negotiation. In their negotiation protocol, when an agent cannot satisfy its own goal, the agent considers the other agent"s goal and searches for conditions under which the goal is acceptable. Our present approach differs from theirs in the following points. First, they use abduction to seek conditions to support arguments, while we use abduction to seek conditions for proposals to accept. Second, in their negotiation protocol, counter-proposals are chosen among candidates based on preference knowledge of an agent at meta-level, which represents policy under which an agent uses its object-level decision rules according to situations. In our framework, counter-proposals are newly constructed using abduction and relaxation. The method of construction is independent of particular negotiation protocols. As [2, 10, 14], abduction or abductive logic programming used in negotiation is mostly based on normal abduction. In contrast, our approach is based on extended abduction which can not only introduce hypotheses but remove them from a program. This is another important difference. Relaxation and neighborhood query answering are devised to make databases cooperative with their users [4, 6]. In this sense, those techniques have the spirit similar to cooperative problem solving in multi-agent systems. As far as the authors know, however, there is no study which applies those technique to agent negotiation. 6. CONCLUSION In this paper we proposed a logical framework for negotiating agents. To construct proposals in the process of negotiation, we combined the techniques of extended abduction and relaxation. It was shown that these two operations are used for general inference rules in producing proposals. We developed a negotiation protocol between two agents based on exchange of proposals and critiques, and provided procedures for computing proposals in abductive logic programming. This enables us to realize automated negotiation on top of the existing answer set solvers. The present framework does not have a mechanism of selecting an optimal (counter-)proposal among different alternatives. To compare and evaluate proposals, an agent must have preference knowledge of candidate proposals. Further elaboration to maximize the utility of agents is left for future study. 7. REFERENCES [1] L. Amgoud, S. Parsons, and N. Maudet. Arguments, dialogue, and negotiation. In: Proc. ECAI-00, pp. 338-342, IOS Press, 2000. [2] A. Bracciali and P. Torroni. A new framework for knowledge revision of abductive agents through their interaction. In: Proc. CLIMA-IV, Computational Logic in Multi-Agent Systems, LNAI 3259, pp. 159-177, 2004. [3] W. Chen, M. Zhang, and N. Foo. Repeated negotiation of logic programs. In: Proc. 7th Workshop on Nonmonotonic Reasoning, Action and Change, 2006. [4] W. W. Chu, Q. Chen, and R.-C. Lee. Cooperative query answering via type abstraction hierarchy. In: Cooperating Knowledge Based Systems, S. M. Deen ed., pp. 271-290, Springer, 1990. [5] N. Foo, T. Meyer, Y. Zhang, and D. Zhang. Negotiating logic programs. In: Proc. 6th Workshop on Nonmonotonic Reasoning, Action and Change, 2005. [6] T. Gaasterland, P. Godfrey, and J. Minker. Relaxation as a platform for cooperative answering. Journal of Intelligence Information Systems 1(3/4):293-321, 1992. [7] M. Gelfond and V. Lifschitz. Classical negation in logic programs and disjunctive databases. New Generation Computing 9:365-385, 1991. [8] K. Inoue and C. Sakama. Abductive framework for nonmonotonic theory change. In: Proc. IJCAI-95, pp. 204-210, Morgan Kaufmann. [9] A. C. Kakas, R. A. Kowalski, and F. Toni, The role of abduction in logic programming. In: Handbook of Logic in AI and Logic Programming, D. M. Gabbay, et al. (eds), vol. 5, pp. 235-324, Oxford University Press, 1998. [10] A. C. Kakas and P. Moraitis. Adaptive agent negotiation via argumentation. In: Proc. AAMAS-06, pp. 384-391, ACM Press. [11] T. Meyer, N. Foo, R. Kwok, and D. Zhang. Logical foundation of negotiation: outcome, concession and adaptation. In: Proc. AAAI-04, pp. 293-298, MIT Press. [12] R. S. Michalski. A theory and methodology of inductive learning. In: Machine Learning: An Artificial Intelligence Approach, R. S. Michalski, et al. (eds), pp. 83-134, Morgan Kaufmann, 1983. [13] S. Parsons, C. Sierra and N. Jennings. Agents that reason and negotiate by arguing. Journal of Logic and Computation, 8(3):261-292, 1988. [14] F. Sadri, F. Toni, and P. Torroni, An abductive logic programming architecture for negotiating agents. In: Proc. 8th European Conf. on Logics in AI, LNAI 2424, pp. 419-431, Springer, 2002. [15] C. Sakama and K. Inoue. An abductive framework for computing knowledge base updates. Theory and Practice of Logic Programming 3(6):671-715, 2003. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1029

relaxation;logic program;anti-instantiation;abductive program;abductive framework;dropping condition;specific meta-knowledge;inductive generalization;one-to-one negotiation;minimal explanation;conditional proposal;integrity constraint;negotiation;extend abduction;automated negotiation;multi-agent system;alternative proposal

train_I-77

The LOGIC Negotiation Model

Successful negotiators prepare by determining their position along five dimensions: Legitimacy, Options, Goals, Independence, and Commitment, (LOGIC). We introduce a negotiation model based on these dimensions and on two primitive concepts: intimacy (degree of closeness) and balance (degree of fairness). The intimacy is a pair of matrices that evaluate both an agent"s contribution to the relationship and its opponent"s contribution each from an information view and from a utilitarian view across the five LOGIC dimensions. The balance is the difference between these matrices. A relationship strategy maintains a target intimacy for each relationship that an agent would like the relationship to move towards in future. The negotiation strategy maintains a set of Options that are in-line with the current intimacy level, and then tactics wrap the Options in argumentation with the aim of attaining a successful deal and manipulating the successive negotiation balances towards the target intimacy.

1. INTRODUCTION In this paper we propose a new negotiation model to deal with long term relationships that are founded on successive negotiation encounters. The model is grounded on results from business and psychological studies [1, 16, 9], and acknowledges that negotiation is an information exchange process as well as a utility exchange process [15, 14]. We believe that if agents are to succeed in real application domains they have to reconcile both views: informational and gametheoretical. Our aim is to model trading scenarios where agents represent their human principals, and thus we want their behaviour to be comprehensible by humans and to respect usual human negotiation procedures, whilst being consistent with, and somehow extending, game theoretical and information theoretical results. In this sense, agents are not just utility maximisers, but aim at building long lasting relationships with progressing levels of intimacy that determine what balance in information and resource sharing is acceptable to them. These two concepts, intimacy and balance are key in the model, and enable us to understand competitive and co-operative game theory as two particular theories of agent relationships (i.e. at different intimacy levels). These two theories are too specific and distinct to describe how a (business) relationship might grow because interactions have some aspects of these two extremes on a continuum in which, for example, agents reveal increasing amounts of private information as their intimacy grows. We don"t follow the "Co-Opetition" aproach [4] where co-operation and competition depend on the issue under negotiation, but instead we belief that the willingness to co-operate/compete affect all aspects in the negotiation process. Negotiation strategies can naturally be seen as procedures that select tactics used to attain a successful deal and to reach a target intimacy level. It is common in human settings to use tactics that compensate for unbalances in one dimension of a negotiation with unbalances in another dimension. In this sense, humans aim at a general sense of fairness in an interaction. In Section 2 we outline the aspects of human negotiation modelling that we cover in this work. Then, in Section 3 we introduce the negotiation language. Section 4 explains in outline the architecture and the concepts of intimacy and balance, and how they influence the negotiation. Section 5 contains a description of the different metrics used in the agent model including intimacy. Finally, Section 6 outlines how strategies and tactics use the LOGIC framework, intimacy and balance. 2. HUMAN NEGOTIATION Before a negotiation starts human negotiators prepare the dialogic exchanges that can be made along the five LOGIC dimensions [7]: • Legitimacy. What information is relevant to the negotiation process? What are the persuasive arguments about the fairness of the options? 1030 978-81-904262-7-5 (RPS) c 2007 IFAAMAS • Options. What are the possible agreements we can accept? • Goals. What are the underlying things we need or care about? What are our goals? • Independence. What will we do if the negotiation fails? What alternatives have we got? • Commitment. What outstanding commitments do we have? Negotiation dialogues, in this context, exchange dialogical moves, i.e. messages, with the intention of getting information about the opponent or giving away information about us along these five dimensions: request for information, propose options, inform about interests, issue promises, appeal to standards . . . A key part of any negotiation process is to build a model of our opponent(s) along these dimensions. All utterances agents make during a negotiation give away information about their current LOGIC model, that is, about their legitimacy, options, goals, independence, and commitments. Also, several utterances can have a utilitarian interpretation in the sense that an agent can associate a preferential gain to them. For instance, an offer may inform our negotiation opponent about our willingness to sign a contract in the terms expressed in the offer, and at the same time the opponent can compute what is its associated expected utilitarian gain. These two views: informationbased and utility-based, are central in the model proposed in this paper. 2.1 Intimacy and Balance in relationships There is evidence from psychological studies that humans seek a balance in their negotiation relationships. The classical view [1] is that people perceive resource allocations as being distributively fair (i.e. well balanced) if they are proportional to inputs or contributions (i.e. equitable). However, more recent studies [16, 17] show that humans follow a richer set of norms of distributive justice depending on their intimacy level: equity, equality, and need. Equity being the allocation proportional to the effort (e.g. the profit of a company goes to the stock holders proportional to their investment), equality being the allocation in equal amounts (e.g. two friends eat the same amount of a cake cooked by one of them), and need being the allocation proportional to the need for the resource (e.g. in case of food scarcity, a mother gives all food to her baby). For instance, if we are in a purely economic setting (low intimacy) we might request equity for the Options dimension but could accept equality in the Goals dimension. The perception of a relation being in balance (i.e. fair) depends strongly on the nature of the social relationships between individuals (i.e. the intimacy level). In purely economical relationships (e.g., business), equity is perceived as more fair; in relations where joint action or fostering of social relationships are the goal (e.g. friends), equality is perceived as more fair; and in situations where personal development or personal welfare are the goal (e.g. family), allocations are usually based on need. We believe that the perception of balance in dialogues (in negotiation or otherwise) is grounded on social relationships, and that every dimension of an interaction between humans can be correlated to the social closeness, or intimacy, between the parties involved. According to the previous studies, the more intimacy across the five LOGIC dimensions the more the need norm is used, and the less intimacy the more the equity norm is used. This might be part of our social evolution. There is ample evidence that when human societies evolved from a hunter-gatherer structure1 to a shelterbased one2 the probability of survival increased when food was scarce. In this context, we can clearly see that, for instance, families exchange not only goods but also information and knowledge based on need, and that few families would consider their relationships as being unbalanced, and thus unfair, when there is a strong asymmetry in the exchanges (a mother explaining everything to her children, or buying toys, does not expect reciprocity). In the case of partners there is some evidence [3] that the allocations of goods and burdens (i.e. positive and negative utilities) are perceived as fair, or in balance, based on equity for burdens and equality for goods. See Table 1 for some examples of desired balances along the LOGIC dimensions. The perceived balance in a negotiation dialogue allows negotiators to infer information about their opponent, about its LOGIC stance, and to compare their relationships with all negotiators. For instance, if we perceive that every time we request information it is provided, and that no significant questions are returned, or no complaints about not receiving information are given, then that probably means that our opponent perceives our social relationship to be very close. Alternatively, we can detect what issues are causing a burden to our opponent by observing an imbalance in the information or utilitarian senses on that issue. 3. COMMUNICATION MODEL 3.1 Ontology In order to define a language to structure agent dialogues we need an ontology that includes a (minimum) repertoire of elements: a set of concepts (e.g. quantity, quality, material) organised in a is-a hierarchy (e.g. platypus is a mammal, Australian-dollar is a currency), and a set of relations over these concepts (e.g. price(beer,AUD)).3 We model ontologies following an algebraic approach [8] as: An ontology is a tuple O = (C, R, ≤, σ) where: 1. C is a finite set of concept symbols (including basic data types); 2. R is a finite set of relation symbols; 3. ≤ is a reflexive, transitive and anti-symmetric relation on C (a partial order) 4. σ : R → C+ is the function assigning to each relation symbol its arity 1 In its purest form, individuals in these societies collect food and consume it when and where it is found. This is a pure equity sharing of the resources, the gain is proportional to the effort. 2 In these societies there are family units, around a shelter, that represent the basic food sharing structure. Usually, food is accumulated at the shelter for future use. Then the food intake depends more on the need of the members. 3 Usually, a set of axioms defined over the concepts and relations is also required. We will omit this here. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1031 Element A new trading partner my butcher my boss my partner my children Legitimacy equity equity equity equality need Options equity equity equity mixeda need Goals equity need equity need need Independence equity equity equality need need Commitment equity equity equity mixed need a equity on burden, equality on good Table 1: Some desired balances (sense of fairness) examples depending on the relationship. where ≤ is the traditional is-a hierarchy. To simplify computations in the computing of probability distributions we assume that there is a number of disjoint is-a trees covering different ontological spaces (e.g. a tree for types of fabric, a tree for shapes of clothing, and so on). R contains relations between the concepts in the hierarchy, this is needed to define ‘objects" (e.g. deals) that are defined as a tuple of issues. The semantic distance between concepts within an ontology depends on how far away they are in the structure defined by the ≤ relation. Semantic distance plays a fundamental role in strategies for information-based agency. How signed contracts, Commit(·), about objects in a particular semantic region, and their execution, Done(·), affect our decision making process about signing future contracts in nearby semantic regions is crucial to modelling the common sense that human beings apply in managing trading relationships. A measure [10] bases the semantic similarity between two concepts on the path length induced by ≤ (more distance in the ≤ graph means less semantic similarity), and the depth of the subsumer concept (common ancestor) in the shortest path between the two concepts (the deeper in the hierarchy, the closer the meaning of the concepts). Semantic similarity is then defined as: Sim(c, c ) = e−κ1l · eκ2h − e−κ2h eκ2h + e−κ2h where l is the length (i.e. number of hops) of the shortest path between the concepts, h is the depth of the deepest concept subsuming both concepts, and κ1 and κ2 are parameters scaling the contributions of the shortest path length and the depth respectively. 3.2 Language The shape of the language that α uses to represent the information received and the content of its dialogues depends on two fundamental notions. First, when agents interact within an overarching institution they explicitly or implicitly accept the norms that will constrain their behaviour, and accept the established sanctions and penalties whenever norms are violated. Second, the dialogues in which α engages are built around two fundamental actions: (i) passing information, and (ii) exchanging proposals and contracts. A contract δ = (a, b) between agents α and β is a pair where a and b represent the actions that agents α and β are responsible for respectively. Contracts signed by agents and information passed by agents, are similar to norms in the sense that they oblige agents to behave in a particular way, so as to satisfy the conditions of the contract, or to make the world consistent with the information passed. Contracts and Information can thus be thought of as normative statements that restrict an agent"s behaviour. Norms, contracts, and information have an obvious temporal dimension. Thus, an agent has to abide by a norm while it is inside an institution, a contract has a validity period, and a piece of information is true only during an interval in time. The set of norms affecting the behaviour of an agent defines the context that the agent has to take into account. α"s communication language has two fundamental primitives: Commit(α, β, ϕ) to represent, in ϕ, the world that α aims at bringing about and that β has the right to verify, complain about or claim compensation for any deviations from, and Done(μ) to represent the event that a certain action μ4 has taken place. In this way, norms, contracts, and information chunks will be represented as instances of Commit(·) where α and β can be individual agents or institutions. C is: μ ::= illoc(α, β, ϕ, t) | μ; μ | Let context In μ End ϕ ::= term | Done(μ) | Commit(α, β, ϕ) | ϕ ∧ ϕ | ϕ ∨ ϕ | ¬ϕ | ∀v.ϕv | ∃v.ϕv context ::= ϕ | id = ϕ | prolog clause | context; context where ϕv is a formula with free variable v, illoc is any appropriate set of illocutionary particles, ‘;" means sequencing, and context represents either previous agreements, previous illocutions, the ontological working context, that is a projection of the ontological trees that represent the focus of the conversation, or code that aligns the ontological differences between the speakers needed to interpret an action a. Representing an ontology as a set predicates in Prolog is simple. The set term contains instances of the ontology concepts and relations.5 For example, we can represent the following offer: If you spend a total of more than e100 in my shop during October then I will give you a 10% discount on all goods in November, as: Offer( α, β,spent(β, α, October, X) ∧ X ≥ e100 → ∀ y. Done(Inform(ξ, α, pay(β, α, y), November)) → Commit(α, β, discount(y,10%))) ξ is an institution agent that reports the payment. 4 Without loss of generality we will assume that all actions are dialogical. 5 We assume the convention that C(c) means that c is an instance of concept C and r(c1, . . . , cn) implicitly determines that ci is an instance of the concept in the i-th position of the relation r. 1032 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) Figure 1: The LOGIC agent architecture 4. AGENT ARCHITECTURE A multiagent system {α, β1, . . . , βn, ξ, θ1, . . . , θt}, contains an agent α that interacts with other argumentation agents, βi, information providing agents, θj, and an institutional agent, ξ, that represents the institution where we assume the interactions happen [2]. The institutional agent reports promptly and honestly on what actually occurs after an agent signs a contract, or makes some other form of commitment. In Section 4.1 this enables us to measure the difference between an utterance and a subsequent observation. The communication language C introduced in Section 3.2 enables us both to structure the dialogues and to structure the processing of the information gathered by agents. Agents have a probabilistic first-order internal language L used to represent a world model, Mt . A generic information-based architecture is described in detail in [15]. The LOGIC agent architecture is shown in Figure 1. Agent α acts in response to a need that is expressed in terms of the ontology. A need may be exogenous such as a need to trade profitably and may be triggered by another agent offering to trade, or endogenous such as α deciding that it owns more wine than it requires. Needs trigger α"s goal/plan proactive reasoning, while other messages are dealt with by α"s reactive reasoning.6 Each plan prepares for the negotiation by assembling the contents of a ‘LOGIC briefcase" that the agent ‘carries" into the negotiation7 . The relationship strategy determines which agent to negotiate with for a given need; it uses risk management analysis to preserve a strategic set of trading relationships for each mission-critical need - this is not detailed here. For each trading relationship this strategy generates a relationship target that is expressed in the LOGIC framework as a desired level of intimacy to be achieved in the long term. Each negotiation consists of a dialogue, Ψt , between two agents with agent α contributing utterance μ and the part6 Each of α"s plans and reactions contain constructors for an initial world model Mt . Mt is then maintained from percepts received using update functions that transform percepts into constraints on Mt - for details, see [14, 15]. 7 Empirical evidence shows that in human negotiation, better outcomes are achieved by skewing the opening Options in favour of the proposer. We are unaware of any empirical investigation of this hypothesis for autonomous agents in real trading scenarios. ner β contributing μ using the language described in Section 3.2. Each dialogue, Ψt , is evaluated using the LOGIC framework in terms of the value of Ψt to both α and β - see Section 5.2. The negotiation strategy then determines the current set of Options {δi}, and then the tactics, guided by the negotiation target, decide which, if any, of these Options to put forward and wraps them in argumentation dialogue - see Section 6. We now describe two of the distributions in Mt that support offer exchange. Pt (acc(α, β, χ, δ)) estimates the probability that α should accept proposal δ in satisfaction of her need χ, where δ = (a, b) is a pair of commitments, a for α and b for β. α will accept δ if: Pt (acc(α, β, χ, δ)) > c, for level of certainty c. This estimate is compounded from subjective and objective views of acceptability. The subjective estimate takes account of: the extent to which the enactment of δ will satisfy α"s need χ, how much δ is ‘worth" to α, and the extent to which α believes that she will be in a position to execute her commitment a [14, 15]. Sα(β, a) is a random variable denoting α"s estimate of β"s subjective valuation of a over some finite, numerical evaluation space. The objective estimate captures whether δ is acceptable on the open market, and variable Uα(b) denotes α"s open-market valuation of the enactment of commitment b, again taken over some finite numerical valuation space. We also consider needs, the variable Tα(β, a) denotes α"s estimate of the strength of β"s motivating need for the enactment of commitment a over a valuation space. Then for δ = (a, b): Pt (acc(α, β, χ, δ)) = Pt „ Tα(β, a) Tα(α, b) «h × „ Sα(α, b) Sα(β, a) «g × Uα(b) Uα(a) ≥ s ! (1) where g ∈ [0, 1] is α"s greed, h ∈ [0, 1] is α"s degree of altruism, and s ≈ 1 is derived from the stance8 described in Section 6. The parameters g and h are independent. We can imagine a relationship that begins with g = 1 and h = 0. Then as the agents share increasing amounts of their information about their open market valuations g gradually reduces to 0, and then as they share increasing amounts of information about their needs h increases to 1. The basis for the acceptance criterion has thus developed from equity to equality, and then to need. Pt (acc(β, α, δ)) estimates the probability that β would accept δ, by observing β"s responses. For example, if β sends the message Offer(δ1) then α derives the constraint: {Pt (acc(β, α, δ1)) = 1} on the distribution Pt (β, α, δ), and if this is a counter offer to a former offer of α"s, δ0, then: {Pt (acc(β, α, δ0)) = 0}. In the not-atypical special case of multi-issue bargaining where the agents" preferences over the individual issues only are known and are complementary to each other"s, maximum entropy reasoning can be applied to estimate the probability that any multi-issue δ will be acceptable to β by enumerating the possible worlds that represent β"s limit of acceptability [6]. 4.1 Updating the World Model Mt α"s world model consists of probability distributions that represent its uncertainty in the world state. α is interested 8 If α chooses to inflate her opening Options then this is achieved in Section 6 by increasing the value of s. If s 1 then a deal may not be possible. This illustrates the wellknown inefficiency of bilateral bargaining established analytically by Myerson and Satterthwaite in 1983. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1033 in the degree to which an utterance accurately describes what will subsequently be observed. All observations about the world are received as utterances from an all-truthful institution agent ξ. For example, if β communicates the goal I am hungry and the subsequent negotiation terminates with β purchasing a book from α (by ξ advising α that a certain amount of money has been credited to α"s account) then α may conclude that the goal that β chose to satisfy was something other than hunger. So, α"s world model contains probability distributions that represent its uncertain expectations of what will be observed on the basis of utterances received. We represent the relationship between utterance, ϕ, and subsequent observation, ϕ , by Pt (ϕ |ϕ) ∈ Mt , where ϕ and ϕ may be ontological categories in the interest of computational feasibility. For example, if ϕ is I will deliver a bucket of fish to you tomorrow then the distribution P(ϕ |ϕ) need not be over all possible things that β might do, but could be over ontological categories that summarise β"s possible actions. In the absence of in-coming utterances, the conditional probabilities, Pt (ϕ |ϕ), should tend to ignorance as represented by a decay limit distribution D(ϕ |ϕ). α may have background knowledge concerning D(ϕ |ϕ) as t → ∞, otherwise α may assume that it has maximum entropy whilst being consistent with the data. In general, given a distribution, Pt (Xi), and a decay limit distribution D(Xi), Pt (Xi) decays by: Pt+1 (Xi) = Δi(D(Xi), Pt (Xi)) (2) where Δi is the decay function for the Xi satisfying the property that limt→∞ Pt (Xi) = D(Xi). For example, Δi could be linear: Pt+1 (Xi) = (1 − νi) × D(Xi) + νi × Pt (Xi), where νi < 1 is the decay rate for the i"th distribution. Either the decay function or the decay limit distribution could also be a function of time: Δt i and Dt (Xi). Suppose that α receives an utterance μ = illoc(α, β, ϕ, t) from agent β at time t. Suppose that α attaches an epistemic belief Rt (α, β, μ) to μ - this probability takes account of α"s level of personal caution. We model the update of Pt (ϕ |ϕ) in two cases, one for observations given ϕ, second for observations given φ in the semantic neighbourhood of ϕ. 4.2 Update of Pt (ϕ |ϕ) given ϕ First, if ϕk is observed then α may set Pt+1 (ϕk|ϕ) to some value d where {ϕ1, ϕ2, . . . , ϕm} is the set of all possible observations. We estimate the complete posterior distribution Pt+1 (ϕ |ϕ) by applying the principle of minimum relative entropy9 as follows. Let p(μ) be the distribution: 9 Given a probability distribution q, the minimum relative entropy distribution p = (p1, . . . , pI ) subject to a set of J linear constraints g = {gj(p) = aj · p − cj = 0}, j = 1, . . . , J (that must include the constraint P i pi − 1 = 0) is: p = arg minr P j rj log rj qj . This may be calculated by introducing Lagrange multipliers λ: L(p, λ) = P j pj log pj qj + λ · g. Minimising L, { ∂L ∂λj = gj(p) = 0}, j = 1, . . . , J is the set of given constraints g, and a solution to ∂L ∂pi = 0, i = 1, . . . , I leads eventually to p. Entropy-based inference is a form of Bayesian inference that is convenient when the data is sparse [5] and encapsulates common-sense reasoning [12]. arg minx P j xj log xj Pt(ϕ |ϕ)j that satisfies the constraint p(μ)k = d. Then let q(μ) be the distribution: q(μ) = Rt (α, β, μ) × p(μ) + (1 − Rt (α, β, μ)) × Pt (ϕ |ϕ) and then let: r(μ) = ( q(μ) if q(μ) is more interesting than Pt (ϕ |ϕ) Pt (ϕ |ϕ) otherwise A general measure of whether q(μ) is more interesting than Pt (ϕ |ϕ) is: K(q(μ) D(ϕ |ϕ)) > K(Pt (ϕ |ϕ) D(ϕ |ϕ)), where K(x y) = P j xj ln xj yj is the Kullback-Leibler distance between two probability distributions x and y [11]. Finally incorporating Eqn. 2 we obtain the method for updating a distribution Pt (ϕ |ϕ) on receipt of a message μ: Pt+1 (ϕ |ϕ) = Δi(D(ϕ |ϕ), r(μ)) (3) This procedure deals with integrity decay, and with two probabilities: first, the probability z in the utterance μ, and second the belief Rt (α, β, μ) that α attached to μ. 4.3 Update of Pt (φ |φ) given ϕ The sim method: Given as above μ = illoc(α, β, ϕ, t) and the observation ϕk we define the vector t by ti = Pt (φi|φ) + (1− | Sim(ϕk, ϕ) − Sim(φi, φ) |) · Sim(ϕk, φ) with {φ1, φ2, . . . , φp} the set of all possible observations in the context of φ and i = 1, . . . , p. t is not a probability distribution. The multiplying factor Sim(ϕ , φ) limits the variation of probability to those formulae whose ontological context is not too far away from the observation. The posterior Pt+1 (φ |φ) is obtained with Equation 3 with r(μ) defined to be the normalisation of t. The valuation method: For a given φk, wexp (φk) =Pm j=1 Pt (φj|φk) · w(φj) is α"s expectation of the value of what will be observed given that β has stated that φk will be observed, for some measure w. Now suppose that, as before, α observes ϕk after agent β has stated ϕ. α revises the prior estimate of the expected valuation wexp (φk) in the light of the observation ϕk to: (wrev (φk) | (ϕk|ϕ)) = g(wexp (φk), Sim(φk, ϕ), w(φk), w(ϕ), wi(ϕk)) for some function g - the idea being, for example, that if the execution, ϕk, of the commitment, ϕ, to supply cheese was devalued then α"s expectation of the value of a commitment, φ, to supply wine should decrease. We estimate the posterior by applying the principle of minimum relative entropy as for Equation 3, where the distribution p(μ) = p(φ |φ) satisfies the constraint: p X j=1 p(ϕ ,ϕ)j · wi(φj) = g(wexp (φk), Sim(φk, ϕ), w(φk), w(ϕ), wi(ϕk)) 5. SUMMARY MEASURES A dialogue, Ψt , between agents α and β is a sequence of inter-related utterances in context. A relationship, Ψ∗t , is a sequence of dialogues. We first measure the confidence that an agent has for another by observing, for each utterance, the difference between what is said (the utterance) and what 1034 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) subsequently occurs (the observation). Second we evaluate each dialogue as it progresses in terms of the LOGIC framework - this evaluation employs the confidence measures. Finally we define the intimacy of a relationship as an aggregation of the value of its component dialogues. 5.1 Confidence Confidence measures generalise what are commonly called trust, reliability and reputation measures into a single computational framework that spans the LOGIC categories. In Section 5.2 confidence measures are applied to valuing fulfilment of promises in the Legitimacy category - we formerly called this honour [14], to the execution of commitments - we formerly called this trust [13], and to valuing dialogues in the Goals category - we formerly called this reliability [14]. Ideal observations. Consider a distribution of observations that represent α"s ideal in the sense that it is the best that α could reasonably expect to observe. This distribution will be a function of α"s context with β denoted by e, and is Pt I (ϕ |ϕ, e). Here we measure the relative entropy between this ideal distribution, Pt I (ϕ |ϕ, e), and the distribution of expected observations, Pt (ϕ |ϕ). That is: C(α, β, ϕ) = 1 − X ϕ Pt I (ϕ |ϕ, e) log Pt I (ϕ |ϕ, e) Pt(ϕ |ϕ) (4) where the 1 is an arbitrarily chosen constant being the maximum value that this measure may have. This equation measures confidence for a single statement ϕ. It makes sense to aggregate these values over a class of statements, say over those ϕ that are in the ontological context o, that is ϕ ≤ o: C(α, β, o) = 1 − P ϕ:ϕ≤o Pt β(ϕ) [1 − C(α, β, ϕ)] P ϕ:ϕ≤o Pt β(ϕ) where Pt β(ϕ) is a probability distribution over the space of statements that the next statement β will make to α is ϕ. Similarly, for an overall estimate of β"s confidence in α: C(α, β) = 1 − X ϕ Pt β(ϕ) [1 − C(α, β, ϕ)] Preferred observations. The previous measure requires that an ideal distribution, Pt I (ϕ |ϕ, e), has to be specified for each ϕ. Here we measure the extent to which the observation ϕ is preferable to the original statement ϕ. Given a predicate Prefer(c1, c2, e) meaning that α prefers c1 to c2 in environment e. Then if ϕ ≤ o: C(α, β, ϕ) = X ϕ Pt (Prefer(ϕ , ϕ, o))Pt (ϕ |ϕ) and: C(α, β, o) = P ϕ:ϕ≤o Pt β(ϕ)C(α, β, ϕ) P ϕ:ϕ≤o Pt β(ϕ) Certainty in observation. Here we measure the consistency in expected acceptable observations, or the lack of expected uncertainty in those possible observations that are better than the original statement. If ϕ ≤ o let: Φ+(ϕ, o, κ) =˘ ϕ | Pt (Prefer(ϕ , ϕ, o)) > κ ¯ for some constant κ, and: C(α, β, ϕ) = 1 + 1 B∗ · X ϕ ∈Φ+(ϕ,o,κ) Pt +(ϕ |ϕ) log Pt +(ϕ |ϕ) where Pt +(ϕ |ϕ) is the normalisation of Pt (ϕ |ϕ) for ϕ ∈ Φ+(ϕ, o, κ), B∗ = ( 1 if |Φ+(ϕ, o, κ)| = 1 log |Φ+(ϕ, o, κ)| otherwise As above we aggregate this measure for observations in a particular context o, and measure confidence as before. Computational Note. The various measures given above involve extensive calculations. For example, Eqn. 4 containsP ϕ that sums over all possible observations ϕ . We obtain a more computationally friendly measure by appealing to the structure of the ontology described in Section 3.2, and the right-hand side of Eqn. 4 may be approximated to: 1 − X ϕ :Sim(ϕ ,ϕ)≥η Pt η,I (ϕ |ϕ, e) log Pt η,I (ϕ |ϕ, e) Pt η(ϕ |ϕ) where Pt η,I (ϕ |ϕ, e) is the normalisation of Pt I (ϕ |ϕ, e) for Sim(ϕ , ϕ) ≥ η, and similarly for Pt η(ϕ |ϕ). The extent of this calculation is controlled by the parameter η. An even tighter restriction may be obtained with: Sim(ϕ , ϕ) ≥ η and ϕ ≤ ψ for some ψ. 5.2 Valuing negotiation dialogues Suppose that a negotiation commences at time s, and by time t a string of utterances, Φt = μ1, . . . , μn has been exchanged between agent α and agent β. This negotiation dialogue is evaluated by α in the context of α"s world model at time s, Ms , and the environment e that includes utterances that may have been received from other agents in the system including the information sources {θi}. Let Ψt = (Φt , Ms , e), then α estimates the value of this dialogue to itself in the context of Ms and e as a 2 × 5 array Vα(Ψt ) where: Vx(Ψt ) = „ IL x (Ψt ) IO x (Ψt ) IG x (Ψt ) II x(Ψt ) IC x (Ψt ) UL x (Ψt ) UO x (Ψt ) UG x (Ψt ) UI x(Ψt ) UC x (Ψt ) « where the I(·) and U(·) functions are information-based and utility-based measures respectively as we now describe. α estimates the value of this dialogue to β as Vβ(Ψt ) by assuming that β"s reasoning apparatus mirrors its own. In general terms, the information-based valuations measure the reduction in uncertainty, or information gain, that the dialogue gives to each agent, they are expressed in terms of decrease in entropy that can always be calculated. The utility-based valuations measure utility gain are expressed in terms of some suitable utility evaluation function U(·) that can be difficult to define. This is one reason why the utilitarian approach has no natural extension to the management of argumentation that is achieved here by our informationbased approach. For example, if α receives the utterance Today is Tuesday then this may be translated into a constraint on a single distribution, and the resulting decrease in entropy is the information gain. Attaching a utilitarian measure to this utterance may not be so simple. We use the term 2 × 5 array loosely to describe Vα in that the elements of the array are lists of measures that will be determined by the agent"s requirements. Table 2 shows a sample measure for each of the ten categories, in it the dialogue commences at time s and terminates at time t. In that Table, U(·) is a suitable utility evaluation function, needs(β, χ) means agent β needs the need χ, cho(β, χ, γ) means agent β satisfies need χ by choosing to negotiate The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1035 with agent γ, N is the set of needs chosen from the ontology at some suitable level of abstraction, Tt is the set of offers on the table at time t, com(β, γ, b) means agent β has an outstanding commitment with agent γ to execute the commitment b where b is defined in the ontology at some suitable level of abstraction, B is the number of such commitments, and there are n + 1 agents in the system. 5.3 Intimacy and Balance The balance in a negotiation dialogue, Ψt , is defined as: Bαβ(Ψt ) = Vα(Ψt ) Vβ(Ψt ) for an element-by-element difference operator that respects the structure of V (Ψt ). The intimacy between agents α and β, I∗t αβ, is the pattern of the two 2 × 5 arrays V ∗t α and V ∗t β that are computed by an update function as each negotiation round terminates, I∗t αβ = ` V ∗t α , V ∗t β ´ . If Ψt terminates at time t: V ∗t+1 x = ν × Vx(Ψt ) + (1 − ν) × V ∗t x (5) where ν is the learning rate, and x = α, β. Additionally, V ∗t x continually decays by: V ∗t+1 x = τ × V ∗t x + (1 − τ) × Dx, where x = α, β; τ is the decay rate, and Dx is a 2 × 5 array being the decay limit distribution for the value to agent x of the intimacy of the relationship in the absence of any interaction. Dx is the reputation of agent x. The relationship balance between agents α and β is: B∗t αβ = V ∗t α V ∗t β . In particular, the intimacy determines values for the parameters g and h in Equation 1. As a simple example, if both IO α (Ψ∗t ) and IO β (Ψ∗t ) increase then g decreases, and as the remaining eight information-based LOGIC components increase, h increases. The notion of balance may be applied to pairs of utterances by treating them as degenerate dialogues. In simple multi-issue bargaining the equitable information revelation strategy generalises the tit-for-tat strategy in single-issue bargaining, and extends to a tit-for-tat argumentation strategy by applying the same principle across the LOGIC framework. 6. STRATEGIES AND TACTICS Each negotiation has to achieve two goals. First it may be intended to achieve some contractual outcome. Second it will aim to contribute to the growth, or decline, of the relationship intimacy. We now describe in greater detail the contents of the Negotiation box in Figure 1. The negotiation literature consistently advises that an agent"s behaviour should not be predictable even in close, intimate relationships. The required variation of behaviour is normally described as varying the negotiation stance that informally varies from friendly guy to tough guy. The stance is shown in Figure 1, it injects bounded random noise into the process, where the bound tightens as intimacy increases. The stance, St αβ, is a 2 × 5 matrix of randomly chosen multipliers, each ≈ 1, that perturbs α"s actions. The value in the (x, y) position in the matrix, where x = I, U and y = L, O, G, I, C, is chosen at random from [ 1 l(I∗t αβ ,x,y) , l(I∗t αβ, x, y)] where l(I∗t αβ, x, y) is the bound, and I∗t αβ is the intimacy. The negotiation strategy is concerned with maintaining a working set of Options. If the set of options is empty then α will quit the negotiation. α perturbs the acceptance machinery (see Section 4) by deriving s from the St αβ matrix such as the value at the (I, O) position. In line with the comment in Footnote 7, in the early stages of the negotiation α may decide to inflate her opening Options. This is achieved by increasing the value of s in Equation 1. The following strategy uses the machinery described in Section 4. Fix h, g, s and c, set the Options to the empty set, let Dt s = {δ | Pt (acc(α, β, χ, δ) > c}, then: • repeat the following as many times as desired: add δ = arg maxx{Pt (acc(β, α, x)) | x ∈ Dt s} to Options, remove {y ∈ Dt s | Sim(y, δ) < k} for some k from Dt s By using Pt (acc(β, α, δ)) this strategy reacts to β"s history of Propose and Reject utterances. Negotiation tactics are concerned with selecting some Options and wrapping them in argumentation. Prior interactions with agent β will have produced an intimacy pattern expressed in the form of ` V ∗t α , V ∗t β ´ . Suppose that the relationship target is (T∗t α , T∗t β ). Following from Equation 5, α will want to achieve a negotiation target, Nβ(Ψt ) such that: ν · Nβ(Ψt ) + (1 − ν) · V ∗t β is a bit on the T∗t β side of V ∗t β : Nβ(Ψt ) = ν − κ ν V ∗t β ⊕ κ ν T∗t β (6) for small κ ∈ [0, ν] that represents α"s desired rate of development for her relationship with β. Nβ(Ψt ) is a 2 × 5 matrix containing variations in the LOGIC dimensions that α would like to reveal to β during Ψt (e.g. I"ll pass a bit more information on options than usual, I"ll be stronger in concessions on options, etc.). It is reasonable to expect β to progress towards her target at the same rate and Nα(Ψt ) is calculated by replacing β by α in Equation 6. Nα(Ψt ) is what α hopes to receive from β during Ψt . This gives a negotiation balance target of: Nα(Ψt ) Nβ(Ψt ) that can be used as the foundation for reactive tactics by striving to maintain this balance across the LOGIC dimensions. A cautious tactic could use the balance to bound the response μ to each utterance μ from β by the constraint: Vα(μ ) Vβ(μ) ≈ St αβ ⊗ (Nα(Ψt ) Nβ(Ψt )), where ⊗ is element-by-element matrix multiplication, and St αβ is the stance. A less neurotic tactic could attempt to achieve the target negotiation balance over the anticipated complete dialogue. If a balance bound requires negative information revelation in one LOGIC category then α will contribute nothing to it, and will leave this to the natural decay to the reputation D as described above. 7. DISCUSSION In this paper we have introduced a novel approach to negotiation that uses information and game-theoretical measures grounded on business and psychological studies. It introduces the concepts of intimacy and balance as key elements in understanding what is a negotiation strategy and tactic. Negotiation is understood as a dialogue that affect five basic dimensions: Legitimacy, Options, Goals, Independence, and Commitment. Each dialogical move produces a change in a 2×5 matrix that evaluates the dialogue along five information-based measures and five utility-based measures. The current Balance and intimacy levels and the desired, or target, levels are used by the tactics to determine what to say next. We are currently exploring the use of this model as an extension of a currently widespread eProcurement software commercialised by iSOCO, a spin-off company of the laboratory of one of the authors. 1036 The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) IL α(Ψt ) = X ϕ∈Ψt Ct (α, β, ϕ) − Cs (α, β, ϕ) UL α (Ψt ) = X ϕ∈Ψt X ϕ Pt β(ϕ |ϕ) × Uα(ϕ ) IO α (Ψt ) = P δ∈T t Hs (acc(β, α, δ)) − P δ∈T t Ht (acc(β, α, δ)) |Tt| UO α (Ψt ) = X δ∈T t Pt (acc(β, α, δ)) × X δ Pt (δ |δ)Uα(δ ) IG α (Ψt ) = P χ∈N Hs (needs(β, χ)) − Ht (needs(β, χ)) |N| UG α (Ψt ) = X χ∈N Pt (needs(β, χ)) × Et (Uα(needs(β, χ))) II α(Ψt ) = Po i=1 P χ∈N Hs (cho(β, χ, βi)) − Ht (cho(β, χ, βi)) n × |N| UI α(Ψt ) = oX i=1 X χ∈N Ut (cho(β, χ, βi)) − Us (cho(β, χ, βi)) IC α (Ψt ) = Po i=1 P δ∈B Hs (com(β, βi, b)) − Ht (com(β, βi, b)) n × |B| UC α (Ψt ) = oX i=1 X δ∈B Ut (com(β, βi, b)) − Us (com(β, βi, b)) Table 2: Sample measures for each category in Vα(Ψt ). (Similarly for Vβ(Ψt ).) Acknowledgements Carles Sierra is partially supported by the OpenKnowledge European STREP project and by the Spanish IEA Project. 8. REFERENCES [1] Adams, J. S. Inequity in social exchange. In Advances in experimental social psychology, L. Berkowitz, Ed., vol. 2. New York: Academic Press, 1965. [2] Arcos, J. L., Esteva, M., Noriega, P., Rodr´ıguez, J. A., and Sierra, C. Environment engineering for multiagent systems. Journal on Engineering Applications of Artificial Intelligence 18 (2005). [3] Bazerman, M. H., Loewenstein, G. F., and White, S. B. Reversal of preference in allocation decisions: judging an alternative versus choosing among alternatives. Administration Science Quarterly, 37 (1992), 220-240. [4] Brandenburger, A., and Nalebuff, B. Co-Opetition : A Revolution Mindset That Combines Competition and Cooperation. Doubleday, New York, 1996. [5] Cheeseman, P., and Stutz, J. Bayesian Inference and Maximum Entropy Methods in Science and Engineering. American Institute of Physics, Melville, NY, USA, 2004, ch. On The Relationship between Bayesian and Maximum Entropy Inference, pp. 445461. [6] Debenham, J. Bargaining with information. In Proceedings Third International Conference on Autonomous Agents and Multi Agent Systems AAMAS-2004 (July 2004), N. Jennings, C. Sierra, L. Sonenberg, and M. Tambe, Eds., ACM Press, New York, pp. 664 - 671. [7] Fischer, R., Ury, W., and Patton, B. Getting to Yes: Negotiating agreements without giving in. Penguin Books, 1995. [8] Kalfoglou, Y., and Schorlemmer, M. IF-Map: An ontology-mapping method based on information-flow theory. In Journal on Data Semantics I, S. Spaccapietra, S. March, and K. Aberer, Eds., vol. 2800 of Lecture Notes in Computer Science. Springer-Verlag: Heidelberg, Germany, 2003, pp. 98-127. [9] Lewicki, R. J., Saunders, D. M., and Minton, J. W. Essentials of Negotiation. McGraw Hill, 2001. [10] Li, Y., Bandar, Z. A., and McLean, D. An approach for measuring semantic similarity between words using multiple information sources. IEEE Transactions on Knowledge and Data Engineering 15, 4 (July / August 2003), 871 - 882. [11] MacKay, D. Information Theory, Inference and Learning Algorithms. Cambridge University Press, 2003. [12] Paris, J. Common sense and maximum entropy. Synthese 117, 1 (1999), 75 - 93. [13] Sierra, C., and Debenham, J. An information-based model for trust. In Proceedings Fourth International Conference on Autonomous Agents and Multi Agent Systems AAMAS-2005 (Utrecht, The Netherlands, July 2005), F. Dignum, V. Dignum, S. Koenig, S. Kraus, M. Singh, and M. Wooldridge, Eds., ACM Press, New York, pp. 497 - 504. [14] Sierra, C., and Debenham, J. Trust and honour in information-based agency. In Proceedings Fifth International Conference on Autonomous Agents and Multi Agent Systems AAMAS-2006 (Hakodate, Japan, May 2006), P. Stone and G. Weiss, Eds., ACM Press, New York, pp. 1225 - 1232. [15] Sierra, C., and Debenham, J. Information-based agency. In Proceedings of Twentieth International Joint Conference on Artificial Intelligence IJCAI-07 (Hyderabad, India, January 2007), pp. 1513-1518. [16] Sondak, H., Neale, M. A., and Pinkley, R. The negotiated allocations of benefits and burdens: The impact of outcome valence, contribution, and relationship. Organizational Behaviour and Human Decision Processes, 3 (December 1995), 249-260. [17] Valley, K. L., Neale, M. A., and Mannix, E. A. Friends, lovers, colleagues, strangers: The effects of relationships on the process and outcome of negotiations. In Research in Negotiation in Organizations, R. Bies, R. Lewicki, and B. Sheppard, Eds., vol. 5. JAI Press, 1995, pp. 65-94. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 07) 1037

set predicate;confidence measure;ontology;view of acceptability;acceptability view;acceptance criterion;component dialogue;long term relationship;utilitarian interpretation;utterance;successive negotiation encounter;negotiation;logic agent architecture;multiagent system;negotiation strategy

train_J-33

Bid Expressiveness and Clearing Algorithms in Multiattribute Double Auctions

We investigate the space of two-sided multiattribute auctions, focusing on the relationship between constraints on the offers traders can express through bids, and the resulting computational problem of determining an optimal set of trades. We develop a formal semantic framework for characterizing expressible offers, and show conditions under which the allocation problem can be separated into first identifying optimal pairwise trades and subsequently optimizing combinations of those trades. We analyze the bilateral matching problem while taking into consideration relevant results from multiattribute utility theory. Network flow models we develop for computing global allocations facilitate classification of the problem space by computational complexity, and provide guidance for developing solution algorithms. Experimental trials help distinguish tractable problem classes for proposed solution techniques.

1. BACKGROUND A multiattribute auction is a market-based mechanism where goods are described by vectors of features, or attributes [3, 5, 8, 19]. Such mechanisms provide traders with the ability to negotiate over a multidimensional space of potential deals, delaying commitment to specific configurations until the most promising candidates are identified. For example, in a multiattribute auction for computers, the good may be defined by attributes such as processor speed, memory, and hard disk capacity. Agents have varying preferences (or costs) associated with the possible configurations. For example, a buyer may be willing to purchase a computer with a 2 GHz processor, 500 MB of memory, and a 50 GB hard disk for a price no greater than $500, or the same computer with 1GB of memory for a price no greater than $600. Existing research in multiattribute auctions has focused primarily on one-sided mechanisms, which automate the process whereby a single agent negotiates with multiple potential trading partners [8, 7, 19, 5, 23, 22]. Models of procurement typically assume the buyer has a value function, v, ranging over the possible configurations, X, and that each seller i can similarly be associated with a cost function ci over this domain. The role of the auction is to elicit these functions (possibly approximate or partial versions), and identify the surplus-maximizing deal. In this case, such an outcome would be arg maxi,x v(x) − ci(x). This problem can be translated into the more familiar auction for a single good without attributes by computing a score for each attribute vector based on the seller valuation function, and have buyers bid scores. Analogs of the classic first- and second-price auctions correspond to firstand second-score auctions [8, 7]. In the absence of a published buyer scoring function, agents on both sides may provide partial specifications of the deals they are willing to engage. Research on such auctions has, for example, produced iterative mechanisms for eliciting cost functions incrementally [19]. Other efforts focus on the optimization problem facing the bid taker, for example considering side constraints on the combination of trades comprising an overall deal [4]. Side constraints have also been analyzed in the context of combinatorial auctions [6, 20]. Our emphasis is on two-sided multiattribute auctions, where multiple buyers and sellers submit bids, and the objective is to construct a set of deals maximizing overall surplus. Previous research on such auctions includes works by Fink et al. [12] and Gong [14], both of which consider a matching problem for continuous double auctions (CDAs), where deals are struck whenever a pair of compatible bids is identified. In a call market, in contrast, bids accumulate until designated times (e.g., on a periodic or scheduled basis) at which the auction clears by determining a comprehensive match over the entire set of bids. Because the optimization is performed over an aggregated scope, call markets often enjoy liquidity and efficiency advantages over CDAs [10].1 Clearing a multiattribute CDA is much like clearing a one-sided multiattribute auction. Because nothing happens between bids, the problem is to match a given new bid (say, an offer to buy) with the existing bids on the other (sell) side. Multiattribute call markets are potentially much more complex. Constructing an optimal overall matching may require consideration of many different combina1 In the interim between clears, call markets may also disseminate price quotes providing summary information about the state of the auction [24]. Such price quotes are often computed based on hypothetical clears, and so the clearing algorithm may be invoked more frequently than actual market clearing operations. 110 tions of trades, among the various potential trading-partner pairings. The problem can be complicated by restrictions on overall assignments, as expressed in side constraints [16]. The goal of the present work is to develop a general framework for multiattribute call markets, to enable investigation of design issues and possibilities. In particular, we use the framework to explore tradeoffs between expressive power of agent bids and computational properties of auction clearing. We conduct our exploration independent of any consideration of strategic issues bearing on mechanism design. As with analogous studies of combinatorial auctions [18], we intend that tradeoffs quantified in this work can be combined with incentive factors within a comprehensive overall approach to multiattribute auction design. We provide the formal semantics of multiattribute offers in our framework in the next section. We abstract, where appropriate, from the specific language used to express offers, characterizing expressiveness semantically in terms of what deals may be offered. This enables us to identify some general conditions under which the problem of multilateral matching can be decomposed into bilateral matching problems. We then develop a family of network flow problems that capture corresponding classes of multiattribute call market optimizations. Experimental trials provide preliminary confirmation that the network formulations provide useful structure for implementing clearing algorithms. 2. MULTIATTRIBUTE OFFERS 2.1 Basic Definitions The distinguishing feature of a multiattribute auction is that the goods are defined by vectors of attributes, x = (x1, . . . , xm), xj ∈ Xj . A configuration is a particular attribute vector, x ∈ X = Qm j=1 Xj . The outcome of the auction is a set of bilateral trades. Trade t takes the form t = (x, q, b, s, π), signifying that agent b buys q > 0 units of configuration x from seller s, for payment π > 0. For convenience, we use the notation xt to denote the configuration associated with trade t (and similarly for other elements of t). For a set of trades T, we denote by Ti that subset of T involving agent i (i.e., b = i or s = i). Let T denote the set of all possible trades. A bid expresses an agent"s willingness to participate in trades. We specify the semantics of a bid in terms of offer sets. Let OT i ⊆ Ti denote agent i"s trade offer set. Intuitively, this represents the trades in which i is willing to participate. However, since the outcome of the auction is a set of trades, several of which may involve agent i, we must in general consider willingness to engage in trade combinations. Accordingly, we introduce the combination offer set of agent i, OC i ⊆ 2Ti . 2.2 Specifying Offer Sets A fully expressive bid language would allow specification of arbitrary combination offer sets. We instead consider a more limited class which, while restrictive, still captures most forms of multiattribute bidding proposed in the literature. Our bids directly specify part of the agent"s trade offer set, and include further directives controlling how this can be extended to the full trade and combination offer sets. For example, one way to specify a trade (buy) offer set would be to describe a set of configurations and quantities, along with the maximal payment one would exchange for each (x, q) specified. This description could be by enumeration, or any available means of defining such a mapping. An explicit set of trades in the offer set generally entails inclusion of many more implicit trades. We assume payment monotonicity, which means that agents always prefer more money. That is, for π > π > 0, (x, q, i, s, π) ∈ OT i ⇒ (x, q, i, s, π ) ∈ OT i , (x, q, b, i, π ) ∈ OT i ⇒ (x, q, b, i, π) ∈ OT i . We also assume free disposal, which dictates that for all i, q > q > 0, (x, q , i, s, π) ∈ OT i ⇒ (x, q, i, s, π) ∈ OT i , (x, q, b, i, π) ∈ OT i ⇒ (x, q , b, i, π) ∈ OT i . Note that the conditions for agents in the role of buyers and sellers are analogous. Henceforth, for expository simplicity, we present all definitions with respect to buyers only, leaving the definition for sellers as understood. Allowing agents" bids to comprise offers from both buyer and seller perspectives is also straightforward. An assertion that offers are divisible entails further implicit members in the trade offer set. DEFINITION 1 (DIVISIBLE OFFER). Agent i"s offer is divisible down to q iff ∀q < q < q. (x, q, i, s, π) ∈ OT i ⇒ (x, q , i, s, q q π) ∈ OT i . We employ the shorthand divisible to mean divisible down to 0. The definition above specifies arbitrary divisibility. It would likewise be possible to define divisibility with respect to integers, or to any given finite granularity. Note that when offers are divisible, it suffices to specify one offer corresponding to the maximal quantity one is willing to trade for any given configuration, trading partner, and per-unit payment (called the price). At the extreme of indivisibility are all-or-none offers. DEFINITION 2 (AON OFFER). Agent i"s offer is all-or-none (AON) iff (x, q, i, s, π) ∈ OT i ∧ (x, q , i, s, π ) ∈ OT i ⇒ [q = q ∨ π = π ]. In many cases, the agent will be indifferent with respect to different trading partners. In that event, it may omit the partner element from trades directly specified in its offer set, and simply assert that its offer is anonymous. DEFINITION 3 (ANONYMITY). Agent i"s offer is anonymous iff ∀s, s , b, b . (x, q, i, s, π) ∈ OT i ⇔ (x, q, i, s , π) ∈ OT i ∧ (x, q, b, i, π) ∈ OT i ⇔ (x, q, b , i, π) ∈ OT i . Because omitting trading partner qualifications simplifies the exposition, we generally assume in the following that all offers are anonymous unless explicitly specified otherwise. Extending to the non-anonymous case is conceptually straightforward. We employ the wild-card symbol ∗ in place of an agent identifier to indicate that any agent is acceptable. To specify a trade offer set, a bidder directly specifies a set of willing trades, along with any regularity conditions (e.g., divisibility, anonymity) that implicitly extend the set. The full trade offer set is then defined by the closure of this direct set with respect to payment monotonicity, free disposal, and any applicable divisibility assumptions. We next consider the specification of combination offer sets. Without loss of generality, we restrict each trade set T ∈ OC i to include at most one trade for any combination of configuration and trading partner (multiple such trades are equivalent to one net trade aggregating the quantities and payments). The key question is to what extent the agent is willing to aggregate deals across configurations or trading partners. One possibility is disallowing any aggregation. 111 DEFINITION 4 (NO AGGREGATION). The no-aggregation combinations are given by ONA i = {∅} ∪ {{t} | t ∈ OT i }. Agent i"s offer exhibits non-aggregation iff OC i = ONA i . We require in general that OC i ⊇ ONA i . A more flexible policy is to allow aggregation across trading partners, keeping configuration constant. DEFINITION 5 (PARTNER AGGREGATION). Suppose a particular trade is offered in the same context (set of additional trades, T) with two different sellers, s and s . That is, {(x, q, i, s, π)} ∪ T ∈ OC i ∧ {(x, q, i, s , π)} ∪ T ∈ OC i . Agent i"s offer allows seller aggregation iff in all such cases, {(x, q , i, s, π ), (x, q − q , i, s , π − π )} ∪ T ∈ OC i . In other words, we may create new trade offer combinations by splitting the common trade (quantity and payment, not necessarily proportionately) between the two sellers. In some cases, it might be reasonable to form combinations by aggregating different configurations. DEFINITION 6 (CONFIGURATION AGGREGATION). Suppose agent i offers, in the same context, the same quantity of two (not necessarily different) configurations, x and x . That is, {(x, q, i, ∗, π)} ∪ T ∈ OC i ∧ {(x , q, i, ∗, π )} ∪ T ∈ OC i . Agent i"s offer allows configuration aggregation iff in all such cases (and analogously when it is a seller), {(x, q , i, ∗, q q π), (x , q − q , i, ∗, q − q q π )} ∪ T ∈ OC i . Note that combination offer sets can accommodate offerings of configuration bundles. However, classes of bundles formed by partner or configuration aggregation are highly regular, covering only a specific type of bundle formed by splitting a desired quantity across configurations. This is quite restrictive compared to the general combinatorial case. 2.3 Willingness to Pay An agent"s offer trade set implicitly defines the agent"s willingness to pay for any given configuration and quantity. We assume anonymity to avoid conditioning our definitions on trading partner. DEFINITION 7 (WILLINGNESS TO PAY). Agent i"s willingness to pay for quantity q of configuration x is given by ˆuB i (x, q) = max π s.t. (x, q, i, ∗, π) ∈ OT i . We use the symbol ˆu to recognize that willingness to pay can be viewed as a proxy for the agent"s utility function, measured in monetary units. The superscript B distinguishes the buyer"s willingnessto-pay function from, a seller"s willingness to accept, ˆuS i (x, q), defined as the minimum payment seller i will accept for q units of configuration x. We omit the superscript where the distinction is inessential or clear from context. DEFINITION 8 (TRADE QUANTITY BOUNDS). Agent i"s minimum trade quantity for configuration x is given by qi(x) = min q s.t. ∃π. (x, q, i, ∗, π) ∈ OT i . The agent"s maximum trade quantity for x is ¯qi(x) = max q s.t. ∃π. (x, q, i, ∗, π) ∈ OT i ∧ ¬∃q < q. (x, q , i, ∗, π) ∈ OT i . When the agent has no offers involving x, we take qi(x) = ¯qi(x) = 0. It is useful to define a special case where all configurations are offered in the same quantity range. DEFINITION 9 (CONFIGURATION PARITY). Agent i"s offers exhibit configuration parity iff qi(x) > 0 ∧ qi(x ) > 0 ⇒ qi(x) = qi(x ) ∧ ¯qi(x) = ¯qi(x ). Under configuration parity we drop the arguments from trade quantity bounds, yielding the constants ¯q and q which apply to all offers. DEFINITION 10 (LINEAR PRICING). Agent i"s offers exhibit linear pricing iff for all qi(x) ≤ q ≤ ¯qi(x), ˆui(x, q) = q ¯qi(x) ˆui(x, ¯qi(x)). Note that linear pricing assumes divisibility down to qi(x). Given linear pricing, we can define the unit willingness to pay, ˆui(x) = ˆui(x, ¯qi(x))/¯qi(x), and take ˆui(x, q) = qˆui(x) for all qi(x) ≤ q ≤ ¯qi(x). In general, an agent"s willingness to pay may depend on a context of other trades the agent is engaging in. DEFINITION 11 (WILLINGNESS TO PAY IN CONTEXT). Agent i"s willingness to pay for quantity q of configuration x in the context of other trades T is given by ˆuB i (x, q; T) = max π s.t. {(x, q, i, s, π)} ∪ Ti ∈ OC i . LEMMA 1. If OC i is either non aggregating, or exhibits linear pricing, then ˆuB i (x, q; T) = ˆuB i (x, q). 3. MULTIATTRIBUTE ALLOCATION DEFINITION 12 (TRADE SURPLUS). The surplus of trade t = (x, q, b, s, π) is given by σ(t) = ˆuB b (x, q) − ˆuS s (x, q). Note that the trade surplus does not depend on the payment, which is simply a transfer from buyer to seller. DEFINITION 13 (TRADE UNIT SURPLUS). The unit surplus of trade t = (x, q, b, s, π) is given by σ1 (t) = σ(t)/q. Under linear pricing, we can equivalently write σ1 (t) = ˆuB b (x) − ˆuS s (x). DEFINITION 14 (SURPLUS OF A TRADE IN CONTEXT). The surplus of trade t = (x, q, b, s, π) in the context of other trades T, σ(t; T), is given by ˆuB b (x, q; T) − ˆuS s (x, q; T). DEFINITION 15 (GMAP). The Global Multiattribute Allocation Problem (GMAP) is to find the set of acceptable trades maximizing total surplus, max T ∈2T X t∈T σ(t; T \ {t}) s.t. ∀i. Ti ∈ OC i . DEFINITION 16 (MMP). The Multiattribute Matching Problem (MMP) is to find a best trade for a given pair of traders, MMP(b, s) = arg max t∈OT b ∩OT s σ(t). If OT b ∩ OT s is empty, we say that MMP has no solution. 112 Proofs of all the following results are provided in an extended version of this paper available from the authors. THEOREM 2. Suppose all agents" offers exhibit no aggregation (Definition 4). Then the solution to GMAP consists of a set of trades, each of which is a solution to MMP for its specified pair of traders. THEOREM 3. Suppose that each agent"s offer set satisfies one of the following (not necessarily the same) sets of conditions. 1. No aggregation and configuration parity (Definitions 4 and 9). 2. Divisibility, linear pricing, and configuration parity (Definitions 1, 10, and 9), with combination offer set defined as the minimal set consistent with configuration aggregation (Definition 6).2 Then the solution to GMAP consists of a set of trades, each of which employs a configuration that solves MMP for its specified pair of traders. Let MMPd (b, s) denote a modified version of MMP, where OT b and OT s are extended to assume divisibility (i.e., the offer sets are taken to be their closures under Definition 1). Then we can extend Theorem 3 to allow aggregating agents to maintain AON or minquantity offers as follows. THEOREM 4. Suppose offer sets as in Theorem 3, except that agents i satisfying configuration aggregation need be divisible only down to qi, rather than down to 0. Then the solution to GMAP consists of a set of trades, each of which employs the same configuration as a solution to MMPd for its specified pair of traders. THEOREM 5. Suppose agents b and s exhibit configuration parity, divisibility, and linear pricing, and there exists configuration x such that ˆub(x) − ˆus(x) > 0. Then t ∈ MMPd (b, s) iff xt = arg max x {ˆub(x) − ˆus(x)} qt = min(¯qb, ¯qs). (1) The preceding results signify that under certain conditions, we can divide the global optimization problem into two parts: first find a bilateral trade that maximizes unit surplus for each pair of traders (or total surplus in the non-aggregation case), and then use the results to find a globally optimal set of trades. In the following two sections we investigate each of these subproblems. 4. UTILITY REPRESENTATION AND MMP We turn next to consider the problem of finding a best deal between pairs of traders. The complexity of MMP depends pivotally on the representation by bids of offer sets, an issue we have postponed to this point. Note that issues of utility representation and MMP apply to a broad class of multiattribute mechanisms, beyond the multiattribute call markets we emphasize. For example, the complexity results contained in this section apply equally to the bidding problem faced by sellers in reverse auctions, given a published buyer scoring function. The simplest representation of an offer set is a direct enumeration of configurations and associated quantities and payments. This approach treats the configurations as atomic entities, making no use 2 That is, for such an agent i, OC i is the closure under configuration aggregation of ONA i . of attribute structure. A common and inexpensive enhancement is to enable a trader to express sets of configurations, by specifying subsets of the domains of component attributes. Associating a single quantity and payment with a set of configurations expresses indifference among them; hence we refer to such a set as an indifference range.3 Indifference ranges include the case of attributes with a natural ordering, in which a bid specifies a minimum or maximum acceptable attribute level. The use of indifference ranges can be convenient for MMP. The compatibility of two indifference ranges is simply found by testing set intersection for each attribute, as demonstrated by the decision-tree algorithm of Fink et al. [12]. Alternatively, bidders may specify willingness-to-pay functions ˆu in terms of compact functional forms. Enumeration based representations, even when enhanced with indifference ranges, are ultimately limited by the exponential size of attribute space. Functional forms may avoid this explosion, but only if ˆu reflects structure among the attributes. Moreover, even given a compact specification of ˆu, we gain computational benefits only if we can perform the matching without expanding the ˆu values of an exponential number of configuration points. 4.1 Additive Forms One particularly useful multiattribute representation is known as the additive scoring function. Though this form is widely used in practice and in the academic literature, it is important to stress the assumptions behind it. The theory of multiattribute representation is best developed in the context where ˆu is interpreted as a utility function representing an underlying preference order [17]. We present the premises of additive utility theory in this section, and discuss some generalizations in the next. DEFINITION 17. A set of attributes Y ⊂ X is preferentially independent (PI) of its complement Z = X \ Y if the conditional preference order over Y given a fixed level Z0 of Z is the same regardless of the choice of Z0 . In other words, the preference order over the projection of X on the attributes in Y is the same for any instantiation of the attributes in Z. DEFINITION 18. X = {x1, . . . , xm} is mutually preferentially independent (MPI) if any subset of X is preferentially independent of its complement. THEOREM 6 ([9]). A preference order over set of attributes X has an additive utility function representation u(x1, . . . , xm) = mX i=1 ui(xi) iff X is mutually preferential independent. A utility function over outcomes including money is quasi-linear if the function can be represented as a function over non-monetary attributes plus payments, π. Interpreting ˆu as a utility function over non-monetary attributes is tantamount to assuming quasi-linearity. Even when quasi-linearity is assumed, however, MPI over nonmonetary attributes is not sufficient for the quasi-linear utility function to be additive. For this, we also need that each of the pairs (π, Xi) for any attribute Xi would be PI of the rest of the attributes. 3 These should not be mistaken with indifference curves, which express dependency between the attributes. Indifference curves can be expressed by the more elaborate utility representations discussed below. 113 This (by MAUT) in turn implies that the set of attributes including money is MPI and the utility function can be represented as u(x1, . . . , xm, π) = mX i=1 ui(xi) + π. Given that form, a willingness-to-pay function reflecting u can be represented additively, as ˆu(x) = mX i=1 ui(xi) In many cases the additivity assumption provides practically crucial simplification of offer set elicitation. In addition to compactness, additivity dramatically simplifies MMP. If both sides provide additive ˆu representations, the globally optimal match reduces to finding the optimal match separately for each attribute. A common scenario in procurement has the buyer define an additive scoring function, while suppliers submit enumerated offer points or indifference ranges. This model is still very amenable to MMP: for each element in a supplier"s enumerated set, we optimize each attribute by finding the point in the supplier"s allowable range that is most preferred by the buyer. A special type of scoring (more particularly, cost) function was defined by Bichler and Kalagnanam [4] and called a configurable offer. This idea is geared towards procurement auctions: assuming suppliers are usually comfortable with expressing their preferences in terms of cost that is quasi-linear in every attribute, they can specify a price for a base offer, and additional cost for every change in a specific attribute level. This model is essentially a pricing out approach [17]. For this case, MMP can still be optimized on a per-attribute basis. A similar idea has been applied to one-sided iterative mechanisms [19], in which sellers refine prices on a perattribute basis at each iteration. 4.2 Multiattribute Utility Theory Under MPI, the tradeoffs between the attributes in each subset cannot be affected by the value of other attributes. For example, when buying a PC, a weaker CPU may increase the importance of the RAM compared to, say, the type of keyboard. Such relationships cannot be expressed under an additive model. Multiattribute utility theory (MAUT) develops various compact representations of utility functions that are based on weaker structural assumptions [17, 2]. There are several challenges in adapting these techniques to multiattribute bidding. First, as noted above, the theory is developed for utility functions, which may behave differently from willingness-to-pay functions. Second, computational efficiency of matching has not been an explicit goal of most work in the area. Third, adapting such representations to iterative mechanisms may be more challenging. One representation that employs somewhat weaker assumptions than additivity, yet retains the summation structure is the generalized additive (GA) decomposition: u(x) = JX j=1 fj(xj ), xj ∈ Xj , (2) where the Xj are potentially overlapping sets of attributes, together exhausting the space X. A key point from our perspective is that the complexity of the matching is similar to the complexity of optimizing a single function, since the sum function is in the form (2) as well. Recent work by Gonzales and Perny [15] provides an elicitation process for GA decomposable preferences under certainty, as well as an optimization algorithm for the GA decomposed function. The complexity of exact optimization is exponential in the induced width of the graph. However, to become operational for multiattribute bidding this decomposition must be detectable and verifiable by statements over preferences with respect to price outcomes. We are exploring this topic in ongoing work [11]. 5. SOLVING GMAP UNDER ALLOCATION CONSTRAINTS Theorems 2, 3, and 4 establish conditions under which GMAP solutions must comprise elements from constituent MMP solutions. In Sections 5.1 and 5.2, we show how to compute these GMAP solutions, given the MMP solutions, under these conditions. In these settings, traders that aggregate partners also aggregate configurations; hence we refer to them simply as aggregating or nonaggregating. Section 5.3 suggests a means to relax the linear pricing restriction employed in these constructions. Section 5.4 provides strategies for allowing traders to aggregate partners and restrict configuration aggregation at the same time. 5.1 Notation and Graphical Representation Our clearing algorithms are based on network flow formulations of the underlying optimization problem [1]. The network model is based on a bipartite graph, in which nodes on the left side represent buyers, and nodes on the right represent sellers. We denote the sets of buyers and sellers by B and S, respectively. We define two graph families, one for the case of non-aggregating traders (called single-unit), and the other for the case of aggregating traders (called multi-unit).4 For both types, a single directed arc is placed from a buyer i ∈ B to a seller j ∈ S if and only if MMP(i, j) is nonempty. We denote by T(i) the set of potential trading partners of trader i (i.e., the nodes connected to buyer or seller i in the bipartite graph. In the single-unit case, we define the weight of an arc (i, j) as wij = σ(MMP(i, j)). Note that free disposal lets a buy offer receive a larger quantity than desired (and similarly for sell offers). For the multi-unit case, the weights are wij = σ1 (MMP(i, j)), and we associate the quantity ¯qi with the node for trader i. We also use the notation qij for the mathematical formulations to denote partial fulfillment of qt for t = MMP(i, j). 5.2 Handling Indivisibility and Aggregation Constraints Under the restrictions of Theorems 2, 3, or 4, and when the solution to MMP is given, GMAP exhibits strong similarity to the problem of clearing double auctions with assignment constraints [16]. A match in our bipartite representation corresponds to a potential trade in which assignment constraints are satisfied. Network flow formulations have been shown to model this problem under the assumption of indivisibility and aggregation for all traders. The novelty in this part of our work is the use of generalized network flow formulations for more complex cases where aggregation and divisibility may be controlled by traders. Initially we examine the simple case of no aggregation (Theorem 2). Observe that the optimal allocation is simply the solution to the well known weighted assignment problem [1] on the singleunit bipartite graph described above. The set of matches that maximizes the total weight of arcs corresponds to the set of trades that maximizes total surplus. Note that any form of (in)divisibility can 4 In the next section, we introduce a hybrid form of graph accommodating mixes of the two trader categories. 114 also be accommodated in this model via the constituent MMP subproblems. The next formulation solves the case in which all traders fall under case 2 of Theorem 3-that is, all traders are aggregating and divisible, and exhibit linear pricing. This case can be represented using the following linear program, corresponding to our multi-unit graph: max X i∈B,j∈S wij qij s.t. X i∈T (j) qij ≤ ¯qj j ∈ S X j∈T (i) qij ≤ ¯qi i ∈ B qij ≥ 0 j ∈ S, i ∈ B Recall that the qij variables in the solution represent the number of units that buyer i procures from seller j. This formulation is known as the network transportation problem with inequality constraints, for which efficient algorithms are available [1]. It is a well known property of the transportation problem (and flow problems on pure networks in general) that given integer input values, the optimal solution is guaranteed to be integer as well. Figure 1 demonstrates the transformation of a set of bids to a transportation problem instance. Figure 1: Multi-unit matching with two boolean attributes. (a) Bids, with offers to buy in the left column and offers to sell at right. q@p indicates an offer to trade q units at price p per unit. Configurations are described in terms of constraints on attribute values. (b) Corresponding multi-unit assignment model. W represents arc weights (unit surplus), s represents source (exogenous) flow, and t represents sink quantity. The problem becomes significantly harder when aggregation is given as an option to bidders, requiring various enhancements to the basic multi-unit bipartite graph described above. In general, we consider traders that are either aggregating or not, with either divisible or AON offers. Initially we examine a special case, which at the same time demonstrates the hardness of the problem but still carries computational advantages. We designate one side (e.g., buyers) as restrictive (AON and non-aggregating), and the other side (sellers) as unrestrictive (divisible and aggregating). This problem can be represented using the following integer programming formulation: max X i∈B,j∈S wij qij s.t. X i∈T (j) ¯qiqij ≤ ¯qj j ∈ S X j∈T (i) qij ≤ 1 i ∈ B qij ∈ {0, 1} j ∈ S, i ∈ B (3) This formulation is a restriction of the generalized assignment problem (GAP) [13]. Although GAP is known to be NP-hard, it can be solved relatively efficiently by exact or approximate algorithms. GAP is more general than the formulation above as it allows buyside quantities (¯qi above) to be different for each potential seller. That this formulation is NP-hard as well (even the case of a single seller corresponds to the knapsack problem), illustrates the drastic increase in complexity when traders with different constraints are admitted to the same problem instance. Other than the special case above, we found no advantage in limiting AON constraints when traders may specify aggregation constraints. Therefore, the next generalization allows any combination of the two boolean constraints, that is, any trader chooses among four bid types: NI Bid AON and not aggregating. AD Bid allows aggregation and divisibility. AI Bid AON, allows aggregation (quantity can be aggregated across configurations, as long as it sums to the whole amount). ND No aggregation, divisibility (one trade, but smaller quantities are acceptable). To formulate an integer programming representation for the problem, we introduce the following variables. Boolean (0/1) variables ri and rj indicate whether buyer i and seller j participate in the solution (used for AON traders). Another indicator variable, yij , applied to non-aggregating buyer i and seller j, is one iff i trades with j. For aggregating traders, yij is not constrained. max X i∈B,j∈S Wij qij (4a) s.t. X j∈T (i) qij = ¯qiri i ∈ AIb (4b) X j∈T (i) qij ≤ ¯qiri i ∈ ADb (4c) X i∈T (j) qij = ¯qirj j ∈ AIs (4d) X i∈T (j) qij ≤ qj rj j ∈ ADs (4e) xij ≤ ¯qiyij i ∈ NDb , j ∈ T(i) (4f) xij ≤ ¯qj yij j ∈ NIs , i ∈ T(j) (4g) X j∈T (i) yij ≤ ri i ∈ NIb ∪ NDb (4h) X i∈T (j) yij ≤ rj j ∈ NIs ∪ NDs (4i) int qij (4j) yij , rj, ri ∈ {0, 1} (4k) 115 Figure 2: Generalized network flow model. B1 is a buyer in AD, B2 ∈ NI, B3 ∈ AI, B4 ∈ ND. V 1 is a seller in ND, V 2 ∈ AI, V 4 ∈ AD. The g values represent arc gains. Problem (4) has additional structure as a generalized min-cost flow problem with integral flow.5 A generalized flow network is a network in which each arc may have a gain factor, in addition to the pure network parameters (which are flow limits and costs). Flow in an arc is then multiplied by its gain factor, so that the flow that enters the end node of an arc equals the flow that entered from its start node, multiplied by the gain factor of the arc. The network model can in turn be translated into an IP formulation that captures such structure. The generalized min-cost flow problem is well-studied and has a multitude of efficient algorithms [1]. The faster algorithms are polynomial in the number of arcs and the logarithm of the maximal gain, that is, performance is not strongly polynomial but is polynomial in the size of the input. The main benefit of this graphical formulation to our matching problem is that it provides a very efficient linear relaxation. Integer programming algorithms such as branch-and-bound use solutions to the linear relaxation instance to bound the optimal integer solution. Since network flow algorithms are much faster than arbitrary linear programs (generalized network flow simplex algorithms have been shown to run in practice only 2 or 3 times slower than pure network min-cost flow [1]), we expect a branch-and-bound solver for the matching problem to show improved performance when taking advantage of network flow modeling. The network flow formulation is depicted in Figure 2. Nonrestrictive traders are treated as in Figure 1. For a non-aggregating buyer, a single unit from the source will saturate up to one of the yij for all j, and be multiplied by ¯qi. If i ∈ ND, the end node of yij will function as a sink that may drain up to ¯qi of the entering flow. For i ∈ NI we use an indicator (0/1) arc ri, on which the flow is multiplied by ¯qi. Trader i trades the full quantity iff ri = 1. At the seller side, the end node of a qij arc functions as a source for sellers j ∈ ND, in order to let the flow through yij arcs be 0 or ¯qj. The flow is then multiplied by 1 ¯qj so 0/1 flows enter an end node which can drain either 1 or 0 units. for sellers j ∈ NI arcs rj ensure AON similarly to arcs rj for buyers. Having established this framework, we are ready to accommo5 Constraint (4j) could be omitted (yielding computational savings) if non-integer quantities are allowed. Here and henceforth we assume the harder problem, where divisibility is with respect to integers. date more flexible versions of side constraints. The first generalization is to replace the boolean AON constraint with divisibility down to q, the minimal quantity. In our network flow instance we simply need to turn the node of the constrained trader i (e.g., the node B3 in Figure 2) to a sink that can drain up to ¯qi − qi units of flow. The integer program (4) can be also easily changed to accommodate this extension. Using gains, we can also apply batch size constraints. If a trader specifies a batch size β, we change the gain on the r arcs to β, and set the available flow of its origin to the maximal number of batches ¯qi/β. 5.3 Nonlinear Pricing A key assumption in handling aggregation up to this point is linear pricing, which enables us to limit attention to a single unit price. Divisibility without linear pricing allows expression of concave willingness-to-pay functions, corresponding to convex preference relations. Bidders may often wish to express non-convex offer sets, for example, due to fixed costs or switching costs in production settings [21]. We consider nonlinear pricing in the form of enumerated payment schedules-that is, defining values ˆu(x, q) for a select set of quantities q. For the indivisible case, these points are distinguished in the offer set by satisfying the following: ∃π. (x, q, i, ∗, π) ∈ OT i ∧ ¬∃q < q. (x, q , i, ∗, π) ∈ OT i . (cf. Definition 8, which defines the maximum quantity, ¯q, as the largest of these.) For the divisible case, the distinguished quantities are those where the unit price changes, which can be formalized similarly. To handle nonlinear pricing, we augment the network to include flow possibilities corresponding to each of the enumerated quantities, plus additional structure to enforce exclusivity among them. In other words, the network treats the offer for a given quantity as in Section 5.2, and embeds this in an XOR relation to ensure that each trader picks only one of these quantities. Since for each such quantity choice we can apply Theorem 3 or 4, the solution we get is in fact the solution to GMAP. The network representation of the XOR relation (which can be embedded into the network of Figure 2) is depicted in Figure 3. For a trader i with K XOR quantity points, we define dummy variables, zk i , k = 1, . . . , K. Since we consider trades between every pair of quantity points we also have qk ij , k = 1, . . . , K. For buyer i ∈ AI with XOR points at quantities ¯qk i , we replace (4b) with the following constraints: X j∈T (i) qk ij = ¯qk i zk i k = 1, . . . , K KX k=1 zk i = ri zk i ∈ {0, 1} k = 1, . . . , K (5) 5.4 Homogeneity Constraints The model (4) handles constraints over the aggregation of quantities from different trading partners. When aggregation is allowed, the formulation permits trades involving arbitrary combinations of configurations. A homogeneity constraint [4] restricts such combinations, by requiring that configurations aggregated in an overall deal must agree on some or all attributes. 116 Figure 3: Extending the network flow model to express an XOR over quantities. B2 has 3 XOR points for 6, 3, or 5 units. In the presence of homogeneity constraints, we can no longer apply the convenient separation of GMAP into MMP plus global bipartite optimization, as the solution to GMAP may include trades not part of any MMP solution. For example, let buyer b specify an offer for maximum quantity 10 of various acceptable configurations, with a homogeneity constraint over the attribute color. This means b is willing to aggregate deals over different trading partners and configurations, as long as all are the same color. If seller s can provide 5 blue units or 5 green units, and seller s can provide only 5 green units, we may prefer that b and s trade on green units, even if the local surplus of a blue trade is greater. Let {x1, . . . , xH} be attributes that some trader constrains to be homogeneous. To preserve the network flow framework, we need to consider, for each trader, every point in the product domain of these homogeneous attributes. Thus, for every assignment ˆx to the homogeneous attributes, we compute MMP(b, s) under the constraint that configurations are consistent with ˆx. We apply the same approach as in Section 5.3: solve the global optimization, such that the alternative ˆx assignments for each trader are combined under XOR semantics, thus enforcing homogeneity constraints. The size of this network is exponential in the number of homogeneous attributes, since we need a node for each point in the product domain of all the homogeneous attributes of each trader.6 Hence this solution method will only be tractable in applications were the traders can be limited to a small number of homogeneous attributes. It is important to note that the graph needs to include a node only for each point that potentially matches a point of the other side. It is therefore possible to make the problem tractable by limiting one of the sides to a less expressive bidding language, and by that limit the set of potential matches. For example, if sellers submit bounded sets of XOR points, we only need to consider the points in the combined set offered by the sellers, and the reduction to network flow is polynomial regardless of the number of homogeneous attributes. If such simplifications do not apply, it may be preferable to solve the global problem directly as a single optimization problem. We provide the formulation for the special case of divisibility (with respect to integers) and configuration parity. Let i index buyers, j sellers, and H homogeneous attributes. Variable xh ij ∈ Xh represents the value of attribute Xh in the trade between buyer i and seller j. Integer variable qij represents the quantity of the trade (zero for no trade) between i and j. 6 If traders differ on which attributes they express such constraints, we can limit consideration to the relevant alternatives. The complexity will still be exponential, but in the maximum number of homogeneous attributes for any pair of traders. max X i∈B,j∈S [ˆuB i (xij , qij ) − ˆuS j (xij , qij )] X j∈S qij ≤ ¯qi i ∈ B X i∈B qij ≤ ¯qj j ∈ S xh 1j = xh 2j = · · · = x|B|j j ∈ S, h ∈ {1, . . . , H} xh i1 = xh i2 = · · · = xi|S| i ∈ B, h ∈ {1, . . . , H} (6) Table 1 summarizes the mapping we presented from allocation constraints to the complexity of solving GMAP. Configuration parity is assumed for all cases but the first. 6. EXPERIMENTAL RESULTS We approach the experimental aspect of this work with two objectives. First, we seek a general idea of the sizes and types of clearing problems that can be solved under given time constraints. We also look to compare the performance of a straightforward integer program as in (4) with an integer program that is based on the network formulations developed here. Since we used CPLEX, a commercial optimization tool, the second objective could be achieved to the extent that CPLEX can take advantage of network structure present in a model. We found that in addition to the problem size (in terms of number of traders), the number of aggregating traders plays a crucial role in determining complexity. When most of the traders are aggregating, problems of larger sizes can be solved quickly. For example, our IP model solved instances with 600 buyers and 500 sellers, where 90% of them are aggregating, in less than two minutes. When the aggregating ratio was reduced to 80% for the same data, solution time was just under five minutes. These results motivated us to develop a new network model. Rather than treat non-aggregating traders as a special case, the new model takes advantage of the single-unit nature of non-aggregating trades (treating the aggregating traders as a special case). This new model outperformed our other models on most problem instances, exceptions being those where aggregating traders constitute a vast majority (at least 80%). This new model (Figure 4) has a single node for each non aggregating trader, with a single-unit arc designating a match to another non-aggregating trader. An aggregating trader has a node for each potential match, connected (via y arcs) to a mutual source node. Unlike the previous model we allow fractional flow for this case, representing the traded fraction of the buyer"s total quantity.7 We tested all three models on random data in the form of bipartite graphs encoding MMP solutions. In our experiments, each trader has a maximum quantity uniformly distributed over [30, 70], and minimum quantity uniformly distributed from zero to maximal quantity. Each buyer/seller pair is selected as matching with probability 0.75, with matches assigned a surplus uniformly distributed over [10, 70]. Whereas the size of the problem is defined by the number of traders on each side, the problem complexity depends on the product |B| × |S|. The tests depicted in Figures 5-7 are for the worst case |B| = |S|, with each data point averaged over six samples. In the figures, the direct IP (4) is designated SW, our first network model (Figure 2) NW, and our revised network model (Figure 4) NW 2. 7 Traded quantity remains integer. 117 Aggregation Hom. attr. Divisibility linear pricing Technique Complexity No aggregation N/A Any Not required Assignment problem Polynomial All aggregate None Down to 0 Required Transpor. problem Polynomial One side None Aggr side div. Aggr. side GAP NP-hard Optional None Down to q, batch Required Generalized ntwrk flow NP-hard Optional Bounded Down to q, batch Bounded size schdl. Generalized ntwrk flow NP-hard Optional Not bounded Down to q, batch Not required Nonlinear opt Depends on ˆu(x, q) Table 1: Mapping from combinations of allocation constraints to the solution methods of GMAP. One Side means that one side aggregates and divisible, and the other side is restrictive. Batch means that traders may submit batch sizes. Figure 4: Generalized network flow model. B1 is a buyer in AD, B2 ∈ AI, B3 ∈ NI, B4 ∈ ND. V 1 is a seller in AD, V 2 ∈ AI, V 4 ∈ ND. The g values represent arc gains, and W values represent weights. Figure 5: Average performance of models when 30% of traders aggregate. Figure 6: Average performance of models when 50% of traders aggregate. Figure 7: Average performance of models when 70% of traders aggregate. 118 Figure 8: Performance of models when varying percentage of aggregating traders Figure 8 shows how the various models are affected by a change in the percentage of aggregating traders, holding problem size fixed.8 Due to the integrality constraints, we could not test available algorithms specialized for network-flow problems on our test problems. Thus, we cannot fully evaluate the potential gain attributable to network structure. However, the model we built based on the insight from the network structure clearly provided a significant speedup, even without using a special-purpose algorithm. Model NW 2 provided speedups of a factor of 4-10 over the model SW. This was consistent throughout the problem sizes, including the smaller sizes for which the speedup is not visually apparent on the chart. 7. CONCLUSIONS The implementation and deployment of market exchanges requires the development of bidding languages, information feedback policies, and clearing algorithms that are suitable for the target domain, while paying heed to the incentive properties of the resulting mechanisms. For multiattribute exchanges, the space of feasible such mechanisms is constrained by computational limitations imposed by the clearing process. The extent to which the space of feasible mechanisms may be quantified a priori will facilitate the search for such exchanges in the full mechanism design problem. In this work, we investigate the space of two-sided multiattribute auctions, focusing on the relationship between constraints on the offers traders can express through bids, and the resulting computational problem of determining an optimal set of trades. We developed a formal semantic framework for characterizing expressible offers, and introduced some basic classes of restrictions. Our key technical results identify sets of conditions under which the overall matching problem can be separated into first identifying optimal pairwise trades and subsequently optimizing combinations of those trades. Based on these results, we developed network flow models for the overall clearing problem, which facilitate classification of problem versions by computational complexity, and provide guidance for developing solution algorithms and relaxing bidding constraints. 8. ACKNOWLEDGMENTS This work was supported in part by NSF grant IIS-0205435, and the STIET program under NSF IGERT grant 0114368. We are 8 All tests were performed on Intel 3.4 GHz processors with 2048 KB cache. Test that did not complete by the one-hour time limit were recorded as 4000 seconds. grateful to comments from an anonymous reviewer. Some of the underlying ideas were developed while the first two authors worked at TradingDynamics Inc. and Ariba Inc. in 1999-2001 (cf. US Patent 6,952,682). We thank Yoav Shoham, Kumar Ramaiyer, and Gopal Sundaram for fruitful discussions about multiattribute auctions in that time frame. 9. REFERENCES [1] R. K. Ahuja, T. L. Magnanti, and J. B. Orlin. Network Flows. Prentice-Hall, 1993. [2] F. Bacchus and A. Grove. Graphical models for preference and utility. In Eleventh Conference on Uncertainty in Artificial Intelligence, pages 3-10, Montreal, 1995. [3] M. Bichler. The Future of e-Markets: Multi-Dimensional Market Mechanisms. Cambridge U. Press, New York, NY, USA, 2001. [4] M. Bichler and J. Kalagnanam. Configurable offers and winner determination in multi-attribute auctions. European Journal of Operational Research, 160:380-394, 2005. [5] M. Bichler, M. Kaukal, and A. Segev. Multi-attribute auctions for electronic procurement. In Proceedings of the 1st IBM IAC Workshop on Internet Based Negotiation Technologies, 1999. [6] C. Boutilier, T. Sandholm, and R. Shields. Eliciting bid taker non-price preferences in (combinatorial) auctions. In Nineteenth Natl. Conf. on Artificial Intelligence, pages 204-211, San Jose, 2004. [7] F. Branco. The design of multidimensional auctions. RAND Journal of Economics, 28(1):63-81, 1997. [8] Y.-K. Che. Design competition through multidimensional auctions. RAND Journal of Economics, 24(4):668-680, 1993. [9] G. Debreu. Topological methods in cardinal utility theory. In K. Arrow, S. Karlin, and P. Suppes, editors, Mathematical Methods in the Social Sciences. Stanford University Press, 1959. [10] N. Economides and R. A. Schwartz. Electronic call market trading. Journal of Portfolio Management, 21(3), 1995. [11] Y. Engel and M. P. Wellman. Multiattribute utility representation for willingness-to-pay functions. Tech. report, Univ. of Michigan, 2006. [12] E. Fink, J. Johnson, and J. Hu. Exchange market for complex goods: Theory and experiments. Netnomics, 6(1):21-42, 2004. [13] M. L. Fisher, R. Jaikumar, and L. N. Van Wassenhove. A multiplier adjustment method for the generalized assignment problem. Management Science, 32(9):1095-1103, 1986. [14] J. Gong. Exchanges for complex commodities: Search for optimal matches. Master"s thesis, University of South Florida, 2002. [15] C. Gonzales and P. Perny. GAI networks for decision making under certainty. In IJCAI-05 workshop on preferences, Edinburgh, 2005. [16] J. R. Kalagnanam, A. J. Davenport, and H. S. Lee. Computational aspects of clearing continuous call double auctions with assignment constraints and indivisible demand. Electronic Commerce Research, 1(3):221-238, 2001. [17] R. L. Keeney and H. Raiffa. Decisions with Multiple Objectives: Preferences and Value Tradeoffs. Wiley, 1976. [18] N. Nisan. Bidding and allocation in combinatorial auctions. In Second ACM Conference on Electronic Commerce, pages 1-12, Minneapolis, MN, 2000. [19] D. C. Parkes and J. Kalagnanam. Models for iterative multiattribute procurement auctions. Management Science, 51:435-451, 2005. [20] T. Sandholm and S. Suri. Side constraints and non-price attributes in markets. In IJCAI-01 Workshop on Distributed Constraint Reasoning, Seattle, 2001. [21] L. J. Schvartzman and M. P. Wellman. Market-based allocation with indivisible bids. In AAMAS-05 Workshop on Agent-Mediated Electronic Commerce, Utrecht, 2005. [22] J. Shachat and J. T. Swarthout. Procurement auctions for differentiated goods. Technical Report 0310004, Economics Working Paper Archive at WUSTL, Oct. 2003. [23] A. V. Sunderam and D. C. Parkes. Preference elicitation in proxied multiattribute auctions. In Fourth ACM Conference on Electronic Commerce, pages 214-215, San Diego, 2003. [24] P. R. Wurman, M. P. Wellman, and W. E. Walsh. A parametrization of the auction design space. Games and Economic Behavior, 35:304-338, 2001. 119

bid;constraint;one-sided mechanism;partial specification;multiattribute auction;combinatorial auction;auction;global allocation;seller valuation function;preference;continuous double auction;multiattribute utility theory;semantic framework

train_J-34

(In)Stability Properties of Limit Order Dynamics

We study the stability properties of the dynamics of the standard continuous limit-order mechanism that is used in modern equity markets. We ask whether such mechanisms are susceptible to butterfly effects - the infliction of large changes on common measures of market activity by only small perturbations of the order sequence. We show that the answer depends strongly on whether the market consists of absolute traders (who determine their prices independent of the current order book state) or relative traders (who determine their prices relative to the current bid and ask). We prove that while the absolute trader model enjoys provably strong stability properties, the relative trader model is vulnerable to great instability. Our theoretical results are supported by large-scale experiments using limit order data from INET, a large electronic exchange for NASDAQ stocks.

1. INTRODUCTION In recent years there has been an explosive increase in the automation of modern equity markets. This increase has taken place both in the exchanges, which are increasingly computerized and offer sophisticated interfaces for order placement and management, and in the trading activity itself, which is ever more frequently undertaken by software. The so-called Electronic Communication Networks (or ECNs) that dominate trading in NASDAQ stocks are a common example of the automation of the exchanges. On the trading side, computer programs now are entrusted not only with the careful execution of large block trades for clients (sometimes referred to on Wall Street as program trading), but with the autonomous selection of stocks, direction (long or short) and volumes to trade for profit (commonly referred to as statistical arbitrage). The vast majority of equity trading is done via the standard limit order market mechanism. In this mechanism, continuous trading takes place via the arrival of limit orders specifying whether the party wishes to buy or sell, the volume desired, and the price offered. Arriving limit orders that are entirely or partially executable with the best offers on the other side are executed immediately, with any volume not immediately executable being placed in an queue (or book) ordered by price on the appropriate side (buy or sell). (A detailed description of the limit order mechanism is given in Section 3.) While traders have always been able to view the prices at the top of the buy and sell books (known as the bid and ask), a relatively recent development in certain exchanges is the real-time revelation of the entire order book - the complete distribution of orders, prices and volumes on both sides of the exchange. With this revelation has come the opportunity - and increasingly, the needfor modeling and exploiting limit order data and dynamics. It is fair to say that market microstructure, as this area is generally known, is a topic commanding great interest both in the real markets and in the academic finance literature. The opportunities and needs span the range from the optimized execution of large trades to the creation of stand-alone proprietary strategies that attempt to profit from high-frequency microstructure signals. In this paper we investigate a previously unexplored but fundamental aspect of limit order microstructure: the stability properties of the dynamics. Specifically, we are interested in the following natural question: To what extent are simple models of limit order markets either susceptible or immune to butterfly effects - that is, the infliction of large changes in important activity statistics (such as the 120 number of shares traded or the average price per share) by only minor perturbations of the order sequence? To examine this question, we consider two stylized but natural models of the limit order arrival process. In the absolute price model, buyers and sellers arrive with limit order prices that are determined independently of the current state of the market (as represented by the order books), though they may depend on all manner of exogenous information or shocks, such as time, news events, announcements from the company whose shares are being traded, private signals or state of the individual traders, etc. This process models traditional fundamentals-based trading, in which market participants each have some inherent but possibly varying valuation for the good that in turn determines their limit price. In contrast, in the relative price model, traders express their limit order prices relative to the best price offered in their respective book (buy or sell). Thus, a buyer would encode their limit order price as an offset ∆ (which may be positive, negative, or zero) from the current bid pb, which is then translated to the limit price pb +∆. Again, in addition to now depending on the state of the order books, prices may also depend on all manner of exogenous information. The relative price model can be viewed as modeling traders who, in addition to perhaps incorporating fundamental external information on the stock, may also position their orders strategically relative to the other orders on their side of the book. A common example of such strategic behavior is known as penny-jumping on Wall Street, in which a trader who has in interest in buying shares quickly, but still at a discount to placing a market order, will deliberately position their order just above the current bid. More generally, the entire area of modern execution optimization [9, 10, 8] has come to rely heavily on the careful positioning of limit orders relative to the current order book state. Note that such positioning may depend on more complex features of the order books than just the current bid and ask, but the relative model is a natural and simplified starting point. We remark that an alternate view of the two models is that all traders behave in a relative manner, but with absolute traders able to act only on a considerably slower time scale than the faster relative traders. How do these two models differ? Clearly, given any fixed sequence of arriving limit order prices, we can choose to express these prices either as their original (absolute) values, or we can run the order book dynamical process and transform each order into a relative difference with the top of its book, and obtain identical results. The differences arise when we consider the stability question introduced above. Intuitively, in the absolute model a small perturbation in the arriving limit price sequence should have limited (but still some) effects on the subsequent evolution of the order books, since prices are determined independently. For the relative model this intuition is less clear. It seems possible that a small perturbation could (for example) slightly modify the current bid, which in turn could slightly modify the price of the next arriving order, which could then slightly modify the price of the subsequent order, and so on, leading to an amplifying sequence of events. Our main results demonstrate that these two models do indeed have dramatically different stability properties. We first show that for any fixed sequence of prices in the absolute model, the modification of a single order has a bounded and extremely limited impact on the subsequent evolution of the books. In particular, we define a natural notion of distance between order books and show that small modifications can result in only constant distance to the original books for all subsequent time steps. We then show that this implies that for almost any standard statistic of market activity - the executed volume, the average price execution price, and many others - the statistic can be influenced only infinitesimally by small perturbations. In contrast, we show that the relative model enjoys no such stability properties. After giving specific (worst-case) relative price sequences in which small perturbations generate large changes in basic statistics (for example, altering the number of shares traded by a factor of two), we proceed to demonstrate that the difference in stability properties of the two models is more than merely theoretical. Using extensive INET (a major ECN for NASDAQ stocks) limit order data and order book reconstruction code, we investigate the empirical stability properties when the data is interpreted as containing either absolute prices, relative prices, or mixtures of the two. The theoretical predictions of stability and instability are strongly borne out by the subsequent experiments. In addition to stability being of fundamental interest in any important dynamical system, we believe that the results described here provide food for thought on the topics of market impact and the backtesting of quantitative trading strategies (the attempt to determine hypothetical past performance using historical data). They suggest that one"s confidence that trading quietly and in small volumes will have minimal market impact is linked to an implicit belief in an absolute price model. Our results and the fact that in the real markets there is a large and increasing amount of relative behavior such as penny-jumping would seem to cast doubts on such beliefs. Similarly, in a purely or largely relative-price world, backtesting even low-frequency, low-volume strategies could result in historical estimates of performance that are not only unrelated to future performance (the usual concern), but are not even accurate measures of a hypothetical past. The outline of the paper follows. In Section 2 we briefly review the large literature on market microstructure. In Section 3 we describe the limit order mechanism and our formal models. Section 4 presents our most important theoretical results, the 1-Modification Theorem for the absolute price model. This theorem is applied in Section 5 to derive a number of strong stability properties in the absolute model. Section 6 presents specific examples establishing the worstcase instability of the relative model. Section 7 contains the simulation studies that largely confirm our theoretical findings on INET market data. 2. RELATED WORK As was mentioned in the Introduction, market microstructure is an important and timely topic both in academic finance and on Wall Street, and consequently has a large and varied recent literature. Here we have space only to summarize the main themes of this literature and to provide pointers to further readings. To our knowledge the stability properties of detailed limit order microstructure dynamics have not been previously considered. (However, see Farmer and Joshi [6] for an example and survey of other price dynamic stability studies.) 121 On the more theoretical side, there is a rich line of work examining what might be considered the game-theoretic properties of limit order markets. These works model traders and market-makers (who provide liquidity by offering both buy and sell quotes, and profit on the difference) by utility functions incorporating tolerance for risks of price movement, large positions and other factors, and examine the resulting equilibrium prices and behaviors. Common findings predict negative price impacts for large trades, and price effects for large inventory holdings by market-makers. An excellent and comprehensive survey of results in this area can be found in [2]. There is a similarly large body of empirical work on microstructure. Major themes include the measurement of price impacts, statistical properties of limit order books, and attempts to establish the informational value of order books [4]. A good overview of the empirical work can be found in [7]. Of particular note for our interests is [3], which empirically studies the distribution of arriving limit order prices in several prominent markets. This work takes a view of arriving prices analogous to our relative model, and establishes a power-law form for the resulting distributions. There is also a small but growing number of works examining market microstructure topics from a computer science perspective, including some focused on the use of microstructure in algorithms for optimized trade execution. Kakade et al. [9] introduced limit order dynamics in competitive analysis for one-way and volume-weighted average price (VWAP) trading. Some recent papers have applied reinforcement learning methods to trade execution using order book properties as state variables [1, 5, 10]. 3. MICROSTRUCTURE PRELIMINARIES The following expository background material is adapted from [9]. The market mechanism we examine in this paper is driven by the simple and standard concept of a limit order. Suppose we wish to purchase 1000 shares of Microsoft (MSFT) stock. In a limit order, we specify not only the desired volume (1000 shares), but also the desired price. Suppose that MSFT is currently trading at roughly $24.07 a share (see Figure 1, which shows an actual snapshot of an MSFT order book on INET), but we are only willing to buy the 1000 shares at $24.04 a share or lower. We can choose to submit a limit order with this specification, and our order will be placed in a queue called the buy order book, which is ordered by price, with the highest offered unexecuted buy price at the top (often referred to as the bid). If there are multiple limit orders at the same price, they are ordered by time of arrival (with older orders higher in the book). In the example provided by Figure 1, our order would be placed immediately after the extant order for 5,503 shares at $24.04; though we offer the same price, this order has arrived before ours. Similarly, a sell order book for sell limit orders is maintained, this time with the lowest sell price offered (often referred to as the ask) at its top. Thus, the order books are sorted from the most competitive limit orders at the top (high buy prices and low sell prices) down to less competitive limit orders. The bid and ask prices together are sometimes referred to as the inside market, and the difference between them as the spread. By definition, the order books always consist exclusively of unexecuted orders - they are queues of orders hopefully waiting for the price to move in their direction. Figure 1: Sample INET order books for MSFT. How then do orders get (partially) executed? If a buy (sell, respectively) limit order comes in above the ask (below the bid, respectively) price, then the order is matched with orders on the opposing books until either the incoming order"s volume is filled, or no further matching is possible, in which case the remaining incoming volume is placed in the books. For instance, suppose in the example of Figure 1 a buy order for 2000 shares arrived with a limit price of $24.08. This order would be partially filled by the two 500-share sell orders at $24.069 in the sell books, the 500-share sell order at $24.07, and the 200-share sell order at $24.08, for a total of 1700 shares executed. The remaining 300 shares of the incoming buy order would become the new bid of the buy book at $24.08. It is important to note that the prices of executions are the prices specified in the limit orders already in the books, not the prices of the incoming order that is immediately executed. Thus in this example, the 1700 executed shares would be at different prices. Note that this also means that in a pure limit order exchange such as INET, market orders can be simulated by limit orders with extreme price values. In exchanges such as INET, any order can be withdrawn or canceled by the party that placed it any time prior to execution. Every limit order arrives atomically and instantaneously - there is a strict temporal sequence in which orders arrive, and two orders can never arrive simultaneously. This gives rise to the definition of the last price of the exchange, which is simply the last price at which the exchange executed an order. It is this quantity that is usually meant when people casually refer to the (ticker) price of a stock. 3.1 Formal Definitions We now provide a formal model for the limit order pro122 cess described above. In this model, limit orders arrive in a temporal sequence, with each order specifying its limit price and an indication of its type (buy or sell). Like the actual exchanges, we also allow cancellation of a standing (unexecuted) order in the books any time prior to its execution. Without loss of generality we limit attention to a model in which every order is for a single share; large order volumes can be represented by 1-share sequences. Definition 3.1. Let Σ = σ1, ...σn be a sequence of limit orders, where each σi has the form ni, ti, vi . Here ni is an order identifier, ti is the order type (buy, sell, or cancel), and vi is the limit order value. In the case that ti is a cancel, ni matches a previously placed order and vi is ignored. We have deliberately called vi in the definition above the limit order value rather than price, since our two models will differ in their interpretation of vi (as being absolute or relative). In the absolute model, we do indeed interpret vi as simply being the price of the limit order. In the relative model, if the current order book configuration is (A, B) (where A is the sell and B the buy book), the price of the order is ask(A) + vi if ti is sell, and bid(B) + vi if ti is buy, where by ask(X) and bid(X) we denote the price of the order at the top of the book X. (Note vi can be negative.) Our main interest in this paper is the effects that the modification of a small number of limit orders can have on the resulting dynamics. For simplicity we consider only modifications to the limit order values, but our results generalize to any modification. Definition 3.2. A k-modification of Σ is a sequence Σ such that for exactly k indices i1, ..., ik vij = vij , tij = tij , and nij = nij . For every = ij , j ∈ {1, . . . , k} σ = σ . We now define the various quantities whose stability properties we examine in the absolute and relative models. All of these are standard quantities of common interest in financial markets. • volume(Σ): Number of shares executed (traded) in the sequence Σ. • average(Σ): Average execution price. • close(Σ): Price of the last (closing) execution. • lastbid(Σ): Bid at the end of the sequence. • lastask(Σ): Ask at end of the sequence. 4. THE 1-MODIFICATION THEOREM In this section we provide our most important technical result. It shows that in the absolute model, the effects that the modification of a single order has on the resulting evolution of the order books is extremely limited. We then apply this result to derive strong stability results for all of the aforementioned quantities in the absolute model. Throughout this section, we consider an arbitrary order sequence Σ in the absolute model, and any 1-modification Σ of Σ. At any point (index) i in the two sequences we shall use (A1, B1) to denote the sell and buy books (respectively) in Σ, and (A2, B2) to denote the sell and buy books in Σ ; for notational convenience we omit explicitly superscripting by the current index i. We will shortly establish that at all times i, (A1, B1) and (A2, B2) are very close. Although the order books are sorted by price, we will use (for example) A1 ∪ {a2} = A2 to indicate that A2 contains an order at some price a2 that is not present in A1, but that otherwise A1 and A2 are identical; thus deleting the order at a2 in A2 would render the books the same. Similarly, B1 ∪ {b2} = B2 ∪ {b1} means B1 contains an order at price b1 not present in B2, B2 contains an order at price b2 not present in B1, and that otherwise B1 and B2 are identical. Using this notation, we now define a set of stable system states, where each state is composed from the order books of the original and the modified sequences. Shortly we show that if we change only one order"s value (price), we remain in this set for any sequence of limit orders. Definition 4.1. Let ab be the set of all states (A1, B1) and (A2, B2) such that A1 = A2 and B1 = B2. Let ¯ab be the set of states such that A1 ∪ {a2} = A2 ∪ {a1}, where a1 = a2, and B1 = B2. Let a¯b be the set of states such that B1∪{b2} = B2∪{b1}, where b1 = b2, and A1 = A2. Let ¯a¯b be the set of states in which A1 = A2∪{a1} and B1 = B2∪{b1}, or in which A2 = A1 ∪ {a2} and B2 = B1 ∪ {b2}. Finally we define S = ab ∪ ¯ab ∪ ¯ba ∪ ¯a¯b as the set of stable states. Theorem 4.1. (1-Modification Theorem) Consider any sequence of orders Σ and any 1-modification Σ of Σ. Then the order books (A1, B1) and (A2, B2) determined by Σ and Σ lie in the set S of stable states at all times. ab ¯a¯b a¯b¯ab Figure 2: Diagram representing the set S of stable states and the possible movements transitions in it after the change. The idea of the proof of this theorem is contained in Figure 2, which shows a state transition diagram labeled by the categories of stable states. This diagram describes all transitions that can take place after the arrival of the order on which Σ and Σ differ. The following establishes that immediately after the arrival of this differing order, the state lies in S. Lemma 4.2. If at any time the current books (A1, B1) and (A2, B2) are in the set ab (and thus identical), then modifying the price of the next order keeps the state in S. Proof. Suppose the arriving order is a sell order and we change it from a1 to a2; assume without loss of generality that a1 > a2. If neither order is executed immediately, then we move to state ¯ab; if both of them are executed then we stay in state ab; and if only a2 is executed then we move to state ¯a¯b. The analysis of an arriving buy order is similar. Following the arrival of their only differing order, Σ and Σ are identical. We now give a sequence of lemmas showing 123 Executed with two orders Not executed in both Arrivng buy order Arriving buy order Arriving buy order Arriving sell order ¯ab ab ¯a¯b Executed only with a1 (not a1 and a2) Executed with a1 and a2 Figure 3: The state diagram when starting at state ¯ab. This diagram provides the intuition of Lemma 4.3 that following the initial difference covered by Lemma 4.2, the state remains in S forever on the remaining (identical) sequence. We first show that from state ¯ab we remain in S regardless the next order. The intuition of this lemma is demonstrated in Figure 3. Lemma 4.3. If the current state is in the set ¯ab, then for any order the state will remain in S. Proof. We first provide the analysis for the case of an arriving sell order. Note that in ¯ab the buy books are identical (B1 = B2). Thus either the arriving sell order is executed with the same buy order in both buy books, or it is not executed in both buy books. For the first case, the buy books remain identical (the bid is executed in both) and the sell books remain unchanged. For the second case, the buy books remain unchanged and identical, and the sell books have the new sell order added to both of them (and thus still differ by one order). Next we provide an analysis of the more subtle case where the arriving item is a buy order. For this case we need to take care of several different scenarios. The first is when the top of both sell books (the ask) is identical. Then regardless of whether the new buy order is executed or not, the state remains in ¯ab (the analysis is similar to an arriving sell order). We are left to deal with case where ask(A1) and ask(A2) are different. Here we discuss two subcases: (a) ask(A1) = a1 and ask(A2) = a2, and (b) ask(A1) = a1 and ask(A2) = a . Here a1 and a2 are as in the definition of ¯ab in Definition 4.1, and a is some other price. For subcase (a), by our assumption a1 < a2, then either (1) both asks get executed, the sell books become identical, and we move to state ab; (2) neither ask is executed and we remain in state ¯ab; or (3) only ask(A1) = a1 is executed, in which case we move to state ¯a¯b with A2 = A1 ∪ {a2} and B2 = B1 ∪ {b2}, where b2 is the arriving buy order price. For subcase (b), either (1) buy order is executed in neither sell book we remain in state ¯ab; or (2) the buy order is executed in both sell books and stay in state ¯ab with A1 ∪ {a } = A2 ∪ {a2}; or (3) only ask(A1) = a1 is executed and we move to state ¯a¯b. Lemma 4.4. If the current state is in the set a¯b, then for any order the state will remain in S. Lemma 4.5. If the current configuration is in the set ¯a¯b, then for any order the state will remain in S The proofs of these two lemmas are omitted, but are similar in spirit to that of Lemma 4.3. The next and final lemma deals with cancellations. Lemma 4.6. If the current order book state lies in S, then following the arrival of a cancellation it remains in S. Proof. When a cancellation order arrives, one of the following possibilities holds: (1) the order is still in both sets of books, (2) it is not in either of them and (3) it is only in one of them. For the first two cases it is easy to see that the cancellation effect is identical on both sets of books, and thus the state remains unchanged. For the case when the order appears only in one set of books, without loss of generality we assume that the cancellation cancels a buy order at b1. Rather than removing b1 from the book we can change it to have price 0, meaning this buy order will never be executed and is effectively canceled. Now regardless the state that we were in, b1 is still only in one buy book (but with a different price), and thus we remain in the same state in S. The proof of Theorem 4.1 follows from the above lemmas. 5. ABSOLUTE MODEL STABILITY In this section we apply the 1-Modification Theorem to show strong stability properties for the absolute model. We begin with an examination of the executed volume. Lemma 5.1. Let Σ be any sequence and Σ be any 1modification of Σ. Then the set of the executed orders (ID numbers) generated by the two sequences differs by at most 2. Proof. By Theorem 4.1 we know that at each stage the books differ by at most two orders. Now since the union of the IDs of the executed orders and the order books is always identical for both sequences, this implies that the executed orders can differ by at most two. Corollary 5.2. Let Σ be any sequence and Σ be any kmodification of Σ. Then the set of the executed orders (ID numbers) generated by the two sequences differs by at most 2k. An order sequence Σ is a k-extension of Σ if Σ can be obtained by deleting any k orders in Σ . Lemma 5.3. Let Σ be any sequence and let Σ be any kextension of Σ. Then the set of the executed orders generated by Σ and Σ differ by at most 2k. This lemma is the key to obtain our main absolute model volume result below. We use edit(Σ, Σ ) to denote the standard edit distance between the sequences Σ and Σ - the minimal number of substitutions, insertions and deletions or orders needed to change Σ to Σ . Theorem 5.4. Let Σ and Σ be any absolute model order sequences. Then if edit(Σ, Σ ) ≤ k, the set of the executed orders generated by Σ and Σ differ by at most 4k. In particular, |volume(Σ) − volume(Σ )| ≤ 4k. Proof. We first define the sequence ˜Σ which is the intersection of Σ and Σ . Since Σ and Σ are at most k apart,we have that by k insertions we change ˜Σ to either Σ or Σ , and by Lemma 5.3 its set of executed orders is at most 2k from each. Thus the set of executed orders in Σ and Σ is at most 4k apart. 124 5.1 Spread Bounds Theorem 5.4 establishes strong stability for executed volume in the absolute model. We now turn to the quantities that involve execution prices as opposed to volume alone - namely, average(Σ), close(Σ), lastbid(Σ) and lastask(Σ). For these results, unlike executed volume, a condition must hold on Σ in order for stability to occur. This condition is expressed in terms of a natural measure of the spread of the market, or the gap between the buyers and sellers. We motivate this condition by first showing that without it, by changing one order, we can change average(Σ) by any positive value x. Lemma 5.5. There exists Σ such that for any x ≥ 0, there is a 1-modification Σ of Σ such that average(Σ ) = average(Σ) + x. Proof. Let Σ be a sequence of alternating sell and buy orders in which each seller offers p and each buyer p + x, and the first order is a sell. Then all executions take place at the ask, which is always p, and thus average(Σ) = p. Now suppose we modify only the first sell order to be at price p+1+x. This initial sell order will never be executed, and now all executions take place at the bid, which is always p + x. Similar instability results can be shown to hold for the other price-based quantities. This motivates the introduction of a quantity we call the second spread of the order books, which is defined as the difference between the prices of the second order in the sell book and the second order in the buy book (as opposed to the bid-ask difference, which is commonly called the spread). We note that in a liquid stock, such as those we examine experimentally in Section 7, the second spread will typically be quite small and in fact almost always equal to the spread. In this subsection we consider changes in the sequence only after an initialization period, and sequences such that the second spread is always defined after the time we make a change. We define s2(Σ) to be the maximum second spread in the sequence Σ following the change. Theorem 5.6. Let Σ be a sequence and let Σ be any 1modification of Σ. Then 1. |lastbid(Σ) − lastbid(Σ )| ≤ s2(Σ) 2. |lastask(Σ) − lastask(Σ )| ≤ s2(Σ) where s2(Σ) is the maximum over the second spread in Σ following the 1-modification. Proof. We provide the proof for the last bid; the proof for the last ask is similar. The proof relies on Theorem 4.1 and considers states in the stable set S. For states ab and ¯ab, we have that the bid is identical. Let bid(X), sb(X), ask(X), be the bid, the second highest buy order, and the ask of a sequence X. Now recall that in state a¯b we have that the sell books are identical, and that the two buy books are identical except one different order. Thus bid(Σ)+s2(Σ) ≥ sb(Σ)+s2(Σ) ≥ ask(Σ) = ask(Σ ) ≥ bid(Σ ). Now it remains to bound bid(Σ). Here we use the fact that the bid of the modified sequence is at least the second highest buy order in the original sequence, due to the fact that the books are different only in one order. Since bid(Σ ) ≥ sb(Σ) ≥ ask(Σ) − s2(Σ) ≥ bid(Σ) − s2(Σ) we have that |bid(Σ) − bid(Σ )| ≤ s2(Σ) as desired. In state ¯a¯b we have that for one sequence the books contain an additional buy order and an additional sell order. First suppose that the books containing the additional orders are the original sequence Σ. Now if the bid is not the additional order we are done, otherwise we have the following: bid(Σ) ≤ ask(Σ) ≤ sb(Σ) + s2(Σ) = bid(Σ ) + s2(Σ), where sb(Σ) ≤ bid(Σ ) since the original buy book has only one additional order. Now assume that the books with the additional orders are for the modified sequence Σ . We have bid(Σ) + s2(Σ) ≥ ask(Σ) ≥ ask(Σ ) ≥ bid(Σ ), where we used the fact that ask(Σ) ≥ ask(Σ ) since the modified sequence has an additional order. Similarly we have that bid(Σ) ≤ bid(Σ ) since the modified buy book contains an additional order. We note that the proof of Theorem 5.6 actually establishes that the bid and ask of the original and modified sequences are within s2(Σ) at all times. Next we provide a technical lemma which relates the (first) spread of the modified sequence to the second spread of the original sequence. Lemma 5.7. Let Σ be a sequence and let Σ be any 1modification of Σ. Then the spread of Σ is bounded by s2(Σ). Proof. By the 1-Modification Theorem, we know that the books of the modified sequence and the original sequence can differ by at most one order in each book (buy and sell). Therefore, the second-highest buy order in the original sequence is always at most the bid in the modified sequence, and the second-lowest sell order in the original sequence is always at least the ask of the modified sequence. We are now ready to state a stability result for the average execution price in the absolute model. It establishes that in highly liquid markets, where the executed volume is large and the spread small, the average price is highly stable. Theorem 5.8. Let Σ be a sequence and let Σ be any 1modification of Σ. Then |average(Σ) − average(Σ )| ≤ 2(pmax + s2(Σ)) volume(Σ) + s2(Σ) where pmax is the highest execution price in Σ. Proof. The proof will show that every execution in Σ besides the execution of the modified order and the last execution has a matching execution in Σ with a price different by at most s2(Σ), and will use the fact that pmax + s2(Σ) is a bound on the price in Σ . Referring to the proof of the 1-Modification Theorem, suppose we are in state ¯a¯b, where we have in one sequence (which can be either Σ or Σ ) an additional buy order b and an additional sell order a. Without loss of generality we assume that the sequence with the additional orders is Σ. If the next execution does not involve a or b then clearly we have the same execution in both Σ and Σ . Suppose that it involves a; there are two possibilities. Either a is the modified order, in which case we change the average price 125 difference by (pmax +s2(Σ))/volume(Σ), and this can happen only once; or a was executed before in Σ and the executions both involve an order whose limit price is a. By Lemma 5.7 the spread of both sequences is bounded by s2(Σ), which implies that the price of the execution in Σ was at most a + s2(Σ), while execution is in Σ is at price a, and thus the prices are different by at most s2(Σ). In states ¯ab, a¯b as long as we have concurrent executions in the two sequences, we know that the prices can differ by at most s2(Σ). If we have an execution only in one sequence, we either match it in state ¯a¯b, or charge it by (pmax + s2(Σ))/volume(Σ) if we end at state ¯a¯b. If we end in state ab, ¯ab or a¯b, then every execution in states ¯ab or a¯b were matched to an execution in state ¯a¯b. If we end up in state ¯a¯b, we have the one execution that is not matched and thus we charge it (pmax +s2(Σ))/volume(Σ). We next give a stability result for the closing price. We first provide a technical lemma regarding the prices of consecutive executions. Lemma 5.9. Let Σ be any sequence. Then the prices of two consecutive executions in Σ differ by at most s2(Σ). Proof. Suppose the first execution is taken at time t; its price is bounded below by the current bid and above by the current ask. Now after this execution the bid is at least the second highest buy order at time t, if the former bid was executed and no higher buy orders arrived, and higher otherwise. Similarly, the ask is at most the second lowest sell order at time t. Therefore, the next execution price is at least the second bid at time t and at most the second ask at time t, which is at most s2(Σ) away from the bid/ask at time t. Lemma 5.10. Let Σ be any sequence and let Σ be a 1modification of Σ. If the volume(Σ) ≥ 2, then |close(Σ) − close(Σ )| ≤ s2(Σ) Proof. We first deal with case where the last execution occurs in both sequences simultaneously. By Theorem 5.6, both the ask and the bid of Σ and Σ are at most s2(Σ) apart at every time t. Since the price of the last execution is their asks (bids) at time t we are done. Next we deal with the case where the last execution among the two sequences occurs only in Σ. In this case we know that either the previous execution happened simultaneously in both sequences at time t, and thus all three executions are within the second spread of Σ at time t (the first execution in Σ by definition, the execution at Σ from identical arguments as in the former case, and the third by Lemma 5.9). Otherwise the previous execution happened only in Σ at time t, in which case the two executions are within the the spread of Σ at time t (the execution of Σ from the same arguments as before, and the execution in Σ must be inside its spread in time t). If the last execution happens only in Σ we know that the next execution of Σ will be at most s2(Σ) away from its previous execution by Lemma 5.9. Together with the fact that if an execution happens only in one sequence it implies that the order is in the spread of the second sequence as long as the sequences are 1-modification, the proof is completed. 5.2 Spread Bounds for k-Modifications As in the case of executed volume, we would like to extend the absolute model stability results for price-based quantities to the case where multiple orders are modified. Here our results are weaker and depend on the k-spread, the distance between the kth highest buy order and the kth lowest sell order, instead of the second spread. (Looking ahead to Section 7, we note that in actual market data for liquid stocks, this quantity is often very small as well.) We use sk(Σ) to denote the k-spread. As before, we assume that the k-spread is always defined after an initialization period. We first state the following generalization of Lemma 5.7. Lemma 5.11. Let Σ be a sequence and let Σ be any 1modification of Σ. For ≥ 1, if s +1(Σ) is always defined after the change, then s (Σ ) ≤ s +1(Σ). The proof is similar to the proof of Lemma 5.7 and omitted. A simple application of this lemma is the following: Let Σ be any sequence which is an -modification of Σ. Then we have s2(Σ ) ≤ s +2(Σ). Now using the above lemma and by simple induction we can obtain the following theorem. Theorem 5.12. Let Σ be a sequence and let Σ be any k-modification of Σ. Then 1. |lastbid(Σ) − lastbid(Σ )| ≤ Pk =1 s +1(Σ) ≤ ksk+1(Σ) 2. |lastask(Σ)−lastask(Σ )| ≤ Pk =1 s +1(Σ) ≤ ksk+1(Σ) 3. |close(Σ) − close(Σ )| ≤ Pk =1 s +1(Σ) ≤ ksk+1(Σ) 4. |average(Σ) − average(Σ )| ≤ Pk =1 2(pmax +s +1(Σ)) volume(Σ) + s +1(Σ) where s (Σ) is the maximum over the -spread in Σ following the first modification. We note that while these bounds depend on deeper measures of spread for more modifications, we are working in a 1-share order model. Thus in an actual market, where single orders contain hundreds or thousands of shares, the k-spread even for large k might be quite small and close to the standard 1-spread in liquid stocks. 6. RELATIVE MODEL INSTABILITY In the relative model the underlying assumption is that traders try to exploit their knowledge of the books to strategically place their orders. Thus if a trader wants her buy order to be executed quickly, she may position it above the current bid and be the first in the queue; if the trader is patient and believes that the price trend is going to be downward she will place orders deeper in the buy book, and so on. While in the previous sections we showed stability results for the absolute model, here we provide simple examples which show instability in the relative model for the executed volume, last bid, last ask, average execution price and the last execution price. In Section 7 we provide many simulations on actual market data that demonstrate that this instability is inherent to the relative model, and not due to artificial constructions. In the relative model we assume that for every sequence the ask and bid are always defined, so the books have a non-empty initial configuration. 126 We begin by showing that in the relative model, even a single modification can double the number of shares executed. Theorem 6.1. There is a sequence Σ and a 1-modification Σ of Σ such that volume(Σ ) ≥ 2volume(Σ). Proof. For concreteness we assume that at the beginning the ask is 10 and the bid is 8. The sequence Σ is composed from n buy orders with ∆ = 0, followed by n sell orders with ∆ = 0, and finally an alternating sequence of buy orders with ∆ = +1 and sell orders with ∆ = −1 of length 2n. Since the books before the alternating sequence contain n + 1 sell orders at 10 and n + 1 buy orders at 8, we have that each pair of buy sell order in the alternating part is matched and executed, but none of the initial 2n orders is executed, and thus volume(Σ) = n. Now we change the first buy order to have ∆ = +1. After the first 2n orders there are still no executions; however, the books are different. Now there are n + 1 sell orders at 10, n buy orders at 9 and one buy order at 8. Now each order in the alternating sequence is executed with one of the former orders and we have volume(Σ ) = 2n. The next theorem shows that the spread-based stability results of Section 5.1 do not also hold in the relative model. Before providing the proof, we give its intuition. At the beginning the sell book contains only two prices which are far apart and both contain only two orders, now several buy orders arrive, at the original sequence they are not being executed, while in the modified sequence they will be executed and leave the sell book with only the orders at the high price. Now many sell orders followed by many buy orders will arrive, such that in the original sequence they will be executed only at the low price and in the modified sequence they will executed at the high price. Theorem 6.2. For any positive numbers s and x, there is sequence Σ such that s2(Σ) = s and a 1-modification Σ of Σ such that • |close(Σ) − close(Σ )| ≥ x • |average(Σ) − average(Σ )| ≥ x • |lastbid(Σ) − lastbid(Σ )| ≥ x • |lastask(Σ) − lastask(Σ )| ≥ x Proof. Without loss of generality let us consider sequences in which all prices are integer-valued, in which case the smallest possible value for the second spread is 1; we provide the proof for the case s2(Σ) = 2, but the s2(Σ) = 1 case is similar. We consider a sequence Σ such that after an initialization period there have been no executions, the buy book has 2 orders at price 10, and the sell book has two orders at price 12 and 2 orders with value 12+y, where y is a positive integer that will be determined by the analysis. The original sequence Σ is a buy order with ∆ = 0, followed by two buy orders with ∆ = +1, then 2y sell orders with ∆ = 0, and then 2y buy orders with ∆ = +1. We first note that s2(Σ) = 2, there are 2y executions, all at price 12, the last bid is 11 and the last ask is 12. Next we analyze a modified sequence. We change the first buy order from ∆ = 0 to ∆ = +1. Therefore, the next two buy orders with ∆ = +1 are executed, and afterwards we have that the bid is 11 and the ask is 12 + y. Now the 2y sell orders are accumulated at 12+y, and after the next y buy orders the bid is at 12+y−1. Therefore, at the end we have that lastbid(Σ ) = 12 + y − 1, lastask(Σ ) = 12 + y, close(Σ ) = 12 + y, and average(Σ ) = y y+2 (12 + y) + 2 y+2 (12). Setting y = x + 2, we obtain the lemma for every property. We note that while this proof was based on the fact that there are two consecutive orders in the books which are far (y) apart, we can provide a slightly more complicated example in which all orders are close (at most 2 apart), yet still one change results in large differences. 7. SIMULATION STUDIES The results presented so far paint a striking contrast between the absolute and relative price models: while the absolute model enjoys provably strong stability over any fixed event sequence, there exist at least specific sequences demonstrating great instability in the relative model. The worstcase nature of these results raises the question of the extent to which such differences could actually occur in real markets. In this section we provide indirect evidence on this question by presenting simulation results exploiting a rich source of real-market historical limit order sequence data. By interpreting arriving limit order prices as either absolute values, or by transforming them into differences with the current bid and ask (relative model), we can perform small modifications on the sequences and examine how different various outcomes (volume traded, average price, etc.) would be from what actually occurred in the market. These simulations provide an empirical counterpart to the theory we have developed. We emphasize that all such simulations interpret the actual historical data as falling into either the absolute or relative model, and are meaningful only within the confines of such an interpretation. Nevertheless, we feel they provide valuable empirical insight into the potential (in)stability properties of modern equity limit order markets, and demonstrate that one"s belief or hope in stability largely relies on an absolute model interpretation. We also investigate the empirical behavior of mixtures of absolute and relative prices. 7.1 Data The historical data used in our simulations is commercially available limit order data from INET, the previously mentioned electronic exchange for NASDAQ stocks. Broadly speaking, this data consists of practically every single event on INET regarding the trading of an individual stockevery arriving limit order (price, volume, and sequence ID number), every execution, and every cancellation of a standing order - all timestamped in milliseconds. It is data sufficient to recreate the precise INET order book in a given stock on a given day and time. We will report stability properties for three stocks: Amazon, Nvidia, and Qualcomm (identified in the sequel by their tickers, AMZN, NVDA and QCOM). These three provide some range of liquidities (with QCOM having the greatest and NVDA the least liquidity on INET) and other trading properties. We note that the qualitative results of our simulations were similar for several other stocks we examined. 127 7.2 Methodology For our simulations we employed order-book reconstruction code operating on the underlying raw data. The basic format of each experiment was the following: 1. Run the order book reconstruction code on the original INET data and compute the quantity of interest (volume traded, average price, etc.) 2. Make a small modification to a single order, and recompute the resulting value of the quantity of interest. In the absolute model case, Step 2 is as simple as modifying the order in the original data and re-running the order book reconstruction. For the relative model, we must first pre-process the raw data and convert its prices to relative values, then make the modification and re-run the order book reconstruction on the relative values. The type of modification we examined was extremely small compared to the volume of orders placed in these stocks: namely, the deletion of a single randomly chosen order from the sequence. Although a deletion is not 1-modification, its edit distance is 1 and we can apply Theorem 5.4. For each trading day examined,this single deleted order was selected among those arriving between 10 AM and 3 PM, and the quantities of interest were measured and compared at 3 PM. These times were chosen to include the busiest part of the trading day but avoid the half hour around the opening and closing of the official NASDAQ market (9:30 AM and 3:30 PM respectively), which are known to have different dynamics than the central portion of the day. We run the absolute and relative model simulations on both the raw INET data and on a cleaned version of this data. In the cleaned we remove all limit orders that were canceled in the actual market prior to their execution (along with the cancellations themselves). The reason is that such cancellations may often be the first step in the repositioning of orders - that is, cancellations of the order that are followed by the submission of a replacement order at a different price. Not removing canceled orders allows the possibility of modified simulations in which the same order 1 is executed twice, which may magnify instability effects. Again, it is clear that neither the raw nor the cleaned data can perfectly reflect what would have happened under the deleted orders in the actual market. However, the results both from the raw data and the clean data are qualitatively similar. The results mainly differ, as expected, in the executed volume, where the instability results for the relative model are much more dramatic in the raw data. 7.3 Results We begin with summary statistics capturing our overall stability findings. Each row of the tables below contains a ticker (e.g. AMZN) followed by either -R (for the uncleaned or raw data) or -C (for the data with canceled orders removed). For each of the approximately 250 trading days in 2003, 1000 trials were run in which a randomly selected order was deleted from the INET event sequence. For each quantity of interest (volume executed, average price, closing price and last bid), we show for the both the absolute and 1 Here same is in quotes since the two orders will actually have different sequence ID numbers, which is what makes such repositioning activity impossible to reliably detect in the data. relative model the average percentage change in the quantity induced by the deletion. The results confirm rather strikingly the qualitative conclusions of the theory we have developed. In virtually every case (stock, raw or cleaned data, and quantity) the percentage change induced by a single deletion in the relative model is many orders of magnitude greater than in the absolute model, and shows that indeed butterfly effects may occur in a relative model market. As just one specific representative example, notice that for QCOM on the cleaned data, the relative model effect of just a single deletion on the closing price is in excess of a full percentage point. This is a variety of market impact entirely separate from the more traditional and expected kind generated by trading a large volume of shares. Stock Date volume average Rel Abs Rel Abs AMZN-R 2003 15.1% 0.04% 0.3% 0.0002% AMZN-C 2003 0.69% 0.087% 0.36% 0.0007% NVDA-R 2003 9.09% 0.05 % 0.17% 0.0003% NVDA-C 2003 0.73% 0.09 % 0.35% 0.001% QCOM-R 2003 16.94% 0.035% 0.21% 0.0002% QCOM-C 2003 0.58% 0.06% 0.35% 0.0005% Stock Date close lastbid Rel Abs Rel Abs AMZN-R 2003 0.78% 0.0001% 0.78% 0.0007% AMZN-C 2003 1.10% 0.077% 1.11% 0.001% NVDA-R 2003 1.17% 0.002 % 1.18 % 0.08% NVDA-C 2003 0.45% 0.0003% 0.45% 0.0006% QCOM-R 2003 0.58% 0.0001% 0.58% 0.0004% QCOM-C 2003 1.05% 0.0006% 1.05% 0.06% In Figure 4 we examine how the change to one the quantities, the average execution price, grows with the introduction of greater perturbations of the event sequence in the two models. Rather than deleting only a single order between 10 AM and 3 PM, in these experiments a growing number of randomly chosen deletions was performed, and the percentage change to the average price measured. As suggested by the theory we have developed, for the absolute model the change to the average price grows linearly with the number of deletions and remains very small (note the vastly different scales of the y-axis in the panels for the absolute and relative models in the figure). For the relative model, it is interesting to note that while small numbers of changes have large effects (often causing average execution price changes well in excess of 0.1 percent), the effects of large numbers of changes levels off quite rapidly and consistently. We conclude with an examination of experiments with a mixture model. Even if one accepts a world in which traders behave in either an absolute or relative manner, one would be likely to claim that the market contains a mixture of both. We thus ran simulations in which each arriving order in the INET event streams was treated as an absolute price with probability α, and as a relative price with probability 1−α. Representative results for the average execution price in this mixture model are shown in Figure 5 for AMZN and NVDA. Perhaps as expected, we see a monotonic decrease in the percentage change (instability) as the fraction of absolute traders increases, with most of the reduction already being realized by the introduction of just a small population of absolute traders. Thus even in a largely relative-price world, a 128 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5 x 10 −3 QCOM−R June 2004: Absolute Number of changes Averageprice 0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 QCOM−R June 2004: Relative Number of changes Averageprice Figure 4: Percentage change to the average execution price (y-axis) as a function of the number of deletions to the sequence (x-axis). The left panel is for the absolute model, the right panel for the relative model, and each curve corresponds to a single day of QCOM trading in June 2004. Curves represent averages over 1000 trials. small minority of absolute traders can have a greatly stabilizing effect. Similar behavior is found for closing price and last bid. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 AMZN−R Feburary 2004 α Averageprice 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 NVDA−R June 2004 α Averageprice Figure 5: Percentage change to the average execution price (y-axis) vs. probability of treating arriving INET orders as absolute prices (x-axis). Each curve corresponds to a single day of trading during a month of 2004. Curves represent averages over 1000 trials. For the executed volume in the mixture model, however, the findings are more curious. In Figure 6, we show how the percentage change to the executed volume varies with the absolute trader fraction α, for NVDA data that is both raw and cleaned of cancellations. We first see that for this quantity, unlike the others, the difference induced by the cleaned and uncleaned data is indeed dramatic, as already suggested by the summary statistics table above. But most intriguing is the fact that the stability is not monotonically increasing with α for either the cleaned or uncleaned datathe market with maximum instability is not a pure relative price market, but occurs at some nonzero value for α. It was in fact not obvious to us that sequences with this property could even be artificially constructed, much less that they would occur as actual market data. We have yet to find a satisfying explanation for this phenomenon and leave it to future research. 8. ACKNOWLEDGMENTS We are grateful to Yuriy Nevmyvaka of Lehman Brothers in New York for the use of his INET order book reconstruction code, and for valuable comments on the work presented 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 NVDA−C June 2004 α Volume 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 NVDA−R June 2004 α Volume Figure 6: Percentage change to the executed volume (y-axis) vs. probability of treating arriving INET orders as absolute prices (x-axis). The left panel is for NVDA using the raw data that includes cancellations, while the right panel is on the cleaned data. Each curve corresponds to a single day of trading during June 2004. Curves represent averages over 1000 trials. here. Yishay Mansour was supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778, by a grant from the Israel Science Foundation and an IBM faculty award. 9. REFERENCES [1] D. Bertsimas and A. Lo. Optimal control of execution costs. Journal of Financial Markets, 1:1-50, 1998. [2] B. Biais, L. Glosten, and C. Spatt. Market microstructure: a survey of microfoundations, empirical results and policy implications. Journal of Financial Markets, 8:217-264, 2005. [3] J.-P. Bouchaud, M. Mezard, and M. Potters. Statistical properties of stock order books: empirical results and models. Quantitative Finance, 2:251-256, 2002. [4] C. Cao, O.Hansch, and X. Wang. The informational content of an open limit order book, 2004. AFA 2005 Philadelphia Meetings, EFA Maastricht Meetings Paper No. 4311. [5] R. Coggins, A. Blazejewski, and M. Aitken. Optimal trade execution of equities in a limit order market. In International Conference on Computational Intelligence for Financial Engineering, pages 371-378, March 2003. [6] D. Farmer and S. Joshi. The price dynamics of common trading strategies. Journal of Economic Behavior and Organization, 29:149-171, 2002. [7] J. Hasbrouck. Empirical market microstructure: Economic and statistical perspectives on the dynamics of trade in securities markets, 2004. Course notes, Stern School of Business, New York University. [8] R. Kissell and M. Glantz. Optimal Trading Strategies. Amacom, 2003. [9] S.Kakade, M. Kearns, Y. Mansour, and L. Ortiz. Competitive algorithms for VWAP and limit order trading. In Proceedings of the ACM Conference on Electronic Commerce, pages 189-198, 2004. [10] Y.Nevmyvaka, Y. Feng, and M. Kearns. Reinforcement learning for optimized trade execution, 2006. Preprint. 129

bid;market microstructure;absolute trader model;computational finance;penny-jumping;standard continuous limit-order mechanism;modern execution optimization;relative trader model;modern equity market;high-frequency microstructure signal;relative price model;quantitative trading strategy;electronic communication network

train_J-35

Efficiency and Nash Equilibria in a Scrip System for P2P Networks

A model of providing service in a P2P network is analyzed. It is shown that by adding a scrip system, a mechanism that admits a reasonable Nash equilibrium that reduces free riding can be obtained. The effect of varying the total amount of money (scrip) in the system on efficiency (i.e., social welfare) is analyzed, and it is shown that by maintaining the appropriate ratio between the total amount of money and the number of agents, efficiency is maximized. The work has implications for many online systems, not only P2P networks but also a wide variety of online forums for which scrip systems are popular, but formal analyses have been lacking.

1. INTRODUCTION A common feature of many online distributed systems is that individuals provide services for each other. Peer-topeer (P2P) networks (such as Kazaa [25] or BitTorrent [3]) have proved popular as mechanisms for file sharing, and applications such as distributed computation and file storage are on the horizon; systems such as Seti@home [24] provide computational assistance; systems such as Slashdot [21] provide content, evaluations, and advice forums in which people answer each other"s questions. Having individuals provide each other with service typically increases the social welfare: the individual utilizing the resources of the system derives a greater benefit from it than the cost to the individual providing it. However, the cost of providing service can still be nontrivial. For example, users of Kazaa and BitTorrent may be charged for bandwidth usage; in addition, in some filesharing systems, there is the possibility of being sued, which can be viewed as part of the cost. Thus, in many systems there is a strong incentive to become a free rider and benefit from the system without contributing to it. This is not merely a theoretical problem; studies of the Gnutella [22] network have shown that almost 70 percent of users share no files and nearly 50 percent of responses are from the top 1 percent of sharing hosts [1]. Having relatively few users provide most of the service creates a point of centralization; the disappearance of a small percentage of users can greatly impair the functionality of the system. Moreover, current trends seem to be leading to the elimination of the altruistic users on which these systems rely. These heavy users are some of the most expensive customers ISPs have. Thus, as the amount of traffic has grown, ISPs have begun to seek ways to reduce this traffic. Some universities have started charging students for excessive bandwidth usage; others revoke network access for it [5]. A number of companies have also formed whose service is to detect excessive bandwidth usage [19]. These trends make developing a system that encourages a more equal distribution of the work critical for the continued viability of P2P networks and other distributed online systems. A significant amount of research has gone into designing reputation systems to give preferential treatment to users who are sharing files. Some of the P2P networks currently in use have implemented versions of these techniques. However, these approaches tend to fall into one of two categories: either they are barter-like or reputational. By barter-like, we mean that each agent bases its decisions only on information it has derived from its own interactions. Perhaps the best-known example of a barter-like system is BitTorrent, where clients downloading a file try to find other clients with parts they are missing so that they can trade, thus creating a roughly equal amount of work. Since the barter is restricted to users currently interested in a single file, this works well for popular files, but tends to have problems maintaining availability of less popular ones. An example of a barter-like system built on top of a more traditional file-sharing system is the credit system used by eMule 140 [8]. Each user tracks his history of interactions with other users and gives priority to those he has downloaded from in the past. However, in a large system, the probability that a pair of randomly-chosen users will have interacted before is quite small, so this interaction history will not be terribly helpful. Anagnostakis and Greenwald [2] present a more sophisticated version of this approach, but it still seems to suffer from similar problems. A number of attempts have been made at providing general reputation systems (e.g. [12, 13, 17, 27]). The basic idea is to aggregate each user"s experience into a global number for each individual that intuitively represents the system"s view of that individual"s reputation. However, these attempts tend to suffer from practical problems because they implicitly view users as either good or bad, assume that the good users will act according to the specified protocol, and that there are relatively few bad users. Unfortunately, if there are easy ways to game the system, once this information becomes widely available, rational users are likely to make use of it. We cannot count on only a few users being bad (in the sense of not following the prescribed protocol). For example, Kazaa uses a measure of the ratio of the number of uploads to the number of downloads to identify good and bad users. However, to avoid penalizing new users, they gave new users an average rating. Users discovered that they could use this relatively good rating to free ride for a while and, once it started to get bad, they could delete their stored information and effectively come back as a new user, thus circumventing the system (see [2] for a discussion and [11] for a formal analysis of this whitewashing). Thus Kazaa"s reputation system is ineffective. This is a simple case of a more general vulnerability of such systems to sybil attacks [6], where a single user maintains multiple identities and uses them in a coordinated fashion to get better service than he otherwise would. Recent work has shown that most common reputation systems are vulnerable (in the worst case)to such attacks [4]; however, the degree of this vulnerability is still unclear. The analyses of the practical vulnerabilities and the existence of such systems that are immune to such attacks remains an area of active research (e.g., [4, 28, 14]). Simple economic systems based on a scrip or money seem to avoid many of these problems, are easy to implement and are quite popular (see, e.g., [13, 15, 26]). However, they have a different set of problems. Perhaps the most common involve determining the amount of money in the system. Roughly speaking, if there is too little money in the system relative to the number of agents, then relatively few users can afford to make request. On the other hand, if there is too much money, then users will not feel the need to respond to a request; they have enough money already. A related problem involves handling newcomers. If newcomers are each given a positive amount of money, then the system is open to sybil attacks. Perhaps not surprisingly, scrip systems end up having to deal with standard economic woes such as inflation, bubbles, and crashes [26]. In this paper, we provide a formal model in which to analyze scrip systems. We describe a simple scrip system and show that, under reasonable assumptions, for each fixed amount of money there is a nontrivial Nash equilibrium involving threshold strategies, where an agent accepts a request if he has less than $k for some threshold k.1 An interesting aspect of our analysis is that, in equilibrium, the distribution of users with each amount of money is the distribution that maximizes entropy (subject to the money supply constraint). This allows us to compute the money supply that maximizes efficiency (social welfare), given the number of agents. It also leads to a solution for the problem of dealing with newcomers: we simply assume that new users come in with no money, and adjust the price of service (which is equivalent to adjusting the money supply) to maintain the ratio that maximizes efficiency. While assuming that new users come in with no money will not work in all settings, we believe the approach will be widely applicable. In systems where the goal is to do work, new users can acquire money by performing work. It should also work in Kazaalike system where a user can come in with some resources (e.g., a private collection of MP3s). The rest of the paper is organized as follows. In Section 2, we present our formal model and observe that it can be used to understand the effect of altruists. In Section 3, we examine what happens in the game under nonstrategic play, if all agents use the same threshold strategy. We show that, in this case, the system quickly converges to a situation where the distribution of money is characterized by maximum entropy. Using this analysis, we show in Section 4 that, under minimal assumptions, there is a nontrivial Nash equilibrium in the game where all agents use some threshold strategy. Moreover, we show in Section 5 that the analysis leads to an understanding of how to choose the amount of money in the system (or, equivalently, the cost to fulfill a request) so as to maximize efficiency, and also shows how to handle new users. In Section 6, we discuss the extent to which our approach can handle sybils and collusion. We conclude in Section 7. 2. THE MODEL To begin, we formalize providing service in a P2P network as a non-cooperative game. Unlike much of the modeling in this area, our model will model the asymmetric interactions in a file sharing system in which the matching of players (those requesting a file with those who have that particular file) is a key part of the system. This is in contrast with much previous work which uses random matching in a prisoner"s dilemma. Such models were studied in the economics literature [18, 7] and first applied to online reputations in [11]; an application to P2P is found in [9]. This random-matching model fails to capture some salient aspects of a number of important settings. When a request is made, there are typically many people in the network who can potentially satisfy it (especially in a large P2P network), but not all can. For example, some people may not have the time or resources to satisfy the request. The randommatching process ignores the fact that some people may not be able to satisfy the request. Presumably, if the person matched with the requester could not satisfy the match, he would have to defect. Moreover, it does not capture the fact that the decision as to whether to volunteer to satisfy the request should be made before the matching process, not after. That is, the matching process does not capture 1 Although we refer to our unit of scrip as the dollar, these are not real dollars nor do we view them as convertible to dollars. 141 the fact that if someone is unwilling to satisfy the request, there will doubtless be others who can satisfy it. Finally, the actions and payoffs in the prisoner"s dilemma game do not obviously correspond to actual choices that can be made. For example, it is not clear what defection on the part of the requester means. In our model we try to deal with all these issues. Suppose that there are n agents. At each round, an agent is picked uniformly at random to make a request. Each other agent is able to satisfy this request with probability β > 0 at all times, independent of previous behavior. The term β is intended to capture the probability that an agent is busy, or does not have the resources to fulfill the request. Assuming that β is time-independent does not capture the intution that being an unable to fulfill a request at time t may well be correlated with being unable to fulfill it at time t+1. We believe that, in large systems, we should be able to drop the independence assumption, but we leave this for future work. In any case, those agents that are able to satisfy the request must choose whether or not to volunteer to satisfy it. If at least one agent volunteers, the requester gets a benefit of 1 util (the job is done) and one of volunteers is chosen at random to fulfill the request. The agent that fulfills the request pays a cost of α < 1. As is standard in the literature, we assume that agents discount future payoffs by a factor of δ per time unit. This captures the intuition that a util now is worth more than a util tomorrow, and allows us to compute the total utility derived by an agent in an infinite game. Lastly, we assume that with more players requests come more often. Thus we assume that the time between rounds is 1/n. This captures the fact that the systems we want to model are really processing many requests in parallel, so we would expect the number of concurrent requests to be proportional to the number of users.2 Let G(n, δ, α, β) denote this game with n agents, a discount factor of δ, a cost to satisfy requests of α, and a probability of being able to satisfy requests of β. When the latter two parameters are not relevant, we sometimes write G(n, δ). We use the following notation throughout the paper: • pt denotes the agent chosen in round t. • Bt i ∈ {0, 1} denotes whether agent i can satisfy the request in round t. Bt i = 1 with probability β > 0 and Bt i is independent of Bt i for all t = t. • V t i ∈ {0, 1} denotes agent i"s decision about whether to volunteer in round t; 1 indicates volunteering. V t i is determined by agent i"s strategy. • vt ∈ {j | V t j Bt j = 1} denotes the agent chosen to satisfy the request. This agent is chosen uniformly at random from those who are willing (V t j = 1) and able (Bt j = 1) to satisfy the request. • ut i denotes agent i"s utility in round t. A standard agent is one whose utility is determined as discussed in the introduction; namely, the agent gets 2 For large n, our model converges to one in which players make requests in real time, and the time between a player"s requests are exponentially distributed with mean 1. In addition, the time between requests served by a single player is also exponentially distributed. a utility of 1 for a fulfilled request and utility −α for fulfilling a request. Thus, if i is a standard agent, then ut i = 8 < : 1 if i = pt and P j=i V t j Bt j > 0 −α if i = vt 0 otherwise. • Ui = P∞ t=0 δt/n ut i denotes the total utility for agent i. It is the discounted total of agent i"s utility in each round. Note that the effective discount factor is δ1/n since an increase in n leads to a shortening of the time between rounds. Now that we have a model of making and satisfying requests, we use it to analyze free riding. Take an altruist to be someone who always fulfills requests. Agent i might rationally behave altruistically if agent i"s utility function has the following form, for some α > 0: ut i = 8 < : 1 if i = pt and P j=i V t j Bt j > 0 α if i = vt 0 otherwise. Thus, rather than suffering a loss of utility when satisfying a request, an agent derives positive utility from satisfying it. Such a utility function is a reasonable representation of the pleasure that some people get from the sense that they provide the music that everyone is playing. For such altruistic agents, playing the strategy that sets V t i = 1 for all t is dominant. While having a nonstandard utility function might be one reason that a rational agent might use this strategy, there are certainly others. For example a naive user of filesharing software with a good connection might well follow this strategy. All that matters for the following discussion is that there are some agents that use this strategy, for whatever reason. As we have observed, such users seem to exist in some large systems. Suppose that our system has a altruists. Intuitively, if a is moderately large, they will manage to satisfy most of the requests in the system even if other agents do no work. Thus, there is little incentive for any other agent to volunteer, because he is already getting full advantage of participating in the system. Based on this intuition, it is a relatively straightforward calculation to determine a value of a that depends only on α, β, and δ, but not the number n of players in the system, such that the dominant strategy for all standard agents i is to never volunteer to satisfy any requests (i.e., V t i = 0 for all t). Proposition 2.1. There exists an a that depends only on α, β, and δ such that, in G(n, δ, α, β) with at least a altruists, not volunteering in every round is a dominant strategy for all standard agents. Proof. Consider the strategy for a standard player j in the presence of a altruists. Even with no money, player j will get a request satisfied with probability 1 − (1 − β)a just through the actions of these altruists. Thus, even if j is chosen to make a request in every round, the most additional expected utility he can hope to gain by having money isP∞ k=1(1 − β)a δk = (1 − β)a /(1 − δ). If (1 − β)a /(1 − δ) > α or, equivalently, if a > log1−β(α(1 − δ)), never volunteering is a dominant strategy. Consider the following reasonable values for our parameters: β = .01 (so that each player can satisfy 1% of the requests), α = .1 (a low but non-negligible cost), δ = .9999/day 142 (which corresponds to a yearly discount factor of approximately 0.95), and an average of 1 request per day per player. Then we only need a > 1145. While this is a large number, it is small relative to the size of a large P2P network. Current systems all have a pool of users behaving like our altruists. This means that attempts to add a reputation system on top of an existing P2P system to influence users to cooperate will have no effect on rational users. To have a fair distribution of work, these systems must be fundamentally redesigned to eliminate the pool of altruistic users. In some sense, this is not a problem at all. In a system with altruists, the altruists are presumably happy, as are the standard agents, who get almost all their requests satisfied without having to do any work. Indeed, current P2P network work quite well in terms of distributing content to people. However, as we said in the introduction, there is some reason to believe these altruists may not be around forever. Thus, it is worth looking at what can be done to make these systems work in their absence. For the rest of this paper we assume that all agents are standard, and try to maximize expected utility. We are interested in equilibria based on a scrip system. Each time an agent has a request satisfied he must pay the person who satisfied it some amount. For now, we assume that the payment is fixed; for simplicity, we take the amount to be $1. We denote by M the total amount of money in the system. We assume that M > 0 (otherwise no one will ever be able to get paid). In principle, agents are free to adopt a very wide variety of strategies. They can make decisions based on the names of other agents or use a strategy that is heavily history dependant, and mix these strategies freely. To aid our analysis, we would like to be able to restrict our attention to a simpler class of strategies. The class of strategies we are interested in is easy to motivate. The intuitive reason for wanting to earn money is to cater for the possibility that an agent will run out before he has a chance to earn more. On the other hand, a rational agent with plenty of mone would not want to work, because by the time he has managed to spend all his money, the util will have less value than the present cost of working. The natural balance between these two is a threshold strategy. Let Sk be the strategy where an agent volunteers whenever he has less than k dollars and not otherwise. Note that S0 is the strategy where the agent never volunteers. While everyone playing S0 is a Nash equilibrium (nobody can do better by volunteering if no one else is willing to), it is an uninteresting one. As we will show in Section 4, it is sufficient to restrict our attention to this class of strategies. We use Kt i to denote the amount of money agent i has at time t. Clearly Kt+1 i = Kt i unless agent i has a request satisfied, in which case Kt+1 i = Kt+1 i − 1 or agent i fulfills a request, in which case Kt+1 i = Kt+1 i + 1. Formally, Kt+1 i = 8 < : Kt i − 1 if i = pt , P j=i V t j Bt j > 0, and Kt i > 0 Kt i + 1 if i = vt and Kt pt > 0 Kt i otherwise. The threshold strategy Sk is the strategy such that V t i =  1 if Kt pt > 0 and Kt i < k 0 otherwise. 3. THE GAME UNDER NONSTRATEGIC PLAY Before we consider strategic play, we examine what happens in the system if everyone just plays the same strategy Sk. Our overall goal is to show that there is some distribution over money (i.e., the fraction of people with each amount of money) such that the system converges to this distribution in a sense to be made precise shortly. Suppose that everyone plays Sk. For simplicity, assume that everyone has at most k dollars. We can make this assumption with essentially no loss of generality, since if someone has more than k dollars, he will just spend money until he has at most k dollars. After this point he will never acquire more than k. Thus, eventually the system will be in such a state. If M ≥ kn, no agent will ever be willing to work. Thus, for the purposes of this section we assume that M < kn. From the perspective of a single agent, in (stochastic) equilibrium, the agent is undergoing a random walk. However, the parameters of this random walk depend on the random walks of the other agents and it is quite complicated to solve directly. Thus we consider an alternative analysis based on the evolution of the system as a whole. If everyone has at most k dollars, then the amount of money that an agent has is an element of {0, . . . , k}. If there are n agents, then the state of the game can be described by identifying how much money each agent has, so we can represent it by an element of Sk,n = {0, . . . , k}{1,...,n} . Since the total amount of money is constant, not all of these states can arise in the game. For example the state where each player has $0 is impossible to reach in any game with money in the system. Let mS(s) = P i∈{1...n} s(i) denote the total mount of money in the game at state s, where s(i) is the number of dollars that agent i has in state s. We want to consider only those states where the total money in the system is M, namely Sk,n,M = {s ∈ Sk,n | mS(s) = M}. Under the assumption that all agents use strategy Sk, the evolution of the system can be treated as a Markov chain Mk,n,M over the state space Sk,n,M . It is possible to move from one state to another in a single round if by choosing a particular agent to make a request and a particular agent to satisfy it, the amounts of money possesed by each agent become those in the second state. Therefore the probability of a transition from a state s to t is 0 unless there exist two agents i and j such that s(i ) = t(i ) for all i /∈ {i, j}, t(i) = s(i) + 1, and t(j) = s(j) − 1. In this case the probability of transitioning from s to t is the probability of j being chosen to spend a dollar and has someone willing and able to satisfy his request ((1/n)(1 − (1 − β)|{i |s(i )=k}|−Ij ) multiplied by the probability of i being chosen to satisfy his request (1/(|({i | s(i ) = k}| − Ij )). Ij is 0 if j has k dollars and 1 otherwise (it is just a correction for the fact that j cannot satisfy his own request.) Let ∆k denote the set of probability distributions on {0, . . . , k}. We can think of an element of ∆k as describing the fraction of people with each amount of money. This is a useful way of looking at the system, since we typically don"t care who has each amount of money, but just the fraction of people that have each amount. As before, not all elements of ∆k are possible, given our constraint that the total amount of 143 money is M. Rather than thinking in terms of the total amount of money in the system, it will prove more useful to think in terms of the average amount of money each player has. Of course, the total amount of money in a system with n agents is M iff the average amount that each player has is m = M/n. Let ∆k m denote all distributions d ∈ ∆k such that E(d) = m (i.e., Pk j=0 d(j)j = m). Given a state s ∈ Sk,n,M , let ds ∈ ∆k m denote the distribution of money in s. Our goal is to show that, if n is large, then there is a distribution d∗ ∈ ∆k m such that, with high probability, the Markov chain Mk,n,M will almost always be in a state s such that ds is close to d∗ . Thus, agents can base their decisions about what strategy to use on the assumption that they will be in such a state. We can in fact completely characterize the distribution d∗ . Given a distribution d ∈ ∆k , let H(d) = − X {j:d(j)=0} d(j) log(d(j)) denote the entropy of d. If ∆ is a closed convex set of distributions, then it is well known that there is a unique distribution in ∆ at which the entropy function takes its maximum value in ∆. Since ∆k m is easily seen to be a closed convex set of distributions, it follows that there is a unique distribution in ∆k m that we denote d∗ k,m whose entropy is greater than that of all other distributions in ∆k m. We now show that, for n sufficiently large, the Markov chain Mk,n,M is almost surely in a state s such that ds is close to d∗ k,M/n. The statement is correct under a number of senses of close. For definiteness, we consider the Euclidean distance. Given > 0, let Sk,n,m, denote the set of states s in Sk,n,mn such that Pk j=0 |ds (j) − d∗ k,m|2 < . Given a Markov chain M over a state space S and S ⊆ S, let Xt,s,S be the random variable that denotes that M is in a state of S at time t, when started in state s. Theorem 3.1. For all > 0, all k, and all m, there exists n such that for all n > n and all states s ∈ Sk,n,mn, there exists a time t∗ (which may depend on k, n, m, and ) such that for t > t∗ , we have Pr(Xt,s,Sk,n,m, ) > 1 − . Proof. (Sketch) Suppose that at some time t, Pr(Xt,s,s ) is uniform for all s . Then the probability of being in a set of states is just the size of the set divided by the total number of states. A standard technique from statistical mechanics is to show that there is a concentration phenomenon around the maximum entropy distribution [16]. More precisely, using a straightforward combinatorial argument, it can be shown that the fraction of states not in Sk,n,m, is bounded by p(n)/ecn , where p is a polynomial. This fraction clearly goes to 0 as n gets large. Thus, for sufficiently large n, Pr(Xt,s,Sk,n,m, ) > 1 − if Pr(Xt,s,s ) is uniform. It is relatively straightforward to show that our Markov Chain has a limit distribution π over Sk,n,mn, such that for all s, s ∈ Sk,n,mn, limt→∞ Pr(Xt,s,s ) = πs . Let Pij denote the probability of transitioning from state i to state j. It is easily verified by an explicit computation of the transition probabilities that Pij = Pji for all states i and j. It immediatly follows from this symmetry that πs = πs , so π is uniform. After a sufficient amount of time, the distribution will be close enough to π, that the probabilities are again bounded by constant, which is sufficient to complete the theorem. 0 0.002 0.004 0.006 0.008 0.01 Euclidean Distance 2000 2500 3000 3500 4000 NumberofSteps Figure 1: Distance from maximum-entropy distribution with 1000 agents. 5000 10000 15000 20000 25000 Number of Agents 0.001 0.002 0.003 0.004 0.005 MaximumDistance Figure 2: Maximum distance from maximumentropy distribution over 106 timesteps. 0 5000 10000 15000 20000 25000 Number of Agents 0 20000 40000 60000 TimetoDistance.001 Figure 3: Average time to get within .001 of the maximum-entropy distribution. 144 We performed a number of experiments that show that the maximum entropy behavior described in Theorem 3.1 arises quickly for quite practical values of n and t. The first experiment showed that, even if n = 1000, we reach the maximum-entropy distribution quickly. We averaged 10 runs of the Markov chain for k = 5 where there is enough money for each agent to have $2 starting from a very extreme distribution (every agent has either $0 or $5) and considered the average time needed to come within various distances of the maximum entropy distribution. As Figure 1 shows, after 2,000 steps, on average, the Euclidean distance from the average distribution of money to the maximum-entropy distribution is .008; after 3,000 steps, the distance is down to .001. Note that this is really only 3 real time units since with 1000 players we have 1000 transactions per time unit. We then considered how close the distribution stays to the maximum entropy distribution once it has reached it. To simplify things, we started the system in a state whose distribution was very close to the maximum-entropy distribution and ran it for 106 steps, for various values of n. As Figure 2 shows, the system does not move far from the maximum-entropy distribution once it is there. For example, if n = 5000, the system is never more than distance .001 from the maximum-entropy distribution; if n = 25, 000, it is never more than .0002 from the maximum-entropy distribution. Finally, we considered how more carefully how quickly the system converges to the maximum-entropy distribution for various values of n. There are approximately kn possible states, so the convergence time could in principle be quite large. However, we suspect that the Markov chain that arises here is rapidly mixing, which means that it will converge significantly faster (see [20] for more details about rapid mixing). We believe that the actually time needed is O(n). This behavior is illustrated in Figure 3, which shows that for our example chain (again averaged over 10 runs), after 3n steps, the Euclidean distance between the actual distribution of money in the system and the maximum-entropy distribution is less than .001. 4. THE GAME UNDER STRATEGIC PLAY We have seen that the system is well behaved if the agents all follow a threshold strategy; we now want to show that there is a nontrivial Nash equilibrium where they do so (that is, a Nash equilibrium where all the agents use Sk for some k > 0.) This is not true in general. If δ is small, then agents have no incentive to work. Intuitively, if future utility is sufficiently discounted, then all that matters is the present, and there is no point in volunteering to work. With small δ, S0 is the only equilibrium. However, we show that for δ sufficiently large, there is another equilibrium in threshold strategies. We do this by first showing that, if every other agent is playing a threshold strategy, then there is a best response that is also a threshold strategy (although not necessarily the same one). We then show that there must be some (mixed) threshold strategy for which this best response is the same strategy. It follows that this tuple of threshold strategies is a Nash equilibrium. As a first step, we show that, for all k, if everyone other than agent i is playing Sk, then there is a threshold strategy Sk that is a best response for agent i. To prove this, we need to assume that the system is close to the steadystate distribution (i.e., the maximum-entropy distribution). However, as long as δ is sufficiently close to 1, we can ignore what happens during the period that the system is not in steady state.3 We have thus far considered threshold strategies of the form Sk, where k is a natural number; this is a discrete set of strategies. For a later proof, it will be helpful to have a continuous set of strategies. If γ = k + γ , where k is a natural number and 0 ≤ γ < 1, let Sγ be the strategy that performs Sk with probability 1 − γ and Sk+1 with probability γ. (Note that we are not considering arbitrary mixed threshold strategies here, but rather just mixing between adjacent strategies for the sole purpose of making out strategies continuous in a natural way.) Theorem 3.1 applies to strategies Sγ (the same proof goes through without change), where γ is an arbitrary nonnegative real number. Theorem 4.1. Fix a strategy Sγ and an agent i. There exists δ∗ < 1 and n∗ such that if δ > δ∗ , n > n∗ , and every agent other than i is playing Sγ in game G(n, δ), then there is an integer k such that the best response for agent i is Sk . Either k is unique (that is, there is a unique best response that is also a threshold strategy), or there exists an integer k such that Sγ is a best response for agent i for all γ in the interval [k , k +1] (and these are the only best responses among threshold strategies). Proof. (Sketch:) If δ is sufficiently large, we can ignore what happens before the system converges to the maximumentropy distribution. If n is sufficiently large, then the strategy played by one agent will not affect the distribution of money significantly. Thus, the probability of i moving from one state (dollar amount) to another depends only on i"s strategy (since we can take the probability that i will be chosen to make a request and the probability that i will be chosen to satisfy a request to be constant). Thus, from i"s point of view, the system is a Markov decision process (MDP), and i needs to compute the optimal policy (strategy) for this MDP. It follows from standard results [23, Theorem 6.11.6] that there is an optimal policy that is a threshold policy. The argument that the best response is either unique or there is an interval of best responses follows from a more careful analysis of the value function for the MDP. We remark that there may be best responses that are not threshold strategies. All that Theorem 4.1 shows is that, among best responses, there is at least one that is a threshold strategy. Since we know that there is a best response that is a threshold strategy, we can look for a Nash equilibrium in the space of threshold strategies. Theorem 4.2. For all M, there exists δ∗ < 1 and n∗ such that if δ > δ∗ and n > n∗ , there exists a Nash equilibrium in the game G(n, δ) where all agents play Sγ for some integer γ > 0. Proof. It follows easily from the proof Theorem 4.1 that if br(δ, γ) is the minimal best response threshold strategy if all the other agents are playing Sγ and the discount factor is δ then, for fixed δ, br(δ, ·) is a step function. It also follows 3 Formally, we need to define the strategies when the system is far from equilibrium. However, these far from (stochastic) equilibrium strategies will not affect the equilibrium behavior when n is large and deviations from stochastic equilibrium are extremely rare. 145 from the theorem that if there are two best responses, then a mixture of them is also a best response. Therefore, if we can join the steps by a vertical line, we get a best-response curve. It is easy to see that everywhere that this bestresponse curve crosses the diagonal y = x defines a Nash equilibrium where all agents are using the same threshold strategy. As we have already observed, one such equilibrium occurs at 0. If there are only $M in the system, we can restrict to threshold strategies Sk where k ≤ M + 1. Since no one can have more than $M, all strategies Sk for k > M are equivalent to SM ; these are just the strategies where the agent always volunteers in response to request made by someone who can pay. Clearly br(δ, SM ) ≤ M for all δ, so the best response function is at or below the equilibrium at M. If k ≤ M/n, every player will have at least k dollars and so will be unwilling to work and the best response is just 0. Consider k∗ , the smallest k such that k > M/n. It is not hard to show that for k∗ there exists a δ∗ such that for all δ ≥ δ∗ , br(δ, k∗ ) ≥ k∗ . It follows by continuity that if δ ≥ δ∗ , there must be some γ such that br(δ, γ) = γ. This is the desired Nash equilibrium. This argument also shows us that we cannot in general expect fixed points to be unique. If br(δ, k∗ ) = k∗ and br(δ, k + 1) > k + 1 then our argument shows there must be a second fixed point. In general there may be multiple fixed points even when br(δ, k∗ ) > k∗ , as illustrated in the Figure 4 with n = 1000 and M = 3000. 0 5 10 15 20 25 Strategy of Rest of Agents 0 5 10 15 20 25 BestResponse Figure 4: The best response function for n = 1000 and M = 3000. Theorem 4.2 allows us to restrict our design to agents using threshold strategies with the confidence that there will be a nontrivial equilibrium. However, it does not rule out the possibility that there may be other equilibria that do not involve threshold stratgies. It is even possible (although it seems unlikely) that some of these equilibria might be better. 5. SOCIAL WELFARE AND SCALABITY Our theorems show that for each value of M and n, for sufficiently large δ, there is a nontrivial Nash equilibrium where all the agents use some threshold strategy Sγ(M,n). From the point of view of the system designer, not all equilibria are equally good; we want an equilibrium where as few as possible agents have $0 when they get a chance to make a request (so that they can pay for the request) and relatively few agents have more than the threshold amount of money (so that there are always plenty of agents to fulfill the request). There is a tension between these objectives. It is not hard to show that as the fraction of agents with $0 increases in the maximum entropy distribution, the fraction of agents with the maximum amount of money decreases. Thus, our goal is to understand what the optimal amount of money should be in the system, given the number of agents. That is, we want to know the amount of money M that maximizes efficiency, i.e., the total expected utility if all the agents use Sγ(M,n). 4 We first observe that the most efficient equilibrium depends only on the ratio of M to n, not on the actual values of M and n. Theorem 5.1. There exists n∗ such that for all games G(n1, δ) and G(n2, δ) where n1, n2 > n∗ , if M1/n1 = M2/n2, then Sγ(M1,n1) = Sγ(M2,n2). Proof. Fix M/n = r. Theorem 3.1 shows that the maximum-entropy distribution depends only on k and the ratio M/n, not on M and n separately. Thus, given r, for each choice of k, there is a unique maximum entropy distribution dk,r. The best response br(δ, k) depends only on the distribution dk,r, not M or n. Thus, the Nash equilibrium depends only on the ratio r. That is, for all choices of M and n such that n is sufficiently large (so that Theorem 3.1 applies) and M/n = r, the equilibrium strategies are the same. In light of Theorem 5.1, the system designer should ensure that there is enough money M in the system so that the ratio between M/n is optimal. We are currently exploring exactly what the optimal ratio is. As our very preliminary results for β = 1 show in Figure 5, the ratio appears to be monotone increasing in δ, which matches the intuition that we should provide more patient agents with the opportunity to save more money. Additionally, it appears to be relatively smooth, which suggests that it may have a nice analytic solution. 0.9 0.91 0.92 0.93 0.94 0.95 Discount Rate ∆ 5 5.5 6 6.5 7 OptimalRatioofMn Figure 5: Optimal average amount of money to the nearest .25 for β = 1 We remark that, in practice, it may be easier for the designer to vary the price of fulfilling a request rather than 4 If there are multiple equilibria, we take Sγ(M,n) to be the Nash equilibrium that has highest efficiency for fixed M and n. 146 injecting money in the system. This produces the same effect. For example, changing the cost of fulfilling a request from $1 to $2 is equivalent to halving the amount of money that each agent has. Similarly, halving the the cost of fulfilling a request is equivalent to doubling the amount of money that everyone has. With a fixed amount of money M, there is an optimal product nc of the number of agents and the cost c of fulfilling a request. Theorem 5.1 also tells us how to deal with a dynamic pool of agents. Our system can handle newcomers relatively easily: simply allow them to join with no money. This gives existing agents no incentive to leave and rejoin as newcomers. We then change the price of fulfilling a request so that the optimal ratio is maintained. This method has the nice feature that it can be implemented in a distributed fashion; if all nodes in the system have a good estimate of n then they can all adjust prices automatically. (Alternatively, the number of agents in the system can be posted in a public place.) Approaches that rely on adjusting the amount of money may require expensive system-wide computations (see [26] for an example), and must be carefully tuned to avoid creating incentives for agents to manipulate the system by which this is done. Note that, in principle, the realization that the cost of fulfilling a request can change can affect an agent"s strategy. For example, if an agent expects the cost to increase, then he may want to defer volunteering to fulfill a request. However, if the number of agents in the system is always increasing, then the cost always decreases, so there is never any advantage in waiting. There may be an advantage in delaying a request, but it is far more costly, in terms of waiting costs than in providing service, since we assume the need for a service is often subject to real waiting costs, while the need to supply the service is merely to augment a money supply. (Related issues are discussed in [10].) We ultimately hope to modify the mechanism so that the price of a job can be set endogenously within the system (as in real-world economies), with agents bidding for jobs rather than there being a fixed cost set externally. However, we have not yet explored the changes required to implement this change. Thus, for now, we assume that the cost is set as a function of the number of agents in the system (and that there is no possibility for agents to satisfy a request for less than the official cost or for requesters to offer to pay more than it). 6. SYBILS AND COLLUSION In a naive sense, our system is essentially sybil-proof. To get d dollars, his sybils together still have to perform d units of work. Moreover, since newcomers enter the system with $0, there is no benefit to creating new agents simply to take advantage of an initial endowment. Nevertheless, there are some less direct ways that an agent could take advantage of sybils. First, by having more identities he will have a greater probability of getting chosen to make a request. It is easy to see that this will lead to the agent having higher total utility. However, this is just an artifact of our model. To make our system simple to analyze, we have assumed that request opportunities came uniformly at random. In practice, requests are made to satisfy a desire. Our model implicitly assumed that all agents are equally likely to have a desire at any particular time. Having sybils should not increase the need to have a request satisfied. Indeed, it would be reasonable to assume that sybils do not make requests at all. Second, having sybils makes it more likely that one of the sybils will be chosen to fulfill a request. This can allow a user to increase his utility by setting a lower threshold; that is, to use a strategy Sk where k is smaller than the k used by the Nash equilibrium strategy. Intuitively, the need for money is not as critical if money is easier to obtain. Unlike the first concern, this seems like a real issue. It seems reasonable to believe that when people make a decision between a number of nodes to satisfy a request they do so at random, at least to some extent. Even if they look for advertised node features to help make this decision, sybils would allow a user to advertise a wide range of features. Third, an agent can drive down the cost of fulfilling a request by introducing many sybils. Similarly, he could increase the cost (and thus the value of his money) by making a number of sybils leave the system. Concievably he could alternate between these techniques to magnify the effects of work he does. We have not yet calculated the exact effect of this change (it interacts with the other two effects of having sybils that we have already noted). Given the number of sybils that would be needed to cause a real change in the perceived size of a large P2P network, the practicality of this attack depends heavily on how much sybils cost an attacker and what resources he has available. The second point raised regarding sybils also applies to collusion if we allow money to be loaned. If k agents collude, they can agree that, if one runs out of money, another in the group will loan him money. By pooling their money in this way, the k agents can again do better by setting a higher threshold. Note that the loan mechanism doesn"t need to be built into the system; the agents can simply use a fake transaction to transfer the money. These appear to be the main avenues for collusive attacks, but we are still exploring this issue. 7. CONCLUSION We have given a formal analysis of a scrip system and have shown that the existence of a Nash equilibrium where all agents use a threshold strategy. Moreover, we can compute efficiency of equilibrium strategy and optimize the price (or money supply) to maximize efficiency. Thus, our analysis provides a formal mechanisms for solving some important problems in implementing scrip systems. It tells us that with a fixed population of rational users, such systems are very unlikely to become unstable. Thus if this stability is common belief among the agents we would not expect inflation, bubbles, or crashes because of agent speculation. However, we cannot rule out the possibility that that agents may have other beliefs that will cause them to speculate. Our analysis also tells us how to scale the system to handle an influx of new users without introducing these problems: scale the money supply to keep the average amount of money constant (or equivalently adjust prices to achieve the same goal). There are a number of theoretical issues that are still open, including a characterization of the multiplicity of equilibria - are there usually 2? In addition, we expect that one should be able to compute analytic estimates for the best response function and optimal pricing which would allow us to understand the relationship between pricing and various parameters in the model. 147 It would also be of great interest to extend our analysis to handle more realistic settings. We mention a few possible extensions here: • We have assumed that the world is homogeneous in a number of ways, including request frequency, utility, and ability to satisfy requests. It would be interesting to examine how relaxing any of these assumptions would alter our results. • We have assumed that there is no cost to an agent to be a member of the system. Suppose instead that we imposed a small cost simply for being present in the system to reflect the costs of routing messages and overlay maintainance. This modification could have a significant impact on sybil attacks. • We have described a scrip system that works when there are no altruists and have shown that no system can work once there there are sufficiently many altruists. What happens between these extremes? • One type of irrational behavior encountered with scrip systems is hoarding. There are some similarities between hoarding and altruistic behavior. While an altruist provide service for everyone, a hoarder will volunteer for all jobs (in order to get more money) and rarely request service (so as not to spend money). It would be interesting to investigate the extent to which our system is robust against hoarders. Clearly with too many hoarders, there may not be enough money remaining among the non-hoarders to guarantee that, typically, a non-hoarder would have enough money to satisfy a request. • Finally, in P2P filesharing systems, there are overlapping communities of various sizes that are significantly more likely to be able to satisfy each other"s requests. It would be interesting to investigate the effect of such communities on the equilibrium of our system. There are also a number of implementation issues that would have to be resolved in a real system. For example, we need to worry about the possibility of agents counterfeiting money or lying about whether service was actually provided. Karma [26] provdes techniques for dealing with both of these issues and a number of others, but some of Karma"s implementation decisions point to problems for our model. For example, it is prohibitively expensive to ensure that bank account balances can never go negative, a fact that our model does not capture. Another example is that Karma has nodes serve as bookkeepers for other nodes account balances. Like maintaining a presence in the network, this imposes a cost on the node, but unlike that, responsibility it can be easily shirked. Karma suggests several ways to incentivize nodes to perform these duties. We have not investigated whether these mechanisms be incorporated without disturbing our equilibrium. 8. ACKNOWLEDGEMENTS We would like to thank Emin Gun Sirer, Shane Henderson, Jon Kleinberg, and 3 anonymous referees for helpful suggestions. EF, IK and JH are supported in part by NSF under grant ITR-0325453. JH is also supported in part by NSF under grants CTC-0208535 and IIS-0534064, by ONR under grant N00014-01-10-511, by the DoD Multidisciplinary University Research Initiative (MURI) program administered by the ONR under grants N00014-01-1-0795 and N00014-04-1-0725, and by AFOSR under grant F49620-021-0101. 9. REFERENCES [1] E. Adar and B. A. Huberman. Free riding on Gnutella. First Monday, 5(10), 2000. [2] K. G. Anagnostakis and M. Greenwald. Exchange-based incentive mechanisms for peer-to-peer file sharing. In International Conference on Distributed Computing Systems (ICDCS), pages 524-533, 2004. [3] BitTorrent Inc. BitTorrent web site. http://www.bittorent.com. [4] A. Cheng and E. Friedman. Sybilproof reputation mechanisms. In Workshop on Economics of Peer-to-Peer Systems (P2PECON), pages 128-132, 2005. [5] Cornell Information Technologies. Cornell"s ccommodity internet usage statistics. http://www.cit.cornell.edu/computer/students/ bandwidth/charts.html. [6] J. R. Douceur. The sybil attack. In International Workshop on Peer-to-Peer Systems (IPTPS), pages 251-260, 2002. [7] G. Ellison. Cooperation in the prisoner"s dilemma with anonymous random matching. Review of Economic Studies, 61:567-588, 1994. [8] eMule Project. eMule web site. http://www.emule-project.net/. [9] M. Feldman, K. Lai, I. Stoica, and J. Chuang. Robust incentive techniques for peer-to-peer networks. In ACM Conference on Electronic Commerce (EC), pages 102-111, 2004. [10] E. J. Friedman and D. C. Parkes. Pricing wifi at starbucks: issues in online mechanism design. In EC "03: Proceedings of the 4th ACM Conference on Electronic Commerce, pages 240-241. ACM Press, 2003. [11] E. J. Friedman and P. Resnick. The social cost of cheap pseudonyms. Journal of Economics and Management Strategy, 10(2):173-199, 2001. [12] R. Guha, R. Kumar, P. Raghavan, and A. Tomkins. Propagation of trust and distrust. In Conference on the World Wide Web(WWW), pages 403-412, 2004. [13] M. Gupta, P. Judge, and M. H. Ammar. A reputation system for peer-to-peer networks. In Network and Operating System Support for Digital Audio and Video(NOSSDAV), pages 144-152, 2003. [14] Z. Gyongi, P. Berkhin, H. Garcia-Molina, and J. Pedersen. Link spam detection based on mass estimation. Technical report, Stanford University, 2005. [15] J. Ioannidis, S. Ioannidis, A. D. Keromytis, and V. Prevelakis. Fileteller: Paying and getting paid for file storage. In Financial Cryptography, pages 282-299, 2002. [16] E. T. Jaynes. Where do we stand on maximum entropy? In R. D. Levine and M. Tribus, editors, The Maximum Entropy Formalism, pages 15-118. MIT Press, Cambridge, Mass., 1978. 148 [17] S. D. Kamvar, M. T. Schlosser, and H. Garcia-Molina. The Eigentrust algorithm for reputation management in P2P networks. In Conference on the World Wide Web (WWW), pages 640-651, 2003. [18] M. Kandori. Social norms and community enforcement. Review of Economic Studies, 59:63-80, 1992. [19] LogiSense Corporation. LogiSense web site. http://www.logisense.com/tm p2p.html. [20] L. Lovasz and P. Winkler. Mixing of random walks and other diffusions on a graph. In Surveys in Combinatorics, 1993, Walker (Ed.), London Mathematical Society Lecture Note Series 187, Cambridge University Press. 1995. [21] Open Source Technology Group. Slashdot FAQcomments and moderation. http://slashdot.org/faq/com-mod.shtml#cm700. [22] OSMB LLC. Gnutella web site. http://www.gnutella.com/. [23] M. L. Puterman. Markov Decision Processes. Wiley, 1994. [24] SETI@home. SETI@home web page. http://setiathome.ssl.berkeley.edu/. [25] Sharman Networks Ltd. Kazaa web site. http://www.kazaa.com/. [26] V. Vishnumurthy, S. Chandrakumar, and E. Sirer. Karma: A secure economic framework for peer-to-peer resource sharing. In Workshop on Economics of Peer-to-Peer Systems (P2PECON), 2003. [27] L. Xiong and L. Liu. Building trust in decentralized peer-to-peer electronic communities. In Internation Conference on Electronic Commerce Research (ICECR), 2002. [28] H. Zhang, A. Goel, R. Govindan, K. Mason, and B. V. Roy. Making eigenvector-based reputation systems robust to collusion. In Workshop on Algorithms and Models for the Web-Graph(WAW), pages 92-104, 2004. 149

nash equilibrium;game theory;gnutellum network;scrip system;agent;threshold strategy;reputation system;social welfare;game;online system;maximum entropy;p2p network;bittorrent;emule

train_J-36

Playing Games in Many Possible Worlds

In traditional game theory, players are typically endowed with exogenously given knowledge of the structure of the game-either full omniscient knowledge or partial but fixed information. In real life, however, people are often unaware of the utility of taking a particular action until they perform research into its consequences. In this paper, we model this phenomenon. We imagine a player engaged in a questionand-answer session, asking questions both about his or her own preferences and about the state of reality; thus we call this setting Socratic game theory. In a Socratic game, players begin with an a priori probability distribution over many possible worlds, with a different utility function for each world. Players can make queries, at some cost, to learn partial information about which of the possible worlds is the actual world, before choosing an action. We consider two query models: (1) an unobservable-query model, in which players learn only the response to their own queries, and (2) an observable-query model, in which players also learn which queries their opponents made. The results in this paper consider cases in which the underlying worlds of a two-player Socratic game are either constant-sum games or strategically zero-sum games, a class that generalizes constant-sum games to include all games in which the sum of payoffs depends linearly on the interaction between the players. When the underlying worlds are constant sum, we give polynomial-time algorithms to find Nash equilibria in both the observable- and unobservable-query models. When the worlds are strategically zero sum, we give efficient algorithms to find Nash equilibria in unobservablequery Socratic games and correlated equilibria in observablequery Socratic games.

1. INTRODUCTION Late October 1960. A smoky room. Democratic Party strategists huddle around a map. How should the Kennedy campaign allocate its remaining advertising budget? Should it focus on, say, California or New York? The Nixon campaign faces the same dilemma. Of course, neither campaign knows the effectiveness of its advertising in each state. Perhaps Californians are susceptible to Nixon"s advertising, but are unresponsive to Kennedy"s. In light of this uncertainty, the Kennedy campaign may conduct a survey, at some cost, to estimate the effectiveness of its advertising. Moreover, the larger-and more expensive-the survey, the more accurate it will be. Is the cost of a survey worth the information that it provides? How should one balance the cost of acquiring more information against the risk of playing a game with higher uncertainty? In this paper, we model situations of this type as Socratic games. As in traditional game theory, the players in a Socratic game choose actions to maximize their payoffs, but we model players with incomplete information who can make costly queries to reduce their uncertainty about the state of the world before they choose their actions. This approach contrasts with traditional game theory, in which players are usually modeled as having fixed, exogenously given information about the structure of the game and its payoffs. (In traditional games of incomplete and imperfect information, there is information that the players do not have; in Socratic games, unlike in these games, the players have a chance to acquire the missing information, at some cost.) A number of related models have been explored by economists and computer scientists motivated by similar situations, often with a focus on mechanism design and auctions; a sampling of this research includes the work of Larson and Sandholm [41, 42, 43, 44], Parkes [59], Fong [22], Compte and Jehiel [12], Rezende [63], Persico and Matthews [48, 60], Cr´emer and Khalil [15], Rasmusen [62], and Bergemann and V¨alim¨aki [4, 5]. The model of Bergemann and V¨alim¨aki is similar in many regards to the one that we explore here; see Section 7 for some discussion. A Socratic game proceeds as follows. A real world is cho150 sen randomly from a set of possible worlds according to a common prior distribution. Each player then selects an arbitrary query from a set of available costly queries and receives a corresponding piece of information about the real world. Finally each player selects an action and receives a payoff-a function of the players" selected actions and the identity of the real world-less the cost of the query that he or she made. Compared to traditional game theory, the distinguishing feature of our model is the introduction of explicit costs to the players for learning arbitrary partial information about which of the many possible worlds is the real world. Our research was initially inspired by recent results in psychology on decision making, but it soon became clear that Socratic game theory is also a general tool for understanding the exploitation versus exploration tradeoff, well studied in machine learning, in a strategic multiplayer environment. This tension between the risk arising from uncertainty and the cost of acquiring information is ubiquitous in economics, political science, and beyond. Our results. We consider Socratic games under two models: an unobservable-query model where players learn only the response to their own queries and an observable-query model where players also learn which queries their opponents made. We give efficient algorithms to find Nash equilibriai.e., tuples of strategies from which no player has unilateral incentive to deviate-in broad classes of two-player Socratic games in both models. Our first result is an efficient algorithm to find Nash equilibria in unobservable-query Socratic games with constant-sum worlds, in which the sum of the players" payoffs is independent of their actions. Our techniques also yield Nash equilibria in unobservable-query Socratic games with strategically zero-sum worlds. Strategically zero-sum games generalize constant-sum games by allowing the sum of the players" payoffs to depend on individual players" choices of strategy, but not on any interaction of their choices. Our second result is an efficient algorithm to find Nash equilibria in observable-query Socratic games with constant-sum worlds. Finally, we give an efficient algorithm to find correlated equilibria-a weaker but increasingly well-studied solution concept for games [2, 3, 32, 56, 57]-in observable-query Socratic games with strategically zero-sum worlds. Like all games, Socratic games can be viewed as a special case of extensive-form games, which represent games by trees in which internal nodes represent choices made by chance or by the players, and the leaves represent outcomes that correspond to a vector of payoffs to the players. Algorithmically, the generality of extensive-form games makes them difficult to solve efficiently, and the special cases that are known to be efficiently solvable do not include even simple Socratic games. Every (complete-information) classical game is a trivial Socratic game (with a single possible world and a single trivial query), and efficiently finding Nash equilibria in classical games has been shown to be hard [10, 11, 13, 16, 17, 27, 54, 55]. Therefore we would not expect to find a straightforward polynomial-time algorithm to compute Nash equilibria in general Socratic games. However, it is well known that Nash equilibria can be found efficiently via an LP for two-player constant-sum games [49, 71] (and strategically zero-sum games [51]). A Socratic game is itself a classical game, so one might hope that these results can be applied to Socratic games with constant-sum (or strategically zero-sum) worlds. We face two major obstacles in extending these classical results to Socratic games. First, a Socratic game with constant-sum worlds is not itself a constant-sum classical game-rather, the resulting classical game is only strategically zero sum. Worse yet, a Socratic game with strategically zero-sum worlds is not itself classically strategically zero sum-indeed, there are no known efficient algorithmic techniques to compute Nash equilibria in the resulting class of classical games. (Exponential-time algorithms like Lemke/Howson, of course, can be used [45].) Thus even when it is easy to find Nash equilibria in each of the worlds of a Socratic game, we require new techniques to solve the Socratic game itself. Second, even when the Socratic game itself is strategically zero sum, the number of possible strategies available to each player is exponential in the natural representation of the game. As a result, the standard linear programs for computing equilibria have an exponential number of variables and an exponential number of constraints. For unobservable-query Socratic games with strategically zero-sum worlds, we address these obstacles by formulating a new LP that uses only polynomially many variables (though still an exponential number of constraints) and then use ellipsoid-based techniques to solve it. For observablequery Socratic games, we handle the exponentiality by decomposing the game into stages, solving the stages separately, and showing how to reassemble the solutions efficiently. To solve the stages, it is necessary to find Nash equilibria in Bayesian strategically zero-sum games, and we give an explicit polynomial-time algorithm to do so. 2. GAMES AND SOCRATIC GAMES In this section, we review background on game theory and formally introduce Socratic games. We present these models in the context of two-player games, but the multiplayer case is a natural extension. Throughout the paper, boldface variables will be used to denote a pair of variables (e.g., a = ai, aii ). Let Pr[x ← π] denote the probability that a particular value x is drawn from the distribution π, and let Ex∼π[g(x)] denote the expectation of g(x) when x is drawn from π. 2.1 Background on Game Theory Consider two players, Player I and Player II, each of whom is attempting to maximize his or her utility (or payoff). A (two-player) game is a pair A, u , where, for i ∈ {i,ii}, • Ai is the set of pure strategies for Player i, and A = Ai, Aii ; and • ui : A → R is the utility function for Player i, and u = ui, uii . We require that A and u be common knowledge. If each Player i chooses strategy ai ∈ Ai, then the payoffs to Players I and II are ui(a) and uii(a), respectively. A game is constant sum if, for all a ∈ A, we have that ui(a) + uii(a) = c for some fixed c independent of a. Player i can also play a mixed strategy αi ∈ Ai, where Ai denotes the space of probability measures over the set Ai. Payoff functions are generalized as ui (α) = ui (αi, αii) := Ea∼α[ui (a)] = P a∈A α(a)ui (a), where the quantity α(a) = 151 αi(ai) · αii(aii) denotes the joint probability of the independent events that each Player i chooses action ai from the distribution αi. This generalization to mixed strategies is known as von Neumann/Morgenstern utility [70], in which players are indifferent between a guaranteed payoff x and an expected payoff of x. A Nash equilibrium is a pair α of mixed strategies so that neither player has an incentive to change his or her strategy unilaterally. Formally, the strategy pair α is a Nash equilibrium if and only if both ui(αi, αii) = maxαi∈Ai ui(αi, αii) and uii(αi, αii) = maxαii∈Aii uii(αi, αii); that is, the strategies αi and αii are mutual best responses. A correlated equilibrium is a distribution ψ over A that obeys the following: if a ∈ A is drawn randomly according to ψ and Player i learns ai, then no Player i has incentive to deviate unilaterally from playing ai. (A Nash equilibrium is a correlated equilibrium in which ψ(a) = αi(ai) · αii(aii) is a product distribution.) Formally, in a correlated equilibrium, for every a ∈ A we must have that ai is a best response to a randomly chosen ˆaii ∈ Aii drawn according to ψ(ai, ˆaii), and the analogous condition must hold for Player II. 2.2 Socratic Games In this section, we formally define Socratic games. A Socratic game is a 7-tuple A, W, u, S, Q, p, δ , where, for i ∈ {i,ii}: • Ai is, as before, the set of pure strategies for Player i. • W is a set of possible worlds, one of which is the real world wreal. • ui = {uw i : A → R | w ∈ W} is a set of payoff functions for Player i, one for each possible world. • S is a set of signals. • Qi is a set of available queries for Player i. When Player i makes query qi : W → S, he or she receives the signal qi(wreal). When Player i receives signal qi(wreal) in response to query qi, he or she can infer that wreal ∈ {w : qi(w) = qi(wreal)}, i.e., the set of possible worlds from which query qi cannot distinguish wreal. • p : W → [0, 1] is a probability distribution over the possible worlds. • δi : Qi → R≥0 gives the query cost for each available query for Player i. Initially, the world wreal is chosen according to the probability distribution p, but the identity of wreal remains unknown to the players. That is, it is as if the players are playing the game A, uwreal but do not know wreal. The players make queries q ∈ Q, and Player i receives the signal qi(wreal). We consider both observable queries and unobservable queries. When queries are observable, each player learns which query was made by the other player, and the results of his or her own query-that is, each Player i learns qi, qii, and qi(wreal). For unobservable queries, Player i learns only qi and qi(wreal). After learning the results of the queries, the players select strategies a ∈ A and receive as payoffs u wreal i (a) − δi(qi). In the Socratic game, a pure strategy for Player i consists of a query qi ∈ Qi and a response function mapping any result of the query qi to a strategy ai ∈ Ai to play. A player"s state of knowledge after a query is a point in R := Q × S or Ri := Qi × S for observable or unobservable queries, respectively. Thus Player i"s response function maps R or Ri to Ai. Note that the number of pure strategies is exponential, as there are exponentially many response functions. A mixed strategy involves both randomly choosing a query qi ∈ Qi and randomly choosing an action ai ∈ Ai in response to the results of the query. Formally, we will consider a mixed-strategy-function profile f = fquery , fresp to have two parts: • a function fquery i : Qi → [0, 1], where fquery i (qi) is the probability that Player i makes query qi. • a function fresp i that maps R or Ri to a probability distribution over actions. Player i chooses an action ai ∈ Ai according to the probability distribution fresp i (q, qi(w)) for observable queries, and according to fresp i (qi, qi(w)) for unobservable queries. (With unobservable queries, for example, the probability that Player I plays action ai conditioned on making query qi in world w is given by Pr[ai ← fresp i (qi, qi(w))].) Mixed strategies are typically defined as probability distributions over the pure strategies, but here we represent a mixed strategy by a pair fquery , fresp , which is commonly referred to as a behavioral strategy in the game-theory literature. As in any game with perfect recall, one can easily map a mixture of pure strategies to a behavioral strategy f = fquery , fresp that induces the same probability of making a particular query qi or playing a particular action after making a query qi in a particular world. Thus it suffices to consider only this representation of mixed strategies. For a strategy-function profile f for observable queries, the (expected) payoff to Player i is given by X q∈Q,w∈W,a∈A 2 6 6 4 fquery i (qi) · fquery ii (qii) · p(w) · Pr[ai ← fresp i (q, qi(w))] · Pr[aii ← fresp ii (q, qii(w))] · (uw i (a) − δi(qi)) 3 7 7 5 . The payoffs for unobservable queries are analogous, with fresp j (qj, qj(w)) in place of fresp j (q, qj(w)). 3. STRATEGICALLY ZERO-SUM GAMES We can view a Socratic game G with constant-sum worlds as an exponentially large classical game, with pure strategies make query qi and respond according to fi. However, this classical game is not constant sum. The sum of the players" payoffs varies depending upon their strategies, because different queries incur different costs. However, this game still has significant structure: the sum of payoffs varies only because of varying query costs. Thus the sum of payoffs does depend on players" choice of strategies, but not on the interaction of their choices-i.e., for fixed functions gi and gii, we have ui(q, f) + uii(q, f) = gi(qi, fi) + gii(qii, fii) for all strategies q, f . Such games are called strategically zero sum and were introduced by Moulin and Vial [51], who describe a notion of strategic equivalence and define strategically zero-sum games as those strategically equivalent to zero-sum games. It is interesting to note that two Socratic games with the same queries and strategically equivalent worlds are not necessarily strategically equivalent. A game A, u is strategically zero sum if there exist labels (i, ai) for every Player i and every pure strategy ai ∈ Ai 152 such that, for all mixed-strategy profiles α, we have that the sum of the utilities satisfies ui(α)+uii(α) = X ai∈Ai αi(ai)· (i, ai)+ X aii∈Aii αii(aii)· (ii, aii). Note that any constant-sum game is strategically zero sum as well. It is not immediately obvious that one can efficiently decide if a given game is strategically zero sum. For completeness, we give a characterization of classical strategically zero-sum games in terms of the rank of a simple matrix derived from the game"s payoffs, allowing us to efficiently decide if a given game is strategically zero sum and, if it is, to compute the labels (i, ai). Theorem 3.1. Consider a game G = A, u with Ai = {a1 i , . . . , ani i }. Let MG be the ni-by-nii matrix whose i, j th entry MG (i,j) satisfies log2 MG (i,j) = ui(ai i , aj ii) + uii(ai i , aj ii). Then the following are equivalent: (i) G is strategically zero sum; (ii) there exist labels (i, ai) for every player i ∈ {i,ii} and every pure strategy ai ∈ Ai such that, for all pure strategies a ∈ A, we have ui(a) + uii(a) = (i, ai) + (ii, aii); and (iii) rank(MG ) = 1. Proof Sketch. (i ⇒ ii) is immediate; every pure strategy is a trivially mixed strategy. For (ii ⇒ iii), let ci be the n-element column vector with jth component 2 (i,a j i ) ; then ci · cii T = MG . For (iii ⇒ i), if rank(MG ) = 1, then MG = u · vT . We can prove that G is strategically zero sum by choosing labels (i, aj i ) := log2 uj and (ii, aj ii) := log2 vj. 4. SOCRATIC GAMES WITH UNOBSERVABLE QUERIES We begin with Socratic games with unobservable queries, where a player"s choice of query is not revealed to her opponent. We give an efficient algorithm to solve unobservablequery Socratic games with strategically zero-sum worlds. Our algorithm is based upon the LP shown in Figure 1, whose feasible points are Nash equilibria for the game. The LP has polynomially many variables but exponentially many constraints. We give an efficient separation oracle for the LP, implying that the ellipsoid method [28, 38] yields an efficient algorithm. This approach extends the techniques of Koller and Megiddo [39] (see also [40]) to solve constant-sum games represented in extensive form. (Recall that their result does not directly apply in our case; even a Socratic game with constant-sum worlds is not a constant-sum classical game.) Lemma 4.1. Let G = A, W, u, S, Q, p, δ be an arbitrary unobservable-query Socratic game with strategically zero-sum worlds. Any feasible point for the LP in Figure 1 can be efficiently mapped to a Nash equilibrium for G, and any Nash equilibrium for G can be mapped to a feasible point for the program. Proof Sketch. We begin with a description of the correspondence between feasible points for the LP and Nash equilibria for G. First, suppose that strategy profile f = fquery , fresp forms a Nash equilibrium for G. Then the following setting for the LP variables is feasible: yi qi = fquery i (qi) xi ai,qi,w = Pr[ai ← fresp i (qi, qi(w))] · yi qi ρi = P w,q∈Q,a∈A p(w) · xi ai,qi,w · xii aii,qii,w · [uw i (a) − δi(qi)]. (We omit the straightforward calculations that verify feasibility.) Next, suppose xi ai,qi,w, yi qi , ρi is feasible for the LP. Let f be the strategy-function profile defined as fquery i : qi → yi qi fresp i (qi, qi(w)) : ai → xi ai,qi,w/yi qi . Verifying that this strategy profile is a Nash equilibrium requires checking that fresp i (qi, qi(w)) is a well-defined function (from constraint VI), that fquery i and fresp i (qi, qi(w)) are probability distributions (from constraints III and IV), and that each player is playing a best response to his or her opponent"s strategy (from constraints I and II). Finally, from constraints I and II, the expected payoff to Player i is at most ρi. Because the right-hand side of constraint VII is equal to the expected sum of the payoffs from f and is at most ρi + ρii, the payoffs are correct and imply the lemma. We now give an efficient separation oracle for the LP in Figure 1, thus allowing the ellipsoid method to solve the LP in polynomial time. Recall that a separation oracle is a function that, given a setting for the variables in the LP, either returns feasible or returns a particular constraint of the LP that is violated by that setting of the variables. An efficient, correct separation oracle allows us to solve the LP efficiently via the ellipsoid method. Lemma 4.2. There exists a separation oracle for the LP in Figure 1 that is correct and runs in polynomial time. Proof. Here is a description of the separation oracle SP. On input xi ai,qi,w, yi qi , ρi : 1. Check each of the constraints (III), (IV), (V), (VI), and (VII). If any one of these constraints is violated, then return it. 2. Define the strategy profile f as follows: fquery i : qi → yi qi fresp i (qi, qi(w)) : ai → xi ai,qi,w/yi qi For each query qi, we will compute a pure best-response function ˆf qi i for Player I to strategy fii after making query qi. More specifically, given fii and the result qi(wreal) of the query qi, it is straightforward to compute the probability that, conditioned on the fact that the result of query qi is qi(w), the world is w and Player II will play action aii ∈ Aii. Therefore, for each query qi and response qi(w), Player I can compute the expected utility of each pure response ai to the induced mixed strategy over Aii for Player II. Player I can then select the ai maximizing this expected payoff. Let ˆfi be the response function such that ˆfi(qi, qi(w)) = ˆf qi i (qi(w)) for every qi ∈ Qi. Similarly, compute ˆfii. 153 Player i does not prefer ‘make query qi, then play according to the function fi" : ∀qi ∈ Qi, fi : Ri → Ai : ρi ≥ P w∈W,aii∈Aii,qii∈Qii,ai=fi(qi,qi(w)) ` p(w) · xii aii,qii,w · [uw i (a) − δi(qi)] ´ (I) ∀qii ∈ Qii, fii : Rii → Aii : ρii ≥ P w∈W,ai∈Ai,qi∈Qi,aii=fii(qii,qii(w)) ` p(w) · xi ai,qi,w · [uw ii (a) − δii(qii)] ´ (II) Every player"s choices form a probability distribution in every world: ∀i ∈ {i,ii}, w ∈ W : 1 = P ai∈Ai,qi∈Qi xi ai,qi,w (III) ∀i ∈ {i,ii}, w ∈ W : 0 ≤ xi ai,qi,w (IV) Queries are independent of the world, and actions depend only on query output: ∀i ∈ {i,ii}, qi ∈ Qi, w ∈ W, w ∈ W such that qi(w) = qi(w ) : yi qi = P ai∈Ai xi ai,qi,w (V) xi ai,qi,w = xi ai,qi,w (VI) The payoffs are consistent with the labels (i, ai, w): ρi + ρii = P i∈{i,ii} P w∈W,qi∈Qi,ai∈Ai ` p(w) · xi ai,qi,w · [ (i, ai, w) − δi(qi)] ´ (VII) Figure 1: An LP to find Nash equilibria in unobservable-query Socratic games with strategically zero-sum worlds. The input is a Socratic game A, W, u, S, Q, p, δ so that world w is strategically zero sum with labels (i, ai, w). Player i makes query qi ∈ Qi with probability yi qi and, when the actual world is w ∈ W, makes query qi and plays action ai with probability xi ai,qi,w. The expected payoff to Player i is given by ρi. 3. Let ˆρ qi i be the expected payoff to Player I using the strategy make query qi and play response function ˆfi if Player II plays according to fii. Let ˆρi = maxqi∈Qq ˆρ qi i and let ˆqi = arg maxqi∈Qq ˆρ qi i . Similarly, define ˆρ qii ii , ˆρii, and ˆqii. 4. For the ˆfi and ˆqi defined in Step 3, return constraint (I-ˆqi- ˆfi) or (II-ˆqii- ˆfii) if either is violated. If both are satisfied, then return feasible. We first note that the separation oracle runs in polynomial time and then prove its correctness. Steps 1 and 4 are clearly polynomial. For Step 2, we have described how to compute the relevant response functions by examining every action of Player I, every world, every query, and every action of Player II. There are only polynomially many queries, worlds, query results, and pure actions, so the running time of Steps 2 and 3 is thus polynomial. We now sketch the proof that the separation oracle works correctly. The main challenge is to show that if any constraint (I-qi-fi ) is violated then (I-ˆqi- ˆfi) is violated in Step 4. First, we observe that, by construction, the function ˆfi computed in Step 3 must be a best response to Player II playing fii, no matter what query Player I makes. Therefore the strategy make query ˆqi, then play response function ˆfi must be a best response to Player II playing fii, by definition of ˆqi. The right-hand side of each constraint (I-qi-fi ) is equal to the expected payoff that Player I receives when playing the pure strategy make query qi and then play response function fi against Player II"s strategy of fii. Therefore, because the pure strategy make query ˆqi and then play response function ˆfi is a best response to Player II playing fii, the right-hand side of constraint (I-ˆqi- ˆfi) is at least as large as the right hand side of any constraint (I-ˆqi-fi ). Therefore, if any constraint (I-qi-fi ) is violated, constraint (I-ˆqi- ˆfi) is also violated. An analogous argument holds for Player II. These lemmas and the well-known fact that Nash equilibria always exist [52] imply the following theorem: Theorem 4.3. Nash equilibria can be found in polynomial time for any two-player unobservable-query Socratic game with strategically zero-sum worlds. 5. SOCRATIC GAMES WITH OBSERVABLE QUERIES In this section, we give efficient algorithms to find (1) a Nash equilibrium for observable-query Socratic games with constant-sum worlds and (2) a correlated equilibrium in the broader class of Socratic games with strategically zero-sum worlds. Recall that a Socratic game G = A, W, u, S, Q, p, δ with observable queries proceeds in two stages: Stage 1: The players simultaneously choose queries q ∈ Q. Player i receives as output qi, qii, and qi(wreal). Stage 2: The players simultaneously choose strategies a ∈ A. The payoff to Player i is u wreal i (a) − δi(qi). Using backward induction, we first solve Stage 2 and then proceed to the Stage-1 game. For a query q ∈ Q, we would like to analyze the Stage-2 game ˆGq resulting from the players making queries q in Stage 1. Technically, however, ˆGq is not actually a game, because at the beginning of Stage 2 the players have different information about the world: Player I knows qi(wreal), and 154 Player II knows qii(wreal). Fortunately, the situation in which players have asymmetric private knowledge has been well studied in the game-theory literature. A Bayesian game is a quadruple A, T, r, u , where: • Ai is the set of pure strategies for Player i. • Ti is the set of types for Player i. • r is a probability distribution over T; r(t) denotes the probability that Player i has type ti for all i. • ui : A × T → R is the payoff function for Player i. If the players have types t and play pure strategies a, then ui(a, t) denotes the payoff for Player i. Initially, a type t is drawn randomly from T according to the distribution r. Player i learns his type ti, but does not learn any other player"s type. Player i then plays a mixed strategy αi ∈ Ai-that is, a probability distribution over Ai-and receives payoff ui(α, t). A strategy function is a function hi : Ti → Ai; Player i plays the mixed strategy hi(ti) ∈ Ai when her type is ti. A strategy-function profile h is a Bayesian Nash equilibrium if and only if no Player i has unilateral incentive to deviate from hi if the other players play according to h. For a two-player Bayesian game, if α = h(t), then the profile h is a Bayesian Nash equilibrium exactly when the following condition and its analogue for Player II hold: Et∼r[ui(α, t)] = maxhi Et∼r[ui( hi(ti), αii , t)]. These conditions hold if and only if, for all ti ∈ Ti occurring with positive probability, Player i"s expected utility conditioned on his type being ti is maximized by hi(ti). A Bayesian game is constant sum if for all a ∈ A and all t ∈ T, we have ui(a, t) + uii(a, t) = ct, for some constant ct independent of a. A Bayesian game is strategically zero sum if the classical game A, u(·, t) is strategically zero sum for every t ∈ T. Whether a Bayesian game is strategically zero sum can be determined as in Theorem 3.1. (For further discussion of Bayesian games, see [25, 31].) We now formally define the Stage-2 game as a Bayesian game. Given a Socratic game G = A, W, u, S, Q, p, δ and a query profile q ∈ Q, we define the Stage-2 Bayesian game Gstage2(q) := A, Tq , pstage2(q) , ustage2(q) , where: • Ai, the set of pure strategies for Player i, is the same as in the original Socratic game; • Tq i = {qi(w) : w ∈ W}, the set of types for Player i, is the set of signals that can result from query qi; • pstage2(q) (t) = Pr[q(w) = t | w ← p]; and • u stage2(q) i (a, t) = P w∈W Pr[w ← p | q(w) = t] · uw i (a). We now define the Stage-1 game in terms of the payoffs for the Stage-2 games. Fix any algorithm alg that finds a Bayesian Nash equilibrium hq,alg := alg(Gstage2(q)) for each Stage-2 game. Define valuealg i (Gstage2(q)) to be the expected payoff received by Player i in the Bayesian game Gstage2(q) if each player plays according to hq,alg , that is, valuealg i (Gstage2(q)) := P w∈W p(w) · u stage2(q) i (hq,alg (q(w)), q(w)). Define the game Galg stage1 := Astage1 , ustage1(alg) , where: • Astage1 := Q, the set of available queries in the Socratic game; and • u stage1(alg) i (q) := valuealg i (Gstage2(q)) − δi(qi). I.e., players choose queries q and receive payoffs corresponding to valuealg (Gstage2(q)), less query costs. Lemma 5.1. Consider an observable-query Socratic game G = A, W, u, S, Q, p, δ . Let Gstage2(q) be the Stage-2 games for all q ∈ Q, let alg be an algorithm finding a Bayesian Nash equilibrium in each Gstage2(q), and let Galg stage1 be the Stage-1 game. Let α be a Nash equilibrium for Galg stage1, and let hq,alg := alg(Gstage2(q)) be a Bayesian Nash equilibrium for each Gstage2(q). Then the following strategy profile is a Nash equilibrium for G: • In Stage 1, Player i makes query qi with probability αi(qi). (That is, set fquery (q) := α(q).) • In Stage 2, if q is the query in Stage 1 and qi(wreal) denotes the response to Player i"s query, then Player i chooses action ai with probability hq,alg i (qi(wreal)). (In other words, set fresp i (q, qi(w)) := hq,alg i (qi(w)).) We now find equilibria in the stage games for Socratic games with constant- or strategically zero-sum worlds. We first show that the stage games are well structured in this setting: Lemma 5.2. Consider an observable-query Socratic game G = A, W, u, S, Q, p, δ with constant-sum worlds. Then the Stage-1 game Galg stage1 is strategically zero sum for every algorithm alg, and every Stage-2 game Gstage2(q) is Bayesian constant sum. If the worlds of G are strategically zero sum, then every Gstage2(q) is Bayesian strategically zero sum. We now show that we can efficiently compute equilibria for these well-structured stage games. Theorem 5.3. There exists a polynomial-time algorithm BNE finding Bayesian Nash equilibria in strategically zerosum Bayesian (and thus classical strategically zero-sum or Bayesian constant-sum) two-player games. Proof Sketch. Let G = A, T, r, u be a strategically zero-sum Bayesian game. Define an unobservable-query Socratic game G∗ with one possible world for each t ∈ T, one available zero-cost query qi for each Player i so that qi reveals ti, and all else as in G. Bayesian Nash equilibria in G correspond directly to Nash equilibria in G∗ , and the worlds of G∗ are strategically zero sum. Thus by Theorem 4.3 we can compute Nash equilibria for G∗ , and thus we can compute Bayesian Nash equilibria for G. (LP"s for zero-sum two-player Bayesian games have been previously developed and studied [61].) Theorem 5.4. We can compute a Nash equilibrium for an arbitrary two-player observable-query Socratic game G = A, W, u, S, Q, p, δ with constant-sum worlds in polynomial time. Proof. Because each world of G is constant sum, Lemma 5.2 implies that the induced Stage-2 games Gstage2(q) are all Bayesian constant sum. Thus we can use algorithm BNE to compute a Bayesian Nash equilibrium hq,BNE := BNE(Gstage2(q)) for each q ∈ Q, by Theorem 5.3. Furthermore, again by Lemma 5.2, the induced Stage-1 game GBNE stage1 is classical strategically zero sum. Therefore we can again use algorithm BNE to compute a Nash equilibrium α := BNE(GBNE stage1), again by Theorem 5.3. Therefore, by Lemma 5.1, we can assemble α and the hq,BNE "s into a Nash equilibrium for the Socratic game G. 155 We would like to extend our results on observable-query Socratic games to Socratic games with strategically zerosum worlds. While we can still find Nash equilibria in the Stage-2 games, the resulting Stage-1 game is not in general strategically zero sum. Thus, finding Nash equilibria in observable-query Socratic games with strategically zerosum worlds seems to require substantially new techniques. However, our techniques for decomposing observable-query Socratic games do allow us to find correlated equilibria in this case. Lemma 5.5. Consider an observable-query Socratic game G = A, W, u, S, Q, p, δ . Let alg be an arbitrary algorithm that finds a Bayesian Nash equilibrium in each of the derived Stage-2 games Gstage2(q), and let Galg stage1 be the derived Stage1 game. Let φ be a correlated equilibrium for Galg stage1, and let hq,alg := alg(Gstage2(q)) be a Bayesian Nash equilibrium for each Gstage2(q). Then the following distribution over pure strategies is a correlated equilibrium for G: ψ(q, f) := φ(q) Y i∈{i,ii} Y s∈S Pr h fi(q, s) ← hq,alg i (s) i . Thus to find a correlated equilibrium in an observable-query Socratic game with strategically zero-sum worlds, we need only algorithm BNE from Theorem 5.3 along with an efficient algorithm for finding a correlated equilibrium in a general game. Such an algorithm exists (the definition of correlated equilibria can be directly translated into an LP [3]), and therefore we have the following theorem: Theorem 5.6. We can provide both efficient oracle access and efficient sampling access to a correlated equilibrium for any observable-query two-player Socratic game with strategically zero-sum worlds. Because the support of the correlated equilibrium may be exponentially large, providing oracle and sampling access is the natural way to represent the correlated equilibrium. By Lemma 5.5, we can also compute correlated equilibria in any observable-query Socratic game for which Nash equilibria are computable in the induced Gstage2(q) games (e.g., when Gstage2(q) is of constant size). Another potentially interesting model of queries in Socratic games is what one might call public queries, in which both the choice and outcome of a player"s query is observable by all players in the game. (This model might be most appropriate in the presence of corporate espionage or media leaks, or in a setting in which the queries-and thus their results-are done in plain view.) The techniques that we have developed in this section also yield exactly the same results as for observable queries. The proof is actually simpler: with public queries, the players" payoffs are common knowledge when Stage 2 begins, and thus Stage 2 really is a complete-information game. (There may still be uncertainty about the real world, but all players use the observed signals to infer exactly the same set of possible worlds in which wreal may lie; thus they are playing a complete-information game against each other.) Thus we have the same results as in Theorems 5.4 and 5.6 more simply, by solving Stage 2 using a (non-Bayesian) Nash-equilibrium finder and solving Stage 1 as before. Our results for observable queries are weaker than for unobservable: in Socratic games with worlds that are strategically zero sum but not constant sum, we find only a correlated equilibrium in the observable case, whereas we find a Nash equilibrium in the unobservable case. We might hope to extend our unobservable-query techniques to observable queries, but there is no obvious way to do so. The fundamental obstacle is that the LP"s payoff constraint becomes nonlinear if there is any dependence on the probability that the other player made a particular query. This dependence arises with observable queries, suggesting that observable Socratic games with strategically zero-sum worlds may be harder to solve. 6. RELATED WORK Our work was initially motivated by research in the social sciences indicating that real people seem (irrationally) paralyzed when they are presented with additional options. In this section, we briefly review some of these social-science experiments and then discuss technical approaches related to Socratic game theory. Prima facie, a rational agent"s happiness given an added option can only increase. However, recent research has found that more choices tend to decrease happiness: for example, students choosing among extra-credit options are more likely to do extra credit if given a small subset of the choices and, moreover, produce higher-quality work [35]. (See also [19].) The psychology literature explores a number of explanations: people may miscalculate their opportunity cost by comparing their choice to a component-wise maximum of all other options instead of the single best alternative [65], a new option may draw undue attention to aspects of the other options [67], and so on. The present work explores an economic explanation of this phenomenon: information is not free. When there are more options, a decision-maker must spend more time to achieve a satisfactory outcome. See, e.g., the work of Skyrms [68] for a philosophical perspective on the role of deliberation in strategic situations. Finally, we note the connection between Socratic games and modal logic [34], a formalism for the logic of possibility and necessity. The observation that human players typically do not play rational strategies has inspired some attempts to model partially rational players. The typical model of this socalled bounded rationality [36, 64, 66] is to postulate bounds on computational power in computing the consequences of a strategy. The work on bounded rationality [23, 24, 53, 58] differs from the models that we consider here in that instead of putting hard limitations on the computational power of the agents, we instead restrict their a priori knowledge of the state of the world, requiring them to spend time (and therefore money/utility) to learn about it. Partially observable stochastic games (POSGs) are a general framework used in AI to model situations of multi-agent planning in an evolving, unknown environment, but the generality of POSGs seems to make them very difficult [6]. Recent work has been done in developing algorithms for restricted classes of POSGs, most notably classes of cooperative POSGs-e.g., [20, 30]-which are very different from the competitive strategically zero-sum games we address in this paper. The fundamental question in Socratic games is deciding on the comparative value of making a more costly but more informative query, or concluding the data-gathering phase and picking the best option, given current information. This tradeoff has been explored in a variety of other contexts; a sampling of these contexts includes aggregating results 156 from delay-prone information sources [8], doing approximate reasoning in intelligent systems [72], deciding when to take the current best guess of disease diagnosis from a beliefpropagation network and when to let it continue inference [33], among many others. This issue can also be viewed as another perspective on the general question of exploration versus exploitation that arises often in AI: when is it better to actively seek additional information instead of exploiting the knowledge one already has? (See, e.g., [69].) Most of this work differs significantly from our own in that it considers single-agent planning as opposed to the game-theoretic setting. A notable exception is the work of Larson and Sandholm [41, 42, 43, 44] on mechanism design for interacting agents whose computation is costly and limited. They present a model in which players must solve a computationally intractable valuation problem, using costly computation to learn some hidden parameters, and results for auctions and bargaining games in this model. 7. FUTURE DIRECTIONS Efficiently finding Nash equilibria in Socratic games with non-strategically zero-sum worlds is probably difficult because the existence of such an algorithm for classical games has been shown to be unlikely [10, 11, 13, 16, 17, 27, 54, 55]. There has, however, been some algorithmic success in finding Nash equilibria in restricted classical settings (e.g., [21, 46, 47, 57]); we might hope to extend our results to analogous Socratic games. An efficient algorithm to find correlated equilibria in general Socratic games seems more attainable. Suppose the players receive recommended queries and responses. The difficulty is that when a player considers a deviation from his recommended query, he already knows his recommended response in each of the Stage-2 games. In a correlated equilibrium, a player"s expected payoff generally depends on his recommended strategy, and thus a player may deviate in Stage 1 so as to land in a Stage-2 game where he has been given a better than average recommended response. (Socratic games are succinct games of superpolynomial type, so Papadimitriou"s results [56] do not imply correlated equilibria for them.) Socratic games can be extended to allow players to make adaptive queries, choosing subsequent queries based on previous results. Our techniques carry over to O(1) rounds of unobservable queries, but it would be interesting to compute equilibria in Socratic games with adaptive observable queries or with ω(1) rounds of unobservable queries. Special cases of adaptive Socratic games are closely related to single-agent problems like minimum latency [1, 7, 26], determining strategies for using priced information [9, 29, 37], and an online version of minimum test cover [18, 50]. Although there are important technical distinctions between adaptive Socratic games and these problems, approximation techniques from this literature may apply to Socratic games. The question of approximation raises interesting questions even in non-adaptive Socratic games. An -approximate Nash equilibrium is a strategy profile α so that no player can increase her payoff by an additive by deviating from α. Finding approximate Nash equilibria in both adaptive and non-adaptive Socratic games is an interesting direction to pursue. Another natural extension is the model where query results are stochastic. In this paper, we model a query as deterministically partitioning the possible worlds into subsets that the query cannot distinguish. However, one could instead model a query as probabilistically mapping the set of possible worlds into the set of signals. With this modification, our unobservable-query model becomes equivalent to the model of Bergemann and V¨alim¨aki [4, 5], in which the result of a query is a posterior distribution over the worlds. Our techniques allow us to compute equilibria in such a stochastic-query model provided that each query is represented as a table that, for each world/signal pair, lists the probability that the query outputs that signal in that world. It is also interesting to consider settings in which the game"s queries are specified by a compact representation of the relevant probability distributions. (For example, one might consider a setting in which the algorithm has only a sampling oracle for the posterior distributions envisioned by Bergemann and V¨alim¨aki.) Efficiently finding equilibria in such settings remains an open problem. Another interesting setting for Socratic games is when the set Q of available queries is given by Q = P(Γ)-i.e., each player chooses to make a set q ∈ P(Γ) of queries from a specified groundset Γ of queries. Here we take the query cost to be a linear function, so that δ(q) = P γ∈q δ({γ}). Natural groundsets include comparison queries (if my opponent is playing strategy aii, would I prefer to play ai or ˆai?), strategy queries (what is my vector of payoffs if I play strategy ai?), and world-identity queries (is the world w ∈ W the real world?). When one can infer a polynomial bound on the number of queries made by a rational player, then our results yield efficient solutions. (For example, we can efficiently solve games in which every groundset element γ ∈ Γ has δ({γ}) = Ω(M − M), where M and M denote the maximum and minimum payoffs to any player in any world.) Conversely, it is NP-hard to compute a Nash equilibrium for such a game when every δ({γ}) ≤ 1/|W|2 , even when the worlds are constant sum and Player II has only a single available strategy. Thus even computing a best response for Player I is hard. (This proof proceeds by reduction from set cover; intuitively, for sufficiently low query costs, Player I must fully identify the actual world through his queries. Selecting a minimum-sized set of these queries is hard.) Computing Player I"s best response can be viewed as maximizing a submodular function, and thus a best response can be (1 − 1/e) ≈ 0.63 approximated greedily [14]. An interesting open question is whether this approximate best-response calculation can be leveraged to find an approximate Nash equilibrium. 8. ACKNOWLEDGEMENTS Part of this work was done while all authors were at MIT CSAIL. We thank Erik Demaine, Natalia Hernandez Gardiol, Claire Monteleoni, Jason Rennie, Madhu Sudan, and Katherine White for helpful comments and discussions. 9. REFERENCES [1] Aaron Archer and David P. Williamson. Faster approximation algorithms for the minimum latency problem. In Proceedings of the Symposium on Discrete Algorithms, pages 88-96, 2003. [2] R. J. Aumann. Subjectivity and correlation in randomized strategies. J. Mathematical Economics, 1:67-96, 1974. 157 [3] Robert J. Aumann. Correlated equilibrium as an expression of Bayesian rationality. Econometrica, 55(1):1-18, January 1987. [4] Dick Bergemann and Juuso V¨alim¨aki. Information acquisition and efficient mechanism design. Econometrica, 70(3):1007-1033, May 2002. [5] Dick Bergemann and Juuso V¨alim¨aki. Information in mechanism design. Technical Report 1532, Cowles Foundation for Research in Economics, 2005. [6] Daniel S. Bernstein, Shlomo Zilberstein, and Neil Immerman. The complexity of decentralized control of Markov Decision Processes. Mathematics of Operations Research, pages 819-840, 2002. [7] Avrim Blum, Prasad Chalasani, Don Coppersmith, Bill Pulleyblank, Prabhakar Raghavan, and Madhu Sudan. The minimum latency problem. In Proceedings of the Symposium on the Theory of Computing, pages 163-171, 1994. [8] Andrei Z. Broder and Michael Mitzenmacher. Optimal plans for aggregation. In Proceedings of the Principles of Distributed Computing, pages 144-152, 2002. [9] Moses Charikar, Ronald Fagin, Venkatesan Guruswami, Jon Kleinberg, Prabhakar Raghavan, and Amit Sahai. Query strategies for priced information. J. Computer and System Sciences, 64(4):785-819, June 2002. [10] Xi Chen and Xiaotie Deng. 3-NASH is PPAD-complete. In Electronic Colloquium on Computational Complexity, 2005. [11] Xi Chen and Xiaotie Deng. Settling the complexity of 2-player Nash-equilibrium. 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[17] Konstantinos Daskalakis and Christos H. Papadimitriou. Three-player games are hard. In Electronic Colloquium on Computational Complexity, 2005. [18] K. M. J. De Bontridder, B. V. Halld´orsson, M. M. Halld´orsson, C. A. J. Hurkens, J. K. Lenstra, R. Ravi, and L. Stougie. Approximation algorithms for the test cover problem. Mathematical Programming, 98(1-3):477-491, September 2003. [19] Ap Dijksterhuis, Maarten W. Bos, Loran F. Nordgren, and Rick B. van Baaren. On making the right choice: The deliberation-without-attention effect. Science, 311:1005-1007, 17 February 2006. [20] Rosemary Emery-Montemerlo, Geoff Gordon, Jeff Schneider, and Sebastian Thrun. Approximate solutions for partially observable stochastic games with common payoffs. In Autonomous Agents and Multi-Agent Systems, 2004. [21] Alex Fabrikant, Christos Papadimitriou, and Kunal Talwar. The complexity of pure Nash equilibria. In Proceedings of the Symposium on the Theory of Computing, 2004. [22] Kyna Fong. Multi-stage Information Acquisition in Auction Design. Senior thesis, Harvard College, 2003. [23] Lance Fortnow and Duke Whang. Optimality and domination in repeated games with bounded players. In Proceedings of the Symposium on the Theory of Computing, pages 741-749, 1994. [24] Yoav Freund, Michael Kearns, Yishay Mansour, Dana Ron, Ronitt Rubinfeld, and Robert E. Schapire. Efficient algorithms for learning to play repeated games against computationally bounded adversaries. In Proceedings of the Foundations of Computer Science, pages 332-341, 1995. [25] Drew Fudenberg and Jean Tirole. Game Theory. MIT, 1991. [26] Michel X. Goemans and Jon Kleinberg. An improved approximation ratio for the minimum latency problem. Mathematical Programming, 82:111-124, 1998. [27] Paul W. Goldberg and Christos H. Papadimitriou. Reducibility among equilibrium problems. In Electronic Colloquium on Computational Complexity, 2005. [28] M. Grotschel, L. Lovasz, and A. Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1:70-89, 1981. [29] Anupam Gupta and Amit Kumar. Sorting and selection with structured costs. In Proceedings of the Foundations of Computer Science, pages 416-425, 2001. [30] Eric A. Hansen, Daniel S. Bernstein, and Shlomo Zilberstein. Dynamic programming for partially observable stochastic games. In National Conference on Artificial Intelligence (AAAI), 2004. [31] John C. Harsanyi. Games with incomplete information played by Bayesian players. Management Science, 14(3,5,7), 1967-1968. [32] Sergiu Hart and David Schmeidler. Existence of correlated equilibria. Mathematics of Operations Research, 14(1):18-25, 1989. [33] Eric Horvitz and Geoffrey Rutledge. Time-dependent utility and action under uncertainty. In Uncertainty in Artificial Intelligence, pages 151-158, 1991. [34] G. E. Hughes and M. J. Cresswell. A New Introduction to Modal Logic. Routledge, 1996. [35] Sheena S. Iyengar and Mark R. Lepper. When choice is demotivating: Can one desire too much of a good thing? J. Personality and Social Psychology, 79(6):995-1006, 2000. [36] Ehud Kalai. Bounded rationality and strategic complexity in repeated games. Game Theory and Applications, pages 131-157, 1990. 158 [37] Sampath Kannan and Sanjeev Khanna. Selection with monotone comparison costs. In Proceedings of the Symposium on Discrete Algorithms, pages 10-17, 2003. [38] L.G. Khachiyan. A polynomial algorithm in linear programming. Dokklady Akademiia Nauk SSSR, 244, 1979. [39] Daphne Koller and Nimrod Megiddo. The complexity of two-person zero-sum games in extensive form. Games and Economic Behavior, 4:528-552, 1992. [40] Daphne Koller, Nimrod Megiddo, and Bernhard von Stengel. Efficient computation of equilibria for extensive two-person games. Games and Economic Behavior, 14:247-259, 1996. [41] Kate Larson. Mechanism Design for Computationally Limited Agents. PhD thesis, CMU, 2004. [42] Kate Larson and Tuomas Sandholm. Bargaining with limited computation: Deliberation equilibrium. Artificial Intelligence, 132(2):183-217, 2001. [43] Kate Larson and Tuomas Sandholm. Costly valuation computation in auctions. In Proceedings of the Theoretical Aspects of Rationality and Knowledge, July 2001. [44] Kate Larson and Tuomas Sandholm. Strategic deliberation and truthful revelation: An impossibility result. In Proceedings of the ACM Conference on Electronic Commerce, May 2004. [45] C. E. Lemke and J. T. Howson, Jr. Equilibrium points of bimatrix games. J. Society for Industrial and Applied Mathematics, 12, 1964. [46] Richard J. Lipton, Evangelos Markakis, and Aranyak Mehta. Playing large games using simple strategies. In Proceedings of the ACM Conference on Electronic Commerce, pages 36-41, 2003. [47] Michael L. Littman, Michael Kearns, and Satinder Singh. An efficient exact algorithm for singly connected graphical games. In Proceedings of Neural Information Processing Systems, 2001. [48] Steven A. Matthews and Nicola Persico. Information acquisition and the excess refund puzzle. Technical Report 05-015, Department of Economics, University of Pennsylvania, March 2005. [49] Richard D. McKelvey and Andrew McLennan. Computation of equilibria in finite games. In H. Amman, D. A. Kendrick, and J. Rust, editors, Handbook of Compututational Economics, volume 1, pages 87-142. Elsevier, 1996. [50] B.M.E. Moret and H. D. Shapiro. On minimizing a set of tests. SIAM J. Scientific Statistical Computing, 6:983-1003, 1985. [51] H. Moulin and J.-P. Vial. Strategically zero-sum games: The class of games whose completely mixed equilibria cannot be improved upon. International J. Game Theory, 7(3/4), 1978. [52] John F. Nash, Jr. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, 36:48-49, 1950. [53] Abraham Neyman. Finitely repeated games with finite automata. Mathematics of Operations Research, 23(3):513-552, August 1998. [54] Christos Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. J. Computer and System Sciences, 48:498-532, 1994. [55] Christos Papadimitriou. Algorithms, games, and the internet. In Proceedings of the Symposium on the Theory of Computing, pages 749-753, 2001. [56] Christos H. Papadimitriou. Computing correlated equilibria in multi-player games. In Proceedings of the Symposium on the Theory of Computing, 2005. [57] Christos H. Papadimitriou and Tim Roughgarden. Computing equilibria in multiplayer games. In Proceedings of the Symposium on Discrete Algorithms, 2005. [58] Christos H. Papadimitriou and Mihalis Yannakakis. On bounded rationality and computational complexity. In Proceedings of the Symposium on the Theory of Computing, pages 726-733, 1994. [59] David C. Parkes. Auction design with costly preference elicitation. Annals of Mathematics and Artificial Intelligence, 44:269-302, 2005. [60] Nicola Persico. Information acquisition in auctions. Econometrica, 68(1):135-148, 2000. [61] Jean-Pierre Ponssard and Sylvain Sorin. The LP formulation of finite zero-sum games with incomplete information. International J. Game Theory, 9(2):99-105, 1980. [62] Eric Rasmussen. Strategic implications of uncertainty over one"s own private value in auctions. Technical report, Indiana University, 2005. [63] Leonardo Rezende. Mid-auction information acquisition. Technical report, University of Illinois, 2005. [64] Ariel Rubinstein. Modeling Bounded Rationality. MIT, 1988. [65] Barry Schwartz. The Paradox of Choice: Why More is Less. Ecco, 2004. [66] Herbert Simon. Models of Bounded Rationality. MIT, 1982. [67] I. Simonson and A. Tversky. Choice in context: Tradeoff contrast and extremeness aversion. J. Marketing Research, 29:281-295, 1992. [68] Brian Skyrms. Dynamic models of deliberation and the theory of games. In Proceedings of the Theoretical Aspects of Rationality and Knowledge, pages 185-200, 1990. [69] Richard Sutton and Andrew Barto. 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algorithm;game theory;nash equilibrium;strategic multiplayer environment;constant-sum game;correlate equilibrium;auction;arbitrary partial information;questionand-answer session;information acquisition;socratic game;priori probability distribution;observable-query model;missing information;game-either full omniscient knowledge;unobservable-query model

train_J-37

Finding Equilibria in Large Sequential Games of Imperfect Information

Finding an equilibrium of an extensive form game of imperfect information is a fundamental problem in computational game theory, but current techniques do not scale to large games. To address this, we introduce the ordered game isomorphism and the related ordered game isomorphic abstraction transformation. For a multi-player sequential game of imperfect information with observable actions and an ordered signal space, we prove that any Nash equilibrium in an abstracted smaller game, obtained by one or more applications of the transformation, can be easily converted into a Nash equilibrium in the original game. We present an algorithm, GameShrink, for abstracting the game using our isomorphism exhaustively. Its complexity is ˜O(n2 ), where n is the number of nodes in a structure we call the signal tree. It is no larger than the game tree, and on nontrivial games it is drastically smaller, so GameShrink has time and space complexity sublinear in the size of the game tree. Using GameShrink, we find an equilibrium to a poker game with 3.1 billion nodes-over four orders of magnitude more than in the largest poker game solved previously. We discuss several electronic commerce applications for GameShrink. To address even larger games, we introduce approximation methods that do not preserve equilibrium, but nevertheless yield (ex post) provably close-to-optimal strategies.

1. INTRODUCTION In environments with more than one agent, an agent"s outcome is generally affected by the actions of the other agent(s). Consequently, the optimal action of one agent can depend on the others. Game theory provides a normative framework for analyzing such strategic situations. In particular, it provides solution concepts that define what rational behavior is in such settings. The most famous and important solution concept is that of Nash equilibrium [36]. It is a strategy profile (one strategy for each agent) in which no agent has incentive to deviate to a different strategy. However, for the concept to be operational, we need algorithmic techniques for finding an equilibrium. Games can be classified as either games of perfect information or imperfect information. Chess and Go are examples of the former, and, until recently, most game playing work has been on games of this type. To compute an optimal strategy in a perfect information game, an agent traverses the game tree and evaluates individual nodes. If the agent is able to traverse the entire game tree, she simply computes an optimal strategy from the bottom-up, using the principle of backward induction.1 In computer science terms, this is done using minimax search (often in conjunction with αβ-pruning to reduce the search tree size and thus enhance speed). Minimax search runs in linear time in the size of the game tree.2 The differentiating feature of games of imperfect information, such as poker, is that they are not fully observable: when it is an agent"s turn to move, she does not have access to all of the information about the world. In such games, the decision of what to do at a point in time cannot generally be optimally made without considering decisions at all other points in time (including ones on other paths of play) because those other decisions affect the probabilities of being at different states at the current point in time. Thus the algorithms for perfect information games do not solve games of imperfect information. For sequential games with imperfect information, one could try to find an equilibrium using the normal (matrix) form, where every contingency plan of the agent is a pure strategy for the agent.3 Unfortunately (even if equivalent strategies 1 This actually yields a solution that satisfies not only the Nash equilibrium solution concept, but a stronger solution concept called subgame perfect Nash equilibrium [45]. 2 This type of algorithm still does not scale to huge trees (such as in chess or Go), but effective game-playing agents can be developed even then by evaluating intermediate nodes using a heuristic evaluation and then treating those nodes as leaves. 3 An -equilibrium in a normal form game with any 160 are replaced by a single strategy [27]) this representation is generally exponential in the size of the game tree [52]. By observing that one needs to consider only sequences of moves rather than pure strategies [41, 46, 22, 52], one arrives at a more compact representation, the sequence form, which is linear in the size of the game tree.4 For 2-player games, there is a polynomial-sized (in the size of the game tree) linear programming formulation (linear complementarity in the non-zero-sum case) based on the sequence form such that strategies for players 1 and 2 correspond to primal and dual variables. Thus, the equilibria of reasonable-sized 2-player games can be computed using this method [52, 24, 25].5 However, this approach still yields enormous (unsolvable) optimization problems for many real-world games, such as poker. 1.1 Our approach In this paper, we take a different approach to tackling the difficult problem of equilibrium computation. Instead of developing an equilibrium-finding method per se, we instead develop a methodology for automatically abstracting games in such a way that any equilibrium in the smaller (abstracted) game corresponds directly to an equilibrium in the original game. Thus, by computing an equilibrium in the smaller game (using any available equilibrium-finding algorithm), we are able to construct an equilibrium in the original game. The motivation is that an equilibrium for the smaller game can be computed drastically faster than for the original game. To this end, we introduce games with ordered signals (Section 2), a broad class of games that has enough structure which we can exploit for abstraction purposes. Instead of operating directly on the game tree (something we found to be technically challenging), we instead introduce the use of information filters (Section 2.1), which coarsen the information each player receives. They are used in our analysis and abstraction algorithm. By operating only in the space of filters, we are able to keep the strategic structure of the game intact, while abstracting out details of the game in a way that is lossless from the perspective of equilibrium finding. We introduce the ordered game isomorphism to describe strategically symmetric situations and the ordered game isomorphic abstraction transformation to take advantange of such symmetries (Section 3). As our main equilibrium result we have the following: constant number of agents can be constructed in quasipolynomial time [31], but finding an exact equilibrium is PPAD-complete even in a 2-player game [8]. The most prevalent algorithm for finding an equilibrium in a 2-agent game is Lemke-Howson [30], but it takes exponentially many steps in the worst case [44]. For a survey of equilibrium computation in 2-player games, see [53]. Recently, equilibriumfinding algorithms that enumerate supports (i.e., sets of pure strategies that are played with positive probability) have been shown efficient on many games [40], and efficient mixed integer programming algorithms that search in the space of supports have been developed [43]. For more than two players, many algorithms have been proposed, but they currently only scale to very small games [19, 34, 40]. 4 There were also early techniques that capitalized in different ways on the fact that in many games the vast majority of pure strategies are not played in equilibrium [54, 23]. 5 Recently this approach was extended to handle computing sequential equilibria [26] as well [35]. Theorem 2 Let Γ be a game with ordered signals, and let F be an information filter for Γ. Let F be an information filter constructed from F by one application of the ordered game isomorphic abstraction transformation, and let σ be a Nash equilibrium strategy profile of the induced game ΓF (i.e., the game Γ using the filter F ). If σ is constructed by using the corresponding strategies of σ , then σ is a Nash equilibrium of ΓF . The proof of the theorem uses an equivalent characterization of Nash equilibria: σ is a Nash equilibrium if and only if there exist beliefs μ (players" beliefs about unknown information) at all points of the game reachable by σ such that σ is sequentially rational (i.e., a best response) given μ, where μ is updated using Bayes" rule. We can then use the fact that σ is a Nash equilibrium to show that σ is a Nash equilibrium considering only local properties of the game. We also give an algorithm, GameShrink, for abstracting the game using our isomorphism exhaustively (Section 4). Its complexity is ˜O(n2 ), where n is the number of nodes in a structure we call the signal tree. It is no larger than the game tree, and on nontrivial games it is drastically smaller, so GameShrink has time and space complexity sublinear in the size of the game tree. We present several algorithmic and data structure related speed improvements (Section 4.1), and we demonstrate how a simple modification to our algorithm yields an approximation algorithm (Section 5). 1.2 Electronic commerce applications Sequential games of imperfect information are ubiquitous, for example in negotiation and in auctions. Often aspects of a player"s knowledge are not pertinent for deciding what action the player should take at a given point in the game. On the trivial end, some aspects of a player"s knowledge are never pertinent (e.g., whether it is raining or not has no bearing on the bidding strategy in an art auction), and such aspects can be completely left out of the model specification. However, some aspects can be pertinent in certain states of the game while they are not pertinent in other states, and thus cannot be left out of the model completely. Furthermore, it may be highly non-obvious which aspects are pertinent in which states of the game. Our algorithm automatically discovers which aspects are irrelevant in different states, and eliminates those aspects of the game, resulting in a more compact, equivalent game representation. One broad application area that has this property is sequential negotiation (potentially over multiple issues). Another broad application area is sequential auctions (potentially over multiple goods). For example, in those states of a 1-object auction where bidder A can infer that his valuation is greater than that of bidder B, bidder A can ignore all his other information about B"s signals, although that information would be relevant for inferring B"s exact valuation. Furthermore, in some states of the auction, a bidder might not care which exact other bidders have which valuations, but cares about which valuations are held by the other bidders in aggregate (ignoring their identities). Many open-cry sequential auction and negotiation mechanisms fall within the game model studied in this paper (specified in detail later), as do certain other games in electronic commerce, such as sequences of take-it-or-leave-it offers [42]. Our techniques are in no way specific to an application. The main experiment that we present in this paper is on 161 a recreational game. We chose a particular poker game as the benchmark problem because it yields an extremely complicated and enormous game tree, it is a game of imperfect information, it is fully specified as a game (and the data is available), and it has been posted as a challenge problem by others [47] (to our knowledge no such challenge problem instances have been proposed for electronic commerce applications that require solving sequential games). 1.3 Rhode Island Hold"em poker Poker is an enormously popular card game played around the world. The 2005 World Series of Poker had over $103 million dollars in total prize money, including $56 million for the main event. Increasingly, poker players compete in online casinos, and television stations regularly broadcast poker tournaments. Poker has been identified as an important research area in AI due to the uncertainty stemming from opponents" cards, opponents" future actions, and chance moves, among other reasons [5]. Almost since the field"s founding, game theory has been used to analyze different aspects of poker [28; 37; 3; 51, pp. 186-219]. However, this work was limited to tiny games that could be solved by hand. More recently, AI researchers have been applying the computational power of modern hardware to computing game theory-based strategies for larger games. Koller and Pfeffer determined solutions to poker games with up to 140,000 nodes using the sequence form and linear programming [25]. Large-scale approximations have been developed [4], but those methods do not provide any guarantees about the performance of the computed strategies. Furthermore, the approximations were designed manually by a human expert. Our approach yields an automated abstraction mechanism along with theoretical guarantees on the strategies" performance. Rhode Island Hold"em was invented as a testbed for computational game playing [47]. It was designed so that it was similar in style to Texas Hold"em, yet not so large that devising reasonably intelligent strategies would be impossible. (The rules of Rhode Island Hold"em, as well as a discussion of how Rhode Island Hold"em can be modeled as a game with ordered signals, that is, it fits in our model, is available in an extended version of this paper [13].) We applied the techniques developed in this paper to find an exact (minimax) solution to Rhode Island Hold"em, which has a game tree exceeding 3.1 billion nodes. Applying the sequence form to Rhode Island Hold"em directly without abstraction yields a linear program with 91,224,226 rows, and the same number of columns. This is much too large for (current) linear programming algorithms to handle. We used our GameShrink algorithm to reduce this with lossless abstraction, and it yielded a linear program with 1,237,238 rows and columns-with 50,428,638 non-zero coefficients. We then applied iterated elimination of dominated strategies, which further reduced this to 1,190,443 rows and 1,181,084 columns. (Applying iterated elimination of dominated strategies without GameShrink yielded 89,471,986 rows and 89,121,538 columns, which still would have been prohibitively large to solve.) GameShrink required less than one second to perform the shrinking (i.e., to compute all of the ordered game isomorphic abstraction transformations). Using a 1.65GHz IBM eServer p5 570 with 64 gigabytes of RAM (the linear program solver actually needed 25 gigabytes), we solved it in 7 days and 17 hours using the interior-point barrier method of CPLEX version 9.1.2. We recently demonstrated our optimal Rhode Island Hold"em poker player at the AAAI-05 conference [14], and it is available for play on-line at http://www.cs.cmu.edu/ ~gilpin/gsi.html. While others have worked on computer programs for playing Rhode Island Hold"em [47], no optimal strategy has been found before. This is the largest poker game solved to date by over four orders of magnitude. 2. GAMES WITH ORDERED SIGNALS We work with a slightly restricted class of games, as compared to the full generality of the extensive form. This class, which we call games with ordered signals, is highly structured, but still general enough to capture a wide range of strategic situations. A game with ordered signals consists of a finite number of rounds. Within a round, the players play a game on a directed tree (the tree can be different in different rounds). The only uncertainty players face stems from private signals the other players have received and from the unknown future signals. In other words, players observe each others" actions, but potentially not nature"s actions. In each round, there can be public signals (announced to all players) and private signals (confidentially communicated to individual players). For simplicity, we assume-as is the case in most recreational games-that within each round, the number of private signals received is the same across players (this could quite likely be relaxed). We also assume that the legal actions that a player has are independent of the signals received. For example, in poker, the legal betting actions are independent of the cards received. Finally, the strongest assumption is that there is a partial ordering over sets of signals, and the payoffs are increasing (not necessarily strictly) in these signals. For example, in poker, this partial ordering corresponds exactly to the ranking of card hands. Definition 1. A game with ordered signals is a tuple Γ = I, G, L, Θ, κ, γ, p, , ω, u where: 1. I = {1, . . . , n} is a finite set of players. 2. G = G1 , . . . , Gr , Gj = ` V j , Ej ´ , is a finite collection of finite directed trees with nodes V j and edges Ej . Let Zj denote the leaf nodes of Gj and let Nj (v) denote the outgoing neighbors of v ∈ V j . Gj is the stage game for round j. 3. L = L1 , . . . , Lr , Lj : V j \ Zj → I indicates which player acts (chooses an outgoing edge) at each internal node in round j. 4. Θ is a finite set of signals. 5. κ = κ1 , . . . , κr and γ = γ1 , . . . , γr are vectors of nonnegative integers, where κj and γj denote the number of public and private signals (per player), respectively, revealed in round j. Each signal θ ∈ Θ may only be revealed once, and in each round every player receives the same number of private signals, so we require Pr j=1 κj + nγj ≤ |Θ|. The public information revealed in round j is αj ∈ Θκj and the public information revealed in all rounds up through round j is ˜αj = ` α1 , . . . , αj ´ . The private information revealed to player i ∈ I in round j is βj i ∈ Θγj and the private information revaled to player i ∈ I in all rounds up through round j is ˜βj i = ` β1 i , . . . , βj i ´ . We 162 also write ˜βj = ˜βj 1, . . . , ˜βj n to represent all private information up through round j, and ˜β j i , ˜βj −i = ˜βj 1, . . . , ˜βj i−1, ˜β j i , ˜βj i+1, . . . , ˜βj n is ˜βj with ˜βj i replaced with ˜β j i . The total information revealed up through round j, ˜αj , ˜βj , is said to be legal if no signals are repeated. 6. p is a probability distribution over Θ, with p(θ) > 0 for all θ ∈ Θ. Signals are drawn from Θ according to p without replacement, so if X is the set of signals already revealed, then p(x | X) = ( p(x)P y /∈X p(y) if x /∈ X 0 if x ∈ X. 7. is a partial ordering of subsets of Θ and is defined for at least those pairs required by u. 8. ω : rS j=1 Zj → {over, continue} is a mapping of terminal nodes within a stage game to one of two values: over, in which case the game ends, or continue, in which case the game continues to the next round. Clearly, we require ω(z) = over for all z ∈ Zr . Note that ω is independent of the signals. Let ωj over = {z ∈ Zj | ω(z) = over} and ωj cont = {z ∈ Zj | ω(z) = continue}. 9. u = (u1 , . . . , ur ), uj : j−1 k=1 ωk cont × ωj over × j k=1 Θκk × n i=1 j k=1 Θγk → Rn is a utility function such that for every j, 1 ≤ j ≤ r, for every i ∈ I, and for every ˜z ∈ j−1 k=1 ωk cont × ωj over, at least one of the following two conditions holds: (a) Utility is signal independent: uj i (˜z, ϑ) = uj i (˜z, ϑ ) for all legal ϑ, ϑ ∈ j k=1 Θκk × n i=1 j k=1 Θγk . (b) is defined for all legal signals (˜αj , ˜βj i ), (˜αj , ˜β j i ) through round j and a player"s utility is increasing in her private signals, everything else equal: ˜αj , ˜βj i ˜αj , ˜β j i =⇒ ui ˜z, ˜αj , ˜βj i , ˜βj −i ≥ ui ˜z, ˜αj , ˜β j i , ˜βj −i . We will use the term game with ordered signals and the term ordered game interchangeably. 2.1 Information filters In this subsection, we define an information filter for ordered games. Instead of completely revealing a signal (either public or private) to a player, the signal first passes through this filter, which outputs a coarsened signal to the player. By varying the filter applied to a game, we are able to obtain a wide variety of games while keeping the underlying action space of the game intact. We will use this when designing our abstraction techniques. Formally, an information filter is as follows. Definition 2. Let Γ = I, G, L, Θ, κ, γ, p, , ω, u be an ordered game. Let Sj ⊆ j k=1 Θκk × j k=1 Θγk be the set of legal signals (i.e., no repeated signals) for one player through round j. An information filter for Γ is a collection F = F1 , . . . , Fr where each Fj is a function Fj : Sj → 2Sj such that each of the following conditions hold: 1. (Truthfulness) (˜αj , ˜βj i ) ∈ Fj (˜αj , ˜βj i ) for all legal (˜αj , ˜βj i ). 2. (Independence) The range of Fj is a partition of Sj . 3. (Information preservation) If two values of a signal are distinguishable in round k, then they are distinguishable fpr each round j > k. Let mj = Pj l=1 κl +γl . We require that for all legal (θ1, . . . , θmk , . . . , θmj ) ⊆ Θ and (θ1, . . . , θmk , . . . , θmj ) ⊆ Θ: (θ1, . . . , θmk ) /∈ Fk (θ1, . . . , θmk ) =⇒ (θ1, . . . , θmk , . . . , θmj ) /∈ Fj (θ1, . . . , θmk , . . . , θmj ). A game with ordered signals Γ and an information filter F for Γ defines a new game ΓF . We refer to such games as filtered ordered games. We are left with the original game if we use the identity filter Fj ˜αj , ˜βj i = n ˜αj , ˜βj i o . We have the following simple (but important) result: Proposition 1. A filtered ordered game is an extensive form game satisfying perfect recall. A simple proof proceeds by constructing an extensive form game directly from the ordered game, and showing that it satisfies perfect recall. In determining the payoffs in a game with filtered signals, we take the average over all real signals in the filtered class, weighted by the probability of each real signal occurring. 2.2 Strategies and Nash equilibrium We are now ready to define behavior strategies in the context of filtered ordered games. Definition 3. A behavior strategy for player i in round j of Γ = I, G, L, Θ, κ, γ, p, , ω, u with information filter F is a probability distribution over possible actions, and is defined for each player i, each round j, and each v ∈ V j \Zj for Lj (v) = i: σj i,v : j−1 k=1 ωk cont×Range Fj → Δ n w ∈ V j | (v, w) ∈ Ej o . (Δ(X) is the set of probability distributions over a finite set X.) A behavior strategy for player i in round j is σj i = (σj i,v1 , . . . , σj i,vm ) for each vk ∈ V j \ Zj where Lj (vk) = i. A behavior strategy for player i in Γ is σi = ` σ1 i , . . . , σr i ´ . A strategy profile is σ = (σ1, . . . , σn). A strategy profile with σi replaced by σi is (σi, σ−i) = (σ1, . . . , σi−1, σi, σi+1, . . . , σn). By an abuse of notation, we will say player i receives an expected payoff of ui(σ) when all players are playing the strategy profile σ. Strategy σi is said to be player i"s best response to σ−i if for all other strategies σi for player i we have ui(σi, σ−i) ≥ ui(σi, σ−i). σ is a Nash equilibrium if, for every player i, σi is a best response for σ−i. A Nash equilibrium always exists in finite extensive form games [36], and one exists in behavior strategies for games with perfect recall [29]. Using these observations, we have the following corollary to Proposition 1: 163 Corollary 1. For any filtered ordered game, a Nash equilibrium exists in behavior strateges. 3. EQUILIBRIUM-PRESERVING ABSTRACTIONS In this section, we present our main technique for reducing the size of games. We begin by defining a filtered signal tree which represents all of the chance moves in the game. The bold edges (i.e. the first two levels of the tree) in the game trees in Figure 1 correspond to the filtered signal trees in each game. Definition 4. Associated with every ordered game Γ = I, G, L, Θ, κ, γ, p, , ω, u and information filter F is a filtered signal tree, a directed tree in which each node corresponds to some revealed (filtered) signals and edges correspond to revealing specific (filtered) signals. The nodes in the filtered signal tree represent the set of all possible revealed filtered signals (public and private) at some point in time. The filtered public signals revealed in round j correspond to the nodes in the κj levels beginning at level Pj−1 k=1 ` κk + nγk ´ and the private signals revealed in round j correspond to the nodes in the nγj levels beginning at level Pj k=1 κk + Pj−1 k=1 nγk . We denote children of a node x as N(x). In addition, we associate weights with the edges corresponding to the probability of the particular edge being chosen given that its parent was reached. In many games, there are certain situations in the game that can be thought of as being strategically equivalent to other situations in the game. By melding these situations together, it is possible to arrive at a strategically equivalent smaller game. The next two definitions formalize this notion via the introduction of the ordered game isomorphic relation and the ordered game isomorphic abstraction transformation. Definition 5. Two subtrees beginning at internal nodes x and y of a filtered signal tree are ordered game isomorphic if x and y have the same parent and there is a bijection f : N(x) → N(y), such that for w ∈ N(x) and v ∈ N(y), v = f(w) implies the weights on the edges (x, w) and (y, v) are the same and the subtrees beginning at w and v are ordered game isomorphic. Two leaves (corresponding to filtered signals ϑ and ϑ up through round r) are ordered game isomorphic if for all ˜z ∈ r−1 j=1 ωj cont × ωr over, ur (˜z, ϑ) = ur (˜z, ϑ ). Definition 6. Let Γ = I, G, L, Θ, κ, γ, p, , ω, u be an ordered game and let F be an information filter for Γ. Let ϑ and ϑ be two nodes where the subtrees in the induced filtered signal tree corresponding to the nodes ϑ and ϑ are ordered game isomorphic, and ϑ and ϑ are at either levelPj−1 k=1 ` κk + nγk ´ or Pj k=1 κk + Pj−1 k=1 nγk for some round j. The ordered game isomorphic abstraction transformation is given by creating a new information filter F : F j ˜αj , ˜βj i = 8 < : Fj ˜αj , ˜βj i if ˜αj , ˜βj i /∈ ϑ ∪ ϑ ϑ ∪ ϑ if ˜αj , ˜βj i ∈ ϑ ∪ ϑ . Figure 1 shows the ordered game isomorphic abstraction transformation applied twice to a tiny poker game. Theorem 2, our main equilibrium result, shows how the ordered game isomorphic abstraction transformation can be used to compute equilibria faster. Theorem 2. Let Γ = I, G, L, Θ, κ, γ, p, , ω, u be an ordered game and F be an information filter for Γ. Let F be an information filter constructed from F by one application of the ordered game isomorphic abstraction transformation. Let σ be a Nash equilibrium of the induced game ΓF . If we take σj i,v ˜z, Fj ˜αj , ˜βj i = σ j i,v ˜z, F j ˜αj , ˜βj i , σ is a Nash equilibrium of ΓF . Proof. For an extensive form game, a belief system μ assigns a probability to every decision node x such thatP x∈h μ(x) = 1 for every information set h. A strategy profile σ is sequentially rational at h given belief system μ if ui(σi, σ−i | h, μ) ≥ ui(τi, σ−i | h, μ) for all other strategies τi, where i is the player who controls h. A basic result [33, Proposition 9.C.1] characterizing Nash equilibria dictates that σ is a Nash equilibrium if and only if there is a belief system μ such that for every information set h with Pr(h | σ) > 0, the following two conditions hold: (C1) σ is sequentially rational at h given μ; and (C2) μ(x) = Pr(x | σ) Pr(h | σ) for all x ∈ h. Since σ is a Nash equilibrium of Γ , there exists such a belief system μ for ΓF . Using μ , we will construct a belief system μ for Γ and show that conditions C1 and C2 hold, thus supporting σ as a Nash equilibrium. Fix some player i ∈ I. Each of i"s information sets in some round j corresponds to filtered signals Fj ˜α∗j , ˜β∗j i , history in the first j − 1 rounds (z1, . . . , zj−1) ∈ j−1 k=1 ωk cont, and history so far in round j, v ∈ V j \ Zj . Let ˜z = (z1, . . . , zj−1, v) represent all of the player actions leading to this information set. Thus, we can uniquely specify this information set using the information Fj ˜α∗j , ˜β∗j i , ˜z . Each node in an information set corresponds to the possible private signals the other players have received. Denote by ˜β some legal (Fj (˜αj , ˜βj 1), . . . , Fj (˜αj , ˜βj i−1), Fj (˜αj , ˜βj i+1), . . . , Fj (˜αj , ˜βj n)). In other words, there exists (˜αj , ˜βj 1, . . . , ˜βj n) such that (˜αj , ˜βj i ) ∈ Fj (˜α∗j , ˜β∗j i ), (˜αj , ˜βj k) ∈ Fj (˜αj , ˜βj k) for k = i, and no signals are repeated. Using such a set of signals (˜αj , ˜βj 1, . . . , ˜βj n), let ˆβ denote (F j (˜αj , ˜βj 1), . . . , F j (˜αj , ˜βj i−1), F j (˜αj , ˜βj i+1), . . . , F j (˜αj , ˜βj n). (We will abuse notation and write F j −i ˆβ = ˆβ .) We can now compute μ directly from μ : μ ˆβ | Fj ˜αj , ˜βj i , ˜z = 8 >>>>>>< >>>>>>: μ ˆβ | F j ˜αj , ˜βj i , ˜z if Fj ˜αj , ˜βj i = F j ˜αj , ˜βj i or ˆβ = ˆβ p∗ μ ˆβ | F j ˜αj , ˜βj i , ˜z if Fj ˜αj , ˜βj i = F j ˜αj , ˜βj i and ˆβ = ˆβ 164 J1 J2 J2 K1 K1 K2 K2 c b C B F B f b c b C B F B f b c b C f b B BF c b C f b B BF c b C B F B f b c b C BF B f b c b C f b B BF c b C f b B BF c b C f b B BF c b C f b B BF c b C B F B f b c b C B F B f b 0 0 0-1 -1 -1 -1 -1 -1 -1 -1-1 -1 -1 -1 -1 -1 -1 -1 -1 -1-1 -1 -1 -1 -1 -10 0 0 0 0 0 0 0 0 -1 -2 -2 -1 -2 -2 -1 -2 -2 -1 -2 -2 1 2 2 1 2 2 1 2 2 1 2 2 J1 K1 K2 J1 J2 K2 J1 J2 K1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 {{J1}, {J2}, {K1}, {K2}} {{J1,J2}, {K1}, {K2}} c b C BF B f b c b C f b B BF c b C B F B f b J1,J2 K1 K2 1 1 c b C f b B BF c b C BF B f b c b C BF B f b c b C B F B f b J1,J2 K1 K2 1 1 1 1 J1,J2 K2 J1,J2 K1 0 0 0-1 -1 -1 -1 -1 -1 -1 -2 -2 -1 -2 -2 2 2 2 2 2 2 -1 -1-1 -1 0 0 0 1 2 2 -1 -1-1 -1 0 0 0 1 2 2 c b C B F B f b -1 -10 0 0 c b B F B f b -1 -1-1 -2 -2 c b C BF B f b 0 0 0-1 -1 c b C BF B f b J1,J2 J1,J2 J1,J2K1,K2 K1,K2 K1,K2 -1 -1 1 2 2 2 2 2 2 {{J1,J2}, {K1,K2}} 1 1 1 1 1/4 1/4 1/4 1/4 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/4 1/41/2 1/3 1/3 1/3 1/32/3 1/32/3 1/2 1/2 1/3 2/3 2/3 1/3 Figure 1: GameShrink applied to a tiny two-person four-card (two Jacks and two Kings) poker game. Next to each game tree is the range of the information filter F. Dotted lines denote information sets, which are labeled by the controlling player. Open circles are chance nodes with the indicated transition probabilities. The root node is the chance node for player 1"s card, and the next level is for player 2"s card. The payment from player 2 to player 1 is given below each leaf. In this example, the algorithm reduces the game tree from 53 nodes to 19 nodes. where p∗ = Pr(ˆβ | F j (˜αj , ˜β j i )) Pr(ˆβ | F j (˜αj , ˜β j i )) . The following three claims show that μ as calculated above supports σ as a Nash equilibrium. Claim 1. μ is a valid belief system for ΓF . Claim 2. For all information sets h with Pr(h | σ) > 0, μ(x) = Pr(x | σ) Pr(h | σ) for all x ∈ h. Claim 3. For all information sets h with Pr(h | σ) > 0, σ is sequentially rational at h given μ. The proofs of Claims 1-3 are in an extended version of this paper [13]. By Claims 1 and 2, we know that condition C2 holds. By Claim 3, we know that condition C1 holds. Thus, σ is a Nash equilibrium. 3.1 Nontriviality of generalizing beyond this model Our model does not capture general sequential games of imperfect information because it is restricted in two ways (as discussed above): 1) there is a special structure connecting the player actions and the chance actions (for one, the players are assumed to observe each others" actions, but nature"s actions might not be publicly observable), and 2) there is a common ordering of signals. In this subsection we show that removing either of these conditions can make our technique invalid. First, we demonstrate a failure when removing the first assumption. Consider the game in Figure 2.6 Nodes a and b are in the same information set, have the same parent (chance) node, have isomorphic subtrees with the same payoffs, and nodes c and d also have similar structural properties. By merging the subtrees beginning at a and b, we get the game on the right in Figure 2. In this game, player 1"s only Nash equilibrium strategy is to play left. But in the original game, player 1 knows that node c will never be reached, and so should play right in that information set. 1/4 1/4 1/4 1/4 2 2 2 1 1 1 2 1 2 3 0 3 0 -10 10 1/2 1/4 1/4 2 2 2 1 1 2 3 0 3 0 a b 2 2 2 10-10 c d Figure 2: Example illustrating difficulty in developing a theory of equilibrium-preserving abstractions for general extensive form games. Removing the second assumption (that the utility functions are based on a common ordering of signals) can also cause failure. Consider a simple three-card game with a deck containing two Jacks (J1 and J2) and a King (K), where player 1"s utility function is based on the ordering 6 We thank Albert Xin Jiang for providing this example. 165 K J1 ∼ J2 but player 2"s utility function is based on the ordering J2 K J1. It is easy to check that in the abstracted game (where Player 1 treats J1 and J2 as being equivalent) the Nash equilibrium does not correspond to a Nash equilibrium in the original game.7 4. GAMESHRINK: AN EFFICIENT ALGORITHM FOR COMPUTING ORDERED GAME ISOMORPHIC ABSTRACTION TRANSFORMATIONS This section presents an algorithm, GameShrink, for conducting the abstractions. It only needs to analyze the signal tree discussed above, rather than the entire game tree. We first present a subroutine that GameShrink uses. It is a dynamic program for computing the ordered game isomorphic relation. Again, it operates on the signal tree. Algorithm 1. OrderedGameIsomorphic? (Γ, ϑ, ϑ ) 1. If ϑ and ϑ have different parents, then return false. 2. If ϑ and ϑ are both leaves of the signal tree: (a) If ur (ϑ | ˜z) = ur (ϑ | ˜z) for all ˜z ∈ r−1 j=1 ωj cont × ωr over, then return true. (b) Otherwise, return false. 3. Create a bipartite graph Gϑ,ϑ = (V1, V2, E) with V1 = N(ϑ) and V2 = N(ϑ ). 4. For each v1 ∈ V1 and v2 ∈ V2: If OrderedGameIsomorphic? (Γ, v1, v2) Create edge (v1, v2) 5. Return true if Gϑ,ϑ has a perfect matching; otherwise, return false. By evaluating this dynamic program from bottom to top, Algorithm 1 determines, in time polynomial in the size of the signal tree, whether or not any pair of equal depth nodes x and y are ordered game isomorphic. We can further speed up this computation by only examining nodes with the same parent, since we know (from step 1) that no nodes with different parents are ordered game isomorphic. The test in step 2(a) can be computed in O(1) time by consulting the relation from the specification of the game. Each call to OrderedGameIsomorphic? performs at most one perfect matching computation on a bipartite graph with O(|Θ|) nodes and O(|Θ|2 ) edges (recall that Θ is the set of signals). Using the Ford-Fulkerson algorithm [12] for finding a maximal matching, this takes O(|Θ|3 ) time. Let S be the maximum number of signals possibly revealed in the game (e.g., in Rhode Island Hold"em, S = 4 because each of the two players has one card in the hand plus there are two cards on the table). The number of nodes, n, in the signal tree is O(|Θ|S ). The dynamic program visits each node in the signal tree, with each visit requiring O(|Θ|2 ) calls to the OrderedGameIsomorphic? routine. So, it takes O(|Θ|S |Θ|3 |Θ|2 ) = O(|Θ|S+5 ) time to compute the entire ordered game isomorphic relation. While this is exponential in the number of revealed signals, we now show that it is polynomial in the size of the signal tree-and thus polynomial in the size of the game tree 7 We thank an anonymous person for this example. because the signal tree is smaller than the game tree. The number of nodes in the signal tree is n = 1 + SX i=1 iY j=1 (|Θ| − j + 1) (Each term in the summation corresponds to the number of nodes at a specific depth of the tree.) The number of leaves is SY j=1 (|Θ| − j + 1) = |Θ| S ! S! which is a lower bound on the number of nodes. For large |Θ| we can use the relation `n k ´ ∼ nk k! to get |Θ| S ! S! ∼ „ |Θ|S S! « S! = |Θ|S and thus the number of leaves in the signal tree is Ω(|Θ|S ). Thus, O(|Θ|S+5 ) = O(n|Θ|5 ), which proves that we can indeed compute the ordered game isomorphic relation in time polynomial in the number of nodes, n, of the signal tree. The algorithm often runs in sublinear time (and space) in the size of the game tree because the signal tree is significantly smaller than the game tree in most nontrivial games. (Note that the input to the algorithm is not an explicit game tree, but a specification of the rules, so the algorithm does not need to read in the game tree.) See Figure 1. In general, if an ordered game has r rounds, and each round"s stage game has at least b nonterminal leaves, then the size of the signal tree is at most 1 br of the size of the game tree. For example, in Rhode Island Hold"em, the game tree has 3.1 billion nodes while the signal tree only has 6,632,705. Given the OrderedGameIsomorphic? routine for determining ordered game isomorphisms in an ordered game, we are ready to present the main algorithm, GameShrink. Algorithm 2. GameShrink (Γ) 1. Initialize F to be the identity filter for Γ. 2. For j from 1 to r: For each pair of sibling nodes ϑ, ϑ at either levelPj−1 k=1 ` κk + nγk ´ or Pj k=1 κk + Pj−1 k=1 nγk in the filtered (according to F) signal tree: If OrderedGameIsomorphic?(Γ, ϑ, ϑ ), then Fj (ϑ) ← Fj (ϑ ) ← Fj (ϑ) ∪ Fj (ϑ ). 3. Output F. Given as input an ordered game Γ, GameShrink applies the shrinking ideas presented above as aggressively as possible. Once it finishes, there are no contractible nodes (since it compares every pair of nodes at each level of the signal tree), and it outputs the corresponding information filter F. The correctness of GameShrink follows by a repeated application of Theorem 2. Thus, we have the following result: Theorem 3. GameShrink finds all ordered game isomorphisms and applies the associated ordered game isomorphic abstraction transformations. Furthermore, for any Nash equilibrium, σ , of the abstracted game, the strategy profile constructed for the original game from σ is a Nash equilibrium. The dominating factor in the run time of GameShrink is in the rth iteration of the main for-loop. There are at most 166 `|Θ| S ´ S! nodes at this level, where we again take S to be the maximum number of signals possibly revealed in the game. Thus, the inner for-loop executes O „`|Θ| S ´ S! 2 « times. As discussed in the next subsection, we use a union-find data structure to represent the information filter F. Each iteration of the inner for-loop possibly performs a union operation on the data structure; performing M operations on a union-find data structure containing N elements takes O(α(M, N)) amortized time per operation, where α(M, N) is the inverse Ackermann"s function [1, 49] (which grows extremely slowly). Thus, the total time for GameShrink is O „`|Θ| S ´ S! 2 α „`|Θ| S ´ S! 2 , |Θ|S «« . By the inequality `n k ´ ≤ nk k! , this is O ` (|Θ|S )2 α ` (|Θ|S )2 , |Θ|S ´´ . Again, although this is exponential in S, it is ˜O(n2 ), where n is the number of nodes in the signal tree. Furthermore, GameShrink tends to actually run in sublinear time and space in the size of the game tree because the signal tree is significantly smaller than the game tree in most nontrivial games, as discussed above. 4.1 Efficiency enhancements We designed several speed enhancement techniques for GameShrink, and all of them are incorporated into our implementation. One technique is the use of the union-find data structure for storing the information filter F. This data structure uses time almost linear in the number of operations [49]. Initially each node in the signalling tree is its own set (this corresponds to the identity information filter); when two nodes are contracted they are joined into a new set. Upon termination, the filtered signals for the abstracted game correspond exactly to the disjoint sets in the data structure. This is an efficient method of recording contractions within the game tree, and the memory requirements are only linear in the size of the signal tree. Determining whether two nodes are ordered game isomorphic requires us to determine if a bipartite graph has a perfect matching. We can eliminate some of these computations by using easy-to-check necessary conditions for the ordered game isomorphic relation to hold. One such condition is to check that the nodes have the same number of chances as being ranked (according to ) higher than, lower than, and the same as the opponents. We can precompute these frequencies for every game tree node. This substantially speeds up GameShrink, and we can leverage this database across multiple runs of the algorithm (for example, when trying different abstraction levels; see next section). The indices for this database depend on the private and public signals, but not the order in which they were revealed, and thus two nodes may have the same corresponding database entry. This makes the database significantly more compact. (For example in Texas Hold"em, the database is reduced by a factor `50 3 ´`47 1 ´`46 1 ´ / `50 5 ´ = 20.) We store the histograms in a 2-dimensional database. The first dimension is indexed by the private signals, the second by the public signals. The problem of computing the index in (either) one of the dimensions is exactly the problem of computing a bijection between all subsets of size r from a set of size n and integers in ˆ 0, . . . , `n r ´ − 1 ˜ . We efficiently compute this using the subsets" colexicographical ordering [6]. Let {c1, . . . , cr}, ci ∈ {0, . . . , n − 1}, denote the r signals and assume that ci < ci+1. We compute a unique index for this set of signals as follows: index(c1, . . . , cr) = Pr i=1 `ci i ´ . 5. APPROXIMATION METHODS Some games are too large to compute an exact equilibrium, even after using the presented abstraction technique. This section discusses general techniques for computing approximately optimal strategy profiles. For a two-player game, we can always evaluate the worst-case performance of a strategy, thus providing some objective evaluation of the strength of the strategy. To illustrate this, suppose we know player 2"s planned strategy for some game. We can then fix the probabilities of player 2"s actions in the game tree as if they were chance moves. Then player 1 is faced with a single-agent decision problem, which can be solved bottomup, maximizing expected payoff at every node. Thus, we can objectively determine the expected worst-case performance of player 2"s strategy. This will be most useful when we want to evaluate how well a given strategy performs when we know that it is not an equilibrium strategy. (A variation of this technique may also be applied in n-person games where only one player"s strategies are held fixed.) This technique provides ex post guarantees about the worst-case performance of a strategy, and can be used independently of the method that is used to compute the strategies. 5.1 State-space approximations By slightly modifying GameShrink, we can obtain an algorithm that yields even smaller game trees, at the expense of losing the equilibrium guarantees of Theorem 2. Instead of requiring the payoffs at terminal nodes to match exactly, we can instead compute a penalty that increases as the difference in utility between two nodes increases. There are many ways in which the penalty function could be defined and implemented. One possibility is to create edge weights in the bipartite graphs used in Algorithm 1, and then instead of requiring perfect matchings in the unweighted graph we would instead require perfect matchings with low cost (i.e., only consider two nodes to be ordered game isomorphic if the corresponding bipartite graph has a perfect matching with cost below some threshold). Thus, with this threshold as a parameter, we have a knob to turn that in one extreme (threshold = 0) yields an optimal abstraction and in the other extreme (threshold = ∞) yields a highly abstracted game (this would in effect restrict players to ignoring all signals, but still observing actions). This knob also begets an anytime algorithm. One can solve increasingly less abstracted versions of the game, and evaluate the quality of the solution at every iteration using the ex post method discussed above. 5.2 Algorithmic approximations In the case of two-player zero-sum games, the equilibrium computation can be modeled as a linear program (LP), which can in turn be solved using the simplex method. This approach has inherent features which we can leverage into desirable properties in the context of solving games. In the LP, primal solutions correspond to strategies of player 2, and dual solutions correspond to strategies of player 1. There are two versions of the simplex method: the primal simplex and the dual simplex. The primal simplex maintains primal feasibility and proceeds by finding better and better primal solutions until the dual solution vector is feasible, 167 at which point optimality has been reached. Analogously, the dual simplex maintains dual feasibility and proceeds by finding increasingly better dual solutions until the primal solution vector is feasible. (The dual simplex method can be thought of as running the primal simplex method on the dual problem.) Thus, the primal and dual simplex methods serve as anytime algorithms (for a given abstraction) for players 2 and 1, respectively. At any point in time, they can output the best strategies found so far. Also, for any feasible solution to the LP, we can get bounds on the quality of the strategies by examining the primal and dual solutions. (When using the primal simplex method, dual solutions may be read off of the LP tableau.) Every feasible solution of the dual yields an upper bound on the optimal value of the primal, and vice versa [9, p. 57]. Thus, without requiring further computation, we get lower bounds on the expected utility of each agent"s strategy against that agent"s worst-case opponent. One problem with the simplex method is that it is not a primal-dual algorithm, that is, it does not maintain both primal and dual feasibility throughout its execution. (In fact, it only obtains primal and dual feasibility at the very end of execution.) In contrast, there are interior-point methods for linear programming that maintain primal and dual feasibility throughout the execution. For example, many interiorpoint path-following algorithms have this property [55, Ch. 5]. We observe that running such a linear programming method yields a method for finding -equilibria (i.e., strategy profiles in which no agent can increase her expected utility by more than by deviating). A threshold on can also be used as a termination criterion for using the method as an anytime algorithm. Furthermore, interior-point methods in this class have polynomial-time worst-case run time, as opposed to the simplex algorithm, which takes exponentially many steps in the worst case. 6. RELATED RESEARCH Functions that transform extensive form games have been introduced [50, 11]. In contrast to our work, those approaches were not for making the game smaller and easier to solve. The main result is that a game can be derived from another by a sequence of those transformations if and only if the games have the same pure reduced normal form. The pure reduced normal form is the extensive form game represented as a game in normal form where duplicates of pure strategies (i.e., ones with identical payoffs) are removed and players essentially select equivalence classes of strategies [27]. An extension to that work shows a similar result, but for slightly different transformations and mixed reduced normal form games [21]. Modern treatments of this prior work on game transformations exist [38, Ch. 6], [10]. The recent notion of weak isomorphism in extensive form games [7] is related to our notion of restricted game isomorphism. The motivation of that work was to justify solution concepts by arguing that they are invariant with respect to isomorphic transformations. Indeed, the author shows, among other things, that many solution concepts, including Nash, perfect, subgame perfect, and sequential equilibrium, are invariant with respect to weak isomorphisms. However, that definition requires that the games to be tested for weak isomorphism are of the same size. Our focus is totally different: we find strategically equivalent smaller games. Also, their paper does not provide algorithms. Abstraction techniques have been used in artificial intelligence research before. In contrast to our work, most (but not all) research involving abstraction has been for singleagent problems (e.g. [20, 32]). Furthermore, the use of abstraction typically leads to sub-optimal solutions, unlike the techniques presented in this paper, which yield optimal solutions. A notable exception is the use of abstraction to compute optimal strategies for the game of Sprouts [2]. However, a significant difference to our work is that Sprouts is a game of perfect information. One of the first pieces of research to use abstraction in multi-agent settings was the development of partition search, which is the algorithm behind GIB, the world"s first expertlevel computer bridge player [17, 18]. In contrast to other game tree search algorithms which store a particular game position at each node of the search tree, partition search stores groups of positions that are similar. (Typically, the similarity of two game positions is computed by ignoring the less important components of each game position and then checking whether the abstracted positions are similar-in some domain-specific expert-defined sense-to each other.) Partition search can lead to substantial speed improvements over α-β-search. However, it is not game theory-based (it does not consider information sets in the game tree), and thus does not solve for the equilibrium of a game of imperfect information, such as poker.8 Another difference is that the abstraction is defined by an expert human while our abstractions are determined automatically. There has been some research on the use of abstraction for imperfect information games. Most notably, Billings et al [4] describe a manually constructed abstraction for Texas Hold"em poker, and include promising results against expert players. However, this approach has significant drawbacks. First, it is highly specialized for Texas Hold"em. Second, a large amount of expert knowledge and effort was used in constructing the abstraction. Third, the abstraction does not preserve equilibrium: even if applied to a smaller game, it might not yield a game-theoretic equilibrium. Promising ideas for abstraction in the context of general extensive form games have been described in an extended abstract [39], but to our knowledge, have not been fully developed. 7. CONCLUSIONS AND DISCUSSION We introduced the ordered game isomorphic abstraction transformation and gave an algorithm, GameShrink, for abstracting the game using the isomorphism exhaustively. We proved that in games with ordered signals, any Nash equilibrium in the smaller abstracted game maps directly to a Nash equilibrium in the original game. The complexity of GameShrink is ˜O(n2 ), where n is the number of nodes in the signal tree. It is no larger than the game tree, and on nontrivial games it is drastically smaller, so GameShrink has time and space complexity sublinear in 8 Bridge is also a game of imperfect information, and partition search does not find the equilibrium for that game either. Instead, partition search is used in conjunction with statistical sampling to simulate the uncertainty in bridge. There are also other bridge programs that use search techniques for perfect information games in conjunction with statistical sampling and expert-defined abstraction [48]. Such (non-game-theoretic) techniques are unlikely to be competitive in poker because of the greater importance of information hiding and bluffing. 168 the size of the game tree. Using GameShrink, we found a minimax equilibrium to Rhode Island Hold"em, a poker game with 3.1 billion nodes in the game tree-over four orders of magnitude more than in the largest poker game solved previously. To further improve scalability, we introduced an approximation variant of GameShrink, which can be used as an anytime algorithm by varying a parameter that controls the coarseness of abstraction. We also discussed how (in a two-player zero-sum game), linear programming can be used in an anytime manner to generate approximately optimal strategies of increasing quality. The method also yields bounds on the suboptimality of the resulting strategies. We are currently working on using these techniques for full-scale 2-player limit Texas Hold"em poker, a highly popular card game whose game tree has about 1018 nodes. That game tree size has required us to use the approximation version of GameShrink (as well as round-based abstraction) [16, 15]. 8. REFERENCES [1] W. Ackermann. Zum Hilbertschen Aufbau der reellen Zahlen. Math. Annalen, 99:118-133, 1928. [2] D. Applegate, G. Jacobson, and D. Sleator. Computer analysis of sprouts. Technical Report CMU-CS-91-144, 1991. [3] R. Bellman and D. Blackwell. Some two-person games involving bluffing. PNAS, 35:600-605, 1949. [4] D. Billings, N. Burch, A. Davidson, R. Holte, J. Schaeffer, T. Schauenberg, and D. Szafron. 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Gilpin. Sequences of take-it-or-leave-it offers: Near-optimal auctions without full valuation revelation. In AAMAS, Hakodate, Japan, 2006. [43] T. Sandholm, A. Gilpin, and V. Conitzer. Mixed-integer programming methods for finding Nash equilibria. In AAAI, pages 495-501, Pittsburgh, PA, USA, 2005. [44] R. Savani and B. von Stengel. Exponentially many steps for finding a Nash equilibrium in a bimatrix game. In FOCS, pages 258-267, 2004. [45] R. Selten. Spieltheoretische behandlung eines oligopolmodells mit nachfragetr¨agheit. Zeitschrift f¨ur die gesamte Staatswissenschaft, 12:301-324, 1965. [46] R. Selten. Evolutionary stability in extensive two-person games - correction and further development. Mathematical Social Sciences, 16:223-266, 1988. [47] J. Shi and M. Littman. Abstraction methods for game theoretic poker. In Computers and Games, pages 333-345. Springer-Verlag, 2001. [48] S. J. J. Smith, D. S. Nau, and T. Throop. Computer bridge: A big win for AI planning. 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imperfect information;ordered game isomorphism;related ordered game isomorphic abstraction transformation;nash equilibrium;game theory;strategy profile;computer poker;sequential game;computational game theory;equilibrium find;ordered signal space;observable action;gameshrink;rational behavior;signal tree;automate abstraction;equilibrium;sequential game of imperfect information;normative framework

train_J-38

Multi-Attribute Coalitional Games∗

We study coalitional games where the value of cooperation among the agents are solely determined by the attributes the agents possess, with no assumption as to how these attributes jointly determine this value. This framework allows us to model diverse economic interactions by picking the right attributes. We study the computational complexity of two coalitional solution concepts for these gamesthe Shapley value and the core. We show how the positive results obtained in this paper imply comparable results for other games studied in the literature.

1. INTRODUCTION When agents interact with one another, the value of their contribution is determined by what they can do with their skills and resources, rather than simply their identities. Consider the problem of forming a soccer team. For a team to be successful, a team needs some forwards, midfielders, defenders, and a goalkeeper. The relevant attributes of the players are their skills at playing each of the four positions. The value of a team depends on how well its players can play these positions. At a finer level, we can extend the model to consider a wider range of skills, such as passing, shooting, and tackling, but the value of a team remains solely a function of the attributes of its players. Consider an example from the business world. Companies in the metals industry are usually vertically-integrated and diversified. They have mines for various types of ores, and also mills capable of processing and producing different kinds of metal. They optimize their production profile according to the market prices for their products. For example, when the price of aluminum goes up, they will allocate more resources to producing aluminum. However, each company is limited by the amount of ores it has, and its capacities in processing given kinds of ores. Two or more companies may benefit from trading ores and processing capacities with one another. To model the metal industry, the relevant attributes are the amount of ores and the processing capacities of the companies. Given the exogenous input of market prices, the value of a group of companies will be determined by these attributes. Many real-world problems can be likewise modeled by picking the right attributes. As attributes apply to both individual agents and groups of agents, we propose the use of coalitional game theory to understand what groups may form and what payoffs the agents may expect in such models. Coalitional game theory focuses on what groups of agents can achieve, and thus connects strongly with e-commerce, as the Internet economies have significantly enhanced the abilities of business to identify and capitalize on profitable opportunities of cooperation. Our goal is to understand the computational aspects of computing the solution concepts (stable and/or fair distribution of payoffs, formally defined in Section 3) for coalitional games described using attributes. Our contributions can be summarized as follows: • We define a formal representation for coalitional games based on attributes, and relate this representation to others proposed in the literature. We show that when compared to other representations, there exists games for which a multi-attribute description can be exponentially more succinct, and for no game it is worse. • Given the generality of the model, positive results carry over to other representations. We discuss two positive results in the paper, one for the Shapley value and one for the core, and show how these imply related results in the literature. 170 • We study an approximation heuristic for the Shapley value when its exact values cannot be found efficiently. We provide an explicit bound on the maximum error of the estimate, and show that the bound is asymptotically tight. We also carry out experiments to evaluate how the heuristic performs on random instances.1 2. RELATED WORK Coalitional game theory has been well studied in economics [9, 10, 14]. A vast amount of literature have focused on defining and comparing solution concepts, and determining their existence and properties. The first algorithmic study of coalitional games, as far as we know, is performed by Deng and Papadimitriou in [5]. They consider coalitional games defined on graphs, where the players are the vertices and the value of coalition is determined by the sum of the weights of the edges spanned by these players. This can be efficiently modeled and generalized using attributes. As a formal representation, multi-attribute coalitional games is closely related to the multi-issue representation of Conitzer and Sandholm [3] and our work on marginal contribution networks [7]. Both of these representations are based on dividing a coalitional game into subgames (termed issues in [3] and rules in [7]), and aggregating the subgames via linear combination. The key difference in our work is the unrestricted aggregation of subgames: the aggregation could be via a polynomial function of the attributes, or even by treating the subgames as input to another computational problem such as a min-cost flow problem. The relationship of these models will be made clear after we define the multiattribute representation in Section 4. Another representation proposed in the literature is one specialized for superadditive games by Conitzer and Sandholm [2]. This representation is succinct, but to find the values of some coalitions may require solving an NP-hard problem. While it is possible for multi-attribute coalitional games to efficiently represent these games, it necessarily requires the solution to an NP-hard problem in order to find out the values of some coalitions. In this paper, we stay within the boundary of games that admits efficient algorithm for determining the value of coalitions. We will therefore not make further comparisons with [2]. The model of coalitional games with attributes has been considered in the works of Shehory and Kraus. They model the agents as possessing capabilities that indicates their proficiencies in different areas, and consider how to efficiently allocate tasks [12] and the dynamics of coalition formation [13]. Our work differs significantly as our focus is on reasoning about solution concepts. Our model also covers a wider scope as attributes generalize the notion of capabilities. Yokoo et al. have also considered a model of coalitional games where agents are modeled by sets of skills, and these skills in turn determine the value of coalitions [15]. There are two major differences between their work and ours. Firstly, Yokoo et al. assume that each skill is fundamentally different from another, hence no two agents may possess the same skill. Also, they focus on developing new solution concepts that are robust with respect to manipulation by agents. Our focus is on reasoning about traditional solution concepts. 1 We acknowledge that random instances may not be typical of what happens in practice, but given the generality of our model, it provides the most unbiased view. Our work is also related to the study of cooperative games with committee control [4]. In these games, there is usually an underlying set of resources each controlled by a (possibly overlapping) set of players known as the committee, engaged in a simple game (defined in Section 3). multiattribute coalitional games generalize these by considering relationship between the committee and the resources beyond simple games. We note that when restricted to simple games, we derive similar results to that in [4]. 3. PRELIMINARIES In this section, we will review the relevant concepts of coalitional game theory and its two most important solution concepts - the Shapley value and the core. We will then define the computational questions that will be studied in the second half of the paper. 3.1 Coalitional Games Throughout this paper, we assume that payoffs to groups of agents can be freely distributed among its members. This transferable utility assumption is commonly made in coalitional game theory. The canonical representation of a coalitional game with transferable utility is its characteristic form. Definition 1. A coalition game with transferable utility in characteristic form is denoted by the pair N, v , where • N is the set of agents; and • v : 2N → R is a function that maps each group of agents S ⊆ N to a real-valued payoff. A group of agents in a game is known as a coalition, and the entire set of agents is known as the grand coalition. An important class of coalitional games is the class of monotonic games. Definition 2. A coalitional game is monotonic if for all S ⊂ T ⊆ N, v(S) ≤ v(T). Another important class of coalitional games is the class of simple games. In a simple game, a coalition either wins, in which case it has a value of 1, or loses, in which case it has a value of 0. It is often used to model voting situations. Simple games are often assumed to be monotonic, i.e., if S wins, then for all T ⊇ S, T also wins. This coincides with the notion of using simple games as a model for voting. If a simple game is monotonic, then it is fully described by the set of minimal winning coalitions, i.e., coalitions S for which v(S) = 1 but for all coalitions T ⊂ S, v(T) = 0. An outcome in a coalitional game specifies the utilities the agents receive. A solution concept assigns to each coalitional game a set of reasonable outcomes. Different solution concepts attempt to capture in some way outcomes that are stable and/or fair. Two of the best known solution concepts are the Shapley value and the core. The Shapley value is a normative solution concept that prescribes a fair way to divide the gains from cooperation when the grand coalition is formed. The division of payoff to agent i is the average marginal contribution of agent i over all possible permutations of the agents. Formally, Definition 3. The Shapley value of agent i, φi(v), in game N, v is given by the following formula φi(v) = S⊆N\{i} |S|!(|N| − |S| − 1)! |N|! (v(S ∪ {i}) − v(S)) 171 The core is a descriptive solution concept that focuses on outcomes that are stable. Stability under core means that no set of players can jointly deviate to improve their payoffs. Definition 4. An outcome x ∈ R|N| is in the core of the game N, v if for all S ⊆ N, i∈S xi ≥ v(S) Note that the core of a game may be empty, i.e., there may not exist any payoff vector that satisfies the stability requirement for the given game. 3.2 Computational Problems We will study the following three problems related to solution concepts in coalitional games. Problem 1. (Shapley Value) Given a description of the coalitional game and an agent i, compute the Shapley value of agent i. Problem 2. (Core Membership) Given a description of the coalitional game and a payoff vector x such that È i∈N xi = v(N), determine if È i∈S xi ≥ v(S) for all S ⊆ N. Problem 3. (Core Non-emptiness) Given a description of the coalitional game, determine if there exists any payoff vector x such that È i∈S xi ≥ V (S) for all S ⊆ N, andÈ i∈N xi = v(N). Note that the complexity of the above problems depends on the how the game is described. All these problems will be easy if the game is described by its characteristic form, but only so because the description takes space exponential in the number of agents, and hence simple brute-force approach takes time polynomial to the input description. To properly understand the computational complexity questions, we have to look at compact representation. 4. FORMAL MODEL In this section, we will give a formal definition of multiattribute coalitional games, and show how it is related to some of the representations discussed in the literature. We will also discuss some limitations to our proposed approach. 4.1 Multi-Attribute Coalitional Games A multi-attribute coalitional game (MACG) consists of two parts: a description of the attributes of the agents, which we termed an attribute model, and a function that assigns values to combination of attributes. Together, they induce a coalitional game over the agents. We first define the attribute model. Definition 5. An attribute model is a tuple N, M, A , where • N denotes the set of agents, of size n; • M denotes the set of attributes, of size m; • A ∈ Rm×n , the attribute matrix, describes the values of the attributes of the agents, with Aij denoting the value of attribute i for agent j. We can directly define a function that maps combinations of attributes to real values. However, for many problems, we can describe the function more compactly by computing it in two steps: we first compute an aggregate value for each attribute, then compute the values of combination of attributes using only the aggregated information. Formally, Definition 6. An aggregating function (or aggregator) takes as input a row of the attribute matrix and a coalition S, and summarizes the attributes of the agents in S with a single number. We can treat it as a mapping from Rn × 2N → R. Aggregators often perform basic arithmetic or logical operations. For example, it may compute the sum of the attributes, or evaluate a Boolean expression by treating the agents i ∈ S as true and j /∈ S as false. Analogous to the notion of simple games, we call an aggregator simple if its range is {0, 1}. For any aggregator, there is a set of relevant agents, and a set of irrelevant agents. An agent i is irrelevant to aggregator aj if aj (S ∪ {i}) = aj (S) for all S ⊆ N. A relevant agent is one not irrelevant. Given the attribute matrix, an aggregator assigns a value to each coalition S ⊆ N. Thus, each aggregator defines a game over N. For aggregator aj , we refer to this induced game as the game of attribute j, and denote it with aj (A). When the attribute matrix is clear from the context, we may drop A and simply denote the game as aj . We may refer to the game as the aggregator when no ambiguities arise. We now define the second step of the computation with the help of aggregators. Definition 7. An aggregate value function takes as input the values of the aggregators and maps these to a real value. In this paper, we will focus on having one aggregator per attribute. Therefore, in what follows, we will refer to the aggregate value function as a function over the attributes. Note that when all aggregators are simple, the aggregate value function implicitly defines a game over the attributes, as it assigns a value to each set of attributes T ⊆ M. We refer to this as the game among attributes. We now define multi-attribute coalitional game. Definition 8. A multi-attribute coalitional game is defined by the tuple N, M, A, a, w , where • N, M, A is an attribute model; • a is a set of aggregators, one for each attribute; we can treat the set together as a vector function, mapping Rm×n × 2N → Rm • w : Rm → R is an aggregate value function. This induces a coalitional game with transferable payoffs N, v with players N and the value function defined by v(S) = w(a(A, S)) Note that MACG as defined is fully capable of representing any coalitional game N, v . We can simply take the set of attributes as equal to the set of agents, i.e., M = N, an identity matrix for A, aggregators of sums, and the aggregate value function w to be v. 172 4.2 An Example Let us illustrate how MACG can be used to represent a game with a simple example. Suppose there are four types of resources in the world: gold, silver, copper, and iron, that each agent is endowed with some amount of these resources, and there is a fixed price for each of the resources in the market. This game can be described using MACG with an attribute matrix A, where Aij denote the amount of resource i that agent j is endowed. For each resource, the aggregator sums together the amount of resources the agents have. Finally, the aggregate value function takes the dot product between the market price vector and the aggregate vector. Note the inherent flexibility in the model: only limited work would be required to update the game as the market price changes, or when a new agent arrives. 4.3 Relationship with Other Representations As briefly discussed in Section 2, MACG is closely related to two other representations in the literature, the multiissue representation of Conitzer and Sandholm [3], and our work on marginal contribution nets [7]. To make their relationships clear, we first review these two representations. We have changed the notations from the original papers to highlight their similarities. Definition 9. A multi-issue representation is given as a vector of coalitional games, (v1, v2, . . . vm), each possibly with a varying set of agents, say N1, . . . , Nm. The coalitional game N, v induced by multi-issue representation has player set N = Ëm i=1 Ni, and for each coalition S ⊆ N, v(S) = Èm i=1 v(S ∩ Ni). The games vi are assumed to be represented in characteristic form. Definition 10. A marginal contribution net is given as a set of rules (r1, r2, . . . , rm), where rule ri has a weight wi, and a pattern pi that is a conjunction over literals (positive or negative). The agents are represented as literals. A coalition S is said to satisfy the pattern pi, if we treat the agents i ∈ S as true, an agent j /∈ S as false, pi(S) evaluates to true. Denote the set of literals involved in rule i by Ni. The coalitional game N, v induced by a marginal contribution net has player set N = Ëm i=1 Ni, and for each coalition S ⊆ N, v(S) = È i:pi(S)=true wi. From these definitions, we can see the relationships among these three representations clearly. An issue of a multi-issue representation corresponds to an attribute in MACG. Similarly, a rule of a marginal contribution net corresponds to an attribute in MACG. The aggregate value functions are simple sums and weighted sums for the respective representations. Therefore, it is clear that MACG will be no less succinct than either representation. However, MACG differs in two important way. Firstly, there is no restriction on the operations performed by the aggregate value function over the attributes. This is an important generalization over the linear combination of issues or rules in the other two approaches. In particular, there are games for which MACG can be exponentially more compact. The proof of the following proposition can be found in the Appendix. Proposition 1. Consider the parity game N, v where coalition S ⊆ N has value v(S) = 1 if |S| is odd, and v(S) = 0 otherwise. MACG can represent the game in O(n) space. Both multi-issue representation and marginal contribution nets requires O(2n ) space. A second important difference of MACG is that the attribute model and the value function is cleanly separated. As suggested in the example in Section 4.2, this often allows us more efficient update of the values of the game as it changes. Also, the same attribute model can be evaluated using different value functions, and the same value function can be used to evaluate different attribute model. Therefore, MACG is very suitable for representing multiple games. We believe the problems of updating games and representing multiple games are interesting future directions to explore. 4.4 Limitation of One Aggregator per Attribute Before focusing on one aggregator per attribute for the rest of the paper, it is natural to wonder if any is lost per such restriction. The unfortunate answer is yes, best illustrated by the following. Consider again the problem of forming a soccer team discussed in the introduction, where we model the attributes of the agents as their ability to take the four positions of the field, and the value of a team depends on the positions covered. If we first aggregate each of the attribute individually, we will lose the distributional information of the attributes. In other words, we will not be able to distinguish between two teams, one of which has a player for each position, the other has one player who can play all positions, but the rest can only play the same one position. This loss of distributional information can be recovered by using aggregators that take as input multiple rows of the attribute matrix rather than just a single row. Alternatively, if we leave such attributes untouched, we can leave the burden of correctly evaluating these attributes to the aggregate value function. However, for many problems that we found in the literature, such as the transportation domain of [12] and the flow game setting of [4], the distribution of attributes does not affect the value of the coalitions. In addition, the problem may become unmanageably complex as we introduce more complicated aggregators. Therefore, we will focus on the representation as defined in Definition 8. 5. SHAPLEY VALUE In this section, we focus on computational issues of finding the Shapley value of a player in MACG. We first set up the problem with the use of oracles to avoid complexities arising from the aggregators. We then show that when attributes are linearly separable, the Shapley value can be efficiently computed. This generalizes the proofs of related results in the literature. For the non-linearly separable case, we consider a natural heuristic for estimating the Shapley value, and study the heuristic theoretically and empirically. 5.1 Problem Setup We start by noting that computing the Shapley value for simple aggregators can be hard in general. In particular, we can define aggregators to compute weighted majority over its input set of agents. As noted in [6], finding the Shapley value of a weighted majority game is #P-hard. Therefore, discussion of complexity of Shapley value for MACG with unrestricted aggregators is moot. Instead of placing explicit restriction on the aggregator, we assume that the Shapley value of the aggregator can be 173 answered by an oracle. For notation, let φi(u) denote the Shapley value for some game u. We make the following assumption: Assumption 1. For each aggregator aj in a MACG, there is an associated oracle that answers the Shapley value of the game of attribute j. In other words, φi(aj ) is known. For many aggregators that perform basic operations over its input, polynomial time oracle for Shapley value exists. This include operations such as sum, and symmetric functions when the attributes are restricted to {0, 1}. Also, when only few agents have an effect on the aggregator, brute-force computation for Shapley value is feasible. Therefore, the above assumption is reasonable for many settings. In any case, such abstraction allows us to focus on the aggregate value function. 5.2 Linearly Separable Attributes When the aggregate value function can be written as a linear function of the attributes, the Shapley value of the game can be efficiently computed. Theorem 1. Given a game N, v represented as a MACG N, M, A, a, w , if the aggregate value function can be written as a linear function of its attributes, i.e., w(a(A, S)) = m j=1 cj aj (A, S) The Shapley value of agent i in N, v is given by φi(v) = m j=1 cj φi(aj ) (1) Proof. First, we note that Shapley value satisfies an additivity axiom [11]. The Shapley value satisfies additivity, namely, φi(a + b) = φi(a) + φi(b), where N, a + b is a game defined to be (a + b)(S) = a(S) + b(S) for all S ⊆ N. It is also clear that Shapley value satisfies scaling, namely φi(αv) = αφi(v) where (αv)(S) = αv(S) for all S ⊆ N. Since the aggregate value function can be expressed as a weighted sum of games of attributes, φi(v) = φi(w(a)) = φi( m j=1 cjaj ) = m j=1 cjφi(aj ) Many positive results regarding efficient computation of Shapley value in the literature depends on some form of linearity. Examples include the edge-spanning game on graphs by Deng and Papadimitriou [5], the multi-issue representation of [3], and the marginal contribution nets of [7]. The key to determine if the Shapley value can be efficiently computed depends on the linear separability of attributes. Once this is satisfied, as long as the Shapley value of the game of attributes can be efficiently determined, the Shapley value of the entire game can be efficiently computed. Corollary 1. The Shapley value for the edge-spanning game of [5], games in multi-issue representation [3], and games in marginal contribution nets [7], can be computed in polynomial time. 5.3 Polynomial Combination of Attributes When the aggregate value function cannot be expressed as a linear function of its attributes, computing the Shapley value exactly is difficult. Here, we will focus on aggregate value function that can be expressed as some polynomial of its attributes. If we do not place a limit on the degree of the polynomial, and the game N, v is not necessarily monotonic, the problem is #P-hard. Theorem 2. Computing the Shapley value of a MACG N, M, A, a, w , when w can be an arbitrary polynomial of the aggregates a, is #P-hard, even when the Shapley value of each aggregator can be efficiently computed. The proof is via reduction from three-dimensional matching, and details can be found in the Appendix. Even if we restrict ourselves to monotonic games, and non-negative coefficients for the polynomial aggregate value function, computing the exact Shapley value can still be hard. For example, suppose there are two attributes. All agents in some set B ⊆ N possess the first attribute, and all agents in some set C ⊆ N possess the second, and B and C are disjoint. For a coalition S ⊆ N, the aggregator for the first evaluates to 1 if and only if |S ∩ B| ≥ b , and similarly, the aggregator for the second evaluates to 1 if and only if |S ∩ C| ≥ c . Let the cardinality of the sets B and C be b and c. We can verify that the Shapley value of an agent i in B equals φi = 1 b b −1 i=0 b i ¡  c c −1 ¡ b+c c +i−1 ¡ c − c + 1 b + c − c − i + 1 The equation corresponds to a weighted sum of probability values of hypergeometric random variables. The correspondence with hypergeometric distribution is due to sampling without replacement nature of Shapley value. As far as we know, there is no close-form formula to evaluate the sum above. In addition, as the number of attributes involved increases, we move to multi-variate hypergeometric random variables, and the number of summands grow exponentially in the number of attributes. Therefore, it is highly unlikely that the exact Shapley value can be determined efficiently. Therefore, we look for approximation. 5.3.1 Approximation First, we need a criteria for evaluating how well an estimate, ˆφ, approximates the true Shapley value, φ. We consider the following three natural criteria: • Maximum underestimate: maxi φi/ˆφi • Maximum overestimate: maxi ˆφi/φi • Total variation: 1 2 È i |φi − ˆφi|, or alternatively maxS | È i∈S φi − È i∈S ˆφi| The total variation criterion is more meaningful when we normalize the game to having a value of 1 for the grand coalition, i.e., v(N) = 1. We can also define additive analogues of the under- and overestimates, especially when the games are normalized. 174 We will assume for now that the aggregate value function is a polynomial over the attributes with non-negative coefficients. We will also assume that the aggregators are simple. We will evaluate a specific heuristic that is analogous to Equation (1). Suppose the aggregate function can be written as a polynomial with p terms w(a(A, S)) = p j=1 cj aj(1) (A, S)aj(2) (A, S) · · · aj(kj ) (A, S) (2) For term j, the coefficient of the term is cj , its degree kj , and the attributes involved in the term are j(1), . . . , j(kj ). We compute an estimate ˆφ to the Shapley value as ˆφi = p j=1 kj l=1 cj kj φi(aj(l) ) (3) The idea behind the estimate is that for each term, we divide the value of the term equally among all its attributes. This is represented by the factor cj kj . Then for for each attribute of an agent, we assign the player a share of value from the attribute. This share is determined by the Shapley value of the simple game of that attribute. Without considering the details of the simple games, this constitutes a fair (but blind) rule of sharing. 5.3.2 Theoretical analysis of heuristic We can derive a simple and tight bound for the maximum (multiplicative) underestimate of the heuristic estimate. Theorem 3. Given a game N, v represented as a MACG N, M, A, a, w , suppose w can be expressed as a polynomial function of its attributes (cf Equation (2)). Let K = maxjkj, i.e., the maximum degree of the polynomial. Let ˆφ denote the estimated Shapley value using Equation (3), and φ denote the true Shapley value. For all i ∈ N, φi ≥ K ˆφi. Proof. We bound the maximum underestimate term-byterm. Let tj be the j-th term of the polynomial. We note that the term can be treated as a game among attributes, as it assigns a value to each coalition S ⊆ N. Without loss of generality, renumber attributes j(1) through j(kj ) as 1 through kj. tj (S) = cj kj l=1 al (A, S) To make the equations less cluttered, let B(N, S) = |S|!(|N| − |S| − 1)! |N|! and for a game a, contribution of agent i to group S : i /∈ S, ∆i(a, S) = a(S ∪ {i}) − a(S) The true Shapley value of the game tj is φi(tj) = cj S⊆N\{i} B(N, S)∆i(tj, S) For each coalition S, i /∈ S, ∆i(tj , S) = 1 if and only if for at least one attribute, say l∗ , ∆i(al∗ , S) = 1. Therefore, if we sum over all the attributes, we would have included l∗ for sure. φi(tj) ≤ cj kj j=1 S⊆N\{i} B(N, S)∆i(aj , S) = kj kj j=1 cj kj φi(aj ) = kj ˆφi(T) Summing over the terms, we see that the worst case underestimate is by the maximum degree. Without loss of generality, since the bound is multiplicative, we can normalize the game to having v(N) = 1. As a corollary, because we cannot overestimate any set by more than K times, we obtain a bound on the total variation: Corollary 2. The total variation between the estimated Shapley value and the true Shapley value, for K-degree bounded polynomial aggregate value function, is K−1 K . We can show that this bound is tight. Example 1. Consider a game with n players and K attributes. Let the first (n−1) agents be a member of the first (K − 1) attributes, and that the corresponding aggregator returns 1 if any one of the first (K − 1) agents is present. Let the n-th agent be the sole member of the K-th attribute. The estimated Shapley will assign a value of K−1 K 1 n−1 to the first (n − 1) agents and 1 K to the n-th agent. However, the true Shapley value of the n-th agent tends to 1 as n → ∞, and the total variation approaches K−1 K . In general, we cannot bound how much ˆφ may overestimate the true Shapley value. The problem is that ˆφi may be non-zero for agent i even though may have no influence over the outcome of a game when attributes are multiplied together, as illustrated by the following example. Example 2. Consider a game with 2 players and 2 attributes, and let the first agent be a member of both attributes, and the other agent a member of the second attribute. For a coalition S, the first aggregator evaluates to 1 if agent 1 ∈ S, and the second aggregator evaluates to 1 if both agents are in S. While agent 2 is not a dummy with respect to the second attribute, it is a dummy with respect to the product of the attributes. Agent 2 will be assigned a value of 1 4 by the estimate. As mentioned, for simple monotonic games, a game is fully described by its set of minimal winning coalitions. When the simple aggregators are represented as such, it is possible to check, in polynomial time, for agents turning dummies after attributes are multiplied together. Therefore, we can improve the heuristic estimate in this special case. 5.3.3 Empirical evaluation Due to a lack of benchmark problems for coalitional games, we have tested the heuristic on random instances. We believe more meaningful results can be obtained when we have real instances to test this heuristic on. Our experiment is set up as follows. We control three parameters of the experiment: the number of players (6 − 10), 175 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 6 7 8 9 10 No. of Players TotalVariationDistance 2 3 4 5 (a) Effect of Max Degree 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 6 7 8 9 10 No. of Players TotalVariationDistance 4 5 6 (b) Effect of Number of Attributes Figure 1: Experimental results the number of attributes (3 − 8), and the maximum degree of the polynomial (2 − 5). For each attribute, we randomly sample one to three minimal winning coalitions. We then randomly generate a polynomial of the desired maximum degree with a random number (3 − 12) of terms, each with a random positive weight. We normalize each game to have v(N) = 1. The results of the experiments are shown in Figure 1. The y-axis of the graphs shows the total variation, and the x-axis the number of players. Each datapoint is an average of approximately 700 random samples. Figure 1(a) explores the effect of the maximum degree and the number of players when the number of attributes is fixed (at six). As expected, the total variation increases as the maximum degree increases. On the other hand, there is only a very small increase in error as the number of players increases. The error is nowhere near the theoretical worstcase bound of 1 2 to 4 5 for polynomials of degrees 2 to 5. Figure 1(b) explores the effect of the number of attributes and the number of players when the maximum degree of the polynomial is fixed (at three). We first note that these three lines are quite tightly clustered together, suggesting that the number of attributes has relatively little effect on the error of the estimate. As the number of attributes increases, the total variation decreases. We think this is an interesting phenomenon. This is probably due to the precise construct required for the worst-case bound, and so as more attributes are available, we have more diverse terms in the polynomial, and the diversity pushes away from the worst-case bound. 6. CORE-RELATED QUESTIONS In this section, we look at the complexity of the two computational problems related to the core: Core Nonemptiness and Core Membership. We show that the nonemptiness of core of the game among attributes and the cores of the aggregators imply non-emptiness of the core of the game induced by the MACG. We also show that there appears to be no such general relationship that relates the core memberships of the game among attributes, games of attributes, and game induced by MACG. 6.1 Problem Setup There are many problems in the literature for which the questions of Core Non-emptiness and Core Membership are known to be hard [1]. For example, for the edgespanning game that Deng and Papadimitriou studied [5], both of these questions are coNP-complete. As MACG can model the edge-spanning game in the same amount of space, these hardness results hold for MACG as well. As in the case for computing Shapley value, we attempt to find a way around the hardness barrier by assuming the existence of oracles, and try to build algorithms with these oracles. First, we consider the aggregate value function. Assumption 2. For a MACG N, M, A, a, w , we assume there are oracles that answers the questions of Core Nonemptiness, and Core Membership for the aggregate value function w. When the aggregate value function is a non-negative linear function of its attributes, the core is always non-empty, and core membership can be determined efficiently. The concept of core for the game among attributes makes the most sense when the aggregators are simple games. We will further assume that these simple games are monotonic. Assumption 3. For a MACG N, M, A, a, w , we assume all aggregators are monotonic and simple. We also assume there are oracles that answers the questions of Core Nonemptiness, and Core Membership for the aggregators. We consider this a mild assumption. Recall that monotonic simple games are fully described by their set of minimal winning coalitions (cf Section 3). If the aggregators are represented as such, Core Non-emptiness and Core Membership can be checked in polynomial time. This is due to the following well-known result regarding simple games: Lemma 1. A simple game N, v has a non-empty core if and only if it has a set of veto players, say V , such that v(S) = 0 for all S ⊇ V . Further, A payoff vector x is in the core if and only if xi = 0 for all i /∈ V . 6.2 Core Non-emptiness There is a strong connection between the non-emptiness of the cores of the games among attributes, games of the attributes, and the game induced by a MACG. Theorem 4. Given a game N, v represented as a MACG N, M, A, a, w , if the core of the game among attributes, 176 M, w , is non-empty, and the cores of the games of attributes are non-empty, then the core of N, v is non-empty. Proof. Let u be an arbitrary payoff vector in the core of the game among attributes, M, w . For each attribute j, let θj be an arbitrary payoff vector in the core of the game of attribute j. By Lemma 1, each attribute j must have a set of veto players; let this set be denoted by Pj . For each agent i ∈ N, let yi = È j ujθj i . We claim that this vector y is in the core of N, v . Consider any coalition S ⊆ N, v(S) = w(a(A, S)) ≤ j:S⊇P j uj (4) This is true because an aggregator cannot evaluate to 1 without all members of the veto set. For any attribute j, by Lemma 1, È i∈P j θj i = 1. Therefore, j:S⊇P j uj = j:S⊇P j uj i∈P j θj i = i∈S j:S⊇P j ujθj i ≤ i∈S yi Note that the proof is constructive, and hence if we are given an element in the core of the game among attributes, we can construct an element of the core of the coalitional game. From Theorem 4, we can obtain the following corollaries that have been previously shown in the literature. Corollary 3. The core of the edge-spanning game of [5] is non-empty when the edge weights are non-negative. Proof. Let the players be the vertices, and their attributes the edges incident on them. For each attribute, there is a veto set - namely, both endpoints of the edges. As previously observed, an aggregate value function that is a non-negative linear function of its aggregates has non-empty core. Therefore, the precondition of Theorem 4 is satisfied, and the edge-spanning game with non-negative edge weights has a non-empty core. Corollary 4 (Theorem 1 of [4]). The core of a flow game with committee control, where each edge is controlled by a simple game with a veto set of players, is non-empty. Proof. We treat each edge of the flow game as an attribute, and so each attribute has a veto set of players. The core of a flow game (without committee) has been shown to be non-empty in [8]. We can again invoke Theorem 4 to show the non-emptiness of core for flow games with committee control. However, the core of the game induced by a MACG may be non-empty even when the core of the game among attributes is empty, as illustrated by the following example. Example 3. Suppose the minimal winning coalition of all aggregators in a MACG N, M, A, a, w is N, then v(S) = 0 for all coalitions S ⊂ N. As long as v(N) ≥ 0, any nonnegative vector x that satisfies È i∈N xi = v(N) is in the core of N, v . Complementary to the example above, when all the aggregators have empty cores, the core of N, v is also empty. Theorem 5. Given a game N, v represented as a MACG N, M, A, a, w , if the cores of all aggregators are empty, v(N) > 0, and for each i ∈ N, v({i}) ≥ 0, then the core of N, v is empty. Proof. Suppose the core of N, v is non-empty. Let x be a member of the core, and pick an agent i such that xi > 0. However, for each attribute, since the core is empty, by Lemma 1, there are at least two disjoint winning coalitions. Pick the winning coalition Sj that does not include i for each attribute j. Let S∗ = Ë j Sj . Because S∗ is winning for all coalitions, v(S∗ ) = v(N). However, v(N) = j∈N xj = xi + j /∈N xj ≥ xi + j∈S∗ xj > j∈S∗ xj Therefore, v(S∗ ) > È j∈S∗ xj, contradicting the fact that x is in the core of N, v . We do not have general results regarding the problem of Core Non-emptiness when some of the aggregators have non-empty cores while others have empty cores. We suspect knowledge about the status of the cores of the aggregators alone is insufficient to decide this problem. 6.3 Core Membership Since it is possible for the game induced by the MACG to have a non-empty core when the core of the aggregate value function is empty (Example 3), we try to explore the problem of Core Membership assuming that the core of both the game among attributes, M, w , and the underlying game, N, v , is known to be non-empty, and see if there is any relationship between their members. One reasonable requirement is whether a payoff vector x in the core of N, v can be decomposed and re-aggregated to a payoff vector y in the core of M, w . Formally, Definition 11. We say that a vector x ∈ Rn ≥0 can be decomposed and re-aggregated into a vector y ∈ Rm ≥0 if there exists Z ∈ Rm×n ≥0 , such that yi = n j=1 Zij for all i xj = m i=1 Zij for all j We may refer Z as shares. When there is no restriction on the entries of Z, it is always possible to decompose a payoff vector x in the core of N, v to a payoff vector y in the core of M, w . However, it seems reasonable to restrict that if an agent j is irrelevant to the aggregator i, i.e., i never changes the outcome of aggregator j, then Zij should be restricted to be 0. Unfortunately, this restriction is already too strong. Example 4. Consider a MACG N, M, A, a, w with two players and three attributes. Suppose agent 1 is irrelevant to attribute 1, and agent 2 is irrelevant to attributes 2 and 3. For any set of attributes T ⊆ M, let w be defined as w(T) = 0 if |T| = 0 or 1 6 if |T| = 2 10 if |T| = 3 177 Since the core of a game with a finite number of players forms a polytope, we can verify that the set of vectors (4, 4, 2), (4, 2, 4), and (2, 4, 4), fully characterize the core C of M, w . On the other hand, the vector (10, 0) is in the core of N, v . This vector cannot be decomposed and re-aggregated to a vector in C under the stated restriction. Because of the apparent lack of relationship among members of the core of N, v and that of M, w , we believe an algorithm for testing Core Membership will require more input than just the veto sets of the aggregators and the oracle of Core Membership for the aggregate value function. 7. CONCLUDING REMARKS Multi-attribute coalitional games constitute a very natural way of modeling problems of interest. Its space requirement compares favorably with other representations discussed in the literature, and hence it serves well as a prototype to study computational complexity of coalitional game theory for a variety of problems. Positive results obtained under this representation can easily be translated to results about other representations. Some of these corollary results have been discussed in Sections 5 and 6. An important direction to explore in the future is the question of efficiency in updating a game, and how to evaluate the solution concepts without starting from scratch. As pointed out at the end of Section 4.3, MACG is very naturally suited for updates. Representation results regarding efficiency of updates, and algorithmic results regarding how to compute the different solution concepts from updates, will both be very interesting. Our work on approximating the Shapley value when the aggregate value function is a non-linear function of the attributes suggests more work to be done there as well. Given the natural probabilistic interpretation of the Shapley value, we believe that a random sampling approach may have significantly better theoretical guarantees. 8. REFERENCES [1] J. M. Bilbao, J. R. Fern´andez, and J. J. L´opez. Complexity in cooperative game theory. http://www.esi.us.es/~mbilbao. [2] V. Conitzer and T. Sandholm. Complexity of determining nonemptiness of the core. In Proc. 18th Int. Joint Conf. on Artificial Intelligence, pages 613-618, 2003. [3] V. Conitzer and T. Sandholm. Computing Shapley values, manipulating value division schemes, and checking core membership in multi-issue domains. In Proc. 19th Nat. Conf. on Artificial Intelligence, pages 219-225, 2004. [4] I. J. Curiel, J. J. Derks, and S. H. Tijs. On balanced games and games with committee control. OR Spectrum, 11:83-88, 1989. [5] X. Deng and C. H. Papadimitriou. On the complexity of cooperative solution concepts. Math. Oper. Res., 19:257-266, May 1994. [6] M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York, 1979. [7] S. Ieong and Y. Shoham. Marginal contribution nets: A compact representation scheme for coalitional games. In Proc. 6th ACM Conf. on Electronic Commerce, pages 193-202, 2005. [8] E. Kalai and E. Zemel. Totally balanced games and games of flow. Math. Oper. Res., 7:476-478, 1982. [9] A. Mas-Colell, M. D. Whinston, and J. R. Green. Microeconomic Theory. Oxford University Press, New York, 1995. [10] M. J. Osborne and A. Rubinstein. A Course in Game Theory. The MIT Press, Cambridge, Massachusetts, 1994. [11] L. S. Shapley. A value for n-person games. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games II, number 28 in Annals of Mathematical Studies, pages 307-317. Princeton University Press, 1953. [12] O. Shehory and S. Kraus. Task allocation via coalition formation among autonomous agents. In Proc. 14th Int. Joint Conf. on Artificial Intelligence, pages 31-45, 1995. [13] O. Shehory and S. Kraus. A kernel-oriented model for autonomous-agent coalition-formation in general environments: Implentation and results. In Proc. 13th Nat. Conf. on Artificial Intelligence, pages 134-140, 1996. [14] J. von Neumann and O. Morgenstern. Theory of Games and Economic Behvaior. Princeton University Press, 1953. [15] M. Yokoo, V. Conitzer, T. Sandholm, N. Ohta, and A. Iwasaki. Coalitional games in open anonymous environments. In Proc. 20th Nat. Conf. on Artificial Intelligence, pages 509-515, 2005. Appendix We complete the missing proofs from the main text here. To prove Proposition 1, we need the following lemma. Lemma 2. Marginal contribution nets when all coalitions are restricted to have values 0 or 1 have the same representation power as an AND/OR circuit with negation at the literal level (i.e., AC0 circuit) of depth two. Proof. If a rule assigns a negative value in the marginal contribution nets, we can write the rule by a corresponding set of at most n rules, where n is the number of agents, such that each of which has positive values through application of De Morgan"s Law. With all values of the rules non-negative, we can treat the weighted summation step of marginal contribution nets can be viewed as an OR, and each rule as a conjunction over literals, possibly negated. This exactly match up with an AND/OR circuit of depth two. Proof (Proposition 1). The parity game can be represented with a MACG using a single attribute, aggregator of sum, and an aggregate value function that evaluates that sum modulus two. As a Boolean function, parity is known to require an exponential number of prime implicants. By Lemma 2, a prime implicant is the exact analogue of a pattern in a rule of marginal contribution nets. Therefore, to represent the parity function, a marginal contribution nets must be an exponential number of rules. Finally, as shown in [7], a marginal contribution net is at worst a factor of O(n) less compact than multi-issue representation. Therefore, multi-issue representation will also 178 take exponential space to represent the parity game. This is assuming that each issue in the game is represented in characteristic form. Proof (Theorem 2). An instance of three-dimensional matching is as follows [6]: Given set P ⊆ W × X × Y , where W , X, Y are disjoint sets having the same number q of elements, does there exist a matching P ⊆ P such that |P | = q and no two elements of P agree in any coordinate. For notation, let P = {p1, p2, . . . , pK}. We construct a MACG N, M, A, a, w as follows: • M: Let attributes 1 to q correspond to elements in W , (q+1) to 2q correspond to elements in X, (2q+1) to 3q corresponds to element in Y , and let there be a special attribute (3q + 1). • N: Let player i corresponds to pi, and let there be a special player . • A: Let Aji = 1 if the element corresponding to attribute j is in pi. Thus, for the first K columns, there are exactly three non-zero entries. We also set A(3q+1) = 1. • a: for each aggregator j, aj (A(S)) = 1 if and only if sum of row j of A(S) equals 1. • w: product over all aj . In the game N, v that corresponds to this construction, v(S) = 1 if and only if all attributes are covered exactly once. Therefore, for /∈ T ⊆ N, v(T ∪ { }) − v(T) = 1 if and only if T covers attributes 1 to 3q exactly once. Since all such T, if exists, must be of size q, the number of threedimensional matchings is given by φ (v) (K + 1)! q!(K − q)! 179

cooperation;polynomial function min-cost flow problem;diverse economic interaction;coalitional game theory;coalitional game;linear combination;min-cost flow problem;graph;agent;unrestricted aggregation of subgame;shapley value;multi-issue representation;superadditive game;compact representation;computational complexity;core;multi-attribute model;multi-attribute coalitional game

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The Sequential Auction Problem on eBay: An Empirical Analysis and a Solution

Bidders on eBay have no dominant bidding strategy when faced with multiple auctions each offering an item of interest. As seen through an analysis of 1,956 auctions on eBay for a Dell E193FP LCD monitor, some bidders win auctions at prices higher than those of other available auctions, while others never win an auction despite placing bids in losing efforts that are greater than the closing prices of other available auctions. These misqueues in strategic behavior hamper the efficiency of the system, and in so doing limit the revenue potential for sellers. This paper proposes a novel options-based extension to eBay"s proxy-bidding system that resolves this strategic issue for buyers in commoditized markets. An empirical analysis of eBay provides a basis for computer simulations that investigate the market effects of the options-based scheme, and demonstrates that the options-based scheme provides greater efficiency than eBay, while also increasing seller revenue.

1. INTRODUCTION Electronic markets represent an application of information systems that has generated significant new trading opportunities while allowing for the dynamic pricing of goods. In addition to marketplaces such as eBay, electronic marketplaces are increasingly used for business-to-consumer auctions (e.g. to sell surplus inventory [19]). Many authors have written about a future in which commerce is mediated by online, automated trading agents [10, 25, 1]. There is still little evidence of automated trading in e-markets, though. We believe that one leading place of resistance is in the lack of provably optimal bidding strategies for any but the simplest of market designs. Without this, we do not expect individual consumers, or firms, to be confident in placing their business in the hands of an automated agent. One of the most common examples today of an electronic marketplace is eBay, where the gross merchandise volume (i.e., the sum of all successfully closed listings) during 2005 was $44B. Among items listed on eBay, many are essentially identical. This is especially true in the Consumer Electronics category [9], which accounted for roughly $3.5B of eBay"s gross merchandise volume in 2005. This presence of essentially identical items can expose bidders, and sellers, to risks because of the sequential auction problem. For example, Alice may want an LCD monitor, and could potentially bid in either a 1 o"clock or 3 o"clock eBay auction. While Alice would prefer to participate in whichever auction will have the lower winning price, she cannot determine beforehand which auction that may be, and could end up winning the wrong auction. This is a problem of multiple copies. Another problem bidders may face is the exposure problem. As investigated by Bykowsky et al. [6], exposure problems exist when buyers desire a bundle of goods but may only participate in single-item auctions.1 For example, if Alice values a video game console by itself for $200, a video game by itself for $30, and both a console and game for $250, Alice must determine how much of the $20 of synergy value she might include in her bid for the console alone. Both problems arise in eBay as a result of sequential auctions of single items coupled with patient bidders with substitutes or complementary valuations. Why might the sequential auction problem be bad? Complex games may lead to bidders employing costly strategies and making mistakes. Potential bidders who do not wish to bear such costs may choose not to participate in the 1 The exposure problem has been primarily investigated by Bykowsky et al. in the context of simultaneous single-item auctions. The problem is also a familiar one of online decision making. 180 market, inhibiting seller revenue opportunities. Additionally, among those bidders who do choose to participate, the mistakes made may lead to inefficient allocations, further limiting revenue opportunities. We are interested in creating modifications to eBay-style markets that simplify the bidder problem, leading to simple equilibrium strategies, and preferably better efficiency and revenue properties. 1.1 Options + Proxies: A Proposed Solution Retail stores have developed policies to assist their customers in addressing sequential purchasing problems. Return policies alleviate the exposure problem by allowing customers to return goods at the purchase price. Price matching alleviates the multiple copies problem by allowing buyers to receive from sellers after purchase the difference between the price paid for a good and a lower price found elsewhere for the same good [7, 15, 18]. Furthermore, price matching can reduce the impact of exactly when a seller brings an item to market, as the price will in part be set by others selling the same item. These two retail policies provide the basis for the scheme proposed in this paper.2 We extend the proxy bidding technology currently employed by eBay. Our super-proxy extension will take advantage of a new, real options-based, market infrastructure that enables simple, yet optimal, bidding strategies. The extensions are computationally simple, handle temporal issues, and retain seller autonomy in deciding when to enter the market and conduct individual auctions. A seller sells an option for a good, which will ultimately lead to either a sale of the good or the return of the option. Buyers interact through a proxy agent, defining a value on all possible bundles of goods in which they have interest together with the latest time period in which they are willing to wait to receive the good(s). The proxy agents use this information to determine how much to bid for options, and follow a dominant bidding strategy across all relevant auctions. A proxy agent exercises options held when the buyer"s patience has expired, choosing options that maximize a buyer"s payoff given the reported valuation. All other options are returned to the market and not exercised. The options-based protocol makes truthful and immediate revelation to a proxy a dominant strategy for buyers, whatever the future auction dynamics. We conduct an empirical analysis of eBay, collecting data on over four months of bids for Dell LCD screens (model E193FP) starting in the Summer of 2005. LCD screens are a high-ticket item, for which we demonstrate evidence of the sequential bidding problem. We first infer a conservative model for the arrival time, departure time and value of bidders on eBay for LCD screens during this period. This model is used to simulate the performance of the optionsbased infrastructure, in order to make direct comparisons to the actual performance of eBay in this market. We also extend the work of Haile and Tamer [11] to estimate an upper bound on the distribution of value of eBay bidders, taking into account the sequential auction problem when making the adjustments. Using this estimate, one can approximate how much greater a bidder"s true value is 2 Prior work has shown price matching as a potential mechanism for colluding firms to set monopoly prices. However, in our context, auction prices will be matched, which are not explicitly set by sellers but rather by buyers" bids. from the maximum bid they were observed to have placed on eBay. Based on this approximation, revenue generated in a simulation of the options-based scheme exceeds revenue on eBay for the comparable population and sequence of auctions by 14.8%, while the options-based scheme demonstrates itself as being 7.5% more efficient. 1.2 Related Work A number of authors [27, 13, 28, 29] have analyzed the multiple copies problem, often times in the context of categorizing or modeling sniping behavior for reasons other than those first brought forward by Ockenfels and Roth [20]. These papers perform equilibrium analysis in simpler settings, assuming bidders can participate in at most two auctions. Peters & Severinov [21] extend these models to allow buyers to consider an arbitrary number of auctions, and characterize a perfect Bayesian equilibrium. However, their model does not allow auctions to close at distinct times and does not consider the arrival and departure of bidders. Previous work have developed a data-driven approach toward developing a taxonomy of strategies employed by bidders in practice when facing multi-unit auctions, but have not considered the sequential bidding problem [26, 2]. Previous work has also sought to provide agents with smarter bidding strategies [4, 3, 5, 1]. Unfortunately, it seems hard to design artificial agents with equilibrium bidding strategies, even for a simple simultaneous ascending price auction. Iwasaki et al. [14] have considered the role of options in the context of a single, monolithic, auction design to help bidders with marginal-increasing values avoid exposure in a multi-unit, homogeneous item auction problem. In other contexts, options have been discussed for selling coal mine leases [23], or as leveled commitment contracts for use in a decentralized market place [24]. Most similar to our work, Gopal et al. [9] use options for reducing the risks of buyers and sellers in the sequential auction problem. However, their work uses costly options and does not remove the sequential bidding problem completely. Work on online mechanisms and online auctions [17, 12, 22] considers agents that can dynamically arrive and depart across time. We leverage a recent price-based characterization by Hajiaghayi et al. [12] to provide a dominant strategy equilibrium for buyers within our options-based protocol. The special case for single-unit buyers is equivalent to the protocol of Hajiaghayi et al., albeit with an options-based interpretation. Jiang and Leyton-Brown [16] use machine learning techniques for bid identification in online auctions. 2. EBAY AND THE DELL E193FP The most common type of auction held on eBay is a singleitem proxy auction. Auctions open at a given time and remain open for a set period of time (usually one week). Bidders bid for the item by giving a proxy a value ceiling. The proxy will bid on behalf of the bidder only as much as is necessary to maintain a winning position in the auction, up to the ceiling received from the bidder. Bidders may communicate with the proxy multiple times before an auction closes. In the event that a bidder"s proxy has been outbid, a bidder may give the proxy a higher ceiling to use in the auction. eBay"s proxy auction implements an incremental version of a Vickrey auction, with the item sold to the highest bidder for the second-highest bid plus a small increment. 181 10 0 10 1 10 2 10 3 10 4 10 0 10 1 10 2 10 3 10 4 Number of Auctions NumberofBidders Auctions Available Auctions in Which Bid Figure 1: Histogram of number of LCD auctions available to each bidder and number of LCD auctions in which a bidder participates. The market analyzed in this paper is that of a specific model of an LCD monitor, a 19 Dell LCD model E193FP. This market was selected for a variety of reasons including: • The mean price of the monitor was $240 (with standard deviation $32), so we believe it reasonable to assume that bidders on the whole are only interested in acquiring one copy of the item on eBay.3 • The volume transacted is fairly high, at approximately 500 units sold per month. • The item is not usually bundled with other items. • The item is typically sold as new, and so suitable for the price-matching of the options-based scheme. Raw auction information was acquired via a PERL script. The script accesses the eBay search engine,4 and returns all auctions containing the terms ‘Dell" and ‘LCD" that have closed within the past month.5 Data was stored in a text file for post-processing. To isolate the auctions in the domain of interest, queries were made against the titles of eBay auctions that closed between 27 May, 2005 through 1 October, 2005.6 Figure 1 provides a general sense of how many LCD auctions occur while a bidder is interested in pursuing a monitor.7 8,746 bidders (86%) had more than one auction available between when they first placed a bid on eBay and the 3 For reference, Dell"s October 2005 mail order catalogue quotes the price of the monitor as being $379 without a desktop purchase, and $240 as part of a desktop purchase upgrade. 4 http://search.ebay.com 5 The search is not case-sensitive. 6 Specifically, the query found all auctions where the title contained all of the following strings: ‘Dell," ‘LCD" and ‘E193FP," while excluding all auctions that contained any of the following strings: ‘Dimension," ‘GHZ," ‘desktop," ‘p4" and ‘GB." The exclusion terms were incorporated so that the only auctions analyzed would be those selling exclusively the LCD of interest. For example, the few bundled auctions selling both a Dell Dimension desktop and the E193FP LCD are excluded. 7 As a reference, most auctions close on eBay between noon and midnight EDT, with almost two auctions for the Dell LCD monitor closing each hour on average during peak time periods. Bidders have an average observed patience of 3.9 days (with a standard deviation of 11.4 days). latest closing time of an auction in which they bid (with an average of 78 auctions available). Figure 1 also illustrates the number of auctions in which each bidder participates. Only 32.3% of bidders who had more than one auction available are observed to bid in more than one auction (bidding in 3.6 auctions on average). A simple regression analysis shows that bidders tend to submit maximal bids to an auction that are $1.22 higher after spending twice as much time in the system, as well as bids that are $0.27 higher in each subsequent auction. Among the 508 bidders that won exactly one monitor and participated in multiple auctions, 201 (40%) paid more than $10 more than the closing price of another auction in which they bid, paying on average $35 more (standard deviation $21) than the closing price of the cheapest auction in which they bid but did not win. Furthermore, among the 2,216 bidders that never won an item despite participating in multiple auctions, 421 (19%) placed a losing bid in one auction that was more than $10 higher than the closing price of another auction in which they bid, submitting a losing bid on average $34 more (standard deviation $23) than the closing price of the cheapest auction in which they bid but did not win. Although these measures do not say a bidder that lost could have definitively won (because we only consider the final winning price and not the bid of the winner to her proxy), or a bidder that won could have secured a better price, this is at least indicative of some bidder mistakes. 3. MODELING THE SEQUENTIAL AUCTION PROBLEM While the eBay analysis was for simple bidders who desire only a single item, let us now consider a more general scenario where people may desire multiple goods of different types, possessing general valuations over those goods. Consider a world with buyers (sometimes called bidders) B and K different types of goods G1...GK . Let T = {0, 1, ...} denote time periods. Let L denote a bundle of goods, represented as a vector of size K, where Lk ∈ {0, 1} denotes the quantity of good type Gk in the bundle.8 The type of a buyer i ∈ B is (ai, di, vi), with arrival time ai ∈ T, departure time di ∈ T, and private valuation vi(L) ≥ 0 for each bundle of goods L received between ai and di, and zero value otherwise. The arrival time models the period in which a buyer first realizes her demand and enters the market, while the departure time models the period in which a buyer loses interest in acquiring the good(s). In settings with general valuations, we need an additional assumption: an upper bound on the difference between a buyer"s arrival and departure, denoted ΔMax. Buyers have quasi-linear utilities, so that the utility of buyer i receiving bundle L and paying p, in some period no later than di, is ui(L, p) = vi(L) − p. Each seller j ∈ S brings a single item kj to the market, has no intrinsic value and wants to maximize revenue. Seller j has an arrival time, aj, which models the period in which she is first interested in listing the item, while the departure time, dj, models the latest period in which she is willing to consider having an auction for the item close. A seller will receive payment by the end of the reported departure of the winning buyer. 8 We extend notation whereby a single item k of type Gk refers to a vector L : Lk = 1. 182 We say an individual auction in a sequence is locally strategyproof (LSP) if truthful bidding is a dominant strategy for a buyer that can only bid in that auction. Consider the following example to see that LSP is insufficient for the existence of a dominant bidding strategy for buyers facing a sequence of auctions. Example 1. Alice values one ton of Sand with one ton of Stone at $2, 000. Bob holds a Vickrey auction for one ton of Sand on Monday and a Vickrey auction for one ton of Stone on Tuesday. Alice has no dominant bidding strategy because she needs to know the price for Stone on Tuesday to know her maximum willingness to pay for Sand on Monday. Definition 1. The sequential auction problem. Given a sequence of auctions, despite each auction being locally strategyproof, a bidder has no dominant bidding strategy. Consider a sequence of auctions. Generally, auctions selling the same item will be uncertainly-ordered, because a buyer will not know the ordering of closing prices among the auctions. Define the interesting bundles for a buyer as all bundles that could maximize the buyer"s profit for some combination of auctions and bids of other buyers.9 Within the interesting bundles, say that an item has uncertain marginal value if the marginal value of an item depends on the other goods held by the buyer.10 Say that an item is oversupplied if there is more than one auction offering an item of that type. Say two bundles are substitutes if one of those bundles has the same value as the union of both bundles.11 Proposition 1. Given locally strategyproof single-item auctions, the sequential auction problem exists for a bidder if and only if either of the following two conditions is true: (1) within the set of interesting bundles (a) there are two bundles that are substitutes, (b) there is an item with uncertain marginal value, or (c) there is an item that is over-supplied; (2) a bidder faces competitors" bids that are conditioned on the bidder"s past bids. Proof. (Sketch.)(⇐) A bidder does not have a dominant strategy when (a) she does not know which bundle among substitutes to pursue, (b) she faces the exposure problem, or (c) she faces the multiple copies problem. Additionally, a bidder does not have a dominant strategy when she does not how to optimally influence the bids of competitors.(⇒) By contradiction. A bidder has a dominant strategy to bid its constant marginal value for a given item in each auction available when conditions (1) and (2) are both false. For example, the following buyers all face the sequential auction problem as a result of condition (a), (b) and (c) respectively: a buyer who values one ton of Sand for $1,000, or one ton of Stone for $2,000, but not both Sand and Stone; a buyer who values one ton of Sand for $1,000, one ton of Stone for $300, and one ton of Sand and one ton of Stone for $1,500, and can participate in an auction for Sand before an auction for Stone; a buyer who values one ton of Sand for $1,000 and can participate in many auctions selling Sand. 9 Assume that the empty set is an interesting bundle. 10 Formally, an item k has uncertain marginal value if |{m : m = vi(Q) − vi(Q − k), ∀Q ⊆ L ∈ InterestingBundle, Q ⊇ k}| > 1. 11 Formally, two bundles A and B are substitutes if vi(A ∪ B) = max(vi(A), vi(B)), where A ∪ B = L where Lk = max(Ak, Bk). 4. SUPER PROXIES AND OPTIONS The novel solution proposed in this work to resolve the sequential auction problem consists of two primary components: richer proxy agents, and options with price matching. In finance, a real option is a right to acquire a real good at a certain price, called the exercise price. For instance, Alice may obtain from Bob the right to buy Sand from him at an exercise price of $1, 000. An option provides the right to purchase a good at an exercise price but not the obligation. This flexibility allows buyers to put together a collection of options on goods and then decide which to exercise. Options are typically sold at a price called the option price. However, options obtained at a non-zero option price cannot generally support a simple, dominant bidding strategy, as a buyer must compute the expected value of an option to justify the cost [8]. This computation requires a model of the future, which in our setting requires a model of the bidding strategies and the values of other bidders. This is the very kind of game-theoretic reasoning that we want to avoid. Instead, we consider costless options with an option price of zero. This will require some care as buyers are weakly better off with a costless option than without one, whatever its exercise price. However, multiple bidders pursuing options with no intention of exercising them would cause the efficiency of an auction for options to unravel. This is the role of the mandatory proxy agents, which intermediate between buyers and the market. A proxy agent forces a link between the valuation function used to acquire options and the valuation used to exercise options. If a buyer tells her proxy an inflated value for an item, she runs the risk of having the proxy exercise options at a price greater than her value. 4.1 Buyer Proxies 4.1.1 Acquiring Options After her arrival, a buyer submits her valuation ˆvi (perhaps untruthfully) to her proxy in some period ˆai ≥ ai, along with a claim about her departure time ˆdi ≥ ˆai. All transactions are intermediated via proxy agents. Each auction is modified to sell an option on that good to the highest bidding proxy, with an initial exercise price set to the second-highest bid received.12 When an option in which a buyer is interested becomes available for the first time, the proxy determines its bid by computing the buyer"s maximum marginal value for the item, and then submits a bid in this amount. A proxy does not bid for an item when it already holds an option. The bid price is: bidt i(k) = max L [ˆvi(L + k) − ˆvi(L)] (1) By having a proxy compute a buyer"s maximum marginal value for an item and then bidding only that amount, a buyer"s proxy will win any auction that could possibly be of benefit to the buyer and only lose those auctions that could never be of value to the buyer. 12 The system can set a reserve price for each good, provided that the reserve is universal for all auctions selling the same item. Without a universal reserve price, price matching is not possible because of the additional restrictions on prices that individual sellers will accept. 183 Buyer Type Monday Tuesday Molly (Mon, Tues, $8) 6Nancy 6Nancy → 4Polly Nancy (Mon, Tues, $6) - 4Polly Polly (Mon, Tues, $4) -Table 1: Three-buyer example with each wanting a single item and one auction occurring on Monday and Tuesday. XY implies an option with exercise price X and bookkeeping that a proxy has prevented Y from currently possessing an option. → is the updating of exercise price and bookkeeping. When a proxy wins an auction for an option, the proxy will store in its local memory the identity (which may be a pseudonym) of the proxy not holding an option because of the proxy"s win (i.e., the proxy that it ‘bumped" from winning, if any). This information will be used for price matching. 4.1.2 Pricing Options Sellers agree by joining the market to allow the proxy representing a buyer to adjust the exercise price of an option that it holds downwards if the proxy discovers that it could have achieved a better price by waiting to bid in a later auction for an option on the same good. To assist in the implementation of the price matching scheme each proxy tracks future auctions for an option that it has already won and will determine who would be bidding in that auction had the proxy delayed its entry into the market until this later auction. The proxy will request price matching from the seller that granted it an option if the proxy discovers that it could have secured a lower price by waiting. To reiterate, the proxy does not acquire more than one option for any good. Rather, it reduces the exercise price on its already issued option if a better deal is found. The proxy is able to discover these deals by asking each future auction to report the identities of the bidders in that auction together with their bids. This needs to be enforced by eBay, as the central authority. The highest bidder in this later auction, across those whose identity is not stored in the proxy"s memory for the given item, is exactly the bidder against whom the proxy would be competing had it delayed its entry until this auction. If this high bid is lower than the current option price held, the proxy price matches down to this high bid price. After price matching, one of two adjustments will be made by the proxy for bookkeeping purposes. If the winner of the auction is the bidder whose identity has been in the proxy"s local memory, the proxy will replace that local information with the identity of the bidder whose bid it just price matched, as that is now the bidder the proxy has prevented from obtaining an option. If the auction winner"s identity is not stored in the proxy"s local memory the memory may be cleared. In this case, the proxy will simply price match against the bids of future auction winners on this item until the proxy departs. Example 2 (Table 1). Molly"s proxy wins the Monday auction, submitting a bid of $8 and receiving an option for $6. Molly"s proxy adds Nancy to its local memory as Nancy"s proxy would have won had Molly"s proxy not bid. On Tuesday, only Nancy"s and Polly"s proxy bid (as Molly"s proxy holds an option), with Nancy"s proxy winning an opBuyer Type Monday Tuesday Truth: Molly (Mon, Mon, $8) 6NancyNancy (Mon, Tues, $6) - 4Polly Polly (Mon, Tues, $4) -Misreport: Molly (Mon, Mon, $8) -Nancy (Mon, Tues, $10) 8Molly 8Molly → 4φ Polly (Mon, Tues, $4) - 0φ Misreport & match low: Molly (Mon, Mon, $8) -Nancy (Mon, Tues, $10) 8 8 → 0 Polly (Mon, Tues, $4) - 0 Table 2: Examples demonstrating why bookkeeping will lead to a truthful system whereas simply matching to the lowest winning price will not. tion for $4 and noting that it bumped Polly"s proxy. At this time, Molly"s proxy will price match its option down to $4 and replace Nancy with Polly in its local memory as per the price match algorithm, as Polly would be holding an option had Molly never bid. 4.1.3 Exercising Options At the reported departure time the proxy chooses which options to exercise. Therefore, a seller of an option must wait until period ˆdw for the option to be exercised and receive payment, where w was the winner of the option.13 For bidder i, in period ˆdi, the proxy chooses the option(s) that maximize the (reported) utility of the buyer: θ∗ t = argmax θ⊆Θ (ˆvi(γ(θ)) − π(θ)) (2) where Θ is the set of all options held, γ(θ) are the goods corresponding to a set of options, and π(θ) is the sum of exercise prices for a set of options. All other options are returned.14 No options are exercised when no combination of options have positive utility. 4.1.4 Why bookkeep and not match winning price? One may believe that an alternative method for implementing a price matching scheme could be to simply have proxies match the lowest winning price they observe after winning an option. However, as demonstrated in Table 2, such a simple price matching scheme will not lead to a truthful system. The first scenario in Table 2 demonstrates the outcome if all agents were to truthfully report their types. Molly 13 While this appears restrictive on the seller, we believe it not significantly different than what sellers on eBay currently endure in practice. An auction on eBay closes at a specific time, but a seller must wait until a buyer relinquishes payment before being able to realize the revenue, an amount of time that could easily be days (if payment is via a money order sent through courier) to much longer (if a buyer is slow but not overtly delinquent in remitting her payment). 14 Presumably, an option returned will result in the seller holding a new auction for an option on the item it still possesses. However, the system will not allow a seller to re-auction an option until ΔMax after the option had first been issued in order to maintain a truthful mechanism. 184 would win the Monday auction and receive an option with an exercise price of $6 (subsequently exercising that option at the end of Monday), and Nancy would win the Tuesday auction and receive an option with an exercise price of $4 (subsequently exercising that option at the end of Tuesday). The second scenario in Table 2 demonstrates the outcome if Nancy were to misreport her value for the good by reporting an inflated value of $10, using the proposed bookkeeping method. Nancy would win the Monday auction and receive an option with an exercise price of $8. On Tuesday, Polly would win the auction and receive an option with an exercise price of $0. Nancy"s proxy would observe that the highest bid submitted on Tuesday among those proxies not stored in local memory is Polly"s bid of $4, and so Nancy"s proxy would price match the exercise price of its option down to $4. Note that the exercise price Nancy"s proxy has obtained at the end of Tuesday is the same as when she truthfully revealed her type to her proxy. The third scenario in Table 2 demonstrates the outcome if Nancy were to misreport her value for the good by reporting an inflated value of $10, if the price matching scheme were for proxies to simply match their option price to the lowest winning price at any time while they are in the system. Nancy would win the Monday auction and receive an option with an exercise price of $8. On Tuesday, Polly would win the auction and receive an option with an exercise price of $0. Nancy"s proxy would observe that the lowest price on Tuesday was $0, and so Nancy"s proxy would price match the exercise price of its option down to $0. Note that the exercise price Nancy"s proxy has obtained at the end of Tuesday is lower than when she truthfully revealed her type to the proxy. Therefore, a price matching policy of simply matching the lowest price paid may not elicit truthful information from buyers. 4.2 Complexity of Algorithm An XOR-valuation of size M for buyer i is a set of M terms, < L1 , v1 i > ...< LM , vM i >, that maps distinct bundles to values, where i is interested in acquiring at most one such bundle. For any bundle S, vi(S) = maxLm⊆S(vm i ). Theorem 1. Given an XOR-valuation which possesses M terms, there is an O(KM2 ) algorithm for computing all maximum marginal values, where K is the number of different item types in which a buyer may be interested. Proof. For each item type, recall Equation 1 which defines the maximum marginal value of an item. For each bundle L in the M-term valuation, vi(L + k) may be found by iterating over the M terms. Therefore, the number of terms explored to determine the maximum marginal value for any item is O(M2 ), and so the total number of bundle comparisons to be performed to calculate all maximum marginal values is O(KM2 ). Theorem 2. The total memory required by a proxy for implementing price matching is O(K), where K is the number of distinct item types. The total work performed by a proxy to conduct price matching in each auction is O(1). Proof. By construction of the algorithm, the proxy stores one maximum marginal value for each item for bidding, of which there are O(K); at most one buyer"s identity for each item, of which there are O(K); and one current option exercise price for each item, of which there are O(K). For each auction, the proxy either submits a precomputed bid or price matches, both of which take O(1) work. 4.3 Truthful Bidding to the Proxy Agent Proxies transform the market into a direct revelation mechanism, where each buyer i interacts with the proxy only once,15 and does so by declaring a bid, bi, which is defined as an announcement of her type, (ˆai, ˆdi, ˆvi), where the announcement may or may not be truthful. We denote all received bids other than i"s as b−i. Given bids, b = (bi, b−i), the market determines allocations, xi(b), and payments, pi(b) ≥ 0, to each buyer (using an online algorithm). A dominant strategy equilibrium for buyers requires that vi(xi(bi, b−i))−pi(bi, b−i) ≥ vi(xi(bi, b−i))−pi(bi, b−i), ∀bi = bi, ∀b−i. We now establish that it is a dominant strategy for a buyer to reveal her true valuation and true departure time to her proxy agent immediately upon arrival to the system. The proof builds on the price-based characterization of strategyproof single-item online auctions in Hajiaghayi et al. [12]. Define a monotonic and value-independent price function psi(ai, di, L, v−i) which can depend on the values of other agents v−i. Price psi(ai, di, L, v−i) will represent the price available to agent i for bundle L in the mechanism if it announces arrival time ai and departure time di. The price is independent of the value vi of agent i, but can depend on ai, di and L as long as it satisfies a monotonicity condition. Definition 2. Price function psi(ai, di, L, v−i) is monotonic if psi(ai, di, L , v−i) ≤ psi(ai, di, L, v−i) for all ai ≤ ai, all di ≥ di, all bundles L ⊆ L and all v−i. Lemma 1. An online combinatorial auction will be strategyproof (with truthful reports of arrival, departure and value a dominant strategy) when there exists a monotonic and value-independent price function, psi(ai, di, L, v−i), such that for all i and all ai, di ∈ T and all vi, agent i is allocated bundle L∗ = argmaxL [vi(L) − psi(ai, di, L, v−i)] in period di and makes payment psi(ai, di, L∗ , v−i). Proof. Agent i cannot benefit from reporting a later departure ˆdi because the allocation is made in period ˆdi and the agent would have no value for this allocation. Agent i cannot benefit from reporting a later arrival ˆai ≥ ai or earlier departure ˆdi ≤ di because of price monotonicity. Finally, the agent cannot benefit from reporting some ˆvi = vi because its reported valuation does not change the prices it faces and the mechanism maximizes its utility given its reported valuation and given the prices. Lemma 2. At any given time, there is at most one buyer in the system whose proxy does not hold an option for a given item type because of buyer i"s presence in the system, and the identity of that buyer will be stored in i"s proxy"s local memory at that time if such a buyer exists. Proof. By induction. Consider the first proxy that a buyer prevents from winning an option. Either (a) the 15 For analysis purposes, we view the mechanism as an opaque market so that the buyer cannot condition her bid on bids placed by others. 185 bumped proxy will leave the system having never won an option, or (b) the bumped proxy will win an auction in the future. If (a), the buyer"s presence prevented exactly that one buyer from winning an option, but will have not prevented any other proxies from winning an option (as the buyer"s proxy will not bid on additional options upon securing one), and will have had that bumped proxy"s identity in its local memory by definition of the algorithm. If (b), the buyer has not prevented the bumped proxy from winning an option after all, but rather has prevented only the proxy that lost to the bumped proxy from winning (if any), whose identity will now be stored in the proxy"s local memory by definition of the algorithm. For this new identity in the buyer"s proxy"s local memory, either scenario (a) or (b) will be true, ad infinitum. Given this, we show that the options-based infrastructure implements a price-based auction with a monotonic and value-independent price schedule to every agent. Theorem 3. Truthful revelation of valuation, arrival and departure is a dominant strategy for a buyer in the optionsbased market. Proof. First, define a simple agent-independent price function pk i (t, v−i) as the highest bid by the proxies not holding an option on an item of type Gk at time t, not including the proxy representing i herself and not including any proxies that would have already won an option had i never entered the system (i.e., whose identity is stored in i"s proxy"s local memory)(∞ if no supply at t). This set of proxies is independent of any declaration i makes to its proxy (as the set explicitly excludes the at most one proxy (see Lemma 2) that i has prevented from holding an option), and each bid submitted by a proxy within this set is only a function of their own buyer"s declared valuation (see Equation 1). Furthermore, i cannot influence the supply she faces as any options returned by bidders due to a price set by i"s proxy"s bid will be re-auctioned after i has departed the system. Therefore, pk i (t, v−i) is independent of i"s declaration to its proxy. Next, define psk i (ˆai, ˆdi, v−i) = minˆai≤τ≤ ˆdi [pk i (τ, v−i)] (possibly ∞) as the minimum price over pk i (t, v−i), which is clearly monotonic. By construction of price matching, this is exactly the price obtained by a proxy on any option that it holds at departure. Define psi(ˆai, ˆdi, L, v−i) = Èk=K k=1 psk i (ˆai, ˆdi, v−i)Lk, which is monotonic in ˆai, ˆdi and L since psk i (ˆai, ˆdi, v−i) is monotonic in ˆai and ˆdi and (weakly) greater than zero for each k. Given the set of options held at ˆdi, which may be a subset of those items with non-infinite prices, the proxy exercises options to maximize the reported utility. Left to show is that all bundles that could not be obtained with options held are priced sufficiently high as to not be preferred. For each such bundle, either there is an item priced at ∞ (in which case the bundle would not be desired) or there must be an item in that bundle for which the proxy does not hold an option that was available. In all auctions for such an item there must have been a distinct bidder with a bid greater than bidt i(k), which subsequently results in psk i (ˆai, ˆdi, v−i) > bidt i(k), and so the bundle without k would be preferred to the bundle. Theorem 4. The super proxy, options-based scheme is individually-rational for both buyers and sellers. Price σ(Price) Value Surplus eBay $240.24 $32 $244 $4 Options $239.66 $12 $263 $23 Table 3: Average price paid, standard deviation of prices paid, average bidder value among winners, and average winning bidder surplus on eBay for Dell E193FP LCD screens as well as the simulated options-based market using worst-case estimates of bidders" true value. Proof. By construction, the proxy exercises the profit maximizing set of options obtained, or no options if no set of options derives non-negative surplus. Therefore, buyers are guaranteed non-negative surplus by participating in the scheme. For sellers, the price of each option is based on a non-negative bid or zero. 5. EVALUATING THE OPTIONS / PROXY INFRASTRUCTURE A goal of the empirical benchmarking and a reason to collect data from eBay is to try and build a realistic model of buyers from which to estimate seller revenue and other market effects under the options-based scheme. We simulate a sequence of auctions that match the timing of the Dell LCD auctions on eBay.16 When an auction successfully closes on eBay, we simulate a Vickrey auction for an option on the item sold in that period. Auctions that do not successfully close on eBay are not simulated. We estimate the arrival, departure and value of each bidder on eBay from their observed behavior.17 Arrival is estimated as the first time that a bidder interacts with the eBay proxy, while departure is estimated as the latest closing time among eBay auctions in which a bidder participates. We initially adopt a particularly conservative estimate for bidder value, estimating it as the highest bid a bidder was observed to make on eBay. Table 3 compares the distribution of closing prices on eBay and in the simulated options scheme. While the average revenue in both schemes is virtually the same ($239.66 in the options scheme vs. $240.24 on eBay), the winners in the options scheme tend to value the item won 7% more than the winners on eBay ($263 in the options scheme vs. $244 on eBay). 5.1 Bid Identification We extend the work of Haile and Tamer [11] to sequential auctions to get a better view of underlying bidder values. Rather than assume for bidders an equilibrium behavior as in standard econometric techniques, Haile and Tamer do not attempt to model how bidders" true values get mapped into a bid in any given auction. Rather, in the context of repeated 16 When running the simulations, the results of the first and final ten days of auctions are not recorded to reduce edge effects that come from viewing a discrete time window of a continuous process. 17 For the 100 bidders that won multiple times on eBay, we have each one bid a constant marginal value for each additional item in each auction until the number of options held equals the total number of LCDs won on eBay, with each option available for price matching independently. This bidding strategy is not a dominant strategy (falling outside the type space possible for buyers on which the proof of truthfulness has been built), but is believed to be the most appropriate first order action for simulation. 186 0 100 200 300 400 500 0 0.2 0.4 0.6 0.8 1 Value ($) CDF Observed Max Bids Upper Bound of True Value Figure 2: CDF of maximum bids observed and upper bound estimate of the bidding population"s distribution for maximum willingness to pay. The true population distribution lies below the estimated upper bound. single-item auctions with distinct bidder populations, Haile and Tamer make only the following two assumptions when estimating the distribution of true bidder values: 1. Bidders do not bid more than they are willing to pay. 2. Bidders do not allow an opponent to win at a price they are willing to beat. From the first of their two assumptions, given the bids placed by each bidder in each auction, Haile and Tamer derive a method for estimating an upper bound of the bidding population"s true value distribution (i.e., the bound that lies above the true value distribution). From the second of their two assumptions, given the winning price of each auction, Haile and Tamer derive a method for estimating a lower bound of the bidding population"s true value distribution. It is only the upper-bound of the distribution that we utilize in our work. Haile and Tamer assume that bidders only participate in a single auction, and require independence of the bidding population from auction to auction. Neither assumption is valid here: the former because bidders are known to bid in more than one auction, and the latter because the set of bidders in an auction is in all likelihood not a true i.i.d. sampling of the overall bidding population. In particular, those who win auctions are less likely to bid in successive auctions, while those who lose auctions are more likely to remain bidders in future auctions. In applying their methods we make the following adjustments: • Within a given auction, each individual bidder"s true willingness to pay is assumed weakly greater than the maximum bid that bidder submits across all auctions for that item (either past or future). • When estimating the upper bound of the value distribution, if a bidder bids in more than one auction, randomly select one of the auctions in which the bidder bid, and only utilize that one observation during the estimation.18 18 In current work, we assume that removing duplicate bidders is sufficient to make the buying populations independent i.i.d. draws from auction to auction. If one believes that certain portions of the population are drawn to cerPrice σ(Price) Value Surplus eBay $240.24 $32 $281 $40 Options $275.80 $14 $302 $26 Table 4: Average price paid, standard deviation of prices paid, average bidder value among winners, and average winning bidder surplus on eBay for Dell E193FP LCD screens as well as in the simulated options-based market using an adjusted Haile and Tamer estimate of bidders" true values being 15% higher than their maximum observed bid. Figure 2 provides the distribution of maximum bids placed by bidders on eBay as well as the estimated upper bound of the true value distribution of bidders based on the extended Haile and Tamer method.19 As can be seen, the smallest relative gap between the two curves meaningfully occurs near the 80th percentile, where the upper bound is 1.17 times the maximum bid. Therefore, adopted as a less conservative model of bidder values is a uniform scaling factor of 1.15. We now present results from this less conservative analysis. Table 4 shows the distribution of closing prices in auctions on eBay and in the simulated options scheme. The mean price in the options scheme is now significantly higher, 15% greater, than the prices on eBay ($276 in the options scheme vs. $240 on eBay), while the standard deviation of closing prices is lower among the options scheme auctions ($14 in the options scheme vs. $32 on eBay). Therefore, not only is the expected revenue stream higher, but the lower variance provides sellers a greater likelihood of realizing that higher revenue. The efficiency of the options scheme remains higher than on eBay. The winners in the options scheme now have an average estimated value 7.5% higher at $302. In an effort to better understand this efficiency, we formulated a mixed integer program (MIP) to determine a simple estimate of the allocative efficiency of eBay. The MIP computes the efficient value of the offline problem with full hindsight on all bids and all supply.20 Using a scaling of 1.15, the total value allocated to eBay winners is estimated at $551,242, while the optimal value (from the MIP) is $593,301. This suggests an allocative efficiency of 92.9%: while the typical value of a winner on eBay is $281, an average value of $303 was possible.21 Note the options-based tain auctions, then further adjustments would be required in order to utilize these techniques. 19 The estimation of the points in the curve is a minimization over many variables, many of which can have smallnumbers bias. Consequently, Haile and Tamer suggest using a weighted average over all terms yi of È i yi exp(yiρ)Èj exp(yj ρ) to approximate the minimum while reducing the small number effects. We used ρ = −1000 and removed observations of auctions with 17 bidders or more as they occurred very infrequently. However, some small numbers bias still demonstrated itself with the plateau in our upper bound estimate around a value of $300. 20 Buyers who won more than one item on eBay are cloned so that they appear to be multiple bidders of identical type. 21 As long as one believes that every bidder"s true value is a constant factor α away from their observed maximum bid, the 92.9% efficiency calculation holds for any value of α. In practice, this belief may not be reasonable. For example, if losing bidders tend to have true values close to their observed 187 scheme comes very close to achieving this level of efficiency [at 99.7% efficient in this estimate] even though it operates without the benefit of hindsight. Finally, although the typical winning bidder surplus decreases between eBay and the options-based scheme, some surplus redistribution would be possible because the total market efficiency is improved.22 6. DISCUSSION The biggest concern with our scheme is that proxy agents who may be interested in many different items may acquire many more options than they finally exercise. This can lead to efficiency loss. Notice that this is not an issue when bidders are only interested in a single item (as in our empirical study), or have linear-additive values on items. To fix this, we would prefer to have proxy agents use more caution in acquiring options and use a more adaptive bidding strategy than that in Equation 1. For instance, if a proxy is already holding an option with an exercise price of $3 on some item for which it has value of $10, and it values some substitute item at $5, the proxy could reason that in no circumstance will it be useful to acquire an option on the second item. We formulate a more sophisticated bidding strategy along these lines. Let Θt be the set of all options a proxy for bidder i already possesses at time t. Let θt ⊆ Θt, be a subset of those options, the sum of whose exercise prices are π(θt), and the goods corresponding to those options being γ(θt). Let Π(θt) = ˆvi(γ(θt)) − π(θt) be the (reported) available surplus associated with a set of options. Let θ∗ t be the set of options currently held that would maximize the buyer"s surplus; i.e., θ∗ t = argmaxθt⊆Θt Π(θt). Let the maximal willingness to pay for an item k represent a price above which the agent knows it would never exercise an option on the item given the current options held. This can be computed as follows: bidt i(k) = max L [0, min[ˆvi(L + k) − Π(θ∗ t ), ˆvi(L + k) − ˆvi(L)]] (3) where ˆvi(L+k)−Π(θ∗ t ) considers surplus already held, ˆvi(L+ k)−ˆvi(L) considers the marginal value of a good, and taking the max[0, .] considers the overall use of pursuing the good. However, and somewhat counter intuitively, we are not able to implement this bidding scheme without forfeiting truthfulness. The Π(θ∗ t ) term in Equation 3 (i.e., the amount of guaranteed surplus bidder i has already obtained) can be influenced by proxy j"s bid. Therefore, bidder j may have the incentive to misrepresent her valuation to her proxy if she believes doing so will cause i to bid differently in the future in a manner beneficial to j. Consider the following example where the proxy scheme is refined to bid the maximum willingness to pay. Example 3. Alice values either one ton of Sand or one ton of Stone for $2,000. Bob values either one ton of Sand or one ton of Stone for $1,500. All bidders have a patience maximum bids while eBay winners have true values much greater than their observed maximum bids then downward bias is introduced in the efficiency calculation at present. 22 The increase in eBay winner surplus between Tables 3 and 4 is to be expected as the α scaling strictly increases the estimated value of the eBay winners while holding the prices at which they won constant. of 2 days. On day one, a Sand auction is held, where Alice"s proxy bids $2,000 and Bob"s bids $1,500. Alice"s proxy wins an option to purchase Sand for $1,500. On day two, a Stone auction is held, where Alice"s proxy bids $1,500 [as she has already obtained a guaranteed $500 of surplus from winning a Sand option, and so reduces her Stone bid by this amount], and Bob"s bids $1,500. Either Alice"s proxy or Bob"s proxy will win the Stone option. At the end of the second day, Alice"s proxy holds an option with an exercise price of $1,500 to obtain a good valued for $2,000, and so obtains $500 in surplus. Now, consider what would have happened had Alice declared that she valued only Stone. Example 4. Alice declares valuing only Stone for $2,000. Bob values either one ton of Sand or one ton of Stone for $1,500. All bidders have a patience of 2 days. On day one, a Sand auction is held, where Bob"s proxy bids $1,500. Bob"s proxy wins an option to purchase Sand for $0. On day two, a Stone auction is held, where Alice"s proxy bids $2,000, and Bob"s bids $0 [as he has already obtained a guaranteed $1,500 of surplus from winning a Sand option, and so reduces his Stone bid by this amount]. Alice"s proxy wins the Stone option for $0. At the end of the second day, Alice"s proxy holds an option with an exercise price of $0 to obtain a good valued for $2,000, and so obtains $2,000 in surplus. By misrepresenting her valuation (i.e., excluding her value of Sand), Alice was able to secure higher surplus by guiding Bob"s bid for Stone to $0. An area of immediate further work by the authors is to develop a more sophisticated proxy agent that can allow for bidding of maximum willingness to pay (Equation 3) while maintaining truthfulness. An additional, practical, concern with our proxy scheme is that we assume an available, trusted, and well understood method to characterize goods (and presumably the quality of goods). We envision this happening in practice by sellers defining a classification for their item upon entering the market, for instance via a UPC code. Just as in eBay, this would allow an opportunity for sellers to improve revenue by overstating the quality of their item (new vs. like new), and raises the issue of how well a reputation scheme could address this. 7. CONCLUSIONS We introduced a new sales channel, consisting of an optionsbased and proxied auction protocol, to address the sequential auction problem that exists when bidders face multiple auctions for substitutes and complements goods. Our scheme provides bidders with a simple, dominant and truthful bidding strategy even though the market remains open and dynamic. In addition to exploring more sophisticated proxies that bid in terms of maximum willingness to pay, future work should aim to better model seller incentives and resolve the strategic problems facing sellers. For instance, does the options scheme change seller incentives from what they currently are on eBay? Acknowledgments We would like to thank Pai-Ling Yin. Helpful comments have been received from William Simpson, attendees at Har188 vard University"s EconCS and ITM seminars, and anonymous reviewers. Thank you to Aaron L. Roth and KangXing Jin for technical support. All errors and omissions remain our own. 8. REFERENCES [1] P. Anthony and N. R. Jennings. Developing a bidding agent for multiple heterogeneous auctions. ACM Trans. On Internet Technology, 2003. [2] R. Bapna, P. Goes, A. Gupta, and Y. Jin. User heterogeneity and its impact on electronic auction market design: An empirical exploration. MIS Quarterly, 28(1):21-43, 2004. [3] D. Bertsimas, J. Hawkins, and G. Perakis. Optimal bidding in on-line auctions. Working Paper, 2002. [4] C. Boutilier, M. Goldszmidt, and B. Sabata. Sequential auctions for the allocation of resources with complementarities. In Proc. 16th International Joint Conference on Artificial Intelligence (IJCAI-99), pages 527-534, 1999. [5] A. Byde, C. Preist, and N. R. Jennings. Decision procedures for multiple auctions. In Proc. 1st Int. Joint Conf. on Autonomous Agents and Multiagent Systems (AAMAS-02), 2002. [6] M. M. Bykowsky, R. J. Cull, and J. O. Ledyard. Mutually destructive bidding: The FCC auction design problem. Journal of Regulatory Economics, 17(3):205-228, 2000. [7] Y. Chen, C. Narasimhan, and Z. J. Zhang. Consumer heterogeneity and competitive price-matching guarantees. Marketing Science, 20(3):300-314, 2001. [8] A. K. Dixit and R. S. Pindyck. Investment under Uncertainty. Princeton University Press, 1994. [9] R. Gopal, S. Thompson, Y. A. Tung, and A. B. Whinston. Managing risks in multiple online auctions: An options approach. Decision Sciences, 36(3):397-425, 2005. [10] A. Greenwald and J. O. Kephart. Shopbots and pricebots. In Proc. 16th International Joint Conference on Artificial Intelligence (IJCAI-99), pages 506-511, 1999. [11] P. A. Haile and E. Tamer. Inference with an incomplete model of English auctions. Journal of Political Economy, 11(1), 2003. [12] M. T. Hajiaghayi, R. Kleinberg, M. Mahdian, and D. C. Parkes. Online auctions with re-usable goods. In Proc. ACM Conf. on Electronic Commerce, 2005. [13] K. Hendricks, I. Onur, and T. Wiseman. Preemption and delay in eBay auctions. University of Texas at Austin Working Paper, 2005. [14] A. Iwasaki, M. Yokoo, and K. Terada. A robust open ascending-price multi-unit auction protocol against false-name bids. Decision Support Systems, 39:23-40, 2005. [15] E. G. James D. Hess. Price-matching policies: An empirical case. Managerial and Decision Economics, 12(4):305-315, 1991. [16] A. X. Jiang and K. Leyton-Brown. Estimating bidders" valuation distributions in online auctions. In Workshop on Game Theory and Decision Theory (GTDT) at IJCAI, 2005. [17] R. Lavi and N. Nisan. Competitive analysis of incentive compatible on-line auctions. In Proc. 2nd ACM Conf. on Electronic Commerce (EC-00), 2000. [18] Y. J. Lin. Price matching in a model of equilibrium price dispersion. Southern Economic Journal, 55(1):57-69, 1988. [19] D. Lucking-Reiley and D. F. Spulber. Business-to-business electronic commerce. Journal of Economic Perspectives, 15(1):55-68, 2001. [20] A. Ockenfels and A. Roth. Last-minute bidding and the rules for ending second-price auctions: Evidence from eBay and Amazon auctions on the Internet. American Economic Review, 92(4):1093-1103, 2002. [21] M. Peters and S. Severinov. Internet auctions with many traders. Journal of Economic Theory (Forthcoming), 2005. [22] R. Porter. Mechanism design for online real-time scheduling. In Proceedings of the 5th ACM conference on Electronic commerce, pages 61-70. ACM Press, 2004. [23] M. H. Rothkopf and R. Engelbrecht-Wiggans. Innovative approaches to competitive mineral leasing. Resources and Energy, 14:233-248, 1992. [24] T. Sandholm and V. Lesser. Leveled commitment contracts and strategic breach. Games and Economic Behavior, 35:212-270, 2001. [25] T. W. Sandholm and V. R. Lesser. Issues in automated negotiation and electronic commerce: Extending the Contract Net framework. In Proc. 1st International Conference on Multi-Agent Systems (ICMAS-95), pages 328-335, 1995. [26] H. S. Shah, N. R. Joshi, A. Sureka, and P. R. Wurman. Mining for bidding strategies on eBay. Lecture Notes on Artificial Intelligence, 2003. [27] M. Stryszowska. Late and multiple bidding in competing second price Internet auctions. EuroConference on Auctions and Market Design: Theory, Evidence and Applications, 2003. [28] J. T.-Y. Wang. Is last minute bidding bad? UCLA Working Paper, 2003. [29] R. Zeithammer. An equilibrium model of a dynamic auction marketplace. Working Paper, University of Chicago, 2005. 189

business-to-consumer auction;empirical analysis;automated trading agent;electronic marketplace;computer simulation;market effect;option;trading opportunity;options-based extension;ebay;sequential auction problem;bidding strategy;strategic behavior;commoditized market;proxy-bidding system;multiple auction;proxy bid;online auction

train_J-40

Networks Preserving Evolutionary Equilibria and the Power of Randomization

We study a natural extension of classical evolutionary game theory to a setting in which pairwise interactions are restricted to the edges of an undirected graph or network. We generalize the definition of an evolutionary stable strategy (ESS), and show a pair of complementary results that exhibit the power of randomization in our setting: subject to degree or edge density conditions, the classical ESS of any game are preserved when the graph is chosen randomly and the mutation set is chosen adversarially, or when the graph is chosen adversarially and the mutation set is chosen randomly. We examine natural strengthenings of our generalized ESS definition, and show that similarly strong results are not possible for them.

1. INTRODUCTION In this paper, we introduce and examine a natural extension of classical evolutionary game theory (EGT) to a setting in which pairwise interactions are restricted to the edges of an undirected graph or network. This extension generalizes the classical setting, in which all pairs of organisms in an infinite population are equally likely to interact. The classical setting can be viewed as the special case in which the underlying network is a clique. There are many obvious reasons why one would like to examine more general graphs, the primary one being in that many scenarios considered in evolutionary game theory, all interactions are in fact not possible. For example, geographical restrictions may limit interactions to physically proximate pairs of organisms. More generally, as evolutionary game theory has become a plausible model not only for biological interaction, but also economic and other kinds of interaction in which certain dynamics are more imitative than optimizing (see [2, 16] and chapter 4 of [19]), the network constraints may come from similarly more general sources. Evolutionary game theory on networks has been considered before, but not in the generality we will do so here (see Section 4). We generalize the definition of an evolutionary stable strategy (ESS) to networks, and show a pair of complementary results that exhibit the power of randomization in our setting: subject to degree or edge density conditions, the classical ESS of any game are preserved when the graph is chosen randomly and the mutation set is chosen adversarially, or when the graph is chosen adversarially and the mutation set is chosen randomly. We examine natural strengthenings of our generalized ESS definition, and show that similarly strong results are not possible for them. The work described here is part of recent efforts examining the relationship between graph topology or structure and properties of equilibrium outcomes. Previous works in this line include studies of the relationship of topology to properties of correlated equilibria in graphical games [11], and studies of price variation in graph-theoretic market exchange models [12]. More generally, this work contributes to the line of graph-theoretic models for game theory investigated in both computer science [13] and economics [10]. 2. CLASSICAL EGT The fundamental concept of evolutionary game theory is the evolutionarily stable strategy (ESS). Intuitively, an ESS is a strategy such that if all the members of a population adopt it, then no mutant strategy could invade the population [17]. To make this more precise, we describe the basic model of evolutionary game theory, in which the notion of an ESS resides. The standard model of evolutionary game theory considers an infinite population of organisms, each of which plays a strategy in a fixed, 2-player, symmetric game. The game is defined by a fitness function F. All pairs of members of the infinite population are equally likely to interact with one another. If two organisms interact, one playing strategy s 200 and the other playing strategy t, the s-player earns a fitness of F(s|t) while the t-player earns a fitness of F(t|s). In this infinite population of organisms, suppose there is a 1 − fraction who play strategy s, and call these organisms incumbents; and suppose there is an fraction who play t, and call these organisms mutants. Assume two organisms are chosen uniformly at random to play each other. The strategy s is an ESS if the expected fitness of an organism playing s is higher than that of an organism playing t, for all t = s and all sufficiently small . Since an incumbent will meet another incumbent with probability 1 − and it will meet a mutant with probability , we can calculate the expected fitness of an incumbent, which is simply (1 − )F(s|s) + F(s|t). Similarly, the expected fitness of a mutant is (1 − )F(t|s) + F(t|t). Thus we come to the formal definition of an ESS [19]. Definition 2.1. A strategy s is an evolutionarily stable strategy (ESS) for the 2-player, symmetric game given by fitness function F, if for every strategy t = s, there exists an t such that for all 0 < < t, (1 − )F(s|s) + F(s|t) > (1 − )F(t|s) + F(t|t). A consequence of this definition is that for s to be an ESS, it must be the case that F(s|s) ≥ F(t|s), for all strategies t. This inequality means that s must be a best response to itself, and thus any ESS strategy s must also be a Nash equilibrium. In general the notion of ESS is more restrictive than Nash equilibrium, and not all 2-player, symmetric games have an ESS. In this paper our interest is to examine what kinds of network structure preserve the ESS strategies for those games that do have a standard ESS. First we must of course generalize the definition of ESS to a network setting. 3. EGT ON GRAPHS In our setting, we will no longer assume that two organisms are chosen uniformly at random to interact. Instead, we assume that organisms interact only with those in their local neighborhood, as defined by an undirected graph or network. As in the classical setting (which can be viewed as the special case of the complete network or clique), we shall assume an infinite population, by which we mean we examine limiting behavior in a family of graphs of increasing size. Before giving formal definitions, some comments are in order on what to expect in moving from the classical to the graph-theoretic setting. In the classical (complete graph) setting, there exist many symmetries that may be broken in moving to the the network setting, at both the group and individual level. Indeed, such asymmetries are the primary interest in examining a graph-theoretic generalization. For example, at the group level, in the standard ESS definition, one need not discuss any particular set of mutants of population fraction . Since all organisms are equally likely to interact, the survival or fate of any specific mutant set is identical to that of any other. In the network setting, this may not be true: some mutant sets may be better able to survive than others due to the specific topologies of their interactions in the network. For instance, foreshadowing some of our analysis, if s is an ESS but F(t|t) is much larger than F(s|s) and F(s|t), a mutant set with a great deal of internal interaction (that is, edges between mutants) may be able to survive, whereas one without this may suffer. At the level of individuals, in the classical setting, the assertion that one mutant dies implies that all mutants die, again by symmetry. In the network setting, individual fates may differ within a group all playing a common strategy. These observations imply that in examining ESS on networks we face definitional choices that were obscured in the classical model. If G is a graph representing the allowed pairwise interactions between organisms (vertices), and u is a vertex of G playing strategy su, then the fitness of u is given by F(u) = P v∈Γ(u) F(su|sv) |Γ(u)| . Here sv is the strategy being played by the neighbor v, and Γ(u) = {v ∈ V : (u, v) ∈ E}. One can view the fitness of u as the average fitness u would obtain if it played each if its neighbors, or the expected fitness u would obtain if it were assigned to play one of its neighbors chosen uniformly at random. Classical evolutionary game theory examines an infinite, symmetric population. Graphs or networks are inherently finite objects, and we are specifically interested in their asymmetries, as discussed above. Thus all of our definitions shall revolve around an infinite family G = {Gn}∞ n=0 of finite graphs Gn over n vertices, but we shall examine asymptotic (large n) properties of such families. We first give a definition for a family of mutant vertex sets in such an infinite graph family to contract. Definition 3.1. Let G = {Gn}∞ n=0 be an infinite family of graphs, where Gn has n vertices. Let M = {Mn}∞ n=0 be any family of subsets of vertices of the Gn such that |Mn| ≥ n for some constant > 0. Suppose all the vertices of Mn play a common (mutant) strategy t, and suppose the remaining vertices in Gn play a common (incumbent) strategy s. We say that Mn contracts if for sufficiently large n, for all but o(n) of the j ∈ Mn, j has an incumbent neighbor i such that F(j) < F(i). A reasonable alternative would be to ask that the condition above hold for all mutants rather than all but o(n). Note also that we only require that a mutant have one incumbent neighbor of higher fitness in order to die; one might considering requiring more. In Sections 6.1 and 6.2 we consider these stronger conditions and demonstrate that our results can no longer hold. In order to properly define an ESS for an infinite family of finite graphs in a way that recovers the classical definition asymptotically in the case of the family of complete graphs, we first must give a definition that restricts attention to families of mutant vertices that are smaller than some invasion threshold n, yet remain some constant fraction of the population. This prevents invasions that survive merely by constituting a vanishing fraction of the population. Definition 3.2. Let > 0, and let G = {Gn}∞ n=0 be an infinite family of graphs, where Gn has n vertices. Let M = {Mn}∞ n=0 be any family of (mutant) vertices in Gn. We say that M is -linear if there exists an , > > 0, such that for all sufficiently large n, n > |Mn| > n. We can now give our definition for a strategy to be evolutionarily stable when employed by organisms interacting with their neighborhood in a graph. 201 Definition 3.3. Let G = {Gn}∞ n=0 be an infinite family of graphs, where Gn has n vertices. Let F be any 2-player, symmetric game for which s is a strategy. We say that s is an ESS with respect to F and G if for all mutant strategies t = s, there exists an t > 0 such that for any t-linear family of mutant vertices M = {Mn}∞ n=0 all playing t, for n sufficiently large, Mn contracts. Thus, to violate the ESS property for G, one must witness a family of mutations M in which each Mn is an arbitrarily small but nonzero constant fraction of the population of Gn, but does not contract (i.e. every mutant set has a subset of linear size that survives all of its incumbent interactions). In Section A.1 we show that the definition given coincides with the classical one in the case where G is the family of complete graphs, in the limit of large n. We note that even in the classical model, small sets of mutants were allowed to have greater fitness than the incumbents, as long as the size of the set was o(n) [18]. In the definition above there are three parameters: the game F, the graph family G and the mutation family M. Our main results will hold for any 2-player, symmetric game F. We will also study two rather general settings for G and M: that in which G is a family of random graphs and M is arbitrary, and that in which G is nearly arbitrary and M is randomly chosen. In both cases, we will see that, subject to conditions on degree or edge density (essentially forcing connectivity of G but not much more), for any 2-player, symmetric game, the ESS of the classical settings, and only those strategies, are always preserved. Thus a common theme of these results is the power of randomization: as long as either the network itself is chosen randomly, or the mutation set is chosen randomly, classical ESS are preserved. 4. RELATED WORK There has been previous work that analyzes which strategies are resilient to mutant invasions with respect to various types of graphs. What sets our work apart is that the model we consider encompasses a significantly more general class of games and graph topologies. We will briefly survey this literature and point out the differences in the previous models and ours. In [8], [3], and [4], the authors consider specific families of graphs, such as cycles and lattices, where players play specific games, such as 2 × 2-games or k × k-coordination games. In these papers the authors specify a simple, local dynamic for players to improve their payoffs by changing strategies, and analyze what type of strategies will grow to dominate the population. The model we propose is more general than both of these, as it encompasses a larger class of graphs as well as a richer set of games. Also related to our work is that of [14], where the authors propose two models. The first assumes organisms interact according to a weighted, undirected graph. However, the fitness of each organism is simply assigned and does not depend on the actions of each organism"s neighborhood. The second model has organisms arranged around a directed cycle, where neighbors play a 2 × 2-game. With probability proportional to its fitness, an organism is chosen to reproduce by placing a replica of itself in its neighbors position, thereby killing the neighbor. We consider more general games than the first model and more general graphs than the second. Finally, the works most closely related to ours are [7], [15], and [6]. The authors consider 2-action, coordination games played by players in a general undirected graph. In these three works, the authors specify a dynamic for a strategy to reproduce, and analyze properties of the graph that allow a strategy to overrun the population. Here again, one can see that our model is more general than these, as it allows for organisms to play any 2-player, symmetric game. 5. NETWORKS PRESERVING ESS We now proceed to state and prove two complementary results in the network ESS model defined in Section 3. First, we consider a setting where the graphs are generated via the Gn,p model of Erd˝os and R´enyi [5]. In this model, every pair of vertices are joined by an edge independently and with probability p (where p may depend on n). The mutant set, however, will be constructed adversarially (subject to the linear size constraint given by Definition 3.3). For these settings, we show that for any 2-player, symmetric game, s is a classical ESS of that game, if and only if s is an ESS for {Gn,p}∞ n=0, where p = Ω(1/nc ) and 0 ≤ c < 1, and any mutant family {Mn}∞ n=0, where each Mn has linear size. We note that under these settings, if we let c = 1 − γ for small γ > 0, the expected number of edges in Gn is n1+γ or larger - that is, just superlinear in the number of vertices and potentially far smaller than O(n2 ). It is easy to convince oneself that once the graphs have only a linear number of edges, we are flirting with disconnectedness, and there may simply be large mutant sets that can survive in isolation due to the lack of any incumbent interactions in certain games. Thus in some sense we examine the minimum plausible edge density. The second result is a kind of dual to the first, considering a setting where the graphs are chosen arbitrarily (subject to conditions) but the mutant sets are chosen randomly. It states that for any 2-player, symmetric game, s is a classical ESS for that game, if and only if s is an ESS for any {Gn = (Vn, En)}∞ n=0 in which for all v ∈ Vn, deg(v) = Ω(nγ ) (for any constant γ > 0), and a family of mutant sets {Mn}∞ n=0, that is chosen randomly (that is, in which each organism is labeled a mutant with constant probability > 0). Thus, in this setting we again find that classical ESS are preserved subject to edge density restrictions. Since the degree assumption is somewhat strong, we also prove another result which only assumes that |En| ≥ n1+γ , and shows that there must exist at least 1 mutant with an incumbent neighbor of higher fitness (as opposed to showing that all but o(n) mutants have an incumbent neighbor of higher fitness). As will be discussed, this rules out stationary mutant invasions. 5.1 Random Graphs, Adversarial Mutations Now we state and prove a theorem which shows that if s is a classical ESS, then s will be an ESS for random graphs, where a linear sized set of mutants is chosen by an adversary. Theorem 5.1. Let F be any 2-player, symmetric game, and suppose s is a classical ESS of F. Let the infinite graph family {Gn}∞ n=0 be drawn according to Gn,p, where p = Ω(1/nc ) and 0 ≤ c < 1. Then with probability 1, s is an ESS. The main idea of the proof is to divide mutants into 2 categories, those with normal fitness and those with ab202 normal fitness. First, we show all but o(n) of the population (incumbent or mutant) have an incumbent neighbor of normal fitness. This will imply that all but o(n) of the mutants of normal fitness have an incumbent neighbor of higher fitness. The vehicle for proving this is Theorem 2.15 of [5], which gives an upper bound on the number of vertices not connected to a sufficiently large set. This theorem assumes that the size of this large set is known with equality, which necessitates the union bound argument below. Secondly, we show that there can be at most o(n) mutants with abnormal fitness. Since there are so few of them, even if none of them have an incumbent neighbor of higher fitness, s will still be an ESS with respect to F and G. Proof. (Sketch) Let t = s be the mutant strategy. Since s is a classical ESS, there exists an t such that (1− )F(s|s)+ F(s|t) > (1 − )F(t|s) + F(t|t), for all 0 < < t. Let M be any mutant family that is t-linear. Thus for any fixed value of n that is sufficiently large, there exists an such that |Mn| = n and t > > 0. Also, let In = Vn \ Mn and let I ⊆ In be the set of incumbents that have fitness in the range (1 ± τ)[(1 − )F(s|s) + F(s|t)] for some constant τ, 0 < τ < 1/6. Lemma 5.1 below shows (1 − )n ≥ |I | ≥ (1 − )n − 24 log n τ2p . Finally, let TI = {x ∈ V \ I : Γ(x) ∩ I = ∅}. (For the sake of clarity we suppress the subscript n on the sets I and T.) The union bound gives us Pr(|TI | ≥ δn) ≤ (1− )n X i=(1− )n− 24 log n τ2p Pr(|TI | ≥ δn and |I | = i) (1) Letting δ = n−γ for some γ > 0 gives δn = o(n). We will apply Theorem 2.15 of [5] to the summand on the right hand side of Equation 1. If we let γ = (1−c)/2, and combine this with the fact that 0 ≤ c < 1, all of the requirements of this theorem will be satisfied (details omitted). Now when we apply this theorem to Equation 1, we get Pr(|TI | ≥ δn) ≤ (1− )n X i=(1− )n− 24 log n τ2p exp „ − 1 6 Cδn « (2) = o(1) This is because equation 2 has only 24 log n τ2p terms, and Theorem 2.15 of [5] gives us that C ≥ (1 − )n1−c − 24 log n τ2 . Thus we have shown, with probability tending to 1 as n → ∞, at most o(n) individuals are not attached to an incumbent which has fitness in the range (1 ± τ)[(1 − )F(s|s) + F(s|t)]. This implies that the number of mutants of approximately normal fitness, not attached to an incumbent of approximately normal fitness, is also o(n). Now those mutants of approximately normal fitness that are attached to an incumbent of approximately normal fitness have fitness in the range (1±τ)[(1− )F(t|s)+ F(t|t)]. The incumbents that they are attached to have fitness in the range (1±τ)[(1− )F(s|s)+ F(s|t)]. Since s is an ESS of F, we know (1− )F(s|s)+ F(s|t) > (1− )F(t|s)+ F(t|t), thus if we choose τ small enough, we can ensure that all but o(n) mutants of normal fitness have a neighboring incumbent of higher fitness. Finally by Lemma 5.1, we know there are at most o(n) mutants of abnormal fitness. So even if all of them are more fit than their respective incumbent neighbors, we have shown all but o(n) of the mutants have an incumbent neighbor of higher fitness. We now state and prove the lemma used in the proof above. Lemma 5.1. For almost every graph Gn,p with (1 − )n incumbents, all but 24 log n δ2p incumbents have fitness in the range (1±δ)[(1− )F(s|s)+ F(s|t)], where p = Ω(1/nc ) and , δ and c are constants satisfying 0 < < 1, 0 < δ < 1/6, 0 ≤ c < 1. Similarly, under the same assumptions, all but 24 log n δ2p mutants have fitness in the range (1 ± δ)[(1 − )F(t|s) + F(t|t)]. Proof. We define the mutant degree of a vertex to be the number of mutant neighbors of that vertex, and incumbent degree analogously. Observe that the only way for an incumbent to have fitness far from its expected value of (1− )F(s|s)+ F(s|t) is if it has a fraction of mutant neighbors either much higher or much lower than . Theorem 2.14 of [5] gives us a bound on the number of such incumbents. It states that the number of incumbents with mutant degree outside the range (1 ± δ)p|M| is at most 12 log n δ2p . By the same theorem, the number of incumbents with incumbent degree outside the range (1 ± δ)p|I| is at most 12 log n δ2p . From the linearity of fitness as a function of the fraction of mutant or incumbent neighbors, one can show that for those incumbents with mutant and incumbent degree in the expected range, their fitness is within a constant factor of (1 − )F(s|s) + F(s|t), where that constant goes to 1 as n tends to infinity and δ tends to 0. The proof for the mutant case is analogous. We note that if in the statement of Theorem 5.1 we let c = 0, then p = 1. This, in turn, makes G = {Kn}∞ n=0, where Kn is a clique of n vertices. Then for any Kn all of the incumbents will have identical fitness and all of the mutants will have identical fitness. Furthermore, since s was an ESS for G, the incumbent fitness will be higher than the mutant fitness. Finally, one can show that as n → ∞, the incumbent fitness converges to (1 − )F(s|s) + F(s|t), and the mutant fitness converges to (1 − )F(t|s) + F(t|t). In other words, s must be a classical ESS, providing a converse to Theorem 5.1. We rigorously present this argument in Section A.1. 5.2 Adversarial Graphs, Random Mutations We now move on to our second main result. Here we show that if the graph family, rather than being chosen randomly, is arbitrary subject to a minimum degree requirement, and the mutation sets are randomly chosen, classical ESS are again preserved. A modified notion of ESS allows us to considerably weaken the degree requirement to a minimum edge density requirement. Theorem 5.2. Let G = {Gn = (Vn, En)}∞ n=0 be an infinite family of graphs in which for all v ∈ Vn, deg(v) = Ω(nγ ) (for any constant γ > 0). Let F be any 2-player, symmetric game, and suppose s is a classical ESS of F. Let t be any mutant strategy, and let the mutant family M = {Mn}∞ n=0 be chosen randomly by labeling each vertex a mutant with constant probability , where t > > 0. Then with probability 1, s is an ESS with respect to F, G and M. 203 Proof. Let t = s be the mutant strategy and let X be the event that every incumbent has fitness within the range (1 ± τ)[(1 − )F(s|s) + F(s|t)], for some constant τ > 0 to be specified later. Similarly, let Y be the event that every mutant has fitness within the range (1 ± τ)[(1 − )F(t|s) + F(t|t)]. Since Pr(X ∩ Y ) = 1 − Pr(¬X ∪ ¬Y ), we proceed by showing Pr(¬X ∪ ¬Y ) = o(1). ¬X is the event that there exists an incumbent with fitness outside the range (1±τ)[(1− )F(s|s)+ F(s|t)]. If degM (v) denotes the number of mutant neighbors of v, similarly, degI (v) denotes the number of incumbent neighbors of v, then an incumbent i has fitness degI (i) deg(i) F(s|s)+ degM (i) deg(i) F(s|t). Since F(s|s) and F(s|t) are fixed quantities, the only variation in an incumbents fitness can come from variation in the terms degI (i) deg(i) and degM (i) deg(i) . One can use the Chernoff bound followed by the union bound to show that for any incumbent i, Pr(F(i) /∈ (1 ± τ)[(1 − )F(s|s) + F(s|t)]) < 4 exp „ − deg(i)τ2 3 « . Next one can use the union bound again to bound the probability of the event ¬X, Pr(¬X) ≤ 4n exp „ − diτ2 3 « where di = mini∈V \M deg(i), 0 < ≤ 1/2. An analogous argument can be made to show Pr(¬Y ) < 4n exp(− dj τ2 3 ), where dj = minj∈M deg(j) and 0 < ≤ 1/2. Thus, by the union bound, Pr(¬X ∪ ¬Y ) < 8n exp „ − dτ2 3 « where d = minv∈V deg(v), 0 < ≤ 1/2. Since deg(v) = Ω(nγ ), for all v ∈ V , and , τ and γ are all constants greater than 0, lim n→∞ 8n exp ( dτ2/3) = 0, so Pr(¬X∪¬Y ) = o(1). Thus, we can choose τ small enough such that (1 + τ)[(1 − )F(t|s) + F(t|t)] < (1 − τ)[(1 − )F(s|s)+ F(s|t)], and then choose n large enough such that with probability 1 − o(1), every incumbent will have fitness in the range (1±τ)[(1− )F(s|s)+F(s|t)], and every mutant will have fitness in the range (1 ± τ)[(1 − )F(t|s) + F(t|t)]. So with high probability, every incumbent will have a higher fitness than every mutant. By arguments similar to those following the proof of Theorem 5.1, if we let G = {Kn}∞ n=0, each incumbent will have the same fitness and each mutant will have the same fitness. Furthermore, since s is an ESS for G, the incumbent fitness must be higher than the mutant fitness. Here again, one has to show show that as n → ∞, the incumbent fitness converges to (1 − )F(s|s) + F(s|t), and the mutant fitness converges to (1 − )F(t|s) + F(t|t). Observe that the exact fraction mutants of Vn is now a random variable. So to prove this convergence we use an argument similar to one that is used to prove that sequence of random variables that converges in probability also converges in distribution (details omitted). This in turn establishes that s must be a classical ESS, and we thus obtain a converse to Theorem 5.2. This argument is made rigorous in Section A.2. The assumption on the degree of each vertex of Theorem 5.2 is rather strong. The following theorem relaxes this requirement and only necessitates that every graph have n1+γ edges, for some constant γ > 0, in which case it shows there will alway be at least 1 mutant with an incumbent neighbor of higher fitness. A strategy that is an ESS in this weakened sense will essentially rule out stable, static sets of mutant invasions, but not more complex invasions. An example of more complex invasions are mutant sets that survive, but only by perpetually migrating through the graph under some natural evolutionary dynamics, akin to gliders in the well-known Game of Life [1]. Theorem 5.3. Let F be any game, and let s be a classical ESS of F, and let t = s be a mutant strategy. For any graph family G = {Gn = (Vn, En)}∞ n=0 in which |En| ≥ n1+γ (for any constant γ > 0), and any mutant family M = {Mn}∞ n=0 which is determined by labeling each vertex a mutant with probability , where t > > 0, the probability that there exists a mutant with an incumbent neighbor of higher fitness approaches 1 as n → ∞. Proof. (Sketch) The main idea behind the proof is to show that with high probability, over only the choice of mutants, there will be an incumbent-mutant edge in which both vertices have high degree. If their degree is high enough, we can show that close to an fraction of their neighbors are mutants, and thus their fitnesses are very close to what we expect them to be in the classical case. Since s is an ESS, the fitness of the incumbent will be higher than the mutant. We call an edge (i, j) ∈ En a g(n)-barbell if deg(i) ≥ g(n) and deg(j) ≥ g(n). Suppose Gn has at most h(n) edges that are g(n)-barbells. This means there are at least |En| − h(n) edges in which at least one vertex has degree at most g(n). We call these vertices light vertices. Let (n) be the number of light vertices in Gn. Observe that |En|−h(n) ≤ (n)g(n). This is because each light vertex is incident on at most g(n) edges. This gives us that |En| ≤ h(n) + (n)g(n) ≤ h(n) + ng(n). So if we choose h(n) and g(n) such that h(n) + ng(n) = o(n1+γ ), then |En| = o(n1+γ ). This contradicts the assumption that |En| = Ω(n1+γ ). Thus, subject to the above constraint on h(n) and g(n), Gn must contain at least h(n) edges that are g(n)-barbells. Now let Hn denote the subgraph induced by the barbell edges of Gn. Note that regardless of the structure of Gn, there is no reason that Hn should be connected. Thus, let m be the number of connected components of Hn, and let c1, c2, . . . , cm be the number of vertices in each of these connected components. Note that since Hn is an edge-induced subgraph we have ck ≥ 2 for all components k. Let us choose the mutant set by first flipping the vertices in Hn only. We now show that the probability, with respect to the random mutant set, that none of the components of Hn have an incumbent-mutant edge is exponentially small in n. Let An be the event that every component of Hn contains only mutants or only incumbents. Then algebraic manipulations can establish that Pr[An] = Πm k=1( ck + (1 − )ck ) ≤ (1 − )(1− β2 2 ) Pm k=1 ck 204 where β is a constant. Thus for sufficiently small the bound decreases exponentially with Pm k=1 ck. Furthermore, sincePm k=1 `ck 2 ´ ≥ h(n) (with equality achieved by making each component a clique), one can show that Pm k=1 ck ≥ p h(n). Thus, as long as h(n) → ∞ with n, the probability that all components are uniformly labeled will go to 0. Now assuming that there exists a non-uniformly labeled component, by construction that component contains an edge (i, j) where i is an incumbent and j is a mutant, that is a g(n)-barbell. We also assume that the h(n) vertices already labeled have been done so arbitrarily, but that the remaining g(n) − h(n) vertices neighboring i and j are labeled mutants independently with probability . Then via a standard Chernoff bound argument, one can show that with high probability, the fraction of mutants neighboring i and the fraction of mutants neighboring j is in the range (1 ± τ)(g(n)−h(n)) g(n) . Similarly, one can show that the fraction of incumbents neighboring i and the fraction of mutants neighboring j is in the range 1 − (1 ± τ)(g(n)−h(n)) g(n) . Since s is an ESS, there exists a ζ > 0 such that (1 − )F(s|s) + F(s|t) = (1 − )F(t|s) + F(t|t) + ζ. If we choose g(n) = nγ , and h(n) = o(g(n)), we can choose n large enough and τ small enough to force F(i) > F(j), as desired. 6. LIMITATIONS OF STRONGER MODELS In this section we show that if one tried to strengthen the model described in Section 3 in two natural ways, one would not be able to prove results as strong as Theorems 5.1 and 5.2, which hold for every 2-player, symmetric game. 6.1 Stronger Contraction for the Mutant Set In Section 3 we alluded to the fact that we made certain design decisions in arriving at Definitions 3.1, 3.2 and 3.3. One such decision was to require that all but o(n) mutants have incumbent neighbors of higher fitness. Instead, we could have required that all mutants have an incumbent neighbor of higher fitness. The two theorems in this subsection show that if one were to strengthen our notion of contraction for the mutant set, given by Definition 3.1, in this way, it would be impossible to prove theorems analogous to Theorems 5.1 and 5.3. Recall that Definition 3.1 gave the notion of contraction for a linear sized subset of mutants. In what follows, we will say an edge (i, j) contracts if i is an incumbent, j is a mutant, and F(i) > F(j). Also, recall that Theorem 5.1 stated that if s is a classical ESS, then it is an ESS for random graphs with adversarial mutations. Next, we prove that if we instead required every incumbent-mutant edge to contract, this need not be the case. Theorem 6.1. Let F be a 2-player, symmetric game that has a classical ESS s for which there exists a mutant strategy t = s with F(t|t) > F(s|s) and F(t|t) > F(s|t). Let G = {Gn}∞ n=0 be an infinite family of random graphs drawn according to Gn,p, where p = Ω(1/nc ) for any constant 0 ≤ c < 1. Then with probability approaching 1 as n → ∞, there exists a mutant family M = {Mn}∞ n=0, where tn > |Mn| > n and t, > 0, in which there is an edge that does not contract. Proof. (Sketch) With probability approaching 1 as n → ∞, there exists a vertex j where deg(j) is arbitrarily close to n. So label j mutant, label one of its neighbors incumbent, denoted i, and label the rest of j"s neighborhood mutant. Also, label all of i"s neighbors incumbent, with the exception of j and j"s neighbors (which were already labeled mutant). In this setting, one can show that F(j) will be arbitrarily close to F(t|t) and F(i) will be a convex combination of F(s|s) and F(s|t), which are both strictly less than F(t|t). Theorem 5.3 stated that if s is a classical ESS, then for graphs where |En| ≥ n1+γ , for some γ > 0, and where each organism is labeled a mutant with probability , one edge must contract. Below we show that, for certain graphs and certain games, there will always exist one edge that will not contract. Theorem 6.2. Let F be a 2-player, symmetric game that has a classical ESS s, such that there exists a mutant strategy t = s where F(t|s) > F(s|t). There exists an infinite family of graphs {Gn = (Vn, En)}∞ n=0, where |En| = Θ(n2 ), such that for a mutant family M = {Mn}∞ n=0, which is determined by labeling each vertex a mutant with probability > 0, the probability there exists an edge in En that does not contract approaches 1 as n → ∞. Proof. (Sketch) Construct Gn as follows. Pick n/4 vertices u1, u2, . . . , un/4 and add edges such that they from a clique. Then, for each ui, i ∈ [n/4] add edges (ui, vi), (vi, wi) and (wi, xi). With probability 1 as n → ∞, there exists an i such that ui and wi are mutants and vi and xi are incumbents. Observe that F(vi) = F(xi) = F(s|t) and F(wi) = F(t|s). 6.2 Stronger Contraction for Individuals The model of Section 3 requires that for an edge (i, j) to contract, the fitness of i must be greater than the fitness of j. One way to strengthen this notion of contraction would be to require that the maximum fitness incumbent in the neighborhood of j be more fit than the maximum fitness mutant in the neighborhood of j. This models the idea that each organism is trying to take over each place in its neighborhood, but only the most fit organism in the neighborhood of a vertex gets the privilege of taking it. If we assume that we adopt this notion of contraction for individual mutants, and require that all incumbent-mutant edges contract, we will next show that Theorems 6.1 and 6.2 still hold, and thus it is still impossible to get results such as Theorems 5.1 and 5.3 which hold for every 2-player, symmetric game. In the proof of Theorem 6.1 we proved that F(i) is strictly less than F(j). Observe that maximum fitness mutant in the neighborhood of j must have fitness at least F(j). Also observe that there is only 1 incumbent in the neighborhood of j, namely i. So under this stronger notion of contraction, the edge (i, j) will not contract. Similarly, in the proof of Theorem 6.2, observe that the only mutant in the neighborhood of wi is wi itself, which has fitness F(t|s). Furthermore, the only incumbents in the neighborhood of wi are vi and xi, both of which have fitness F(s|t). By assumption, F(t|s) > F(s|t), thus, under this stronger notion of contraction, neither of the incumbentmutant edges, (vi, wi) and (xi, wi), will contract. 7. REFERENCES [1] Elwyn R. Berlekamp, John Horton Conway, and Richard K. Guy. Winning Ways for Your 205 Mathematical Plays, volume 4. AK Peters, Ltd, March 2004. [2] Jonas Bj¨ornerstedt and Karl H. Schlag. On the evolution of imitative behavior. Discussion Paper B-378, University of Bonn, 1996. [3] L. E. Blume. The statistical mechanics of strategic interaction. Games and Economic Behavior, 5:387-424, 1993. [4] L. E. Blume. The statistical mechanics of best-response strategy revision. Games and Economic Behavior, 11(2):111-145, November 1995. [5] B. Bollob´as. Random Graphs. Cambridge University Press, 2001. [6] Michael Suk-Young Chwe. Communication and coordination in social networks. Review of Economic Studies, 67:1-16, 2000. [7] Glenn Ellison. Learning, local interaction, and coordination. Econometrica, 61(5):1047-1071, Sept. 1993. [8] I. Eshel, L. Samuelson, and A. Shaked. Altruists, egoists, and hooligans in a local interaction model. The American Economic Review, 88(1), 1998. [9] Geoffrey R. Grimmett and David R. Stirzaker. Probability and Random Processes. Oxford University Press, 3rd edition, 2001. [10] M. Jackson. A survey of models of network formation: Stability and efficiency. In Group Formation in Economics; Networks, Clubs and Coalitions. Cambridge University Press, 2004. [11] S. Kakade, M. Kearns, J. Langford, and L. Ortiz. Correlated equilibria in graphical games. ACM Conference on Electronic Commerce, 2003. [12] S. Kakade, M. Kearns, L. Ortiz, R. Pemantle, and S. Suri. Economic properties of social networks. Neural Information Processing Systems, 2004. [13] M. Kearns, M. Littman, and S. Singh. Graphical models for game theory. Conference on Uncertainty in Artificial Intelligence, pages 253-260, 2001. [14] E. Lieberman, C. Hauert, and M. A. Nowak. Evolutionary dynamics on graphs. Nature, 433:312-316, 2005. [15] S. Morris. Contagion. Review of Economic Studies, 67(1):57-78, 2000. [16] Karl H. Schlag. Why imitate and if so, how? Journal of Economic Theory, 78:130-156, 1998. [17] J. M. Smith. Evolution and the Theory of Games. Cambridge University Press, 1982. [18] William L. Vickery. How to cheat against a simple mixed strategy ESS. Journal of Theoretical Biology, 127:133-139, 1987. [19] J¨orgen W. Weibull. Evolutionary Game Theory. The MIT Press, 1995. APPENDIX A. GRAPHICAL AND CLASSICAL ESS In this section we explore the conditions under which a graphical ESS is also a classical ESS. To do so, we state and prove two theorems which provide converses to each of the major theorems in Section 3. A.1 Random Graphs, Adversarial Mutations Theorem 5.2 states that if s is a classical ESS and G = {Gn,p}, where p = Ω(1/nc ) and 0 ≤ c < 1, then with probability 1 as n → ∞, s is an ESS with respect to G. Here we show that if s is an ESS with respect to G, then s is a classical ESS. In order to prove this theorem, we do not need the full generality of s being an ESS for G when p = Ω(1/nc ) where 0 ≤ c < 1. All we need is s to be an ESS for G when p = 1. In this case there are no more probabilistic events in the theorem statement. Also, since p = 1 each graph in G is a clique, so if one incumbent has a higher fitness than one mutant, then all incumbents have higher fitness than all mutants. This gives rise to the following theorem. Theorem A.1. Let F be any 2-player, symmetric game, and suppose s is a strategy for F and t = s is a mutant strategy. Let G = {Kn}∞ n=0. If, as n → ∞, for any t-linear family of mutants M = {Mn}∞ n=0, there exists an incumbent i and a mutant j such that F(i) > F(j), then s is a classical ESS of F. The proof of this theorem analyzes the limiting behavior of the mutant population as the size of the cliques in G tends to infinity. It also shows how the definition of ESS given in Section 5 recovers the classical definition of ESS. Proof. Since each graph in G is a clique, every incumbent will have the same number of incumbent and mutant neighbors, and every mutant will have the same number of incumbent and mutant neighbors. Thus, all incumbents will have identical fitness and all mutants will have identical fitness. Next, one can construct an t-linear mutant family M, where the fraction of mutants converges to for any , where t > > 0. So for n large enough, the number of mutants in Kn will be arbitrarily close to n. Thus, any mutant subset of size n will result in all incumbents having fitness (1 − n n−1 )F(s|s) + n n−1 F(s|t), and all mutants having fitness (1 − n−1 n−1 )F(t|s) + n−1 n−1 F(t|t). Furthermore, by assumption the incumbent fitness must be higher than the mutant fitness. This implies, lim n→∞ „ (1 − n n − 1 )F(s|s) + n n − 1 F(s|t) > (1 − n − 1 n − 1 )F(t|s) + n − 1 n − 1 F(t|t) « = 1. This implies, (1− )F(s|s)+ F(s|t) > (1− )F(t|s)+ F(t|t), for all , where t > > 0. A.2 Adversarial Graphs, Random Mutations Theorem 5.2 states that if s is a classical ESS for a 2player, symmetric game F, where G is chosen adversarially subject to the constraint that the degree of each vertex is Ω(nγ ) (for any constant γ > 0), and mutants are chosen with probability , then s is an ESS with respect to F, G, and M. Here we show that if s is an ESS with respect to F, G, and M then s is a classical ESS. All we will need to prove this is that s is an ESS with respect to G = {Kn}∞ n=0, that is when each vertex has degree n − 1. As in Theorem A.1, since the graphs are cliques, if one incumbent has higher fitness than one mutant, then all incumbents have higher fitness than all mutants. Thus, the theorem below is also a converse to Theorem 5.3. (Recall that Theorem 5.3 uses a weaker notion of contraction that 206 requires only one incumbent to have higher fitness than one mutant.) Theorem A.2. Let F be any 2-player symmetric game, and suppose s is an incumbent strategy for F and t = s is a mutant strategy. Let G = {Kn}∞ n=0. If with probability 1 as n → ∞, s is an ESS for G and a mutant family M = {Mn}∞ n=0, which is determined by labeling each vertex a mutant with probability , where t > > 0, then s is a classical ESS of F. This proof also analyzes the limiting behavior of the mutant population as the size of the cliques in G tends to infinity. Since the mutants are chosen randomly we will use an argument similar to the proof that a sequence of random variables that converges in probability, also converge in distribution. In this case the sequence of random variables will be actual fraction of mutants in each Kn. Proof. Fix any value of , where n > > 0, and construct each Mn by labeling a vertex a mutant with probability . By the same argument as in the proof of Theorem A.1, if the actual number of mutants in Kn is denoted by nn, any mutant subset of size nn will result in all incumbents having fitness (1 − nn n−1 )F(s|s) + nn n−1 F(s|t), and in all mutants having fitness (1 − nn−1 n−1 )F(t|s) + nn−1 n−1 F(t|t). This implies lim n→∞ Pr(s is an ESS for Gn w.r.t. nn mutants) = 1 ⇒ lim n→∞ Pr „ (1 − nn n − 1 )F(s|s) + nn n − 1 F(s|t) > (1 − nn − 1 n − 1 )F(t|s) + nn − 1 n − 1 F(t|t) « = 1 ⇔ lim n→∞ Pr „ n > F(t|s) − F(s|s) F(s|t) − F(s|s) − F(t|t) + F(t|s) + F(s|s) − F(t|t) n « = 1 (3) By two simple applications of the Chernoff bound and an application of the union bound, one can show the sequence of random variables { n}∞ n=0 converges to in probability. Next, if we let Xn = − n, X = − , b = −F(s|s) + F(t|t), and a = − F (t|s)−F (s|s) F (s|t)−F (s|s)−F (t|t)+F (t|s) , by Theorem A.3 below, we get that limn→∞ Pr(Xn < a + b/n) = Pr(X < a). Combining this with equation 3, Pr( > −a) = 1. The proof of the following theorem is very similar to the proof that a sequence of random variables that converges in probability, also converge in distribution. A good explanation of this can be found in [9], which is the basis for the argument below. Theorem A.3. If {Xn}∞ n=0 is a sequence of random variables that converge in probability to the random variable X, and a and b are constants, then limn→∞ Pr(Xn < a+b/n) = Pr(X < a). Proof. By Lemma A.1 (see below) we have the following two inequalities, Pr(X < a + b/n − τ) ≤ Pr(Xn < a + b/n) + Pr(|X − Xn| > τ), Pr(Xn < a + b/n) ≤ Pr(X < a + b/n + τ) + Pr(|X − Xn| > τ). Combining these gives, Pr(X < a + b/n − τ) − Pr(|X − Xn| > τ) ≤ Pr(Xn < a + b/n) ≤ Pr(X < a + b/n + τ) + Pr(|X − Xn| > τ). There exists an n0 such that for all n > n0, |b/n| < τ, so the following statement holds for all n > n0. Pr(X < a − 2τ) − Pr(|X − Xn| > τ) ≤ Pr(Xn < a + b/n) ≤ Pr(X < a + 2τ) + Pr(|X − Xn| > τ). Take the limn→∞ of both sides of both inequalities, and since Xn converges in probability to X, Pr(X < a − 2τ) ≤ lim n→∞ Pr(Xn < a + b/n) (4) ≤ Pr(X < a + 2τ). (5) Recall that X is a continuous random variable representing the fraction of mutants in an infinite sized graph. So if we let FX (a) = Pr(X < a), we see that FX (a) is a cumulative distribution function of a continuous random variable, and is therefore continuous from the right. So lim τ↓0 FX (a − τ) = lim τ↓0 FX (a + τ) = FX (a). Thus if we take the limτ↓0 of inequalities 4 and 5 we get Pr(X < a) = lim n→∞ Pr(Xn < a + b/n). The following lemma is quite useful, as it expresses the cumulative distribution of one random variable Y , in terms of the cumulative distribution of another random variable X and the difference between X and Y . Lemma A.1. If X and Y are random variables, c ∈ and τ > 0, then Pr(Y < c) ≤ Pr(X < c + τ) + Pr(|Y − X| > τ). Proof. Pr(Y < c) = Pr(Y < c, X < c + τ) + Pr(Y < c, X ≥ c + τ) ≤ Pr(Y < c | X < c + τ) Pr(X < c + τ) + Pr(|Y − X| > τ) ≤ Pr(X < c + τ) + Pr(|Y − X| > τ) 207

nash equilibrium;game theory;randomization power;evolutionary stable strategy;natural strengthening;geographical restriction;equilibrium outcome;power of randomization;undirected graph;edge density condition;graph topology;mutation set;evolutionary game theory;graph-theoretic model;relationship of topology;network;topology relationship;pairwise interaction

train_J-41

An Analysis of Alternative Slot Auction Designs for Sponsored Search

Billions of dollars are spent each year on sponsored search, a form of advertising where merchants pay for placement alongside web search results. Slots for ad listings are allocated via an auction-style mechanism where the higher a merchant bids, the more likely his ad is to appear above other ads on the page. In this paper we analyze the incentive, efficiency, and revenue properties of two slot auction designs: rank by bid (RBB) and rank by revenue (RBR), which correspond to stylized versions of the mechanisms currently used by Yahoo! and Google, respectively. We also consider first- and second-price payment rules together with each of these allocation rules, as both have been used historically. We consider both the short-run incomplete information setting and the long-run complete information setting. With incomplete information, neither RBB nor RBR are truthful with either first or second pricing. We find that the informational requirements of RBB are much weaker than those of RBR, but that RBR is efficient whereas RBB is not. We also show that no revenue ranking of RBB and RBR is possible given an arbitrary distribution over bidder values and relevance. With complete information, we find that no equilibrium exists with first pricing using either RBB or RBR. We show that there typically exists a multitude of equilibria with second pricing, and we bound the divergence of (economic) value in such equilibria from the value obtained assuming all merchants bid truthfully.

1. INTRODUCTION Today, Internet giants Google and Yahoo! boast a combined market capitalization of over $150 billion, largely on the strength of sponsored search, the fastest growing component of a resurgent online advertising industry. PricewaterhouseCoopers estimates that 2004 industry-wide sponsored search revenues were $3.9 billion, or 40% of total Internet advertising revenues.1 Industry watchers expect 2005 revenues to reach or exceed $7 billion.2 Roughly 80% of Google"s estimated $4 billion in 2005 revenue and roughly 45% of Yahoo!"s estimated $3.7 billion in 2005 revenue will likely be attributable to sponsored search.3 A number of other companies-including LookSmart, FindWhat, InterActiveCorp (Ask Jeeves), and eBay (Shopping.com)-earn hundreds of millions of dollars of sponsored search revenue annually. Sponsored search is a form of advertising where merchants pay to appear alongside web search results. For example, when a user searches for used honda accord san diego in a web search engine, a variety of commercial entities (San Diego car dealers, Honda Corp, automobile information portals, classified ad aggregators, eBay, etc...) may bid to to have their listings featured alongside the standard algorithmic search listings. Advertisers bid for placement on the page in an auction-style format where the higher they bid the more likely their listing will appear above other ads on the page. By convention, sponsored search advertisers generally pay per click, meaning that they pay only when a user clicks on their ad, and do not pay if their ad is displayed but not clicked. Though many people claim to systematically ignore sponsored search ads, Majestic Research reports that 1 www.iab.net/resources/adrevenue/pdf/IAB PwC 2004full.pdf 2 battellemedia.com/archives/002032.php 3 These are rough back of the envelope estimates. Google and Yahoo! 2005 revenue estimates were obtained from Yahoo! Finance. We assumed $7 billion in 2005 industry-wide sponsored search revenues. We used Nielsen/NetRatings estimates of search engine market share in the US, the most monetized market: wired-vig.wired.com/news/technology/0,1282,69291,00.html Using comScore"s international search engine market share estimates would yield different estimates: www.comscore.com/press/release.asp?press=622 218 as many as 17% of Google searches result in a paid click, and that Google earns roughly nine cents on average for every search query they process.4 Usually, sponsored search results appear in a separate section of the page designated as sponsored above or to the right of the algorithmic results. Sponsored search results are displayed in a format similar to algorithmic results: as a list of items each containing a title, a text description, and a hyperlink to a corresponding web page. We call each position in the list a slot. Generally, advertisements that appear in a higher ranked slot (higher on the page) garner more attention and more clicks from users. Thus, all else being equal, merchants generally prefer higher ranked slots to lower ranked slots. Merchants bid for placement next to particular search queries; for example, Orbitz and Travelocity may bid for las vegas hotel while Dell and HP bid for laptop computer. As mentioned, bids are expressed as a maximum willingness to pay per click. For example, a forty-cent bid by HostRocket for web hosting means HostRocket is willing to pay up to forty cents every time a user clicks on their ad.5 The auctioneer (the search engine6 ) evaluates the bids and allocates slots to advertisers. In principle, the allocation decision can be altered with each new incoming search query, so in effect new auctions clear continuously over time as search queries arrive. Many allocation rules are plausible. In this paper, we investigate two allocation rules, roughly corresponding to the two allocation rules used by Yahoo! and Google. The rank by bid (RBB) allocation assigns slots in order of bids, with higher ranked slots going to higher bidders. The rank by revenue (RBR) allocation assigns slots in order of the product of bid times expected relevance, where relevance is the proportion of users that click on the merchant"s ad after viewing it. In our model, we assume that an ad"s expected relevance is known to the auctioneer and the advertiser (but not necessarily to other advertisers), and that clickthrough rate decays monotonically with lower ranked slots. In practice, the expected clickthrough rate depends on a number of factors, including the position on the page, the ad text (which in turn depends on the identity of the bidder), the nature and intent of the user, and the context of other ads and algorithmic results on the page, and must be learned over time by both the auctioneer and the bidder [13]. As of this writing, to a rough first-order approximation, Yahoo! employs a RBB allocation and Google employs a RBR allocation, though numerous caveats apply in both cases when it comes to the vagaries of real-world implementations.7 Even when examining a one-shot version of a slot auction, the mechanism differs from a standard multi-item auc4 battellemedia.com/archives/001102.php 5 Usually advertisers also set daily or monthly budget caps; in this paper we do not model budget constraints. 6 In the sponsored search industry, the auctioneer and search engine are not always the same entity. For example Google runs the sponsored search ads for AOL web search, with revenue being shared. Similarly, Yahoo! currently runs the sponsored search ads for MSN web search, though Microsoft will begin independent operations soon. 7 Here are two among many exceptions to the Yahoo! = RBB and Google = RBR assertion: (1) Yahoo! excludes ads deemed insufficiently relevant either by a human editor or due to poor historical click rate; (2) Google sets differing reserve prices depending on Google"s estimate of ad quality. tion in subtle ways. First, a single bid per merchant is used to allocate multiple non-identical slots. Second, the bid is communicated not as a direct preference over slots, but as a preference for clicks that depend stochastically on slot allocation. We investigate a number of economic properties of RBB and RBR slot auctions. We consider the short-run incomplete information case in Section 3, adapting and extending standard analyses of single-item auctions. In Section 4 we turn to the long-run complete information case; our characterization results here draw on techniques from linear programming. Throughout, important observations are highlighted as claims supported by examples. Our contributions are as follows: • We show that with multiple slots, bidders do not reveal their true values with either RBB or RBR, and with either first- or second-pricing. • With incomplete information, we find that the informational requirements of playing the equilibrium bid are much weaker for RBB than for RBR, because bidders need not know any information about each others" relevance (or even their own) with RBB. • With incomplete information, we prove that RBR is efficient but that RBB is not. • We show via a simple example that no general revenue ranking of RBB and RBR is possible. • We prove that in a complete-information setting, firstprice slot auctions have no pure strategy Nash equilibrium, but that there always exists a pure-strategy equilibrium with second pricing. • We provide a constant-factor bound on the deviation from efficiency that can occur in the equilibrium of a second-price slot auction. In Section 2 we specify our model of bidders and the various slot auction formats. In Section 3.1 we study the incentive properties of each format, asking in which cases agents would bid truthfully. There is possible confusion here because the second-price design for slot auctions is reminiscent of the Vickrey auction for a single item; we note that for slot auctions the Vickrey mechanism is in fact very different from the second-price mechanism, and so they have different incentive properties.8 In Section 3.2 we derive the Bayes-Nash equilibrium bids for the various auction formats. This is useful for the efficiency and revenue results in later sections. It should become clear in this section that slot auctions in our model are a straightforward generalization of single-item auctions. Sections 3.3 and 3.4 address questions of efficiency and revenue under incomplete information, respectively. In Section 4.1 we determine whether pure-strategy equilibria exist for the various auction formats, under complete information. In Section 4.2 we derive bounds on the deviation from efficiency in the pure-strategy equilibria of secondprice slot auctions. Our approach is positive rather than normative. We aim to clarify the incentive, efficiency, and revenue properties of two slot auction designs currently in use, under settings of 8 Other authors have also made this observation [5, 6]. 219 incomplete and complete information. We do not attempt to derive the optimal mechanism for a slot auction. Related work. Feng et al. [7] compare the revenue performance of various ranking mechanisms for slot auctions in a model with incomplete information, much as we do in Section 3.4, but they obtain their results via simulations whereas we perform an equilibrium analysis. Liu and Chen [12] study properties of slot auctions under incomplete information. Their setting is essentially the same as ours, except they restrict their attention to a model with a single slot and a binary type for bidder relevance (high or low). They find that RBR is efficient, but that no general revenue ranking of RBB and RBR is possible, which agrees with our results. They also take a design approach and show how the auctioneer should assign relevance scores to optimize its revenue. Edelman et al. [6] model the slot auction problem both as a static game of complete information and a dynamic game of incomplete information. They study the locally envyfree equilibria of the static game of complete information; this is a solution concept motivated by certain bidding behaviors that arise due to the presence of budget constraints. They do not view slot auctions as static games of incomplete information as we do, but do study them as dynamic games of incomplete information and derive results on the uniqueness and revenue properties of the resulting equilibria. They also provide a nice description of the evolution of the market for sponsored search. Varian [18] also studies slot auctions under a setting of complete information. He focuses on symmetric equilibria, which are a refinement of Nash equilibria appropriate for slot auctions. He provides bounds on the revenue obtained in equilibrium. He also gives bounds that can be used to infer bidder values given their bids, and performs some empirical analysis using these results. In contrast, we focus instead on efficiency and provide bounds on the deviation from efficiency in complete-information equilibria. 2. PRELIMINARIES We focus on a slot auction for a single keyword. In a setting of incomplete information, a bidder knows only distributions over others" private information (value per click and relevance). With complete information, a bidder knows others" private information, and so does not need to rely on distributions to strategize. We first describe the model for the case with incomplete information, and drop the distributional information from the model when we come to the complete-information case in Section 4. 2.1 The Model There is a fixed number K of slots to be allocated among N bidders. We assume without loss of generality that K ≤ N, since superfluous slots can remain blank. Bidder i assigns a value of Xi to each click received on its advertisement, regardless of this advertisement"s rank.9 The probability that i"s advertisement will be clicked if viewed is Ai ∈ [0, 1]. We refer to Ai as bidder i"s relevance. We refer to Ri = AiXi as bidder i"s revenue. The Xi, Ai, and Ri are random 9 Indeed Kitts et al. [10] find that in their sample of actual click data, the correlation between rank and conversion rate is not statistically significant. However, for the purposes of our model it is also important that bidders believe that conversion rate does not vary with rank. variables and we denote their realizations by xi, αi, and ri respectively. The probability that an advertisement will be viewed if placed in slot j is γj ∈ [0, 1]. We assume γ1 > γ2 > . . . > γK. Hence bidder i"s advertisement will have a clickthrough rate of γjαi if placed in slot j. Of course, an advertisement does not receive any clicks if it is not allocated a slot. Each bidder"s value and relevance pair (Xi, Ai) is independently and identically distributed on [0, ¯x] × [0, 1] according to a continuous density function f that has full support on its domain. The density f and slot probabilities γ1, . . . , γK are common knowledge. Only bidder i knows the realization xi of its value per click Xi. Both bidder i and the seller know the realization αi of Ai, but this realization remains unobservable to the other bidders. We assume that bidders have quasi-linear utility functions. That is, the expected utility to bidder i of obtaining the slot of rank j at a price of b per click is ui(j, b) = γjαi(xi − b) If the advertising firms bidding in the slot auction are riskneutral and have ample liquidity, quasi-linearity is a reasonable assumption. The assumptions of independence, symmetry, and riskneutrality made above are all quite standard in single-item auction theory [11, 19]. The assumption that clickthrough rate decays monotonically with lower slots-by the same factors for each agent-is unique to the slot auction problem. We view it as a main contribution of our work to show that this assumption allows for tractable analysis of the slot auction problem using standard tools from singleitem auction theory. It also allows for interesting results in the complete information case. A common model of decaying clickthrough rate is the exponential decay model, where γk = 1 δk−1 with decay δ > 1. Feng et al. [7] state that their actual clickthrough data is fitted extremely well by an exponential decay model with δ = 1.428. Our model lacks budget constraints, which are an important feature of real slot auctions. With budget constraints keyword auctions cannot be considered independently of one another, because the budget must be allocated across multiple keywords-a single advertiser typically bids on multiple keywords relevant to his business. Introducing this element into the model is an important next step for future work.10 2.2 Auction Formats In a slot auction a bidder provides to the seller a declared value per click ˜xi(xi, αi) which depends on his true value and relevance. We often denote this declared value (bid) by ˜xi for short. Since a bidder"s relevance αi is observable to the seller, the bidder cannot misrepresent it. We denote the kth highest of the N declared values by ˜x(k) , and the kth highest of the N declared revenues by ˜r(k) , where the declared revenue of bidder i is ˜ri = αi ˜xi. We consider two types of allocation rules, rank by bid (RBB) and rank by revenue (RBR): 10 Models with budget constraints have begun to appear in this research area. Abrams [1] and Borgs et al. [3] design multi-unit auctions for budget-constrained bidders, which can be interpreted as slot auctions, with a focus on revenue optimization and truthfulness. Mehta et al. [14] address the problem of matching user queries to budget-constrained advertisers so as to maximize revenue. 220 RBB. Slot k goes to bidder i if and only if ˜xi = ˜x(k) . RBR. Slot k goes to bidder i if and only if ˜ri = ˜r(k) . We will commonly represent an allocation by a one-to-one function σ : [K] → [N], where [n] is the set of integers {1, 2, . . . , n}. Hence slot k goes to bidder σ(k). We also consider two different types of payment rules. Note that no matter what the payment rule, a bidder that is not allocated a slot will pay 0 since his listing cannot receive any clicks. First-price. The bidder allocated slot k, namely σ(k), pays ˜xσ(k) per click under both the RBB and RBR allocation rules. Second-price. If k < N, bidder σ(k) pays ˜xσ(k+1) per click under the RBB rule, and pays ˜rσ(k+1)/ασ(k) per click under the RBR rule. If k = N, bidder σ(k) pays 0 per click.11 Intuitively, a second-price payment rule sets a bidder"s payment to the lowest bid it could have declared while maintaining the same ranking, given the allocation rule used. Overture introduced the first slot auction design in 1997, using a first-price RBB scheme. Google then followed in 2000 with a second-price RBR scheme. In 2002, Overture (at this point acquired by Yahoo!) then switched to second pricing but still allocates using RBB. One possible reason for the switch is given in Section 4. We assume that ties are broken as follows in the event that two agents make the exact same bid or declare the same revenue. There is a permutation of the agents κ : [N] → [N] that is fixed beforehand. If the bids of agents i and j are tied, then agent i obtains a higher slot if and only if κ(i) < κ(j). This is consistent with the practice in real slot auctions where ties are broken by the bidders" order of arrival. 3. INCOMPLETE INFORMATION 3.1 Incentives It should be clear that with a first-price payment rule, truthful bidding is neither a dominant strategy nor an ex post Nash equilibrium using either RBB or RBR, because this guarantees a payoff of 0. There is always an incentive to shade true values with first pricing. The second-price payment rule is reminiscent of the secondprice (Vickrey) auction used for selling a single item, and in a Vickrey auction it is a dominant strategy for a bidder to reveal his true value for the item [19]. However, using a second-price rule in a slot auction together with either allocation rule above does not yield an incentive-compatible mechanism, either in dominant strategies or ex post Nash equilibrium.12 With a second-price rule there is no incentive for a bidder to bid higher than his true value per click using either RBB or RBR: this either leads to no change 11 We are effectively assuming a reserve price of zero, but in practice search engines charge a non-zero reserve price per click. 12 Unless of course there is only a single slot available, since this is the single-item case. With a single slot both RBB and RBR with a second-price payment rule are dominantstrategy incentive-compatible. in the outcome, or a situation in which he will have to pay more than his value per click for each click received, resulting in a negative payoff.13 However, with either allocation rule there may be an incentive to shade true values with second pricing. Claim 1. With second pricing and K ≥ 2, truthful bidding is not a dominant strategy nor an ex post Nash equilibrium for either RBB or RBR. Example. There are two agents and two slots. The agents have relevance α1 = α2 = 1, whereas γ1 = 1 and γ2 = 1/2. Agent 1 has a value of x1 = 6 per click, and agent 2 has a value of x2 = 4 per click. Let us first consider the RBB rule. Suppose agent 2 bids truthfully. If agent 1 also bids truthfully, he wins the first slot and obtains a payoff of 2. However, if he shades his bid down below 4, he obtains the second slot at a cost of 0 per click yielding a payoff of 3. Since the agents have equal relevance, the exact same situation holds with the RBR rule. Hence truthful bidding is not a dominant strategy in either format, and neither is it an ex post Nash equilibrium. To find payments that make RBB and RBR dominantstrategy incentive-compatible, we can apply Holmstrom"s lemma [9] (see also chapter 3 in Milgrom [15]). Under the restriction that a bidder with value 0 per click does not pay anything (even if he obtains a slot, which can occur if there are as many slots as bidders), this lemma implies that there is a unique payment rule that achieves dominant-strategy incentive compatibility for either allocation rule. For RBB, the bidder allocated slot k is charged per click KX i=k+1 (γi−1 − γi)˜x(i) + γK ˜x(K+1) (1) Note that if K = N, ˜x(K+1) = 0 since there is no K + 1th bidder. For RBR, the bidder allocated slot k is charged per click 1 ασ(k) KX i=k+1 (γi−1 − γi)˜r(i) + γK ˜r(K+1) ! (2) Using payment rule (2) and RBR, the auctioneer is aware of the true revenues of the bidders (since they reveal their values truthfully), and hence ranks them according to their true revenues. We show in Section 3.3 that this allocation is in fact efficient. Since the VCG mechanism is the unique mechanism that is efficient, truthful, and ensures bidders with value 0 pay nothing (by the Green-Laffont theorem [8]), the RBR rule and payment scheme (2) constitute exactly the VCG mechanism. In the VCG mechanism an agent pays the externality he imposes on others. To understand payment (2) in this sense, note that the first term is the added utility (due to an increased clickthrough rate) agents in slots k + 1 to K would receive if they were all to move up a slot; the last term is the utility that the agent with the K +1st revenue would receive by obtaining the last slot as opposed to nothing. The leading coefficient simply reduces the agent"s expected payment to a payment per click. 13 In a dynamic setting with second pricing, there may be an incentive to bid higher than one"s true value in order to exhaust competitors" budgets. This phenomenon is commonly called bid jamming or antisocial bidding [4]. 221 3.2 Equilibrium Analysis To understand the efficiency and revenue properties of the various auction formats, we must first understand which rankings of the bidders occur in equilibrium with different allocation and payment rule combinations. The following lemma essentially follows from the Monotonic Selection Theorem by Milgrom and Shannon [16]. Lemma 1. In a RBB (RBR) auction with either a firstor second-price payment rule, the symmetric Bayes-Nash equilibrium bid is strictly increasing with value ( revenue). As a consequence of this lemma, we find that RBB and RBR auctions allocate the slots greedily by the true values and revenues of the agents, respectively (whether using firstor second-price payment rules). This will be relevant in Section 3.3 below. For a first-price payment rule, we can explicitly derive the symmetric Bayes-Nash equilibrium bid functions for RBB and RBR auctions. The purpose of this exercise is to lend qualitative insights into the parameters that influence an agent"s bidding, and to derive formulae for the expected revenue in RBB and RBR auctions in order to make a revenue ranking of these two allocation rules (in Section 3.4). Let G(y) be the expected resulting clickthrough rate, in a symmetric equilibrium of the RBB auction (with either payment rule), to a bidder with value y and relevance α = 1. Let H(y) be the analogous quantity for a bidder with revenue y and relevance 1 in a RBR auction. By Lemma 1, a bidder with value y will obtain slot k in a RBB auction if y is the kth highest of the true realized values. The same applies in a RBR auction when y is the kth highest of the true realized revenues. Let FX (y) be the distribution function for value, and let FR(y) be the distribution function for revenue. The probability that y is the kth highest out of N values is N − 1 k − 1 ! (1 − FX (y))k−1 FX (y)N−k whereas the probability that y is the kth highest out of N revenues is the same formula with FR replacing FX . Hence we have G(y) = KX k=1 γk N − 1 k − 1 ! (1 − FX (y))k−1 FX (y)N−k The H function is analogous to G with FR replacing FX . In the two propositions that follow, g and h are the derivatives of G and H respectively. We omit the proof of the next proposition, because it is almost identical to the derivation of the equilibrium bid in the single-item case (see Krishna [11], Proposition 2.2). Proposition 1. The symmetric Bayes-Nash equilibrium strategies in a first-price RBB auction are given by ˜xB (x, α) = 1 G(x) Z x 0 y g(y)dy The first-price equilibrium above closely parallels the firstprice equilibrium in the single-item model. With a single item g is the density of the second highest value among all N agent values, whereas in a slot auction it is a weighted combination of the densities for the second, third, etc. highest values. Note that the symmetric Bayes-Nash equilibrium bid in a first-price RBB auction does not depend on a bidder"s relevance α. To see clearly why, note that a bidder chooses a bid b so as to maximize the objective αG(˜x−1 (b))(x − b) and here α is just a leading constant factor. So dropping it does not change the set of optimal solutions. Hence the equilibrium bid depends only on the value x and function G, and G in turn depends only on the marginal cumulative distribution of value FX . So really only the latter needs to be common knowledge to the bidders. On the other hand, we will now see that information about relevance is needed for bidders to play the equilibrium in the first-price RBR auction. So the informational requirements for a first-price RBB auction are much weaker than for a first-price RBR auction: in the RBB auction a bidder need not know his own relevance, and need not know any distributional information over others" relevance in order to play the equilibrium. Again we omit the next proposition"s proof since it is so similar to the one above. Proposition 2. The symmetric Bayes-Nash equilibrium strategies in a first-price RBR auction are given by ˜xR (x, α) = 1 αH(αx) Z αx 0 y h(y) dy Here it can be seen that the equilibrium bid is increasing with x, but not necessarily with α. This should not be much of a concern to the auctioneer, however, because in any case the declared revenue in equilibrium is always increasing in the true revenue. It would be interesting to obtain the equilibrium bids when using a second-price payment rule, but it appears that the resulting differential equations for this case do not have a neat analytical solution. Nonetheless, the same conclusions about the informational requirements of the RBB and RBR rules still hold, as can be seen simply by inspecting the objective function associated with an agent"s bidding problem for the second-price case. 3.3 Efficiency A slot auction is efficient if in equilibrium the sum of the bidders" revenues from their allocated slots is maximized. Using symmetry as our equilibrium selection criterion, we find that the RBB auction is not efficient with either payment rule. Claim 2. The RBB auction is not efficient with either first or second pricing. Example. There are two agents and one slot, with γ1 = 1. Agent 1 has a value of x1 = 6 per click and relevance α1 = 1/2. Agent 2 has a value of x2 = 4 per click and relevance α2 = 1. By Lemma 1, agents are ranked greedily by value. Hence agent 1 obtains the lone slot, for a total revenue of 3 to the agents. However, it is most efficient to allocate the slot to agent 2, for a total revenue of 4. Examples with more agents or more slots are simple to construct along the same lines. On the other hand, under our assumptions on how clickthrough rate decreases with lower rank, the RBR auction is efficient with either payment rule. 222 Theorem 1. The RBR auction is efficient with either first- or second-price payments rules. Proof. Since by Lemma 1 the agents" equilibrium bids are increasing functions of their revenues in the RBR auction, slots are allocated greedily according to true revenues. Let σ be a non-greedy allocation. Then there are slots s, t with s < t and rσ(s) < rσ(t). We can switch the agents in slots s and t to obtain a new allocation, and the difference between the total revenue in this new allocation and the original allocation"s total revenue is ` γtrσ(s) + γsrσ(t) ´ − ` γsrσ(s) + γtrσ(t) ´ = (γs − γt) ` rσ(t) − rσ(s) ´ Both parenthesized terms above are positive. Hence the switch has increased the total revenue to the bidders. If we continue to perform such switches, we will eventually reach a greedy allocation of greater revenue than the initial allocation. Since the initial allocation was arbitrary, it follows that a greedy allocation is always efficient, and hence the RBR auction"s allocation is efficient. Note that the assumption that clickthrough rate decays montonically by the same factors γ1, . . . , γK for all agents is crucial to this result. A greedy allocation scheme does not necessarily find an efficient solution if the clickthrough rates are monotonically decreasing in an independent fashion for each agent. 3.4 Revenue To obtain possible revenue rankings for the different auction formats, we first note that when the allocation rule is fixed to RBB, then using either a first-price, second-price, or truthful payment rule leads to the same expected revenue in a symmetric, increasing Bayes-Nash equilibrium. Because a RBB auction ranks agents by their true values in equilibrium for any of these payment rules (by Lemma 1), it follows that expected revenue is the same for all these payment rules, following arguments that are virtually identical to those used to establish revenue equivalence in the singleitem case (see e.g. Proposition 3.1 in Krishna [11]). The same holds for RBR auctions; however, the revenue ranking of the RBB and RBR allocation rules is still unclear. Because of this revenue equivalence principle, we can choose whichever payment rule is most convenient for the purpose of making revenue comparisons. Using Propositions 1 and 2, it is a simple matter to derive formulae for the expected revenue under both allocation rules. The payment of an agent in a RBB auction is mB (x, α) = αG(x)˜xV (x, α) The expected revenue is then N · E ˆ mV (X, A) ˜ , where the expectation is taken with respect to the joint density of value and relevance. The expected revenue formula for RBR auctions is entirely analogous using ˜xR (x, α) and the H function. With these in hand we can obtain revenue rankings for specific numbers of bidders and slots, and specific distributions over values and relevance. Claim 3. For fixed K, N, and fixed γ1, . . . , γK, no revenue ranking of RBB and RBR is possible for an arbitrary density f. Example. Assume there are 2 bidders, 2 slots, and that γ1 = 1, γ2 = 1/2. Assume that value-relevance pairs are uniformly distributed over [0, 1]× [0, 1]. For such a distribution with a closed-form formula, it is most convenient to use the revenue formulae just derived. RBB dominates RBR in terms of revenue for these parameters. The formula for the expected revenue in a RBB auction yields 1/12, whereas for RBR auctions we have 7/108. Assume instead that with probability 1/2 an agent"s valuerelevance pair is (1, 1/2), and that with probability 1/2 it is (1/2, 1). In this scenario it is more convenient to appeal to formulae (1) and (2). In a truthful auction the second agent will always pay 0. According to (1), in a truthful RBB auction the first agent makes an expected payment of E ˆ (γ1 − γ2)Aσ(1)Xσ(2) ˜ = 1 2 E ˆ Aσ(1) ˜ E ˆ Xσ(2) ˜ where we have used the fact that value and relevance are independently distributed for different agents. The expected relevance of the agent with the highest value is E ˆ Aσ(1) ˜ = 5/8. The expected second highest value is also E ˆ Xσ(2) ˜ = 5/8. The expected revenue for a RBB auction here is then 25/128. According to (2), in a truthful RBR auction the first agent makes an expected payment of E ˆ (γ1 − γ2)Rσ(2) ˜ = 1 2 E ˆ Rσ(2) ˜ In expectation the second highest revenue is E ˆ Rσ(2) ˜ = 1/2, so the expected revenue for a RBR auction is 1/4. Hence in this case the RBR auction yields higher expected revenue.1415 This example suggests the following conjecture: when value and relevance are either uncorrelated or positively correlated, RBB dominates RBR in terms of revenue. When value and relevance are negatively correlated, RBR dominates. 4. COMPLETE INFORMATION In typical slot auctions such as those run by Yahoo! and Google, bidders can adjust their bids up or down at any time. As B¨orgers et al. [2] and Edelman et al. [6] have noted, this can be viewed as a continuous-time process in which bidders learn each other"s bids. If the process stabilizes the result can then be modeled as a Nash equilibrium in pure strategies of the static one-shot game of complete information, since each bidder will be playing a best-response to the others" bids.16 This argument seems especially appropriate for Yahoo!"s slot auction design where all bids are 14 To be entirely rigorous and consistent with our initial assumptions, we should have constructed a continuous probability density with full support over an appropriate domain. Taking the domain to be e.g. [0, 1] × [0, 1] and a continuous density with full support that is sufficiently concentrated around (1, 1/2) and (1/2, 1), with roughly equal mass around both, would yield the same conclusion. 15 Claim 3 should serve as a word of caution, because Feng et al. [7] find through their simulations that with a bivariate normal distribution over value-relevance pairs, and with 5 slots, 15 bidders, and δ = 2, RBR dominates RBB in terms of revenue for any level of correlation between value and relevance. However, they assume that bidding behavior in a second-price slot auction can be well approximated by truthful bidding. 16 We do not claim that bidders will actually learn each others" private information (value and relevance), just that for a stable set of bids there is a corresponding equilibrium of the complete information game. 223 made public. Google keeps bids private, but experimentation can allow one to discover other bids, especially since second pricing automatically reveals to an agent the bid of the agent ranked directly below him. 4.1 Equilibrium Analysis In this section we ask whether a pure-strategy Nash equilibrium exists in a RBB or RBR slot auction, with either first or second pricing. Before dealing with the first-price case there is a technical issue involving ties. In our model we allow bids to be nonnegative real numbers for mathematical convenience, but this can become problematic because there is then no bid that is just higher than another. We brush over such issues by assuming that an agent can bid infinitesimally higher than another. This is imprecise but allows us to focus on the intuition behind the result that follows. See Reny [17] for a full treatment of such issues. For the remainder of the paper, we assume that there are as many slots as bidders. The following result shows that there can be no pure-strategy Nash equilibrium with first pricing.17 Note that the argument holds for both RBB and RBR allocation rules. For RBB, bids should be interpreted as declared values, and for RBR as declared revenues. Theorem 2. There exists no complete information Nash equilibrium in pure strategies in the first-price slot auction, for any possible values of the agents, whether using a RBB or RBR allocation rule. Proof. Let σ : [K] → [N] be the allocation of slots to the agents resulting from their bids. Let ri and bi be the revenue and bid of the agent ranked ith , respectively. Note that we cannot have bi > bi+1, or else the agent in slot i can make a profitable deviation by instead bidding bi − > bi+1 for small enough > 0. This does not change its allocation, but increases its profit. Hence we must have bi = bi+1 (i.e. with one bidder bidding infinitesimally higher than the other). Since this holds for any two consecutive bidders, it follows that in a Nash equilibrium all bidders must be bidding 0 (since the bidder ranked last matches the bid directly below him, which is 0 by default because there is no such bid). But this is impossible: consider the bidder ranked last. The identity of this bidder is always clear given the deterministic tie-breaking rule. This bidder can obtain the top spot and increase his revenue by (γ1 −γK)rK > 0 by bidding some > 0, and for small enough this is necessarily a profitable deviation. Hence there is no Nash equilibrium in pure strategies. On the other hand, we find that in a second-price slot auction there can be a multitude of pure strategy Nash equilibria. The next two lemmas give conditions that characterize the allocations that can occur as a result of an equilibrium profile of bids, given fixed agent values and revenues. Then if we can exhibit an allocation that satisfies these conditions, there must exist at least one equilibrium. We first consider the RBR case. 17 B¨orgers et al. [2] have proven this result in a model with three bidders and three slots, and we generalize their argument. Edelman et al. [6] also point out this non-existence phenomenon. They only illustrate the fact with an example because the result is quite immediate. Lemma 2. Given an allocation σ, there exists a Nash equilibrium profile of bids b leading to σ in a second-price RBR slot auction if and only if „ 1 − γi γj+1 « rσ(i) ≤ rσ(j) for 1 ≤ j ≤ N − 2 and i ≥ j + 2. Proof. There exists a desired vector b which constitutes a Nash equilibrium if and only if the following set of inequalities can be satisfied (the variables are the πi and bj): πi ≥ γj(rσ(i) − bj) ∀i, ∀j < i (3) πi ≥ γj(rσ(i) − bj+1) ∀i, ∀j > i (4) πi = γi(rσ(i) − bi+1) ∀i (5) bi ≥ bi+1 1 ≤ i ≤ N − 1 (6) πi ≥ 0, bi ≥ 0 ∀i Here rσ(i) is the revenue of the agent allocated slot i, and πi and bi may be interpreted as this agent"s surplus and declared revenue, respectively. We first argue that constraints (6) can be removed, because the inequalities above can be satisfied if and only if the inequalities without (6) can be satisfied. The necessary direction is immediate. Assume we have a vector (π, b) which satisfies all inequalities above except (6). Then there is some i for which bi < bi+1. Construct a new vector (π, b ) identical to the original except with bi+1 = bi. We now have bi = bi+1. An agent in slot k < i sees the price of slot i decrease from bi+1 to bi+1 = bi, but this does not make i more preferred than k to this agent because we have πk ≥ γi−1(rσ(k) − bi) ≥ γi(rσ(k) − bi) = γi(rσ(k) −bi+1) (i.e. because the agent in slot k did not originally prefer slot i − 1 at price bi, he will not prefer slot i at price bi). A similar argument applies for agents in slots k > i + 1. The agent in slot i sees the price of this slot go down, which only makes it more preferred. Finally, the agent in slot i + 1 sees no change in the price of any slot, so his slot remains most preferred. Hence inequalities (3)-(5) remain valid at (π, b ). We first make this change to the bi+1 where bi < bi+1 and index i is smallest. We then recursively apply the change until we eventually obtain a vector that satisfies all inequalities. We safely ignore inequalities (6) from now on. By the Farkas lemma, the remaining inequalities can be satisfied if and only if there is no vector z such that X i,j (γj rσ(i)) zσ(i)j > 0 X i>j γjzσ(i)j + X i<j γj−1zσ(i)j−1 ≤ 0 ∀j (7) X j zσ(i)j ≤ 0 ∀i (8) zσ(i)j ≥ 0 ∀i, ∀j = i zσ(i)i free ∀i Note that a variable of the form zσ(i)i appears at most once in a constraint of type (8), so such a variable can never be positive. Also, zσ(i)1 = 0 for all i = 1 by constraint (7), since such variables never appear with another of the form zσ(i)i. Now if we wish to raise zσ(i)j above 0 by one unit for j = i, we must lower zσ(i)i by one unit because of the constraint of type (8). Because γjrσ(i) ≤ γirσ(i) for i < j, raising 224 zσ(i)j with i < j while adjusting other variables to maintain feasibility cannot make the objective P i,j(γjrσ(i))zσ(i)j positive. If this objective is positive, then this is due to some component zσ(i)j with i > j being positive. Now for the constraints of type (7), if i > j then zσ(i)j appears with zσ(j−1)j−1 (for 1 < j < N). So to raise the former variable γ−1 j units and maintain feasibility, we must (I) lower zσ(i)i by γ−1 j units, and (II) lower zσ(j−1)j−1 by γ−1 j−1 units. Hence if the following inequalities hold: rσ(i) ≤ „ γi γj « rσ(i) + rσ(j−1) (9) for 2 ≤ j ≤ N − 1 and i > j, raising some zσ(i)j with i > j cannot make the objective positive, and there is no z that satisfies all inequalities above. Conversely, if some inequality (9) does not hold, the objective can be made positive by raising the corresponding zσ(i)j and adjusting other variables so that feasibility is just maintained. By a slight reindexing, inequalities (9) yield the statement of the theorem. The RBB case is entirely analogous. Lemma 3. Given an allocation σ, there exists a Nash equilibrium profile of bids b leading to σ in a second-price RBB slot auction if and only if „ 1 − γi γj+1 « xσ(i) ≤ xσ(j) for 1 ≤ j ≤ N − 2 and i ≥ j + 2. Proof Sketch. The proof technique is the same as in the previous lemma. The desired Nash equilibrium exists if and only if a related set of inequalities can be satisfied; by the Farkas lemma, this occurs if and only if an alternate set of inequalities cannot be satisfied. The conditions that determine whether the latter holds are given in the statement of the lemma. The two lemmas above immediately lead to the following result. Theorem 3. There always exists a complete information Nash equilibrium in pure strategies in the second-price RBB slot auction. There always exists an efficient complete information Nash equilibrium in pure strategies in the secondprice RBR slot auction. Proof. First consider RBB. Suppose agents are ranked according to their true values. Since xσ(i) ≤ xσ(j) for i > j, the system of inequalities in Lemma 3 is satisfied, and the allocation is the result of some Nash equilibrium bid profile. By the same type of argument but appealing to Lemma 2 for RBR, there exists a Nash equilibrium bid profile such that bidders are ranked according to their true revenues. By Theorem 1, this latter allocation is efficient. This theorem establishes existence but not uniqueness. Indeed we expect that in many cases there will be multiple allocations (and hence equilibria) which satisfy the conditions of Lemmas 2 and 3. In particular, not all equilibria of a second-price RBR auction will be efficient. For instance, according to Lemma 2, with two agents and two slots any allocation can arise in a RBR equilibrium because no constraints apply. Theorems 2 and 3 taken together provide a possible explanation for Yahoo!"s switch from first to second pricing. We saw in Section 3.1 that this does not induce truthfulness from bidders. With first pricing, there will always be some bidder that feels compelled to adjust his bid. Second pricing is more convenient because an equilibrium can be reached, and this reduces the cost of bid management. 4.2 Efficiency For a given allocation rule, we call the allocation that would result if the bidders reported their values truthfully the standard allocation. Hence in the standard RBB allocation bidders are ranked by true values, and in the standard RBR allocation they are ranked by true revenues. According to Lemmas 2 and 3, a ranking that results from a Nash equilibrium profile can only deviate from the standard allocation by having agents with relatively similar values or revenues switch places. That is, if ri > rj then with RBR agent j can be ranked higher than i only if the ratio rj/ri is sufficiently large; similarly for RBB. This suggests that the value of an equilibrium allocation cannot differ too much from the value obtained in the standard allocation, and the following theorems confirms this. For an allocation σ of slots to agents, we denote its total value by f(σ) = PN i=1 γirσ(i). We denote by g(σ) = PN i=1 γixσ(i) allocation σ"s value when assuming all agents have identical relevance, normalized to 1. Let L = min i=1,...,N−1 min  γi+1 γi , 1 − γi+2 γi+1 ff (where by default γN+1 = 0). Let ηx and ηr be the standard allocations when using RBB and RBR, respectively. Theorem 4. For an allocation σ that results from a purestrategy Nash equilibrium of a second-price RBR slot auction, we have f(σ) ≥ Lf(ηr). Proof. We number the agents so that agent i has the ith highest revenue, so r1 ≥ r2 ≥ . . . ≥ rN . Hence the standard allocation has value f(ηr) = PN i=1 γiri. To prove the theorem, we will make repeated use of the fact thatP k akP k bk ≥ mink ak bk when the ak and bk are positive. Note that according to Lemma 2, if agent i lies at least two slots below slot j, then rσ(j) ≥ ri 1 − γj+2 γj+1 . It may be the case that for some slot i, we have σ(i) > i and for slots k > i + 1 we have σ(k) > i. We then say that slot i is inverted. Let S be the set of agents with indices at least i + 1; there are N − i of these. If slot i is inverted, it is occupied by some agent from S. Also all slots strictly lower than i + 1 must be occupied by the remaining agents from S, since σ(k) > i for k ≥ i + 2. The agent in slot i + 1 must then have an index σ(i + 1) ≤ i (note this means slot i + 1 cannot be inverted). Now there are two cases. In the first case we have σ(i) = i + 1. Then γirσ(i) + γi+1rσ(i+1) γiri + γi+1ri+1 ≥ γi+1ri + γiri+1 γiri + γi+1ri+1 ≥ min  γi+1 γi , γi γi+1 ff = γi+1 γi In the second case we have σ(i) > i+1. Then since all agents in S except the one in slot i lie strictly below slot i + 1, and 225 the agent in slot i is not agent i + 1, it must be that agent i+1 is in a slot strictly below slot i+1. This means that it is at least two slots below the agent that actually occupies slot i, and by Lemma 2 we then have rσ(i) ≥ ri+1 1 − γi+2 γi+1 . Thus, γirσ(i) + γi+1rσ(i+1) γiri + γi+1ri+1 ≥ γi+1ri + γirσ(i) γiri + γi+1ri+1 ≥ min  γi+1 γi , 1 − γi+2 γi+1 ff If slot i is not inverted, then on one hand we may have σ(i) ≤ i, in which case rσ(i)/ri ≥ 1. On the other hand we may have σ(i) > i but there is some agent with index j ≤ i that lies at least two slots below slot i. Then by Lemma 2, rσ(i) ≥ rj 1 − γi+2 γi+1 ≥ ri 1 − γi+2 γi+1 . We write i ∈ I if slot i is inverted, and i ∈ I if neither i nor i − 1 are inverted. By our arguments above two consecutive slots cannot be inverted, so we can write f(σ) f(γr) = P i∈I ` γirσ(i) + γi+1rσ(i+1) ´ + P i∈I γirσ(i) P i∈I (γiri + γi+1ri+1) + P i∈I γiri ≥ min  min i∈I  γirσ(i) + γi+1rσ(i+1) γiri + γi+1ri+1 ff , min i∈I  γirσ(i) γiri ffff ≥ L and this completes the proof. Note that for RBR, the standard value is also the efficient value by Theorem 1. Also note that for an exponential decay model, L = min ˘1 δ , 1 − 1 δ ¯ . With δ = 1.428 (see Section 2.1), the factor is L ≈ 1/3.34, so the total value in a pure-strategy Nash equilibrium of a second-price RBR slot auction is always within a factor of 3.34 of the efficient value with such a discount. Again for RBB we have an analogous result. Theorem 5. For an allocation σ that results from a purestrategy Nash equilibrium of a second-price RBB slot auction, we have g(σ) ≥ Lg(ηx). Proof Sketch. Simply substitute bidder values for bidder revenues in the proof of Theorem 4, and appeal to Lemma 3. 5. CONCLUSIONS This paper analyzed stylized versions of the slot auction designs currently used by Yahoo! and Google, namely rank by bid (RBB) and rank by revenue (RBR), respectively. We also considered first and second pricing rules together with each of these allocation rules, since both have been used historically. We first studied the short-run setting with incomplete information, corresponding to the case where agents have just approached the mechanism. Our equilibrium analysis revealed that RBB has much weaker informational requirements than RBR, because bidders need not know any information about relevance (even their own) to play the Bayes-Nash equilibrium. However, RBR leads to an efficient allocation in equilibrium, whereas RBB does not. We showed that for an arbitrary distribution over value and relevance, no revenue ranking of RBB and RBR is possible. We hope that the tools we used to establish these results (revenue equivalence, the form of first-price equilibria, the truthful payments rules) will help others wanting to pursue further analyses of slot auctions. We also studied the long-run case where agents have experimented with their bids and each settled on one they find optimal. We argued that a stable set of bids in this setting can be modeled as a pure-strategy Nash equilibrium of the static game of complete information. We showed that no pure-strategy equilibrium exists with either RBB or RBR using first pricing, but that with second pricing there always exists such an equilibrium (in the case of RBR, an efficient equilibrium). In general second pricing allows for multiple pure-strategy equilibria, but we showed that the value of such equilibria diverges by only a constant factor from the value obtained if all agents bid truthfully (which in the case of RBR is the efficient value). 6. FUTURE WORK Introducing budget constraints into the model is a natural next step for future work. The complication here lies in the fact that budgets are often set for entire campaigns rather than single keywords. Assuming that the optimal choice of budget can be made independent of the choice of bid for a specific keyword, it can be shown that it is a dominant-strategy to report this optimal budget with one"s bid. The problem is then to ascertain that bids and budgets can indeed be optimized separately, or to find a plausible model where deriving equilibrium bids and budgets together is tractable. Identifying a condition on the distribution over value and relevance that actually does yield a revenue ranking of RBB and RBR (such as correlation between value and relevance, perhaps) would yield a more satisfactory characterization of their relative revenue properties. Placing bounds on the revenue obtained in a complete information equilibrium is also a relevant question. Because the incomplete information case is such a close generalization of the most basic single-item auction model, it would be interesting to see which standard results from single-item auction theory (e.g. results with risk-averse bidders, an endogenous number of bidders, asymmetries, etc...) automatically generalize and which do not, to fully understand the structural differences between single-item and slot auctions. Acknowledgements David Pennock provided valuable guidance throughout this project. I would also like to thank David Parkes for helpful comments. 7. REFERENCES [1] Z. Abrams. Revenue maximization when bidders have budgets. In Proc. the ACM-SIAM Symposium on Discrete Algorithms, 2006. [2] T. B¨orgers, I. Cox, and M. Pesendorfer. Personal Communication. [3] C. Borgs, J. Chayes, N. Immorlica, M. Mahdian, and A. Saberi. Multi-unit auctions with budget-constrained bidders. In Proc. the Sixth ACM Conference on Electronic Commerce, Vancouver, BC, 2005. [4] F. Brandt and G. Weiß. Antisocial agents and Vickrey auctions. In J.-J. C. Meyer and M. Tambe, editors, 226 Intelligent Agents VIII, volume 2333 of Lecture Notes in Artificial Intelligence. Springer Verlag, 2001. [5] B. Edelman and M. Ostrovsky. Strategic bidder behavior in sponsored search auctions. In Workshop on Sponsored Search Auctions, ACM Electronic Commerce, 2005. [6] B. Edelman, M. Ostrovsky, and M. Schwarz. Internet advertising and the generalized second price auction: Selling billions of dollars worth of keywords. NBER working paper 11765, November 2005. [7] J. Feng, H. K. Bhargava, and D. M. Pennock. Implementing sponsored search in web search engines: Computational evaluation of alternative mechanisms. INFORMS Journal on Computing, 2005. Forthcoming. [8] J. Green and J.-J. Laffont. Characterization of satisfactory mechanisms for the revelation of preferences for public goods. Econometrica, 45:427-438, 1977. [9] B. Holmstrom. Groves schemes on restricted domains. Econometrica, 47(5):1137-1144, 1979. [10] B. Kitts, P. Laxminarayan, B. LeBlanc, and R. Meech. A formal analysis of search auctions including predictions on click fraud and bidding tactics. In Workshop on Sponsored Search Auctions, ACM Electronic Commerce, 2005. [11] V. Krishna. Auction Theory. Academic Press, 2002. [12] D. Liu and J. Chen. Designing online auctions with past performance information. Decision Support Systems, 2005. Forthcoming. [13] C. Meek, D. M. Chickering, and D. B. Wilson. Stochastic and contingent payment auctions. In Workshop on Sponsored Search Auctions, ACM Electronic Commerce, 2005. [14] A. Mehta, A. Saberi, U. Vazirani, and V. Vazirani. Adwords and generalized on-line matching. In Proc. 46th IEEE Symposium on Foundations of Computer Science, 2005. [15] P. Milgrom. Putting Auction Theory to Work. Cambridge University Press, 2004. [16] P. Milgrom and C. Shannon. Monotone comparative statics. Econometrica, 62(1):157-180, 1994. [17] P. J. Reny. On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica, 67(5):1029-1056, 1999. [18] H. R. Varian. Position auctions. Working Paper, February 2006. [19] W. Vickrey. Counterspeculation, auctions and competitive sealed tenders. Journal of Finance, 16:8-37, 1961. 227

rank by bid;second pricing;incomplete information;web search engine;resurgent online advertising industry;second-price payment rule;alternative slot auction design;rank by revenue;pay per click;divergence of value;divergence of economic value;auction theory;combined market capitalization;multitude of equilibrium;sponsor search;sponsored search;slot allocation;auction-style mechanism;search engine;ad listing;equilibrium multitude

train_J-42

The Dynamics of Viral Marketing ∗

We present an analysis of a person-to-person recommendation network, consisting of 4 million people who made 16 million recommendations on half a million products. We observe the propagation of recommendations and the cascade sizes, which we explain by a simple stochastic model. We then establish how the recommendation network grows over time and how effective it is from the viewpoint of the sender and receiver of the recommendations. While on average recommendations are not very effective at inducing purchases and do not spread very far, we present a model that successfully identifies product and pricing categories for which viral marketing seems to be very effective.

1. INTRODUCTION With consumers showing increasing resistance to traditional forms of advertising such as TV or newspaper ads, marketers have turned to alternate strategies, including viral marketing. Viral marketing exploits existing social networks by encouraging customers to share product information with their friends. Previously, a few in depth studies have shown that social networks affect the adoption of individual innovations and products (for a review see [15] or [16]). But until recently it has been difficult to measure how influential person-to-person recommendations actually are over a wide range of products. We were able to directly measure and model the effectiveness of recommendations by studying one online retailer"s incentivised viral marketing program. The website gave discounts to customers recommending any of its products to others, and then tracked the resulting purchases and additional recommendations. Although word of mouth can be a powerful factor influencing purchasing decisions, it can be tricky for advertisers to tap into. Some services used by individuals to communicate are natural candidates for viral marketing, because the product can be observed or advertised as part of the communication. Email services such as Hotmail and Yahoo had very fast adoption curves because every email sent through them contained an advertisement for the service and because they were free. Hotmail spent a mere $50,000 on traditional marketing and still grew from zero to 12 million users in 18 months [7]. Google"s Gmail captured a significant part of market share in spite of the fact that the only way to sign up for the service was through a referral. Most products cannot be advertised in such a direct way. At the same time the choice of products available to consumers has increased manyfold thanks to online retailers who can supply a much wider variety of products than traditional brick-and-mortar stores. Not only is the variety of products larger, but one observes a ‘fat tail" phenomenon, where a large fraction of purchases are of relatively obscure items. On Amazon.com, somewhere between 20 to 40 percent of unit sales fall outside of its top 100,000 ranked products [2]. Rhapsody, a streaming-music service, streams more tracks outside than inside its top 10,000 tunes [1]. Effectively advertising these niche products using traditional advertising approaches is impractical. Therefore using more targeted marketing approaches is advantageous both to the merchant and the consumer, who would benefit from learning about new products. The problem is partly addressed by the advent of online product and merchant reviews, both at retail sites such as EBay and Amazon, and specialized product comparison sites such as Epinions and CNET. Quantitative marketing techniques have been proposed [12], and the rating of products and merchants has been shown to effect the likelihood of an item being bought [13, 4]. Of further help to the consumer are collaborative filtering recommendations of the form people who bought x also bought y feature [11]. These refinements help consumers discover new products 228 and receive more accurate evaluations, but they cannot completely substitute personalized recommendations that one receives from a friend or relative. It is human nature to be more interested in what a friend buys than what an anonymous person buys, to be more likely to trust their opinion, and to be more influenced by their actions. Our friends are also acquainted with our needs and tastes, and can make appropriate recommendations. A Lucid Marketing survey found that 68% of individuals consulted friends and relatives before purchasing home electronics - more than the half who used search engines to find product information [3]. Several studies have attempted to model just this kind of network influence. Richardson and Domingos [14] used Epinions" trusted reviewer network to construct an algorithm to maximize viral marketing efficiency assuming that individuals" probability of purchasing a product depends on the opinions on the trusted peers in their network. Kempe, Kleinberg and Tardos [8] evaluate the efficiency of several algorithms for maximizing the size of influence set given various models of adoption. While these models address the question of maximizing the spread of influence in a network, they are based on assumed rather than measured influence effects. In contrast, in our study we are able to directly observe the effectiveness of person to person word of mouth advertising for hundreds of thousands of products for the first time. We find that most recommendation chains do not grow very large, often terminating with the initial purchase of a product. However, occasionally a product will propagate through a very active recommendation network. We propose a simple stochastic model that seems to explain the propagation of recommendations. Moreover, the characteristics of recommendation networks influence the purchase patterns of their members. For example, individuals" likelihood of purchasing a product initially increases as they receive additional recommendations for it, but a saturation point is quickly reached. Interestingly, as more recommendations are sent between the same two individuals, the likelihood that they will be heeded decreases. We also propose models to identify products for which viral marketing is effective: We find that the category and price of product plays a role, with recommendations of expensive products of interest to small, well connected communities resulting in a purchase more often. We also observe patterns in the timing of recommendations and purchases corresponding to times of day when people are likely to be shopping online or reading email. We report on these and other findings in the following sections. 2. THE RECOMMENDATION NETWORK 2.1 Dataset description Our analysis focuses on the recommendation referral program run by a large retailer. The program rules were as follows. Each time a person purchases a book, music, or a movie he or she is given the option of sending emails recommending the item to friends. The first person to purchase the same item through a referral link in the email gets a 10% discount. When this happens the sender of the recommendation receives a 10% credit on their purchase. The recommendation dataset consists of 15,646,121 recommendations made among 3,943,084 distinct users. The data was collected from June 5 2001 to May 16 2003. In total, 548,523 products were recommended, 99% of them belonging to 4 main product groups: Books, DVDs, Music and Videos. In addition to recommendation data, we also crawled the retailer"s website to obtain product categories, reviews and ratings for all products. Of the products in our data set, 5813 (1%) were discontinued (the retailer no longer provided any information about them). Although the data gives us a detailed and accurate view of recommendation dynamics, it does have its limitations. The only indication of the success of a recommendation is the observation of the recipient purchasing the product through the same vendor. We have no way of knowing if the person had decided instead to purchase elsewhere, borrow, or otherwise obtain the product. The delivery of the recommendation is also somewhat different from one person simply telling another about a product they enjoy, possibly in the context of a broader discussion of similar products. The recommendation is received as a form email including information about the discount program. Someone reading the email might consider it spam, or at least deem it less important than a recommendation given in the context of a conversation. The recipient may also doubt whether the friend is recommending the product because they think the recipient might enjoy it, or are simply trying to get a discount for themselves. Finally, because the recommendation takes place before the recommender receives the product, it might not be based on a direct observation of the product. Nevertheless, we believe that these recommendation networks are reflective of the nature of word of mouth advertising, and give us key insights into the influence of social networks on purchasing decisions. 2.2 Recommendation network statistics For each recommendation, the dataset included the product and product price, sender ID, receiver ID, the sent date, and a buy-bit, indicating whether the recommendation resulted in a purchase and discount. The sender and receiver ID"s were shadowed. We represent this data set as a directed multi graph. The nodes represent customers, and a directed edge contains all the information about the recommendation. The edge (i, j, p, t) indicates that i recommended product p to customer j at time t. The typical process generating edges in the recommendation network is as follows: a node i first buys a product p at time t and then it recommends it to nodes j1, . . . , jn. The j nodes can they buy the product and further recommend it. The only way for a node to recommend a product is to first buy it. Note that even if all nodes j buy a product, only the edge to the node jk that first made the purchase (within a week after the recommendation) will be marked by a buy-bit. Because the buy-bit is set only for the first person who acts on a recommendation, we identify additional purchases by the presence of outgoing recommendations for a person, since all recommendations must be preceded by a purchase. We call this type of evidence of purchase a buyedge. Note that buy-edges provide only a lower bound on the total number of purchases without discounts. It is possible for a customer to not be the first to act on a recommendation and also to not recommend the product to others. Unfortunately, this was not recorded in the data set. We consider, however, the buy-bits and buy-edges as proxies for the total number of purchases through recommendations. For each product group we took recommendations on all products from the group and created a network. Table 1 229 0 1 2 3 4 x 10 6 0 2 4 6 8 10 12 x 10 4 number of nodes sizeofgiantcomponent by month quadratic fit 0 10 20 0 2 4 x 10 6 m (month) n # nodes 1.7*10 6 m 10 0 10 1 10 2 10 3 10 1 10 2 10 3 10 4 10 5 10 6 kp (recommendations by a person for a product) N(x>=k p ) level 0 γ = 2.6 level 1 γ = 2.0 level 2 γ = 1.5 level 3 γ = 1.2 level 4 γ = 1.2 (a) Network growth (b) Recommending by level Figure 1: (a) The size of the largest connected component of customers over time. The inset shows the linear growth in the number of customers n over time. (b) The number of recommendations sent by a user with each curve representing a different depth of the user in the recommendation chain. A power law exponent γ is fitted to all but the tail. (first 7 columns) shows the sizes of various product group recommendation networks with p being the total number of products in the product group, n the total number of nodes spanned by the group recommendation network and e the number of edges (recommendations). The column eu shows the number of unique edges - disregarding multiple recommendations between the same source and recipient. In terms of the number of different items, there are by far the most music CDs, followed by books and videos. There is a surprisingly small number of DVD titles. On the other hand, DVDs account for more half of all recommendations in the dataset. The DVD network is also the most dense, having about 10 recommendations per node, while books and music have about 2 recommendations per node and videos have only a bit more than 1 recommendation per node. Music recommendations reached about the same number of people as DVDs but used more than 5 times fewer recommendations to achieve the same coverage of the nodes. Book recommendations reached by far the most people - 2.8 million. Notice that all networks have a very small number of unique edges. For books, videos and music the number of unique edges is smaller than the number of nodes - this suggests that the networks are highly disconnected [5]. Figure 1(a) shows the fraction of nodes in largest weakly connected component over time. Notice the component is very small. Even if we compose a network using all the recommendations in the dataset, the largest connected component contains less than 2.5% (100,420) of the nodes, and the second largest component has only 600 nodes. Still, some smaller communities, numbering in the tens of thousands of purchasers of DVDs in categories such as westerns, classics and Japanese animated films (anime), had connected components spanning about 20% of their members. The insert in figure 1(a) shows the growth of the customer base over time. Surprisingly it was linear, adding on average 165,000 new users each month, which is an indication that the service itself was not spreading epidemically. Further evidence of non-viral spread is provided by the relatively high percentage (94%) of users who made their first recommendation without having previously received one. Back to table 1: given the total number of recommendations e and purchases (bb + be) influenced by recommendations we can estimate how many recommendations need to be independently sent over the network to induce a new purchase. Using this metric books have the most influential recommendations followed by DVDs and music. For books one out of 69 recommendations resulted in a purchase. For DVDs it increases to 108 recommendations per purchase and further increases to 136 for music and 203 for video. Even with these simple counts we can make the first few observations. It seems that some people got quite heavily involved in the recommendation program, and that they tended to recommend a large number of products to the same set of friends (since the number of unique edges is so small). This shows that people tend to buy more DVDs and also like to recommend them to their friends, while they seem to be more conservative with books. One possible reason is that a book is bigger time investment than a DVD: one usually needs several days to read a book, while a DVD can be viewed in a single evening. One external factor which may be affecting the recommendation patterns for DVDs is the existence of referral websites (www.dvdtalk.com). On these websites people, who want to buy a DVD and get a discount, would ask for recommendations. This way there would be recommendations made between people who don"t really know each other but rather have an economic incentive to cooperate. We were not able to find similar referral sharing sites for books or CDs. 2.3 Forward recommendations Not all people who make a purchase also decide to give recommendations. So we estimate what fraction of people that purchase also decide to recommend forward. To obtain this information we can only use the nodes with purchases that resulted in a discount. The last 3 columns of table 1 show that only about a third of the people that purchase also recommend the product forward. The ratio of forward recommendations is much higher for DVDs than for other kinds of products. Videos also have a higher ratio of forward recommendations, while books have the lowest. This shows that people are most keen on recommending movies, while more conservative when recommending books and music. Figure 1(b) shows the cumulative out-degree distribution, that is the number of people who sent out at least kp recommendations, for a product. It shows that the deeper an individual is in the cascade, if they choose to make recommendations, they tend to recommend to a greater number of people on average (the distribution has a higher variance). This effect is probably due to only very heavily recommended products producing large enough cascades to reach a certain depth. We also observe that the probability of an individual making a recommendation at all (which can only occur if they make a purchase), declines after an initial increase as one gets deeper into the cascade. 2.4 Identifying cascades As customers continue forwarding recommendations, they contribute to the formation of cascades. In order to identify cascades, i.e. the causal propagation of recommendations, we track successful recommendations as they influence purchases and further recommendations. We define a recommendation to be successful if it reached a node before its first purchase. We consider only the first purchase of an item, because there are many cases when a person made multiple 230 Group p n e eu bb be Purchases Forward Percent Book 103,161 2,863,977 5,741,611 2,097,809 65,344 17,769 65,391 15,769 24.2 DVD 19,829 805,285 8,180,393 962,341 17,232 58,189 16,459 7,336 44.6 Music 393,598 794,148 1,443,847 585,738 7,837 2,739 7,843 1,824 23.3 Video 26,131 239,583 280,270 160,683 909 467 909 250 27.6 Total 542,719 3,943,084 15,646,121 3,153,676 91,322 79,164 90,602 25,179 27.8 Table 1: Product group recommendation statistics. p: number of products, n: number of nodes, e: number of edges (recommendations), eu: number of unique edges, bb: number of buy bits, be: number of buy edges. Last 3 columns of the table: Fraction of people that purchase and also recommend forward. Purchases: number of nodes that purchased. Forward: nodes that purchased and then also recommended the product. 973 938 (a) Medical book (b) Japanese graphic novel Figure 2: Examples of two product recommendation networks: (a) First aid study guide First Aid for the USMLE Step, (b) Japanese graphic novel (manga) Oh My Goddess!: Mara Strikes Back. 10 0 10 5 10 0 10 2 10 4 10 6 10 8 Number of recommendations Count = 3.4e6 x−2.30 R2 =0.96 10 0 10 1 10 2 10 3 10 4 10 0 10 2 10 4 10 6 10 8 Number of purchases Count = 4.1e6 x−2.49 R2 =0.99 (a) Recommendations (b) Purchases Figure 3: Distribution of the number of recommendations and number of purchases made by a node. purchases of the same product, and in between those purchases she may have received new recommendations. In this case one cannot conclude that recommendations following the first purchase influenced the later purchases. Each cascade is a network consisting of customers (nodes) who purchased the same product as a result of each other"s recommendations (edges). We delete late recommendations - all incoming recommendations that happened after the first purchase of the product. This way we make the network time increasing or causal - for each node all incoming edges (recommendations) occurred before all outgoing edges. Now each connected component represents a time obeying propagation of recommendations. Figure 2 shows two typical product recommendation networks: (a) a medical study guide and (b) a Japanese graphic novel. Throughout the dataset we observe very similar patters. Most product recommendation networks consist of a large number of small disconnected components where we do not observe cascades. Then there is usually a small number of relatively small components with recommendations successfully propagating. This observation is reflected in the heavy tailed distribution of cascade sizes (see figure 4), having a power-law exponent close to 1 for DVDs in particular. We also notice bursts of recommendations (figure 2(b)). Some nodes recommend to many friends, forming a star like pattern. Figure 3 shows the distribution of the recommendations and purchases made by a single node in the recommendation network. Notice the power-law distributions and long flat tails. The most active person made 83,729 recommendations and purchased 4,416 different items. Finally, we also sometimes observe ‘collisions", where nodes receive recommendations from two or more sources. A detailed enumeration and analysis of observed topological cascade patterns for this dataset is made in [10]. 2.5 The recommendation propagation model A simple model can help explain how the wide variance we observe in the number of recommendations made by individuals can lead to power-laws in cascade sizes (figure 4). The model assumes that each recipient of a recommendation will forward it to others if its value exceeds an arbitrary threshold that the individual sets for herself. Since exceeding this value is a probabilistic event, let"s call pt the probability that at time step t the recommendation exceeds the thresh231 10 0 10 1 10 2 10 0 10 2 10 4 10 6 = 1.8e6 x−4.98 R2 =0.99 10 0 10 1 10 2 10 3 10 0 10 2 10 4 = 3.4e3 x−1.56 R2 =0.83 10 0 10 1 10 2 10 0 10 2 10 4 = 4.9e5 x−6.27 R2 =0.97 10 0 10 1 10 2 10 0 10 2 10 4 = 7.8e4 x−5.87 R2 =0.97 (a) Book (b) DVD (c) Music (d) Video Figure 4: Size distribution of cascades (size of cascade vs. count). Bold line presents a power-fit. old. In that case the number of recommendations Nt+1 at time (t + 1) is given in terms of the number of recommendations at an earlier time by Nt+1 = ptNt (1) where the probability pt is defined over the unit interval. Notice that, because of the probabilistic nature of the threshold being exceeded, one can only compute the final distribution of recommendation chain lengths, which we now proceed to do. Subtracting from both sides of this equation the term Nt and diving by it we obtain N(t+1) − Nt Nt = pt − 1 (2) Summing both sides from the initial time to some very large time T and assuming that for long times the numerator is smaller than the denominator (a reasonable assumption) we get dN N = pt (3) The left hand integral is just ln(N), and the right hand side is a sum of random variables, which in the limit of a very large uncorrelated number of recommendations is normally distributed (central limit theorem). This means that the logarithm of the number of messages is normally distributed, or equivalently, that the number of messages passed is log-normally distributed. In other words the probability density for N is given by P(N) = 1 N √ 2πσ2 exp −(ln(N) − μ)2 2σ2 (4) which, for large variances describes a behavior whereby the typical number of recommendations is small (the mode of the distribution) but there are unlikely events of large chains of recommendations which are also observable. Furthermore, for large variances, the lognormal distribution can behave like a power law for a range of values. In order to see this, take the logarithms on both sides of the equation (equivalent to a log-log plot) and one obtains ln(P(N)) = − ln(N) − ln( √ 2πσ2) − (ln (N) − μ)2 2σ2 (5) So, for large σ, the last term of the right hand side goes to zero, and since the the second term is a constant one obtains a power law behavior with exponent value of minus one. There are other models which produce power-law distributions of cascade sizes, but we present ours for its simplicity, since it does not depend on network topology [6] or critical thresholds in the probability of a recommendation being accepted [18]. 3. SUCCESS OF RECOMMENDATIONS So far we only looked into the aggregate statistics of the recommendation network. Next, we ask questions about the effectiveness of recommendations in the recommendation network itself. First, we analyze the probability of purchasing as one gets more and more recommendations. Next, we measure recommendation effectiveness as two people exchange more and more recommendations. Lastly, we observe the recommendation network from the perspective of the sender of the recommendation. Does a node that makes more recommendations also influence more purchases? 3.1 Probability of buying versus number of incoming recommendations First, we examine how the probability of purchasing changes as one gets more and more recommendations. One would expect that a person is more likely to buy a product if she gets more recommendations. On the other had one would also think that there is a saturation point - if a person hasn"t bought a product after a number of recommendations, they are not likely to change their minds after receiving even more of them. So, how many recommendations are too many? Figure 5 shows the probability of purchasing a product as a function of the number of incoming recommendations on the product. As we move to higher numbers of incoming recommendations, the number of observations drops rapidly. For example, there were 5 million cases with 1 incoming recommendation on a book, and only 58 cases where a person got 20 incoming recommendations on a particular book. The maximum was 30 incoming recommendations. For these reasons we cut-off the plot when the number of observations becomes too small and the error bars too large. Figure 5(a) shows that, overall, book recommendations are rarely followed. Even more surprisingly, as more and more recommendations are received, their success decreases. We observe a peak in probability of buying at 2 incoming recommendations and then a slow drop. For DVDs (figure 5(b)) we observe a saturation around 10 incoming recommendations. This means that after a person gets 10 recommendations on a particular DVD, they become immune to them - their probability of buying does not increase anymore. The number of observations is 2.5 million at 1 incoming recommendation and 100 at 60 incoming recommendations. The maximal number of received recommendations is 172 (and that person did not buy) 232 2 4 6 8 10 0 0.01 0.02 0.03 0.04 0.05 0.06 Incoming Recommendations ProbabilityofBuying 10 20 30 40 50 60 0 0.02 0.04 0.06 0.08 Incoming Recommendations ProbabilityofBuying (a) Books (b) DVD Figure 5: Probability of buying a book (DVD) given a number of incoming recommendations. 5 10 15 20 25 30 35 40 4 6 8 10 12 x 10 −3 Exchanged recommendations Probabilityofbuying 5 10 15 20 25 30 35 40 0.02 0.03 0.04 0.05 0.06 0.07 Exchanged recommendations Probabilityofbuying (a) Books (b) DVD Figure 6: The effectiveness of recommendations with the total number of exchanged recommendations. 3.2 Success of subsequent recommendations Next, we analyze how the effectiveness of recommendations changes as two persons exchange more and more recommendations. A large number of exchanged recommendations can be a sign of trust and influence, but a sender of too many recommendations can be perceived as a spammer. A person who recommends only a few products will have her friends" attention, but one who floods her friends with all sorts of recommendations will start to loose her influence. We measure the effectiveness of recommendations as a function of the total number of previously exchanged recommendations between the two nodes. We construct the experiment in the following way. For every recommendation r on some product p between nodes u and v, we first determine how many recommendations were exchanged between u and v before recommendation r. Then we check whether v, the recipient of recommendation, purchased p after recommendation r arrived. For the experiment we consider only node pairs (u, v), where there were at least a total of 10 recommendations sent from u to v. We perform the experiment using only recommendations from the same product group. Figure 6 shows the probability of buying as a function of the total number of exchanged recommendations between two persons up to that point. For books we observe that the effectiveness of recommendation remains about constant up to 3 exchanged recommendations. As the number of exchanged recommendations increases, the probability of buying starts to decrease to about half of the original value and then levels off. For DVDs we observe an immediate and consistent drop. This experiment shows that recommendations start to lose effect after more than two or three are passed between two people. We performed the experiment also for video and music, but the number of observations was too low and the measurements were noisy. 3.3 Success of outgoing recommendations In previous sections we examined the data from the viewpoint of the receiver of the recommendation. Now we look from the viewpoint of the sender. The two interesting questions are: how does the probability of getting a 10% credit change with the number of outgoing recommendations; and given a number of outgoing recommendations, how many purchases will they influence? One would expect that recommendations would be the most effective when recommended to the right subset of friends. If one is very selective and recommends to too few friends, then the chances of success are slim. One the other hand, recommending to everyone and spamming them with recommendations may have limited returns as well. The top row of figure 7 shows how the average number of purchases changes with the number of outgoing recommendations. For books, music, and videos the number of purchases soon saturates: it grows fast up to around 10 outgoing recommendations and then the trend either slows or starts to drop. DVDs exhibit different behavior, with the expected number of purchases increasing throughout. But if we plot the probability of getting a 10% credit as a function of the number of outgoing recommendations, as in the bottom row of figure 7, we see that the success of DVD recommendations saturates as well, while books, videos and music have qualitatively similar trends. The difference in the curves for DVD recommendations points to the presence of collisions in the dense DVD network, which has 10 recommendations per node and around 400 per product - an order of magnitude more than other product groups. This means that many different individuals are recommending to the same person, and after that person makes a purchase, even though all of them made a ‘successful recommendation" 233 10 20 30 40 50 60 0 0.1 0.2 0.3 0.4 0.5 Outgoing Recommendations NumberofPurchases 20 40 60 80 100 120 140 0 1 2 3 4 5 6 7 Outgoing Recommendations NumberofPurchases 5 10 15 20 0 0.05 0.1 0.15 0.2 Outgoing Recommendations NumberofPurchases 2 4 6 8 10 12 0 0.05 0.1 0.15 0.2 0.25 Outgoing Recommendations NumberofPurchases 10 20 30 40 50 60 70 80 0 0.05 0.1 0.15 0.2 0.25 Outgoing Recommendations ProbabilityofCredit 10 20 30 40 50 60 70 80 0 0.02 0.04 0.06 0.08 0.1 0.12 Outgoing Recommendations ProbabilityofCredit 5 10 15 20 0 0.02 0.04 0.06 0.08 0.1 Outgoing Recommendations ProbabilityofCredit 2 4 6 8 10 12 14 0 0.02 0.04 0.06 0.08 Outgoing Recommendations ProbabilityofCredit (a) Books (b) DVD (c) Music (d) Video Figure 7: Top row: Number of resulting purchases given a number of outgoing recommendations. Bottom row: Probability of getting a credit given a number of outgoing recommendations. 1 2 3 4 5 6 7 > 7 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Lag [day] ProportionofPurchases 1 2 3 4 5 6 7 > 7 0 0.1 0.2 0.3 0.4 0.5 Lag [day] ProportionofPurchases (a) Books (b) DVD Figure 8: The time between the recommendation and the actual purchase. We use all purchases. by our definition, only one of them receives a credit. 4. TIMING OF RECOMMENDATIONS AND PURCHASES The recommendation referral program encourages people to purchase as soon as possible after they get a recommendation, since this maximizes the probability of getting a discount. We study the time lag between the recommendation and the purchase of different product groups, effectively how long it takes a person to both receive a recommendation, consider it, and act on it. We present the histograms of the thinking time, i.e. the difference between the time of purchase and the time the last recommendation was received for the product prior to the purchase (figure 8). We use a bin size of 1 day. Around 35%40% of book and DVD purchases occurred within a day after the last recommendation was received. For DVDs 16% purchases occur more than a week after last recommendation, while this drops to 10% for books. In contrast, if we consider the lag between the purchase and the first recommendation, only 23% of DVD purchases are made within a day, while the proportion stays the same for books. This reflects a greater likelihood for a person to receive multiple recommendations for a DVD than for a book. At the same time, DVD recommenders tend to send out many more recommendations, only one of which can result in a discount. Individuals then often miss their chance of a discount, which is reflected in the high ratio (78%) of recommended DVD purchases that did not a get discount (see table 1, columns bb and be). In contrast, for books, only 21% of purchases through recommendations did not receive a discount. We also measure the variation in intensity by time of day for three different activities in the recommendation system: recommendations (figure 9(a)), all purchases (figure 9(b)), and finally just the purchases which resulted in a discount (figure 9(c)). Each is given as a total count by hour of day. The recommendations and purchases follow the same pattern. The only small difference is that purchases reach a sharper peak in the afternoon (after 3pm Pacific Time, 6pm Eastern time). The purchases that resulted in a discount look like a negative image of the first two figures. This means that most of discounted purchases happened in the morning when the traffic (number of purchases/recommendations) on the retailer"s website was low. This makes a lot of sense since most of the recommendations happened during the day, and if the person wanted to get the discount by being the first one to purchase, she had the highest chances when the traffic on the website was the lowest. 5. RECOMMENDATION EFFECTIVENESS BY BOOK CATEGORY Social networks are a product of the contexts that bring people together. Some contexts result in social ties that are more effective at conducting an action. For example, in small world experiments, where participants attempt to reach a target individual through their chain of acquaintances, profession trumped geography, which in turn was more useful in locating a target than attributes such as religion or hobbies [9, 17]. In the context of product recommendations, we can ask whether a recommendation for a work of fiction, which may be made by any friend or neighbor, is 234 0 5 10 15 20 25 0 2 4 6 8 10 x 10 5 Hour of the Day Recommendtions 0 5 10 15 20 25 0 0.5 1 1.5 2 x 10 4 Hour of the Day AllPurchases 0 5 10 15 20 25 0 1000 2000 3000 4000 5000 6000 7000 Hour of the Day DiscountedPurchases (a) Recommendations (b) Purchases (c) Purchases with Discount Figure 9: Time of day for purchases and recommendations. (a) shows the distribution of recommendations over the day, (b) shows all purchases and (c) shows only purchases that resulted in getting discount. more or less influential than a recommendation for a technical book, which may be made by a colleague at work or school. Table 2 shows recommendation trends for all top level book categories by subject. An analysis of other product types can be found in the extended version of the paper. For clarity, we group the results by 4 different category types: fiction, personal/leisure, professional/technical, and nonfiction/other. Fiction encompasses categories such as Sci-Fi and Romance, as well as children"s and young adult books. Personal/Leisure encompasses everything from gardening, photography and cooking to health and religion. First, we compare the relative number of recommendations to reviews posted on the site (column cav/rp1 of table 2). Surprisingly, we find that the number of people making personal recommendations was only a few times greater than the number of people posting a public review on the website. We observe that fiction books have relatively few recommendations compared to the number of reviews, while professional and technical books have more recommendations than reviews. This could reflect several factors. One is that people feel more confident reviewing fiction than technical books. Another is that they hesitate to recommend a work of fiction before reading it themselves, since the recommendation must be made at the point of purchase. Yet another explanation is that the median price of a work of fiction is lower than that of a technical book. This means that the discount received for successfully recommending a mystery novel or thriller is lower and hence people have less incentive to send recommendations. Next, we measure the per category efficacy of recommendations by observing the ratio of the number of purchases occurring within a week following a recommendation to the number of recommenders for each book subject category (column b of table 2). On average, only 2% of the recommenders of a book received a discount because their recommendation was accepted, and another 1% made a recommendation that resulted in a purchase, but not a discount. We observe marked differences in the response to recommendation for different categories of books. Fiction in general is not very effectively recommended, with only around 2% of recommenders succeeding. The efficacy was a bit higher (around 3%) for non-fiction books dealing with personal and leisure pursuits, but is significantly higher in the professional and technical category. Medical books have nearly double the average rate of recommendation acceptance. This could be in part attributed to the higher median price of medical books and technical books in general. As we will see in Section 6, a higher product price increases the chance that a recommendation will be accepted. Recommendations are also more likely to be accepted for certain religious categories: 4.3% for Christian living and theology and 4.8% for Bibles. In contrast, books not tied to organized religions, such as ones on the subject of new age (2.5%) and occult (2.2%) spirituality, have lower recommendation effectiveness. These results raise the interesting possibility that individuals have greater influence over one another in an organized context, for example through a professional contact or a religious one. There are exceptions of course. For example, Japanese anime DVDs have a strong following in the US, and this is reflected in their frequency and success in recommendations. Another example is that of gardening. In general, recommendations for books relating to gardening have only a modest chance of being accepted, which agrees with the individual prerogative that accompanies this hobby. At the same time, orchid cultivation can be a highly organized and social activity, with frequent ‘shows" and online communities devoted entirely to orchids. Perhaps because of this, the rate of acceptance of orchid book recommendations is twice as high as those for books on vegetable or tomato growing. 6. MODELING THE RECOMMENDATION SUCCESS We have examined the properties of recommendation network in relation to viral marketing, but one question still remains: what determines the product"s viral marketing success? We present a model which characterizes product categories for which recommendations are more likely to be accepted. We use a regression of the following product attributes to correlate them with recommendation success: • r: number of recommendations • ns: number of senders of recommendations • nr: number of recipients of recommendations • p: price of the product • v: number of reviews of the product • t: average product rating 235 category np n cc rp1 vav cav/ pm b ∗ 100 rp1 Books general 370230 2,860,714 1.87 5.28 4.32 1.41 14.95 3.12 Fiction Children"s Books 46,451 390,283 2.82 6.44 4.52 1.12 8.76 2.06** Literature & Fiction 41,682 502,179 3.06 13.09 4.30 0.57 11.87 2.82* Mystery and Thrillers 10,734 123,392 6.03 20.14 4.08 0.36 9.60 2.40** Science Fiction & Fantasy 10,008 175,168 6.17 19.90 4.15 0.64 10.39 2.34** Romance 6,317 60,902 5.65 12.81 4.17 0.52 6.99 1.78** Teens 5,857 81,260 5.72 20.52 4.36 0.41 9.56 1.94** Comics & Graphic Novels 3,565 46,564 11.70 4.76 4.36 2.03 10.47 2.30* Horror 2,773 48,321 9.35 21.26 4.16 0.44 9.60 1.81** Personal/Leisure Religion and Spirituality 43,423 441,263 1.89 3.87 4.45 1.73 9.99 3.13 Health Mind and Body 33,751 572,704 1.54 4.34 4.41 2.39 13.96 3.04 History 28,458 28,3406 2.74 4.34 4.30 1.27 18.00 2.84 Home and Garden 19,024 180,009 2.91 1.78 4.31 3.48 15.37 2.26** Entertainment 18,724 258,142 3.65 3.48 4.29 2.26 13.97 2.66* Arts and Photography 17,153 179,074 3.49 1.56 4.42 3.85 20.95 2.87 Travel 12,670 113,939 3.91 2.74 4.26 1.87 13.27 2.39** Sports 10,183 120,103 1.74 3.36 4.34 1.99 13.97 2.26** Parenting and Families 8,324 182,792 0.73 4.71 4.42 2.57 11.87 2.81 Cooking Food and Wine 7,655 146,522 3.02 3.14 4.45 3.49 13.97 2.38* Outdoors & Nature 6,413 59,764 2.23 1.93 4.42 2.50 15.00 3.05 Professional/Technical Professional & Technical 41,794 459,889 1.72 1.91 4.30 3.22 32.50 4.54** Business and Investing 29,002 476,542 1.55 3.61 4.22 2.94 20.99 3.62** Science 25,697 271,391 2.64 2.41 4.30 2.42 28.00 3.90** Computers and Internet 18,941 375,712 2.22 4.51 3.98 3.10 34.95 3.61** Medicine 16,047 175,520 1.08 1.41 4.40 4.19 39.95 5.68** Engineering 10,312 107,255 1.30 1.43 4.14 3.85 59.95 4.10** Law 5,176 53,182 2.64 1.89 4.25 2.67 24.95 3.66* Nonfiction-other Nonfiction 55,868 560,552 2.03 3.13 4.29 1.89 18.95 3.28** Reference 26,834 371,959 1.94 2.49 4.19 3.04 17.47 3.21 Biographies and Memoirs 18,233 277,356 2.80 7.65 4.34 0.90 14.00 2.96 Table 2: Statistics by book category: np:number of products in category, n number of customers, cc percentage of customers in the largest connected component, rp1 av. # reviews in 2001 - 2003, rp2 av. # reviews 1st 6 months 2005, vav average star rating, cav average number of people recommending product, cav/rp1 ratio of recommenders to reviewers, pm median price, b ratio of the number of purchases resulting from a recommendation to the number of recommenders. The symbol ** denotes statistical significance at the 0.01 level, * at the 0.05 level. From the original set of half a million products, we compute a success rate s for the 48,218 products that had at least one purchase made through a recommendation and for which a price was given. In section 5 we defined recommendation success rate s as the ratio of the total number purchases made through recommendations and the number of senders of the recommendations. We decided to use this kind of normalization, rather than normalizing by the total number of recommendations sent, in order not to penalize communities where a few individuals send out many recommendations (figure 2(b)). Since the variables follow a heavy tailed distribution, we use the following model: s = exp( i βi log(xi) + i) where xi are the product attributes (as described on previous page), and i is random error. We fit the model using least squares and obtain the coefficients βi shown on table 3. With the exception of the average rating, they are all significant. The only two attributes with a positive coefficient are the number of recommendations and price. This shows that more expensive and more recommended products have a higher success rate. The number of senders and receivers have large negative coefficients, showing that successfully recommended products are more likely to be not so widely popular. They have relatively many recommendations with a small number of senders and receivers, which suggests a very dense recommendation network where lots of recommendations were exchanged between a small community of people. These insights could be to marketers - personal recommendations are most effective in small, densely connected communities enjoying expensive products. 236 Variable Coefficient βi const -0.940 (0.025)** r 0.426 (0.013)** ns -0.782 (0.004)** nr -1.307 (0.015)** p 0.128 (0.004)** v -0.011 (0.002)** t -0.027 (0.014)* R2 0.74 Table 3: Regression using the log of the recommendation success rate, ln(s), as the dependent variable. For each coefficient we provide the standard error and the statistical significance level (**:0.01, *:0.1). 7. DISCUSSION AND CONCLUSION Although the retailer may have hoped to boost its revenues through viral marketing, the additional purchases that resulted from recommendations are just a drop in the bucket of sales that occur through the website. Nevertheless, we were able to obtain a number of interesting insights into how viral marketing works that challenge common assumptions made in epidemic and rumor propagation modeling. Firstly, it is frequently assumed in epidemic models that individuals have equal probability of being infected every time they interact. Contrary to this we observe that the probability of infection decreases with repeated interaction. Marketers should take heed that providing excessive incentives for customers to recommend products could backfire by weakening the credibility of the very same links they are trying to take advantage of. Traditional epidemic and innovation diffusion models also often assume that individuals either have a constant probability of ‘converting" every time they interact with an infected individual or that they convert once the fraction of their contacts who are infected exceeds a threshold. In both cases, an increasing number of infected contacts results in an increased likelihood of infection. Instead, we find that the probability of purchasing a product increases with the number of recommendations received, but quickly saturates to a constant and relatively low probability. This means individuals are often impervious to the recommendations of their friends, and resist buying items that they do not want. In network-based epidemic models, extremely highly connected individuals play a very important role. For example, in needle sharing and sexual contact networks these nodes become the super-spreaders by infecting a large number of people. But these models assume that a high degree node has as much of a probability of infecting each of its neighbors as a low degree node does. In contrast, we find that there are limits to how influential high degree nodes are in the recommendation network. As a person sends out more and more recommendations past a certain number for a product, the success per recommendation declines. This would seem to indicate that individuals have influence over a few of their friends, but not everybody they know. We also presented a simple stochastic model that allows for the presence of relatively large cascades for a few products, but reflects well the general tendency of recommendation chains to terminate after just a short number of steps. We saw that the characteristics of product reviews and effectiveness of recommendations vary by category and price, with more successful recommendations being made on technical or religious books, which presumably are placed in the social context of a school, workplace or place of worship. Finally, we presented a model which shows that smaller and more tightly knit groups tend to be more conducive to viral marketing. So despite the relative ineffectiveness of the viral marketing program in general, we found a number of new insights which we hope will have general applicability to marketing strategies and to future models of viral information spread. 8. REFERENCES [1] Anonymous. Profiting from obscurity: What the long tail means for the economics of e-commerce. Economist, 2005. [2] E. Brynjolfsson, Y. Hu, and M. D. Smith. Consumer surplus in the digital economy: Estimating the value of increased product variety at online booksellers. Management Science, 49(11), 2003. [3] K. Burke. As consumer attitudes shift, so must marketing strategies. 2003. [4] J. Chevalier and D. Mayzlin. The effect of word of mouth on sales: Online book reviews. 2004. [5] P. Erd¨os and A. R´enyi. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci., 1960. [6] D. Gruhl, R. Guha, D. Liben-Nowell, and A. Tomkins. Information diffusion through blogspace. In WWW "04, 2004. [7] S. Jurvetson. What exactly is viral marketing? Red Herring, 78:110-112, 2000. [8] D. Kempe, J. Kleinberg, and E. Tardos. Maximizing the spread of infuence in a social network. In ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), 2003. [9] P. Killworth and H. Bernard. Reverse small world experiment. Social Networks, 1:159-192, 1978. [10] J. Leskovec, A. Singh, and J. Kleinberg. Patterns of influence in a recommendation network. In Pacific-Asia Conference on Knowledge Discovery and Data Mining (PAKDD), 2006. [11] G. Linden, B. Smith, and J. York. Amazon.com recommendations: item-to-item collaborative filtering. IEEE Internet Computing, 7(1):76-80, 2003. [12] A. L. Montgomery. Applying quantitative marketing techniques to the internet. Interfaces, 30:90-108, 2001. [13] P. Resnick and R. Zeckhauser. Trust among strangers in internet transactions: Empirical analysis of ebays reputation system. In The Economics of the Internet and E-Commerce. Elsevier Science, 2002. [14] M. Richardson and P. Domingos. Mining knowledge-sharing sites for viral marketing. In ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), 2002. [15] E. M. Rogers. Diffusion of Innovations. Free Press, New York, fourth edition, 1995. [16] D. Strang and S. A. Soule. Diffusion in organizations and social movements: From hybrid corn to poison pills. Annual Review of Sociology, 24:265-290, 1998. [17] J. Travers and S. Milgram. An experimental study of the small world problem. Sociometry, 1969. [18] D. Watts. A simple model of global cascades on random networks. PNAS, 99(9):4766-5771, Apr 2002. 237

directed multi graph;product;consumer;recommender system;viral market;recommendation network;viral marketing;advertisement;pricing category;probability;e-commerce;purchase;stochastic model;connected individual

train_J-44

Scouts, Promoters, and Connectors: The Roles of Ratings in Nearest Neighbor Collaborative Filtering

Recommender systems aggregate individual user ratings into predictions of products or services that might interest visitors. The quality of this aggregation process crucially affects the user experience and hence the effectiveness of recommenders in e-commerce. We present a novel study that disaggregates global recommender performance metrics into contributions made by each individual rating, allowing us to characterize the many roles played by ratings in nearestneighbor collaborative filtering. In particular, we formulate three roles-scouts, promoters, and connectors-that capture how users receive recommendations, how items get recommended, and how ratings of these two types are themselves connected (resp.). These roles find direct uses in improving recommendations for users, in better targeting of items and, most importantly, in helping monitor the health of the system as a whole. For instance, they can be used to track the evolution of neighborhoods, to identify rating subspaces that do not contribute (or contribute negatively) to system performance, to enumerate users who are in danger of leaving, and to assess the susceptibility of the system to attacks such as shilling. We argue that the three rating roles presented here provide broad primitives to manage a recommender system and its community.

1. INTRODUCTION Recommender systems have become integral to e-commerce, providing technology that suggests products to a visitor based on previous purchases or rating history. Collaborative filtering, a common form of recommendation, predicts a user"s rating for an item by combining (other) ratings of that user with other users" ratings. Significant research has been conducted in implementing fast and accurate collaborative filtering algorithms [2, 7], designing interfaces for presenting recommendations to users [1], and studying the robustness of these algorithms [8]. However, with the exception of a few studies on the influence of users [10], little attention has been paid to unraveling the inner workings of a recommender in terms of the individual ratings and the roles they play in making (good) recommendations. Such an understanding will give an important handle to monitoring and managing a recommender system, to engineer mechanisms to sustain the recommender, and thereby ensure its continued success. Our motivation here is to disaggregate global recommender performance metrics into contributions made by each individual rating, allowing us to characterize the many roles played by ratings in nearest-neighbor collaborative filtering. We identify three possible roles: (scouts) to connect the user into the system to receive recommendations, (promoters) to connect an item into the system to be recommended, and (connectors) to connect ratings of these two kinds. Viewing ratings in this way, we can define the contribution of a rating in each role, both in terms of allowing recommendations to occur, and in terms of influence on the quality of recommendations. In turn, this capability helps support scenarios such as: 1. Situating users in better neighborhoods: A user"s ratings may inadvertently connect the user to a neighborhood for which the user"s tastes may not be a perfect match. Identifying ratings responsible for such bad recommendations and suggesting new items to rate can help situate the user in a better neighborhood. 2. Targeting items: Recommender systems suffer from lack of user participation, especially in cold-start scenarios [13] involving newly arrived items. Identifying users who can be encouraged to rate specific items helps ensure coverage of the recommender system. 3. Monitoring the evolution of the recommender system and its stakeholders: A recommender system is constantly under change: growing with new users and 250 items, shrinking with users leaving the system, items becoming irrelevant, and parts of the system under attack. Tracking the roles of a rating and its evolution over time provides many insights into the health of the system, and how it could be managed and improved. These include being able to identify rating subspaces that do not contribute (or contribute negatively) to system performance, and could be removed; to enumerate users who are in danger of leaving, or have left the system; and to assess the susceptibility of the system to attacks such as shilling [5]. As we show, the characterization of rating roles presented here provides broad primitives to manage a recommender system and its community. The rest of the paper is organized as follows. Background on nearest-neighbor collaborative filtering and algorithm evaluation is discussed in Section 2. Section 3 defines and discusses the roles of a rating, and Section 4 defines measures of the contribution of a rating in each of these roles. In Section 5, we illustrate the use of these roles to address the goals outlined above. 2. BACKGROUND 2.1 Algorithms Nearest-neighbor collaborative filtering algorithms either use neighborhoods of users or neighborhoods of items to compute a prediction. An algorithm of the first kind is called user-based, and one of the second kind is called itembased [12]. In both families of algorithms, neighborhoods are formed by first computing the similarity between all pairs of users (for user-based) or items (for item-based). Predictions are then computed by aggregating ratings, which in a user-based algorithm involves aggregating the ratings of the target item by the user"s neighbors and, in an item-based algorithm, involves aggregating the user"s ratings of items that are neighbors of the target item. Algorithms within these families differ in the definition of similarity, formation of neighborhoods, and the computation of predictions. We consider a user-based algorithm based on that defined for GroupLens [11] with variations from Herlocker et al. [2], and an item-based algorithm similar to that of Sarwar et al. [12]. The algorithm used by Resnick et al. [11] defines the similarity of two users u and v as the Pearson correlation of their common ratings: sim(u, v) = P i∈Iu∩Iv (ru,i − ¯ru)(rv,i − ¯rv) qP i∈Iu (ru,i − ¯ru)2 qP i∈Iv (rv,i − ¯rv)2 , where Iu is the set of items rated by user u, ru,i is user u"s rating for item i, and ¯ru is the average rating of user u (similarly for v). Similarity computed in this manner is typically scaled by a factor proportional to the number of common ratings, to reduce the chance of making a recommendation made on weak connections: sim (u, v) = max(|Iu ∩ Iv|, γ) γ · sim(u, v), where γ ≈ 5 is a constant used as a lower limit in scaling [2]. These new similarities are then used to define a static neighborhood Nu for each user u consisting of the top K users most similar to user u. A prediction for user u and item i is computed by a weighted average of the ratings by the neighbors pu,i = ¯ru + P v∈V sim (u, v)(rv,i − ¯rv) P v∈V sim (u, v) (1) where V = Nu ∩ Ui is the set of users most similar to u who have rated i. The item-based algorithm we use is the one defined by Sarwar et al. [12]. In this algorithm, similarity is defined as the adjusted cosine measure sim(i, j) = P u∈Ui∩Uj (ru,i − ¯ru)(ru,j − ¯ru) qP u∈Ui (ru,i − ¯ru)2 qP u∈Uj (ru,j − ¯ru)2 (2) where Ui is the set of users who have rated item i. As for the user-based algorithm, the similarity weights are adjusted proportionally to the number of users that have rated the items in common sim (i, j) = max(|Ui ∩ Uj|, γ) γ · sim(i, j). (3) Given the similarities, the neighborhood Ni of an item i is defined as the top K most similar items for that item. A prediction for user u and item i is computed as the weighted average pu,i = ¯ri + P j∈J sim (i, j)(ru,j − ¯rj) P j∈J sim (i, j) (4) where J = Ni ∩ Iu is the set of items rated by u that are most similar to i. 2.2 Evaluation Recommender algorithms have typically been evaluated using measures of predictive accuracy and coverage [3]. Studies on recommender algorithms, notably Herlocker et al. [2] and Sarwar et al. [12], typically compute predictive accuracy by dividing a set of ratings into training and test sets, and compute the prediction for an item in the test set using the ratings in the training set. A standard measure of predictive accuracy is mean absolute error (MAE), which for a test set T = {(u, i)} is defined as, MAE = P (u,i)∈T |pu,i − ru,i| |T | . (5) Coverage has a number of definitions, but generally refers to the proportion of items that can be predicted by the algorithm [3]. A practical issue with predictive accuracy is that users typically are presented with recommendation lists, and not individual numeric predictions. Recommendation lists are lists of items in decreasing order of prediction (sometimes stated in terms of star-ratings), and so predictive accuracy may not be reflective of the accuracy of the list. So, instead we can measure recommendation or rank accuracy, which indicates the extent to which the list is in the correct order. Herlocker et al. [3] discuss a number of rank accuracy measures, which range from Kendall"s Tau to measures that consider the fact that users tend to only look at a prefix of the list [5]. Kendall"s Tau measures the number of inversions when comparing ordered pairs in the true user ordering of 251 Jim Tom Jeff My Cousin Vinny The Matrix Star Wars The Mask Figure 1: Ratings in simple movie recommender. items and the recommended order, and is defined as τ = C − D p (C + D + TR)(C + D + TP) (6) where C is the number of pairs that the system predicts in the correct order, D the number of pairs the system predicts in the wrong order, TR the number of pairs in the true ordering that have the same ratings, and TP is the number of pairs in the predicted ordering that have the same ratings [3]. A shortcoming of the Tau metric is that it is oblivious to the position in the ordered list where the inversion occurs [3]. For instance, an inversion toward the end of the list is given the same weight as one in the beginning. One solution is to consider inversions only in the top few items in the recommended list or to weight inversions based on their position in the list. 3. ROLES OF A RATING Our basic observation is that each rating plays a different role in each prediction in which it is used. Consider a simplified movie recommender system with three users Jim, Jeff, and Tom and their ratings for a few movies, as shown in Fig. 1. (For this initial discussion we will not consider the rating values involved.) The recommender predicts whether Tom will like The Mask using the other already available ratings. How this is done depends on the algorithm: 1. An item-based collaborative filtering algorithm constructs a neighborhood of movies around The Mask by using the ratings of users who rated The Mask and other movies similarly (e.g., Jim"s ratings of The Matrix and The Mask; and Jeff"s ratings of Star Wars and The Mask). Tom"s ratings of those movies are then used to make a prediction for The Mask. 2. A user-based collaborative filtering algorithm would construct a neighborhood around Tom by tracking other users whose rating behaviors are similar to Tom"s (e.g., Tom and Jeff have rated Star Wars; Tom and Jim have rated The Matrix). The prediction of Tom"s rating for The Mask is then based on the ratings of Jeff and Tim. Although the nearest-neighbor algorithms aggregate the ratings to form neighborhoods used to compute predictions, we can disaggregate the similarities to view the computation of a prediction as simultaneously following parallel paths of ratings. So, irrespective of the collaborative filtering algorithm used, we can visualize the prediction of Tom"s rating of The Mask as walking through a sequence of ratings. In Jim Tom Jeff The Matrix Star Wars The Mask q1 q2 q3 p1 p2 p3 Figure 2: Ratings used to predict The Mask for Tom. Jim Tom Jeff The Matrix Star Wars The Mask q1 q2 q3 p1 p2 p3 Jerry r2 r3 Figure 3: Prediction of The Mask for Tom in which a rating is used more than once. this example, two paths were used for this prediction as depicted in Fig. 2: (p1, p2, p3) and (q1, q2, q3). Note that these paths are undirected, and are all of length 3. Only the order in which the ratings are traversed is different between the item-based algorithm (e.g., (p3, p2, p1), (q3, q2, q1)) and the user-based algorithm (e.g., (p1, p2, p3), (q1, q2, q3).) A rating can be part of many paths for a single prediction as shown in Fig. 3, where three paths are used for a prediction, two of which follow p1: (p1, p2, p3) and (p1, r2, r3). Predictions in a collaborative filtering algorithms may involve thousands of such walks in parallel, each playing a part in influencing the predicted value. Each prediction path consists of three ratings, playing roles that we call scouts, promoters, and connectors. To illustrate these roles, consider the path (p1, p2, p3) in Fig. 2 used to make a prediction of The Mask for Tom: 1. The rating p1 (Tom → Star Wars) makes a connection from Tom to other ratings that can be used to predict Tom"s rating for The Mask. This rating serves as a scout in the bipartite graph of ratings to find a path that leads to The Mask. 2. The rating p2 (Jeff → Star Wars) helps the system recommend The Mask to Tom by connecting the scout to the promoter. 3. The rating p3 (Jeff → The Mask) allows connections to The Mask, and, therefore, promotes this movie to Tom. Formally, given a prediction pu,a of a target item a for user u, a scout for pu,a is a rating ru,i such that there exists a user v with ratings rv,a and rv,i for some item i; a promoter for pu,a is a rating rv,a for some user v, such that there exist ratings rv,i and ru,i for an item i, and; a connector for pu,a 252 Jim Tom Jeff Jerry My Cousin Vinny The Matrix Star Wars The Mask Jurasic Park Figure 4: Scouts, promoters, and connectors. is a rating rv,i by some user v and rating i, such that there exists ratings ru,i and rv,a. The scouts, connectors, and promoters for the prediction of Tom"s rating of The Mask are p1 and q1, p2 and q2, and p3 and q3 (respectively). Each of these roles has a value in the recommender to the user, the user"s neighborhood, and the system in terms of allowing recommendations to be made. 3.1 Roles in Detail Ratings that act as scouts tend to help the recommender system suggest more movies to the user, though the extent to which this is true depends on the rating behavior of other users. For example, in Fig. 4 the rating Tom → Star Wars helps the system recommend only The Mask to him, while Tom → The Matrix helps recommend The Mask, Jurassic Park, and My Cousin Vinny. Tom makes a connection to Jim who is a prolific user of the system, by rating The Matrix. However, this does not make The Matrix the best movie to rate for everyone. For example, Jim is benefited equally by both The Mask and The Matrix, which allow the system to recommend Star Wars to him. His rating of The Mask is the best scout for Jeff, and Jerry"s only scout is his rating of Star Wars. This suggests that good scouts allow a user to build similarity with prolific users, and thereby ensure they get more from the system. While scouts represent beneficial ratings from the perspective of a user, promoters are their duals, and are of benefit to items. In Fig. 4, My Cousin Vinny benefits from Jim"s rating, since it allows recommendations to Jeff and Tom. The Mask is not so dependent on just one rating, since the ratings by Jim and Jeff help it. On the other hand, Jerry"s rating of Star Wars does not help promote it to any other user. We conclude that a good promoter connects an item to a broader neighborhood of other items, and thereby ensures that it is recommended to more users. Connectors serve a crucial role in a recommender system that is not as obvious. The movies My Cousin Vinny and Jurassic Park have the highest recommendation potential since they can be recommended to Jeff, Jerry and Tom based on the linkage structure illustrated in Fig. 4. Beside the fact that Jim rated these movies, these recommendations are possible only because of the ratings Jim → The Matrix and Jim → The Mask, which are the best connectors. A connector improves the system"s ability to make recommendations with no explicit gain for the user. Note that every rating can be of varied benefit in each of these roles. The rating Jim → My Cousin Vinny is a poor scout and connector, but is a very good promoter. The rating Jim → The Mask is a reasonably good scout, a very good connector, and a good promoter. Finally, the rating Jerry → Star Wars is a very good scout, but is of no value as a connector or promoter. As illustrated here, a rating can have different value in each of the three roles in terms of whether a recommendation can be made or not. We could measure this value by simply counting the number of times a rating is used in each role, which alone would be helpful in characterizing the behavior of a system. But we can also measure the contribution of each rating to the quality of recommendations or health of the system. Since every prediction is a combined effort of several recommendation paths, we are interested in discerning the influence of each rating (and, hence, each path) in the system towards the system"s overall error. We can understand the dynamics of the system at a finer granularity by tracking the influence of a rating according to the role played. The next section describes the approach to measuring the values of a rating in each role. 4. CONTRIBUTIONS OF RATINGS As we"ve seen, a rating may play different roles in different predictions and, in each prediction, contribute to the quality of a prediction in different ways. Our approach can use any numeric measure of a property of system health, and assigns credit (or blame) to each rating proportional to its influence in the prediction. By tracking the role of each rating in a prediction, we can accumulate the credit for a rating in each of the three roles, and also track the evolution of the roles of rating over time in the system. This section defines the methodology for computing the contribution of ratings by first defining the influence of a rating, and then instantiating the approach for predictive accuracy, and then rank accuracy. We also demonstrate how these contributions can be aggregated to study the neighborhood of ratings involved in computing a user"s recommendations. Note that although our general formulation for rating influence is algorithm independent, due to space considerations, we present the approach for only item-based collaborative filtering. The definition for user-based algorithms is similar and will be presented in an expanded version of this paper. 4.1 Influence of Ratings Recall that an item-based approach to collaborative filtering relies on building item neighborhoods using the similarity of ratings by the same user. As described earlier, similarity is defined by the adjusted cosine measure (Equations (2) and (3)). A set of the top K neighbors is maintained for all items for space and computational efficiency. A prediction of item i for a user u is computed as the weighted deviation from the item"s mean rating as shown in Equation (4). The list of recommendations for a user is then the list of items sorted in descending order of their predicted values. We first define impact(a, i, j), the impact a user a has in determining the similarity between two items i and j. This is the change in the similarity between i and j when a"s rating is removed, and is defined as impact(a, i, j) = |sim (i, j) − sim¯a(i, j)| P w∈Cij |sim (i, j) − sim ¯w(i, j)| where Cij = {u ∈ U | ∃ ru,i, ru,j ∈ R(u)} is the set of coraters 253 of items i and j (users who rate both i and j), R(u) is the set of ratings provided by user u, and sim¯a(i, j) is the similarity of i and j when the ratings of user a are removed sim¯a(i, j) = P v∈U\{a} (ru,i − ¯ru)(ru,j − ¯ru) qP u∈U\{a}(ru,i − ¯ru)2 qP u∈U\{a}(ru,j − ¯ru)2 , and adjusted for the number of raters sim¯a(i, j) = max(|Ui ∩ Uj| − 1, γ) γ · sim(i, j). If all coraters of i and j rate them identically, we define the impact as impact(a, i, j) = 1 |Cij| since P w∈Cij |sim (i, j) − sim ¯w(i, j)| = 0. The influence of each path (u, j, v, i) = [ru,j, rv,j, rv,i] in the prediction of pu,i is given by influence(u, j, v, i) = sim (i, j) P l∈Ni∩Iu sim (i, l) · impact(v, i, j) It follows that the sum of influences over all such paths, for a given set of endpoints, is 1. 4.2 Role Values for Predictive Accuracy The value of a rating in each role is computed from the influence depending on the evaluation measure employed. Here we illustrate the approach using predictive accuracy as the evaluation metric. In general, the goodness of a prediction decides whether the ratings involved must be credited or discredited for their role. For predictive accuracy, the error in prediction e = |pu,i − ru,i| is mapped to a comfort level using a mapping function M(e). Anecdotal evidence suggests that users are unable to discern errors less than 1.0 (for a rating scale of 1 to 5) [4], and so an error less than 1.0 is considered acceptable, but anything larger is not. We hence define M(e) as (1 − e) binned to an appropriate value in [−1, −0.5, 0.5, 1]. For each prediction pu,i, M(e) is attributed to all the paths that assisted the computation of pu,i, proportional to their influences. This tribute, M(e)∗influence(u, j, v, i), is in turn inherited by each of the ratings in the path [ru,j, rv,j, rv,i], with the credit/blame accumulating to the respective roles of ru,j as a scout, rv,j as a connector, and rv,i as a promoter. In other words, the scout value SF(ru,j), the connector value CF(rv,j) and the promoter value PF(rv,i) are all incremented by the tribute amount. Over a large number of predictions, scouts that have repeatedly resulted in big error rates have a big negative scout value, and vice versa (similarly with the other roles). Every rating is thus summarized by its triple [SF, CF, PF]. 4.3 Role Values for Rank Accuracy We now define the computation of the contribution of ratings to observed rank accuracy. For this computation, we must know the user"s preference order for a set of items for which predictions can be computed. We assume that we have a test set of the users" ratings of the items presented in the recommendation list. For every pair of items rated by a user in the test data, we check whether the predicted order is concordant with his preference. We say a pair (i, j) is concordant (with error ) whenever one of the following holds: • if (ru,i < ru,j) then (pu,i − pu,j < ); • if (ru,i > ru,j) then (pu,i − pu,j > ); or • if (ru,i = ru,j) then (|pu,i − pu,j| ≤ ). Similarly, a pair (i, j) is discordant (with error ) if it is not concordant. Our experiments described below use an error tolerance of = 0.1. All paths involved in the prediction of the two items in a concordant pair are credited, and the paths involved in a discordant pair are discredited. The credit assigned to a pair of items (i, j) in the recommendation list for user u is computed as c(i, j) = ( t T · 1 C+D if (i, j) are concordant − t T · 1 C+D if (i, j) are discordant (7) where t is the number of items in the user"s test set whose ratings could be predicted, T is the number of items rated by user u in the test set, C is the number of concordances and D is the number of discordances. The credit c is then divided among all paths responsible for predicting pu,i and pu,j proportional to their influences. We again add the role values obtained from all the experiments to form a triple [SF, CF, PF] for each rating. 4.4 Aggregating rating roles After calculating the role values for individual ratings, we can also use these values to study neighborhoods and the system. Here we consider how we can use the role values to characterize the health of a neighborhood. Consider the list of top recommendations presented to a user at a specific point in time. The collaborative filtering algorithm traversed many paths in his neighborhood through his scouts and other connectors and promoters to make these recommendations. We call these ratings the recommender neighborhood of the user. The user implicitly chooses this neighborhood of ratings through the items he rates. Apart from the collaborative filtering algorithm, the health of this neighborhood completely influences a user"s satisfaction with the system. We can characterize a user"s recommender neighborhood by aggregating the individual role values of the ratings involved, weighted by the influence of individual ratings in determining his recommended list. Different sections of the user"s neighborhood wield varied influence on his recommendation list. For instance, ratings reachable through highly rated items have a bigger say in the recommended items. Our aim is to study the system and classify users with respect to their positioning in a healthy or unhealthy neighborhood. A user can have a good set of scouts, but may be exposed to a neighborhood with bad connectors and promoters. He can have a good neighborhood, but his bad scouts may ensure the neighborhood"s potential is rendered useless. We expect that users with good scouts and good neighborhoods will be most satisfied with the system in the future. A user"s neighborhood is characterized by a triple that represents the weighted sum of the role values of individual ratings involved in making recommendations. Consider a user u and his ordered list of recommendations L. An item i 254 in the list is weighted inversely, as K(i), depending on its position in the list. In our studies we use K(i) = p position(i). Several paths of ratings [ru,j, rv,j, rv,i] are involved in predicting pu,i which ultimately decides its position in L, each with influence(u, j, v, i). The recommender neighborhood of a user u is characterized by the triple, [SFN(u), CFN(u), PFN(u)] where SFN(u) = X i∈L P [ru,j ,rv,j ,rv,i] SF(ru,j)influence(u, j, v, i) K(i) ! CFN(u) and PFN(u) are defined similarly. This triple estimates the quality of u"s recommendations based on the past track record of the ratings involved in their respective roles. 5. EXPERIMENTATION As we have seen, we can assign role values to each rating when evaluating a collaborative filtering system. In this section, we demonstrate the use of this approach to our overall goal of defining an approach to monitor and manage the health of a recommender system through experiments done on the MovieLens million rating dataset. In particular, we discuss results relating to identifying good scouts, promoters, and connectors; the evolution of rating roles; and the characterization of user neighborhoods. 5.1 Methodology Our experiments use the MovieLens million rating dataset, which consists of ratings by 6040 users of 3952 movies. The ratings are in the range 1 to 5, and are labeled with the time the rating was given. As discussed before, we consider only the item-based algorithm here (with item neighborhoods of size 30) and, due to space considerations, only present role value results for rank accuracy. Since we are interested in the evolution of the rating role values over time, the model of the recommender system is built by processing ratings in their arrival order. The timestamping provided by MovieLens is hence crucial for the analyses presented here. We make assessments of rating roles at intervals of 10,000 ratings and processed the first 200,000 ratings in the dataset (giving rise to 20 snapshots). We incrementally update the role values as the time ordered ratings are merged into the model. To keep the experiment computationally manageable, we define a test dataset for each user. As the time ordered ratings are merged into the model, we label a small randomly selected percentage (20%) as test data. At discrete epochs, i.e., after processing every 10,000 ratings, we compute the predictions for the ratings in the test data, and then compute the role values for the ratings used in the predictions. One potential criticism of this methodology is that the ratings in the test set are never evaluated for their roles. We overcome this concern by repeating the experiment, using different random seeds. The probability that every rating is considered for evaluation is then considerably high: 1 − 0.2n , where n is the number of times the experiment is repeated with different random seeds. The results here are based on n = 4 repetitions. The item-based collaborative filtering algorithm"s performance was ordinary with respect to rank accuracy. Fig. 5 shows a plot of the precision and recall as ratings were merged in time order into the model. The recall was always high, but the average precision was just about 53%. 0 0.2 0.4 0.6 0.8 1 1.2 10000 30000 50000 70000 90000110000130000150000 Ratings merged into model Value Precision Recall Figure 5: Precision and recall for the item-based collaborative filtering algorithm. 5.2 Inducing good scouts The ratings of a user that serve as scouts are those that allow the user to receive recommendations. We claim that users with ratings that have respectable scout values will be happier with the system than those with ratings with low scout values. Note that the item-based algorithm discussed here produces recommendation lists with nearly half of the pairs in the list discordant from the user"s preference. Whether all of these discordant pairs are observable by the user is unclear, however, certainly this suggests that there is a need to be able to direct users to items whose ratings would improve the lists. The distribution of the scout values for most users" ratings are Gaussian with mean zero. Fig. 6 shows the frequency distribution of scout values for a sample user at a given snapshot. We observe that a large number of ratings never serve as scouts for their users. A relatable scenario is when Amazon"s recommender makes suggestions of books or items based on other items that were purchased as gifts. With simple relevance feedback from the user, such ratings can be isolated as bad scouts and discounted from future predictions. Removing bad scouts was found to be worthwhile for individual users but the overall performance improvement was only marginal. An obvious question is whether good scouts can be formed by merely rating popular movies as suggested by Rashid et al. [9]. They show that a mix of popularity and rating entropy identifies the best items to suggest to new users when evaluated using MAE. Following their intuition, we would expect to see a higher correlation between popularityentropy and good scouts. We measured the Pearson correlation coefficient between aggregated scout values for a movie with the popularity of a movie (number of times it is rated); and with its popularity*variance measure at different snapshots of the system. Note that the scout values were initially anti-correlated with popularity (Fig. 7), but became moderately correlated as the system evolved. Both popularity and popularity*variance performed similarly. A possible explanation is that there has been insufficient time for the popular movies to accumulate ratings. 255 -10 0 10 20 30 40 50 60 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 Scout Value Frequency Figure 6: Distribution of scout values for a sample user. -0.4 -0.2 0 0.2 0.4 0.6 0.8 30000 60000 90000 120000 150000 180000 Popularity Pop*Var Figure 7: Correlation between aggregated scout value and item popularity (computed at different intervals). -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 30000 60000 90000 120000 150000 180000 Figure 8: Correlation between aggregated promoter value and user prolificity (computed at different intervals). Table 1: Movies forming the best scouts. Best Scouts Conf. Pop. Being John Malkovich (1999) 1.00 445 Star Wars: Episode IV - A New Hope (1977) 0.92 623 Princess Bride, The (1987) 0.85 477 Sixth Sense, The (1999) 0.85 617 Matrix, The (1999) 0.77 522 Ghostbusters (1984) 0.77 441 Casablanca (1942) 0.77 384 Insider, The (1999) 0.77 235 American Beauty (1999) 0.69 624 Terminator 2: Judgment Day (1991) 0.69 503 Fight Club (1999) 0.69 235 Shawshank Redemption, The (1994) 0.69 445 Run Lola Run (Lola rennt) (1998) 0.69 220 Terminator, The (1984) 0.62 450 Usual Suspects, The (1995) 0.62 326 Aliens (1986) 0.62 385 North by Northwest (1959) 0.62 245 Fugitive, The (1993) 0.62 402 End of Days (1999) 0.62 132 Raiders of the Lost Ark (1981) 0.54 540 Schindler"s List (1993) 0.54 453 Back to the Future (1985) 0.54 543 Toy Story (1995) 0.54 419 Alien (1979) 0.54 415 Abyss, The (1989) 0.54 345 2001: A Space Odyssey (1968) 0.54 358 Dogma (1999) 0.54 228 Little Mermaid, The (1989) 0.54 203 Table 2: Movies forming the worst scouts. Worst scouts Conf. Pop. Harold and Maude (1971) 0.46 141 Grifters, The (1990) 0.46 180 Sting, The (1973) 0.38 244 Godfather: Part III, The (1990) 0.38 154 Lawrence of Arabia (1962) 0.38 167 High Noon (1952) 0.38 84 Women on the Verge of a... (1988) 0.38 113 Grapes of Wrath, The (1940) 0.38 115 Duck Soup (1933) 0.38 131 Arsenic and Old Lace (1944) 0.38 138 Midnight Cowboy (1969) 0.38 137 To Kill a Mockingbird (1962) 0.31 195 Four Weddings and a Funeral (1994) 0.31 271 Good, The Bad and The Ugly, The (1966) 0.31 156 It"s a Wonderful Life (1946) 0.31 146 Player, The (1992) 0.31 220 Jackie Brown (1997) 0.31 118 Boat, The (Das Boot) (1981) 0.31 210 Manhattan (1979) 0.31 158 Truth About Cats & Dogs, The (1996) 0.31 143 Ghost (1990) 0.31 227 Lone Star (1996) 0.31 125 Big Chill, The (1983) 0.31 184 256 By studying the evolution of scout values, we can identify movies that consistently feature in good scouts over time. We claim these movies will make viable scouts for other users. We found the aggregated scout values for all movies in intervals of 10,000 ratings each. A movie is said to induce a good scout if the movie was in the top 100 of the sorted list, and to induce a bad scout if it was in bottom 100 of the same list. Movies appearing consistently high over time are expected to remain up there in the future. The effective confidence in a movie m is Cm = Tm − Bm N (8) where Tm is the number of times it appeared in the top 100, Bm the number of times it appeared in the bottom 100, and N is the number of intervals considered. Using this measure, the top few movies expected to induce the best scouts are shown in Table 1. Movies that would be bad scout choices are shown in Table 2 with their associated confidences. The popularities of the movies are also displayed. Although more popular movies appear in the list of good scouts, these tables show that a blind choice of scout based on popularity alone can be potentially dangerous. Interestingly, the best scout-‘Being John Malkovich"-is about a puppeteer who discovers a portal into a movie star, a movie that has been described variously on amazon.com as ‘makes you feel giddy," ‘seriously weird," ‘comedy with depth," ‘silly," ‘strange," and ‘inventive." Indicating whether someone likes this movie or not goes a long way toward situating the user in a suitable neighborhood, with similar preferences. On the other hand, several factors may have made a movie a bad scout, like the sharp variance in user preferences in the neighborhood of a movie. Two users may have the same opinion about Lawrence of Arabia, but they may differ sharply about how they felt about the other movies they saw. Bad scouts ensue when there is deviation in behavior around a common synchronization point. 5.3 Inducing good promoters Ratings that serve to promote items in a collaborative filtering system are critical to allowing a new item be recommended to users. So, inducing good promoters is important for cold-start recommendation. We note that the frequency distribution of promoter values for a sample movie"s ratings is also Gaussian (similar to Fig. 6). This indicates that the promotion of a movie is benefited most by the ratings of a few users, and are unaffected by the ratings of most users. We find a strong correlation between a user"s number of ratings and his aggregated promoter value. Fig. 8 depicts the evolution of the Pearson correlation co-efficient between the prolificity of a user (number of ratings) versus his aggregated promoter value. We expect that conspicuous shills, by recommending wrong movies to users, will be discredited with negative aggregate promoter values and should be identifiable easily. Given this observation, the obvious rule to follow when introducing a new movie is to have it rated directly by prolific users who posses high aggregated promoter values. A new movie is thus cast into the neighborhood of many other movies improving its visibility. Note, though, that a user may have long stopped using the system. Tracking promoter values consistently allows only the most active recent users to be considered. 5.4 Inducing good connectors Given the way scouts, connectors, and promoters are characterized, it follows that the movies that are part of the best scouts are also part of the best connectors. Similarly, the users that constitute the best promoters are also part of the best connectors. Good connectors are induced by ensuring a user with a high promoter value rates a movie with a high scout value. In our experiments, we find that a rating"s longest standing role is often as a connector. A rating with a poor connector value is often seen due to its user being a bad promoter, or its movie being a bad scout. Such ratings can be removed from the prediction process to bring marginal improvements to recommendations. In some selected experiments, we observed that removing a set of badly behaving connectors helped improve the system"s overall performance by 1.5%. The effect was even higher on a few select users who observed an improvement of above 10% in precision without much loss in recall. 5.5 Monitoring the evolution of rating roles One of the more significant contributions of our work is the ability to model the evolution of recommender systems, by studying the changing roles of ratings over time. The role and value of a rating can change depending on many factors like user behavior, redundancy, shilling effects or properties of the collaborative filtering algorithm used. Studying the dynamics of rating roles in terms of transitions between good, bad, and negligible values can provide insights into the functioning of the recommender system. We believe that a continuous visualization of these transitions will improve the ability to manage a recommender system. We classify different rating states as good, bad, or negligible. Consider a user who has rated 100 movies in a particular interval, of which 20 are part of the test set. If a scout has a value greater than 0.005, it indicates that it is uniquely involved in at least 2 concordant predictions, which we will say is good. Thus, a threshold of 0.005 is chosen to bin a rating as good, bad or negligible in terms of its scout, connector and promoter value. For instance, a rating r, at time t with role value triple [0.1, 0.001, −0.01] is classified as [scout +, connector 0, promoter −], where + indicates good, 0 indicates negligible, and − indicates bad. The positive credit held by a rating is a measure of its contribution to the betterment of the system, and the discredit is a measure of its contribution to the detriment of the system. Even though the positive roles (and the negative roles) make up a very small percentage of all ratings, their contribution supersedes their size. For example, even though only 1.7% of all ratings were classified as good scouts, they hold 79% of all positive credit in the system! Similarly, the bad scouts were just 1.4% of all ratings but hold 82% of all discredit. Note that good and bad scouts, together, comprise only 1.4% + 1.7% = 3.1% of the ratings, implying that the majority of the ratings are negligible role players as scouts (more on this later). Likewise, good connectors were 1.2% of the system, and hold 30% of all positive credit. The bad connectors (0.8% of the system) hold 36% of all discredit. Good promoters (3% of the system) hold 46% of all credit, while bad promoters (2%) hold 50% of all discredit. This reiterates that a few ratings influence most of the system"s performance. Hence it is important to track transitions between them regardless of their small numbers. 257 Across different snapshots, a rating can remain in the same state or change. A good scout can become a bad scout, a good promoter can become a good connector, good and bad scouts can become vestigial, and so on. It is not practical to expect a recommender system to have no ratings in bad roles. However, it suffices to see ratings in bad roles either convert to good or vestigial roles. Similarly, observing a large number of good roles become bad ones is a sign of imminent failure of the system. We employ the principle of non-overlapping episodes [6] to count such transitions. A sequence such as: [+, 0, 0] → [+, 0, 0] → [0, +, 0] → [0, 0, 0] is interpreted as the transitions [+, 0, 0] ; [0, +, 0] : 1 [+, 0, 0] ; [0, 0, 0] : 1 [0, +, 0] ; [0, 0, 0] : 1 instead of [+, 0, 0] ; [0, +, 0] : 2 [+, 0, 0] ; [0, 0, 0] : 2 [0, +, 0] ; [0, 0, 0] : 1. See [6] for further details about this counting procedure. Thus, a rating can be in one of 27 possible states, and there are about 272 possible transitions. We make a further simplification and utilize only 9 states, indicating whether the rating is a scout, promoter, or connector, and whether it has a positive, negative, or negligible role. Ratings that serve multiple purposes are counted using multiple episode instantiations but the states themselves are not duplicated beyond the 9 restricted states. In this model, a transition such as [+, 0, +] ; [0, +, 0] : 1 is counted as [scout+] ; [scout0] : 1 [scout+] ; [connector+] : 1 [scout+] ; [promoter0] : 1 [connector0] ; [scout0] : 1 [connector0] ; [scout+] : 1 [connector0] ; [promoter0] : 1 [promoter+] ; [scout0] : 1 [promoter+] ; [connector+] : 1 [promoter+] ; [promoter0] : 1 Of these, transitions like [pX] ; [q0] where p = q, X ∈ {+, 0, −} are considered uninteresting, and only the rest are counted. Fig. 9 depicts the major transitions counted while processing the first 200,000 ratings from the MovieLens dataset. Only transitions with frequency greater than or equal to 3% are shown. The percentages for each state indicates the number of ratings that were found to be in those states. We consider transitions from any state to a good state as healthy, from any state to a bad state as unhealthy, and from any state to a vestigial state as decaying. From Fig. 9, we can observe: • The bulk of the ratings have negligible values, irrespective of their role. The majority of the transitions involve both good and bad ratings becoming negligible. Scout + (2%) Scout(1.5%) Scout 0 (96.5%) Connector + (1.2%) Connector(0.8%) Connector 0 (98%) Promoter + (3%) Promoter(2%) Promoter 0 (95%) 84% 84% 81% 74% 10% 6% 11% 77% 8% 7% 8% 82% 4% 86% 4% 68% 15% 13% 5% 5% 77% 11% 7% 5% 4% 3% 3% 3% Healthy Unhealthy Decaying Figure 9: Transitions among rating roles. • The number of good ratings is comparable to the bad ratings, and ratings are seen to switch states often, except in the case of scouts (see below). • The negative and positive scout states are not reachable through any transition, indicating that these ratings must begin as such, and cannot be coerced into these roles. • Good promoters and good connectors have a much longer survival period than scouts. Transitions that recur to these states have frequencies of 10% and 15% when compared to just 4% for scouts. Good connectors are the slowest to decay whereas (good) scouts decay the fastest. • Healthy percentages are seen on transitions between promoters and connectors. As indicated earlier, there are hardly any transitions from promoters/connectors to scouts. This indicates that, over the long run, a user"s rating is more useful to others (movies or other users) than to the user himself. • The percentages of healthy transitions outweigh the unhealthy ones - this hints that the system is healthy, albeit only marginally. Note that these results are conditioned by the static nature of the dataset, which is a set of ratings over a fixed window of time. However a diagram such as Fig. 9 is clearly useful for monitoring the health of a recommender system. For instance, acceptable limits can be imposed on different types of transitions and, if a transition fails to meet the threshold, the recommender system or a part of it can be brought under closer scrutiny. Furthermore, the role state transition diagram would also be the ideal place to study the effects of shilling, a topic we will consider in future research. 5.6 Characterizing neighborhoods Earlier we saw that we can characterize the neighborhood of ratings involved in creating a recommendation list L for 258 a user. In our experiment, we consider lists of length 30, and sample the lists of about 5% of users through the evolution of the model (at intervals of 10,000 ratings each) and compute their neighborhood characteristics. To simplify our presentation, we consider the percentage of the sample that fall into one of the following categories: 1. Inactive user: (SFN(u) = 0) 2. Good scouts, Good neighborhood: (SFN(u) > 0) ∧ (CFN(u) > 0 ∧ PFN(u) > 0) 3. Good scouts, Bad neighborhood: (SFN(u) > 0) ∧ (CFN(u) < 0 ∨ PFN(u) < 0) 4. Bad scouts, Good neighborhood: (SFN(u) < 0) ∧ (CFN(u) > 0 ∧ PFN(u) > 0) 5. Bad scouts, Bad neighborhood: (SFN(u) < 0) ∧ (CFN(u) < 0 ∨ PFN(u) < 0) From our sample set of 561 users, we found that 476 users were inactive. Of the remaining 85 users, we found 26 users had good scouts and a good neighborhood, 6 had bad scouts and a good neighborhood, 29 had good scouts and a bad neighborhood, and 24 had bad scouts and a bad neighborhood. Thus, we conjecture that 59 users (29+24+6) are in danger of leaving the system. As a remedy, users with bad scouts and a good neighborhood can be asked to reconsider rating of some movies hoping to improve the system"s recommendations. The system can be expected to deliver more if they engineer some good scouts. Users with good scouts and a bad neighborhood are harder to address; this situation might entail selectively removing some connector-promoter pairs that are causing the damage. Handling users with bad scouts and bad neighborhoods is a more difficult challenge. Such a classification allows the use of different strategies to better a user"s experience with the system depending on his context. In future work, we intend to conduct field studies and study the improvement in performance of different strategies for different contexts. 6. CONCLUSIONS To further recommender system acceptance and deployment, we require new tools and methodologies to manage an installed recommender and develop insights into the roles played by ratings. A fine-grained characterization in terms of rating roles such as scouts, promoters, and connectors, as done here, helps such an endeavor. Although we have presented results on only the item-based algorithm with list rank accuracy as the metric, the same approach outlined here applies to user-based algorithms and other metrics. In future research, we plan to systematically study the many algorithmic parameters, tolerances, and cutoff thresholds employed here and reason about their effects on the downstream conclusions. We also aim to extend our formulation to other collaborative filtering algorithms, study the effect of shilling in altering rating roles, conduct field studies, and evaluate improvements in user experience by tweaking ratings based on their role values. Finally, we plan to develop the idea of mining the evolution of rating role patterns into a reporting and tracking system for all aspects of recommender system health. 7. REFERENCES [1] Cosley, D., Lam, S., Albert, I., Konstan, J., and Riedl, J. Is Seeing Believing?: How Recommender System Interfaces Affect User"s Opinions. In Proc. CHI (2001), pp. 585-592. [2] Herlocker, J. L., Konstan, J. A., Borchers, A., and Riedl, J. An Algorithmic Framework for Performing Collaborative Filtering. In Proc. SIGIR (1999), pp. 230-237. [3] Herlocker, J. L., Konstan, J. A., Terveen, L. G., and Riedl, J. T. Evaluating Collaborative Filtering Recommender Systems. ACM Transactions on Information Systems Vol. 22, 1 (2004), pp. 5-53. [4] Konstan, J. A. Personal communication. 2003. [5] Lam, S. K., and Riedl, J. Shilling Recommender Systems for Fun and Profit. In Proceedings of the 13th International World Wide Web Conference (2004), ACM Press, pp. 393-402. [6] Laxman, S., Sastry, P. S., and Unnikrishnan, K. P. Discovering Frequent Episodes and Learning Hidden Markov Models: A Formal Connection. IEEE Transactions on Knowledge and Data Engineering Vol. 17, 11 (2005), 1505-1517. [7] McLaughlin, M. R., and Herlocker, J. L. A Collaborative Filtering Algorithm and Evaluation Metric that Accurately Model the User Experience. In Proceedings of the 27th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval (2004), pp. 329 - 336. [8] O"Mahony, M., Hurley, N. J., Kushmerick, N., and Silvestre, G. Collaborative Recommendation: A Robustness Analysis. ACM Transactions on Internet Technology Vol. 4, 4 (Nov 2004), pp. 344-377. [9] Rashid, A. M., Albert, I., Cosley, D., Lam, S., McNee, S., Konstan, J. A., and Riedl, J. Getting to Know You: Learning New User Preferences in Recommender Systems. In Proceedings of the 2002 Conference on Intelligent User Interfaces (IUI 2002) (2002), pp. 127-134. [10] Rashid, A. M., Karypis, G., and Riedl, J. Influence in Ratings-Based Recommender Systems: An Algorithm-Independent Approach. In Proc. of the SIAM International Conference on Data Mining (2005). [11] Resnick, P., Iacovou, N., Sushak, M., Bergstrom, P., and Riedl, J. GroupLens: An Open Architecture for Collaborative Filtering of Netnews. In Proceedings of the Conference on Computer Supported Collaborative Work (CSCW"94) (1994), ACM Press, pp. 175-186. [12] Sarwar, B., Karypis, G., Konstan, J., and Reidl, J. Item-Based Collaborative Filtering Recommendation Algorithms. In Proceedings of the Tenth International World Wide Web Conference (WWW"10) (2001), pp. 285-295. [13] Schein, A., Popescu, A., Ungar, L., and Pennock, D. Methods and Metrics for Cold-Start Recommendation. In Proc. SIGIR (2002), pp. 253-260. 259

opinion;list rank accuracy;recommender system;neighborhood;nearest neighbor;rating;recommender;promoter;aggregation process;scout;purchase;collaborative filtering algorithm;collaborative filter;user-base and item-base algorithm;connector

train_J-45

Empirical Mechanism Design: Methods, with Application to a Supply-Chain Scenario

Our proposed methods employ learning and search techniques to estimate outcome features of interest as a function of mechanism parameter settings. We illustrate our approach with a design task from a supply-chain trading competition. Designers adopted several rule changes in order to deter particular procurement behavior, but the measures proved insufficient. Our empirical mechanism analysis models the relation between a key design parameter and outcomes, confirming the observed behavior and indicating that no reasonable parameter settings would have been likely to achieve the desired effect. More generally, we show that under certain conditions, the estimator of optimal mechanism parameter setting based on empirical data is consistent.

1. MOTIVATION We illustrate our problem with an anecdote from a supply chain research exercise: the 2003 and 2004 Trading Agent Competition (TAC) Supply Chain Management (SCM) game. TAC/SCM [1] defines a scenario where agents compete to maximize their profits as manufacturers in a supply chain. The agents procure components from the various suppliers and assemble finished goods for sale to customers, repeatedly over a simulated year. As it happened, the specified negotiation behavior of suppliers provided a great incentive for agents to procure large quantities of components on day 0: the very beginning of the simulation. During the early rounds of the 2003 SCM competition, several agent developers discovered this, and the apparent success led to most agents performing the majority of their purchasing on day 0. Although jockeying for day-0 procurement turned out to be an interesting strategic issue in itself [19], the phenomenon detracted from other interesting problems, such as adapting production levels to varying demand (since component costs were already sunk), and dynamic management of production, sales, and inventory. Several participants noted that the predominance of day-0 procurement overshadowed other key research issues, such as factory scheduling [2] and optimizing bids for customer orders [13]. After the 2003 tournament, there was a general consensus in the TAC community that the rules should be changed to deter large day-0 procurement. The task facing game organizers can be viewed as a problem in mechanism design. The designers have certain game features under their control, and a set of objectives regarding game outcomes. Unlike most academic treatments of mechanism design, the objective is a behavioral feature (moderate day-0 procurement) rather than an allocation feature like economic efficiency, and the allowed mechanisms are restricted to those judged to require only an incremental modification of the current game. Replacing the supplychain negotiation procedures with a one-shot direct mechanism, for example, was not an option. We believe that such operational restrictions and idiosyncratic objectives are actually quite typical of practical mechanism design settings, where they are perhaps more commonly characterized as incentive engineering problems. In response to the problem, the TAC/SCM designers adopted several rule changes intended to penalize large day-0 orders. These included modifications to supplier pricing policies and introduction of storage costs assessed on inventories of components and finished goods. Despite the changes, day-0 procurement was very high in the early rounds of the 2004 competition. In a drastic measure, the GameMaster imposed a fivefold increase of storage costs midway through the tournament. Even this did not stem the tide, and day-0 procurement in the final rounds actually increased (by some measures) from 2003 [9]. The apparent difficulty in identifying rule modifications that effect moderation in day-0 procurement is quite striking. Although the designs were widely discussed, predictions for the effects of various proposals were supported primarily by intuitive arguments or at best by back-of-the-envelope calculations. Much of the difficulty, of course, is anticipating the agents" (and their developers") responses without essentially running a gaming exercise for this purpose. The episode caused us to consider whether new ap306 proaches or tools could enable more systematic analysis of design options. Standard game-theoretic and mechanism design methods are clearly relevant, although the lack of an analytic description of the game seems to be an impediment. Under the assumption that the simulator itself is the only reliable source of outcome computation, we refer to our task as empirical mechanism design. In the sequel, we develop some general methods for empirical mechanism design and apply them to the TAC/SCM redesign problem. Our analysis focuses on the setting of storage costs (taking other game modifications as fixed), since this is the most direct deterrent to early procurement adopted. Our results confirm the basic intuition that incentives for day-0 purchasing decrease as storage costs rise. We also confirm that the high day-0 procurement observed in the 2004 tournament is a rational response to the setting of storage costs used. Finally, we conclude from our data that it is very unlikely that any reasonable setting of storage costs would result in acceptable levels of day-0 procurement, so a different design approach would have been required to eliminate this problem. Overall, we contribute a formal framework and a set of methods for tackling indirect mechanism design problems in settings where only a black-box description of players" utilities is available. Our methods incorporate estimation of sets of Nash equilibria and sample Nash equilibria, used in conjuction to support general claims about the structure of the mechanism designer"s utility, as well as a restricted probabilistic analysis to assess the likelihood of conclusions. We believe that most realistic problems are too complex to be amenable to exact analysis. Consequently, we advocate the approach of gathering evidence to provide indirect support of specific hypotheses. 2. PRELIMINARIES A normal form game2 is denoted by [I, {Ri}, {ui(r)}], where I refers to the set of players and m = |I| is the number of players. Ri is the set of strategies available to player i ∈ I, with R = R1 ×. . .×Rm representing the set of joint strategies of all players. We designate the set of pure strategies available to player i by Ai, and denote the joint set of pure strategies of all players by A = A1 ×. . .×Am. It is often convenient to refer to a strategy of player i separately from that of the remaining players. To accommodate this, we use a−i to denote the joint strategy of all players other than player i. Let Si be the set of all probability distributions (mixtures) over Ai and, similarly, S be the set of all distributions over A. An s ∈ S is called a mixed strategy profile. When the game is finite (i.e., A and I are both finite), the probability that a ∈ A is played under s is written s(a) = s(ai, a−i). When the distribution s is not correlated, we can simply say si(ai) when referring to the probability player i plays ai under s. Next, we define the payoff (utility) function of each player i by ui : A1 ×· · ·×Am → R, where ui(ai, a−i) indicates the payoff to player i to playing pure strategy ai when the remaining players play a−i. We can extend this definition to mixed strategies by assuming that ui are von Neumann-Morgenstern (vNM) utilities as follows: ui(s) = Es[ui], where Es is the expectation taken with respect to the probability distribution of play induced by the players" mixed strategy s. 2 By employing the normal form, we model agents as playing a single action, with decisions taken simultaneously. This is appropriate for our current study, which treats strategies (agent programs) as atomic actions. We could capture finer-grained decisions about action over time in the extensive form. Although any extensive game can be recast in normal form, doing so may sacrifice compactness and blur relevant distinctions (e.g., subgame perfection). Occasionally, we write ui(x, y) to mean that x ∈ Ai or Si and y ∈ A−i or S−i depending on context. We also express the set of utility functions of all players as u(·) = {u1(·), . . . , um(·)}. We define a function, : R → R, interpreted as the maximum benefit any player can obtain by deviating from its strategy in the specified profile. (r) = max i∈I max ai∈Ai [ui(ai, r−i) − ui(r)], (1) where r belongs to some strategy set, R, of either pure or mixed strategies. Faced with a game, an agent would ideally play its best strategy given those played by the other agents. A configuration where all agents play strategies that are best responses to the others constitutes a Nash equilibrium. DEFINITION 1. A strategy profile r = (r1, . . . , rm) constitutes a Nash equilibrium of game [I, {Ri}, {ui(r)}] if for every i ∈ I, ri ∈ Ri, ui(ri, r−i) ≥ ui(ri, r−i). When r ∈ A, the above defines a pure strategy Nash equilibrium; otherwise the definition describes a mixed strategy Nash equilibrium. We often appeal to the concept of an approximate, or -Nash equilibrium, where is the maximum benefit to any agent for deviating from the prescribed strategy. Thus, (r) as defined above (1) is such that profile r is an -Nash equilibrium iff (r) ≤ . In this study we devote particular attention to games that exhibit symmetry with respect to payoffs, rendering agents strategically identical. DEFINITION 2. A game [I, {Ri}, {ui(r)}] is symmetric if for all i, j ∈ I, (a) Ri = Rj and (b) ui(ri, r−i) = uj (rj, r−j) whenever ri = rj and r−i = r−j 3. THE MODEL We model the strategic interactions between the designer of the mechanism and its participants as a two-stage game. The designer moves first by selecting a value, θ, from a set of allowable mechanism settings, Θ. All the participant agents observe the mechanism parameter θ and move simultaneously thereafter. For example, the designer could be deciding between a first-price and second-price sealed-bid auction mechanisms, with the presumption that after the choice has been made, the bidders will participate with full awareness of the auction rules. Since the participants play with full knowledge of the mechanism parameter, we define a game between them in the second stage as Γθ = [I, {Ri}, {ui(r, θ)}]. We refer to Γθ as a game induced by θ. Let N(θ) be the set of strategy profiles considered solutions of the game Γθ.3 Suppose that the goal of the designer is to optimize the value of some welfare function, W (r, θ), dependent on the mechanism parameter and resulting play, r. We define a pessimistic measure, W ( ˆR, θ) = inf{W (r, θ) : r ∈ ˆR}, representing the worst-case welfare of the game induced by θ, assuming that agents play some joint strategy in ˆR. Typically we care about W (N(θ), θ), the worst-case outcome of playing some solution.4 On some problems we can gain considerable advantage by using an aggregation function to map the welfare outcome of a game 3 We generally adopt Nash equilibrium as the solution concept, and thus take N(θ) to be the set of equilibria. However, much of the methodology developed here could be employed with alternative criteria for deriving agent behavior from a game definition. 4 Again, alternatives are available. For example, if one has a probability distribution over the solution set N(θ), it would be natural to take the expectation of W (r, θ) instead. 307 specified in terms of agent strategies to an equivalent welfare outcome specified in terms of a lower-dimensional summary. DEFINITION 3. A function φ : R1 × · · · × Rm → Rq is an aggregation function if m ≥ q and W (r, θ) = V (φ(r), θ) for some function V . We overload the function symbol to apply to sets of strategy profiles: φ( ˆR) = {φ(r) : r ∈ ˆR}. For convenience of exposition, we write φ∗ (θ) to mean φ(N(θ)). Using an aggregation function yields a more compact representation of strategy profiles. For example, suppose-as in our application below-that an agent"s strategy is defined by a numeric parameter. If all we care about is the total value played, we may take φ(a) = Pm i=1 ai. If we have chosen our aggregator carefully, we may also capture structure not obvious otherwise. For example, φ∗ (θ) could be decreasing in θ, whereas N(θ) might have a more complex structure. Given a description of the solution correspondence N(θ) (equivalently, φ∗ (θ)), the designer faces a standard optimization problem. Alternatively, given a simulator that could produce an unbiased sample from the distribution of W (N(θ), θ) for any θ, the designer would be faced with another much appreciated problem in the literature: simulation optimization [12]. However, even for a game Γθ with known payoffs it may be computationally intractable to solve for Nash equilibria, particularly if the game has large or infinite strategy sets. Additionally, we wish to study games where the payoffs are not explicitly given, but must be determined from simulation or other experience with the game.5 Accordingly, we assume that we are given a (possibly noisy) data set of payoff realizations: Do = {(θ1 , a1 , U1 ), . . . , (θk , ak , Uk )}, where for every data point θi is the observed mechanism parameter setting, ai is the observed pure strategy profile of the participants, and Ui is the corresponding realization of agent payoffs. We may also have additional data generated by a (possibly noisy) simulator: Ds = {(θk+1 , ak+1 , Uk+1 ), . . . , (θk+l , ak+l , Uk+l )}. Let D = {Do, Ds} be the combined data set. (Either Do or Ds may be null for a particular problem.) In the remainder of this paper, we apply our modeling approach, together with several empirical game-theoretic methods, in order to answer questions regarding the design of the TAC/SCM scenario. 4. EMPIRICAL DESIGN ANALYSIS Since our data comes in the form of payoff experience and not as the value of an objective function for given settings of the control variable, we can no longer rely on the methods for optimizing functions using simulations. Indeed, a fundamental aspect of our design problem involves estimating the Nash equilibrium correspondence. Furthermore, we cannot rely directly on the convergence results that abound in the simulation optimization literature, and must establish probabilistic analysis methods tailored for our problem setting. 4.1 TAC/SCM Design Problem We describe our empirical design analysis methods by presenting a detailed application to the TAC/SCM scenario introduced above. Recall that during the 2004 tournament, the designers of the supplychain game chose to dramatically increase storage costs as a measure aimed at curbing day-0 procurement, to little avail. Here we systematically explore the relationship between storage costs and 5 This is often the case for real games of interest, where natural language or algorithmic descriptions may substitute for a formal specification of strategy and payoff functions. the aggregate quantity of components procured on day 0 in equilibrium. In doing so, we consider several questions raised during and after the tournament. First, does increasing storage costs actually reduce day-0 procurement? Second, was the excessive day-0 procurement that was observed during the 2004 tournament rational? And third, could increasing storage costs sufficiently have reduced day-0 procurement to an acceptable level, and if so, what should the setting of storage costs have been? It is this third question that defines the mechanism design aspect of our analysis.6 To apply our methods, we must specify the agent strategy sets, the designer"s welfare function, the mechanism parameter space, and the source of data. We restrict the agent strategies to be a multiplier on the quantity of the day-0 requests by one of the finalists, Deep Maize, in the 2004 TAC/SCM tournament. We further restrict it to the set [0,1.5], since any strategy below 0 is illegal and strategies above 1.5 are extremely aggressive (thus unlikely to provide refuting deviations beyond those available from included strategies, and certainly not part of any desirable equilibrium). All other behavior is based on the behavior of Deep Maize and is identical for all agents. This choice can provide only an estimate of the actual tournament behavior of a typical agent. However, we believe that the general form of the results should be robust to changes in the full agent behavior. We model the designer"s welfare function as a threshold on the sum of day-0 purchases. Let φ(a) = P6 i=1 ai be the aggregation function representing the sum of day-0 procurement of the six agents participating in a particular supply-chain game (for mixed strategy profiles s, we take expectation of φ with respect to the mixture). The designer"s welfare function W (N(θ), θ) is then given by I{sup{φ∗ (θ)} ≤ α}, where α is the maximum acceptable level of day-0 procurement and I is the indicator function. The designer selects a value θ of storage costs, expressed as an annual percentage of the baseline value of components in the inventory (charged daily), from the set Θ = R+ . Since the designer"s decision depends only on φ∗ (θ), we present all of our results in terms of the value of the aggregation function. 4.2 Estimating Nash Equilibria The objective of TAC/SCM agents is to maximize profits realized over a game instance. Thus, if we fix a strategy for each agent at the beginning of the simulation and record the corresponding profits at the end, we will have obtained a data point in the form (a, U(a)). If we also have fixed the parameter θ of the simulator, the resulting data point becomes part of our data set D. This data set, then, contains data only in the form of pure strategies of players and their corresponding payoffs, and, consequently, in order to formulate the designer"s problem as optimization, we must first determine or approximate the set of Nash equilibria of each game Γθ. Thus, we need methods for approximating Nash equilibria for infinite games. Below, we describe the two methods we used in our study. The first has been explored empirically before, whereas the second is introduced here as the method specifically designed to approximate a set of Nash equilibria. 4.2.1 Payoff Function Approximation The first method for estimating Nash equilibria based on data uses supervised learning to approximate payoff functions of mech6 We do not address whether and how other measures (e.g., constraining procurement directly) could have achieved design objectives. Our approach takes as given some set of design options, in this case defined by the storage cost parameter. In principle our methods could be applied to a different or larger design space, though with corresponding complexity growth. 308 anism participants from a data set of game experience [17]. Once approximate payoff functions are available for all players, the Nash equilibria may be either found analytically or approximated using numerical techniques, depending on the learning model. In what follows, we estimate only a sample Nash equilibrium using this technique, although this restriction can be removed at the expense of additional computation time. One advantage of this method is that it can be applied to any data set and does not require the use of a simulator. Thus, we can apply it when Ds = ∅. If a simulator is available, we can generate additional data to build confidence in our initial estimates.7 We tried the following methods for approximating payoff functions: quadratic regression (QR), locally weighted average (LWA), and locally weighted linear regression (LWLR). We also used control variates to reduce the variance of payoff estimates, as in our previous empirical game-theoretic analysis of TAC/SCM-03 [19]. The quadratic regression model makes it possible to compute equilibria of the learned game analytically. For the other methods we applied replicator dynamics [7] to a discrete approximation of the learned game. The expected total day-0 procurement in equilibrium was taken as the estimate of an outcome. 4.2.2 Search in Strategy Profile Space When we have access to a simulator, we can also use directed search through profile space to estimate the set of Nash equilibria, which we describe here after presenting some additional notation. DEFINITION 4. A strategic neighbor of a pure strategy profile a is a profile that is identical to a in all but one strategy. We define Snb(a, D) as the set of all strategic neighbors of a available in the data set D. Similarly, we define Snb(a, ˜D) to be all strategic neighbors of a not in D. Finally, for any a ∈ Snb(a, D) we define the deviating agent as i(a, a ). DEFINITION 5. The -bound, ˆ, of a pure strategy profile a is defined as maxa ∈Snb(a,D) max{ui(a,a )(a )−ui(a,a )(a), 0}. We say that a is a candidate δ-equilibrium for δ ≥ ˆ. When Snb(a, ˜D) = ∅ (i.e., all strategic neighbors are represented in the data), a is confirmed as an ˆ-Nash equilibrium. Our search method operates by exploring deviations from candidate equilibria. We refer to it as BestFirstSearch, as it selects with probability one a strategy profile a ∈ Snb(a, ˜D) that has the smallest ˆin D. Finally we define an estimator for a set of Nash equilibria. DEFINITION 6. For a set K, define Co(K) to be the convex hull of K. Let Bδ be the set of candidates at level δ. We define ˆφ∗ (θ) = Co({φ(a) : a ∈ Bδ}) for a fixed δ to be an estimator of φ∗ (θ). In words, the estimate of a set of equilibrium outcomes is the convex hull of all aggregated strategy profiles with -bound below some fixed δ. This definition allows us to exploit structure arising from the aggregation function. If two profiles are close in terms of aggregation values, they may be likely to have similar -bounds. In particular, if one is an equilibrium, the other may be as well. We present some theoretical support for this method of estimating the set of Nash equilibria below. Since the game we are interested in is infinite, it is necessary to terminate BestFirstSearch before exploring the entire space of strat7 For example, we can use active learning techniques [5] to improve the quality of payoff function approximation. In this work, we instead concentrate on search in strategy profile space. egy profiles. We currently determine termination time in a somewhat ad-hoc manner, based on observations about the current set of candidate equilibria.8 4.3 Data Generation Our data was collected by simulating TAC/SCM games on a local version of the 2004 TAC/SCM server, which has a configuration setting for the storage cost. Agent strategies in simulated games were selected from the set {0, 0.3, 0.6, . . . , 1.5} in order to have positive probability of generating strategic neighbors.9 A baseline data set Do was generated by sampling 10 randomly generated strategy profiles for each θ ∈ {0, 50, 100, 150, 200}. Between 5 and 10 games were run for each profile after discarding games that had various flaws.10 We used search to generate a simulated data set Ds, performing between 12 and 32 iterations of BestFirstSearch for each of the above settings of θ. Since simulation cost is extremely high (a game takes nearly 1 hour to run), we were able to run a total of 2670 games over the span of more than six months. For comparison, to get the entire description of an empirical game defined by the restricted finite joint strategy space for each value of θ ∈ {0, 50, 100, 150, 200} would have required at least 23100 games (sampling each profile 10 times). 4.4 Results 4.4.1 Analysis of the Baseline Data Set We applied the three learning methods described above to the baseline data set Do. Additionally, we generated an estimate of the Nash equilibrium correspondence, ˆφ∗ (θ), by applying Definition 6 with δ =2.5E6. The results are shown in Figure 1. As we can see, the correspondence ˆφ∗ (θ) has little predictive power based on Do, and reveals no interesting structure about the game. In contrast, all three learning methods suggest that total day-0 procurement is a decreasing function of storage costs. 0 1 2 3 4 5 6 7 8 9 10 0 50 100 150 200 Storage Cost TotalDay-0Procurement LWA LWLR QR BaselineMin BaselineMax Figure 1: Aggregate day-0 procurement estimates based on Do. The correspondence ˆφ∗ (θ) is the interval between BaselineMin and BaselineMax. 8 Generally, search is terminated once the set of candidate equilibria is small enough to draw useful conclusions about the likely range of equilibrium strategies in the game. 9 Of course, we do not restrict our Nash equilibrium estimates to stay in this discrete subset of [0,1.5]. 10 For example, if we detected that any agent failed during the game (failures included crashes, network connectivity problems, and other obvious anomalies), the game would be thrown out. 309 4.4.2 Analysis of Search Data To corroborate the initial evidence from the learning methods, we estimated ˆφ∗ (θ) (again, using δ =2.5E6) on the data set D = {Do, Ds}, where Ds is data generated through the application of BestFirstSearch. The results of this estimate are plotted against the results of the learning methods trained on Do 11 in Figure 2. First, we note that the addition of the search data narrows the range of potential equilibria substantially. Furthermore, the actual point predictions of the learning methods and those based on -bounds after search are reasonably close. Combining the evidence gathered from these two very different approaches to estimating the outcome correspondence yields a much more compelling picture of the relationship between storage costs and day-0 procurement than either method used in isolation. 0 1 2 3 4 5 6 7 8 9 10 0 50 100 150 200 Storage Cost TotayDay-0Procurement LWA LWLR QR SearchMin SearchMax Figure 2: Aggregate day-0 procurement estimates based on search in strategy profile space compared to function approximation techniques trained on Do. The correspondence ˆφ∗ (θ) for D = {Do, Ds} is the interval between SearchMin and SearchMax. This evidence supports the initial intuition that day-0 procurement should be decreasing with storage costs. It also confirms that high levels of day-0 procurement are a rational response to the 2004 tournament setting of average storage cost, which corresponds to θ = 100. The minimum prediction for aggregate procurement at this level of storage costs given by any experimental methods is approximately 3. This is quite high, as it corresponds to an expected commitment of 1/3 of the total supplier capacity for the entire game. The maximum prediction is considerably higher at 4.5. In the actual 2004 competition, aggregate day-0 procurement was equivalent to 5.71 on the scale used here [9]. Our predictions underestimate this outcome to some degree, but show that any rational outcome was likely to have high day-0 procurement. 4.4.3 Extrapolating the Solution Correspondence We have reasonably strong evidence that the outcome correspondence is decreasing. However, the ultimate goal is to be able to either set the storage cost parameter to a value that would curb day-0 procurement in equilibrium or conclude that this is not possible. To answer this question directly, suppose that we set a conservative threshold α = 2 on aggregate day-0 procurement.12 Linear 11 It is unclear how meaningful the results of learning would be if Ds were added to the training data set. Indeed, the additional data may actually increase the learning variance. 12 Recall that designer"s objective is to incentivize aggergate day-0 procurement that is below the threshold α. Our threshold here still represents a commitment of over 20% of the suppliers" capacity for extrapolation of the maximum of the outcome correspondence estimated from D yields θ = 320. The data for θ = 320 were collected in the same way as for other storage cost settings, with 10 randomly generated profiles followed by 33 iterations of BestFirstSearch. Figure 3 shows the detailed -bounds for all profiles in terms of their corresponding values of φ. 0.00E+00 5.00E+06 1.00E+07 1.50E+07 2.00E+07 2.50E+07 3.00E+07 3.50E+07 4.00E+07 4.50E+07 5.00E+07 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 5.1 5.4 5.7 6 6.3 6.6 6.9 7.2 Total Day-0 Procurement ε−boundFigure 3: Values of ˆ for profiles explored using search when θ = 320. Strategy profiles explored are presented in terms of the corresponding values of φ(a). The gray region corresponds to ˆφ∗ (320) with δ =2.5M. The estimated set of aggregate day-0 outcomes is very close to that for θ = 200, indicating that there is little additional benefit to raising storage costs above 200. Observe, that even the lower bound of our estimated set of Nash equilibria is well above the target day-0 procurement of 2. Furthermore, payoffs to agents are almost always negative at θ = 320. Consequently, increasing the costs further would be undesirable even if day-0 procurement could eventually be curbed. Since we are reasonably confident that φ∗ (θ) is decreasing in θ, we also do not expect that setting θ somewhere between 200 and 320 will achieve the desired result. We conclude that it is unlikely that day-0 procurement could ever be reduced to a desirable level using any reasonable setting of the storage cost parameter. That our predictions tend to underestimate tournament outcomes reinforces this conclusion. To achieve the desired reduction in day-0 procurement requires redesigning other aspects of the mechanism. 4.5 Probabilistic Analysis Our empirical analysis has produced evidence in support of the conclusion that no reasonable setting of storage cost was likely to sufficiently curb excessive day-0 procurement in TAC/SCM "04. All of this evidence has been in the form of simple interpolation and extrapolation of estimates of the Nash equilibrium correspondence. These estimates are based on simulating game instances, and are subject to sampling noise contributed by the various stochastic elements of the game. In this section, we develop and apply methods for evaluating the sensitivity of our -bound calculations to such stochastic effects. Suppose that all agents have finite (and small) pure strategy sets, A. Thus, it is feasible to sample the entire payoff matrix of the game. Additionally, suppose that noise is additive with zero-mean the entire game on average, so in practice we would probably want the threshold to be even lower. 310 and finite variance, that is, Ui(a) = ui(a) + ˜ξi(a), where Ui(a) is the observed payoff to i when a was played, ui(a) is the actual corresponding payoff, and ˜ξi(a) is a mean-zero normal random variable. We designate the known variance of ˜ξi(a) by σ2 i (a). Thus, we assume that ˜ξi(a) is normal with distribution N(0, σ2 i (a)). We take ¯ui(a) to be the sample mean over all Ui(a) in D, and follow Chang and Huang [3] to assume that we have an improper prior over the actual payoffs ui(a) and sampling was independent for all i and a. We also rely on their result that ui(a)|¯ui(a) = ¯ui(a)−Zi(a)/[σi(a)/ p ni(a)] are independent with posterior distributions N(¯ui(a), σ2 i (a)/ni(a)), where ni(a) is the number of samples taken of payoffs to i for pure profile a, and Zi(a) ∼ N(0, 1). We now derive a generic probabilistic bound that a profile a ∈ A is an -Nash equilibrium. If ui(·)|¯ui(·) are independent for all i ∈ I and a ∈ A, we have the following result (from this point on we omit conditioning on ¯ui(·) for brevity): PROPOSITION 1. Pr „ max i∈I max b∈Ai ui(b, a−i) − ui(a) ≤ « = = Y i∈I Z R Y b∈Ai\ai Pr(ui(b, a−i) ≤ u + )fui(a)(u)du, (2) where fui(a)(u) is the pdf of N(¯ui(a), σi(a)). The proofs of this and all subsequent results are in the Appendix. The posterior distribution of the optimum mean of n samples, derived by Chang and Huang [3], is Pr (ui(a) ≤ c) = 1 − Φ "p ni(a)(¯ui(a) − c) σi(a) # , (3) where a ∈ A and Φ(·) is the N(0, 1) distribution function. Combining the results (2) and (3), we obtain a probabilistic confidence bound that (a) ≤ γ for a given γ. Now, we consider cases of incomplete data and use the results we have just obtained to construct an upper bound (restricted to profiles represented in data) on the distribution of sup{φ∗ (θ)} and inf{φ∗ (θ)} (assuming that both are attainable): Pr{sup{φ∗ (θ)} ≤ x} ≤D Pr{∃a ∈ D : φ(a) ≤ x ∧ a ∈ N(θ)} ≤ X a∈D:φ(a)≤x Pr{a ∈ N(θ)} = X a∈D:φ(a)≤x Pr{ (a) = 0}, where x is a real number and ≤D indicates that the upper bound accounts only for strategies that appear in the data set D. Since the events {∃a ∈ D : φ(a) ≤ x ∧ a ∈ N(θ)} and {inf{φ∗ (θ)} ≤ x} are equivalent, this also defines an upper bound on the probability of {inf{φ∗ (θ)} ≤ x}. The values thus derived comprise the Tables 1 and 2. φ∗ (θ) θ = 0 θ = 50 θ = 100 <2.7 0.000098 0 0.146 <3 0.158 0.0511 0.146 <3.9 0.536 0.163 1 <4.5 1 1 1 Table 1: Upper bounds on the distribution of inf{φ∗ (θ)} restricted to D for θ ∈ {0, 50, 100} when N(θ) is a set of Nash equilibria. φ∗ (θ) θ = 150 θ = 200 θ = 320 <2.7 0 0 0.00132 <3 0.0363 0.141 1 <3.9 1 1 1 <4.5 1 1 1 Table 2: Upper bounds on the distribution of inf{φ∗ (θ)} restricted to D for θ ∈ {150, 200, 320} when N(θ) is a set of Nash equilibria. Tables 1 and 2 suggest that the existence of any equilibrium with φ(a) < 2.7 is unlikely for any θ that we have data for, although this judgment, as we mentioned, is only with respect to the profiles we have actually sampled. We can then accept this as another piece of evidence that the designer could not find a suitable setting of θ to achieve his objectives-indeed, the designer seems unlikely to achieve his objective even if he could persuade participants to play a desirable equilibrium! Table 1 also provides additional evidence that the agents in the 2004 TAC/SCM tournament were indeed rational in procuring large numbers of components at the beginning fo the game. If we look at the third column of this table, which corresponds to θ = 100, we can gather that no profile a in our data with φ(a) < 3 is very likely to be played in equilibrium. The bounds above provide some general evidence, but ultimately we are interested in a concrete probabilistic assessment of our conclusion with respect to the data we have sampled. Particularly, we would like to say something about what happens for the settings of θ for which we have no data. To derive an approximate probabilistic bound on the probability that no θ ∈ Θ could have achieved the designer"s objective, let ∪J j=1Θj, be a partition of Θ, and assume that the function sup{φ∗ (θ)} satisfies the Lipschitz condition with Lipschitz constant Aj on each subset Θj.13 Since we have determined that raising the storage cost above 320 is undesirable due to secondary considerations, we restrict attention to Θ = [0, 320]. We now define each subset j to be the interval between two points for which we have produced data. Thus, Θ = [0, 50] [ (50, 100] [ (100, 150] [ (150, 200] [ (200, 320], with j running between 1 and 5, corresponding to subintervals above. We will further denote each Θj by (aj , bj].14 Then, the following Proposition gives us an approximate upper bound15 on the probability that sup{φ∗ (θ)} ≤ α. PROPOSITION 2. Pr{ _ θ∈Θ sup{φ(θ)} ≤ α} ≤D 5X j=1 X y,z∈D:y+z≤cj 0 @ X a:φ(a)=z Pr{ (a) = 0} 1 A × × 0 @ X a:φ(a)=y Pr{ (a) = 0} 1 A , where cj = 2α + Aj(bj − aj) and ≤D indicates that the upper bound only accounts for strategies that appear in the data set D. 13 A function that satisfies the Lipschitz condition is called Lipschitz continuous. 14 The treatment for the interval [0,50] is identical. 15 It is approximate in a sense that we only take into account strategies that are present in the data. 311 Due to the fact that our bounds are approximate, we cannot use them as a conclusive probabilistic assessment. Instead, we take this as another piece of evidence to complement our findings. Even if we can assume that a function that we approximate from data is Lipschitz continuous, we rarely actually know the Lipschitz constant for any subset of Θ. Thus, we are faced with a task of estimating it from data. Here, we tried three methods of doing this. The first one simply takes the highest slope that the function attains within the available data and uses this constant value for every subinterval. This produces the most conservative bound, and in many situations it is unlikely to be informative. An alternative method is to take an upper bound on slope obtained within each subinterval using the available data. This produces a much less conservative upper bound on probabilities. However, since the actual upper bound is generally greater for each subinterval, the resulting probabilistic bound may be deceiving. A final method that we tried is a compromise between the two above. Instead of taking the conservative upper bound based on data over the entire function domain Θ, we take the average of upper bounds obtained at each Θj. The bound at an interval is then taken to be the maximum of the upper bound for this interval and the average upper bound for all intervals. The results of evaluating the expression for Pr{ _ θ∈Θ sup{φ∗ (θ)} ≤ α} when α = 2 are presented in Table 3. In terms of our claims in maxj Aj Aj max{Aj ,ave(Aj)} 1 0.00772 0.00791 Table 3: Approximate upper bound on probability that some setting of θ ∈ [0, 320] will satisfy the designer objective with target α = 2. Different methods of approximating the upper bound on slope in each subinterval j are used. this work, the expression gives an upper bound on the probability that some setting of θ (i.e., storage cost) in the interval [0,320] will result in total day-0 procurement that is no greater in any equilibrium than the target specified by α and taken here to be 2. As we had suspected, the most conservative approach to estimating the upper bound on slope, presented in the first column of the table, provides us little information here. However, the other two estimation approaches, found in columns two and three of Table 3, suggest that we are indeed quite confident that no reasonable setting of θ ∈ [0, 320] would have done the job. Given the tremendous difficulty of the problem, this result is very strong.16 Still, we must be very cautious in drawing too heroic a conclusion based on this evidence. Certainly, we have not checked all the profiles but only a small proportion of them (infinitesimal, if we consider the entire continuous domain of θ and strategy sets). Nor can we expect ever to obtain enough evidence to make completely objective conclusions. Instead, the approach we advocate here is to collect as much evidence as is feasible given resource constraints, and make the most compelling judgment based on this evidence, if at all possible. 5. CONVERGENCE RESULTS At this point, we explore abstractly whether a design parameter choice based on payoff data can be asymptotically reliable. 16 Since we did not have all the possible deviations for any profile available in the data, the true upper bounds may be even lower. As a matter of convenience, we will use notation un,i(a) to refer to a payoff function of player i based on an average over n i.i.d. samples from the distribution of payoffs. We also assume that un,i(a) are independent for all a ∈ A and i ∈ I. We will use the notation Γn to refer to the game [I, R, {ui,n(·)}], whereas Γ will denote the underlying game, [I, R, {ui(·)}]. Similarly, we define n(r) to be (r) with respect to the game Γn. In this section, we show that n(s) → (s) a.s. uniformly on the mixed strategy space for any finite game, and, furthermore, that all mixed strategy Nash equilibria in empirical games eventually become arbitrarily close to some Nash equilibrium strategies in the underlying game. We use these results to show that under certain conditions, the optimal choice of the design parameter based on empirical data converges almost surely to the actual optimum. THEOREM 3. Suppose that |I| < ∞, |A| < ∞. Then n(s) → (s) a.s. uniformly on S. Recall that N is a set of all Nash equilibria of Γ. If we define Nn,γ = {s ∈ S : n(s) ≤ γ}, we have the following corollary to Theorem 3: COROLLARY 4. For every γ > 0, there is M such that ∀n ≥ M, N ⊂ Nn,γ a.s. PROOF. Since (s) = 0 for every s ∈ N, we can find M large enough such that Pr{supn≥M sups∈N n(s) < γ} = 1. By the Corollary, for any game with a finite set of pure strategies and for any > 0, all Nash equilibria lie in the set of empirical -Nash equilibria if enough samples have been taken. As we now show, this provides some justification for our use of a set of profiles with a non-zero -bound as an estimate of the set of Nash equilibria. First, suppose we conclude that for a particular setting of θ, sup{ˆφ∗ (θ)} ≤ α. Then, since for any fixed > 0, N(θ) ⊂ Nn, (θ) when n is large enough, sup{φ∗ (θ)} = sup s∈N (θ) φ(s) ≤ sup s∈Nn, (θ) φ(s) = sup{ˆφ∗ (θ)} ≤ α for any such n. Thus, since we defined the welfare function of the designer to be I{sup{φ∗ (θ)} ≤ α} in our domain of interest, the empirical choice of θ satisfies the designer"s objective, thereby maximizing his welfare function. Alternatively, suppose we conclude that inf{ˆφ∗ (θ)} > α for every θ in the domain. Then, α < inf{ˆφ∗ (θ)} = inf s∈Nn, (θ) φ(s) ≤ inf s∈N (θ) φ(s) ≤ ≤ sup s∈N (θ) φ(s) = sup{φ∗ (θ)}, for every θ, and we can conclude that no setting of θ will satisfy the designer"s objective. Now, we will show that when the number of samples is large enough, every Nash equilibrium of Γn is close to some Nash equilibrium of the underlying game. This result will lead us to consider convergence of optimizers based on empirical data to actual optimal mechanism parameter settings. We first note that the function (s) is continuous in a finite game. LEMMA 5. Let S be a mixed strategy set defined on a finite game. Then : S → R is continuous. 312 For the exposition that follows, we need a bit of additional notation. First, let (Z, d) be a metric space, and X, Y ⊂ Z and define directed Hausdorff distance from X to Y to be h(X, Y ) = sup x∈X inf y∈Y d(x, y). Observe that U ⊂ X ⇒ h(U, Y ) ≤ h(X, Y ). Further, define BS(x, δ) to be an open ball in S ⊂ Z with center x ∈ S and radius δ. Now, let Nn denote all Nash equilibria of the game Γn and let Nδ = [ x∈N BS(x, δ), that is, the union of open balls of radius δ with centers at Nash equilibria of Γ. Note that h(Nδ, N) = δ. We can then prove the following general result. THEOREM 6. Suppose |I| < ∞ and |A| < ∞. Then almost surely h(Nn, N) converges to 0. We will now show that in the special case when Θ and A are finite and each Γθ has a unique Nash equilibrium, the estimates ˆθ of optimal designer parameter converge to an actual optimizer almost surely. Let ˆθ = arg maxθ∈Θ W (Nn(θ), θ), where n is the number of times each pure profile was sampled in Γθ for every θ, and let θ∗ = arg maxθ∈Θ W (N(θ), θ). THEOREM 7. Suppose |N(θ)| = 1 for all θ ∈ Θ and suppose that Θ and A are finite. Let W (s, θ) be continuous at the unique s∗ (θ) ∈ N(θ) for each θ ∈ Θ. Then ˆθ is a consistent estimator of θ∗ if W (N(θ), θ) is defined as a supremum, infimum, or expectation over the set of Nash equilibria. In fact, ˆθ → θ∗ a.s. in each of these cases. The shortcoming of the above result is that, within our framework, the designer has no way of knowing or ensuring that Γθ do, indeed, have unique equilibria. However, it does lend some theoretical justification for pursuing design in this manner, and, perhaps, will serve as a guide for more general results in the future. 6. RELATED WORK The mechanism design literature in Economics has typically explored existence of a mechanism that implements a social choice function in equilibrium [10]. Additionally, there is an extensive literature on optimal auction design [10], of which the work by Roger Myerson [11] is, perhaps, the most relevant. In much of this work, analytical results are presented with respect to specific utility functions and accounting for constraints such as incentive compatibility and individual rationality. Several related approaches to search for the best mechanism exist in the Computer Science literature. Conitzer and Sandholm [6] developed a search algorithm when all the relevant game parameters are common knowledge. When payoff functions of players are unknown, a search using simulations has been explored as an alternative. One approach in that direction, taken in [4] and [15], is to co-evolve the mechanism parameter and agent strategies, using some notion of social utility and agent payoffs as fitness criteria. An alternative to co-evolution explored in [16] was to optimize a well-defined welfare function of the designer using genetic programming. In this work the authors used a common learning strategy for all agents and defined an outcome of a game induced by a mechanism parameter as the outcome of joint agent learning. Most recently, Phelps et al. [14] compared two mechanisms based on expected social utility with expectation taken over an empirical distribution of equilibria in games defined by heuristic strategies, as in [18]. 7. CONCLUSION In this work we spent considerable effort developing general tactics for empirical mechanism design. We defined a formal gametheoretic model of interaction between the designer and the participants of the mechanism as a two-stage game. We also described in some generality the methods for estimating a sample Nash equilibrium function when the data is extremely scarce, or a Nash equilibrium correspondence when more data is available. Our techniques are designed specifically to deal with problems in which both the mechanism parameter space and the agent strategy sets are infinite and only a relatively small data set can be acquired. A difficult design issue in the TAC/SCM game which the TAC community has been eager to address provides us with a setting to test our methods. In applying empirical game analysis to the problem at hand, we are fully aware that our results are inherently inexact. Thus, we concentrate on collecting evidence about the structure of the Nash equilibrium correspondence. In the end, we can try to provide enough evidence to either prescribe a parameter setting, or suggest that no setting is possible that will satisfy the designer. In the case of TAC/SCM, our evidence suggests quite strongly that storage cost could not have been effectively adjusted in the 2004 tournament to curb excessive day-0 procurement without detrimental effects on overall profitability. The success of our analysis in this extremely complex environment with high simulation costs makes us optimistic that our methods can provide guidance in making mechanism design decisions in other challenging domains. The theoretical results confirm some intuitions behind the empirical mechanism design methods we have introduced, and increases our confidence that our framework can be effective in estimating the best mechanism parameter choice in relatively general settings. Acknowledgments We thank Terence Kelly, Matthew Rudary, and Satinder Singh for helpful comments on earlier drafts of this work. This work was supported in part by NSF grant IIS-0205435 and the DARPA REAL strategic reasoning program. 8. REFERENCES [1] R. Arunachalam and N. M. Sadeh. The supply chain trading agent competition. Electronic Commerce Research and Applications, 4:63-81, 2005. [2] M. Benisch, A. Greenwald, V. Naroditskiy, and M. Tschantz. A stochastic programming approach to scheduling in TAC SCM. In Fifth ACM Conference on Electronic Commerce, pages 152-159, New York, 2004. [3] Y.-P. Chang and W.-T. Huang. Generalized confidence intervals for the largest value of some functions of parameters under normality. Statistica Sinica, 10:1369-1383, 2000. [4] D. Cliff. Evolution of market mechanism through a continuous space of auction-types. In Congress on Evolutionary Computation, 2002. [5] D. A. Cohn, Z. Ghahramani, and M. I. Jordan. Active learning with statistical models. Journal of Artificial Intelligence Research, 4:129-145, 1996. [6] V. Conitzer and T. Sandholm. An algorithm for automatically designing deterministic mechanisms without payments. In 313 Third International Joint Conference on Autonomous Agents and Multi-Agent Systems, pages 128-135, 2004. [7] D. Friedman. Evolutionary games in economics. Econometrica, 59(3):637-666, May 1991. [8] R. Keener. Statistical Theory: A Medley of Core Topics. University of Michigan Department of Statistics, 2004. [9] C. Kiekintveld, Y. Vorobeychik, and M. P. Wellman. An analysis of the 2004 supply chain management trading agent competition. In IJCAI-05 Workshop on Trading Agent Design and Analysis, Edinburgh, 2005. [10] A. Mas-Colell, M. Whinston, and J. Green. Microeconomic Theory. Oxford University Press, 1995. [11] R. B. Myerson. Optimal auction design. Mathematics of Operations Research, 6(1):58-73, February 1981. [12] S. Olafsson and J. Kim. Simulation optimization. In E. Yucesan, C.-H. Chen, J. Snowdon, and J. Charnes, editors, 2002 Winter Simulation Conference, 2002. [13] D. Pardoe and P. Stone. TacTex-03: A supply chain management agent. SIGecom Exchanges, 4(3):19-28, 2004. [14] S. Phelps, S. Parsons, and P. McBurney. Automated agents versus virtual humans: an evolutionary game theoretic comparison of two double-auction market designs. In Workshop on Agent Mediated Electronic Commerce VI, 2004. [15] S. Phelps, S. Parsons, P. McBurney, and E. Sklar. Co-evolution of auction mechanisms and trading strategies: towards a novel approach to microeconomic design. In ECOMAS 2002 Workshop, 2002. [16] S. Phelps, S. Parsons, E. Sklar, and P. McBurney. Using genetic programming to optimise pricing rules for a double-auction market. In Workshop on Agents for Electronic Commerce, 2003. [17] Y. Vorobeychik, M. P. Wellman, and S. Singh. Learning payoff functions in infinite games. In Nineteenth International Joint Conference on Artificial Intelligence, pages 977-982, 2005. [18] W. E. Walsh, R. Das, G. Tesauro, and J. O. Kephart. Analyzing complex strategic interactions in multi-agent systems. In AAAI-02 Workshop on Game Theoretic and Decision Theoretic Agents, 2002. [19] M. P. Wellman, J. J. Estelle, S. Singh, Y. Vorobeychik, C. Kiekintveld, and V. Soni. Strategic interactions in a supply chain game. Computational Intelligence, 21(1):1-26, February 2005. APPENDIX A. PROOFS A.1 Proof of Proposition 1 Pr „ max i∈I max b∈Ai\ai ui(b, a−i) − ui(a) ≤ « = = Y i∈I Eui(a) » Pr( max b∈Ai\ai ui(b, a−i) − ui(a) ≤ |ui(a)) = = Y i∈I Z R Y b∈Ai\ai Pr(ui(b, a−i) ≤ u + )fui(a)(u)du. A.2 Proof of Proposition 2 First, let us suppose that some function, f(x) defined on [ai, bi], satisfy Lipschitz condition on (ai, bi] with Lipschitz constant Ai. Then the following claim holds: Claim: infx∈(ai,bi] f(x) ≥ 0.5(f(ai) + f(bi) − Ai(bi − ai). To prove this claim, note that the intersection of lines at f(ai) and f(bi) with slopes −Ai and Ai respectively will determine the lower bound on the minimum of f(x) on [ai, bi] (which is a lower bound on infimum of f(x) on (ai, bj ]). The line at f(ai) is determined by f(ai) = −Aiai + cL and the line at f(bi) is determined by f(bi) = Aibi +cR. Thus, the intercepts are cL = f(ai)+Aiai and cR = f(bi) + Aibi respectively. Let x∗ be the point at which these lines intersect. Then, x∗ = − f(x∗ ) − cR A = f(x∗ ) − cL A . By substituting the expressions for cR and cL, we get the desired result. Now, subadditivity gives us Pr{ _ θ∈Θ sup{φ∗ (θ)} ≤ α} ≤ 5X j=1 Pr{ _ θ∈Θj sup{φ∗ (θ)} ≤ α}, and, by the claim, Pr{ _ θ∈Θj sup{φ∗ (θ)} ≤ α} = 1 − Pr{ inf θ∈Θj sup{φ∗ (θ)} > α} ≤ Pr{sup{φ∗ (aj)} + sup{φ∗ (bj)} ≤ 2α + Aj(bj − aj )}. Since we have a finite number of points in the data set for each θ, we can obtain the following expression: Pr{sup{φ∗ (aj)} + sup{φ∗ (bj)} ≤ cj } =D X y,z∈D:y+z≤cj Pr{sup{φ∗ (bj )} = y} Pr{sup{φ∗ (aj)} = z}. We can now restrict attention to deriving an upper bound on Pr{sup{φ∗ (θ)} = y} for a fixed θ. To do this, observe that Pr{sup{φ∗ (θ)} = y} ≤D Pr{ _ a∈D:φ(a)=y (a) = 0} ≤ X a∈D:φ(a)=y Pr{ (a) = 0} by subadditivity and the fact that a profile a is a Nash equilibrium if and only if (a) = 0. Putting everything together yields the desired result. A.3 Proof of Theorem 3 First, we will need the following fact: Claim: Given a function fi(x) and a set X, | maxx∈X f1(x) − maxx∈X f2(x)| ≤ maxx∈X |f1(x) − f2(x)|. To prove this claim, observe that | max x∈X f1(x) − max x∈X f2(x)| =  maxx f1(x) − maxx f2(x) if maxx f1(x) ≥ maxx f2(x) maxx f2(x) − maxx f1(x) if maxx f2(x) ≥ maxx f1(x) In the first case, max x∈X f1(x) − max x∈X f2(x) ≤ max x∈X (f1(x) − f2(x)) ≤ ≤ max x∈X |f1(x) − f2(x)|. 314 Similarly, in the second case, max x∈X f2(x) − max x∈X f1(x) ≤ max x∈X (f2(x) − f1(x)) ≤ ≤ max x∈X |f2(x) − f1(x)| = max x∈X |f1(x) − f2(x)|. Thus, the claim holds. By the Strong Law of Large Numbers, un,i(a) → ui(a) a.s. for all i ∈ I, a ∈ A. That is, Pr{ lim n→∞ un,i(a) = ui(a)} = 1, or, equivalently [8], for any α > 0 and δ > 0, there is M(i, a) > 0 such that Pr{ sup n≥M(i,a) |un,i(a) − ui(a)| < δ 2|A| } ≥ 1 − α. By taking M = maxi∈I maxa∈A M(i, a), we have Pr{max i∈I max a∈A sup n≥M |un,i(a) − ui(a)| < δ 2|A| } ≥ 1 − α. Thus, by the claim, for any n ≥ M, sup n≥M | n(s) − (s)| ≤ max i∈I max ai∈Ai sup n≥M |un,i(ai, s−i) − ui(ai, s−i)|+ + sup n≥M max i∈I |un,i(s) − ui(s)| ≤ max i∈I max ai∈Ai X b∈A−i sup n≥M |un,i(ai, b) − ui(ai, b)|s−i(b)+ + max i∈I X b∈A sup n≥M |un,i(b) − ui(b)|s(b) ≤ max i∈I max ai∈Ai X b∈A−i sup n≥M |un,i(ai, b) − ui(ai, b)|+ + max i∈I X b∈A sup n≥M |un,i(b) − ui(b)| < max i∈I max ai∈Ai X b∈A−i ( δ 2|A| ) + max i∈I X b∈A ( δ 2|A| ) ≤ δ with probability at least 1 − α. Note that since s−i(a) and s(a) are bounded between 0 and 1, we were able to drop them from the expressions above to obtain a bound that will be valid independent of the particular choice of s. Furthermore, since the above result can be obtained for an arbitrary α > 0 and δ > 0, we have Pr{limn→∞ n(s) = (s)} = 1 uniformly on S. A.4 Proof of Lemma 5 We prove the result using uniform continuity of ui(s) and preservation of continuity under maximum. Claim: A function f : Rk → R defined by f(t) = Pk i=1 ziti, where zi are constants in R, is uniformly continuous in t. The claim follows because |f(t)−f(t )| = | Pk i=1 zi(ti−ti)| ≤ Pk i=1 |zi||ti − ti|. An immediate result of this for our purposes is that ui(s) is uniformly continuous in s and ui(ai, s−i) is uniformly continuous in s−i. Claim: Let f(a, b) be uniformly continuous in b ∈ B for every a ∈ A, with |A| < ∞. Then V (b) = maxa∈A f(a, b) is uniformly continuous in b. To show this, take γ > 0 and let b, b ∈ B such that b − b < δ(a) ⇒ |f(a, b) − f(a, b )| < γ. Now take δ = mina∈A δ(a). Then, whenever b − b < δ, |V (b) − V (b )| = | max a∈A f(a, b) − max a∈A f(a, b )| ≤ max a∈A |f(a, b) − f(a, b )| < γ. Now, recall that (s) = maxi[maxai∈Ai ui(ai, s−i) − ui(s)]. By the claims above, maxai∈Ai ui(ai, s−i) is uniformly continuous in s−i and ui(s) is uniformly continuous in s. Since the difference of two uniformly continuous functions is uniformly continuous, and since this continuity is preserved under maximum by our second claim, we have the desired result. A.5 Proof of Theorem 6 Choose δ > 0. First, we need to ascertain that the following claim holds: Claim: ¯ = mins∈S\Nδ (s) exists and ¯ > 0. Since Nδ is an open subset of compact S, it follows that S \ Nδ is compact. As we had also proved in Lemma 5 that (s) is continuous, existence follows from the Weierstrass theorem. That ¯ > 0 is clear since (s) = 0 if and only if s is a Nash equilibrium of Γ. Now, by Theorem 3, for any α > 0 there is M such that Pr{ sup n≥M sup s∈S | n(s) − (s)| < ¯} ≥ 1 − α. Consequently, for any δ > 0, Pr{ sup n≥M h(Nn, Nδ) < δ} ≥ Pr{∀n ≥ M Nn ⊂ Nδ} ≥ Pr{ sup n≥M sup s∈N (s) < ¯} ≥ Pr{ sup n≥M sup s∈S | n(s) − (s)| < ¯} ≥ 1 − α. Since this holds for an arbitrary α > 0 and δ > 0, the desired result follows. A.6 Proof of Theorem 7 Fix θ and choose δ > 0. Since W (s, θ) is continuous at s∗ (θ), given > 0 there is δ > 0 such that for every s that is within δ of s∗ (θ), |W (s , θ) − W (s∗ (θ), θ)| < . By Theorem 6, we can find M(θ) large enough such that all s ∈ Nn are within δ of s∗ (θ) for all n ≥ M(θ) with probability 1. Consequently, for any > 0 we can find M(θ) large enough such that with probability 1 we have supn≥M(θ) sups ∈Nn |W (s , θ) − W (s∗ (θ), θ)| < . Let us assume without loss of generality that there is a unique optimal choice of θ. Now, since the set Θ is finite, there is also the second-best choice of θ (if there is only one θ ∈ Θ this discussion is moot anyway): θ∗∗ = arg max Θ\θ∗ W (s∗ (θ), θ). Suppose w.l.o.g. that θ∗∗ is also unique and let ∆ = W (s∗ (θ∗ ), θ∗ ) − W (s∗ (θ∗∗ ), θ∗∗ ). Then if we let < ∆/2 and let M = maxθ∈Θ M(θ), where each M(θ) is large enough such that supn≥M(θ) sups ∈Nn |W (s , θ)− W (s∗ (θ), θ)| < a.s., the optimal choice of θ based on any empirical equilibrium will be θ∗ with probability 1. Thus, in particular, given any probability distribution over empirical equilibria, the best choice of θ will be θ∗ with probability 1 (similarly, if we take supremum or infimum of W (Nn(θ), θ) over the set of empirical equilibria in constructing the objective function). 315

analysis;nash equilibrium;participant;game theory;observed behavior;outcome feature of interest;parameter setting;player;gametheoretic model;supply-chain trading;two-stage game;empirical mechanism design;interest outcome feature;empirical mechanism

train_J-47

On the Computational Power of Iterative Auctions∗

We embark on a systematic analysis of the power and limitations of iterative combinatorial auctions. Most existing iterative combinatorial auctions are based on repeatedly suggesting prices for bundles of items, and querying the bidders for their demand under these prices. We prove a large number of results showing the boundaries of what can be achieved by auctions of this kind. We first focus on auctions that use a polynomial number of demand queries, and then we analyze the power of different kinds of ascending-price auctions.

1. INTRODUCTION Combinatorial auctions have recently received a lot of attention. In a combinatorial auction, a set M of m nonidentical items are sold in a single auction to n competing bidders. The bidders have preferences regarding the bundles of items that they may receive. The preferences of bidder i are specified by a valuation function vi : 2M → R+ , where vi(S) denotes the value that bidder i attaches to winning the bundle of items S. We assume free disposal, i.e., that the vi"s are monotone non-decreasing. The usual goal of the auctioneer is to optimize the social welfare P i vi(Si), where the allocation S1...Sn must be a partition of the items. Applications include many complex resource allocation problems and, in fact, combinatorial auctions may be viewed as the common abstraction of many complex resource allocation problems. Combinatorial auctions face both economic and computational difficulties and are a central problem in the recently active border of economic theory and computer science. A forthcoming book [11] addresses many of the issues involved in the design and implementation of combinatorial auctions. The design of a combinatorial auction involves many considerations. In this paper we focus on just one central issue: the communication between bidders and the allocation mechanism - preference elicitation. Transferring all information about bidders" preferences requires an infeasible (exponential in m) amount of communication. Thus, direct revelation auctions in which bidders simply declare their preferences to the mechanism are only practical for very small auction sizes or for very limited families of bidder preferences. We have therefore seen a multitude of suggested iterative auctions in which the auction protocol repeatedly interacts with the different bidders, aiming to adaptively elicit enough information about the bidders" preferences as to be able to find a good (optimal or close to optimal) allocation. Most of the suggested iterative auctions proceed by maintaining temporary prices for the bundles of items and repeatedly querying the bidders as to their preferences between the bundles under the current set of prices, and then updating the set of bundle prices according to the replies received (e.g., [22, 12, 17, 37, 3]). Effectively, such an iterative auction accesses the bidders" preferences by repeatedly making the following type of demand query to bidders: Query to bidder i: a vector of bundle prices p = {p(S)}S⊆M ; Answer: a bundle of items S ⊆ M that maximizes vi(S) − p(S).. These types of queries are very natural in an economic setting as they capture the revealed preferences of the bidders. Some auctions, called item-price or linear-price auctions, specify a price pi for each item, and the price of any given bundle S is always linear, p(S) = P i∈S pi. Other auctions, called bundle-price auctions, allow specifying arbitrary (non-linear) prices p(S) for bundles. Another important differentiation between models of iterative auctions is 29 based on whether they use anonymous or non-anonymous prices: In some auctions the prices that are presented to the bidders are always the same (anonymous prices). In other auctions (non-anonymous), different bidders may face different (discriminatory) vectors of prices. In ascending-price auctions, forcing prices to be anonymous may be a significant restriction. In this paper, we embark on a systematic analysis of the computational power of iterative auctions that are based on demand queries. We do not aim to present auctions for practical use but rather to understand the limitations and possibilities of these kinds of auctions. In the first part of this paper, our main question is what can be done using a polynomial number of these types of queries? That is, polynomial in the main parameters of the problem: n, m and the number of bits t needed for representing a single value vi(S). Note that from an algorithmic point of view we are talking about sub-linear time algorithms: the input size here is really n(2m − 1) numbers - the descriptions of the valuation functions of all bidders. There are two aspects to computational efficiency in these settings: the first is the communication with the bidders, i.e., the number of queries made, and the second is the usual computational tractability. Our lower bounds will depend only on the number of queriesand hold independently of any computational assumptions like P = NP. Our upper bounds will always be computationally efficient both in terms of the number of queries and in terms of regular computation. As mentioned, this paper concentrates on the single aspect of preference elicitation and on its computational consequences and does not address issues of incentives. This strengthens our lower bounds, but means that the upper bounds require evaluation from this perspective also before being used in any real combinatorial auction.1 The second part of this paper studies the power of ascending -price auctions. Ascending auctions are iterative auctions where the published prices cannot decrease in time. In this work, we try to systematically analyze what do the differences between various models of ascending auctions mean. We try to answer the following questions: (i) Which models of ascending auctions can find the optimal allocation, and for which classes of valuations? (ii) In cases where the optimal allocation cannot be determined by ascending auctions, how well can such auctions approximate the social welfare? (iii) How do the different models for ascending auctions compare? Are some models computationally stronger than others? Ascending auctions have been extensively studied in the literature (see the recent survey by Parkes [35]). Most of this work presented "upper bounds", i.e., proposed mechanisms with ascending prices and analyzed their properties. A result which is closer in spirit to ours, is by Gul and Stacchetti [17], who showed that no item-price ascending auction can always determine the VCG prices, even for substitutes valuations.2 Our framework is more general than the traditional line of research that concentrates on the final allocation and 1 We do observe however that some weak incentive property comes for free in demand-query auctions since myopic players will answer all demand queries truthfully. We also note that in some cases (but not always!) the incentives issues can be handled orthogonally to the preference elicitation issues, e.g., by using Vickrey-Clarke-Groves (VCG) prices (e.g., [4, 34]). 2 We further discuss this result in Section 5.3. Iterative auctions Demand auctions Item-price auctions Anonymous price auctions Ascending auctions 1 2 3 4 5 6 97 8 10 Figure 1: The diagram classifies the following auctions according to their properties: (1) The adaptation [12] for Kelso & Crawford"s [22] auction. (2) The Proxy Auction [3] by Ausubel & Milgrom. (3) iBundle(3) by Parkes & Ungar [34]. (4) iBundle(2) by Parkes & Ungar [37]. (5) Our descending adaptation for the 2-approximation for submodular valuations by [25] (see Subsection 5.4). (6) Ausubel"s [4] auction for substitutes valuations. (7) The adaptation by Nisan & Segal [32] of the O( √ m) approximation by [26]. (8) The duplicate-item auction by [5]. (9) Auction for Read-Once formulae by [43]. (10) The AkBA Auction by Wurman & Wellman [42]. payments and in particular, on reaching "Walrasian equilibria" or "Competitive equilibria". A Walrasian equilibrium3 is known to exist in the case of Substitutes valuations, and is known to be impossible for any wider class of valuations [16]. This does not rule out other allocations by ascending auctions: in this paper we view the auctions as a computational process where the outcome - both the allocation and the payments - can be determined according to all the data elicited throughout the auction; This general framework strengthens our negative results.4 We find the study of ascending auctions appealing for various reasons. First, ascending auctions are widely used in many real-life settings from the FCC spectrum auctions [15] to almost any e-commerce website (e.g., [2, 1]). Actually, this is maybe the most straightforward way to sell items: ask the bidders what would they like to buy under certain prices, and increase the prices of over-demanded goods. Ascending auctions are also considered more intuitive for many bidders, and are believed to increase the trust of the bidders in the auctioneer, as they see the result gradually emerging from the bidders" responses. Ascending auctions also have other desired economic properties, e.g., they incur smaller information revelation (consider, for example, English auctions vs. second-price sealed bid auctions). 1.1 Extant Work Many iterative combinatorial auction mechanisms rely on demand queries (see the survey in [35]). Figure 1 summa3 A Walrasian equilibrium is vector of item prices for which all the items are sold when each bidder receives a bundle in his demand set. 4 In few recent auction designs (e.g., [4, 28]) the payments are not necessarily the final prices of the auctions. 30 Valuation family Upper bound Reference Lower bound Reference General min(n, O( √ m)) [26], Section 4.2 min(n, m1/2− ) [32] Substitutes 1 [32] Submodular 2 [25], 1+ 1 2m , 1-1 e (*) [32],[23] Subadditive O(logm) [13] 2 [13] k-duplicates O(m1/k+1 ) [14] O(m1/k+1 ) [14] Procurement ln m [32] (log m)/2 [29, 32] Figure 2: The best approximation factors currently achievable by computationally-efficient combinatorial auctions, for several classes of valuations. All lower bounds in the table apply to all iterative auctions (except the one marked by *); all upper bounds in the table are achieved with item-price demand queries. rizes the basic classes of auctions implied by combinations of the above properties and classifies some of the auctions proposed in the literature according to this classification. For our purposes, two families of these auctions serve as the main motivating starting points: the first is the ascending item-price auctions of [22, 17] that with computational efficiency find an optimal allocation among (gross) substitutes valuations, and the second is the ascending bundleprice auctions of [37, 3] that find an optimal allocation among general valuations - but not necessarily with computational efficiency. The main lower bound in this area, due to [32], states that indeed, due to inherent communication requirements, it is not possible for any iterative auction to find the optimal allocation among general valuations with sub-exponentially many queries. A similar exponential lower bound was shown in [32] also for even approximating the optimal allocation to within a factor of m1/2− . Several lower bounds and upper bounds for approximation are known for some natural classes of valuations - these are summarized in Figure 2. In [32], the universal generality of demand queries is also shown: any non-deterministic communication protocol for finding an allocation that optimizes the social welfare can be converted into one that only uses demand queries (with bundle prices). In [41] this was generalized also to nondeterministic protocols for finding allocations that satisfy other natural types of economic requirements (e.g., approximate social efficiency, envy-freeness). However, in [33] it was demonstrated that this completeness of demand queries holds only in the nondeterministic setting, while in the usual deterministic setting, demand queries (even with bundle prices) may be exponentially weaker than general communication. Bundle-price auctions are a generalization of (the more natural and intuitive) item-price auctions. It is known that indeed item-price auctions may be exponentially weaker: a nice example is the case of valuations that are a XOR of k bundles5 , where k is small (say, polynomial). Lahaie and Parkes [24] show an economically-efficient bundle-price auction that uses a polynomial number of queries whenever k is polynomial. In contrast, [7] show that there exist valuations that are XORs of k = √ m bundles such that any item-price auction that finds an optimal allocation between them requires exponentially many queries. These results are part of a recent line of research ([7, 43, 24, 40]) that study the preference elicitation problem in combinatorial auctions and its relation to the full elicitation problem (i.e., learn5 These are valuations where bidders have values for k specific packages, and the value of each bundle is the maximal value of one of these packages that it contains. ing the exact valuations of the bidders). These papers adapt methods from machine-learning theory to the combinatorialauction setting. The preference elicitation problem and the full elicitation problem relate to a well studied problem in microeconomics known as the integrability problem (see, e.g., [27]). This problem studies if and when one can derive the utility function of a consumer from her demand function. Paper organization: Due to the relatively large number of results we present, we start with a survey of our new results in Section 2. After describing our formal model in Section 3, we present our results concerning the power of demand queries in Section 4. Then, we describe the power of item-price ascending auctions (Section 5) and bundle-price ascending auctions (Section 6). Readers who are mainly interested in the self-contained discussion of ascending auctions can skip Section 4. Missing proofs from Section 4 can be found in part I of the full paper ([8]). Missing proofs from Sections 5 and 6 can be found in part II of the full paper ([9]). 2. A SURVEY OF OUR RESULTS Our systematic analysis is composed of the combination of a rather large number of results characterizing the power and limitations of various classes of auctions. In this section, we will present an exposition describing our new results. We first discuss the power of demand-query iterative auctions, and then we turn our attention to ascending auctions. Figure 3 summarizes some of our main results. 2.1 Demand Queries Comparison of query types We first ask what other natural types of queries could we imagine iterative auctions using? Here is a list of such queries that are either natural, have been used in the literature, or that we found useful. 1. Value query: The auctioneer presents a bundle S, the bidder reports his value v(S) for this bundle. 2. Marginal-value query: The auctioneer presents a bundle A and an item j, the bidder reports how much he is willing to pay for j, given that he already owns A, i.e., v(j|A) = v(A ∪ {j}) − v(A). 3. Demand query (with item prices): The auctioneer presents a vector of item prices p1...pm; the bidder reports his demand under these prices, i.e., some set S that maximizes v(S) − P i∈S pi.6 6 A tie breaking rule should be specified. All of our results 31 Communication Constraint Can find an optimal allocation? Upper bound for welfare approx. Lower bound for welfare approx. Item-Price Demand Queries Yes 1 1 Poly. Communication No [32] min(n, O(m1/2 )) [26] min(n, m1/2− ) [32] Poly. Item-Price Demand Queries No [32] min(n, O(m1/2 )) min(n, m1/2− ) [32] Poly. Value Queries No [32] O( m√ log m ) [19] O( m log m ) Anonymous Item-Price AA No - min(O(n), O(m1/2− )) Non-anonymous Item-Price AA No -Anonymous Bundle-Price AA No - min(O(n), O(m1/2− )) Non-anonymous Bundle-Price AA Yes [37] 1 1 Poly Number of Item-Price AA No min(n, O(m1/2 ))Figure 3: This paper studies the economic efficiency of auctions that follow certain communication constraints. For each class of auctions, the table shows whether the optimal allocation can be achieved, or else, how well can it be approximated (both upper bounds and lower bounds). New results are highlighted. Abbreviations: Poly. (Polynomial number/size), AA (ascending auctions). - means that nothing is currently known except trivial solutions. 4. Indirect-utility query: The auctioneer presents a set of item prices p1...pm, and the bidder responds with his indirect-utility under these prices, that is, the highest utility he can achieve from a bundle under these prices: maxS⊆M (v(S) − P i∈S pi).7 5. Relative-demand query: the auctioneer presents a set of non-zero prices p1...pm, and the bidder reports the bundle that maximizes his value per unit of money, i.e., some set that maximizes v(S)P i∈S pi .8 Theorem: Each of these queries can be efficiently (i.e., in time polynomial in n, m, and the number of bits of precision t needed to represent a single value vi(S)) simulated by a sequence of demand queries with item prices. In particular this shows that demand queries can elicit all information about a valuation by simulating all 2m −1 value queries. We also observe that value queries and marginalvalue queries can simulate each other in polynomial time and that demand queries and indirect-utility queries can also simulate each other in polynomial time. We prove that exponentially many value queries may be needed in order to simulate a single demand query. It is interesting to note that for the restricted class of substitutes valuations, demand queries may be simulated by polynomial number of value queries [6]. Welfare approximation The next question that we ask is how well can a computationally-efficient auction that uses only demand queries approximate the optimal allocation? Two separate obstacles are known: In [32], a lower bound of min(n, m1/2− ), for any fixed > 0, was shown for the approximation factor apply for any fixed tie breaking rule. 7 This is exactly the utility achieved by the bundle which would be returned in a demand query with the same prices. This notion relates to the Indirect-utility function studied in the Microeconomic literature (see, e.g., [27]). 8 Note that when all the prices are 1, the bidder actually reports the bundle with the highest per-item price. We found this type of query useful, for example, in the design of the approximation algorithm described in Figure 5 in Section 4.2. obtained using any polynomial amount of communication. A computational bound with the same value applies even for the case of single-minded bidders, but under the assumption of NP = ZPP [39]. As noted in [32], the computationallyefficient greedy algorithm of [26] can be adapted to become a polynomial-time iterative auction that achieves a nearly matching approximation factor of min(n, O( √ m)). This iterative auction may be implemented with bundle-price demand queries but, as far as we see, not as one with item prices. Since in a single bundle-price demand query an exponential number of prices can be presented, this algorithm can have an exponential communication cost. In Section 4.2, we describe a different item-price auction that achieves the same approximation factor with a polynomial number of queries (and thus with a polynomial communication). Theorem: There exists a computationally-efficient iterative auction with item-price demand queries that finds an allocation that approximates the optimal welfare between arbitrary valuations to within a factor of min(n, O( √ m)). One may then attempt obtaining such an approximation factor using iterative auctions that use only the weaker value queries. However, we show that this is impossible: Theorem: Any iterative auction that uses a polynomial (in n and m) number of value queries can not achieve an approximation factor that is better than O( m log m ).9 Note however that auctions with only value queries are not completely trivial in power: the bundling auctions of Holzman et al. [19] can easily be implemented by a polynomial number of value queries and can achieve an approximation factor of O( m√ log m ) by using O(log m) equi-sized bundles. We do not know how to close the (tiny) gap between this upper bound and our lower bound. Representing bundle-prices We then deal with a critical issue with bundle-price auctions that was side-stepped by our model, as well as by all previous works that used bundle-price auctions: how are 9 This was also proven independently by Shahar Dobzinski and Michael Schapira. 32 the bundle prices represented? For item-price auctions this is not an issue since a query needs only to specify a small number, m, of prices. In bundle-price auctions that situation is more difficult since there are exponentially many bundles that require pricing. Our basic model (like all previous work that used bundle prices, e.g., [37, 34, 3]), ignores this issue, and only requires that the prices be determined, somehow, by the protocol. A finer model would fix a specific language for denoting bundle prices, force the protocol to represent the bundle-prices in this language, and require that the representations of the bundle-prices also be polynomial. What could such a language for denoting prices for all bundles look like? First note that specifying a price for each bundle is equivalent to specifying a valuation. Second, as noted in [31], most of the proposed bidding languages are really just languages for representing valuations, i.e., a syntactic representation of valuations - thus we could use any of them. This point of view opens up the general issue of which language should be used in bundle-price auctions and what are the implications of this choice. Here we initiate this line of investigation. We consider bundle-price auctions where the prices must be given as a XOR-bid, i.e., the protocol must explicitly indicate the price of every bundle whose value is different than that of all of its proper subsets. Note that all bundle-price auctions that do not explicitly specify a bidding language must implicitly use this language or a weaker one, since without a specific language one would need to list prices for all bundles, perhaps except for trivial ones (those with value 0, or more generally, those with a value that is determined by one of their proper subsets.) We show that once the representation length of bundle prices is taken into account (using the XOR-language), bundle-price auctions are no more strictly stronger than item-price auctions. Define the cost of an iterative auction as the total length of the queries and answers used throughout the auction (in the worst case). Theorem: For some class of valuations, bundle price auctions that use the XOR-language require an exponential cost for finding the optimal allocation. In contrast, item-price auctions can find the optimal allocation for this class within polynomial cost.10 This put doubts on the applicability of bundle-price auctions like [3, 37], and it may justify the use of hybrid pricing methods such as Ausubel, Cramton and Milgrom"s Clock-Proxy auction ([10]). Demand queries and linear programs The winner determination problem in combinatorial auctions may be formulated as an integer program. In many cases solving the linear-program relaxation of this integer program is useful: for some restricted classes of valuations it finds the optimum of the integer program (e.g., substitute valuations [22, 17]) or helps approximating the optimum (e.g., by randomized rounding [13, 14]). However, the linear program has an exponential number of variables. Nisan and Segal [32] observed the surprising fact that despite the ex10 Our proof relies on the sophisticated known lower bounds for constant depth circuits. We were not able to find an elementary proof. ponential number of variables, this linear program may be solved within polynomial communication. The basic idea is to solve the dual program using the Ellipsoid method (see, e.g., [20]). The dual program has a polynomial number of variables, but an exponential number of constraints. The Ellipsoid algorithm runs in polynomial time even on such programs, provided that a separation oracle is given for the set of constraints. Surprisingly, such a separation oracle can be implemented using a single demand query (with item prices) to each of the bidders. The treatment of [32] was somewhat ad-hoc to the problem at hand (the case of substitute valuations). Here we give a somewhat more general form of this important observation. Let us call the following class of linear programs generalized-winner-determination-relaxation (GWDR) LPs: Maximize X i∈N,S⊆M wi xi,S vi(S) s.t. X i∈N, S|j∈S xi,S ≤ qj ∀j ∈ M X S⊆M xi,S ≤ di ∀i ∈ N xi,S ≥ 0 ∀i ∈ N, S ⊆ M The case where wi = 1, di = 1, qj = 1 (for every i, j) is the usual linear relaxation of the winner determination problem. More generally, wi may be viewed as the weight given to bidder i"s welfare, qj may be viewed as the quantity of units of good j, and di may be viewed as duplicity of the number of bidders of type i. Theorem: Any GWDR linear program may be solved in polynomial time (in n, m, and the number of bits of precision t) using only demand queries with item prices.11 2.2 Ascending Auctions Ascending item-price auctions: It is well known that the item-price ascending auctions of Kelso and Crawford [22] and its variants [12, 16] find the optimal allocation as long as all players" valuations have the substitutes property. The obvious question is whether the optimal allocation can be found for a larger class of valuations. Our main result here is a strong negative result: Theorem: There is a 2-item 2-player problem where no ascending item-price auction can find the optimal allocation. This is in contrast to both the power of bundle-price ascending auctions and to the power of general item-price demand queries (see above), both of which can always find the optimal allocation and in fact even provide full preference elicitation. The same proof proves a similar impossibility result for other types of auctions (e.g., descending auctions, non-anonymous auctions). More extension of this result: • Eliciting some classes of valuations requires an exponential number of ascending item-price trajectories. 11 The produced optimal solution will have polynomial support and thus can be listed fully. 33 • At least k − 1 ascending item-price trajectories are needed to elicit XOR formulae with k terms. This result is in some sense tight, since we show that any k-term XOR formula can be fully elicited by k−1 nondeterministic (i.e., when some exogenous teacher instructs the auctioneer on how to increase the prices) ascending auctions.12 We also show that item-price ascending auctions and iterative auctions that are limited to a polynomial number of queries (of any kind, not necessarily ascending) are incomparable in their power: ascending auctions, with small enough increments, can elicit the preferences in cases where any polynomial number of queries cannot. Motivated by several recent papers that studied the relation between eliciting and fully-eliciting the preferences in combinatorial auctions (e.g., [7, 24]), we explore the difference between these problems in the context of ascending auctions. We show that although a single ascending auction can determine the optimal allocation among any number of bidders with substitutes valuations, it cannot fully-elicit such a valuation even for a single bidder. While it was shown in [25] that the set of substitutes valuations has measure zero in the space of general valuations, its dimension is not known, and in particular it is still open whether a polynomial amount of information suffices to describe a substitutes valuation. While our result may be a small step in that direction (a polynomial full elicitation may still be possible with other communication protocols), we note that our impossibility result also holds for valuations in the class OXS defined by [25], valuations that we are able to show have a compact representation. We also give several results separating the power of different models for ascending combinatorial auctions that use item-prices: we prove, not surprisingly, that adaptive ascending auctions are more powerful than oblivious ascending auctions and that non-deterministic ascending auctions are more powerful than deterministic ascending auctions. We also compare different kinds of non-anonymous auctions (e.g., simultaneous or sequential), and observe that anonymous bundle-price auctions and non-anonymous item-price auctions are incomparable in their power. Finally, motivated by Dutch auctions, we consider descending auctions, and how they compare to ascending ones; we show classes of valuations that can be elicited by ascending item-price auctions but not by descending item-price auctions, and vice versa. Ascending bundle-price auctions: All known ascending bundle-price auctions that are able to find the optimal allocation between general valuations (with free disposal) use non-anonymous prices. Anonymous ascending-price auctions (e.g., [42, 21, 37]) are only known to be able to find the optimal allocation among superadditive valuations or few other simple classes ([36]). We show that this is no mistake: Theorem: No ascending auction with anonymous prices can find the optimal allocation between general valuations. 12 Non-deterministic computation is widely used in CS and also in economics (e.g, a Walrasian equilibrium or [38]). In some settings, deterministic and non-deterministic models have equal power (e.g., computation with finite automata). This bound is regardless of the running time, and it also holds for descending auctions and non-deterministic auctions. We strengthen this result significantly by showing that anonymous ascending auctions cannot produce a better than O( √ m) approximation - the approximation ratio that can be achieved with a polynomial number of queries ([26, 32]) and, as mentioned, with a polynomial number of item-price demand queries. The same lower bound clearly holds for anonymous item-price ascending auctions since such auctions can be simulated by anonymous bundle-price ascending auctions. We currently do not have any lower bound on the approximation achievable by non-anonymous item-price ascending auctions. Finally, we study the performance of the existing computationally-efficient ascending auctions. These protocols ([37, 3]) require exponential time in the worst case, and this is unavoidable as shown by [32]. However, we also observe that these auctions, as well as the whole class of similar ascending bundle-price auctions, require an exponential time even for simple additive valuations. This is avoidable and indeed the ascending item-price auctions of [22] can find the optimal allocation for these simple valuations with polynomial communication. 3. THE MODEL 3.1 Discrete Auctions for Continuous Values Our model aims to capture iterative auctions that operate on real-valued valuations. There is a slight technical difficulty here in bridging the gap between the discrete nature of an iterative auction, and the continuous nature of the valuations. This is exactly the same problem as in modeling a simple English auction. There are three standard formal ways to model it: 1. Model the auction as a continuous process and study its trajectory in time. For example, the so-called Japanese auction is basically a continuous model of an English model.13 2. Model the auction as discrete and the valuations as continuously valued. In this case we introduce a parameter and usually require the auction to produce results that are -close to optimal. 3. Model the valuations as discrete. In this case we will assume that all valuations are integer multiples of some small fixed quantity δ, e.g., 1 penny. All communication in this case is then naturally finite. In this paper we use the latter formulation and assume that all values are multiples of some δ. Thus, in some parts of the paper we assume without loss of generality that δ = 1, hence all valuations are integral. Almost all (if not all) of our results can be translated to the other two models with little effort. 3.2 Valuations A single auctioneer is selling m indivisible non-homogeneous items in a single auction, and let M be the set of these 13 Another similar model is the moving knives model in the cake-cutting literature. 34 items and N be the set of bidders. Each one of the n bidders in the auction has a valuation function vi : 2m → {0, δ, 2δ, ..., L}, where for every bundle of items S ⊆ M, vi(S) denotes the value of bidder i for the bundle S and is a multiple of δ in the range 0...L. We will sometimes denote the number of bits needed to represent such values in the range 0...L by t = log L. We assume free disposal, i.e., S ⊆ T implies vi(S) ≤ vi(T) and that vi(∅) = 0 for all bidders. We will mention the following classes of valuations: • A valuation is called sub-modular if for all sets of items A and B we have that v(A ∪ B) + v(A ∩ B) ≤ v(A) + v(B). • A valuation is called super-additive if for all disjoint sets of items A and B we have that v(A∪B) ≥ v(A)+ v(B). • A valuation is called a k-bundle XOR if it can be represented as a XOR combination of at most k atomic bids [30], i.e., if there are at most k bundles Si and prices pi such that for all S, v(S) = maxi|S⊇Si pi. Such valuations will be denoted by v = (S1 : p1) ⊕ (S2 : p2) ⊕ . . . ⊕ (Sk : pk).14 3.3 Iterative Auctions The auctioneer sets up a protocol (equivalently an algorithm), where at each stage of the protocol some information q - termed the query - is sent to some bidder i, and then bidder i replies with some reply that depends on the query as well as on his own valuation. In this paper, we assume that we have complete control over the bidders" behavior, and thus the protocol also defines a reply function ri(q, vi) that specifies bidder i"s reply to query q. The protocol may be adaptive: the query value as well as the queried bidder may depend on the replies received for past queries. At the end of the protocol, an allocation S1...Sn must be declared, where Si ∩ Sj = ∅ for i = j. We say that the auction finds an optimal allocation if it finds the allocation that maximizes the social welfareP i vi(Si). We say that it finds a c-approximation if P i vi(Si) ≥ P i vi(Ti)/c where T1...Tn is an optimal allocation. The running time of the auction on a given instance of the bidders" valuations is the total number of queries made on this instance. The running time of a protocol is the worst case cost over all instances. Note that we impose no computational limitations on the protocol or on the players.15 This of course only strengthens our hardness results. Yet, our positive results will not use this power and will be efficient also in the usual computational sense. Our goal will be to design computationally-efficient protocols. We will deem a protocol computationally-efficient if its cost is polynomial in the relevant parameters: the number of bidders n, the number of items m, and t = log L, where L is the largest possible value of a bundle. However, when we discuss ascending-price auctions and their variants, a computationally-efficient protocol will be required to be 14 For example, v = (abcd : 5) ⊕ (ab : 3) ⊕ (c : 4) denotes the XOR valuation with the terms abcd, ab, c and prices 5, 3, 4 respectively. For this valuation, v(abcd) = 5, v(abd) = 3, v(abc) = 4. 15 The running time really measures communication costs and not computational running time. pseudo-polynomial, i.e., it should ask a number of queries which is polynomial in m, n and L δ . This is because that ascending auctions can usually not achieve such running times (consider even the English auction on a single item).16 Note that all of our results give concrete bounds, where the dependence on the parameters is given explicitly; we use the standard big-Oh notation just as a shorthand. We say than an auction elicits some class V of valuations, if it determines the optimal allocation for any profile of valuations drawn from V ; We say that an auction fully elicits some class of valuations V , if it can fully learn any single valuation v ∈ V (i.e., learn v(S) for every S). 3.4 Demand Queries and Ascending Auctions Most of the paper will be concerned with a common special case of iterative auctions that we term demand auctions. In such auctions, the queries that are sent to bidders are demand queries: the query specifies a price p(S) ∈ + for each bundle S. The reply of bidder i is simply the set most desired - demanded - under these prices. Formally, a set S that maximizes vi(S) − p(S). It may happen that more than one set S maximizes this value. In which case, ties are broken according to some fixed tie-breaking rule, e.g., the lexicographically first such set is returned. All of our results hold for any fixed tie-breaking rule. Ascending auctions are iterative auctions with non-decreasing prices: Definition 1. In an ascending auction, the prices in the queries to the same bidder can only increase in time. Formally, let p be a query made for bidder i, and q be a query made for bidder i at a later stage in the protocol. Then for all sets S, q(S) ≥ p(S). A similar variant, which we also study and that is also common in real life, is descending auctions, in which prices can only decrease in time. Note that the term ascending auction refers to an auction with a single ascending trajectory of prices. It may be useful to define multi-trajectory ascending auctions, in which the prices maybe reset to zero a number of times (see, e.g., [4]). We consider two main restrictions on the types of allowed demand queries: Definition 2. Item Prices: The prices in each query are given by prices pj for each item j. The price of a set S is additive: p(S) = P j∈S pj. Definition 3. Anonymous prices: The prices seen by the bidders at any stage in the auction are the same, i.e. whenever a query is made to some bidder, the same query is also made to all other bidders (with the prices unchanged). In auctions with non-anonymous (discriminatory) prices, each bidder i has personalized prices denoted by pi (S).17 In this paper, all auctions are anonymous unless otherwise specified. Note that even though in our model valuations are integral (or multiples of some δ), we allow the demand query to 16 Most of the auctions we present may be adapted to run in time polynomial in log L, using a binary-search-like procedure, losing their ascending nature. 17 Note that a non-anonymous auction can clearly be simulated by n parallel anonymous auctions. 35 use arbitrary real numbers in +. That is, we assume that the increment we use in the ascending auctions may be significantly smaller than δ. All our hardness results hold for any , even for continuous price increments. A practical issue here is how will the query be specified: in the general case, an exponential number of prices needs to be sent in a single query. Formally, this is not a problem as the model does not limit the length of queries in any way - the protocol must simply define what the prices are in terms of the replies received for previous queries. We look into this issue further in Section 4.3. 4. THE POWER OF DEMAND QUERIES In this section, we study the power of iterative auctions that use demand queries (not necessarily ascending). We start by comapring demand queries to other types of queries. Then, we discuss how well can one approximate the optimal welfare using a polynomial number of demand queries. We also initiate the study of the representation of bundle-price demand queries, and finally, we show how demand queries help solving the linear-programming relaxation of combinatorial auctions in polynomial time. 4.1 The Power of Different Types of Queries In this section we compare the power of the various types of queries defined in Section 2. We will present computationally -efficient simulations of these query types using item-price demand queries. In Section 5.1 we show that these simulations can also be done using item-price ascending auctions. Lemma 4.1. A value query can be simulated by m marginalvalue queries. A marginal-value query can be simulated by two value queries. Lemma 4.2. A value query can be simulated by mt demand queries (where t = log L is the number of bits needed to represent a single bundle value).18 As a direct corollary we get that demand auctions can always fully elicit the bidders" valuations by simulating all possible 2m − 1 queries and thus elicit enough information for determining the optimal allocation. Note, however, that this elicitation may be computationally inefficient. The next lemma shows that demand queries can be exponentially more powerful than value queries. Lemma 4.3. An exponential number of value queries may be required for simulating a single demand query. Indirect utility queries are, however, equivalent in power to demand queries: Lemma 4.4. An indirect-utility query can be simulated by mt + 1 demand queries. A demand query can be simulated by m + 1 indirect-utility queries. Demand queries can also simulate relative-demand queries:19 18 Note that t bundle-price demand queries can easily simulate a value query by setting the prices of all the bundles except S (the bundle with the unknown value) to be L, and performing a binary search on the price of S. 19 Note: although in our model values are integral (our multiples of δ), we allow the query prices to be arbitrary real numV MV D IU RD V 1 2 exp exp exp MV m 1 exp exp exp D mt poly 1 mt+1 poly IU 1 2 m+1 1 poly RD - - - - 1 Figure 4: Each entry in the table specifies how many queries of this row are needed to simulate a query from the relevant column. Abbreviations: V (value query), MV (marginal-value query), D (demand query), IU (Indirect-utility query), RD (relative demand query). Lemma 4.5. Relative-demand queries can be simulated by a polynomial number of demand queries. According to our definition of relative-demand queries, they clearly cannot simulate even value queries. Figure 4 summarizes the relations between these query types. 4.2 Approximating the Social Welfare with Value and Demand Queries We know from [32] that iterative combinatorial auctions that only use a polynomial number of queries can not find an optimal allocation among general valuations and in fact can not even approximate it to within a factor better than min{n, m1/2− }. In this section we ask how well can this approximation be done using demand queries with item prices, or using the weaker value queries. We show that, using demand queries, the lower bound can be matched, while value queries can only do much worse. Figure 5 describes a polynomial-time algorithm that achieves a min(n, O( √ m)) approximation ratio. This algorithm greedily picks the bundles that maximize the bidders" per-item value (using relative-demand queries, see Section 4.1). As a final step, it allocates all the items to a single bidder if it improves the social welfare (this can be checked using value queries). Since both value queries and relative-demand queries can be simulated by a polynomial number of demand queries with item prices (Lemmas 4.2 and 4.5), this algorithm can be implemented by a polynomial number of demand queries with item prices.20 Theorem 4.6. The auction described in Figure 5 can be implemented by a polynomial number of demand queries and achieves a min{n, 4 √ m}-approximation for the social welfare. We now ask how well can the optimal welfare be approximated by a polynomial number of value queries. First we note that value queries are not completely powerless: In [19] it is shown that if the m items are split into k fixed bundles of size m/k each, and these fixed bundles are auctioned as though each was indivisible, then the social welfare bers, thus we may have bundles with arbitrarily close relative demands. In this sense the simulation above is only up to any given (and the number of queries is O(log L+log 1 )). When the relative-demand query prices are given as rational numbers, exact simulation is implied when log is linear in the input length. 20 In the full paper [8], we observe that this algorithm can be implemented by two descending item-price auctions (where we allow removing items along the auction). 36 generated by such an auction is at least m√ k -approximation of that possible in the original auction. Notice that such an auction can be implemented by 2k − 1 value queries to each bidder - querying the value of each bundle of the fixed bundles. Thus, if we choose k = log m bundles we get an m√ log m -approximation while still using a polynomial number of queries. The following lemma shows that not much more is possible using value queries: Lemma 4.7. Any iterative auction that uses only value queries and distinguishes between k-tuples of 0/1 valuations where the optimal allocation has value 1, and those where the optimal allocation has value k requires at least 2 m k queries. Proof. Consider the following family of valuations: for every S, such that |S| > m/2, v(S) = 1, and there exists a single set T, such that for |S| ≤ m/2, v(S) = 1 iff T ⊆ S and v(S) = 0 otherwise. Now look at the behavior of the protocol when all valuations vi have T = {1...m}. Clearly in this case the value of the best allocation is 1 since no set of size m 2 or lower has non-zero value for any player. Fix the sequence of queries and answers received on this k-tuple of valuations. Now consider the k-tuple of valuations chosen at random as follows: a partition of the m items into k sets T1...Tk each of size m k each is chosen uniformly at random among all such partitions. Now consider the k-tuple of valuations from our family that correspond to this partition - clearly Ti can be allocated to i, for each i, getting a total value of k. Now look at the protocol when running on these valuations and compare its behavior to the original case. Note that the answer to a query S to player i differs between the case of Ti and the original case of T = {1...m} only if |S| ≤ m 2 and Ti ⊆ S. Since Ti is distributed uniformly among all sets of size exactly m k , we have that for any fixed query S to player i, where |S| ≤ m 2 , Pr[Ti ⊆ S] ≤ „ |S| m «|Ti| ≤ 2− m k Using the union-bound, if the original sequence of queries was of length less than 2 m k , then with positive probability none of the queries in the sequence would receive a different answer than for the original input tuple. This is forbidden since the protocol must distinguish between this case and the original case - which cannot happen if all queries receive the same answer. Hence there must have been at least 2 m k queries for the original tuple of valuations. We conclude that a polynomial time protocol that uses only value queries cannot obtain a better than O( m log m ) approximation of the welfare: Theorem 4.8. An iterative auction that uses a polynomial number of value queries cannot achieve better than O( m log m )-approximation for the social welfare. Proof. Immediate from Lemma 4.7: achieving any approximation ratio k which is asymptotically greater than m log m needs an exponential number of value queries. An Approximation Algorithm: Initialization: Let T ← M be the current items for sale. Let K ← N be the currently participating bidders. Let S∗ 1 ← ∅, ..., S∗ n ← ∅ be the provisional allocation. Repeat until T = ∅ or K = ∅: Ask each bidder i in K for the bundle Si that maximizes her per-item value, i.e., Si ∈ argmaxS⊆T vi(S) |S| . Let i be the bidder with the maximal per-item value, i.e., i ∈ argmaxi∈K vi(Si) |Si| . Set: s∗ i = si, K = K \ i, M = M \ Si Finally: Ask the bidders for their values vi(M) for the grand bundle. If allocating all the items to some bidder i improves the social welfare achieved so far (i.e., ∃i ∈ N such that vi(M) > P i∈N vi(S∗ i )), then allocate all items to this bidder i. Figure 5: This algorithm achieves a min{n, 4 √ m}approximation for the social welfare, which is asymptotically the best worst-case approximation possible with polynomial communication. This algorithm can be implemented with a polynomial number of demand queries. 4.3 The Representation of Bundle Prices In this section we explicitly fix the language in which bundle prices are presented to the bidders in bundle-price auctions. This language requires the algorithm to explicitly list the price of every bundle with a non-trivial price. Trivial in this context is a price that is equal to that of one of its proper subsets (which was listed explicitly). This representation is equivalent to the XOR-language for expressing valuations. Formally, each query q is given by an expression: q = (S1 : p1) ⊕ (S2 : p2) ⊕ ... ⊕ (Sl : pl). In this representation, the price demanded for every set S is simply p(S) = max{k=1...l|Sk⊆S}pk. Definition 4. The length of the query q = (S1 : p1) ⊕ (S2 : p2) ⊕ ... ⊕ (Sl : pl) is l. The cost of an algorithm is the sum of the lengths of the queries asked during the operation of the algorithm on the worst case input. Note that under this definition, bundle-price auctions are not necessarily stronger than item-price auctions. An itemprice query that prices each item for 1, is translated to an exponentially long bundle-price query that needs to specify the price |S| for each bundle S. But perhaps bundle-price auctions can still find optimal allocations whenever itemprice auction can, without directly simulating such queries? We show that this is not the case: indeed, when the representation length is taken into account, bundle price auctions are sometimes seriously inferior to item price auctions. Consider the following family of valuations: Each item is valued at 3, except that for some single set S, its value is a bit more: 3|S| + b, where b ∈ {0, 1, 2}. Note that an item price auction can easily find the optimal allocation between any two such valuations: Set the prices of each item to 3+ ; if the demand sets of the two players are both empty, then b = 0 for both valuations, and an arbitrary allocation is fine. If one of them is empty and the other non-empty, allocate the non-empty demand set to its bidder, and the rest to the other. If both demand sets are non-empty then, unless they form an exact partition, we need to see which b is larger, which we can do by increasing the price of a single item in each demand set. 37 We will show that any bundle-price auction that uses only the XOR-language to describe bundle prices requires an exponential cost (which includes the sum of all description lengths of prices) to find an optimal allocation between any two such valuations. Lemma 4.9. Every bundle-price auction that uses XORexpressions to denote bundle prices requires 2Ω( √ m) cost in order to find the optimal allocation among two valuations from the above family. The complication in the proof stems from the fact that using XOR-expressions, the length of the price description depends on the number of bundles whose price is strictly larger than each of their subsets - this may be significantly smaller than the number of bundles that have a non-zero price. (The proof becomes easy if we require the protocol to explicitly name every bundle with non-zero price.) We do not know of any elementary proof for this lemma (although we believe that one can be found). Instead we reduce the problem to a well known lower bound in boolean circuit complexity [18] stating that boolean circuits of depth 3 that compute the majority function on m variables require 2Ω( √ m) size. 4.4 Demand Queries and Linear Programming Consider the following linear-programming relaxation for the generalized winner-determination problem in combinatorial auctions (the primal program): Maximize X i∈N,S⊆M wi xi,S vi(S) s.t. X i∈N, S|j∈S xi,S ≤ qj ∀j ∈ M X S⊆M xi,S ≤ di ∀i ∈ N xi,S ≥ 0 ∀i ∈ N, S ⊆ M Note that the primal program has an exponential number of variables. Yet, we will be able to solve it in polynomial time using demand queries to the bidders. The solution will have a polynomial size support (non-zero values for xi,S), and thus we will be able to describe it in polynomial time. Here is its dual: Minimize X j∈M qjpj + X i∈N diui s.t. ui + X j∈S pj ≥ wivi(S) ∀i ∈ N, S ⊆ M pi ≥ 0, uj ≥ 0 ∀i ∈ M, j ∈ N Notice that the dual problem has exactly n + m variables but an exponential number of constraints. Thus, the dual can be solved using the Ellipsoid method in polynomial time - if a separation oracle can be implemented in polynomial time. Recall that a separation oracle, when given a possible solution, either confirms that it is a feasible solution, or responds with a constraint that is violated by the possible solution. We construct a separation oracle for solving the dual program, using a single demand query to each of the bidders. Consider a possible solution (u, p) for the dual program. We can re-write the constraints of the dual program as: ui/wi ≥ vi(S) − X j∈S pj/wi Now a demand query to bidder i with prices pj/wi reveals exactly the set S that maximizes the RHS of the previous inequality. Thus, in order to check whether (u, p) is feasible it suffices to (1) query each bidder i for his demand Di under the prices pj/wi; (2) check only the n constraints ui + P j∈Di pj ≥ wivi(Di) (where vi(Di) can be simulated using a polynomial sequence of demand queries as shown in Lemma 4.2). If none of these is violated then we are assured that (u, p) is feasible; otherwise we get a violated constraint. What is left to be shown is how the primal program can be solved. (Recall that the primal program has an exponential number of variables.) Since the Ellipsoid algorithm runs in polynomial time, it encounters only a polynomial number of constraints during its operation. Clearly, if all other constraints were removed from the dual program, it would still have the same solution (adding constraints can only decrease the space of feasible solutions). Now take the reduced dual where only the constraints encountered exist, and look at its dual. It will have the same solution as the original dual and hence of the original primal. However, look at the form of this dual of the reduced dual. It is just a version of the primal program with a polynomial number of variables - those corresponding to constraints that remained in the reduced dual. Thus, it can be solved in polynomial time, and this solution clearly solves the original primal program, setting all other variables to zero. 5. ITEM-PRICE ASCENDING AUCTIONS In this section we characterize the power of ascending item-price auctions. We first show that this power is not trivial: such auctions can in general elicit an exponential amount of information. On the other hand, we show that the optimal allocation cannot always be determined by a single ascending auction, and in some cases, nor by an exponential number of ascending-price trajectories. Finally, we separate the power of different models of ascending auctions. 5.1 The Power of Item-Price Ascending Auctions We first show that if small enough increments are allowed, a single ascending trajectory of item-prices can elicit preferences that cannot be elicited with polynomial communication. As mentioned, all our hardness results hold for any increment, even infinitesimal. Theorem 5.1. Some classes of valuations can be elicited by item-price ascending auctions, but cannot be elicited by a polynomial number of queries of any kind. Proof. (sketch) Consider two bidders with v(S) = 1 if |S| > n 2 , v(S) = 0 if |S| < n 2 and every S such that |S| = n 2 has an unknown value from {0, 1}. Due to [32], determining the optimal allocation here requires exponential communication in the worst case. Nevertheless, we show (see [9]) that an item-price ascending auction can do it, as long as it can use exponentially small increments. We now describe another positive result for the power of item-price ascending auctions. In section 4.1, we showed 38 v(ab) v(a) v(b) Bidder 1 2 α ∈ (0, 1) β ∈ (0, 1) Bidder 2 2 2 2 Figure 6: No item-price ascending auctions can determine the optimal allocation for this class of valuations. that a value query can be simulated with a (truly) polynomial number of item-price demand queries. Here, we show that every value query can be simulated by a (pseudo) polynomial number of ascending item-price demand queries. (In the next subsection, we show that we cannot always simulate even a pair of value queries using a single item-price ascending auction.) In the full paper (part II,[9]), we show that we can simulate other types of queries using item-price ascending auctions. Proposition 5.2. A value query can be simulated by an item-price ascending auction. This simulation requires a polynomial number of queries. Actually, the proof for Proposition 5.2 proves a stronger useful result regarding the information elicited by iterative auctions. It says that in any iterative auction in which the changes of prices are small enough in each stage (pseudocontinuous auctions), the value of all bundles demanded during the auction can be computed. The basic idea is that when the bidder moves from demanding some bundle Ti to demanding another bundle Ti+1, there is a point in which she is indifferent between these two bundles. Thus, knowing the value of some demanded bundle (e.g., the empty set) enables computing the values of all other demanded bundles. We say that an auction is pseudo-continuous, if it only uses demand queries, and in each step, the price of at most one item is changed by (for some ∈ (0, δ]) with respect to the previous query. Proposition 5.3. Consider any pseudo-continuous auction (not necessarily ascending), in which bidder i demands the empty set at least once along the auction. Then, the value of every bundle demanded by bidder i throughout the auction can be calculated at the end of the auction. 5.2 Limitations of Item-Price Ascending Auctions Although we observed that demand queries can solve any combinatorial auction problem, when the queries are restricted to be ascending, some classes of valuations cannot be elicited nor fully-elicited. An example for such class of valuations is given in Figure 6. Theorem 5.4. There are classes of valuations that cannot be elicited nor fully elicited by any item-price ascending auction. Proof. Let bidder 1 have the valuation described in the first row of Figure 6, where α and β are unknown values in (0, 1). First, we prove that this class cannot be fully elicited by a single ascending auction. Specifically, an ascending auction cannot reveal the values of both α and β. As long as pa and pb are both below 1, the bidder will always demand the whole bundle ab: her utility from ab is strictly greater than the utility from either a or b separately. For example, we show that u1(ab) > u1(a): u1(ab) = 2 − (pa + pb) = 1 − pa + 1 − pb > vA(a) − pa + 1 − pb > u1(a) Thus, in order to gain any information about α or β, the price of one of the items should become at least 1, w.l.o.g. pa ≥ 1. But then, the bundle a will not be demanded by bidder 1 throughout the auction, thus no information at all will be gained about α. Now, assume that bidder 2 is known to have the valuation described in the second row of Figure 6. The optimal allocation depends on whether α is greater than β (in bidder 1"s valuation), and we proved that an ascending auction cannot determine this. The proof of the theorem above shows that for an unknown value to be revealed, the price of one item should be greater than 1, and the other price should be smaller than 1. Therefore, in a price-monotonic trajectory of prices, only one of these values can be revealed. An immediate conclusion is that this impossibility result also holds for item-price descending auctions. Since no such trajectory exists, then the same conclusion even holds for non-deterministic itemprice auctions (in which exogenous data tells us how to increase the prices). Also note that since the hardness stems from the impossibility to fully-elicit a valuation of a single bidder, this result also holds for non-anonymous ascending item-price auctions. 5.3 Limitations of Multi-Trajectory Ascending Auctions According to Theorem 5.4, no ascending item-price auction can always elicit the preferences (we prove a similar result for bundle prices in section 6). But can two ascending trajectories do the job? Or a polynomial number of ascending trajectories? We give negative answers for such suggestions. We define a k-trajectory ascending auction as a demandquery iterative auction in which the demand queries can be partitioned to k sets of queries, where the prices published in each set only increase in time. Note that we use a general definition; It allows the trajectories to run in parallel or sequentially, and to use information elicited in some trajectories for determining the future queries in other trajectories. The power of multiple-trajectory auctions can be demonstrated by the negative result of Gul and Stacchetti [17] who showed that even for an auction among substitutes valuations, an anonymous ascending item-price auction cannot compute VCG prices for all players.21 Ausubel [4] overcame this impossibility result and designed auctions that do compute VCG prices by organizing the auction as a sequence of n + 1 ascending auctions. Here, we prove that one cannot elicit XOR valuations with k terms by less than k − 1 ascending trajectories. On the other hand, we show that an XOR formula can be fully elicited by k−1 non-deterministic ascending auctions (or by k−1 deterministic ascending auctions if the auctioneer knows the atomic bundles).22 21 A recent unpublished paper by Mishra and Parkes extends this result, and shows that non-anonymous prices with bundle-prices are necessary in order that an ascending auction will end up with a universal-competitive-equilibrium (that leads to VCG payments). 22 This result actually separates the power of deterministic 39 Proposition 5.5. XOR valuations with k terms cannot be elicited (or fully elicited) by any (k-2)-trajectory itemprice ascending auction, even when the atomic bundles are known to the elicitor. However, these valuations can be elicited (and fully elicited) by (k-1)-trajectory non-deterministic non-anonymous item-price ascending auctions. Moreover, an exponential number of trajectories is required for eliciting some classes of valuations: Proposition 5.6. Elicitation and full-elicitation of some classes of valuations cannot be done by any k-trajectory itemprice ascending auction, where k = o(2m ). Proof. (sketch) Consider the following class of valuations: For |S| < m 2 , v(S) = 0 and for |S| > m 2 , v(S) = 2; every bundle S of size m 2 has some unknown value in (0, 1). We show ([9]) that a single item-price ascending auction can reveal the value of at most one bundle of size n 2 , and therefore an exponential number of ascending trajectories is needed in order to elicit such valuations. We observe that the algorithm we presented in Section 4.2 can be implemented by a polynomial number of ascending auctions (each item-price demand query can be considered as a separate ascending auction), and therefore a min(n, 4 √ m)-approximation can be achieved by a polynomial number of ascending auctions. We do not currently have a better upper bound or any lower bound. 5.4 Separating the Various Models of Ascending Auctions Various models for ascending auctions have been suggested in the literature. In this section, we compare the power of the different models. As mentioned, all auctions are considered anonymous and deterministic, unless specified otherwise. Ascending vs. Descending Auctions: We begin the discussion of the relation between ascending auctions and descending auctions with an example. The algorithm by Lehmann, Lehmann and Nisan [25] can be implemented by a simple item-price descending auction (see the full paper for details [9]). This algorithm guarantees at least half of the optimal efficiency for submodular valuations. However, we are not familiar with any ascending auction that guarantees a similar fraction of the efficiency. This raises a more general question: can ascending auctions solve any combinatorialauction problem that is solvable using a descending auction (and vice versa)? We give negative answers to these questions. The idea behind the proofs is that the information that the auctioneer can get for free at the beginning of each type of auction is different.23 and non-deterministic iterative auctions: our proof shows that a non-deterministic iterative auction can elicit the kterm XOR valuations with a polynomial number of demand queries, and [7] show that this elicitation must take an exponential number of demand queries. 23 In ascending auctions, the auctioneer can reveal the most valuable bundle (besides M) before she starts raising the prices, thus she can use this information for adaptively choose the subsequent queries. In descending auctions, one can easily find the bundle with the highest average per-item price, keeping all other bundles with non-positive utilities, and use this information in the adaptive price change. Proposition 5.7. There are classes that cannot be elicited (fully elicited) by ascending item-price auctions, but can be elicited (resp. fully elicited) with a descending item-price auction. Proposition 5.8. There are classes that cannot be elicited (fully elicited) by item-price descending auctions, but can be elicited (resp. fully elicited) by item-price ascending auctions. Deterministic vs. Non-Deterministic Auctions: Nondeterministic ascending auctions can be viewed as auctions where some benevolent teacher that has complete information guides the auctioneer on how she should raise the prices. That is, preference elicitation can be done by a non-deterministic ascending auction, if there is some ascending trajectory that elicits enough information for determining the optimal allocation (and verifying that it is indeed optimal). We show that non-deterministic ascending auctions are more powerful than deterministic ascending auctions: Proposition 5.9. Some classes can be elicited (fully elicited) by an item-price non-deterministic ascending auction, but cannot be elicited (resp. fully elicited) by item-price deterministic ascending auctions. Anonymous vs. Non-Anonymous Auctions: As will be shown in Section 6, the power of anonymous and nonanonymous bundle-price ascending auctions differs significantly. Here, we show that a difference also exists for itemprice ascending auctions. Proposition 5.10. Some classes cannot be elicited by anonymous item-price ascending auctions, but can be elicited by a non-anonymous item-price ascending auction. Sequential vs. Simultaneous Auctions: A non-anonymous auction is called simultaneous if at each stage, the price of some item is raised by for every bidder. The auctioneer can use the information gathered until each stage, in all the personalized trajectories, to determine the next queries. A non-anonymous auction is called sequential if the auctioneer performs an auction for each bidder separately, in sequential order. The auctioneer can determine the next query based on the information gathered in the trajectories completed so far and on the history of the current trajectory. Proposition 5.11. There are classes that cannot be elicited by simultaneous non-anonymous item-price ascending auctions, but can be elicited by a sequential non-anonymous item-price ascending auction. Adaptive vs. Oblivious Auctions: If the auctioneer determines the queries regardless of the bidders" responses (i.e., the queries are predefined) we say that the auction is oblivious. Otherwise, the auction is adaptive. We prove that an adaptive behaviour of the auctioneer may be beneficial. Proposition 5.12. There are classes that cannot be elicited (fully elicited) using oblivious item-price ascending auctions, but can be elicited (resp. fully elicited) by an adaptive item-price ascending auction. 40 5.5 Preference Elicitation vs. Full Elicitation Preference elicitation and full elicitation are closely related problems. If full elicitation is easy (e.g., in polynomial time) then clearly elicitation is also easy (by a nonanonymous auction, simply by learning all the valuations separately24 ). On the other hand, there are examples where preference elicitation is considered easy but learning is hard (typically, elicitation requires smaller amount of information; some examples can be found in [7]). The tatonnement algorithms by [22, 12, 16] end up with the optimal allocation for substitutes valuations.25 We prove that we cannot fully elicit substitutes valuations (or even their sub-class of OXS valuations defined in [25]), even for a single bidder, by an item-price ascending auction (although the optimal allocation can be found by an ascending auction for any number of bidders!). Theorem 5.13. Substitute valuations cannot be fully elicited by ascending item-price auctions. Moreover, they cannot be fully elicited by any m 2 ascending trajectories (m > 3). Whether substitutes valuations have a compact representation (i.e., polynomial in the number of goods) is an important open question. As a step in this direction, we show that its sub-class of OXS valuations does have a compact representation: every OXS valuation can be represented by at most m2 values.26 Lemma 5.14. Any OXS valuation can be represented by no more than m2 values. 6. BUNDLE-PRICE ASCENDING AUCTIONS All the ascending auctions in the literature that are proved to find the optimal allocation for unrestricted valuations are non-anonymous bundle-price auctions (iBundle(3) by Parkes and Ungar [37] and the Proxy Auction by Ausubel and Milgrom [3]). Yet, several anonymous ascending auctions have been suggested (e.g., AkBA [42], [21] and iBundle(2) [37]). In this section, we prove that anonymous bundle-price ascending auctions achieve poor results in the worst-case. We also show that the family of non-anonymous bundleprice ascending auctions can run exponentially slower than simple item-price ascending auctions. 6.1 Limitations of Anonymous Bundle-Price Ascending Auctions We present a class of valuations that cannot be elicited by anonymous bundle-price ascending auctions. These valuations are described in Figure 7. The basic idea: for determining some unknown value of one bidder we must raise 24 Note that an anonymous ascending auction cannot necessarily elicit a class that can be fully elicited by an ascending auction. 25 Substitute valuations are defined, e.g., in [16]. Roughly speaking, a bidder with a substitute valuation will continue demand a certain item after the price of some other items was increased. For completeness, we present in the full paper [9] a proof for the efficiency of such auctions for substitutes valuations. 26 A unit-demand valuation is an XOR valuation in which all the atomic bundles are singletons. OXS valuations can be interpreted as an aggregation (OR) of any number of unit-demand bidders. Bid. 1 v1(ac) = 2 v1(bd) = 2 v1(cd) = α ∈ (0, 1) Bid. 2 v2(ab) = 2 v2(cd) = 2 v2(bd) = β ∈ (0, 1) Figure 7: Anonymous ascending bundle-price auctions cannot determine the optimal allocation for this class of valuations. a price of a bundle that should be demanded by the other bidder in the future. Theorem 6.1. Some classes of valuations cannot be elicited by anonymous bundle-price ascending auctions. Proof. Consider a pair of XOR valuations as described in Figure 7. For finding the optimal allocation we must know which value is greater between α and β.27 However, we cannot learn the value of both α and β by a single ascending trajectory: assume w.l.o.g. that bidder 1 demands cd before bidder 2 demands bd (no information will be elicited if none of these happens). In this case, the price for bd must be greater than 1 (otherwise, bidder 1 prefers bd to cd). Thus, bidder 2 will never demand the bundle bd, and no information will be elicited about β. The valuations described in the proof of Theorem 6.1 can be easily elicited by a non-anonymous item-price ascending auction. On the other hand, the valuations in Figure 6 can be easily elicited by an anonymous bundle-price ascending auction. We conclude that the power of these two families of ascending auctions is incomparable. We strengthen the impossibility result above by showing that anonymous bundle-price auctions cannot even achieve better than a min{O(n), O( √ m)}-approximation for the social welfare. This approximation ratio can be achieved with polynomial communication, and specifically with a polynomial number of item-price demand queries.28 Theorem 6.2. An anonymous bundle-price ascending auction cannot guarantee better than a min{ n 2 , √ m 2 } approximation for the optimal welfare. Proof. (Sketch) Assume we have n bidders and n2 items for sale, and that n is prime. We construct n2 distinct bundles with the following properties: for each bidder, we define a partition Si = (Si 1, ..., Si n) of the n2 items to n bundles, such that any two bundles from different partitions intersect. In the full paper, part II [9] we show an explicit construction using the properties of linear functions over finite fields. The rest of the proof is independent of the specific construction. Using these n2 bundles we construct a hard-to-elicit class. Every bidder has an atomic bid, in his XOR valuation, for each of these n2 bundles. A bidder i has a value of 2 for any bundle Si j in his partition. For all bundles in the other partitions, he has a value of either 0 or of 1 − δ, and these values are unknown to the auctioneer. Since every pair of bundles from different partitions intersect, only one bidder can receive a bundle with a value of 2. 27 If α > β, the optimal allocation will allocate cd to bidder 1 and ab to bidder 2. Otherwise, we give bd to bidder 2 and ac to bidder 1. Note that both bidders cannot gain a value of 2 in the same allocation, due to the intersections of the high-valued bundles. 28 Note that bundle-price queries may use exponential communication, thus the lower bound of [32] does not hold. 41 Non-anonymous Bundle-Price Economically-Efficient Ascending Auctions: Initialization: All prices are initialized to zero (non-anonymous bundle prices). Repeat: - Each bidder submits a bundle that maximizes his utility under his current personalized prices. - The auctioneer calculates a provisional allocation that maximizes his revenue under the current prices. - The prices of bundles that were demanded by losing bidders are increased by . Finally: Terminate when the provisional allocation assigns to each bidder the bundle he demanded. Figure 8: Auctions from this family (denoted by NBEA auctions) are known to achieve the optimal welfare. No bidder will demand a low-valued bundle, as long as the price of one of his high-valued bundles is below 1 (and thus gain him a utility greater than 1). Therefore, for eliciting any information about the low-valued bundles, the auctioneer should first arbitrarily choose a bidder (w.l.o.g bidder 1) and raise the prices of all the bundles (S1 1 , ..., S1 n) to be greater than 1. Since the prices cannot decrease, the other bidders will clearly never demand these bundles in future stages. An adversary may choose the values such that the low values of all the bidders for the bundles not in bidder 1"s partition are zero (i.e., vi(S1 j ) = 0 for every i = 1 and every j), however, allocating each bidder a different bundle from bidder 1"s partition, might achieve a welfare of n+1−(n−1)δ (bidder 1"s valuation is 2, and 1 − δ for all other bidders); If these bundles were wrongly allocated, only a welfare of 2 might be achieved (2 for bidder 1"s high-valued bundle, 0 for all other bidders). At this point, the auctioneer cannot have any information about the identity of the bundles with the non-zero values. Therefore, an adversary can choose the values of the bundles received by bidders 2, ..., n in the final allocation to be zero. We conclude that anonymous bundleprice auctions cannot guarantee a welfare greater than 2 for this class, where the optimal welfare can be arbitrarily close to n + 1. 6.2 Bundle Prices vs. Item Prices The core of the auctions in [37, 3] is the scheme described in Figure 8 (in the spirit of [35]) for auctions with nonanonymous bundle prices. Auctions from this scheme end up with the optimal allocation for any class of valuations. We denote this family of ascending auctions as NBEA auctions29 . NBEA auctions can elicit k-term XOR valuations by a polynomial (in k) number of steps , although the elicitation of such valuations may require an exponential number of item-price queries ([7]), and item-price ascending auctions cannot do it at all (Theorem 5.4). Nevertheless, we show that NBEA auctions (and in particular, iBundle(3) and the proxy auction) are sometimes inferior to simple item-price demand auctions. This may justify the use of hybrid auctions that use both linear and non-linear prices (e.g., the clock-proxy auction [10]). We show that auctions from this 29 Non-anonymous Bundle-price economically Efficient Ascending auctions. For completeness, we give in the full paper [9] a simple proof for the efficiency (up to an ) of auctions of this scheme . family may use an exponential number of queries even for determining the optimal allocation among two bidders with additive valuations30 , where such valuations can be elicited by a simple item-price ascending auction. We actually prove this property for a wider class of auctions we call conservative auctions. We also observe that in conservative auctions, allowing the bidders to submit all the bundles in their demand sets ensures that the auction runs a polynomial number of steps - if L is not too high (but with exponential communication, of course). An ascending auction is called conservative if it is nonanonymous, uses bundle prices initialized to zero and at every stage the auctioneer can only raise prices of bundles demanded by the bidders until this stage. In addition, each bidder can only receive bundles he demanded during the auction. Note that NBEA auctions are by definition conservative. Proposition 6.3. If every bidder demands a single bundle in each step of the auction, conservative auctions may run for an exponential number of steps even for additive valuations. 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[24] Sebastien Lahaie and David C. Parkes. Applying learning algorithms to preference elicitation. In EC 04. [25] Benny Lehmann, Daniel Lehmann, and Noam Nisan. Combinatorial auctions with decreasing marginal utilities. In ACM conference on electronic commerce. To appear, Games and Economic Behaviour., 2001. [26] D. Lehmann, L. O"Callaghan, and Y. Shoham. Truth revelation in approximately efficient combinatorial auctions. JACM, 49(5):577-602, Sept. 2002. [27] A. Mas-Collel, W. Whinston, and J. Green. Microeconomic Theory. Oxford university press, 1995. [28] Debasis Mishra and David Parkes. Ascending price vickrey auctions using primal-dual algorithms., 2004. Working paper, Harvard University. [29] Noam Nisan. The communication complexity of approximate set packing and covering. In ICALP 2002. [30] Noam Nisan. Bidding and allocation in combinatorial auctions. In ACM Conference on Electronic Commerce, 2000. [31] Noam Nisan. In P. Cramton and Y. Shoham and R. Steinberg (Editors), Combinatorial Auctions. Chapter 1. Bidding Languages. MIT Press. Forthcoming, 2005. [32] Noam Nisan and Ilya Segal. The communication requirements of efficient allocations and supporting prices, 2003. Working paper. Available from http://www.cs.huji.ac.il/˜noam/mkts.html Forthcoming in the Journal of Economic Theory. [33] Noam Nisan and Ilya Segal. Exponential communication inefficiency of demand queries, 2004. Working paper. Available from http://www.stanford.edu/ isegal/queries1.pdf. [34] D. C. Parkes and L. H. Ungar. An ascending-price generalized vickrey auction. Tech. Rep., Harvard University, 2002. [35] David Parkes. In P. Cramton and Y. Shoham and R. Steinberg (Editors), Combinatorial Auctions. Chapter 3. Iterative Combinatorial Auctions. MIT Press. Forthcoming, 2005. [36] David C. Parkes. Iterative combinatorial auctions: Achieving economic and computational efficiency. Ph.D. Thesis, Department of Computer and Information Science, University of Pennsylvania., 2001. [37] David C. Parkes and Lyle H. Ungar. Iterative combinatorial auctions: Theory and practice. In AAAI/IAAI, pages 74-81, 2000. [38] Ariel Rubinstein. Why are certain properties of binary relations relatively more common in natural languages. Econometrica, 64:343-356, 1996. [39] Tuomas Sandholm. Algorithm for optimal winner determination in combinatorial auctions. In Artificial Intelligence, volume 135, pages 1-54, 2002. [40] P. Santi, V. Conitzer, and T. Sandholm. Towards a characterization of polynomial preference elicitation with value queries in combinatorial auctions. In The 17th Annual Conference on Learning Theory, 2004. [41] Ilya Segal. The communication requirements of social choice rules and supporting budget sets, 2004. Working paper. Available from http://www.stanford.edu/ isegal/rules.pdf. [42] P.R. Wurman and M.P. Wellman. 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bound;ascend auction;combinatorial auction;price;optimal allocation;ascending-price auction;approximation factor;demand query;bidder;polynomial demand;preference elicitation;communication complexity

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Information Markets vs. Opinion Pools: An Empirical Comparison

In this paper, we examine the relative forecast accuracy of information markets versus expert aggregation. We leverage a unique data source of almost 2000 people"s subjective probability judgments on 2003 US National Football League games and compare with the market probabilities given by two different information markets on exactly the same events. We combine assessments of multiple experts via linear and logarithmic aggregation functions to form pooled predictions. Prices in information markets are used to derive market predictions. Our results show that, at the same time point ahead of the game, information markets provide as accurate predictions as pooled expert assessments. In screening pooled expert predictions, we find that arithmetic average is a robust and efficient pooling function; weighting expert assessments according to their past performance does not improve accuracy of pooled predictions; and logarithmic aggregation functions offer bolder predictions than linear aggregation functions. The results provide insights into the predictive performance of information markets, and the relative merits of selecting among various opinion pooling methods.

1. INTRODUCTION Forecasting is a ubiquitous endeavor in human societies. For decades, scientists have been developing and exploring various forecasting methods, which can be roughly divided into statistical and non-statistical approaches. Statistical approaches require not only the existence of enough historical data but also that past data contains valuable information about the future event. When these conditions can not be met, non-statistical approaches that rely on judgmental information about the future event could be better choices. One widely used non-statistical method is to elicit opinions from experts. Since experts are not generally in agreement, many belief aggregation methods have been proposed to combine expert opinions together and form a single prediction. These belief aggregation methods are called opinion pools, which have been extensively studied in statistics [20, 24, 38], and management sciences [8, 9, 30, 31], and applied in many domains such as group decision making [29] and risk analysis [12]. With the fast growth of the Internet, information markets have recently emerged as a promising non-statistical forecasting tool. Information markets (sometimes called prediction markets, idea markets, or event markets) are markets designed for aggregating information and making predictions about future events. To form the predictions, information markets tie payoffs of securities to outcomes of events. For example, in an information market to predict the result of a US professional National Football League (NFL) game, say New England vs Carolina, the security pays a certain amount of money per share to its holders if and only if New England wins the game. Otherwise, it pays off nothing. The security price before the game reflects the consensus expectation of market traders about the probability of New England winning the game. Such markets are becoming very popular. The Iowa Electronic Markets (IEM) [2] are real-money futures markets to predict economic and political events such as elections. The Hollywood Stock Exchange (HSX) [3] is a virtual (play-money) exchange for trading securities to forecast future box office proceeds of new movies, and the outcomes of entertainment awards, etc. TradeSports.com [7], a real-money betting exchange registered in Ireland, hosts markets for sports, political, entertainment, and financial events. The Foresight Exchange (FX) [4] allows traders to wager play money on unresolved scientific questions or other claims of public interest, and NewsFutures.com"s World News Exchange [1] has 58 popular sports and financial betting markets, also grounded in a play-money currency. Despite the popularity of information markets, one of the most important questions to ask is: how accurately can information markets predict? Previous research in general shows that information markets are remarkably accurate. The political election markets at IEM predict the election outcomes better than polls [16, 17, 18, 19]. Prices in HSX and FX have been found to give as accurate or more accurate predictions than judgment of individual experts [33, 34, 37]. However, information markets have not been calibrated against opinion pools, except for Servan-Schreiber et. al [36], in which the authors compare two information markets against arithmetic average of expert opinions. Since information markets, in nature, offer an adaptive and selforganized mechanism to aggregate opinions of market participants, it is interesting to compare them with existing opinion pooling methods, to evaluate the performance of information markets from another perspective. The comparison will provide beneficial guidance for practitioners to choose the most appropriate method for their needs. This paper contributes to the literature in two ways: (1) As an initial attempt to compare information markets with opinion pools of multiple experts, it leads to a better understanding of information markets and their promise as an alternative institution for obtaining accurate forecasts; (2) In screening opinion pools to be used in the comparison, we cast insights into relative performances of different opinion pools. In terms of prediction accuracy, we compare two information markets with several linear and logarithmic opinion pools (LinOP and LogOP) at predicting the results of NFL games. Our results show that at the same time point ahead of the game, information markets provide as accurate predictions as our carefully selected opinion pools. In selecting the opinion pools to be used in our comparison, we find that arithmetic average is a robust and efficient pooling function; weighting expert assessments according to their past performances does not improve the prediction accuracy of opinion pools; and LogOP offers bolder predictions than LinOP. The remainder of the paper is organized as follows. Section 2 reviews popular opinion pooling methods. Section 3 introduces the basics of information markets. Data sets and our analysis methods are described in Section 4. We present results and analysis in Section 5, followed by conclusions in Section 6. 2. REVIEW OF OPINION POOLS Clemen and Winkler [12] classify opinion pooling methods into two broad categories: mathematical approaches and behavioral approaches. In mathematical approaches, the opinions of individual experts are expressed as subjective probability distributions over outcomes of an uncertain event. They are combined through various mathematical methods to form an aggregated probability distribution. Genest and Zidek [24] and French [20] provide comprehensive reviews of mathematical approaches. Mathematical approaches can be further distinguished into axiomatic approaches and Bayesian approaches. Axiomatic approaches apply prespecified functions that map expert opinions, expressed as a set of individual probability distributions, to a single aggregated probability distribution. These pooling functions are justified using axioms or certain desirable properties. Two of the most common pooling functions are the linear opinion pool (LinOP) and the logarithmic opinion pool (LogOP). Using LinOP, the aggregate probability distribution is a weighted arithmetic mean of individual probability distributions: p(θ) = n i=1 wipi(θ), (1) where pi(θ) is expert i"s probability distribution of uncertain event θ, p(θ) represents the aggregate probability distribution, wi"s are weights for experts, which are usually nonnegative and sum to 1, and n is the number of experts. Using LogOP, the aggregate probability distribution is a weighted geometric mean of individual probability distributions: p(θ) = k n i=1 pi(θ)wi , (2) where k is a normalization constant to ensure that the pooled opinion is a probability distribution. Other axiomatic pooling methods often are extensions of LinOP [22], LogOP [23], or both [13]. Winkler [39] and Morris [29, 30] establish the early framework of Bayesian aggregation methods. Bayesian approaches assume as if there is a decision maker who has a prior probability distribution over event θ and a likelihood function over expert opinions given the event. This decision maker takes expert opinions as evidence and updates its priors over the event and opinions according to Bayes rule. The resulted posterior probability distribution of θ is the pooled opinion. Behavioral approaches have been widely studied in the field of group decision making and organizational behavior. The important assumption of behavioral approaches is that, through exchanging opinions or information, experts can eventually reach an equilibrium where further interaction won"t change their opinions. One of the best known behavioral approaches is the Delphi technique [28]. Typically, this method and its variants do not allow open discussion, but each expert has chance to judge opinions of other experts, and is given feedback. Experts then can reassess their opinions and repeat the process until a consensus or a smaller spread of opinions is achieved. Some other behavioral methods, such as the Nominal Group technique [14], promote open discussions in controlled environments. Each approach has its pros and cons. Axiomatic approaches are easy to use. But they don"t have a normative basis to choose weights. In addition, several impossibility results (e.g., Genest [21]) show that no aggregation function can satisfy all desired properties of an opinion pool, unless the pooled opinion degenerates to a single individual opinion, which effectively implies a dictator. Bayesian approaches are nicely based on the normative Bayesian framework. However, it is sometimes frustratingly difficult to apply because it requires either (1) constructing an obscenely complex joint prior over the event and opinions (often impractical even in terms of storage / space complexity, not to mention from an elicitation standpoint) or (2) making strong assumptions about the prior, like conditional independence of experts. Behavior approaches allow experts to dynamically improve their information and revise their opinions during interactions, but many of them are not fixed or completely specified, and can"t guarantee convergence or repeatability. 59 3. HOW INFORMATION MARKETS WORK Much of the enthusiasm for information markets stems from Hayek hypothesis [26] and efficient market hypothesis [15]. Hayek, in his classic critique of central planning in 1940"s, claims that the price system in a competitive market is a very efficient mechanism to aggregate dispersed information among market participants. The efficient market hypothesis further states that, in an efficient market, the price of a security almost instantly incorporates all available information. The market price summarizes all relevant information across traders, hence is the market participants" consensus expectation about the future value of the security. Empirical evidence supports both hypotheses to a large extent [25, 27, 35]. Thus, when associating the value of a security with the outcome of an uncertain future event, market price, by revealing the consensus expectation of the security value, can indirectly predict the outcome of the event. This idea gives rise to information markets. For example, if we want to predict which team will win the NFL game between New England and Carolina, an information market can trade a security $100 if New England defeats Carolina, whose payoff per share at the end of the game is specified as follow: $100 if New England wins the game; $0 otherwise. The security price should roughly equal the expected payoff of the security in an efficient market. The time value of money usually can be ignored because durations of most information markets are short. Assuming exposure to risk is roughly equal for both outcomes, or that there are sufficient effectively risk-neutral speculators in the market, the price should not be biased by the risk attitudes of various players in the market. Thus, p = Pr(Patriots win) × 100 + [1 − Pr(Patriots win)] × 0, where p is the price of the security $100 if New England defeats Carolina and Pr(Patriots win) is the probability that New England will win the game. Observing the security price p before the game, we can derive Pr(Patriots win), which is the market participants" collective prediction about how likely it is that New England will win the game. The above security is a winner-takes-all contract. It is used when the event to be predicted is a discrete random variable with disjoint outcomes (in this case binary). Its price predicts the probability that a specific outcome will be realized. When the outcome of a prediction problem can be any value in a continuous interval, we can design a security that pays its holder proportional to the realized value. This kind of security is what Wolfers and Zitzewitz [40] called an index contract. It predicts the expected value of a future outcome. Many other aspects of a future event such as median value of outcome can also be predicted in information markets by designing and trading different securities. Wolfers and Zitzewitz [40] provide a summary of the main types of securities traded in information markets and what statistical properties they can predict. In practice, conceiving a security for a prediction problem is only one of the many decisions in designing an effective information market. Spann and Skiera [37] propose an initial framework for designing information markets. 4. DESIGN OF ANALYSIS 4.1 Data Sets Our data sets cover 210 NFL games held between September 28th, 2003 and December 28th, 2003. NFL games are very suitable for our purposes because: (1) two online exchanges and one online prediction contest already exist that provide data on both information markets and the opinions of self-identified experts for the same set of games; (2) the popularity of NFL games in the United States provides natural incentives for people to participate in information markets and/or the contest, which increases liquidity of information markets and improves the quality and number of opinions in the contest; (3) intense media coverage and analysis of the profiles and strengths of teams and individual players provide the public with much information so that participants of information markets and the contest can be viewed as knowledgeable regarding to the forecasting goal. Information market data was acquired, by using a specially designed crawler program, from TradeSports.com"s Football-NFL markets [7] and NewsFutures.com"s Sports Exchange [1]. For each NFL game, both TradeSports and NewsFutures have a winner-takes-all information market to predict the game outcome. We introduce the design of the two markets according to Spann and Skiera"s three steps for designing an information market [37] as below. • Choice of forecasting goal: Markets at both TradeSports and NewsFutures aim at predicting which one of the two teams will win a NFL football game. They trade similar winner-takes-all securities that pay off 100 if a team wins the game and 0 if it loses the game. Small differences exist in how they deal with ties. In the case of a tie, TradeSports will unwind all trades that occurred and refund all exchange fees, but the security is worth 50 in NewsFutures. Since the probability of a tie is usually very low (much less the 1%), prices at both markets effectively represent the market participants" consensus assessment of the probability that the team will win. • Incentive for participation and information revelation: TradeSports and NewsFutures use different incentives for participation and information revelation. TradeSports is a real-money exchange. A trader needs to open and fund an account with a minimum of $100 to participate in the market. Both profits and losses can occur as a result of trading activity. On the contrary, a trader can register at NewsFutures for free and get 2000 units of Sport Exchange virtual money at the time of registration. Traders at NewsFutures will not incur any real financial loss. They can accumulate virtual money by trading securities. The virtual money can then be used to bid for a few real prizes at NewsFutures" online shop. • Financial market design: Both markets at TradeSports and NewsFutures use the continuous double auction as their trading mechanism. TradeSports charges a small fee on each security transaction and expiry, while NewsFutures does not. We can see that the main difference between two information markets is real money vs. virtual money. Servan-Schreiber 60 et. al [36] have compared the effect of money on the performance of the two information markets and concluded that the prediction accuracy of the two markets are at about the same level. Not intending to compare these two markets, we still use both markets in our analysis to ensure that our findings are not accidental. We obtain the opinions of 1966 self-identified experts for NFL games from the ProbabilityFootball online contest [5], one of several ProbabilitySports contests [6]. The contest is free to enter. Participants of the contest are asked to enter their subjective probability that a team will win a game by noon on the day of the game. Importantly, the contest evaluates the participants" performance via the quadratic scoring rule: s = 100 − 400 × Prob Lose2 , (3) where s represents the score that a participant earns for the game, and Prob Lose is the probability that the participant assigns to the actual losing team. The quadratic score is one of a family of so-called proper scoring rules that have the property that an expert"s expected score is maximized when the expert reports probabilities truthfully. For example, for a game team A vs. team B, if a player assigns 0.5 to both team A and B, his/her score for the game is 0 no matter which team wins. If he/she assigns 0.8 to team A and 0.2 to team B, showing that he is confident in team A"s winning, he/she will score 84 points for the game if team A wins, and lose 156 points if team B wins. This quadratic scoring rule rewards bold predictions that are right, but penalizes bold predictions that turn out to be wrong. The top players, measured by accumulated scores over all games, win the prizes of the contest. The suggested strategy at the contest website is to make picks for each game that match, as closely as possible, the probabilities that each team will win. This strategy is correct if the participant seeks to maximize expected score. However, as prizes are awarded only to the top few winners, participants" goals are to maximize the probability of winning, not maximize expected score, resulting in a slightly different and more risk-seeking optimization.1 Still, as far as we are aware, this data offer the closest thing available to true subjective probability judgments from so many people over so many public events that have corresponding information markets. 4.2 Methods of Analysis In order to compare the prediction accuracy of information markets and that of opinion pools, we proceed to derive predictions from market data of TradeSports and NewsFutures, form pooled opinions using expert data from ProbabilityFootball contest, and specify the performance measures to be used. 4.2.1 Deriving Predictions For information markets, deriving predictions is straightforward. We can take the security price and divide it by 100 to get the market"s prediction of the probability that a team will win. To match the time when participants at the ProbabilityFootball contest are required to report their probability assessments, we derive predictions using the last trade price before noon on the day of the game. For more 1 Ideally, prizes would be awarded by lottery in proportion to accumulated score. than half of the games, this time is only about an hour earlier than the game starting time, while it is several hours earlier for other games. Two sets of market predictions are derived: • NF: Prediction equals NewsFutures" last trade price before noon of the game day divided by 100. • TS: Prediction equals TradeSports" last trade price before noon of the game day divided by 100. We apply LinOP and LogOP to ProbabilityFootball data to obtain aggregate expert predictions. The reason that we do not consider other aggregation methods include: (1) data from ProbabilityFootball is only suitable for mathematical pooling methods-we can rule out behavioral approaches, (2) Bayesian aggregation requires us to make assumptions about the prior probability distribution of game outcomes and the likelihood function of expert opinions: given the large number of games and participants, making reasonable assumptions is difficult, and (3) for axiomatic approaches, previous research has shown that simpler aggregation methods often perform better than more complex methods [12]. Because the output of LogOP is indeterminate if there are probability assessments of both 0 and 1 (and because assessments of 0 and 1 are dictatorial using LogOP), we add a small number 0.01 to an expert opinion if it is 0, and subtract 0.01 from it if it is 1. In pooling opinions, we consider two influencing factors: weights of experts and number of expert opinions to be pooled. For weights of experts, we experiment with equal weights and performance-based weights. The performancebased weights are determined according to previous accumulated score in the contest. The score for each game is calculated according to equation 3, the scoring rule used in the ProbabilityFootball contest. For the first week, since no previous scores are available, we choose equal weights. For later weeks, we calculate accumulated past scores for each player. Because the cumulative scores can be negative, we shift everyone"s score if needed to ensure the weights are non-negative. Thus, wi = cumulative scorei + shift n j=1(cumulative scorej + shift) . (4) where shift equals 0 if the smallest cumulative scorej is non-negative, and equals the absolute value of the smallest cumulative scorej otherwise. For simplicity, we call performance-weighted opinion pool as weighted, and equally weighted opinion pool as unweighted. We will use them interchangeably in the remaining of the paper. As for the number of opinions used in an opinion pool, we form different opinion pools with different number of experts. Only the best performing experts are selected. For example, to form an opinion pool with 20 expert opinions, we choose the top 20 participants. Since there is no performance record for the first week, we use opinions of all participants in the first week. For week 2, we select opinions of 20 individuals whose scores in the first week are among the top 20. For week 3, 20 individuals whose cumulative scores of week 1 and 2 are among the top 20s are selected. Experts are chosen in a similar way for later weeks. Thus, the top 20 participants can change from week to week. The possible opinion pools, varied in pooling functions, weighting methods, and number of expert opinions, are shown 61 Table 1: Pooled Expert Predictions # Symbol Description 1 Lin-All-u Unweighted (equally weighted) LinOP of all experts. 2 Lin-All-w Weighted (performance-weighted) LinOP of all experts. 3 Lin-n-u Unweighted (equally weighted) LinOP with n experts. 4 Lin-n-w Weighted (performance-weighted) LinOP with n experts. 5 Log-All-u Unweighted (equally weighted) LogOP of all experts. 6 Log-All-w Weighted (performance-weighted) LogOP of all experts. 7 Log-n-u Unweighted (equally weighted) LogOP with n experts. 8 Log-n-w Weighted (performance-weighted) LogOP with n experts. in Table 1. Lin represents linear, and Log represents Logarithmic. n is the number of expert opinions that are pooled, and All indicates that all opinions are combined. We use u to symbolize unweighted (equally weighted) opinion pools. w is used for weighted (performance-weighted) opinion pools. Lin-All-u, the equally weighted LinOP with all participants, is basically the arithmetic mean of all participants" opinions. Log-All-u is simply the geometric mean of all opinions. When a participant did not enter a prediction for a particular game, that participant was removed from the opinion pool for that game. This contrasts with the ProbabilityFootball average reported on the contest website and used by Servan-Schreiber et. al [36], where unreported predictions were converted to 0.5 probability predictions. 4.2.2 Performance Measures We use three common metrics to assess prediction accuracy of information markets and opinion pools. These measures have been used by Servan-Schreiber et. al [36] in evaluating the prediction accuracy of information markets. 1. Absolute Error = Prob Lose, where Prob Lose is the probability assigned to the eventual losing team. Absolute error simply measures the difference between a perfect prediction (1 for winning team) and the actual prediction. A prediction with lower absolute error is more accurate. 2. Quadratic Score = 100 − 400 × (Prob Lose2 ). Quadratic score is the scoring function that is used in the ProbabilityFootball contest. It is a linear transformation of squared error, Prob Lose2 , which is one of the mostly used metrics in evaluating forecasting accuracy. Quadratic score can be negative. A prediction with higher quadratic score is more accurate. 3. Logarithmic Score = log(Prob W in), where Prob W in is the probability assigned to the eventual winning team. The logarithmic score, like the quadratic score, is a proper scoring rule. A prediction with higher (less negative) logarithmic score is more accurate. 5. EMPIRICAL RESULTS 5.1 Performance of Opinion Pools Depending on how many opinions are used, there can be numerous different opinion pools. We first examine the effect of number of opinions on prediction accuracy by forming opinion pools with the number of expert opinions varying from 1 to 960. In the ProbabilityFootball Competition, not all 1966 registered participants provide their probability assessments for every game. 960 is the smallest number of participants for all games. For each game, we sort experts according to their accumulated quadratic score in previous weeks. Predictions of the best performing n participants are picked to form an opinion pool with n experts. Figure 1 shows the prediction accuracy of LinOP and LogOP in terms of mean values of the three performance measures across all 210 games. We can see the following trends in the figure. 1. Unweighted opinion pools and performance-weighted opinion pools have similar levels of prediction accuracy, especially for LinOP. 2. For LinOP, increasing the number of experts in general increases or keeps the same the level of prediction accuracy. When there are more than 200 experts, the prediction accuracy of LinOP is stable regarding the number of experts. 3. LogOP seems more accurate than LinOP in terms of mean absolute error. But, using all other performance measures, LinOP outperforms LogOP. 4. For LogOP, increasing the number of experts increases the prediction accuracy at the beginning. But the curves (including the points with all experts) for mean quadratic score, and mean logarithmic score have slight bell-shapes, which represent a decrease in prediction accuracy when the number of experts is very large. The curves for mean absolute error, on the other hand, show a consistent increase of accuracy. The first and second trend above imply that when using LinOP, the simplest way, which has good prediction accuracy, is to average the opinions of all experts. Weighting does not seem to improve performance. Selecting experts according to past performance also does not help. It is a very interesting observation that even if many participants of the ProbabilityFootball contest do not provide accurate individual predictions (they have negative quadratic scores in the contest), including their opinions into the opinion pool still increases the prediction accuracy. One explanation of this phenomena could be that biases of individual judgment can offset with each other when opinions are diverse, which makes the pooled prediction more accurate. The third trend presents a controversy. The relative prediction accuracy of LogOP and LinOP flips when using different accuracy measures. To investigate this disagreement, we plot the absolute error of Log-All-u and Lin-All-u for each game in Figure 2. When the absolute error of an opinion 62 0 100 200 300 400 500 600 700 800 900 All 0.4 0.405 0.41 0.415 0.42 0.425 0.43 0.435 0.44 0.445 0.45 Number of Expert Opinions MeanAbsoluteError Unweighted Linear Weighted Linear Unweighted Logarithmic Weighted Logarithmic Lin−All−u Lin−All−w Log−All−u Log−All−w (a) Mean Absolute Error 0 100 200 300 400 500 600 700 800 900 All 4 5 6 7 8 9 10 11 12 13 14 Number of Expert Opinions MeanQuadraticScore Unweighted Linear Weighted Linear Unweighted Logarithmic Weighted Logarithmic Lin−All−u Lin−All−w Log−All−u Log−All−w 0 100 200 300 400 500 600 700 800 900 All −0.71 −0.7 −0.69 −0.68 −0.67 −0.66 −0.65 −0.64 −0.63 −0.62 Number of Expert Opinions MeanLogarithmicScore Unweighted Linear Weighted Linear Unweighted Logarithmic Weighted Logarithmic Lin−All−u Lin−All−w Log−All−u Log−All−w (b) Mean Quadratic Score (c) Mean Logarithmic Score Figure 1: Prediction Accuracy of Opinion Pools pool for a game is less than 0.5, it means that the team favored by the opinion pool wins the game. If it is greater than 0.5, the underdog wins. Compared with Lin-All-u, Log-All-u has lower absolute error when it is less than 0.5, and greater absoluter error when it is greater than 0.5, which indicates that predictions of Log-All-u are bolder, more close to 0 or 1, than those of Lin-All-u. This is due to the nature of linear and logarithmic aggregating functions. Because quadratic score and logarithmic score penalize bold predictions that are wrong, LogOP is less accurate when measured in these terms. Similar reasoning accounts for the fourth trend. When there are more than 500 experts, increasing number of experts used in LogOP improves the prediction accuracy measured by absolute error, but worsens the accuracy measured by the other two metrics. Examining expert opinions, we find that participants who rank lower are more frequent in offering extreme predictions (0 or 1) than those ranking high in the list. When we increase the number of experts in an opinion pool, we are incorporating more extreme predictions into it. The resulting LogOP is bolder, and hence has lower mean quadratic score and mean logarithmic score. 5.2 Comparison of Information Markets and Opinion Pools Through the first screening of various opinion pools, we select Lin-All-u, Log-All-u, Log-All-w, and Log-200-u to compare with predictions from information markets. Lin-All-u as shown in Figure 1 can represent what LinOP can achieve. However, the performance of LogOP is not consistent when evaluated using different metrics. Log-All-u and Log-All-w offer either the best or the worst predictions. Log-200-u, the LogOP with the 200 top performing experts, provides more stable predictions. We use all of the three to stand for the performance of LogOP in our later comparison. If a prediction of the probability that a team will win a game, either from an opinion pool or an information market, is higher than 0.5, we say that the team is the predicted favorite for the game. Table 2 presents the number and percentage of games that predicted favorites actually win, out of a total of 210 games. All four opinion pools correctly predict a similar number and percentage of games as NF and TS. Since NF, TS, and the four opinion pools form their predictions using information available at noon of the game 63 Table 2: Number and Percentage of Games that Predicted Favorites Win NF TS Lin-All-u Log-All-u Log-All-w Log-200-u Number 142 137 144 144 143 141 Percentage 67.62% 65.24% 68.57% 68.57% 68.10% 67.14% Table 3: Mean of Prediction Accuracy Measures Absolute Error Quadratic Score Logarithmic Score NF 0.4253 15.4352 -0.6136 (0.0121) (4.6072) (0.0258) TS 0.4275 15.2739 -0.6121 (0.0118) (4.3982) (0.0241) Lin-All-u 0.4292 13.0525 -0.6260 (0.0126) (4.8088) (0.0268) Log-All-u 0.4024 10.0099 -0.6546 (0.0173) (6.6594) (0.0418) Log-All-w 0.4059 10.4491 -0.6497 (0.0168) (6.4440) (0.0398) Log-200-u 0.4266 12.3868 -0.6319 (0.0133) (5.0764) (0.0295) *Numbers in parentheses are standard errors. *Best value for each metric is shown in bold. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Absolute Error of Lin−All−u AbsoluteErrorofLog−All−u 45 Degree Line Absolute Error Figure 2: Absolute Error: Lin-All-u vs. Log-All-u day, information markets and opinion pools have comparable potential at the same time point. We then take a closer look at prediction accuracy of information markets and opinion pools using the three performance measures. Table 3 displays mean values of these measures over 210 games. Numbers in parentheses are standard errors, which estimate the standard deviation of the mean. To take into consideration of skewness of distributions, we also report median values of accuracy measures in Table 4. Judged by the mean values of accuracy measures in Table 3, all methods have similar accuracy levels, with NF and TS slightly better than the opinion pools. However, the median values of accuracy measures indicate that LogAll-u and Log-All-w opinion pools are more accurate than all other predictions. We employ the randomization test [32] to study whether the differences in prediction accuracy presented in Table 3 and Table 4 are statistically significant. The basic idea of randomization test is that, by randomly swapping predictions of two methods numerous times, an empirical distribution for the difference of prediction accuracy can be constructed. Using this empirical distribution, we are then able to evaluate that at what confidence level the observed difference reflects a real difference. For example, the mean absolute error of NF is higher than that of Log-All-u by 0.0229, as shown in Table 3. To test whether this difference is statistically significant, we shuffle predictions from two methods, randomly label half of predictions as NF and the other half as Log-All-u, and compute the difference of mean absolute error of the newly formed NF and Log-All-u data. The above procedure is repeated 10,000 times. The 10,000 differences of mean absolute error results in an empirical distribution of the difference. Comparing our observed difference, 0.0229, with this distribution, we find that the observed difference is greater than 75.37% of the empirical differences. This leads us to conclude that the difference of mean absolute error between NF and Log-All-u is not statistically significant, if we choose the level of significance to be 0.05. Table 5 and Table 6 are results of randomization test for mean and median differences respectively. Each cell of the table is for two different prediction methods, represented by name of the row and name of the column. The first lines of table cells are results for absolute error. The second and third lines are dedicated to quadratic score and logarithmic score respectively. We can see that, in terms of mean values of accuracy measures, the differences of all methods are not statistically significant to any reasonable degree. When it 64 Table 4: Median of Prediction Accuracy Measures Absolute Error Quadratic Score Logarithmic Score NF 0.3800 42.2400 -0.4780 TS 0.4000 36.0000 -0.5108 Lin-All-u 0.3639 36.9755 -0.5057 Log-All-u 0.3417 53.2894 -0.4181 Log-All-w 0.3498 51.0486 -0.4305 Log-200-u 0.3996 36.1300 -0.5101 *Best value for each metric is shown in bold. Table 5: Statistical Confidence of Mean Differences in Prediction Accuracy TS Lin-All-u Log-All-u Log-All-w Log-200-u NF 8.92% 22.07% 75.37% 66.47% 7.76% 2.38% 26.60% 50.74% 44.26% 32.24% 2.99% 22.81% 59.35% 56.21% 33.26% TS 10.13% 77.79% 68.15% 4.35% 27.25% 53.65% 44.90% 28.30% 32.35% 57.89% 60.69% 38.84% Lin-All-u 82.19% 68.86% 9.75% 28.91% 23.92% 6.81% 44.17% 43.01% 17.36% Log-All-u 11.14% 72.49% 3.32% 18.89% 5.25% 39.06% Log-All-w 69.89% 18.30% 30.23% *In each table cell, row 1 accounts for absolute error, row 2 for quadratic score, and row 3 for logarithmic score. comes to median values of prediction accuracy, Log-All-u outperforms Lin-All-u at a high confidence level. These results indicate that differences in prediction accuracy between information markets and opinion pools are not statistically significant. This may seem to contradict the result of Servan-Schreiber et. al [36], in which NewsFutures"s information markets have been shown to provide statistically significantly more accurate predictions than the (unweighted) average of all ProbabilityFootball opinions. The discrepancy emerges in dealing with missing data. Not all 1966 registered ProbabilityFootball participants offer probability assessments for each game. When a participant does not provide a probability assessment for a game, the contest considers their prediction as 0.5.. This makes sense in the context of the contest, since 0.5 always yields 0 quadratic score. The ProbabilityFootball average reported on the contest website and used by Servan-Schreiber et. al includes these 0.5 estimates. Instead, we remove participants from games that they do not provide assessments, pooling only the available opinions together. Our treatment increases the prediction accuracy of Lin-All-u significantly. 6. CONCLUSIONS With the fast growth of the Internet, information markets have recently emerged as an alternative tool for predicting future events. Previous research has shown that information markets give as accurate or more accurate predictions than individual experts and polls. However, information markets, as an adaptive mechanism to aggregate different opinions of market participants, have not been calibrated against many belief aggregation methods. In this paper, we compare prediction accuracy of information markets with linear and logarithmic opinion pools (LinOP and LogOP) using predictions from two markets and 1966 individuals regarding the outcomes of 210 American football games during the 2003 NFL season. In screening for representative opinion pools to compare with information markets, we investigate the effect of weights and number of experts on prediction accuracy. Our results on both the comparison of information markets and opinion pools and the relative performance of different opinion pools are summarized as below. 1. At the same time point ahead of the events, information markets offer as accurate predictions as our selected opinion pools. We have selected four opinion pools to represent the prediction accuracy level that LinOP and LogOP can achieve. With all four performance metrics, our two information markets obtain similar prediction accuracy as the four opinion pools. 65 Table 6: Statistical Confidence of Median Differences in Prediction Accuracy TS Lin-All-u Log-All-u Log-All-w Log-200-u NF 48.85% 47.3% 84.8% 77.9% 65.36% 45.26% 44.55% 85.27% 75.65% 66.75% 44.89% 46.04% 84.43% 77.16% 64.78% TS 5.18% 94.83% 94.31% 0% 5.37% 92.08% 92.53% 0% 7.41% 95.62% 91.09% 0% Lin-All-u 95.11% 91.37% 7.31% 96.10% 92.69% 9.84% 95.45% 95.12% 7.79% Log-All-u 23.47% 95.89% 26.68% 93.85% 22.47% 96.42% Log-All-w 91.3% 91.4% 90.37% *In each table cell, row 1 accounts for absolute error, row 2 for quadratic score, and row 3 for logarithmic score. *Confidence above 95% is shown in bold. 2. The arithmetic average of all opinions (Lin-All-u) is a simple, robust, and efficient opinion pool. Simply averaging across all experts seems resulting in better predictions than individual opinions and opinion pools with a few experts. It is quite robust in the sense that even if the included individual predictions are less accurate, averaging over all opinions still gives better (or equally good) predictions. 3. Weighting expert opinions according to past performance does not seem to significantly improve prediction accuracy of either LinOP or LogOP. Comparing performance-weighted opinion pools with equally weighted opinion pools, we do not observe much difference in terms of prediction accuracy. Since we only use one performance-weighting method, calculating the weights according to past accumulated quadratic score that participants earned, this might due to the weighting method we chose. 4. LogOP yields bolder predictions than LinOP. LogOP yields predictions that are closer to the extremes, 0 or 1. An information markets is a self-organizing mechanism for aggregating information and making predictions. Compared with opinion pools, it is less constrained by space and time, and can eliminate the efforts to identify experts and decide belief aggregation methods. But the advantages do not compromise their prediction accuracy to any extent. On the contrary, information markets can provide real-time predictions, which are hardly achievable through resorting to experts. In the future, we are interested in further exploring: • Performance comparison of information markets with other opinion pools and mathematical aggregation procedures. In this paper, we only compare information markets with two simple opinion pools, linear and logarithmic. It will be meaningful to investigate their relative prediction accuracy with other belief aggregation methods such as Bayesian approaches. There are also a number of theoretical expert algorithms with proven worst-case performance bounds [10] whose average-case or practical performance would be instructive to investigate. • Whether defining expertise more narrowly can improve predictions of opinion pools. In our analysis, we broadly treat participants of the ProbabilityFootball contest as experts in all games. If we define expertise more narrowly, selecting experts in certain football teams to predict games involving these teams, will the predictions of opinion pools be more accurate? • The possibility of combining information markets with other forecasting methods to achieve better prediction accuracy. Chen, Fine, and Huberman [11] use an information market to determine the risk attitude of participants, and then perform a nonlinear aggregation of their predictions based on their risk attitudes. The nonlinear aggregation mechanism is shown to outperform both the market and the best individual participants. It is worthy of more attention whether information markets, as an alternative forecasting method, can be used together with other methods to improve our predictions. 7. ACKNOWLEDGMENTS We thank Brian Galebach, the owner and operator of the ProbabilitySports and ProbabilityFootball websites, for providing us with such unique and valuable data. We thank Varsha Dani, Lance Fortnow, Omid Madani, Sumit Sang66 hai, and the anonymous reviewers for useful insights and pointers. The authors acknowledge the support of The Penn State eBusiness Research Center. 8. REFERENCES [1] http://us.newsfutures.com [2] http://www.biz.uiowa.edu/iem/ [3] http://www.hsx.com/ [4] http://www.ideosphere.com/fx/ [5] http://www.probabilityfootball.com/ [6] http://www.probabilitysports.com/ [7] http://www.tradesports.com/ [8] A. H. Ashton and R. H. Ashton. Aggregating subjective forecasts: Some empirical results. Management Science, 31:1499-1508, 1985. [9] R. P. Batchelor and P. Dua. Forecaster diversity and the benefits of combining forecasts. Management Science, 41:68-75, 1995. [10] N. Cesa-Bianchi, Y. Freund, D. Haussler, D. P. Helmbold, R. E. Schapire, and M. K. Warmuth. How to use expert advice. Journal of the ACM, 44(3):427-485, 1997. [11] K. Chen, L. Fine, and B. Huberman. Predicting the future. Information System Frontier, 5(1):47-61, 2003. [12] R. T. Clemen and R. L. Winkler. Combining probability distributions from experts in risk analysis. Risk Analysis, 19(2):187-203, 1999. [13] R. M. Cook. Experts in Uncertainty: Opinion and Subjective Probability in Science. Oxford University Press, New York, 1991. [14] A. L. Delbecq, A. H. Van de Ven, and D. H. Gustafson. Group Techniques for Program Planners: A Guide to Nominal Group and Delphi Processes. Scott Foresman and Company, Glenview, IL, 1975. [15] E. F. Fama. Efficient capital market: A review of theory and empirical work. Journal of Finance, 25:383-417, 1970. [16] R. Forsythe and F. Lundholm. Information aggregation in an experimental market. Econometrica, 58:309-47, 1990. [17] R. Forsythe, F. Nelson, G. R. Neumann, and J. Wright. Forecasting elections: A market alternative to polls. In T. R. Palfrey, editor, Contemporary Laboratory Experiments in Political Economy, pages 69-111. University of Michigan Press, Ann Arbor, MI, 1991. [18] R. Forsythe, F. Nelson, G. R. Neumann, and J. Wright. Anatomy of an experimental political stock market. American Economic Review, 82(5):1142-1161, 1992. [19] R. Forsythe, T. A. Rietz, and T. W. Ross. Wishes, expectations, and actions: A survey on price formation in election stock markets. Journal of Economic Behavior and Organization, 39:83-110, 1999. [20] S. French. Group consensus probability distributions: a critical survey. Bayesian Statistics, 2:183-202, 1985. [21] C. Genest. A conflict between two axioms for combining subjective distributions. Journal of the Royal Statistical Society, 46(3):403-405, 1984. [22] C. Genest. Pooling operators with the marginalization property. Canadian Journal of Statistics, 12(2):153-163, 1984. [23] C. Genest, K. J. McConway, and M. J. Schervish. Characterization of externally Bayesian pooling operators. Annals of Statistics, 14(2):487-501, 1986. [24] C. Genest and J. V. Zidek. Combining probability distributions: A critique and an annotated bibliography. Statistical Science, 1(1):114-148, 1986. [25] S. J. Grossman. An introduction to the theory of rational expectations under asymmetric information. Review of Economic Studies, 48(4):541-559, 1981. [26] F. A. Hayek. The use of knowledge in society. American Economic Review, 35(4):519-530, 1945. [27] J. C. Jackwerth and M. Rubinstein. Recovering probability distribution from options prices. Journal of Finance, 51(5):1611-1631, 1996. [28] H. A. Linstone and M. Turoff. The Delphi Method: Techniques and Applications. Addison-Wesley, Reading, MA, 1975. [29] P. A. Morris. Decision analysis expert use. Management Science, 20(9):1233-1241, 1974. [30] P. A. Morris. Combining expert judgments: A bayesian approach. Management Science, 23(7):679-693, 1977. [31] P. A. Morris. An axiomatic approach to expert resolution. Management Science, 29(1):24-32, 1983. [32] E. W. Noreen. Computer-Intensive Methods for Testing Hypotheses: An Introduction. Wiley and Sons, Inc., New York, 1989. [33] D. M. Pennock, S. Lawrence, C. L. Giles, and F. A. Nielsen. The real power of artificial markets. Science, 291:987-988, February 2002. [34] D. M. Pennock, S. Lawrence, F. A. Nielsen, and C. L. Giles. Extracting collective probabilistic forecasts from web games. In Proceedings of the 7th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 174-183, San Francisco, CA, 2001. [35] C. Plott and S. Sunder. Rational expectations and the aggregation of diverse information in laboratory security markets. Econometrica, 56:1085-118, 1988. [36] E. Servan-Schreiber, J. Wolfers, D. M. Pennock, and B. Galebach. Prediction markets: Does money matter? Electronic Markets, 14(3):243-251, 2004. [37] M. Spann and B. Skiera. Internet-based virtual stock markets for business forecasting. Management Science, 49(10):1310-1326, 2003. [38] M. West. Bayesian aggregation. Journal of the Royal Statistical Society. Series A. General, 147(4):600-607, 1984. [39] R. L. Winkler. The consensus of subjective probability distributions. Management Science, 15(2):B61-B75, 1968. [40] J. Wolfers and E. Zitzewitz. Prediction markets. Journal of Economic Perspectives, 18(2):107-126, 2004. 67

prediction accuracy;pooled prediction;future event;expertise;price;contract;information market;market probability;opinion pool;expert aggregation;forecast;expert opinion

train_J-50

Communication Complexity of Common Voting Rules∗

We determine the communication complexity of the common voting rules. The rules (sorted by their communication complexity from low to high) are plurality, plurality with runoff, single transferable vote (STV), Condorcet, approval, Bucklin, cup, maximin, Borda, Copeland, and ranked pairs. For each rule, we first give a deterministic communication protocol and an upper bound on the number of bits communicated in it; then, we give a lower bound on (even the nondeterministic) communication requirements of the voting rule. The bounds match for all voting rules except STV and maximin.

1. INTRODUCTION One key factor in the practicality of any preference aggregation rule is its communication burden. To successfully aggregate the agents" preferences, it is usually not necessary for all the agents to report all of their preference information. Clever protocols that elicit the agents" preferences partially and sequentially have the potential to dramatically reduce the required communication. This has at least the following advantages: • It can make preference aggregation feasible in settings where the total amount of preference information is too large to communicate. • Even when communicating all the preference information is feasible, reducing the communication requirements lessens the burden placed on the agents. This is especially true when the agents, rather than knowing all their preferences in advance, need to invest effort (such as computation or information gathering) to determine their preferences [16]. • It preserves (some of) the agents" privacy. Most of the work on reducing the communication burden in preference aggregation has focused on resource allocation settings such as combinatorial auctions, in which an auctioneer auctions off a number of (possibly distinct) items in a single event. Because in a combinatorial auction, bidders can have separate valuations for each of an exponential number of possible bundles of items, this is a setting in which reducing the communication burden is especially crucial. This can be accomplished by supplementing the auctioneer with an elicitor that incrementally elicits parts of the bidders" preferences on an as-needed basis, based on what the bidders have revealed about their preferences so far, as suggested by Conen and Sandholm [5]. For example, the elicitor can ask for a bidder"s value for a specific bundle (value queries), which of two bundles the bidder prefers (order queries), which bundle he ranks kth or what the rank of a given bundle is (rank queries), which bundle he would purchase given a particular vector of prices (demand queries), etc.-until (at least) the final allocation can be determined. Experimentally, this yields drastic savings in preference revelation [11]. Furthermore, if the agents" valuation functions are drawn from certain natural subclasses, the elicitation problem can be solved using only polynomially many queries even in the worst case [23, 4, 13, 18, 14]. For a review of preference elicitation in combinatorial auctions, see [17]. Ascending combinatorial auctions are a well-known special form of preference elicitation, where the elicitor asks demand queries with increasing prices [15, 21, 1, 9]. Finally, resource 78 allocation problems have also been studied from a communication complexity viewpoint, thereby deriving lower bounds on the required communication. For example, Nisan and Segal show that exponential communication is required even to obtain a surplus greater than that obtained by auctioning off all objects as a single bundle [14]. Segal also studies social choice rules in general, and shows that for a large class of social choice rules, supporting budget sets must be revealed such that if every agent prefers the same outcome in her budget set, this proves the optimality of that outcome. Segal then uses this characterization to prove bounds on the communication required in resource allocation as well as matching settings [20]. In this paper, we will focus on the communication requirements of a generally applicable subclass of social choice rules, commonly known as voting rules. In a voting setting, there is a set of candidate outcomes over which the voters express their preferences by submitting a vote (typically, a ranking of the candidates), and the winner (that is, the chosen outcome) is determined based on these votes. The communication required by voting rules can be large either because the number of voters is large (such as, for example, in national elections), or because the number of candidates is large (for example, the agents can vote over allocations of a number of resources), or both. Prior work [8] has studied elicitation in voting, studying how computationally hard it is to decide whether a winner can be determined with the information elicited so far, as well as how hard it is to find the optimal sequence of queries given perfect suspicions about the voters" preferences. In addition, that paper discusses strategic (game-theoretic) issues introduced by elicitation. In contrast, in this paper, we are concerned with the worst-case number of bits that must be communicated to execute a given voting rule, when nothing is known in advance about the voters" preferences. We determine the communication complexity of the common voting rules. For each rule, we first give an upper bound on the (deterministic) communication complexity by providing a communication protocol for it and analyzing how many bits need to be transmitted in this protocol. (Segal"s results [20] do not apply to most voting rules because most voting rules are not intersectionmonotonic (or even monotonic).1 ) For many of the voting rules under study, it turns out that one cannot do better than simply letting each voter immediately communicate all her (potentially relevant) information. However, for some rules (such as plurality with runoff, STV and cup) there is a straightforward multistage communication protocol that, with some analysis, can be shown to significantly outperform the immediate communication of all (potentially relevant) information. Finally, for some rules (such as the Condorcet and Bucklin rules), we need to introduce a more complex communication protocol to achieve the best possible upper 1 For two of the rules that we study that are intersectionmonotonic, namely the approval and Condorcet rules, Segal"s results can in fact be used to give alternative proofs of our lower bounds. We only give direct proofs for these rules here because 1) these direct proofs are among the easier ones in this paper, 2) the alternative proofs are nontrivial even given Segal"s results, and 3) a space constraint applies. However, we hope to also include the alternative proofs in a later version. bound. After obtaining the upper bounds, we show that they are tight by giving matching lower bounds on (even the nondeterministic) communication complexity of each voting rule. There are two exceptions: STV, for which our upper and lower bounds are apart by a factor log m; and maximin, for which our best deterministic upper bound is also a factor log m above the (nondeterministic) lower bound, although we give a nondeterministic upper bound that matches the lower bound. 2. REVIEW OF VOTING RULES In this section, we review the common voting rules that we study in this paper. A voting rule2 is a function mapping a vector of the n voters" votes (i.e. preferences over candidates) to one of the m candidates (the winner) in the candidate set C. In some cases (such as the Condorcet rule), the rule may also declare that no winner exists. We do not concern ourselves with what happens in case of a tie between candidates (our lower bounds hold regardless of how ties are broken, and the communication protocols used for our upper bounds do not attempt to break the ties). All of the rules that we study are rank-based rules, which means that a vote is defined as an ordering of the candidates (with the exception of the plurality rule, for which a vote is a single candidate, and the approval rule, for which a vote is a subset of the candidates). We will consider the following voting rules. (For rules that define a score, the candidate with the highest score wins.) • scoring rules. Let α = α1, . . . , αm be a vector of integers such that α1 ≥ α2 . . . ≥ αm. For each voter, a candidate receives α1 points if it is ranked first by the voter, α2 if it is ranked second etc. The score sα of a candidate is the total number of points the candidate receives. The Borda rule is the scoring rule with α = m−1, m−2, . . . , 0 . The plurality rule is the scoring rule with α = 1, 0, . . . , 0 . • single transferable vote (STV). The rule proceeds through a series of m − 1 rounds. In each round, the candidate with the lowest plurality score (that is, the least number of voters ranking it first among the remaining candidates) is eliminated (and each of the votes for that candidate transfer to the next remaining candidate in the order given in that vote). The winner is the last remaining candidate. • plurality with run-off. In this rule, a first round eliminates all candidates except the two with the highest plurality scores. Votes are transferred to these as in the STV rule, and a second round determines the winner from these two. • approval. Each voter labels each candidate as either approved or disapproved. The candidate approved by the greatest number of voters wins. • Condorcet. For any two candidates i and j, let N(i, j) be the number of voters who prefer i to j. If there is a candidate i that is preferred to any other candidate by a majority of the voters (that is, N(i, j) > N(j, i) for all j = i-that is, i wins every pairwise election), then candidate i wins. 2 The term voting protocol is often used to describe the same concept, but we seek to draw a sharp distinction between the rule mapping preferences to outcomes, and the communication/elicitation protocol used to implement this rule. 79 • maximin (aka. Simpson). The maximin score of i is s(i) = minj=i N(i, j)-that is, i"s worst performance in a pairwise election. The candidate with the highest maximin score wins. • Copeland. For any two distinct candidates i and j, let C(i, j) = 1 if N(i, j) > N(j, i), C(i, j) = 1/2 if N(i, j) = N(j, i) and C(i, j) = 0 if N(i, j) < N(j, i). The Copeland score of candidate i is s(i) = j=i C(i, j). • cup (sequential binary comparisons). The cup rule is defined by a balanced3 binary tree T with one leaf per candidate, and an assignment of candidates to leaves (each leaf gets one candidate). Each non-leaf node is assigned the winner of the pairwise election of the node"s children; the candidate assigned to the root wins. • Bucklin. For any candidate i and integer l, let B(i, l) be the number of voters that rank candidate i among the top l candidates. The winner is arg mini(min{l : B(i, l) > n/2}). That is, if we say that a voter approves her top l candidates, then we repeatedly increase l by 1 until some candidate is approved by more than half the voters, and this candidate is the winner. • ranked pairs. This rule determines an order on all the candidates, and the winner is the candidate at the top of this order. Sort all ordered pairs of candidates (i, j) by N(i, j), the number of voters who prefer i to j. Starting with the pair (i, j) with the highest N(i, j), we lock in the result of their pairwise election (i j). Then, we move to the next pair, and we lock the result of their pairwise election. We continue to lock every pairwise result that does not contradict the ordering established so far. We emphasize that these definitions of voting rules do not concern themselves with how the votes are elicited from the voters; all the voting rules, including those that are suggestively defined in terms of rounds, are in actuality just functions mapping the vector of all the voters" votes to a winner. Nevertheless, there are always many different ways of eliciting the votes (or the relevant parts thereof) from the voters. For example, in the plurality with runoff rule, one way of eliciting the votes is to ask every voter to declare her entire ordering of the candidates up front. Alternatively, we can first ask every voter to declare only her most preferred candidate; then, we will know the two candidates in the runoff, and we can ask every voter which of these two candidates she prefers. Thus, we distinguish between the voting rule (the mapping from vectors of votes to outcomes) and the communication protocol (which determines how the relevant parts of the votes are actually elicited from the voters). The goal of this paper is to give efficient communication protocols for the voting rules just defined, and to prove that there do not exist any more efficient communication protocols. It is interesting to note that the choice of the communication protocol may affect the strategic behavior of the voters. Multistage communication protocols may reveal to the voters some information about how the other voters are voting (for example, in the two-stage communication protocol just given for plurality with runoff, in the second stage voters 3 Balanced means that the difference in depth between two leaves can be at most one. will know which two candidates have the highest plurality scores). In general, when the voters receive such information, it may give them incentives to vote differently than they would have in a single-stage communication protocol in which all voters declare their entire votes simultaneously. Of course, even the single-stage communication protocol is not strategy-proof4 for any reasonable voting rule, by the Gibbard-Satterthwaite theorem [10, 19]. However, this does not mean that we should not be concerned about adding even more opportunities for strategic voting. In fact, many of the communication protocols introduced in this paper do introduce additional opportunities for strategic voting, but we do not have the space to discuss this here. (In prior work [8], we do give an example where an elicitation protocol for the approval voting rule introduces strategic voting, and give principles for designing elicitation protocols that do not introduce strategic problems.) Now that we have reviewed voting rules, we move on to a brief review of communication complexity theory. 3. REVIEW OF SOME COMMUNICATION COMPLEXITY THEORY In this section, we review the basic model of a communication problem and the lower-bounding technique of constructing a fooling set. (The basic model of a communication problem is due to Yao [22]; for an overview see Kushilevitz and Nisan [12].) Each player 1 ≤ i ≤ n knows (only) input xi. Together, they seek to compute f(x1, x2, . . . , xn). In a deterministic protocol for computing f, in each stage, one of the players announces (to all other players) a bit of information based on her own input and the bits announced so far. Eventually, the communication terminates and all players know f(x1, x2, . . . , xn). The goal is to minimize the worst-case (over all input vectors) number of bits sent. The deterministic communication complexity of a problem is the worstcase number of bits sent in the best (correct) deterministic protocol for it. In a nondeterministic protocol, the next bit to be sent can be chosen nondeterministically. For the purposes of this paper, we will consider a nondeterministic protocol correct if for every input vector, there is some sequence of nondeterministic choices the players can make so that the players know the value of f when the protocol terminates. The nondeterministic communication complexity of a problem is the worst-case number of bits sent in the best (correct) nondeterministic protocol for it. We are now ready to give the definition of a fooling set. Definition 1. A fooling set is a set of input vectors {(x1 1, x1 2, . . . , x1 n), (x2 1, x2 2, . . . , x2 n), . . . , (xk 1 , xk 2 , . . . , xk n) such that for any i, f(xi 1, xi 2, . . . , xi n) = f0 for some constant f0, but for any i = j, there exists some vector (r1, r2, . . . , rn) ∈ {i, j}n such that f(xr1 1 , xr2 2 , . . . , xrn n ) = f0. (That is, we can mix the inputs from the two input vectors to obtain a vector with a different function value.) It is known that if a fooling set of size k exists, then log k is a lower bound on the communication complexity (even the nondeterministic communication complexity) [12]. 4 A strategy-proof protocol is one in which it is in the players" best interest to report their preferences truthfully. 80 For the purposes of this paper, f is the voting rule that maps the votes to the winning candidate, and xi is voter i"s vote (the information that the voting rule would require from the voter if there were no possibility of multistage communication-i.e. the most preferred candidate (plurality), the approved candidates (approval), or the ranking of all the candidates (all other protocols)). However, when we derive our lower bounds, f will only signify whether a distinguished candidate a wins. (That is, f is 1 if a wins, and 0 otherwise.) This will strengthen our lower bound results (because it implies that even finding out whether one given candidate wins is hard).5 Thus, a fooling set in our context is a set of vectors of votes so that a wins (does not win) with each of them; but for any two different vote vectors in the set, there is a way of taking some voters" votes from the first vector and the others" votes from the second vector, so that a does not win (wins). To simplify the proofs of our lower bounds, we make assumptions such as the number of voters n is odd in many of these proofs. Therefore, technically, we do not prove the lower bound for (number of candidates, number of voters) pairs (m, n) that do not satisfy these assumptions (for example, if we make the above assumption, then we technically do not prove the lower bound for any pair (m, n) in which n is even). Nevertheless, we always prove the lower bound for a representative set of (m, n) pairs. For example, for every one of our lower bounds it is the case that for infinitely many values of m, there are infinitely many values of n such that the lower bound is proved for the pair (m, n). 4. RESULTS We are now ready to present our results. For each voting rule, we first give a deterministic communication protocol for determining the winner to establish an upper bound. Then, we give a lower bound on the nondeterministic communication complexity (even on the complexity of deciding whether a given candidate wins, which is an easier question). The lower bounds match the upper bounds in all but two cases: the STV rule (upper bound O(n(log m)2 ); lower bound Ω(n log m)) and the maximin rule (upper bound O(nm log m), although we do give a nondeterministic protocol that is O(nm); lower bound Ω(nm)). When we discuss a voting rule in which the voters rank the candidates, we will represent a ranking in which candidate c1 is ranked first, c2 is ranked second, etc. as c1 c2 . . . cm. 5 One possible concern is that in the case where ties are possible, it may require much communication to verify whether a specific candidate a is among the winners, but little communication to produce one of the winners. However, all the fooling sets we use in the proofs have the property that if a wins, then a is the unique winner. Therefore, in these fooling sets, if one knows any one of the winners, then one knows whether a is a winner. Thus, computing one of the winners requires at least as much communication as verifying whether a is among the winners. In general, when a communication problem allows multiple correct answers for a given vector of inputs, this is known as computing a relation rather than a function [12]. However, as per the above, we can restrict our attention to a subset of the domain where the voting rule truly is a (single-valued) function, and hence lower bounding techniques for functions rather than relations will suffice. Sometimes for the purposes of a proof the internal ranking of a subset of the candidates does not matter, and in this case we will not specify it. For example, if S = {c2, c3}, then c1 S c4 indicates that either the ranking c1 c2 c3 c4 or the ranking c1 c3 c2 c4 can be used for the proof. We first give a universal upper bound. Theorem 1. The deterministic communication complexity of any rank-based voting rule is O(nm log m). Proof. This bound is achieved by simply having everyone communicate their entire ordering of the candidates (indicating the rank of an individual candidate requires only O(log m) bits, so each of the n voters can simply indicate the rank of each of the m candidates). The next lemma will be useful in a few of our proofs. Lemma 1. If m divides n, then log(n!)−m log((n/m)!) ≥ n(log m − 1)/2. Proof. If n/m = 1 (that is, n = m), then this expression simplifies to log(n!). We have log(n!) = n i=1 log i ≥ n x=1 log(i)dx, which, using integration by parts, is equal to n log n − (n − 1) > n(log n − 1) = n(log m − 1) > n(log m − 1)/2. So, we can assume that n/m ≥ 2. We observe that log(n!) = n i=1 log i = n/m−1 i=0 m j=1 log(im+j) ≥ n/m−1 i=1 m j=1 log(im) = m n/m−1 i=1 log(im), and that m log((n/m)!) = m n/m i=1 log(i). Therefore, log(n!) − m log((n/m)!) ≥ m n/m−1 i=1 log(im) − m n/m i=1 log(i) = m(( n/m−1 i=1 log(im/i))−log(n/m)) = m((n/m− 1) log m−log n+log m) = n log m−m log n. Now, using the fact that n/m ≥ 2, we have m log n = n(m/n) log m(n/m) = n(m/n)(log m + log(n/m)) ≤ n(1/2)(log m + log 2). Thus, log(n!) − m log((n/m)!) ≥ n log m − m log n ≥ n log m − n(1/2)(log m + log 2) = n(log m − 1)/2. Theorem 2. The deterministic communication complexity of the plurality rule is O(n log m). Proof. Indicating one of the candidates requires only O(log m) bits, so each voter can simply indicate her most preferred candidate. Theorem 3. The nondeterministic communication complexity of the plurality rule is Ω(n log m) (even to decide whether a given candidate a wins). Proof. We will exhibit a fooling set of size n ! (( n m )!)m where n = (n−1)/2. Taking the logarithm of this gives log(n !)− m log((n /m)!), so the result follows from Lemma 1. The fooling set will consist of all vectors of votes satisfying the following constraints: • For any 1 ≤ i ≤ n , voters 2i−1 and 2i vote the same. 81 • Every candidate receives equally many votes from the first 2n = n − 1 voters. • The last voter (voter n) votes for a. Candidate a wins with each one of these vote vectors because of the extra vote for a from the last voter. Given that m divides n , let us see how many vote vectors there are in the fooling set. We need to distribute n voter pairs evenly over m candidates, for a total of n /m voter pairs per candidate; and there are precisely n ! (( n m )!)m ways of doing this.6 All that remains to show is that for any two distinct vectors of votes in the fooling set, we can let each of the voters vote according to one of these two vectors in such a way that a loses. Let i be a number such that the two vote vectors disagree on the candidate for which voters 2i − 1 and 2i vote. Without loss of generality, suppose that in the first vote vector, these voters do not vote for a (but for some other candidate, b, instead). Now, construct a new vote vector by taking votes 2i − 1 and 2i from the first vote vector, and the remaining votes from the second vote vector. Then, b receives 2n /m + 2 votes in this newly constructed vote vector, whereas a receives at most 2n /m+1 votes. So, a is not the winner in the newly constructed vote vector, and hence we have a correct fooling set. Theorem 4. The deterministic communication complexity of the plurality with runoff rule is O(n log m). Proof. First, let every voter indicate her most preferred candidate using log m bits. After this, the two candidates in the runoff are known, and each voter can indicate which one she prefers using a single additional bit. Theorem 5. The nondeterministic communication complexity of the plurality with runoff rule is Ω(n log m) (even to decide whether a given candidate a wins). Proof. We will exhibit a fooling set of size n ! (( n m )!)m where m = m/2 and n = (n − 2)/4. Taking the logarithm of this gives log(n !) − m log((n /m )!), so the result follows from Lemma 1. Divide the candidates into m pairs: (c1, d1), (c2, d2), . . . , (cm , dm ) where c1 = a and d1 = b. The fooling set will consist of all vectors of votes satisfying the following constraints: • For any 1 ≤ i ≤ n , voters 4i − 3 and 4i − 2 rank the candidates ck(i) a C − {a, ck(i)}, for some candidate ck(i). (If ck(i) = a then the vote is simply a C − {a}.) • For any 1 ≤ i ≤ n , voters 4i − 1 and 4i rank the candidates dk(i) a C − {a, dk(i)} (that is, their most preferred candidate is the candidate that is paired with the candidate that the previous two voters vote for). 6 An intuitive proof of this is the following. We can count the number of permutations of n elements as follows. First, divide the elements into m buckets of size n /m, so that if x is placed in a lower-indexed bucket than y, then x will be indexed lower in the eventual permutation. Then, decide on the permutation within each bucket (for which there are (n /m)! choices per bucket). It follows that n ! equals the number of ways to divide n elements into m buckets of size n /m, times ((n /m)!)m . • Every candidate is ranked at the top of equally many of the first 4n = n − 2 votes. • Voter 4n +1 = n−1 ranks the candidates a C−{a}. • Voter 4n + 2 = n ranks the candidates b C − {b}. Candidate a wins with each one of these vote vectors: because of the last two votes, candidates a and b are one vote ahead of all the other candidates and continue to the runoff, and at this point all the votes that had another candidate ranked at the top transfer to a, so that a wins the runoff. Given that m divides n , let us see how many vote vectors there are in the fooling set. We need to distribute n groups of four voters evenly over the m pairs of candidates, and (as in the proof of Theorem 3) there are n ! (( n m )!)m ways of doing this. All that remains to show is that for any two distinct vectors of votes in the fooling set, we can let each of the voters vote according to one of these two vectors in such a way that a loses. Let i be a number such that ck(i) is not the same in both of these two vote vectors, that is, c1 k(i) (ck(i) in the first vote vector) is not equal to c2 k(i) (ck(i) in the second vote vector). Without loss of generality, suppose c1 k(i) = a. Now, construct a new vote vector by taking votes 4i − 3, 4i − 2, 4i − 1, 4i from the first vote vector, and the remaining votes from the second vote vector. In this newly constructed vote vector, c1 k(i) and d1 k(i) each receive 4n /m+2 votes in the first round, whereas a receives at most 4n /m+1 votes. So, a does not continue to the runoff in the newly constructed vote vector, and hence we have a correct fooling set. Theorem 6. The nondeterministic communication complexity of the Borda rule is Ω(nm log m) (even to decide whether a given candidate a wins). Proof. We will exhibit a fooling set of size (m !)n where m = m−2 and n = (n−2)/4. This will prove the theorem because m ! is Ω(m log m), so that log((m !)n ) = n log(m !) is Ω(nm log m). For every vector (π1, π2, . . . , πn ) consisting of n orderings of all candidates other than a and another fixed candidate b (technically, the orderings take the form of a one-to-one function πi : {1, 2, . . . , m } → C − {a, b} with πi(j) = c indicating that candidate c is the jth in the order represented by πi), let the following vector of votes be an element of the fooling set: • For 1 ≤ i ≤ n , let voters 4i − 3 and 4i − 2 rank the candidates a b πi(1) πi(2) . . . πi(m ). • For 1 ≤ i ≤ n , let voters 4i − 1 and 4i rank the candidates πi(m ) πi(m − 1) . . . πi(1) b a. • Let voter 4n + 1 = n − 1 rank the candidates a b π0(1) π0(2) . . . π0(m ) (where π0 is an arbitrary order of the candidates other than a and b which is the same for every element of the fooling set). • Let voter 4n + 2 = n rank the candidates π0(m ) π0(m − 1) . . . π0(1) a b. We observe that this fooling set has size (m !)n , and that candidate a wins in each vector of votes in the fooling set (to 82 see why, we observe that for any 1 ≤ i ≤ n , votes 4i−3 and 4i − 2 rank the candidates in the exact opposite way from votes 4i − 1 and 4i, which under the Borda rule means they cancel out; and the last two votes give one more point to a than to any other candidate-besides b who gets two fewer points than a). All that remains to show is that for any two distinct vectors of votes in the fooling set, we can let each of the voters vote according to one of these two vectors in such a way that a loses. Let the first vote vector correspond to the vector (π1 1, π1 2, . . . , π1 n ), and let the second vote vector correspond to the vector (π2 1, π2 2, . . . , π2 n ). For some i, we must have π1 i = π2 i , so that for some candidate c /∈ {a, b}, (π1 i )−1 (c) < (π2 i )−1 (c) (that is, c is ranked higher in π1 i than in π2 i ). Now, construct a new vote vector by taking votes 4i−3 and 4i−2 from the first vote vector, and the remaining votes from the second vote vector. a"s Borda score remains unchanged. However, because c is ranked higher in π1 i than in π2 i , c receives at least 2 more points from votes 4i−3 and 4i − 2 in the newly constructed vote vector than it did in the second vote vector. It follows that c has a higher Borda score than a in the newly constructed vote vector. So, a is not the winner in the newly constructed vote vector, and hence we have a correct fooling set. Theorem 7. The nondeterministic communication complexity of the Copeland rule is Ω(nm log m) (even to decide whether a given candidate a wins). Proof. We will exhibit a fooling set of size (m !)n where m = (m − 2)/2 and n = (n − 2)/2. This will prove the theorem because m ! is Ω(m log m), so that log((m !)n ) = n log(m !) is Ω(nm log m). We write the set of candidates as the following disjoint union: C = {a, b} ∪ L ∪ R where L = {l1, l2, . . . , lm } and R = {r1, r2, . . . , rm }. For every vector (π1, π2, . . . , πn ) consisting of n permutations of the integers 1 through m (πi : {1, 2, . . . , m } → {1, 2, . . . , m }), let the following vector of votes be an element of the fooling set: • For 1 ≤ i ≤ n , let voter 2i − 1 rank the candidates a b lπi(1) rπi(1) lπi(2) rπi(2) . . . lπi(m ) rπi(m ). • For 1 ≤ i ≤ n , let voter 2i rank the candidates rπi(m ) lπi(m ) rπi(m −1) lπi(m −1) . . . rπi(1) lπi(1) b a. • Let voter n − 1 = 2n + 1 rank the candidates a b l1 r1 l2 r2 . . . lm rm . • Let voter n = 2n +2 rank the candidates rm lm rm −1 lm −1 . . . r1 l1 a b. We observe that this fooling set has size (m !)n , and that candidate a wins in each vector of votes in the fooling set (every pair of candidates is tied in their pairwise election, with the exception that a defeats b, so that a wins the election by half a point). All that remains to show is that for any two distinct vectors of votes in the fooling set, we can let each of the voters vote according to one of these two vectors in such a way that a loses. Let the first vote vector correspond to the vector (π1 1, π1 2, . . . , π1 n ), and let the second vote vector correspond to the vector (π2 1, π2 2, . . . , π2 n ). For some i, we must have π1 i = π2 i , so that for some j ∈ {1, 2, . . . , m }, we have (π1 i )−1 (j) < (π2 i )−1 (j). Now, construct a new vote vector by taking vote 2i−1 from the first vote vector, and the remaining votes from the second vote vector. a"s Copeland score remains unchanged. Let us consider the score of lj. We first observe that the rank of lj in vote 2i − 1 in the newly constructed vote vector is at least 2 higher than it was in the second vote vector, because (π1 i )−1 (j) < (π2 i )−1 (j). Let D1 (lj) be the set of candidates in L ∪ R that voter 2i − 1 ranked lower than lj in the first vote vector (D1 (lj) = {c ∈ L ∪ R : lj 1 2i−1 c}), and let D2 (lj) be the set of candidates in L ∪ R that voter 2i − 1 ranked lower than lj in the second vote vector (D2 (lj) = {c ∈ L ∪ R : lj 2 2i−1 c}). Then, it follows that in the newly constructed vote vector, lj defeats all the candidates in D1 (lj) − D2 (lj) in their pairwise elections (because lj receives an extra vote in each one of these pairwise elections relative to the second vote vector), and loses to all the candidates in D2 (lj) − D1 (lj) (because lj loses a vote in each one of these pairwise elections relative to the second vote vector), and ties with everyone else. But |D1 (lj)|−|D2 (lj)| ≥ 2, and hence |D1 (lj) − D2 (lj)| − |D2 (lj) − D1 (lj)| ≥ 2. Hence, in the newly constructed vote vector, lj has at least two more pairwise wins than pairwise losses, and therefore has at least 1 more point than if lj had tied all its pairwise elections. Thus, lj has a higher Copeland score than a in the newly constructed vote vector. So, a is not the winner in the newly constructed vote vector, and hence we have a correct fooling set. Theorem 8. The nondeterministic communication complexity of the maximin rule is O(nm). Proof. The nondeterministic protocol will guess which candidate w is the winner, and, for each other candidate c, which candidate o(c) is the candidate against whom c receives its lowest score in a pairwise election. Then, let every voter communicate the following: • for each candidate c = w, whether she prefers c to w; • for each candidate c = w, whether she prefers c to o(c). We observe that this requires the communication of 2n(m− 1) bits. If the guesses were correct, then, letting N(d, e) be the number of voters preferring candidate d to candidate e, we should have N(c, o(c)) < N(w, c ) for any c = w, c = w, which will prove that w wins the election. Theorem 9. The nondeterministic communication complexity of the maximin rule is Ω(nm) (even to decide whether a given candidate a wins). Proof. We will exhibit a fooling set of size 2n m where m = m − 2 and n = (n − 1)/4. Let b be a candidate other than a. For every vector (S1, S2, . . . , Sn ) consisting of n subsets Si ⊆ C − {a, b}, let the following vector of votes be an element of the fooling set: • For 1 ≤ i ≤ n , let voters 4i − 3 and 4i − 2 rank the candidates Si a C − (Si ∪ {a, b}) b. • For 1 ≤ i ≤ n , let voters 4i − 1 and 4i rank the candidates b C − (Si ∪ {a, b}) a Si. 83 • Let voter 4n + 1 = n rank the candidates a b C − {a, b}. We observe that this fooling set has size (2m )n = 2n m , and that candidate a wins in each vector of votes in the fooling set (in every one of a"s pairwise elections, a is ranked higher than its opponent by 2n +1 = (n+1)/2 > n/2 votes). All that remains to show is that for any two distinct vectors of votes in the fooling set, we can let each of the voters vote according to one of these two vectors in such a way that a loses. Let the first vote vector correspond to the vector (S1 1 , S1 2 , . . . , S1 n ), and let the second vote vector correspond to the vector (S2 1 , S2 2 , . . . , S2 n ). For some i, we must have S1 i = S2 i , so that either S1 i S2 i or S2 i S1 i . Without loss of generality, suppose S1 i S2 i , and let c be some candidate in S1 i − S2 i . Now, construct a new vote vector by taking votes 4i − 3 and 4i − 2 from the first vote vector, and the remaining votes from the second vote vector. In this newly constructed vote vector, a is ranked higher than c by only 2n −1 voters, for the following reason. Whereas voters 4i−3 and 4i − 2 do not rank c higher than a in the second vote vector (because c /∈ S2 i ), voters 4i − 3 and 4i − 2 do rank c higher than a in the first vote vector (because c ∈ S1 i ). Moreover, in every one of b"s pairwise elections, b is ranked higher than its opponent by at least 2n voters. So, a has a lower maximin score than b, therefore a is not the winner in the newly constructed vote vector, and hence we have a correct fooling set. Theorem 10. The deterministic communication complexity of the STV rule is O(n(log m)2 ). Proof. Consider the following communication protocol. Let each voter first announce her most preferred candidate (O(n log m) communication). In the remaining rounds, we will keep track of each voter"s most preferred candidate among the remaining candidates, which will be enough to implement the rule. When candidate c is eliminated, let each of the voters whose most preferred candidate among the remaining candidates was c announce their most preferred candidate among the candidates remaining after c"s elimination. If candidate c was the ith candidate to be eliminated (that is, there were m − i + 1 candidates remaining before c"s elimination), it follows that at most n/(m − i + 1) voters had candidate c as their most preferred candidate among the remaining candidates, and thus the number of bits to be communicated after the elimination of the ith candidate is O((n/(m−i+1)) log m).7 Thus, the total communication in this communication protocol is O(n log m + m−1 i=1 (n/(m − i + 1)) log m). Of course, m−1 i=1 1/(m − i + 1) = m i=2 1/i, which is O(log m). Substituting into the previous expression, we find that the communication complexity is O(n(log m)2 ). Theorem 11. The nondeterministic communication complexity of the STV rule is Ω(n log m) (even to decide whether a given candidate a wins). Proof. We omit this proof because of space constraint. 7 Actually, O((n/(m − i + 1)) log(m − i + 1)) is also correct, but it will not improve the bound. Theorem 12. The deterministic communication complexity of the approval rule is O(nm). Proof. Approving or disapproving of a candidate requires only one bit of information, so every voter can simply approve or disapprove of every candidate for a total communication of nm bits. Theorem 13. The nondeterministic communication complexity of the approval rule is Ω(nm) (even to decide whether a given candidate a wins). Proof. We will exhibit a fooling set of size 2n m where m = m − 1 and n = (n − 1)/4. For every vector (S1, S2, . . . , Sn ) consisting of n subsets Si ⊆ C − {a}, let the following vector of votes be an element of the fooling set: • For 1 ≤ i ≤ n , let voters 4i − 3 and 4i − 2 approve Si ∪ {a}. • For 1 ≤ i ≤ n , let voters 4i − 1 and 4i approve C − (Si ∪ {a}). • Let voter 4n + 1 = n approve {a}. We observe that this fooling set has size (2m )n = 2n m , and that candidate a wins in each vector of votes in the fooling set (a is approved by 2n + 1 voters, whereas each other candidate is approved by only 2n voters). All that remains to show is that for any two distinct vectors of votes in the fooling set, we can let each of the voters vote according to one of these two vectors in such a way that a loses. Let the first vote vector correspond to the vector (S1 1 , S1 2 , . . . , S1 n ), and let the second vote vector correspond to the vector (S2 1 , S2 2 , . . . , S2 n ). For some i, we must have S1 i = S2 i , so that either S1 i S2 i or S2 i S1 i . Without loss of generality, suppose S1 i S2 i , and let b be some candidate in S1 i − S2 i . Now, construct a new vote vector by taking votes 4i − 3 and 4i − 2 from the first vote vector, and the remaining votes from the second vote vector. In this newly constructed vote vector, a is still approved by 2n + 1 votes. However, b is approved by 2n + 2 votes, for the following reason. Whereas voters 4i−3 and 4i−2 do not approve b in the second vote vector (because b /∈ S2 i ), voters 4i − 3 and 4i − 2 do approve b in the first vote vector (because b ∈ S1 i ). It follows that b"s score in the newly constructed vote vector is b"s score in the second vote vector (2n ), plus two. So, a is not the winner in the newly constructed vote vector, and hence we have a correct fooling set. Interestingly, an Ω(m) lower bound can be obtained even for the problem of finding a candidate that is approved by more than one voter [20]. Theorem 14. The deterministic communication complexity of the Condorcet rule is O(nm). Proof. We maintain a set of active candidates S which is initialized to C. At each stage, we choose two of the active candidates (say, the two candidates with the lowest indices), and we let each voter communicate which of the two candidates she prefers. (Such a stage requires the communication of n bits, one per voter.) The candidate preferred by fewer 84 voters (the loser of the pairwise election) is removed from S. (If the pairwise election is tied, both candidates are removed.) After at most m − 1 iterations, only one candidate is left (or zero candidates are left, in which case there is no Condorcet winner). Let a be the remaining candidate. To find out whether candidate a is the Condorcet winner, let each voter communicate, for every candidate c = a, whether she prefers a to c. (This requires the communication of at most n(m − 1) bits.) This is enough to establish whether a won each of its pairwise elections (and thus, whether a is the Condorcet winner). Theorem 15. The nondeterministic communication complexity of the Condorcet rule is Ω(nm) (even to decide whether a given candidate a wins). Proof. We will exhibit a fooling set of size 2n m where m = m − 1 and n = (n − 1)/2. For every vector (S1, S2, . . . , Sn ) consisting of n subsets Si ⊆ C − {a}, let the following vector of votes be an element of the fooling set: • For 1 ≤ i ≤ n , let voter 2i − 1 rank the candidates Si a C − Si. • For 1 ≤ i ≤ n , let voter 2i rank the candidates C − Si a Si. • Let voter 2n +1 = n rank the candidates a C −{a}. We observe that this fooling set has size (2m )n = 2n m , and that candidate a wins in each vector of votes in the fooling set (a wins each of its pairwise elections by a single vote). All that remains to show is that for any two distinct vectors of votes in the fooling set, we can let each of the voters vote according to one of these two vectors in such a way that a loses. Let the first vote vector correspond to the vector (S1 1 , S1 2 , . . . , S1 n ), and let the second vote vector correspond to the vector (S2 1 , S2 2 , . . . , S2 n ). For some i, we must have S1 i = S2 i , so that either S1 i S2 i or S2 i S1 i . Without loss of generality, suppose S1 i S2 i , and let b be some candidate in S1 i − S2 i . Now, construct a new vote vector by taking vote 2i − 1 from the first vote vector, and the remaining votes from the second vote vector. In this newly constructed vote vector, b wins its pairwise election against a by one vote (vote 2i − 1 ranks b above a in the newly constructed vote vector because b ∈ S1 i , whereas in the second vote vector vote 2i − 1 ranked a above b because b /∈ S2 i ). So, a is not the Condorcet winner in the newly constructed vote vector, and hence we have a correct fooling set. Theorem 16. The deterministic communication complexity of the cup rule is O(nm). Proof. Consider the following simple communication protocol. First, let all the voters communicate, for every one of the matchups in the first round, which of its two candidates they prefer. After this, the matchups for the second round are known, so let all the voters communicate which candidate they prefer in each matchup in the second round-etc. Because communicating which of two candidates is preferred requires only one bit per voter, and because there are only m − 1 matchups in total, this communication protocol requires O(nm) communication. Theorem 17. The nondeterministic communication complexity of the cup rule is Ω(nm) (even to decide whether a given candidate a wins). Proof. We will exhibit a fooling set of size 2n m where m = (m − 1)/2 and n = (n − 7)/2. Given that m + 1 is a power of 2, so that one candidate gets a bye (that is, does not face an opponent) in the first round, let a be the candidate with the bye. Of the m first-round matchups, let lj denote the one (left) candidate in the jth matchup, and let rj be the other (right) candidate. Let L = {lj : 1 ≤ j ≤ m } and R = {rj : 1 ≤ j ≤ m }, so that C = L ∪ R ∪ {a}. . . . . . . . . . l r l r l r a m"1 1 2 2 m" Figure 1: The schedule for the cup rule used in the proof of Theorem 17. For every vector (S1, S2, . . . , Sn ) consisting of n subsets Si ⊆ R, let the following vector of votes be an element of the fooling set: • For 1 ≤ i ≤ n , let voter 2i − 1 rank the candidates Si L a R − Si. • For 1 ≤ i ≤ n , let voter 2i rank the candidates R − Si L a Si. • Let voters 2n +1 = n−6, 2n +2 = n−5, 2n +3 = n−4 rank the candidates L a R. • Let voters 2n + 4 = n − 3, 2n + 5 = n − 2 rank the candidates a r1 l1 r2 l2 . . . rm lm . • Let voters 2n + 6 = n − 1, 2n + 7 = n rank the candidates rm lm rm −1 lm −1 . . . r1 l1 a. We observe that this fooling set has size (2m )n = 2n m . Also, candidate a wins in each vector of votes in the fooling set, for the following reasons. Each candidate rj defeats its opponent lj in the first round. (For any 1 ≤ i ≤ n , the net effect of votes 2i − 1 and 2i on the pairwise election between rj and lj is zero; votes n − 6, n − 5, n − 4 prefer lj to rj, but votes n − 3, n − 2, n − 1, n all prefer rj to lj.) Moreover, a defeats every rj in their pairwise election. (For any 1 ≤ i ≤ n , the net effect of votes 2i − 1 and 2i on the pairwise election between a and rj is zero; votes n − 1, n prefer rj to a, but votes n − 6, n − 5, n − 4, n − 3, n − 2 all prefer a to rj.) It follows that a will defeat all the candidates that it faces. All that remains to show is that for any two distinct vectors of votes in the fooling set, we can let each of the voters vote according to one of these two vectors in such a way that a loses. Let the first vote vector correspond to the vector 85 (S1 1 , S1 2 , . . . , S1 n ), and let the second vote vector correspond to the vector (S2 1 , S2 2 , . . . , S2 n ). For some i, we must have S1 i = S2 i , so that either S1 i S2 i or S2 i S1 i . Without loss of generality, suppose S1 i S2 i , and let rj be some candidate in S1 i − S2 i . Now, construct a new vote vector by taking vote 2i from the first vote vector, and the remaining votes from the second vote vector. We note that, whereas in the second vote vector vote 2i preferred rj to lj (because rj ∈ R−S2 i ), in the newly constructed vote vector this is no longer the case (because rj ∈ S1 i ). It follows that, whereas in the second vote vector, rj defeated lj in the first round by one vote, in the newly constructed vote vector, lj defeats rj in the first round. Thus, at least one lj advances to the second round after defeating its opponent rj. Now, we observe that in the newly constructed vote vector, any lk wins its pairwise election against any rq with q = k. This is because among the first 2n votes, at least n − 1 prefer lk to rq; votes n − 6, n − 5, n − 4 prefer lk to rq; and, because q = k, either votes n − 3, n − 2 prefer lk to rq (if k < q), or votes n − 1, n prefer lk to rq (if k > q). Thus, at least n + 4 = (n + 1)/2 > n/2 votes prefer lk to rq. Moreover, any lk wins its pairwise election against a. This is because only votes n − 3 and n − 2 prefer a to lk. It follows that, after the first round, any surviving candidate lk can only lose a matchup against another surviving lk , so that one of the lk must win the election. So, a is not the winner in the newly constructed vote vector, and hence we have a correct fooling set. Theorem 18. The deterministic communication complexity of the Bucklin rule is O(nm). Proof. Let l be the minimum integer for which there is a candidate who is ranked among the top l candidates by more than half the votes. We will do a binary search for l. At each point, we will have a lower bound lL which is smaller than l (initialized to 0), and an upper bound lH which is at least l (initialized to m). While lH − lL > 1, we continue by finding out whether (lH − l)/2 is smaller than l, after which we can update the bounds. To find out whether a number k is smaller than l, we determine every voter"s k most preferred candidates. Every voter can communicate which candidates are among her k most preferred candidates using m bits (for each candidate, indicate whether the candidate is among the top k or not), but because the binary search requires log m iterations, this gives us an upper bound of O((log m)nm), which is not strong enough. However, if lL < k < lH , and we already know a voter"s lL most preferred candidates, as well as her lH most preferred candidates, then the voter no longer needs to communicate whether the lL most preferred candidates are among her k most preferred candidates (because they must be), and she no longer needs to communicate whether the m−lH least preferred candidates are among her k most preferred candidates (because they cannot be). Thus the voter needs to communicate only m−lL −(m−lH ) = lH −lL bits in any given stage. Because each stage, lH − lL is (roughly) halved, each voter in total communicates only (roughly) m + m/2 + m/4 + . . . ≤ 2m bits. Theorem 19. The nondeterministic communication complexity of the Bucklin rule is Ω(nm) (even to decide whether a given candidate a wins). Proof. We will exhibit a fooling set of size 2n m where m = (m−1)/2 and n = n/2. We write the set of candidates as the following disjoint union: C = {a} ∪ L ∪ R where L = {l1, l2, . . . , lm } and R = {r1, r2, . . . , rm }. For any subset S ⊆ {1, 2, . . . , m }, let L(S) = {li : i ∈ S} and let R(S) = {ri : i ∈ S}. For every vector (S1, S2, . . . , Sn ) consisting of n sets Si ⊆ {1, 2, . . . , m }, let the following vector of votes be an element of the fooling set: • For 1 ≤ i ≤ n , let voter 2i − 1 rank the candidates L(Si) R − R(Si) a L − L(Si) R(Si). • For 1 ≤ i ≤ n , let voter 2i rank the candidates L − L(Si) R(Si) a L(Si) R − R(Si). We observe that this fooling set has size (2m )n = 2n m , and that candidate a wins in each vector of votes in the fooling set, for the following reason. Each candidate in C − {a} is ranked among the top m candidates by exactly half the voters (which is not enough to win). Thus, we need to look at the voters" top m +1 candidates, and a is ranked m +1th by all voters. All that remains to show is that for any two distinct vectors of votes in the fooling set, we can let each of the voters vote according to one of these two vectors in such a way that a loses. Let the first vote vector correspond to the vector (S1 1 , S1 2 , . . . , S1 n ), and let the second vote vector correspond to the vector (S2 1 , S2 2 , . . . , S2 n ). For some i, we must have S1 i = S2 i , so that either S1 i S2 i or S2 i S1 i . Without loss of generality, suppose S1 i S2 i , and let j be some integer in S1 i − S2 i . Now, construct a new vote vector by taking vote 2i − 1 from the first vote vector, and the remaining votes from the second vote vector. In this newly constructed vote vector, a is still ranked m + 1th by all votes. However, lj is ranked among the top m candidates by n + 1 = n/2 + 1 votes. This is because whereas vote 2i − 1 does not rank lj among the top m candidates in the second vote vector (because j /∈ S2 i , we have lj /∈ L(S2 i )), vote 2i − 1 does rank lj among the top m candidates in the first vote vector (because j ∈ S1 i , we have lj ∈ L(S1 i )). So, a is not the winner in the newly constructed vote vector, and hence we have a correct fooling set. Theorem 20. The nondeterministic communication complexity of the ranked pairs rule is Ω(nm log m) (even to decide whether a given candidate a wins). Proof. We omit this proof because of space constraint. 5. DISCUSSION One key obstacle to using voting for preference aggregation is the communication burden that an election places on the voters. By lowering this burden, it may become feasible to conduct more elections over more issues. In the limit, this could lead to a shift from representational government to a system in which most issues are decided by referenda-a veritable e-democracy. In this paper, we analyzed the communication complexity of the common voting rules. Knowing which voting rules require little communication is especially important when the issue to be voted on is of low enough importance that the following is true: the parties involved are willing to accept a rule that tends 86 to produce outcomes that are slightly less representative of the voters" preferences, if this rule reduces the communication burden on the voters significantly. The following table summarizes the results we obtained. Rule Lower bound Upper bound plurality Ω(n log m) O(n log m) plurality w/ runoff Ω(n log m) O(n log m) STV Ω(n log m) O(n(log m)2) Condorcet Ω(nm) O(nm) approval Ω(nm) O(nm) Bucklin Ω(nm) O(nm) cup Ω(nm) O(nm) maximin Ω(nm) O(nm) Borda Ω(nm log m) O(nm log m) Copeland Ω(nm log m) O(nm log m) ranked pairs Ω(nm log m) O(nm log m) Communication complexity of voting rules, sorted from low to high. All of the upper bounds are deterministic (with the exception of maximin, for which the best deterministic upper bound we proved is O(nm log m)). All of the lower bounds hold even for nondeterministic communication and even just for determining whether a given candidate a is the winner. One area of future research is to study what happens when we restrict our attention to communication protocols that do not reveal any strategically useful information. This restriction may invalidate some of the upper bounds that we derived using multistage communication protocols. Also, all of our bounds are worst-case bounds. It may be possible to outperform these bounds when the distribution of votes has additional structure. When deciding which voting rule to use for an election, there are many considerations to take into account. The voting rules that we studied in this paper are the most common ones that have survived the test of time. One way to select among these rules is to consider recent results on complexity. The table above shows that from a communication complexity perspective, plurality, plurality with runoff, and STV are preferable. However, plurality has the undesirable property that it is computationally easy to manipulate by voting strategically [3, 7]. Plurality with runoff is NP-hard to manipulate by a coalition of weighted voters, or by an individual that faces correlated uncertainty about the others" votes [7, 6]. STV is NP-hard to manipulate in those settings as well [7], but also by an individual with perfect knowledge of the others" votes (when the number of candidates is unbounded) [2]. Therefore, STV is more robust, although it may require slightly more worst-case communication as per the table above. Yet other selection criteria are the computational complexity of determining whether enough information has been elicited to declare a winner, and that of determining the optimal sequence of queries [8]. 6. REFERENCES [1] Lawrence Ausubel and Paul Milgrom. Ascending auctions with package bidding. Frontiers of Theoretical Economics, 1, 2002. No. 1, Article 1. [2] John Bartholdi, III and James Orlin. Single transferable vote resists strategic voting. Social Choice and Welfare, 8(4):341-354, 1991. [3] John Bartholdi, III, Craig Tovey, and Michael Trick. The computational difficulty of manipulating an election. Social Choice and Welfare, 6(3):227-241, 1989. [4] Avrim Blum, Jeffrey Jackson, Tuomas Sandholm, and Martin Zinkevich. Preference elicitation and query learning. Journal of Machine Learning Research, 5:649-667, 2004. [5] Wolfram Conen and Tuomas Sandholm. Preference elicitation in combinatorial auctions: Extended abstract. In Proceedings of the ACM Conference on Electronic Commerce (ACM-EC), pages 256-259, 2001. [6] Vincent Conitzer, Jerome Lang, and Tuomas Sandholm. How many candidates are needed to make elections hard to manipulate? In Theoretical Aspects of Rationality and Knowledge (TARK), pages 201-214, 2003. [7] Vincent Conitzer and Tuomas Sandholm. Complexity of manipulating elections with few candidates. In Proceedings of the National Conference on Artificial Intelligence (AAAI), pages 314-319, 2002. [8] Vincent Conitzer and Tuomas Sandholm. Vote elicitation: Complexity and strategy-proofness. In Proceedings of the National Conference on Artificial Intelligence (AAAI), pages 392-397, 2002. [9] Sven de Vries, James Schummer, and Rakesh V. Vohra. On ascending auctions for heterogeneous objects, 2003. Draft. [10] Allan Gibbard. Manipulation of voting schemes. Econometrica, 41:587-602, 1973. [11] Benoit Hudson and Tuomas Sandholm. Effectiveness of query types and policies for preference elicitation in combinatorial auctions. In International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), pages 386-393, 2004. [12] E Kushilevitz and N Nisan. Communication Complexity. Cambridge University Press, 1997. [13] Sebasti´en Lahaie and David Parkes. Applying learning algorithms to preference elicitation. In Proceedings of the ACM Conference on Electronic Commerce, 2004. [14] Noam Nisan and Ilya Segal. The communication requirements of efficient allocations and supporting prices. Journal of Economic Theory, 2005. Forthcoming. [15] David Parkes. iBundle: An efficient ascending price bundle auction. In Proceedings of the ACM Conference on Electronic Commerce (ACM-EC), pages 148-157, 1999. [16] Tuomas Sandholm. An implementation of the contract net protocol based on marginal cost calculations. In Proceedings of the National Conference on Artificial Intelligence (AAAI), pages 256-262, 1993. [17] Tuomas Sandholm and Craig Boutilier. Preference elicitation in combinatorial auctions. In Peter Cramton, Yoav Shoham, and Richard Steinberg, editors, Combinatorial Auctions, chapter 10. MIT Press, 2005. [18] Paolo Santi, Vincent Conitzer, and Tuomas Sandholm. Towards a characterization of polynomial preference elicitation with value queries in combinatorial auctions. In Conference on Learning Theory (COLT), pages 1-16, 2004. [19] Mark Satterthwaite. Strategy-proofness and Arrow"s conditions: existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory, 10:187-217, 1975. [20] Ilya Segal. The communication requirements of social choice rules and supporting budget sets, 2004. Draft. Presented at the DIMACS Workshop on Computational Issues in Auction Design, Rutgers University, New Jersey, USA. [21] Peter Wurman and Michael Wellman. AkBA: A progressive, anonymous-price combinatorial auction. In Proceedings of the ACM Conference on Electronic Commerce (ACM-EC), pages 21-29, 2000. [22] A. C. Yao. Some complexity questions related to distributed computing. In Proceedings of the 11th ACM symposium on theory of computing (STOC), pages 209-213, 1979. [23] Martin Zinkevich, Avrim Blum, and Tuomas Sandholm. On polynomial-time preference elicitation with value queries. In Proceedings of the ACM Conference on Electronic Commerce (ACM-EC), pages 176-185, 2003. 87

communication;protocol;preference aggregation;maximin;voting rule;complexity;preference;stv;vote;resource allocation;communication complexity;elicitation problem

train_J-51

Complexity of (Iterated) Dominance∗

We study various computational aspects of solving games using dominance and iterated dominance. We first study both strict and weak dominance (not iterated), and show that checking whether a given strategy is dominated by some mixed strategy can be done in polynomial time using a single linear program solve. We then move on to iterated dominance. We show that determining whether there is some path that eliminates a given strategy is NP-complete with iterated weak dominance. This allows us to also show that determining whether there is a path that leads to a unique solution is NP-complete. Both of these results hold both with and without dominance by mixed strategies. (A weaker version of the second result (only without dominance by mixed strategies) was already known [7].) Iterated strict dominance, on the other hand, is path-independent (both with and without dominance by mixed strategies) and can therefore be done in polynomial time. We then study what happens when the dominating strategy is allowed to place positive probability on only a few pure strategies. First, we show that finding the dominating strategy with minimum support size is NP-complete (both for strict and weak dominance). Then, we show that iterated strict dominance becomes path-dependent when there is a limit on the support size of the dominating strategies, and that deciding whether a given strategy can be eliminated by iterated strict dominance under this restriction is NP-complete (even when the limit on the support size is 3). Finally, we study Bayesian games. We show that, unlike in normal form games, deciding whether a given pure strategy is dominated by another pure strategy in a Bayesian game is NP-complete (both with strict and weak dominance); however, deciding whether a strategy is dominated by some mixed strategy can still be done in polynomial time with a single linear program solve (both with strict and weak ∗ This material is based upon work supported by the National Science Foundation under ITR grants IIS-0121678 and IIS-0427858, and a Sloan Fellowship. dominance). Finally, we show that iterated dominance using pure strategies can require an exponential number of iterations in a Bayesian game (both with strict and weak dominance).

1. INTRODUCTION In multiagent systems with self-interested agents, the optimal action for one agent may depend on the actions taken by other agents. In such settings, the agents require tools from game theory to rationally decide on an action. Game theory offers various formal models of strategic settings-the best-known of which is a game in normal (or matrix) form, specifying a utility (payoff) for each agent for each combination of strategies that the agents choose-as well as solution concepts, which, given a game, specify which outcomes are reasonable (under various assumptions of rationality and common knowledge). Probably the best-known (and certainly the most-studied) solution concept is that of Nash equilibrium. A Nash equilibrium specifies a strategy for each player, in such a way that no player has an incentive to (unilaterally) deviate from the prescribed strategy. Recently, numerous papers have studied computing Nash equilibria in various settings [9, 4, 12, 3, 13, 14], and the complexity of constructing a Nash equilibrium in normal form games has been labeled one of the two most important open problems on the boundary of P today [20]. The problem of computing solutions according to the perhaps more elementary solution concepts of dominance and iterated dominance has received much less attention. (After an early short paper on an easy case [11], the main computational study of these concepts has actually taken place in a paper in the game theory community [7].1 ) A strategy strictly dominates another strategy if it performs strictly 1 This is not to say that computer scientists have ignored 88 better against all vectors of opponent strategies, and weakly dominates it if it performs at least as well against all vectors of opponent strategies, and strictly better against at least one. The idea is that dominated strategies can be eliminated from consideration. In iterated dominance, the elimination proceeds in rounds, and becomes easier as more strategies are eliminated: in any given round, the dominating strategy no longer needs to perform better than or as well as the dominated strategy against opponent strategies that were eliminated in earlier rounds. Computing solutions according to (iterated) dominance is important for at least the following reasons: 1) it can be computationally easier than computing (for instance) a Nash equilibrium (and therefore it can be useful as a preprocessing step in computing a Nash equilibrium), and 2) (iterated) dominance requires a weaker rationality assumption on the players than (for instance) Nash equilibrium, and therefore solutions derived according to it are more likely to occur. In this paper, we study some fundamental computational questions concerning dominance and iterated dominance, including how hard it is to check whether a given strategy can be eliminated by each of the variants of these notions. The rest of the paper is organized as follows. In Section 2, we briefly review definitions and basic properties of normal form games, strict and weak dominance, and iterated strict and weak dominance. In the remaining sections, we study computational aspects of dominance and iterated dominance. In Section 3, we study one-shot (not iterated) dominance. In Section 4, we study iterated dominance. In Section 5, we study dominance and iterated dominance when the dominating strategy can only place probability on a few pure strategies. Finally, in Section 6, we study dominance and iterated dominance in Bayesian games. 2. DEFINITIONS AND BASIC PROPERTIES In this section, we briefly review normal form games, as well as dominance and iterated dominance (both strict and weak). An n-player normal form game is defined as follows. Definition 1. A normal form game is given by a set of players {1, 2, . . . , n}; and, for each player i, a (finite) set of pure strategies Σi and a utility function ui : Σ1 × Σ2 × . . . × Σn → R (where ui(σ1, σ2, . . . , σn) denotes player i"s utility when each player j plays action σj). The two main notions of dominance are defined as follows. Definition 2. Player i"s strategy σi is said to be strictly dominated by player i"s strategy σi if for any vector of strategies σ−i for the other players, ui(σi, σ−i) > ui(σi, σ−i). Player i"s strategy σi is said to be weakly dominated by player i"s strategy σi if for any vector of strategies σ−i for the other players, ui(σi, σ−i) ≥ ui(σi, σ−i), and for at least one vector of strategies σ−i for the other players, ui(σi, σ−i) > ui(σi, σ−i). In this definition, it is sometimes allowed for the dominating strategy σi to be a mixed strategy, that is, a probability distribution over pure strategies. In this case, the utilities in dominance altogether. For example, simple dominance checks are sometimes used as a subroutine in searching for Nash equilibria [21]. the definition are the expected utilities.2 There are other notions of dominance, such as very weak dominance (in which no strict inequality is required, so two strategies can dominate each other), but we will not study them here. When we are looking at the dominance relations for player i, the other players (−i) can be thought of as a single player.3 Therefore, in the rest of the paper, when we study one-shot (not iterated) dominance, we will focus without loss of generality on two-player games.4 In two-player games, we will generally refer to the players as r (row) and c (column) rather than 1 and 2. In iterated dominance, dominated strategies are removed from the game, and no longer have any effect on future dominance relations. Iterated dominance can eliminate more strategies than dominance, as follows. σr may originally not dominate σr because the latter performs better against σc; but then, once σc is removed because it is dominated by σc, σr dominates σr, and the latter can be removed. For example, in the following game, R can be removed first, after which D is dominated. L R U 1, 1 0, 0 D 0, 1 1, 0 Either strict or weak dominance can be used in the definition of iterated dominance. We note that the process of iterated dominance is never helped by removing a dominated mixed strategy, for the following reason. If σi gives player i a higher utility than σi against mixed strategy σj for player j = i (and strategies σ−{i,j} for the other players), then for at least one pure strategy σj that σj places positive probability on, σi must perform better than σi against σj (and strategies σ−{i,j} for the other players). Thus, removing the mixed strategy σj does not introduce any new dominances. More detailed discussions and examples can be found in standard texts on microeconomics or game theory [17, 5]. We are now ready to move on to the core of this paper. 3. DOMINANCE (NOT ITERATED) In this section, we study the notion of one-shot (not iterated) dominance. As a first observation, checking whether a given strategy is strictly (weakly) dominated by some pure strategy is straightforward, by checking, for every pure strategy for that player, whether the latter strategy performs strictly better against all the opponent"s strategies (at least as well against all the opponent"s strategies, and strictly 2 The dominated strategy σi is, of course, also allowed to be mixed, but this has no technical implications for the paper: when we study one-shot dominance, we ask whether a given strategy is dominated, and it does not matter whether the given strategy is pure or mixed; when we study iterated dominance, there is no use in eliminating mixed strategies, as we will see shortly. 3 This player may have a very large strategy space (one pure strategy for every vector of pure strategies for the players that are being replaced). Nevertheless, this will not result in an increase in our representation size, because the original representation already had to specify utilities for each of these vectors. 4 We note that a restriction to two-player games would not be without loss of generality for iterated dominance. This is because for iterated dominance, we need to look at the dominated strategies of each individual player, so we cannot merge any players. 89 better against at least one).5 Next, we show that checking whether a given strategy is dominated by some mixed strategy can be done in polynomial time by solving a single linear program. (Similar linear programs have been given before [18]; we present the result here for completeness, and because we will build on the linear programs given below in Theorem 6.) Proposition 1. Given the row player"s utilities, a subset Dr of the row player"s pure strategies Σr, and a distinguished strategy σ∗ r for the row player, we can check in time polynomial in the size of the game (by solving a single linear program of polynomial size) whether there exists some mixed strategy σr, that places positive probability only on strategies in Dr and dominates σ∗ r , both for strict and for weak dominance. Proof. Let pdr be the probability that σr places on dr ∈ Dr. We will solve a single linear program in each of our algorithms; linear programs can be solved in polynomial time [10]. For strict dominance, the question is whether the pdr can be set so that for every pure strategy for the column player σc ∈ Σc, dr∈Dr pdr ur(dr, σc) > ur(σ∗ r , σc). Because the inequality must be strict, we cannot solve this directly by linear programming. We proceed as follows. Because the game is finite, we may assume without loss of generality that all utilities are positive (if not, simply add a constant to all utilities.) Solve the following linear program: minimize dr∈Dr pdr such that for any σc ∈ Σc, dr∈Dr pdr ur(dr, σc) ≥ ur(σ∗ r , σc). If σ∗ r is strictly dominated by some mixed strategy, this linear program has a solution with objective value < 1. (The dominating strategy is a feasible solution with objective value exactly 1. Because no constraint is binding for this solution, we can reduce one of the probabilities slightly without affecting feasibility, thereby obtaining a solution with objective value < 1.) Moreover, if this linear program has a solution with objective value < 1, there is a mixed strategy strictly dominating σ∗ r , which can be obtained by taking the LP solution and adding the remaining probability to any strategy (because all the utilities are positive, this will add to the left side of any inequality, so all inequalities will become strict). For weak dominance, we can solve the following linear program: maximize σc∈Σc (( dr∈Dr pdr ur(dr, σc)) − ur(σ∗ r , σc)) such that for any σc ∈ Σc, dr∈Dr pdr ur(dr, σc) ≥ ur(σ∗ r , σc); dr∈Dr pdr = 1. If σ∗ r is weakly dominated by some mixed strategy, then that mixed strategy is a feasible solution to this program with objective value > 0, because for at least one strategy σc ∈ Σc we have ( dr∈Dr pdr ur(dr, σc)) − ur(σ∗ r , σc) > 0. On the other hand, if this program has a solution with objective value > 0, then for at least one strategy σc ∈ Σc we 5 Recall that the assumption of a single opponent (that is, the assumption of two players) is without loss of generality for one-shot dominance. must have ( dr∈Dr pdr ur(dr, σc)) − ur(σ∗ r , σc) > 0, and thus the linear program"s solution is a weakly dominating mixed strategy. 4. ITERATED DOMINANCE We now move on to iterated dominance. It is well-known that iterated strict dominance is path-independent [6, 19]that is, if we remove dominated strategies until no more dominated strategies remain, in the end the remaining strategies for each player will be the same, regardless of the order in which strategies are removed. Because of this, to see whether a given strategy can be eliminated by iterated strict dominance, all that needs to be done is to repeatedly remove strategies that are strictly dominated, until no more dominated strategies remain. Because we can check in polynomial time whether any given strategy is dominated (whether or not dominance by mixed strategies is allowed, as described in Section 3), this whole procedure takes only polynomial time. In the case of iterated dominance by pure strategies with two players, Knuth et al. [11] slightly improve on (speed up) the straightforward implementation of this procedure by keeping track of, for each ordered pair of strategies for a player, the number of opponent strategies that prevent the first strategy from dominating the second. Hereby the runtime for an m × n game is reduced from O((m + n)4 ) to O((m + n)3 ). (Actually, they only study very weak dominance (for which no strict inequalities are required), but the approach is easily extended.) In contrast, iterated weak dominance is known to be pathdependent.6 For example, in the following game, using iterated weak dominance we can eliminate M first, and then D, or R first, and then U. L M R U 1, 1 0, 0 1, 0 D 1, 1 1, 0 0, 0 Therefore, while the procedure of removing weakly dominated strategies until no more weakly dominated strategies remain can certainly be executed in polynomial time, which strategies survive in the end depends on the order in which we remove the dominated strategies. We will investigate two questions for iterated weak dominance: whether a given strategy is eliminated in some path, and whether there is a path to a unique solution (one pure strategy left per player). We will show that both of these problems are computationally hard. Definition 3. Given a game in normal form and a distinguished strategy σ∗ , IWD-STRATEGY-ELIMINATION asks whether there is some path of iterated weak dominance that eliminates σ∗ . Given a game in normal form, IWDUNIQUE-SOLUTION asks whether there is some path of iterated weak dominance that leads to a unique solution (one strategy left per player). The following lemma shows a special case of normal form games in which allowing for weak dominance by mixed strategies (in addition to weak dominance by pure strategies) does 6 There is, however, a restriction of weak dominance called nice weak dominance which is path-independent [15, 16]. For an overview of path-independence results, see Apt [1]. 90 not help. We will prove the hardness results in this setting, so that they will hold whether or not dominance by mixed strategies is allowed. Lemma 1. Suppose that all the utilities in a game are in {0, 1}. Then every pure strategy that is weakly dominated by a mixed strategy is also weakly dominated by a pure strategy. Proof. Suppose pure strategy σ is weakly dominated by mixed strategy σ∗ . If σ gets a utility of 1 against some opponent strategy (or vector of opponent strategies if there are more than 2 players), then all the pure strategies that σ∗ places positive probability on must also get a utility of 1 against that opponent strategy (or else the expected utility would be smaller than 1). Moreover, at least one of the pure strategies that σ∗ places positive probability on must get a utility of 1 against an opponent strategy that σ gets 0 against (or else the inequality would never be strict). It follows that this pure strategy weakly dominates σ. We are now ready to prove the main results of this section. Theorem 1. IWD-STRATEGY-ELIMINATION is NPcomplete, even with 2 players, and with 0 and 1 being the only utilities occurring in the matrix-whether or not dominance by mixed strategies is allowed. Proof. The problem is in NP because given a sequence of strategies to be eliminated, we can easily check whether this is a valid sequence of eliminations (even when dominance by mixed strategies is allowed, using Proposition 1). To show that the problem is NP-hard, we reduce an arbitrary satisfiability instance (given by a nonempty set of clauses C over a nonempty set of variables V , with corresponding literals L = {+v : v ∈ V } ∪ {−v : v ∈ V }) to the following IWD-STRATEGY-ELIMINATION instance. (In this instance, we will specify that certain strategies are uneliminable. A strategy σr can be made uneliminable, even when 0 and 1 are the only allowed utilities, by adding another strategy σr and another opponent strategy σc, so that: 1. σr and σr are the only strategies that give the row player a utility of 1 against σc. 2. σr and σr always give the row player the same utility. 3. σc is the only strategy that gives the column player a utility of 1 against σr, but otherwise σc always gives the column player utility 0. This makes it impossible to eliminate any of these three strategies. We will not explicitly specify the additional strategies to make the proof more legible.) In this proof, we will denote row player strategies by s, and column player strategies by t, to improve legibility. Let the row player"s pure strategy set be given as follows. For every variable v ∈ V , the row player has corresponding strategies s1 +v, s2 +v, s1 −v, s2 −v. Additionally, the row player has the following 2 strategies: s1 0 and s2 0, where s2 0 = σ∗ r (that is, it is the strategy we seek to eliminate). Finally, for every clause c ∈ C, the row player has corresponding strategies s1 c (uneliminable) and s2 c. Let the column player"s pure strategy set be given as follows. For every variable v ∈ V , the column player has a corresponding strategy tv. For every clause c ∈ C, the column player has a corresponding strategy tc, and additionally, for every literal l ∈ L that occurs in c, a strategy tc,l. For every variable v ∈ V , the column player has corresponding strategies t+v, t−v (both uneliminable). Finally, the column player has three additional strategies: t1 0 (uneliminable), t2 0, and t1. The utility function for the row player is given as follows: • ur(s1 +v, tv) = 0 for all v ∈ V ; • ur(s2 +v, tv) = 1 for all v ∈ V ; • ur(s1 −v, tv) = 1 for all v ∈ V ; • ur(s2 −v, tv) = 0 for all v ∈ V ; • ur(s1 +v, t1) = 1 for all v ∈ V ; • ur(s2 +v, t1) = 0 for all v ∈ V ; • ur(s1 −v, t1) = 0 for all v ∈ V ; • ur(s2 −v, t1) = 1 for all v ∈ V ; • ur(sb +v, t+v) = 1 for all v ∈ V and b ∈ {1, 2}; • ur(sb −v, t−v) = 1 for all v ∈ V and b ∈ {1, 2}; • ur(sl, t) = 0 otherwise for all l ∈ L and t ∈ S2; • ur(s1 0, tc) = 0 for all c ∈ C; • ur(s2 0, tc) = 1 for all c ∈ C; • ur(sb 0, t1 0) = 1 for all b ∈ {1, 2}; • ur(s1 0, t2 0) = 1; • ur(s2 0, t2 0) = 0; • ur(sb 0, t) = 0 otherwise for all b ∈ {1, 2} and t ∈ S2; • ur(sb c, t) = 0 otherwise for all c ∈ C and b ∈ {1, 2}; and the row player"s utility is 0 in every other case. The utility function for the column player is given as follows: • uc(s, tv) = 0 for all v ∈ V and s ∈ S1; • uc(s, t1) = 0 for all s ∈ S1; • uc(s2 l , tc) = 1 for all c ∈ C and l ∈ L where l ∈ c (literal l occurs in clause c); • uc(s2 l2 , tc,l1 ) = 1 for all c ∈ C and l1, l2 ∈ L, l1 = l2 where l2 ∈ c; • uc(s1 c, tc) = 1 for all c ∈ C; • uc(s2 c, tc) = 0 for all c ∈ C; • uc(sb c, tc,l) = 1 for all c ∈ C, l ∈ L, and b ∈ {1, 2}; • uc(s2, tc) = uc(s2, tc,l) = 0 otherwise for all c ∈ C and l ∈ L; and the column player"s utility is 0 in every other case. We now show that the two instances are equivalent. First, suppose there is a solution to the satisfiability instance: that is, a truth-value assignment to the variables in V such that all clauses are satisfied. Then, consider the following sequence of eliminations in our game: 1. For every variable v that is set to true in the assignment, eliminate tv (which gives the column player utility 0 everywhere). 2. Then, for every variable v that is set to true in the assignment, eliminate s2 +v using s1 +v (which is possible because tv has been eliminated, and because t1 has not been eliminated (yet)). 3. Now eliminate t1 (which gives the column player utility 0 everywhere). 4. Next, for every variable v that is set to false in the assignment, eliminate s2 −v using s1 −v (which is possible because t1 has been eliminated, and because tv has not been eliminated (yet)). 5. For every clause c which has the variable corresponding to one of its positive literals l = +v set to true in the assignment, eliminate tc using tc,l (which is possible because s2 l has been eliminated, and s2 c has not been eliminated (yet)). 6. For every clause c which has the variable corresponding to one of its negative literals l = −v set to false in the assignment, eliminate tc using tc,l 91 (which is possible because s2 l has been eliminated, and s2 c has not been eliminated (yet)). 7. Because the assignment satisfied the formula, all the tc have now been eliminated. Thus, we can eliminate s2 0 = σ∗ r using s1 0. It follows that there is a solution to the IWD-STRATEGY-ELIMINATION instance. Now suppose there is a solution to the IWD-STRATEGYELIMINATION instance. By Lemma 1, we can assume that all the dominances are by pure strategies. We first observe that only s1 0 can eliminate s2 0 = σ∗ r , because it is the only other strategy that gets the row player a utility of 1 against t1 0, and t1 0 is uneliminable. However, because s2 0 performs better than s1 0 against the tc strategies, it follows that all of the tc strategies must be eliminated. For each c ∈ C, the strategy tc can only be eliminated by one of the strategies tc,l (with the same c), because these are the only other strategies that get the column player a utility of 1 against s1 c, and s1 c is uneliminable. But, in order for some tc,l to eliminate tc, s2 l must be eliminated first. Only s1 l can eliminate s2 l , because it is the only other strategy that gets the row player a utility of 1 against tl, and tl is uneliminable. We next show that for every v ∈ V only one of s2 +v, s2 −v can be eliminated. This is because in order for s1 +v to eliminate s2 +v, tv needs to have been eliminated and t1, not (so tv must be eliminated before t1); but in order for s1 −v to eliminate s2 −v, t1 needs to have been eliminated and tv, not (so t1 must be eliminated before tv). So, set v to true if s2 +v is eliminated, and to false otherwise Because by the above, for every clause c, one of the s2 l with l ∈ c must be eliminated, it follows that this is a satisfying assignment to the satisfiability instance. Using Theorem 1, it is now (relatively) easy to show that IWD-UNIQUE-SOLUTION is also NP-complete under the same restrictions. Theorem 2. IWD-UNIQUE-SOLUTION is NP-complete, even with 2 players, and with 0 and 1 being the only utilities occurring in the matrix-whether or not dominance by mixed strategies is allowed. Proof. Again, the problem is in NP because we can nondeterministically choose the sequence of eliminations and verify whether it is correct. To show NP-hardness, we reduce an arbitrary IWD-STRATEGY-ELIMINATION instance to the following IWD-UNIQUE-SOLUTION instance. Let all the strategies for each player from the original instance remain part of the new instance, and let the utilities resulting from the players playing a pair of these strategies be the same. We add three additional strategies σ1 r , σ2 r , σ3 r for the row player, and three additional strategies σ1 c , σ2 c , σ3 c for the column player. Let the additional utilities be as follows: • ur(σr, σj c) = 1 for all σr /∈ {σ1 r , σ2 r , σ3 r } and j ∈ {2, 3}; • ur(σi r, σc) = 1 for all i ∈ {1, 2, 3} and σc /∈ {σ2 c , σ3 c }; • ur(σi r, σ2 c ) = 1 for all i ∈ {2, 3}; • ur(σ1 r , σ3 c ) = 1; • and the row player"s utility is 0 in all other cases involving a new strategy. • uc(σ3 r , σc) = 1 for all σc /∈ {σ1 c , σ2 c , σ3 c }; • uc(σ∗ r , σj c) = 1 for all j ∈ {2, 3} (σ∗ r is the strategy to be eliminated in the original instance); • uc(σi r, σ1 c ) = 1 for all i ∈ {1, 2}; • ur(σ1 r , σ2 c ) = 1; • ur(σ2 r , σ3 c ) = 1; • and the column player"s utility is 0 in all other cases involving a new strategy. We proceed to show that the two instances are equivalent. First suppose there exists a solution to the original IWDSTRATEGY-ELIMINATION instance. Then, perform the same sequence of eliminations to eliminate σ∗ r in the new IWD-UNIQUE-SOLUTION instance. (This is possible because at any stage, any weak dominance for the row player in the original instance is still a weak dominance in the new instance, because the two strategies" utilities for the row player are the same when the column player plays one of the new strategies; and the same is true for the column player.) Once σ∗ r is eliminated, let σ1 c eliminate σ2 c . (It performs better against σ2 r .) Then, let σ1 r eliminate all the other remaining strategies for the row player. (It always performs better against either σ1 c or σ3 c .) Finally, σ1 c is the unique best response against σ1 r among the column player"s remaining strategies, so let it eliminate all the other remaining strategies for the column player. Thus, there exists a solution to the IWD-UNIQUE-SOLUTION instance. Now suppose there exists a solution to the IWD-UNIQUESOLUTION instance. By Lemma 1, we can assume that all the dominances are by pure strategies. We will show that none of the new strategies (σ1 r , σ2 r , σ3 r , σ1 c , σ2 c , σ3 c ) can either eliminate another strategy, or be eliminated before σ∗ r is eliminated. Thus, there must be a sequence of eliminations ending in the elimination of σ∗ r , which does not involve any of the new strategies, and is therefore a valid sequence of eliminations in the original game (because all original strategies perform the same against each new strategy). We now show that this is true by exhausting all possibilities for the first elimination before σ∗ r is eliminated that involves a new strategy. None of the σi r can be eliminated by a σr /∈ {σ1 r , σ2 r , σ3 r }, because the σi r perform better against σ1 c . σ1 r cannot eliminate any other strategy, because it always performs poorer against σ2 c . σ2 r and σ3 r are equivalent from the row player"s perspective (and thus cannot eliminate each other), and cannot eliminate any other strategy because they always perform poorer against σ3 c . None of the σj c can be eliminated by a σc /∈ {σ1 c , σ2 c , σ3 c }, because the σj c always perform better against either σ1 r or σ2 r . σ1 c cannot eliminate any other strategy, because it always performs poorer against either σ∗ r or σ3 r . σ2 c cannot eliminate any other strategy, because it always performs poorer against σ2 r or σ3 r . σ3 c cannot eliminate any other strategy, because it always performs poorer against σ1 r or σ3 r . Thus, there exists a solution to the IWDSTRATEGY-ELIMINATION instance. A slightly weaker version of the part of Theorem 2 concerning dominance by pure strategies only is the main result of Gilboa et al. [7]. (Besides not proving the result for dominance by mixed strategies, the original result was weaker because it required utilities {0, 1, 2, 3, 4, 5, 6, 7, 8} rather than just {0, 1} (and because of this, our Lemma 1 cannot be applied to it to get the result for mixed strategies).) 5. (ITERATED) DOMINANCE USING MIXED STRATEGIES WITH SMALL SUPPORTS When showing that a strategy is dominated by a mixed strategy, there are several reasons to prefer exhibiting a 92 dominating strategy that places positive probability on as few pure strategies as possible. First, this will reduce the number of bits required to specify the dominating strategy (and thus the proof of dominance can be communicated quicker): if the dominating mixed strategy places positive probability on only k strategies, then it can be specified using k real numbers for the probabilities, plus k log m (where m is the number of strategies for the player under consideration) bits to indicate which strategies are used. Second, the proof of dominance will be cleaner: for a dominating mixed strategy, it is typically (always in the case of strict dominance) possible to spread some of the probability onto any unused pure strategy and still have a dominating strategy, but this obscures which pure strategies are the ones that are key in making the mixed strategy dominating. Third, because (by the previous) the argument for eliminating the dominated strategy is simpler and easier to understand, it is more likely to be accepted. Fourth, the level of risk neutrality required for the argument to work is reduced, at least in the extreme case where dominance by a single pure strategy can be exhibited (no risk neutrality is required here). This motivates the following problem. Definition 4 (MINIMUM-DOMINATING-SET). We are given the row player"s utilities of a game in normal form, a distinguished strategy σ∗ for the row player, a specification of whether the dominance should be strict or weak, and a number k. We are asked whether there exists a mixed strategy σ for the row player that places positive probability on at most k pure strategies, and dominates σ∗ in the required sense. Unfortunately, this problem is NP-complete. Theorem 3. MINIMUM-DOMINATING-SET is NPcomplete, both for strict and for weak dominance. Proof. The problem is in NP because we can nondeterministically choose a set of at most k strategies to give positive probability, and decide whether we can dominate σ∗ with these k strategies as described in Proposition 1. To show NP-hardness, we reduce an arbitrary SET-COVER instance (given a set S, subsets S1, S2, . . . , Sr, and a number t, can all of S be covered by at most t of the subsets?) to the following MINIMUM-DOMINATING-SET instance. For every element s ∈ S, there is a pure strategy σs for the column player. For every subset Si, there is a pure strategy σSi for the row player. Finally, there is the distinguished pure strategy σ∗ for the row player. The row player"s utilities are as follows: ur(σSi , σs) = t + 1 if s ∈ Si; ur(σSi , σs) = 0 if s /∈ Si; ur(σ∗ , σs) = 1 for all s ∈ S. Finally, we let k = t. We now proceed to show that the two instances are equivalent. First suppose there exists a solution to the SET-COVER instance. Without loss of generality, we can assume that there are exactly k subsets in the cover. Then, for every Si that is in the cover, let the dominating strategy σ place exactly 1 k probability on the corresponding pure strategy σSi . Now, if we let n(s) be the number of subsets in the cover containing s (we observe that that n(s) ≥ 1), then for every strategy σs for the column player, the row player"s expected utility for playing σ when the column player is playing σs is u(σ, σs) = n(s) k (k + 1) ≥ k+1 k > 1 = u(σ∗ , σs). So σ strictly (and thus also weakly) dominates σ∗ , and there exists a solution to the MINIMUM-DOMINATING-SET instance. Now suppose there exists a solution to the MINIMUMDOMINATING-SET instance. Consider the (at most k) pure strategies of the form σSi on which the dominating mixed strategy σ places positive probability, and let T be the collection of the corresponding subsets Si. We claim that T is a cover. For suppose there is some s ∈ S that is not in any of the subsets in T . Then, if the column player plays σs, the row player (when playing σ) will always receive utility 0-as opposed to the utility of 1 the row player would receive for playing σ∗ , contradicting the fact that σ dominates σ∗ (whether this dominance is weak or strict). It follows that there exists a solution to the SET-COVER instance. On the other hand, if we require that the dominating strategy only places positive probability on a very small number of pure strategies, then it once again becomes easy to check whether a strategy is dominated. Specifically, to find out whether player i"s strategy σ∗ is dominated by a strategy that places positive probability on only k pure strategies, we can simply check, for every subset of k of player i"s pure strategies, whether there is a strategy that places positive probability only on these k strategies and dominates σ∗ , using Proposition 1. This requires only O(|Σi|k ) such checks. Thus, if k is a constant, this constitutes a polynomial-time algorithm. A natural question to ask next is whether iterated strict dominance remains computationally easy when dominating strategies are required to place positive probability on at most k pure strategies, where k is a small constant. (We have already shown in Section 4 that iterated weak dominance is hard even when k = 1, that is, only dominance by pure strategies is allowed.) Of course, if iterated strict dominance were path-independent under this restriction, computational easiness would follow as it did in Section 4. However, it turns out that this is not the case. Observation 1. If we restrict the dominating strategies to place positive probability on at most two pure strategies, iterated strict dominance becomes path-dependent. Proof. Consider the following game: 7, 1 0, 0 0, 0 0, 0 7, 1 0, 0 3, 0 3, 0 0, 0 0, 0 0, 0 3, 1 1, 0 1, 0 1, 0 Let (i, j) denote the outcome in which the row player plays the ith row and the column player plays the jth column. Because (1, 1), (2, 2), and (4, 3) are all Nash equilibria, none of the column player"s pure strategies will ever be eliminated, and neither will rows 1, 2, and 4. We now observe that randomizing uniformly over rows 1 and 2 dominates row 3, and randomizing uniformly over rows 3 and 4 dominates row 5. However, if we eliminate row 3 first, it becomes impossible to dominate row 5 without randomizing over at least 3 pure strategies. Indeed, iterated strict dominance turns out to be hard even when k = 3. Theorem 4. If we restrict the dominating strategies to place positive probability on at most three pure strategies, it becomes NP-complete to decide whether a given strategy can be eliminated using iterated strict dominance. 93 Proof. The problem is in NP because given a sequence of strategies to be eliminated, we can check in polynomial time whether this is a valid sequence of eliminations (for any strategy to be eliminated, we can check, for every subset of three other strategies, whether there is a strategy placing positive probability on only these three strategies that dominates the strategy to be eliminated, using Proposition 1). To show that the problem is NP-hard, we reduce an arbitrary satisfiability instance (given by a nonempty set of clauses C over a nonempty set of variables V , with corresponding literals L = {+v : v ∈ V } ∪ {−v : v ∈ V }) to the following two-player game. For every variable v ∈ V , the row player has strategies s+v, s−v, s1 v, s2 v, s3 v, s4 v, and the column player has strategies t1 v, t2 v, t3 v, t4 v. For every clause c ∈ C, the row player has a strategy sc, and the column player has a strategy tc, as well as, for every literal l occurring in c, an additional strategy tl c. The row player has two additional strategies s1 and s2. (s2 is the strategy that we are seeking to eliminate.) Finally, the column player has one additional strategy t1. The utility function for the row player is given as follows (where is some sufficiently small number): • ur(s+v, tj v) = 4 if j ∈ {1, 2}, for all v ∈ V ; • ur(s+v, tj v) = 1 if j ∈ {3, 4}, for all v ∈ V ; • ur(s−v, tj v) = 1 if j ∈ {1, 2}, for all v ∈ V ; • ur(s−v, tj v) = 4 if j ∈ {3, 4}, for all v ∈ V ; • ur(s+v, t) = ur(s−v, t) = 0 for all v ∈ V and t /∈ {t1 v, t2 v, t3 v, t4 v}; • ur(si v, ti v) = 13 for all v ∈ V and i ∈ {1, 2, 3, 4}; • ur(si v, t) = for all v ∈ V , i ∈ {1, 2, 3, 4}, and t = ti v; • ur(sc, tc) = 2 for all c ∈ C; • ur(sc, t) = 0 for all c ∈ C and t = tc; • ur(s1, t1) = 1 + ; • ur(s1, t) = for all t = t1; • ur(s2, t1) = 1; • ur(s2, tc) = 1 for all c ∈ C; • ur(s2, t) = 0 for all t /∈ {t1} ∪ {tc : c ∈ C}. The utility function for the column player is given as follows: • uc(si v, ti v) = 1 for all v ∈ V and i ∈ {1, 2, 3, 4}; • uc(s, ti v) = 0 for all v ∈ V , i ∈ {1, 2, 3, 4}, and s = si v; • uc(sc, tc) = 1 for all c ∈ C; • uc(sl, tc) = 1 for all c ∈ C and l ∈ L occurring in c; • uc(s, tc) = 0 for all c ∈ C and s /∈ {sc} ∪ {sl : l ∈ c}; • uc(sc, tl c) = 1 + for all c ∈ C; • uc(sl , tl c) = 1 + for all c ∈ C and l = l occurring in c; • uc(s, tl c) = for all c ∈ C and s /∈ {sc} ∪ {sl : l ∈ c, l = l }; • uc(s2, t1) = 1; • uc(s, t1) = 0 for all s = s2. We now show that the two instances are equivalent. First, suppose that there is a solution to the satisfiability instance. Then, consider the following sequence of eliminations in our game: 1. For every variable v that is set to true in the satisfying assignment, eliminate s+v with the mixed strategy σr that places probability 1/3 on s−v, probability 1/3 on s1 v, and probability 1/3 on s2 v. (The expected utility of playing σr against t1 v or t2 v is 14/3 > 4; against t3 v or t4 v, it is 4/3 > 1; and against anything else it is 2 /3 > 0. Hence the dominance is valid.) 2. Similarly, for every variable v that is set to false in the satisfying assignment, eliminate s−v with the mixed strategy σr that places probability 1/3 on s+v, probability 1/3 on s3 v, and probability 1/3 on s4 v. (The expected utility of playing σr against t1 v or t2 v is 4/3 > 1; against t3 v or t4 v, it is 14/3 > 4; and against anything else it is 2 /3 > 0. Hence the dominance is valid.) 3. For every c ∈ C, eliminate tc with any tl c for which l was set to true in the satisfying assignment. (This is a valid dominance because tl c performs better than tc against any strategy other than sl, and we eliminated sl in step 1 or in step 2.) 4. Finally, eliminate s2 with s1. (This is a valid dominance because s1 performs better than s2 against any strategy other than those in {tc : c ∈ C}, which we eliminated in step 3.) Hence, there is an elimination path that eliminates s2. Now, suppose that there is an elimination path that eliminates s2. The strategy that eventually dominates s2 must place most of its probability on s1, because s1 is the only other strategy that performs well against t1, which cannot be eliminated before s2. But, s1 performs significantly worse than s2 against any strategy tc with c ∈ C, so it follows that all these strategies must be eliminated first. Each strategy tc can only be eliminated by a strategy that places most of its weight on the corresponding strategies tl c with l ∈ c, because they are the only other strategies that perform well against sc, which cannot be eliminated before tc. But, each strategy tl c performs significantly worse than tc against sl, so it follows that for every clause c, for one of the literals l occurring in it, sl must be eliminated first. Now, strategies of the form tj v will never be eliminated because they are the unique best responses to the corresponding strategies sj v (which are, in turn, the best responses to the corresponding tj v). As a result, if strategy s+v (respectively, s−v) is eliminated, then its opposite strategy s−v (respectively, s+v) can no longer be eliminated, for the following reason. There is no other pure strategy remaining that gets a significant utility against more than one of the strategies t1 v, t2 v, t3 v, t4 v, but s−v (respectively, s+v) gets significant utility against all 4, and therefore cannot be dominated by a mixed strategy placing positive probability on at most 3 strategies. It follows that for each v ∈ V , at most one of the strategies s+v, s−v is eliminated, in such a way that for every clause c, for one of the literals l occurring in it, sl must be eliminated. But then setting all the literals l such that sl is eliminated to true constitutes a solution to the satisfiability instance. In the next section, we return to the setting where there is no restriction on the number of pure strategies on which a dominating mixed strategy can place positive probability. 6. (ITERATED) DOMINANCE IN BAYESIAN GAMES So far, we have focused on normal form games that are flatly represented (that is, every matrix entry is given ex94 plicitly). However, for many games, the flat representation is too large to write down explicitly, and instead, some representation that exploits the structure of the game needs to be used. Bayesian games, besides being of interest in their own right, can be thought of as a useful structured representation of normal form games, and we will study them in this section. In a Bayesian game, each player first receives privately held preference information (the player"s type) from a distribution, which determines the utility that that player receives for every outcome of (that is, vector of actions played in) the game. After receiving this type, the player plays an action based on it.7 Definition 5. A Bayesian game is given by a set of players {1, 2, . . . , n}; and, for each player i, a (finite) set of actions Ai, a (finite) type space Θi with a probability distribution πi over it, and a utility function ui : Θi × A1 × A2 × . . . × An → R (where ui(θi, a1, a2, . . . , an) denotes player i"s utility when i"s type is θi and each player j plays action aj). A pure strategy in a Bayesian game is a mapping from types to actions, σi : Θi → Ai, where σi(θi) denotes the action that player i plays for type θi. Any vector of pure strategies in a Bayesian game defines an (expected) utility for each player, and therefore we can translate a Bayesian game into a normal form game. In this normal form game, the notions of dominance and iterated dominance are defined as before. However, the normal form representation of the game is exponentially larger than the Bayesian representation, because each player i has |Ai||Θi| distinct pure strategies. Thus, any algorithm for Bayesian games that relies on expanding the game to its normal form will require exponential time. Specifically, our easiness results for normal form games do not directly transfer to this setting. In fact, it turns out that checking whether a strategy is dominated by a pure strategy is hard in Bayesian games. Theorem 5. In a Bayesian game, it is NP-complete to decide whether a given pure strategy σr : Θr → Ar is dominated by some other pure strategy (both for strict and weak dominance), even when the row player"s distribution over types is uniform. Proof. The problem is in NP because it is easy to verify whether a candidate dominating strategy is indeed a dominating strategy. To show that the problem is NP-hard, we reduce an arbitrary satisfiability instance (given by a set of clauses C using variables from V ) to the following Bayesian game. Let the row player"s action set be Ar = {t, f, 0} and let the column player"s action set be Ac = {ac : c ∈ C}. Let the row player"s type set be Θr = {θv : v ∈ V }, with a distribution πr that is uniform. Let the row player"s utility function be as follows: • ur(θv, 0, ac) = 0 for all v ∈ V and c ∈ C; • ur(θv, b, ac) = |V | for all v ∈ V , c ∈ C, and b ∈ {t, f} such that setting v to b satisfies c; • ur(θv, b, ac) = −1 for all v ∈ V , c ∈ C, and b ∈ {t, f} such that setting v to b does not satisfy c. 7 In general, a player can also receive a signal about the other players" preferences, but we will not concern ourselves with that here. Let the pure strategy to be dominated be the one that plays 0 for every type. We show that the strategy is dominated by a pure strategy if and only if there is a solution to the satisfiability instance. First, suppose there is a solution to the satisfiability instance. Then, let σd r be given by: σd r (θv) = t if v is set to true in the solution to the satisfiability instance, and σd r (θv) = f otherwise. Then, against any action ac by the column player, there is at least one type θv such that either +v ∈ c and σd r (θv) = t, or −v ∈ c and σd r (θv) = f. Thus, the row player"s expected utility against action ac is at least |V | |V | − |V |−1 |V | = 1 |V | > 0. So, σd r is a dominating strategy. Now, suppose there is a dominating pure strategy σd r . This dominating strategy must play t or f for at least one type. Thus, against any ac by the column player, there must at least be some type θv for which ur(θv, σd r (θv), ac) > 0. That is, there must be at least one variable v such that setting v to σd r (θv) satifies c. But then, setting each v to σd r (θv) must satisfy all the clauses. So a satisfying assignment exists. However, it turns out that we can modify the linear programs from Proposition 1 to obtain a polynomial time algorithm for checking whether a strategy is dominated by a mixed strategy in Bayesian games. Theorem 6. In a Bayesian game, it can be decided in polynomial time whether a given (possibly mixed) strategy σr is dominated by some other mixed strategy, using linear programming (both for strict and weak dominance). Proof. We can modify the linear programs presented in Proposition 1 as follows. For strict dominance, again assuming without loss of generality that all the utilities in the game are positive, use the following linear program (in which pσr r (θr, ar) is the probability that σr, the strategy to be dominated, places on ar for type θr): minimize θr∈Θr ar∈Ar pr(ar) such that for any ac ∈ Ac, θr∈Θr ar∈Ar π(θr)ur(θr, ar, ac)pr(θr, ar) ≥ θr∈Θr ar∈Ar π(θr)ur(θr, ar, ac)pσr r (θr, ar); for any θr ∈ Θr, ar∈Ar pr(θr, ar) ≤ 1. Assuming that π(θr) > 0 for all θr ∈ Θr, this program will return an objective value smaller than |Θr| if and only if σr is strictly dominated, by reasoning similar to that done in Proposition 1. For weak dominance, use the following linear program: maximize ac∈Ac ( θr∈Θr ar∈Ar π(θr)ur(θr, ar, ac)pr(θr, ar)− θr∈Θr ar∈Ar π(θr)ur(θr, ar, ac)pσr r (θr, ar)) such that for any ac ∈ Ac, θr∈Θr ar∈Ar π(θr)ur(θr, ar, ac)pr(θr, ar) ≥ θr∈Θr ar∈Ar π(θr)ur(θr, ar, ac)pσr r (θr, ar); for any θr ∈ Θr, ar∈Ar pr(θr, ar) = 1. This program will return an objective value greater than 0 if and only if σr is weakly dominated, by reasoning similar to that done in Proposition 1. We now turn to iterated dominance in Bayesian games. Na¨ıvely, one might argue that iterated dominance in Bayesian 95 games always requires an exponential number of steps when a significant fraction of the game"s pure strategies can be eliminated, because there are exponentially many pure strategies. However, this is not a very strong argument because oftentimes we can eliminate exponentially many pure strategies in one step. For example, if for some type θr ∈ Θr we have, for all ac ∈ Ac, that u(θr, a1 r, ac) > u(θr, a2 r, ac), then any pure strategy for the row player which plays action a2 r for type θr is dominated (by the strategy that plays action a1 r for type θr instead)-and there are exponentially many (|Ar||Θr|−1 ) such strategies. It is therefore conceivable that we need only polynomially many eliminations of collections of a player"s strategies. However, the following theorem shows that this is not the case, by giving an example where an exponential number of iterations (that is, alternations between the players in eliminating strategies) is required. (We emphasize that this is not a result about computational complexity.) Theorem 7. Even in symmetric 3-player Bayesian games, iterated dominance by pure strategies can require an exponential number of iterations (both for strict and weak dominance), even with only three actions per player. Proof. Let each player i ∈ {1, 2, 3} have n + 1 types θ1 i , θ2 i , . . . , θn+1 i . Let each player i have 3 actions ai, bi, ci, and let the utility function of each player be defined as follows. (In the below, i + 1 and i + 2 are shorthand for i + 1(mod 3) and i + 2(mod 3) when used as player indices. Also, −∞ can be replaced by a sufficiently negative number. Finally, δ and should be chosen to be very small (even compared to 2−(n+1) ), and should be more than twice as large as δ.) • ui(θ1 i ; ai, ci+1, ci+2) = −1; • ui(θ1 i ; ai, si+1, si+2) = 0 for si+1 = ci+1 or si+2 = ci+2; • ui(θ1 i ; bi, si+1, si+2) = − for si+1 = ai+1 and si+2 = ai+2; • ui(θ1 i ; bi, si+1, si+2) = −∞ for si+1 = ai+1 or si+2 = ai+2; • ui(θ1 i ; ci, si+1, si+2) = −∞ for all si+1, si+2; • ui(θj i ; ai, si+1, si+2) = −∞ for all si+1, si+2 when j > 1; • ui(θj i ; bi, si+1, si+2) = − for all si+1, si+2 when j > 1; • ui(θj i ; ci, si+1, ci+2) = δ − − 1/2 for all si+1 when j > 1; • ui(θj i ; ci, si+1, si+2) = δ− for all si+1 and si+2 = ci+2 when j > 1. Let the distribution over each player"s types be given by p(θj i ) = 2−j (with the exception that p(θ2 i ) = 2−2 +2−(n+1) ). We will be interested in eliminating strategies of the following form: play bi for type θ1 i , and play one of bi or ci otherwise. Because the utility function is the same for any type θj i with j > 1, these strategies are effectively defined by the total probability that they place on ci,8 which is t2 i (2−2 + 2−(n+1) ) + n+1 j=3 tj i 2−j where tj i = 1 if player i 8 Note that the strategies are still pure strategies; the probability placed on an action by a strategy here is simply the sum of the probabilities of the types for which the strategy chooses that action. plays ci for type θj i , and 0 otherwise. This probability is different for any two different strategies of the given form, and we have exponentially many different strategies of the given form. For any probability q which can be expressed as t2(2−2 + 2−(n+1) ) + n+1 j=3 tj2−j (with all tj ∈ {0, 1}), let σi(q) denote the (unique) strategy of the given form for player i which places a total probability of q on ci. Any strategy that plays ci for type θ1 i or ai for some type θj i with j > 1 can immediately be eliminated. We will show that, after that, we must eliminate the strategies σi(q) with high q first, slowly working down to those with lower q. Claim 1: If σi+1(q ) and σi+2(q ) have not yet been eliminated, and q < q , then σi(q) cannot yet be eliminated. Proof: First, we show that no strategy σi(q ) can eliminate σi(q). Against σi+1(q ), σi+2(q ), the utility of playing σi(p) is − + p · δ − p · q /2. Thus, when q = 0, it is best to set p as high as possible (and we note that σi+1(0) and σi+2(0) have not been eliminated), but when q > 0, it is best to set p as low as possible because δ < q /2. Thus, whether q > q or q < q , σi(q) will always do strictly better than σi(q ) against some remaining opponent strategies. Hence, no strategy σi(q ) can eliminate σi(q). The only other pure strategies that could dominate σi(q) are strategies that play ai for type θ1 i , and bi or ci for all other types. Let us take such a strategy and suppose that it plays c with probability p. Against σi+1(q ), σi+2(q ) (which have not yet been eliminated), the utility of playing this strategy is −(q )2 /2 − /2 + p · δ − p · q /2. On the other hand, playing σi(q) gives − + q · δ − q · q /2. Because q > q, we have −(q )2 /2 < −q · q /2, and because δ and are small, it follows that σi(q) receives a higher utility. Therefore, no strategy dominates σi(q), proving the claim. Claim 2: If for all q > q, σi+1(q ) and σi+2(q ) have been eliminated, then σi(q) can be eliminated. Proof: Consider the strategy for player i that plays ai for type θ1 i , and bi for all other types (call this strategy σi); we claim σi dominates σi(q). First, if either of the other players k plays ak for θ1 k, then σi performs better than σi(q) (which receives −∞ in some cases). Because the strategies for player k that play ck for type θ1 k, or ak for some type θj k with j > 1, have already been eliminated, all that remains to check is that σi performs better than σi(q) whenever both of the other two players play strategies of the following form: play bk for type θ1 k, and play one of bk or ck otherwise. We note that among these strategies, there are none left that place probability greater than q on ck. Letting qk denote the probability with which player k plays ck, the expected utility of playing σi is −qi+1 · qi+2/2 − /2. On the other hand, the utility of playing σi(q) is − + q · δ − q · qi+2/2. Because qi+1 ≤ q, the difference between these two expressions is at least /2 − δ, which is positive. It follows that σi dominates σi(q). From Claim 2, it follows that all strategies of the form σi(q) will eventually be eliminated. However, Claim 1 shows that we cannot go ahead and eliminate multiple such strategies for one player, unless at least one other player simultaneously keeps up in the eliminated strategies: every time a σi(q) is eliminated such that σi+1(q) and σi+2(q) have not yet been eliminated, we need to eliminate one of the latter two strategies before any σi(q ) with q > q can be eliminated-that is, we need to alternate between players. Because there are exponentially many strategies of the form σi(q), it follows that iterated elimination will require exponentially many iterations to complete. 96 It follows that an efficient algorithm for iterated dominance (strict or weak) by pure strategies in Bayesian games, if it exists, must somehow be able to perform (at least part of) many iterations in a single step of the algorithm (because if each step only performed a single iteration, we would need exponentially many steps). Interestingly, Knuth et al. [11] argue that iterated dominance appears to be an inherently sequential problem (in light of their result that iterated very weak dominance is P-complete, that is, apparently not efficiently parallelizable), suggesting that aggregating many iterations may be difficult. 7. CONCLUSIONS While the Nash equilibrium solution concept is studied more and more intensely in our community, the perhaps more elementary concept of (iterated) dominance has received much less attention. In this paper we studied various computational aspects of this concept. We first studied both strict and weak dominance (not iterated), and showed that checking whether a given strategy is dominated by some mixed strategy can be done in polynomial time using a single linear program solve. We then moved on to iterated dominance. We showed that determining whether there is some path that eliminates a given strategy is NP-complete with iterated weak dominance. This allowed us to also show that determining whether there is a path that leads to a unique solution is NP-complete. Both of these results hold both with and without dominance by mixed strategies. (A weaker version of the second result (only without dominance by mixed strategies) was already known [7].) Iterated strict dominance, on the other hand, is path-independent (both with and without dominance by mixed strategies) and can therefore be done in polynomial time. We then studied what happens when the dominating strategy is allowed to place positive probability on only a few pure strategies. First, we showed that finding the dominating strategy with minimum support size is NP-complete (both for strict and weak dominance). Then, we showed that iterated strict dominance becomes path-dependent when there is a limit on the support size of the dominating strategies, and that deciding whether a given strategy can be eliminated by iterated strict dominance under this restriction is NP-complete (even when the limit on the support size is 3). Finally, we studied dominance and iterated dominance in Bayesian games, as an example of a concise representation language for normal form games that is interesting in its own right. We showed that, unlike in normal form games, deciding whether a given pure strategy is dominated by another pure strategy in a Bayesian game is NP-complete (both with strict and weak dominance); however, deciding whether a strategy is dominated by some mixed strategy can still be done in polynomial time with a single linear program solve (both with strict and weak dominance). Finally, we showed that iterated dominance using pure strategies can require an exponential number of iterations in a Bayesian game (both with strict and weak dominance). There are various avenues for future research. First, there is the open question of whether it is possible to complete iterated dominance in Bayesian games in polynomial time (even though we showed that an exponential number of alternations between the players in eliminating strategies is sometimes required). Second, we can study computational aspects of (iterated) dominance in concise representations of normal form games other than Bayesian games-for example, in graphical games [9] or local-effect/action graph games [12, 2]. (How to efficiently perform iterated very weak dominance has already been studied for partially observable stochastic games [8].) Finally, we can ask whether some of the algorithms we described (such as the one for iterated strict dominance with mixed strategies) can be made faster. 8. REFERENCES [1] Krzysztof R. Apt. Uniform proofs of order independence for various strategy elimination procedures. Contributions to Theoretical Economics, 4(1), 2004. [2] Nivan A. R. Bhat and Kevin Leyton-Brown. Computing Nash equilibria of action-graph games. In UAI, 2004. [3] Ben Blum, Christian R. Shelton, and Daphne Koller. A continuation method for Nash equilibria in structured games. In IJCAI, 2003. [4] Vincent Conitzer and Tuomas Sandholm. Complexity results about Nash equilibria. In IJCAI, pages 765-771, 2003. [5] Drew Fudenberg and Jean Tirole. Game Theory. MIT Press, 1991. [6] Itzhak Gilboa, Ehud Kalai, and Eitan Zemel. On the order of eliminating dominated strategies. Operations Research Letters, 9:85-89, 1990. [7] Itzhak Gilboa, Ehud Kalai, and Eitan Zemel. The complexity of eliminating dominated strategies. Mathematics of Operation Research, 18:553-565, 1993. [8] Eric A. Hansen, Daniel S. Bernstein, and Shlomo Zilberstein. Dynamic programming for partially observable stochastic games. In AAAI, pages 709-715, 2004. [9] Michael Kearns, Michael Littman, and Satinder Singh. Graphical models for game theory. In UAI, 2001. [10] Leonid Khachiyan. A polynomial algorithm in linear programming. Soviet Math. Doklady, 20:191-194, 1979. [11] Donald E. Knuth, Christos H. Papadimitriou, and John N Tsitsiklis. A note on strategy elimination in bimatrix games. Operations Research Letters, 7(3):103-107, 1988. [12] Kevin Leyton-Brown and Moshe Tennenholtz. Local-effect games. In IJCAI, 2003. [13] Richard Lipton, Evangelos Markakis, and Aranyak Mehta. Playing large games using simple strategies. In ACM-EC, pages 36-41, 2003. [14] Michael Littman and Peter Stone. A polynomial-time Nash equilibrium algorithm for repeated games. In ACM-EC, pages 48-54, 2003. [15] Leslie M. Marx and Jeroen M. Swinkels. Order independence for iterated weak dominance. Games and Economic Behavior, 18:219-245, 1997. [16] Leslie M. Marx and Jeroen M. Swinkels. Corrigendum, order independence for iterated weak dominance. Games and Economic Behavior, 31:324-329, 2000. [17] Andreu Mas-Colell, Michael Whinston, and Jerry R. Green. Microeconomic Theory. Oxford University Press, 1995. [18] Roger Myerson. Game Theory: Analysis of Conflict. Harvard University Press, Cambridge, 1991. [19] Martin J Osborne and Ariel Rubinstein. A Course in Game Theory. MIT Press, 1994. [20] Christos Papadimitriou. Algorithms, games and the Internet. In STOC, pages 749-753, 2001. [21] Ryan Porter, Eugene Nudelman, and Yoav Shoham. Simple search methods for finding a Nash equilibrium. In AAAI, pages 664-669, 2004. 97

game theory;nash equilibrium;bayesian game;dominance;normal form game;elimination;iterated dominance;self-interested agent;strategy;optimal action;multiagent system

train_J-52

Hidden-Action in Multi-Hop Routing

In multi-hop networks, the actions taken by individual intermediate nodes are typically hidden from the communicating endpoints; all the endpoints can observe is whether or not the end-to-end transmission was successful. Therefore, in the absence of incentives to the contrary, rational (i.e., selfish) intermediate nodes may choose to forward packets at a low priority or simply not forward packets at all. Using a principal-agent model, we show how the hidden-action problem can be overcome through appropriate design of contracts, in both the direct (the endpoints contract with each individual router) and recursive (each router contracts with the next downstream router) cases. We further demonstrate that per-hop monitoring does not necessarily improve the utility of the principal or the social welfare in the system. In addition, we generalize existing mechanisms that deal with hidden-information to handle scenarios involving both hidden-information and hidden-action.

1. INTRODUCTION Endpoints wishing to communicate over a multi-hop network rely on intermediate nodes to forward packets from the sender to the receiver. In settings where the intermediate nodes are independent agents (such as individual nodes in ad-hoc and peer-topeer networks or autonomous systems on the Internet), this poses an incentive problem; the intermediate nodes may incur significant communication and computation costs in the forwarding of packets without deriving any direct benefit from doing so. Consequently, a rational (i.e., utility maximizing) intermediate node may choose to forward packets at a low priority or not forward the packets at all. This rational behavior may lead to suboptimal system performance. The endpoints can provide incentives, e.g., in the form of payments, to encourage the intermediate nodes to forward their packets. However, the actions of the intermediate nodes are often hidden from the endpoints. In many cases, the endpoints can only observe whether or not the packet has reached the destination, and cannot attribute failure to a specific node on the path. Even if some form of monitoring mechanism allows them to pinpoint the location of the failure, they may still be unable to attribute the cause of failure to either the deliberate action of the intermediate node, or to some external factors beyond the control of the intermediate node, such as network congestion, channel interference, or data corruption. The problem of hidden action is hardly unique to networks. Also known as moral hazard, this problem has long been of interest in the economics literature concerning information asymmetry, incentive and contract theory, and agency theory. We follow this literature by formalizing the problem as a principal-agent model, where multiple agents making sequential hidden actions [17, 27]. Our results are threefold. First, we show that it is possible to design contracts to induce cooperation when intermediate nodes can choose to forward or drop packets, as well as when the nodes can choose to forward packets with different levels of quality of service. If the path and transit costs are known prior to transmission, the principal achieves first best solution, and can implement the contracts either directly with each intermediate node or recursively through the network (each node making a contract with the following node) without any loss in utility. Second, we find that introducing per-hop monitoring has no impact on the principal"s expected utility in equilibrium. For a principal who wishes to induce an equilibrium in which all intermediate nodes cooperate, its expected total payment is the same with or without monitoring. However, monitoring provides a dominant strategy equilibrium, which is a stronger solution concept than the Nash equilibrium achievable in the absence of monitoring. Third, we show that in the absence of a priori information about transit costs on the packet forwarding path, it is possible to generalize existing mechanisms to overcome scenarios that involve both hidden-information and hidden-action. In these scenarios, the principal pays a premium compared to scenarios with known transit costs. 2. BASELINE MODEL We consider a principal-agent model, where the principal is a pair of communication endpoints who wish to communicate over a multi-hop network, and the agents are the intermediate nodes capable of forwarding packets between the endpoints. The principal (who in practice can be either the sender, the receiver, or 117 both) makes individual take-it-or-leave-it offers (contracts) to the agents. If the contracts are accepted, the agents choose their actions sequentially to maximize their expected payoffs based on the payment schedule of the contract. When necessary, agents can in turn make subsequent take-it-or-leave-it offers to their downstream agents. We assume that all participants are risk neutral and that standard assumptions about the global observability of the final outcome and the enforceability of payments by guaranteeing parties hold. For simplicity, we assume that each agent has only two possible actions; one involving significant effort and one involving little effort. We denote the action choice of agent i by ai ∈ {0, 1}, where ai = 0 and ai = 1 stand for the low-effort and high-effort actions, respectively. Each action is associated with a cost (to the agent) C(ai), and we assume: C(ai = 1) > C(ai = 0) At this stage, we assume that all nodes have the same C(ai) for presentation clarity, but we relax this assumption later. Without loss of generality we normalize the C(ai = 0) to be zero, and denote the high-effort cost by c, so C(ai = 0) = 0 and C(ai = 1) = c. The utility of agent i, denoted by ui, is a function of the payment it receives from the principal (si), the action it takes (ai), and the cost it incurs (ci), as follows: ui(si, ci, ai) = si − aici The outcome is denoted by x ∈ {xG , xB }, where xG stands for the Good outcome in which the packet reaches the destination, and xB stands for the Bad outcome in which the packet is dropped before it reaches the destination. The outcome is a function of the vector of actions taken by the agents on the path, a = (a1, ..., an) ∈ {0, 1}n , and the loss rate on the channels, k. The benefit of the sender from the outcome is denoted by w(x), where: w(xG ) = wG ; and w(xB ) = wB = 0 The utility of the sender is consequently: u(x, S) = w(x) − S where: S = Pn i=1 si A sender who wishes to induce an equilibrium in which all nodes engage in the high-effort action needs to satisfy two constraints for each agent i: (IR) Individual rationality (participation constraint)1 : the expected utility from participation should (weakly) exceed its reservation utility (which we normalize to 0). (IC) Incentive compatibility: the expected utility from exerting high-effort should (weakly) exceed its expected utility from exerting low-effort. In some network scenarios, the topology and costs are common knowledge. That is, the sender knows in advance the path that its packet will take and the costs on that path. In other routing scenarios, the sender does not have this a priori information. We show that our model can be applied to both scenarios with known and unknown topologies and costs, and highlight the implications of each scenario in the context of contracts. We also distinguish between direct contracts, where the principal signs an individual contract 1We use the notion of ex ante individual rationality, in which the agents choose to participate before they know the state of the system. S Dn1 Source Destination n intermediate nodes Figure 1: Multi-hop path from sender to destination. Figure 2: Structure of the multihop routing game under known topology and transit costs. with each node, and recursive contracts, where each node enters a contractual relationship with its downstream node. The remainder of this paper is organized as follows. In Section 3 we consider agents who decide whether to drop or forward packets with and without monitoring when the transit costs are common knowledge. In Section 4, we extend the model to scenarios with unknown transit costs. In Section 5, we distinguish between recursive and direct contracts and discuss their relationship. In Section 6, we show that the model applies to scenarios in which agents choose between different levels of quality of service. We consider Internet routing as a case study in Section 7. In Section 8 we present related work, and Section 9 concludes the paper. 3. KNOWN TRANSIT COSTS In this section we analyze scenarios in which the principal knows in advance the nodes on the path to the destination and their costs, as shown in figure 1. We consider agents who decide whether to drop or forward packets, and distinguish between scenarios with and without monitoring. 3.1 Drop versus Forward without Monitoring In this scenario, the agents decide whether to drop (a = 0) or forward (a = 1) packets. The principal uses no monitoring to observe per-hop outcomes. Consequently, the principal makes the payment schedule to each agent contingent on the final outcome, x, as follows: si(x) = (sB i , sG i ) where: sB i = si(x = xB ) sG i = si(x = xG ) The timeline of this scenario is shown in figure 2. Given a perhop loss rate of k, we can express the probability that a packet is successfully delivered from node i to its successor i + 1 as: Pr(xG i→i+1|ai) = (1 − k)ai (1) where xG i→j denotes a successful transmission from node i to j. PROPOSITION 3.1. Under the optimal contract that induces high-effort behavior from all intermediate nodes in the Nash Equi118 librium2 , the expected payment to each node is the same as its expected cost, with the following payment schedule: sB i = si(x = xB ) = 0 (2) sG i = si(x = xG ) = c (1 − k)n−i+1 (3) PROOF. The principal needs to satisfy the IC and IR constraints for each agent i, which can be expressed as follows: (IC)Pr(xG |aj≥i = 1)sG i + Pr(xB |aj≥i = 1)sB i − c ≥ Pr(xG |ai = 0, aj>i = 1)sG i + Pr(xB |ai = 0, aj>i = 1)sB i (4) This constraint says that the expected utility from forwarding is greater than or equal to its expected utility from dropping, if all subsequent nodes forward as well. (IR)Pr(xG S→i|aj<i = 1)(Pr(xG |aj≥i = 1)sG i + Pr(xB |aj≥i = 1)sB i − c) + Pr(xB S→i|aj<i = 1)sB i ≥ 0 (5) This constraint says that the expected utility from participating is greater than or equal to zero (reservation utility), if all other nodes forward. The above constraints can be expressed as follows, based on Eq. 1: (IC) : (1 − k)n−i+1 sG i + (1 − (1 − k)n−i+1 )sB i − c ≥ sB i (IR) : (1−k)i ((1−k)n−i+1 sG i +(1−(1−k)n−i+1 )sB i −c)+ (1 − (1 − k)i )sB i ≥ 0 It is a standard result that both constraints bind at the optimal contract (see [23]). Solving the two equations, we obtain the solution that is presented in Eqs. 2 and 3. We next prove that the expected payment to a node equals its expected cost in equilibrium. The expected cost of node i is its transit cost multiplied by the probability that it faces this cost (i.e., the probability that the packet reaches node i), which is: (1 − k)i c. The expected payment that node i receives is: Pr(xG )sG i + Pr(xB )sB i = (1 − k)n+1 c (1 − k)n−i+1 = (1 − k)i c Note that the expected payment to a node decreases as the node gets closer to the destination due to the asymmetric distribution of risk. The closer the node is to the destination, the lower the probability that a packet will fail to reach the destination, resulting in the low payment being made to the node. The expected payment by the principal is: E[S] = (1 − k)n+1 nX i=1 sG i + (1 − (1 − k)n+1 ) nX i=1 sB i = (1 − k)n+1 nX i=1 ci (1 − k)n−i+1 (6) The expected payment made by the principal depends not only on the total cost, but also the number of nodes on the path. PROPOSITION 3.2. Given two paths with respective lengths of n1 and n2 hops, per-hop transit costs of c1 and c2, and per-hop loss rates of k1 and k2, such that: 2Since transit nodes perform actions sequentially, this is really a subgameperfect equilibrium (SPE), but we will refer to it as Nash equilibrium in the remainder of the paper. Figure 3: Two paths of equal total costs but different lengths and individual costs. • c1n1 = c2n2 (equal total cost) • (1 − k1)n1+1 = (1 − k2)n2+1 (equal expected benefit) • n1 < n2 (path 1 is shorter than path 2) the expected total payment made by the principal is lower on the shorter path. PROOF. The expected payment in path j is E[S]j = nj X i=1 cj (1 − kj )i = cj (1 − kj ) 1 − (1 − kj)nj kj So, we have to show that: c1(1 − k1) 1 − (1 − k1)n1 k1 > c2(1 − k2) 1 − (1 − k2)n2 k2 Let M = c1n1 = c2n2 and N = (1 − k1)n1+1 = (1 − k2)n2+1 . We have to show that MN 1 n1+1 (1 − N n1 n1+1 ) n1(1 − N 1 n1+1 ) < MN 1 n2+1 (1 − N n2 n2+1 ) n2(1 − N 1 n2+1 ) (7) Let f = N 1 n+1 (1 − N n n+1 ) n(1 − N 1 n+1 ) Then, it is enough to show that f is monotonically increasing in n ∂f ∂n = g(N, n) h(N, n) where: g(N, n) = −((ln(N)n − (n + 1)2 )(N 1 n+1 − N n+2 n+1 ) − (n + 1)2 (N + N 2 n+1 )) and h(N, n) = (n + 1)2 n2 (−1 + N 1 n+1 )2 but h(N, n) > 0 ∀N, n, therefore, it is enough to show that g(N, n) > 0. Because N ∈ (0, 1): (i) ln(N) < 0, and (ii) N 1 n+1 > N n+2 n+1 . Therefore, g(N, n) > 0 ∀N, n. This means that, ceteris paribus, shorter paths should always be preferred over longer ones. For example, consider the two topologies presented in Figure 3. While the paths are of equal total cost, the total expected payment by the principal is different. Based on Eqs. 2 and 3, the expected total payment for the top path is: E[S] = Pr(xG )(sG A + sG B) = „ c1 (1 − k1)2 + c1 1 − k1 « (1 − k1)3 (8) 119 while the expect total payment for the bottom path is: E[S] = Pr(xG )(sG A + sG B + sG C ) = ( c2 (1 − k2)3 + c2 (1 − k2)2 + c2 1 − k2 )(1 − k2)4 For n1 = 2, c1 = 1.5, k1 = 0.5, n2 = 3, c2 = 1, k2 = 0.405, we have equal total cost and equal expected benefit, but E[S]1 = 0.948 and E[S]2 = 1.313. 3.2 Drop versus Forward with Monitoring Suppose the principal obtains per-hop monitoring information.3 Per-hop information broadens the set of mechanisms the principal can use. For example, the principal can make the payment schedule contingent on arrival to the next hop instead of arrival to the final destination. Can such information be of use to a principal wishing to induce an equilibrium in which all intermediate nodes forward the packet? PROPOSITION 3.3. In the drop versus forward model, the principal derives the same expected utility whether it obtains per-hop monitoring information or not. PROOF. The proof to this proposition is already implied in the findings of the previous section. We found that in the absence of per-hop information, the expected cost of each intermediate node equals its expected payment. In order to satisfy the IR constraint, it is essential to pay each intermediate node an expected amount of at least its expected cost; otherwise, the node would be better-off not participating. Therefore, no other payment scheme can reduce the expected payment from the principal to the intermediate nodes. In addition, if all nodes are incentivized to forward packets, the probability that the packet reaches the destination is the same in both scenarios, thus the expected benefit of the principal is the same. Indeed, we have found that even in the absence of per-hop monitoring information, the principal achieves first-best solution. To convince the reader that this is indeed the case, we provide an example of a mechanism that conditions payments on arrival to the next hop. This is possible only if per-hop monitoring information is provided. In the new mechanism, the principal makes the payment schedule contingent on whether the packet has reached the next hop or not. That is, the payment to node i is sG i if the packet has reached node i + 1, and sB i otherwise. We assume costless monitoring, giving us the best case scenario for the use of monitoring. As before, we consider a principal who wishes to induce an equilibrium in which all intermediate nodes forward the packet. The expected utility of the principal is the difference between its expected benefit and its expected payment. Because the expected benefit when all nodes forward is the same under both scenarios, we only need to show that the expected total payment is identical as well. Under the monitoring mechanism, the principal has to satisfy the following constraints: (IC)Pr(xG i→i+1|ai = 1)sG + Pr(xB i→i+1|ai = 1)sB − c ≥ Pr(xG i→i+1|ai = 0)sG + Pr(xB i→i+1|ai = 0)sB (9) (IR)Pr(xG S→i|aj<i = 1)(Pr(xG i→i+1|ai = 1)sG + Pr(xB i→i+1|ai = 1)sB − c) ≥ 0 (10) 3For a recent proposal of an accountability framework that provides such monitoring information see [4]. These constraints can be expressed as follows: (IC) : (1 − k)sG + ksB − c ≥ s0 (IR) : (1 − k)i ((1 − k)sG + ksB − c) ≥ 0 The two constraints bind at the optimal contract as before, and we get the following payment schedule: sB = 0 sG = c 1 − k The expected total payment under this scenario is: E[S] = nX i=1 ((1 − k)i (sB + (i − 1)sG )) + (1 − k)n+1 nsG = (1 − k)n+1 nX i=1 ci (1 − k)n−i+1 as in the scenario without monitoring (see Equation 6.) While the expected total payment is the same with or without monitoring, there are some differences between the two scenarios. First, the payment structure is different. If no per-hop monitoring is used, the payment to each node depends on its location (i). In contrast, monitoring provides us with n identical contracts. Second, the solution concept used is different. If no monitoring is used, the strategy profile of ai = 1 ∀i is a Nash equilibrium, which means that no agent has an incentive to deviate unilaterally from the strategy profile. In contrast, with the use of monitoring, the action chosen by node i is independent of the other agents" forwarding behavior. Therefore, monitoring provides us with dominant strategy equilibrium, which is a stronger solution concept than Nash equilibrium. [15], [16] discuss the appropriateness of different solution concepts in the context of online environments. 4. UNKNOWN TRANSIT COSTS In certain network settings, the transit costs of nodes along the forwarding path may not be common knowledge, i.e., there exists the problem of hidden information. In this section, we address the following questions: 1. Is it possible to design contracts that induce cooperative behavior in the presence of both hidden-action and hiddeninformation? 2. What is the principal"s loss due to the lack of knowledge of the transit costs? In hidden-information problems, the principal employs mechanisms to induce truthful revelation of private information from the agents. In the routing game, the principal wishes to extract transit cost information from the network routers in order to determine the lowest cost path (LCP) for a given source-destination pair. The network routers act strategically and declare transit costs to maximize their profit. Mechanisms that have been proposed in the literature for the routing game [24, 13] assume that once the transit costs have been obtained, and the LCP has been determined, the nodes on the LCP obediently forward all packets, and that there is no loss in the network, i.e., k = 0. In this section, we consider both hidden information and hidden action, and generalize these mechanisms to induce both truth revelation and high-effort action in equilibrium, where nodes transmit over a lossy communication channel, i.e., k ≥ 0. 4.1 V CG Mechanism In their seminal paper [24], Nisan and Ronen present a VCG mechanism that induces truthful revelation of transit costs by edges 120 Figure 4: Game structure for F P SS, where only hidden-information is considered. Figure 5: Game structure for F P SS , where both hiddeninformation and hidden-action are considered. in a biconnected network, such that lowest cost paths can be chosen. Like all VCG mechanisms, it is a strategyproof mechanism, meaning that it induces truthful revelation in a dominant strategy equilibrium. In [13] (FPSS), Feigenbaum et al. slightly modify the model to have the routers as the selfish agents instead of the edges, and present a distributed algorithm that computes the VCG payments. The timeline of the FPSS game is presented in figure 4. Under FPSS, transit nodes keep track of the amount of traffic routed through them via counters, and payments are periodically transferred from the principals to the transit nodes based on the counter values. FPSS assumes that transit nodes are obedient in packet forwarding behavior, and will not update the counters without exerting high effort in packet forwarding. In this section, we present FPSS , which generalizes FPSS to operate in an environment with lossy communication channels (i.e., k ≥ 0) and strategic behavior in terms of packet forwarding. We will show that FPSS induces an equilibrium in which all nodes truthfully reveal their transit costs and forward packets if they are on the LCP. Figure 5 presents the timeline of FPSS . In the first stage, the sender declares two payment functions, (sG i , sB i ), that will be paid upon success or failure of packet delivery. Given these payments, nodes have incentive to reveal their costs truthfully, and later to forward packets. Payments are transferred based on the final outcome. In FPSS , each node i submits a bid bi, which is its reported transit cost. Node i is said to be truthful if bi = ci. We write b for the vector (b1, . . . , bn) of bids submitted by all transit nodes. Let Ii(b) be the indicator function for the LCP given the bid vector b such that Ii(b) =  1 if i is on the LCP; 0 otherwise. Following FPSS [13], the payment received by node i at equilibrium is: pi = biIi(b) + [ X r Ir(b|i ∞)br − X r Ir(b)br] = X r Ir(b|i ∞)br − X r=i Ir(b)br (11) where the expression b|i x means that (b|i x)j = cj for all j = i, and (b|i x)i = x. In FPSS , we compute sB i and sG i as a function of pi, k, and n. First, we recognize that sB i must be less than or equal to zero in order for the true LCP to be chosen. Otherwise, strategic nodes may have an incentive to report extremely small costs to mislead the principal into believing that they are on the LCP. Then, these nodes can drop any packets they receive, incur zero transit cost, collect a payment of sB i > 0, and earn positive profit. PROPOSITION 4.1. Let the payments of FPSS be: sB i = 0 sG i = pi (1 − k)n−i+1 Then, FPSS has a Nash equilibrium in which all nodes truthfully reveal their transit costs and all nodes on the LCP forward packets. PROOF. In order to prove the proposition above, we have to show that nodes have no incentive to engage in the following misbehaviors: 1. truthfully reveal cost but drop packet, 2. lie about cost and forward packet, 3. lie about cost and drop packet. If all nodes truthfully reveal their costs and forward packets, the expected utility of node i on the LCP is: E[u]i = Pr(xG S→i)(E[si] − ci) + Pr(xB S→i)sB i = (1 − k)i (1 − k)n−i+1 sG i + (1 − (1 − k)n−i+1 )sB i − ci + (1 − (1 − k)i )sB i = (1 − k)i (1 − k)n−i+1 pi (1 − k)n−i+1 − (1 − k)i ci = (1 − k)i (pi − ci) ≥ 0 (12) The last inequality is derived from the fact that FPSS is a truthful mechanism, thus pi ≥ ci. The expected utility of a node not on the LCP is 0. A node that drops a packet receives sB i = 0, which is smaller than or equal to E[u]i for i ∈ LCP and equals E[u]i for i /∈ LCP. Therefore, nodes cannot gain utility from misbehaviors (1) or (3). We next show that nodes cannot gain utility from misbehavior (2). 1. if i ∈ LCP, E[u]i > 0. (a) if it reports bi > ci: i. if bi < P r Ir(b|i ∞)br − P r=i Ir(b)br, it is still on the LCP, and since the payment is independent of bi, its utility does not change. ii. if bi > P r Ir(b|i ∞)br − P r=i Ir(b)br, it will not be on the LCP and obtain E[u]i = 0, which is less than its expected utility if truthfully revealing its cost. 121 (b) if it reports bi < ci, it is still on the LCP, and since the payment is independent of bi, its utility does not change. 2. if i /∈ LCP, E[u]i = 0. (a) if it reports bi > ci, it remains out of the LCP, so its utility does not change. (b) if it reports bi < ci: i. if bi < P r Ir(b|i ∞)br − P r=i Ir(b)br, it joins the LCP, and gains an expected utility of E[u]i = (1 − k)i (pi − ct) However, if i /∈ LCP, it means that ci > X r Ir(c|i ∞)cr − X r=i Ir(c)cr But if all nodes truthfully reveal their costs, pi = X r Ir(c|i ∞)cr − X r=i Ir(c)cr < ci therefore, E[u]i < 0 ii. if bi > P r Ir(b|i ∞)br − P r=i Ir(b)br, it remains out of the LCP, so its utility does not change. Therefore, there exists an equilibrium in which all nodes truthfully reveal their transit costs and forward the received packets. We note that in the hidden information only context, FPSS induces truthful revelation as a dominant strategy equilibrium. In the current setting with both hidden information and hidden action, FPSS achieves a Nash equilibrium in the absence of per-hop monitoring, and a dominant strategy equilibrium in the presence of per-hop monitoring, consistent with the results in section 3 where there is hidden action only. In particular, with per-hop monitoring, the principal declares the payments sB i and sG i to each node upon failure or success of delivery to the next node. Given the payments sB i = 0 and sG i = pi/(1 − k), it is a dominant strategy for the nodes to reveal costs truthfully and forward packets. 4.2 Discussion More generally, for any mechanism M that induces a bid vector b in equilibrium by making a payment of pi(b) to node i on the LCP, there exists a mechanism M that induces an equilibrium with the same bid vector and packet forwarding by making a payment of: sB i = 0 sG i = pi(b) (1 − k)n−i+1 . A sketch of the proof would be as follows: 1. IM i (b) = IM i (b)∀i, since M uses the same choice metric. 2. The expected utility of a LCP node is E[u]i = (1 − k)i (pi(b) − ci) ≥ 0 if it forwards and 0 if it drops, and the expected utility of a non-LCP node is 0. 3. From 1 and 2, we get that if a node i can increase its expected utility by deviating from bi under M , it can also increase its utility by deviating from bi in M, but this is in contradiction to bi being an equilibrium in M. 4. Nodes have no incentive to drop packets since they derive an expected utility of 0 if they do. In addition to the generalization of FPSS into FPSS , we can also consider the generalization of the first-price auction (FPA) mechanism, where the principal determines the LCP and pays each node on the LCP its bid, pi(b) = bi. First-price auctions achieve Nash equilibrium as opposed to dominant strategy equilibrium. Therefore, we should expect the generalization of FPA to achieve Nash equilibrium with or without monitoring. We make two additional comments concerning this class of mechanisms. First, we find that the expected total payment made by the principal under the proposed mechanisms is E[S] = nX i=1 (1 − k)i pi(b) and the expected benefit realized by the principal is E[w] = (1 − k)n+1 wG where Pn i=1 pi and wG are the expected payment and expected benefit, respectively, when only the hidden-information problem is considered. When hidden action is also taken into consideration, the generalized mechanism handles strategic forwarding behavior by conditioning payments upon the final outcome, and accounts for lossy communication channels by designing payments that reflect the distribution of risk. The difference between expected payment and benefit is not due to strategic forwarding behavior, but to lossy communications. Therefore, in a lossless network, we should not see any gap between expected benefits and payments, independent of strategic or non-strategic forwarding behavior. Second, the loss to the principal due to unknown transit costs is also known as the price of frugality, and is an active field of research [2, 12]. This price greatly depends on the network topology and on the mechanism employed. While it is simple to characterize the principal"s loss in some special cases, it is not a trivial problem in general. For example, in topologies with parallel disjoint paths from source to destination, we can prove that under first-price auctions, the loss to the principal is the difference between the cost of the shortest path and the second-shortest path, and the loss is higher under the FPSS mechanism. 5. RECURSIVE CONTRACTS In this section, we distinguish between direct and recursive contracts. In direct contracts, the principal contracts directly with each node on the path and pays it directly. In recursive payment, the principal contracts with the first node on the path, which in turn contracts with the second, and so on, such that each node contracts with its downstream node and makes the payment based on the final result, as demonstrated in figure 6. With direct payments, the principal needs to know the identity and cost of each node on the path and to have some communication channel with the node. With recursive payments, every node needs to communicate only with its downstream node. Several questions arise in this context: • What knowledge should the principal have in order to induce cooperative behavior through recursive contracts? • What should be the structure of recursive contracts that induce cooperative behavior? • What is the relation between the total expected payment under direct and recursive contracts? • Is it possible to design recursive contracts in scenarios of unknown transit costs? 122 Figure 6: Structure of the multihop routing game under known topology and recursive contracts. In order to answer the questions outlined above, we look at the IR and IC constraints that the principal needs to satisfy when contracting with the first node on the path. When the principal designs a contract with the first node, he should take into account the incentives that the first node should provide to the second node, and so on all the way to the destination. For example, consider the topology given in figure 3 (a). When the principal comes to design a contract with node A, he needs to consider the subsequent contract that A should sign with B, which should satisfy the following constrints. (IR) :Pr(xG A→B|aA = 1)(E[s|aB = 1] − c)+ Pr(xB A→B|aA = 1)sB A→B ≥ 0 (IC) :E[s|aB = 1] − c ≥ E[s|aB = 0] where: E[s|aB = 1] = Pr(xG B→D|aB = 1)sG A→B + Pr(xB B→D|aB = 1)sB A→B and E[s|aB = 0] = Pr(xG B→D|aB = 0)sG A→B + Pr(xB B→D|aB = 0)sB A→B These (binding) constraints yield the values of sB A→B and sG A→B: sB A→B = 0 sG A→B = c/(1 − k) Based on these values, S can express the constraints it should satisfy in a contract with A. (IR) :Pr(xG S→A|aS = 1)(E[sS→A − sA→B|ai = 1∀i] − c) + Pr(xB S→A|aS = 1)sB S→A ≥ 0 (IC) : E[sS→A − sA→B|ai = 1∀i] − c ≥ E[sS→A − sA→B|aA = 0, aB = 1] where: E[sS→A − sA→B|ai = 1∀i] = Pr(xG A→D|ai = 1∀i)(sG S→A − sG A→B) +Pr(xB A→D|ai = 1∀i)(sB S→A − sB A→B) and E[sS→A − sA→B|aA = 0, aB = 1] = Pr(xG A→D|aA = 0, aB = 1)(sG S→A − sG A→B) +Pr(xB A→D|aA = 0, aB = 1)(sB S→A − sB A→B) Solving for sB S→A and sG S→A, we get: sB S→A = 0 sG S→A = c(2 − k) 1 − 2k + k2 The expected total payment is E[S] = sG S→APr(xG S→D) + sB S→APr(xB S→D) = c(2 − k)(1 − k) (13) which is equal to the expected total payment under direct contracts (see Eq. 8). PROPOSITION 5.1. The expected total payments by the principal under direct and recursive contracts are equal. PROOF. In order to calculate the expected total payment, we have to find the payment to the first node on the path that will induce appropriate behavior. Because sB i = 0 in the drop / forward model, both constraints can be reduced to: Pr(xG i→R|aj = 1∀j)(sG i − sG i+1) − ci = 0 ⇔ (1 − k)n−i+1 (sG i − sG i+1) − ci = 0 which yields, for all 1 ≤ i ≤ n: sG i = ci (1 − k)n−i+1 + sG i+1 Thus, sG n = cn 1 − k sG n−1 = cn−1 (1 − k)2 + sG n = cn−1 (1 − k)2 + cn 1 − k · · · sG 1 = c1 (1 − k)n + sG 2 = . . . = nX i=1 ci (1 − k)i and the expected total payment is E[S] = (1 − k)n+1 sG 1 = (1 − k)n+1 nX i=1 ci (1 − k)n−i+1 which equals the total expected payment in direct payments, as expressed in Eq. 6. Because the payment is contingent on the final outcome, and the expected payment to a node equals its expected cost, nodes have no incentive to offer their downstream nodes lower payment than necessary, since if they do, their downstream nodes will not forward the packet. What information should the principal posess in order to implement recursive contracts? Like in direct payments, the expected payment is not affected solely by the total payment on the path, but also by the topology. Therefore, while the principal only needs to communicate with the first node on the forwarding path and does not have to know the identities of the other nodes, it still needs to know the number of nodes on the path and their individual transit costs. Finally, is it possible to design recursive contracts under unknown transit costs, and, if so, what should be the structure of such contracts? Suppose the principal has implemented the distributed algorithm that calculates the necessary payments, pi for truthful 123 revelation, would the following payment schedule to the first node induce cooperative behavior? sB 1 = 0 sG 1 = nX i=1 pi (1 − k)i The answer is not clear. Unlike contracts in known transit costs, the expected payment to a node usually exceeds its expected cost. Therefore, transit nodes may not have the appropriate incentive to follow the principal"s guarantee during the payment phase. For example, in FPSS , the principal guarantees to pay each node an expected payment of pi > ci. We assume that payments are enforceable if made by the same entity that pledge to pay. However, in the case of recursive contracts, the entity that pledges to pay in the cost discovery stage (the principal) is not the same as the entity that defines and executes the payments in the forwarding stage (the transit nodes). Transit nodes, who design the contracts in the second stage, know that their downstream nodes will forward the packet as long as the expected payment exceeds the expected cost, even if it is less than the promised amount. Thus, every node has incentive to offer lower payments than promised and keep the profit. Transit nodes, who know this is a plausible scenario, may no longer truthfully reveal their cost. Therefore, while recursive contracts under known transit costs are strategically equivalent to direct contracts, it is not clear whether this is the case under unknown transit costs. 6. HIGH-QUALITY VERSUS LOW-QUALITY FORWARDING So far, we have considered the agents" strategy space to be limited to the drop (a = 0) and forward (a = 1) actions. In this section, we consider a variation of the model where the agents choose between providing a low-quality service (a = 0) and a high-quality service (a = 1). This may correspond to a service-differentiated service model where packets are forwarded on a best-effort or a priority basis [6]. In contrast to drop versus forward, a packet may still reach the next hop (albeit with a lower probability) even if the low-effort action is taken. As a second example, consider the practice of hot-potato routing in inter-domain routing of today"s Internet. Individual autonomous systems (AS"s) can either adopt hot-potato routing or early exit routing (a = 0), where a packet is handed off to the downstream AS at the first possible exit, or late exit routing (a = 1), where an AS carries the packet longer than it needs to, handing off the packet at an exit closer to the destination. In the absence of explicit incentives, it is not surprising that AS"s choose hot-potato routing to minimize their costs, even though it leads to suboptimal routes [28, 29]. In both examples, in the absence of contracts, a rational node would exert low-effort, resulting in lower performance. Nevertheless, this behavior can be avoided with an appropriate design of contracts. Formally, the probability that a packet successfully gets from node i to node i + 1 is: Pr(xG i→i+1|ai) = 1 − (k − qai) (14) where: q ∈ (0, 1] and k ∈ (q, 1] In the drop versus forward model, a low-effort action by any node results in a delivery failure. In contrast, a node in the high/low scenario may exert low-effort and hope to free-ride on the higheffort level exerted by the other agents. PROPOSITION 6.1. In the high-quality versus low-quality forwarding model, where transit costs are common knowledge, the principal derives the same expected utility whether it obtains perhop monitoring information or not. PROOF. The IC and IR constraints are the same as specified in the proof of proposition 3.1, but their values change, based on Eq. 14 to reflect the different model: (IC) : (1−k +q)n−i+1 sG i +(1−(1−k +q)n−i+1 )sB i −c ≥ (1 − k)(1 − k + q)n−i sG i + (1 − (1 − k)(1 − k + q)n−i )sB i (IR) : (1 − k + q)i ((1 − k + q)n−i+1 sG i +(1 − (1 − k + q)n−i+1 )sB i − c) + (1 − (1 − k + q)i )sB i ≥ 0 For this set of constraints, we obtain the following solution: sB i = (1 − k + q)i c(k − 1) q (15) sG i = (1 − k + q)i c(k − 1 + (1 − k + q)−n ) q (16) We observe that in this version, both the high and the low payments depend on i. If monitoring is used, we obtain the following constraints: (IC) : (1 − k + q)sG i + (k − q)sB i − c ≥ (1 − k)sG i + (k)sB i (IR) : (1 − k + q)i ((1 − k + q)sG i + (k − q)sB i − c) ≥ 0 and we get the solution: sB i = c(k − 1) q sG i = ck q The expected payment by the principal with or without forwarding is the same, and equals: E[S] = c(1 − k + q)(1 − (1 − k + q)n ) k − q (17) and this concludes the proof. The payment structure in the high-quality versus low-quality forwarding model is different from that in the drop versus forward model. In particular, at the optimal contract, the low-outcome payment sB i is now less than zero. A negative payment means that the agent must pay the principal in the event that the packet fails to reach the destination. In some settings, it may be necessary to impose a limited liability constraint, i.e., si ≥ 0. This prevents the first-best solution from being achieved. PROPOSITION 6.2. In the high-quality versus low-quality forwarding model, if negative payments are disallowed, the expected payment to each node exceeds its expected cost under the optimal contract. PROOF. The proof is a direct outcome of the following statements, which are proved above: 1. The optimal contract is the contract specified in equations 15 and 16 2. Under the optimal contract, E[si] equals node i s expected cost 3. Under the optimal contract, sB i = (1−k+q)i c(k−1) q < 0 Therefore, under any other contract the sender will have to compensate each node with an expected payment that is higher than its expected transit cost. 124 There is an additional difference between the two models. In drop versus forward, a principal either signs a contract with all n nodes along the path or with none. This is because a single node dropping the packet determines a failure. In contrast, in high versus low-quality forwarding, a success may occur under the low effort actions as well, and payments are used to increase the probability of success. Therefore, it may be possible for the principal to maximize its utility by contracting with only m of the n nodes along the path. While the expected outcome depends on m, it is independent of which specific m nodes are induced. At the same time, the individual expected payments decrease in i (see Eq. 16). Therefore, a principal who wishes to sign a contract with only m out of the n nodes should do so with the nodes that are closest to the destination; namely, nodes (n − m + 1, ..., n − 1, n). Solving for the high-quality versus low-quality forwarding model with unknown transit costs is left for future work. 7. CASE STUDY: INTERNET ROUTING We can map different deployed and proposed Internet routing schemes to the various models we have considered in this work. Border Gateway Protocol (BGP), the current inter-domain routing protocol in the Internet, computes routes based on path vectors. Since the protocol reveals only the autonomous systems (AS"s) along a route but not the cost associated to them, the current BGP routing is best characterized by lack of a priori information about transit costs. In this case, the principal (e.g., a multi-homed site or a tier-1 AS) can implement one of the mechanisms proposed in Section 4 by contracting with individual nodes on the path. Such contracts involve paying some premium over the real cost, and it is not clear whether recursive contacts can be implemented in this scenario. In addition, the current protocol does not have the infrastructure to support implementation of direct contracts between endpoints and the network. Recently, several new architectures have been proposed in the context of the Internet to provide the principal not only with a set of paths from which it can chose (like BGP does) but also with the performance along those paths and the network topology. One approach to obtain such information is through end-to-end probing [1]. Another approach is to have the edge networks perform measurements and discover the network topology [32]. Yet another approach is to delegate the task of obtaining topology and performance information to a third-party, like in the routing-as-a-service proposal [21]. These proposals are quite different in nature, but they are common in their attempt to provide more visibility and transparency into the network. If information about topology and transit costs is obtained, the scenario is mapped to the known transit costs model (Section 3). In this case, first-best contracts can be achieved through individual contracts with nodes along the path. However, as we have shown in Section 5, as long as each agent can chose the next hop, the principal can gain full benefit by contracting with only the first hop (through the implementation of recursive contracts). However, the various proposals for acquiring network topology and performance information do not deal with strategic behavior by the intermediate nodes. With the realization that the information collected may be used by the principal in subsequent contractual relationships, the intermediate nodes may behave strategically, misrepresenting their true costs to the entities that collect and aggregate such information. One recent approach that can alleviate this problem is to provide packet obituaries by having each packet to confirm its delivery or report its last successful AS hop [4]. Another approach is to have third parties like Keynote independently monitor the network performance. 8. RELATED WORK The study of non-cooperative behavior in communication networks, and the design of incentives, has received significant attention in the context of wireless ad-hoc routing. [22] considers the problem of malicious behavior, where nodes respond positively to route requests but then fail to forward the actual packets. It proposes to mitigate it by detection and report mechanisms that will essentially help to route around the malicious nodes. However, rather than penalizing nodes that do not forward traffic, it bypasses the misbehaving nodes, thereby relieving their burden. Therefore, such a mechanism is not effective against selfish behavior. In order to mitigate selfish behavior, some approaches [7, 8, 9] require reputation exchange between nodes, or simply first-hand observations [5]. Other approaches propose payment schemes [10, 20, 31] to encourage cooperation. [31] is the closest to our work in that it designs payment schemes in which the sender pays the intermediate nodes in order to prevent several types of selfish behavior. In their approach, nodes are supposed to send receipts to a thirdparty entity. We show that this type of per-hop monitoring may not be needed. In the context of Internet routing, [4] proposes an accountability framework that provide end hosts and service providers after-thefact audits on the fate of their packets. This proposal is part of a broader approach to provide end hosts with greater control over the path of their packets [3, 30]. If senders have transit cost information and can fully control the path of their packets, they can design contracts that yield them with first-best utility. The accountability framework proposed in [4] can serve two main goals: informing nodes of network conditions to help them make informed decision, and helping entities to establish whether individual ASs have performed their duties adequately. While such a framework can be used for the first task, we propose a different approach to the second problem without the need of per-hop auditing information. Research in distributed algorithmic mechanism design (DAMD) has been applied to BGP routing [13, 14]. These works propose mechanisms to tackle the hidden-information problem, but ignore the problem of forwarding enforcement. Inducing desired behavior is also the objective in [26], which attempts to respond to the challenge of distributed AMD raised in [15]: if the same agents that seek to manipulate the system also run the mechanism, what prevents them from deviating from the mechanism"s proposed rules to maximize their own welfare? They start with the proposed mechanism presented in [13] and use mostly auditing mechanisms to prevent deviation from the algorithm. The focus of this work is the design of a payment scheme that provides the appropriate incentives within the context of multi-hop routing. Like other works in this field, we assume that all the accounting services are done using out-of-band mechanisms. Security issues within this context, such as node authentication or message encryption, are orthogonal to the problem presented in this paper, and can be found, for example, in [18, 19, 25]. The problem of information asymmetry and hidden-action (also known as moral hazard) is well studied in the economics literature [11, 17, 23, 27]. [17] identifies the problem of moral hazard in production teams, and shows that it is impossible to design a sharing rule which is efficient and budget-balanced. [27] shows that this task is made possible when production takes place sequentially. 9. CONCLUSIONS AND FUTURE DIRECTIONS In this paper we show that in a multi-hop routing setting, where the actions of the intermediate nodes are hidden from the source 125 and/or destination, it is possible to design payment schemes to induce cooperative behavior from the intermediate nodes. We conclude that monitoring per-hop outcomes may not improve the utility of the participants or the network performace. In addition, in scenarios of unknown transit costs, it is also possible to design mechanisms that induce cooperative behavior in equilibrium, but the sender pays a premium for extracting information from the transit nodes. Our model and results suggest several natural and intriguing research avenues: • Consider manipulative or collusive behaviors which may arise under the proposed payment schemes. • Analyze the feasibility of recursive contracts under hiddeninformation of transit costs. • While the proposed payment schemes sustain cooperation in equilibrium, it is not a unique equilibrium. We plan to study under what mechanisms this strategy profile may emerge as a unique equilibrium (e.g., penalty by successor nodes). • Consider the effect of congestion and capacity constraints on the proposed mechanisms. Our preliminary results show that when several senders compete for a single transit node"s capacity, the sender with the highest demand pays a premium even if transit costs are common knowledge. The premium can be expressed as a function of the second-highest demand. In addition, if congestion affects the probability of successful delivery, a sender with a lower cost alternate path may end up with a lower utility level than his rival with a higher cost alternate path. • Fully characterize the full-information Nash equilibrium in first price auctions, and use this characterization to derive its overcharging compared to truthful mechaisms. 10. ACKNOWLEDGEMENTS We thank Hal Varian for his useful comments. This work is supported in part by the National Science Foundation under ITR awards ANI-0085879 and ANI-0331659, and Career award ANI0133811. 11. REFERENCES [1] ANDERSEN, D. G., BALAKRISHNAN, H., KAASHOEK, M. F., AND MORRIS, R. Resilient Overlay Networks. In 18th ACM SOSP (2001). [2] ARCHER, A., AND TARDOS, E. Frugal path mechanisms. [3] ARGYRAKI, K., AND CHERITON, D. Loose Source Routing as a Mechanism for Traffic Policies. In Proceedings of SIGCOMM FDNA (August 2004). [4] ARGYRAKI, K., MANIATIS, P., CHERITON, D., AND SHENKER, S. Providing Packet Obituaries. In Third Workshop on Hot Topics in Networks (HotNets) (November 2004). [5] BANSAL, S., AND BAKER, M. Observation-based cooperation enforcement in ad-hoc networks. Technical report, Stanford university (2003). [6] BLAKE, S., BLACK, D., CARLSON, M., DAVIES, E., WANG, Z., AND WEISS, W. An Architecture for Differentiated Service. RFC 2475, 1998. [7] BUCHEGGER, S., AND BOUDEC, J.-Y. L. Performance Analysis of the CONFIDANT Protocol: Cooperation of Nodes - Fairness in Dynamic ad-hoc Networks. In IEEE/ACM Symposium on Mobile Ad Hoc Networking and Computing (MobiHOC) (2002). [8] BUCHEGGER, S., AND BOUDEC, J.-Y. L. Coping with False Accusations in Misbehavior Reputation Systems For Mobile ad-hoc Networks. In EPFL, Technical report (2003). [9] BUCHEGGER, S., AND BOUDEC, J.-Y. L. The effect of rumor spreading in reputation systems for mobile ad-hoc networks. In WiOpt"03: Modeling and Optimization in Mobile ad-hoc and Wireless Networks (2003). [10] BUTTYAN, L., AND HUBAUX, J. Stimulating Cooperation in Self-Organizing Mobile ad-hoc Networks. ACM/Kluwer Journal on Mobile Networks and Applications (MONET) (2003). [11] CAILLAUD, B., AND HERMALIN, B. Hidden Action and Incentives. Teaching Notes. U.C. Berkeley. [12] ELKIND, E., SAHAI, A., AND STEIGLITZ, K. Frugality in path auctions, 2004. [13] FEIGENBAUM, J., PAPADIMITRIOU, C., SAMI, R., AND SHENKER, S. A BGP-based Mechanism for Lowest-Cost Routing. In Proceedings of the ACM Symposium on Principles of Distributed Computing (2002). [14] FEIGENBAUM, J., SAMI, R., AND SHENKER, S. Mechanism Design for Policy Routing. In Yale University, Technical Report (2003). [15] FEIGENBAUM, J., AND SHENKER, S. Distributed Algorithmic Mechanism Design: Recent Results and Future Directions. In Proceedings of the International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications (2002). [16] FRIEDMAN, E., AND SHENKER, S. Learning and implementation on the internet. In Manuscript. New Brunswick: Rutgers University, Department of Economics (1997). [17] HOLMSTROM, B. Moral Hazard in Teams. Bell Journal of Economics 13 (1982), 324-340. [18] HU, Y., PERRIG, A., AND JOHNSON, D. Ariadne: A Secure On-Demand Routing Protocol for ad-hoc Networks. In Eighth Annual International Conference on Mobile Computing and Networking (Mobicom) (2002), pp. 12-23. [19] HU, Y., PERRIG, A., AND JOHNSON, D. SEAD: Secure Efficient Distance Vector Routing for Mobile ad-hoc Networks. In 4th IEEE Workshop on Mobile Computing Systems and Applications (WMCSA) (2002). [20] JAKOBSSON, M., HUBAUX, J.-P., AND BUTTYAN, L. A Micro-Payment Scheme Encouraging Collaboration in Multi-Hop Cellular Networks. In Financial Cryptography (2003). [21] LAKSHMINARAYANAN, K., STOICA, I., AND SHENKER, S. Routing as a service. In UCB Technical Report No. UCB/CSD-04-1327 (January 2004). [22] MARTI, S., GIULI, T. J., LAI, K., AND BAKER, M. Mitigating Routing Misbehavior in Mobile ad-hoc Networks. In Proceedings of MobiCom (2000), pp. 255-265. [23] MASS-COLELL, A., WHINSTON, M., AND GREEN, J. Microeconomic Theory. Oxford University Press, 1995. [24] NISAN, N., AND RONEN, A. Algorithmic Mechanism Design. In Proceedings of the 31st Symposium on Theory of Computing (1999). [25] SANZGIRI, K., DAHILL, B., LEVINE, B., SHIELDS, C., AND BELDING-ROYER, E. A Secure Routing Protocol for ad-hoc Networks. In International Conference on Network Protocols (ICNP) (2002). [26] SHNEIDMAN, J., AND PARKES, D. C. Overcoming rational manipulation in mechanism implementation, 2004. [27] STRAUSZ, R. Moral Hazard in Sequential Teams. Departmental Working Paper. Free University of Berlin (1996). [28] TEIXEIRA, R., GRIFFIN, T., SHAIKH, A., AND VOELKER, G. Network sensitivity to hot-potato disruptions. In Proceedings of ACM SIGCOMM (September 2004). [29] TEIXEIRA, R., SHAIKH, A., GRIFFIN, T., AND REXFORD, J. Dynamics of hot-potato routing in IP networks. In Proceedings of ACM SIGMETRICS (June 2004). [30] YANG, X. NIRA: A New Internet Routing Architecture. In Proceedings of SIGCOMM FDNA (August 2003). [31] ZHONG, S., CHEN, J., AND YANG, Y. R. Sprite: A Simple, Cheat-Proof, Credit-Based System for Mobile ad-hoc Networks. In 22nd Annual Joint Conference of the IEEE Computer and Communications Societies (2003). [32] ZHU, D., GRITTER, M., AND CHERITON, D. Feedback-based Routing. In Proc Hotnets-I (2002). 126

mechanism;moralhazard;route;multi-hop;endpoint;hidden action;contract;hidden-action;failure cause;cost;multi-hop network;principal-agent model;intermediate node;moral hazard;cause of failure;priority;incentive;mechanism design

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A Price-Anticipating Resource Allocation Mechanism for Distributed Shared Clusters

In this paper we formulate the fixed budget resource allocation game to understand the performance of a distributed marketbased resource allocation system. Multiple users decide how to distribute their budget (bids) among multiple machines according to their individual preferences to maximize their individual utility. We look at both the efficiency and the fairness of the allocation at the equilibrium, where fairness is evaluated through the measures of utility uniformity and envy-freeness. We show analytically and through simulations that despite being highly decentralized, such a system converges quickly to an equilibrium and unlike the social optimum that achieves high efficiency but poor fairness, the proposed allocation scheme achieves a nice balance of high degrees of efficiency and fairness at the equilibrium.

1. INTRODUCTION The primary advantage of distributed shared clusters like the Grid [7] and PlanetLab [1] is their ability to pool together shared computational resources. This allows increased throughput because of statistical multiplexing and the bursty utilization pattern of typical users. Sharing nodes that are dispersed in the network allows lower delay because applications can store data close to users. Finally, sharing allows greater reliability because of redundancy in hosts and network connections. However, resource allocation in these systems remains the major challenge. The problem is how to allocate a shared resource both fairly and efficiently (where efficiency is the ratio of the achieved social welfare to the social optimal) with the presence of strategic users who act in their own interests. Several non-economic allocation algorithms have been proposed, but these typically assume that task values (i.e., their importance) are the same, or are inversely proportional to the resources required, or are set by an omniscient administrator. However, in many cases, task values vary significantly, are not correlated to resource requirements, and are difficult and time-consuming for an administrator to set. Instead, we examine a market-based resource allocation system (others are described in [2, 4, 6, 21, 26, 27]) that allows users to express their preferences for resources through a bidding mechanism. In particular, we consider a price-anticipating [12] scheme in which a user bids for a resource and receives the ratio of his bid to the sum of bids for that resource. This proportional scheme is simpler, more scalable, and more responsive [15] than auction-based schemes [6, 21, 26]. Previous work has analyzed price-anticipating schemes in the context of allocating network capacity for flows for users with unlimited budgets. In this work, we examine a price-anticipating scheme in the context of allocating computational capacity for users with private preferences and limited budgets, resulting in a qualitatively different game (as discussed in Section 6). In this paper, we formulate the fixed budget resource allocation game and study the existence and performance of the Nash equilibria of this game. For evaluating the Nash equilibria, we consider both their efficiency, measuring how close the social welfare at equilibrium is to the social optimum, and fairness, measuring how different the users" utilities are. Although rarely considered in previous game theoretical study, we believe fairness is a critical metric for a resource allocation schemes because the perception of unfairness will cause some users to reject a system with more efficient, but less fair resource allocation in favor of one with less efficient, more fair resource allocation. We use both utility uniformity and envy-freeness to measure fairness. Utility uniformity, which is common in Computer Science work, measures the closeness of utilities of different users. Envyfreeness, which is more from the Economic perspective, measures the happiness of users with their own resources compared to the resources of others. Our contributions are as follows: • We analyze the existence and performance of 127 Nash equilibria. Using analysis, we show that there is always a Nash equilibrium in the fixed budget game if the utility functions satisfy a fairly weak and natural condition of strong competitiveness. We also show the worst case performance bounds: for m players the efficiency at equilibrium is Ω(1/ √ m), the utility uniformity is ≥ 1/m, and the envyfreeness ≥ 2 √ 2−2 ≈ 0.83. Although these bounds are quite low, the simulations described below indicate these bounds are overly pessimistic. • We describe algorithms that allow strategic users to optimize their utility. As part of the fixed budget game analysis, we show that strategic users with linear utility functions can calculate their bids using a best response algorithm that quickly results in an allocation with high efficiency with little computational and communication overhead. We present variations of the best response algorithm for both finite and infinite parallelism tasks. In addition, we present a local greedy adjustment algorithm that converges more slowly than best response, but allows for non-linear or unformulatable utility functions. • We show that the price-anticipating resource allocation mechanism achieves a high degree of efficiency and fairness. Using simulation, we find that although the socially optimal allocation results in perfect efficiency, it also results in very poor fairness. Likewise, allocating according to only users" preference weights results in a high fairness, but a mediocre efficiency. Intuition would suggest that efficiency and fairness are exclusive. Surprisingly, the Nash equilibrium, reached by each user iteratively applying the best response algorithm to adapt his bids, achieves nearly the efficiency of the social optimum and nearly the fairness of the weight-proportional allocation: the efficiency is ≥ 0.90, the utility uniformity is ≥ 0.65, and the envyfreeness is ≥ 0.97, independent of the number of users in the system. In addition, the time to converge to the equilibrium is ≤ 5 iterations when all users use the best response strategy. The local adjustment algorithm performs similarly when there is sufficient competitiveness, but takes 25 to 90 iterations to stabilize. As a result, we believe that shared distributed systems based on the fixed budget game can be highly decentralized, yet achieve a high degree of efficiency and fairness. The rest of the paper is organized as follows. We describe the model in Section 2 and derive the performance at the Nash equilibria for the infinite parallelism model in Section 3. In Section 4, we describe algorithms for users to optimize their own utility in the fixed budget game. In Section 5, we describe our simulator and simulation results. We describe related work in Section 6. We conclude by discussing some limit of our model and future work in Section 7. 2. THE MODEL Price-Anticipating Resource Allocation. We study the problem of allocating a set of divisible resources (or machines). Suppose that there are m users and n machines. Each machine can be continuously divided for allocation to multiple users. An allocation scheme ω = (r1, . . . , rm), where ri = (ri1, · · · , rin) with rij representing the share of machine j allocated to user i, satisfies that for any 1 ≤ i ≤ m and 1 ≤ j ≤ n, rij ≥ 0 and Pm i=1 rij ≤ 1. Let Ω denote the set of all the allocation schemes. We consider the price anticipating mechanism in which each user places a bid to each machine, and the price of the machine is determined by the total bids placed. Formally, suppose that user i submits a non-negative bid xij to machine j. The price of machine j is then set to Yj = Pn i=1 xij, the total bids placed on the machine j. Consequently, user i receives a fraction of rij = xij Yj of j. When Yj = 0, i.e. when there is no bid on a machine, the machine is not allocated to anyone. We call xi = (xi1, . . . , xin) the bidding vector of user i. The additional consideration we have is that each user i has a budget constraint Xi. Therefore, user i"s total bids have to sum up to his budget, i.e. Pn j=1 xij = Xi. The budget constraints come from the fact that the users do not have infinite budget. Utility Functions. Each user i"s utility is represented by a function Ui of the fraction (ri1, . . . , rin) the user receives from each machine. Given the problem domain we consider, we assume that each user has different and relatively independent preferences for different machines. Therefore, the basic utility function we consider is the linear utility function: Ui(ri1, · · · , rin) = wi1ri1 +· · ·+winrin, where wij ≥ 0 is user i"s private preference, also called his weight, on machine j. For example, suppose machine 1 has a faster CPU but less memory than machine 2, and user 1 runs CPU bounded applications, while user 2 runs memory bounded applications. As a result, w11 > w12 and w21 < w22. Our definition of utility functions corresponds to the user having enough jobs or enough parallelism within jobs to utilize all the machines. Consequently, the user"s goal is to grab as much of a resource as possible. We call this the infinite parallelism model. In practice, a user"s application may have an inherent limit on parallelization (e.g., some computations must be done sequentially) or there may be a system limit (e.g., the application"s data is being served from a file server with limited capacity). To model this, we also consider the more realistic finite parallelism model, where the user"s parallelism is bounded by ki, and the user"s utility Ui is the sum of the ki largest wijrij. In this model, the user only submits bids to up to ki machines. Our abstraction is to capture the essense of the problem and facilitate our analysis. In Section 7, we discuss the limit of the above definition of utility functions. Best Response. As typically, we assume the users are selfish and strategic - they all act to maximize their own utility, defined by their utility functions. From the perspective of user i, if the total bids of the other users placed on each machine j is yj, then the best response of user i to the system is the solution of the following optimization problem: maximize Ui( xij xij +yj ) subject to Pn j=1 xij = Xi, and xij ≥ 0. The difficulty of the above optimization problem depends on the formulation of Ui. We will show later how to solve it for the infinite parallelism model and provide a heuristic for finite parallelism model. Nash Equilibrium. By the assumption that the user is selfish, each user"s bidding vector is the best response to the system. The question we are most interested in is whether there exists a collection of bidding vectors, one for each user, such that each user"s bidding vector is the best response to those of the other users. Such a state is known as the Nash equilibrium, a central concept in Game Theory. Formally, the bidding vectors x1, . . . , xm is a Nash equilibrium if for 128 any 1 ≤ i ≤ m, xi is the best response to the system, or, for any other bidding vector xi, Ui(x1, . . . , xi, . . . , xm) ≥ Ui(x1, . . . , xi, . . . , xm) . The Nash equilibrium is desirable because it is a stable state at which no one has incentive to change his strategy. But a game may not have an equilibrium. Indeed, a Nash equilibrium may not exist in the price anticipating scheme we define above. This can be shown by a simple example of two players and two machines. For example, let U1(r1, r2) = r1 and U2(r1, r2) = r1 + r2. Then player 1 should never bid on machine 2 because it has no value to him. Now, player 2 has to put a positive bid on machine 2 to claim the machine, but there is no lower limit, resulting in the non-existence of the Nash equilibrium. We should note that even the mixed strategy equilibrium does not exist in this example. Clearly, this happens whenever there is a resource that is wanted by only one player. To rule out this case, we consider those strongly competitive games.1 Under the infinite parallelism model, a game is called strongly competitive if for any 1 ≤ j ≤ n, there exists an i = k such that wij, wkj > 0. Under such a condition, we have that (see [5] for a proof), Theorem 1. There always exists a pure strategy Nash equilibrium in a strongly competitive game. Given the existence of the Nash equilibrium, the next important question is the performance at the Nash equilibrium, which is often measured by its efficiency and fairness. Efficiency (Price of Anarchy). For an allocation scheme ω ∈ Ω, denote by U(ω) = P i Ui(ri) the social welfare under ω. Let U∗ = maxω∈Ω U(ω) denote the optimal social welfare - the maximum possible aggregated user utilities. The efficiency at an allocation scheme ω is defined as π(ω) = U(ω) U∗ . Let Ω0 denote the set of the allocation at the Nash equilibrium. When there exists Nash equilibrium, i.e. Ω0 = ∅, define the efficiency of a game Q to be π(Q) = minω∈Ω0 π(ω). It is usually the case that π < 1, i.e. there is an efficiency loss at a Nash equilibrium. This is the price of anarchy [18] paid for not having central enforcement of the user"s good behavior. This price is interesting because central control results in the best possible outcome, but is not possible in most cases. Fairness. While the definition of efficiency is standard, there are multiple ways to define fairness. We consider two metrics. One is by comparing the users" utilities. The utility uniformity τ(ω) of an allocation scheme ω is defined to be mini Ui(ω) maxi Ui(ω) , the ratio of the minimum utility and the maximum utility among the users. Such definition (or utility discrepancy defined similarly as maxi Ui(ω) mini Ui(ω) ) is used extensively in Computer Science literature. Under this definition, the utility uniformity τ(Q) of a game Q is defined to be τ(Q) = minω∈Ω0 τ(ω). The other metric extensively studied in Economics is the concept of envy-freeness [25]. Unlike the utility uniformity metric, the enviness concerns how the user perceives the value of the share assigned to him, compared to the shares other users receive. Within such a framework, define the envy-freeness of an allocation scheme ω by ρ(ω) = mini,j Ui(ri) Ui(rj ) . 1Alternatives include adding a reservation price or limiting the lowest allowable bid to each machine. These alternatives, however, introduce the problem of coming up with the right price or limit. When ρ(ω) ≥ 1, the scheme is known as an envy-free allocation scheme. Likewise, the envy-freeness ρ(Q) of a game Q is defined to be ρ(Q) = minω∈Ω0 ρ(ω). 3. NASH EQUILIBRIUM In this section, we present some theoretical results regarding the performance at Nash equilibrium under the infinite parallelism model. We assume that the game is strongly competitive to guarantee the existence of equilibria. For a meaningful discussion of efficiency and fairness, we assume that the users are symmetric by requiring that Xi = 1 andPn j=1 wij = 1 for all the 1 ≤ i ≤ m. Or informally, we require all the users have the same budget, and they have the same utility when they own all the resources. This precludes the case when a user has an extremely high budget, resulting in very low efficiency or low fairness at equilibrium. We first provide a characterization of the equilibria. By definition, the bidding vectors x1, . . . , xm is a Nash equilibrium if and only if each player"s strategy is the best response to the group"s bids. Since Ui is a linear function and the domain of each users bids {(xi1, . . . , xin)| P j xij = Xi , and xij ≥ 0} is a convex set, the optimality condition is that there exists λi > 0 such that ∂Ui ∂xij = wij Yj − xij Y 2 j  = λi if xij > 0, and < λi if xij = 0. (1) Or intuitively, at an equilibrium, each user has the same marginal value on machines where they place positive bids and has lower marginal values on those machines where they do not bid. Under the infinite parallelism model, it is easy to compute the social optimum U∗ as it is achieved when we allocate each machine wholly to the person who has the maximum weight on the machine, i.e. U∗ = Pn j=1 max1≤i≤m wij. 3.1 Two-player Games We first show that even in the simplest nontrivial case when there are two users and two machines, the game has interesting properties. We start with two special cases to provide some intuition about the game. The weight matrices are shown in figure 1(a) and (b), which correspond respectively to the equal-weight and opposite-weight games. Let x and y denote the respective bids of users 1 and 2 on machine 1. Denote by s = x + y and δ = (2 − s)/s. Equal-weight game. In Figure 1, both users have equal valuations for the two machines. By the optimality condition, for the bid vectors to be in equilibrium, they need to satisfy the following equations according to (1) α y (x + y)2 = (1 − α) 1 − y (2 − x − y)2 α x (x + y)2 = (1 − α) 1 − x (2 − x − y)2 By simplifying the above equations, we obtain that δ = 1 − 1/α and x = y = α. Thus, there exists a unique Nash equilibrium of the game where the two users have the same bidding vector. At the equilibrium, the utility of each user is 1/2, and the social welfare is 1. On the other hand, the social optimum is clearly 1. Thus, the equal-weight game is ideal as the efficiency, utility uniformity, and the envyfreeness are all 1. 129 m1 m2 u1 α 1 − α u2 α 1 − α m1 m2 u1 α 1 − α u2 1 − α α (a) equal weight game (b) opposite weight game Figure 1: Two special cases of two-player games. Opposite-weight game. The situation is different for the opposite game in which the two users put the exact opposite weights on the two machines. Assume that α ≥ 1/2. Similarly, for the bid vectors to be at the equilibrium, they need to satisfy α y (x + y)2 = (1 − α) 1 − y (2 − x − y)2 (1 − α) x (x + y)2 = α 1 − x (2 − x − y)2 By simplifying the above equations, we have that each Nash equilibrium corresponds to a nonnegative root of the cubic equation f(δ) = δ3 − cδ2 + cδ − 1 = 0, where c = 1 2α(1−α) − 1. Clearly, δ = 1 is a root of f(δ). When δ = 1, we have that x = α, y = 1 − α, which is the symmetric equilibrium that is consistent with our intuition - each user puts a bid proportional to his preference of the machine. At this equilibrium, U = 2 − 4α(1 − α), U∗ = 2α, and U/U∗ = (2α + 1 α ) − 2, which is minimized when α = √ 2 2 with the minimum value of 2 √ 2 − 2 ≈ 0.828. However, when α is large enough, there exist two other roots, corresponding to less intuitive asymmetric equilibria. Intuitively, the asymmetric equilibrium arises when user 1 values machine 1 a lot, but by placing even a relatively small bid on machine 1, he can get most of the machine because user 2 values machine 1 very little, and thus places an even smaller bid. In this case, user 1 gets most of machine 1 and almost half of machine 2. The threshold is at when f (1) = 0, i.e. when c = 1 2α(1−α) = 4. This solves to α0 = 2+ √ 2 4 ≈ 0.854. Those asymmetric equilibria at δ = 1 are bad as they yield lower efficiency than the symmetric equilibrium. Let δ0 be the minimum root. When α → 0, c → +∞, and δ0 = 1/c + o(1/c) → 0. Then, x, y → 1. Thus, U → 3/2, U∗ → 2, and U/U∗ → 0.75. From the above simple game, we already observe that the Nash equilibrium may not be unique, which is different from many congestion games in which the Nash equilibrium is unique. For the general two player game, we can show that 0.75 is actually the worst efficiency bound with a proof in [5]. Further, at the asymmetric equilibrium, the utility uniformity approaches 1/2 when α → 1. This is the worst possible for two player games because as we show in Section 3.2, a user"s utility at any Nash equilibrium is at least 1/m in the m-player game. Another consequence is that the two player game is always envy-free. Suppose that the two user"s shares are r1 = (r11, . . . , r1n) and r2 = (r21, . . . , r2n) respectively. Then U1(r1) + U1(r2) = U1(r1 + r2) = U1(1, . . . , 1) = 1 because ri1 + ri2 = 1 for all 1 ≤ i ≤ n. Again by that U1(r1) ≥ 1/2, we have that U1(r1) ≥ U1(r2), i.e. any equilibrium allocation is envy-free. Theorem 2. For a two player game, π(Q) ≥ 3/4, τ(Q) ≥ 0.5, and ρ(Q) = 1. All the bounds are tight in the worst case. 3.2 Multi-player Game For large numbers of players, the loss in social welfare can be unfortunately large. The following example shows the worst case bound. Consider a system with m = n2 + n players and n machines. Of the players, there are n2 who have the same weights on all the machines, i.e. 1/n on each machine. The other n players have weight 1, each on a different machine and 0 (or a sufficiently small ) on all the other machines. Clearly, U∗ = n. The following allocation is an equilibrium: the first n2 players evenly distribute their money among all the machines, the other n player invest all of their money on their respective favorite machine. Hence, the total money on each machine is n + 1. At this equilibrium, each of the first n2 players receives 1 n 1/n n+1 = 1 n2(n+1) on each machine, resulting in a total utility of n3 · 1 n2(n+1) < 1. The other n players each receives 1 n+1 on their favorite machine, resulting in a total utility of n · 1 n+1 < 1. Therefore, the total utility of the equilibrium is < 2, while the social optimum is n = Θ( √ m). This bound is the worst possible. What about the utility uniformity of the multi-player allocation game? We next show that the utility uniformity of the m-player allocation game cannot exceed m. Let (S1, . . . , Sn) be the current total bids on the n machines, excluding user i. User i can ensure a utility of 1/m by distributing his budget proportionally to the current bids. That is, user i, by bidding sij = Xi/ Pn i=1 Si on machine j, obtains a resource level of: rij = sij sij + Sj = Sj/ Pn i=1 Si Sj/ Pn i=1 Si + Sj = 1 1 + Pn i=1 Si , where Pn j=1 Sj = Pm j=1 Xj − Xi = m − 1. Therefore, rij = 1 1+m−1 = 1 m . The total utility of user i is nX j=1 rijwij = (1/m) nX j=1 wij = 1/m . Since each user"s utility cannot exceed 1, the minimal possible uniformity is 1/m. While the utility uniformity can be small, the envy-freeness, on the other hand, is bounded by a constant of 2 √ 2 − 2 ≈ 0.828, as shown in [29]. To summarize, we have that Theorem 3. For the m-player game Q, π(Q) = Ω(1/ √ m), τ(Q) ≥ 1/m, and ρ(Q) ≥ 2 √ 2 − 2. All of these bounds are tight in the worst case. 4. ALGORITHMS In the previous section, we present the performance bounds of the game under the infinite parallelism model. However, the more interesting questions in practice are how the equilibrium can be reached and what is the performance at the Nash equilibrium for the typical distribution of utility functions. In particular, we would like to know if the intuitive strategy of each player constantly re-adjusting his bids according to the best response algorithm leads to the equilibrium. To answer these questions, we resort to simulations. In this section, we present the algorithms that we use to compute or approximate the best response and the social optimum in our experiments. We consider both the infinite parallelism and finite parallelism model. 130 4.1 Infinite Parallelism Model As we mentioned before, it is easy to compute the social optimum under the infinite parallelism model - we simply assign each machine to the user who likes it the most. We now present the algorithm for computing the best response. Recall that for weights w1, . . . , wn, total bids y1, . . . , yn, and the budget X, the best response is to solve the following optimization problem maximize U = Pn j=1 wj xj xj +yj subject to Pn j=1 xj = X, and xj ≥ 0. To compute the best response, we first sort wj yj in decreasing order. Without loss of generality, suppose that w1 y1 ≥ w2 y2 ≥ . . . wn yn . Suppose that x∗ = (x∗ 1, . . . , x∗ n) is the optimum solution. We show that if x∗ i = 0, then for any j > i, x∗ j = 0 too. Suppose this were not true. Then ∂U ∂xj (x∗ ) = wj yj (x∗ j + yj)2 < wj yj y2 j = wj yj ≤ wi yi = ∂U ∂xi (x∗ ) . Thus it contradicts with the optimality condition (1). Suppose that k = max{i|x∗ i > 0}. Again, by the optimality condition, there exists λ such that wi yi (x∗ i +yi)2 = λ for 1 ≤ i ≤ k, and x∗ i = 0 for i > k. Equivalently, we have that: x∗ i = r wiyi λ − yi , for 1 ≤ i ≤ k, and x∗ i = 0 for i > k. Replacing them in the equation Pn i=1 x∗ i = X, we can solve for λ = ( Pk i=1 √ wiyi)2 (X+ Pk i=1 yi)2 . Thus, x∗ i = √ wiyi Pk i=1 √ wiyi (X + kX i=1 yi) − yi . The remaining question is how to determine k. It is the largest value such that x∗ k > 0. Thus, we obtain the following algorithm to compute the best response of a user: 1. Sort the machines according to wi yi in decreasing order. 2. Compute the largest k such that √ wkyk Pk i=1 √ wiyi (X + kX i=1 yi) − yk ≥ 0. 3. Set xj = 0 for j > k, and for 1 ≤ j ≤ k, set: xj = √ wjyj Pk i=1 √ wiyi (X + kX i=1 yi) − yj. The computational complexity of this algorithm is O(n log n), dominated by the sorting. In practice, the best response can be computed infrequently (e.g. once a minute), so for a typically powerful modern host, this cost is negligible. The best response algorithm must send and receive O(n) messages because each user must obtain the total bids from each host. In practice, this is more significant than the computational cost. Note that hosts only reveal to users the sum of the bids on them. As a result, hosts do not reveal the private preferences and even the individual bids of one user to another. 4.2 Finite Parallelism Model Recall that in the finite parallelism model, each user i only places bids on at most ki machines. Of course, the infinite parallelism model is just a special case of finite parallelism model in which ki = n for all the i"s. In the finite parallelism model, computing the social optimum is no longer trivial due to bounded parallelism. It can instead be computed by using the maximum matching algorithm. Consider the weighted complete bipartite graph G = U × V , where U = {ui |1 ≤ i ≤ m , and 1 ≤ ≤ ki}, V = {1, 2, . . . , n} with edge weight wij assigned to the edge (ui , vj). A matching of G is a set of edges with disjoint nodes, and the weight of a matching is the total weights of the edges in the matching. As a result, the following lemma holds. Lemma 1. The social optimum is the same as the maximum weight matching of G. Thus, we can use the maximum weight matching algorithm to compute the social optimum. The maximum weight matching is a classical network problem and can be solved in polynomial time [8, 9, 14]. We choose to implement the Hungarian algorithm [14, 19] because of its simplicity. There may exist a more efficient algorithm for computing the maximum matching by exploiting the special structure of G. This remains an interesting open question. However, we do not know an efficient algorithm to compute the best response under the finite parallelism model. Instead, we provide the following local search heuristic. Suppose we again have n machines with weights w1, . . . , wn and total bids y1, . . . , yn. Let the user"s budget be X and the parallelism bound be k. Our goal is to compute an allocation of X to up to k machines to maximize the user"s utility. For a subset of machines A, denote by x(A) the best response on A without parallelism bound and by U(A) the utility obtained by the best response algorithm. The local search works as follows: 1. Set A to be the k machines with the highest wi/yi. 2. Compute U(A) by the infinite parallelism best response algorithm (Sec 4.1) on A. 3. For each i ∈ A and each j /∈ A, repeat 4. Let B = A − {i} + {j}, compute U(B). 5. If(U(B) > U(A)), let A ← B, and goto 2. 6. Output x(A). Intuitively, by the local search heuristic, we test if we can swap a machine in A for one not in A to improve the best response utility. If yes, we swap the machines and repeat the process. Otherwise, we have reached a local maxima and output that value. We suspect that the local maxima that this algorithm finds is also the global maximum (with respect to an individual user) and that this process stop after a few number of iterations, but we are unable to establish it. However, in our simulations, this algorithm quickly converges to a high (≥ .7) efficiency. 131 4.3 Local Greedy Adjustment The above best response algorithms only work for the linear utility functions described earlier. In practice, utility functions may have more a complicated form, or even worse, a user may not have a formulation of his utility function. We do assume that the user still has a way to measure his utility, which is the minimum assumption necessary for any market-based resource allocation mechanism. In these situations, users can use a more general strategy, the local greedy adjustment method, which works as follows. A user finds the two machines that provide him with the highest and lowest marginal utility. He then moves a fixed small amount of money from the machine with low marginal utility to the machine with the higher one. This strategy aims to adjust the bids so that the marginal values at each machine being bid on are the same. This condition guarantees the allocation is the optimum when the utility function is concave. The tradeoff for local greedy adjustment is that it takes longer to stabilize than best-response. 5. SIMULATION RESULTS While the analytic results provide us with worst-case analysis for the infinite parallelism model, in this section we employ simulations to study the properties of the Nash equilibria in more realistic scenarios and for the finite parallelism model. First, we determine whether the user bidding process converges, and if so, what the rate of convergence is. Second, in cases of convergence, we look at the performance at equilibrium, using the efficiency and fairness metrics defined above. Iterative Method. In our simulations, each user starts with an initial bid vector and then iteratively updates his bids until a convergence criterion (described below) is met. The initial bid is set proportional to the user"s weights on the machines. We experiment with two update methods, the best response methods, as described in Section 4.1 and 4.2, and the local greedy adjustment method, as described in Section 4.3. Convergence Criteria. Convergence time measures how quickly the system reaches equilibrium. It is particularly important in the highly dynamic environment of distributed shared clusters, in which the system"s conditions may change before reaching the equilibrium. Thus, a high convergence rate may be more significant than the efficiency at the equilibrium. There are several different criteria for convergence. The strongest criterion is to require that there is only negligible change in the bids of each user. The problem with this criterion is that it is too strict: users may see negligible change in their utilities, but according to this definition the system has not converged. The less strict utility gap criterion requires there to be only negligible change in the users" utility. Given users" concern for utility, this is a more natural definition. Indeed, in practice, the user is probably not willing to re-allocate their bids dramatically for a small utility gain. Therefore, we use the utility gap criterion to measure convergence time for the best response update method, i.e. we consider that the system has converged if the utility gap of each user is smaller than (0.001 in our experiments). However, this criterion does not work for the local greedy adjustment method because users of that method will experience constant fluctuations in utility as they move money around. For this method, we use the marginal utility gap criterion. We compare the highest and lowest utility margins on the machines. If the difference is negligible, then we consider the system to be converged. In addition to convergence to the equilibrium, we also consider the criterion from the system provider"s view, the social welfare stabilization criterion. Under this criterion, a system has stabilized if the change in social welfare is ≤ . Individual users" utility may not have converged. This criterion is useful to evaluate how quickly the system as a whole reaches a particular efficiency level. User preferences. We experiment with two models of user preferences, random distribution and correlated distribution. With random distribution, users" weights on the different machines are independently and identically distributed, according the uniform distribution. In practice, users" preferences are probably correlated based on factors like the hosts" location and the types of applications that users run. To capture these correlations, we associate with each user and machine a resource profile vector where each dimension of the vector represents one resource (e.g., CPU, memory, and network bandwidth). For a user i with a profile pi = (pi1, . . . , pi ), pik represents user i"s need for resource k. For machine j with profile qj = (qj1, . . . , qj ), qjk represents machine j"s strength with respect to resource k. Then, wij is the dot product of user i"s and machine j"s resource profiles, i.e. wij = pi · qj = P k=1 pikqjk. By using these profiles, we compress the parameter space and introduce correlations between users and machines. In the following simulations, we fix the number of machines to 100 and vary the number of users from 5 to 250 (but we only report the results for the range of 5 − 150 users since the results remain similar for a larger number of users). Sections 5.1 and 5.2 present the simulation results when we apply the infinite parallelism and finite parallelism models, respectively. If the system converges, we report the number of iterations until convergence. A convergence time of 200 iterations indicates non-convergence, in which case we report the efficiency and fairness values at the point we terminate the simulation. 5.1 Infinite parallelism In this section, we apply the infinite parallelism model, which assumes that users can use an unlimited number of machines. We present the efficiency and fairness at the equilibrium, compared to two baseline allocation methods: social optimum and weight-proportional, in which users distribute their bids proportionally to their weights on the machines (which may seem a reasonable distribution method intuitively). We present results for the two user preference models. With uniform preferences, users" weights for the different machines are independently and identically distributed according to the uniform distribution, U ∼ (0, 1) (and are normalized thereafter). In correlated preferences, each user"s and each machine"s resource profile vector has three dimensions, and their values are also taken from the uniform distribution, U ∼ (0, 1). Convergence Time. Figure 2 shows the convergence time, efficiency and fairness of the infinite parallelism model under uniform (left) and correlated (right) preferences. Plots (a) and (b) show the convergence and stabilization time of the best-response and local greedy adjustment methods. 132 0 50 100 150 200 0 20 40 60 80 100 120 140 160 Convergencetime(#iterations) Number of Users Uniform preferences (a) Best-Response Greedy (convergence) Greedy (stabilization) 0 50 100 150 200 0 20 40 60 80 100 120 140 160 Number of Users Correlated preferences (b) Best-response Greedy (convergence) Greedy (stabilization) 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 160 Efficiency Number of Users (c) Nash equilibrium Weight-proportional Social Optimum 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 160 Number of Users (d) Nash equilibrium Weight-proportional Social optimum 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120 140 160 Utilityuniformity Number of Users (e) Nash equilibrium Weight-proportional Social optimum 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120 140 160 Number of Users (f) Nash equilibrium Weight proportional Social optimum 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 160 Envy-freeness Number of Users (g) Nash equilibrium Weight proportional Social optimum 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 160 Number of Users (h) Nash equilibrium Weight proportional Social optimum Figure 2: Efficiency, utility uniformity, enviness and convergence time as a function of the number of users under the infinite parallelism model, with uniform and correlated preferences. n = 100. 133 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Efficiency Iteration number Best-Response Greedy Figure 3: Efficiency level over time under the infinite parallelism model. number of users = 40. n = 100. The best-response algorithm converges within a few number of iterations for any number of users. In contrast, the local greedy adjustment algorithm does not converge even within 500 iterations when the number of users is smaller than 60, but does converge for a larger number of users. We believe that for small numbers of users, there are dependency cycles among the users that prevent the system from converging because one user"s decisions affects another user, whose decisions affect another user, etc. Regardless, the local greedy adjustment method stabilizes within 100 iterations. Figure 3 presents the efficiency over time for a system with 40 users. It demonstrates that while both adjustment methods reach the same social welfare, the best-response algorithm is faster. In the remainder of this paper, we will refer to the (Nash) equilibrium, independent of the adjustment method used to reach it. Efficiency. Figure 2 (c) and (d) present the efficiency as a function of the number of users. We present the efficiency at equilibrium, and use the social optimum and the weightproportional static allocation methods for comparison. Social optimum provides an efficient allocation by definition. For both user preference models, the efficiency at the equilibrium is approximately 0.9, independent of the number of users, which is only slightly worse than the social optimum. The efficiency at the equilibrium is ≈ 50% improvement over the weight-proportional allocation method for uniform preferences, and ≈ 30% improvement for correlated preferences. Fairness. Figure 2(e) and (f) present the utility uniformity as a function of the number of users, and figures (g) and (h) present the envy-freeness. While the social optimum yields perfect efficiency, it has poor fairness. The weightproportional method achieves the highest fairness among the three allocation methods, but the fairness at the equilibrium is close. The utility uniformity is slightly better at the equilibrium under uniform preferences (> 0.7) than under correlated preferences (> 0.6), since when users" preferences are more aligned, users" happiness is more likely going to be at the expense of each other. Although utility uniformity decreases in the number of users, it remains reasonable even for a large number of users, and flattens out at some point. At the social optimum, utility uniformity can be infinitely poor, as some users may be allocated no resources at all. The same is true with respect to envy-freeness. The difference between uniform and correlated preferences is best demonstrated in the social optimum results. When the number of users is small, it may be possible to satisfy all users to some extent if their preferences are not aligned, but if they are aligned, even with a very small number of users, some users get no resources, thus both utility uniformity and envy-freeness go to zero. As the number of users increases, it becomes almost impossible to satisfy all users independent of the existence of correlation. These results demonstrate the tradeoff between the different allocation methods. The efficiency at the equilibrium is lower than the social optimum, but it performs much better with respect to fairness. The equilibrium allocation is completely envy-free under uniform preferences and almost envy-free under correlated preferences. 5.2 Finite parallelism 0 50 100 150 200 0 10 20 30 40 50 60 70 80 90 Convergencetime(#iterations) Number of Users 5 machines/user 20 machines/user Figure 4: Convergence time under the finite parallelism model. n = 100. 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 Efficiency Iteration number 5-machines/user (40 users) 20-machines/user (10 users) Figure 5: Efficiency level over time under the finite parallelism model with local search algorithm. n = 100. We also consider the finite parallelism model and use the local search algorithm, as described in Section 4.2, to adjust user"s bids. We again experimented with both the uniform and correlated preferences distributions and did not find significant differences in the results so we present the simulation results for only the uniform distribution. In our experiments, the local search algorithm stops quickly - it usually discovers a local maximum within two iterations. As mentioned before, we cannot prove that a local maximum is the global maximum, but our experiments indicate that the local search heuristic leads to high efficiency. 134 Convergence time. Let ∆ denote the parallelism bound that limits the maximum number of machines each user can bid on. We experiment with ∆ = 5 and ∆ = 20. In both cases, we use 100 machines and vary the number of users. Figure 4 shows that the system does not always converge, but if it does, the convergence happens quickly. The nonconvergence occurs when the number of users is between 20 and 40 for ∆ = 5, between 5 and 10 for ∆ = 20. We believe that the non-convergence is caused by moderate competition. No competition allows the system to equilibrate quickly because users do not have to change their bids in reaction to changes in others" bids. High competition also allows convergence because each user"s decision has only a small impact on other users, so the system is more stable and can gradually reach convergence. However, when there is moderate competition, one user"s decisions may cause dramatic changes in another"s decisions and cause large fluctuations in bids. In both cases of non-convergence, the ratio of competitors per machine, δ = m×∆/n for m users and n machines, is in the interval [1, 2]. Although the system does not converge in these bad ranges, the system nontheless achieves and maintains a high level of overall efficiency after a few iterations (as shown in Figure 5). Performance. In Figure 6, we present the efficiency, utility uniformity, and envy-freeness at the Nash equilibrium for the finite parallelism model. When the system does not converge, we measure performance by taking the minimum value we observe after running for many iterations. When ∆ = 5, there is a performance drop, in particular with respect to the fairness metrics, in the range between 20 and 40 users (where it does not converge). For a larger number of users, the system converges and achieves a lower level of utility uniformity, but a high degree of efficiency and envy-freeness, similar to those under the infinite parallelism model. As described above, this is due the competition ratio falling into the head-to-head range. When the parallelism bound is large (∆ = 20), the performance is closer to the infinite parallelism model, and we do not observe this drop in performance. 6. RELATED WORK There are two main groups of related work in resource allocation: those that incorporate an economic mechanism, and those that do not. One non-economic approach is scheduling (surveyed by Pinedo [20]). Examples of this approach are queuing in first-come, first-served (FCFS) order, queueing using the resource consumption of tasks (e.g., [28]), and scheduling using combinatorial optimization [19]. These all assume that the values and resource consumption of tasks are reported accurately, which does not apply in the presence of strategic users. We view scheduling and resource allocation as two separate functions. Resource allocation divides a resource among different users while scheduling takes a given allocation and orders a user"s jobs. Examples of the economic approach are Spawn [26]), work by Stoica, et al. [24]., the Millennium resource allocator [4], work by Wellman, et al. [27], Bellagio [2]), and Tycoon [15]). Spawn and the work by Wellman, et al. uses a reservation abstraction similar to the way airline seats are allocated. Unfortunately, reservations have a high latency to acquire resources, unlike the price-anticipating scheme we consider. The tradeoff of the price-anticipating schemes is that users have uncertainty about exactly how much of the resources they will receive. Bellagio[3] uses the SHARE centralized allocator. SHARE allocates resources using a centralized combinatorial auction that allows users to express preferences with complementarities. Solving the NP-complete combinatorial auction problem provides an optimally efficient allocation. The priceanticipating scheme that we consider does not explicitly operate on complementarities, thereby possibly losing some efficiency, but it also avoids the complexity and overhead of combinatorial auctions. There have been several analyses [10, 11, 12, 13, 23] of variations of price-anticipating allocation schemes in the context of allocation of network capacity for flows. Their methodology follows the study of congestion (potential) games [17, 22] by relating the Nash equilibrium to the solution of a (usually convex) global optimization problem. But those techniques no longer apply to our game because we model users as having fixed budgets and private preferences for machines. For example, unlike those games, there may exist multiple Nash equilibria in our game. Milchtaich [16] studied congestion games with private preferences but the technique in [16] is specific to the congestion game. 7. CONCLUSIONS This work studies the performance of a market-based mechanism for distributed shared clusters using both analyatical and simulation methods. We show that despite the worst case bounds, the system can reach a high performance level at the Nash equilibrium in terms of both efficiency and fairness metrics. In addition, with a few exceptions under the finite parallelism model, the system reaches equilibrium quickly by using the best response algorithm and, when the number of users is not too small, by the greedy local adjustment method. While our work indicates that the price-anticipating scheme may work well for resource allocation for shared clusters, there are many interesting directions for future work. One direction is to consider more realistic utility functions. For example, we assume that there is no parallelization cost, and there is no performance degradation when multiple users share the same machine. In practice, both assumptions may not be correct. For examples, the user must copy code and data to a machine before running his application there, and there is overhead for multiplexing resources on a single machine. When the job size is large enough and the degree of multiplexing is sufficiently low, we can probably ignore those effects, but those costs should be taken into account for a more realistic modeling. Another assumption is that users have infinite work, so the more resources they can acquire, the better. In practice, users have finite work. One approach to address this is to model the user"s utility according to the time to finish a task rather than the amount of resources he receives. Another direction is to study the dynamic properties of the system when the users" needs change over time, according to some statistical model. In addition to the usual questions concerning repeated games, it would also be important to understand how users should allocate their budgets wisely over time to accomodate future needs. 135 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 160 Number of Users (a) Limit: 5 machines/user Efficiency Utility uniformity Envy-freeness 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 Number of Users (b) Limit: 20 machines/user Efficiency Utility uniformity Envy-freeness Figure 6: Efficiency, utility uniformity and envy-freeness under the finite parallelism model. n = 100. 8. ACKNOWLEDGEMENTS We thank Bernardo Huberman, Lars Rasmusson, Eytan Adar and Moshe Babaioff for fruitful discussions. 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Spawn: A Distributed Computational Economy. IEEE Transactions on Software Engineering, 18(2):103-117, February 1992. [27] M. P. Wellman, W. E. Walsh, P. R. Wurman, and J. K. MacKie-Mason. Auction Protocols for Decentralized Scheduling. Games and Economic Behavior, 35:271-303, 2001. [28] A. Wierman and M. Harchol-Balter. Classifying Scheduling Policies with respect to Unfairness in an M/GI/1. In Proceedings of the ACM SIGMETRICS 2003 Conference on Measurement and Modeling of Computer Systems, 2003. [29] L. Zhang. On the Efficiency and Fairness of a Fixed Budget Resource Allocation Game. Manuscript, 2004. 136

algorithm;nash equilibrium;distributed shared cluster;fairness;parallelism;efficiency;anarchy price;simulation;price-anticipate mechanism;utility;bidding mechanism;resource allocation;price-anticipating scheme;price of anarchy

train_J-55

From Optimal Limited To Unlimited Supply Auctions

We investigate the class of single-round, sealed-bid auctions for a set of identical items to bidders who each desire one unit. We adopt the worst-case competitive framework defined by [9, 5] that compares the profit of an auction to that of an optimal single-price sale of least two items. In this paper, we first derive an optimal auction for three items, answering an open question from [8]. Second, we show that the form of this auction is independent of the competitive framework used. Third, we propose a schema for converting a given limited-supply auction into an unlimited supply auction. Applying this technique to our optimal auction for three items, we achieve an auction with a competitive ratio of 3.25, which improves upon the previously best-known competitive ratio of 3.39 from [7]. Finally, we generalize a result from [8] and extend our understanding of the nature of the optimal competitive auction by showing that the optimal competitive auction occasionally offers prices that are higher than all bid values.

1. INTRODUCTION The research area of optimal mechanism design looks at designing a mechanism to produce the most desirable outcome for the entity running the mechanism. This problem is well studied for the auction design problem where the optimal mechanism is the one that brings the seller the most profit. Here, the classical approach is to design such a mechanism given the prior distribution from which the bidders" preferences are drawn (See e.g., [12, 4]). Recently Goldberg et al. [9] introduced the use of worst-case competitive analysis (See e.g., [3]) to analyze the performance of auctions that have no knowledge of the prior distribution. The goal of such work is to design an auction that achieves a large constant fraction of the profit attainable by an optimal mechanism that knows the prior distribution in advance. Positive results in this direction are fueled by the observation that in auctions for a number of identical units, much of the distribution from which the bidders are drawn can be deduced on the fly by the auction as it is being run [9, 14, 2]. The performance of an auction in such a worst-case competitive analysis is measured by its competitive ratio, the ratio between a benchmark performance and the auction"s performance on the input distribution that maximizes this ratio. The holy grail of the worstcase competitive analysis of auctions is the auction that achieves the optimal competitive ratio (as small as possible). Since [9] this search has led to improved understanding of the nature of the optimal auction, the techniques for on-the-fly pricing in these scenarios, and the competitive ratio of the optimal auction [5, 7, 8]. In this paper we continue this line of research by improving in all of these directions. Furthermore, we give evidence corroborating the conjecture that the form of the optimal auction is independent of the benchmark used in the auction"s competitive analysis. This result further validates the use of competitive analysis in gauging auction performance. We consider the single item, multi-unit, unit-demand auction problem. In such an auction there are many units of a single item available for sale to bidders who each desire only one unit. Each bidder has a valuation representing how much the item is worth to him. The auction is performed by soliciting a sealed bid from each of the bidders and deciding on the allocation of units to bidders and the prices to be paid by the bidders. The bidders are assumed to bid so as to maximize their personal utility, the difference between their valuation and the price they pay. To handle the problem of designing and analyzing auctions where bidders may falsely declare their valuations to get a better deal, we will adopt the solution concept of truthful mechanism design (see, e.g., [9, 15, 13]). In a truthful auction, revealing one"s true valuation as one"s bid is an optimal strategy for each bidder regardless of the bids of the other bidders. In this paper, we will restrict our attention to truthful (a.k.a., incentive compatible or strategyproof) auctions. A particularly interesting special case of the auction problem is the unlimited supply case. In this case the number of units for sale is at least the number of bidders in the auction. This is natural for the sale of digital goods where there is negligible cost for duplicating 175 and distributing the good. Pay-per-view television and downloadable audio files are examples of such goods. The competitive framework introduced in [9] and further refined in [5] uses the profit of the optimal omniscient single priced mechanism that sells at least two units as the benchmark for competitive analysis. The assumption that two or more units are sold is necessary because in the worst case it is impossible to obtain a constant fraction of the profit of the optimal mechanism when it sells only one unit [9]. In this framework for competitive analysis, an auction is said to be β-competitive if it achieves a profit that is within a factor of β ≥ 1 of the benchmark profit on every input. The optimal auction is the one which is β-competitive with the minimum value of β. Previous to this work, the best known auction for the unlimited supply case had a competitive ratio of 3.39 [7] and the best lower bound known was 2.42 [8]. For the limited supply case, auctions can achieve substantially better competitive ratios. When there are only two units for sale, the optimal auction gives a competitive ratio of 2, which matches the lower bound for two units. When there are three units for sale, the best previously known auction had a competitive ratio of 2.3, compared with a lower bound of 13/6 ≈ 2.17 [8]. The results of this paper are as follows: • We give the auction for three units that is optimally competitive against the profit of the omniscient single priced mechanism that sells at least two units. This auction achieves a competitive ratio of 13/6, matching the lower bound from [8] (Section 3). • We show that the form of the optimal auction is independent of the benchmark used in competitive analysis. In doing so, we give an optimal three bidder auction for generalized benchmarks (Section 4). • We give a general technique for converting a limited supply auction into an unlimited supply auction where it is possible to use the competitive ratio of the limited supply auction to obtain a bound on the competitive ratio of the unlimited supply auction. We refer to auctions derived from this framework as aggregation auctions (Section 5). • We improve on the best known competitive ratio by proving that the aggregation auction constructed from our optimal three-unit auction is 3.25-competitive (Section 5.1). • Assuming that the conjecture that the optimal -unit auction has a competitive ratio that matches the lower bound proved in [8], we show that this optimal auction for ≥ 3 on some inputs will occasionally offer prices that are higher than any bid in that input (Section 6). For the three-unit case where we have shown that the lower bound of [8] is tight, this observation led to our construction of the optimal three-unit auction. 2. DEFINITIONS AND BACKGROUND We consider single-round, sealed-bid auctions for a set of identical units of an item to bidders who each desire one unit. As mentioned in the introduction, we adopt the game-theoretic solution concept of truthful mechanism design. A useful simplification of the problem of designing truthful auctions is obtained through the following algorithmic characterization [9]. Related formulations to this one have appeared in numerous places in recent literature (e.g., [1, 14, 5, 10]). DEFINITION 1. Given a bid vector of n bids, b = (b1, . . . , bn), let b-i denote the vector of with bi replaced with a ‘?", i.e., b-i = (b1, . . . , bi−1, ?, bi+1, . . . , bn). DEFINITION 2. Let f be a function from bid vectors (with a ‘?") to prices (non-negative real numbers). The deterministic bidindependent auction defined by f, BIf , works as follows. For each bidder i: 1. Set ti = f(b-i). 2. If ti < bi, bidder i wins at price ti. 3. If ti > bi, bidder i loses. 4. Otherwise, (ti = bi) the auction can either accept the bid at price ti or reject it. A randomized bid-independent auction is a distribution over deterministic bid-independent auctions. The proof of the following theorem can be found, for example, in [5]. THEOREM 1. An auction is truthful if and only if it is equivalent to a bid-independent auction. Given this equivalence, we will use the the terminology bidindependent and truthful interchangeably. For a randomized bid-independent auction, f(b-i) is a random variable. We denote the probability density of f(b-i) at z by ρb-i (z). We denote the profit of a truthful auction A on input b as A(b). The expected profit of the auction, E[A(b)], is the sum of the expected payments made by each bidder, which we denote by pi(b) for bidder i. Clearly, the expected payment of each bid satisfies pi(b) = bi 0 xρb-i (x)dx. 2.1 Competitive Framework We now review the competitive framework from [5]. In order to evaluate the performance of auctions with respect to the goal of profit maximization, we introduce the optimal single price omniscient auction F and the related omniscient auction F(2) . DEFINITION 3. Give a vector b = (b1, . . . , bn), let b(i) represent the i-th largest value in b. The optimal single price omniscient auction, F, is defined as follows. Auction F on input b determines the value k such that kb(k) is maximized. All bidders with bi ≥ b(k) win at price b(k); all remaining bidders lose. The profit of F on input b is thus F(b) = max1≤k≤n kb(k). In the competitive framework of [5] and subsequent papers, the performance of a truthful auction is gauged in comparison to F(2) , the optimal singled priced auction that sells at least two units. The profit of F(2) is max2≤k≤n kb(k) There are a number of reasons to choose this benchmark for comparison, interested readers should see [5] or [6] for a more detailed discussion. Let A be a truthful auction. We say that A is β-competitive against F(2) (or just β-competitive) if for all bid vectors b, the expected profit of A on b satisfies E[A(b)] ≥ F(2) (b) β . In Section 4 we generalize this framework to other profit benchmarks. 176 2.2 Scale Invariant and Symmetric Auctions A symmetric auction is one where the auction outcome is unchanged when the input bids arrive in a different permutation. Goldberg et al. [8] show that a symmetric auction achieves the optimal competitive ratio. This is natural as the profit benchmark we consider is symmetric, and it allows us to consider only symmetric auctions when looking for the one with the optimal competitive ratio. An auction defined by bid-independent function f is scale invariant if, for all i and all z, Pr[f(b-i) ≥ z] = Pr[f(cb-i) ≥ cz]. It is conjectured that the assumption of scale invariance is without loss of generality. Thus, we are motivated to consider symmetric scale-invariant auctions. When specifying a symmetric scaleinvariant auction we can assume that f is only a function of the relative magnitudes of the n − 1 bids in b-i and that one of the bids, bj = 1. It will be convenient to specify such auctions via the density function of f(b-i), ρb-i (z). It is enough to specify such a density function of the form ρ1,z1,...,zn−1 (z) with 1 ≤ zi ≤ zi+1. 2.3 Limited Supply Versus Unlimited Supply Following [8], throughout the remainder of this paper we will be making the assumption that n = , i.e., the number of bidders is equal to the number of units for sale. This is without loss of generality as (a) any lower bound that applies to the n = case also extends to the case where n ≥ [8], and (b) there is a reduction from the unlimited supply auction problem to the limited supply auction problem that takes an unlimited supply auction that is β-competitive with F(2) and constructs a limited supply auction parameterized by that is β-competitive with F(2, ) , the optimal omniscient auction that sells between 2 and units [6]. Henceforth, we will assume that we are in the unlimited supply case, and we will examine lower bounds for limited supply problems by placing a restriction on the number of bidders in the auction. 2.4 Lower Bounds and Optimal Auctions Frequently in this paper, we will refer to the best known lower bound on the competitive ratio of truthful auctions: THEOREM 2. [8] The competitive ratio of any auction on n bidders is at least 1 − n i=2 −1 n i−1 i i − 1 n − 1 i − 1 . DEFINITION 4. Let Υn denote the n-bidder auction that achieves the optimal competitive ratio. This bound is derived by analyzing the performance of any auction on the following distribution B. In each random bid vector B, each bid Bi is drawn i.i.d. from the distribution such that Pr[Bi ≥ s] ≤ 1/s for all s ∈ S. In the two-bidder case, this lower bound is 2. This is achieved by Υ2 which is the 1-unit Vickrey auction.1 In the three-bidder case, this lower bound is 13/6. In the next section, we define the auction Υ3 which matches this lower bound. In the four-bidder case, this lower bound is 96/215. In the limit as the number of bidders grows, this lower bound approaches a number which is approximately 2.42. It is conjectured that this lower bound is tight for any number of bidders and the optimal auction, Υn, matches it. 1 The 1-unit Vickrey auction sells to the highest bidder at the second highest bid value. 2.5 Profit Extraction In this section we review the truthful profit extraction mechanism ProfitExtractR. This mechanism is a special case of a general cost-sharing schema due to Moulin and Shenker [11]. The goal of profit extraction is, given bids b, to extract a target value R of profit from some subset of the bidders. ProfitExtractR: Given bids b, find the largest k such that the highest k bidders can equally share the cost R. Charge each of these bidders R/k. If no subset of bidders can cover the cost, the mechanism has no winners. Important properties of this auction are as follows: • ProfitExtractR is truthful. • If R ≤ F(b), ProfitExtractR(b) = R; otherwise it has no winners and no revenue. We will use this profit extraction mechanism in Section 5 with the following intuition. Such a profit extractor makes it possible to treat this subset of bidders as a single bid with value F(S). Note that given a single bid, b, a truthful mechanism might offer it price t and if t ≤ b then the bidder wins and pays t; otherwise the bidder pays nothing (and loses). Likewise, a mechanism can offer the set of bidders S a target revenue R. If R ≤ F(2) (S), then ProfitExtractR raises R from S; otherwise, the it raises no revenue from S. 3. AN OPTIMAL AUCTION FOR THREE BIDDERS In this section we define the optimal auction for three bidders, Υ3, and prove that it indeed matches the known lower bound of 13/6. We follow the definition and proof with a discussion of how this auction was derived. DEFINITION 5. Υ3 is scale-invariant and symmetric and given by the bid-independent function with density function ρ1,x(z) = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ For x ≤ 3/2 1 with probability 9/13 z with probability density g(z) for z > 3/2 For x > 3/2⎧ ⎪⎨ ⎪⎩ 1 with probability 9/13 − x 3/2 zg(z)dz x with probability x 3/2 (z + 1)g(z)dz z with probability density g(z) for z > x where g(x) = 2/13 (x−1)3 . THEOREM 3. The Υ3 auction has a competitive ratio of 13/6 ≈ 2.17, which is optimal. Furthermore, the auction raises exactly 6 13 F(2) on every input with non-identical bids. PROOF. Consider the bids 1, x, y, with 1 < x < y. There are three cases. CASE 1 (x < y ≤ 3/2): F(2) = 3. The auction must raise expected revenue of at least 18/13 on these bids. The bidder with valuation x will pay 1 with 9/13, and the bidder with valuation y will pay 1 with probability 9/13. Therefore Υ3 raises 18/13 on these bids. CASE 2 (x ≤ 3/2 < y): F(2) = 3. The auction must raise expected revenue of at least 18/13 on these bids. The bidder with 177 valuation x will pay 9/13 − y 3/2 zg(z)dz in expectation. The bidder with valuation y will pay 9/13 + y 3/2 zg(z)dz in expectation. Therefore Υ3 raises 18/13 on these bids. CASE 3 (3/2 < x ≤ y): F(2) = 2x. The auction must raise expected revenue of at least 12x/13 on these bids. Consider the revenue raised from all three bidders: E[Υ3(b)] = p(1, x, y) + p(x, 1, y) + p(y, 1, x) = 0 + 9/13 − y 3/2 zg(z)dz + 9/13 − x 3/2 zg(z)dz + x x 3/2 (z + 1)g(z)dz + y x zg(z)dz = 18/13 + (x − 2) x 3/2 zg(z)dz + x x 3/2 g(z)dz = 12x/13. The final equation comes from substituting in g(x) = 2/13 (x−1)3 and expanding the integrals. Note that the fraction of F(2) raised on every input is identical. If any of the inequalities 1 ≤ x ≤ y are not strict, the same proof applies giving a lower bound on the auction"s profit; however, this bound may no longer be tight. Motivation for Υ3 In this section, we will conjecture that a particular input distribution is worst-case, and show, as a consequence, that all inputs are worstcase in the optimal auction. By applying this consequence, we will derive an optimal auction for three bidders. A truthful, randomized auction on n bidders can be represented by a randomized function f : Rn−1 × n → R that maps masked bid vectors to prices in R. By normalization, we can assume that the lowest possible bid is 1. Recall that ρb-i (z) = Pr[f(b-i) = z]. The optimal auction for the finite auction problem can be found by the following optimization problem in which the variables are ρb-i (z): maximize r subject to n i=1 bi z=1 zρb-i (z)dz ≥ rF(2) (b) ∞ z=1 ρb-i (z)dz = 1 ρb-i (z) ≥ 0 This set of integral inequalities is difficult to maximize over. However, by guessing which constraints are tight and which are slack at the optimum, we will be able to derive a set of differential equations for which any feasible solution is an optimal auction. As we discuss in Section 2.4, in [8], the authors define a distribution and use it to find a lower bound on the competitive ratio of the optimal auction. For two bidders, this bid distribution is the worst-case input distribution. We guess (and later verify) that this distribution is the worst-case input distribution for three bidders as well. Since this distribution has full support over the set of all bid vectors and a worst-case distribution puts positive probability only on worst-case inputs, we can therefore assume that all but a measure zero set of inputs is worst-case for the optimal auction. In the optimal two-bidder auction, all inputs with non-identical bids are worst-case, so we will assume the same for three bidders. The guess that these constraints are tight allows us to transform the optimization problem into a feasibility problem constrained by differential equations. If the solution to these equations has value matching the lower bound obtained from the worst-case distribution, then this solution is the optimal auction and that our conjectured choice of worst-case distribution is correct. In Section 6 we show that the optimal auction must sometimes place probability mass on sale prices above the highest bid. This motivates considering symmetric scale-invariant auctions for three bidders with probability density function, ρ1,x(z), of the following form: ρ1,x(z) = ⎧ ⎪⎨ ⎪⎩ 1 with discrete probability a(x) x with discrete probability b(x) z with probability density g(z) for z > x In this auction, the sale price for the first bidder is either one of the latter two bids, or higher than either bid with a probability density which is independent of the input. The feasibility problem which arises from the linear optimization problem by assuming the constraints are tight is as follows: a(y) + a(x) + xb(x) + y x zg(z)dz = r max(3, 2x) ∀x < y a(x) + b(x) + ∞ x g(z)dz = 1 a(x) ≥ 0 b(x) ≥ 0 g(z) ≥ 0 Solving this feasibility problem gives the auction Υ3 proposed above. The proof of its optimality validates its proposed form. Finding a simple restriction on the form of n-bidder auctions for n > 3 under which the optimal auction can be found analytically as above remains an open problem. 4. GENERALIZED PROFIT BENCHMARKS In this section, we widen our focus beyond auctions that compete with F(2) to consider other benchmarks for an auction"s profit. We will show that, for three bidders, the form of the optimal auction is essentially independent of the benchmark profit used. This results strongly corroborates the worst-case competitive analysis of auctions by showing that our techniques allow us to derive auctions which are competitive against a broad variety of reasonable benchmarks rather than simply against F(2) . Previous work in competitive analysis of auctions has focused on the question of designing the auction with the best competitive ratio against F(2) , the profit of the optimal omniscient single-priced mechanism that sells at least two items. However, it is reasonable to consider other benchmarks. For instance, one might wish to compete against V∗ , the profit of the k-Vickrey auction with optimal-inhindsight choice of k.2 Alternatively, if an auction is being used as a subroutine in a larger mechanism, one might wish to choose the auction which is optimally competitive with a benchmark specific to that purpose. Recall that F(2) (b) = max2≥k≥n kb(k). We can generalize this definition to Gs, parameterized by s = (s2, . . . , sn) and defined as: Gs(b) = max 2≤k≤n skb(k). When considering Gs we assume without loss of generality that si < si+1 as otherwise the constraint imposed by si+1 is irrelevant. Note that F(2) is the special case of Gs with si = i, and that V∗ = Gs with si = i − 1. 2 Recall that the k-Vickrey auction sells a unit to each of the highest k bidders at a price equal to the k + 1st highest bid, b(k+1), achieving a profit of kb(k+1). 178 Competing with Gs We will now design a three-bidder auction Υs,t 3 that achieves the optimal competitive ratio against Gs,t. As before, we will first find a lower bound on the competitive ratio and then design an auction to meet that bound. We can lower bound the competitive ratio of Υs,t 3 using the same worst-case distribution from [8] that we used against F(2) . Evaluating the performance of any auction competing against Gs,t on this distribution will yield the following theorem. We denote the optimal auction for three bidders against Gs,t as Υs,t 3 . THEOREM 4. The optimal three-bidder auction, Υs,t 3 , competing against Gs,t(b) = max(sb(2), tb(3)) has a competitive ratio of at least s2 +t2 2t . The proof can be found in the appendix. Similarly, we can find the optimal auction against Gs,t using the same technique we used to solve for the three bidder auction with the best competitive ratio against F(2) . DEFINITION 6. Υs,t 3 is scale-invariant and symmetric and given by the bid-independent function with density function ρ1,x(z) = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ For x ≤ t s 1 with probability t2 s2+t2 z with probability density g(z) for z > t s For x > t s⎧ ⎪⎪⎨ ⎪⎪⎩ 1 with probability t2 s2+t2 − x t s zg(z)dz x with probability x t s (z + 1)g(z)dz z with probability density g(z) for z > x where g(x) = 2(t−s)2 /(s2 +t2 ) (x−1)3 . THEOREM 5. Υs,t 3 is s2 +t2 2t -competitive with Gs,t. This auction, like Υ3, can be derived by reducing the optimization problem to a feasibility problem, guessing that the optimal solution has the same form as Υs,t 3 , and solving. The auction is optimal because it matches the lower bound found above. Note that the form of Υs,t 3 is essentially the same as for Υ3, but that the probability of each price is scaled depending on the values of s and t. That our auction for three bidders matches the lower bound computed by the input distribution used in [8] is strong evidence that this input distribution is the worst-case input distribution for any number of bidders and any generalized profit benchmark. Furthermore, we strongly suspect that for any number of bidders, the form of the optimal auction will be independent of the benchmark used. 5. AGGREGATION AUCTIONS We have seen that optimal auctions for small cases of the limitedsupply model can be found analytically. In this section, we will construct a schema for turning limited supply auctions into unlimited supply auctions with a good competitive ratio. As discussed in Section 2.5, the existence of a profit extractor, ProfitExtractR, allows an auction to treat a set of bids S as a single bid with value F(S). Given n bidders and an auction, Am, for m < n bidders, we can convert the m-bidder auction into an n-bidder auction by randomly partitioning the bidders into m subsets and then treating each subset as a single bidder (via ProfitExtractR) and running the m-bidder auction. DEFINITION 7. Given a truthful m-bidder auction, Am, the m-aggregation auction for Am, AggAm , works as follows: 1. Cast each bid uniformly at random into one of m bins, resulting in bid vectors b(1) , . . . , b(m) . 2. For each bin j, compute the aggregate bid Bj = F(b(j) ). Let B be the vector of aggregate bids, and B−j be the aggregate bids for all bins other than j. 3. Compute the aggregate price Tj = f(B−j), where f is the bid-independent function for Am. 4. For each bin j, run ProfitExtractTj on b(j) . Since Am and ProfitExtractR are truthful, Tj is computed independently of any bid in bin j and thus the price offered any bidder in b(j) is independent of his bid; therefore, THEOREM 6. If Am is truthful, the m-aggregation auction for Am, AggAm , is truthful. Note that this schema yields a new way of understanding the Random Sampling Profit Extraction (RSPE) auction [5] as the simplest case of an aggregation auction. It is the 2-aggregation auction for Υ2, the 1-unit Vickrey auction. To analyze AggAm , consider throwing k balls into m labeled bins. Let k represent a configuration of balls in bins, so that ki is equal to the number of balls in bin i, and k(i) is equal to the number of balls in the ith largest bin. Let Km,k represent the set of all possible configurations of k balls in m bins. We write the multinomial coefficient of k as k k . The probability that a particular configuration k arises by throwing balls into bins uniformly at random is k k m−k . THEOREM 7. Let Am be an auction with competitive ratio β. Then the m-aggregation auction for Am, AggAm , raises the following fraction of the optimal revenue F(2) (b): E AggAm (b) F(2) ≥ min k≥2 k∈Km,k F(2) (k) k k βkmk PROOF. By definition, F(2) sells to k ≥ 2 bidders at a single price p. Let kj be the number of such bidders in b(j) . Clearly, F(b(j) ) ≥ pkj. Therefore, F(2) (F(b(1) ), . . . , F(b(n) )) F(2)(b) ≥ F(2) (pk1, . . . , pkn) pk = F(2) (k1, . . . , kn) k The inequality follows from the monotonicity of F(2) , and the equality from the homogeneity of F(2) . ProfitExtractTj will raise Tj if Tj ≤ Bj , and no profit otherwise. Thus, E AggAm (b) ≥ E F(2) (B)/β . The theorem follows by rewriting this expectation as a sum over all k in Km,k. 5.1 A 3.25 Competitive Auction We apply the aggregation auction schema to Υ3, our optimal auction for three bidders, to achieve an auction with competitive ratio 3.25. This improves on the previously best known auction which is 3.39-competitive [7]. THEOREM 8. The aggregation auction for Υ3 has competitive ratio 3.25. 179 PROOF. By theorem 7, E AggΥ3 (b) F(2)(b) ≥ min k≥2 k i=1 k−i j=1 F(2) (i, j, k − i − j) k i,j,k−i−j βk3k For k = 2 and k = 3, E AggΥ3 (b) = 2 3 k/β. As k increases, E AggΥ3 (b) /F(2) increases as well. Since we do not expect to find a closed-form formula for the revenue, we lower bound F(2) (b) by 3b(3). Using large deviation bounds, one can show that this lower bound is greater than 2 3 k/β for large-enough k, and the remainder can be shown by explicit calculation. Plugging in β = 13/6, the competitive ratio is 13/4. As k increases, the competitive ratio approaches 13/6. Note that the above bound on the competitive ratio of AggΥ3 is tight. To see this, consider the bid vector with two very large and non-identical bids of h and h + with the remaining bids 1. Given that the competitive ratio of Υ3 is tight on this example, the expected revenue of this auction on this input will be exactly 13/4. 5.2 A Gs,t-based Aggregation Auction In this section we show that Υ3 is not the optimal auction to use in an aggregation auction. One can do better by choosing the auction that is optimally competitive against a specially tailored benchmark. To see why this might be the case, notice (Table 1) that the fraction of F(2) (b) raised for when there are k = 2 and k = 3 winning bidders in F(2) (b) is substantially smaller than the fraction of F(2) (b) raised when there are more winners. This occurs because the expected ratio between F(2) (B) and F(2) (b) is lower in this case while the competitive ratio of Υ3 is constant. If we chose a three bidder auction that performed better when F(2) has smaller numbers of winners, our aggregation auction would perform better in the worst case. One approach is to compete against a different benchmark that puts more weight than F(2) on solutions with a small number of winners. Recall that F(2) is the instance of Gs,t with s = 2 and t = 3. By using the auction that competes optimally against Gs,t with s > 2, while holding t = 3, we will raise a higher fraction of revenue on smaller numbers of winning bidders and a lower fraction of revenue on large numbers of winning bidders. We can numerically optimize the values of s and t in Gs,t(b) in order to achieve the best competitive ratio for the aggregation auction. In fact, this will allow us to improve our competitive ratio slightly. THEOREM 9. For an optimal choice of s and t, the aggregation auction for Υs,t 3 is 3.243-competitive. The proof follows the outline of Theorem 7 and 8 with the optimal choice of s = 2.162 (while t is held constant at 3). 5.3 Further Reducing the Competitive Ratio There are a number of ways we might attempt to use this aggregation auction schema to continue to push the competitive ratio down. In this section, we give a brief discussion of several attempts. 5.3.1 AggΥm for m > 3 If the aggregation auction for Υ2 has a competitive ratio of 4 and the aggregation auction for Υ3 has a competitive ratio of 3.25, can we improve the competitive ratio by aggregating Υ4 or Υm for larger m? We conjecture in the negative: for m > 3, the aggregation auction for Υm has a larger competitive ratio than the aggregation auction for Υ3. The primary difficulty in proving this k m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 2 0.25 0.3077 0.3349 0.3508 0.3612 0.3686 3 0.25 0.3077 0.3349 0.3508 0.3612 0.3686 4 0.3125 0.3248 0.3349 0.3438 0.3512 0.3573 5 0.3125 0.3191 0.3244 0.3311 0.3378 0.3439 6 0.3438 0.321 0.3057 0.3056 0.311 0.318 7 0.3438 0.333 0.3081 0.3009 0.3025 0.3074 8 0.3633 0.3229 0.3109 0.3022 0.3002 0.3024 9 0.3633 0.3233 0.3057 0.2977 0.2927 0.292 10 0.377 0.3328 0.308 0.2952 0.2866 0.2837 11 0.377 0.3319 0.3128 0.298 0.2865 0.2813 12 0.3872 0.3358 0.3105 0.3001 0.2894 0.2827 13 0.3872 0.3395 0.3092 0.2976 0.2905 0.2841 14 0.3953 0.3391 0.312 0.2961 0.2888 0.2835 15 0.3953 0.3427 0.3135 0.2973 0.2882 0.2825 16 0.4018 0.3433 0.3128 0.298 0.2884 0.2823 17 0.4018 0.3428 0.3129 0.2967 0.2878 0.282 18 0.4073 0.3461 0.3133 0.2959 0.2859 0.2808 19 0.4073 0.3477 0.3137 0.2962 0.2844 0.2789 20 0.4119 0.3486 0.3148 0.2973 0.2843 0.2777 21 0.4119 0.3506 0.3171 0.298 0.2851 0.2775 22 0.4159 0.3519 0.3189 0.2986 0.2863 0.2781 23 0.4159 0.3531 0.3202 0.2995 0.2872 0.2791 24 0.4194 0.3539 0.3209 0.3003 0.2878 0.2797 25 0.4194 0.3548 0.3218 0.3012 0.2886 0.2801 Table 1: E A(b)/F(2) (b) for AggΥm as a function of k, the optimal number of winners in F(2) (b). The lowest value for each column is printed in bold. conjecture lies in the difficulty of finding a closed-form solution for the formula of Theorem 7. We can, however, evaluate this formula numerically for different values of m and k, assuming that the competitive ratio for Υm matches the lower bound for m given by Theorem 2. Table 1 shows, for each value of m and k, the fraction of F(2) raised by the aggregation auction for AggΥm when there are k winning bidders, assuming the lower bound of Theorem 2 is tight. 5.3.2 Convex combinations of AggΥm As can be seen in Table 1, when m > 3, the worst-case value of k is no longer 2 and 3, but instead an increasing function of m. An aggregation auction for Υm outperforms the aggregation auction for Υ3 when there are two or three winning bidders, while the aggregation auction for Υ3 outperforms the other when there are at least six winning bidders. Thus, for instance, an auction which randomizes between aggregation auctions for Υ3 and Υ4 will have a worst-case which is better than that of either auction alone. Larger combinations of auctions will allow more room to optimize the worst-case. However, we suspect that no convex combination of aggregation auctions will have a competitive ratio lower than 3. Furthermore, note that we cannot yet claim the existence of a good auction via this technique as the optimal auction Υn for n > 3 is not known and it is only conjectured that the bound given by Theorem 2 and represented in Table 1 is correct for Υn. 6. A LOWER BOUND FOR CONSERVATIVE AUCTIONS In this section, we define a class of auctions that never offer a sale price which is higher than any bid in the input and prove a lower bound on the competitive ratio of these auctions. As this 180 lower bound is stronger than the lower bound of Theorem 2 for n ≥ 3, it shows that the optimal auction must occasionally charge a sales price higher than any bid in the input. Specifically, this result partially explains the form of the optimal three bidder auction. DEFINITION 8. We say an auction BIf is conservative if its bidindependent function f satisfies f(b-i) ≤ max(b-i). We can now state our lower bound for conservative auctions. THEOREM 10. Let A be a conservative auction for n bidders. Then the competitive ratio of A is at least 3n−2 n . COROLLARY 1. The competitive ratio of any conservative auction for an arbitrary number of bidders is at least three. For a two-bidder auction, this restriction does not prevent optimality. Υ2, the 1-unit Vickrey auction, is conservative. For larger numbers of bidders, however, the restriction to conservative auctions does affect the competitive ratio. For the three-bidder case, Υ3 has competitive ratio 2.17, while the best conservative auction is no better than 2.33-competitive. The k-Vickrey auction and the Random Sampling Optimal Price auction [9] are conservative auctions. The Random Sampling Profit Extraction auction [5] and the CORE auction [7], on the other hand, use the ProfitExtractR mechanism as a subroutine and thus sometimes offer a sale price which is higher than the highest input bid value. In [8], the authors define a restricted auction as one on which, for any input, the sale prices are drawn from the set of input bid values. The class of conservative auctions can be viewed as a generalization of the class of restricted auctions and therefore our result below gives lower bounds on the performance of a broader class of auctions. We will prove Theorem 10 with the aid of the following lemma: LEMMA 1. Let A be a conservative auction with competitive ratio 1/r for n bidders. Let h n. Let h0 = 1 and hk = kh otherwise. Then, for all k and H ≥ kh, Pr[f(1, 1, . . . , 1, H) ≤ hk] ≥ nr−1 n−1 + k(3nr−2r−n n−1 ). PROOF. The lemma is proved by strong induction on k. First some notation that will be convenient. For any particular k and H we will be considering the bid vector b = (1, . . . , 1, hk, H) and placing bounds on ρb-i (z). Since we can assume without loss of generality that the auction is symmetric, we will notate b-1 as b with any one of the 1-valued bids masked. Similarly we notate b-hk (resp. b-H ) as b with the hk-valued bid (resp. H-valued bid) masked. We will also let p1(b), phk (b), and pH (b) represent the expected payment of a 1-valued, hk-valued, and H-valued bidder in A on b, respectively (note by symmetry the expected payment for all 1-valued bidders is the same). Base case (k = 0, hk = 1): A must raise revenue of at least rn on b = (1, . . . , 1, 1, H): rn ≤ pH (b) + (n − 1)p1(b) = 1 + (n − 1) 1 0 xρb-1 (x)dx ≤ 1 + (n − 1) 1 0 ρb-1 (x)dx The second inequality follows from the conservatism of the underlying auction. The base case follows trivially from the final inequality. Inductive case (k > 0, hk = kh): Let b = (1, . . . , 1, hk, H). First, we will find an upper bound on pH(b) pH (b) = 1 0 xρb-H (x)dx + k i=1 hi hi−1 xρb-H (x)dx (1) ≤ 1 + k i=1 hi hi hi−1 ρb-H (x)dx ≤ 1 + 3nr − 2r − n n − 1 k−1 i=1 ih + kh 1 − nr − 1 n − 1 − (k − 1) 3nr − 2r − n n − 1 (2) = kh n(1 − r) n − 1 + (k − 1) 2 3nr − 2r − n n − 1 + 1. (3) Equation (1) follows from the conservatism of A and (2) is from invoking the strong inductive hypothesis with H = kh and the observation that the maximum possible revenue will be found by placing exactly enough probability at each multiple of h to satisfy the constraints of the inductive hypothesis and placing the remaining probability at kh. Next, we will find a lower bound on phk (b) by considering the revenue raised by the bids b. Recall that A must obtain a profit of at least rF(2) (b) = 2rkh. Given upper-bounds on the profit from the H-valued, equation bid (3), and the 1-valued bids, the profit from the hk-valued bid must be at least: phk (b) ≥ 2rkh − (n − 2)p1(b) − pH(b) ≥ kh 2r − n(1 − r) n − 1 + (k − 1) 2 3nr − 2r − n n − 1 − O(n). (4) In order to lower bound Pr[f(b-hk ) ≤ kh], consider the auction that minimizes it and is consistent with the lower bounds obtained by the strong inductive hypothesis on Pr[f(b-hk ) ≤ ih]. To minimize the constraints implied by the strong inductive hypothesis, we place the minimal amount of probability mass required each price level. This gives ρhk (b) with nr−1 n−1 probability at 1 and exactly 3nr−2r−n n−1 at each hi for 1 ≤ i < k. Thus, the profit from offering prices at most hk−1 is nr−1 n−1 −kh(k−1)3nr−2r−n n−1 . In order to satisfy our lower bound, (4), on phk (b), it must put at least 3nr−2r−n n−1 on hk. Therefore, the probability that the sale price will be no more than kh on masked bid vector on bid vector b = (1, . . . , 1, kh, H) must be at least nr−1 n−1 + k(3nr−2r−n n−1 ). Given Lemma 1, Theorem 10 is simple to prove. PROOF. Let A be a conservative auction. Suppose 3nr−2r−n n−1 = > 0. Let k = 1/ , H ≥ kh, and h n. By Lemma 1, Pr[f(1, . . . , 1, kh, H) ≤ hk] ≥ nr−1 n−1 + k > 1. But this is a contradiction, so 3nr−2r−n n−1 ≤ 0. Thus, r ≤ n 3n−2 . The theorem follows. 7. CONCLUSIONS AND FUTURE WORK We have found the optimal auction for the three-unit limitedsupply case, and shown that its structure is essentially independent of the benchmark used in its competitive analysis. We have then used this auction to derive the best known auction for the unlimited supply case. Our work leaves many interesting open questions. We found that the lower bound of [8] is matched by an auction for three bidders, 181 even when competing against generalized benchmarks. The most interesting open question from our work is whether the lower bound from Theorem 2 can be matched by an auction for more than three bidders. We conjecture that it can. Second, we consider whether our techniques can be extended to find optimal auctions for greater numbers of bidders. The use of our analytic solution method requires knowledge of a restricted class of auctions which is large enough to contain an optimal auction but small enough that the optimal auction in this class can be found explicitly through analytic methods. No class of auctions which meets these criteria is known even for the four bidder case. Also, when the number of bidders is greater than three, it might be the case that the optimal auction is not expressible in terms of elementary functions. Another interesting set of open questions concerns aggregation auctions. As we show, the aggregation auction for Υ3 outperforms the aggregation auction for Υ2 and it appears that the aggregation auction for Υ3 is better than Υm for m > 3. We leave verification of this conjecture for future work. We also show that Υ3 is not the best three-bidder auction for use in an aggregation auction, but the auction that beats it is able to reduce the competitive ratio of the overall auction only a little bit. It would be interesting to know whether for any m there is an m-aggregation auction that substantially improves on the competitive ratio of AggΥm . Finally, we remark that very little is known about the structure of the optimal competitive auction. In our auction Υ3, the sales price for a given bidder is restricted either to be one of the other bid values or to be higher than all other bid values. The optimal auction for two bidders, the 1-unit Vickrey auction, also falls within this class of auctions, as its sales prices are restricted to bid values. We conjecture that an optimal auction for any number of bidders lies within this class. Our paper provides partial evidence for this conjecture: the lower bound of Section 6 on conservative auctions shows that the optimal auction must offer sales prices higher than any bid value if the lower bound of Theorem 2 is tight, as is conjectured. It remains to show that optimal auctions otherwise only offer sales prices at bid values. 8. ACKNOWLEDGEMENTS The authors wish to thank Yoav Shoham and Noga Alon for helpful discussions. 9. REFERENCES [1] A. Archer and E. Tardos. Truthful mechanisms for one-parameter agents. In Proc. of the 42nd IEEE Symposium on Foundations of Computer Science, 2001. [2] S. Baliga and R. Vohra. Market research and market design. Advances in Theoretical Economics, 3, 2003. [3] A. Borodin and R. El-Yaniv. Online Computation and Competitive Analysis. Cambridge University Press, 1998. [4] J. Bulow and J. Roberts. The Simple Economics of Optimal Auctions. The Journal of Political Economy, 97:1060-90, 1989. [5] A. Fiat, A. V. Goldberg, J. D. Hartline, and A. R. Karlin. Competitive generalized auctions. In Proc. 34th ACM Symposium on the Theory of Computing, pages 72-81. ACM, 2002. [6] A. Goldberg, J. Hartline, A. Karlin, M. Saks, and A. Wright. Competitive auctions and digital goods. Games and Economic Behavior, 2002. Submitted for publication. An earlier version available as InterTrust Technical Report at URL http://www.star-lab.com/tr/tr-99-01.html. [7] A. V. Goldberg and J. D. Hartline. Competitiveness via consensus. In Proc. 14th Symposium on Discrete Algorithms, pages 215-222. ACM/SIAM, 2003. [8] A. V. Goldberg, J. D. Hartline, A. R. Karlin, and M. E. Saks. A lower bound on the competitive ratio of truthful auctions. In Proc. 21st Symposium on Theoretical Aspects of Computer Science, pages 644-655. Springer, 2004. [9] A. V. Goldberg, J. D. Hartline, and A. Wright. Competitive auctions and digital goods. In Proc. 12th Symposium on Discrete Algorithms, pages 735-744. ACM/SIAM, 2001. [10] D. Lehmann, L. I. O"Callaghan, and Y. Shoham. Truth Revelation in Approximately Efficient Combinatorial Auctions. In Proc. of 1st ACM Conf. on E-Commerce, pages 96-102. ACM Press, New York, 1999. [11] H. Moulin and S. Shenker. Strategyproof Sharing of Submodular Costs: Budget Balance Versus Efficiency. Economic Theory, 18:511-533, 2001. [12] R. Myerson. Optimal Auction Design. Mathematics of Operations Research, 6:58-73, 1981. [13] N. Nisan and A. Ronen. Algorithmic Mechanism Design. In Proc. of 31st Symp. on Theory of Computing, pages 129-140. ACM Press, New York, 1999. [14] I. Segal. Optimal pricing mechanisms with unknown demand. American Economic Review, 16:50929, 2003. [15] W. Vickrey. Counterspeculation, Auctions, and Competitive Sealed Tenders. J. of Finance, 16:8-37, 1961. APPENDIX A. PROOF OF THEOREM 4 We wish to prove that Υs,t 3 , the optimal auction for three bidders against Gs,t, has competitive ratio at least s2 +t2 2t . Our proof follows the outline of the proof of Lemma 5 and Theorem 1 from [8]; however, our case is simpler because we only looking for a bound when n = 3. Define the random bid vector B = (B1, B2, B3) with Pr[Bi > z] = 1/z. We compute EB[Gs,t(B)] by integrating Pr[Gs,t(B) > z]. Then we use the fact that no auction can have expected profit greater than 3 on B to find a lower bound on the competitive ratio against Gs,t for any auction. For the input distribution B defined above, let B(i) be the ith largest bid. Define the disjoint events H2 = B(2) ≥ z/s ∧ B(3) < z/t, and H3 = B(3) ≥ z/t. Intuitively, H3 corresponds to the event that all three bidders win in Gs,t, while H2 corresponds to the event that only the top two bidders win. Gs,t(B) will be greater than z if either event occurs: Pr[Gs,t(B) > z] = Pr[H2] + Pr[H3] (5) = 3 s z 2 1 − t z + t z 3 (6) Using the identity defined for non-negative continuous random variables that E[X] = ∞ 0 Pr[X > x] dx, we have EB[Gs,t(B)] = t + ∞ t 3 s z 2 1 − t z + t z 3 dz (7) = 3 s2 + t2 2t (8) Given that, for any auction A, EB[EA[A(B)]] ≤ 3 [8], it is clear that EB[Gs,t(B)] EB[EA[A(B)]] ≥ s2 +t2 2t . Therefore, there exists some input b for each auction A such that Gs,t(b) EA[A(b)] ≥ s2+t2 2t . 182

unlimited supply;ratio;benchmark;bound;distribution;competitive analysis;auction;preference;aggregation auction;mechanism design

train_J-56

Robust Solutions for Combinatorial Auctions

Bids submitted in auctions are usually treated as enforceable commitments in most bidding and auction theory literature. In reality bidders often withdraw winning bids before the transaction when it is in their best interests to do so. Given a bid withdrawal in a combinatorial auction, finding an alternative repair solution of adequate revenue without causing undue disturbance to the remaining winning bids in the original solution may be difficult or even impossible. We have called this the Bid-taker"s Exposure Problem. When faced with such unreliable bidders, it is preferable for the bid-taker to preempt such uncertainty by having a solution that is robust to bid withdrawal and provides a guarantee that possible withdrawals may be repaired easily with a bounded loss in revenue. In this paper, we propose an approach to addressing the Bidtaker"s Exposure Problem. Firstly, we use the Weighted Super Solutions framework [13], from the field of constraint programming, to solve the problem of finding a robust solution. A weighted super solution guarantees that any subset of bids likely to be withdrawn can be repaired to form a new solution of at least a given revenue by making limited changes. Secondly, we introduce an auction model that uses a form of leveled commitment contract [26, 27], which we have called mutual bid bonds, to improve solution reparability by facilitating backtracking on winning bids by the bid-taker. We then examine the trade-off between robustness and revenue in different economically motivated auction scenarios for different constraints on the revenue of repair solutions. We also demonstrate experimentally that fewer winning bids partake in robust solutions, thereby reducing any associated overhead in dealing with extra bidders. Robust solutions can also provide a means of selectively discriminating against distrusted bidders in a measured manner.

1. INTRODUCTION A combinatorial auction (CA) [5] provides an efficient means of allocating multiple distinguishable items amongst bidders whose perceived valuations for combinations of items differ. Such auctions are gaining in popularity and there is a proliferation in their usage across various industries such as telecoms, B2B procurement and transportation [11, 19]. Revenue is the most obvious optimization criterion for such auctions, but another desirable attribute is solution robustness. In terms of combinatorial auctions, a robust solution is one that can withstand bid withdrawal (a break) by making changes easily to form a repair solution of adequate revenue. A brittle solution to a CA is one in which an unacceptable loss in revenue is unavoidable if a winning bid is withdrawn. In such situations the bid-taker may be left with a set of items deemed to be of low value by all other bidders. These bidders may associate a higher value for these items if they were combined with items already awarded to others, hence the bid-taker is left in an undesirable local optimum in which a form of backtracking is required to reallocate the items in a manner that results in sufficient revenue. We have called this the Bid-taker"s Exposure Problem that bears similarities to the Exposure Problem faced by bidders seeking multiple items in separate single-unit auctions but holding little or no value for a subset of those items. However, reallocating items may be regarded as disruptive to a solution in many real-life scenarios. Consider a scenario where procurement for a business is conducted using a CA. It would be highly undesirable to retract contracts from a group of suppliers because of the failure of a third party. A robust solution that is tolerant of such breaks is preferable. Robustness may be regarded as a preventative measure protecting against future uncertainty by sacrificing revenue in place of solution stability and reparability. We assume a probabilistic approach whereby the bid-taker has knowledge of the reliability of bidders from which the likelihood of an incomplete transaction may be inferred. Repair solutions are required for bids that are seen as brittle (i.e. likely to break). Repairs may also be required for sets of bids deemed brittle. We propose the use of the Weighted Super 183 Solutions (WSS) framework [13] for constraint programming, that is ideal for establishing such robust solutions. As we shall see, this framework can enforce constraints on solutions so that possible breakages are reparable. This paper is organized as follows. Section 2 presents the Winner Determination Problem (WDP) for combinatorial auctions, outlines some possible reasons for bid withdrawal and shows how simply maximizing expected revenue can lead to intolerable revenue losses for risk-averse bid-takers. This motivates the use of robust solutions and Section 3 introduces a constraint programming (CP) framework, Weighted Super Solutions [13], that finds such solutions. We then propose an auction model in Section 4 that enhances reparability by introducing mandatory mutual bid bonds, that may be seen as a form of leveled commitment contract [26, 27]. Section 5 presents an extensive empirical evaluation of the approach presented in this paper, in the context of a number of well-known combinatorial auction distributions, with very encouraging results. Section 6 discusses possible extensions and questions raised by our research that deserve future work. Finally, in Section 7 a number of concluding remarks are made. 2. COMBINATORIAL AUCTIONS Before presenting the technical details of our solution to the Bid-taker"s Exposure Problem, we shall present a brief survey of combinatorial auctions and existing techniques for handling bid withdrawal. Combinatorial auctions involve a single bid-taker allocating multiple distinguishable items amongst a group of bidders. The bidtaker has a set of m items for sale, M = {1, 2, . . . , m}, and bidders submit a set of bids, B = {B1, B2, . . . , Bn}. A bid is a tuple Bj = Sj, pj where Sj ⊆ M is a subset of the items for sale and pj ≥ 0 is a price. The WDP for a CA is to label all bids as either winning or losing so as to maximize the revenue from winning bids without allocating any item to more than one bid. The following is the integer programming formulation for the WDP: max n j=1 pjxj s.t. j|i∈Sj xj ≤ 1, ∀i ∈ {1 . . . m}, xj ∈ {0, 1}. This problem is NP-complete [23] and inapproximable [25], and is otherwise known as the Set Packing Problem. The above problem formulation assumes the notion of free disposal. This means that the optimal solution need not necessarily sell all of the items. If the auction rules stipulate that all items must be sold, the problem becomes a Set Partition Problem [5]. The WDP has been extensively studied in recent years. The fastest search algorithms that find optimal solutions (e.g. CABOB [25]) can, in practice, solve very large problems involving thousands of bids very quickly. 2.1 The Problem of Bid Withdrawal We assume an auction protocol with a three stage process involving the submission of bids, winner determination, and finally a transaction phase. We are interested in bid withdrawals that occur between the announcement of winning bids and the end of the transaction phase. All bids are valid until the transaction is complete, so we anticipate an expedient transaction process1 . 1 In some instances the transaction period may be so lengthy that consideration of non-winning bids as still being valid may not be fair. Breaks that occur during a lengthy transaction phase are more difficult to remedy and may require a subsequent auction. For example, if the item is a service contract for a given period of time and the break occurs after partial fulfilment of this contract, the other An example of a winning bid withdrawal occurred in an FCC spectrum auction [32]. Withdrawals, or breaks, may occur for various reasons. Bid withdrawal may be instigated by the bid-taker when Quality of Service agreements are broken or payment deadlines are not met. We refer to bid withdrawal by the bid-taker as item withdrawal in this paper to distinguish between the actions of a bidder and the bid-taker. Harstad and Rothkopf [8] outlined several possibilities for breaks in single item auctions that include: 1. an erroneous initial valuation/bid; 2. unexpected events outside the winning bidder"s control; 3. a desire to have the second-best bid honored; 4. information obtained or events that occurred after the auction but before the transaction that reduces the value of an item; 5. the revelation of competing bidders" valuations infers reduced profitability, a problem known as the Winner"s Curse. Kastner et al. [15] examined how to handle perturbations given a solution whilst minimizing necessary changes to that solution. These perturbations may include bid withdrawals, change of valuation/items of a bid or the submission of a new bid. They looked at the problem of finding incremental solutions to restructure a supply chain whose formation is determined using combinatorial auctions [30]. Following a perturbation in the optimal solution they proceed to impose involuntary item withdrawals from winning bidders. They formulated an incremental integer linear program (ILP) that sought to maximize the valuation of the repair solution whilst preserving the previous solution as much as possible. 2.2 Being Proactive against Bid Withdrawal When a bid is withdrawn there may be constraints on how the solution can be repaired. If the bid-taker was freely able to revoke the awarding of items to other bidders then the solution could be repaired easily by reassigning all the items to the optimal solution without the withdrawn bid. Alternatively, the bidder who reneged upon a bid may have all his other bids disqualified and the items could be reassigned based on the optimum solution without that bidder present. However, the bid-taker is often unable to freely reassign the items already awarded to other bidders. When items cannot be withdrawn from winning bidders, following the failure of another bidder to honor his bid, repair solutions are restricted to the set of bids whose items only include those in the bid(s) that were reneged upon. We are free to award items to any of the previously unsuccessful bids when finding a repair solution. When faced with uncertainty over the reliability of bidders a possible approach is to maximize expected revenue. This approach does not make allowances for risk-averse bid-takers who may view a small possibility of very low revenue as unacceptable. Consider the example in Table 1, and the optimal expected revenue in the situation where a single bid may be withdrawn. There are three submitted bids for items A and B, the third being a combination bid for the pair of items at a value of 190. The optimal solution has a value of 200, with the first and second bids as winners. When we consider the probabilities of failure, in the fourth column, the problem of which solution to choose becomes more difficult. Computing the expected revenue for the solution with the first and second bids winning the items, denoted 1, 1, 0 , gives: (200×0.9×0.9)+(2×100×0.9×0.1)+(190×0.1×0.1) = 181.90. bidders" valuations for the item may have decreased in a non-linear fashion. 184 Table 1: Example Combinatorial Auction. Items Bids A B AB Withdrawal prob x1 100 0 0 0.1 x2 0 100 0 0.1 x3 0 0 190 0.1 If a single bid is withdrawn there is probability of 0.18 of a revenue of 100, given the fact that we cannot withdraw an item from the other winning bidder. The expected revenue for 0, 0, 1 is: (190 × 0.9) + (200 × 0.1) = 191.00. We can therefore surmise that the second solution is preferable to the first based on expected revenue. Determining the maximum expected revenue in the presence of such uncertainty becomes computationally infeasible however, as the number of brittle bids grows. A WDP needs to be solved for all possible combinations of bids that may fail. The possible loss in revenue for breaks is also not tightly bounded using this approach, therefore a large loss may be possible for a small number of breaks. Consider the previous example where the bid amount for x3 becomes 175. The expected revenue of 1, 1, 0 (181.75) becomes greater than that of 0, 0, 1 (177.50). There are some bid-takers who may prefer the latter solution because the revenue is never less than 175, but the former solution returns revenue of only 100 with probability 0.18. A risk-averse bid-taker may not tolerate such a possibility, preferring to sacrifice revenue for reduced risk. If we modify our repair search so that a solution of at least a given revenue is guaranteed, the search for a repair solution becomes a satisfiability test rather than an optimization problem. The approaches described above are in contrast to that which we propose in the next section. Our approach can be seen as preventative in that we find an initial allocation of items to bidders which is robust to bid withdrawal. Possible losses in revenue are bounded by a fixed percentage of the true optimal allocation. Perturbations to the original solution are also limited so as to minimize disruption. We regard this as the ideal approach for real-world combinatorial auctions. DEFINITION 1 (ROBUST SOLUTION FOR A CA). A robust solution for a combinatorial auction is one where any subset of successful bids whose probability of withdrawal is greater than or equal to α can be repaired by reassigning items at a cost of at most β to other previously losing bids to form a repair solution. Constraints on acceptable revenue, e.g. being a minimum percentage of the optimum, are defined in the problem model and are thus satisfied by all solutions. The maximum cost of repair, β, may be a fixed value that may be thought of as a fund for compensating winning bidders whose items are withdrawn from them when creating a repair solution. Alternatively, β may be a function of the bids that were withdrawn. Section 4 will give an example of such a mechanism. In the following section we describe an ideal constraint-based framework for the establishment of such robust solutions. 3. FINDING ROBUST SOLUTIONS In constraint programming [4] (CP), a constraint satisfaction problem (CSP) is modeled as a set of n variables X = {x1, . . . , xn}, a set of domains D = {D(x1), . . . , D(xn)}, where D(xi) is the set of finite possible values for variable xi and a set C = {C1, . . . , Cm} of constraints, each restricting the assignments of some subset of the variables in X. Constraint satisfaction involves finding values for each of the problem variables such that all constraints are satisfied. Its main advantages are its declarative nature and flexibility in tackling problems with arbitrary side constraints. Constraint optimization seeks to find a solution to a CSP that optimizes some objective function. A common technique for solving constraint optimization problems is to use branch-and-bound techniques that avoid exploring sub-trees that are known not to contain a better solution than the best found so far. An initial bound can be determined by finding a solution that satisfies all constraints in C or by using some heuristic methods. A classical super solution (SS) is a solution to a CSP in which, if a small number of variables lose their values, repair solutions are guaranteed with only a few changes, thus providing solution robustness [9, 10]. It is a generalization of both fault tolerance in CP [31] and supermodels in propositional satisfiability (SAT) [7]. An (a,b)-super solution is one in which if at most a variables lose their values, a repair solution can be found by changing at most b other variables [10]. Super solutions for combinatorial auctions minimize the number of bids whose status needs to be changed when forming a repair solution [12]. Only a particular set of variables in the solution may be subject to change and these are said to be members of the breakset. For each combination of brittle assignments in the break-set, a repair-set is required that comprises the set of variables whose values must change to provide another solution. The cardinality of the repair set is used to measure the cost of repair. In reality, changing some variable assignments in a repair solution incurs a lower cost than others thereby motivating the use of a different metric for determining the legality of repair sets. The Weighted Super Solution (WSS) framework [13] considers the cost of repair required, rather than simply the number of assignments modified, to form an alternative solution. For CAs this may be a measure of the compensation penalties paid to winning bidders to break existing agreements. Robust solutions are particularly desirable for applications where unreliability is a problem and potential breakages may incur severe penalties. Weighted super solutions offer a means of expressing which variables are easily re-assigned and those that incur a heavy cost [13]. Hebrard et al. [9] describe how some variables may fail (such as machines in a job-shop problem) and others may not. A WSS generalizes this approach so that there is a probability of failure associated with each assignment and sets of variables whose assignments have probabilities of failure greater than or equal to a threshold value, α, require repair solutions. A WSS measures the cost of repairing, or reassigning, other variables using inertia as a metric. Inertia is a measure of a variable"s aversion to change and depends on its current assignment, future assignment and the breakage variable(s). It may be desirable to reassign items to different bidders in order to find a repair solution of satisfactory revenue. Compensation may have to be paid to bidders who lose items during the formation of a repair solution. The inertia of a bid reflects the cost of changing its state. For winning bids this may reflect the necessary compensation penalty for the bid-taker to break the agreement (if such breaches are permitted), whereas for previously losing bids this is a free operation. The total amount of compensation payable to bidders may depend upon other factors, such as the cause of the break. There is a limit to how much these overall repair costs should be, and this is given by the value β. This value may not be known in advance and 185 Algorithm 1: WSS(int level, double α, double β):Boolean begin if level > number of variables then return true choose unassigned variable x foreach value v in the domain of x do assign x : v if problem is consistent then foreach combination of brittle assignments, A do if ¬reparable(A, β) then return false; if WSS(level+1) then return true unassign x return false end may depend upon the break. Therefore, β may be viewed as the fund used to compensate winning bidders for the unilateral withdrawal of their bids by the bid-taker. In summary, an (α,β)-WSS allows any set of variables whose probability of breaking is greater than or equal to α be repaired with changes to the original robust solution with a cost of at most β. The depth-first search for a WSS (see pseudo-code description in Algorithm 1) maintains arc-consistency [24] at each node of the tree. As search progresses, the reparability of each previous assignment is verified at each node by extending a partial repair solution to the same depth as the current partial solution. This may be thought of as maintaining concurrent search trees for repairs. A repair solution is provided for every possible set of break variables, A. The WSS algorithm attempts to extend the current partial assignment by choosing a variable and assigning it a value. Backtracking may then occur for one of two reasons: we cannot extend the assignment to satisfy the given constraints, or the current partial assignment cannot be associated with a repair solution whose cost of repair is less than β should a break occur. The procedure reparable searches for partial repair solutions using backtracking and attempts to extend the last repair found, just as in (1,b)super solutions [9]; the differences being that a repair is provided for a set of breakage variables rather than a single variable and the cost of repair is considered. A summation operator is used to determine the overall cost of repair. If a fixed bound upon the size of any potential break-set can be formed, the WSS algorithm is NPcomplete. For a more detailed description of the WSS search algorithm, the reader is referred to [13], since a complete description of the algorithm is beyond the scope of this paper. EXAMPLE 1. We shall step through the example given in Table 1 when searching for a WSS. Each bid is represented by a single variable with domain values of 0 and 1, the former representing bid-failure and the latter bid-success. The probability of failure of the variables are 0.1 when they are assigned to 1 and 0.0 otherwise. The problem is initially solved using an ILP solver such as lp_solve [3] or CPLEX, and the optimal revenue is found to be 200. A fixed percentage of this revenue can be used as a threshold value for a robust solution and its repairs. The bid-taker wishes to have a robust solution so that if a single winning bid is withdrawn, a repair solution can be formed without withdrawing items from any other winning bidder. This example may be seen as searching for a (0.1,0)-weighted super solution, β is 0 because no funds are available to compensate the withdrawal of items from winning bidders. The bid-taker is willing to compromise on revenue, but only by 5%, say, of the optimal value. Bids 1 and 3 cannot both succeed, since they both require item A, so a constraint is added precluding the assignment in which both variables take the value 1. Similarly, bids 2 and 3 cannot both win so another constraint is added between these two variables. Therefore, in this example the set of CSP variables is V = {x1, x2, x3}, whose domains are all {0, 1}. The constraints are x1 + x3 ≤ 1, x2 + x3 ≤ 1 and xi∈V aixi ≥ 190, where ai reflects the relevant bid-amounts for the respective bid variables. In order to find a robust solution of optimal revenue we seek to maximize the sum of these amounts, max xi∈V aixi. When all variables are set to 0 (see Figure 1(a) branch 3), this is not a solution because the minimum revenue of 190 has not been met, so we try assigning bid3 to 1 (branch 4). This is a valid solution but this variable is brittle because there is a 10% chance that this bid may be withdrawn (see Table 1). Therefore we need to determine if a repair can be formed should it break. The search for a repair begins at the first node, see Figure 1(b). Notice that value 1 has been removed from bid3 because this search tree is simulating the withdrawal of this bid. When bid1 is set to 0 (branch 4.1), the maximum revenue solution in the remaining subtree has revenue of only 100, therefore search is discontinued at that node of the tree. Bid1 and bid2 are both assigned to 1 (branches 4.2 and 4.4) and the total cost of both these changes is still 0 because no compensation needs to be paid for bids that change from losing to winning. With bid3 now losing (branch 4.5), this gives a repair solution of 200. Hence 0, 0, 1 is reparable and therefore a WSS. We continue our search in Figure 1(a) however, because we are seeking a robust solution of optimal revenue. When bid1 is assigned to 1 (branch 6) we seek a partial repair for this variable breaking (branch 5 is not considered since it offers insufficient revenue). The repair search sets bid1 to 0 in a separate search tree, (not shown), and control is returned to the search for a WSS. Bid2 is set to 0 (branch 7), but this solution would not produce sufficient revenue so bid2 is then set to 1 (branch 8). We then attempt to extend the repair for bid1 (not shown). This fails because the repair for bid1 cannot assign bid2 to 0 because the cost of repairing such an assignment would be ∞, given that the auction rules do not permit the withdrawal of items from winning bids. A repair for bid1 breaking is therefore not possible because items have already been awarded to bid2. A repair solution with bid2 assigned to 1 does not produce sufficient revenue when bid1 is assigned to 0. The inability to withdraw items from winning bids implies that 1, 1, 0 is an irreparable solution when the minimum tolerable revenue is greater than 100. The italicized comments and dashed line in Figure 1(a) illustrate the search path for a WSS if both of these bids were deemed reparable. Section 4 introduces an alternative auction model that will allow the bid-taker to receive compensation for breakages and in turn use this payment to compensate other bidders for withdrawal of items from winning bids. This will enable the reallocation of items and permit the establishment of 1, 1, 0 as a second WSS for this example. 4. MUTUAL BID BONDS: A BACKTRACKING MECHANISM Some auction solutions are inherently brittle and it may be impossible to find a robust solution. If we can alter the rules of an auction so that the bid-taker can retract items from winning bidders, then the reparability of solutions to such auctions may be improved. In this section we propose an auction model that permits bid and item withdrawal by the bidders and bid-taker, respectively. We propose a model that incorporates mutual bid bonds to enable solution reparability for the bid-taker, a form of insurance against 186 0 0 0 0 0 0 0 1 1 1 1 1 1 1 Insufficient revenue Find repair solution for bid 3 breakage Find partial repair for bid 1 breakage Insufficient revenue (a) Extend partial repair for bid 1 breakage (b) Find partial repair for bid 2 breakage Bid 1 Bid 2 Bid 3 Find repair solutions for bid 1 & 2 breakages [0] [190] [100] [100] [200] 1 2 3 4 5 6 7 8 9 Insufficient revenue (a) Search for WSS. 0 0 0 0 0 0 0 1 1 1 1 1 1 1 Insufficient revenue Insufficient revenue Bid 1 Bid 2 Bid 3 inertia=0 inertia=0 inertia=0 4.1 4.2 4.3 4.4 4.5 (b) Search for a repair for bid 3 breakage. Figure 1: Search Tree for a WSS without item withdrawal. the winner"s curse for the bidder whilst also compensating bidders in the case of item withdrawal from winning bids. We propose that such Winner"s Curse & Bid-taker"s Exposure insurance comprise a fixed percentage, κ, of the bid amount for all bids. Such mutual bid bonds are mandatory for each bid in our model2 . The conditions attached to the bid bonds are that the bid-taker be allowed to annul winning bids (item withdrawal) when repairing breaks elsewhere in the solution. In the interests of fairness, compensation is paid to bidders from whom items are withdrawn and is equivalent to the penalty that would have been imposed on the bidder should he have withdrawn the bid. Combinatorial auctions impose a heavy computational burden on the bidder so it is important that the hedging of risk should be a simple and transparent operation for the bidder so as not to further increase this burden unnecessarily. We also contend that it is imperative that the bidder knows the potential penalty for withdrawal in advance of bid submission. This information is essential for bidders when determining how aggressive they should be in their bidding strategy. Bid bonds are commonplace in procurement for construction projects. Usually they are mandatory for all bids, are a fixed percentage, κ, of the bid amount and are unidirectional in that item withdrawal by the bid-taker is not permitted. Mutual bid bonds may be seen as a form of leveled commitment contract in which both parties may break the contract for the same fixed penalty. Such contracts permit unilateral decommitment for prespecified penalties. Sandholm et al. showed that this can increase the expected payoffs of all parties and enables deals that would be impossible under full commitment [26, 28, 29]. In practice a bid bond typically ranges between 5 and 20% of the 2 Making the insurance optional may be beneficial in some instances. If a bidder does not agree to the insurance, it may be inferred that he may have accurately determined the valuation for the items and therefore less likely to fall victim to the winner"s curse. The probability of such a bid being withdrawn may be less, so a repair solution may be deemed unnecessary for this bid. On the other hand it decreases the reparability of solutions. bid amount [14, 18]. If the decommitment penalties are the same for both parties in all bids, κ does not influence the reparability of a given set of bids. It merely influences the levels of penalties and compensation transacted by agents. Low values of κ incur low bid withdrawal penalties and simulate a dictatorial bid-taker who does not adequately compensate bidders for item withdrawal. Andersson and Sandholm [1] found that myopic agents reach a higher social welfare quicker if they act selfishly rather than cooperatively when penalties in leveled commitment contracts are low. Increased levels of bid withdrawal are likely when the penalties are low also. High values of κ tend towards full-commitment and reduce the advantages of such Winner"s Curse & Bid-taker"s Exposure insurance. The penalties paid are used to fund a reassignment of items to form a repair solution of sufficient revenue by compensating previously successful bidders for withdrawal of the items from them. EXAMPLE 2. Consider the example given in Table 1 once more, where the bids also comprise a mutual bid bond of 5% of the bid amount. If a bid is withdrawn, the bidder forfeits this amount and the bid-taker can then compensate winning bidders whose items are withdrawn when trying to form a repair solution later. The search for repair solutions for breaks to bid1 and bid2 appear in Figures 2(a) and 2(b), respectively3 . When bid1 breaks, there is a compensation penalty paid to the bid-taker equal to 5 that can be used to fund a reassignment of the items. We therefore set β to 5 and this becomes the maximum expenditure allowed to withdraw items from winning bidders. β may also be viewed as the size of the fund available to facilitate backtracking by the bid-taker. When we extend the partial repair for bid1 so that bid2 loses an item (branch 8.1), the overall cost of repair increases to 5, due to this item withdrawal by the bid-taker, 3 The actual implementation of WSS search checks previous solutions to see if they can repair breaks before searching for a new repair solution. 0, 0, 1 is a solution that has already been found so the search for a repair in this example is not strictly necessary but is described for pedagogical reasons. 187 0 0 0 1 1 Bid 1 Bid 2 Bid 3 Insufficient revenue inertia=5 =5 inertia=0 =5 inertia=5 =5 1 6.1 8.1 9.1 9.2 (a) Search for a repair for bid 1 breakage. 0 0 0 1 1 Bid 1 Bid 2 Bid 3 Insufficient revenue inertia=10 =10 inertia=10 =10 inertia=10 =10 1 8.2 8.3 9.3 9.4 (b) Search for a repair for bid 2 breakage. Figure 2: Repair Search Tree for breaks 1 and 2, κ = 0.05. and is just within the limit given by β. In Figure 1(a) the search path follows the dashed line and sets bid3 to be 0 (branch 9). The repair solutions for bids 1 and 2 can be extended further by assigning bid3 to 1 (branches 9.2 and 9.4). Therefore, 1, 1, 0 may be considered a robust solution. Recall, that previously this was not the case. Using mutual bid bonds thus increases reparability and allows a robust solution of revenue 200 as opposed to 190, as was previously the case. 5. EXPERIMENTS We have used the Combinatorial Auction Test Suite (CATS) [16] to generate sample auction data. We generated 100 instances of problems in which there are 20 items for sale and 100-2000 bids that may be dominated in some instances4 . Such dominated bids can participate in repair solutions although they do not feature in optimal solutions. CATS uses economically motivated bidding patterns to generate auction data in various scenarios. To motivate the research presented in this paper we use sensitivity analysis to examine the brittleness of optimal solutions and hence determine the types of auctions most likely to benefit from a robust solution. We then establish robust solutions for CAs using the WSS framework. 5.1 Sensitivity Analysis for the WDP We have performed sensitivity analysis of the following four distributions: airport take-off/landing slots (matching), electronic components (arbitrary), property/spectrum-rights (regions) and transportation (paths). These distributions were chosen because they describe a broad array of bidding patterns in different application domains. The method used is as follows. We first of all determined the optimal solution using lp_solve, a mixed integer linear program solver [3]. We then simulated a single bid withdrawal and re-solved the problem with the other winning bids remaining fixed, i.e. there were no involuntary dropouts. The optimal repair solution was then determined. This process is repeated for all winning bids in the overall optimal solution, thus assuming that all bids are brittle. Figure 3 shows the average revenue of such repair solutions as a percentage of the optimum. Also shown is the average worst-case scenario over 100 auctions. We also implemented an auction rule that disallows bids from the reneging bidder participate in a repair5 . Figure 3(a) illustrates how the paths distribution is inherently the most robust distribution since when any winning bid is withdrawn the solution can be repaired to achieve over 98.5% of the 4 The CATS flags included int prices with the bid alpha parameter set to 1000. 5 We assumed that all bids in a given XOR bid with the same dummy item were from the same bidder. optimal revenue on average for auctions with more than 250 bids. There are some cases however when such withdrawals result in solutions whose revenue is significantly lower than optimum. Even in auctions with as many as 2000 bids there are occasions when a single bid withdrawal can result in a drop in revenue of over 5%, although the average worst-case drop in revenue is only 1%. Figure 3(b) shows how the matching distribution is more brittle on average than paths and also has an inferior worst-case revenue on average. This trend continues as the regions-npv (Figure 3(c)) and arbitrary-npv (Figure 3(d)) distributions are more brittle still. These distributions are clearly sensitive to bid withdrawal when no other winning bids in the solution may be involuntarily withdrawn by the bid-taker. 5.2 Robust Solutions using WSS In this section we focus upon both the arbitrary-npv and regions-npv distributions because the sensitivity analysis indicated that these types of auctions produce optimal solutions that tend to be most brittle, and therefore stand to benefit most from solution robustness. We ignore the auctions with 2000 bids because the sensitivity analysis has indicated that these auctions are inherently robust with a very low average drop in revenue following a bid withdrawal. They would also be very computationally expensive, given the extra complexity of finding robust solutions. A pure CP approach needs to be augmented with global constraints that incorporate operations research techniques to increase pruning sufficiently so that thousands of bids may be examined. Global constraints exploit special-purpose filtering algorithms to improve performance [21]. There are a number of ways to speed up the search for a weighted super solution in a CA, although this is not the main focus of our current work. Polynomial matching algorithms may be used in auctions whose bid length is short, such as those for airport landing/take-off slots for example. The integer programming formulation of the WDP stipulates that a bid either loses or wins. If we relax this constraint so that bids can partially win, this corresponds to the linear relaxation of the problem and is solvable in polynomial time. At each node of the search tree we can quickly solve the linear relaxation of the remaining problem in the subtree below the current node to establish an upper bound on remaining revenue. If this upper bound plus revenue in the parent tree is less than the current lower bound on revenue, search at that node can cease. The (continuous) LP relaxation thus provides a vital speed-up in the search for weighted super solutions, which we have exploited in our implementation. The LP formulation is as follows: max xi∈V aixi 188 100 95 90 85 80 75 250 500 750 1000 1250 1500 1750 2000 Revenue(%ofoptimum) Bids Average Repair Solution Revenue Worst-case Repair Solution Revenue (a) paths 100 95 90 85 80 75 250 500 750 1000 1250 1500 1750 2000 Revenue(%ofoptimum) Bids Average Repair Solution Revenue Worst-case Repair Solution Revenue (b) matching 100 95 90 85 80 75 250 500 750 1000 1250 1500 1750 2000 Revenue(%ofoptimum) Bids Average Repair Solution Revenue Worst-case Repair Solution Revenue (c) regions-npv 100 95 90 85 80 75 250 500 750 1000 1250 1500 1750 2000 Revenue(%ofoptimum) Bids Average Repair Solution Revenue Worst-case Repair Solution Revenue (d) arbitrary-npv Figure 3: Sensitivity of bid distributions to single bid withdrawal. s.t. j|i∈Sj xj ≤ 1, ∀i ∈ {1 . . . m}, xj ≥ 0, xj ∈ R. Additional techniques, that are outlined in [25], can aid the scalability of a CP approach but our main aim in these experiments is to examine the robustness of various auction distributions and consider the tradeoff between robustness and revenue. The WSS solver we have developed is an extension of the super solution solver presented in [9, 10]. This solver is, in turn, based upon the EFC constraint solver [2]. Combinatorial auctions are easily modeled as a constraint optimization problems. We have chosen the branch-on-bids formulation because in tests it worked faster than a branch-on-items formulation for the arbitrary-npv and regions-npv distributions. All variables are binary and our search mechanism uses a reverse lexicographic value ordering heuristic. This complements our dynamic variable ordering heuristic that selects the most promising unassigned variable as the next one in the search tree. We use the product of the solution of the LP relaxation and the degree of a variable to determine the likelihood of its participation in a robust solution. High values in the LP solution are a strong indication of variables most likely to form a high revenue solution whilst the a variable"s degree reflects the number of other bids that overlap in terms of desired items. Bids for large numbers of items tend to be more robust, which is why we weight our robust solution search in this manner. We found this heuristic to be slightly more effective than the LP solution alone. As the number of bids in the auction increases however, there is an increase in the inherent robustness of solutions so the degree of a variable loses significance as the auction size increases. 5.3 Results Our experiments simulate three different constraints on repair solutions. The first is that no winning bids are withdrawn by the bid-taker and a repair solution must return a revenue of at least 90% of the optimal overall solution. Secondly, we relaxed the revenue constraint to 85% of optimum. Thirdly, we allowed backtracking by the bid-taker on winning bids using mutual bid bonds but maintaining the revenue constraint at 90% of optimum. Prior to finding a robust solution we solved the WDP optimally using lp_solve [3]. We then set the minimum tolerable revenue for a solution to be 90% (then 85%) of the revenue of this optimal solution. We assumed that all bids were brittle, thus a repair solution is required for every bid in the solution. Initially we assume that no backtracking was permitted on assignments of items to other winning bids given a bid withdrawal elsewhere in the solution. Table 2 shows the percentage of optimal solutions that are robust for minimum revenue constraints for repair solutions of 90% and 85% of optimal revenue. Relaxing the revenue constraint on repair solutions to 85% of the optimum revenue greatly increases the number of optimal solutions that are robust. We also conducted experiments on the same auctions in which backtracking by the bid-taker is permitted using mutual bid bonds. This significantly improves the reparability of optimal solutions whilst still maintaining repair solutions of 90% of optimum. An interesting feature of the arbitrary-npv distribution is that optimal solutions can become more brittle as the number of bids increases. The reason for this is that optimal solutions for larger auctions have more winning bids. Some of the optimal solutions for the smallest auctions with 100 bids have only one winning bidder. If this bid is withdrawn it is usually easy to find a new repair solution within 90% of the previous optimal revenue. Also, repair solutions for bids that contain a small number of items may be made difficult by the fact that a reduced number of bids cover only a subset of those items. A mitigating factor is that such bids form a smaller percentage of the revenue of the optimal solution on average. We also implemented a rule stipulating that any losing bids from 189 Table 2: Optimal Solutions that are Inherently Robust (%). #Bids Min Revenue 100 250 500 1000 2000 arbitrary-npv repair ≥ 90% 21 5 3 37 93 repair ≥ 85% 26 15 40 87 100 MBB & repair ≥ 90% 41 35 60 94 ≥ 93 regions-npv repair ≥ 90% 30 33 61 91 98 repair ≥ 85% 50 71 95 100 100 MBB & repair ≥ 90% 60 78 96 99 ≥ 98 Table 3: Occurrence of Robust Solutions (%). #Bids Min Revenue 100 250 500 1000 arbitrary-npv repair ≥ 90% 58 39 51 98 repair ≥ 85% 86 88 94 99 MBB & repair ≥ 90% 78 86 98 100 regions-npv repair ≥ 90% 61 70 97 100 repair ≥ 85% 89 99 99 100 MBB & repair ≥ 90% 83 96 100 100 a withdrawing bidder cannot participate in a repair solution. This acts as a disincentive for strategic withdrawal and was also used previously in the sensitivity analysis. In some auctions, a robust solution may not exist. Table 3 shows the percentage of auctions that support robust solutions for the arbitrary-npv and regions -npv distributions. It is clear that finding robust solutions for the former distribution is particularly difficult for auctions with 250 and 500 bids when revenue constraints are 90% of optimum. This difficulty was previously alluded to by the low percentage of optimal solutions that were robust for these auctions. Relaxing the revenue constraint helps increase the percentage of auctions in which robust solutions are achievable to 88% and 94%, respectively. This improves the reparability of all solutions thereby increasing the average revenue of the optimal robust solution. It is somewhat counterintuitive to expect a reduction in reparability of auction solutions as the number of bids increases because there tends to be an increased number of solutions above a revenue threshold in larger auctions. The MBB auction model performs very well however, and ensures that robust solutions are achievable for such inherently brittle auctions without sacrificing over 10% of optimal revenue to achieve repair solutions. Figure 4 shows the average revenue of the optimal robust solution as a percentage of the overall optimum. Repair solutions found for a WSS provide a lower bound on possible revenue following a bid withdrawal. Note that in some instances it is possible for a repair solution to have higher revenue than the original solution. When backtracking on winning bids by the bid-taker is disallowed, this can only happen when the repair solution includes two or more bids that were not in the original. Otherwise the repair bids would participate in the optimal robust solution in place of the bid that was withdrawn. A WSS guarantees minimum levels of revenue for repair solutions but this is not to say that repair solutions cannot be improved upon. It is possible to use an incremental algorithm to 100 98 96 94 92 250 500 750 1000 1250 1500 1750 2000 Revenue(%ofoptimum) Bids Repair Revenue: Min 90% Optimal Repair Revenue: Min 85% Optimal MBB: Repair Revenue: Min 90% Optimal (a) regions-npv 100 98 96 94 92 250 500 750 1000 1250 1500 1750 2000 Revenue(%ofoptimum) Bids Repair Revenue: Min 90% Optimal Repair Revenue: Min 85% Optimal MBB: Repair Revenue: Min 90% Optimal (b) arbitrary-npv Figure 4: Revenue of optimal robust solutions. determine an optimal repair solution following a break, whilst safe in the knowledge that in advance of any possible bid withdrawal we can establish a lower bound on the revenue of a repair. Kastner et al. have provided such an incremental ILP formulation [15]. Mutual bid bonds facilitate backtracking by the bid-taker on already assigned items. This improves the reparability of all possible solutions thus increasing the revenue of the optimal robust solution on average. Figure 4 shows the increase in revenue of robust solutions in such instances. The revenues of repair solutions are bounded by at least 90% of the optimum in our experiments thereby allowing a direct comparison with robust solutions already found using the same revenue constraint but not providing for backtracking. It is immediately obvious that such a mechanism can significantly increase revenue whilst still maintaining solution robustness. Table 4 shows the number of winning bids participating in optimal and optimal robust solutions given the three different constraints on repairing solutions listed at the beginning of this section. As the number of bids increases, more of the optimal overall solutions are robust. This leads to a convergence in the number of winning bids. The numbers in brackets are derived from the sensitivity analysis of optimal solutions that reveals the fact that almost all optimal solutions for auctions of 2000 bids are robust. We can therefore infer that the average number of winning bids in revenuemaximizing robust solutions converges towards that of the optimal overall solutions. A notable side-effect of robust solutions is that fewer bids participate in the solutions. It can be clearly seen from Table 4 that when revenue constraints on repair solutions are tight, there are fewer winning bids in the optimal robust solution on average. This is particularly pronounced for smaller auctions in both distributions. This can win benefits for the bid-taker such as reduced overheads in dealing with fewer suppliers. Although MBBs aid solution repara190 Table 4: Number of winning bids. #Bids Solution 100 250 500 1000 2000 arbitrary-npv Optimal 3.31 5.60 7.17 9.31 10.63 Repair ≥ 90% 1.40 2.18 6.10 9.03 (≈ 10.63) Repair ≥ 85% 1.65 3.81 6.78 9.31 (10.63) MBB (≥ 90%) 2.33 5.49 7.33 9.34 (≈ 10.63) regions-npv Optimal 4.34 7.05 9.10 10.67 12.76 Repair ≥ 90% 3.03 5.76 8.67 10.63 (≈ 12.76) Repair ≥ 85% 3.45 6.75 9.07 (10.67) (12.76) MBB (≥ 90%) 3.90 6.86 9.10 10.68 (≈ 12.76) bility, the number of bids in the solutions increases on average. This is to be expected because a greater fraction of these solutions are in fact optimal, as we saw in Table 2. 6. DISCUSSION AND FUTURE WORK Bidding strategies can become complex in non-incentive-compatible mechanisms where winner determination is no longer necessarily optimal. The perceived reparability of a bid may influence the bid amount, with reparable bids reaching a lower equilibrium point and perceived irreparable bids being more aggressive. Penalty payments for bid withdrawal also create an incentive for more aggressive bidding by providing a form of insurance against the winner"s curse [8]. If a winning bidder"s revised valuation for a set of items drops by more than the penalty for withdrawal of the bid, then it is in his best interests to forfeit the item(s) and pay the penalty. Should the auction rules state that the bid-taker will refuse to sell the items to any of the remaining bidders in the event of a withdrawal, then insurance against potential losses will stimulate more aggressive bidding. However, in our case we are seeking to repair the solution with the given bids. A side-effect of such a policy is to offset the increased aggressiveness by incentivizing reduced valuations in expectation that another bidder"s successful bid is withdrawn. Harstad and Rothkopf [8] examined the conditions required to ensure an equilibrium position in which bidding was at least as aggressive as if no bid withdrawal was permitted, given this countervailing incentive to under-estimate a valuation. Three major results arose from their study of bid withdrawal in a single item auction: 1. Equilibrium bidding is more aggressive with withdrawal for sufficiently small probabilities of an award to the second highest bidder in the event of a bid withdrawal; 2. Equilibrium bidding is more aggressive with withdrawal if the number of bidders is large enough; 3. For many distributions of costs and estimates, equilibrium bidding is more aggressive with withdrawal if the variability of the estimating distribution is sufficiently large. It is important that mutual bid bonds do not result in depressed bidding in equilibrium. An analysis of the resultant behavior of bidders must incorporate the possibility of a bidder winning an item and having it withdrawn in order for the bid-taker to formulate a repair solution after a break elsewhere. Harstad and Rothkopf have analyzed bidder aggressiveness [8] using a strictly game-theoretic model in which the only reason for bid withdrawal is the winner"s curse. They assumed all bidders were risk-neutral, but surmised that it is entirely possible for the bid-taker to collect a risk premium from risk-averse bidders with the offer of such insurance. Combinatorial auctions with mutual bid bonds add an extra incentive to bid aggressively because of the possibility of being compensated for having a winning bid withdrawn by a bid-taker. This is militated against by the increased probability of not having items withdrawn in a repair solution. We leave an in-depth analysis of the sufficient conditions for more aggressive bidding for future work. Whilst the WSS framework provides ample flexibility and expressiveness, scalability becomes a problem for larger auctions. Although solutions to larger auctions tend to be naturally more robust, some bid-takers in such auctions may require robustness. A possible extension of our work in this paper may be to examine the feasibility of reformulating integer linear programs so that the solutions are robust. Hebrard et al. [10] examined reformulation of CSPs for finding super solutions. Alternatively, it may be possible to use a top-down approach by looking at the k-best solutions sequentially, in terms of revenue, and performing sensitivity analysis upon each solution until a robust one is found. In procurement settings the principle of free disposal is often discounted and all items must be sold. This reduces the number of potential solutions and thereby reduces the reparability of each solution. The impact of such a constraint on revenue of robust solutions is also left for future work. There is another interesting direction this work may take, namely robust mechanism design. Porter et al. introduced the notion of fault tolerant mechanism design in which agents have private information regarding costs for task completion, but also their probabilities of failure [20]. When the bid-taker has combinatorial valuations for task completions it may be desirable to assign the same task to multiple agents to ensure solution robustness. It is desirable to minimize such potentially redundant task assignments but not to the detriment of completed task valuations. This problem could be modeled using the WSS framework in a similar manner to that of combinatorial auctions. In the case where no robust solutions are found, it is possible to optimize robustness, instead of revenue, by finding a solution of at least a given revenue that minimizes the probability of an irreparable break. In this manner the least brittle solution of adequate revenue may be chosen. 7. CONCLUSION Fairness is often cited as a reason for choosing the optimal solution in terms of revenue only [22]. Robust solutions militate against bids deemed brittle, therefore bidders must earn a reputation for being reliable to relax the reparability constraint attached to their bids. This may be seen as being fair to long-standing business partners whose reliability is unquestioned. Internet-based auctions are often seen as unwelcome price-gouging exercises by suppliers in many sectors [6, 17]. Traditional business partnerships are being severed by increased competition amongst suppliers. Quality of Service can suffer because of the increased focus on short-term profitability to the detriment of the bid-taker in the long-term. Robust solutions can provide a means of selectively discriminating against distrusted bidders in a measured manner. As combinatorial auction deployment moves from large value auctions with a small pool of trusted bidders (e.g. spectrum-rights sales) towards lower value auctions with potentially unknown bidders (e.g. Supply Chain Management [30]), solution robustness becomes more relevant. As well as being used to ensure that the bid-taker is not left vulnerable to bid withdrawal, it may also be used to cement relationships with preferred, possibly incumbent, suppliers. 191 We have shown that it is possible to attain robust solutions for CAs with only a small loss in revenue. We have also illustrated how such solutions tend to have fewer winning bids than overall optimal solutions, thereby reducing any overheads associated with dealing with more bidders. We have also demonstrated that introducing mutual bid bonds, a form of leveled commitment contract, can significantly increase the revenue of optimal robust solutions by improving reparability. We contend that robust solutions using such a mechanism can allow a bid-taker to offer the possibility of bid withdrawal to bidders whilst remaining confident about postrepair revenue and also facilitating increased bidder aggressiveness. 8. REFERENCES [1] Martin Andersson and Tuomas Sandholm. Leveled commitment contracts with myopic and strategic agents. Journal of Economic Dynamics and Control, 25:615-640, 2001. Special issue on Agent-Based Computational Economics. [2] Fahiem Bacchus and George Katsirelos. EFC solver. www.cs.toronto.edu/˜gkatsi/efc/efc.html. [3] Michael Berkelaar, Kjell Eikland, and Peter Notebaert. lp solve version 5.0.10.0. http://groups.yahoo.com/group/lp_solve/. [4] Rina Dechter. Constraint Processing. Morgan Kaufmann, 2003. [5] Sven DeVries and Rakesh Vohra. Combinatorial auctions: A survey. INFORMS Journal on Computing, pages 284-309, 2003. [6] Jim Ericson. Reverse auctions: Bad idea. Line 56, Sept 2001. [7] Matthew L. Ginsberg, Andrew J. Parkes, and Amitabha Roy. Supermodels and Robustness. In Proceedings of AAAI-98, pages 334-339, Madison, WI, 1998. [8] Ronald M. Harstad and Michael H. Rothkopf. Withdrawable bids as winner"s curse insurance. Operations Research, 43(6):982-994, November-December 1995. [9] Emmanuel Hebrard, Brahim Hnich, and Toby Walsh. Robust solutions for constraint satisfaction and optimization. In Proceedings of the European Conference on Artificial Intelligence, pages 186-190, 2004. [10] Emmanuel Hebrard, Brahim Hnich, and Toby Walsh. Super solutions in constraint programming. In Proceedings of CP-AI-OR 2004, pages 157-172, 2004. [11] Gail Hohner, John Rich, Ed Ng, Grant Reid, Andrew J. Davenport, Jayant R. Kalagnanam, Ho Soo Lee, and Chae An. Combinatorial and quantity-discount procurement auctions benefit Mars Incorporated and its suppliers. Interfaces, 33(1):23-35, 2003. [12] Alan Holland and Barry O"Sullivan. Super solutions for combinatorial auctions. In Ercim-Colognet Constraints Workshop (CSCLP 04). Springer LNAI, Lausanne, Switzerland, 2004. [13] Alan Holland and Barry O"Sullivan. Weighted super solutions for constraint programs, December 2004. Technical Report: No. UCC-CS-2004-12-02. [14] Selective Insurance. Business insurance. http://www.selectiveinsurance.com/psApps /Business/Ins/bonds.asp?bc=13.16.127. [15] Ryan Kastner, Christina Hsieh, Miodrag Potkonjak, and Majid Sarrafzadeh. On the sensitivity of incremental algorithms for combinatorial auctions. In WECWIS, pages 81-88, June 2002. [16] Kevin Leyton-Brown, Mark Pearson, and Yoav Shoham. Towards a universal test suite for combinatorial auction algorithms. In ACM Conference on Electronic Commerce, pages 66-76, 2000. [17] Associated General Contractors of America. Associated general contractors of America white paper on reverse auctions for procurement of construction. http://www.agc.org/content/public/pdf /Member_Resources/ ReverseAuctionWhitePaper.pdf, 2003. [18] National Society of Professional Engineers. A basic guide to surety bonds. http://www.nspe.org/pracdiv /76-02surebond.asp. [19] Martin Pesendorfer and Estelle Cantillon. Combination bidding in multi-unit auctions. Harvard Business School Working Draft, 2003. [20] Ryan Porter, Amir Ronen, Yoav Shoham, and Moshe Tennenholtz. Mechanism design with execution uncertainty. In Proceedings of UAI-02, pages 414-421, 2002. [21] Jean-Charles R´egin. Global constraints and filtering algorithms. In Constraint and Integer ProgrammingTowards a Unified Methodology, chapter 4, pages 89-129. Kluwer Academic Publishers, 2004. [22] Michael H. Rothkopf and Aleksandar Peke˘c. Combinatorial auction design. Management Science, 4(11):1485-1503, November 2003. [23] Michael H. Rothkopf, Aleksandar Peke˘c, and Ronald M. Harstad. Computationally manageable combinatorial auctions. Management Science, 44(8):1131-1147, 1998. [24] Daniel Sabin and Eugene C. Freuder. Contradicting conventional wisdom in constraint satisfaction. In A. Cohn, editor, Proceedings of ECAI-94, pages 125-129, 1994. [25] Tuomas Sandholm. Algorithm for optimal winner determination in combinatorial auctions. Artificial Intelligence, 135(1-2):1-54, 2002. [26] Tuomas Sandholm and Victor Lesser. Leveled Commitment Contracts and Strategic Breach. Games and Economic Behavior, 35:212-270, January 2001. [27] Tuomas Sandholm and Victor Lesser. Leveled commitment contracting: A backtracking instrument for multiagent systems. AI Magazine, 23(3):89-100, 2002. [28] Tuomas Sandholm, Sandeep Sikka, and Samphel Norden. Algorithms for optimizing leveled commitment contracts. In Proceedings of the IJCAI-99, pages 535-541. Morgan Kaufmann Publishers Inc., 1999. [29] Tuomas Sandholm and Yunhong Zhou. Surplus equivalence of leveled commitment contracts. Artificial Intelligence, 142:239-264, 2002. [30] William E. Walsh, Michael P. Wellman, and Fredrik Ygge. Combinatorial auctions for supply chain formation. In ACM Conference on Electronic Commerce, pages 260-269, 2000. [31] Rainier Weigel and Christian Bliek. On reformulation of constraint satisfaction problems. In Proceedings of ECAI-98, pages 254-258, 1998. [32] Margaret W. Wiener. Access spectrum bid withdrawal. http://wireless.fcc.gov/auctions/33 /releases/da011719.pdf, July 2001. 192

bid;bid withdrawal;winner determination problem;mutual bid bond;robustness;mandatory mutual bid bond;constraint programming;combinatorial auction;bid-taker's exposure problem;set partition problem;weighted super solution;constraint program;exposure problem;enforceable commitment;weight super solution

train_J-57

Marginal Contribution Nets: A Compact Representation Scheme for Coalitional Games

We present a new approach to representing coalitional games based on rules that describe the marginal contributions of the agents. This representation scheme captures characteristics of the interactions among the agents in a natural and concise manner. We also develop efficient algorithms for two of the most important solution concepts, the Shapley value and the core, under this representation. The Shapley value can be computed in time linear in the size of the input. The emptiness of the core can be determined in time exponential only in the treewidth of a graphical interpretation of our representation.

1. INTRODUCTION Agents can often benefit by coordinating their actions. Coalitional games capture these opportunities of coordination by explicitly modeling the ability of the agents to take joint actions as primitives. As an abstraction, coalitional games assign a payoff to each group of agents in the game. This payoff is intended to reflect the payoff the group of agents can secure for themselves regardless of the actions of the agents not in the group. These choices of primitives are in contrast to those of non-cooperative games, of which agents are modeled independently, and their payoffs depend critically on the actions chosen by the other agents. 1.1 Coalitional Games and E-Commerce Coalitional games have appeared in the context of e-commerce. In [7], Kleinberg et al. use coalitional games to study recommendation systems. In their model, each individual knows about a certain set of items, is interested in learning about all items, and benefits from finding out about them. The payoffs to groups of agents are the total number of distinct items known by its members. Given this coalitional game setting, Kleinberg et al. compute the value of the private information of the agents is worth to the system using the solution concept of the Shapley value (definition can be found in section 2). These values can then be used to determine how much each agent should receive for participating in the system. As another example, consider the economics behind supply chain formation. The increased use of the Internet as a medium for conducting business has decreased the costs for companies to coordinate their actions, and therefore coalitional game is a good model for studying the supply chain problem. Suppose that each manufacturer purchases his raw materials from some set of suppliers, and that the suppliers offer higher discount with more purchases. The decrease in communication costs will let manufacturers find others interested in the same set of suppliers cheaper, and facilitates formation of coalitions to bargain with the suppliers. Depending on the set of suppliers and how much from each supplier each coalition purchases, we can assign payoffs to the coalitions depending on the discount it receives. The resulting game can be analyzed using coalitional game theory, and we can answer questions such as the stability of coalitions, and how to fairly divide the benefits among the participating manufacturers. A similar problem, combinatorial coalition formation, has previously been studied in [8]. 1.2 Evaluation Criteria for Coalitional Game Representation To capture the coalitional games described above and perform computations on them, we must first find a representation for these games. The na¨ıve solution is to enumerate the payoffs to each set of agents, therefore requiring space 193 exponential in the number of agents in the game. For the two applications described, the number of agents in the system can easily exceed a hundred; this na¨ıve approach will not be scalable to such problems. Therefore, it is critical to find good representation schemes for coalitional games. We believe that the quality of a representation scheme should be evaluated by four criteria. Expressivity: the breadth of the class of coalitional games covered by the representation. Conciseness: the space requirement of the representation. Efficiency: the efficiency of the algorithms we can develop for the representation. Simplicity: the ease of use of the representation by users of the system. The ideal representation should be fully expressive, i.e., it should be able to represent any coalitional games, use as little space as possible, have efficient algorithms for computation, and be easy to use. The goal of this paper is to develop a representation scheme that has properties close to the ideal representation. Unfortunately, given that the number of degrees of freedom of coalitional games is O(2n ), not all games can be represented concisely using a single scheme due to information theoretic constraints. For any given class of games, one may be able to develop a representation scheme that is tailored and more compact than a general scheme. For example, for the recommendation system game, a highly compact representation would be one that simply states which agents know of which products, and let the algorithms that operate on the representation to compute the values of coalitions appropriately. For some problems, however, there may not be efficient algorithms for customized representations. By having a general representation and efficient algorithms that go with it, the representation will be useful as a prototyping tool for studying new economic situations. 1.3 Previous Work The question of coalitional game representation has only been sparsely explored in the past [2, 3, 4]. In [4], Deng and Papadimitriou focused on the complexity of different solution concepts on coalitional games defined on graphs. While the representation is compact, it is not fully expressive. In [2], Conitzer and Sandholm looked into the problem of determining the emptiness of the core in superadditive games. They developed a compact representation scheme for such games, but again the representation is not fully expressive either. In [3], Conitzer and Sandholm developed a fully expressive representation scheme based on decomposition. Our work extends and generalizes the representation schemes in [3, 4] through decomposing the game into a set of rules that assign marginal contributions to groups of agents. We will give a more detailed review of these papers in section 2.2 after covering the technical background. 1.4 Summary of Our Contributions • We develop the marginal contribution networks representation, a fully expressive representation scheme whose size scales according to the complexity of the interactions among the agents. We believe that the representation is also simple and intuitive. • We develop an algorithm for computing the Shapley value of coalitional games under this representation that runs in time linear in the size of the input. • Under the graphical interpretation of the representation, we develop an algorithm for determining the whether a payoff vector is in the core and the emptiness of the core in time exponential only in the treewidth of the graph. 2. PRELIMINARIES In this section, we will briefly review the basics of coalitional game theory and its two primary solution concepts, the Shapley value and the core.1 We will also review previous work on coalitional game representation in more detail. Throughout this paper, we will assume that the payoff to a group of agents can be freely distributed among its members. This assumption is often known as the transferable utility assumption. 2.1 Technical Background We can represent a coalition game with transferable utility by the pair N, v , where • N is the set of agents; and • v : 2N → R is a function that maps each group of agents S ⊆ N to a real-valued payoff. This representation is known as the characteristic form. As there are exponentially many subsets, it will take space exponential in the number of agents to describe a coalitional game. An outcome in a coalitional game specifies the utilities the agents receive. A solution concept assigns to each coalitional game a set of reasonable outcomes. Different solution concepts attempt to capture in some way outcomes that are stable and/or fair. Two of the best known solution concepts are the Shapley value and the core. The Shapley value is a normative solution concept. It prescribes a fair way to divide the gains from cooperation when the grand coalition (i.e., N) is formed. The division of payoff to agent i is the average marginal contribution of agent i over all possible permutations of the agents. Formally, let φi(v) denote the Shapley value of i under characteristic function v, then2 φi(v) = S⊂N s!(n − s − 1)! n! (v(S ∪ {i}) − v(S)) (1) The Shapley value is a solution concept that satisfies many nice properties, and has been studied extensively in the economic and game theoretic literature. It has a very useful axiomatic characterization. Efficiency (EFF) A total of v(N) is distributed to the agents, i.e., i∈N φi(v) = v(N). Symmetry (SYM) If agents i and j are interchangeable, then φi(v) = φj(v). 1 The materials and terminology are based on the textbooks by Mas-Colell et al. [9] and Osborne and Rubinstein [11]. 2 As a notational convenience, we will use the lower-case letter to represent the cardinality of a set denoted by the corresponding upper-case letter. 194 Dummy (DUM) If agent i is a dummy player, i.e., his marginal contribution to all groups S are the same, φi(v) = v({i}). Additivity (ADD) For any two coalitional games v and w defined over the same set of agents N, φi(v + w) = φi(v) + φi(w) for all i ∈ N, where the game v + w is defined as (v + w)(S) = v(S) + w(S) for all S ⊆ N. We will refer to these axioms later in our proof of correctness of the algorithm for computing the Shapley value under our representation in section 4. The core is another major solution concept for coalitional games. It is a descriptive solution concept that focuses on outcomes that are stable. Stability under core means that no set of players can jointly deviate to improve their payoffs. Formally, let x(S) denote i∈S xi. An outcome x ∈ Rn is in the core if ∀S ⊆ N x(S) ≥ v(S) (2) The core was one of the first proposed solution concepts for coalitional games, and had been studied in detail. An important question for a given coalitional game is whether the core is empty. In other words, whether there is any outcome that is stable relative to group deviation. For a game to have a non-empty core, it must satisfy the property of balancedness, defined as follows. Let 1S ∈ Rn denote the characteristic vector of S given by (1S)i = 1 if i ∈ S 0 otherwise Let (λS)S⊆N be a set of weights such that each λS is in the range between 0 and 1. This set of weights, (λS)S⊆N , is a balanced collection if for all i ∈ N, S⊆N λS(1S)i = 1 A game is balanced if for all balanced collections of weights, S⊆N λSv(S) ≤ v(N) (3) By the Bondereva-Shapley theorem, the core of a coalitional game is non-empty if and only if the game is balanced. Therefore, we can use linear programming to determine whether the core of a game is empty. maximize λ∈R2n S⊆N λSv(S) subject to S⊆N λS1S = 1 ∀i ∈ N λS ≥ 0 ∀S ⊆ N (4) If the optimal value of (4) is greater than the value of the grand coalition, then the core is empty. Unfortunately, this program has an exponential number of variables in the number of players in the game, and hence an algorithm that operates directly on this program would be infeasible in practice. In section 5.4, we will describe an algorithm that answers the question of emptiness of core that works on the dual of this program instead. 2.2 Previous Work Revisited Deng and Papadimitriou looked into the complexity of various solution concepts on coalitional games played on weighted graphs in [4]. In their representation, the set of agents are the nodes of the graph, and the value of a set of agents S is the sum of the weights of the edges spanned by them. Notice that this representation is concise since the space required to specify such a game is O(n2 ). However, this representation is not general; it will not be able to represent interactions among three or more agents. For example, it will not be able to represent the majority game, where a group of agents S will have value of 1 if and only if s > n/2. On the other hand, there is an efficient algorithm for computing the Shapley value of the game, and for determining whether the core is empty under the restriction of positive edge weights. However, in the unrestricted case, determining whether the core is non-empty is coNP-complete. Conitzer and Sandholm in [2] considered coalitional games that are superadditive. They described a concise representation scheme that only states the value of a coalition if the value is strictly superadditive. More precisely, the semantics of the representation is that for a group of agents S, v(S) = max {T1,T2,...,Tn}∈Π i v(Ti) where Π is the set of all possible partitions of S. The value v(S) is only explicitly specified for S if v(S) is greater than all partitioning of S other than the trivial partition ({S}). While this representation can represent all games that are superadditive, there are coalitional games that it cannot represent. For example, it will not be able to represent any games with substitutability among the agents. An example of a game that cannot be represented is the unit game, where v(S) = 1 as long as S = ∅. Under this representation, the authors showed that determining whether the core is non-empty is coNP-complete. In fact, even determining the value of a group of agents is NP-complete. In a more recent paper, Conitzer and Sandholm described a representation that decomposes a coalitional game into a number of subgames whose sum add up to the original game [3]. The payoffs in these subgames are then represented by their respective characteristic functions. This scheme is fully general as the characteristic form is a special case of this representation. For any given game, there may be multiple ways to decompose the game, and the decomposition may influence the computational complexity. For computing the Shapley value, the authors showed that the complexity is linear in the input description; in particular, if the largest subgame (as measured by number of agents) is of size n and the number of subgames is m, then their algorithm runs in O(m2n ) time, where the input size will also be O(m2n ). On the other hand, the problem of determining whether a certain outcome is in the core is coNP-complete. 3. MARGINAL CONTRIBUTION NETS In this section, we will describe the Marginal Contribution Networks representation scheme. We will show that the idea is flexible, and we can easily extend it to increase its conciseness. We will also show how we can use this scheme to represent the recommendation game from the introduction. Finally, we will show that this scheme is fully expressive, and generalizes the representation schemes in [3, 4]. 3.1 Rules and MarginalContributionNetworks The basic idea behind marginal contribution networks (MC-nets) is to represent coalitional games using sets of rules. The rules in MC-nets have the following syntactic 195 form: Pattern → value A rule is said to apply to a group of agents S if S meets the requirement of the Pattern. In the basic scheme, these patterns are conjunctions of agents, and S meets the requirement of the given pattern if S is a superset of it. The value of a group of agents is defined to be the sum over the values of all rules that apply to the group. For example, if the set of rules are {a ∧ b} → 5 {b} → 2 then v({a}) = 0, v({b}) = 2, and v({a, b}) = 5 + 2 = 7. MC-nets is a very flexible representation scheme, and can be extended in different ways. One simple way to extend it and increase its conciseness is to allow a wider class of patterns in the rules. A pattern that we will use throughout the remainder of the paper is one that applies only in the absence of certain agents. This is useful for expressing concepts such as substitutability or default values. Formally, we express such patterns by {p1 ∧ p2 ∧ . . . ∧ pm ∧ ¬n1 ∧ ¬n2 ∧ . . . ∧ ¬nn} which has the semantics that such rule will apply to a group S only if {pi}m i=1 ∈ S and {nj}n j=1 /∈ S. We will call the {pi}m i=1 in the above pattern the positive literals, and {nj}n j=1 the negative literals. Note that if the pattern of a rule consists solely of negative literals, we will consider that the empty set of agents will also satisfy such pattern, and hence v(∅) may be non-zero in the presence of negative literals. To demonstrate the increase in conciseness of representation, consider the unit game described in section 2.2. To represent such a game without using negative literals, we will need 2n rules for n players: we need a rule of value 1 for each individual agent, a rule of value −1 for each pair of agents to counter the double-counting, a rule of value 1 for each triplet of agents, etc., similar to the inclusion-exclusion principle. On the other hand, using negative literals, we only need n rules: value 1 for the first agent, value 1 for the second agent in the absence of the first agent, value 1 for the third agent in the absence of the first two agents, etc. The representational savings can be exponential in the number of agents. Given a game represented as a MC-net, we can interpret the set of rules that make up the game as a graph. We call this graph the agent graph. The nodes in the graph will represent the agents in the game, and for each rule in the MCnet, we connect all the agents in the rule together and assign a value to the clique formed by the set of agents. Notice that to accommodate negative literals, we will need to annotate the clique appropriately. This alternative view of MC-nets will be useful in our algorithm for Core-Membership in section 5. We would like to end our discussion of the representation scheme by mentioning a trade-off between the expressiveness of patterns and the space required to represent them. To represent a coalitional game in characteristic form, one would need to specify all 2n − 1 values. There is no overhead on top of that since there is a natural ordering of the groups. For MC-nets, however, specification of the rules requires specifying both the patterns and the values. The patterns, if not represented compactly, may end up overwhelming the savings from having fewer values to specify. The space required for the patterns also leads to a tradeoff between the expressiveness of the allowed patterns and the simplicity of representing them. However, we believe that for most naturally arising games, there should be sufficient structure in the problem such that our representation achieves a net saving over the characteristic form. 3.2 Example: Recommendation Game As an example, we will use MC-net to represent the recommendation game discussed in the introduction. For each product, as the benefit of knowing about the product will count only once for each group, we need to capture substitutability among the agents. This can be captured by a scaled unit game. Suppose the value of the knowledge about product i is vi, and there are ni agents, denoted by {xj i }, who know about the product, the game for product i can then be represented as the following rules: {x1 i } → vi {x2 i ∧ ¬x1 i } → vi ... {xni i ∧ ¬xni−1 i ∧ · · · ∧ ¬x1 i } → vi The entire game can then be built up from the sets of rules of each product. The space requirement will be O(mn∗ ), where m is the number of products in the system, and n∗ is the maximum number of agents who knows of the same product. 3.3 Representation Power We will discuss the expressiveness and conciseness of our representation scheme and compare it with the previous works in this subsection. Proposition 1. Marginal contribution networks constitute a fully expressive representation scheme. Proof. Consider an arbitrary coalitional game N, v in characteristic form representation. We can construct a set of rules to describe this game by starting from the singleton sets and building up the set of rules. For any singleton set {i}, we create a rule {i} → v(i). For any pair of agents {i, j}, we create a rule {i ∧ j} → v({i, j}) − v({i}) − v({j}. We can continue to build up rules in a manner similar to the inclusion-exclusion principle. Since the game is arbitrary, MC-nets are fully expressive. Using the construction outlined in the proof, we can show that our representation scheme can simulate the multi-issue representation scheme of [3] in almost the same amount of space. Proposition 2. Marginal contribution networks use at most a linear factor (in the number of agents) more space than multi-issue representation for any game. Proof. Given a game in multi-issue representation, we start by describing each of the subgames, which are represented in characteristic form in [3], with a set of rules. 196 We then build up the grand game by including all the rules from the subgames. Note that our representation may require a space larger by a linear factor due to the need to describe the patterns for each rule. On the other hand, our approach may have fewer than exponential number of rules for each subgame, depending on the structure of these subgames, and therefore may be more concise than multi-issue representation. On the other hand, there are games that require exponentially more space to represent under the multi-issue scheme compared to our scheme. Proposition 3. Marginal contribution networks are exponentially more concise than multi-issue representation for certain games. Proof. Consider a unit game over all the agents N. As explained in 3.1, this game can be represented in linear space using MC-nets with negative literals. However, as there is no decomposition of this game into smaller subgames, it will require space O(2n ) to represent this game under the multiissue representation. Under the agent graph interpretation of MC-nets, we can see that MC-nets is a generalization of the graphical representation in [4], namely from weighted graphs to weighted hypergraphs. Proposition 4. Marginal contribution networks can represent any games in graphical form (under [4]) in the same amount of space. Proof. Given a game in graphical form, G, for each edge (i, j) with weight wij in the graph, we create a rule {i, j} → wij. Clearly this takes exactly the same space as the size of G, and by the additive semantics of the rules, it represents the same game as G. 4. COMPUTING THE SHAPLEY VALUE Given a MC-net, we have a simple algorithm to compute the Shapley value of the game. Considering each rule as a separate game, we start by computing the Shapley value of the agents for each rule. For each agent, we then sum up the Shapley values of that agent over all the rules. We first show that this final summing process correctly computes the Shapley value of the agents. Proposition 5. The Shapley value of an agent in a marginal contribution network is equal to the sum of the Shapley values of that agent over each rule. Proof. For any group S, under the MC-nets representation, v(S) is defined to be the sum over the values of all the rules that apply to S. Therefore, considering each rule as a game, by the (ADD) axiom discussed in section 2, the Shapley value of the game created from aggregating all the rules is equal to the sum of the Shapley values over the rules. The remaining question is how to compute the Shapley values of the rules. We can separate the analysis into two cases, one for rules with only positive literals and one for rules with mixed literals. For rules that have only positive literals, the Shapley value of the agents is v/m, where v is the value of the rule and m is the number of agents in the rule. This is a direct consequence of the (SYM) axiom of the Shapley value, as the agents in a rule are indistinguishable from each other. For rules that have both positive and negative literals, we can consider the positive and the negative literals separately. For a given positive literal i, the rule will apply only if i occurs in a given permutation after the rest of the positive literals but before any of the negative literals. Formally, let φi denote the Shapley value of i, p denote the cardinality of the positive set, and n denote the cardinality of the negative set, then φi = (p − 1)!n! (p + n)! v = v p p+n n For a given negative literal j, j will be responsible for cancelling the application of the rule if all positive literals come before the negative literals in the ordering, and j is the first among the negative literals. Therefore, φj = p!(n − 1)! (p + n)! (−v) = −v n p+n p By the (SYM) axiom, all positive literals will have the value of φi and all negative literals will have the value of φj. Note that the sum over all agents in rules with mixed literals is 0. This is to be expected as these rules contribute 0 to the grand coalition. The fact that these rules have no effect on the grand coalition may appear odd at first. But this is because the presence of such rules is to define the values of coalitions smaller than the grand coalition. In terms of computational complexity, given that the Shapley value of any agent in a given rule can be computed in time linear in the pattern of the rule, the total running time of the algorithm for computing the Shapley value of the game is linear in the size of the input. 5. ANSWERING CORE-RELATED QUESTIONS There are a few different but related computational problems associated with the solution concept of the core. We will focus on the following two problems: Definition 1. (Core-Membership) Given a coalitional game and a payoff vector x, determine if x is in the core. Definition 2. (Core-Non-Emptiness) Given a coalitional game, determine if the core is non-empty. In the rest of the section, we will first show that these two problems are coNP-complete and coNP-hard respectively, and discuss some complexity considerations about these problems. We will then review the main ideas of tree decomposition as it will be used extensively in our algorithm for Core-Membership. Next, we will present the algorithm for Core-Membership, and show that the algorithm runs in polynomial time for graphs of bounded treewidth. We end by extending this algorithm to answer the question of CoreNon-Emptiness in polynomial time for graphs of bounded treewidth. 5.1 Computational Complexity The hardness of Core-Membership and Core-NonEmptiness follows directly from the hardness results of games over weighted graphs in [4]. 197 Proposition 6. Core-Membership for games represented as marginal contribution networks is coNP-complete. Proof. Core-Membership in MC-nets is in the class of coNP since any set of agents S of which v(S) > x(S) will serve as a certificate to show that x does not belong to the core. As for its hardness, given any instance of CoreMembership for a game in graphical form of [4], we can encode the game in exactly the same space using MC-net due to Proposition 4. Since Core-Membership for games in graphical form is coNP-complete, Core-Membership in MC-nets is coNP-hard. Proposition 7. Core-Non-Emptiness for games represented as marginal contribution networks is coNP-hard. Proof. The same argument for hardness between games in graphical frm and MC-nets holds for the problem of CoreNon-Emptiness. We do not know of a certificate to show that Core-NonEmptiness is in the class of coNP as of now. Note that the obvious certificate of a balanced set of weights based on the Bondereva-Shapley theorem is exponential in size. In [4], Deng and Papadimitriou showed the coNP-completeness of Core-Non-Emptiness via a combinatorial characterization, namely that the core is non-empty if and only if there is no negative cut in the graph. In MC-nets, however, there need not be a negative hypercut in the graph for the core to be empty, as demonstrated by the following game (N = {1, 2, 3, 4}): v(S) =    1 if S = {1, 2, 3, 4} 3/4 if S = {1, 2}, {1, 3}, {1, 4}, or {2, 3, 4} 0 otherwise (5) Applying the Bondereva-Shapley theorem, if we let λ12 = λ13 = λ14 = 1/3, and λ234 = 2/3, this set of weights demonstrates that the game is not balanced, and hence the core is empty. On the other hand, this game can be represented with MC-nets as follows (weights on hyperedges): w({1, 2}) = w({1, 3}) = w({1, 4}) = 3/4 w({1, 2, 3}) = w({1, 2, 4}) = w({1, 3, 4}) = −6/4 w({2, 3, 4}) = 3/4 w({1, 2, 3, 4}) = 10/4 No matter how the set is partitioned, the sum over the weights of the hyperedges in the cut is always non-negative. To overcome the computational hardness of these problems, we have developed algorithms that are based on tree decomposition techniques. For Core-Membership, our algorithm runs in time exponential only in the treewidth of the agent graph. Thus, for graphs of small treewidth, such as trees, we have a tractable solution to determine if a payoff vector is in the core. By using this procedure as a separation oracle, i.e., a procedure for returning the inequality violated by a candidate solution, to solving a linear program that is related to Core-Non-Emptiness using the ellipsoid method, we can obtain a polynomial time algorithm for Core-Non-Emptiness for graphs of bounded treewidth. 5.2 Review of Tree Decomposition As our algorithm for Core-Membership relies heavily on tree decomposition, we will first briefly review the main ideas in tree decomposition and treewidth.3 Definition 3. A tree decomposition of a graph G = (V, E) is a pair (X, T), where T = (I, F) is a tree and X = {Xi | i ∈ I} is a family of subsets of V , one for each node of T, such that • i∈I Xi = V ; • For all edges (v, w) ∈ E, there exists an i ∈ I with v ∈ Xi and w ∈ Xi; and • (Running Intersection Property) For all i, j, k ∈ I: if j is on the path from i to k in T, then Xi ∩ Xk ⊆ Xj. The treewidth of a tree decomposition is defined as the maximum cardinality over all sets in X, less one. The treewidth of a graph is defined as the minimum treewidth over all tree decompositions of the graph. Given a tree decomposition, we can convert it into a nice tree decomposition of the same treewidth, and of size linear in that of T. Definition 4. A tree decomposition T is nice if T is rooted and has four types of nodes: Leaf nodes i are leaves of T with |Xi| = 1. Introduce nodes i have one child j such that Xi = Xj ∪ {v} of some v ∈ V . Forget nodes i have one child j such that Xi = Xj \ {v} for some v ∈ Xj. Join nodes i have two children j and k with Xi = Xj = Xk. An example of a (partial) nice tree decomposition together with a classification of the different types of nodes is in Figure 1. In the following section, we will refer to nodes in the tree decomposition as nodes, and nodes in the agent graph as agents. 5.3 Algorithm for Core Membership Our algorithm for Core-Membership takes as an input a nice tree decomposition T of the agent graph and a payoff vector x. By definition, if x belongs to the core, then for all groups S ⊆ N, x(S) ≥ v(S). Therefore, the difference x(S)−v(S) measures how close the group S is to violating the core condition. We call this difference the excess of group S. Definition 5. The excess of a coalition S, e(S), is defined as x(S) − v(S). A brute-force approach to determine if a payoff vector belongs to the core will have to check that the excesses of all groups are non-negative. However, this approach ignores the structure in the agent graph that will allow an algorithm to infer that certain groups have non-negative excesses due to 3 This is based largely on the materials from a survey paper by Bodlaender [1]. 198 i j k l nm Introduce Node: Xj = {1, 4} Xk = {1, 4} Forget Node: Xl = {1, 4} Introduce Node: Xm = {1, 2, 4} Xn = {4} Leaf Node: Join Node: Xi = {1, 3, 4} Join Node: Figure 1: Example of a (partial) nice tree decomposition the excesses computed elsewhere in the graph. Tree decomposition is the key to take advantage of such inferences in a structured way. For now, let us focus on rules with positive literals. Suppose we have already checked that the excesses of all sets R ⊆ U are non-negative, and we would like to check if the addition of an agent i to the set U will create a group with negative excess. A na¨ıve solution will be to compute the excesses of all sets that include i. The excess of the group (R ∪ {i}) for any group R can be computed as follows e(R ∪ {i}) = e(R) + xi − v(c) (6) where c is the cut between R and i, and v(c) is the sum of the weights of the edges in the cut. However, suppose that from the tree decomposition, we know that i is only connected to a subset of U, say S, which we will call the entry set to U. Ideally, because i does not share any edges with members of ¯U = (U \ S), we would hope that an algorithm can take advantage of this structure by checking only sets that are subsets of (S ∪ {i}). This computational saving may be possible since (xi −v(c)) in the update equation of (6) does not depend on ¯U. However, we cannot simply ignore ¯U as members of ¯U may still influence the excesses of groups that include agent i through group S. Specifically, if there exists a group T ⊃ S such that e(T) < e(S), then even when e(S ∪ {i}) has non-negative excess, e(T ∪{i}) may have negative excess. In other words, the excess available at S may have been drained away due to T. This motivates the definition of the reserve of a group. Definition 6. The reserve of a coalition S relative to a coalition U is the minimum excess over all coalitions between S and U, i.e., all T : S ⊆ T ⊆ U. We denote this value by r(S, U). We will refer to the group T that has the minimum excess as arg r(S, U). We will also call U the limiting set of the reserve and S the base set of the reserve. Our algorithm works by keeping track of the reserves of all non-empty subsets that can be formed by the agents of a node at each of the nodes of the tree decomposition. Starting from the leaves of the tree and working towards the root, at each node i, our algorithm computes the reserves of all groups S ⊆ Xi, limited by the set of agents in the subtree rooted at i, Ti, except those in (Xi\S). The agents in (Xi\S) are excluded to ensure that S is an entry set. Specifically, S is the entry set to ((Ti \ Xi) ∪ S). To accomodate for negative literals, we will need to make two adjustments. Firstly, the cut between an agent m and a set S at node i now refers to the cut among agent m, set S, and set ¬(Xi \ S), and its value must be computed accordingly. Also, when an agent m is introduced to a group at an introduce node, we will also need to consider the change in the reserves of groups that do not include m due to possible cut involving ¬m and the group. As an example of the reserve values we keep track of at a tree node, consider node i of the tree in Figure 1. At node i, we will keep track of the following: r({1}, {1, 2, . . .}) r({3}, {2, 3, . . .}) r({4}, {2, 4, . . .}) r({1, 3}, {1, 2, 3, . . .}) r({1, 4}, {1, 2, 4, . . .}) r({3, 4}, {2, 3, 4, . . .}) r({1, 3, 4}, {1, 2, 3, 4, . . .} where the dots . . . refer to the agents rooted under node m. For notational use, we will use ri(S) to denote r(S, U) at node i where U is the set of agents in the subtree rooted at node i excluding agents in (Xi \ S). We sometimes refer to these values as the r-values of a node. The details of the r-value computations are in Algorithm 1. To determine if the payoff vector x is in the core, during the r-value computation at each node, we can check if all of the r-values are non-negative. If this is so for all nodes in the tree, the payoff vector x is in the core. The correctness of the algorithm is due to the following proposition. Proposition 8. The payoff vector x is not in the core if and only if the r-values at some node i for some group S is negative. Proof. (⇐) If the reserve at some node i for some group S is negative, then there exists a coalition T for which e(T) = x(T) − v(T) < 0, hence x is not in the core. (⇒) Suppose x is not in the core, then there exists some group R∗ such that e(R∗ ) < 0. Let Xroot be the set of nodes at the root. Consider any set S ∈ Xroot, rroot(S) will have the base set of S and the limiting set of ((N \ Xroot) ∪ S). The union over all of these ranges includes all sets U for which U ∩ Xroot = ∅. Therefore, if R∗ is not disjoint from Xroot, the r-value for some group in the root is negative. If R∗ is disjoint from U, consider the forest {Ti} resulting from removal of all tree nodes that include agents in Xroot. 199 Algorithm 1 Subprocedures for Core Membership Leaf-Node(i) 1: ri(Xi) ← e(Xi) Introduce-Node(i) 2: j ← child of i 3: m ← Xi \ Xj {the introduced node} 4: for all S ⊆ Xj, S = ∅ do 5: C ← all hyperedges in the cut of m, S, and ¬(Xi \ S) 6: ri(S ∪ {x}) ← rj(S) + xm − v(C) 7: C ← all hyperedges in the cut of ¬m, S, and ¬(Xi \S) 8: ri(S) ← rj(S) − v(C) 9: end for 10: r({m}) ← e({m}) Forget-Node(i) 11: j ← child of i 12: m ← Xj \ Xi {the forgotten node} 13: for all S ⊆ Xi, S = ∅ do 14: ri(S) = min(rj(S), rj(S ∪ {m})) 15: end for Join-Node(i) 16: {j, k} ← {left, right} child of i 17: for all S ⊆ Xi, S = ∅ do 18: ri(S) ← rj(S) + rk(S) − e(S) 19: end for By the running intersection property, the sets of nodes in the trees Ti"s are disjoint. Thus, if the set R∗ = i Si for some Si ∈ Ti, e(R∗ ) = i e(Si) < 0 implies some group S∗ i has negative excess as well. Therefore, we only need to check the r-values of the nodes on the individual trees in the forest. But for each tree in the forest, we can apply the same argument restricted to the agents in the tree. In the base case, we have the leaf nodes of the original tree decomposition, say, for agent i. If R∗ = {i}, then r({i}) = e({i}) < 0. Therefore, by induction, if e(R∗ ) < 0, some reserve at some node would be negative. We will next explain the intuition behind the correctness of the computations for the r-values in the tree nodes. A detailed proof of correctness of these computations can be found in the appendix under Lemmas 1 and 2. Proposition 9. The procedure in Algorithm 1 correctly compute the r-values at each of the tree nodes. Proof. (Sketch) We can perform a case analysis over the four types of tree nodes in a nice tree decomposition. Leaf nodes (i) The only reserve value to be computed is ri(Xi), which equals r(Xi, Xi), and therefore it is just the excess of group Xi. Forget nodes (i with child j) Let m be the forgotten node. For any subset S ⊆ Xi, arg ri(S) must be chosen between the groups of S and S ∪ {m}, and hence we choose between the lower of the two from the r-values at node j. Introduce nodes (i with child j) Let m be the introduced node. For any subset T ⊆ Xi that includes m, let S denote (T \ {m}). By the running intersection property, there are no rules that involve m and agents of the subtree rooted at node i except those involving m and agents in Xi. As both the base set and the limiting set of the r-values of node j and node i differ by {m}, for any group V that lies between the base set and the limiting set of node i, the excess of group V will differ by a constant amount from the corresponding group (V \ {m}) at node j. Therefore, the set arg ri(T) equals the set arg rj(S) ∪ {m}, and ri(T) = rj(S) + xm − v(cut), where v(cut) is the value of the rules in the cut between m and S. For any subset S ⊂ Xi that does not include m, we need to consider the values of rules that include ¬m as a literal in the pattern. Also, when computing the reserve, the payoff xm will not contribute to group S. Therefore, together with the running intersection property as argued above, we can show that ri(S) = rj(S) − v(cut). Join nodes (i with left child j and right child k) For any given set S ⊆ Xi, consider the r-values of that set at j and k. If arg rj(S) or arg rk(S) includes agents not in S, then argrj(S) and argrk(S) will be disjoint from each other due to the running intersection property. Therefore, we can decompose arg ri(S) into three sets, (arg rj(S) \ S) on the left, S in the middle, and (arg rk(S) \ S) on the right. The reserve rj(S) will cover the excesses on the left and in the middle, whereas the reserve rk(S) will cover those on the right and in the middle, and so the excesses in the middle is double-counted. We adjust for the double-counting by subtracting the excesses in the middle from the sum of the two reserves rj(S) and rk(S). Finally, note that each step in the computation of the rvalues of each node i takes time at most exponential in the size of Xi, hence the algorithm runs in time exponential only in the treewidth of the graph. 5.4 Algorithm for Core Non-emptiness We can extend the algorithm for Core-Membership into an algorithm for Core-Non-Emptiness. As described in section 2, whether the core is empty can be checked using the optimization program based on the balancedness condition (3). Unfortunately, that program has an exponential number of variables. On the other hand, the dual of the program has only n variables, and can be written as follows: minimize x∈Rn n i=1 xi subject to x(S) ≥ v(S), ∀S ⊆ N (7) By strong duality, optimal value of (7) is equal to optimal value of (4), the primal program described in section 2. Therefore, by the Bondereva-Shapley theorem, if the optimal value of (7) is greater than v(N), the core is empty. We can solve the dual program using the ellipsoid method with Core-Membership as a separation oracle, i.e., a procedure for returning a constraint that is violated. Note that a simple extension to the Core-Membership algorithm will allow us to keep track of the set T for which e(T) < 0 during the r-values computation, and hence we can return the inequality about T as the constraint violated. Therefore, Core-Non-Emptiness can run in time polynomial in the running time of Core-Membership, which in turn runs in 200 time exponential only in the treewidth of the graph. Note that when the core is not empty, this program will return an outcome in the core. 6. CONCLUDING REMARKS We have developed a fully expressive representation scheme for coalitional games of which the size depends on the complexity of the interactions among the agents. Our focus on general representation is in contrast to the approach taken in [3, 4]. We have also developed an efficient algorithm for the computation of the Shapley values for this representation. While Core-Membership for MC-nets is coNP-complete, we have developed an algorithm for CoreMembership that runs in time exponential only in the treewidth of the agent graph. We have also extended the algorithm to solve Core-Non-Emptiness. Other than the algorithm for Core-Non-Emptiness in [4] under the restriction of non-negative edge weights, and that in [2] for superadditive games when the value of the grand coalition is given, we are not aware of any explicit description of algorithms for core-related problems in the literature. The work in this paper is related to a number of areas in computer science, especially in artificial intelligence. For example, the graphical interpretation of MC-nets is closely related to Markov random fields (MRFs) of the Bayes nets community. They both address the issue of of conciseness of representation by using the combinatorial structure of weighted hypergraphs. In fact, Kearns et al. first apply these idea to games theory by introducing a representation scheme derived from Bayes net to represent non-cooperative games [6]. The representational issues faced in coalitional games are closely related to the problem of expressing valuations in combinatorial auctions [5, 10]. The OR-bid language, for example, is strongly related to superadditivity. The question of the representation power of different patterns is also related to Boolean expression complexity [12]. We believe that with a better understanding of the relationships among these related areas, we may be able to develop more efficient representations and algorithms for coalitional games. Finally, we would like to end with some ideas for extending the work in this paper. One direction to increase the conciseness of MC-nets is to allow the definition of equivalent classes of agents, similar to the idea of extending Bayes nets to probabilistic relational models. The concept of symmetry is prevalent in games, and the use of classes of agents will allow us to capture symmetry naturally and concisely. This will also address the problem of unpleasing assymetric representations of symmetric games in our representation. Along the line of exploiting symmetry, as the agents within the same class are symmetric with respect to each other, we can extend the idea above by allowing functional description of marginal contributions. More concretely, we can specify the value of a rule as dependent on the number of agents of each relevant class. The use of functions will allow concise description of marginal diminishing returns (MDRs). Without the use of functions, the space needed to describe MDRs among n agents in MC-nets is O(n). With the use of functions, the space required can be reduced to O(1). Another idea to extend MC-nets is to augment the semantics to allow constructs that specify certain rules cannot be applied simultaneously. This is useful in situations where a certain agent represents a type of exhaustible resource, and therefore rules that depend on the presence of the agent should not apply simultaneously. For example, if agent i in the system stands for coal, we can either use it as fuel for a power plant or as input to a steel mill for making steel, but not for both at the same time. Currently, to represent such situations, we have to specify rules to cancel out the effects of applications of different rules. The augmented semantics can simplify the representation by specifying when rules cannot be applied together. 7. ACKNOWLEDGMENT The authors would like to thank Chris Luhrs, Bob McGrew, Eugene Nudelman, and Qixiang Sun for fruitful discussions, and the anonymous reviewers for their helpful comments on the paper. 8. REFERENCES [1] H. L. Bodlaender. Treewidth: Algorithmic techniques and results. In Proc. 22nd Symp. on Mathematical Foundation of Copmuter Science, pages 19-36. Springer-Verlag LNCS 1295, 1997. [2] V. Conitzer and T. Sandholm. Complexity of determining nonemptiness of the core. In Proc. 18th Int. Joint Conf. on Artificial Intelligence, pages 613-618, 2003. [3] V. Conitzer and T. Sandholm. Computing Shapley values, manipulating value division schemes, and checking core membership in multi-issue domains. In Proc. 19th Nat. Conf. on Artificial Intelligence, pages 219-225, 2004. [4] X. Deng and C. H. Papadimitriou. On the complexity of cooperative solution concepts. Math. Oper. Res., 19:257-266, May 1994. [5] Y. Fujishima, K. Leyton-Brown, and Y. Shoham. Taming the computational complexity of combinatorial auctions: Optimal and approximate approaches. In Proc. 16th Int. Joint Conf. on Artificial Intelligence, pages 548-553, 1999. [6] M. Kearns, M. L. Littman, and S. Singh. Graphical models for game theory. In Proc. 17th Conf. on Uncertainty in Artificial Intelligence, pages 253-260, 2001. [7] J. Kleinberg, C. H. Papadimitriou, and P. Raghavan. On the value of private information. In Proc. 8th Conf. on Theoretical Aspects of Rationality and Knowledge, pages 249-257, 2001. [8] C. Li and K. Sycara. Algoirthms for combinatorial coalition formation and payoff division in an electronic marketplace. Technical report, Robotics Insititute, Carnegie Mellon University, November 2001. [9] A. Mas-Colell, M. D. Whinston, and J. R. Green. Microeconomic Theory. Oxford University Press, New York, 1995. [10] N. Nisan. Bidding and allocation in combinatorial auctions. In Proc. 2nd ACM Conf. on Electronic Commerce, pages 1-12, 2000. [11] M. J. Osborne and A. Rubinstein. A Course in Game Theory. The MIT Press, Cambridge, Massachusetts, 1994. [12] I. Wegener. The Complexity of Boolean Functions. John Wiley & Sons, New York, October 1987. 201 APPENDIX We will formally show the correctness of the r-value computation in Algorithm 1 of introduce nodes and join nodes. Lemma 1. The procedure for computing the r-values of introduce nodes in Algorithm 1 is correct. Proof. Let node m be the newly introduced agent at i. Let U denote the set of agents in the subtree rooted at i. By the running intersection property, all interactions (the hyperedges) between m and U must be in node i. For all S ⊆ Xi : m ∈ S, let R denote (U \ Xi) ∪ S), and Q denote (R \ {m}). ri(S) = r(S, R) = min T :S⊆T ⊆R e(T) = min T :S⊆T ⊆R x(T) − v(T) = min T :S⊆T ⊆R x(T \ {m}) + xm − v(T \ {m}) − v(cut) = min T :S\{m}⊆T ⊆Q e(T ) + xm − v(cut) = rj(S) + xm − v(cut) The argument for sets S ⊆ Xi : m /∈ S is symmetric except xm will not contribute to the reserve due to the absence of m. Lemma 2. The procedure for computing the r-values of join nodes in Algorithm 1 is correct. Proof. Consider any set S ⊆ Xi. Let Uj denote the subtree rooted at the left child, Rj denote ((Uj \ Xj) ∪ S), and Qj denote (Uj \ Xj). Let Uk, Rk, and Qk be defined analogously for the right child. Let R denote (U \ Xi) ∪ S). ri(S) = r(S, R) = min T :S⊆T ⊆R x(T) − v(T) = min T :S⊆T ⊆R x(S) + x(T ∩ Qj) + x(T ∩ Qk) − v(S) − v(cut(S, T ∩ Qj) − v(cut(S, T ∩ Qk) = min T :S⊆T ⊆R x(T ∩ Qj) − v(cut(S, T ∩ Qj)) + min T :S⊆T ⊆R x(T ∩ Qk) − v(cut(S, T ∩ Qk)) + (x(S) − v(S)) (*) = min T :S⊆T ⊆R x(T ∩ Qj) + x(S) − v(cut(S, T ∩ Qj)) − v(S) + min T :S⊆T ⊆R x(T ∩ Qk) + x(S) − v(cut(S, T ∩ Qk)) − v(S) − (x(S) − v(S)) = min T :S⊆T ⊆R e(T ∩ Rj) + min T :S⊆T ⊆R e(T ∩ Rk) − e(S) = min T :S⊆T ⊆Rj e(T ) + min T :S⊆T ⊆Rk e(T ) − e(S) = rj(S) + rk(S) − e(S) where (*) is true as T ∩ Qj and T ∩ Qk are disjoint due to the running intersection property of tree decomposition, and hence the minimum of the sum can be decomposed into the sum of the minima. 202

marginal contribution;mc-net;coalitional game theory;coalitional game;treewidth;coremembership;agent;markov random field;shapley value;representation;interaction;core;marginal diminishing return;compact representation scheme

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Towards Truthful Mechanisms for Binary Demand Games: A General Framework

The family of Vickrey-Clarke-Groves (VCG) mechanisms is arguably the most celebrated achievement in truthful mechanism design. However, VCG mechanisms have their limitations. They only apply to optimization problems with a utilitarian (or affine) objective function, and their output should optimize the objective function. For many optimization problems, finding the optimal output is computationally intractable. If we apply VCG mechanisms to polynomial-time algorithms that approximate the optimal solution, the resulting mechanisms may no longer be truthful. In light of these limitations, it is useful to study whether we can design a truthful non-VCG payment scheme that is computationally tractable for a given allocation rule O. In this paper, we focus our attention on binary demand games in which the agents" only available actions are to take part in the a game or not to. For these problems, we prove that a truthful mechanism M = (O, P) exists with a proper payment method P iff the allocation rule O satisfies a certain monotonicity property. We provide a general framework to design such P. We further propose several general composition-based techniques to compute P efficiently for various types of output. In particular, we show how P can be computed through or/and combinations, round-based combinations, and some more complex combinations of the outputs from subgames.

1. INTRODUCTION In recent years, with the rapid development of the Internet, many protocols and algorithms have been proposed to make the Internet more efficient and reliable. The Internet is a complex distributed system where a multitude of heterogeneous agents cooperate to achieve some common goals, and the existing protocols and algorithms often assume that all agents will follow the prescribed rules without deviation. However, in some settings where the agents are selfish instead of altruistic, it is more reasonable to assume these agents are rational - maximize their own profits - according to the neoclassic economics, and new models are needed to cope with the selfish behavior of such agents. Towards this end, Nisan and Ronen [14] proposed the framework of algorithmic mechanism design and applied VCG mechanisms to some fundamental problems in computer science, including shortest paths, minimum spanning trees, and scheduling on unrelated machines. The VCG mechanisms [5, 11, 21] are applicable to mechanism design problems whose outputs optimize the utilitarian objective function, which is simply the sum of all agents" valuations. Unfortunately, some objective functions are not utilitarian; even for those problems with a utilitarian objective function, sometimes it is impossible to find the optimal output in polynomial time unless P=NP. Some mechanisms other than VCG mechanism are needed to address these issues. Archer and Tardos [2] studied a scheduling problem where it is NP-Hard to find the optimal output. They pointed out that a certain monotonicity property of the output work load is a necessary and sufficient condition for the existence of a truthful mechanism for their scheduling problem. Auletta et al. [3] studied a similar scheduling problem. They provided a family of deterministic truthful (2 + )-approximation mechanisms for any fixed number of machines and several (1 + )-truthful mechanisms for some NP-hard restrictions of their scheduling problem. Lehmann et al. [12] studied the single-minded combinatorial auction and gave a√ m-approximation truthful mechanism, where m is the number of goods. They also pointed out that a certain monotonicity in the allocation rule can lead to a truthful mechanism. The work of Mu"alem and Nisan [13] is the closest in spirit to our work. They characterized all truthful mechanisms based on a certain monotonicity property in a single-minded auction setting. They also showed how to used MAX and IF-THEN-ELSE to combine outputs from subproblems. As shown in this paper, the MAX and IF-THEN-ELSE combinations are special cases of the composition-based techniques that we present in this paper for computing the payments in polynomial time under mild assumptions. More generally, we study how to design truthful mechanisms for binary demand games where the allocation of an agent is either selected or not selected. We also assume that the valuations 213 of agents are uncorrelated, i.e., the valuation of an agent only depends on its own allocation and type. Recall that a mechanism M = (O, P) consists of two parts, an allocation rule O and a payment scheme P. Previously, it is often assumed that there is an objective function g and an allocation rule O, that either optimizes g exactly or approximately. In contrast to the VCG mechanisms, we do not require that the allocation should optimize the objective function. In fact, we do not even require the existence of an objective function. Given any allocation rule O for a binary demand game, we showed that a truthful mechanism M = (O, P) exists for the game if and only if O satisfies a certain monotonicity property. The monotonicity property only guarantees the existence of a payment scheme P such that (O, P) is truthful. We complement this existence theorem with a general framework to design such a payment scheme P. Furthermore, we present general techniques to compute the payment when the output is a composition of the outputs of subgames through the operators or and and; through round-based combinations; or through intermediate results, which may be themselves computed from other subproblems. The remainder of the paper is organized as follows. In Section 2, we discuss preliminaries and previous works, define binary demand games and discuss the basic assumptions about binary demand games. In Section 3, we show that O satisfying a certain monotonicity property is a necessary and sufficient condition for the existence of a truthful mechanism M = (O, P). A framework is then proposed in Section 4 to compute the payment P in polynomial time for several types of allocation rules O. In Section 5, we provide several examples to demonstrate the effectiveness of our general framework. We conclude our paper in Section 6 with some possible future directions. 2. PRELIMINARIES 2.1 Mechanism Design As usually done in the literatures about the designing of algorithms or protocols with inputs from individual agents, we adopt the assumption in neoclassic economics that all agents are rational, i.e., they respond to well-defined incentives and will deviate from the protocol only if the deviation improves their gain. A standard model for mechanism design is as follows. There are n agents 1, . . . , n and each agent i has some private information ti, called its type, only known to itself. For example, the type ti can be the cost that agent i incurs for forwarding a packet in a network or can be a payment that the agent is willing to pay for a good in an auction. The agents" types define the type vector t = (t1, t2, . . . , tn). Each agent i has a set of strategies Ai from which it can choose. For each input vector a = (a1, . . . , an) where agent i plays strategy ai ∈ Ai, the mechanism M = (O, P) computes an output o = O(a) and a payment vector p(a) = (p1(a), . . . , pn(a)). Here the payment pi(·) is the money given to agent i and depends on the strategies used by the agents. A game is defined as G = (S, M), where S is the setting for the game G. Here, S consists the parameters of the game that are set before the game starts and do not depend on the players" strategies. For example, in a unicast routing game [14], the setting consists of the topology of the network, the source node and the destination node. Throughout this paper, unless explicitly mentioned otherwise, the setting S of the game is fixed and we are only interested in how to design P for a given allocation rule O. A valuation function v(ti, o) assigns a monetary amount to agent i for each possible output o. Everything about a game S, M , including the setting S, the allocation rule O and the payment scheme P, is public knowledge except the agent i"s actual type ti, which is private information to agent i. Let ui(ti, o) denote the utility of agent i at the outcome of the game o, given its preferences ti. Here, following a common assumption in the literature, we assume the utility for agent i is quasi-linear, i.e., ui(ti, o) = v(ti, o) + Pi(a). Let a|i ai = (a1, · · · , ai−1, ai, ai+1, · · · , an), i.e., each agent j = i plays an action aj except that the agent i plays ai. Let a−i = (a1, · · · , ai−1, ai+1, · · · , an) denote the actions of all agents except i. Sometimes, we write (a−i, bi) as a|i bi. An action ai is called dominant for i if it (weakly) maximizes the utility of i for all possible strategies b−i of other agents, i.e., ui(ti, O(b−i, ai)) ≥ ui(ti, O(b−i, ai)) for all ai = ai and b−i. A direct-revelation mechanism is a mechanism in which the only actions available to each agent are to report its private type either truthfully or falsely to the mechanism. An incentive compatible (IC) mechanism is a direct-revelation mechanism in which if an agent reports its type ti truthfully, then it will maximize its utility. Then, in a direct-revelation mechanism satisfying IC, the payment scheme should satisfy the property that, for each agent i, v(ti, O(t)) + pi(t) ≥ v(ti, O(t|i ti)) + pi(t|i ti). Another common requirement in the literature for mechanism design is so called individual rationality or voluntary participation: the agent"s utility of participating in the output of the mechanism is not less than the utility of the agent of not participating. A direct-revelation mechanism is strategproof if it satisfies both IC and IR properties. Arguably the most important positive result in mechanism design is the generalized Vickrey-Clarke-Groves (VCG) mechanism by Vickrey [21], Clarke [5], and Groves [11]. The VCG mechanism applies to (affine) maximization problems where the objective function is utilitarian g(o, t) = P i v(ti, o) (i.e., the sum of all agents" valuations) and the set of possible outputs is assumed to be finite. A direct revelation mechanism M = (O(t), P(t)) belongs to the VCG family if (1) the allocation O(t) maximizesP i v(ti, o), and (2) the payment to agent i is pi(t) = P j=i vj(tj, O(t))+ hi (t−i), where hi () is an arbitrary function of t−i. Under mild assumptions, VCG mechanisms are the only truthful implementations for utilitarian problems [10]. The allocation rule of a VCG mechanism is required to maximize the objective function in the range of the allocation function. This makes the mechanism computationally intractable in many cases. Furthermore, replacing an optimal algorithm for computing the output with an approximation algorithm usually leads to untruthful mechanisms if a VCG payment scheme is used. In this paper, we study how to design a truthful mechanism that does not optimize a utilitarian objective function. 2.2 Binary Demand Games A binary demand game is a game G = (S, M), where M = (O, P) and the range of O is {0, 1}n . In other words, the output is a n-tuple vector O(t) = (O1(t), O2(t), . . . , On(t)), where Oi(t) = 1 (respectively, 0) means that agent i is (respectively, is not) selected. Examples of binary demand games include: unicast [14, 22, 9] and multicast [23, 24, 8] (generally subgraph construction by selecting some links/nodes to satisfy some property), facility location [7], and a certain auction [12, 2, 13]. Hereafter, we make the following further assumptions. 1. The valuation of the agents are not correlated, i.e., v(ti, o) is a function of v(ti, oi) only is denoted as v(ti, oi). 2. The valuation v(ti, oi) is a publicly known value and is normalized to 0. This assumption is needed to guarantee the IR property. Thus, throughout his paper, we only consider these direct-revelation mechanisms in which every agent only needs to reveal its valuation vi = v(ti, 1). 214 Notice that in applications where agents providing service and receiving payment, e.g., unicast and job scheduling, the valuation vi of an agent i is usually negative. For the convenience of presentation, we define the cost of agent as ci = −v(ti, 1), i.e., it costs agent i ci to provide the service. Throughout this paper, we will use ci instead of vi in our analysis. All our results can apply to the case where the agents receive the service rather than provide by setting ci to negative, as in auction. In a binary demand game, if we want to optimize an objective function g(o, t), then we call it a binary optimization demand game. The main differences between the binary demand games and those problems that can be solved by VCG mechanisms are: 1. The objective function is utilitarian (or affine maximization problem) for a problem solvable by VCG while there is no restriction on the objective function for a binary demand game. 2. The allocation rule O studied here does not necessarily optimize an objective function, while a VCG mechanism only uses the output that optimizes the objective function. We even do not require the existence of an objective function. 3. We assume that the agents" valuations are not correlated in a binary demand game, while the agents" valuations may be correlated in a VCG mechanism. In this paper, we assume for technical convenience that the objective function g(o, c), if exists, is continuous with respect to the cost ci, but most of our results are directly applicable to the discrete case without any modification. 2.3 Previous Work Lehmann et al. [12] studied how to design an efficient truthful mechanism for single-minded combinatorial auction. In a singleminded combinatorial auction, each agent i (1 ≤ i ≤ n) only wants to buy a subset Si ⊆ S with private price ci. A single-minded bidder i declares a bid bi = Si, ai with Si ⊆ S and ai ∈ R+ . In [12], it is assumed that the set of goods allocated to an agent i is either Si or ∅, which is known as exactness. Lehmann et al. gave a greedy round-based allocation algorithm, based on the rank ai |Si|1/2 , that has an approximation ratio √ m, where m is the number of goods in S. Based on the approximation algorithm, they gave a truthful payment scheme. For an allocation rule satisfying (1) exactness: the set of goods allocated to an agent i is either Si or ∅; (2) monotonicity: proposing more money for fewer goods cannot cause a bidder to lose its bid, they proposed a truthful payment scheme as follows: (1) charge a winning bidder a certain amount that does not depend on its own bidding; (2) charge a losing bidder 0. Notice the assumption of exactness reveals that the single minded auction is indeed a binary demand game. Their payment scheme inspired our payment scheme for binary demand game. In [1], Archer et al. studied the combinatorial auctions where multiple copies of many different items are on sale, and each bidder i desires only one subset Si. They devised a randomized rounding method that is incentive compatible and gave a truthful mechanism for combinatorial auctions with single parameter agents that approximately maximizes the social value of the auction. As they pointed out, their method is strongly truthful in sense that it is truthful with high probability 1 − , where is an error probability. On the contrary, in this paper, we study how to design a deterministic mechanism that is truthful based on some given allocation rules. In [2], Archer and Tardos showed how to design truthful mechanisms for several combinatorial problems where each agent"s private information is naturally expressed by a single positive real number, which will always be the cost incurred per unit load. The mechanism"s output could be arbitrary real number but their valuation is a quasi-linear function t · w, where t is the private per unit cost and w is the work load. Archer and Tardos characterized that all truthful mechanism should have decreasing work curves w and that the truthful payment should be Pi(bi) = Pi(0) + biwi(bi) − R bi 0 wi(u)du Using this model, Archer and Tardos designed truthful mechanisms for several scheduling related problems, including minimizing the span, maximizing flow and minimizing the weighted sum of completion time problems. Notice when the load of the problems is w = {0, 1}, it is indeed a binary demand game. If we apply their characterization of the truthful mechanism, their decreasing work curves w implies exactly the monotonicity property of the output. But notice that their proof is heavily based on the assumption that the output is a continuous function of the cost, thus their conclusion can"t directly apply to binary demand games. The paper of Ahuva Mu"alem and Noam Nisan [13] is closest in spirit to our work. They clearly stated that we only discussed a limited class of bidders, single minded bidders, that was introduced by [12]. They proved that all truthful mechanisms should have a monotonicity output and their payment scheme is based on the cut value. With a simple generalization, we get our conclusion for general binary demand game. They proposed several combination methods including MAX, IF-THEN-ELSE construction to perform partial search. All of their methods required the welfare function associated with the output satisfying bitonic property. Distinction between our contributions and previous results: It has been shown in [2, 6, 12, 13] that for the single minded combinatorial auction, there exists a payment scheme which results in a truthful mechanism if the allocation rule satisfies a certain monotonicity property. Theorem 4 also depends on the monotonicity property, but it is applicable to a broader setting than the single minded combinatorial auction. In addition, the binary demand game studied here is different from the traditional packing IP"s: we only require that the allocation to each agent is binary and the allocation rule satisfies a certain monotonicity property; we do not put any restrictions on the objective function. Furthermore, the main focus of this paper is to design some general techniques to find the truthful payment scheme for a given allocation rule O satisfying a certain monotonicity property. 3. GENERAL APPROACHES 3.1 Properties of Strategyproof Mechanisms We discuss several properties that mechanisms need to satisfy in order to be truthful. THEOREM 1. If a mechanism M = (O, P) satisfies IC, then ∀i, if Oi(t|i ti1 ) = Oi(t|i ti2 ), then pi(t|i ti1 ) = pi(t|i ti2 ). COROLLARY 2. For any strategy-proof mechanism for a binary demand game G with setting S, if we fix the cost c−i of all agents other than i, the payment to agent i is a constant p1 i if Oi(c) = 1, and it is another constant p0 i if Oi(c) = 0. THEOREM 3. Fixed the setting S for a binary demand game, if mechanism M = (O, P) satisfies IC, then mechanism M = (O, P ) with the same output method O and pi(c) = pi(c) − δi(c−i) for any function δi(c−i) also satisfies IC. The proofs of above theorems are straightforward and thus omitted due to space limit. This theorem implies that for the binary demand games we can always normalize the payment to an agent i such that the payment to the agent is 0 when it is not selected. Hereafter, we will only consider normalized payment schemes. 215 3.2 Existence of Strategyproof Mechanisms Notice, given the setting S, a mechanism design problem is composed of two parts: the allocation rule O and a payment scheme P. In this paper, given an allocation rule O we focus our attention on how to design a truthful payment scheme based on O. Given an allocation rule O for a binary demand game, we first present a sufficient and necessary condition for the existence of a truthful payment scheme P. DEFINITION 1 (MONOTONE NON-INCREASING PROPERTY (MP)). An output method O is said to satisfy the monotone non-increasing property if for every agent i and two of its possible costs ci1 < ci2 , Oi(c|i ci2 ) ≤ Oi(c|i ci1 ). This definition is not restricted only to binary demand games. For binary demand games, this definition implies that if Oi(c|i ci2 ) = 1 then Oi(c|i ci1 ) = 1. THEOREM 4. Fix the setting S, c−i in a binary demand game G with the allocation rule O, the following three conditions are equivalent: 1. There exists a value κi(O, c−i)(which we will call a cut value, such that Oi(c) = 1 if ci < κi(O, c−i) and Oi(c) = 0 if ci > κi(O, c−i). When ci = κi(O, c−i), Oi(c) can be either 0 or 1 depending on the tie-breaker of the allocation rule O. Hereafter, we will not consider the tie-breaker scenario in our proofs. 2. The allocation rule O satisfies MP. 3. There exists a truthful payment scheme P for this binary demand game. PROOF. The proof that Condition 2 implies Condition is straightforward and is omitted here. We then show Condition 3 implies Condition 2. The proof of this is similar to a proof in [13]. To prove this direction, we assume there exists an agent i and two valuation vectors c|i ci1 and c|i ci2 , where ci1 < ci2 , Oi(c|i ci2 ) = 1 and Oi(c|i ci1 ) = 0. From corollary 2, we know that pi(c|i ci1 ) = p0 i and pi(c|i ci2 ) = p1 i . Now fix c−i, the utility for i when ci = ci1 is ui(ci1 ) = p0 i . When agent i lies its valuation to ci2 , its utility is p1 i − ci1 . Since M = (O, P) is truthful, we have p0 i > p1 i − ci1 . Now consider the scenario when the actual valuation of agent i is ci = ci2 . Its utility is p1 i − ci2 when it reports its true valuation. Similarly, if it lies its valuation to ci1 , its utility is p0 i . Since M = (O, P) is truthful, we have p0 i < p1 i − ci2 . Consequently, we have p1 i −ci2 > p0 i > p1 i −ci1 . This inequality implies that ci1 > ci2 , which is a contradiction. We then show Condition 1 implies Condition 3. We prove this by constructing a payment scheme and proving that this payment scheme is truthful. The payment scheme is: If Oi(c) = 1, then agent i gets payment pi(c) = κi(O, c−i); else it gets payment pi(c) = 0. From condition 1, if Oi(c) = 1 then ci > κi(O, c−i). Thus, its utility is κi(O, c−i) − ci > 0, which implies that the payment scheme satisfies the IR. In the following we prove that this payment scheme also satisfies IC property. There are two cases here. Case 1: ci < κ(O, c−i). In this case, when i declares its true cost ci, its utility is κi(O, c−i) − ci > 0. Now consider the situation when i declares a cost di = ci. If di < κi(O, c−i), then i gets the same payment and utility since it is still selected. If di > κi(O, c−i), then its utility becomes 0 since it is not selected anymore. Thus, it has no incentive to lie in this case. Case 2: ci ≥ κ(O, c−i). In this case, when i reveals its true valuation, its payment is 0 and the utility is 0. Now consider the situation when i declares a valuation di = ci. If di > κi(O, c−i), then i gets the same payment and utility since it is still not selected. If di ≤ κi(O, c−i), then its utility becomes κi(O, c−i) − ci ≤ 0 since it is selected now. Thus, it has no incentive to lie. The equivalence of the monotonicity property of the allocation rule O and the existence of a truthful mechanism using O can be extended to games beyond binary demand games. The details are omitted here due to space limit. We now summarize the process to design a truthful payment scheme for a binary demand game based on an output method O. General Framework 1 Truthful mechanism design for a binary demand game Stage 1: Check whether the allocation rule O satisfies MP. If it does not, then there is no payment scheme P such that mechanism M = (O, P) is truthful. Otherwise, define the payment scheme P as follows. Stage 2: Based on the allocation rule O, find the cut value κi(O, c−i) for agent i such that Oi(c|i di) = 1 when di < κi(O, c−i), and Oi(c|i di) = 0 when di > κi(O, c−i). Stage 3: The payment for agent i is 0 if Oi(c) = 0; the payment is κi(O, c−i) if Oi(c) = 1. THEOREM 5. The payment defined by our general framework is minimum among all truthful payment schemes using O as output. 4. COMPUTING CUT VALUE FUNCTIONS To find the truthful payment scheme by using General Framework 1, the most difficult stage seems to be the stage 2. Notice that binary search does not work generally since the valuations of agents may be continuous. We give some general techniques that can help with finding the cut value function under certain circumstances. Our basic approach is as follows. First, we decompose the allocation rule into several allocation rules. Next find the cut value function for each of these new allocation rules. Then, we compute the original cut value function by combining these cut value functions of the new allocation rules. 4.1 Simple Combinations In this subsection, we introduce techniques to compute the cut value function by combining multiple allocation rules with conjunctions or disconjunctions. For simplicity, given an allocation rule O, we will use κ(O, c) to denote a n-tuple vector (κ1(O, c−1), κ2(O, c−2), . . . , κn(O, c−n)). Here, κi(O, c−i) is the cut value for agent i when the allocation rule is O and the costs c−i of all other agents are fixed. THEOREM 6. With a fixed setting S of a binary demand game, assume that there are m allocation rules O1 , O2 , · · · , Om satisfying the monotonicity property, and κ(Oi , c) is the cut value vector for Oi . Then the allocation rule O(c) = Wm i=1 Oi (c) satisfies the monotonicity property. Moreover, the cut value for O is κ(O, c) = maxm i=1{κ(Oi , c)} Here κ(O, c) = maxm i=1{κ(Oi , c)} means, ∀j ∈ [1, n], κj(O, c−j) = maxm i=1{κj(Oi , c−j)} and O(c) =Wm i=1 Oi (c) means, ∀j ∈ [1, n], Oj(c) = O1 j (c) ∨ O2 j (c) ∨ · · · ∨ Om j (c). PROOF. Assume that ci > ci and Oi(c) = 1. Without loss of generality, we assume that Ok i (c) = 1 for some k, 1 ≤ k ≤ m. From the assumption that Ok i (c) satisfies MP, we obtain that 216 Ok i (c|i ci) = 1. Thus, Oi(c|i ci) = Wm j=1 Oj (c) = 1. This proves that O(c) satisfies MP. The correctness of the cut value function follows directly from Theorem 4. Many algorithms indeed fall into this category. To demonstrate the usefulness of Theorem 6, we discuss a concrete example here. In a network, sometimes we want to deliver a packet to a set of nodes instead of one. This problem is known as multicast. The most commonly used structure in multicast routing is so called shortest path tree (SPT). Consider a network G = (V, E, c), where V is the set of nodes, and vector c is the actual cost of the nodes forwarding the data. Assume that the source node is s and the receivers are Q ⊂ V . For each receiver qi ∈ Q, we compute the shortest path (least cost path), denoted by LCP(s, qi, d), from the source s to qi under the reported cost profile d. The union of all such shortest paths forms the shortest path tree. We then use General Framework 1 to design the truthful payment scheme P when the SPT structure is used as the output for multicast, i.e., we design a mechanism M = (SPT, P). Notice that VCG mechanisms cannot be applied here since SPT is not an affine maximization. We define LCP(s,qi) as the allocation corresponds to the path LCP(s, qi, d), i.e., LCP (s,qi) k (d) = 1 if and only if node vk is in LCP(s, qi, d). Then the output SPT is defined as W qi∈Q LCP(s,qi) . In other words, SPTk(d) = 1 if and only if qk is selected in some LCP(s, qi, d). The shortest path allocation rule is a utilitarian and satisfies MP. Thus, from Theorem 6, SPT also satisfies MP, and the cut value function vector for SPT can be calculated as κ(SPT, c) = maxqi∈Q κ(LCP(s,qi) , c), where κ(LCP(s,qi) , c) is the cut value function vector for the shortest path LCP(s, qi, c). Consequently, the payment scheme above is truthful and the minimum among all truthful payment schemes when the allocation rule is SPT. THEOREM 7. Fixed the setting S of a binary demand game, assume that there are m output methods O1 , O2 , · · · , Om satisfying MP, and κ(Oi , c) are the cut value functions respectively for Oi where i = 1, 2, · · · , m. Then the allocation rule O(c) =Vm i=1 Oi (c) satisfies MP. Moreover, the cut value function for O is κ(O, c) = minm i=1{κ(Oi , c)}. We show that our simple combination generalizes the IF-THENELSE function defined in [13]. For an agent i, assume that there are two allocation rules O1 and O2 satisfying MP. Let κi(O1 , c−i), κi(O2 , c−i) be the cut value functions for O1 , O2 respectively. Then the IF-THEN-ELSE function Oi(c) is actually Oi(c) = [(ci ≤ κi(O1 , c−i) + δ1(c−i)) ∧ O2 (c−i, ci)] ∨ (ci < κi(O1 , c−i) − δ2(c−i)) where δ1(c−i) and δ2(c−i) are two positive functions. By applying Theorems 6 and 7, we know that the allocation rule O satisfies MP and consequently κi(O, c−i) = max{min(κi(O1 , c−i)+ δ1(c−i), κi(O2 , c−i)), κi(O1 , c−i) − δ2(c−i))}. 4.2 Round-Based Allocations Some approximation algorithms are round-based, where each round of an algorithm selects some agents and updates the setting and the cost profile if necessary. For example, several approximation algorithms for minimum weight vertex cover [19], maximum weight independent set, minimum weight set cover [4], and minimum weight Steiner [18] tree fall into this category. As an example, we discuss the minimum weighted vertex cover problem (MWVC) [16, 15] to show how to compute the cut value for a round-based output. Given a graph G = (V, E), where the nodes v1, v2, . . . , vn are the agents and each agent vi has a weight ci, we want to find a node set V ⊆ V such that for every edge (u, v) ∈ E at least one of u and v is in V . Such V is called a vertex cover of G. The valuation of a node i is −ci if it is selected; otherwise its valuation is 0. For a subset of nodes V ∈ V , we define its weight as c(V ) = P i∈V ci. We want to find a vertex cover with the minimum weight. Hence, the objective function to be implemented is utilitarian. To use the VCG mechanism, we need to find the vertex cover with the minimum weight, which is NP-hard [16]. Since we are interested in mechanisms that can be computed in polynomial time, we must use polynomial-time computable allocation rules. Many algorithms have been proposed in the literature to approximate the optimal solution. In this paper, we use a 2-approximation algorithm given in [16]. For the sake of completeness, we briefly review this algorithm here. The algorithm is round-based. Each round selects some vertices and discards some vertices. For each node i, w(i) is initialized to its weight ci, and when w(i) drops to 0, i is included in the vertex cover. To make the presentation clear, we say an edge (i1, j1) is lexicographically smaller than edge (i2, j2) if (1) min(i1, j1) < min(i2, j2), or (2) min(i1, j1) = min(i2, j2) and max(i1, j1) < max(i2, j2). Algorithm 2 Approximate Minimum Weighted Vertex Cover Input: A node weighted graph G = (V, E, c). Output: A vertex cover V . 1: Set V = ∅. For each i ∈ V , set w(i) = ci. 2: while V is not a vertex cover do 3: Pick an uncovered edge (i, j) with the least lexicographic order among all uncovered edges. 4: Let m = min(w(i), w(j)). 5: Update w(i) to w(i) − m and w(j) to w(j) − m. 6: If w(i) = 0, add i to V . If w(j) = 0, add j to V . Notice, selecting an edge using the lexicographic order is crutial to guarantee the monotonicity property. Algorithm 2 outputs a vertex cover V whose weight is within 2 times of the optimum. For convenience, we use VC(c) to denote the vertex cover computed by Algorithm 2 when the cost vector of vertices is c. Below we generalize Algorithm 2 to a more general scenario. Typically, a round-based output can be characterized as follows (Algorithm 3). DEFINITION 2. An updating rule Ur is said to be crossingindependent if, for any agent i not selected in round r, (1) Sr+1 and cr+1 −i do not depend on cr j (2) for fixed cr −i, cr i1 ≤ cr i2 implies that cr+1 i1 ≤ cr+1 i2 . We have the following theorem about the existence of a truthful payment using a round based allocation rule A. THEOREM 8. A round-based output A, with the framework defined in Algorithm 3, satisfies MP if the output methods Or satisfy MP and all updating rules Ur are crossing-independent. PROOF. Consider an agent i and fixed c−i. We prove that when an agent i is selected with cost ci, then it is also selected with cost di < ci. Assume that i is selected in round r with cost ci. Then under cost di, if agent i is selected in a round before r, our claim holds. Otherwise, consider in round r. Clearly, the setting Sr and the costs of all other agents are the same as what if agent i had cost ci since i is not selected in the previous rounds due to the crossindependent property. Since i is selected in round r with cost ci, i is also selected in round r with di < ci due to the reason that Or satisfies MP. This finishes the proof. 217 Algorithm 3 A General Round-Based Allocation Rule A 1: Set r = 0, c0 = c, and G0 = G initially. 2: repeat 3: Compute an output or using a deterministic algorithm Or : Sr × cr → {0, 1}n . Here Or , cr and Sr are allocation rule, cost vector and game setting in game Gr , respectively. Remark: Or is often a simple greedy algorithm such as selecting the agents that minimize some utilitarian function. For the example of vertex cover, Or will always select the light-weighted node on the lexicographically least uncovered edge (i, j). 4: Let r = r + 1. Update the game Gr−1 to obtain a new game Gr with setting Sr and cost vector cr according to some rule Ur : Or−1 × (Sr−1 , cr−1 ) → (Sr , cr ). Here we updates the cost and setting of the game. Remark: For the example of vertex cover, the updating rule will decrease the weight of vertices i and j by min(w(i), w(j)). 5: until a valid output is found 6: Return the union of the set of selected players of each round as the final output. For the example of vertex cover, it is the union of nodes selected in all rounds. Algorithm 4 Compute Cut Value for Round-Based Algorithms Input: A round-based output A, a game G1 = G, and a updating function vector U. Output: The cut value x for agent k. 1: Set r = 0 and ck = ζ. Recall that ζ is a value that can guarantee Ak = 0 when an agent reports the cost ζ. 2: repeat 3: Compute an output or using a deterministic algorithm based on setting Sr using allocation rule Or : Sr ×cr → {0, 1}n . 4: Find the cut value for agent k based on the allocation rule Or for costs cr −k. Let r = κk(Or , cr −k) be the cut value. 5: Set r = r + 1 and obtain a new game Gr from Gr−1 and or according to the updating rule Ur . 6: Let cr be the new cost vector for game Gr . 7: until a valid output is found. 8: Let gi(x) be the cost of ci k when the original cost vector is c|k x. 9: Find the minimum value x such that 8 >>>>>< >>>>>: g1(x) ≥ 1; g2(x) ≥ 2; ... gt−1(x) ≥ t−1; gt(x) ≥ t. Here, t is the total number of rounds. 10: Output the value x as the cut value. If the round-based output satisfies monotonicity property, the cut-value always exists. We then show how to find the cut value for a selected agent k in Algorithm 4. The correctness of Algorithm 4 is straightforward. To compute the cut value, we assume that (1) the cut value r for each round r can be computed in polynomial time; (2) we can solve the equation gr(x) = r to find x in polynomial time when the cost vector c−i and b are given. Now we consider the vertex cover problem. For each round r, we select a vertex with the least weight and that is incident on the lexicographically least uncovered edge. The output satisfies MP. For agent i, we update its cost to cr i − cr j iff edge (i, j) is selected. It is easy to verify this updating rule is crossing-independent, thus we can apply Algorithm 4 to compute the cut value for the set cover game as shown in Algorithm 5. Algorithm 5 Compute Cut Value for MVC. Input: A node weighted graph G = (V, E, c) and a node k selected by Algorithm 2. Output: The cut value κk(V C, c−k). 1: For each i ∈ V , set w(i) = ci. 2: Set w(k) = ∞, pk = 0 and V = ∅. 3: while V is not a vertex cover do 4: Pick an uncovered edge (i, j) with the least lexicographic order among all uncovered edges. 5: Set m = min(w(i), w(j)). 6: Update w(i) = w(i) − m and w(j) = w(j) − m. 7: If w(i) = 0, add i to V ; else add j to V . 8: If i == k or j == k then set pk = pk + m. 9: Output pk as the cut value κk(V C, c−k). 4.3 Complex Combinations In subsection 4.1, we discussed how to find the cut value function when the output of the binary demand game is a simple combination of some outputs, whose cut values can be computed through other means (typically VCG). However, some algorithms cannot be decomposed in the way described in subsection 4.1. Next we present a more complex way to combine allocation rules, and as we may expected, the way to find the cut value is also more complicated. Assume that there are n agents 1 ≤ i ≤ n with cost vector c, and there are m binary demand games Gi with objective functions fi(o, c), setting Si and allocation rule ψi where i = 1, 2, · · · , m. There is another binary demand game with setting S and allocation rule O, whose input is a cost vector d = (d1, d2, · · · , dm). Let f be the function vector (f1, f2, · · · , fm), ψ be the allocation rule vector (ψ1 , ψ2 , · · · , ψm ) and ∫ be the setting vector (S1, S2, · · · , Sm). For notation simplicity, we define Fi(c) = fi(ψi (c), c), for each 1 ≤ i ≤ m, and F(c) = (F1(c), F2(c), · · · , Fm(c)). Let us see a concrete example of these combinations. Consider a link weighted graph G = (V, E, c), and a subset of q nodes Q ⊆ V . The Steiner tree problem is to find a set of links with minimum total cost to connect Q. One way to find an approximation of the Steiner tree is as follows: (1) we build a virtual complete graph H using Q as its vertices, and the cost of each edge (i, j) is the cost of LCP(i, j, c) in graph G; (2) build the minimum spanning tree of H, denoted as MST(H); (3) an edge of G is selected iff it is selected in some LCP(i, j, c) and edge (i, j) of H is selected to MST(H). In this game, we define q(q − 1)/2 games Gi,j, where i, j ∈ Q, with objective functions fi,j(o, c) being the minimum cost of 218 connecting i and j in graph G, setting Si being the original graph G and allocation rule is LCP(i, j, c). The game G corresponds to the MST game on graph H. The cost of the pair-wise q(q − 1)/2 shortest paths defines the input vector d = (d1, d2, · · · , dm) for game MST. More details will be given in Section 5.2. DEFINITION 3. Given an allocation rule O and setting S, an objective function vector f, an allocation rule vector ψ and setting vector ∫, we define a compound binary demand game with setting S and output O ◦ F as (O ◦ F)i(c) = Wm j=1(Oj(F(c)) ∧ ψj i (c)). The allocation rule of the above definition can be interpreted as follows. An agent i is selected if and only if there is a j such that (1) i is selected in ψj (c), and (2) the allocation rule O will select index j under cost profile F(c). For simplicity, we will use O ◦ F to denote the output of this compound binary demand game. Notice that a truthful payment scheme using O ◦ F as output exists if and only if it satisfies the monotonicity property. To study when O ◦F satisfies MP, several necessary definitions are in order. DEFINITION 4. Function Monotonicity Property (FMP) Given an objective function g and an allocation rule O, a function H(c) = g(O(c), c) is said to satisfy the function monotonicity property, if, given fixed c−i, it satisfies: 1. When Oi(c) = 0, H(c) does not increase over ci. 2. When Oi(c) = 1, H(c) does not decrease over ci. DEFINITION 5. Strong Monotonicity Property (SMP) An allocation rule O is said to satisfy the strong monotonicity property if O satisfies MP, and for any agent i with Oi(c) = 1 and agent j = i, Oi(c|j cj) = 1 if cj ≥ cj or Oj(c|j cj) = 0. LEMMA 1. For a given allocation rule O satisfying SMP and cost vectors c, c with ci = ci, if Oi(c) = 1 and Oi(c ) = 0, then there must exist j = i such that cj < cj and Oj(c ) = 1. From the definition of the strong monotonicity property, we have Lemma 1 directly. We now can give a sufficient condition when O ◦ F satisfies the monotonicity property. THEOREM 9. If ∀i ∈ [1, m], Fi satisfies FMP, ψi satisfies MP, and the output O satisfies SMP, then O ◦ F satisfies MP. PROOF. Assuming for cost vector c we have (O ◦ F)i(c) = 1, we should prove for any cost vector c = c|i ci with ci < ci, (O ◦ F)i(c ) = 1. Noticing that (O ◦ F)i(c) = 1, without loss of generality, we assume that Ok(F(c)) = 1 and ψk i (c) = 1 for some index 1 ≤ k ≤ m. Now consider the output O with the cost vector F(c )|k Fk(c). There are two scenarios, which will be studied one by one as follows. One scenario is that index k is not chosen by the output function O. From Lemma 1, there must exist j = k such that Fj(c ) < Fj(c) (1) Oj(F(c )|k Fk(c)) = 1 (2) We then prove that agent i will be selected in the output ψj (c ), i.e., ψj i (c ) = 1. If it is not, since ψj (c) satisfies MP, we have ψj i (c) = ψj i (c ) = 0 from ci < ci. Since Fj satisfies FMP, we know Fj(c ) ≥ Fj(c), which is a contradiction to the inequality (1). Consequently, we have ψj i (c ) = 1. From Equation (2), the fact that index k is not selected by allocation rule O and the definition of SMP, we have Oj(F(c )) = 1, Thus, agent i is selected by O ◦ F because of Oj(F(c )) = 1 and ψj i (c ) = 1. The other scenario is that index k is chosen by the output function O. First, agent i is chosen in ψk (c ) since the output ψk (c) satisfies the monotonicity property and ci < ci and ψk i (c) = 1. Secondly, since the function Fk satisfies FMP, we know that Fk(c ) ≤ Fk(c). Remember that output O satisfies the SMP, thus we can obtain Ok(F(c )) = 1 from the fact that Ok(F(c )|k Fk(c)) = 1 and Fk(c ) ≤ Fk(c). Consequently, agent i will also be selected in the final output O ◦ F. This finishes our proof. This theorem implies that there is a cut value for the compound output O ◦ F. We then discuss how to find the cut value for this output. Below we will give an algorithm to calculate κi(O ◦ F) when (1) O satisfies SMP, (2) ψj satisfies MP, and (3) for fixed c−i, Fj(c) is a constant, say hj, when ψj i (c) = 0, and Fj(c) increases when ψj i (c) = 1. Notice that here hj can be easily computed by setting ci = ∞ since ψj satisfies the monotonicity property. When given i and fixed c−i, we define (Fi j )−1 (y) as the smallest x such that Fj(c|i x) = y. For simplicity, we denote (Fi j )−1 as F−1 j if no confusion is caused when i is a fixed agent. In this paper, we assume that given any y, we can find such x in polynomial time. Algorithm 6 Find Cut Value for Compound Method O ◦ F Input: allocation rule O, objective function vector F and inverse function vector F−1 = {F−1 1 , · · · , F−1 m }, allocation rule vector ψ and fixed c−i. Output: Cut value for agent i based on O ◦ F. 1: for 1 ≤ j ≤ m do 2: Compute the outputs ψj (ci). 3: Compute hj = Fj(c|i ∞). 4: Use h = (h1, h2, · · · , hm) as the input for the output function O. Denote τj = κj(O, h−j) as the cut value function of output O based on input h. 5: for 1 ≤ j ≤ m do 6: Set κi,j = F−1 j (min{τj, hj}). 7: The cut value for i is κi(O ◦ F, c−i) = maxm j=1 κi,j. THEOREM 10. Algorithm 6 computes the correct cut value for agent i based on the allocation rule O ◦ F. PROOF. In order to prove the correctness of the cut value function calculated by Algorithm 6, we prove the following two cases. For our convenience, we will use κi to represent κi(O ◦ F, c−i) if no confusion caused. First, if di < κi then (O ◦ F)i(c|i di) = 1. Without loss of generality, we assume that κi = κi,j for some j. Since function Fj satisfies FMP and ψj i (c|i di) = 1, we have Fj(c|i di) < Fj(κi). Notice di < κi,j, from the definition of κi,j = F−1 j (min{τj, hj}) we have (1) ψj i (c|i di) = 1, (2) Fj(c|i di) < τj due to the fact that Fj(x) is a non-decreasing function when j is selected. Thus, from the monotonicity property of O and τj is the cut value for output O, we have Oj(h|j Fj(c|i di)) = 1. (3) If Oj(F(c|i di)) = 1 then (O◦F)i(c|i di) = 1. Otherwise, since O satisfies SMP, Lemma 1 and equation 3 imply that there exists at least one index k such that Ok(F(c|i di)) = 1 and Fk(c|i di) < hk. Note Fk(c|i di) < hk implies that i is selected in ψk (c|i di) since hk = Fk(ci|i ∞). In other words, agent i is selected in O◦F. 219 Second, if di ≥ κi(O ◦ F, c−i) then (O ◦ F)i(c|i di) = 0. Assume for the sake of contradiction that (O ◦ F)i(c|i di) = 1. Then there exists an index 1 ≤ j ≤ m such that Oj(F(c|i di)) = 1 and ψj i (c|i di) = 1. Remember that hk ≥ Fk(c|i di) for any k. Thus, from the fact that O satisfies SMP, when changing the cost vector from F(c|i di) to h|j Fj(c|i di), we still have Oj(h|j Fj(c|i di)) = 1. This implies that Fj(c|i di) < τj. Combining the above inequality and the fact that Fj(c|i c|i di) < hj, we have Fj(c|i di) < min{hj, τj}. This implies di < F−1 j (min{hj, τj}) = κi,j < κi(O ◦ F, c−i). which is a contradiction. This finishes our proof. In most applications, the allocation rule ψj implements the objective function fj and fj is utilitarian. Thus, we can compute the inverse of F−1 j efficiently. Another issue is that it seems the conditions when we can apply Algorithm 6 are restrictive. However, lots of games in practice satisfy these properties and here we show how to deduct the MAX combination in [13]. Assume A1 and A2 are two allocation rules for single minded combinatorial auction, then the combination MAX(A1, A2) returns the allocation with the larger welfare. If algorithm A1 and A2 satisfy MP and FMP, the operation max(x, y) which returns the larger element of x and y satisfies SMP. From Theorem 9 we obtain that combination MAX(A1, A2) also satisfies MP. Further, the cut value of the MAX combination can be found by Algorithm 6. As we will show in Section 5, the complex combination can apply to some more complicated problems. 5. CONCRETE EXAMPLES 5.1 Set Cover In the set cover problem, there is a set U of m elements needed to be covered, and each agent 1 ≤ i ≤ n can cover a subset of elements Si with a cost ci. Let S = {S1, S2, · · · , Sn} and c = (c1, c2, · · · , cn). We want to find a subset of agents D such that U ⊆ S i∈D Si. The selected subsets is called the set cover for U. The social efficiency of the output D is defined as P i∈D ci, which is the objective function to be minimized. Clearly, this is a utilitarian and thus VCG mechanism can be applied if we can find the subset of S that covers U with the minimum cost. It is well-known that finding the optimal solution is NP-hard. In [4], an algorithm of approximation ratio of Hm has been proposed and it has been proved that this is the best ratio possible for the set cover problem. For the completeness of presentation, we review their method here. Algorithm 7 Greedy Set Cover (GSC) Input: Agent i"s subset Si covered and cost ci. (1 ≤ i ≤ n). Output: A set of agents that can cover all elements. 1: Initialize r = 1, T0 = ∅, and R = ∅. 2: while R = U do 3: Find the set Sj with the minimum density cj |Sj −Tr| . 4: Set Tr+1 = Tr S Sj and R = R S j. 5: r = r + 1 6: Output R. Let GSC(S) be the sets selected by the Algorithm 7.Notice that the output set is a function of S and c. Some works assume that the type of an agent could be ci, i.e., Si is assumed to be a public knowledge. Here, we consider a more general case in which the type of an agent is (Si, ci). In other words, we assume that every agent i can not only lie about its cost ci but also can lie about the set Si. This problem now looks similar to the combinatorial auction with single minded bidder studied in [12], but with the following differences: in the set cover problem we want to cover all the elements and the sets chosen can have some overlap while in combinatorial auction the chosen sets are disjoint. We can show that the mechanism M = (GSC, PV CG ), using Algorithm 7 to find a set cover and apply VCG mechanism to compute the payment to the selected agents, is not truthful. Obviously, the set cover problem is a binary demand game. For the moment, we assume that agent i won"t be able to lie about Si. We will drop this assumption later. We show how to design a truthful mechanism by applying our general framework. 1. Check the monotonicity property: The output of Algorithm 7 is a round-based output. Thus, for an agent i, we first focus on the output of one round r. In round r, if i is selected by Algorithm 7, then it has the minimum ratio ci |Si−Tr| among all remaining agents. Now consider the case when i lies its cost to ci < ci, obviously ci |Si−Tr| is still minimum among all remaining agents. Consequently, agent i is still selected in round r, which means the output of round r satisfies MP. Now we look into the updating rules. For every round, we only update the Tr+1 = Tr S Sj and R = R S j, which is obviously cross-independent. Thus, by applying Theorem 8, we know the output by Algorithm 7 satisfies MP. 2. Find the cut value: To calculate the cut value for agent i with fixed cost vector c−i, we follow the steps in Algorithm 4. First, we set ci = ∞ and apply Algorithm 7. Let ir be the agent selected in round r and T−i r+1 be the corresponding set. Then the cut value of round r is r = cir |Sir − T−i r | · |Si − T−i r |. Remember the updating rule only updates the game setting but not the cost of the agent, thus we have gr(x) = x ≥ r for 1 ≤ r ≤ t. Therefore, the final cut value for agent i is κi(GSC, c−i) = max r { cir |Sir − T−i r | · |Si − T−i r |} The payment to an agent i is κi if i is selected; otherwise its payment is 0. We now consider the scenario when agent i can lie about Si. Assume that agent i cannot lie upward, i.e., it can only report a set Si ⊆ Si. We argue that agent i will not lie about its elements Si. Notice that the cut value computed for round r is r = cir |Sir −T −i r | · |Si − T−i r |. Obviously |Si − T−i r | ≤ |Si − T−i r | for any Si ⊆ Si. Thus, lying its set as Si will not increase the cut value for each round. Thus lying about Si will not improve agent i"s utility. 5.2 Link Weighted Steiner Trees Consider any link weighted network G = (V, E, c), where E = {e1, e2, · · · , em} are the set of links and ci is the weight of the link ei. The link weighted Steiner tree problem is to find a tree rooted at source node s spanning a given set of nodes Q = {q1, q2, · · · , qk} ⊂ V . For simplicity, we assume that qi = vi, for 1 ≤ i ≤ k. Here the links are agents. The total cost of links in a graph H ⊆ G is called the weight of H, denoted as ω(H). It is NP-hard to find the minimum cost multicast tree when given an arbitrary link weighted 220 graph G [17, 20]. The currently best polynomial time method has approximation ratio 1 + ln 3 2 [17]. Here, we review and discuss the first approximation method by Takahashi and Matsuyama [20]. Algorithm 8 Find LinkWeighted SteinerTree (LST) Input: Network G = (V, E, c) where c is the cost vector for link set E. Source node s and receiver set Q. Output: A tree LST rooted at s and spanned all receivers. 1: Set r = 1, G1 = G, Q1 = Q and s1 = s. 2: repeat 3: In graph Gr, find the receiver, say qi, that is closest to the source s, i.e., LCP(s, qi, c) has the least cost among the shortest paths from s to all receivers in Qr . 4: Select all links on LCP(s, qi, c) as relay links and set their cost to 0. The new graph is denoted as Gr+1. 5: Set tr as qi and Pr = LCP(s, qi, c). 6: Set Qr+1 = Qr \qi and r = r + 1. 7: until all receivers are spanned. Hereafter, let LST(G) be the final tree constructed using the above method. It is shown in [24] that mechanism M = (LST, pV CG ) is not truthful, where pV CG is the payment calculated based on VCG mechanism. We then show how to design a truthful payment scheme using our general framework. Observe that the output Pr, for any round r, satisfies MP, and the update rule for every round satisfies crossing-independence. Thus, from Theorem 8, the roundbased output LST satisfies MP. In round r, the cut value for a link ei can be obtained by using the VCG mechanism. Now we set ci = ∞ and execute Algorithm 8. Let w−i r (ci) be the cost of the path Pr(ci) selected in the rth round and Πi r(ci) be the shortest path selected in round r if the cost of ci is temporarily set to −∞. Then the cut value for round r is r = wi r(c−i) − |Πi r(c−i)| where |Πi r(c−i)| is the cost of the path Πi r(c−i) excluding node vi. Using Algorithm 4, we obtain the final cut value for agent i: κi(LST, c−i) = maxr{ r}. Thus, the payment to a link ei is κi(LST, c−i) if its reported cost is di < κi(LST, d−i); otherwise, its payment is 0. 5.3 Virtual Minimal Spanning Trees To connect the given set of receivers to the source node, besides the Steiner tree constructed by the algorithms described before, a virtual minimum spanning tree is also often used. Assume that Q is the set of receivers, including the sender. Assume that the nodes in a node-weighted graph are all agents. The virtual minimum spanning tree is constructed as follows. Algorithm 9 Construct VMST 1: for all pairs of receivers qi, qj ∈ Q do 2: Calculate the least cost path LCP(qi, qj, d). 3: Construct a virtual complete link weighted graph K(d) using Q as its node set, where the link qiqj corresponds to the least cost path LCP(qi, qj, d), and its weight is w(qiqj) = |LCP(qi, qj, d)|. 4: Build the minimum spanning tree on K(d), denoted as V MST(d). 5: for every virtual link qiqj in V MST(d) do 6: Find the corresponding least cost path LCP(qi, qj, d) in the original network. 7: Mark the agents on LCP(qi, qj, d) selected. The mechanism M = (V MST, pV CG ) is not truthful [24], where the payment pV CG to a node is based on the VCG mechanism. We then show how to design a truthful mechanism based on the framework we described. 1. Check the monotonicity property: Remember that in the complete graph K(d), the weight of a link qiqj is |LCP(qi, qj, d)|. In other words, we implicitly defined |Q|(|Q| − 1)/2 functions fi,j, for all i < j and qi ∈ Q and qj ∈ Q, with fi,j(d) = |LCP(qi, qj, d)|. We can show that the function fi,j(d) = |LCP(qi, qj, d)| satisfies FMP, LCP satisfies MP, and the output MST satisfies SMP. From Theorem 9, the allocation rule VMST satisfies the monotonicity property. 2. Find the cut value: Notice VMST is the combination of MST and function fi,j, so cut value for VMST can be computed based on Algorithm 6 as follows. (a) Given a link weighted complete graph K(d) on Q, we should find the cut value function for edge ek = (qi, qj) based on MST. Given a spanning tree T and a pair of terminals p and q, clearly there is a unique path connecting them on T. We denote this path as ΠT (p, q), and the edge with the maximum length on this path as LE(p, q, T). Thus, the cut value can be represented as κk(MST, d) = LE(qi, qj, MST(d|k ∞)) (b) We find the value-cost function for LCP. Assume vk ∈ LCP(qi, qj, d), then the value-cost function is xk = yk − |LCPvk (qi, qj, d|k 0)|. Here, LCPvk (qi, qj, d) is the least cost path between qi and qj with node vk on this path. (c) Remove vk and calculate the value K(d|k ∞). Set h(i,j) = |LCP(qi, qj, d|∞ ))| for every pair of node i = j and let h = {h(i,j)} be the vector. Then it is easy to show that τ(i,j) = |LE(qi, qj, MST(h|(i,j) ∞))| is the cut value for output VMST. It easy to verify that min{h(i,j), τ(i,j)} = |LE(qi, qj, MST(h)|. Thus, we know κ (i,j) k (V MST, d) is |LE(qi, qj, MST(h)|− |LCPvk (qi, qj, d|k 0)|. The cut value for agent k is κk(V MST, d−k) = max0≤i,j≤r κij k (V MST, d−k). 3. We pay agent k κk(V MST, d−k) if and only if k is selected in V MST(d); else we pay it 0. 5.4 Combinatorial Auctions Lehmann et al. [12] studied how to design an efficient truthful mechanism for single-minded combinatorial auction. In a singleminded combinatorial auction, there is a set of items S to be sold and there is a set of agents 1 ≤ i ≤ n who wants to buy some of the items: agent i wants to buy a subset Si ⊆ S with maximum price mi. A single-minded bidder i declares a bid bi = Si, ai with Si ⊆ S and ai ∈ R+ . Two bids Si, ai and Sj, aj conflict if Si ∩ Sj = ∅. Given the bids b1, b2, · · · , bn, they gave a greedy round-based algorithm as follows. First the bids are sorted by some criterion ( ai |Si|1/2 is used in[12]) in an increasing order and let L be the list of sorted bids. The first bid is granted. Then the algorithm exams each bid of L in order and grants the bid if it does not conflict with any of the bids previously granted. If it does, it is denied. They proved that this greedy allocation scheme using criterion ai |Si|1/2 approximates the optimal allocation within a factor of √ m, where m is the number of goods in S. In the auction settings, we have ci = −ai. It is easy to verify the output of the greedy algorithm is a round-based output. Remember after bidder j is selected for round r, every bidder has conflict 221 with j will not be selected in the rounds after. This equals to update the cost of every bidder having conflict with j to 0, which satisfies crossing-independence. In addition, in any round, if bidder i is selected with ai then it will still be selected when it declares ai > ai. Thus, for every round, it satisfies MP and the cut value is |Si|1/2 · ajr |Sjr |1/2 where jr is the bidder selected in round r if we did not consider the agent i at all. Notice ajr |Sjr |1/2 does not increase when round r increases, so the final cut value is |Si|1/2 · aj |Sj |1/2 where bj is the first bid that has been denied but would have been selected were it not only for the presence of bidder i. Thus, the payment by agent i is |Si|1/2 · aj |Sj |1/2 if ai ≥ |Si|1/2 · aj |Sj |1/2 , and 0 otherwise. This payment scheme is exactly the same as the payment scheme in [12]. 6. CONCLUSIONS In this paper, we have studied how to design a truthful mechanism M = (O, P) for a given allocation rule O for a binary demand game. We first showed that the allocation rule O satisfying the MP is a necessary and sufficient condition for a truthful mechanism M to exist. We then formulate a general framework for designing payment P such that the mechanism M = (O, P) is truthful and computable in polynomial time. We further presented several general composition-based techniques to compute P efficiently for various allocation rules O. Several concrete examples were discussed to demonstrate our general framework for designing P and for composition-based techniques of computing P in polynomial time. In this paper, we have concentrated on how to compute P in polynomial time. Our algorithms do not necessarily have the optimal running time for computing P given O. It would be of interest to design algorithms to compute P in optimal time. We have made some progress in this research direction in [22] by providing an algorithm to compute the payments for unicast in a node weighted graph in optimal O(n log n + m) time. Another research direction is to design an approximation allocation rule O satisfying MP with a good approximation ratio for a given binary demand game. Many works [12, 13] in the mechanism design literature are in this direction. We point out here that the goal of this paper is not to design a better allocation rule for a problem, but to design an algorithm to compute the payments efficiently when O is given. It would be of significance to design allocation rules with good approximation ratios such that a given binary demand game has a computationally efficient payment scheme. In this paper, we have studied mechanism design for binary demand games. However, some problems cannot be directly formulated as binary demand games. The job scheduling problem in [2] is such an example. For this problem, a truthful payment scheme P exists for an allocation rule O if and only if the workload assigned by O is monotonic in a certain manner. It wound be of interest to generalize our framework for designing a truthful payment scheme for a binary demand game to non-binary demand games. Towards this research direction, Theorem 4 can be extended to a general allocation rule O, whose range is R+ . The remaining difficulty is then how to compute the payment P under mild assumptions about the valuations if a truthful mechanism M = (O, P) does exist. Acknowledgements We would like to thank Rakesh Vohra, Tuomas Sandholm, and anonymous reviewers for helpful comments and discussions. 7. REFERENCES [1] A. ARCHER, C. PAPADIMITRIOU, K. T., AND TARDOS, E. An approximate truthful mechanism for combinatorial auctions with single parameter agents. In ACM-SIAM SODA (2003), pp. 205-214. [2] ARCHER, A., AND TARDOS, E. Truthful mechanisms for one-parameter agents. In Proceedings of the 42nd IEEE FOCS (2001), IEEE Computer Society, p. 482. [3] AULETTA, V., PRISCO, R. D., PENNA, P., AND PERSIANO, P. Deterministic truthful approximation schemes for scheduling related machines. [4] CHVATAL, V. A greedy heuristic for the set covering problem. Mathematics of Operations Research 4, 3 (1979), 233-235. [5] CLARKE, E. H. Multipart pricing of public goods. Public Choice (1971), 17-33. [6] R. Muller, and R. V. Vohra. On Dominant Strategy Mechanisms. Working paper, 2003. [7] DEVANUR, N. R., MIHAIL, M., AND VAZIRANI, V. V. Strategyproof cost-sharing mechanisms for set cover and facility location games. In ACM Electronic Commerce (EC03) (2003). [8] FEIGENBAUM, J., KRISHNAMURTHY, A., SAMI, R., AND SHENKER, S. Approximation and collusion in multicast cost sharing (abstract). In ACM Economic Conference (2001). [9] FEIGENBAUM, J., PAPADIMITRIOU, C., SAMI, R., AND SHENKER, S. A BGP-based mechanism for lowest-cost routing. In Proceedings of the 2002 ACM Symposium on Principles of Distributed Computing. (2002), pp. 173-182. [10] GREEN, J., AND LAFFONT, J. J. Characterization of satisfactory mechanisms for the revelation of preferences for public goods. Econometrica (1977), 427-438. [11] GROVES, T. Incentives in teams. Econometrica (1973), 617-631. [12] LEHMANN, D., OCALLAGHAN, L. I., AND SHOHAM, Y. Truth revelation in approximately efficient combinatorial auctions. Journal of ACM 49, 5 (2002), 577-602. [13] MU"ALEM, A., AND NISAN, N. Truthful approximation mechanisms for restricted combinatorial auctions: extended abstract. In 18th National Conference on Artificial intelligence (2002), American Association for Artificial Intelligence, pp. 379-384. [14] NISAN, N., AND RONEN, A. Algorithmic mechanism design. In Proc. 31st Annual ACM STOC (1999), pp. 129-140. [15] E. Halperin. Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 329-337, 2000. [16] R. Bar-Yehuda and S. Even. A local ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics, Volume 25: Analysis and Design of Algorithms for Combinatorial Problems, pages 27-46, 1985. Editor: G. Ausiello and M. Lucertini [17] ROBINS, G., AND ZELIKOVSKY, A. Improved steiner tree approximation in graphs. In Proceedings of the 11th annual ACM-SIAM SODA (2000), pp. 770-779. [18] A. Zelikovsky. An 11/6-approximation algorithm for the network Steiner problem. Algorithmica, 9(5):463-470, 1993. [19] D. S. Hochbaum. Efficient bounds for the stable set, vertex cover, and set packing problems, Discrete Applied Mathematics, 6:243-254, 1983. [20] TAKAHASHI, H., AND MATSUYAMA, A. An approximate solution for the steiner problem in graphs. Math. Japonica 24 (1980), 573-577. [21] VICKREY, W. Counterspeculation, auctions and competitive sealed tenders. Journal of Finance (1961), 8-37. [22] WANG, W., AND LI, X.-Y. Truthful low-cost unicast in selfish wireless networks. In 4th IEEE Transactions on Mobile Computing (2005), to appear. [23] WANG, W., LI, X.-Y., AND SUN, Z. Design multicast protocols for non-cooperative networks. IEEE INFOCOM 2005, to appear. [24] WANG, W., LI, X.-Y., AND WANG, Y. Truthful multicast in selfish wireless networks. ACM MobiCom, 2005. 222

mechanism design;cut value function;objective function;monotonicity property;selfish wireless network;price;selfish agent;composition-based technique;demand game;truthful mechanism;pricing;vickrey-clarke-grove;combination;binary demand game

train_J-59

Cost Sharing in a Job Scheduling Problem Using the Shapley Value

A set of jobs need to be served by a single server which can serve only one job at a time. Jobs have processing times and incur waiting costs (linear in their waiting time). The jobs share their costs through compensation using monetary transfers. We characterize the Shapley value rule for this model using fairness axioms. Our axioms include a bound on the cost share of jobs in a group, efficiency, and some independence properties on the the cost share of a job.

1. INTRODUCTION A set of jobs need to be served by a server. The server can process only one job at a time. Each job has a finite processing time and a per unit time waiting cost. Efficient ordering of this queue directs us to serve the jobs in increasing order of the ratio of per unit time waiting cost and processing time. To compensate for waiting by jobs, monetary transfers to jobs are allowed. How should the jobs share the cost equitably amongst themselves (through transfers)? The problem of fair division of costs among agents in a queue has many practical applications. For example, computer programs are regularly scheduled on servers, data are scheduled to be transmitted over networks, jobs are scheduled in shop-floor on machines, and queues appear in many public services (post offices, banks). Study of queueing problems has attracted economists for a long time [7, 17]. Cost sharing is a fundamental problem in many settings on the Internet. Internet can be seen as a common resource shared by many users and the cost incured by using the resource needs to be shared in an equitable manner. The current surge in cost sharing literature from computer scientists validate this claim [8, 11, 12, 6, 24]. Internet has many settings in which our model of job scheduling appears and the agents waiting in a queue incur costs (jobs scheduled on servers, queries answered from a database, data scheduled to be transmitted over a fixed bandwidth network etc.). We hope that our analysis will give new insights on cost sharing problems of this nature. Recently, there has been increased interest in cost sharing methods with submodular cost functions [11, 12, 6, 24]. While many settings do have submodular cost functions (for example, multi-cast transmission games [8]), while the cost function of our game is supermodular. Also, such literature typically does not assume budget-balance (transfers adding up to zero), while it is an inherent feature of our model. A recent paper by Maniquet [15] is the closest to our model and is the motivation behind our work 1 . Maniquet [15] studies a model where he assumes all processing times are unity. For such a model, he characterizes the Shapley value rule using classical fairness axioms. Chun [1] interprets the worth of a coalition of jobs in a different manner for the same model and derives a reverse rule. Chun characterizes this rule using similar fairness axioms. Chun [2] also studies the envy properties of these rules. Moulin [22, 21] studies the queueing problem from a strategic point view when per unit waiting costs are unity. Moulin introduces new concepts in the queueing settings such as splitting and merging of jobs, and ways to prevent them. Another stream of literature is on sequencing games, first introduced by Curiel et al. [4]. For a detailed survey, refer to Curiel et al. [3]. Curiel et al. [4] defined sequencing games similar to our model, but in which an initial ordering of jobs is given. Besides, their notion of worth of a coalition is very different from the notions studied in Maniquet [15] and Chun [1] (these are the notions used in our work too). The particular notion of the worth of a coalition makes the sequencing game of Curiel et al. [4] convex, whereas our game is not convex and does not assume the presence of any initial order. In summary, the focus of this stream of 1 The authors thank Fran¸cois Maniquet for several fruitful discussions. 232 research is how to share the savings in costs from the initial ordering to the optimal ordering amongst jobs (also see Hamers et al. [9], Curiel et al. [5]). Recently, Klijn and S´anchez [13, 14] considered sequencing games without any initial ordering of jobs. They take two approaches to define the worth of coalitions. One of their approaches, called the tail game, is related to the reverse rule of Chun [1]. In the tail game, jobs in a coalition are served after the jobs not in the coalition are served. Klijn and S´anchez [14] showed that the tail game is balanced. Further, they provide expressions for the Shapley value in tail game in terms of marginal vectors and reversed marginal vectors. We provide a simpler expression of the Shapley value in the tail game, generalizing the result in Chun [1]. Klijn and S´anchez [13] study the core of this game in detail. Strategic aspects of queueing problems have also been researched. Mitra [19] studies the first best implementation in queueing models with generic cost functions. First best implementation means that there exists an efficient mechanism in which jobs in the queue have a dominant strategy to reveal their true types and their transfers add up to zero. Suijs [27] shows that if waiting costs of jobs are linear then first best implementation is possible. Mitra [19] shows that among a more general class of queueing problems first best implementation is possible if and only if the cost is linear. For another queueing model, Mitra [18] shows that first best implementation is possible if and only if the cost function satisfies a combinatorial property and an independence property. Moulin [22, 21] studies strategic concepts such as splitting and merging in queueing problems with unit per unit waiting costs. The general cost sharing literature is vast and has a long history. For a good survey, we refer to [20]. From the seminal work of Shapley [25] to recent works on cost sharing in multi-cast transmission and optimization problems [8, 6, 23] this area has attracted economists, computer scientists, and operations researchers. 1.1 Our Contribution Ours is the first model which considers cost sharing when both processing time and per unit waiting cost of jobs are present. We take a cooperative game theory approach and apply the classical Shapley value rule to the problem. We show that the Shapley value rule satisfies many intuitive fairness axioms. Due to two dimensional nature of our model and one dimensional nature of Maniquet"s model [15], his axioms are insufficient to characterize the Shapley value in our setting. We introduce axioms such as independece of preceding jobs" unit waiting cost and independence of following jobs" processing time. A key axiom that we introduce gives us a bound on cost share of a job in a group of jobs which have the same ratio of unit time waiting cost and processing time (these jobs can be ordered in any manner between themseleves in an efficient ordering). If such a group consists of just one job, then the axiom says that such a job should at least pay his own processing cost (i.e., the cost it would have incurred if it was the only job in the queue). If there are multiple jobs in such a group, the probability of any two jobs from such a group inflicting costs on each other is same (1 2 ) in an efficient ordering. Depending on the ordering selected, one job inflicts cost on the other. Our fairness axiom says that each job should at least bear such expected costs. We characterize the Shapley value rule using these fairness axioms. We also extend the envy results in [2] to our setting and discuss a class of reasonable cost sharing mechanisms. 2. THE MODEL There are n jobs that need to be served by one server which can process only one job at a time. The set of jobs are denoted as N = {1, . . . , n}. σ : N → N is an ordering of jobs in N and σi denotes the position of job i in the ordering σ. Given an ordering σ, define Fi(σ) = {j ∈ N : σi < σj} and Pi(σ) = {j ∈ N : σi > σj}. Every job i is identified by two parameters: (pi, θi). pi is the processing time and θi is the cost per unit waiting time of job i. Thus, a queueing problem is defined by a list q = (N, p, θ) ∈ Q, where Q is the set of all possible lists. We will denote γi = θi pi . Given an ordering of jobs σ, the cost incurred by job i is given by ci(σ) = piθi + θi j∈Pi(σ) pj. The total cost incurred by all jobs due to an ordering σ can be written in two ways: (i) by summing the cost incurred by every job and (ii) by summing the costs inflicted by a job on other jobs with their own processing cost. C(N, σ) = i∈N ci(σ) = i∈N piθi + i∈N ¡θi j∈Pi(σ) pj¢. = i∈N piθi + i∈N ¡pi j∈Fi(σ) θj¢. An efficient ordering σ∗ is the one which minimizes the total cost incurred by all jobs. So, C(N, σ∗ ) ≤ C(N, σ) ∀ σ ∈ Σ. To achieve notational simplicity, we will write the total cost in an efficient ordering of jobs from N as C(N) whenever it is not confusing. Sometimes, we will deal with only a subset of jobs S ⊆ N. The ordering σ will then be defined on jobs in S only and we will write the total cost from an efficient ordering of jobs in S as C(S). The following lemma shows that jobs are ordered in decreasing γ in an efficient ordering. This is also known as the weighted shortest processing time rule, first introduced by Smith [26]. Lemma 1. For any S ⊆ N, let σ∗ be an efficient ordering of jobs in S. For every i = j, i, j ∈ S, if σ∗ i > σ∗ j , then γi ≤ γj. Proof. Assume for contradiction that the statment of the lemma is not true. This means, we can find two consecutive jobs i, j ∈ S (σ∗ i = σ∗ j + 1) such that γi > γj. Define a new ordering σ by interchanging i and j in σ∗ . The costs to jobs in S \ {i, j} is not changed from σ∗ to σ. The difference between total costs in σ∗ and σ is given by, C(S, σ) − C(S, σ∗ ) = θjpi − θipj. From efficiency we get θjpi − θipj ≥ 0. This gives us γj ≥ γi, which is a contradiction. An allocation for q = (N, p, θ) ∈ Q has two components: an ordering σ and a transfer ti for every job i ∈ N. ti denotes the payment received by job i. Given a transfer ti and an ordering σ, the cost share of job i is defined as, πi = ci(σ) − ti = θi j∈N:σj ≤σi pj − ti. 233 An allocation (σ, t) is efficient for q = (N, p, θ) whenever σ is an efficient ordering and £i∈N ti = 0. The set of efficient orderings of q is denoted as Σ∗ (q) and σ∗ (q) will be used to refer to a typical element of the set. The following straightforward lemma says that for two different efficient orderings, the cost share in one efficient allocation is possible to achieve in the other by appropriately modifying the transfers. Lemma 2. Let (σ, t) be an efficient allocation and π be the vector of cost shares of jobs from this allocation. If σ∗ = σ be an efficient ordering and t∗ i = ci(σ∗ ) − πi ∀ i ∈ N, then (σ∗ , t∗ ) is also an efficient allocation. Proof. Since (σ, t) is efficient, £i∈N ti = 0. This gives £i∈N πi = C(N). Since σ∗ is an efficient ordering, £i∈N ci(σ∗ ) = C(N). This means, £i∈N t∗ i = £i∈N [ci(σ∗ ) − πi] = 0. So, (σ∗ , t∗ ) is an efficient allocation. Depending on the transfers, the cost shares in different efficient allocations may differ. An allocation rule ψ associates with every q ∈ Q a non-empty subset ψ(q) of allocations. 3. COST SHARING USING THE SHAPLEY VALUE In this section, we define the coalitional cost of this game and analyze the solution proposed by the Shapley value. Given a queue q ∈ Q, the cost of a coalition of S ⊆ N jobs in the queue is defined as the cost incurred by jobs in S if these are the only jobs served in the queue using an efficient ordering. Formally, the cost of a coalition S ⊆ N is, C(S) = i∈S j∈S:σ∗ j ≤σ∗ i θjpj, where σ∗ = σ∗ (S) is an efficient ordering considering jobs from S only. The worth of a coalition of S jobs is just −C(S). Maniquet [15] observes another equivalent way to define the worth of a coalition is using the dual function of the cost function C(·). Other interesting ways to define the worth of a coalition in such games is discussed by Chun [1], who assume that a coalition of jobs are served after the jobs not in the coalition are served. The Shapley value (or cost share) of a job i is defined as, SVi = S⊆N\{i} |S|!(|N| − |S| − 1)! |N|! ¡C(S∪{i})−C(S)¢. (1) The Shapley value allocation rule says that jobs are ordered using an efficient ordering and transfers are assigned to jobs such that the cost share of job i is given by Equation 1. Lemma 3. Let σ∗ be an efficient ordering of jobs in set N. For all i ∈ N, the Shapley value is given by, SVi = piθi + 1 2 ¡Li + Ri¢, where Li = θi £j∈Pi(σ∗) pj and Ri = pi £j∈Fi(σ∗) θj. Proof. Another way to write the Shapley value formula is the following [10], SVi = S⊆N:i∈S ∆(S) |S| , where ∆(S) = C(S) if |S| = 1 and ∆(S) = C(S)−£T S ∆(T). This gives ∆({i}) = C({i}) = piθi ∀i ∈ N. For any i, j ∈ N with i = j, we have ∆({i, j}) = C({i, j}) − C({i}) − C({j}) = min(piθi + pjθj + pjθi, piθi + pjθj + piθj) − piθi − pjθj = min(pjθi, piθj). We will show by induction that ∆(S) = 0 if |S| > 2. For |S| = 3, let S = {i, j, k}. Without loss of generality, assume θi pi ≥ θj pj ≥ θk pk . So, ∆(S) = C(S) − ∆({i, j}) − ∆({j, k}) − ∆({i, k})−∆({i})−∆({j})−∆({k}) = C(S)−piθj −pjθk − piθk − piθi − pjθj − pkθk = C(S) − C(S) = 0. Now, assume for T S, ∆(T) = 0 if |T| > 2. Without loss of generality assume that σ to be the identity mapping. Now, ∆(S) = C(S) − T S ∆(T) = C(S) − i∈S j∈S:j<i ∆({i, j}) − i∈S ∆({i}) = C(S) − i∈S j∈S:j<i pjθi − i∈S piθi = C(S) − C(S) = 0. This proves that ∆(S) = 0 if |S| > 2. Using the Shapley value formula now, SVi = S⊆N:i∈S ∆(S) |S| = ∆({i}) + 1 2 j∈N:j=i ∆({i, j}) = piθi + 1 2 ¡ j<i ∆({i, j}) + j>i ∆({i, j})¢ = piθi + 1 2 ¡ j<i pjθi + j>i piθj¢= piθi + 1 2 ¡Li + Ri¢. 4. AXIOMATICCHARACTERIZATIONOF THE SHAPLEY VALUE In this section, we will define serveral axioms on fairness and characterize the Shapley value using them. For a given q ∈ Q, we will denote ψ(q) as the set of allocations from allocation rule ψ. Also, we will denote the cost share vector associated with an allocation rule (σ, t) as π and that with allocation rule (σ , t ) as π etc. 4.1 The Fairness Axioms We will define three types of fairness axioms: (i) related to efficiency, (ii) related to equity, and (iii) related to independence. Efficiency Axioms We define two types of efficiency axioms. One related to efficiency which states that an efficient ordering should be selected and the transfers of jobs should add up to zero (budget balance). Definition 1. An allocation rule ψ satisfies efficiency if for every q ∈ Q and (σ, t) ∈ ψ(q), (σ, t) is an efficient allocation. 234 The second axiom related to efficiency says that the allocation rule should not discriminate between two allocations which are equivalent to each other in terms of cost shares of jobs. Definition 2. An allocation rule ψ satisfies Pareto indifference if for every q ∈ Q, (σ, t) ∈ ψ(q), and (σ , t ) ∈ Σ(q), we have ¡πi = πi ∀ i ∈ N¢⇒ ¡(σ , t ) ∈ ψ(q)¢. An implication of Pareto indifference axiom and Lemma 2 is that for every efficient ordering there is some set of transfers of jobs such that it is part of an efficient rule and the cost share of a job in all these allocations are same. Equity Axioms How should the cost be shared between two jobs if the jobs have some kind of similarity between them? Equity axioms provide us with fairness properties which help us answer this question. We provide five such axioms. Some of these axioms (for example anonymity, equal treatment of equals) are standard in the literature, while some are new. We start with a well known equity axiom called anonymity. Denote ρ : N → N as a permutation of elements in N. Let ρ(σ, t) denote the allocation obtained by permuting elements in σ and t according to ρ. Similarly, let ρ(p, θ) denote the new list of (p, θ) obtained by permuting elements of p and θ according to ρ. Our first equity axiom states that allocation rules should be immune to such permutation of data. Definition 3. An allocation rule ψ satisfies anonymity if for all q ∈ Q, (σ, t) ∈ ψ(q) and every permutation ρ, we then ρ(σ, t) ∈ ψ(N, ρ(q)). The next equity axiom is classical in literature and says that two similar jobs should be compensated such that their cost shares are equal. This implies that if all the jobs are of same type, then jobs should equally share the total system cost. Definition 4. An allocation rule ψ satisfies equal treatment of equals (ETE) if for all q ∈ Q, (σ, t) ∈ ψ(q), i, j ∈ N, then ¡pi = pj; θi = θj¢⇒ ¡πi = πj¢. ETE directs us to share costs equally between jobs if they are of the same per unit waiting cost and processing time. But it is silent about the cost shares of two jobs i and j which satisfy θi pi = θj pj . We introduce a new axiom for this. If an efficient rule chooses σ such that σi < σj for some i, j ∈ N, then job i is inflicting a cost of piθj on job j and job j is inflicting zero cost on job i. Define for some γ ≥ 0, S(γ) = {i ∈ N : γi = γ}. In an efficient rule, the elements in S(γ) can be ordered in any manner (in |S(γ)|! ways). If i, j ∈ S(γ) then we have pjθi = piθj. Probability of σi < σj is 1 2 and so is the probability of σi > σj. The expected cost i inflicts on j is 1 2 piθj and j inflicts on i is 1 2 pjθi. Our next fairness axiom says that i and j should each be responsible for their own processing cost and this expected cost they inflict on each other. Arguing for every pair of jobs i, j ∈ S(γ), we establish a bound on the cost share of jobs in S(γ). We impose this as an equity axiom below. Definition 5. An allocation rule satisfies expected cost bound (ECB) if for all q ∈ Q, (σ, t) ∈ ψ(q) with π being the resulting cost share, for any γ ≥ 0, and for every i ∈ S(γ), we have πi ≥ piθi + 1 2 ¡ j∈S(γ):σj <σi pjθi + j∈S(γ):σj >σi piθj¢. The central idea behind this axiom is that of expected cost inflicted. If an allocation rule chooses multiple allocations, we can assign equal probabilities of selecting one of the allocations. In that case, the expected cost inflicted by a job i on another job j in the allocation rule can be calculated. Our axiom says that the cost share of a job should be at least its own processing cost and the total expected cost it inflicts on others. Note that the above bound poses no constraints on how the costs are shared among different groups. Also observe that if S(γ) contains just one job, ECB says that job should at least bear its own processing cost. A direct consequence of ECB is the following lemma. Lemma 4. Let ψ be an efficient rule which satisfies ECB. For a q ∈ Q if S(γ) = N, then for any (σ, t) ∈ ψ(q) which gives a cost share of π, πi = piθi + 1 2 ¡Li + Ri¢∀ i ∈ N. Proof. From ECB, we get πi ≥ piθi+1 2 ¡Li+Ri¢∀ i ∈ N. Assume for contradiction that there exists j ∈ N such that πj > pjθj + 1 2 ¡Li + Ri¢. Using efficiency and the fact that £i∈N Li = £i∈N Ri, we get £i∈N πi = C(N) > £i∈N piθi + 1 2 £i∈N ¡Li + Ri¢ = C(N). This gives us a contradiction. Next, we introduce an axiom about sharing the transfer of a job between a set of jobs. In particular, if the last job quits the system, then the ordering need not change. But the transfer to the last job needs to be shared between the other jobs. This should be done in proportion to their processing times because every job influenced the last job based on its processing time. Definition 6. An allocation rule ψ satisfies proportionate responsibility of p (PRp) if for all q ∈ Q, for all (σ, t) ∈ ψ(q), k ∈ N such that σk = |N|, q = (N \ {k}, p , θ ) ∈ Q, such that for all i ∈ N\{k}: θi = θi, pi = pi, there exists (σ , t ) ∈ ψ(q ) such that for all i ∈ N \ {k}: σi = σi and ti = ti + tk pi £j=k pj . An analogous fairness axiom results if we remove the job from the beginning of the queue. Since the presence of the first job influenced each job depending on their θ values, its transfer needs to be shared in proportion to θ values. Definition 7. An allocation rule ψ satisfies proportionate responsibility of θ (PRθ) if for all q ∈ Q, for all (σ, t) ∈ ψ(q), k ∈ N such that σk = 1, q = (N \{k}, p , θ ) ∈ Q, such that for all i ∈ N \{k}: θi = θi, pi = pi, there exists (σ , t ) ∈ ψ(q ) such that for all i ∈ N \ {k}: σi = σi and ti = ti + tk θi £j=k θj . The proportionate responsibility axioms are generalizations of equal responsibility axioms introduced by Maniquet [15]. 235 Independence Axioms The waiting cost of a job does not depend on the per unit waiting cost of its preceding jobs. Similarly, the waiting cost inflicted by a job to its following jobs is independent of the processing times of the following jobs. These independence properties should be carried over to the cost sharing rules. This gives us two independence axioms. Definition 8. An allocation rule ψ satisfies independence of preceding jobs" θ (IPJθ) if for all q = (N, p, θ), q = (N, p , θ ) ∈ Q, (σ, t) ∈ ψ(q), (σ , t ) ∈ ψ(q ), if for all i ∈ N \ {k}: θi = θi, pi = pi and γk < γk, pk = pk, then for all j ∈ N such that σj > σk: πj = πj, where π is the cost share in (σ, t) and π is the cost share in (σ , t ). Definition 9. An allocation rule ψ satisfies independence of following jobs" p (IFJp) if for all q = (N, p, θ), q = (N, p , θ ) ∈ Q, (σ, t) ∈ ψ(q), (σ , t ) ∈ ψ(q ), if for all i ∈ N \ {k}: θi = θi, pi = pi and γk > γk, θk = θk, then for all j ∈ N such that σj < σk: πj = πj, where π is the cost share in (σ, t) and π is the cost share in (σ , t ). 4.2 The Characterization Results Having stated the fairness axioms, we propose three different ways to characterize the Shapley value rule using these axioms. All our characterizations involve efficiency and ECB. But if we have IPJθ, we either need IFJp or PRp. Similarly if we have IFJp, we either need IPJθ or PRθ. Proposition 1. Any efficient rule ψ that satisfies ECB, IPJθ, and IFJp is a rule implied by the Shapley value rule. Proof. Define for any i, j ∈ N, θi j = γipj and pi j = θj γi . Assume without loss of generality that σ is an efficient ordering with σi = i ∀ i ∈ N. Consider the following q = (N, p , θ ) corresponding to job i with pj = pj if j ≤ i and pj = pi j if j > i, θj = θi j if j < i and θj = θj if j ≥ i. Observe that all jobs have the same γ: γi. By Lemma 2 and efficiency, (σ, t ) ∈ ψ(q ) for some set of transfers t . Using Lemma 4, we get cost share of i from (σ, t ) as πi = piθi + 1 2 ¡Li + Ri¢. Now, for any j < i, if we change θj to θj without changing processing time, the new γ of j is γj ≥ γi. Applying IPJθ, the cost share of job i should not change. Similarly, for any job j > i, if we change pj to pj without changing θj, the new γ of j is γj ≤ γi. Applying IFJp, the cost share of job i should not change. Applying this procedure for every j < i with IPJθ and for every j > i with IFJp, we reach q = (N, p, θ) and the payoff of i does not change from πi. Using this argument for every i ∈ N and using the expression for the Shapley value in Lemma 3, we get the Shapley value rule. It is possible to replace one of the independence axioms with an equity axiom on sharing the transfer of a job. This is shown in Propositions 2 and 3. Proposition 2. Any efficient rule ψ that satisfies ECB, IPJθ, and PRp is a rule implied by the Shapley value rule. Proof. As in the proof of Proposition 1, define θi j = γipj ∀ i, j ∈ N. Assume without loss of generality that σ is an efficient ordering with σi = i ∀ i ∈ N. Consider a queue with jobs in set K = {1, . . . , i, i + 1}, where i < n. Define q = (K, p, θ ), where θj = θi+1 j ∀ j ∈ K. Define σj = σj ∀ j ∈ K. σ is an efficient ordering for q . By ECB and Lemma 4 the cost share of job i + 1 in any allocation rule in ψ must be πi+1 = pi+1θi+1 + 1 2 ¡£j<i+1 pjθi+1¢. Now, consider q = (K, p, θ ) such that θj = θi j ∀ j ≤ i and θi+1 = θi+1. σ remains an efficient ordering in q and by IPJθ the cost share of i + 1 remains πi+1. In q = (K \ {i + 1}, p, θ ), we can calculate the cost share of job i using ECB and Lemma 4 as πi = piθi + 1 2 £j<i pjθi. So, using PRp we get the new cost share of job i in q as πi = πi + ti+1 pi j<i+1 pj = piθi + 1 2 ¡£j<i pjθi + piθi+1¢. Now, we can set K = K ∪ {i + 2}. As before, we can find cost share of i + 2 in this queue as πi+2 = pi+2θi+2 + 1 2 ¡£j<i+2 pjθi+2¢. Using PRp we get the new cost share of job i in the new queue as πi = piθi + 1 2 ¡£j<i pjθi + piθi+1 + piθi+2¢. This process can be repeated till we add job n at which point cost share of i is piθi + 1 2 ¡£j<i pjθi + £j>i piθj¢. Then, we can adjust the θ of preceding jobs of i to their original value and applying IPJθ, the payoffs of jobs i through n will not change. This gives us the Shapley values of jobs i through n. Setting i = 1, we get cost shares of all the jobs from ψ as the Shapley value. Proposition 3. Any efficient rule ψ that satisfies ECB, IFJp, and PRθ is a rule implied by the Shapley value rule. Proof. The proof mirrors the proof of Proposition 2. We provide a short sketch. Analogous to the proof of Proposition 2, θs are kept equal to original data and processing times are initialized to pi+1 j . This allows us to use IFJp. Also, contrast to Proposition 2, we consider K = {i, i + 1, . . . , n} and repeatedly add jobs to the beginning of the queue maintaining the same efficient ordering. So, we add the cost components of preceding jobs to the cost share of jobs in each iteration and converge to the Shapley value rule. The next proposition shows that the Shapley value rule satisfies all the fairness axioms discussed. Proposition 4. The Shapley value rule satisfies efficiency, pareto indifference, anonymity, ETE, ECB, IPJθ, IFJp, PRp, and PRθ. Proof. The Shapley value rule chooses an efficient ordering and by definition the payments add upto zero. So, it satisfies efficiency. The Shapley value assigns same cost share to a job irrespective of the efficient ordering chosen. So, it is pareto indifferent. The Shapley value is anonymous because the particular index of a job does not effect his ordering or cost share. For ETE, consider two jobs i, j ∈ N such that pi = pj and θi = θj. Without loss of generality assume the efficient ordering to be 1, . . . , i, . . . , j, . . . , n. Now, the Shapley value of job i is 236 SVi = piθi + 1 2 ¡Li + Ri¢(From Lemma 3) = pjθj + 1 2 ¡Lj + Rj¢− 1 2 ¡Li − Lj + Ri − Rj¢ = SVj − 1 2 ¡ i<k≤j piθk − i≤k<j pkθi¢ = SVj − 1 2 i<k≤j (piθk − pkθi) (Using pi = pj and θi = θj) = SVj (Using θk pk = θi pi for all i ≤ k ≤ j). The Shapley value satisfies ECB by its expression in Lemma 3. Consider any job i, in an efficient ordering σ, if we increase the value of γj for some j = i such that σj > σi, then the set Pi ( preceding jobs) does not change in the new efficient ordering. If γj is changed such that pj remains the same, then the expression £j∈Pi θipj is unchanged. If (p, θ) values of no other jobs are changed, then the Shapley value is unchanged by increasing γj for some j ∈ Pi while keeping pj unchanged. Thus, the Shapley value rule satisfies IPJθ. An analogous argument shows that the Shapley value rule satisfies IFJp. For PRp, assume without loss of generality that jobs are ordered 1, . . . , n in an efficient ordering. Denote the transfer of job i = n due to the Shapley value with set of jobs N and set of jobs N \ {n} as ti and ti respectively. Transfer of last job is tn = 1 2 θn £j<n pj. Now, ti = 1 2 ¡θi j<i pj − pi j>i θj¢ = 1 2 ¡θi j<i pj − pi j>i:j=n θj¢− 1 2 piθn = ti − 1 2 θn j<n pj pi £j<n pj = ti − tn pi £j<n pj . A similar argument shows that the Shapley value rule satisfies PRθ. These series of propositions lead us to our main result. Theorem 1. Let ψ be an allocation rule. The following statements are equivalent: 1) For each q ∈ Q, ψ(q) selects all the allocation assigning jobs cost shares implied by the Shapley value. 2) ψ satisfies efficiency, ECB, IFJp, and IPJθ. 3) ψ satisfies efficiency, ECB, IFJp, and PRθ. 4) ψ satisfies efficiency, ECB, PRp, and IPJθ. Proof. The proof follows from Propositions 1, 2, 3, and 4. 5. DISCUSSIONS 5.1 A Reasonable Class of Cost Sharing Mechanisms In this section, we will define a reasonable class of cost sharing mechanisms. We will show how these reasonable mechanisms lead to the Shapley value mechanism. Definition 10. An allocation rule ψ is reasonable if for all q ∈ Q and (σ, t) ∈ ψ(q) we have for all i ∈ N, ti = α ¡θi j∈Pi(σ) pj − pi j∈Fi(σ) θj¢∀ i ∈ N, where 0 ≤ α ≤ 1. The reasonable cost sharing mechanism says that every job should be paid a constant fraction of the difference between the waiting cost he incurs and the waiting cost he inflicts on other jobs. If α = 0, then every job bears its own cost. If α = 1, then every job gets compensated for its waiting cost but compensates others for the cost he inflicts on others. The Shapley value rule comes as a result of ETE as shown in the following proposition. Proposition 5. Any efficient and reasonable allocation rule ψ that satisfies ETE is a rule implied by the Shapley value rule. Proof. Consider a q ∈ Q in which pi = pj and θi = θj. Let (σ, t) ∈ ψ(q) and π be the resulting cost shares. From ETE, we get, πi = πj ⇒ ci(σ) − ti = cj(σ) − tj ⇒ piθi + (1 − α)Li + αRi = pjθj + (1 − α)Lj + αRj (Since ψ is efficient and reasonable) ⇒ (1 − α)(Li − Lj) = α(Rj − Ri) (Using pi = pj, θi = θj) ⇒ 1 − α = α (Using Li − Lj = Rj − Ri = 0) ⇒ α = 1 2 . This gives us the Shapley value rule by Lemma 3. 5.2 Results on Envy Chun [2] discusses a fariness condition called no-envy for the case when processing times of all jobs are unity. Definition 11. An allocation rule satisfies no-envy if for all q ∈ Q, (σ, t) ∈ ψ(q), and i, j ∈ N, we have πi ≤ ci(σij ) − tj, where π is the cost share from allocation rule (σ, t) and σij is the ordering obtaining by swapping i and j. From the result in [2], the Shapley value rule does not satisfy no-envy in our model also. To overcome this, Chun [2] introduces the notion of adjusted no-envy, which he shows is satisfied in the Shapley value rule when processing times of all jobs are unity. Here, we show that adjusted envy continues to hold in the Shapley value rule in our model (when processing times need not be unity). As before denote σij be an ordering where the position of i and j is swapped from an ordering σ. For adjusted noenvy, if (σ, t) is an allocation for some q ∈ Q, let tij be the 237 transfer of job i when the transfer of i is calculated with respect to ordering σij . Observe that an allocation may not allow for calculation of tij . For example, if ψ is efficient, then tij cannot be calculated if σij is also not efficient. For simplicity, we state the definition of adjusted no-envy to apply to all such rules. Definition 12. An allocation rule satisfies adjusted noenvy if for all q ∈ Q, (σ, t) ∈ ψ(q), and i, j ∈ N, we have πi ≤ ci(σij ) − tij i . Proposition 6. The Shapley value rule satisfies adjusted no-envy. Proof. Without loss of generality, assume efficient ordering of jobs is: 1, . . . , n. Consider two jobs i and i + k. From Lemma 3, SVi = piθi + 1 2 ¡ j<i θipj + j>i θjpi¢. Let ˆπi be the cost share of i due to adjusted transfer tii+k i in the ordering σii+k . ˆπi = ci(σii+k ) − tii+k i = piθi + 1 2 ¡ j<i θipj + θipi+k + i<j<i+k θipj + j>i θjpi − θi+kpi − i<j<i+k θjpi¢ = SVi + 1 2 i<j≤i+k ¡θipj − θjpi¢ ≥ SVi (Using the fact that θi pi ≥ θj pj for i < j). 6. CONCLUSION We studied the problem of sharing costs for a job scheduling problem on a single server, when jobs have processing times and unit time waiting costs. We took a cooperative game theory approach and show that the famous the Shapley value rule satisfies many nice fairness properties. We characterized the Shapley value rule using different intuitive fairness axioms. In future, we plan to further simplify some of the fairness axioms. Some initial simplifications already appear in [16], where we provide an alternative axiom to ECB and also discuss the implication of transfers between jobs (in stead of transfers from jobs to a central server). We also plan to look at cost sharing mechanisms other than the Shapley value. Investigating the strategic power of jobs in such mechanisms is another line of future research. 7. REFERENCES [1] Youngsub Chun. A Note on Maniquet"s Characterization of the Shapley Value in Queueing Problems. Working Paper, Rochester University, 2004. [2] Youngsub Chun. No-envy in Queuing Problems. Working Paper, Rochester University, 2004. [3] Imma Curiel, Herbert Hamers, and Flip Klijn. Sequencing Games: A Survey. In Peter Borm and Hans Peters, editors, Chapter in Game Theory. Theory and Decision Library, Kulwer Academic Publishers, 2002. [4] Imma Curiel, Giorgio Pederzoli, and Stef Tijs. Sequencing Games. European Journal of Operational Research, 40:344-351, 1989. [5] Imma Curiel, Jos Potters, Rajendra Prasad, Stef Tijs, and Bart Veltman. Sequencing and Cooperation. Operations Research, 42(3):566-568, May-June 1994. [6] Nikhil R. Devanur, Milena Mihail, and Vijay V. Vazirani. Strategyproof Cost-sharing Mechanisms for Set Cover and Facility Location Games. In Proceedings of Fourth Annual ACM Conferece on Electronic Commerce, 2003. [7] Robert J. Dolan. Incentive Mechanisms for Priority Queueing Problems. Bell Journal of Economics, 9:421-436, 1978. [8] Joan Feigenbaum, Christos Papadimitriou, and Scott Shenker. Sharing the Cost of Multicast Transmissions. In Proceedings of Thirty-Second Annual ACM Symposium on Theory of Computing, 2000. [9] Herbert Hamers, Jeroen Suijs, Stef Tijs, and Peter Borm. The Split Core for Sequencing Games. Games and Economic Behavior, 15:165-176, 1996. [10] John C. Harsanyi. Contributions to Theory of Games IV, chapter A Bargaining Model for Cooperative n-person Games. Princeton University Press, 1959. Editors: A. W. Tucker, R. D. Luce. [11] Kamal Jain and Vijay Vazirani. Applications of Approximate Algorithms to Cooperative Games. In Proceedings of 33rd Symposium on Theory of Computing (STOC "01), 2001. [12] Kamal Jain and Vijay Vazirani. Equitable Cost Allocations via Primal-Dual Type Algorithms. In Proceedings of 34th Symposium on Theory of Computing (STOC "02), 2002. [13] Flip Klijn and Estela S´anchez. Sequencing Games without a Completely Specified Initial Order. Report in Statistics and Operations Research, pages 1-17, 2002. Report 02-04. [14] Flip Klijn and Estela S´anchez. Sequencing Games without Initial Order. Working Paper, Universitat Aut´onoma de Barcelona, July 2004. [15] Franois Maniquet. A Characterization of the Shapley Value in Queueing Problems. Journal of Economic Theory, 109:90-103, 2003. [16] Debasis Mishra and Bharath Rangarajan. Cost sharing in a job scheduling problem. Working Paper, CORE, 2005. [17] Manipushpak Mitra. Essays on First Best Implementable Incentive Problems. Ph.D. Thesis, Indian Statistical Institute, New Delhi, 2000. [18] Manipushpak Mitra. Mechanism design in queueing problems. Economic Theory, 17(2):277-305, 2001. [19] Manipushpak Mitra. Achieving the first best in sequencing problems. Review of Economic Design, 7:75-91, 2002. [20] Herv´e Moulin. Handbook of Social Choice and Welfare, chapter Axiomatic Cost and Surplus Sharing. North-Holland, 2002. Publishers: Arrow, Sen, Suzumura. [21] Herv´e Moulin. On Scheduling Fees to Prevent 238 Merging, Splitting and Transferring of Jobs. Working Paper, Rice University, 2004. [22] Herv´e Moulin. Split-proof Probabilistic Scheduling. Working Paper, Rice University, 2004. [23] Herv´e Moulin and Rakesh Vohra. Characterization of Additive Cost Sharing Methods. Economic Letters, 80:399-407, 2003. [24] Martin P´al and ´Eva Tardos. Group Strategyproof Mechanisms via Primal-Dual Algorithms. In Proceedings of the 44th Annual IEEE Symposium on the Foundations of Computer Science (FOCS "03), 2003. [25] Lloyd S. Shapley. Contributions to the Theory of Games II, chapter A Value for n-person Games, pages 307-317. Annals of Mathematics Studies, 1953. Ediors: H. W. Kuhn, A. W. Tucker. [26] Wayne E. Smith. Various Optimizers for Single-Stage Production. Naval Research Logistics Quarterly, 3:59-66, 1956. [27] Jeroen Suijs. On incentive compatibility and budget balancedness in public decision making. Economic Design, 2, 2002. 239

job scheduling;cooperative game theory approach;unit waiting cost;processing time;expected cost bound;agent;monetary transfer;queueing problem;shapley value;job schedule;queue problem;fairness axiom;cost sharing;allocation rule;cost share

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On Decentralized Incentive Compatible Mechanisms for Partially Informed Environments

Algorithmic Mechanism Design focuses on Dominant Strategy Implementations. The main positive results are the celebrated Vickrey-Clarke-Groves (VCG) mechanisms and computationally efficient mechanisms for severely restricted players (single-parameter domains). As it turns out, many natural social goals cannot be implemented using the dominant strategy concept [35, 32, 22, 20]. This suggests that the standard requirements must be relaxed in order to construct general-purpose mechanisms. We observe that in many common distributed environments computational entities can take advantage of the network structure to collect and distribute information. We thus suggest a notion of partially informed environments. Even if the information is recorded with some probability, this enables us to implement a wider range of social goals, using the concept of iterative elimination of weakly dominated strategies. As a result, cooperation is achieved independent of agents" belief. As a case study, we apply our methods to derive Peer-to-Peer network mechanism for file sharing.

1. INTRODUCTION Recently, global networks have attracted widespread study. The emergence of popular scalable shared networks with self-interested entities - such as peer-to-peer systems over the Internet and mobile wireless communication ad-hoc networks - poses fundamental challenges. Naturally, the study of such giant decentralized systems involves aspects of game theory [32, 34]. In particular, the subfield of Mechanism Design deals with the construction of mechanisms: for a given social goal the challenge is to design rules for interaction such that selfish behavior of the agents will result in the desired social goal [23, 33]. Algorithmic Mechanism Design (AMD) focuses on efficiently computable constructions [32]. Distributed Algorithmic Mechanism Design (DAMD) studies mechanism design in inherently decentralized settings [30, 12]. The standard model assumes rational agents with quasi-linear utilities and private information, playing dominant strategies. The solution concept of dominant strategies - in which each player has a best response strategy regardless of the strategy played by any other player - is well suited to the assumption of private information, in which each player is not assumed to have knowledge or beliefs regarding the other players. The appropriateness of this set-up stems from the strength of the solution concept, which complements the weak information assumption. Many mechanisms have been constructed using this set-up, e.g., [1, 4, 6, 11, 14, 22]. Most of these apply to severely-restricted cases (e.g., single-item auctions with no externalities) in which a player"s preference is described by only one parameter (single-parameter domains). To date, Vickrey-Clarke-Groves (VCG) mechanisms are the only known general method for designing dominant strategy mechanisms for general domains of preferences. However, in distributed settings without available subsidies from outside sources, VCG mechanisms cannot be accepted as valid solutions due to a serious lack of budget balance. Additionally, for some domains of preferences, VCG mechanisms and weighted VCG mechanisms are faced with computational hardness [22, 20]. Further limitations of the set-up are discussed in subsection 1.3. In most distributed environments, players can take advantage of the network structure to collect and distribute information about other players. This paper thus studies the effects of relaxing the private information assumption. 240 One model that has been extensively studied recently is the Peer-to-Peer (P2P) network. A P2P network is a distributed network with no centralized authority, in which the participants share their individual resources (e.g., processing power, storage capacity, bandwidth and content). The aggregation of such resources provides inexpensive computational platforms. The most popular P2P networks are those for sharing media files, such as Napster, Gnutella, and Kazaa. Recent work on P2P Incentives include micropayment methods [15] and reputation-based methods [9, 13]. The following description of a P2P network scenario illustrates the relevance of our relaxed informational assumption. Example 1. Consider a Peer-to-Peer network for file sharing. Whenever agent B uploads a file from agent A, all peers along the routing path know that B has loaded the file. They can record this information about agent B. In addition, they can distribute this information. However, it is impossible to record all the information everywhere. First, such duplication induces huge costs. Second, as agents dynamically enter and exit from the network, the information might not be always available. And so it is seems natural to consider environments in which the information is locally recorded, that is, the information is recorded in the closest neighborhood with some probability p. In this paper we shall see that if the information is available with some probability, then this enables us to implement a wider range of social goals. As a result, cooperation is achieved independent of agents" belief. This demonstrates that in some computational contexts our approach is far less demanding than the Bayesian approach (that assumes that players" types are drawn according to some identified probability density function). 1.1 Implementations in Complete Information Set-ups In complete information environments, each agent is informed about everyone else. That is, each agent observes his own preference and the preferences of all other agents. However, no outsider can observe this information. Specifically, neither the mechanism designer nor the court. Many positive results were shown for such arguably realistic settings. For recent surveys see [25, 27, 18]. Moore and Repullo implement a large class of social goals using sequential mechanisms with a small number of rounds [28]. The concept they used is subgame-perfect implementations (SPE). The SPE-implementability concept seems natural for the following reasons: the designed mechanisms usually have non-artificial constructs and a small strategy space. As a result, it is straightforward for a player to compute his strategy.1 Second, sequential mechanisms avoid simultaneous moves, and thus can be considered for distributed networks. Third, the constructed mechanisms are often decentralized (i.e., lacking a centralized authority or designer) 1 Interestingly, in real life players do not always use their subgame perfect strategies. One such widely studied case is the Ultimatum Bargaining 2-person game. In this simple game, the proposer first makes an offer of how to divide a certain known sum of money, and the responder either agrees or refuses, in the latter case both players earn zero. Somewhat surprisingly, experiments show that the responder often rejects the suggested offer, even if it is bounded away from zero and the game is played only once (see e.g. [38]). and budget-balanced (i.e., transfers always sum up to zero). This happens essentially if there are at least three players, and a direct network link between any two agents. Finally, Moore and Repullo observed that they actually use a relaxed complete information assumption: it is only required that for every player there exists only one other player who is informed about him. 1.2 Implementations in Partially Informed Set-ups and Our Results The complete information assumption is realistic for small groups of players, but not in general. In this paper we consider players that are informed about each other with some probability. More formally, we say that agent B is p-informed about agent A, if B knows the type of A with probability p. For such partially-informed environments, we show how to use the solution concept of iterative elimination of weakly dominated strategies. We demonstrate this concept through some motivating examples that (i) seem natural in distributed settings and (ii) cannot be implemented in dominant strategies even if there is an authorized center with a direct connection to every agent or even if players have single-parameter domains. 1. We first show how the subgame perfect techniques of Moore and Repullo [28] can be applied to p-informed environments and further adjusted to the concept of iterative elimination of weakly dominated strategies (for large enough p). 2. We then suggest a certificate based challenging method that is more natural in computerized p-informed environments and different from the one introduced by Moore and Repullo [28] (for p ∈ (0, 1]). 3. We consider implementations in various network structures. As a case study we apply our methods to derive: (1) Simplified Peer-to-Peer network for file sharing with no payments in equilibrium. Our approach is (agent, file)-specific. (2) Web-cache budget-balanced and economically efficient mechanism. Our mechanisms use reasonable punishments that inversely depend on p. And so, if the fines are large then small p is enough to induce cooperation. Essentially, large p implies a large amount of recorded information. 1.2.1 Malicious Agents Decentralized mechanisms often utilize punishing outcomes. As a result, malicious players might cause severe harm to others. We suggest a quantified notion of malicious player, who benefits from his own gained surplus and from harm caused to others. [12] suggests several categories to classify non-cooperating players. Our approach is similar to [7] (and the references therein), who considered independently such players in different context. We show a simple decentralized mechanism in which q-malicious players cooperate and in particular, do not use their punishing actions in equilibrium. 241 1.3 Dominant Strategy Implementations In this subsection we shall refer to some recent results demonstrating that the set-up of private information with the concept of dominant strategies is restrictive in general. First, Roberts" classical impossibility result shows that if players" preferences are not restricted and there are at least 3 different outcomes, then every dominant-strategy mechanism must be weighted VCG (with the social goal that maximizes the weighted welfare) [35]. For slightly-restricted preference domains, it is not known how to turn efficiently computable algorithms into dominant strategy mechanisms. This was observed and analyzed in [32, 22, 31]. Recently [20] extends Roberts" result to some leading examples. They showed that under mild assumptions any dominant strategy mechanism for variety of Combinatorial Auctions over multi-dimensional domains must be almost weighted VCG. Additionally, it turns out that the dominant strategy requirement implies that the social goal must be monotone [35, 36, 22, 20, 5, 37]. This condition is very restrictive, as many desired natural goals are non-monotone2 . Several recent papers consider relaxations of the dominant strategy concept: [32, 1, 2, 19, 16, 17, 26, 21]. However, most of these positive results either apply to severely restricted cases (e.g., single-parameter, 2 players) or amount to VCG or almost VCG mechanisms (e.g., [19]). Recently, [8, 3] considered implementations for generalized single-parameter players. Organization of this paper: In section 2 we illustrate the concepts of subgame perfect and iterative elimination of weakly dominated strategies in completely-informed and partially-informed environments. In section 3 we show a mechanism for Peer-to-Peer file sharing networks. In section 4 we apply our methods to derive a web cache mechanism. Future work is briefly discussed in section 5. 2. MOTIVATING EXAMPLES In this section we examine the concepts of subgame perfect and iterative elimination of weakly dominated strategies for completely informed and p-informed environments. We also present the notion of q-maliciousness and some other related considerations through two illustrative examples. 2.1 The Fair Assignment Problem Our first example is an adjustment to computerized context of an ancient procedure to ensure that the wealthiest man in Athens would sponsor a theatrical production known as the Choregia [27]. In the fair assignment problem, Alice and Bob are two workers, and there is a new task to be performed. Their goal is to assign the task to the least loaded worker without any monetary transfers. The informational assumption is that Alice and Bob know both loads and the duration of the new task.3 2 E.g., minimizing the makespan within a factor of 2 [32] and Rawls" Rule over some multi-dimensional domains [20]. 3 In first glance one might ask why the completely informed agents could not simply sign a contract, specifying the desired goal. Such a contract is sometimes infeasible due to fact that the true state cannot be observed by outsiders, especially not the court. Claim 1. The fair assignment goal cannot be implemented in dominant strategies.4 2.1.1 Basic Mechanism The following simple mechanism implements this goal in subgame perfect equilibrium. • Stage 1: Alice either agrees to perform the new task or refuses. • Stage 2: If she refuses, Bob has to choose between: - (a) Performing the task himself. - (b) Exchanging his load with Alice and performing the new task as well. Let LT A, LT B be the true loads of Alice and Bob, and let t > 0 be the load of the new task. Assume that load exchanging takes zero time and cost. We shall see that the basic mechanism achieves the goal in a subgame perfect equilibrium. Intuitively this means that in equilibrium each player will choose his best action at each point he might reach, assuming similar behavior of others, and thus every SPE is a Nash equilibrium. Claim 2. ([27]) The task is assigned to the least loaded worker in subgame perfect equilibrium. Proof. By backward induction argument (look forward and reason backward), consider the following cases: 1. LT B ≤ LT A. If stage 2 is reached then Bob will not exchange. 2. LT A < LT B < LT A + t. If stage 2 is reached Bob will exchange, and this is what Alice prefers. 3. LT A + t ≤ LT B. If stage 2 is reached then Bob would exchange, as a result it is strictly preferable by Alice to perform the task. Note that the basic mechanism does not use monetary transfers at all and is decentralized in the sense that no third party is needed to run the procedure. The goal is achieved in equilibrium (ties are broken in favor of Alice). However, in the second case exchange do occur in an equilibrium point. Recall the unrealistic assumption that load exchange takes zero time and cost. Introducing fines, the next mechanism overcomes this drawback. 2.1.2 Elicitation Mechanism In this subsection we shall see a centralized mechanism for the fair assignment goal without load exchange in equilibrium. The additional assumptions are as follows. The cost performing a load of duration d is exactly d. We assume that the duration t of the new task is < T. The payoffs 4 proof: Assume that there exists a mechanism that implements this goal in dominant strategies. Then by the Revelation Principle [23] there exists a mechanism that implements this goal for which the dominant strategy of each player is to report his true load. Clearly, truthfully reporting cannot be a dominant strategy for this goal (if monetary transfers are not available), as players would prefer to report higher loads. 242 of the utility maximizers agents are quasilinear. The following mechanism is an adaptation of Moore and Repullo"s elicitation mechanism [28]5 . • Stage 1: (Elicitation of Alice"s load) Alice announces LA. Bob announces LA ≤ LA. If LA = LA (Bob agrees) goto the next Stage. Otherwise (Bob challenges), Alice is assigned the task. She then has to choose between: - (a) Transferring her original load to Bob and paying him LA − 0.5 · min{ , LA − LA}. Alice pays to the mechanism. Bob pays the fine of T + to the mechanism. - (b) No load transfer. Alice pays to Bob. STOP. • Stage 2: The elicitation of Bob"s load is similar to Stage 1 (switching the roles of Alice and Bob). • Stage 3: If LA < LB Alice is assigned the task, otherwise Bob. STOP. Observe that Alice is assigned the task and fined with whenever Bob challenges. We shall see that the bonus of is paid to a challenging player only in out of equilibria cases. Claim 3. If the mechanism stops at Stage 3, then the payoff of each agent is at least −t and at most 0. Proposition 1. It is a subgame perfect equilibrium of the elicitation mechanism to report the true load, and to challenge with the true load only if the other agent overreports. Proof. Assume w.l.o.g that the elicitation of Alice"s load is done after Bob"s, and that Stage 2 is reached. If Alice truly reports LA = LT A, Bob strictly prefers to agree. Otherwise, if Bob challenges, Alice would always strictly prefer to transfer (as in this case Bob would perform her load for smaller cost), as a result Bob would pay T + to the mechanism. This punishing outcome is less preferable than the normal outcome of Stage 3 achieved had he agreed. If Alice misreports LA > LT A, then Bob can ensure himself the bonus (which is always strictly preferable than reaching Stage 3) by challenging with LA = LT A, and so whenever Bob gets the bonus Alice gains the worst of all payoffs. Reporting a lower load LA < LT A is not beneficial for Alice. In this case, Bob would strictly prefer to agree (and not to announce LA < LA, as he limited to challenge with a smaller load than what she announces). Thus such misreporting can only increase the possibility that she is assigned the task. And so there is no incentive for Alice to do so. All together, Alice would prefer to report the truth in this stage. And so Stage 2 would not abnormally end by STOP, and similarly Stage 1. Observe that the elicitation mechanism is almost balanced: in all outcomes no money comes in or out, except for the non-equilibrium outcome (a), in which both players pay to the mechanism. 5 In [28], if an agent misreport his type then it is always beneficial to the other agent to challenge. In particular, even if the agent reports a lower load. 2.1.3 Elicitation Mechanism for Partially Informed Agents In this subsection we consider partially informed agents. Formally: Definition 1. An agent A is p-informed about agent B, if A knows the type of B with probability p (independently of what B knows). It turns out that a version of the elicitation mechanism works for this relaxed information assumption, if we use the concept of iterative elimination of weakly dominated strategies6 . We replace the fixed fine of in the elicitation mechanism with the fine: βp = max{L, 1 − p 2p − 1 T} + , and assume the bounds LT A, LT B ≤ L. Proposition 2. If all agents are p-informed, p > 0.5, the elicitation mechanism(βp) implements the fair assignment goal with the concept of iterative elimination of weakly dominated strategies. The strategy of each player is to report the true load and to challenge with the true load if the other agent overreport. Proof. Assume w.l.o.g that the elicitation of Alice"s load is done after Bob"s, and that Stage 2 is reached. First observe that underreporting the true value is a dominated strategy, whether Bob is not informed and mistakenly challenges with a lower load (as βp ≥ L) or not, or even if t is very small. Now we shall see that overreporting her value is a dominated strategy, as well. Alice"s expected payoff gained by misreporting ≤ p (payoff if she lies and Bob is informed) +(1 − p) (max payoff if Bob is not informed) ≤ p (−t − βp) < p (−t) + (1 − p) (−t − βp) ≤ p (min payoff of true report if Bob is informed) + (1 − p) (min payoff if Bob is not informed) ≤ Alice"s expected payoff if she truly reports. The term (−t−βp) in the left hand side is due to the fact that if Bob is informed he will always prefer to challenge. In the right hand side, if he is informed, then challenging is a dominated strategy, and if he is not informed the worst harm he can make is to challenge. Thus in stage 2 Alice will report her true load. This implies that challenging without being informed is a dominated strategy for Bob. This argument can be reasoned also for the first stage, when Bob reports his value. Bob knows the maximum payoff he can gain is at most zero since he cannot expect to get the bonus in the next stage. 2.1.4 Extensions The elicitation mechanism for partially informed agents is rather general. As in [28], we need the capability to judge between two distinct declarations in the elicitation rounds, 6 A strategy si of player i is weakly dominated if there exists si such that (i) the payoff gained by si is at least as high as the payoff gained by si, for all strategies of the other players and all preferences, and (ii) there exist a preference and a combination of strategies for the other players such that the payoff gained by si is strictly higher than the payoff gained by si. 243 and upper and lower bounds based on the possible payoffs derived from the last stage. In addition, for p-informed environments, some structure is needed to ensure that underbidding is a dominated strategy. The Choregia-type mechanisms can be applied to more than 2 players with the same number of stages: the player in the first stage can simply points out the name of the wealthiest agent. Similarly, the elicitation mechanisms can be extended in a straightforward manner. These mechanisms can be budget-balanced, as some player might replace the role of the designer, and collect the fines, as observed in [28]. Open Problem 1. Design a decentralized budget balanced mechanism with reasonable fines for independently p-informed n players, where p ≤ 1 − 1/2 1 n−1 . 2.2 Seller and Buyer Scenario A player might cause severe harm to others by choosing a non-equilibrium outcome. In the mechanism for the fair assignment goal, an agent might maliciously challenge even if the other agent truly reports his load. In this subsection we consider such malicious scenarios. For the ease of exposition we present a second example. We demonstrate that equilibria remain unchanged even if players are malicious. In the seller-buyer example there is one item to be traded and two possible future states. The goal is to sell the item for the average low price pl = ls+lb 2 in state L, and the higher price ph = hs+hb 2 in the other state H, where ls is seller"s cost and lb is buyer"s value in state L, and similarly hs, hb in H. The players fix the prices without knowing what will be the future state. Assume that ls < hs < lb < hb, and that trade can occur in both prices (that is, pl, ph ∈ (hs, lb)). Only the players can observe the realization of the true state. The payoffs are of the form ub = xv−tb, us = ts −xvs, where the binary variable x indicates if trade occurred, and tb, ts are the transfers. Consider the following decentralized trade mechanism. • Stage 1: If seller reports H goto Stage 2. Otherwise, trade at the low price pl. STOP. • Stage 2: The buyer has to choose between: - (a) Trade at the high price ph. - (b) No trade and seller pays ∆ to the buyer. Claim 4. Let ∆ = lb−ph+ . The unique subgame perfect equilibrium of the trade mechanism is to report the true state in Stage 1 and trading if Stage 2 is reached. Note that the outcome (b) is never chosen in equilibrium. 2.2.1 Trade Mechanism for Malicious Agents The buyer might maliciously punish the seller by choosing the outcome (b) when the true state is H. The following notion quantifies the consideration that a player is not indifferent to the private surpluses of others. Definition 2. A player is q-malicious if his payoff equals: (1 − q) (his private surplus) − q (summation of others surpluses), q ∈ [0, 1]. This definition appeared independently in [7] in different context. We shall see that the traders would avoid such bad behavior if they are q-malicious, where q < 0.5, that is if their non-indifference impact is bounded by 0.5. Equilibria outcomes remain unchanged, and so cooperation is achieved as in the original case of non-malicious players. Consider the trade mechanism with pl = (1 − q) hs + q lb , ph = q hs + (1 − q) lb , ∆ = (1 − q) (hb − lb − ). Note that pl < ph for q < 0.5. Claim 5. If q < 0.5, then the unique subgame perfect equilibrium for q-malicious players remains unchanged. Proof. By backward induction we consider two cases. In state H, the q-malicious buyer would prefer to trade if (1 − q)(hb − ph) + q(hs − ph) > (1 − q)∆ + q(∆). Indeed, (1 − q)hb + qhs > ∆ + ph. Trivially, the seller prefers to trade at the higher price, (1 − q)(pl − hs) + q(pl − hb) < (1 − q)(ph − hs) + q(ph − hb). In state L the buyer prefers the no trade outcome, as (1−q)(lb −ph)+q(ls −ph) < ∆. The seller prefers to trade at a low price, as (1 − q)(pl − ls) + q(pl − lb) > 0 > −∆. 2.2.2 Discussion No mechanism can Nash-implement this trading goal if the only possible outcomes are trade at pl and trade at ph. To see this, it is enough to consider normal forms (as any extensive form mechanism can be presented as a normal one). Consider a matrix representation, where the seller is the row player and the buyer is the column player, in which every entry includes an outcome. Suppose there is equilibrium entry for the state L. The associate column must be all pl, otherwise the seller would have an incentive to deviate. Similarly, the associate row of the H equilibrium entry must be all ph (otherwise the buyer would deviate), a contradiction. 7 8 The buyer prefers pl and seller ph, and so the preferences are identical in both states. Hence reporting preferences over outcomes is not enough - players must supply additional information. This is captured by outcome (b) in the trade mechanism. Intuitively, if a goal is not Nash-implementable we need to add more outcomes. The drawback is that some new additional equilibria must be ruled out. E.g., additional Nash equilibrium for the trade mechanism is (trade at pl, (b)). That is, the seller chooses to trade at low price at either states, and the buyer always chooses the no trade option that fines the seller, if the second stage is reached. Such buyer"s threat is not credible, because if the mechanism is played only once, and Stage 2 is reached in state H, the buyer would strictly decrease his payoff if he chooses (b). Clearly, this is not a subgame perfect equilibrium. Although each extensive game-form is strategically equivalent to a normal form one, the extensive form representation places more structure and so it seems plausible that the subgame perfect equilibrium will be played.9 7 Formally, this goal is not Maskin monotonic, a necessary condition for Nash-implementability [24]. 8 A similar argument applies for the Fair Assignment Problem. 9 Interestingly, it is a straight forward to construct a sequential mechanism with unique SPE, and additional NE with a strictly larger payoff for every player. 244 3. PEER-TO-PEER NETWORKS In this section we describe a simplified Peer-to-Peer network for file sharing, without payments in equilibrium, using a certificate-based challenging method. In this challenging method - as opposed to [28] - an agent that challenges cannot harm other agents, unless he provides a valid certificate. In general, if agent B copied a file f from agent A, then agent A knows that agent B holds a copy of the file. We denote such information as a certificate(B, f) (we shall omit cryptographic details). Such a certificate can be recorded and distributed along the network, and so we can treat each agent holding the certificate as an informed agent. Assumptions: We assume an homogeneous system with files of equal size. The benefit each agent gains by holding a copy of any file is V . The only cost each agent has is the uploading cost C (induced while transferring a file to an immediate neighbor). All other costs are negligible (e.g., storing the certificates, forwarding messages, providing acknowledgements, digital signatures, etc). Let upA, downA be the numbers of agent A uploads and downloads if he always cooperates. We assume that each agent A enters the system if upA · C < downA · V . Each agent has a quasilinear utility and only cares about his current bandwidth usage. In particular, he ignores future scenarios (e.g., whether forwarding or dropping of a packet might affect future demand). 3.1 Basic Mechanism We start with a mechanism for a network with 3 p-informed agents: B, A1, A2. We assume that B is directly connected to A1 and A2. If B has the certificate(A1, f), then he can apply directly to A1 and request the file (if he refuses, then B can go to court). The following basic sequential mechanism is applicable whenever agent B is not informed and still would like to download the file if it exists in the network. Note that this goal cannot be implemented in dominant strategies without payments (similar to Claim 1, when the type of each agent here is the set of files he holds). Define tA,B to be the monetary amount that agent A should transfer to B. • Stage 1: Agent B requests the file f from A1. - If A1 replies yes then B downloads the file from A1. STOP. - Otherwise, agent B sends A1s no reply to agent A2. ∗ If A2 declares agree then goto the next stage. ∗ Else, A2 sends a certificate(A1, f) to agent B. · If the certificate is correct then tA1,A2 = βp. STOP. · Else tA2,A1 = |C| + . STOP. Stage 2: Agent B requests the file f from A2. Switch the roles of the agents A1, A2. Claim 6. The basic mechanism is budget-balanced (transfers always sum to zero) and decentralized. Theorem 1. Let βp = |C| p + , p ∈ (0, 1]. A strategy that survives iterative elimination of weakly dominated strategies is to reply yes if Ai holds the file, and to challenge only with a valid certificate. As a result, B downloads the file if some agent holds it, in equilibrium. There are no payments or transfers in equilibrium. Proof. Clearly if the mechanism ends without challenging: −C ≤ u(Ai) ≤ 0. And so, challenging with an invalid certificate is always a dominated strategy. Now, when Stage 2 is reached, A2 is the last to report if he has the file. If A2 has the file it is a weakly undominated strategy to misreport, whether A1 is informed or not: A2"s expected payoff gained by misreporting no ≤ p · (−βp) + (1 − p) · 0 < −C ≤ A2"s payoff if she reports yes. This argument can be reasoned also for Stage 1, when A1 reports whether he has the file. A1 knows that A2 will report yes if and only if she has the file in the next stage, and so the maximum payoff he can gain is at most zero since he cannot expect to get a bonus. 3.2 Chain Networks In a chain network, agent B is directly connected to A1, and Ai is directly connected to agent Ai+1. Assume that we have an acknowledgment protocol to confirm the receipt of a particular message. To avoid message dropping, we add the fine (βp +2 ) to be paid by an agent who hasn"t properly forwarded a message. The chain mechanism follows: • Stage i: Agent B forwards a request for the file f to Ai (through {Ak}k≤i). • If Ai reports yes, then B downloads f from Ai. STOP. • Otherwise Ai reports no. If Aj sends a certificate(Ak, f) to B, ( j, k ≤ i), then - If certificate(Ak, f) is correct, then t(Ak, Aj) = βp. STOP. - Else, t(Aj, Ak) = C + . STOP. If Ai reports that he has no copy of the file, then any agent in between might challenge. Using digital signatures and acknowledgements, observe that every agent must forward each message, even if it contains a certificate showing that he himself has misreported. We use the same fine, βp, as in the basic mechanism, because the protocol might end at stage 1 (clearly, the former analysis still applies, since the actual p increases with the number of players). 3.3 Network Mechanism In this subsection we consider general network structures. We need the assumption that there is a ping protocol that checks whether a neighbor agent is on-line or not (that is, an on-line agent cannot hide himself). To limit the amount of information to be recorded, we assume that an agent is committed to keep any downloaded file to at least one hour, and so certificates are valid for a limited amount of time. We assume that each agent has a digitally signed listing of his current immediate neighbors. As in real P2P file sharing applications, we restrict each request for a file to be forwarded at most r times (that is, downloads are possible only inside a neighborhood of radius r). 245 The network mechanism utilizes the chain mechanism in the following way: When agent B requests a file from agent A (at most r − 1 far), then A sends to B the list of his neighbors and the output of the ping protocol to all of these neighbors. As a result, B can explore the network. Remark: In this mechanism we assumed that the environment is p-informed. An important design issue that it is not addressed here is the incentives for the information propagation phase. 4. WEB CACHE Web caches are widely used tool to improve overall system efficiency by allowing fast local access. They were listed in [12] as a challenging application of Distributed Algorithmic Mechanism Design. Nisan [30] considered a single cache shared by strategic agents. In this problem, agent i gains the value vT i if a particular item is loaded to the local shared cache. The efficient goal is to load the item if and only if ΣvT i ≥ C, where C is the loading cost. This goal reduces to the public project problem analyzed by Clarke [10]. However, it is well known that this mechanism is not budget-balanced (e.g., if the valuation of each player is C, then everyone pays zero). In this section we suggest informational and environmental assumptions for which we describe a decentralized budgetbalanced efficient mechanism. We consider environments for which future demand of each agent depends on past demand. The underlying informational and environmental requirements are as follows. 1. An agent can read the content of a message only if he is the target node (even if he has to forward the message as an intermediate node of some routing path). An agent cannot initiate a message on behalf of other agents. 2. An acknowledgement protocol is available, so that every agent can provide a certificate indicating that he handled a certain message properly. 3. Negligible costs: we assume p-informed agents, where p is such that the agent"s induced cost for keeping records of information is negligible. We also assume that the cost incurred by sending and forwarding messages is negligible. 4. Let qi(t) denotes the number of loading requests agent i initiated for the item during the time slot t. We assume that vT i (t), the value for caching the item in the beginning of slot t depends only on most recent slot, formally vT i (t) = max{Vi(qi(t − 1)), C}, where Vi(·) is a non-decreasing real function. In addition, Vi(·) is a common knowledge among the players. 5. The network is homogeneous in the sense that if agent j happens to handle k requests initiated by agent i during the time slot t, then qi(t) = kα, where α depends on the routing protocol and the environment (α might be smaller than 1, if each request is flooded several times). We assume that the only way agent i can affect the true qi(t) is by superficially increasing his demand for the cached item, but not the other way (that is, agent"s loss, incurred by giving up a necessary request for the item, is not negligible). The first requirement is to avoid free riding, and also to avoid the case that an agent superficially increases the demand of others and as a result decreases his own demand. The second requirement is to avoid the case that an agent who gets a routing request for the item, records it and then drops it. The third is to ensure that the environment stays well informed. In addition, if the forwarding cost is negligible each agent cooperates and forwards messages as he would not like to decrease the future demand (that monotonically depends on the current time slot, as assumed in the forth requirement) of some other agent. Given that the payments are increasing with the declared values, the forth and fifth requirements ensure that the agent would not increase his demand superficially and so qi(t) is the true demand. The following Web-Cache Mechanism implements the efficient goal that shares the cost proportionally. For simplicity it is described for two players and w.l.o.g vT i (t) equals the number of requests initiated by i and observed by any informed j (that is, α = 1 and Vi(qi(t − 1)) = qi(t − 1)). • Stage 1: (Elicitation of vT A(t)) Alice announces vA. Bob announces vA ≥ vA. If vA = vA goto the next Stage. Otherwise (Bob challenges): - If Bob provides vA valid records then Alice pays C to finance the loading of the item into the cache. She also pays βp to Bob. STOP. - Otherwise, Bob finances the loading of the item into the cache. STOP. • Stage 2: The elicitation of vT B(t) is done analogously. • Stage 3: If vA + vB < C, then STOP. Otherwise, load the item to the cache, Alice pays pA = vA vA+vB · C, and Bob pays pB = vB vA+vB · C. Claim 7. It is a dominated strategy to overreport the true value. Proof. Let vT A < VA. There are two cases to consider: • If vT A + vB < C and vA + vB ≥ C. We need to show that if the mechanism stops normally Alice would pay more than vT A, that is: vA vA+vB ·C > vT A. Indeed, vA C > vA (vT A + vB) > vT A (vA + vB). • If vT A + vB ≥ C, then clearly, vA vA+vB > vT A vT A +vB . Theorem 2. Let βp = max{0, 1−2p p · C} + , p ∈ (0, 1]. A strategy that survives iterative elimination of weakly dominated strategies is to report the truth and to challenge only when the agent is informed. The mechanism is efficient, budget-balanced, exhibits consumer sovereignty, no positive transfer and individual rationality10 . Proof. Challenging without being informed (that is, without providing enough valid records) is always dominated strategy in this mechanism. Now, assume w.l.o.g. Alice is 10 See [29] or [12] for exact definitions. 246 the last to report her value. Alice"s expected payoff gained by underreporting ≤ p · (−C − βp) + (1 − p) · C < p · 0 + (1 − p) · 0 ≤ Alice"s expected payoff if she honestly reports. The right hand side equals zero as the participation costs are negligible. Reasoning back, Bob cannot expect to get the bonus and so misreporting is dominated strategy for him. 5. CONCLUDING REMARKS In this paper we have seen a new partial informational assumption, and we have demonstrated its suitability to networks in which computational agents can easily collect and distribute information. We then described some mechanisms using the concept of iterative elimination of weakly dominated strategies. Some issues for future work include: • As we have seen, the implementation issue in p-informed environments is straightforward - it is easy to construct incentive compatible mechanisms even for non-singleparameter cases. The challenge is to find more realistic scenarios in which the partial informational assumption is applicable. • Mechanisms for information propagation and maintenance. In our examples we choose p such that the maintenance cost over time is negligible. However, the dynamics of the general case is delicate: an agent can use the recorded information to eliminate data that is not likely to be needed, in order to decrease his maintenance costs. As a result, the probability that the environment is informed decreases, and selfish agents would not cooperate. 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In Jean-Jacques Laffont, editor, Aggregation and Revelation of Preferences. Papers presented at the 1st European Summer Workshop of the Econometric Society, pages 321-349. North-Holland, 1979. [36] Irit Rozenshtrom. Dominant strategy implementation with quasi-linear preferences, 1999. Master"s thesis, Dept. of Economics, The Hebrew University, Jerusalem, Israel. [37] Rakesh Vohra and Rudolf Muller. On dominant strategy mechanisms, 2003. Working paper. [38] Shmuel Zamir. Rationality and emotions in ultimatum bargaining. Annales D" Economie et De Statistique, 61, 2001. 248

iterative elimination of weakly dominated strategy;cooperation;distribute algorithmic mechanism design;p-informed environment;partially informed environment;weakly dominated strategy iterative elimination;agent;dominant strategy implementation;decentralized incentive compatible mechanism;distributed algorithmic mechanism design;peer-to-peer;vickrey-clarke-grove;distributed environment;computational entity

train_J-61

ICE: An Iterative Combinatorial Exchange

We present the first design for an iterative combinatorial exchange (ICE). The exchange incorporates a tree-based bidding language that is concise and expressive for CEs. Bidders specify lower and upper bounds on their value for different trades. These bounds allow price discovery and useful preference elicitation in early rounds, and allow termination with an efficient trade despite partial information on bidder valuations. All computation in the exchange is carefully optimized to exploit the structure of the bid-trees and to avoid enumerating trades. A proxied interpretation of a revealedpreference activity rule ensures progress across rounds. A VCG-based payment scheme that has been shown to mitigate opportunities for bargaining and strategic behavior is used to determine final payments. The exchange is fully implemented and in a validation phase.

1. INTRODUCTION Combinatorial exchanges combine and generalize two different mechanisms: double auctions and combinatorial auctions. In a double auction (DA), multiple buyers and sellers trade units of an identical good [20]. In a combinatorial auction (CA), a single seller has multiple heterogeneous items up for sale [11]. Buyers may have complementarities or substitutabilities between goods, and are provided with an expressive bidding language. A common goal in both market designs is to determine the efficient allocation, which is the allocation that maximizes total value. A combinatorial exchange (CE) [24] is a combinatorial double auction that brings together multiple buyers and sellers to trade multiple heterogeneous goods. For example, in an exchange for wireless spectrum, a bidder may declare that she is willing to pay $1 million for a trade where she obtains licenses for New York City, Boston, and Philadelphia, and loses her license for Washington DC. Thus, unlike a DA, a CE allows all participants to express complex valuations via expressive bids. Unlike a CA, a CE allows for fragmented ownership, with multiple buyers and sellers and agents that are both buying and selling. CEs have received recent attention both in the context of wireless spectrum allocation [18] and for airport takeoff and landing slot allocation [3]. In both of these domains there are incumbents with property rights, and it is important to facilitate a complex multi-way reallocation of resources. Another potential application domain for CEs is to resource allocation in shared distributed systems, such as PlanetLab [13]. The instantiation of our general purpose design to specific domains is a compelling next step in our research. This paper presents the first design for an iterative combinatorial exchange (ICE). The genesis of this project was a class, CS 286r Topics at the Interface between Economics and Computer Science, taught at Harvard University in Spring 2004.1 The entire class was dedicated to the design and prototyping of an iterative CE. The ICE design problem is multi-faceted and quite hard. The main innovation in our design is an expressive yet concise tree-based bidding language (which generalizes known languages such as XOR/OR [23]), and the tight coupling of this language with efficient algorithms for price-feedback to guide bidding, winner-determination to determine trades, and revealed-preference activity rules to ensure progress across rounds. The exchange is iterative: bidders express upper and lower valuations on trades by annotating their bid-tree, and then tighten these bounds in response to price feedback in each round. The Threshold payment rule, introduced by Parkes et al. [24], is used to determine final payments. The exchange has a number of interesting theoretical properties. For instance, when there exist linear prices we establish soundness and completeness: for straightforward bidders that adjust their bounds to meet activity rules while keeping their true value within the bounds, the exchange will terminate with the efficient allocation. In addition, the 1 http://www.eecs.harvard.edu/∼parkes/cs286r/ice.html 249 Truth Agent Act Rule WD ACC FAIR BALClosing RuleVickreyThreshold DONE ! DONE 2,2 +A +10 +B +10 BUYER 2,2 -A -5 -B -5 SELLER 2,2 +A +15 +8 +B +15 +8 BUYER 2,2 -A -2 -6 -B -2 -6 SELLER BUYER, buy AB SELLER, sell AB 12 < PA+PB < 16 PA+PB=14 PA=PB=7 PBUYER = 16 - (4-0) = 12 PSELLER = -12 - (4-0) = -16 PBUYER = 14 PSELLER = -14 Pessim istic O ptim istic = 1 Figure 1: ICE System Flow of Control efficient allocation can often be determined without bidders revealing, or even knowing, their exact value for all trades. This is essential in complex domains where the valuation problem can itself be very challenging for a participant [28]. While we cannot claim that straightforward bidding is an equilibrium of the exchange (and indeed, should not expect to by the Myerson-Satterthwaite impossibility theorem [22]), the Threshold payment rule minimizes the ex post incentive to manipulate across all budget-balanced payment rules. The exchange is implemented in Java and is currently in validation. In describing the exchange we will first provide an overview of the main components and introduce several working examples. Then, we introduce the basic components for a simple one-shot variation in which bidders state their exact values for trades in a single round. We then describe the full iterative exchange, with upper and lower values, price-feedback, activity rules, and termination conditions. We state some theoretical properties of the exchange, and end with a discussion to motivate our main design decisions, and suggest some next steps. 2. AN OVERVIEW OF THE ICE DESIGN The design has four main components, which we will introduce in order through the rest of the paper: • Expressive and concise tree-based bidding language. The language describes values for trades, such as my value for selling AB and buying C is $100, or my value for selling ABC is -$50, with negative values indicating that a bidder must receive a payment for the trade to be acceptable. The language allows bidders to express upper and lower bounds on value, which can be tightened across rounds. • Winner Determination. Winner-determination (WD) is formulated as a mixed-integer program (MIP), with the structure of the bid-trees captured explicitly in the formulation. Comparing the solution at upper and lower values allows for a determination to be made about termination, with progress in intermediate rounds driven by an intermediate valuation and the lower values adopted on termination. • Payments. Payments are computed using the Threshold payment rule [24], with the intermediate valuations adopted in early rounds and lower values adopted on termination. • Price feedback. An approximate price is computed for each item in the exchange in each round, in terms of the intermediate valuations and the provisional trade. The prices are optimized to approximate competitive equilibrium prices, and further optimized to best approximate the current Threshold payments with remaining ties broken to favor prices that are balanced across different items. In computing the prices, we adopt the methods of constraint-generation to exploit the structure of the bidding language and avoid enumerating all feasible trades. The subproblem to generate new constraints is a variation of the WD problem. • Activity rule. A revealed-preference activity rule [1] ensures progress across rounds. In order to remain active, a bidder must tighten bounds so that there is enough information to define a trade that maximizes surplus at the current prices. Another variation on the WD problem is formulated, both to verify that the activity rule is met and also to provide feedback to a bidder to explain how to meet the rule. An outline of the ICE system flow of control is provided in Figure 1. We will return to this example later in the paper. For now, just observe in this two-agent example that the agents state lower and upper bounds that are checked in the activity rule, and then passed to winner-determination (WD), and then through three stages of pricing (accuracy, fairness, balance). On passing the closing rule (in which parameters αeff and αthresh are checked for convergence of the trade and payments), the exchange goes to a last-and-final round. At the end of this round, the trade and payments are finally determined, based on the lower valuations. 2.1 Related Work Many ascending-price one-sided CAs are known in the literature [10, 25, 29]. Direct elicitation approaches have also been proposed for one-sided CAs in which agents respond to explicit queries about their valuations [8, 14, 19]. A number of ascending CAs are designed to work with simple prices on items [12, 17]. The price generation methods that we use in ICE generalize the methods in these earlier papers. Parkes et al. [24] studied sealed-bid combinatorial exchanges and introduced the Threshold payment rule. Subsequently, Krych [16] demonstrated experimentally that the Threshold rule promotes efficient allocations. We are not aware of any previous studies of iterative CEs. Dominant strategy DAs are known for unit demand [20] and also for single-minded agents [2]. No dominant strategy mechanisms are known for the general CE problem. ICE is a hybrid auction design, in that it couples simple item prices to drive bidding in early rounds with combinatorial WD and payments, a feature it shares with the clock-proxy design of Ausubel et al. [1] for one-sided CAs. We adopt a variation on the clock-proxy auctions"s revealedpreference activity rule. The bidding language shares some structural elements with the LGB language of Boutilier and Hoos [7], but has very different semantics. Rothkopf et al. [27] also describe a restricted tree-based bidding language. In LGB, the semantics are those of propositional logic, with the same items in an allocation able to satisfy a tree in multiple places. Although this can make LGB especially concise in some settings, the semantics that we propose appear to provide useful locality, so that the value of one component in a tree can be understood independently from the rest of the tree. The idea of capturing the structure of our bidding language explicitly within a mixed-integer programming formulation follows the developments in Boutilier [6]. 3. PRELIMINARIES In our model, we consider a set of goods, indexed {1, . . . , m} and a set of bidders, indexed {1, . . . , n}. The initial allocation of goods is denoted x0 = (x0 1, . . . , x0 n), with x0 i = (x0 i1, . . . , x0 im) and x0 ij ≥ 0 for good j indicating the number 250 of units of good j held by bidder i. A trade λ = (λ1, . . . , λn) denotes the change in allocation, with λi = (λi1, . . . , λim) where λij ∈ is the change in the number of units of item j to bidder i. So, the final allocation is x1 = x0 + λ. Each bidder has a value vi(λi) ∈ ¡ for a trade λi. This value can be positive or negative, and represents the change in value between the final allocation x0 i +λi and the initial allocation x0 i . Utility is quasi-linear, with ui(λi, p) = vi(λi)−p for trade λi and payment p ∈ ¡ . Price p can be negative, indicating the bidder receives a payment for the trade. We use the term payoff interchangeably with utility. Our goal in the ICE design is to implement the efficient trade. The efficient trade, λ∗ , maximizes the total increase in value across bidders. Definition 1 (Efficient trade). The efficient trade λ∗ solves max (λ1,...,λn) ¢ i vi(λi) s.t. λij + x0 ij ≥ 0, ∀i, ∀j (1) ¢ i λij ≤ 0, ∀j (2) λij ∈ (3) Constraints (1) ensure that no agent sells more items than it has in its initial allocation. Constraints (2) provide free disposal, and allows feasible trades to sell more items than are purchased (but not vice versa). Later, we adopt Feas(x0 ) to denote the set of feasible trades, given these constraints and given an initial allocation x0 = (x0 1, . . . , x0 n). 3.1 Working Examples In this section, we provide three simple examples of instances that we will use to illustrate various components of the exchange. All three examples have only one seller, but this is purely illustrative. Example 1. One seller and one buyer, two goods {A, B}, with the seller having an initial allocation of AB. Changes in values for trades: seller buyer AND(−A, −B) AND(+A, +B) -10 +20 The AND indicates that both the buyer and the seller are only interested in trading both goods as a bundle. Here, the efficient (value-maximizing) trade is for the seller to sell AB to the buyer, denoted λ∗ = ([−1, −1], [+1, +1]). Example 2. One seller and four buyers, four goods {A, B, C, D}, with the seller having an initial allocation of ABCD. Changes in values for trades: seller buyer1 buyer 2 buyer 3 buyer 4 OR(−A, −B, AND(+A, XOR(+A, AND(+C, XOR(+C, −C, −D) +B) +B) +D) +D) 0 +6 +4 +3 +2 The OR indicates that the seller is willing to sell any number of goods. The XOR indicates that buyers 2 and 4 are willing to buy at most one of the two goods in which they are interested. The efficient trade is for bundle AB to go to buyer 1 and bundle CD to buyer 3, denoted λ∗ = ([−1, −1, −1, −1], [+1, +1, 0, 0], [0, 0, 0, 0], [0, 0, +1, +1], [0, 0, 0, 0]). 2,2 +A +10 +B +10 BUYER 2,2 -A -5 -B -5 SELLER Example 1: Example 3: 2,2 +C +D BUYER 2 2,2 +A +B BUYER 1 +11 +84,4 -B SELLER -A -C -D Example 2: 1,1 +A +B BUYER 2 2,2 +A +B BUYER 1 +6 +40,4 -B SELLER -C -D-A 1,1 +C +D BUYER 4 2,2 +C +D +3 +2 BUYER 3 -18 Figure 2: Example Bid Trees. Example 3. One seller and two buyers, four goods {A, B, C, D}, with the seller having an initial allocation of ABCD. Changes in values for trades: seller buyer1 buyer 2 AND(−A, −B, −C, −D) AND(+A, +B) AND(+C, +D) -18 +11 +8 The efficient trade is for bundle AB to go to buyer 1 and bundle CD to go to buyer 2, denoted λ∗ = ([−1, −1, −1, −1], [+1, +1, 0, 0], [0, 0, +1, +1]). 4. A ONE-SHOT EXCHANGE DESIGN The description of ICE is broken down into two sections: one-shot (sealed-bid) and iterative. In this section we abstract away the iterative aspect and introduce a specialization of the tree-based language that supports only exact values on nodes. 4.1 Tree-Based Bidding Language The bidding language is designed to be expressive and concise, entirely symmetric with respect to buyers and sellers, and to extend to capture bids from mixed buyers and sellers, ranging from simple swaps to highly complex trades. Bids are expressed as annotated bid trees, and define a bidder"s value for all possible trades. The language defines changes in values on trades, with leaves annotated with traded items and nodes annotated with changes in values (either positive or negative). The main feature is that it has a general interval-choose logical operator on internal nodes, and that it defines careful semantics for propagating values within the tree. We illustrate the language on each of Examples 1-3 in Figure 2. The language has a tree structure, with trades on items defined on leaves and values annotated on nodes and leaves. The nodes have zero values where no value is indicated. Internal nodes are also labeled with interval-choose (IC) ranges. Given a trade, the semantics of the language define which nodes in the tree can be satisfied, or switched-on. First, if a child is on then its parent must be on. Second, if a parent node is on, then the number of children that are on must be within the IC range on the parent node. Finally, leaves in which the bidder is buying items can only be on if the items are provided in the trade. For instance, in Example 2 we can consider the efficient trade, and observe that in this trade all nodes in the trees of buyers 1 and 3 (and also the seller), but none of the nodes in the trees of buyers 2 and 4, can be on. On the other hand, in 251 the trade in which A goes to buyer 2 and D to buyer 4, then the root and appropriate leaf nodes can be on for buyers 2 and 4, but no nodes can be on for buyers 1 and 3. Given a trade there is often a number of ways to choose the set of satisfied nodes. The semantics of the language require that the nodes that maximize the summed value across satisfied nodes be activated. Consider bid tree Ti from bidder i. This defines nodes β ∈ Ti, of which some are leaves, Leaf (i) ⊆ Ti. Let Child(β) ⊆ Ti denote the children of a node β (that is not itself a leaf). All nodes except leaves are labeled with the interval-choose operator [IC x i (β), ICy i (β)]. Every node is also labeled with a value, viβ ∈ ¡ . Each leaf β is labeled with a trade, qiβ ∈ m (i.e., leaves can define a bundled trade on more than one type of item.) Given a trade λi to bidder i, the interval-choose operators and trades on leaves define which nodes can be satisfied. There will often be a choice. Ties are broken to maximize value. Let satiβ ∈ {0, 1} denote whether node β is satisfied. Solution sati is valid given tree Ti and trade λi, written sati ∈ valid(Ti, λi), if and only if: ¢ β∈Leaf (i) qiβj · satiβ ≤ λij , ∀i, ∀j (4) ICx i (β)satiβ ≤ ¢ β ∈Child(β) satiβ ≤ ICy i (β)satiβ, ∀β /∈ Leaf (i) (5) In words, a set of leaves can only be considered satisfied given trade λi if the total increase in quantity summed across all such leaves is covered by the trade, for all goods (Eq. 4). This works for sellers as well as buyers: for sellers a trade is negative and this requires that the total number of items indicated sold in the tree is at least the total number sold as defined in the trade. We also need upwards-propagation: any time a node other than the root is satisfied then its parent must be satisfied (by β ∈Child(β) satiβ ≤ ICy i (β)satiβ in Eq. 5). Finally, we need downwards-propagation: any time an internal node is satisfied then the appropriate number of children must also be satisfied (Eq. 5). The total value of trade λi, given bid-tree Ti, is defined as: vi(Ti, λi) = max sat∈valid(Ti,λi) ¢ β∈T vβ · satβ (6) The tree-based language generalizes existing languages. For instance: IC(2, 2) on a node with 2 children is equivalent to an AND operator; IC(1, 3) on a node with 3 children is equivalent to an OR operator; and IC(1, 1) on a node with 2 children is equivalent to an XOR operator. Similarly, the XOR/OR bidding languages can be directly expressed as a bid tree in our language.2 4.2 Winner Determination This section defines the winner determination problem, which is formulated as a MIP and solved in our implementation with a commercial solver.3 The solver uses branchand-bound search with dynamic cut generation and branching heuristics to solve large MIPs in economically feasible run times. 2 The OR* language is the OR language with dummy items to provide additional structure. OR* is known to be expressive and concise. However, it is not known whether OR* dominates XOR/OR in terms of conciseness [23]. 3 CPLEX, www.ilog.com In defining the MIP representation we are careful to avoid an XOR-based enumeration of all bundles. A variation on the WD problem is reused many times within the exchange, e.g. for column generation in pricing and for checking revealed preference. Given bid trees T = (T1, . . . , Tn) and initial allocation x0 , the mixed-integer formulation for WD is: WD(T, x0 ) : max λ,sat ¢ i ¢ β∈Ti viβ · satiβ s.t. (1), (2), satiβ ∈ {0, 1}, λij ∈ sati ∈ valid(Ti, λi), ∀i Some goods may go unassigned because free disposal is allowed within the clearing rules of winner determination. These items can be allocated back to agents that sold the items, i.e. for which λij < 0. 4.3 Computing Threshold Payments The Threshold payment rule is based on the payments in the Vickrey-Clarke-Groves (VCG) mechanism [15], which itself is truthful and efficient but does not satisfy budget balance. Budget-balance requires that the total payments to the exchange are equal to the total payments made by the exchange. In VCG, the payment paid by agent i is pvcg,i = ˆv(λ∗ i ) − (V ∗ − V−i) (7) where λ∗ is the efficient trade, V ∗ is the reported value of this trade, and V−i is the reported value of the efficient trade that would be implemented without bidder i. We call ∆vcg,i = V ∗ − V−i the VCG discount. For instance, in Example 1 pvcg,seller = −10 − (+10 − 0) = −20 and pvcg,buyer = +20 − (+10 − 0) = 10, and the exchange would run at a budget deficit of −20 + 10 = −10. The Threshold payment rule [24] determines budgetbalanced payments to minimize the maximal error across all agents to the VCG outcome. Definition 2. The Threshold payment scheme implements the efficient trade λ∗ given bids, and sets payments pthresh,i = ˆvi(λ∗ i ) − ∆i, where ∆ = (∆1, . . . , ∆n) is set to minimize maxi(∆vcg,i − ∆i) subject to ∆i ≤ ∆vcg,i and i ∆i ≤ V ∗ (this gives budget-balance). Example 4. In Example 2, the VCG discounts are (9, 2, 0, 1, 0) to the seller and four buyers respectively, VCG payments are (−9, 4, 0, 2, 0) and the exchange runs at a deficit of -3. In Threshold, the discounts are (8, 1, 0, 0, 0) and the payments are (−8, 5, 0, 3, 0). This minimizes the worst-case error to VCG discounts across all budget-balanced payment schemes. Threshold payments are designed to minimize the maximal ex post incentive to manipulate. Krych [16] confirmed that Threshold promotes allocative efficiency in restricted and approximate Bayes-Nash equilibrium. 5. THE ICE DESIGN We are now ready to introduce the iterative combinatorial exchange (ICE) design. Several new components are introduced, relative to the design for the one-shot exchange. Rather than provide precise valuations, bidders can provide lower and upper valuations and revise this bid information across rounds. The exchange provides price-based feedback 252 to guide bidders in this process, and terminates with an efficient (or approximately-efficient) trade with respect to reported valuations. In each round t ∈ {0, 1, . . .} the current lower and upper bounds, vt and vt , are used to define a provisional valuation profile vα (the α-valuation), together with a provisional trade λt and provisional prices pt = (pt 1, . . . , pt m) on items. The α-valuation is a linear combination of the current upper and lower valuations, with αEFF ∈ [0, 1] chosen endogenously based on the closeness of the optimistic trade (at v) and the pessimistic trade (at v). Prices pt are used to inform an activity rule, and drive progress towards an efficient trade. 5.1 Upper and Lower Valuations The bidding language is extended to allow a bidder i to report a lower and upper value (viβ, viβ) on each node. These take the place of the exact value viβ defined in Section 4.1. Based on these labels, we can define the valuation functions vi(Ti, λi) and vi(Ti, λi), using the exact same semantics as in Eq. (6). We say that such a bid-tree is well-formed if viβ ≤ viβ for all nodes. The following lemma is useful: Lemma 1. Given a well-formed tree, T, then vi(Ti, λi) ≤ vi(Ti, λi) for all trades. Proof. Suppose there is some λi for which vi(Ti, λi) > vi(Ti, λi). Then, maxsat∈valid(Ti,λi) β∈Ti viβ · satβ > maxsat∈valid(Ti,λi) β∈Ti viβ · satβ. But, this is a contradiction because the trade λ that defines vi(Ti, λi) is still feasible with upper bounds vi, and viβ ≥ viβ for all nodes β in a well-formed tree. 5.2 Price Feedback In each round, approximate competitive-equilibrium (CE) prices, pt = (pt 1, . . . , pt m), are determined. Given these provisional prices, the price on trade λi for bidder i is pt (λi) = j≤m pt j · λij. Definition 3 (CE prices). Prices p∗ are competitive equilibrium prices if the efficient trade λ∗ is supported at prices p∗ , so that for each bidder: λ∗ i ∈ arg max λ∈Feas(x0) {vi(λi) − p∗ (λi)} (8) CE prices will not always exist and we will often need to compute approximate prices [5]. We extend ideas due to Rassenti et al. [26], Kwasnica et al. [17] and Dunford et al. [12], and select approximate prices as follows: I: Accuracy. First, we compute prices that minimize the maximal error in the best-response constraints across all bidders. II: Fairness. Second, we break ties to prefer prices that minimize the maximal deviation from Threshold payments across all bidders. III: Balance. Third, we break ties to prefer prices that minimize the maximal price across all items. Taken together, these steps are designed to promote the informativeness of the prices in driving progress across rounds. In computing prices, we explain how to compute approximate (or otherwise) prices for structured bidding languages, and without enumerating all possible trades. For this, we adopt constraint generation to efficient handle an exponential number of constraints. Each step is described in detail below. I: Accuracy. We adopt a definition of price accuracy that generalizes the notions adopted in previous papers for unstructured bidding languages. Let λt denote the current provisional trade and suppose the provisional valuation is vα . To compute accurate CE prices, we consider: min p,δ δ (9) s.t. vα i (λ) − p(λ) ≤ vα i (λt i) − p(λt i) + δ, ∀i, ∀λ (10) δ ≥ 0,pj ≥ 0, ∀j. This linear program (LP) is designed to find prices that minimize the worst-case error across all agents. From the definition of CE prices, it follows that CE prices would have δ = 0 as a solution to (9), at which point trade λt i would be in the best-response set of every agent (with λt i = ∅, i.e. no trade, for all agents with no surplus for trade at the prices.) Example 5. We can illustrate the formulation (9) on Example 2, assuming for simplicity that vα = v (i.e. truth). The efficient trade allocates AB to buyer 1 and CD to buyer 3. Accuracy will seek prices p(A), p(B), p(C) and p(D) to minimize the δ ≥ 0 required to satisfy constraints: p(A) + p(B) + p(C) + p(D) ≥ 0 (seller) p(A) + p(B) ≤ 6 + δ (buyer 1) p(A) + δ ≥ 4, p(B) + δ ≥ 4 (buyer 2) p(C) + p(D) ≤ 3 (buyer 3) p(C) + δ ≥ 2, p(D) + δ ≥ 2 (buyer 4) An optimal solution requires p(A) = p(B) = 10/3, with δ = 2/3, with p(C) and p(D) taking values such as p(C) = p(D) = 3/2. But, (9) has an exponential number of constraints (Eq. 10). Rather than solve it explicitly we use constraint generation [4] and dynamically generate a sufficient subset of constraints. Let i denote a manageable subset of all possible feasible trades to bidder i. Then, a relaxed version of (9) (written ACC) is formulated by substituting (10) with vα i (λ) − p(λ) ≤ vα i (λt i) − p(λt i) + δ, ∀i, ∀λ ∈ i , (11) where i is a set of trades that are feasible for bidder i given the other bids. Fixing the prices p∗ , we then solve n subproblems (one for each bidder), max λ vα i (λi) − p∗ (λi) [R-WD(i)] s.t. λ ∈ Feas(x0 ), (12) to check whether solution (p∗ , δ∗ ) to ACC is feasible in problem (9). In R-WD(i) the objective is to determine a most preferred trade for each bidder at these prices. Let ˆλi denote the solution to R-WD(i). Check condition: vα i (ˆλi) − p∗ (ˆλ) ≤ vα i (λt i) − p∗ (λt i) + δ∗ , (13) and if this condition holds for all bidders i, then solution (p∗ , δ∗ ) is optimal for problem (9). Otherwise, trade ˆλi is added to i for all bidders i for which this constraint is 253 violated and we re-solve the LP with the new set of constraints.4 II: Fairness. Second, we break remaining ties to prefer fair prices: choosing prices that minimize the worst-case error with respect to Threshold payoffs (i.e. utility to bidders with Threshold payments), but without choosing prices that are less accurate.5 Example 6. For example, accuracy in Example 1 (depicted in Figure 1) requires 12 ≤ pA +pB ≤ 16 (for vα = v). At these valuations the Threshold payoffs would be 2 to both the seller and the buyer. This can be exactly achieved in pricing with pA + pB = 14. The fairness tie-breaking method is formulated as the following LP: min p,π π [FAIR] s.t. vα i (λ) − p(λ) ≤ vα i (λt i) − p(λt i) + δ∗ i , ∀i, ∀λ ∈ i (14) π ≥ πvcg,i − (vα i (λt i) − p(λt i)), ∀i (15) π ≥ 0,pj ≥ 0, ∀j, where δ∗ represents the error in the optimal solution, from ACC. The objective here is the same as in the Threshold payment rule (see Section 4.3): minimize the maximal error between bidder payoff (at vα ) for the provisional trade and the VCG payoff (at vα ). Problem FAIR is also solved through constraint generation, using R-WD(i) to add additional violated constraints as necessary. III: Balance. Third, we break remaining ties to prefer balanced prices: choosing prices that minimize the maximal price across all items. Returning again to Example 1, depicted in Figure 1, we see that accuracy and fairness require p(A) + p(B) = 14. Finally, balance sets p(A) = p(B) = 7. Balance is justified when, all else being equal, items are more likely to have similar than dissimilar values.6 The LP for balance is formulated as follows: min p,Y Y [BAL] s.t. vα i (λ) − p(λ) ≤ vα i (λt i) − p(λt i) + δ∗ i , ∀i, ∀λ ∈ i (16) π∗ i ≥ πvcg,i − (vα i (λt i) − p(λt i)), ∀i, (17) Y ≥ pj, ∀j (18) Y ≥ 0, pj ≥ 0, ∀j, where δ∗ represents the error in the optimal solution from ACC and π∗ represents the error in the optimal solution from FAIR. Constraint generation is also used to solve BAL, generating new trades for i as necessary. 4 Problem R-WD(i) is a specialization of the WD problem, in which the objective is to maximize the payoff of a single bidder, rather than the total value across all bidders. It is solved as a MIP, by rewriting the objective in WD(T, x0 ) as max{viβ · satiβ − j p∗ j · λij } for agent i. Thus, the structure of the bid-tree language is exploited in generating new constraints, because this is solved as a concise MIP. The other bidders are kept around in the MIP (but do not appear in the objective), and are used to define the space of feasible trades. 5 The methods of Dunford et al. [12], that use a nucleolus approach, are also closely related. 6 The use of balance was advocated by Kwasnica et al. [17]. Dunford et al. [12] prefer to smooth prices across rounds. Comment 1: Lexicographical Refinement. For all three sub-problems we also perform lexicographical refinement (with respect to bidders in ACC and FAIR, and with respect to goods in BAL). For instance, in ACC we successively minimize the maximal error across all bidders. Given an initial solution we first pin down the error on all bidders for whom a constraint (11) is binding. For such a bidder i, the constraint is replaced with vα i (λ) − p(λ) ≤ vα i (λt i) − p(λt i) + δ∗ i , ∀λ ∈ i , (19) and the error to bidder i no longer appears explicitly in the objective. ACC is then re-solved, and makes progress by further minimizing the maximal error across all bidders yet to be pinned down. This continues, pinning down any new bidders for whom one of constraints (11) is binding, until the error is lexicographically optimized for all bidders.7 The exact same process is repeated for FAIR and BAL, with bidders pinned down and constraints (15) replaced with π∗ i ≥ πvcg,i − (vα i (λt i) − p(λt i)), ∀λ ∈ i , (where π∗ i is the current objective) in FAIR, and items pinned down and constraints (18) replaced with p∗ j ≥ pj (where p∗ j represents the target for the maximal price on that item) in BAL. Comment 2: Computation. All constraints in i are retained, and this set grows across all stages and across all rounds of the exchange. Thus, the computational effort in constraint generation is re-used. In implementation we are careful to address a number of -issues that arise due to floating-point issues. We prefer to err on the side of being conservative in determining whether or not to add another constraint in performing check (13). This avoids later infeasibility issues. In addition, when pinning-down bidders for the purpose of lexicographical refinement we relax the associated bidder-constraints with a small > 0 on the righthand side. 5.3 Revealed-Preference Activity Rules The role of activity rules in the auction is to ensure both consistency and progress across rounds [21]. Consistency in our exchange requires that bidders tighten bounds as the exchange progresses. Activity rules ensure that bidders are active during early rounds, and promote useful elicitation throughout the exchange. We adopt a simple revealed-preference (RP) activity rule. The idea is loosely based around the RP-rule in Ausubel et al. [1], where it is used for one-sided CAs. The motivation is to require more than simply consistency: we need bidders to provide enough information for the system to be able to to prove that an allocation is (approximately) efficient. It is helpful to think about the bidders interacting with proxy agents that will act on their behalf in responding to provisional prices pt−1 determined at the end of round t − 1. The only knowledge that such a proxy has of the valuation of a bidder is through the bid-tree. Suppose a proxy was queried by the exchange and asked which trade the bidder was most interested in at the provisional prices. The RP rule says the following: the proxy must have enough 7 For example, applying this to accuracy on Example 2 we solve once and find bidders 1 and 2 are binding, for error δ∗ = 2/3. We pin these down and then minimize the error to bidders 3 and 4. Finally, this gives p(A) = p(B) = 10/3 and p(C) = p(D) = 5/3, with accuracy 2/3 to bidders 1 and 2 and 1/3 to bidders 3 and 4. 254 information to be able to determine this surplus-maximizing trade at current prices. Consider the following examples: Example 7. A bidder has XOR(+A, +B) and a value of +5 on the leaf +A and a value range of [5,10] on leaf +B. Suppose prices are currently 3 for each of A and B. The RP rule is satisfied because the proxy knows that however the remaining value uncertainty on +B is resolved the bidder will always (weakly) prefer +B to +A. Example 8. A bidder has XOR(+A, +B) and value bounds [5, 10] on the root node and a value of 1 on leaf +A. Suppose prices are currently 3 for each of A and B. The RP rule is satisfied because the bidder will always prefer +A to +B at equal prices, whichever way the uncertain value on the root node is ultimately resolved. Overloading notation, let vi ∈ Ti denote a valuation that is consistent with lower and upper valuations in bid tree Ti. Definition 4. Bid tree Ti satisfies RP at prices pt−1 if and only if there exists some feasible trade L∗ for which, vi(L∗ i ) − pt−1 (L∗ i ) ≥ max λ∈Feas(x0) vi(λi) − pt−1 (λi), ∀vi ∈ Ti. (20) To make this determination for bidder i we solve a sequence of problems, each of which is a variation on the WD problem. First, we construct a candidate lower-bound trade, which is a feasible trade that solves: max λ vi(λi) − pt−1 (λi) [RP1(i)] s.t. λ ∈ Feas(x0 ), (21) The solution π∗ l to RP1(i) represents the maximal payoff that bidder i can achieve across all feasible trades, given its pessimistic valuation. Second, we break ties to find a trade with maximal value uncertainty across all possible solutions to RP1(i): max λ vi(λi) − vi(λi) [RP2(i)] s.t. λ ∈ Feas(x0 ) (22) vi(λi) − pt−1 (λi) ≥ π∗ l (23) We adopt solution L∗ i as our candidate for the trade that may satisfy RP. To understand the importance of this tiebreaking rule consider Example 7. The proxy can prove +B but not +A is a best-response for all vi ∈ Ti, and should choose +B as its candidate. Notice that +B is a counterexample to +A, but not the other way round. Now, we construct a modified valuation ˜vi, by setting ˜viβ = viβ , if β ∈ sat(L∗ i ) viβ , otherwise. (24) where sat(L∗ i ) is the set of nodes that are satisfied in the lower-bound tree for trade L∗ i . Given this modified valuation, we find U∗ to solve: max λ ˜vi(λi) − pt−1 (λi) [RP3(i)] s.t. λ ∈ Feas(x0 ) (25) Let π∗ u denote the payoff from this optimal trade at modified values ˜v. We call trade U∗ i the witness trade. We show in Proposition 1 that the RP rule is satisfied if and only if π∗ l ≥ π∗ u. Constructing the modified valuation as ˜vi recognizes that there is shared uncertainty across trades that satisfy the same nodes in a bid tree. Example 8 helps to illustrate this. Just using vi in RP3(i), we would find L∗ i is buy A with payoff π∗ l = 3 but then find U∗ i is buy B with π∗ u = 7 and fail RP. We must recognize that however the uncertainty on the root node is resolved it will affect +A and +B in exactly the same way. For this reason, we set ˜viβ = viβ = 5 on the root node, which is exactly the same value that was adopted in determining π∗ l . Then, RP3(i) applied to U∗ i gives buy A and the RP test is judged to be passed. Proposition 1. Bid tree Ti satisfies RP given prices pt−1 if and only if any lower-bound trade L∗ i that solves RP1(i) and RP2(i) satisfies: vi(Ti, L∗ i ) − pt−1 (L∗ i ) ≥ ˜vi(Ti, U∗ i ) − pt−1 (U∗ i ), (26) where ˜vi is the modified valuation in Eq. (24). Proof. For sufficiency, notice that the difference in payoff between trade L∗ i and another trade λi is unaffected by the way uncertainty is resolved on any node that is satisfied in both L∗ i and λi. Fixing the values in ˜vi on nodes satisfied in L∗ i has the effect of removing this consideration when a trade U∗ i is selected that satisfies one of these nodes. On the other hand, fixing the values on these nodes has no effect on trades considered in RP3(i) that do not share a node with L∗ i . For the necessary direction, we first show that any trade that satisfies RP must solve RP1(i). Suppose otherwise, that some λi with payoff greater than π∗ l satisfies RP. But, valuation vi ∈ Ti together with L∗ i presents a counterexample to RP (Eq. 20). Now, suppose (for contradiction) that some λi with maximal payoff π∗ l but uncertainty less than L∗ i satisfies RP. Proceed by case analysis. Case a): only one solution to RP1(i) has uncertain value and so λi has certain value. But, this cannot satisfy RP because L∗ i with uncertain value would be a counterexample to RP (Eq. 20). Case b): two or more solutions to RP1(i) have uncertain value. Here, we first argue that one of these trades must satisfy a (weak) superset of all the nodes with uncertain value that are satisfied by all other trades in this set. This is by RP. Without this, then for any choice of trade that solves RP1(i), there is another trade with a disjoint set of uncertain but satisfied nodes that provides a counterexample to RP (Eq. 20). Now, consider the case that some trade contains a superset of all the uncertain satisfied nodes of the other trades. Clearly RP2(i) will choose this trade, L∗ i , and λi must satisfy a subset of these nodes (by assumption). But, we now see that λi cannot satisfy RP because L∗ i would be a counterexample to RP. Failure to meet the activity rule must have some consequence. In the current rules, the default action we choose is to set the upper bounds in valuations down to the maximal value of the provisional price on a node8 and the lowerbound value on that node.9 Such a bidder can remain active 8 The provisional price on a node is defined as the minimal total price across all feasible trades for which the subtree rooted at the tree is satisfied. 9 This is entirely analogous to when a bidder in an ascending clock auction stops bidding at a price: she is not permitted to bid at a higher price again in future rounds. 255 within the exchange, but only with valuations that are consistent with these new bounds. 5.4 Bidder Feedback In each round, our default design provides every bidder with the provisional trade and also with the current provisional prices. See 7 for an additional discussion. We also provide guidance to help a bidder meet the RP rule. Let sat(L∗ i ) and sat(U∗ i ) denote the nodes that are satisfied in trades L∗ i and U∗ i , as computed in RP1-RP3. Lemma 2. When RP fails, a bidder must increase a lower bound on at least one node in sat(L∗ i ) \ sat(U∗ i ) or decrease an upper bound on at least one node in sat(U∗ i ) \ sat(L∗ i ) in order to meet the activity rule. Proof. Changing the upper- or lower- values on nodes that are not satisfied by either trade does not change L∗ i or U∗ i , and does not change the payoff from these trades. Thus, the RP condition will continue to fail. Similarly, changing the bounds on nodes that are satisfied in both trades has no effect on revealed preference. A change to a lower bound on a shared node affects both L∗ i and U∗ i identically because of the use of the modified valuation to determine U∗ i . A change to an upper bound on a shared node has no effect in determining either L∗ i or U∗ i . Note that when sat(U∗ i ) = sat(L∗ i ) then condition (26) is always trivially satisfied, and so the guidance in the lemma is always well-defined when RP fails. This is an elegant feedback mechanism because it is adaptive. Once a bidder makes some changes on some subset of these nodes, the bidder can query the exchange. The exchange can then respond yes, or can revise the set of nodes sat(λ∗ l ) and sat(λ∗ u) as necessary. 5.5 Termination Conditions Once each bidder has committed its new bids (and either met the RP rule or suffered the penalty) then round t closes. At this point, the task is to determine the new α-valuation, and in turn the provisional allocation λt and provisional prices pt . A termination condition is also checked, to determine whether to move the exchange to a last-and-final round. To define the α-valuation we compute the following two quantities: Pessimistic at Pessimistic (PP) Determine an efficient trade, λ∗ l , at pessimistic values, i.e. to solve maxλ i vi(λi), and set PP= i vi(λ∗ li). Pessimistic at Optimistic (PO) Determine an efficient trade, λ∗ u, at optimistic values, i.e. to solve maxλ i vi(λi), and set PO= i vi(λ∗ ui). First, note that PP ≥ PO and PP ≥ 0 by definition, for all bid-trees, although PO can be negative (because the right trade at v is not currently a useful trade at v). Recognizing this, define γeff (PP, PO) = 1 + PP − PO PP , (27) when PP > 0, and observe that γeff (PP, PO) ≥ 1 when this is defined, and that γeff (PP, PO) will start large and then trend towards 1 as the optimistic allocation converges towards the pessimistic allocation. In each round, we define αeff ∈ [0, 1] as: αeff = 0 when PP is 0 1/γeff otherwise (28) which is 0 while PP is 0 and then trends towards 1 once PP> 0 in some round. This is used to define α-valuation vα i = αeff vi + (1 − αeff )vi, ∀i, (29) which is used to define the provisional allocation and provisional prices. The effect is to endogenously define a schedule for moving from optimistic to pessimistic values across rounds, based on how close the trades are to one another. Termination Condition. In moving to the last-and-final round, and finally closing, we also care about the convergence of payments, in addition to the convergence towards an efficient trade. For this we introduce another parameter, αthresh ∈ [0, 1], that trends from 0 to 1 as the Threshold payments at lower and upper valuations converge. Consider the following parameter: γthresh = 1 + ||pthresh(v) − pthresh(v)||2 (PP/Nactive) , (30) which is defined for PP > 0, where pthresh(v) denotes the Threshold payments at valuation profile v, Nactive is the number of bidders that are actively engaged in trade in the PP trade, and || · ||2 is the L2-norm. Note that γthresh is defined for payments and not payoffs. This is appropriate because it is the accuracy of the outcome of the exchange that matters: i.e. the trade and the payments. Given this, we define αthresh = 0 when PP is 0 1/γthresh otherwise (31) which is 0 while PP is 0 and then trends towards 1 as progress is made. Definition 5 (termination). ICE transitions to a lastand-final round when one of the following holds: 1. αeff ≥ CUTOFFeff and αthresh ≥ CUTOFFthresh, 2. there is no trade at the optimistic values, where CUTOFFeff , CUTOFFthresh ∈ (0, 1] determine the accuracy required for termination. At the end of the last-and-final round vα = v is used to define the final trade and the final Threshold payments. Example 9. Consider again Example 1, and consider the upper and lower bounds as depicted in Figure 1. First, if the seller"s bounds were [−20, −4] then there is an optimistic trade but no pessimistic trade, and PO = −4 and PP = 0, and αeff = 0. At the bounds depicted, both the optimistic and the pessimistic trades occur and PO = PP = 4 and αeff = 1. However, we can see the Threshold payments are (17, −17) at v but (14, −14) at v. Evaluating γthresh , we have γthresh = 1 + √ 1/2(32+32) (4/2) = 5/2, and αthresh = 2/5. For CUTOFFthresh < 2/5 the exchange would remain open. On the other hand, if the buyer"s value for +AB was between [18, 24] and the seller"s value for −AB was between [−12, −6], the Threshold payments are (15, −15) at both upper and lower bounds, and αthresh = 1. 256 Component Purpose Lines Agent. Captures strategic behavior and information revelation decisions 762 Model Support Provides XML support to load goods and valuations into world 200 World Keeps track of all agent, good, and valuation details 998 Exchange Driver & Communication Controls exchange, and coordinates remote agent behavior 585 Bidding Language Implements the tree-based bidding language 1119 Activity Rule Engine Implements the revealed preference rule with range support 203 Closing Rule Engine Checks if auction termination condition reached 137 WD Engine Provides WD-related logic 377 Pricing Engine Provides Pricing-related logic 460 MIP Builders Translates logic used by engines into our general optimizer formulation 346 Pricing Builders Used by three pricing stages 256 Winner Determination Builders Used by WD, activity rule, closing rule, and pricing constraint generation 365 Framework Support code; eases modular replacement of above components 510 Table 1: Exchange Component and Code Breakdown. 6. SYSTEMS INFRASTRUCTURE ICE is approximately 6502 lines of Java code, broken up into the functional packages described in Table 1.10 The prototype is modular so that researchers may easily replace components for experimentation. In addition to the core exchange discussed in this paper, we have developed an agent component that allows a user to simulate the behavior and knowledge of other players in the system, better allowing a user to formulate their strategy in advance of actual play. A user specifies a valuation model in an XMLinterpretation of our bidding language, which is revealed to the exchange via the agent"s strategy. Major exchange tasks are handled by engines that dictate the non-optimizer specific logic. These engines drive the appropriate MIP/LP builders. We realized that all of our optimization formulations boil down to two classes of optimization problem. The first, used by winner determination, activity rule, closing rule, and constraint generation in pricing, is a MIP that finds trades that maximize value, holding prices and slacks constant. The second, used by the three pricing stages, is an LP that holds trades constant, seeking to minimize slack, profit, or prices. We take advantage of the commonality of these problems by using common LP/MIP builders that differ only by a few functional hooks to provide the correct variables for optimization. We have generalized our back-end optimization solver interface11 (we currently support CPLEX and the LGPL- licensed LPSolve), and can take advantage of the load-balancing and parallel MIP/LP solving capability that this library provides. 7. DISCUSSION The bidding language was defined to allow for perfect symmetry between buyers and sellers and provide expressiveness in an exchange domain, for instance for mixed bidders interested in executing trades such as swaps. This proved especially challenging. The breakthrough came when we focused on changes in value for trades rather than providing absolute values for allocations. For simplicity, we require the same tree structure for both the upper and lower valuations. 10 Code size is measured in physical source line of code (SLOC), as generated using David A. Wheeler"s SLOC Count. The total of 6502 includes 184 for instrumentation (not shown in the table). The JOpt solver interface is another 1964 lines, and Castor automatically generates around 5200 lines of code for XML file manipulation. 11 http://econcs.eecs.harvard.edu/jopt This allows the language itself to ensure consistency (with the upper value at least the lower value on all trades) and enforce monotonic tightening of these bounds for all trades across rounds. It also provides for an efficient method to check the RP activity rule, because it makes it simple to reason about shared uncertainty between trades. The decision to adopt a direct and proxied approach in which bidders express their upper and lower values to a trusted proxy agent that interacts with the exchange was made early in the design process. In many ways this is the clearest and most immediate way to generalize the design in Parkes et al. [24] and make it iterative. In addition, this removes much opportunity for strategic manipulation: bidders are restricted to making (incremental) statements about their valuations. Another advantage is that it makes the activity rule easy to explain: bidders can always meet the activity rule by tightening bounds such that their true value remains in the support.12 Perhaps most importantly, having explicit information on upper and lower values permits progress in early rounds, even while there is no efficient trade at pessimistic values. Upper and lower bound information also provides guidance about when to terminate. Note that taken by itself, PP = PO does not imply that the current provisional trade is efficient with respect to all values consistent with current value information. The difference in values between different trades, aggregated across all bidders, could be similar at lower and upper bounds but quite different at intermediate values (including truth). Nevertheless, we conjecture that PP = PO will prove an excellent indicator of efficiency in practical settings where the shape of the upper and lower valuations does convey useful information. This is worthy of experimental investigation. Moreover, the use of price and RP activity provides additional guarantees. We adopted linear prices (prices on individual items) rather than non-linear prices (with prices on a trade not equal to the sum of the prices on the component items) early in the design process. The conciseness of this price representation is very important for computational tractability within the exchange and also to promote simplicity and transparency for bidders. The RP activity rule was adopted later, and is a good choice because of its excellent theoretical properties when coupled with CE prices. The following can be easily established: given exact CE prices pt−1 for provisional trade 12 This is in contrast to indirect price-based approaches, such as clock-proxy [1], in which bidders must be able to reason about the RP-constraints implied by bids in each round. 257 λt−1 at valuations vα , then if the upper and lower values at the start of round t already satisfy the RP rule (and without the need for any tie-breaking), the provisional trade is efficient for all valuations consistent with the current bid trees. When linear CE prices exist, this provides for a soundness and completeness statement: if PP = PO, linear CE prices exist, and the RP rule is satisfied, the provisional trade is efficient (soundness); if prices are exact CE prices for the provisional trade at vα , but the trade is inefficient with respect to some valuation profile consistent with the current bid trees, then at least one bidder must fail RP with her current bid tree and progress will be made (completeness). Future work must study convergence experimentally, and extend this theory to allow for approximate prices. Some strategic aspects of our ICE design deserve comment, and further study. First, we do not claim that truthfully responding to the RP rule is an ex post equilibrium.13 However, the exchange is designed to mimic the Threshold rule in its payment scheme, which is known to have useful incentive properties [16]. We must be careful, though. For instance we do not suggest to provide αeff to bidders, because as αeff approaches 1 it would inform bidders that bid values are becoming irrelevant to determining the trade but merely used to determine payments (and bidders would become increasingly reluctant to increase their lower valuations). Also, no consideration has been given in this work to collusion by bidders. This is an issue that deserves some attention in future work. 8. CONCLUSIONS In this work we designed and prototyped a scalable and highly-expressive iterative combinatorial exchange. The design includes many interesting features, including: a new bid-tree language for exchanges, a new method to construct approximate linear prices from expressive languages, and a proxied elicitation method with optimistic and pessimistic valuations with a new method to evaluate a revealed- preference activity rule. The exchange is fully implemented in Java and is in a validation phase. The next steps for our work are to allow bidders to refine the structure of the bid tree in addition to values on the tree. We intend to study the elicitation properties of the exchange and we have put together a test suite of exchange problem instances. In addition, we are beginning to engage in collaborations to apply the design to airline takeoff and landing slot scheduling and to resource allocation in widearea network distributed computational systems. Acknowledgments We would like to dedicate this paper to all of the participants in CS 286r at Harvard University in Spring 2004. This work is supported in part by NSF grant IIS-0238147. 9. REFERENCES [1] L. Ausubel, P. Cramton, and P. Milgrom. The clock-proxy auction: A practical combinatorial auction design. In Cramton et al. [9], chapter 5. [2] M. Babaioff, N. Nisan, and E. Pavlov. Mechanisms for a spatially distributed market. In Proc. 5th ACM Conf. on Electronic Commerce, pages 9-20. ACM Press, 2001. 13 Given the Myerson-Satterthwaite impossibility theorem [22] and the method by which we determine the trade we should not expect this. [3] M. Ball, G. Donohue, and K. Hoffman. Auctions for the safe, efficient, and equitable allocation of airspace system resources. In S. Cramton, Shoham, editor, Combinatorial Auctions. 2004. Forthcoming. [4] D. Bertsimas and J. Tsitsiklis. Introduction to Linear Optimization. Athena Scientific, 1997. [5] S. Bikhchandani and J. M. Ostroy. The package assignment model. Journal of Economic Theory, 107(2):377-406, 2002. [6] C. Boutilier. A pomdp formulation of preference elicitation problems. In Proc. 18th National Conference on Artificial Intelligence (AAAI-02), 2002. [7] C. Boutilier and H. Hoos. Bidding languages for combinatorial auctions. In Proc. 17th International Joint Conference on Artificial Intelligence (IJCAI-01), 2001. [8] W. Conen and T. Sandholm. Preference elicitation in combinatorial auctions. In Proc. 3rd ACM Conf. on Electronic Commerce (EC-01), pages 256-259. ACM Press, New York, 2001. [9] P. Cramton, Y. Shoham, and R. Steinberg, editors. Combinatorial Auctions. MIT Press, 2004. [10] S. de Vries, J. Schummer, and R. V. Vohra. On ascending Vickrey auctions for heterogeneous objects. Technical report, MEDS, Kellogg School, Northwestern University, 2003. [11] S. de Vries and R. V. Vohra. Combinatorial auctions: A survey. Informs Journal on Computing, 15(3):284-309, 2003. [12] M. Dunford, K. Hoffman, D. Menon, R. Sultana, and T. Wilson. Testing linear pricing algorithms for use in ascending combinatorial auctions. Technical report, SEOR, George Mason University, 2003. [13] Y. Fu, J. Chase, B. Chun, S. Schwab, and A. Vahdat. Sharp: an architecture for secure resource peering. In Proceedings of the nineteenth ACM symposium on Operating systems principles, pages 133-148. ACM Press, 2003. [14] B. Hudson and T. Sandholm. Effectiveness of query types and policies for preference elicitation in combinatorial auctions. In Proc. 3rd Int. Joint. Conf. on Autonomous Agents and Multi Agent Systems, pages 386-393, 2004. [15] V. Krishna. Auction Theory. Academic Press, 2002. [16] D. Krych. Calculation and analysis of Nash equilibria of Vickrey-based payment rules for combinatorial exchanges, Harvard College, April 2003. [17] A. M. Kwasnica, J. O. Ledyard, D. Porter, and C. DeMartini. A new and improved design for multi-object iterative auctions. Management Science, 2004. To appear. [18] E. Kwerel and J. Williams. A proposal for a rapid transition to market allocation of spectrum. Technical report, FCC Office of Plans and Policy, Nov 2002. [19] S. M. Lahaie and D. C. Parkes. Applying learning algorithms to preference elicitation. In Proc. ACM Conf. on Electronic Commerce, pages 180-188, 2004. [20] R. P. McAfee. A dominant strategy double auction. J. of Economic Theory, 56:434-450, 1992. [21] P. Milgrom. Putting auction theory to work: The simultaneous ascending auction. J.Pol. Econ., 108:245-272, 2000. [22] R. B. Myerson and M. A. Satterthwaite. Efficient mechanisms for bilateral trading. Journal of Economic Theory, 28:265-281, 1983. [23] N. Nisan. Bidding and allocation in combinatorial auctions. In Proc. 2nd ACM Conf. on Electronic Commerce (EC-00), pages 1-12, 2000. [24] D. C. Parkes, J. R. Kalagnanam, and M. Eso. Achieving budget-balance with Vickrey-based payment schemes in exchanges. In Proc. 17th International Joint Conference on Artificial Intelligence (IJCAI-01), pages 1161-1168, 2001. [25] D. C. Parkes and L. H. Ungar. Iterative combinatorial auctions: Theory and practice. In Proc. 17th National Conference on Artificial Intelligence (AAAI-00), pages 74-81, July 2000. [26] S. J. Rassenti, V. L. Smith, and R. L. Bulfin. A combinatorial mechanism for airport time slot allocation. Bell Journal of Economics, 13:402-417, 1982. [27] M. H. Rothkopf, A. Pekeˇc, and R. M. Harstad. Computationally manageable combinatorial auctions. Management Science, 44(8):1131-1147, 1998. [28] T. Sandholm and C. Boutilier. Preference elicitation in combinatorial auctions. In Cramton et al. [9], chapter 10. [29] P. R. Wurman and M. P. Wellman. AkBA: A progressive, anonymous-price combinatorial auction. In Second ACM Conference on Electronic Commerce, pages 21-29, 2000. 258

winner-determination;double auction;combinatorial auction;price;combinatorial exchange;vcg;bidding;threshold payment;tree-based bidding language;trade;preference elicitation;iterative combinatorial exchange;buyer and seller

train_J-62

Weak Monotonicity Suffices for Truthfulness on Convex Domains

Weak monotonicity is a simple necessary condition for a social choice function to be implementable by a truthful mechanism. Roberts [10] showed that it is sufficient for all social choice functions whose domain is unrestricted. Lavi, Mu"alem and Nisan [6] proved the sufficiency of weak monotonicity for functions over order-based domains and Gui, Muller and Vohra [5] proved sufficiency for order-based domains with range constraints and for domains defined by other special types of linear inequality constraints. Here we show the more general result, conjectured by Lavi, Mu"alem and Nisan [6], that weak monotonicity is sufficient for functions defined on any convex domain.

1. INTRODUCTION Social choice theory centers around the general problem of selecting a single outcome out of a set A of alternative outcomes based on the individual preferences of a set P of players. A method for aggregating player preferences to select one outcome is called a social choice function. In this paper we assume that the range A is finite and that each player"s preference is expressed by a valuation function which assigns to each possible outcome a real number representing the benefit the player derives from that outcome. The ensemble of player valuation functions is viewed as a valuation matrix with rows indexed by players and columns by outcomes. A major difficulty connected with social choice functions is that players can not be required to tell the truth about their preferences. Since each player seeks to maximize his own benefit, he may find it in his interest to misrepresent his valuation function. An important approach for dealing with this problem is to augment a given social choice function with a payment function, which assigns to each player a (positive or negative) payment as a function of all of the individual preferences. By carefully choosing the payment function, one can hope to entice each player to tell the truth. A social choice function augmented with a payment function is called a mechanism 1 and the mechanism is said to implement the social choice function. A mechanism is truthful (or to be strategyproof or to have a dominant strategy) if each player"s best strategy, knowing the preferences of the others, is always to declare his own true preferences. A social choice function is truthfully implementable, or truthful if it has a truthful implementation. (The property of truthful implementability is sometimes called dominant strategy incentive compatibility). This framework leads naturally to the question: which social choice functions are truthful? This question is of the following general type: given a class of functions (here, social choice functions) and a property that holds for some of them (here, truthfulness), characterize the property. The definition of the property itself provides a characterization, so what more is needed? Here are some useful notions of characterization: • Recognition algorithm. Give an algorithm which, given an appropriate representation of a function in the class, determines whether the function has the property. • Parametric representation. Give an explicit parametrized family of functions and show that each function in the 1 The usual definition of mechanism is more general than this (see [8] Chapter 23.C or [9]); the mechanisms we consider here are usually called direct revelation mechanisms. 286 family has the property, and that every function with the property is in the family. A third notion applies in the case of hereditary properties of functions. A function g is a subfunction of function f, or f contains g, if g is obtained by restricting the domain of f. A property P of functions is hereditary if it is preserved under taking subfunctions. Truthfulness is easily seen to be hereditary. • Sets of obstructions. For a hereditary property P, a function g that does not have the property is an obstruction to the property in the sense that any function containing g doesn"t have the property. An obstruction is minimal if every proper subfunction has the property. A set of obstructions is complete if every function that does not have the property contains one of them as a subfunction. The set of all functions that don"t satisfy P is a complete (but trivial and uninteresting) set of obstructions; one seeks a set of small (ideally, minimal) obstructions. We are not aware of any work on recognition algorithms for the property of truthfulness, but there are significant results concerning parametric representations and obstruction characterizations of truthfulness. It turns out that the domain of the function, i.e., the set of allowed valuation matrices, is crucial. For functions with unrestricted domain, i.e., whose domain is the set of all real matrices, there are very good characterizations of truthfulness. For general domains, however, the picture is far from complete. Typically, the domains of social choice functions are specified by a system of constraints. For example, an order constraint requires that one specified entry in some row be larger than another in the same row, a range constraint places an upper or lower bound on an entry, and a zero constraint forces an entry to be 0. These are all examples of linear inequality constraints on the matrix entries. Building on work of Roberts [10], Lavi, Mu"alem and Nisan [6] defined a condition called weak monotonicity (WMON). (Independently, in the context of multi-unit auctions, Bikhchandani, Chatterji and Sen [3] identified the same condition and called it nondecreasing in marginal utilities (NDMU).) The definition of W-MON can be formulated in terms of obstructions: for some specified simple set F of functions each having domains of size 2, a function satisfies W-MON if it contains no function from F. The functions in F are not truthful, and therefore W-MON is a necessary condition for truthfulness. Lavi, Mu"alem and Nisan [6] showed that W-MON is also sufficient for truthfulness for social choice functions whose domain is order-based, i.e., defined by order constraints and zero constraints, and Gui, Muller and Vohra [5] extended this to other domains. The domain constraints considered in both papers are special cases of linear inequality constraints, and it is natural to ask whether W-MON is sufficient for any domain defined by such constraints. Lavi, Mu"alem and Nisan [6] conjectured that W-MON suffices for convex domains. The main result of this paper is an affirmative answer to this conjecture: Theorem 1. For any social choice function having convex domain and finite range, weak monotonicity is necessary and sufficient for truthfulness. Using the interpretation of weak monotonicity in terms of obstructions each having domain size 2, this provides a complete set of minimal obstructions for truthfulness within the class of social choice functions with convex domains. The two hypotheses on the social choice function, that the domain is convex and that the range is finite, can not be omitted as is shown by the examples given in section 7. 1.1 Related Work There is a simple and natural parametrized set of truthful social choice functions called affine maximizers. Roberts [10] showed that for functions with unrestricted domain, every truthful function is an affine maximizer, thus providing a parametrized representation for truthful functions with unrestricted domain. There are many known examples of truthful functions over restricted domains that are not affine maximizers (see [1], [2], [4], [6] and [7]). Each of these examples has a special structure and it seems plausible that there might be some mild restrictions on the class of all social choice functions such that all truthful functions obeying these restrictions are affine maximizers. Lavi, Mu"alem and Nisan [6] obtained a result in this direction by showing that for order-based domains, under certain technical assumptions, every truthful social choice function is almost an affine maximizer. There are a number of results about truthfulness that can be viewed as providing obstruction characterizations, although the notion of obstruction is not explicitly discussed. For a player i, a set of valuation matrices is said to be i-local if all of the matrices in the set are identical except for row i. Call a social choice function i-local if its domain is ilocal and call it local if it is i-local for some i. The following easily proved fact is used extensively in the literature: Proposition 2. The social choice function f is truthful if and only if every local subfunction of f is truthful. This implies that the set of all local non-truthful functions comprises a complete set of obstructions for truthfulness. This set is much smaller than the set of all non-truthful functions, but is still far from a minimal set of obstructions. Rochet [11], Rozenshtrom [12] and Gui, Muller and Vohra [5] identified a necessary and sufficient condition for truthfulness (see lemma 3 below) called the nonnegative cycle property. This condition can be viewed as providing a minimal complete set of non-truthful functions. As is required by proposition 2, each function in the set is local. Furthermore it is one-to-one. In particular its domain has size at most the number of possible outcomes |A|. As this complete set of obstructions consists of minimal non-truthful functions, this provides the optimal obstruction characterization of non-truthful functions within the class of all social choice functions. But by restricting attention to interesting subclasses of social choice functions, one may hope to get simpler sets of obstructions for truthfulness within that class. The condition of weak monotonicity mentioned earlier can be defined by a set of obstructions, each of which is a local function of domain size exactly 2. Thus the results of Lavi, Mu"alem and Nisan [6], and of Gui, Muller and Vohra [5] give a very simple set of obstructions for truthfulness within certain subclasses of social choice functions. Theorem 1 extends these results to a much larger subclass of functions. 287 1.2 Weak Monotonicity and the Nonnegative Cycle Property By proposition 2, a function is truthful if and only if each of its local subfunctions is truthful. Therefore, to get a set of obstructions for truthfulness, it suffices to obtain such a set for local functions. The domain of an i-local function consists of matrices that are fixed on all rows but row i. Fix such a function f and let D ⊆ RA be the set of allowed choices for row i. Since f depends only on row i and row i is chosen from D, we can view f as a function from D to A. Therefore, f is a social choice function having one player; we refer to such a function as a single player function. Associated to any single player function f with domain D we define an edge-weighted directed graph Hf whose vertex set is the image of f. For convenience, we assume that f is surjective and so this image is A. For each a, b ∈ A, x ∈ f−1 (a) there is an edge ex(a, b) from a to b with weight x(a) − x(b). The weight of a set of edges is just the sum of the weights of the edges. We say that f satisfies: • the nonnegative cycle property if every directed cycle has nonnegative weight. • the nonnegative two-cycle property if every directed cycle between two vertices has nonnegative weight. We say a local function g satisfies nonnegative cycle property/nonnegative two-cycle property if its associated single player function f does. The graph Hf has a possibly infinite number of edges between any two vertices. We define Gf to be the edgeweighted directed graph with exactly one edge from a to b, whose weight δab is the infimum (possibly −∞) of all of the edge weights ex(a, b) for x ∈ f−1 (a). It is easy to see that Hf has the nonnegative cycle property/nonnegative two-cycle property if and only if Gf does. Gf is called the outcome graph of f. The weak monotonicity property mentioned earlier can be defined for arbitrary social choice functions by the condition that every local subfunction satisfies the nonnegative two-cycle property. The following result was obtained by Rochet [11] in a slightly different form and rediscovered by Rozenshtrom [12] and Gui, Muller and Vohra [5]: Lemma 3. A local social choice function is truthful if and only if it has the nonnegative cycle property. Thus a social choice function is truthful if and only if every local subfunction satisfies the nonnegative cycle property. In light of this, theorem 1 follows from: Theorem 4. For any surjective single player function f : D −→ A where D is a convex subset of RA and A is finite, the nonnegative two-cycle property implies the nonnegative cycle property. This is the result we will prove. 1.3 Overview of the Proof of Theorem 4 Let D ⊆ RA be convex and let f : D −→ A be a single player function such that Gf has no negative two-cycles. We want to conclude that Gf has no negative cycles. For two vertices a, b, let δ∗ ab denote the minimum weight of any path from a to b. Clearly δ∗ ab ≤ δab. Our proof shows that the δ∗ -weight of every cycle is exactly 0, from which theorem 4 follows. There seems to be no direct way to compute δ∗ and so we proceed indirectly. Based on geometric considerations, we identify a subset of paths in Gf called admissible paths and a subset of admissible paths called straight paths. We prove that for any two outcomes a, b, there is a straight path from a to b (lemma 8 and corollary 10), and all straight paths from a to b have the same weight, which we denote ρab (theorem 12). We show that ρab ≤ δab (lemma 14) and that the ρ-weight of every cycle is 0. The key step to this proof is showing that the ρ-weight of every directed triangle is 0 (lemma 17). It turns out that ρ is equal to δ∗ (corollary 20), although this equality is not needed in the proof of theorem 4. To expand on the above summary, we give the definitions of an admissible path and a straight path. These are somewhat technical and rely on the geometry of f. We first observe that, without loss of generality, we can assume that D is (topologically) closed (section 2). In section 3, for each a ∈ A, we enlarge the set f−1 (a) to a closed convex set Da ⊆ D in such a way that for a, b ∈ A with a = b, Da and Db have disjoint interiors. We define an admissible path to be a sequence of outcomes (a1, . . . , ak) such that each of the sets Ij = Daj ∩ Daj+1 is nonempty (section 4). An admissible path is straight if there is a straight line that meets one point from each of the sets I1, . . . , Ik−1 in order (section 5). Finally, we mention how the hypotheses of convex domain and finite range are used in the proof. Both hypotheses are needed to show: (1) the existence of a straight path from a to b for all a, b (lemma 8). (2) that the ρ-weight of a directed triangle is 0 (lemma 17). The convex domain hypothesis is also needed for the convexity of the sets Da (section 3). The finite range hypothesis is also needed to reduce theorem 4 to the case that D is closed (section 2) and to prove that every straight path from a to b has the same δ-weight (theorem 12). 2. REDUCTION TO CLOSED DOMAIN We first reduce the theorem to the case that D is closed. Write DC for the closure of D. Since A is finite, DC = ∪a∈A(f−1 (a))C . Thus for each v ∈ DC − D, there is an a = a(v) ∈ A such that v ∈ (f−1 (a))C . Extend f to the function g on DC by defining g(v) = a(v) for v ∈ DC − D and g(v) = f(v) for v ∈ D. It is easy to check that δab(g) = δab(f) for all a, b ∈ A and therefore it suffices to show that the nonnegative two-cycle property for g implies the nonnegative cycle property for g. Henceforth we assume D is convex and closed. 3. A DISSECTION OF THE DOMAIN In this section, we construct a family of closed convex sets {Da : a ∈ A} with disjoint interiors whose union is D and satisfying f−1 (a) ⊆ Da for each a ∈ A. Let Ra = {v : ∀b ∈ A, v(a) − v(b) ≥ δab}. Ra is a closed polyhedron containing f−1 (a). The next proposition implies that any two of these polyhedra intersect only on their boundary. Proposition 5. Let a, b ∈ A. If v ∈ Ra ∩Rb then v(a)− v(b) = δab = −δba. 288 Da Db Dc Dd De v w x y z u p Figure 1: A 2-dimensional domain with 5 outcomes. Proof. v ∈ Ra implies v(a) − v(b) ≥ δab and v ∈ Rb implies v(b)−v(a) ≥ δba which, by the nonnegative two-cycle property, implies v(a) − v(b) ≤ δab. Thus v(a) − v(b) = δab and by symmetry v(b) − v(a) = δba. Finally, we restrict the collection of sets {Ra : a ∈ A} to the domain D by defining Da = Ra ∩ D for each a ∈ A. Clearly, Da is closed and convex, and contains f−1 (a). Therefore S a∈A Da = D. Also, by proposition 5, any point v in Da ∩ Db satisfies v(a) − v(b) = δab = −δba. 4. PATHS AND D-SEQUENCES A path of size k is a sequence −→a = (a1, . . . , ak) with each ai ∈ A (possibly with repetition). We call −→a an (a1, ak)path. For a path −→a , we write |−→a | for the size of −→a . −→a is simple if the ai"s are distinct. For b, c ∈ A we write Pbc for the set of (b, c)-paths and SPbc for the set of simple (b, c)-paths. The δ-weight of path −→a is defined by δ(−→a ) = k−1X i=1 δaiai+1 . A D-sequence of order k is a sequence −→u = (u0, . . . , uk) with each ui ∈ D (possibly with repetition). We call −→u a (u0, uk)-sequence. For a D-sequence −→u , we write ord(u) for the order of −→u . For v, w ∈ D we write Svw for the set of (v, w)-sequences. A compatible pair is a pair (−→a , −→u ) where −→a is a path and −→u is a D-sequence satisfying ord(−→u ) = |−→a | and for each i ∈ [k], both ui−1 and ui belong to Dai . We write C(−→a ) for the set of D-sequences −→u that are compatible with −→a . We say that −→a is admissible if C(−→a ) is nonempty. For −→u ∈ C(−→a ) we define ∆−→a (−→u ) = |−→a |−1 X i=1 (ui(ai) − ui(ai+1)). For v, w ∈ D and b, c ∈ A, we define Cvw bc to be the set of compatible pairs (−→a , −→u ) such that −→a ∈ Pbc and −→u ∈ Svw . To illustrate these definitions, figure 1 gives the dissection of a domain, a 2-dimensional plane, into five regions Da, Db, Dc, Dd, De. D-sequence (v, w, x, y, z) is compatible with both path (a, b, c, e) and path (a, b, d, e); D-sequence (v, w, u, y, z) is compatible with a unique path (a, b, d, e). D-sequence (x, w, p, y, z) is compatible with a unique path (b, a, d, e). Hence (a, b, c, e), (a, b, d, e) and (b, a, d, e) are admissible paths. However, path (a, c, d) or path (b, e) is not admissible. Proposition 6. For any compatible pair (−→a , −→u ), ∆−→a (−→u ) = δ(−→a ). Proof. Let k = ord(−→u ) = |−→a |. By the definition of a compatible pair, ui ∈ Dai ∩ Dai+1 for i ∈ [k − 1]. ui(ai) − ui(ai+1) = δaiai+1 from proposition 5. Therefore, ∆−→a (−→u ) = k−1X i=1 (ui(ai) − ui(ai+1)) = k−1X i=1 δaiai+1 = δ(−→a ). Lemma 7. Let b, c ∈ A and let −→a , −→a ∈ Pbc. If C(−→a ) ∩ C(−→a ) = ∅ then δ(−→a ) = δ(−→a ). Proof. Let −→u be a D-sequence in C(−→a ) ∩ C(−→a ). By proposition 6, δ(−→a ) = ∆−→a (−→u ) and δ(−→a ) = ∆−→a (−→u ), it suffices to show ∆−→a (−→u ) = ∆−→a (−→u ). Let k = ord(−→u ) = |−→a | = |−→a |. Since ∆−→a (−→u ) = k−1X i=1 (ui(ai) − ui(ai+1)) = u1(a1) + k−1X i=2 (ui(ai) − ui−1(ai)) − uk−1(ak) = u1(b) + k−1X i=2 (ui(ai) − ui−1(ai)) − uk−1(c), ∆−→a (−→u ) − ∆−→a (−→u ) = k−1X i=2 ((ui(ai) − ui−1(ai)) − (ui(ai) − ui−1(ai))) = k−1X i=2 ((ui(ai) − ui(ai)) − (ui−1(ai) − ui−1(ai))). Noticing both ui−1 and ui belong to Dai ∩ Dai , we have by proposition 5 ui−1(ai) − ui−1(ai) = δaiai = ui(ai) − ui(ai). Hence ∆−→a (−→u ) − ∆−→a (−→u ) = 0. 5. LINEAR D-SEQUENCES AND STRAIGHT PATHS For v, w ∈ D we write vw for the (closed) line segment joining v and w. A D-sequence −→u of order k is linear provided that there is a sequence of real numbers 0 = λ0 ≤ λ1 ≤ . . . ≤ λk = 1 such that ui = (1 − λi)u0 + λiuk. In particular, each ui belongs to u0uk. For v, w ∈ D we write Lvw for the set of linear (v, w)-sequences. For b, c ∈ A and v, w ∈ D we write LCvw bc for the set of compatible pairs (−→a , −→u ) such that −→a ∈ Pbc and −→u ∈ Lvw . For a path −→a , we write L(−→a ) for the set of linear sequences compatible with −→a . We say that −→a is straight if L(−→a ) = ∅. For example, in figure 1, D-sequence (v, w, x, y, z) is linear while (v, w, u, y, z), (x, w, p, y, z), and (x, v, w, y, z) are not. Hence path (a, b, c, e) and (a, b, d, e) are both straight. However, path (b, a, d, e) is not straight. 289 Lemma 8. Let b, c ∈ A and v ∈ Db, w ∈ Dc. There is a simple path −→a and D-sequence −→u such that (−→a , −→u ) ∈ LCvw bc . Furthermore, for any such path −→a , δ(−→a ) ≤ v(b) − v(c). Proof. By the convexity of D, any sequence of points on vw is a D-sequence. If b = c, singleton path −→a = (b) and D-sequence −→u = (v, w) are obviously compatible. δ(−→a ) = 0 = v(b) − v(c). So assume b = c. If Db ∩Dc ∩vw = ∅, we pick an arbitrary x from this set and let −→a = (b, c) ∈ SPbc, −→u = (v, x, w) ∈ Lvw . Again it is easy to check the compatibility of (−→a , −→u ). Since v ∈ Db, v(b) − v(c) ≥ δbc = δ(−→a ). For the remaining case b = c and Db ∩Dc ∩vw = ∅, notice v = w otherwise v = w ∈ Db ∩ Dc ∩ vw. So we can define λx for every point x on vw to be the unique number in [0, 1] such that x = (1 − λx)v + λxw. For convenience, we write x ≤ y for λx ≤ λy. Let Ia = Da ∩ vw for each a ∈ A. Since D = ∪a∈ADa, we have vw = ∪a∈AIa. Moreover, by the convexity of Da and vw, Ia is a (possibly trivial) closed interval. We begin by considering the case that Ib and Ic are each a single point, that is, Ib = {v} and Ic = {w}. Let S be a minimal subset of A satisfying ∪s∈SIs = vw. For each s ∈ S, Is is maximal, i.e., not contained in any other It, for t ∈ S. In particular, the intervals {Is : s ∈ S} have all left endpoints distinct and all right endpoints distinct and the order of the left endpoints is the same as that of the right endpoints. Let k = |S| + 2 and index S as a2, . . . , ak−1 in the order defined by the right endpoints. Denote the interval Iai by [li, ri]. Thus l2 < l3 < . . . < lk−1, r2 < r3 < . . . < rk−1 and the fact that these intervals cover vw implies l2 = v, rk−1 = w and for all 2 ≤ i ≤ k − 2, li+1 ≤ ri which further implies li < ri. Now we define the path −→a = (a1, a2, . . . , ak−1, ak) with a1 = b, ak = c and a2, a3, . . . , ak−1 as above. Define the linear D-sequence −→u = (u0, u1, . . . , uk) by u0 = u1 = v, uk = w and for 2 ≤ i ≤ k−1, ui = ri. It follows immediately that (−→a , −→u ) ∈ LCvw bc . Neither b nor c is in S since lb = rb and lc = rc. Thus −→a is simple. Finally to show δ(−→a ) ≤ v(b) − v(c), we note v(b) − v(c) = v(a1) − v(ak) = k−1X i=1 (v(ai) − v(ai+1)) and δ(−→a ) = ∆−→a (−→u ) = k−1X i=1 (ui(ai) − ui(ai+1)) = v(a1) − v(a2) + k−1X i=2 (ri(ai) − ri(ai+1)). For two outcomes d, e ∈ A, let us define fde(z) = z(d)−z(e) for all z ∈ D. It suffices to show faiai+1 (ri) ≤ faiai+1 (v) for 2 ≤ i ≤ k − 1. Fact 9. For d, e ∈ A, fde(z) is a linear function of z. Furthermore, if x ∈ Dd and y ∈ De with x = y, then fde(x) = x(d) − x(e) ≥ δde ≥ −δed ≥ −(y(e) − y(d)) = fde(y). Therefore fde(z) is monotonically nonincreasing along the line ←→ xy as z moves in the direction from x to y. Applying this fact with d = ai, e = ai+1, x = li and y = ri gives the desired conclusion. This completes the proof for the case that Ib = {v} and Ic = {w}. For general Ib, Ic, rb < lc otherwise Db ∩ Dc ∩ vw = Ib ∩ Ic = ∅. Let v = rb and w = lc. Clearly we can apply the above conclusion to v ∈ Db, w ∈ Dc and get a compatible pair (−→a , −→u ) ∈ LCv w bc with −→a simple and δ(−→a ) ≤ v (b) − v (c). Define the linear D-sequence −→u by u0 = v, uk = w and ui = ui for i ∈ [k − 1]. (−→a , −→u ) ∈ LCvw bc is evident. Moreover, applying the above fact with d = b, e = c, x = v and y = w, we get v(b) − v(c) ≥ v (b) − v (c) ≥ δ(−→a ). Corollary 10. For any b, c ∈ A there is a straight (b, c)path. The main result of this section (theorem 12) says that for any b, c ∈ A, every straight (b, c)-path has the same δ-weight. To prove this, we first fix v ∈ Db and w ∈ Dc and show (lemma 11) that every straight (b, c)-path compatible with some linear (v, w)-sequence has the same δ-weight ρbc(v, w). We then show in theorem 12 that ρbc(v, w) is the same for all choices of v ∈ Db and w ∈ Dc. Lemma 11. For b, c ∈ A, there is a function ρbc : Db × Dc −→ R satisfying that for any (−→a , −→u ) ∈ LCvw bc , δ(−→a ) = ρbc(v, w). Proof. Let (−→a , −→u ), (−→a , −→u ) ∈ LCvw bc . It suffices to show δ(−→a ) = δ(−→a ). To do this we construct a linear (v, w)-sequence −→u and paths −→a ∗ , −→a ∗∗ ∈ Pbc, both compatible with −→u , satisfying δ(−→a ∗ ) = δ(−→a ) and δ(−→a ∗∗ ) = δ(−→a ). Lemma 7 implies δ(−→a ∗ ) = δ(−→a ∗∗ ), which will complete the proof. Let |−→a | = ord(−→u ) = k and |−→a | = ord(−→u ) = l. We select −→u to be any linear (v, w)-sequence (u0, u1, . . . , ut) such that −→u and −→u are both subsequences of −→u , i.e., there are indices 0 = i0 < i1 < · · · < ik = t and 0 = j0 < j1 < · · · < jl = t such that −→u = (ui0 , ui1 , . . . , uik ) and −→u = (uj0 , uj1 , . . . , ujl ). We now construct a (b, c)-path −→a ∗ compatible with −→u such that δ(−→a ∗ ) = δ(−→a ). (An analogous construction gives −→a ∗∗ compatible with −→u such that δ(−→a ∗∗ ) = δ(−→a ).) This will complete the proof. −→a ∗ is defined as follows: for 1 ≤ j ≤ t, a∗ j = ar where r is the unique index satisfying ir−1 < j ≤ ir. Since both uir−1 = ur−1 and uir = ur belong to Dar , uj ∈ Dar for ir−1 ≤ j ≤ ir by the convexity of Dar . The compatibility of (−→a ∗ , −→u ) follows immediately. Clearly, a∗ 1 = a1 = b and a∗ t = ak = c, so −→a ∗ ∈ Pbc. Furthermore, as δa∗ j a∗ j+1 = δarar = 0 for each r ∈ [k], ir−1 < j < ir, δ(−→a ∗ ) = k−1X r=1 δa∗ ir a∗ ir+1 = k−1X r=1 δarar+1 = δ(−→a ). We are now ready for the main theorem of the section: Theorem 12. ρbc is a constant map on Db × Dc. Thus for any b, c ∈ A, every straight (b, c)-path has the same δweight. Proof. For a path −→a , (v, w) is compatible with −→a if there is a linear (v, w)-sequence compatible with −→a . We write CP(−→a ) for the set of pairs (v, w) compatible with −→a . ρbc is constant on CP(−→a ) because for each (v, w) ∈ CP(−→a ), ρbc(v, w) = δ(−→a ). By lemma 8, we also haveS −→a ∈SPbc CP(−→a ) = Db ×Dc. Since A is finite, SPbc, the set of simple paths from b to c, is finite as well. 290 Next we prove that for any path −→a , CP(−→a ) is closed. Let ((vn , wn ) : n ∈ N) be a convergent sequence in CP(−→a ) and let (v, w) be the limit. We want to show that (v, w) ∈ CP(−→a ). For each n ∈ N, since (vn , wn ) ∈ CP(−→a ), there is a linear (vn , wn )-sequence un compatible with −→a , i.e., there are 0 = λn 0 ≤ λn 1 ≤ . . . ≤ λn k = 1 (k = |−→a |) such that un j = (1 − λn j )vn + λn j wn (j = 0, 1, . . . , k). Since for each n, λn = (λn 0 , λn 1 , . . . , λn k ) belongs to the closed bounded set [0, 1]k+1 we can choose an infinite subset I ⊆ N such that the sequence (λn : n ∈ I) converges. Let λ = (λ0, λ1, . . . , λk) be the limit. Clearly 0 = λ0 ≤ λ1 ≤ · · · ≤ λk = 1. Define the linear (v, w)-sequence −→u by uj = (1 − λj )v + λj w (j = 0, 1, . . . , k). Then for each j ∈ {0, . . . , k}, uj is the limit of the sequence (un j : n ∈ I). For j > 0, each un j belongs to the closed set Daj , so uj ∈ Daj . Similarly, for j < k each un j belongs to the closed set Daj+1 , so uj ∈ Daj+1 . Hence (−→a , −→u ) is compatible, implying (v, w) ∈ CP(−→a ). Now we have Db × Dc covered by finitely many closed subsets on each of them ρbc is a constant. Suppose for contradiction that there are (v, w), (v , w ) ∈ Db × Dc such that ρbc(v, w) = ρbc(v , w ). L = {((1 − λ)v + λv , (1 − λ)w + λw ) : λ ∈ [0, 1]} is a line segment in Db ×Dc by the convexity of Db, Dc. Let L1 = {(x, y) ∈ L : ρbc(x, y) = ρbc(v, w)} and L2 = L − L1. Clearly (v, w) ∈ L1, (v , w ) ∈ L2. Let P = {−→a ∈ SPbc : δ(−→a ) = ρbc(v, w)}. L1 = `S −→a ∈P CP(−→a ) ´ ∩ L, L2 = S −→a ∈SPbc−P CP(−→a ) ∩ L are closed by the finiteness of P. This is a contradiction, since it is well known (and easy to prove) that a line segment can not be expressed as the disjoint union of two nonempty closed sets. Summarizing corollary 10, lemma 11 and theorem 12, we have Corollary 13. For any b, c ∈ A, there is a real number ρbc with the property that (1) There is at least one straight (b, c)-path of δ-weight ρbc and (2) Every straight (b, c)-path has δ-weight ρbc. 6. PROOF OF THEOREM 4 Lemma 14. ρbc ≤ δbc for all b, c ∈ A. Proof. For contradiction, suppose ρbc − δbc = > 0. By the definition of δbc, there exists v ∈ f−1 (b) ⊆ Db with v(b) − v(c) < δbc + = ρbc. Pick an arbitrary w ∈ Dc. By lemma 8, there is a compatible pair (−→a , −→u ) ∈ LCvw bc with δ(−→a ) ≤ v(b) − v(c). Since −→a is a straight (b, c)-path, ρbc = δ(−→a ) ≤ v(b) − v(c), leading to a contradiction. Define another edge-weighted complete directed graph Gf on vertex set A where the weight of arc (a, b) is ρab. Immediately from lemma 14, the weight of every directed cycle in Gf is bounded below by its weight in Gf . To prove theorem 4, it suffices to show the zero cycle property of Gf , i.e., every directed cycle has weight zero. We begin by considering two-cycles. Lemma 15. ρbc + ρcb = 0 for all b, c ∈ A. Proof. Let −→a be a straight (b, c)-path compatible with linear sequence −→u . let −→a be the reverse of −→a and −→u the reverse of −→u . Obviously, (−→a , −→u ) is compatible as well and thus −→a is a straight (c, b)-path. Therefore, ρbc + ρcb = δ(−→a ) + δ(−→a ) = k−1X i=1 δaiai+1 + k−1X i=1 δai+1ai = k−1X i=1 (δaiai+1 + δai+1ai ) = 0, where the final equality uses proposition 5. Next, for three cycles, we first consider those compatible with linear triples. Lemma 16. If there are collinear points u ∈ Da, v ∈ Db, w ∈ Dc (a, b, c ∈ A), ρab + ρbc + ρca = 0. Proof. First, we prove for the case where v is between u and w. From lemma 8, there are compatible pairs (−→a , −→u ) ∈ LCuv ab , (−→a , −→u ) ∈ LCvw bc . Let |−→a | = ord(−→u ) = k and |−→a | = ord(−→u ) = l. We paste −→a and −→a together as −→a = (a = a1, a2, . . . , ak−1, ak, a1 , . . . , al = c), −→u and −→u as −→u = (u = u0, u1, . . . , uk = v = u0 , u1 , . . . , ul = w). Clearly (−→a , −→u ) ∈ LCuw ac and δ(−→a ) = k−1X i=1 δaiai+1 + δak a1 + l−1X i=1 δai ai+1 = δ(−→a ) + δbb + δ(−→a ) = δ(−→a ) + δ(−→a ). Therefore, ρac = δ(−→a ) = δ(−→a ) + δ(−→a ) = ρab + ρbc. Moreover, ρac = −ρca by lemma 15, so we get ρab + ρbc + ρca = 0. Now suppose w is between u and v. By the above argument, we have ρac + ρcb + ρba = 0 and by lemma 15, ρab + ρbc + ρca = −ρba − ρcb − ρac = 0. The case that u is between v and w is similar. Now we are ready for the zero three-cycle property: Lemma 17. ρab + ρbc + ρca = 0 for all a, b, c ∈ A. Proof. Let S = {(a, b, c) : ρab + ρbc + ρca = 0} and for contradiction, suppose S = ∅. S is finite. For each a ∈ A, choose va ∈ Da arbitrarily and let T be the convex hull of {va : a ∈ A}. For each (a, b, c) ∈ S, let Rabc = Da × Db × Dc ∩ T3 . Clearly, each Rabc is nonempty and compact. Moreover, by lemma 16, no (u, v, w) ∈ Rabc is collinear. Define f : D3 → R by f(u, v, w) = |v−u|+|w−v|+|u−w|. For (a, b, c) ∈ S, the restriction of f to the compact set Rabc attains a minimum m(a, b, c) at some point (u, v, w) ∈ Rabc by the continuity of f, i.e., there exists a triangle ∆uvw of minimum perimeter within T with u ∈ Da, v ∈ Db, w ∈ Dc. Choose (a∗ , b∗ , c∗ ) ∈ S so that m(a∗ , b∗ , c∗ ) is minimum and let (u∗ , v∗ , w∗ ) ∈ Ra∗b∗c∗ be a triple achieving it. Pick an arbitrary point p in the interior of ∆u∗ v∗ w∗ . By the convexity of domain D, there is d ∈ A such that p ∈ Dd. 291 Consider triangles ∆u∗ pw∗ , ∆w∗ pv∗ and ∆v∗ pu∗ . Since each of them has perimeter less than that of ∆u∗ v∗ w∗ and all three triangles are contained in T, by the minimality of ∆u∗ v∗ w∗ , (a∗ , d, c∗ ), (c∗ , d, b∗ ), (b∗ , d, a∗ ) ∈ S. Thus ρa∗d + ρdc∗ + ρc∗a∗ = 0, ρc∗d + ρdb∗ + ρb∗c∗ = 0, ρb∗d + ρda∗ + ρa∗b∗ = 0. Summing up the three equalities, (ρa∗d + ρdc∗ + ρc∗d + ρdb∗ + ρb∗d + ρda∗ ) +(ρc∗a∗ + ρb∗c∗ + ρa∗b∗ ) = 0, which yields a contradiction ρa∗b∗ + ρb∗c∗ + ρc∗a∗ = 0. With the zero two-cycle and three-cycle properties, the zero cycle property of Gf is immediate. As noted earlier, this completes the proof of theorem 4. Theorem 18. Every directed cycle of Gf has weight zero. Proof. Clearly, zero two-cycle and three-cycle properties imply triangle equality ρab +ρbc = ρac for all a, b, c ∈ A. For a directed cycle C = a1a2 . . . aka1, by inductively applying triangle equality, we have Pk−1 i=1 ρaiai+1 = ρa1ak . Therefore, the weight of C is k−1X i=1 ρaiai+1 + ρaka1 = ρa1ak + ρaka1 = 0. As final remarks, we note that our result implies the following strengthenings of theorem 12: Corollary 19. For any b, c ∈ A, every admissible (b, c)path has the same δ-weight ρbc. Proof. First notice that for any b, c ∈ A, if Db ∩Dc = ∅, δbc = ρbc. To see this, pick v ∈ Db ∩ Dc arbitrarily. Obviously, path −→a = (b, c) is compatible with linear sequence −→u = (v, v, v) and is thus a straight (b, c)-path. Hence ρbc = δ(−→a ) = δbc. Now for any b, c ∈ A and any (b, c)-path −→a with C(−→a ) = ∅, let −→u ∈ C(−→a ). Since ui ∈ Dai ∩ Dai+1 for i ∈ [|−→a | − 1], δ(−→a ) = |−→a |−1 X i=1 δaiai+1 = |−→a |−1 X i=1 ρaiai+1 , which by theorem 18, = −ρa|−→a |a1 = ρa1a|−→a | = ρbc. Corollary 20. For any b, c ∈ A, ρbc is equal to δ∗ bc, the minimum δ-weight over all (b, c)-paths. Proof. Clearly ρbc ≥ δ∗ bc by corollary 13. On the other hand, for every (b, c)-path −→a = (b = a1, a2, . . . , ak = c), by lemma 14, δ(−→a ) = k−1X i=1 δaiai+1 ≥ k−1X i=1 ρaiai+1 , which by theorem 18, = −ρaka1 = ρa1ak = ρbc. Hence ρbc ≤ δ∗ bc, which completes the proof. 7. COUNTEREXAMPLES TO STRONGER FORMS OF THEOREM 4 Theorem 4 applies to social choice functions with convex domain and finite range. We now show that neither of these hypotheses can be omitted. Our examples are single player functions. The first example illustrates that convexity can not be omitted. We present an untruthful single player social choice function with three outcomes a, b, c satisfying W-MON on a path-connected but non-convex domain. The domain is the boundary of a triangle whose vertices are x = (0, 1, −1), y = (−1, 0, 1) and z = (1, −1, 0). x and the open line segment zx is assigned outcome a, y and the open line segment xy is assigned outcome b, and z and the open line segment yz is assigned outcome c. Clearly, δab = −δba = δbc = −δcb = δca = −δac = −1, W-MON (the nonnegative twocycle property) holds. Since there is a negative cycle δab + δbc + δca = −3, by lemma 3, this is not a truthful choice function. We now show that the hypothesis of finite range can not be omitted. We construct a family of single player social choice functions each having a convex domain and an infinite number of outcomes, and satisfying weak monotonicity but not truthfulness. Our examples will be specified by a positive integer n and an n × n matrix M satisfying the following properties: (1) M is non-singular. (2) M is positive semidefinite. (3) There are distinct i1, i2, . . . , ik ∈ [n] satisfying k−1X j=1 (M(ij, ij) − M(ij , ij+1)) + (M(ik, ik) − M(ik, i1)) < 0. Here is an example matrix with n = 3 and (i1, i2, i3) = (1, 2, 3): 0 @ 0 1 −1 −1 0 1 1 −1 0 1 A Let e1, e2, . . . , en denote the standard basis of Rn . Let Sn denote the convex hull of {e1, e2 . . . , en}, which is the set of vectors in Rn with nonnegative coordinates that sum to 1. The range of our social choice function will be the set Sn and the domain D will be indexed by Sn, that is D = {yλ : λ ∈ Sn}, where yλ is defined below. The function f maps yλ to λ. Next we specify yλ. By definition, D must be a set of functions from Sn to R. For λ ∈ Sn, the domain element yλ : Sn −→ R is defined by yλ(α) = λT Mα. The nonsingularity of M guarantees that yλ = yµ for λ = µ ∈ Sn. It is easy to see that D is a convex subset of the set of all functions from Sn to R. The outcome graph Gf is an infinite graph whose vertex set is the outcome set A = Sn. For outcomes λ, µ ∈ A, the edge weight δλµ is equal to δλµ = inf{v(λ) − v(µ) : f(v) = λ} = yλ(λ) − yλ(µ) = λT Mλ − λT Mµ = λT M(λ − µ). We claim that Gf satisfies the nonnegative two-cycle property (W-MON) but has a negative cycle (and hence is not truthful). For outcomes λ, µ ∈ A, δλµ +δµλ = λT M(λ−µ)+µT M(µ−λ) = (λ−µ)T M(λ−µ), 292 which is nonnegative since M is positive semidefinite. Hence the nonnegative two-cycle property holds. Next we show that Gf has a negative cycle. Let i1, i2, . . . , ik be a sequence of indices satisfying property 3 of M. We claim ei1 ei2 . . . eik ei1 is a negative cycle. Since δeiej = eT i M(ei − ej) = M(i, i) − M(i, j) for any i, j ∈ [k], the weight of the cycle k−1X j=1 δeij eij+1 + δeik ei1 = k−1X j=1 (M(ij , ij ) − M(ij, ij+1)) + (M(ik, ik) − M(ik, i1)) < 0, which completes the proof. Finally, we point out that the third property imposed on the matrix M has the following interpretation. Let R(M) = {r1, r2, . . . , rn} be the set of row vectors of M and let hM be the single player social choice function with domain R(M) and range {1, 2, . . . , n} mapping ri to i. Property 3 is equivalent to the condition that the outcome graph GhM has a negative cycle. By lemma 3, this is equivalent to the condition that hM is untruthful. 8. FUTURE WORK As stated in the introduction, the goal underlying the work in this paper is to obtain useful and general characterizations of truthfulness. Let us say that a set D of P × A real valuation matrices is a WM-domain if any social choice function on D satisfying weak monotonicity is truthful. In this paper, we showed that for finite A, any convex D is a WM-domain. Typically, the domains of social choice functions considered in mechanism design are convex, but there are interesting examples with non-convex domains, e.g., combinatorial auctions with unknown single-minded bidders. It is intriguing to find the most general conditions under which a set D of real matrices is a WM-domain. We believe that convexity is the main part of the story, i.e., a WM-domain is, after excluding some exceptional cases, essentially a convex set. Turning to parametric representations, let us say a set D of P × A matrices is an AM-domain if any truthful social choice function with domain D is an affine maximizer. Roberts" theorem says that the unrestricted domain is an AM-domain. What are the most general conditions under which a set D of real matrices is an AM-domain? Acknowledgments We thank Ron Lavi for helpful discussions and the two anonymous referees for helpful comments. 9. REFERENCES [1] A. Archer and E. Tardos. Truthful mechanisms for one-parameter agents. In IEEE Symposium on Foundations of Computer Science, pages 482-491, 2001. [2] Y. Bartal, R. Gonen, and N. Nisan. Incentive compatible multi unit combinatorial auctions. In TARK "03: Proceedings of the 9th conference on Theoretical aspects of rationality and knowledge, pages 72-87. ACM Press, 2003. [3] S. Bikhchandani, S. Chatterjee, and A. Sen. Incentive compatibility in multi-unit auctions. Technical report, UCLA Department of Economics, Dec. 2004. [4] A. Goldberg, J. Hartline, A. Karlin, M. Saks and A. Wright. Competitive Auctions, 2004. [5] H. Gui, R. Muller, and R. Vohra. Dominant strategy mechanisms with multidimensional types. Technical Report 047, Maastricht: METEOR, Maastricht Research School of Economics of Technology and Organization, 2004. available at http://ideas.repec.org/p/dgr/umamet/2004047.html. [6] R. Lavi, A. Mu"alem, and N. Nisan. Towards a characterization of truthful combinatorial auctions. In FOCS "03: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, page 574. IEEE Computer Society, 2003. [7] D. Lehmann, L. O"Callaghan, and Y. Shoham. Truth revelation in approximately efficient combinatorial auctions. J. ACM, 49(5):577-602, 2002. [8] A. Mas-Colell, M. Whinston, and J. Green. Microeconomic Theory. Oxford University Press, 1995. [9] N. Nisan. Algorithms for selfish agents. Lecture Notes in Computer Science, 1563:1-15, 1999. [10] K. Roberts. The characterization of implementable choice rules. Aggregation and Revelation of Preferences, J-J. Laffont (ed.), North Holland Publishing Company. [11] J.-C. Rochet. A necessary and sufficient condition for rationalizability in a quasi-linear context. Journal of Mathematical Economics, 16:191-200, 1987. [12] I. Rozenshtrom. Dominant strategy implementation with quasi-linear preferences. Master"s thesis, Dept. of Economics, The Hebrew University, Jerusalem, Israel, 1999. 293

truthful implementation;weak monotonicity;non-truthful function;strategyproof;individual preference;social choice function;truthful;truthfulness;dominant strategy;recognition algorithm;affine maximizer;convex domain;nonnegative cycle property;mechanism design

train_J-63

Negotiation-Range Mechanisms: Exploring the Limits of Truthful Efficient Markets

This paper introduces a new class of mechanisms based on negotiation between market participants. This model allows us to circumvent Myerson and Satterthwaite"s impossibility result and present a bilateral market mechanism that is efficient, individually rational, incentive compatible and budget balanced in the single-unit heterogeneous setting. The underlying scheme makes this combination of desirable qualities possible by reporting a price range for each buyer-seller pair that defines a zone of possible agreements, while the final price is left open for negotiation.

1. INTRODUCTION In this paper we introduce the concept of negotiation based mechanisms in the context of the theory of efficient truthful markets. A market consists of multiple buyers and sellers who wish to exchange goods. The market"s main objective is to produce an allocation of sellers" goods to buyers as to maximize the total gain from trade. A commonly studied model of participant behavior is taken from the field of economic mechanism design [3, 4, 11]. In this model each player has a private valuation function that assigns real values to each possible allocation. The algorithm motivates players to participate truthfully by handing payments to them. The mechanism in an exchange collects buyer bids and seller bids and clears the exchange by computing:(i) a set of trades, and (ii) the payments made and received by players. In designing a mechanism to compute trades and payments we must consider the bidding strategies of self-interested players, i.e. rational players that follow expected-utility maximizing strategies. We set allocative efficiency as our primary goal. That is the mechanism must compute a set of trades that maximizes gain from trade. In addition we require individual rationality (IR) so that all players have positive expected utility to participate, budget balance (BB) so that the exchange does not run at a loss, and incentive compatibility (IC) so that reporting the truth is a dominant strategy for each player. Unfortunately, Myerson and Satterthwaite"s (1983) well known result demonstrates that in bilateral trade it is impossible to simultaneously achieve perfect efficiency, BB, and IR using an IC mechanism [10]. A unique approach to overcome Myerson and Satterthwaite"s impossibility result was attempted by Parkes, Kalagnanam and Eso [12]. This result designs both a regular and a combinatorial bilateral trade mechanism (which imposes BB and IR) that approximates truth revelation and allocation efficiency. In this paper we circumvent Myerson and Satterthwaite"s impossibility result by introducing a new model of negotiationrange markets. A negotiation-range mechanism does not produce payment prices for the market participants. Rather, is assigns each buyer-seller pair a price range, called Zone Of Possible Agreements (ZOPA). The buyer is provided with the high end of the range and the seller with the low end of the range. This allows the trading parties to engage in negotiation over the final price with guarantee that the deal is beneficial for both of them. The negotiation process is not considered part of the mechanism but left up to the interested parties, or to some external mechanism to perform. In effect a negotiation-range mechanism operates as a mediator between the market participants, offering them the grounds to be able to finalize the terms of the trade by themselves. This concept is natural to many real-world market environments such as the real estate market. 1 We focus on the single-unit heterogeneous setting: n sellers offer one unique good each by placing sealed bids specifying their willingness to sell, and m buyers, interested in buying a single good each, placing sealed bids specifying their willingness to pay for each good they may be interested in. Our main result is a single-unit heterogeneous bilateral trade negotiation-range mechanism (ZOPAS) that is efficient, individually rational, incentive compatible and budget balanced. Our result does not contradict Myerson and Satterthwaite"s important theorem. Myerson-Satterthwaite"s proof relies on a theorem assuring that in two different efficient IC markets; if the sellers have the same upper bound utility then they will receive the same prices in each market and if the buyers have the same lower bound utility then they will receive the same prices in each market. Our single-unit heterogeneous mechanism bypasses Myerson and Satterthwaite"s theorem by producing a price range, defined by a seller"s floor and a buyer"s ceiling, for each pair of matched players. In our market mechanism the seller"s upper bound utility may be the same while the seller"s floor is different and the buyer"s lower bound utility may be the same while the buyer"s ceiling is different. Moreover, the final price is not fixed by the mechanism at all. Instead, it is determined by an independent negotiation between the buyer and seller. More specifically, in a negotiation-range mechanism, the range of prices each matched pair is given is resolved by a negotiation stage where a final price is determined. This negotiation stage is crucial for our mechanism to be IC. Intuitively, a negotiation range mechanism is incentive compatible if truth telling promises the best ZOPA from the point of view of the player in question. That is, he would tell the truth if this strategy maximizes the upper and lower bounds on his utility as expressed by the ZOPA boundaries. Yet, when carefully examined it turns out that it is impossible (by [10]) that this goal will always hold. This is simply because such a mechanism could be easily modified to determine final prices for the players (e.g. by taking the average of the range"s boundaries). Here, the negotiation stage come into play. We show that if the above utility maximizing condition does not hold then it is the case that the player cannot influence the negotiation bound that is assigned to his deal partner no matter what value he declares. This means that the only thing that he may achieve by reporting a false valuation is modifying his own negotiation bound, something that he could alternatively achieve by reporting his true valuation and incorporating the effect of the modified negotiation bound into his negotiation strategy. This eliminates the benefit of reporting false valuations and allows our mechanism to compute the optimal gain from trade according to the players" true values. The problem of computing the optimal allocation which maximizes gain from trade can be conceptualized as the problem of finding the maximum weighted matching in a weighted bipartite graph connecting buyers and sellers, where each edge in the graph is assigned a weight equal to the difference between the respective buyer bid and seller bid. It is well known that this problem can be solved efficiently in polynomial time. VCG IC payment schemes [2, 7, 13] support efficient and IR bilateral trade but not simultaneously BB. Our particular approach adapts the VCG payment scheme to achieve budget balance. The philosophy of the VCG payment schemes in bilateral trade is that the buyer pays the seller"s opportunity cost of not selling the good to another buyer and not keeping the good to herself. The seller is paid in addition to the buyer"s price a compensation for the damage the mechanism did to the seller by not extracting the buyer"s full bid. Our philosophy is a bit different: The seller is paid at least her opportunity cost of not selling the good to another buyer and not keeping the good for herself. The buyer pays at most his alternate seller"s opportunity cost of not selling the good to another buyer and not keeping the alternate good for herself. The rest of this paper is organized as follows. In Section 2 we describe our model and definitions. In section 3 we present the single-unit heterogeneous negotiation-range mechanism and show that it is efficient, IR, IC and BB. Finally, we conclude with a discussion in Section 4. 2. NEGOTIATION MARKETS PRELIMINARIES Let Π denote the set of players, N the set of n selling players, and M the set of m buying players, where Π = N ∪ M. Let Ψ = {1, ..., t} denote the set of goods. Let Ti ∈ {−1, 0, 1}t denote an exchange vector for a trade, such that player i buys goods {A ∈ Ψ|Ti (A) = 1} and sells goods {A ∈ Ψ|Ti (A) = −1}. Let T = (T1, ..., T|Π|) denote the complete trade between all players. We view T as describing the allocation of goods by the mechanism to the buyers and sellers. In the single-unit heterogeneous setting every good belongs to specific seller, and every buyer is interested in buying one good. The buyer may bid for several or all goods. At the end of the auction every good is either assigned to one of the buyers who bid for it or kept unsold by the seller. It is convenient to assume the sets of buyers and sellers are disjoint (though it is not required), i.e. N ∩ M = ∅. Each seller i is associated with exactly one good Ai, for which she has true valuation ci which expresses the price at which it is beneficial for her to sell the good. If the seller reports a false valuation at an attempt to improve the auction results for her, this valuation is denoted ˆci. A buyer has a valuation vector describing his valuation for each of the goods according to their owner. Specifically, vj(k) denotes buyer j"s valuation for good Ak. Similarly, if he reports a false valuation it is denoted ˆvj(k). If buyer j is matched by the mechanism with seller i then Ti(Ai) = −1 and Tj(Ai) = 1. Notice, that in our setting for every k = i, Ti(Ak) = 0 and Tj(Ai) = 0 and also for every z = j, Tz(Ai) = 0. For a matched buyer j - seller i pair, the gain from trade on the deal is defined as vj(i) − ci. Given and allocation T, the gain from trade associated with T is V = j∈M,i∈N (vj(i) − ci) · Tj(Ai). Let T∗ denote the optimal allocation which maximizes the gain from trade, computed according to the players" true valuations. Let V ∗ denote the optimal gain from trade associated with this allocation. When players" report false valuations we use ˆT∗ and ˆV ∗ to denote the optimal allocation and gain from trade, respectively, when computed according to the reported valuations. 2 We are interested in the design of negotiation-range mechanisms. In contrast to a standard auction mechanism where the buyer and seller are provided with the prices they should pay, the goal of a negotiation-range mechanism is to provide the player"s with a range of prices within which they can negotiate the final terms of the deal by themselves. The mechanism would provide the buyer with the upper bound on the range and the seller with the lower bound on the range. This gives each of them a promise that it will be beneficial for them to close the deal, but does not provide information about the other player"s terms of negotiation. Definition 1. Negotiation Range: Zone Of Possible Agreements, ZOPA, between a matched buyer and seller. The ZOPA is a range, (L, H), 0 ≤ L ≤ H, where H is an upper bound (ceiling) price for the buyer and L is a lower bound (floor) price for the seller. Definition 2. Negotiation-Range Mechanism: A mechanism that computes a ZOPA, (L, H), for each matched buyer and seller in T∗ , and provides the buyer with the upper bound H and the seller with the lower bound L. The basic assumption is that participants in the auction are self-interested players. That is their main goal is to maximize their expected utility. The utility for a buyer who does not participate in the trade is 0. If he does win some good, his utility is the surplus between his valuation for that good and the price he pays. For a seller, if she keeps the good unsold, her utility is just her valuation of the good, and the surplus is 0. If she gets to sell it, her utility is the price she is paid for it, and the surplus is the difference between this price and her valuation. Since negotiation-range mechanisms assign bounds on the range of prices rather than the final price, it is useful to define the upper and lower bounds on the player"s utilities defined by the range"s limits. Definition 3. Consider a buyer j - seller i pair matched by a negotiation-range mechanism and let (L, H) be their associated negotiation range. • The buyer"s top utility is: vj(i) − L, and the buyer"s bottom utility is vj(i) − H. • The seller"s top utility is H, with surplus H − ci, and the seller"s bottom utility is L, with surplus L − ci. 3. THE SINGLE-UNIT HETEROGENEOUS MECHANISM (ZOPAS) 3.1 Description of the Mechanism ZOPAS is a negotiation-range mechanism, it finds the optimal allocation T∗ and uses it to define a ZOPA for each buyer-seller pair. The first stage in applying the mechanism is for the buyers and sellers to submit their sealed bids. The mechanism then allocates buyers to sellers by computing the allocation T ∗ , which results in the optimal gain from trade V ∗ , and defines a ZOPA for each buyer-seller pair. Finally, buyers and sellers use the ZOPA to negotiate a final price. Computing T∗ involves solving the maximum weighted bipartite matching problem for the complete bipartite graph Kn,m constructed by placing the buyers on one side of the Find the optimal allocation T ∗ Compute the maximum weighted bipartite matching for the bipartite graph of buyers and sellers, and edge weights equal to the gain from trade. Calculate Sellers" Floors For every buyer j, allocated good Ai Find the optimal allocation (T−j)∗ Li = vj(i) + (V−j)∗ − V ∗ Calculate Buyers" Ceilings For every buyer j, allocated good Ai Find the optimal allocation (T −i )∗ Find the optimal allocation (T −i −j )∗ Hj = vj(i) + (V −i −j )∗ − (V −i )∗ Negotiation Phase For every buyer j and every seller i of good Ai Report to seller i her floor Li and identify her matched buyer j Report to buyer j his ceiling Hj and identify his matched seller i i, j negotiate the good"s final price Figure 1: The ZOPAS mechanism graph, the seller on another and giving the edge between buyer j and seller i weight equal to vj(i) − ci. The maximum weighted matching problem in solvable in polynomial time (e.g., using the Hungarian Method). This results in a matching between buyers and sellers that maximizes gain from trade. The next step is to compute for each buyer-seller pair a seller"s floor, which provides the lower bound of the ZOPA for this pair, and assigns it to the seller. A seller"s floor is computed by calculating the difference between the total gain from trade when the buyer is excluded and the total gain from trade of the other participants when the buyer is included (the VCG principle). Let (T−j)∗ denote the gain from trade of the optimal allocation when buyer j"s bids are discarded. Denote by (V−j)∗ the total gain from trade in the allocation (T−j)∗ . Definition 4. Seller Floor: The lowest price the seller should expect to receive, communicated to the seller by the mechanism. The seller floor for player i who was matched with buyer j on good Ai, i.e., Tj(Ai) = 1, is computed as: Li = vj(i) + (V−j)∗ − V ∗ . The seller is instructed not to accept less than this price from her matched buyer. Next, the mechanism computes for each buyer-seller pair a buyer ceiling, which provides the upper bound of the ZOPA for this pair, and assigns it to the buyer. Each buyer"s ceiling is computed by removing the buyer"s matched seller and calculating the difference between the total gain from trade when the buyer is excluded and the total gain from trade of the other participants when the 3 buyer is included. Let (T−i )∗ denote the gain from trade of the optimal allocation when seller i is removed from the trade. Denote by (V −i )∗ the total gain from trade in the allocation (T−i )∗ . Let (T−i −j )∗ denote the gain from trade of the optimal allocation when seller i is removed from the trade and buyer j"s bids are discarded. Denote by (V −i −j )∗ the total gain from trade in the allocation (T −i −j )∗ . Definition 5. Buyer Ceiling: The highest price the seller should expect to pay, communicated to the buyer by the mechanism. The buyer ceiling for player j who was matched with seller i on good Ai, i.e., Tj(Ai) = 1, is computed as: Hj = vj(i) + (V −i −j )∗ − (V −i )∗ . The buyer is instructed not to pay more than this price to his matched seller. Once the negotiation range lower bound and upper bound are computed for every matched pair, the mechanism reports the lower bound price to the seller and the upper bound price to the buyer. At this point each buyer-seller pair negotiates on the final price and concludes the deal. A schematic description the ZOPAS mechanism is given in Figure 3.1. 3.2 Analysis of the Mechanism In this section we analyze the properties of the ZOPAS mechanism. Theorem 1. The ZOPAS market negotiation-range mechanism is an incentive-compatible bilateral trade mechanism that is efficient, individually rational and budget balanced. Clearly ZOPAS is an efficient polynomial time mechanism. Let us show it satisfies the rest of the properties in the theorem. Claim 1. ZOPAS is individually rational, i.e., the mechanism maintains nonnegative utility surplus for all participants. Proof. If a participant does not trade in the optimal allocation then his utility surplus is zero by definition. Consider a pair of buyer j and seller i which are matched in the optimal allocation T ∗ . Then the buyer"s utility is at least vj(i) − Hj. Recall that Hj = vj(i) + (V −i −j )∗ − (V −i )∗ , so that vj(i) − Hj = (V −i )∗ − (V −i −j )∗ . Since the optimal gain from trade which includes j is higher than that which does not, we have that the utility is nonnegative: vj(i) − Hj ≥ 0. Now, consider the seller i. Her utility surplus is at least Li − ci. Recall that Li = vj(i) + (V−j)∗ − V ∗ . If we removed from the optimal allocation T ∗ the contribution of the buyer j - seller i pair, we are left with an allocation which excludes j, and has value V ∗ − (vj(i) − ci). This implies that (V−j)∗ ≥ V ∗ − vj(i) + ci, which implies that Li − ci ≥ 0. The fact that ZOPAS is a budget-balanced mechanism follows from the following lemma which ensures the validity of the negotiation range, i.e., that every seller"s floor is below her matched buyer"s ceiling. This ensures that they can close the deal at a final price which lies in this range. Lemma 1. For every buyer j- seller i pair matched by the mechanism: Li ≤ Hj. Proof. Recall that Li = vj(i) + (V−j)∗ − V ∗ and Hj = vj(i)+(V −i −j )∗ −(V −i )∗ . To prove that Li ≤ Hj it is enough to show that (V −i )∗ + (V−j)∗ ≤ V ∗ + (V −i −j )∗ . (1) The proof of (1) is based on a method which we apply several times in our analysis. We start with the allocations (T−i )∗ and (T−j)∗ which together have value equal to (V −i )∗ + (V−j)∗ . We now use them to create a pair of new valid allocations, by using the same pairs that were matched in the original allocations. This means that the sum of values of the new allocations is the same as the original pair of allocations. We also require that one of the new allocations does not include buyer j or seller i. This means that the sum of values of these new allocations is at most V ∗ + (V −i −j )∗ , which proves (1). Let G be the bipartite graph where the nodes on one side of G represent the buyers and the nodes on the other side represent the sellers, and edge weights represent the gain from trade for the particular pair. The different allocations represent bipartite matchings in G. It will be convenient for the sake of our argument to think of the edges that belong to each of the matchings as being colored with a specific color representing this matching. Assign color 1 to the edges in the matching (T −i )∗ and assign color 2 to the edges in the matching (T−j)∗ . We claim that these edges can be recolored using colors 3 and 4 so that the new coloring represents allocations T (represented by color 3) and (T−i −j ) (represented by color 4). This implies the that inequality (1) holds. Figure 2 illustrates the graph G and the colorings of the different matchings. Define an alternating path P starting at j. Let S1 be the seller matched to j in (T −i )∗ (if none exists then P is empty). Let B1 be the buyer matched to S1 in (T−j)∗ , S2 be the seller matched to B1 in (T−i )∗ , B2 be the buyer matched to S2 in (T−j)∗ , and so on. This defines an alternating path P, starting at j, whose edges" colors alternate between colors 1 and 2 (starting with 1). This path ends either in a seller who is not matched in (T−j)∗ or in a buyer who is not matched in (T−i )∗ . Since all sellers in this path are matched in (T−i )∗ , we have that seller i does not belong to P. This ensures that edges in P may be colored by alternating colors 3 and 4 (starting with 3). Since except for the first edge, all others do not involve i or j and thus may be colored 4 and be part of an allocation (T −i −j ) . We are left to recolor the edges that do not belong to P. Since none of these edges includes j we have that the edges that were colored 1, which are part of (T −i )∗ , may now be colored 4, and be included in the allocation (T −i −j ) . It is also clear that the edges that were colored 2, which are part of (T−j)∗ , may now be colored 3, and be included in the allocation T . This completes the proof of the lemma. 3.3 Incentive Compatibility The basic requirement in mechanism design is for an exchange mechanism to be incentive compatible. This means that its payment structure enforces that truth-telling is the players" weakly dominant strategy, that is that the strategy by which the player is stating his true valuation results 4 ... jS1 S2 B1 S3 B2 B3S4 S5 B4 S6 B5 S8 B7 S7 B6 ... Figure 2: Alternating path argument for Lemma 1 (Validity of the Negotiation Range) and Claim 2 (part of Buyer"s IC proof) Colors Bidders 1 32 4 UnmatchedMatched Figure 3: Key to Figure 2 in bigger or equal utility as any other strategy. The utility surplus is defined as the absolute difference between the player"s bid and his price. Negotiation-range mechanisms assign bounds on the range of prices rather than the final price and therefore the player"s valuation only influences the minimum and maximum bounds on his utility. For a buyer the minimum (bottom) utility would be based on the top of the negotiation range (ceiling), and the maximum (top) utility would be based on the bottom of the negotiation range (floor). For a seller it"s the other way around. Therefore the basic natural requirement from negotiation-range mechanisms would be that stating the player"s true valuation results in both the higher bottom utility and higher top utility for the player, compared with other strategies. Unfortunately, this requirement is still too strong and it is impossible (by [10]) that this will always hold. Therefore we slightly relax it as follows: we require this holds when the false valuation based strategy changes the player"s allocation. When the allocation stays unchanged we require instead that the player would not be able to change his matched player"s bound (e.g. a buyer cannot change the seller"s floor). This means that the only thing he can influence is his own bound, something that he can alternatively achieve through means of negotiation. The following formally summarizes our incentive compatibility requirements from the negotiation-range mechanism. Buyer"s incentive compatibility: • Let j be a buyer matched with seller i by the mechanism according to valuation vj and the negotiationrange assigned is (Li, Hj). Assume that when the mechanism is applied according to valuation ˆvj, seller k = i is matched with j and the negotiation-range assigned is (ˆLk, ˆHj). Then vj(i) − Hj ≥ vj(k) − ˆHj. (2) vj(i) − Li ≥ vj(k) − ˆLk. (3) • Let j be a buyer not matched by the mechanism according to valuation vj. Assume that when the mechanism is applied according to valuation ˆvj, seller k = i is matched with j and the negotiation-range assigned is (ˆLk, ˆHj). Then vj(k) − ˆHj ≤ vj(k) − ˆLk ≤ 0. (4) • Let j be a buyer matched with seller i by the mechanism according to valuation vj and let the assigned bottom of the negotiation range (seller"s floor) be Li. Assume that when the mechanism is applied according to valuation ˆvj, the matching between i and j remains unchanged and let the assigned bottom of the negotiation range (seller"s floor) be ˆLi. Then, ˆLi = Li. (5) Notice that the first inequality of (4) always holds for a valid negotiation range mechanism (Lemma 1). Seller"s incentive compatibility: • Let i be a seller not matched by the mechanism according to valuation ci. Assume that when the mechanism 5 is applied according to valuation ˆci, buyer z = j is matched with i and the negotiation-range assigned is (ˆLi, ˆHz). Then ˆLi − ci ≤ ˆHz − ci ≤ 0. (6) • Let i be a buyer matched with buyer j by the mechanism according to valuation ci and let the assigned top of the negotiation range (buyer"s ceiling) be Hj. Assume that when the mechanism is applied according to valuation ˆci, the matching between i and j remains unchanged and let the assigned top of the negotiation range (buyer"s ceiling) be ˆHj. Then, ˆHj = Hj. (7) Notice that the first inequality of (6) always holds for a valid negotiation range mechanism (Lemma 1). Observe that in the case of sellers in our setting, the case expressed by requirement (6) is the only case in which the seller may change the allocation to her benefit. In particular, it is not possible for seller i who is matched in T ∗ to change her buyer by reporting a false valuation. This fact simply follows from the observation that reducing the seller"s valuation increases the gain from trade for the current allocation by at least as much than any other allocation, whereas increasing the seller"s valuation decreases the gain from trade for the current allocation by exactly the same amount as any other allocation in which it is matched. Therefore, the only case the optimal allocation may change is when in the new allocation i is not matched in which case her utility surplus is 0. Theorem 2. ZOPAS is an incentive compatible negotiationrange mechanism. Proof. We begin with the incentive compatibility for buyers. Consider a buyer j who is matched with seller i according to his true valuation v. Consider that j is reporting instead a false valuation ˆv which results in a different allocation in which j is matched with seller k = i. The following claim shows that a buyer j which changed his allocation due to a false declaration of his valuation cannot improve his top utility. Claim 2. Let j be a buyer matched to seller i in T ∗ , and let k = i be the seller matched to j in ˆT∗ . Then, vj(i) − Hj ≥ vj(k) − ˆHj. (8) Proof. Recall that Hj = vj(i) + (V −i −j )∗ − (V −i )∗ and ˆHj = ˆvj(k) + ( ˆV −k −j )∗ − ( ˆV −k )∗ . Therefore, vj(i) − Hj = (V −i )∗ − (V −i −j )∗ and vj(k) − ˆHj = vj(k) − ˆvj(k) + ( ˆV −k )∗ − ( ˆV −k −j )∗ . It follows that in order to prove (8) we need to show ( ˆV −k )∗ + (V −i −j )∗ ≤ (V −i )∗ + ( ˆV −k −j )∗ + ˆvj(k) − vj(k). (9) Consider first the case were j is matched to i in ( ˆT−k )∗ . If we remove this pair and instead match j with k we obtain a matching which excludes i, if the gain from trade on the new pair is taken according to the true valuation then we get ( ˆV −k )∗ − (ˆvj(i) − ci) + (vj(k) − ck) ≤ (V −i )∗ . Now, since the optimal allocation ˆT∗ matches j with k rather than with i we have that (V −i −j )∗ + (ˆvj(i) − ci) ≤ ˆV ∗ = ( ˆV −k −j )∗ + (ˆvj(k) − ck), where we have used that ( ˆV −i −j )∗ = (V −i −j )∗ since these allocations exclude j. Adding up these two inequalities implies (9) in this case. It is left to prove (9) when j is not matched to i in ( ˆT−k )∗ . In fact, in this case we prove the stronger inequality ( ˆV −k )∗ + (V −i −j )∗ ≤ (V −i )∗ + ( ˆV −k −j )∗ . (10) It is easy to see that (10) indeed implies (9) since it follows from the fact that k is assigned to j in ˆT∗ that ˆvj(k) ≥ vj(k). The proof of (10) works as follows. We start with the allocations ( ˆT−k )∗ and (T−i −j )∗ which together have value equal to ( ˆV −k )∗ + (V −i −j )∗ . We now use them to create a pair of new valid allocations, by using the same pairs that were matched in the original allocations. This means that the sum of values of the new allocations is the same as the original pair of allocations. We also require that one of the new allocations does not include seller i and is based on the true valuation v, while the other allocation does not include buyer j or seller k and is based on the false valuation ˆv. This means that the sum of values of these new allocations is at most (V −i )∗ + ( ˆV −k −j )∗ , which proves (10). Let G be the bipartite graph where the nodes on one side of G represent the buyers and the nodes on the other side represent the sellers, and edge weights represent the gain from trade for the particular pair. The different allocations represent bipartite matchings in G. It will be convenient for the sake of our argument to think of the edges that belong to each of the matchings as being colored with a specific color representing this matching. Assign color 1 to the edges in the matching ( ˆT−k )∗ and assign color 2 to the edges in the matching (T −i −j )∗ . We claim that these edges can be recolored using colors 3 and 4 so that the new coloring represents allocations (T −i ) (represented by color 3) and ( ˆT−k −j ) (represented by color 4). This implies the that inequality (10) holds. Figure 2 illustrates the graph G and the colorings of the different matchings. Define an alternating path P starting at j. Let S1 = i be the seller matched to j in ( ˆT−k )∗ (if none exists then P is empty). Let B1 be the buyer matched to S1 in (T−i −j )∗ , S2 be the seller matched to B1 in ( ˆT−k )∗ , B2 be the buyer matched to S2 in (T−i −j )∗ , and so on. This defines an alternating path P, starting at j, whose edges" colors alternate between colors 1 and 2 (starting with 1). This path ends either in a seller who is not matched in (T −i −j )∗ or in a buyer who is not matched in ( ˆT−k )∗ . Since all sellers in this path are matched in ( ˆT−k )∗ , we have that seller k does not belong to P. Since in this case S1 = i and the rest of the sellers in P are matched in (T−i −j )∗ we have that seller i as well does not belong to P. This ensures that edges in P may be colored by alternating colors 3 and 4 (starting with 3). Since S1 = i, we may use color 3 for the first edge and thus assign it to the allocation (T−i ) . All other edges, do not involve i, j or k and thus may be either colored 4 and be part of an allocation ( ˆT−k −j ) or colored 3 and be part of an allocation (T−i ) , in an alternating fashion. We are left to recolor the edges that do not belong to P. Since none of these edges includes j we have that the edges 6 that were colored 1, which are part of ( ˆT−k )∗ , may now be colored 4, and be included in the allocation ( ˆT−k −j ) . It is also clear that the edges that were colored 2, which are part of (T−i −j )∗ , may now be colored 3, and be included in the allocation (T−i ) . This completes the proof of (10) and the claim. The following claim shows that a buyer j which changed his allocation due to a false declaration of his valuation cannot improve his bottom utility. The proof is basically the standard VCG argument. Claim 3. Let j be a buyer matched to seller i in T ∗ , and k = i be the seller matched to j in ˆT∗ . Then, vj(i) − Li ≥ vj(k) − ˆLk. (11) Proof. Recall that Li = vj(i) + (V−j)∗ − V ∗ , and ˆLk = ˆvj(k) + ( ˆV−j)∗ − ˆV ∗ = ˆvj(k) + (V−j)∗ − ˆV ∗ . Therefore, vj(i) − Li = V ∗ − (V−j)∗ and vj(k) − ˆLk = vj(k) − ˆvj(k) + ˆV ∗ − (V−j)∗ . It follows that in order to prove (11) we need to show V ∗ ≥ vj(k) − ˆvj(k) + ˆV ∗ . (12) The scenario of this claim occurs when j understates his value for Ai or overstated his value for Ak. Consider these two cases: • ˆvj(k) > vj(k): Since Ak was allocated to j in the allocation ˆT∗ we have that using the allocation of ˆT∗ according to the true valuation gives an allocation of value U satisfying ˆV ∗ − ˆvj(k) + vj(k) ≤ U ≤ V ∗ . • ˆvj(k) = vj(k) and ˆvj(i) < vj(i): In this case (12) reduces to V ∗ ≥ ˆV ∗ . Since j is not allocated i in ˆT∗ we have that ˆT∗ is an allocation that uses only true valuations. From the optimality of T ∗ we conclude that V ∗ ≥ ˆV ∗ . Another case in which a buyer may try to improve his utility is when he does not win any good by stating his true valuation. He may give a false valuation under which he wins some good. The following claim shows that doing this is not beneficial to him. Claim 4. Let j be a buyer not matched in T ∗ , and assume seller k is matched to j in ˆT∗ . Then, vj(k) − ˆLk ≤ 0. Proof. The scenario of this claim occurs if j did not buy in the truth-telling allocation and overstates his value for Ak, ˆvj(k) > vj(k) in his false valuation. Recall that ˆLk = ˆvj(k) + ( ˆV−j)∗ − ˆV ∗ . Thus we need to show that 0 ≥ vj(k) − ˆvj(k) + ˆV ∗ − (V−j)∗ . Since j is not allocated in T∗ then (V−j)∗ = V ∗ . Since j is allocated Ak in ˆT∗ we have that using the allocation of ˆT∗ according to the true valuation gives an allocation of value U satisfying ˆV ∗ − ˆvj(k) + vj(k) ≤ U ≤ V ∗ . Thus we can conclude that 0 ≥ vj(k) − ˆvj(k) + ˆV ∗ − (V−j)∗ . Finally, the following claim ensures that a buyer cannot influence the floor bound of the ZOPA for the good he wins. Claim 5. Let j be a buyer matched to seller i in T ∗ , and assume that ˆT∗ = T∗ , then ˆLi = Li. Proof. Recall that Li = vj(i) + (V−j)∗ − V ∗ , and ˆLi = ˆvj(i) + ( ˆV−j)∗ − ˆV ∗ = ˆvj(i) + (V−j)∗ − ˆV ∗ . Therefore we need to show that ˆV ∗ = V ∗ + ˆvj(i) − vj(i). Since j is allocated Ai in T∗ , we have that using the allocation of T∗ according to the false valuation gives an allocation of value U satisfying V ∗ − vj(i) + ˆvj(i) ≤ U ≤ ˆV ∗ . Similarly since j is allocated Ai in ˆT∗ , we have that using the allocation of ˆT∗ according to the true valuation gives an allocation of value U satisfying ˆV ∗ − ˆvj(i)+vj(i) ≤ U ≤ V ∗ , which together with the previous inequality completes the proof. This completes the analysis of the buyer"s incentive compatibility. We now turn to prove the seller"s incentive compatibility properties of our mechanism. The following claim handles the case where a seller that was not matched in T ∗ falsely understates her valuation such that she gets matched n ˆT∗ . Claim 6. Let i be a seller not matched in T ∗ , and assume buyer z is matched to i in ˆT∗ . Then, ˆHz − ci ≤ 0. Proof. Recall that ˆHz = vz(i) + ( ˆV −i −z )∗ − ( ˆV −i )∗ . Since i is not matched in T ∗ and ( ˆT−i )∗ involves only true valuations we have that ( ˆV −i )∗ = V ∗ . Since i is matched with z in ˆT∗ it can be obtained by adding the buyer z - seller i pair to ( ˆT−i −z)∗ . It follows that ˆV ∗ = ( ˆV −i −z )∗ + vz(i) − ˆci. Thus, we have that ˆHz = ˆV ∗ + ˆci − V ∗ . Now, since i is matched in ˆT∗ , using this allocation according to the true valuation gives an allocation of value U satisfying ˆV ∗ + ˆci − ci ≤ U ≤ V ∗ . Therefore ˆHz −ci = ˆV ∗ +ˆci −V ∗ −ci ≤ 0. Finally, the following simple claim ensures that a seller cannot influence the ceiling bound of the ZOPA for the good she sells. Claim 7. Let i be a seller matched to buyer j in T ∗ , and assume that ˆT∗ = T∗ , then ˆHj = Hj. Proof. Since ( ˆV −i −j )∗ = (V −i −j )∗ and ( ˆV −i )∗ = (V −i )∗ it follows that ˆHj = vj(i)+( ˆV −i −j )∗ −( ˆV −i )∗ = vj(i)+(V −i −j )∗ −(V −i )∗ = Hj. 4. CONCLUSIONS AND EXTENSIONS In this paper we suggest a way to deal with the impossibility of producing mechanisms which are efficient, individually rational, incentive compatible and budget balanced. To this aim we introduce the concept of negotiation-range mechanisms which avoid the problem by leaving the final determination of prices to a negotiation between the buyer and seller. The goal of the mechanism is to provide the initial range (ZOPA) for negotiation in a way that it will be beneficial for the participants to close the proposed deals. We present a negotiation range mechanism that is efficient, individually rational, incentive compatible and budget balanced. The ZOPA produced by our mechanism is based 7 on a natural adaptation of the VCG payment scheme in a way that promises valid negotiation ranges which permit a budget balanced allocation. The basic question that we aimed to tackle seems very exciting: which properties can we expect a market mechanism to achieve ? Are there different market models and requirements from the mechanisms that are more feasible than classic mechanism design goals ? In the context of our negotiation-range model, is natural to further study negotiation based mechanisms in more general settings. A natural extension is that of a combinatorial market. Unfortunately, finding the optimal allocation in a combinatorial setting is NP-hard, and thus the problem of maintaining BB is compounded by the problem of maintaining IC when efficiency is approximated [1, 5, 6, 9, 11]. Applying the approach in this paper to develop negotiationrange mechanisms for combinatorial markets, even in restricted settings, seems a promising direction for research. 5. REFERENCES [1] Y. Bartal, R. Gonen, and N. Nisan. Incentive Compatible Multi-Unit Combinatorial Auctions. Proceeding of 9th TARK 2003, pp. 72-87, June 2003. [2] E. H. Clarke. Multipart Pricing of Public Goods. In journal Public Choice 1971, volume 2, pages 17-33. [3] J.Feigenbaum, C. Papadimitriou, and S. Shenker. Sharing the Cost of Multicast Transmissions. Journal of Computer and System Sciences, 63(1),2001. [4] A. Fiat, A. Goldberg, J. Hartline, and A. Karlin. Competitive Generalized Auctions. Proceeding of 34th ACM Symposium on Theory of Computing,2002. [5] R. Gonen, and D. Lehmann. Optimal Solutions for Multi-Unit Combinatorial Auctions: Branch and Bound Heuristics. Proceeding of ACM Conference on Electronic Commerce EC"00, pages 13-20, October 2000. [6] R. Gonen, and D. Lehmann. Linear Programming helps solving Large Multi-unit Combinatorial Auctions. In Proceeding of INFORMS 2001, November, 2001. [7] T. Groves. Incentives in teams. In journal Econometrica 1973, volume 41, pages 617-631. [8] R. Lavi, A. Mu"alem and N. Nisan. Towards a Characterization of Truthful Combinatorial Auctions. Proceeding of 44th Annual IEEE Symposium on Foundations of Computer Science,2003. [9] D. Lehmann, L. I. O"Callaghan, and Y. Shoham. Truth revelation in rapid, approximately efficient combinatorial auctions. In Proceedings of the First ACM Conference on Electronic Commerce, pages 96-102, November 1999. [10] R. Myerson, M. Satterthwaite. Efficient Mechanisms for Bilateral Trading. Journal of Economic Theory, 28, pages 265-81, 1983. [11] N. Nisan and A. Ronen. Algorithmic Mechanism Design. In Proceeding of 31th ACM Symposium on Theory of Computing, 1999. [12] D.C. Parkes, J. Kalagnanam, and M. Eso. Achieving Budget-Balance with Vickrey-Based Payment Schemes in Exchanges. Proceeding of 17th International Joint Conference on Artificial Intelligence, pages 1161-1168, 2001. [13] W. Vickrey. Counterspeculation, Auctions and Competitive Sealed Tenders. In Journal of Finance 1961, volume 16, pages 8-37. 8

negotiation-range mechanism;real-world market environment;efficient market;impossibility result;negotiationrange market;zone of possible agreement;negotiation based mechanism;incentive compatibility;utility;efficient truthful market;individual rationality;good exchange;possible agreement zone;buyer and seller;mechanism design

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Privacy in Electronic Commerce and the Economics of Immediate Gratification

Dichotomies between privacy attitudes and behavior have been noted in the literature but not yet fully explained. We apply lessons from the research on behavioral economics to understand the individual decision making process with respect to privacy in electronic commerce. We show that it is unrealistic to expect individual rationality in this context. Models of self-control problems and immediate gratification offer more realistic descriptions of the decision process and are more consistent with currently available data. In particular, we show why individuals who may genuinely want to protect their privacy might not do so because of psychological distortions well documented in the behavioral literature; we show that these distortions may affect not only ‘na¨ıve" individuals but also ‘sophisticated" ones; and we prove that this may occur also when individuals perceive the risks from not protecting their privacy as significant.

1. PRIVACY AND ELECTRONIC COMMERCE Privacy remains an important issue for electronic commerce. A PriceWaterhouseCoopers study in 2000 showed that nearly two thirds of the consumers surveyed would shop more online if they knew retail sites would not do anything with their personal information [15]. A Federal Trade Commission study reported in 2000 that sixty-seven percent of consumers were very concerned about the privacy of the personal information provided on-line [11]. More recently, a February 2002 Harris Interactive survey found that the three biggest consumer concerns in the area of on-line personal information security were: companies trading personal data without permission, the consequences of insecure transactions, and theft of personal data [19]. According to a Jupiter Research study in 2002, $24.5 billion in on-line sales will be lost by 2006 - up from $5.5 billion in 2001. Online retail sales would be approximately twenty-four percent higher in 2006 if consumers" fears about privacy and security were addressed effectively [21]. Although the media hype has somewhat diminished, risks and costs have notas evidenced by the increasing volumes of electronic spam and identity theft [16]. Surveys in this field, however, as well as experiments and anecdotal evidence, have also painted a different picture. [36, 10, 18, 21] have found evidence that even privacy concerned individuals are willing to trade-off privacy for convenience, or bargain the release of very personal information in exchange for relatively small rewards. The failure of several on-line services aimed at providing anonymity for Internet users [6] offers additional indirect evidence of the reluctance by most individuals to spend any effort in protecting their personal information. The dichotomy between privacy attitudes and behavior has been highlighted in the literature. Preliminary interpretations of this phenomenon have been provided [2, 38, 33, 40]. Still missing are: an explanation grounded in economic or psychological theories; an empirical validation of the proposed explanation; and, of course, the answer to the most recurring question: should people bother at all about privacy? In this paper we focus on the first question: we formally analyze the individual decision making process with respect to privacy and its possible shortcomings. We focus on individual (mis)conceptions about their handling of risks they face when revealing private information. We do not address the issue of whether people should actually protect themselves. We will comment on that in Section 5, where we will also discuss strategies to empirically validate our theory. We apply lessons from behavioral economics. Traditional economics postulates that people are forward-looking and bayesian updaters: they take into account how current behavior will influence their future well-being and preferences. For example, [5] study rational models of addiction. This approach can be compared to those who see in the decision 21 not to protect one"s privacy a rational choice given the (supposedly) low risks at stake. However, developments in the area of behavioral economics have highlighted various forms of psychological inconsistencies (self-control problems, hyperbolic discounting, present-biases, etc.) that clash with the fully rational view of the economic agent. In this paper we draw from these developments to reach the following conclusions: • We show that it is unlikely that individuals can act rationally in the economic sense when facing privacy sensitive decisions. • We show that alternative models of personal behavior and time-inconsistent preferences are compatible with the dichotomy between attitudes and behavior and can better match current data. For example, they can explain the results presented by [36] at the ACM EC "01 conference. In their experiment, self-proclaimed privacy advocates were found to be willing to reveal varying amounts of personal information in exchange for small rewards. • In particular, we show that individuals may have a tendency to under-protect themselves against the privacy risks they perceive, and over-provide personal information even when wary of (perceived) risks involved. • We show that the magnitude of the perceived costs of privacy under certain conditions will not act as deterrent against behavior the individual admits is risky. • We show, following similar studies in the economics of immediate gratification [31], that even ‘sophisticated" individuals may under certain conditions become ‘privacy myopic." Our conclusion is that simply providing more information and awareness in a self-regulative environment is not sufficient to protect individual privacy. Improved technologies, by lowering costs of adoption and protection, certainly can help. However, more fundamental human behavioral responses must also be addressed if privacy ought to be protected. In the next section we propose a model of rational agents facing privacy sensitive decisions. In Section 3 we show the difficulties that hinder any model of privacy decision making based on full rationality. In Section 4 we show how behavioral models based on immediate gratification bias can better explain the attitudes-behavior dichotomy and match available data. In Section 5 we summarize and discuss our conclusions. 2. A MODEL OF RATIONALITY IN PRIVACY DECISION MAKING Some have used the dichotomy between privacy attitudes and behavior to claim that individuals are acting rationally when it comes to privacy. Under this view, individuals may accept small rewards for giving away information because they expect future damages to be even smaller (when discounted over time and with their probability of occurrence). Here we want to investigate what underlying assumptions about personal behavior would support the hypothesis of full rationality in privacy decision making. Since [28, 37, 29] economists have been interested in privacy, but only recently formal models have started appearing [3, 7, 39, 40]. While these studies focus on market interactions between one agent and other parties, here we are interested in formalizing the decision process of the single individual. We want to see if individuals can be economically rational (forward-lookers, bayesian updaters, utility maximizers, and so on) when it comes to protect their own personal information. The concept of privacy, once intended as the right to be left alone [41], has transformed as our society has become more information oriented. In an information society the self is expressed, defined, and affected through and by information and information technology. The boundaries between private and public become blurred. Privacy has therefore become more a class of multifaceted interests than a single, unambiguous concept. Hence its value may be discussed (if not ascertained) only once its context has also been specified. This most often requires the study of a network of relations between a subject, certain information (related to the subject), other parties (that may have various linkages of interest or association with that information or that subject), and the context in which such linkages take place. To understand how a rational agent could navigate through those complex relations, in Equation 1 we abstract the decision process of an idealized rational economic agent who is facing privacy trade-offs when completing a certain transaction. max d Ut = δ vE (a) , pd (a) + γ vE (t) , pd (t) − cd t (1) In Equation 1, δ and γ are unspecified functional forms that describe weighted relations between expected payoffs from a set of events v and the associated probabilities of occurrence of those events p. More precisely, the utility U of completing a transaction t (the transaction being any action - not necessarily a monetary operation - possibly involving exposure of personal information) is equal to some function of the expected payoff vE (a) from maintaining (or not) certain information private during that transaction, and the probability of maintaining [or not maintaining] that information private when using technology d, pd (a) [1 − pd (a)]; plus some function of the expected payoff vE (t) from completing (or non completing) the transaction (possibly revealing personal information), and the probability of completing [or not completing] that transaction with a certain technology d, pd (t) [1 − pd (t)]; minus the cost of using the technology t: cd t .1 The technology d may or may not be privacy enhancing. Since the payoffs in Equation 1 can be either positive or negative, Equation 1 embodies the duality implicit in privacy issues: there are both costs and benefits gained from revealing or from protecting personal information, and the costs and benefits from completing a transaction, vE (t), might be distinct from the costs and benefits from keeping the associated information private, vE (a). For instance, revealing one"s identity to an on-line bookstore may earn a discount. Viceversa, it may also cost a larger bill, because of price discrimination. Protecting one"s financial privacy by not divulging credit card information on-line may protect against future losses and hassles related to identity theft. But it may 1 See also [1]. 22 make one"s on-line shopping experience more cumbersome, and therefore more expensive. The functional parameters δ and γ embody the variable weights and attitudes an individual may have towards keeping her information private (for example, her privacy sensitivity, or her belief that privacy is a right whose respect should be enforced by the government) and completing certain transactions. Note that vE and p could refer to sets of payoffs and the associated probabilities of occurrence. The payoffs are themselves only expected because, regardless of the probability that the transaction is completed or the information remains private, they may depend on other sets of events and their associated probabilities. vE() and pd (), in other words, can be read as multi-variate parameters inside which are hidden several other variables, expectations, and functions because of the complexity of the privacy network described above. Over time, the probability of keeping certain information private, for instance, will not only depend on the chosen technology d but also on the efforts by other parties to appropriate that information. These efforts may be function, among other things, of the expected value of that information to those parties. The probability of keeping information private will also depend on the environment in which the transaction is taking place. Similarly, the expected benefit from keeping information private will also be a collection over time of probability distributions dependent on several parameters. Imagine that the probability of keeping your financial transactions private is very high when you use a bank in Bermuda: still, the expected value from keeping your financial information confidential will depend on a number of other factors. A rational agent would, in theory, choose the technology d that maximizes her expected payoff in Equation 1. Maybe she would choose to complete the transaction under the protection of a privacy enhancing technology. Maybe she would complete the transaction without protection. Maybe she would not complete the transaction at all (d = 0). For example, the agent may consider the costs and benefits of sending an email through an anonymous MIX-net system [8] and compare those to the costs and benefits of sending that email through a conventional, non-anonymous channel. The magnitudes of the parameters in Equation 1 will change with the chosen technology. MIX-net systems may decrease the expected losses from privacy intrusions. Nonanonymous email systems may promise comparably higher reliability and (possibly) reduced costs of operations. 3. RATIONALITY AND PSYCHOLOGICAL DISTORTIONS IN PRIVACY Equation 1 is a comprehensive (while intentionally generic) road-map for navigation across privacy trade-offs that no human agent would be actually able to use. We hinted to some difficulties as we noted that several layers of complexities are hidden inside concepts such as the expected value of maintaining certain information private, and the probability of succeeding doing so. More precisely, an agent will face three problems when comparing the tradeoffs implicit in Equation 1: incomplete information about all parameters; bounded power to process all available information; no deviation from the rational path towards utilitymaximization. Those three problems are precisely the same issues real people have to deal with on an everyday basis as they face privacy-sensitive decisions. We discuss each problem in detail. 1. Incomplete information. What information has the individual access to as she prepares to take privacy sensitive decisions? For instance, is she aware of privacy invasions and the associated risks? What is her knowledge of the existence and characteristics of protective technologies? Economic transactions are often characterized by incomplete or asymmetric information. Different parties involved may not have the same amount of information about the transaction and may be uncertain about some important aspects of it [4]. Incomplete information will affect almost all parameters in Equation 1, and in particular the estimation of costs and benefits. Costs and benefits associated with privacy protection and privacy intrusions are both monetary and immaterial. Monetary costs may for instance include adoption costs (which are probably fixed) and usage costs (which are variable) of protective technologies - if the individual decides to protect herself. Or they may include the financial costs associated to identity theft, if the individual"s information turns out not to have been adequately protected. Immaterial costs may include learning costs of a protective technology, switching costs between different applications, or social stigma when using anonymizing technologies, and many others. Likewise, the benefits from protecting (or not protecting) personal information may also be easy to quantify in monetary terms (the discount you receive for revealing personal data) or be intangible (the feeling of protection when you send encrypted emails). It is difficult for an individual to estimate all these values. Through information technology, privacy invasions can be ubiquitous and invisible. Many of the payoffs associated with privacy protection or intrusion may be discovered or ascertained only ex post through actual experience. Consider, for instance, the difficulties in using privacy and encrypting technologies described in [43]. In addition, the calculations implicit in Equation 1 depend on incomplete information about the probability distribution of future events. Some of those distributions may be predicted after comparable data - for example, the probability that a certain credit card transaction will result in fraud today could be calculated using existing statistics. The probability distributions of other events may be very difficult to estimate because the environment is too dynamicfor example, the probability of being subject to identity theft 5 years in the future because of certain data you are releasing now. And the distributions of some other events may be almost completely subjective - for example, the probability that a new and practical form of attack on a currently secure cryptosystem will expose all of your encrypted personal communications a few years from now. This leads to a related problem: bounded rationality. 2. Bounded rationality. Is the individual able to calculate all the parameters relevant to her choice? Or is she limited by bounded rationality? In our context, bounded rationality refers to the inability to calculate and compare the magnitudes of payoffs associated with various strategies the individual may choose in privacy-sensitive situations. It also refers to the inability to process all the stochastic information related to risks and probabilities of events leading to privacy costs and benefits. 23 In traditional economic theory, the agent is assumed to have both rationality and unbounded ‘computational" power to process information. But human agents are unable to process all information in their hands and draw accurate conclusions from it [34]. In the scenario we consider, once an individual provides personal information to other parties, she literally loses control of that information. That loss of control propagates through other parties and persists for unpredictable spans of time. Being in a position of information asymmetry with respect to the party with whom she is transacting, decisions must be based on stochastic assessments, and the magnitudes of the factors that may affect the individual become very difficult to aggregate, calculate, and compare.2 Bounded rationality will affect the calculation of the parameters in Equation 1, and in particular δ, γ, vE(), and pt(). The cognitive costs involved in trying to calculate the best strategy could therefore be so high that the individual may just resort to simple heuristics. 3. Psychological distortions. Eventually, even if an individual had access to complete information and could appropriately compute it, she still may find it difficult to follow the rational strategy presented in Equation 1. A vast body of economic and psychological literature has by now confirmed the impact of several forms of psychological distortions on individual decision making. Privacy seems to be a case study encompassing many of those distortions: hyperbolic discounting, under insurance, self-control problems, immediate gratification, and others. The traditional dichotomy between attitude and behavior, observed in several aspects of human psychology and studied in the social psychology literature since [24] and [13], may also appear in the privacy space because of these distortions. For example, individuals have a tendency to discount ‘hyperbolically" future costs or benefits [31, 27]. In economics, hyperbolic discounting implies inconsistency of personal preferences over time - future events may be discounted at different discount rates than near-term events. Hyperbolic discounting may affect privacy decisions, for instance when we heavily discount the (low) probability of (high) future risks such as identity theft.3 Related to hyperbolic discounting is the tendency to underinsure oneself against certain risks [22]. In general, individuals may put constraints on future behavior that limit their own achievement of maximum utility: people may genuinely want to protect themselves, but because of self-control bias, they will not actually take those steps, and opt for immediate gratification instead. People tend to underappreciate the effects of changes in their states, and hence falsely project their current preferences over consumption onto their future preferences. Far more than suggesting merely that people mispredict future tastes, this projection bias posits a systematic pattern in these mispredictions which can lead to systematic errors in dynamicchoice environments [25, p. 2]. 2 The negative utility coming from future potential misuses of somebody"s personal information could be a random shock whose probability and scope are extremely variable. For example, a small and apparently innocuous piece of information might become a crucial asset or a dangerous liability in the right context. 3 A more rigorous description and application of hyperbolic discounting is provided in Section 4. In addition, individuals suffer from optimism bias [42], the misperception that one"s risks are lower than those of other individuals under similar conditions. Optimism bias may lead us to believe that we will not be subject to privacy intrusions. Individuals encounter difficulties when dealing with cumulative risks. [35], for instance, shows that while young smokers appreciate the long term risks of smoking, they do not fully realize the cumulative relation between the low risks of each additional cigarette and the slow building up of a serious danger. Difficulties with dealing with cumulative risks apply to privacy, because our personal information, once released, can remain available over long periods of time. And since it can be correlated to other data, the ‘anonymity sets" [32, 14] in which we wish to remain hidden get smaller. As a result, the whole risk associated with revealing different pieces of personal information is more than the sum of the individual risks associated with each piece of data. Also, it is easier to deal with actions and effects that are closer to us in time. Actions and effects that are in the distant future are difficult to focus on given our limited foresight perspective. As the foresight changes, so does behavior, even when preferences remain the same [20]. This phenomenon may also affects privacy decisions, since the costs of privacy protection may be immediate, but the rewards may be invisible (absence of intrusions) and spread over future periods of time. To summarize: whenever we face privacy sensitive decisions, we hardly have all data necessary for an informed choice. But even if we had, we would be likely unable to process it. And even if we could process it, we may still end behaving against our own better judgment. In what follows, we present a model of privacy attitudes and behavior based on some of these findings, and in particular on the plight of immediate gratification. 4. PRIVACY AND THE ECONOMICS OF IMMEDIATE GRATIFICATION The problem of immediate gratification (which is related to the concepts of time inconsistency, hyperbolic discounting, and self-control bias) is so described by O"Donoghue and Rabin [27, p. 4]: A person"s relative preference for wellbeing at an earlier date over a later date gets stronger as the earlier date gets closer. [...] [P]eople have self-control problems caused by a tendency to pursue immediate gratification in a way that their ‘long-run selves" do not appreciate. For example, if you were given only two alternatives, on Monday you may claim you will prefer working 5 hours on Saturday to 5 hours and half on Sunday. But as Saturday comes, you will be more likely to prefer postponing work until Sunday. This simple observation has rather important consequences in economic theory, where time-consistency of preferences is the dominant model. Consider first the traditional model of utility that agents derive from consumption: the model states that utility discounts exponentially over time: Ut = T τ=t δτ uτ (2) In Equation 2, the cumulative utility U at time t is the discounted sum of all utilities from time t (the present) until time T (the future). δ is the discount factor, with a value 24 Period 1 Period 2 Period 3 Period 4 Benefits from selling period 1 2 0 0 0 Costs from selling period 1 0 1 1 1 Benefits from selling period 2 0 2 0 0 Costs from selling period 2 0 0 1 1 Benefits from selling period 3 0 0 2 0 Costs from selling period 3 0 0 0 1 Table 1: (Fictional) expected payoffs from joining loyalty program. between 0 and 1. A value of 0 would imply that the individual discounts so heavily that the utility from future periods is worth zero today. A value of 1 would imply that the individual is so patient she does not discount future utilities. The discount factor is used in economics to capture the fact that having (say) one dollar one year from now is valuable, but not as much as having that dollar now. In Equation 2, if all uτ were constant - for instance, 10 - and δ was 0.9, then at time t = 0 (that is, now) u0 would be worth 10, but u1 would be worth 9. Modifying the traditional model of utility discounting, [23] and then [31] have proposed a model which takes into account possible time-inconsistency of preferences. Consider Equation 3: Ut(ut, ut+1, ..., uT ) = δt ut + β T τ=t+1 δτ uτ (3) Assume that δ, β ∈ [0, 1]. δ is the discount factor for intertemporal utility as in Equation 2. β is the parameter that captures an individual"s tendency to gratify herself immediately (a form of time-inconsistent preferences). When β is 1, the model maps the traditional time-consistent utility model, and Equation 3 is identical to Equation 2. But when β is zero, the individual does not care for anything but today. In fact, any β smaller than 1 represents self-control bias. The experimental literature has convincingly proved that human beings tend to have self-control problems even when they claim otherwise: we tend to avoid and postpone undesirable activities even when this will imply more effort tomorrow; and we tend to over-engage in pleasant activities even though this may cause suffering or reduced utility in the future. This analytical framework can be applied to the study of privacy attitudes and behavior. Protecting your privacy sometimes means protecting yourself from a clear and present hassle (telemarketers, or people peeping through your window and seeing how you live - see [33]); but sometimes it represents something akin to getting an insurance against future and only uncertain risks. In surveys completed at time t = 0, subjects asked about their attitude towards privacy risks may mentally consider some costs of protecting themselves at a later time t = s and compare those to the avoided costs of privacy intrusions in an even more distant future t = s + n. Their alternatives at survey time 0 are represented in Equation 4. min wrt x DU0 = β[(E(cs,p)δs x) + (E(cs+n,i)δs+n (1 − x))] (4) x is a dummy variable that can take values 0 or 1. It represents the individual"s choice - which costs the individual opts to face: the expected cost of protecting herself at time s, E(cs,p) (in which case x = 1), or the expected costs of being subject to privacy intrusions at a later time s + n, E(cs+n,i). The individual is trying to minimize the disutility DU of these costs with respect to x. Because she discounts the two future events with the same discount factor (although at different times), for certain values of the parameters the individual may conclude that paying to protect herself is worthy. In particular, this will happen when: E(cs,p)δs < E(cs+n,i)δs+n (5) Now, consider what happens as the moment t = s comes. Now a real price should be paid in order to enjoy some form of protection (say, starting to encrypt all of your emails to protect yourself from future intrusions). Now the individual will perceive a different picture: min wrt x DUs = δE(cs,p)x + βE(cn,i)δn (1 − x)] (6) Note that nothing has changed in the equation (certainly not the individual"s perceived risks) except time. If β (the parameter indicating the degree of self-control problems) is less than one, chances are that the individual now will actually choose not to protect herself. This will in fact happen when: δE(cs,p) > βE(cn,i)δn (7) Note that Disequalities 5 and 7 may be simultaneously met for certain β < 1. At survey time the individual honestly claimed she wanted to protect herself in principlethat is, some time in the future. But as she is asked to make an effort to protect herself right now, she chooses to run the risk of privacy intrusion. Similar mathematical arguments can be made for the comparison between immediate costs with immediate benefits (subscribing to a ‘no-call" list to stop telemarketers from harassing you at dinner), and immediate costs with only future expected rewards (insuring yourself against identity theft, or protecting yourself from frauds by never using your credit card on-line), particularly when expected future rewards (or avoided risks) are also intangible: the immaterial consequences of living (or not) in a dossier society, or the chilling effects (or lack thereof) of being under surveillance. The reader will have noticed that we have focused on perceived (expected) costs E(c), rather than real costs. We do not know the real costs and we do not claim that the 25 individual does. But we are able to show that under certain conditions even costs perceived as very high (as during periods of intense privacy debate) will be ignored. We can provide some fictional numerical examples to make the analysis more concrete. We present some scenarios inspired by the calculations in [31]. Imagine an economy with just 4 periods (Table 1). Each individual can enroll in a supermarket"s loyalty program by revealing personal information. If she does so, the individual gets a discount of 2 during the period of enrollment, only to pay one unit each time thereafter because of price discrimination based on the information she revealed (we make no attempt at calibrating the realism of this obviously abstract example; the point we are focusing on is how time inconsistencies may affect individual behavior given the expected costs and benefits of certain actions).4 Depending on which period the individual chooses for ‘selling" her data, we have the undiscounted payoffs represented in Table 1. Imagine that the individual is contemplating these options and discounting them according to Equation 3. Suppose that δ = 1 for all types of individuals (this means that for simplicity we do not consider intertemporal discounting) but β = 1/2 for time-inconsistent individuals and β = 1 for everybody else. The time-consistent individual will choose to join the program at the very last period and rip off a benefit of 2-1=1. The individual with immediate gratification problems, for whom β = 1/2, will instead perceive the benefits from joining now or in period 3 as equivalent (0.5), and will join the program now, thus actually making herself worse off. [31] also suggest that, in addition to the distinction between time-consistent individuals and individuals with timeinconsistent preferences, we should also distinguish timeinconsistent individuals who are na¨ıve from those who are sophisticated. Na¨ıve time-inconsistent individuals are not aware of their self-control problems - for example, they are those who always plan to start a diet next week. Sophisticated time-inconsistent individuals suffer of immediate gratification bias, but are at least aware of their inconsistencies. People in this category choose their behavior today correctly estimating their future time-inconsistent behavior. Now consider how this difference affects decisions in another scenario, represented in Table 2. An individual is considering the adoption of a certain privacy enhancing technology. It will cost her some money both to protect herself and not to protect herself. If she decides to protect herself, the cost will be the amount she pays - for example - for some technology that shields her personal information. If she decides not to protect herself, the cost will be the expected consequences of privacy intrusions. We assume that both these aggregate costs increase over time, although because of separate dynamics. As time goes by, more and more information about the individual has been revealed, and it becomes more costly to be protected against privacy intrusions. At the same time, however, intrusions become more frequent and dangerous. 4 One may claim that loyalty cards keep on providing benefits over time. Here we make the simplifying assumption that such benefits are not larger than the future costs incurred after having revealed one"s tastes. We also assume that the economy ends in period 4 for all individuals, regardless of when they chose to join the loyalty program. In period 1, the individual may protect herself by spending 5, or she may choose to face a risk of privacy intrusion the following period, expected to cost 7. In the second period, assuming that no intrusion has yet taken place, she may once again protect herself by spending a little more, 6; or she may choose to face a risk of privacy intrusion the next (third) period, expected to cost 9. In the third period she could protect herself for 8 or face an expected cost of 15 in the following last period. Here too we make no attempt at calibrating the values in Table 2. Again, we focus on the different behavior driven by heterogeneity in time-consistency and sophistication versus na¨ıvete. We assume that β = 1 for individuals with no self control problems and β = 1/2 for everybody else. We assume for simplicity that δ = 1 for all. The time-consistent individuals will obviously choose to protect themselves as soon as possible. In the first period, na¨ıve time-inconsistent individuals will compare the costs of protecting themselves then or face a privacy intrusion in the second period. Because 5 > 7 ∗ (1/2), they will prefer to wait until the following period to protect themselves. But in the second period they will be comparing 6 > 9 ∗ (1/2) - and so they will postpone their protection again. They will keep on doing so, facing higher and higher risks. Eventually, they will risk to incur the highest perceived costs of privacy intrusions (note again that we are simply assuming that individuals believe there are privacy risks and that they increase over time; we will come back to this concept later on). Time-inconsistent but sophisticated individuals, on the other side, will adopt a protective technology in period 2 and pay 6. By period 2, in fact, they will (correctly) realize that if they wait till period 3 (which they are tempted to do, because 6 > 9 ∗ (1/2)), their self-control bias will lead them to postpone adopting the technology once more (because 8 > 15 ∗ (1/2)). Therefore they predict they would incur the expected cost 15 ∗ (1/2), which is larger than 6the cost of protecting oneself in period 2. In period 1, however, they correctly predict that they will not wait to protect themselves further than period 2. So they wait till period 2, because 5 > 6 ∗ (1/2), at which time they will adopt a protective technology (see also [31]). To summarize, time-inconsistent people tend not to fully appreciate future risks and, if na¨ıve, also their inability to deal with them. This happens even if they are aware of those risks and they are aware that those risks are increasing. As we learnt from the second scenario, time inconsistency can lead individuals to accept higher and higher risks. Individuals may tend to downplay the fact that single actions present low risks, but their repetition forms a huge liability: it is a deceiving aspect of privacy that its value is truly appreciated only after privacy itself is lost. This dynamics captures the essence of privacy and the so-called anonymity sets [32, 14], where each bit of information we reveal can be linked to others, so that the whole is more than the sum of the parts. In addition, [31] show that when costs are immediate, time-inconsistent individuals tend to procrastinate; when benefits are immediate, they tend to preoperate. In our context things are even more interesting because all privacy decisions involve at the same time costs and benefits. So we opt against using eCash [9] in order to save us the costs of switching from credit cards. But we accept the risk that our credit card number on the Internet could be used ma26 Period 1 Period 2 Period 3 Period 4 Protection costs 5 6 8 . Expected intrusion costs . 7 9 15 Table 2: (Fictional) costs of protecting privacy and expected costs of privacy intrusions over time. liciously. And we give away our personal information to supermarkets in order to gain immediate discounts - which will likely turn into price discrimination in due time [3, 26]. We have shown in the second scenario above how sophisticated but time-inconsistent individuals may choose to protect their information only in period 2. Sophisticated people with self-control problems may be at a loss, sometimes even when compared to na¨ıve people with time inconsistency problems (how many privacy advocates do use privacy enhancing technologies all the time?). The reasoning is that sophisticated people are aware of their self-control problems, and rather than ignoring them, they incorporate them into their decision process. This may decrease their own incentive to behave in the optimal way now. Sophisticated privacy advocates might realize that protecting themselves from any possible privacy intrusion is unrealistic, and so they may start misbehaving now (and may get used to that, a form of coherent arbitrariness). This is consistent with the results by [36] presented at the ACM EC "01 conference. [36] found that privacy advocates were also willing to reveal personal information in exchange for monetary rewards. It is also interesting to note that these inconsistencies are not caused by ignorance of existing risks or confusion about available technologies. Individuals in the abstract scenarios we described are aware of their perceived risks and costs. However, under certain conditions, the magnitude of those liabilities is almost irrelevant. The individual will take very slowly increasing risks, which become steps towards huge liabilities. 5. DISCUSSION Applying models of self-control bias and immediate gratification to the study of privacy decision making may offer a new perspective on the ongoing privacy debate. We have shown that a model of rational privacy behavior is unrealistic, while models based on psychological distortions offer a more accurate depiction of the decision process. We have shown why individuals who genuinely would like to protect their privacy may not do so because of psychological distortions well documented in the behavioral economics literature. We have highlighted that these distortions may affect not only na¨ıve individuals but also sophisticated ones. Surprisingly, we have also found that these inconsistencies may occur when individuals perceive the risks from not protecting their privacy as significant. Additional uncertainties, risk aversion, and varying attitudes towards losses and gains may be confounding elements in our analysis. Empirical validation is necessary to calibrate the effects of different factors. An empirical analysis may start with the comparison of available data on the adoption rate of privacy technologies that offer immediate refuge from minor but pressing privacy concerns (for example, ‘do not call" marketing lists), with data on the adoption of privacy technologies that offer less obviously perceivable protection from more dangerous but also less visible privacy risks (for example, identity theft insurances). However, only an experimental approach over different periods of time in a controlled environment may allow us to disentangle the influence of several factors. Surveys alone cannot suffice, since we have shown why survey-time attitudes will rarely match decision-time actions. An experimental verification is part of our ongoing research agenda. The psychological distortions we have discussed may be considered in the ongoing debate on how to deal with the privacy problem: industry self-regulation, users" self protection (through technology or other strategies), or government"s intervention. The conclusions we have reached suggest that individuals may not be trusted to make decisions in their best interests when it comes to privacy. This does not mean that privacy technologies are ineffective. On the contrary, our results, by aiming at offering a more realistic model of user-behavior, could be of help to technologists in their design of privacy enhancing tools. However, our results also imply that technology alone or awareness alone may not address the heart of the privacy problem. Improved technologies (with lower costs of adoption and protection) and more information about risks and opportunities certainly can help. However, more fundamental human behavioral mechanisms must also be addressed. Self-regulation, even in presence of complete information and awareness, may not be trusted to work for the same reasons. A combination of technology, awareness, and regulative policies - calibrated to generate and enforce liabilities and incentives for the appropriate parties - may be needed for privacy-related welfare increase (as in other areas of an economy: see on a related analysis [25]). Observing that people do not want to pay for privacy or do not care about privacy, therefore, is only a half truth. People may not be able to act as economically rational agents when it comes to personal privacy. And the question whether do consumers care? is a different question from does privacy matter? Whether from an economic standpoint privacy ought to be protected or not, is still an open question. It is a question that involves defining specific contexts in which the concept of privacy is being invoked. But the value of privacy eventually goes beyond the realms of economic reasoning and cost benefit analysis, and ends up relating to one"s views on society and freedom. Still, even from a purely economic perspective, anecdotal evidence suggest that the costs of privacy (from spam to identity theft, lost sales, intrusions, and the like [30, 12, 17, 33, 26]) are high and increasing. 6. ACKNOWLEDGMENTS The author gratefully acknowledges Carnegie Mellon University"s Berkman Development Fund, that partially supported this research. The author also wishes to thank Jens Grossklags, Charis Kaskiris, and three anonymous referees for their helpful comments. 27 7. REFERENCES [1] A. Acquisti, R. Dingledine, and P. Syverson. On the economics of anonymity. In Financial CryptographyFC "03, pages 84-102. Springer Verlag, LNCS 2742, 2003. [2] A. Acquisti and J. Grossklags. Losses, gains, and hyperbolic discounting: An experimental approach to information security attitudes and behavior. In 2nd Annual Workshop on Economics and Information Security - WEIS "03, 2003. [3] A. Acquisti and H. R. Varian. Conditioning prices on purchase history. Technical report, University of California, Berkeley, 2001. Presented at the European Economic Association Conference, Venice, IT, August 2002. http://www.heinz.cmu.edu/~acquisti/ papers/privacy.pdf. [4] G. A. Akerlof. The market for ‘lemons:" quality uncertainty and the market mechanism. Quarterly Journal of Economics, 84:488-500, 1970. [5] G. S. Becker and K. M. Murphy. A theory of rational addiction. Journal of Political Economy, 96:675-700, 1988. [6] B. D. Brunk. Understanding the privacy space. First Monday, 7, 2002. http://firstmonday.org/issues/ issue7_10/brunk/index.html. [7] G. Calzolari and A. Pavan. Optimal design of privacy policies. Technical report, Gremaq, University of Toulouse, 2001. [8] D. Chaum. Untraceable electronic mail, return addresses, and digital pseudonyms. Communications of the ACM, 24(2):84-88, 1981. [9] D. Chaum. Blind signatures for untraceable payments. In Advances in Cryptology - Crypto "82, pages 199-203. Plenum Press, 1983. [10] R. K. Chellappa and R. Sin. Personalization versus privacy: An empirical examination of the online consumer"s dilemma. In 2002 Informs Meeting, 2002. [11] F. T. Commission. Privacy online: Fair information practices in the electronic marketplace, 2000. http://www.ftc.gov/reports/privacy2000/ privacy2000.pdf. [12] Community Banker Association of Indiana. Identity fraud expected to triple by 2005, 2001. http://www.cbai.org/Newsletter/December2001/ identity_fraud_de2001.htm. [13] S. Corey. Professional attitudes and actual behavior. Journal of Educational Psychology, 28(1):271 - 280, 1937. [14] C. Diaz, S. Seys, J. Claessens, and B. Preneel. Towards measuring anonymity. In P. Syverson and R. Dingledine, editors, Privacy Enhancing Technologies - PET "02. Springer Verlag, 2482, 2002. [15] ebusinessforum.com. eMarketer: The great online privacy debate, 2000. http://www.ebusinessforum. com/index.asp?doc_id=1785&layout=rich_story. [16] Federal Trade Commission. Identity theft heads the ftc"s top 10 consumer fraud complaints of 2001, 2002. http://www.ftc.gov/opa/2002/01/idtheft.htm. [17] R. Gellman. Privacy, consumers, and costs - How the lack of privacy costs consumers and why business studies of privacy costs are biased and incomplete, 2002. http://www.epic.org/reports/dmfprivacy.html. [18] I.-H. Harn, K.-L. Hui, T. S. Lee, and I. P. L. Png. Online information privacy: Measuring the cost-benefit trade-off. In 23rd International Conference on Information Systems, 2002. [19] Harris Interactive. First major post-9.11 privacy survey finds consumers demanding companies do more to protect privacy; public wants company privacy policies to be independently verified, 2002. http://www.harrisinteractive.com/news/ allnewsbydate.asp?NewsID=429. [20] P. Jehiel and A. Lilico. Smoking today and stopping tomorrow: A limited foresight perspective. Technical report, Department of Economics, UCLA, 2002. [21] Jupiter Research. Seventy percent of US consumers worry about online privacy, but few take protective action, 2002. http: //www.jmm.com/xp/jmm/press/2002/pr_060302.xml. [22] H. Kunreuther. Causes of underinsurance against natural disasters. Geneva Papers on Risk and Insurance, 1984. [23] D. Laibson. Essays on hyperbolic discounting. MIT, Department of Economics, Ph.D. Dissertation, 1994. [24] R. LaPiere. Attitudes versus actions. Social Forces, 13:230-237, 1934. [25] G. Lowenstein, T. O"Donoghue, and M. Rabin. Projection bias in predicting future utility. Technical report, Carnegie Mellon University, Cornell University, and University of California, Berkeley, 2003. [26] A. Odlyzko. Privacy, economics, and price discrimination on the Internet. In Fifth International Conference on Electronic Commerce, pages 355-366. ACM, 2003. [27] T. O"Donoghue and M. Rabin. Choice and procrastination. Quartely Journal of Economics, 116:121-160, 2001. The page referenced in the text refers to the 2000 working paper version. [28] R. A. Posner. An economic theory of privacy. Regulation, pages 19-26, 1978. [29] R. A. Posner. The economics of privacy. American Economic Review, 71(2):405-409, 1981. [30] Privacy Rights Clearinghouse. Nowhere to turn: Victims speak out on identity theft, 2000. http: //www.privacyrights.org/ar/idtheft2000.htm. [31] M. Rabin and T. O"Donoghue. The economics of immediate gratification. Journal of Behavioral Decision Making, 13:233-250, 2000. [32] A. Serjantov and G. Danezis. Towards an information theoretic metric for anonymity. In P. Syverson and R. Dingledine, editors, Privacy Enhancing Technologies - PET "02. Springer Verlag, LNCS 2482, 2002. [33] A. Shostack. Paying for privacy: Consumers and infrastructures. In 2nd Annual Workshop on Economics and Information Security - WEIS "03, 2003. [34] H. A. Simon. Models of bounded rationality. The MIT Press, Cambridge, MA, 1982. 28 [35] P. Slovic. What does it mean to know a cumulative risk? Adolescents" perceptions of short-term and long-term consequences of smoking. Journal of Behavioral Decision Making, 13:259-266, 2000. [36] S. Spiekermann, J. Grossklags, and B. Berendt. E-privacy in 2nd generation e-commerce: Privacy preferences versus actual behavior. In 3rd ACM Conference on Electronic Commerce - EC "01, pages 38-47, 2002. [37] G. J. Stigler. An introduction to privacy in economics and politics. Journal of Legal Studies, 9:623-644, 1980. [38] P. Syverson. The paradoxical value of privacy. In 2nd Annual Workshop on Economics and Information Security - WEIS "03, 2003. [39] C. R. Taylor. Private demands and demands for privacy: Dynamic pricing and the market for customer information. Department of Economics, Duke University, Duke Economics Working Paper 02-02, 2002. [40] T. Vila, R. Greenstadt, and D. Molnar. Why we can"t be bothered to read privacy policies: Models of privacy economics as a lemons market. In 2nd Annual Workshop on Economics and Information SecurityWEIS "03, 2003. [41] S. Warren and L. Brandeis. The right to privacy. Harvard Law Review, 4:193-220, 1890. [42] N. D. Weinstein. Optimistic biases about personal risks. Science, 24:1232-1233, 1989. [43] A. Whitten and J. D. Tygar. Why Johnny can"t encrypt: A usability evaluation of PGP 5.0. In 8th USENIX Security Symposium, 1999. 29

privacy;financial privacy;electronic commerce;psychological inconsistency;personal information protection;rationality;hyperbolic discounting;time-inconsistent preference;individual decision making process;privacy enhancing technology;psychological distortion;self-control problem;immediate gratification;privacy sensitive decision;anonymity;hyperbolic discount

train_J-66

Expressive Negotiation over Donations to Charities∗

When donating money to a (say, charitable) cause, it is possible to use the contemplated donation as negotiating material to induce other parties interested in the charity to donate more. Such negotiation is usually done in terms of matching offers, where one party promises to pay a certain amount if others pay a certain amount. However, in their current form, matching offers allow for only limited negotiation. For one, it is not immediately clear how multiple parties can make matching offers at the same time without creating circular dependencies. Also, it is not immediately clear how to make a donation conditional on other donations to multiple charities, when the donator has different levels of appreciation for the different charities. In both these cases, the limited expressiveness of matching offers causes economic loss: it may happen that an arrangement that would have made all parties (donators as well as charities) better off cannot be expressed in terms of matching offers and will therefore not occur. In this paper, we introduce a bidding language for expressing very general types of matching offers over multiple charities. We formulate the corresponding clearing problem (deciding how much each bidder pays, and how much each charity receives), and show that it is NP-complete to approximate to any ratio even in very restricted settings. We give a mixed-integer program formulation of the clearing problem, and show that for concave bids, the program reduces to a linear program. We then show that the clearing problem for a subclass of concave bids is at least as hard as the decision variant of linear programming. Subsequently, we show that the clearing problem is much easier when bids are quasilinear-for surplus, the problem decomposes across charities, and for payment maximization, a greedy approach is optimal if the bids are concave (although this latter problem is weakly NP-complete when the bids are not concave). For the quasilinear setting, we study the mechanism design question. We show that an ex-post efficient mechanism is impossible even with only one charity and a very restricted class of bids. We also show that there may be benefits to linking the charities from a mechanism design standpoint.

1. INTRODUCTION When money is donated to a charitable (or other) cause (hereafter referred to as charity), often the donating party gives unconditionally: a fixed amount is transferred from the donator to the charity, and none of this transfer is contingent on other events-in particular, it is not contingent on the amount given by other parties. Indeed, this is currently often the only way to make a donation, especially for small donating parties such as private individuals. However, when multiple parties support the same charity, each of them would prefer to see the others give more rather than less to this charity. In such scenarios, it is sensible for a party to use its contemplated donation as negotiating material to induce the others to give more. This is done by making the donation conditional on the others" donations. The following example will illustrate this, and show that the donating parties as well as the charitable cause may simultaneously benefit from the potential for such negotiation. Suppose we have two parties, 1 and 2, who are both supporters of charity A. To either of them, it would be worth $0.75 if A received $1. It follows neither of them will be willing to give unconditionally, because $0.75 < $1. However, if the two parties draw up a contract that says that they will each give $0.5, both the parties have an incentive to accept this contract (rather than have no contract at all): with the contract, the charity will receive $1 (rather than $0 without a contract), which is worth $0.75 to each party, which is greater than the $0.5 that that party will have to give. Effectively, each party has made its donation conditional on the other party"s donation, leading to larger donations and greater happiness to all parties involved. 51 One method that is often used to effect this is to make a matching offer. Examples of matching offers are: I will give x dollars for every dollar donated., or I will give x dollars if the total collected from other parties exceeds y. In our example above, one of the parties can make the offer I will donate $0.5 if the other party also donates at least that much, and the other party will have an incentive to indeed donate $0.5, so that the total amount given to the charity increases by $1. Thus this matching offer implements the contract suggested above. As a real-world example, the United States government has authorized a donation of up to $1 billion to the Global Fund to fight AIDS, TB and Malaria, under the condition that the American contribution does not exceed one third of the total-to encourage other countries to give more [23]. However, there are several severe limitations to the simple approach of matching offers as just described. 1. It is not clear how two parties can make matching offers where each party"s offer is stated in terms of the amount that the other pays. (For example, it is not clear what the outcome should be when both parties offer to match the other"s donation.) Thus, matching offers can only be based on payments made by parties that are giving unconditionally (not in terms of a matching offer)-or at least there can be no circular dependencies.1 2. Given the current infrastructure for making matching offers, it is impractical to make a matching offer depend on the amounts given to multiple charities. For instance, a party may wish to specify that it will pay $100 given that charity A receives a total of $1000, but that it will also count donations made to charity B, at half the rate. (Thus, a total payment of $500 to charity A combined with a total payment of $1000 to charity B would be just enough for the party"s offer to take effect.) In contrast, in this paper we propose a new approach where each party can express its relative preferences for different charities, and make its offer conditional on its own appreciation for the vector of donations made to the different charities. Moreover, the amount the party offers to donate at different levels of appreciation is allowed to vary arbitrarily (it does need to be a dollar-for-dollar (or n-dollarfor-dollar) matching arrangement, or an arrangement where the party offers a fixed amount provided a given (strike) total has been exceeded). Finally, there is a clear interpretation of what it means when multiple parties are making conditional offers that are stated in terms of each other. Given each combination of (conditional) offers, there is a (usually) unique solution which determines how much each party pays, and how much each charity is paid. However, as we will show, finding this solution (the clearing problem) requires solving a potentially difficult optimization problem. A large part of this paper is devoted to studying how difficult this problem is under different assumptions on the structure of the offers, and providing algorithms for solving it. 1 Typically, larger organizations match offers of private individuals. For example, the American Red Cross Liberty Disaster Fund maintains a list of businesses that match their customers" donations [8]. Towards the end of the paper, we also study the mechanism design problem of motivating the bidders to bid truthfully. In short, expressive negotiation over donations to charities is a new way in which electronic commerce can help the world. A web-based implementation of the ideas described in this paper can facilitate voluntary reallocation of wealth on a global scale. Aditionally, optimally solving the clearing problem (and thereby generating the maximum economic welfare) requires the application of sophisticated algorithms. 2. COMPARISON TO COMBINATORIAL AUCTIONS AND EXCHANGES This section discusses the relationship between expressive charity donation and combinatorial auctions and exchanges. It can be skipped, but may be of interest to the reader with a background in combinatorial auctions and exchanges. In a combinatorial auction, there are m items for sale, and bidders can place bids on bundles of one or more items. The auctioneer subsequently labels each bid as winning or losing, under the constraint that no item can be in more than one winning bid, to maximize the sum of the values of the winning bids. (This is known as the clearing problem.) Variants include combinatorial reverse auctions, where the auctioneer is seeking to procure a set of items; and combinatorial exchanges, where bidders can both buy and and sell items (even within the same bid). Other extensions include allowing for side constraints, as well as the specification of attributes of the items in bids. Combinatorial auctions and exchanges have recently become a popular research topic [20, 21, 17, 22, 9, 18, 13, 3, 12, 26, 19, 25, 2]. The problems of clearing expressive charity donation markets and clearing combinatorial auctions or exchanges are very different in formulation. Nevertheless, there are interesting parallels. One of the main reasons for the interest in combinatorial auctions and exchanges is that it allows for expressive bidding. A bidder can express exactly how much each different allocation is worth to her, and thus the globally optimal allocation may be chosen by the auctioneer. Compare this to a bidder having to bid on two different items in two different (one-item) auctions, without any way of expressing that (for instance) one item is worthless if the other item is not won. In this scenario, the bidder may win the first item but not the second (because there was another high bid on the second item that she did not anticipate), leading to economic inefficiency. Expressive bidding is also one of the main benefits of the expressive charity donation market. Here, bidders can express exactly how much they are willing to donate for every vector of amounts donated to charities. This may allow bidders to negotiate a complex arrangement of who gives how much to which charity, which is beneficial to all parties involved; whereas no such arrangement may have been possible if the bidders had been restricted to using simple matching offers on individual charities. Again, expressive bidding is necessary to achieve economic efficiency. Another parallel is the computational complexity of the clearing problem. In order to achieve the full economic efficiency allowed by the market"s expressiveness (or even come close to it), hard computational problems must be solved in combinatorial auctions and exchanges, as well as in the charity donation market (as we will see). 52 3. DEFINITIONS Throughout this paper, we will refer to the offers that the donating parties make as bids, and to the donating parties as bidders. In our bidding framework, a bid will specify, for each vector of total payments made to the charities, how much that bidder is willing to contribute. (The contribution of this bidder is also counted in the vector of paymentsso, the vector of total payments to the charities represents the amount given by all donating parties, not just the ones other than this bidder.) The bidding language is expressive enough that no bidder should have to make more than one bid. The following definition makes the general form of a bid in our framework precise. Definition 1. In a setting with m charities c1, c2, . . . , cm, a bid by bidder bj is a function vj : Rm → R. The interpretation is that if charity ci receives a total amount of πci , then bidder j is willing to donate (up to) vj(πc1 , πc2 , . . . , πcm ). We now define possible outcomes in our model, and which outcomes are valid given the bids that were made. Definition 2. An outcome is a vector of payments made by the bidders (πb1 , πb2 , . . . , πbn ), and a vector of payments received by the charities (πc1 , πc2 , . . . , πcm ). A valid outcome is an outcome where 1. n j=1 πbj ≥ m i=1 πci (at least as much money is collected as is given away); 2. For all 1 ≤ j ≤ n, πbj ≤ vj(πc1 , πc2 , . . . , πcm ) (no bidder gives more than she is willing to). Of course, in the end, only one of the valid outcomes can be chosen. We choose the valid outcome that maximizes the objective that we have for the donation process. Definition 3. An objective is a function from the set of all outcomes to R.2 After all bids have been collected, a valid outcome will be chosen that maximizes this objective. One example of an objective is surplus, given by n j=1 πbj − m i=1 πci . The surplus could be the profits of a company managing the expressive donation marketplace; but, alternatively, the surplus could be returned to the bidders, or given to the charities. Another objective is total amount donated, given by m i=1 πci . (Here, different weights could also be placed on the different charities.) Finding the valid outcome that maximizes the objective is a (nontrivial) computational problem. We will refer to it as the clearing problem. The formal definition follows. Definition 4 (DONATION-CLEARING). We are given a set of n bids over charities c1, c2, . . . , cm. Additionally, we are given an objective function. We are asked to find an objective-maximizing valid outcome. How difficult the DONATION-CLEARING problem is depends on the types of bids used and the language in which they are expressed. This is the topic of the next section. 2 In general, the objective function may also depend on the bids, but the objective functions under consideration in this paper do not depend on the bids. The techniques presented in this paper will typically generalize to objectives that take the bids into account directly. 4. A SIMPLIFIED BIDDING LANGUAGE Specifying a general bid in our framework (as defined above) requires being able to specify an arbitrary real-valued function over Rm . Even if we restricted the possible total payment made to each charity to the set {0, 1, 2, . . . , s}, this would still require a bidder to specify (s+1)m values. Thus, we need a bidding language that will allow the bidders to at least specify some bids more concisely. We will specify a bidding language that only represents a subset of all possible bids, which can be described concisely.3 To introduce our bidding language, we will first describe the bidding function as a composition of two functions; then we will outline our assumptions on each of these functions. First, there is a utility function uj : Rm → R, specifying how much bidder j appreciates a given vector of total donations to the charities. (Note that the way we define a bidder"s utility function, it does not take the payments the bidder makes into account.) Then, there is a donation willingness function wj : R → R, which specifies how much bidder j is willing to pay given her utility for the vector of donations to the charities. We emphasize that this function does not need to be linear, so that utilities should not be thought of as expressible in dollar amounts. (Indeed, when an individual is donating to a large charity, the reason that the individual donates only a bounded amount is typically not decreasing marginal value of the money given to the charity, but rather that the marginal value of a dollar to the bidder herself becomes larger as her budget becomes smaller.) So, we have wj(uj(πc1 , πc2 , . . . , πcm )) = vj(πc1 , πc2 , . . . , πcm ), and we let the bidder describe her functions uj and wj separately. (She will submit these functions as her bid.) Our first restriction is that the utility that a bidder derives from money donated to one charity is independent of the amount donated to another charity. Thus, uj(πc1 , πc2 , . . . , πcm ) = m i=1 ui j(πci ). (We observe that this does not imply that the bid function vj decomposes similarly, because of the nonlinearity of wj.) Furthermore, each ui j must be piecewise linear. An interesting special case which we will study is when each ui j is a line: ui j(πci ) = ai jπci . This special case is justified in settings where the scale of the donations by the bidders is small relative to the amounts the charities receive from other sources, so that the marginal use of a dollar to the charity is not affected by the amount given by the bidders. The only restriction that we place on the payment willingness functions wj is that they are piecewise linear. One interesting special case is a threshold bid, where wj is a step function: the bidder will provide t dollars if her utility exceeds s, and otherwise 0. Another interesting case is when such a bid is partially acceptable: the bidder will provide t dollars if her utility exceeds s; but if her utility is u < s, she is still willing to provide ut s dollars. One might wonder why, if we are given the bidders" utility functions, we do not simply maximize the sum of the utilities rather than surplus or total donated. There are several reasons. First, because affine transformations do not affect utility functions in a fundamental way, it would be possi3 Of course, our bidding language can be trivially extended to allow for fully expressive bids, by also allowing bids from a fully expressive bidding language, in addition to the bids in our bidding language. 53 ble for a bidder to inflate her utility by changing its units, thereby making her bid more important for utility maximization purposes. Second, a bidder could simply give a payment willingness function that is 0 everywhere, and have her utility be taken into account in deciding on the outcome, in spite of her not contributing anything. 5. AVOIDING INDIRECT PAYMENTS In an initial implementation, the approach of having donations made out to a center, and having a center forward these payments to charities, may not be desirable. Rather, it may be preferable to have a partially decentralized solution, where the donating parties write out checks to the charities directly according to a solution prescribed by the center. In this scenario, the center merely has to verify that parties are giving the prescribed amounts. Advantages of this include that the center can keep its legal status minimal, as well as that we do not require the donating parties to trust the center to transfer their donations to the charities (or require some complicated verification protocol). It is also a step towards a fully decentralized solution, if this is desirable. To bring this about, we can still use the approach described earlier. After we clear the market in the manner described before, we know the amount that each donator is supposed to give, and the amount that each charity is supposed to receive. Then, it is straightforward to give some specification of who should give how much to which charity, that is consistent with that clearing. Any greedy algorithm that increases the cash flow from any bidder who has not yet paid enough, to any charity that has not yet received enough, until either the bidder has paid enough or the charity has received enough, will provide such a specification. (All of this is assuming that bj πbj = ci πci . In the case where there is nonzero surplus, that is, bj πbj > ci πci , we can distribute this surplus across the bidders by not requiring them to pay the full amount, or across the charities by giving them more than the solution specifies.) Nevertheless, with this approach, a bidder may have to write out a check to a charity that she does not care for at all. (For example, an environmental activist who was using the system to increase donations to a wildlife preservation fund may be required to write a check to a group supporting a right-wing political party.) This is likely to lead to complaints and noncompliance with the clearing. We can address this issue by letting each bidder specify explicitly (before the clearing) which charities she would be willing to make a check out to. These additional constraints, of course, may change the optimal solution. In general, checking whether a given centralized solution (with zero surplus) can be accomplished through decentralized payments when there are such constraints can be modeled as a MAX-FLOW problem. In the MAX-FLOW instance, there is an edge from the source node s to each bidder bj, with a capacity of πbj (as specified in the centralized solution); an edge from each bidder bj to each charity ci that the bidder is willing to donate money to, with a capacity of ∞; and an edge from each charity ci to the target node t with capacity πci (as specified in the centralized solution). In the remainder of this paper, all our hardness results apply even to the setting where there is no constraint on which bidders can pay to which charity (that is, even the problem as it was specified before this section is hard). We also generalize our clearing algorithms to the partially decentralized case with constraints. 6. HARDNESS OF CLEARING THE MARKET In this section, we will show that the clearing problem is completely inapproximable, even when every bidder"s utility function is linear (with slope 0 or 1 in each charity"s payments), each bidder cares either about at most two charities or about all charities equally, and each bidder"s payment willingness function is a step function. We will reduce from MAX2SAT (given a formula in conjunctive normal form (where each clause has two literals) and a target number of satisfied clauses T, does there exist an assignment of truth values to the variables that makes at least T clauses true?), which is NP-complete [7]. Theorem 1. There exists a reduction from MAX2SAT instances to DONATION-CLEARING instances such that 1. If the MAX2SAT instance has no solution, then the only valid outcome is the zero outcome (no bidder pays anything and no charity receives anything); 2. Otherwise, there exists a solution with positive surplus. Additionally, the DONATION-CLEARING instances that we reduce to have the following properties: 1. Every ui j is a line; that is, the utility that each bidder derives from any charity is linear; 2. All the ui j have slope either 0 or 1; 3. Every bidder either has at most 2 charities that affect her utility (with slope 1), or all charities affect her utility (with slope 1); 4. Every bid is a threshold bid; that is, every bidder"s payment willingness function wj is a step function. Proof. The problem is in NP because we can nondeterministically choose the payments to be made and received, and check the validity and objective value of this outcome. In the following, we will represent bids as follows: ({(ck, ak)}, s, t) indicates that uk j (πck ) = akπck (this function is 0 for ck not mentioned in the bid), and wj(uj) = t for uj ≥ s, wj(uj) = 0 otherwise. To show NP-hardness, we reduce an arbitrary MAX2SAT instance, given by a set of clauses K = {k} = {(l1 k, l2 k)} over a variable set V together with a target number of satisfied clauses T, to the following DONATION-CLEARING instance. Let the set of charities be as follows. For every literal l ∈ L, there is a charity cl. Then, let the set of bids be as follows. For every variable v, there is a bid bv = ({(c+v, 1), (c−v, 1)}, 2, 1 − 1 4|V | ). For every literal l, there is a bid bl = ({(cl, 1)}, 2, 1). For every clause k = {l1 k, l2 k} ∈ K, there is a bid bk = ({(cl1 k , 1), (cl2 k , 1)}, 2, 1 8|V ||K| ). Finally, there is a single bid that values all charities equally: b0 = ({(c1, 1), (c2, 1), . . . , (cm, 1)}, 2|V |+ T 8|V ||K| , 1 4 + 1 16|V ||K| ). We show the two instances are equivalent. First, suppose there exists a solution to the MAX2SAT instance. If in this solution, l is true, then let πcl = 2 + T 8|V |2|K| ; otherwise πcl = 0. Also, the only bids that are not accepted (meaning the threshold is not met) are the bl where l is false, and the bk such that both of l1 k, l2 k are false. First we show that no bidder whose bid is accepted pays more than she is willing to. For each bv, either c+v or c−v receives at least 2, so this bidder"s threshold has been met. 54 For each bl, either l is false and the bid is not accepted, or l is true, cl receives at least 2, and the threshold has been met. For each bk, either both of l1 k, l2 k are false and the bid is not accepted, or at least one of them (say li k) is true (that is, k is satisfied) and cli k receives at least 2, and the threshold has been met. Finally, because the total amount received by the charities is 2|V | + T 8|V ||K| , b0"s threshold has also been met. The total amount that can be extracted from the accepted bids is at least |V |(1− 1 4|V | )+|V |+T 1 8|V ||K| + 1 4 + 1 16|V ||K| ) = 2|V |+ T 8|V ||K| + 1 16|V ||K| > 2|V |+ T 8|V ||K| , so there is positive surplus. So there exists a solution with positive surplus to the DONATION-CLEARING instance. Now suppose there exists a nonzero outcome in the DONATION-CLEARING instance. First we show that it is not possible (for any v ∈ V ) that both b+v and b−v are accepted. For, this would require that πc+v + πc−v ≥ 4. The bids bv, b+v, b−v cannot contribute more than 3, so we need another 1 at least. It is easily seen that for any other v , accepting any subset of {bv , b+v , b−v } would require that at least as much is given to c+v and c−v as can be extracted from these bids, so this cannot help. Finally, all the other bids combined can contribute at most |K| 1 8|V ||K| + 1 4 + 1 16|V ||K| < 1. It follows that we can interpret the outcome in the DONATION-CLEARING instance as a partial assignment of truth values to variables: v is set to true if b+v is accepted, and to false if b−v is accepted. All that is left to show is that this partial assignment satisfies at least T clauses. First we show that if a clause bid bk is accepted, then either bl1 k or bl2 k is accepted (and thus either l1 k or l2 k is set to true, hence k is satisfied). If bk is accepted, at least one of cl1 k and cl2 k must be receiving at least 1; without loss of generality, say it is cl1 k , and say l1 k corresponds to variable v1 k (that is, it is +v1 k or −v1 k). If cl1 k does not receive at least 2, bl1 k is not accepted, and it is easy to check that the bids bv1 k , b+v1 k , b−v1 k contribute (at least) 1 less than is paid to c+v1 k and c+v1 k . But this is the same situation that we analyzed before, and we know it is impossible. All that remains to show is that at least T clause bids are accepted. We now show that b0 is accepted. Suppose it is not; then one of the bv must be accepted. (The solution is nonzero by assumption; if only some bk are accepted, the total payment from these bids is at most |K| 1 8|V ||K| < 1, which is not enough for any bid to be accepted; and if one of the bl is accepted, then the threshold for the corresponding bv is also reached.) For this v, bv1 k , b+v1 k , b−v1 k contribute (at least) 1 4|V | less than the total payments to c+v and c−v. Again, the other bv and bl cannot (by themselves) help to close this gap; and the bk can contribute at most |K| 1 8|V ||K| < 1 4|V | . It follows that b0 is accepted. Now, in order for b0 to be accepted, a total of 2|V |+ T 8|V ||K| must be donated. Because is not possible (for any v ∈ V ) that both b+v and b−v are accepted, it follows that the total payment by the bv and the bl can be at most 2|V | − 1 4 . Adding b0"s payment of 1 4 + 1 16|V ||K| to this, we still need T − 1 2 8|V ||K| from the bk. But each one of them contributes at most 1 8|V ||K| , so at least T of them must be accepted. Corollary 1. Unless P=NP, there is no polynomial-time algorithm for approximating DONATION-CLEARING (with either the surplus or the total amount donated as the objective) within any ratio f(n), where f is a nonzero function of the size of the instance. This holds even if the DONATIONCLEARING structures satisfy all the properties given in Theorem 1. Proof. Suppose we had such a polynomial time algorithm, and applied it to the DONATION-CLEARING instances that were reduced from MAX2SAT instances in Theorem 1. It would return a nonzero solution when the MAX2SAT instance has a solution, and a zero solution otherwise. So we can decide whether arbitrary MAX2SAT instances are satisfiable this way, and it would follow that P=NP. (Solving the problem to optimality is NP-complete in many other (noncomparable or even more restricted) settings as well-we omit such results because of space constraint.) This should not be interpreted to mean that our approach is infeasible. First, as we will show, there are very expressive families of bids for which the problem is solvable in polynomial time. Second, NP-completeness is often overcome in practice (especially when the stakes are high). For instance, even though the problem of clearing combinatorial auctions is NP-complete [20] (even to approximate [21]), they are typically solved to optimality in practice. 7. MIXED INTEGER PROGRAMMING FORMULATION In this section, we give a mixed integer programming (MIP) formulation for the general problem. We also discuss in which special cases this formulation reduces to a linear programming (LP) formulation. In such cases, the problem is solvable in polynomial time, because linear programs can be solved in polynomial time [11]. The variables of the MIP defining the final outcome are the payments made to the charities, denoted by πci , and the payments extracted from the bidders, πbj . In the case where we try to avoid direct payments and let the bidders pay the charities directly, we add variables πci,bj indicating how much bj pays to ci, with the constraints that for each ci, πci ≤ bj πci,bj ; and for each bj, πbj ≥ ci πci,bj . Additionally, there is a constraint πci,bj = 0 whenever bidder bj is unwilling to pay charity ci. The rest of the MIP can be phrased in terms of the πci and πbj . The objectives we have discussed earlier are both linear: surplus is given by n j=1 πbj − m i=1 πci , and total amount donated is given by m i=1 πci (coefficients can be added to represent different weights on the different charities in the objective). The constraint that the outcome should be valid (no deficit) is given simply by: n j=1 πbj ≥ m i=1 πci . For every bidder, for every charity, we define an additional utility variable ui j indicating the utility that this bidder derives from the payment to this charity. The bidder"s total 55 utility is given by another variable uj, with the constraint that uj = m i=1 ui j. Each ui j is given as a function of πci by the (piecewise linear) function provided by the bidder. In order to represent this function in the MIP formulation, we will merely place upper bounding constraints on ui j, so that it cannot exceed the given functions. The MIP solver can then push the ui j variables all the way up to the constraint, in order to extract as much payment from this bidder as possible. In the case where the ui j are concave, this is easy: if (sl, tl) and (sl+1, tl+1) are endpoints of a finite linear segment in the function, we add the constraint that ui j ≤ tl + πci −sl sl+1−sl (tl+1 − tl). If the final (infinite) segment starts at (sk, tk) and has slope d, we add the constraint that ui j ≤ tk + d(πci − sk). Using the fact that the function is concave, for each value of πci , the tightest upper bound on ui j is the one corresponding to the segment above that value of πci , and therefore these constraints are sufficient to force the correct value of ui j. When the function is not concave, we require (for the first time) some binary variables. First, we define another point on the function: (sk+1, tk+1) = (sk + M, tk + dM), where d is the slope of the infinite segment and M is any upper bound on the πcj . This has the effect that we will never be on the infinite segment again. Now, let xi,j l be an indicator variable that should be 1 if πci is below the lth segment of the function, and 0 otherwise. To effect this, first add a constraint k l=0 xi,j l = 1. Now, we aim to represent πci as a weighted average of its two neighboring si,j l . For 0 ≤ l ≤ k + 1, let λi,j l be the weight on si,j l . We add the constraint k+1 l=0 λi,j l = 1. Also, for 0 ≤ l ≤ k + 1, we add the constraint λi,j l ≤ xl−1 +xl (where x−1 and xk+1 are defined to be zero), so that indeed only the two neighboring si,j l have nonzero weight. Now we add the constraint πci = k+1 l=0 si,j l λi,j l , and now the λi,j l must be set correctly. Then, we can set ui j = k+1 l=0 ti,j l λi,j l . (This is a standard MIP technique [16].) Finally, each πbj is bounded by a function of uj by the (piecewise linear) function provided by the bidder (wj). Representing this function is entirely analogous to how we represented ui j as a function of πci . (Again we will need binary variables only if the function is not concave.) Because we only use binary variables when either a utility function ui j or a payment willingness function wj is not concave, it follows that if all of these are concave, our MIP formulation is simply a linear program-which can be solved in polynomial time. Thus: Theorem 2. If all functions ui j and wj are concave (and piecewise linear), the DONATION-CLEARING problem can be solved in polynomial time using linear programming. Even if some of these functions are not concave, we can simply replace each such function by the smallest upper bounding concave function, and use the linear programming formulation to obtain an upper bound on the objectivewhich may be useful in a search formulation of the general problem. 8. WHY ONE CANNOT DO MUCH BETTER THAN LINEAR PROGRAMMING One may wonder if, for the special cases of the DONATIONCLEARING problem that can be solved in polynomial time with linear programming, there exist special purpose algorithms that are much faster than linear programming algorithms. In this section, we show that this is not the case. We give a reduction from (the decision variant of) the general linear programming problem to (the decision variant of) a special case of the DONATION-CLEARING problem (which can be solved in polynomial time using linear programming). (The decision variant of an optimization problem asks the binary question: Can the objective value exceed o?) Thus, any special-purpose algorithm for solving the decision variant of this special case of the DONATIONCLEARING problem could be used to solve a decision question about an arbitrary linear program just as fast. (And thus, if we are willing to call the algorithm a logarithmic number of times, we can solve the optimization version of the linear program.) We first observe that for linear programming, a decision question about the objective can simply be phrased as another constraint in the LP (forcing the objective to exceed the given value); then, the original decision question coincides with asking whether the resulting linear program has a feasible solution. Theorem 3. The question of whether an LP (given by a set of linear constraints4 ) has a feasible solution can be modeled as a DONATION-CLEARING instance with payment maximization as the objective, with 2v charities and v + c bids (where v is the number of variables in the LP, and c is the number of constraints). In this model, each bid bj has only linear ui j functions, and is a partially acceptable threshold bid (wj(u) = tj for u ≥ sj, otherwise wj(u) = utj sj ). The v bids corresponding to the variables mention only two charities each; the c bids corresponding to the constraints mention only two times the number of variables in the corresponding constraint. Proof. For every variable xi in the LP, let there be two charities, c+xi and c−xi . Let H be some number such that if there is a feasible solution to the LP, there is one in which every variable has absolute value at most H. In the following, we will represent bids as follows: ({(ck, ak)}, s, t) indicates that uk j (πck ) = akπck (this function is 0 for ck not mentioned in the bid), and wj(uj) = t for uj ≥ s, wj(uj) = uj t s otherwise. For every variable xi in the LP, let there be a bid bxi = ({(c+xi , 1), (c−xi , 1)}, 2H, 2H − c v ). For every constraint i rj i xi ≤ sj in the linear program, let there be a bid bj = ({(c−xi , rj i )}i:r j i >0 ∪ {(c+xi , −rj i )}i:r j i <0 , ( i |rj i |)H − sj, 1). Let the target total amount donated be 2vH. Suppose there is a feasible solution (x∗ 1, x∗ 2, . . . , x∗ v) to the LP. Without loss of generality, we can suppose that |x∗ i | ≤ H for all i. Then, in the DONATION-CLEARING instance, 4 These constraints must include bounds on the variables (including nonnegativity bounds), if any. 56 for every i, let πc+xi = H + x∗ i , and let πc−xi = H − x∗ i (for a total payment of 2H to these two charities). This allows us to extract the maximum payment from the bids bxi -a total payment of 2vH − c. Additionally, the utility of bidder bj is now i:r j i >0 rj i (H − x∗ i ) + i:r j i <0 −rj i (H + x∗ i ) = ( i |rj i |)H − i rj i x∗ i ≥ ( i |rj i |)H − sj (where the last inequality stems from the fact that constraint j must be satisfied in the LP solution), so it follows we can extract the maximum payment from all the bidders bj, for a total payment of c. It follows that we can extract the required 2vH payment from the bidders, and there exists a solution to the DONATION-CLEARING instance with a total amount donated of at least 2vH. Now suppose there is a solution to the DONATIONCLEARING instance with a total amount donated of at least vH. Then the maximum payment must be extracted from each bidder. From the fact that the maximum payment must be extracted from each bidder bxi , it follows that for each i, πc+xi + πc−xi ≥ 2H. Because the maximum extractable total payment is 2vH, it follows that for each i, πc+xi + πc−xi = 2H. Let x∗ i = πc+xi − H = H − πc−xi . Then, from the fact that the maximum payment must be extracted from each bidder bj, it follows that ( i |rj i |)H − sj ≤ i:r j i >0 rj i πc−xi + i:r j i <0 −rj i πc+xi = i:r j i >0 rj i (H − x∗ i ) + i:r j i <0 −rj i (H + x∗ i ) = ( i |rj i |)H − i rj i x∗ i . Equivalently, i rj i x∗ i ≤ sj. It follows that the x∗ i constitute a feasible solution to the LP. 9. QUASILINEAR BIDS Another class of bids of interest is the class of quasilinear bids. In a quasilinear bid, the bidder"s payment willingness function is linear in utility: that is, wj = uj. (Because the units of utility are arbitrary, we may as well let them correspond exactly to units of money-so we do not need a constant multiplier.) In most cases, quasilinearity is an unreasonable assumption: for example, usually bidders have a limited budget for donations, so that the payment willingness will stop increasing in utility after some point (or at least increase slower in the case of a softer budget constraint). Nevertheless, quasilinearity may be a reasonable assumption in the case where the bidders are large organizations with large budgets, and the charities are a few small projects requiring relatively little money. In this setting, once a certain small amount has been donated to a charity, a bidder will derive no more utility from more money being donated from that charity. Thus, the bidders will never reach a high enough utility for their budget constraint (even when it is soft) to take effect, and thus a linear approximation of their payment willingness function is reasonable. Another reason for studying the quasilinear setting is that it is the easiest setting for mechanism design, which we will discuss shortly. In this section, we will see that the clearing problem is much easier in the case of quasilinear bids. First, we address the case where we are trying to maximize surplus (which is the most natural setting for mechanism design). The key observation here is that when bids are quasilinear, the clearing problem decomposes across charities. Lemma 1. Suppose all bids are quasilinear, and surplus is the objective. Then we can clear the market optimally by clearing the market for each charity individually. That is, for each bidder bj, let πbj = ci πbi j . Then, for each charity ci, maximize ( bj πbi j ) − πci , under the constraint that for every bidder bj, πbi j ≤ ui j(πci ). Proof. The resulting solution is certainly valid: first of all, at least as much money is collected as is given away, because bj πbj − ci πci = bj ci πbi j − ci πci = ci (( bj πbi j ) − πci )-and the terms of this summation are the objectives of the individual optimization problems, each of which can be set at least to 0 (by setting all the variables are set to 0), so it follows that the expression is nonnegative. Second, no bidder bj pays more than she is willing to, because uj −πbj = ci ui j(πci )− ci πbi j = ci (ui j(πci )−πbi j )-and the terms of this summation are nonnegative by the constraints we imposed on the individual optimization problems. All that remains to show is that the solution is optimal. Because in an optimal solution, we will extract as much payment from the bidders as possible given the πci , all we need to show is that the πci are set optimally by this approach. Let π∗ ci be the amount paid to charity πci in some optimal solution. If we change this amount to πci and leave everything else unchanged, this will only affect the payment that we can extract from the bidders because of this particular charity, and the difference in surplus will be bj ui j(πci ) − ui j(π∗ ci ) − πci + π∗ ci . This expression is, of course, 0 if πci = π∗ ci . But now notice that this expression is maximized as a function of πci by the decomposed solution for this charity (the terms without πci in them do not matter, and of course in the decomposed solution we always set πbi j = ui j(πci )). It follows that if we change πci to the decomposed solution, the change in surplus will be at least 0 (and the solution will still be valid). Thus, we can change the πci one by one to the decomposed solution without ever losing any surplus. Theorem 4. When all bids are quasilinear and surplus is the objective, DONATION-CLEARING can be done in linear time. Proof. By Lemma 1, we can solve the problem separately for each charity. For charity ci, this amounts to maximizing ( bj ui j(πci )) − πci as a function of πci . Because all its terms are piecewise linear functions, this whole function is piecewise linear, and must be maximized at one of the points where it is nondifferentiable. It follows that we need only check all the points at which one of the terms is nondifferentiable. Unfortunately, the decomposing lemma does not hold for payment maximization. Proposition 1. When the objective is payment maximization, even when bids are quasilinear, the solution obtained by decomposing the problem across charities is in general not optimal (even with concave bids). 57 Proof. Consider a single bidder b1 placing the following quasilinear bid over two charities c1 and c2: u1 1(πc1 ) = 2πci for 0 ≤ πci ≤ 1, u1 1(πc1 ) = 2 + πci −1 4 otherwise; u2 1(πc2 ) = πci 2 . The decomposed solution is πc1 = 7 3 , πc2 = 0, for a total donation of 7 3 . But the solution πc1 = 1, πc2 = 2 is also valid, for a total donation of 3 > 7 3 . In fact, when payment maximization is the objective, DONATION-CLEARING remains (weakly) NP-complete in general. (In the remainder of the paper, proofs are omitted because of space constraint.) Theorem 5. DONATION-CLEARING is (weakly) NPcomplete when payment maximization is the objective, even when every bid is concerns only one charity (and has a stepfunction utility function for this charity), and is quasilinear. However, when the bids are also concave, a simple greedy clearing algorithm is optimal. Theorem 6. Given a DONATION-CLEARING instance with payment maximization as the objective where all bids are quasilinear and concave, consider the following algorithm. Start with πci = 0 for all charities. Then, letting γci = d bj ui j (πci ) dπci (at nondifferentiable points, these derivatives should be taken from the right), increase πc∗ i (where c∗ i ∈ arg maxci γci ), until either γc∗ i is no longer the highest (in which case, recompute c∗ i and start increasing the corresponding payment), or bj uj = ci πci and γc∗ i < 1. Finally, let πbj = uj. (A similar greedy algorithm works when the objective is surplus and the bids are quasilinear and concave, with as only difference that we stop increasing the payments as soon as γc∗ i < 1.) 10. INCENTIVE COMPATIBILITY Up to this point, we have not discussed the bidders" incentives for bidding any particular way. Specifically, the bids may not truthfully reflect the bidders" preferences over charities because a bidder may bid strategically, misrepresenting her preferences in order to obtain a result that is better to herself. This means the mechanism is not strategy-proof. (We will show some concrete examples of this shortly.) This is not too surprising, because the mechanism described so far is, in a sense, a first-price mechanism, where the mechanism will extract as much payment from a bidder as her bid allows. Such mechanisms (for example, first-price auctions, where winners pay the value of their bids) are typically not strategy-proof: if a bidder reports her true valuation for an outcome, then if this outcome occurs, the payment the bidder will have to make will offset her gains from the outcome completely. Of course, we could try to change the rules of the game-which outcome (payment vector to charities) do we select for which bid vector, and which bidder pays how much-in order to make bidding truthfully beneficial, and to make the outcome better with regard to the bidders" true preferences. This is the field of mechanism design. In this section, we will briefly discuss the options that mechanism design provides for the expressive charity donation problem. 10.1 Strategic bids under the first-price mechanism We first point out some reasons for bidders to misreport their preferences under the first-price mechanism described in the paper up to this point. First of all, even when there is only one charity, it may make sense to underbid one"s true valuation for the charity. For example, suppose a bidder would like a charity to receive a certain amount x, but does not care if the charity receives more than that. Additionally, suppose that the other bids guarantee that the charity will receive at least x no matter what bid the bidder submits (and the bidder knows this). Then the bidder is best off not bidding at all (or submitting a utility for the charity of 0), to avoid having to make any payment. (This is known in economics as the free rider problem [14]. With multiple charities, another kind of manipulation may occur, where the bidder attempts to steer others" payments towards her preferred charity. Suppose that there are two charities, and three bidders. The first bidder bids u1 1(πc1 ) = 1 if πc1 ≥ 1, u1 1(πc1 ) = 0 otherwise; u2 1(πc2 ) = 1 if πc2 ≥ 1, u2 1(πc2 ) = 0 otherwise; and w1(u1) = u1 if u1 ≤ 1, w1(u1) = 1+ 1 100 (u1 −1) otherwise. The second bidder bids u1 2(πc1 ) = 1 if πc1 ≥ 1, u1 1(πc1 ) = 0 otherwise; u2 2(πc2 ) = 0 (always); w2(u2) = 1 4 u2 if u2 ≤ 1, w2(u2) = 1 4 + 1 100 (u2 −1) otherwise. Now, the third bidder"s true preferences are accurately represented5 by the bid u1 3(πc1 ) = 1 if πc1 ≥ 1, u1 3(πc1 ) = 0 otherwise; u2 3(πc2 ) = 3 if πc2 ≥ 1, u2 3(πc1 ) = 0 otherwise; and w3(u3) = 1 3 u3 if u3 ≤ 1, w3(u3) = 1 3 + 1 100 (u3 − 1) otherwise. Now, it is straightforward to check that, if the third bidder bids truthfully, regardless of whether the objective is surplus maximization or total donated, charity 1 will receive at least 1, and charity 2 will receive less than 1. The same is true if bidder 3 does not place a bid at all (as in the previous type of manipulation); hence bidder 2"s utility will be 1 in this case. But now, if bidder 3 reports u1 3(πc1 ) = 0 everywhere; u2 3(πc2 ) = 3 if πc2 ≥ 1, u2 3(πc2 ) = 0 otherwise (this part of the bid is truthful); and w3(u3) = 1 3 u3 if u3 ≤ 1, w3(u3) = 1 3 otherwise; then charity 2 will receive at least 1, and bidder 3 will have to pay at most 1 3 . Because up to this amount of payment, one unit of money corresponds to three units of utility to bidder 3, it follows his utility is now at least 3 − 1 = 2 > 1. We observe that in this case, the strategic bidder is not only affecting how much the bidders pay, but also how much the charities receive. 10.2 Mechanism design in the quasilinear setting There are four reasons why the mechanism design approach is likely to be most successful in the setting of quasilinear preferences. First, historically, mechanism design has been been most successful when the quasilinear assumption could be made. Second, because of this success, some very general mechanisms have been discovered for the quasilinear setting (for instance, the VCG mechanisms [24, 4, 10], or the dAGVA mechanism [6, 1]) which we could apply directly to the expressive charity donation problem. Third, as we saw in Section 9, the clearing problem is much easier in 5 Formally, this means that if the bidder is forced to pay the full amount that his bid allows for a particular vector of payments to charities, the bidder is indifferent between this and not participating in the mechanism at all. (Compare this to bidding truthfully in a first-price auction.) 58 this setting, and thus we are less likely to run into computational trouble for the mechanism design problem. Fourth, as we will show shortly, the quasilinearity assumption in some cases allows for decomposing the mechanism design problem over the charities (as it did for the simple clearing problem). Moreover, in the quasilinear setting (unlike in the general setting), it makes sense to pursue social welfare (the sum of the utilities) as the objective, because now 1) units of utility correspond directly to units of money, so that we do not have the problem of the bidders arbitrarily scaling their utilities; and 2) it is no longer possible to give a payment willingness function of 0 while still affecting the donations through a utility function. Before presenting the decomposition result, we introduce some terms from game theory. A type is a preference profile that a bidder can have and can report (thus, a type report is a bid). Incentive compatibility (IC) means that bidders are best off reporting their preferences truthfully; either regardless of the others" types (in dominant strategies), or in expectation over them (in Bayes-Nash equilibrium). Individual rationality (IR) means agents are at least as well off participating in the mechanism as not participating; either regardless of the others" types (ex-post), or in expectation over them (ex-interim). A mechanism is budget balanced if there is no flow of money into or out of the system-in general (ex-post), or in expectation over the type reports (ex-ante). A mechanism is efficient if it (always) produces the efficient allocation of wealth to charities. Theorem 7. Suppose all agents" preferences are quasilinear. Furthermore, suppose that there exists a single-charity mechanism M that, for a certain subclass P of (quasilinear) preferences, under a given solution concept S (implementation in dominant strategies or Bayes-Nash equilibrium) and a given notion of individual rationality R (ex post, ex interim, or none), satisfies a certain notion of budget balance (ex post, ex ante, or none), and is ex-post efficient. Then there exists such a mechanism for any number of charities. Two mechanisms that satisfy efficiency (and can in fact be applied directly to the multiple-charity problem without use of the previous theorem) are the VCG (which is incentive compatible in dominant strategies) and dAGVA (which is incentive compatible only in Bayes-Nash equilibrium) mechanisms. Each of them, however, has a drawback that would probably make it impractical in the setting of donations to charities. The VCG mechanism is not budget balanced. The dAGVA mechanism does not satisfy ex-post individual rationality. In the next subsection, we will investigate if we can do better in the setting of donations to charities. 10.3 Impossibility of efficiency In this subsection, we show that even in a very restricted setting, and with minimal requirements on IC and IR constraints, it is impossible to create a mechanism that is efficient. Theorem 8. There is no mechanism which is ex-post budget balanced, ex-post efficient, and ex-interim individually rational with Bayes-Nash equilibrium as the solution concept (even with only one charity, only two quasilinear bidders, with identical type distributions (uniform over two types, with either both utility functions being step functions or both utility functions being concave piecewise linear functions)). The case of step-functions in this theorem corresponds exactly to the case of a single, fixed-size, nonexcludable public good (the public good being that the charity receives the desired amount)-for which such an impossibility result is already known [14]. Many similar results are known, probably the most famous of which is the Myerson-Satterthwaite impossibility result, which proves the impossibility of efficient bilateral trade under the same requirements [15]. Theorem 7 indicates that there is no reason to decide on donations to multiple charities under a single mechanism (rather than a separate one for each charity), when an efficient mechanism with the desired properties exists for the single-charity case. However, because under the requirements of Theorem 8, no such mechanism exists, there may be a benefit to bringing the charities under the same umbrella. The next proposition shows that this is indeed the case. Proposition 2. There exist settings with two charities where there exists no ex-post budget balanced, ex-post efficient, and ex-interim individually rational mechanism with Bayes-Nash equilibrium as the solution concept for either charity alone; but there exists an ex-post budget balanced, ex-post efficient, and ex-post individually rational mechanism with dominant strategies as the solution concept for both charities together. (Even when the conditions are the same as in Theorem 8, apart from the fact that there are now two charities.) 11. CONCLUSION We introduced a bidding language for expressing very general types of matching offers over multiple charities. We formulated the corresponding clearing problem (deciding how much each bidder pays, and how much each charity receives), and showed that it is NP-complete to approximate to any ratio even in very restricted settings. We gave a mixed-integer program formulation of the clearing problem, and showed that for concave bids (where utility functions and payment willingness function are concave), the program reduces to a linear program and can hence be solved in polynomial time. We then showed that the clearing problem for a subclass of concave bids is at least as hard as the decision variant of linear programming, suggesting that we cannot do much better than a linear programming implementation for such bids. Subsequently, we showed that the clearing problem is much easier when bids are quasilinear (where payment willingness functions are linear)-for surplus, the problem decomposes across charities, and for payment maximization, a greedy approach is optimal if the bids are concave (although this latter problem is weakly NP-complete when the bids are not concave). For the quasilinear setting, we studied the mechanism design question of making the bidders report their preferences truthfully rather than strategically. We showed that an ex-post efficient mechanism is impossible even with only one charity and a very restricted class of bids. We also showed that even though the clearing problem decomposes over charities in the quasilinear setting, there may be benefits to linking the charities from a mechanism design standpoint. There are many directions for future research. One is to build a web-based implementation of the (first-price) mechanism proposed in this paper. Another is to study the computational scalability of our MIP/LP approach. It is also 59 important to identify other classes of bids (besides concave ones) for which the clearing problem is tractable. Much crucial work remains to be done on the mechanism design problem. Finally, are there good iterative mechanisms for charity donation?6 12. REFERENCES [1] K. Arrow. The property rights doctrine and demand revelation under incomplete information. In M. Boskin, editor, Economics and human welfare. New York Academic Press, 1979. [2] L. M. Ausubel and P. Milgrom. Ascending auctions with package bidding. Frontiers of Theoretical Economics, 1, 2002. No. 1, Article 1. [3] Y. Bartal, R. Gonen, and N. Nisan. Incentive compatible multi-unit combinatorial auctions. In Theoretical Aspects of Rationality and Knowledge (TARK IX), Bloomington, Indiana, USA, 2003. [4] E. H. Clarke. Multipart pricing of public goods. Public Choice, 11:17-33, 1971. [5] V. Conitzer and T. Sandholm. Complexity of mechanism design. In Proceedings of the 18th Annual Conference on Uncertainty in Artificial Intelligence (UAI-02), pages 103-110, Edmonton, Canada, 2002. [6] C. d"Aspremont and L. A. G´erard-Varet. Incentives and incomplete information. Journal of Public Economics, 11:25-45, 1979. [7] M. R. Garey, D. S. Johnson, and L. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1:237-267, 1976. [8] D. Goldburg and S. McElligott. Red cross statement on official donation locations. 2001. Press release, http://www.redcross.org/press/disaster/ds pr/ 011017legitdonors.html. [9] R. Gonen and D. Lehmann. Optimal solutions for multi-unit combinatorial auctions: Branch and bound heuristics. In Proceedings of the ACM Conference on Electronic Commerce (ACM-EC), pages 13-20, Minneapolis, MN, Oct. 2000. [10] T. Groves. Incentives in teams. Econometrica, 41:617-631, 1973. [11] L. Khachiyan. A polynomial algorithm in linear programming. Soviet Math. Doklady, 20:191-194, 1979. [12] R. Lavi, A. Mu"Alem, and N. Nisan. Towards a characterization of truthful combinatorial auctions. In Proceedings of the Annual Symposium on Foundations of Computer Science (FOCS), 2003. [13] D. Lehmann, L. I. O"Callaghan, and Y. Shoham. Truth revelation in rapid, approximately efficient combinatorial auctions. Journal of the ACM, 49(5):577-602, 2002. Early version appeared in ACMEC-99. 6 Compare, for example, iterative mechanisms in the combinatorial auction setting [19, 25, 2]. [14] A. Mas-Colell, M. Whinston, and J. R. Green. Microeconomic Theory. Oxford University Press, 1995. [15] R. Myerson and M. Satterthwaite. Efficient mechanisms for bilateral trading. Journal of Economic Theory, 28:265-281, 1983. [16] G. L. Nemhauser and L. A. Wolsey. Integer and Combinatorial Optimization. John Wiley & Sons, 1999. Section 4, page 11. [17] N. Nisan. Bidding and allocation in combinatorial auctions. In Proceedings of the ACM Conference on Electronic Commerce (ACM-EC), pages 1-12, Minneapolis, MN, 2000. [18] N. Nisan and A. Ronen. Computationally feasible VCG mechanisms. In Proceedings of the ACM Conference on Electronic Commerce (ACM-EC), pages 242-252, Minneapolis, MN, 2000. [19] D. C. Parkes. iBundle: An efficient ascending price bundle auction. In Proceedings of the ACM Conference on Electronic Commerce (ACM-EC), pages 148-157, Denver, CO, Nov. 1999. [20] M. H. Rothkopf, A. Pekeˇc, and R. M. Harstad. Computationally manageable combinatorial auctions. Management Science, 44(8):1131-1147, 1998. [21] T. Sandholm. Algorithm for optimal winner determination in combinatorial auctions. Artificial Intelligence, 135:1-54, Jan. 2002. Conference version appeared at the International Joint Conference on Artificial Intelligence (IJCAI), pp. 542-547, Stockholm, Sweden, 1999. [22] T. Sandholm, S. Suri, A. Gilpin, and D. Levine. CABOB: A fast optimal algorithm for combinatorial auctions. In Proceedings of the Seventeenth International Joint Conference on Artificial Intelligence (IJCAI), pages 1102-1108, Seattle, WA, 2001. [23] J. Tagliabue. Global AIDS Funds Is Given Attention, but Not Money. The New York Times, June 1, 2003. Reprinted on http://www.healthgap.org/press releases/a03/ 060103 NYT HGAP G8 fund.html. [24] W. Vickrey. Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance, 16:8-37, 1961. [25] P. R. Wurman and M. P. Wellman. AkBA: A progressive, anonymous-price combinatorial auction. In Proceedings of the ACM Conference on Electronic Commerce (ACM-EC), pages 21-29, Minneapolis, MN, Oct. 2000. [26] M. Yokoo. The characterization of strategy/false-name proof combinatorial auction protocols: Price-oriented, rationing-free protocol. In Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence (IJCAI), Acapulco, Mexico, Aug. 2003. 60

bidding language;linear programming;expressive negotiation;supporter of charity;combinatorial auction;economic efficiency;charity supporter;expressive charity donation;threshold bid;incentive compatibility;market clear;concave bid;donation to charity;payment willingness function;negotiating material;quasilinearity;bidding framework;donation-clearing;mechanism design

train_J-67

Mechanism Design for Online Real-Time Scheduling

For the problem of online real-time scheduling of jobs on a single processor, previous work presents matching upper and lower bounds on the competitive ratio that can be achieved by a deterministic algorithm. However, these results only apply to the non-strategic setting in which the jobs are released directly to the algorithm. Motivated by emerging areas such as grid computing, we instead consider this problem in an economic setting, in which each job is released to a separate, self-interested agent. The agent can then delay releasing the job to the algorithm, inflate its length, and declare an arbitrary value and deadline for the job, while the center determines not only the schedule, but the payment of each agent. For the resulting mechanism design problem (in which we also slightly strengthen an assumption from the non-strategic setting), we present a mechanism that addresses each incentive issue, while only increasing the competitive ratio by one. We then show a matching lower bound for deterministic mechanisms that never pay the agents.

1. INTRODUCTION We consider the problem of online scheduling of jobs on a single processor. Each job is characterized by a release time, a deadline, a processing time, and a value for successful completion by its deadline. The objective is to maximize the sum of the values of the jobs completed by their respective deadlines. The key challenge in this online setting is that the schedule must be constructed in real-time, even though nothing is known about a job until its release time. Competitive analysis [6, 10], with its roots in [12], is a well-studied approach for analyzing online algorithms by comparing them against the optimal offline algorithm, which has full knowledge of the input at the beginning of its execution. One interpretation of this approach is as a game between the designer of the online algorithm and an adversary. First, the designer selects the online algorithm. Then, the adversary observes the algorithm and selects the sequence of jobs that maximizes the competitive ratio: the ratio of the value of the jobs completed by an optimal offline algorithm to the value of those completed by the online algorithm. Two papers paint a complete picture in terms of competitive analysis for this setting, in which the algorithm is assumed to know k, the maximum ratio between the value densities (value divided by processing time) of any two jobs. For k = 1, [4] presents a 4-competitive algorithm, and proves that this is a lower bound on the competitive ratio for deterministic algorithms. The same paper also generalizes the lower bound to (1 + √ k)2 for any k ≥ 1, and [15] then presents a matching (1 + √ k)2 -competitive algorithm. The setting addressed by these papers is completely nonstrategic, and the algorithm is assumed to always know the true characteristics of each job upon its release. However, in domains such as grid computing (see, for example, [7, 8]) this assumption is invalid, because buyers of processor time choose when and how to submit their jobs. Furthermore, sellers not only schedule jobs but also determine the amount that they charge buyers, an issue not addressed in the non-strategic setting. Thus, we consider an extension of the setting in which each job is owned by a separate, self-interested agent. Instead of being released to the algorithm, each job is now released only to its owning agent. Each agent now has four different ways in which it can manipulate the algorithm: it decides when to submit the job to the algorithm after the true release time, it can artificially inflate the length of the job, and it can declare an arbitrary value and deadline for the job. Because the agents are self-interested, they will choose to manipulate the algorithm if doing so will cause 61 their job to be completed; and, indeed, one can find examples in which agents have incentive to manipulate the algorithms presented in [4] and [15]. The addition of self-interested agents moves the problem from the area of algorithm design to that of mechanism design [17], the science of crafting protocols for self-interested agents. Recent years have seen much activity at the interface of computer science and mechanism design (see, e.g., [9, 18, 19]). In general, a mechanism defines a protocol for interaction between the agents and the center that culminates with the selection of an outcome. In our setting, a mechanism will take as input a job from each agent, and return a schedule for the jobs, and a payment to be made by each agent to the center. A basic solution concept of mechanism design is incentive compatibility, which, in our setting, requires that it is always in each agent"s best interests to immediately submit its job upon release, and to truthfully declare its value, length, and deadline. In order to evaluate a mechanism using competitive analysis, the adversary model must be updated. In the new model, the adversary still determines the sequence of jobs, but it is the self-interested agents who determine the observed input of the mechanism. Thus, in order to achieve a competitive ratio of c, an online mechanism must both be incentive compatible, and always achieve at least 1 c of the value that the optimal offline mechanism achieves on the same sequence of jobs. The rest of the paper is structured as follows. In Section 2, we formally define and review results from the original, non-strategic setting. After introducing the incentive issues through an example, we formalize the mechanism design setting in Section 3. In Section 4 we present our first main result, a ((1 + √ k)2 + 1)-competitive mechanism, and formally prove incentive compatibility and the competitive ratio. We also show how we can simplify this mechanism for the special case in which k = 1 and each agent cannot alter the length of its job. Returning the general setting, we show in Section 5 that this competitive ratio is a lower bound for deterministic mechanisms that do not pay agents. Finally, in Section 6, we discuss related work other than the directly relevant [4] and [15], before concluding with Section 7. 2. NON-STRATEGIC SETTING In this section, we formally define the original, non-strategic setting, and recap previous results. 2.1 Formulation There exists a single processor on which jobs can execute, and N jobs, although this number is not known beforehand. Each job i is characterized by a tuple θi = (ri, di, li, vi), which denotes the release time, deadline, length of processing time required, and value, respectively. The space Θi of possible tuples is the same for each job and consists of all θi such that ri, di, li, vi ∈ + (thus, the model of time is continuous). Each job is released at time ri, at which point its three other characteristics are known. Nothing is known about the job before its arrival. Each deadline is firm (or, hard), which means that no value is obtained for a job that is completed after its deadline. Preemption of jobs is allowed, and it takes no time to switch between jobs. Thus, job i is completed if and only if the total time it executes on the processor before di is at least li. Let θ = (θ1, . . . , θN ) denote the vector of tuples for all jobs, and let θ−i = (θ1, . . . , θi−1, θi+1, . . . , θN ) denote the same vector without the tuple for job i. Thus, (θi, θ−i) denotes a complete vector of tuples. Define the value density ρi = vi li of job i to be the ratio of its value to its length. For an input θ, denote the maximum and minimum value densities as ρmin = mini ρi and ρmax = maxi ρi. The importance ratio is then defined to be ρmax ρmin , the maximal ratio of value densities between two jobs. The algorithm is assumed to always know an upper bound k on the importance ratio. For simplicity, we normalize the range of possible value densities so that ρmin = 1. An online algorithm is a function f : Θ1 × . . . × ΘN → O that maps the vector of tuples (for any number N) to an outcome o. An outcome o ∈ O is simply a schedule of jobs on the processor, recorded by the function S : + → {0, 1, . . . , N}, which maps each point in time to the active job, or to 0 if the processor is idle. To denote the total elapsed time that a job has spent on the processor at time t, we will use the function ei(t) = t 0 µ(S(x) = i)dx, where µ(·) is an indicator function that returns 1 if the argument is true, and zero otherwise. A job"s laxity at time t is defined to be di − t − li + ei(t) , the amount of time that it can remain inactive and still be completed by its deadline. A job is abandoned if it cannot be completed by its deadline (formally, if di −t+ei(t) < li). Also, overload S(·) and ei(·) so that they can also take a vector θ as an argument. For example, S(θ, t) is shorthand for the S(t) of the outcome f(θ), and it denotes the active job at time t when the input is θ. Since a job cannot be executed before its release time, the space of possible outcomes is restricted in that S(θ, t) = i implies ri ≤ t. Also, because the online algorithm must produce the schedule over time, without knowledge of future inputs, it must make the same decision at time t for inputs that are indistinguishable at this time. Formally, let θ(t) denote the subset of the tuples in θ that satisfy ri ≤ t. The constraint is then that θ(t) = θ (t) implies S(θ, t) = S(θ , t). The objective function is the sum of the values of the jobs that are completed by their respective deadlines: W(o, θ) = i vi · µ(ei(θ, di) ≥ li) . Let W∗ (θ) = maxo∈O W(o, θ) denote the maximum possible total value for the profile θ. In competitive analysis, an online algorithm is evaluated by comparing it against an optimal offline algorithm. Because the offline algorithm knows the entire input θ at time 0 (but still cannot start each job i until time ri), it always achieves W∗ (θ). An online algorithm f(·) is (strictly) c-competitive if there does not exist an input θ such that c · W(f(θ), θ) < W∗ (θ). An algorithm that is c-competitive is also said to achieve a competitive ratio of c. We assume that there does not exist an overload period of infinite duration. A period of time [ts , tf ] is overloaded if the sum of the lengths of the jobs whose release time and deadline both fall within the time period exceeds the duration of the interval (formally, if tf −ts ≤ i|(ts≤ri,di≤tf ) li). Without such an assumption, it is not possible to achieve a finite competitive ratio [15]. 2.2 Previous Results In the non-strategic setting, [4] presents a 4-competitive algorithm called TD1 (version 2) for the case of k = 1, while [15] presents a (1+ √ k)2 -competitive algorithm called Dover for the general case of k ≥ 1. Matching lower bounds for deterministic algorithms for both of these cases were shown 62 in [4]. In this section we provide a high-level description of TD1 (version 2) using an example. TD1 (version 2) divides the schedule into intervals, each of which begins when the processor transitions from idle to busy (call this time tb ), and ends with the completion of a job. The first active job of an interval may have laxity; however, for the remainder of the interval, preemption of the active job is only considered when some other job has zero laxity. For example, when the input is the set of jobs listed in Table 1, the first interval is the complete execution of job 1 over the range [0.0, 0.9]. No preemption is considered during this interval, because job 2 has laxity until time 1.5. Then, a new interval starts at tb = 0.9 when job 2 becomes active. Before job 2 can finish, preemption is considered at time 4.8, when job 3 is released with zero laxity. In order to decide whether to preempt the active job, TD1 (version 2) uses two more variables: te and p loss. The former records the latest deadline of a job that would be abandoned if the active job executes to completion (or, if no such job exists, the time that the active job will finish if it is not preempted). In this case, te = 17.0. The value te −tb represents the an upper bound on the amount of possible execution time lost to the optimal offline algorithm due to the completion of the active job. The other variable, p loss, is equal to the length of the first active job of the current interval. Because in general this job could have laxity, the offline algorithm may be able to complete it outside of the range [tb , te ].1 If the algorithm completes the active job and this job"s length is at least te −tb +p loss 4 , then the algorithm is guaranteed to be 4-competitive for this interval (note that k = 1 implies that all jobs have the same value density and thus that lengths can used to compute the competitive ratio). Because this is not case at time 4.8 (since te −tb +p loss 4 = 17.0−0.9+4.0 4 > 4.0 = l2), the algorithm preempts job 2 for job 3, which then executes to completion. Job ri di li vi 1 0.0 0.9 0.9 0.9 2 0.5 5.5 4.0 4.0 3 4.8 17.0 12.2 12.2 01 5 17 6 ? 6 ? 6 ? Table 1: Input used to recap TD1 (version 2) [4]. The up and down arrows represent ri and di, respectively, while the length of the box equals li. 3. MECHANISM DESIGN SETTING However, false information about job 2 would cause TD1 (version 2) to complete this job. For example, if job 2"s deadline were declared as ˆd2 = 4.7, then it would have zero laxity at time 0.7. At this time, the algorithm would preempt job 1 for job 2, because te −tb +p loss 4 = 4.7−0.0+1.0 4 > 0.9 = l1. Job 2 would then complete before the arrival of job 3.2 1 While it would be easy to alter the algorithm to recognize that this is not possible for the jobs in Table 1, our example does not depend on the use of p loss. 2 While we will not describe the significantly more complex In order to address incentive issues such as this one, we need to formalize the setting as a mechanism design problem. In this section we first present the mechanism design formulation, and then define our goals for the mechanism. 3.1 Formulation There exists a center, who controls the processor, and N agents, where the value of N is unknown by the center beforehand. Each job i is owned by a separate agent i. The characteristics of the job define the agent"s type θi ∈ Θi. At time ri, agent i privately observes its type θi, and has no information about job i before ri. Thus, jobs are still released over time, but now each job is revealed only to the owning agent. Agents interact with the center through a direct mechanism Γ = (Θ1, . . . , ΘN , g(·)), in which each agent declares a job, denoted by ˆθi = (ˆri, ˆdi, ˆli, ˆvi), and g : Θ1×. . .×ΘN → O maps the declared types to an outcome o ∈ O. An outcome o = (S(·), p1, . . . , pN ) consists of a schedule and a payment from each agent to the mechanism. In a standard mechanism design setting, the outcome is enforced at the end of the mechanism. However, since the end is not well-defined in this online setting, we choose to model returning the job if it is completed and collecting a payment from each agent i as occurring at ˆdi, which, according to the agent"s declaration, is the latest relevant point of time for that agent. That is, even if job i is completed before ˆdi, the center does not return the job to agent i until that time. This modelling decision could instead be viewed as a decision by the mechanism designer from a larger space of possible mechanisms. Indeed, as we will discuss later, this decision of when to return a completed job is crucial to our mechanism. Each agent"s utility, ui(g(ˆθ), θi) = vi · µ(ei(ˆθ, di) ≥ li) · µ( ˆdi ≤ di) − pi(ˆθ), is a quasi-linear function of its value for its job (if completed and returned by its true deadline) and the payment it makes to the center. We assume that each agent is a rational, expected utility maximizer. Agent declarations are restricted in that an agent cannot declare a length shorter than the true length, since the center would be able to detect such a lie if the job were completed. On the other hand, in the general formulation we will allow agents to declare longer lengths, since in some settings it may be possible add unnecessary work to a job. However, we will also consider a restricted formulation in which this type of lie is not possible. The declared release time ˆri is the time that the agent chooses to submit job i to the center, and it cannot precede the time ri at which the job is revealed to the agent. The agent can declare an arbitrary deadline or value. To summarize, agent i can declare any type ˆθi = (ˆri, ˆdi, ˆli, ˆvi) such that ˆli ≥ li and ˆri ≥ ri. While in the non-strategic setting it was sufficient for the algorithm to know the upper bound k on the ratio ρmax ρmin , in the mechanism design setting we will strengthen this assumption so that the mechanism also knows ρmin (or, equivalently, the range [ρmin, ρmax] of possible value densities).3 Dover , we note that it is similar in its use of intervals and its preference for the active job. Also, we note that the lower bound we will show in Section 5 implies that false information can also benefit a job in Dover . 3 Note that we could then force agent declarations to satisfy ρmin ≤ ˆvi ˆli ≤ ρmax. However, this restriction would not 63 While we feel that it is unlikely that a center would know k without knowing this range, we later present a mechanism that does not depend on this extra knowledge in a restricted setting. The restriction on the schedule is now that S(ˆθ, t) = i implies ˆri ≤ t, to capture the fact that a job cannot be scheduled on the processor before it is declared to the mechanism. As before, preemption of jobs is allowed, and job switching takes no time. The constraints due to the online mechanism"s lack of knowledge of the future are that ˆθ(t) = ˆθ (t) implies S(ˆθ, t) = S(ˆθ , t), and ˆθ( ˆdi) = ˆθ ( ˆdi) implies pi(ˆθ) = pi(ˆθ ) for each agent i. The setting can then be summarized as follows. 1Overview of the Setting: for all t do The center instantiates S(ˆθ, t) ← i, for some i s.t. ˆri ≤ t if ∃i, (ri = t) then θi is revealed to agent i if ∃i, (t ≥ ri) and agent i has not declared a job then Agent i can declare any job ˆθi, s.t. ˆri = t and ˆli ≥ li if ∃i, ( ˆdi = t) ∧ (ei(ˆθ, t) ≥ li) then Completed job i is returned to agent i if ∃i, ( ˆdi = t) then Center sets and collects payment pi(ˆθ) from agent i 3.2 Mechanism Goals Our aim as mechanism designer is to maximize the value of completed jobs, subject to the constraints of incentive compatibility and individual rationality. The condition for (dominant strategy) incentive compatibility is that for each agent i, regardless of its true type and of the declared types of all other agents, agent i cannot increase its utility by unilaterally changing its declaration. Definition 1. A direct mechanism Γ satisfies incentive compatibility (IC) if ∀i, θi, θi, ˆθ−i : ui(g(θi, ˆθ−i), θi) ≥ ui(g(θi, ˆθ−i), θi) From an agent perspective, dominant strategies are desirable because the agent does not have to reason about either the strategies of the other agents or the distribution from the which other agent"s types are drawn. From a mechanism designer perspective, dominant strategies are important because we can reasonably assume that an agent who has a dominant strategy will play according to it. For these reasons, in this paper we require dominant strategies, as opposed to a weaker equilibrium concept such as Bayes-Nash, under which we could improve upon our positive results.4 decrease the lower bound on the competitive ratio. 4 A possible argument against the need for incentive compatibility is that an agent"s lie may actually improve the schedule. In fact, this was the case in the example we showed for the false declaration ˆd2 = 4.7. However, if an agent lies due to incorrect beliefs over the future input, then the lie could instead make the schedule the worse (for example, if job 3 were never released, then job 1 would have been unnecessarily abandoned). Furthermore, if we do not know the beliefs of the agents, and thus cannot predict how they will lie, then we can no longer provide a competitive guarantee for our mechanism. While restricting ourselves to incentive compatible direct mechanisms may seem limiting at first, the Revelation Principle for Dominant Strategies (see, e.g., [17]) tells us that if our goal is dominant strategy implementation, then we can make this restriction without loss of generality. The second goal for our mechanism, individual rationality, requires that agents who truthfully reveal their type never have negative utility. The rationale behind this goal is that participation in the mechanism is assumed to be voluntary. Definition 2. A direct mechanism Γ satisfies individual rationality (IR) if ∀i, θi, ˆθ−i, ui(g(θi, ˆθ−i), θi) ≥ 0. Finally, the social welfare function that we aim to maximize is the same as the objective function of the non-strategic setting: W(o, θ) = i vi · µ(ei(θ, di) ≥ li) . As in the nonstrategic setting, we will evaluate an online mechanism using competitive analysis to compare it against an optimal offline mechanism (which we will denote by Γoffline). An offline mechanism knows all of the types at time 0, and thus can always achieve W∗ (θ).5 Definition 3. An online mechanism Γ is (strictly) ccompetitive if it satisfies IC and IR, and if there does not exist a profile of agent types θ such that c·W(g(θ), θ) < W∗ (θ). 4. RESULTS In this section, we first present our main positive result: a (1+ √ k)2 +1 -competitive mechanism (Γ1). After providing some intuition as to why Γ1 satisfies individual rationality and incentive compatibility, we formally prove first these two properties and then the competitive ratio. We then consider a special case in which k = 1 and agents cannot lie about the length of their job, which allows us to alter this mechanism so that it no longer requires either knowledge of ρmin or the collection of payments from agents. Unlike TD1 (version 2) and Dover , Γ1 gives no preference to the active job. Instead, it always executes the available job with the highest priority: (ˆvi + √ k · ei(ˆθ, t) · ρmin). Each agent whose job is completed is then charged the lowest value that it could have declared such that its job still would have been completed, holding constant the rest of its declaration. By the use of a payment rule similar to that of a secondprice auction, Γ1 satisfies both IC with respect to values and IR. We now argue why it satisfies IC with respect to the other three characteristics. Declaring an improved job (i.e., declaring an earlier release time, a shorter length, or a later deadline) could possibly decrease the payment of an agent. However, the first two lies are not possible in our setting, while the third would cause the job, if it is completed, to be returned to the agent after the true deadline. This is the reason why it is important to always return a completed job at its declared deadline, instead of at the point at which it is completed. 5 Another possibility is to allow only the agents to know their types at time 0, and to force Γoffline to be incentive compatible so that agents will truthfully declare their types at time 0. However, this would not affect our results, since executing a VCG mechanism (see, e.g., [17]) at time 0 both satisfies incentive compatibility and always maximizes social welfare. 64 Mechanism 1 Γ1 Execute S(ˆθ, ·) according to Algorithm 1 for all i do if ei(ˆθ, ˆdi) ≥ ˆli {Agent i"s job is completed} then pi(ˆθ) ← arg minvi≥0(ei(((ˆri, ˆdi, ˆli, vi), ˆθ−i), ˆdi) ≥ ˆli) else pi(ˆθ) ← 0 Algorithm 1 for all t do Avail ← {i|(t ≥ ˆri)∧(ei(ˆθ, t) < ˆli)∧(ei(ˆθ, t)+ ˆdi−t ≥ ˆli)} {Set of all released, non-completed, non-abandoned jobs} if Avail = ∅ then S(ˆθ, t) ← arg maxi∈Avail(ˆvi + √ k · ei(ˆθ, t) · ρmin) {Break ties in favor of lower ˆri} else S(ˆθ, t) ← 0 It remains to argue why an agent does not have incentive to worsen its job. The only possible effects of an inflated length are delaying the completion of the job and causing it to be abandoned, and the only possible effects of an earlier declared deadline are causing to be abandoned and causing it to be returned earlier (which has no effect on the agent"s utility in our setting). On the other hand, it is less obvious why agents do not have incentive to declare a later release time. Consider a mechanism Γ1 that differs from Γ1 in that it does not preempt the active job i unless there exists another job j such that (ˆvi + √ k·li(ˆθ, t)·ρmin) < ˆvj. Note that as an active job approaches completion in Γ1, its condition for preemption approaches that of Γ1. However, the types in Table 2 for the case of k = 1 show why an agent may have incentive to delay the arrival of its job under Γ1. Job 1 becomes active at time 0, and job 2 is abandoned upon its release at time 6, because 10 + 10 = v1 +l1 > v2 = 13. Then, at time 8, job 1 is preempted by job 3, because 10 + 10 = v1 + l1 < v3 = 22. Job 3 then executes to completion, forcing job 1 to be abandoned. However, job 2 had more weight than job 1, and would have prevented job 3 from being executed if it had been the active job at time 8, since 13 + 13 = v2 + l2 > v3 = 22. Thus, if agent 1 had falsely declared ˆr1 = 20, then job 3 would have been abandoned at time 8, and job 1 would have completed over the range [20, 30]. Job ri di li vi 1 0 30 10 10 2 6 19 13 13 3 8 30 22 22 0 6 10 20 30 6 ? 6 ? 6 ? Table 2: Jobs used to show why a slightly altered version of Γ1 would not be incentive compatible with respect to release times. Intuitively, Γ1 avoids this problem because of two properties. First, when a job becomes active, it must have a greater priority than all other available jobs. Second, because a job"s priority can only increase through the increase of its elapsed time, ei(ˆθ, t), the rate of increase of a job"s priority is independent of its characteristics. These two properties together imply that, while a job is active, there cannot exist a time at which its priority is less than the priority that one of these other jobs would have achieved by executing on the processor instead. 4.1 Proof of Individual Rationality and Incentive Compatibility After presenting the (trivial) proof of IR, we break the proof of IC into lemmas. Theorem 1. Mechanism Γ1 satisfies individual rationality. Proof. For arbitrary i, θi, ˆθ−i, if job i is not completed, then agent i pays nothing and thus has a utility of zero; that is, pi(θi, ˆθ−i) = 0 and ui(g(θi, ˆθ−i), θi) = 0. On the other hand, if job i is completed, then its value must exceed agent i"s payment. Formally, ui(g(θi, ˆθ−i), θi) = vi − arg minvi≥0(ei(((ri, di, li, vi), ˆθ−i), di) ≥ li) ≥ 0 must hold, since vi = vi satisfies the condition. To prove IC, we need to show that for an arbitrary agent i, and an arbitrary profile ˆθ−i of declarations of the other agents, agent i can never gain by making a false declaration ˆθi = θi, subject to the constraints that ˆri ≥ ri and ˆli ≥ li. We start by showing that, regardless of ˆvi, if truthful declarations of ri, di, and li do not cause job i to be completed, then worse declarations of these variables (that is, declarations that satisfy ˆri ≥ ri, ˆli ≥ li and ˆdi ≤ di) can never cause the job to be completed. We break this part of the proof into two lemmas, first showing that it holds for the release time, regardless of the declarations of the other variables, and then for length and deadline. Lemma 2. In mechanism Γ1, the following condition holds for all i, θi, ˆθ−i: ∀ ˆvi, ˆli ≥ li, ˆdi ≤ di, ˆri ≥ ri, ei ((ˆri, ˆdi, ˆli, ˆvi), ˆθ−i), ˆdi ≥ ˆli =⇒ ei ((ri, ˆdi, ˆli, ˆvi), ˆθ−i), ˆdi ≥ ˆli Proof. Assume by contradiction that this condition does not hold- that is, job i is not completed when ri is truthfully declared, but is completed for some false declaration ˆri ≥ ri. We first analyze the case in which the release time is truthfully declared, and then we show that job i cannot be completed when agent i delays submitting it to the center. Case I: Agent i declares ˆθi = (ri, ˆdi, ˆli, ˆvi). First, define the following three points in the execution of job i. • Let ts = arg mint S((ˆθi, ˆθ−i), t) = i be the time that job i first starts execution. • Let tp = arg mint>ts S((ˆθi, ˆθ−i), t) = i be the time that job i is first preempted. • Let ta = arg mint ei((ˆθi, ˆθ−i), t) + ˆdi − t < ˆli be the time that job i is abandoned. 65 If ts and tp are undefined because job i never becomes active, then let ts = tp = ta . Also, partition the jobs declared by other agents before ta into the following three sets. • X = {j|(ˆrj < tp ) ∧ (j = i)} consists of the jobs (other than i) that arrive before job i is first preempted. • Y = {j|(tp ≤ ˆrj ≤ ta )∧(ˆvj > ˆvi + √ k·ei((ˆθi, ˆθ−i), ˆrj)} consists of the jobs that arrive in the range [tp , ta ] and that when they arrive have higher priority than job i (note that we are make use of the normalization). • Z = {j|(tp ≤ ˆrj ≤ ta )∧(ˆvj ≤ ˆvi + √ k ·ei((ˆθi, ˆθ−i), ˆrj)} consists of the jobs that arrive in the range [tp , ta ] and that when they arrive have lower priority than job i. We now show that all active jobs during the range (tp , ta ] must be either i or in the set Y . Unless tp = ta (in which case this property trivially holds), it must be the case that job i has a higher priority than an arbitrary job x ∈ X at time tp , since at the time just preceding tp job x was available and job i was active. Formally, ˆvx + √ k · ex((ˆθi, ˆθ−i), tp ) < ˆvi + √ k · ei((ˆθi, ˆθ−i), tp ) must hold.6 We can then show that, over the range [tp , ta ], no job x ∈ X runs on the processor. Assume by contradiction that this is not true. Let tf ∈ [tp , ta ] be the earliest time in this range that some job x ∈ X is active, which implies that ex((ˆθi, ˆθ−i), tf ) = ex((ˆθi, ˆθ−i), tp ). We can then show that job i has a higher priority at time tf as follows: ˆvx+ √ k·ex((ˆθi, ˆθ−i), tf ) = ˆvx+ √ k·ex((ˆθi, ˆθ−i), tp ) < ˆvi + √ k · ei((ˆθi, ˆθ−i), tp ) ≤ ˆvi + √ k · ei((ˆθi, ˆθ−i), tf ), contradicting the fact that job x is active at time tf . A similar argument applies to an arbitrary job z ∈ Z, starting at it release time ˆrz > tp , since by definition job i has a higher priority at that time. The only remaining jobs that can be active over the range (tp , ta ] are i and those in the set Y . Case II: Agent i declares ˆθi = (ˆri, ˆdi, ˆli, ˆvi), where ˆri > ri. We now show that job i cannot be completed in this case, given that it was not completed in case I. First, we can restrict the range of ˆri that we need to consider as follows. Declaring ˆri ∈ (ri, ts ] would not affect the schedule, since ts would still be the first time that job i executes. Also, declaring ˆri > ta could not cause the job to be completed, since di − ta < ˆli holds, which implies that job i would be abandoned at its release. Thus, we can restrict consideration to ˆri ∈ (ts , ta ]. In order for declaring ˆθi to cause job i to be completed, a necessary condition is that the execution of some job yc ∈ Y must change during the range (tp , ta ], since the only jobs other than i that are active during that range are in Y . Let tc = arg mint∈(tp,ta][∃yc ∈ Y, (S((ˆθi, ˆθ−i), t) = yc ) ∧ (S((ˆθi, ˆθ−i), t) = yc )] be the first time that such a change occurs. We will now show that for any ˆri ∈ (ts , ta ], there cannot exist a job with higher priority than yc at time tc , contradicting (S((ˆθi, ˆθ−i), t) = yc ). First note that job i cannot have a higher priority, since there would have to exist a t ∈ (tp , tc ) such that ∃y ∈ 6 For simplicity, when we give the formal condition for a job x to have a higher priority than another job y, we will assume that job x"s priority is strictly greater than job y"s, because, in the case of a tie that favors x, future ties would also be broken in favor of job x. Y, (S((ˆθi, ˆθ−i), t) = y) ∧ (S((ˆθi, ˆθ−i), t) = i), contradicting the definition of tc . Now consider an arbitrary y ∈ Y such that y = yc . In case I, we know that job y has lower priority than yc at time tc ; that is, ˆvy + √ k·ey((ˆθi, ˆθ−i), tc ) < ˆvyc + √ k·eyc ((ˆθi, ˆθ−i), tc ). Thus, moving to case II, job y must replace some other job before tc . Since ˆry ≥ tp , the condition is that there must exist some t ∈ (tp , tc ) such that ∃w ∈ Y ∪{i}, (S((ˆθi, ˆθ−i), t) = w) ∧ (S((ˆθi, ˆθ−i), t) = y). Since w ∈ Y would contradict the definition of tc , we know that w = i. That is, the job that y replaces must be i. By definition of the set Y , we know that ˆvy > ˆvi + √ k · ei((ˆθi, ˆθ−i), ˆry). Thus, if ˆry ≤ t, then job i could not have executed instead of y in case I. On the other hand, if ˆry > t, then job y obviously could not execute at time t, contradicting the existence of such a time t. Now consider an arbitrary job x ∈ X. We know that in case I job i has a higher priority than job x at time ts , or, formally, that ˆvx + √ k · ex((ˆθi, ˆθ−i), ts ) < ˆvi + √ k · ei((ˆθi, ˆθ−i), ts ). We also know that ˆvi + √ k·ei((ˆθi, ˆθ−i), tc ) < ˆvyc + √ k · eyc ((ˆθi, ˆθ−i), tc ). Since delaying i"s arrival will not affect the execution up to time ts , and since job x cannot execute instead of a job y ∈ Y at any time t ∈ (tp , tc ] by definition of tc , the only way for job x"s priority to increase before tc as we move from case I to II is to replace job i over the range (ts , tc ]. Thus, an upper bound on job x"s priority when agent i declares ˆθi is: ˆvx+ √ k· ex((ˆθi, ˆθ−i), ts )+ei((ˆθi, ˆθ−i), tc )−ei((ˆθi, ˆθ−i), ts ) < ˆvi + √ k· ei((ˆθi, ˆθ−i), ts )+ei((ˆθi, ˆθ−i), tc )−ei((ˆθi, ˆθ−i), ts ) = ˆvi + √ k · ei((ˆθi, ˆθ−i), tc ) < ˆvyc + √ k · eyc ((ˆθi, ˆθ−i), tc ). Thus, even at this upper bound, job yc would execute instead of job x at time tc . A similar argument applies to an arbitrary job z ∈ Z, starting at it release time ˆrz. Since the sets {i}, X, Y, Z partition the set of jobs released before ta , we have shown that no job could execute instead of job yc , contradicting the existence of tc , and completing the proof. Lemma 3. In mechanism Γ1, the following condition holds for all i, θi, ˆθ−i: ∀ ˆvi, ˆli ≥ li, ˆdi ≤ di, ei ((ri, ˆdi, ˆli, ˆvi), ˆθ−i), ˆdi ≥ ˆli =⇒ ei ((ri, di, li, ˆvi), ˆθ−i), ˆdi ≥ li Proof. Assume by contradiction there exists some instantiation of the above variables such that job i is not completed when li and di are truthfully declared, but is completed for some pair of false declarations ˆli ≥ li and ˆdi ≤ di. Note that the only effect that ˆdi and ˆli have on the execution of the algorithm is on whether or not i ∈ Avail. Specifically, they affect the two conditions: (ei(ˆθ, t) < ˆli) and (ei(ˆθ, t) + ˆdi − t ≥ ˆli). Because job i is completed when ˆli and ˆdi are declared, the former condition (for completion) must become false before the latter. Since truthfully declaring li ≤ ˆli and di ≥ ˆdi will only make the former condition become false earlier and the latter condition become false later, the execution of the algorithm will not be affected when moving to truthful declarations, and job i will be completed, a contradiction. We now use these two lemmas to show that the payment for a completed job can only increase by falsely declaring worse ˆli, ˆdi, and ˆri. 66 Lemma 4. In mechanism Γ1, the following condition holds for all i, θi, ˆθ−i: ∀ ˆli ≥ li, ˆdi ≤ di, ˆri ≥ ri, arg min vi≥0 ei ((ˆri, ˆdi, ˆli, vi), ˆθ−i), ˆdi ≥ ˆli ≥ arg min vi≥0 ei ((ri, di, li, vi), ˆθ−i), di ≥ li Proof. Assume by contradiction that this condition does not hold. This implies that there exists some value vi such that the condition (ei(((ˆri, ˆdi, ˆli, vi), ˆθ−i), ˆdi) ≥ ˆli) holds, but (ei(((ri, di, li, vi), ˆθ−i), di) ≥ li) does not. Applying Lemmas 2 and 3: (ei(((ˆri, ˆdi, ˆli, vi), ˆθ−i), ˆdi) ≥ ˆli) =⇒ (ei(((ri, ˆdi, ˆli, vi), ˆθ−i), ˆdi) ≥ ˆli) =⇒ (ei(((ri, di, li, vi), ˆθ−i), di) ≥ li), a contradiction. Finally, the following lemma tells us that the completion of a job is monotonic in its declared value. Lemma 5. In mechanism Γ1, the following condition holds for all i, ˆθi, ˆθ−i: ∀ ˆvi ≥ ˆvi, ei ((ˆri, ˆdi, ˆli, ˆvi), ˆθ−i), ˆdi ≥ ˆli =⇒ ei ((ˆri, ˆdi, ˆli, ˆvi), ˆθ−i), ˆdi ≥ ˆli The proof, by contradiction, of this lemma is omitted because it is essentially identical to that of Lemma 2 for ˆri. In case I, agent i declares (ˆri, ˆdi, ˆli, ˆvi) and the job is not completed, while in case II he declares (ˆri, ˆdi, ˆli, ˆvi) and the job is completed. The analysis of the two cases then proceeds as before- the execution will not change up to time ts because the initial priority of job i decreases as we move from case I to II; and, as a result, there cannot be a change in the execution of a job other than i over the range (tp , ta ]. We can now combine the lemmas to show that no profitable deviation is possible. Theorem 6. Mechanism Γ1 satisfies incentive compatibility. Proof. For an arbitrary agent i, we know that ˆri ≥ ri and ˆli ≥ li hold by assumption. We also know that agent i has no incentive to declare ˆdi > di, because job i would never be returned before its true deadline. Then, because the payment function is non-negative, agent i"s utility could not exceed zero. By IR, this is the minimum utility it would achieve if it truthfully declared θi. Thus, we can restrict consideration to ˆθi that satisfy ˆri ≥ ri, ˆli ≥ li, and ˆdi ≤ di. Again using IR, we can further restrict consideration to ˆθi that cause job i to be completed, since any other ˆθi yields a utility of zero. If truthful declaration of θi causes job i to be completed, then by Lemma 4 any such false declaration ˆθi could not decrease the payment of agent i. On the other hand, if truthful declaration does not cause job i to be completed, then declaring such a ˆθi will cause agent i to have negative utility, since vi < arg minvi≥0 ei(((ri, di, li, vi), ˆθ−i), ˆdi) ≥ li ≤ arg minvi≥0 ei(((ˆri, ˆdi, ˆli, vi), ˆθ−i), ˆdi) ≥ ˆli holds by Lemmas 5 and 4, respectively. 4.2 Proof of Competitive Ratio The proof of the competitive ratio, which makes use of techniques adapted from those used in [15], is also broken into lemmas. Having shown IC, we can assume truthful declaration (ˆθ = θ). Since we have also shown IR, in order to prove the competitive ratio it remains to bound the loss of social welfare against Γoffline. Denote by (1, 2, . . . , F) the sequence of jobs completed by Γ1. Divide time into intervals If = (topen f , tclose f ], one for each job f in this sequence. Set tclose f to be the time at which job f is completed, and set topen f = tclose f−1 for f ≥ 2, and topen 1 = 0 for f = 1. Also, let tbegin f be the first time that the processor is not idle in interval If . Lemma 7. For any interval If , the following inequality holds: tclose f − tbegin f ≤ (1 + 1√ k ) · vf Proof. Interval If begins with a (possibly zero length) period of time in which the processor is idle because there is no available job. Then, it continuously executes a sequence of jobs (1, 2, . . . , c), where each job i in this sequence is preempted by job i + 1, except for job c, which is completed (thus, job c in this sequence is the same as job f is the global sequence of completed jobs). Let ts i be the time that job i begins execution. Note that ts 1 = tbegin f . Over the range [tbegin f , tclose f ], the priority (vi+ √ k·ei(θ, t)) of the active job is monotonically increasing with time, because this function linearly increases while a job is active, and can only increase at a point in time when preemption occurs. Thus, each job i > 1 in this sequence begins execution at its release time (that is, ts i = ri), because its priority does not increase while it is not active. We now show that the value of the completed job c exceeds the product of √ k and the time spent in the interval on jobs 1 through c−1, or, more formally, that the following condition holds: vc ≥ √ k c−1 h=1(eh(θ, ts h+1) − eh(θ, ts h)). To show this, we will prove by induction that the stronger condition vi ≥ √ k i−1 h=1 eh(θ, ts h+1) holds for all jobs i in the sequence. Base Case: For i = 1, v1 ≥ √ k 0 h=1 eh(θ, ts h+1) = 0, since the sum is over zero elements. Inductive Step: For an arbitrary 1 ≤ i < c, we assume that vi ≥ √ k i−1 h=1 eh(θ, ts h+1) holds. At time ts i+1, we know that vi+1 ≥ vi + √ k · ei(θ, ts i+1) holds, because ts i+1 = ri+1. These two inequalities together imply that vi+1 ≥√ k i h=1 eh(θ, ts h+1), completing the inductive step. We also know that tclose f − ts c ≤ lc ≤ vc must hold, by the simplifying normalization of ρmin = 1 and the fact that job c"s execution time cannot exceed its length. We can thus bound the total execution time of If by: tclose f − tbegin f = (tclose f −ts c)+ c−1 h=1(eh(θ, ts h+1)−eh(θ, ts h)) ≤ (1+ 1√ k )vf . We now consider the possible execution of uncompleted jobs by Γoffline. Associate each job i that is not completed by Γ1 with the interval during which it was abandoned. All jobs are now associated with an interval, since there are no gaps between the intervals, and since no job i can be abandoned after the close of the last interval at tclose F . Because the processor is idle after tclose F , any such job i would become active at some time t ≥ tclose F , which would lead to the completion of some job, creating a new interval and contradicting the fact that IF is the last one. 67 The following lemma is equivalent to Lemma 5.6 of [15], but the proof is different for our mechanism. Lemma 8. For any interval If and any job i abandoned in If , the following inequality holds: vi ≤ (1 + √ k)vf . Proof. Assume by contradiction that there exists a job i abandoned in If such that vi > (1 + √ k)vf . At tclose f , the priority of job f is vf + √ k · lf < (1 + √ k)vf . Because the priority of the active job monotonically increases over the range [tbegin f , tclose f ], job i would have a higher priority than the active job (and thus begin execution) at some time t ∈ [tbegin f , tclose f ]. Again applying monotonicity, this would imply that the priority of the active job at tclose f exceeds (1 + √ k)vf , contradicting the fact that it is (1 + √ k)vf . As in [15], for each interval If , we give Γoffline the following gift: k times the amount of time in the range [tbegin f , tclose f ] that it does not schedule a job. Additionally, we give the adversary vf , since the adversary may be able to complete this job at some future time, due to the fact that Γ1 ignores deadlines. The following lemma is Lemma 5.10 in [15], and its proof now applies directly. Lemma 9. [15] With the above gifts the total net gain obtained by the clairvoyant algorithm from scheduling the jobs abandoned during If is not greater than (1 + √ k) · vf . The intuition behind this lemma is that the best that the adversary can do is to take almost all of the gift of k ·(tclose f −tbegin f ) (intuitively, this is equivalent to executing jobs with the maximum possible value density over the time that Γ1 is active), and then begin execution of a job abandoned by Γ1 right before tclose f . By Lemma 8, the value of this job is bounded by (1 + √ k) · vf . We can now combine the results of these lemmas to prove the competitive ratio. Theorem 10. Mechanism Γ1 is (1+ √ k)2+1 -competitive. Proof. Using the fact that the way in which jobs are associated with the intervals partitions the entire set of jobs, we can show the competitive ratio by showing that Γ1 is (1+ √ k)2 +1 -competitive for each interval in the sequence (1, . . . , F). Over an arbitrary interval If , the offline algorithm can achieve at most (tclose f −tbegin f )·k+vf +(1+ √ k)vf , from the two gifts and the net gain bounded by Lemma 9. Applying Lemma 7, this quantity is then bounded from above by (1+ 1√ k )·vf ·k+vf +(1+ √ k)vf = ((1+ √ k)2 +1)·vf . Since Γ1 achieves vf , the competitive ratio holds. 4.3 Special Case: Unalterable length and k=1 While so far we have allowed each agent to lie about all four characteristics of its job, lying about the length of the job is not possible in some settings. For example, a user may not know how to alter a computational problem in a way that both lengthens the job and allows the solution of the original problem to be extracted from the solution to the altered problem. Another restriction that is natural in some settings is uniform value densities (k = 1), which was the case considered by [4]. If the setting satisfies these two conditions, then, by using Mechanism Γ2, we can achieve a competitive ratio of 5 (which is the same competitive ratio as Γ1 for the case of k = 1) without knowledge of ρmin and without the use of payments. The latter property may be necessary in settings that are more local than grid computing (e.g., within a department) but in which the users are still self-interested.7 Mechanism 2 Γ2 Execute S(ˆθ, ·) according to Algorithm 2 for all i do pi(ˆθ) ← 0 Algorithm 2 for all t do Avail ← {i|(t ≥ ˆri)∧(ei(ˆθ, t) < li)∧(ei(ˆθ, t)+ ˆdi−t ≥ li)} if Avail = ∅ then S(ˆθ, t) ← arg maxi∈Avail(li + ei(ˆθ, t)) {Break ties in favor of lower ˆri} else S(ˆθ, t) ← 0 Theorem 11. When k = 1, and each agent i cannot falsely declare li, Mechanism Γ2 satisfies individual rationality and incentive compatibility. Theorem 12. When k = 1, and each agent i cannot falsely declare li, Mechanism Γ2 is 5-competitive. Since this mechanism is essentially a simplification of Γ1, we omit proofs of these theorems. Basically, the fact that k = 1 and ˆli = li both hold allows Γ2 to substitute the priority (li +ei(ˆθ, t)) for the priority used in Γ1; and, since ˆvi is ignored, payments are no longer needed to ensure incentive compatibility. 5. COMPETITIVE LOWER BOUND We now show that the competitive ratio of (1 + √ k)2 + 1 achieved by Γ1 is a lower bound for deterministic online mechanisms. To do so, we will appeal to third requirement on a mechanism, non-negative payments (NNP), which requires that the center never pays an agent (formally, ∀i, ˆθ, pi(ˆθi) ≥ 0). Unlike IC and IR, this requirement is not standard in mechanism design. We note, however, that both Γ1 and Γ2 satisfy it trivially, and that, in the following proof, zero only serves as a baseline utility for an agent, and could be replaced by any non-positive function of ˆθ−i. The proof of the lower bound uses an adversary argument similar to that used in [4] to show a lower bound of (1 +√ k)2 in the non-strategic setting, with the main novelty lying in the perturbation of the job sequence and the related incentive compatibility arguments. We first present a lemma relating to the recurrence used for this argument, with the proof omitted due to space constraints. Lemma 13. For any k ≥ 1, for the recurrence defined by li+1 = λ · li − k · i h=1 lh and l1 = 1, where (1 + √ k)2 − 1 < λ < (1 + √ k)2 , there exists an integer m ≥ 1 such that lm+k· m−1 h=1 lh lm > λ. 7 While payments are not required in this setting, Γ2 can be changed to collect a payments without affecting incentive compatibility by charging some fixed fraction of li for each job i that is completed. 68 Theorem 14. There does not exist a deterministic online mechanism that satisfies NNP and that achieves a competitive ratio less than (1 + √ k)2 + 1. Proof. Assume by contradiction that there exists a deterministic online mechanism Γ that satisfies NNP and that achieves a competitive ratio of c = (1 + √ k)2 + 1 − for some > 0 (and, by implication, satisfies IC and IR as well). Since a competitive ratio of c implies a competitive ratio of c + x, for any x > 0, we assume without loss of generality that < 1. First, we will construct a profile of agent types θ using an adversary argument. After possibly slightly perturbing θ to assure that a strictness property is satisfied, we will then use a more significant perturbation of θ to reach a contradiction. We now construct the original profile θ. Pick an α such that 0 < α < , and define δ = α ck+3k . The adversary uses two sequences of jobs: minor and major. Minor jobs i are characterized by li = δ, vi = k · δ, and zero laxity. The first minor job is released at time 0, and ri = di−1 for all i > 1. The sequence stops whenever Γ completes any job. Major jobs also have zero laxity, but they have the smallest possible value ratio (that is, vi = li). The lengths of the major jobs that may be released, starting with i = 1, are determined by the following recurrence relation. li+1 = (c − 1 + α) · li − k · i h=1 lh l1 = 1 The bounds on α imply that (1 + √ k)2 − 1 < c − 1 + α < (1+ √ k)2 , which allows us to apply Lemma 13. Let m be the smallest positive number such that lm+k· m−1 h=1 lh lm > c−1+α. The first major job has a release time of 0, and each major job i > 1 has a release time of ri = di−1 − δ, just before the deadline of the previous job. The adversary releases major job i ≤ m if and only if each major job j < i was executed continuously over the range [ri, ri+1]. No major job is released after job m. In order to achieve the desired competitive ratio, Γ must complete some major job f, because Γoffline can always at least complete major job 1 (for a value of 1), and Γ can complete at most one minor job (for a value of α c+3 < 1 c ). Also, in order for this job f to be released, the processor time preceding rf can only be spent executing major jobs that are later abandoned. If f < m, then major job f + 1 will be released and it will be the final major job. Γ cannot complete job f +1, because rf +lf = df > rf+1. Therefore, θ consists of major jobs 1 through f + 1 (or, f, if f = m), plus minor jobs from time 0 through time df . We now possibly perturb θ slightly. By IR, we know that vf ≥ pf (θ). Since we will later need this inequality to be strict, if vf = pf (θ), then change θf to θf , where rf = rf , but vf , lf , and df are all incremented by δ over their respective values in θf . By IC, job f must still be completed by Γ for the profile (θf , θ−f ). If not, then by IR and NNP we know that pf (θf , θ−f ) = 0, and thus that uf (g(θf , θ−f ), θf ) = 0. However, agent f could then increase its utility by falsely declaring the original type of θf , receiving a utility of: uf (g(θf , θ−f ), θf ) = vf − pf (θ) = δ > 0, violating IC. Furthermore, agent f must be charged the same amount (that is, pf (θf , θ−f ) = pf (θ)), due to a similar incentive compatibility argument. Thus, for the remainder of the proof, assume that vf > pf (θ). We now use a more substantial perturbation of θ to complete the proof. If f < m, then define θf to be identical to θf , except that df = df+1 + lf , allowing job f to be completely executed after job f + 1 is completed. If f = m, then instead set df = df +lf . IC requires that for the profile (θf , θ−f ), Γ still executes job f continuously over the range [rf , rf +lf ], thus preventing job f +1 from being completed. Assume by contradiction that this were not true. Then, at the original deadline of df , job f is not completed. Consider the possible profile (θf , θ−f , θx), which differs from the new profile only in the addition of a job x which has zero laxity, rx = df , and vx = lx = max(df − df , (c + 1) · (lf + lf+1)). Because this new profile is indistinguishable from (θf , θ−f ) to Γ before time df , it must schedule jobs in the same way until df . Then, in order to achieve the desired competitive ratio, it must execute job x continuously until its deadline, which is by construction at least as late as the new deadline df of job f. Thus, job f will not be completed, and, by IR and NNP, it must be the case that pf (θf , θ−f , θx) = 0 and uf (g(θf , θ−f , θx), θf ) = 0. Using the fact that θ is indistinguishable from (θf , θ−f , θx) up to time df , if agent f falsely declared his type to be the original θf , then its job would be completed by df and it would be charged pf (θ). Its utility would then increase to uf (g(θf , θ−f , θx), θf ) = vf − pf (θ) > 0, contradicting IC. While Γ"s execution must be identical for both (θf , θ−f ) and (θf , θ−f ), Γoffline can take advantage of the change. If f < m, then Γ achieves a value of at most lf +δ (the value of job f if it were perturbed), while Γoffline achieves a value of at least k·( f h=1 lh −2δ)+lf+1 +lf by executing minor jobs until rf+1, followed by job f +1 and then job f (we subtract two δ"s instead of one because the last minor job before rf+1 may have to be abandoned). Substituting in for lf+1, the competitive ratio is then at least: k·( f h=1 lh−2δ)+lf+1+lf lf +δ = k·( f h=1 lh)−2k·δ+(c−1+α)·lf −k·( f h=1 lh)+lf lf +δ = c·lf +(α·lf −2k·δ) lf +δ ≥ c·lf +((ck+3k)δ−2k·δ) lf +δ > c. If instead f = m, then Γ achieves a value of at most lm +δ, while Γoffline achieves a value of at least k · ( m h=1 lh − 2δ) + lm by completing minor jobs until dm = rm + lm, and then completing job m. The competitive ratio is then at least: k·( m h=1 lh−2δ)+lm lm+δ = k·( m−1 h=1 lh)−2k·δ+klm+lm lm+δ > (c−1+α)·lm−2k·δ+klm lm+δ = (c+k−1)·lm+(αlm−2k·δ) lm+δ > c. 6. RELATED WORK In this section we describe related work other than the two papers ([4] and [15]) on which this paper is based. Recent work related to this scheduling domain has focused on competitive analysis in which the online algorithm uses a faster processor than the offline algorithm (see, e.g., [13, 14]). Mechanism design was also applied to a scheduling problem in [18]. In their model, the center owns the jobs in an offline setting, and it is the agents who can execute them. The private information of an agent is the time it will require to execute each job. Several incentive compatible mechanisms are presented that are based on approximation algorithms for the computationally infeasible optimization problem. This paper also launched the area of algorithmic mechanism design, in which the mechanism must sat69 isfy computational requirements in addition to the standard incentive requirements. A growing sub-field in this area is multicast cost-sharing mechanism design (see, e.g., [1]), in which the mechanism must efficiently determine, for each agent in a multicast tree, whether the agent receives the transmission and the price it must pay. For a survey of this and other topics in distributed algorithmic mechanism design, see [9]. Online execution presents a different type of algorithmic challenge, and several other papers study online algorithms or mechanisms in economic settings. For example, [5] considers an online market clearing setting, in which the auctioneer matches buy and sells bids (which are assumed to be exogenous) that arrive and expire over time. In [2], a general method is presented for converting an online algorithm into an online mechanism that is incentive compatible with respect to values. Truthful declaration of values is also considered in [3] and [16], which both consider multi-unit online auctions. The main difference between the two is that the former considers the case of a digital good, which thus has unlimited supply. It is pointed out in [16] that their results continue to hold when the setting is extended so that bidders can delay their arrival. The only other paper we are aware of that addresses the issue of incentive compatibility in a real-time system is [11], which considers several variants of a model in which the center allocates bandwidth to agents who declare both their value and their arrival time. A dominant strategy IC mechanism is presented for the variant in which every point in time is essentially independent, while a Bayes-Nash IC mechanism is presented for the variant in which the center"s current decision affects the cost of future actions. 7. CONCLUSION In this paper, we considered an online scheduling domain for which algorithms with the best possible competitive ratio had been found, but for which new solutions were required when the setting is extended to include self-interested agents. We presented a mechanism that is incentive compatible with respect to release time, deadline, length and value, and that only increases the competitive ratio by one. We also showed how this mechanism could be simplified when k = 1 and each agent cannot lie about the length of its job. We then showed a matching lower bound on the competitive ratio that can be achieved by a deterministic mechanism that never pays the agents. Several open problems remain in this setting. One is to determine whether the lower bound can be strengthened by removing the restriction of non-negative payments. Also, while we feel that it is reasonable to strengthen the assumption of knowing the maximum possible ratio of value densities (k) to knowing the actual range of possible value densities, it would be interesting to determine whether there exists a ((1 + √ k)2 + 1)-competitive mechanism under the original assumption. Finally, randomized mechanisms provide an unexplored area for future work. 8. REFERENCES [1] A. Archer, J. Feigenbaum, A. Krishnamurthy, R. Sami, and S. Shenker, Approximation and collusion in multicast cost sharing, Games and Economic Behavior (to appear). [2] B. Awerbuch, Y. Azar, and A. Meyerson, Reducing truth-telling online mechanisms to online optimization, Proceedings on the 35th Symposium on the Theory of Computing, 2003. [3] Z. Bar-Yossef, K. Hildrum, and F. Wu, Incentive-compatible online auctions for digital goods, Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, 2002. [4] S. Baruah, G. Koren, D. Mao, B. Mishra, A. Raghunathan, L. Rosier, D. Shasha, and F. Wang, On the competitiveness of on-line real-time task scheduling, Journal of Real-Time Systems 4 (1992), no. 2, 125-144. [5] A. Blum, T. Sandholm, and M. Zinkevich, Online algorithms for market clearing, Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, 2002. [6] A. Borodin and R. El-Yaniv, Online computation and competitive analysis, Cambridge University Press, 1998. [7] R. Buyya, D. Abramson, J. Giddy, and H. Stockinger, Economic models for resource management and scheduling in grid computing, The Journal of Concurrency and Computation: Practice and Experience 14 (2002), 1507-1542. [8] N. Camiel, S. London, N. Nisan, and O. Regev, The popcorn project: Distributed computation over the internet in java, 6th International World Wide Web Conference, 1997. [9] J. Feigenbaum and S. Shenker, Distributed algorithmic mechanism design: Recent results and future directions, Proceedings of the 6th International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, 2002, pp. 1-13. [10] A. Fiat and G. Woeginger (editors), Online algorithms: The state of the art, Springer Verlag, 1998. [11] E. Friedman and D. Parkes, Pricing wifi at starbucksissues in online mechanism design, EC"03, 2003. [12] R. L. Graham, Bounds for certain multiprocessor anomalies, Bell System Technical Journal 45 (1966), 1563-1581. [13] B. Kalyanasundaram and K. Pruhs, Speed is as powerful as clairvoyance, Journal of the ACM 47 (2000), 617-643. [14] C. Koo, T. Lam, T. Ngan, and K. To, On-line scheduling with tight deadlines, Theoretical Computer Science 295 (2003), 251-261. [15] G. Koren and D. Shasha, D-over: An optimal on-line scheduling algorithm for overloaded real-time systems, SIAM Journal of Computing 24 (1995), no. 2, 318-339. [16] R. Lavi and N. Nisan, Competitive analysis of online auctions, EC"00, 2000. [17] A. Mas-Colell, M. Whinston, and J. Green, Microeconomic theory, Oxford University Press, 1995. [18] N. Nisan and A. Ronen, Algorithmic mechanism design, Games and Economic Behavior 35 (2001), 166-196. [19] C. Papadimitriou, Algorithms, games, and the internet, STOC, 2001, pp. 749-753. 70

deterministic mechanism;game theory;non-strategic setting;online algorithm;profitable deviation;monotonicity;online scheduling of job;competitive ratio;deadline;importance ratio;incentive compatibility;job online scheduling;individual rationality;zero laxity;quasi-linear function;schedule;deterministic algorithm;mechanism design

train_J-69

Robust Incentive Techniques for Peer-to-Peer Networks

Lack of cooperation (free riding) is one of the key problems that confronts today"s P2P systems. What makes this problem particularly difficult is the unique set of challenges that P2P systems pose: large populations, high turnover, asymmetry of interest, collusion, zero-cost identities, and traitors. To tackle these challenges we model the P2P system using the Generalized Prisoner"s Dilemma (GPD), and propose the Reciprocative decision function as the basis of a family of incentives techniques. These techniques are fully distributed and include: discriminating server selection, maxflowbased subjective reputation, and adaptive stranger policies. Through simulation, we show that these techniques can drive a system of strategic users to nearly optimal levels of cooperation.

1. INTRODUCTION Many peer-to-peer (P2P) systems rely on cooperation among selfinterested users. For example, in a file-sharing system, overall download latency and failure rate increase when users do not share their resources [3]. In a wireless ad-hoc network, overall packet latency and loss rate increase when nodes refuse to forward packets on behalf of others [26]. Further examples are file preservation [25], discussion boards [17], online auctions [16], and overlay routing [6]. In many of these systems, users have natural disincentives to cooperate because cooperation consumes their own resources and may degrade their own performance. As a result, each user"s attempt to maximize her own utility effectively lowers the overall A BC Figure 1: Example of asymmetry of interest. A wants service from B, B wants service form C, and C wants service from A. utility of the system. Avoiding this tragedy of the commons [18] requires incentives for cooperation. We adopt a game-theoretic approach in addressing this problem. In particular, we use a prisoners" dilemma model to capture the essential tension between individual and social utility, asymmetric payoff matrices to allow asymmetric transactions between peers, and a learning-based [14] population dynamic model to specify the behavior of individual peers, which can be changed continuously. While social dilemmas have been studied extensively, P2P applications impose a unique set of challenges, including: • Large populations and high turnover: A file sharing system such as Gnutella and KaZaa can exceed 100, 000 simultaneous users, and nodes can have an average life-time of the order of minutes [33]. • Asymmetry of interest: Asymmetric transactions of P2P systems create the possibility for asymmetry of interest. In the example in Figure 1, A wants service from B, B wants service from C, and C wants service from A. • Zero-cost identity: Many P2P systems allow peers to continuously switch identities (i.e., whitewash). Strategies that work well in traditional prisoners" dilemma games such as Tit-for-Tat [4] will not fare well in the P2P context. Therefore, we propose a family of scalable and robust incentive techniques, based upon a novel Reciprocative decision function, to address these challenges and provide different tradeoffs: • Discriminating Server Selection: Cooperation requires familiarity between entities either directly or indirectly. However, the large populations and high turnover of P2P systems makes it less likely that repeat interactions will occur with a familiar entity. We show that by having each peer keep a 102 private history of the actions of other peers toward her, and using discriminating server selection, the Reciprocative decision function can scale to large populations and moderate levels of turnover. • Shared History: Scaling to higher turnover and mitigating asymmetry of interest requires shared history. Consider the example in Figure 1. If everyone provides service, then the system operates optimally. However, if everyone keeps only private history, no one will provide service because B does not know that A has served C, etc. We show that with shared history, B knows that A served C and consequently will serve A. This results in a higher level of cooperation than with private history. The cost of shared history is a distributed infrastructure (e.g., distributed hash table-based storage) to store the history. • Maxflow-based Subjective Reputation: Shared history creates the possibility for collusion. In the example in Figure 1, C can falsely claim that A served him, thus deceiving B into providing service. We show that a maxflow-based algorithm that computes reputation subjectively promotes cooperation despite collusion among 1/3 of the population. The basic idea is that B would only believe C if C had already provided service to B. The cost of the maxflow algorithm is its O(V 3 ) running time, where V is the number of nodes in the system. To eliminate this cost, we have developed a constant mean running time variation, which trades effectiveness for complexity of computation. We show that the maxflow-based algorithm scales better than private history in the presence of colluders without the centralized trust required in previous work [9] [20]. • Adaptive Stranger Policy: Zero-cost identities allows noncooperating peers to escape the consequences of not cooperating and eventually destroy cooperation in the system if not stopped. We show that if Reciprocative peers treat strangers (peers with no history) using a policy that adapts to the behavior of previous strangers, peers have little incentive to whitewash and whitewashing can be nearly eliminated from the system. The adaptive stranger policy does this without requiring centralized allocation of identities, an entry fee for newcomers, or rate-limiting [13] [9] [25]. • Short-term History: History also creates the possibility that a previously well-behaved peer with a good reputation will turn traitor and use his good reputation to exploit other peers. The peer could be making a strategic decision or someone may have hijacked her identity (e.g., by compromising her host). Long-term history exacerbates this problem by allowing peers with many previous transactions to exploit that history for many new transactions. We show that short-term history prevents traitors from disrupting cooperation. The rest of the paper is organized as follows. We describe the model in Section 2 and the reciprocative decision function in Section 3. We then proceed to the incentive techniques in Section 4. In Section 4.1, we describe the challenges of large populations and high turnover and show the effectiveness of discriminating server selection and shared history. In Section 4.2, we describe collusion and demonstrate how subjective reputation mitigates it. In Section 4.3, we present the problem of zero-cost identities and show how an adaptive stranger policy promotes persistent identities. In Section 4.4, we show how traitors disrupt cooperation and how short-term history deals with them. We discuss related work in Section 5 and conclude in Section 6. 2. MODEL AND ASSUMPTIONS In this section, we present our assumptions about P2P systems and their users, and introduce a model that aims to capture the behavior of users in a P2P system. 2.1 Assumptions We assume a P2P system in which users are strategic, i.e., they act rationally to maximize their benefit. However, to capture some of the real-life unpredictability in the behavior of users, we allow users to randomly change their behavior with a low probability (see Section 2.4). For simplicity, we assume a homogeneous system in which all peers issue and satisfy requests at the same rate. A peer can satisfy any request, and, unless otherwise specified, peers request service uniformly at random from the population.1 . Finally, we assume that all transactions incur the same cost to all servers and provide the same benefit to all clients. We assume that users can pollute shared history with false recommendations (Section 4.2), switch identities at zero-cost (Section 4.3), and spoof other users (Section 4.4). We do not assume any centralized trust or centralized infrastructure. 2.2 Model To aid the development and study of the incentive schemes, in this section we present a model of the users" behaviors. In particular, we model the benefits and costs of P2P interactions (the game) and population dynamics caused by mutation, learning, and turnover. Our model is designed to have the following properties that characterize a large set of P2P systems: • Social Dilemma: Universal cooperation should result in optimal overall utility, but individuals who exploit the cooperation of others while not cooperating themselves (i.e., defecting) should benefit more than users who do cooperate. • Asymmetric Transactions: A peer may want service from another peer while not currently being able to provide the service that the second peer wants. Transactions should be able to have asymmetric payoffs. • Untraceable Defections: A peer should not be able to determine the identity of peers who have defected on her. This models the difficulty or expense of determining that a peer could have provided a service, but didn"t. For example, in the Gnutella file sharing system [21], a peer may simply ignore queries despite possessing the desired file, thus preventing the querying peer from identifying the defecting peer. • Dynamic Population: Peers should be able to change their behavior and enter or leave the system independently and continuously. 1The exception is discussed in Section 4.1.1 103 Cooperate Defect Cooperate DefectClient Server sc RR / sc ST / sc PP / sc TS / Figure 2: Payoff matrix for the Generalized Prisoner"s Dilemma. T, R, P, and S stand for temptation, reward, punishment and sucker, respectively. 2.3 Generalized Prisoner"s Dilemma The Prisoner"s Dilemma, developed by Flood, Dresher, and Tucker in 1950 [22] is a non-cooperative repeated game satisfying the social dilemma requirement. Each game consists of two players who can defect or cooperate. Depending how each acts, the players receive a payoff. The players use a strategy to decide how to act. Unfortunately, existing work either uses a specific asymmetric payoff matrix or only gives the general form for a symmetric one [4]. Instead, we use the Generalized Prisoner"s Dilemma (GPD), which specifies the general form for an asymmetric payoff matrix that preserves the social dilemma. In the GPD, one player is the client and one player is the server in each game, and it is only the decision of the server that is meaningful for determining the outome of the transaction. A player can be a client in one game and a server in another. The client and server receive the payoff from a generalized payoff matrix (Figure 2). Rc, Sc, Tc, and Pc are the client"s payoff and Rs, Ss, Ts, and Ps are the server"s payoff. A GPD payoff matrix must have the following properties to create a social dilemma: 1. Mutual cooperation leads to higher payoffs than mutual defection (Rs + Rc > Ps + Pc). 2. Mutual cooperation leads to higher payoffs than one player suckering the other (Rs + Rc > Sc + Ts and Rs + Rc > Ss + Tc). 3. Defection dominates cooperation (at least weakly) at the individual level for the entity who decides whether to cooperate or defect: (Ts ≥ Rs and Ps ≥ Ss and (Ts > Rs or Ps > Ss)) The last set of inequalities assume that clients do not incur a cost regardless of whether they cooperate or defect, and therefore clients always cooperate. These properties correspond to similar properties of the classic Prisoner"s Dilemma and allow any form of asymmetric transaction while still creating a social dilemma. Furthermore, one or more of the four possible actions (client cooperate and defect, and server cooperate and defect) can be untraceable. If one player makes an untraceable action, the other player does not know the identity of the first player. For example, to model a P2P application like file sharing or overlay routing, we use the specific payoff matrix values shown in Figure 3. This satisfies the inequalities specified above, where only the server can choose between cooperating and defecting. In addition, for this particular payoff matrix, clients are unable to trace server defections. This is the payoff matrix that we use in our simulation results. Request Service Don't Request 7 / -1 0 / 0 0 / 0 0 / 0 Provide Service Ignore Request Client Server Figure 3: The payoff matrix for an application like P2P file sharing or overlay routing. 2.4 Population Dynamics A characteristic of P2P systems is that peers change their behavior and enter or leave the system independently and continuously. Several studies [4] [28] of repeated Prisoner"s Dilemma games use an evolutionary model [19] [34] of population dynamics. An evolutionary model is not suitable for P2P systems because it only specifies the global behavior and all changes occur at discrete times. For example, it may specify that a population of 5 100% Cooperate players and 5 100% Defect players evolves into a population with 3 and 7 players, respectively. It does not specify which specific players switched. Furthermore, all the switching occurs at the end of a generation instead of continuously, like in a real P2P system. As a result, evolutionary population dynamics do not accurately model turnover, traitors, and strangers. In our model, entities take independent and continuous actions that change the composition of the population. Time consists of rounds. In each round, every player plays one game as a client and one game as a server. At the end of a round, a player may: 1) mutate 2) learn, 3) turnover, or 4) stay the same. If a player mutates, she switches to a randomly picked strategy. If she learns, she switches to a strategy that she believes will produce a higher score (described in more detail below). If she maintains her identity after switching strategies, then she is referred to as a traitor. If a player suffers turnover, she leaves the system and is replaced with a newcomer who uses the same strategy as the exiting player. To learn, a player collects local information about the performance of different strategies. This information consists of both her personal observations of strategy performance and the observations of those players she interacts with. This models users communicating out-of-band about how strategies perform. Let s be the running average of the performance of a player"s current strategy per round and age be the number of rounds she has been using the strategy. A strategy"s rating is RunningAverage(s ∗ age) RunningAverage(age) . We use the age and compute the running average before the ratio to prevent young samples (which are more likely to be outliers) from skewing the rating. At the end of a round, a player switches to highest rated strategy with a probability proportional to the difference in score between her current strategy and the highest rated strategy. 104 3. RECIPROCATIVE DECISION FUNCTION In this section, we present the new decision function, Reciprocative, that is the basis for our incentive techniques. A decision function maps from a history of a player"s actions to a decision whether to cooperate with or defect on that player. A strategy consists of a decision function, private or shared history, a server selection mechanism, and a stranger policy. Our approach to incentives is to design strategies which maximize both individual and social benefit. Strategic users will choose to use such strategies and thereby drive the system to high levels of cooperation. Two examples of simple decision functions are 100% Cooperate and 100% Defect. 100% Cooperate models a naive user who does not yet realize that she is being exploited. 100% Defect models a greedy user who is intent on exploiting the system. In the absence of incentive techniques, 100% Defect users will quickly dominate the 100% Cooperate users and destroy cooperation in the system. Our requirements for a decision function are that (1) it can use shared and subjective history, (2) it can deal with untraceable defections, and (3) it is robust against different patterns of defection. Previous decision functions such as Tit-for-Tat[4] and Image[28] (see Section 5) do not satisfy these criteria. For example, Tit-for-Tat and Image base their decisions on both cooperations and defections, therefore cannot deal with untraceable defections . In this section and the remaining sections we demonstrate how the Reciprocativebased strategies satisfy all of the requirements stated above. The probability that a Reciprocative player cooperates with a peer is a function of its normalized generosity. Generosity measures the benefit an entity has provided relative to the benefit it has consumed. This is important because entities which consume more services than they provide, even if they provide many services, will cause cooperation to collapse. For some entity i, let pi and ci be the services i has provided and consumed, respectively. Entity i"s generosity is simply the ratio of the service it provides to the service it consumes: g(i) = pi/ci. (1) One possibility is to cooperate with a probability equal to the generosity. Although this is effective in some cases, in other cases, a Reciprocative player may consume more than she provides (e.g., when initially using the Stranger Defect policy in 4.3). This will cause Reciprocative players to defect on each other. To prevent this situation, a Reciprocative player uses its own generosity as a measuring stick to judge its peer"s generosity. Normalized generosity measures entity i"s generosity relative to entity j"s generosity. More concretely, entity i"s normalized generosity as perceived by entity j is gj(i) = g(i)/g(j). (2) In the remainder of this section, we describe our simulation framework, and use it to demonstrate the benefits of the baseline Reciprocative decision function. Parameter Nominal value Section Population Size 100 2.4 Run Time 1000 rounds 2.4 Payoff Matrix File Sharing 2.3 Ratio using 100% Cooperate 1/3 3 Ratio using 100% Defect 1/3 3 Ratio using Reciprocative 1/3 3 Mutation Probability 0.0 2.4 Learning Probability 0.05 2.4 Turnover Probability 0.0001 2.4 Hit Rate 1.0 4.1.1 Table 1: Default simulation parameters. 3.1 Simulation Framework Our simulator implements the model described in Section 2. We use the asymmetric file sharing payoff matrix (Figure 3) with untraceable defections because it models transactions in many P2P systems like file-sharing and packet forwarding in ad-hoc and overlay networks. Our simulation study is composed of different scenarios reflecting the challenges of various non-cooperative behaviors. Table 1 presents the nominal parameter values used in our simulation. The Ratio using rows refer to the initial ratio of the total population using a particular strategy. In each scenario we vary the value range of a specific parameter to reflect a particular situation or attack. We then vary the exact properties of the Reciprocative strategy to defend against that situation or attack. 3.2 Baseline Results 0 20 40 60 80 100 120 0 200 400 600 800 1000 Population Time (a) Total Population: 60 0 20 40 60 80 100 120 0 200 400 600 800 1000 Time (b) Total Population: 120 Defector Cooperator Recip. Private Figure 4: The evolution of strategy populations over time. Time the number of elapsed rounds. Population is the number of players using a strategy. In this section, we present the dynamics of the game for the basic scenario presented in Table 1 to familiarize the reader and set a baseline for more complicated scenarios. Figures 4(a) (60 players) and (b) (120 players) show players switching to higher scoring strategies over time in two separate runs of the simulator. Each point in the graph represents the number of players using a particular strategy at one point in time. Figures 5(a) and (b) show the corresponding mean overall score per round. This measures the degree of cooperation in the system: 6 is the maximum possible (achieved when everybody cooperates) and 0 is the minimum (achieved when everybody defects). From the file sharing payoff matrix, a net of 6 means everyone is able to download a file and a 0 means that no one 105 0 1 2 3 4 5 6 0 200 400 600 800 1000 MeanOverallScore/Round Time (a) Total Population: 60 0 1 2 3 4 5 6 0 200 400 600 800 1000 Time (b) Total Population: 120 Figure 5: The mean overall per round score over time. is able to do so. We use this metric in all later results to evaluate our incentive techniques. Figure 5(a) shows that the Reciprocative strategy using private history causes a system of 60 players to converge to a cooperation level of 3.7, but drops to 0.5 for 120 players. One would expect the 60 player system to reach the optimal level of cooperation (6) because all the defectors are eliminated from the system. It does not because of asymmetry of interest. For example, suppose player B is using Reciprocative with private history. Player A may happen to ask for service from player B twice in succession without providing service to player B in the interim. Player B does not know of the service player A has provided to others, so player B will reject service to player A, even though player A is cooperative. We discuss solutions to asymmetry of interest and the failure of Reciprocative in the 120 player system in Section 4.1. 4. RECIPROCATIVE-BASED INCENTIVE TECHNIQUES In this section we present our incentives techniques and evaluate their behavior by simulation. To make the exposition clear we group our techniques by the challenges they address: large populations and high turnover (Section 4.1), collusions (Section 4.2), zero-cost identities (Section 4.3), and traitors (Section 4.4). 4.1 Large Populations and High Turnover The large populations and high turnover of P2P systems makes it less likely that repeat interactions will occur with a familiar entity. Under these conditions, basing decisions only on private history (records about interactions the peer has been directly involved in) is not effective. In addition, private history does not deal well with asymmetry of interest. For example, if player B has cooperated with others but not with player A himself in the past, player A has no indication of player B"s generosity, thus may unduly defect on him. We propose two mechanisms to alleviate the problem of few repeat transactions: server-selection and shared history. 4.1.1 Server Selection A natural way to increase the probability of interacting with familiar peers is by discriminating server selection. However, the asymmetry of transactions challenges selection mechanisms. Unlike in the prisoner"s dilemma payoff matrix, where players can benefit one another within a single transaction, transactions in GPD are asymmetric. As a result, a player who selects her donor for the second time without contributing to her in the interim may face a defection. In addition, due to untraceability of defections, it is impossible to maintain blacklists to avoid interactions with known defectors. In order to deal with asymmetric transactions, every player holds (fixed size) lists of both past donors and past recipients, and selects a server from one of these lists at random with equal probabilities. This way, users approach their past recipients and give them a chance to reciprocate. In scenarios with selective users we omit the complete availability assumption to prevent players from being clustered into a lot of very small groups; thus, we assume that every player can perform the requested service with probability p (for the results presented in this section, p = .3). In addition, in order to avoid bias in favor of the selective players, all players (including the non-discriminative ones) select servers for games. Figure 6 demonstrates the effectiveness of the proposed selection mechanism in scenarios with large population sizes. We fix the initial ratio of Reciprocative in the population (33%) while varying the population size (between 24 to 1000) (Notice that while in Figures 4(a) and (b), the data points demonstrates the evolution of the system over time, each data point in this figure is the result of an entire simulation for a specific scenario). The figure shows that the Reciprocative decision function using private history in conjunction with selective behavior can scale to large populations. In Figure 7 we fix the population size and vary the turnover rate. It demonstrates that while selective behavior is effective for low turnover rates, as turnover gets higher, selective behavior does not scale. This occurs because selection is only effective as long as players from the past stay alive for long enough such that they can be selected for future games. 4.1.2 Shared history In order to mitigate asymmetry of interest and scale to higher turnover rate, there is a need in shared history. Shared history means that every peer keeps records about all of the interactions that occur in the system, regardless of whether he was directly involved in them or not. It allows players to leverage off of the experiences of others in cases of few repeat transactions. It only requires that someone has interacted with a particular player for the entire population to observe it, thus scales better to large populations and high turnovers, and also tolerates asymmetry of interest. Some examples of shared history schemes are [20] [23] [28]. Figure 7 shows the effectiveness of shared history under high turnover rates. In this figure, we fix the population size and vary the turnover rate. While selective players with private history can only tolerate a moderate turnover, shared history scales to turnovers of up to approximately 0.1. This means that 10% of the players leave the system at the end of each round. In Figure 6 we fix the turnover and vary the population size. It shows that shared history causes the system to converge to optimal cooperation and performance, regardless of the size of the population. These results show that shared history addresses all three challenges of large populations, high turnover, and asymmetry of transactions. Nevertheless, shared history has two disadvantages. First, 106 0 1 2 3 4 5 6 0 50 100 150 200 250 300 350 400 MeanOverallScore/Round NumPlayers Shared Non-Sel Private Non-Sel Private Selective Figure 6: Private vs. Shared History as a function of population size. 0 1 2 3 4 5 6 0.0001 0.001 0.01 0.1 MeanOverallScore/Round Turnover Shared Non-Sel Private Non-Sel Private Selective Figure 7: Performance of selection mechanism under turnover. The x-axis is the turnover rate. The y-axis is the mean overall per round score. while a decentralized implementation of private history is straightforward, implementation of shared-history requires communication overhead or centralization. A decentralized shared history can be implemented, for example, on top of a DHT, using a peer-to-peer storage system [36] or by disseminating information to other entities in a similar way to routing protocols. Second, and more fundamental, shared history is vulnerable to collusion. In the next section we propose a mechanism that addresses this problem. 4.2 Collusion and Other Shared History Attacks 4.2.1 Collusion While shared history is scalable, it is vulnerable to collusion. Collusion can be either positive (e.g. defecting entities claim that other defecting entities cooperated with them) or negative (e.g. entities claim that other cooperative entities defected on them). Collusion subverts any strategy in which everyone in the system agrees on the reputation of a player (objective reputation). An example of objective reputation is to use the Reciprocative decision function with shared history to count the total number of cooperations a player has given to and received from all entities in the system; another example is the Image strategy [28]. The effect of collusion is magnified in systems with zero-cost identities, where users can create fake identities that report false statements. Instead, to deal with collusion, entities can compute reputation subjectively, where player A weighs player B"s opinions based on how much player A trusts player B. Our subjective algorithm is based on maxflow [24] [32]. Maxflow is a graph theoretic problem, which given a directed graph with weighted edges asks what is the greatest rate at which material can be shipped from the source to the target without violating any capacity constraints. For example, in figure 8 each edge is labeled with the amount of traffic that can travel on it. The maxflow algorithm computes the maximum amount of traffic that can go from the source (s) to the target (t) without violating the constraints. In this example, even though there is a loop of high capacity edges, the maxflow between the source and the target is only 2 (the numbers in brackets represent the actual flow on each edge in the solution). 100(0) 1(1) 5(1) s t 10(1) 100(1) 1(1) 100(1) 20(0) Figure 8: Each edge in the graph is labeled with its capacity and the actual flow it carries in brackets. The maxflow between the source and the target in the graph is 2. C C CCCC 100100100100 100 00 0 0 20 20 0 0 A B Figure 9: This graph illustrates the robustness of maxflow in the presence of colluders who report bogus high reputation values. We apply the maxflow algorithm by constructing a graph whose vertices are entities and the edges are the services that entities have received from each other. This information can be stored using the same methods as the shared history. A maxflow is the greatest level of reputation the source can give to the sink without violating reputation capacity constraints. As a result, nodes who dishonestly report high reputation values will not be able to subvert the reputation system. Figure 9 illustrates a scenario in which all the colluders (labeled with C) report high reputation values for each other. When node A computes the subjective reputation of B using the maxflow algorithm, it will not be affected by the local false reputation values, rather the maxflow in this case will be 0. This is because no service has been received from any of the colluders. 107 In our algorithm, the benefit that entity i has received (indirectly) from entity j is the maxflow from j to i. Conversely, the benefit that entity i has provided indirectly to j is the maxflow from i to j. The subjective reputation of entity j as perceived by i is: min maxflow(j to i) maxflow(i to j) , 1 (3) 0 1 2 3 4 5 6 0 100 200 300 400 500 600 700 800 900 1000 MeanOverallScore/Round Population Shared Private Subjective Figure 10: Subjective shared history compared to objective shared history and private history in the presence of colluders. Algorithm 1 CONSTANTTIMEMAXFLOW Bound the mean running time of Maxflow to a constant. method CTMaxflow(self, src, dst) 1: self.surplus ← self.surplus + self.increment {Use the running mean as a prediction.} 2: if random() > (0.5∗self.surplus/self.mean iterations) then 3: return None {Not enough surplus to run.} 4: end if {Get the flow and number of iterations used from the maxflow alg.} 5: flow, iterations ← Maxflow(self.G, src, dst) 6: self.surplus ← self.surplus − iterations {Keep a running mean of the number of iterations used.} 7: self.mean iterations ← self.α ∗ self.mean iterations + (1 − self.α) ∗ iterations 8: return flow The cost of maxflow is its long running time. The standard preflowpush maxflow algorithm has a worst case running time of O(V 3 ). Instead, we use Algorithm 1 which has a constant mean running time, but sometimes returns no flow even though one exists. The essential idea is to bound the mean number of nodes examined during the maxflow computation. This bounds the overhead, but also bounds the effectiveness. Despite this, the results below show that a maxflow-based Reciprocative decision function scales to higher populations than one using private history. Figure 10 compares the effectiveness of subjective reputation to objective reputation in the presence of colluders. In these scenarios, defectors collude by claiming that other colluders that they encounter gave them 100 cooperations for that encounter. Also, the parameters for Algorithm 1 are set as follows: increment = 100, α = 0.9. As in previous sections, Reciprocative with private history results in cooperation up to a point, beyond which it fails. The difference here is that objective shared history fails for all population sizes. This is because the Reciprocative players cooperate with the colluders because of their high reputations. However, subjective history can reach high levels of cooperation regardless of colluders. This is because there are no high weight paths in the cooperation graph from colluders to any non-colluders, so the maxflow from a colluder to any non-colluder is 0. Therefore, a subjective Reciprocative player will conclude that that colluder has not provided any service to her and will reject service to the colluder. Thus, the maxflow algorithm enables Reciprocative to maintain the scalability of shared history without being vulnerable to collusion or requiring centralized trust (e.g., trusted peers). Since we bound the running time of the maxflow algorithm, cooperation decreases as the population size increases, but the key point is that the subjective Reciprocative decision function scales to higher populations than one using private history. This advantage only increases over time as CPU power increases and more cycles can be devoted to running the maxflow algorithm (by increasing the increment parameter). Despite the robustness of the maxflow algorithm to the simple form of collusion described previously, it still has vulnerabilities to more sophisticated attacks. One is for an entity (the mole) to provide service and then lie positively about other colluders. The other colluders can then exploit their reputation to receive service. However, the effectiveness of this attack relies on the amount of service that the mole provides. Since the mole is paying all of the cost of providing service and receiving none of the benefit, she has a strong incentive to stop colluding and try another strategy. This forces the colluders to use mechanisms to maintain cooperation within their group, which may drive the cost of collusion to exceed the benefit. 4.2.2 False reports Another attack is for a defector to lie about receiving or providing service to another entity. There are four possibile actions that can be lied about: providing service, not providing service, receiving service, and not receiving service. Falsely claiming to receive service is the simple collusion attack described above. Falsely claiming not to have provided service provides no benefit to the attacker. Falsely claiming to have provided service or not to have received it allows an attacker to boost her own reputation and/or lower the reputation of another entity. An entity may want to lower another entity"s reputation in order to discourage others from selecting it and exclusively use its service. These false claims are clearly identifiable in the shared history as inconsistencies where one entity claims a transaction occurred and another claims it did not. To limit this attack, we modify the maxflow algorithm so that an entity always believes the entity that is closer to him in the flow graph. If both entities are equally distant, then the disputed edge in the flow is not critical to the evaluation and is ignored. This modification prevents those cases where the attacker is making false claims about an entity that is closer than her to the evaluating entity, which prevents her from boosting her own reputation. The remaining possibilities are for the attacker to falsely claim to have provided service to or not to have received it from a victim entity that is farther from the evalulator than her. In these cases, an attacker can only lower the reputation of the victim. The effectiveness of doing this is limited by the number of services provided and received by the attacker, which makes executing this attack expensive. 108 4.3 Zero-Cost Identities History assumes that entities maintain persistent identities. However, in most P2P systems, identities are zero-cost. This is desirable for network growth as it encourages newcomers to join the system. However, this also allows misbehaving users to escape the consequences of their actions by switching to new identities (i.e., whitewashing). Whitewashers can cause the system to collapse if they are not punished appropriately. Unfortunately, a player cannot tell if a stranger is a whitewasher or a legitimate newcomer. Always cooperating with strangers encourages newcomers to join, but at the same time encourages whitewashing behavior. Always defecting on strangers prevents whitewashing, but discourages newcomers from joining and may also initiate unfavorable cycles of defection. This tension suggests that any stranger policy that has a fixed probability of cooperating with strangers will fail by either being too stingy when most strangers are newcomers or too generous when most strangers are whitewashers. Our solution is the Stranger Adaptive stranger policy. The idea is to be generous to strangers when they are being generous and stingy when they are stingy. Let ps and cs be the number of services that strangers have provided and consumed, respectively. The probability that a player using Stranger Adaptive helps a stranger is ps/cs. However, we do not wish to keep these counts permanently (for reasons described in Section 4.4). Also, players may not know when strangers defect because defections are untraceable (as described in Section 2). Consequently, instead of keeping ps and cs, we assume that k = ps + cs, where k is a constant and we keep the running ratio r = ps/cs. When we need to increment ps or cs, we generate the current values of ps and cs from k and r: cs = k/(1 + r) ps = cs ∗ r We then compute the new r as follows: r = (ps + 1)/cs , if the stranger provided service r = ps/(cs + 1) , if the stranger consumed service This method allows us to keep a running ratio that reflects the recent generosity of strangers without knowing when strangers have defected. 0 1 2 3 4 5 6 0.0001 0.001 0.01 0.1 1 MeanOverallScore/Round Turnover Stranger Cooperate Stranger Defect Stranger Adaptive Figure 11: Different stranger policies for Reciprocative with shared history. The x-axis is the turnover rate on a log scale. The y-axis is the mean overall per round score. Figures 11 and 12 compare the effectiveness of the Reciprocative strategy using different policies toward strangers. Figure 11 0 1 2 3 4 5 6 0.0001 0.001 0.01 0.1 1 MeanOverallScore/Round Turnover Stranger Cooperate Stranger Defect Stranger Adaptive Figure 12: Different stranger policies for Reciprocative with private history. The x-axis is the turnover rate on a log scale. The y-axis is the mean overall per round score. compares different stranger policies for Reciprocative with shared history, while Figure 12 is with private history. In both figures, the players using the 100% Defect strategy change their identity (whitewash) after every transaction and are indistinguishable from legitimate newcomers. The Reciprocative players using the Stranger Cooperate policy completely fail to achieve cooperation. This stranger policy allows whitewashers to maximize their payoff and consequently provides a high incentive for users to switch to whitewashing. In contrast, Figure 11 shows that the Stranger Defect policy is effective with shared history. This is because whitewashers always appear to be strangers and therefore the Reciprocative players will always defect on them. This is consistent with previous work [13] showing that punishing strangers deals with whitewashers. However, Figure 12 shows that Stranger Defect is not effective with private history. This is because Reciprocative requires some initial cooperation to bootstrap. In the shared history case, a Reciprocative player can observe that another player has already cooperated with others. With private history, the Reciprocative player only knows about the other players" actions toward her. Therefore, the initial defection dictated by the Stranger Defect policy will lead to later defections, which will prevent Reciprocative players from ever cooperating with each other. In other simulations not shown here, the Stranger Defect stranger policy fails even with shared history when there are no initial 100% Cooperate players. Figure 11 shows that with shared history, the Stranger Adaptive policy performs as well as Stranger Defect policy until the turnover rate is very high (10% of the population turning over after every transaction). In these scenarios, Stranger Adaptive is using k = 10 and each player keeps a private r. More importantly, it is significantly better than Stranger Defect policy with private history because it can bootstrap cooperation. Although the Stranger Defect policy is marginally more effective than Stranger Adaptive at very high rates of turnover, P2P systems are unlikely to operate there because other services (e.g., routing) also cannot tolerate very high turnover. We conclude that of the stranger policies that we have explored, Stranger Adaptive is the most effective. By using Stranger Adaptive, P2P systems with zero-cost identities and a sufficiently low turnover can sustain cooperation without a centralized allocation of identities. 109 4.4 Traitors Traitors are players who acquire high reputation scores by cooperating for a while, and then traitorously turn into defectors before leaving the system. They model both users who turn deliberately to gain a higher score and cooperators whose identities have been stolen and exploited by defectors. A strategy that maintains longterm history without discriminating between old and recent actions becomes highly vulnerable to exploitation by these traitors. The top two graphs in Figure 13 demonstrate the effect of traitors on cooperation in a system where players keep long-term history (never clear history). In these simulations, we run for 2000 rounds and allow cooperative players to keep their identities when switching to the 100% Defector strategy. We use the default values for the other parameters. Without traitors, the cooperative strategies thrive. With traitors, the cooperative strategies thrive until a cooperator turns traitor after 600 rounds. As this cooperator exploits her reputation to achieve a high score, other cooperative players notice this and follow suit via learning. Cooperation eventually collapses. On the other hand, if we maintain short-term history and/or discounting ancient history vis-a-vis recent history, traitors can be quickly detected, and the overall cooperation level stays high, as shown in the bottom two graphs in Figure 13. 0 20 40 60 80 100 1K 2K Long-TermHistory No Traitors Population 0 20 40 60 80 100 1K 2K Traitors Defector Cooperator Recip. Shared 0 20 40 60 80 100 1K 2K Short-TermHistory Time Population 0 20 40 60 80 100 1K 2K Time Figure 13: Keeping long-term vs. short-term history both with and without traitors. 5. RELATED WORK Previous work has examined the incentive problem as applied to societies in general and more recently to Internet applications and peer-to-peer systems in particular. A well-known phenomenon in this context is the tragedy of the commons [18] where resources are under-provisioned due to selfish users who free-ride on the system"s resources, and is especially common in large networks [29] [3]. The problem has been extensively studied adopting a game theoretic approach. The prisoners" dilemma model provides a natural framework to study the effectiveness of different strategies in establishing cooperation among players. In a simulation environment with many repeated games, persistent identities, and no collusion, Axelrod [4] shows that the Tit-for-Tat strategy dominates. Our model assumes growth follows local learning rather than evolutionary dynamics [14], and also allows for more kinds of attacks. Nowak and Sigmund [28] introduce the Image strategy and demonstrate its ability to establish cooperation among players despite few repeat transactions by the employment of shared history. Players using Image cooperate with players whose global count of cooperations minus defections exceeds some threshold. As a result, an Image player is either vulnerable to partial defectors (if the threshold is set too low) or does not cooperate with other Image players (if the threshold is set too high). In recent years, researchers have used economic mechanism design theory to tackle the cooperation problem in Internet applications. Mechanism design is the inverse of game theory. It asks how to design a game in which the behavior of strategic players results in the socially desired outcome. Distributed Algorithmic Mechanism Design seeks solutions within this framework that are both fully distributed and computationally tractable [12]. [10] and [11] are examples of applying DAMD to BGP routing and multicast cost sharing. More recently, DAMD has been also studied in dynamic environments [38]. In this context, demonstrating the superiority of a cooperative strategy (as in the case of our work) is consistent with the objective of incentivizing the desired behavior among selfish players. The unique challenges imposed by peer-to-peer systems inspired additional body of work [5] [37], mainly in the context of packet forwarding in wireless ad-hoc routing [8] [27] [30] [35], and file sharing [15] [31]. Friedman and Resnick [13] consider the problem of zero-cost identities in online environments and find that in such systems punishing all newcomers is inevitable. Using a theoretical model, they demonstrate that such a system can converge to cooperation only for sufficiently low turnover rates, which our results confirm. [6] and [9] show that whitewashing and collusion can have dire consequences for peer-to-peer systems and are difficult to prevent in a fully decentralized system. Some commercial file sharing clients [1] [2] provide incentive mechanisms which are enforced by making it difficult for the user to modify the source code. These mechanisms can be circumvented by a skilled user or by a competing company releasing a compatible client without the incentive restrictions. Also, these mechanisms are still vulnerable to zero-cost identities and collusion. BitTorrent [7] uses Tit-for-Tat as a method for resource allocation, where a user"s upload rate dictates his download rate. 6. CONCLUSIONS In this paper we take a game theoretic approach to the problem of cooperation in peer-to-peer networks. Addressing the challenges imposed by P2P systems, including large populations, high turnover, asymmetry of interest and zero-cost identities, we propose a family of scalable and robust incentive techniques, based upon the Reciprocative decision function, to support cooperative behavior and improve overall system performance. We find that the adoption of shared history and discriminating server selection techniques can mitigate the challenge of few repeat transactions that arises due to large population size, high turnover and asymmetry of interest. Furthermore, cooperation can be established even in the presence of zero-cost identities through the use of an adaptive policy towards strangers. Finally, colluders and traitors can be kept in check via subjective reputations and short-term history, respectively. 110 7. ACKNOWLEDGMENTS We thank Mary Baker, T.J. 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To Share or Not to Share: An Analysis of Incentives to Contribute in Collaborative File Sharing Environments. In Workshop on Economics of Peer-to-Peer Systems (June 2003). [32] REITER, M. K., AND STUBBLEBINE, S. G. Authentication Metric Analysis and Design. ACM Transactions on Information and System Security 2, 2 (1999), 138-158. [33] SAROIU, S., GUMMADI, P. K., AND GRIBBLE, S. D. A Measurement Study of Peer-to-Peer File Sharing Systems. In Proceedings of Multimedia Computing and Networking 2002 (MMCN "02) (2002). [34] SMITH, J. M. Evolution and the Theory of Games. Cambridge University Press, 1982. [35] URPI, A., BONUCCELLI, M., AND GIORDANO, S. Modeling Cooperation in Mobile ad-hoc Networks: a Formal Description of Selfishness. In Modeling and Optimization in Mobile, ad-hoc and Wireless Networks (2003). [36] VISHNUMURTHY, V., CHANDRAKUMAR, S., AND SIRER, E. G. KARMA : A Secure Economic Framework for P2P Resource Sharing. In Workshop on Economics of Peer-to-Peer Networks (2003). 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generosity;prisoner dilemma;p2p system;reputation;free-ride;selfinterested user;peer-to-peer;maxflow-based algorithm;stranger adaptive;cheap pseudonym;game-theoretic approach;reciprocative peer;mutual cooperation;parameter nominal value;adaptive stranger policy;incentive for cooperation;collusion;whitewasher;reciprocative decision function;stranger defect;asymmetric payoff;whitewash;incentive

train_J-70

Self-interested Automated Mechanism Design and Implications for Optimal Combinatorial Auctions∗

Often, an outcome must be chosen on the basis of the preferences reported by a group of agents. The key difficulty is that the agents may report their preferences insincerely to make the chosen outcome more favorable to themselves. Mechanism design is the art of designing the rules of the game so that the agents are motivated to report their preferences truthfully, and a desirable outcome is chosen. In a recently proposed approach-called automated mechanism design-a mechanism is computed for the preference aggregation setting at hand. This has several advantages, but the downside is that the mechanism design optimization problem needs to be solved anew each time. Unlike the earlier work on automated mechanism design that studied a benevolent designer, in this paper we study automated mechanism design problems where the designer is self-interested. In this case, the center cares only about which outcome is chosen and what payments are made to it. The reason that the agents" preferences are relevant is that the center is constrained to making each agent at least as well off as the agent would have been had it not participated in the mechanism. In this setting, we show that designing optimal deterministic mechanisms is NP-complete in two important special cases: when the center is interested only in the payments made to it, and when payments are not possible and the center is interested only in the outcome chosen. We then show how allowing for randomization in the mechanism makes problems in this setting computationally easy. Finally, we show that the payment-maximizing AMD problem is closely related to an interesting variant of the optimal (revenuemaximizing) combinatorial auction design problem, where the bidders have best-only preferences. We show that here, too, designing an optimal deterministic auction is NPcomplete, but designing an optimal randomized auction is easy.

1. INTRODUCTION In multiagent settings, often an outcome must be chosen on the basis of the preferences reported by a group of agents. Such outcomes could be potential presidents, joint plans, allocations of goods or resources, etc. The preference aggregator generally does not know the agents" preferences a priori. Rather, the agents report their preferences to the coordinator. Unfortunately, an agent may have an incentive to misreport its preferences in order to mislead the mechanism into selecting an outcome that is more desirable to the agent than the outcome that would be selected if the agent revealed its preferences truthfully. Such manipulation is undesirable because preference aggregation mechanisms are tailored to aggregate preferences in a socially desirable way, and if the agents reveal their preferences insincerely, a socially undesirable outcome may be chosen. Manipulability is a pervasive problem across preference aggregation mechanisms. A seminal negative result, the Gibbard-Satterthwaite theorem, shows that under any nondictatorial preference aggregation scheme, if there are at least 3 possible outcomes, there are preferences under which an agent is better off reporting untruthfully [10, 23]. (A preference aggregation scheme is called dictatorial if one of the agents dictates the outcome no matter what preferences the other agents report.) What the aggregator would like to do is design a preference aggregation mechanism so that 1) the self-interested agents are motivated to report their preferences truthfully, and 2) the mechanism chooses an outcome that is desirable from the perspective of some objective. This is the classic setting of mechanism design in game theory. In this paper, we study the case where the designer is self-interested, that is, the designer does not directly care about how the out132 come relates to the agents" preferences, but is rather concerned with its own agenda for which outcome should be chosen, and with maximizing payments to itself. This is the mechanism design setting most relevant to electronic commerce. In the case where the mechanism designer is interested in maximizing some notion of social welfare, the importance of collecting the agents" preferences is clear. It is perhaps less obvious why they should be collected when the designer is self-interested and hence its objective is not directly related to the agents" preferences. The reason for this is that often the agents" preferences impose limits on how the designer chooses the outcome and payments. The most common such constraint is that of individual rationality (IR), which means that the mechanism cannot make any agent worse off than the agent would have been had it not participated in the mechanism. For instance, in the setting of optimal auction design, the designer (auctioneer) is only concerned with how much revenue is collected, and not per se with how well the allocation of the good (or goods) corresponds to the agents" preferences. Nevertheless, the designer cannot force an agent to pay more than its valuation for the bundle of goods allocated to it. Therefore, even a self-interested designer will choose an outcome that makes the agents reasonably well off. On the other hand, the designer will not necessarily choose a social welfare maximizing outcome. For example, if the designer always chooses an outcome that maximizes social welfare with respect to the reported preferences, and forces each agent to pay the difference between the utility it has now and the utility it would have had if it had not participated in the mechanism, it is easy to see that agents may have an incentive to misreport their preferences-and this may actually lead to less revenue being collected. Indeed, one of the counterintuitive results of optimal auction design theory is that sometimes the good is allocated to nobody even when the auctioneer has a reservation price of 0. Classical mechanism design provides some general mechanisms, which, under certain assumptions, satisfy some notion of nonmanipulability and maximize some objective. The upside of these mechanisms is that they do not rely on (even probabilistic) information about the agents" preferences (e.g., the Vickrey-Clarke-Groves (VCG) mechanism [24, 4, 11]), or they can be easily applied to any probability distribution over the preferences (e.g., the dAGVA mechanism [8, 2], the Myerson auction [18], and the Maskin-Riley multi-unit auction [17]). However, the general mechanisms also have significant downsides: • The most famous and most broadly applicable general mechanisms, VCG and dAGVA, only maximize social welfare. If the designer is self-interested, as is the case in many electronic commerce settings, these mechanisms do not maximize the designer"s objective. • The general mechanisms that do focus on a selfinterested designer are only applicable in very restricted settings-such as Myerson"s expected revenue maximizing auction for selling a single item, and Maskin and Riley"s expected revenue maximizing auction for selling multiple identical units of an item. • Even in the restricted settings in which these mechanisms apply, the mechanisms only allow for payment maximization. In practice, the designer may also be interested in the outcome per se. For example, an auctioneer may care which bidder receives the item. • It is often assumed that side payments can be used to tailor the agents" incentives, but this is not always practical. For example, in barter-based electronic marketplaces-such as Recipco, firstbarter.com, BarterOne, and Intagio-side payments are not allowed. Furthermore, among software agents, it might be more desirable to construct mechanisms that do not rely on the ability to make payments, because many software agents do not have the infrastructure to make payments. In contrast, we follow a recent approach where the mechanism is designed automatically for the specific problem at hand. This approach addresses all of the downsides listed above. We formulate the mechanism design problem as an optimization problem. The input is characterized by the number of agents, the agents" possible types (preferences), and the aggregator"s prior distributions over the agents" types. The output is a nonmanipulable mechanism that is optimal with respect to some objective. This approach is called automated mechanism design. The automated mechanism design approach has four advantages over the classical approach of designing general mechanisms. First, it can be used even in settings that do not satisfy the assumptions of the classical mechanisms (such as availability of side payments or that the objective is social welfare). Second, it may allow one to circumvent impossibility results (such as the Gibbard-Satterthwaite theorem) which state that there is no mechanism that is desirable across all preferences. When the mechanism is designed for the setting at hand, it does not matter that it would not work more generally. Third, it may yield better mechanisms (in terms of stronger nonmanipulability guarantees and/or better outcomes) than classical mechanisms because the mechanism capitalizes on the particulars of the setting (the probabilistic information that the designer has about the agents" types). Given the vast amount of information that parties have about each other today, this approach is likely to lead to tremendous savings over classical mechanisms, which largely ignore that information. For example, imagine a company automatically creating its procurement mechanism based on statistical knowledge about its suppliers, rather than using a classical descending procurement auction. Fourth, the burden of design is shifted from humans to a machine. However, automated mechanism design requires the mechanism design optimization problem to be solved anew for each setting. Hence its computational complexity becomes a key issue. Previous research has studied this question for benevolent designers-that wish to maximize, for example, social welfare [5, 6]. In this paper we study the computational complexity of automated mechanism design in the case of a self-interested designer. This is an important setting for automated mechanism design due to the shortage of general mechanisms in this area, and the fact that in most e-commerce settings the designer is self-interested. We also show that this problem is closely related to a particular optimal (revenue-maximizing) combinatorial auction design problem. 133 The rest of this paper is organized as follows. In Section 2, we justify the focus on nonmanipulable mechanisms. In Section 3, we define the problem we study. In Section 4, we show that designing an optimal deterministic mechanism is NP-complete even when the designer only cares about the payments made to it. In Section 5, we show that designing an optimal deterministic mechanism is also NP-complete when payments are not possible and the designer is only interested in the outcome chosen. In Section 6, we show that an optimal randomized mechanism can be designed in polynomial time even in the general case. Finally, in Section 7, we show that for designing optimal combinatorial auctions under best-only preferences, our results on AMD imply that this problem is NP-complete for deterministic auctions, but easy for randomized auctions. 2. JUSTIFYING THE FOCUS ON NONMANIPULABLE MECHANISMS Before we define the computational problem of automated mechanism design, we should justify our focus on nonmanipulable mechanisms. After all, it is not immediately obvious that there are no manipulable mechanisms that, even when agents report their types strategically and hence sometimes untruthfully, still reach better outcomes (according to whatever objective we use) than any nonmanipulable mechanism. This does, however, turn out to be the case: given any mechanism, we can construct a nonmanipulable mechanism whose performance is identical, as follows. We build an interface layer between the agents and the original mechanism. The agents report their preferences (or types) to the interface layer; subsequently, the interface layer inputs into the original mechanism the types that the agents would have strategically reported to the original mechanism, if their types were as declared to the interface layer. The resulting outcome is the outcome of the new mechanism. Since the interface layer acts strategically on each agent"s behalf, there is never an incentive to report falsely to the interface layer; and hence, the types reported by the interface layer are the strategic types that would have been reported without the interface layer, so the results are exactly as they would have been with the original mechanism. This argument is known in the mechanism design literature as the revelation principle [16]. (There are computational difficulties with applying the revelation principle in large combinatorial outcome and type spaces [7, 22]. However, because here we focus on flatly represented outcome and type spaces, this is not a concern here.) Given this, we can focus on truthful mechanisms in the rest of the paper. 3. DEFINITIONS We now formalize the automated mechanism design setting. Definition 1. In an automated mechanism design setting, we are given: • a finite set of outcomes O; • a finite set of N agents; • for each agent i, 1. a finite set of types Θi, 2. a probability distribution γi over Θi (in the case of correlated types, there is a single joint distribution γ over Θ1 × . . . × ΘN ), and 3. a utility function ui : Θi × O → R; 1 • An objective function whose expectation the designer wishes to maximize. There are many possible objective functions the designer might have, for example, social welfare (where the designer seeks to maximize the sum of the agents" utilities), or the minimum utility of any agent (where the designer seeks to maximize the worst utility had by any agent). In both of these cases, the designer is benevolent, because the designer, in some sense, is pursuing the agents" collective happiness. However, in this paper, we focus on the case of a self-interested designer. A self-interested designer cares only about the outcome chosen (that is, the designer does not care how the outcome relates to the agents" preferences, but rather has a fixed preference over the outcomes), and about the net payments made by the agents, which flow to the designer. Definition 2. A self-interested designer has an objective function given by g(o) + N i=1 πi, where g : O → R indicates the designer"s own preference over the outcomes, and πi is the payment made by agent i. In the case where g = 0 everywhere, the designer is said to be payment maximizing. In the case where payments are not possible, g constitutes the objective function by itself. We now define the kinds of mechanisms under study. By the revelation principle, we can restrict attention to truthful, direct revelation mechanisms, where agents report their types directly and never have an incentive to misreport them. Definition 3. We consider the following kinds of mechanism: • A deterministic mechanism without payments consists of an outcome selection function o : Θ1 × Θ2 × . . . × ΘN → O. • A randomized mechanism without payments consists of a distribution selection function p : Θ1 × Θ2 × . . . × ΘN → P(O), where P(O) is the set of probability distributions over O. • A deterministic mechanism with payments consists of an outcome selection function o : Θ1 ×Θ2 ×. . .×ΘN → O and for each agent i, a payment selection function πi : Θ1 × Θ2 × . . . × ΘN → R, where πi(θ1, . . . , θN ) gives the payment made by agent i when the reported types are θ1, . . . , θN . 1 Though this follows standard game theory notation [16], the fact that the agent has both a utility function and a type is perhaps confusing. The types encode the various possible preferences that the agent may turn out to have, and the agent"s type is not known to the aggregator. The utility function is common knowledge, but because the agent"s type is a parameter in the agent"s utility function, the aggregator cannot know what the agent"s utility is without knowing the agent"s type. 134 • A randomized mechanism with payments consists of a distribution selection function p : Θ1 × Θ2 × . . . × ΘN → P(O), and for each agent i, a payment selection function πi : Θ1 × Θ2 × . . . × ΘN → R.2 There are two types of constraint on the designer in building the mechanism. 3.1 Individual rationality (IR) constraints The first type of constraint is the following. The utility of each agent has to be at least as great as the agent"s fallback utility, that is, the utility that the agent would receive if it did not participate in the mechanism. Otherwise that agent would not participate in the mechanism-and no agent"s participation can ever hurt the mechanism designer"s objective because at worst, the mechanism can ignore an agent by pretending the agent is not there. (Furthermore, if no such constraint applied, the designer could simply make the agents pay an infinite amount.) This type of constraint is called an IR (individual rationality) constraint. There are three different possible IR constraints: ex ante, ex interim, and ex post, depending on what the agent knows about its own type and the others" types when deciding whether to participate in the mechanism. Ex ante IR means that the agent would participate if it knew nothing at all (not even its own type). We will not study this concept in this paper. Ex interim IR means that the agent would always participate if it knew only its own type, but not those of the others. Ex post IR means that the agent would always participate even if it knew everybody"s type. We will define the latter two notions of IR formally. First, we need to formalize the concept of the fallback outcome. We assume that each agent"s fallback utility is zero for each one of its types. This is without loss of generality because we can add a constant term to an agent"s utility function (for a given type), without affecting the decision-making behavior of that expected utility maximizing agent [16]. Definition 4. In any automated mechanism design setting with an IR constraint, there is a fallback outcome o0 ∈ O where, for any agent i and any type θi ∈ Θi, we have ui(θi, o0) = 0. (Additionally, in the case of a self-interested designer, g(o0) = 0.) We can now to define the notions of individual rationality. Definition 5. Individual rationality (IR) is defined by: • A deterministic mechanism is ex interim IR if for any agent i, and any type θi ∈ Θi, we have E(θ1,..,θi−1,θi+1,..,θN )|θi [ui(θi, o(θ1, .., θN ))−πi(θ1, .., θN )] ≥ 0. A randomized mechanism is ex interim IR if for any agent i, and any type θi ∈ Θi, we have E(θ1,..,θi−1,θi+1,..,θN )|θi Eo|θ1,..,θn [ui(θi, o)−πi(θ1, .., θN )] ≥ 0. • A deterministic mechanism is ex post IR if for any agent i, and any type vector (θ1, . . . , θN ) ∈ Θ1 × . . . × ΘN , we have ui(θi, o(θ1, . . . , θN )) − πi(θ1, . . . , θN ) ≥ 0. 2 We do not randomize over payments because as long as the agents and the designer are risk neutral with respect to payments, that is, their utility is linear in payments, there is no reason to randomize over payments. A randomized mechanism is ex post IR if for any agent i, and any type vector (θ1, . . . , θN ) ∈ Θ1 × . . . × ΘN , we have Eo|θ1,..,θn [ui(θi, o) − πi(θ1, .., θN )] ≥ 0. The terms involving payments can be left out in the case where payments are not possible. 3.2 Incentive compatibility (IC) constraints The second type of constraint says that the agents should never have an incentive to misreport their type (as justified above by the revelation principle). For this type of constraint, the two most common variants (or solution concepts) are implementation in dominant strategies, and implementation in Bayes-Nash equilibrium. Definition 6. Given an automated mechanism design setting, a mechanism is said to implement its outcome and payment functions in dominant strategies if truthtelling is always optimal even when the types reported by the other agents are already known. Formally, for any agent i, any type vector (θ1, . . . , θi, . . . , θN ) ∈ Θ1 × . . . × Θi × . . . × ΘN , and any alternative type report ˆθi ∈ Θi, in the case of deterministic mechanisms we have ui(θi, o(θ1, . . . , θi, . . . , θN )) − πi(θ1, . . . , θi, . . . , θN ) ≥ ui(θi, o(θ1, . . . , ˆθi, . . . , θN )) − πi(θ1, . . . , ˆθi, . . . , θN ). In the case of randomized mechanisms we have Eo|θ1,..,θi,..,θn [ui(θi, o) − πi(θ1, . . . , θi, . . . , θN )] ≥ Eo|θ1,.., ˆθi,..,θn [ui(θi, o) − πi(θ1, . . . , ˆθi, . . . , θN )]. The terms involving payments can be left out in the case where payments are not possible. Thus, in dominant strategies implementation, truthtelling is optimal regardless of what the other agents report. If it is optimal only given that the other agents are truthful, and given that one does not know the other agents" types, we have implementation in Bayes-Nash equilibrium. Definition 7. Given an automated mechanism design setting, a mechanism is said to implement its outcome and payment functions in Bayes-Nash equilibrium if truthtelling is always optimal to an agent when that agent does not yet know anything about the other agents" types, and the other agents are telling the truth. Formally, for any agent i, any type θi ∈ Θi, and any alternative type report ˆθi ∈ Θi, in the case of deterministic mechanisms we have E(θ1,..,θi−1,θi+1,..,θN )|θi [ui(θi, o(θ1, . . . , θi, . . . , θN ))− πi(θ1, . . . , θi, . . . , θN )] ≥ E(θ1,..,θi−1,θi+1,..,θN )|θi [ui(θi, o(θ1, . . . , ˆθi, . . . , θN ))− πi(θ1, . . . , ˆθi, . . . , θN )]. In the case of randomized mechanisms we have E(θ1,..,θi−1,θi+1,..,θN )|θi Eo|θ1,..,θi,..,θn [ui(θi, o)− πi(θ1, . . . , θi, . . . , θN )] ≥ E(θ1,..,θi−1,θi+1,..,θN )|θi Eo|θ1,.., ˆθi,..,θn [ui(θi, o)− πi(θ1, . . . , ˆθi, . . . , θN )]. The terms involving payments can be left out in the case where payments are not possible. 135 3.3 Automated mechanism design We can now define the computational problem we study. Definition 8. (AUTOMATED-MECHANISM-DESIGN (AMD)) We are given: • an automated mechanism design setting, • an IR notion (ex interim, ex post, or none), • a solution concept (dominant strategies or Bayes-Nash), • whether payments are possible, • whether randomization is possible, • (in the decision variant of the problem) a target value G. We are asked whether there exists a mechanism of the specified kind (in terms of payments and randomization) that satisfies both the IR notion and the solution concept, and gives an expected value of at least G for the objective. An interesting special case is the setting where there is only one agent. In this case, the reporting agent always knows everything there is to know about the other agents" types-because there are no other agents. Since ex post and ex interim IR only differ on what an agent is assumed to know about other agents" types, the two IR concepts coincide here. Also, because implementation in dominant strategies and implementation in Bayes-Nash equilibrium only differ on what an agent is assumed to know about other agents" types, the two solution concepts coincide here. This observation will prove to be a useful tool in proving hardness results: if we prove computational hardness in the singleagent setting, this immediately implies hardness for both IR concepts, for both solution concepts, for any number of agents. 4. PAYMENT-MAXIMIZINGDETERMINISTIC AMD IS HARD In this section we demonstrate that it is NP-complete to design a deterministic mechanism that maximizes the expected sum of the payments collected from the agents. We show that this problem is hard even in the single-agent setting, thereby immediately showing it hard for both IR concepts, for both solution concepts. To demonstrate NPhardness, we reduce from the MINSAT problem. Definition 9 (MINSAT). We are given a formula φ in conjunctive normal form, represented by a set of Boolean variables V and a set of clauses C, and an integer K (K < |C|). We are asked whether there exists an assignment to the variables in V such that at most K clauses in φ are satisfied. MINSAT was recently shown to be NP-complete [14]. We can now present our result. Theorem 1. Payment-maximizing deterministic AMD is NP-complete, even for a single agent, even with a uniform distribution over types. Proof. It is easy to show that the problem is in NP. To show NP-hardness, we reduce an arbitrary MINSAT instance to the following single-agent payment-maximizing deterministic AMD instance. Let the agent"s type set be Θ = {θc : c ∈ C} ∪ {θv : v ∈ V }, where C is the set of clauses in the MINSAT instance, and V is the set of variables. Let the probability distribution over these types be uniform. Let the outcome set be O = {o0} ∪ {oc : c ∈ C} ∪ {ol : l ∈ L}, where L is the set of literals, that is, L = {+v : v ∈ V } ∪ {−v : v ∈ V }. Let the notation v(l) = v denote that v is the variable corresponding to the literal l, that is, l ∈ {+v, −v}. Let l ∈ c denote that the literal l occurs in clause c. Then, let the agent"s utility function be given by u(θc, ol) = |Θ| + 1 for all l ∈ L with l ∈ c; u(θc, ol) = 0 for all l ∈ L with l /∈ c; u(θc, oc) = |Θ| + 1; u(θc, oc ) = 0 for all c ∈ C with c = c ; u(θv, ol) = |Θ| for all l ∈ L with v(l) = v; u(θv, ol) = 0 for all l ∈ L with v(l) = v; u(θv, oc) = 0 for all c ∈ C. The goal of the AMD instance is G = |Θ| + |C|−K |Θ| , where K is the goal of the MINSAT instance. We show the instances are equivalent. First, suppose there is a solution to the MINSAT instance. Let the assignment of truth values to the variables in this solution be given by the function f : V → L (where v(f(v)) = v for all v ∈ V ). Then, for every v ∈ V , let o(θv) = of(v) and π(θv) = |Θ|. For every c ∈ C, let o(θc) = oc; let π(θc) = |Θ| + 1 if c is not satisfied in the MINSAT solution, and π(θc) = |Θ| if c is satisfied. It is straightforward to check that the IR constraint is satisfied. We now check that the agent has no incentive to misreport. If the agent"s type is some θv, then any other report will give it an outcome that is no better, for a payment that is no less, so it has no incentive to misreport. If the agent"s type is some θc where c is a satisfied clause, again, any other report will give it an outcome that is no better, for a payment that is no less, so it has no incentive to misreport. The final case to check is where the agent"s type is some θc where c is an unsatisfied clause. In this case, we observe that for none of the types, reporting it leads to an outcome ol for a literal l ∈ c, precisely because the clause is not satisfied in the MINSAT instance. Because also, no type besides θc leads to the outcome oc, reporting any other type will give an outcome with utility 0, while still forcing a payment of at least |Θ| from the agent. Clearly the agent is better off reporting truthfully, for a total utility of 0. This establishes that the agent never has an incentive to misreport. Finally, we show that the goal is reached. If s is the number of satisfied clauses in the MINSAT solution (so that s ≤ K), the expected payment from this mechanism is |V ||Θ|+s|Θ|+(|C|−s)(|Θ|+1) |Θ| ≥ |V ||Θ|+K|Θ|+(|C|−K)(|Θ|+1) |Θ| = |Θ| + |C|−K |Θ| = G. So there is a solution to the AMD instance. Now suppose there is a solution to the AMD instance, given by an outcome function o and a payment function π. First, suppose there is some v ∈ V such that o(θv) /∈ {o+v, o−v}. Then the utility that the agent derives from the given outcome for this type is 0, and hence, by IR, no payment can be extracted from the agent for this type. Because, again by IR, the maximum payment that can be extracted for any other type is |Θ| + 1, it follows that the maximum expected payment that could be obtained is at most (|Θ|−1)(|Θ|+1) |Θ| < |Θ| < G, contradicting that this is a solution to the AMD instance. It follows that in the solution to the AMD instance, for every v ∈ V , o(θv) ∈ {o+v, o−v}. 136 We can interpret this as an assignment of truth values to the variables: v is set to true if o(θv) = o+v, and to false if o(θv) = o−v. We claim this assignment is a solution to the MINSAT instance. By the IR constraint, the maximum payment we can extract from any type θv is |Θ|. Because there can be no incentives for the agent to report falsely, for any clause c satisfied by the given assignment, the maximum payment we can extract for the corresponding type θc is |Θ|. (For if we extracted more from this type, the agent"s utility in this case would be less than 1; and if v is the variable satisfying c in the assignment, so that o(θv) = ol where l occurs in c, then the agent would be better off reporting θv instead of the truthful report θc, to get an outcome worth |Θ|+1 to it while having to pay at most |Θ|.) Finally, for any unsatisfied clause c, by the IR constraint, the maximum payment we can extract for the corresponding type θc is |Θ| + 1. It follows that the expected payment from our mechanism is at most V |Θ|+s|Θ|+(|C|−s)(|Θ|+1) Θ , where s is the number of satisfied clauses. Because our mechanism achieves the goal, it follows that V |Θ|+s|Θ|+(|C|−s)(|Θ|+1) Θ ≥ G, which by simple algebraic manipulations is equivalent to s ≤ K. So there is a solution to the MINSAT instance. Because payment-maximizing AMD is just the special case of AMD for a self-interested designer where the designer has no preferences over the outcome chosen, this immediately implies hardness for the general case of AMD for a selfinterested designer where payments are possible. However, it does not yet imply hardness for the special case where payments are not possible. We will prove hardness in this case in the next section. 5. SELF-INTERESTED DETERMINISTIC AMD WITHOUT PAYMENTS IS HARD In this section we demonstrate that it is NP-complete to design a deterministic mechanism that maximizes the expectation of the designer"s objective when payments are not possible. We show that this problem is hard even in the single-agent setting, thereby immediately showing it hard for both IR concepts, for both solution concepts. Theorem 2. Without payments, deterministic AMD for a self-interested designer is NP-complete, even for a single agent, even with a uniform distribution over types. Proof. It is easy to show that the problem is in NP. To show NP-hardness, we reduce an arbitrary MINSAT instance to the following single-agent self-interested deterministic AMD without payments instance. Let the agent"s type set be Θ = {θc : c ∈ C} ∪ {θv : v ∈ V }, where C is the set of clauses in the MINSAT instance, and V is the set of variables. Let the probability distribution over these types be uniform. Let the outcome set be O = {o0} ∪ {oc : c ∈ C}∪{ol : l ∈ L}∪{o∗ }, where L is the set of literals, that is, L = {+v : v ∈ V } ∪ {−v : v ∈ V }. Let the notation v(l) = v denote that v is the variable corresponding to the literal l, that is, l ∈ {+v, −v}. Let l ∈ c denote that the literal l occurs in clause c. Then, let the agent"s utility function be given by u(θc, ol) = 2 for all l ∈ L with l ∈ c; u(θc, ol) = −1 for all l ∈ L with l /∈ c; u(θc, oc) = 2; u(θc, oc ) = −1 for all c ∈ C with c = c ; u(θc, o∗ ) = 1; u(θv, ol) = 1 for all l ∈ L with v(l) = v; u(θv, ol) = −1 for all l ∈ L with v(l) = v; u(θv, oc) = −1 for all c ∈ C; u(θv, o∗ ) = −1. Let the designer"s objective function be given by g(o∗ ) = |Θ|+1; g(ol) = |Θ| for all l ∈ L; g(oc) = |Θ| for all c ∈ C. The goal of the AMD instance is G = |Θ| + |C|−K |Θ| , where K is the goal of the MINSAT instance. We show the instances are equivalent. First, suppose there is a solution to the MINSAT instance. Let the assignment of truth values to the variables in this solution be given by the function f : V → L (where v(f(v)) = v for all v ∈ V ). Then, for every v ∈ V , let o(θv) = of(v). For every c ∈ C that is satisfied in the MINSAT solution, let o(θc) = oc; for every unsatisfied c ∈ C, let o(θc) = o∗ . It is straightforward to check that the IR constraint is satisfied. We now check that the agent has no incentive to misreport. If the agent"s type is some θv, it is getting the maximum utility for that type, so it has no incentive to misreport. If the agent"s type is some θc where c is a satisfied clause, again, it is getting the maximum utility for that type, so it has no incentive to misreport. The final case to check is where the agent"s type is some θc where c is an unsatisfied clause. In this case, we observe that for none of the types, reporting it leads to an outcome ol for a literal l ∈ c, precisely because the clause is not satisfied in the MINSAT instance. Because also, no type leads to the outcome oc, there is no outcome that the mechanism ever selects that would give the agent utility greater than 1 for type θc, and hence the agent has no incentive to report falsely. This establishes that the agent never has an incentive to misreport. Finally, we show that the goal is reached. If s is the number of satisfied clauses in the MINSAT solution (so that s ≤ K), then the expected value of the designer"s objective function is |V ||Θ|+s|Θ|+(|C|−s)(|Θ|+1) |Θ| ≥ |V ||Θ|+K|Θ|+(|C|−K)(|Θ|+1) |Θ| = |Θ| + |C|−K |Θ| = G. So there is a solution to the AMD instance. Now suppose there is a solution to the AMD instance, given by an outcome function o. First, suppose there is some v ∈ V such that o(θv) /∈ {o+v, o−v}. The only other outcome that the mechanism is allowed to choose under the IR constraint is o0. This has an objective value of 0, and because the highest value the objective function ever takes is |Θ| + 1, it follows that the maximum expected value of the objective function that could be obtained is at most (|Θ|−1)(|Θ|+1) |Θ| < |Θ| < G, contradicting that this is a solution to the AMD instance. It follows that in the solution to the AMD instance, for every v ∈ V , o(θv) ∈ {o+v, o−v}. We can interpret this as an assignment of truth values to the variables: v is set to true if o(θv) = o+v, and to false if o(θv) = o−v. We claim this assignment is a solution to the MINSAT instance. By the above, for any type θv, the value of the objective function in this mechanism will be |Θ|. For any clause c satisfied by the given assignment, the value of the objective function in the case where the agent reports type θc will be at most |Θ|. (This is because we cannot choose the outcome o∗ for such a type, as in this case the agent would have an incentive to report θv instead, where v is the variable satisfying c in the assignment (so that o(θv) = ol where l occurs in c).) Finally, for any unsatisfied clause c, the maximum value the objective function can take in the case where the agent reports type θc is |Θ| + 1, simply because this is the largest value the function ever takes. It follows that the expected value of the objective function for our mechanism is at most V |Θ|+s|Θ|+(|C|−s)(|Θ|+1) Θ , where s is the number of satisfied 137 clauses. Because our mechanism achieves the goal, it follows that V |Θ|+s|Θ|+(|C|−s)(|Θ|+1) Θ ≥ G, which by simple algebraic manipulations is equivalent to s ≤ K. So there is a solution to the MINSAT instance. Both of our hardness results relied on the constraint that the mechanism should be deterministic. In the next section, we show that the hardness of design disappears when we allow for randomization in the mechanism. 6. RANDOMIZED AMD FOR A SELFINTERESTED DESIGNER IS EASY We now show how allowing for randomization over the outcomes makes the problem of self-interested AMD tractable through linear programming, for any constant number of agents. Theorem 3. Self-interested randomized AMD with a constant number of agents is solvable in polynomial time by linear programming, both with and without payments, both for ex post and ex interim IR, and both for implementation in dominant strategies and for implementation in Bayes-Nash equilibrium-even if the types are correlated. Proof. Because linear programs can be solved in polynomial time [13], all we need to show is that the number of variables and equations in our program is polynomial for any constant number of agents-that is, exponential only in N. Throughout, for purposes of determining the size of the linear program, let T = maxi{|Θi|}. The variables of our linear program will be the probabilities (p(θ1, θ2, . . . , θN ))(o) (at most TN |O| variables) and the payments πi(θ1, θ2, . . . , θN ) (at most NTN variables). (We show the linear program for the case where payments are possible; the case without payments is easily obtained from this by simply omitting all the payment variables in the program, or by adding additional constraints forcing the payments to be 0.) First, we show the IR constraints. For ex post IR, we add the following (at most NTN ) constraints to the LP: • For every i ∈ {1, 2, . . . , N}, and for every (θ1, θ2, . . . , θN ) ∈ Θ1 × Θ2 × . . . × ΘN , we add ( o∈O (p(θ1, θ2, . . . , θN ))(o)u(θi, o)) − πi(θ1, θ2, . . . , θN ) ≥ 0. For ex interim IR, we add the following (at most NT) constraints to the LP: • For every i ∈ {1, 2, . . . , N}, for every θi ∈ Θi, we add θ1,... ,θN γ(θ1, . . . , θN |θi)(( o∈O (p(θ1, θ2, . . . , θN ))(o)u(θi, o))− πi(θ1, θ2, . . . , θN )) ≥ 0. Now, we show the solution concept constraints. For implementation in dominant strategies, we add the following (at most NTN+1 ) constraints to the LP: • For every i ∈ {1, 2, . . . , N}, for every (θ1, θ2, . . . , θi, . . . , θN ) ∈ Θ1 × Θ2 × . . . × ΘN , and for every alternative type report ˆθi ∈ Θi, we add the constraint ( o∈O (p(θ1, θ2, . . . , θi, . . . , θN ))(o)u(θi, o)) − πi(θ1, θ2, . . . , θi, . . . , θN ) ≥ ( o∈O (p(θ1, θ2, . . . , ˆθi, . . . , θN ))(o)u(θi, o)) − πi(θ1, θ2, . . . , ˆθi, . . . , θN ). Finally, for implementation in Bayes-Nash equilibrium, we add the following (at most NT2 ) constraints to the LP: • For every i ∈ {1, 2, ..., N}, for every θi ∈ Θi, and for every alternative type report ˆθi ∈ Θi, we add the constraint θ1,...,θN γ(θ1, ..., θN |θi)(( o∈O (p(θ1, θ2, ..., θi, ..., θN ))(o)u(θi, o)) − πi(θ1, θ2, ..., θi, ..., θN )) ≥ θ1,...,θN γ(θ1, ..., θN |θi)(( o∈O (p(θ1, θ2, ..., ˆθi, ..., θN ))(o)u(θi, o)) − πi(θ1, θ2, ..., ˆθi, ..., θN )). All that is left to do is to give the expression the designer is seeking to maximize, which is: • θ1,...,θN γ(θ1, ..., θN )(( o∈O (p(θ1, θ2, ..., θi, ..., θN ))(o)g(o)) + N i=1 πi(θ1, θ2, ..., θN )). As we indicated, the number of variables and constraints is exponential only in N, and hence the linear program is of polynomial size for constant numbers of agents. Thus the problem is solvable in polynomial time. 7. IMPLICATIONS FOR AN OPTIMAL COMBINATORIAL AUCTION DESIGN PROBLEM In this section, we will demonstrate some interesting consequences of the problem of automated mechanism design for a self-interested designer on designing optimal combinatorial auctions. Consider a combinatorial auction with a set S of items for sale. For any bundle B ⊆ S, let ui(θi, B) be bidder i"s utility for receiving bundle B when the bidder"s type is θi. The optimal auction design problem is to specify the rules of the auction so as to maximize expected revenue to the auctioneer. (By the revelation principle, without loss of generality, we can assume the auction is truthful.) The optimal auction design problem is solved for the case of a single item by the famous Myerson auction [18]. However, designing optimal auctions in combinatorial auctions is a recognized open research problem [3, 25]. The problem is open even if there are only two items for sale. (The twoitem case with a very special form of complementarity and no substitutability has been solved recently [1].) Suppose we have free disposal-items can be thrown away at no cost. Also, suppose that the bidders" preferences have the following structure: whenever a bidder receives a bundle of items, the bidder"s utility for that bundle is determined by the best item in the bundle only. (We emphasize that 138 which item is the best is allowed to depend on the bidder"s type.) Definition 10. Bidder i is said to have best-only preferences over bundles of items if there exists a function vi : Θi × S → R such that for any θi ∈ Θi, for any B ⊆ S, ui(θi, B) = maxs∈B vi(θi, s). We make the following useful observation in this setting: there is no sense in awarding a bidder more than one item. The reason is that if the bidder is reporting truthfully, taking all but the highest valued item away from the bidder will not hurt the bidder; and, by free disposal, doing so can only reduce the incentive for this bidder to falsely report this type, when the bidder actually has another type. We now show that the problem of designing a deterministic optimal auction here is NP-complete, by a reduction from the payment maximizing AMD problem! Theorem 4. Given an optimal combinatorial auction design problem under best-only preferences (given by a set of items S and for each bidder i, a finite type space Θi and a function vi : Θi × S → R such that for any θi ∈ Θi, for any B ⊆ S, ui(θi, B) = maxs∈B vi(θi, s)), designing the optimal deterministic auction is NP-complete, even for a single bidder with a uniform distribution over types. Proof. The problem is in NP because we can nondeterministically generate an allocation rule, and then set the payments using linear programming. To show NP-hardness, we reduce an arbitrary paymentmaximizing deterministic AMD instance, with a single agent and a uniform distribution over types, to the following optimal combinatorial auction design problem instance with a single bidder with best-only preferences. For every outcome o ∈ O in the AMD instance (besides the outcome o0), let there be one item so ∈ S. Let the type space be the same, and let v(θi, so) = ui(θi, o) (where u is as specified in the AMD instance). Let the expected revenue target value be the same in both instances. We show the instances are equivalent. First suppose there exists a solution to the AMD instance, given by an outcome function and a payment function. Then, if the AMD solution chooses outcome o for a type, in the optimal auction solution, allocate {so} to the bidder for this type. (Unless o = o0, in which case we allocate {} to the bidder.) Let the payment functions be the same in both instances. Then, the utility that an agent receives for reporting a type (given the true type) in either solution is the same, so we have incentive compatibility in the optimal auction solution. Moreover, because the type distribution and the payment function are the same, the expected revenue to the auctioneer/designer is the same. It follows that there exists a solution to the optimal auction design instance. Now suppose there exists a solution to the optimal auction design instance. By the at-most-one-item observation, we can assume without loss of generality that the solution never allocates more than one item. Then, if the optimal auction solution allocates item so to the bidder for a type, in the AMD solution, let the mechanism choose outcome o for that type. If the optimal auction solution allocates nothing to the bidder for a type, in the AMD solution, let the mechanism choose outcome o0 for that type. Let the payment functions be the same. Then, the utility that an agent receives for reporting a type (given the true type) in either solution is the same, so we have incentive compatibility in the AMD solution. Moreover, because the type distribution and the payment function are the same, the expected revenue to the designer/auctioneer is the same. It follows that there exists a solution to the AMD instance. Fortunately, we can also carry through the easiness result for randomized mechanisms to this combinatorial auction setting-giving us one of the few known polynomial-time algorithms for an optimal combinatorial auction design problem. Theorem 5. Given an optimal combinatorial auction design problem under best-only preferences (given by a set of items S and for each bidder i, a finite type space Θi and a function vi : Θi × S → R such that for any θi ∈ Θi, for any B ⊆ S, ui(θi, B) = maxs∈B vi(θi, s)), if the number of bidders is a constant k, then the optimal randomized auction can be designed in polynomial time. (For any IC and IR constraints.) Proof. By the at-most-one-item observation, we can without loss of generality restrict ourselves to allocations where each bidder receives at most one item. There are fewer than (|S| + 1)k such allocations-that is, a polynomial number of allocations. Because we can list the outcomes explicitly, we can simply solve this as a payment-maximizing AMD instance, with linear programming. 8. RELATED RESEARCH ON COMPLEXITY IN MECHANISM DESIGN There has been considerable recent interest in mechanism design in computer science. Some of it has focused on issues of computational complexity, but most of that work has strived toward designing mechanisms that are easy to execute (e.g. [20, 15, 19, 9, 12]), rather than studying the complexity of designing the mechanism. The closest piece of earlier work studied the complexity of automated mechanism design by a benevolent designer [5, 6]. Roughgarden has studied the complexity of designing a good network topology for agents that selfishly choose the links they use [21]. This is related to mechanism design, but differs significantly in that the designer only has restricted control over the rules of the game because there is no party that can impose the outcome (or side payments). Also, there is no explicit reporting of preferences. 9. CONCLUSIONS AND FUTURE RESEARCH Often, an outcome must be chosen on the basis of the preferences reported by a group of agents. The key difficulty is that the agents may report their preferences insincerely to make the chosen outcome more favorable to themselves. Mechanism design is the art of designing the rules of the game so that the agents are motivated to report their preferences truthfully, and a desirable outcome is chosen. In a recently emerging approach-called automated mechanism design-a mechanism is computed for the specific preference aggregation setting at hand. This has several advantages, 139 but the downside is that the mechanism design optimization problem needs to be solved anew each time. Unlike earlier work on automated mechanism design that studied a benevolent designer, in this paper we studied automated mechanism design problems where the designer is self-interesteda setting much more relevant for electronic commerce. In this setting, the center cares only about which outcome is chosen and what payments are made to it. The reason that the agents" preferences are relevant is that the center is constrained to making each agent at least as well off as the agent would have been had it not participated in the mechanism. In this setting, we showed that designing an optimal deterministic mechanism is NP-complete in two important special cases: when the center is interested only in the payments made to it, and when payments are not possible and the center is interested only in the outcome chosen. These hardness results imply hardness in all more general automated mechanism design settings with a self-interested designer. The hardness results apply whether the individual rationality (participation) constraints are applied ex interim or ex post, and whether the solution concept is dominant strategies implementation or Bayes-Nash equilibrium implementation. We then showed that allowing randomization in the mechanism makes the design problem in all these settings computationally easy. Finally, we showed that the paymentmaximizing AMD problem is closely related to an interesting variant of the optimal (revenue-maximizing) combinatorial auction design problem, where the bidders have best-only preferences. We showed that here, too, designing an optimal deterministic mechanism is NP-complete even with one agent, but designing an optimal randomized mechanism is easy. Future research includes studying automated mechanism design with a self-interested designer in more restricted settings such as auctions (where the designer"s objective may include preferences about which bidder should receive the good-as well as payments). We also want to study the complexity of automated mechanism design in settings where the outcome and type spaces have special structure so they can be represented more concisely. Finally, we plan to assemble a data set of real-world mechanism design problems-both historical and current-and apply automated mechanism design to those problems. 10. REFERENCES [1] M. Armstrong. Optimal multi-object auctions. Review of Economic Studies, 67:455-481, 2000. [2] K. Arrow. The property rights doctrine and demand revelation under incomplete information. In M. Boskin, editor, Economics and human welfare. New York Academic Press, 1979. [3] C. Avery and T. Hendershott. Bundling and optimal auctions of multiple products. Review of Economic Studies, 67:483-497, 2000. [4] E. H. Clarke. Multipart pricing of public goods. Public Choice, 11:17-33, 1971. [5] V. Conitzer and T. Sandholm. Complexity of mechanism design. In Proceedings of the 18th Annual Conference on Uncertainty in Artificial Intelligence (UAI-02), pages 103-110, Edmonton, Canada, 2002. [6] V. Conitzer and T. Sandholm. Automated mechanism design: Complexity results stemming from the single-agent setting. In Proceedings of the 5th International Conference on Electronic Commerce (ICEC-03), pages 17-24, Pittsburgh, PA, USA, 2003. [7] V. Conitzer and T. Sandholm. Computational criticisms of the revelation principle. In Proceedings of the ACM Conference on Electronic Commerce (ACM-EC), New York, NY, 2004. Short paper. Full-length version appeared in the AAMAS-03 workshop on Agent-Mediated Electronic Commerce (AMEC). [8] C. d"Aspremont and L. A. G´erard-Varet. Incentives and incomplete information. Journal of Public Economics, 11:25-45, 1979. [9] J. Feigenbaum, C. Papadimitriou, and S. Shenker. Sharing the cost of muliticast transmissions. Journal of Computer and System Sciences, 63:21-41, 2001. Early version in Proceedings of the Annual ACM Symposium on Theory of Computing (STOC), 2000. [10] A. Gibbard. Manipulation of voting schemes. Econometrica, 41:587-602, 1973. [11] T. Groves. Incentives in teams. Econometrica, 41:617-631, 1973. [12] J. Hershberger and S. Suri. Vickrey prices and shortest paths: What is an edge worth? In Proceedings of the Annual Symposium on Foundations of Computer Science (FOCS), 2001. [13] L. Khachiyan. A polynomial algorithm in linear programming. Soviet Math. Doklady, 20:191-194, 1979. [14] R. Kohli, R. Krishnamurthi, and P. Mirchandani. The minimum satisfiability problem. SIAM Journal of Discrete Mathematics, 7(2):275-283, 1994. [15] D. Lehmann, L. I. O"Callaghan, and Y. Shoham. Truth revelation in rapid, approximately efficient combinatorial auctions. Journal of the ACM, 49(5):577-602, 2002. Early version appeared in Proceedings of the ACM Conference on Electronic Commerce (ACM-EC), 1999. [16] A. Mas-Colell, M. Whinston, and J. R. Green. Microeconomic Theory. Oxford University Press, 1995. [17] E. S. Maskin and J. Riley. Optimal multi-unit auctions. In F. Hahn, editor, The Economics of Missing Markets, Information, and Games, chapter 14, pages 312-335. Clarendon Press, Oxford, 1989. [18] R. Myerson. Optimal auction design. Mathematics of Operation Research, 6:58-73, 1981. [19] N. Nisan and A. Ronen. Computationally feasible VCG mechanisms. In Proceedings of the ACM Conference on Electronic Commerce (ACM-EC), pages 242-252, Minneapolis, MN, 2000. [20] N. Nisan and A. Ronen. Algorithmic mechanism design. Games and Economic Behavior, 35:166-196, 2001. Early version in Proceedings of the Annual ACM Symposium on Theory of Computing (STOC), 1999. [21] T. Roughgarden. Designing networks for selfish users is hard. In Proceedings of the Annual Symposium on Foundations of Computer Science (FOCS), 2001. [22] T. Sandholm. Issues in computational Vickrey auctions. International Journal of Electronic Commerce, 4(3):107-129, 2000. Special Issue on 140 Applying Intelligent Agents for Electronic Commerce. A short, early version appeared at the Second International Conference on Multi-Agent Systems (ICMAS), pages 299-306, 1996. [23] M. A. Satterthwaite. Strategy-proofness and Arrow"s conditions: existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory, 10:187-217, 1975. [24] W. Vickrey. Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance, 16:8-37, 1961. [25] R. V. Vohra. Research problems in combinatorial auctions. Mimeo, version Oct. 29, 2001. 141

preference aggregator;desirable outcome;statistical knowledge;automated mechanism design;revenue maximization;nonmanipulable mechanism;payment maximizing;complementarity;combinatorial auction;minsat;fallback outcome;manipulability;individual rationality;automate mechanism design;self-interested amd;classical mechanism;mechanism design

train_J-71

A Dynamic Pari-Mutuel Market for Hedging, Wagering, and Information Aggregation

I develop a new mechanism for risk allocation and information speculation called a dynamic pari-mutuel market (DPM). A DPM acts as hybrid between a pari-mutuel market and a continuous double auction (CDA), inheriting some of the advantages of both. Like a pari-mutuel market, a DPM offers infinite buy-in liquidity and zero risk for the market institution; like a CDA, a DPM can continuously react to new information, dynamically incorporate information into prices, and allow traders to lock in gains or limit losses by selling prior to event resolution. The trader interface can be designed to mimic the familiar double auction format with bid-ask queues, though with an addition variable called the payoff per share. The DPM price function can be viewed as an automated market maker always offering to sell at some price, and moving the price appropriately according to demand. Since the mechanism is pari-mutuel (i.e., redistributive), it is guaranteed to pay out exactly the amount of money taken in. I explore a number of variations on the basic DPM, analyzing the properties of each, and solving in closed form for their respective price functions.

1. INTRODUCTION A wide variety of financial and wagering mechanisms have been developed to support hedging (i.e., insuring) against exposure to uncertain events and/or speculative trading on uncertain events. The dominant mechanism used in financial circles is the continuous double auction (CDA), or in some cases the CDA with market maker (CDAwMM). The primary mechanism used for sports wagering is a bookie or bookmaker, who essentially acts exactly as a market maker. Horse racing and jai alai wagering traditionally employ the pari-mutuel mechanism. Though there is no formal or logical separation between financial trading and wagering, the two endeavors are socially considered distinct. Recently, there has been a move to employ CDAs or CDAwMMs for all types of wagering, including on sports, horse racing, political events, world news, and many other uncertain events, and a simultaneous and opposite trend to use bookie systems for betting on financial markets. These trends highlight the interchangeable nature of the mechanisms and further blur the line between investing and betting. Some companies at the forefront of these movements are growing exponentially, with some industry observers declaring the onset of a revolution in the wagering business.1 Each mechanism has pros and cons for the market institution and the participating traders. A CDA only matches willing traders, and so poses no risk whatsoever for the market institution. But a CDA can suffer from illiquidity in the form huge bid-ask spreads or even empty bid-ask queues if trading is light and thus markets are thin. A successful CDA must overcome a chicken-and-egg problem: traders are attracted to liquid markets, but liquid markets require a large number of traders. A CDAwMM and the similar bookie mechanism have built-in liquidity, but at a cost: the market maker itself, usually affiliated with the market institution, is exposed to significant risk of large monetary losses. Both the CDA and CDAwMM offer incentives for traders to leverage information continuously as soon as that information becomes available. As a result, prices are known to capture the current state of information exceptionally well. Pari-mutuel markets effectively have infinite liquidity: anyone can place a bet on any outcome at any time, without the need for a matching offer from another bettor or a market maker. Pari-mutuel markets also involve no risk for the market institution, since they only redistribute money from losing wagers to winning wagers. However, pari-mutuel mar1 http://www.wired.com/news/ebiz/0,1272,61051,00.html 170 kets are not suitable for situations where information arrives over time, since there is a strong disincentive for placing bets until either (1) all information is revealed, or (2) the market is about to close. For this reason, pari-mutuel prices prior to the market"s close cannot be considered a reflection of current information. Pari-mutuel market participants cannot buy low and sell high: they cannot cash out gains (or limit losses) before the event outcome is revealed. Because the process whereby information arrives continuously over time is the rule rather than the exception, the applicability of the standard pari-mutuel mechanism is questionable in a large number of settings. In this paper, I develop a new mechanism suitable for hedging, speculating, and wagering, called a dynamic parimutuel market (DPM). A DPM can be thought of as a hybrid between a pari-mutuel market and a CDA. A DPM is indeed pari-mutuel in nature, meaning that it acts only to redistribute money from some traders to others, and so exposes the market institution to no volatility (no risk). A constant, pre-determined subsidy is required to start the market. The subsidy can in principle be arbitrarily small and might conceivably come from traders (via antes or transaction fees) rather than the market institution, though a nontrivial outside subsidy may actually encourage trading and information aggregation. A DPM has the infinite liquidity of a pari-mutuel market: traders can always purchase shares in any outcome at any time, at some price automatically set by the market institution. A DPM is also able to react to and incorporate information arriving over time, like a CDA. The market institution changes the price for particular outcomes based on the current state of wagering. If a particular outcome receives a relatively large number of wagers, its price increases; if an outcome receives relatively few wagers, its price decreases. Prices are computed automatically using a price function, which can differ depending on what properties are desired. The price function determines the instantaneous price per share for an infinitesimal quantity of shares; the total cost for purchasing n shares is computed as the integral of the price function from 0 to n. The complexity of the price function can be hidden from traders by communicating only the ask prices for various lots of shares (e.g., lots of 100 shares), as is common practice in CDAs and CDAwMMs. DPM prices do reflect current information, and traders can cash out in an aftermarket to lock in gains or limit losses before the event outcome is revealed. While there is always a market maker willing to accept buy orders, there is not a market maker accepting sell orders, and thus no guaranteed liquidity for selling: instead, selling is accomplished via a standard CDA mechanism. Traders can always hedge-sell by purchasing the opposite outcome than they already own. 2. BACKGROUND AND RELATED WORK 2.1 Pari-mutuel markets Pari-mutuel markets are common at horse races [1, 22, 24, 25, 26], dog races, and jai alai games. In a pari-mutuel market people place wagers on which of two or more mutually exclusive and exhaustive outcomes will occur at some time in the future. After the true outcome becomes known, all of the money that is lost by those who bet on the incorrect outcome is redistributed to those who bet on the correct outcome, in direct proportion to the amount they wagered. More formally, if there are k mutually exclusive and exhaustive outcomes (e.g., k horses, exactly one of which will win), and M1, M2, . . . , Mk dollars are bet on each outcome, and outcome i occurs, then everyone who bet on an outcome j = i loses their wager, while everyone who bet on outcome i receives Pk j=1 Mj/Mi dollars for every dollar they wagered. That is, every dollar wagered on i receives an equal share of all money wagered. An equivalent way to think about the redistribution rule is that every dollar wagered on i is refunded, then receives an equal share of all remaining money bet on the losing outcomes, or P j=i Mj/Mi dollars. In practice, the market institution (e.g., the racetrack) first takes a certain percent of the total amount wagered, usually about 20% in the United States, then redistributes whatever money remains to the winners in proportion to their amount bet. Consider a simple example with two outcomes, A and B. The outcomes are mutually exclusive and exhaustive, meaning that Pr(A ∧ B) = 0 and Pr(A) + Pr(B) = 1. Suppose $800 is bet on A and $200 on B. Now suppose that A occurs (e.g., horse A wins the race). People who wagered on B lose their money, or $200 in total. People who wagered on A win and each receives a proportional share of the total $1000 wagered (ignoring fees). Specifically, each $1 wager on A entitles its owner a 1/800 share of the $1000, or $1.25. Every dollar bet in a pari-mutuel market has an equal payoff, regardless of when the wager was placed or how much money was invested in the various outcomes at the time the wager was placed. The only state that matters is the final state: the final amounts wagered on all the outcomes when the market closes, and the identity of the correct outcome. As a result, there is a disincentive to place a wager early if there is any chance that new information might become available. Moreover, there are no guarantees about the payoff rate of a particular bet, except that it will be nonnegative if the correct outcome is chosen. Payoff rates can fluctuate arbitrarily until the market closes. So a second reason not to bet early is to wait to get a better sense of the final payout rates. This is in contrast to CDAs and CDAwMMs, like the stock market, where incentives exist to invest as soon as new information is revealed. Pari-mutuel bettors may be allowed to switch their chosen outcome, or even cancel their bet, prior to the market"s close. However, they cannot cash out of the market early, to either lock in gains or limit losses, if new information favors one outcome over another, as is possible in a CDA or a CDAwMM. If bettors can cancel or change their bets, then an aftermarket to sell existing wagers is not sensible: every dollar wagered is worth exactly $1 up until the market"s close-no one would buy at greater than $1 and no one would sell at less than $1. Pari-mutuel bettors must wait until the outcome is revealed to realize any profit or loss. Unlike a CDA, in a pari-mutuel market, anyone can place a wager of any amount at any time-there is in a sense infinite liquidity for buying. A CDAwMM also has built-in liquidity, but at the cost of significant risk for the market maker. In a pari-mutuel market, since money is only redistributed among bettors, the market institution itself has no risk. The main drawback of a pari-mutuel market is that it is useful only for capturing the value of an uncertain asset at some instant in time. It is ill-suited for situations where information arrives over time, continuously updating the estimated value of the asset-situations common in al171 most all trading and wagering scenarios. There is no notion of buying low and selling high, as occurs in a CDA, where buying when few others are buying (and the price is low) is rewarded more than buying when many others are buying (and the price is high). Perhaps for this reason, in most dynamic environments, financial mechanisms like the CDA that can react in real-time to changing information are more typically employed to facilitate speculating and hedging. Since a pari-mutuel market can estimate the value of an asset at a single instant in time, a repeated pari-mutuel market, where distinct pari-mutuel markets are run at consecutive intervals, could in principle capture changing information dynamics. But running multiple consecutive markets would likely thin out trading in each individual market. Also, in each individual pari-mutuel market, the incentives would still be to wait to bet until just before the ending time of that particular market. This last problem might be mitigated by instituting a random stopping rule for each individual pari-mutuel market. In laboratory experiments, pari-mutuel markets have shown a remarkable ability to aggregate and disseminate information dispersed among traders, at least for a single snapshot in time [17]. A similar ability has been recognized at real racetracks [1, 22, 24, 25, 26]. 2.2 Financial markets In the financial world, wagering on the outcomes of uncertain future propositions is also common. The typical market mechanism used is the continuous double auction (CDA). The term securities market in economics and finance generically encompasses a number of markets where speculating on uncertain events is possible. Examples include stock markets like NASDAQ, options markets like the CBOE [13], futures markets like the CME [21], other derivatives markets, insurance markets, political stock markets [6, 7], idea futures markets [12], decision markets [10] and even market games [3, 15, 16]. Securities markets generally have an economic and social value beyond facilitating speculation or wagering: they allow traders to hedge risk, or to insure against undesirable outcomes. So if a particular outcome has disutility for a trader, he or she can mitigate the risk by wagering for the outcome, to arrange for compensation in case the outcome occurs. In this sense, buying automobile insurance is effectively a bet that an accident or other covered event will occur. Similarly, buying a put option, which is useful as a hedge for a stockholder, is a bet that the underlying stock will go down. In practice, agents engage in a mixture of hedging and speculating, and there is no clear dividing line between the two [14]. Like pari-mutuel markets, often prices in financial markets are excellent information aggregators, yielding very accurate forecasts of future events [5, 18, 19]. A CDA constantly matches orders to buy an asset with orders to sell. If at any time one party is willing to buy one unit of the asset at a bid price of pbid, while another party is willing to sell one unit of the asset at an ask price of pask, and pbid is greater than or equal to pask, then the two parties transact (at some price between pbid and pask). If the highest bid price is less than the lowest ask price, then no transactions occur. In a CDA, the bid and ask prices rapidly change as new information arrives and traders reassess the value of the asset. Since the auctioneer only matches willing bidders, the auctioneer takes on no risk. However, buyers can only buy as many shares as sellers are willing to sell; for any transaction to occur, there must be a counterparty on the other side willing to accept the trade. As a result, when few traders participate in a CDA, it may become illiquid, meaning that not much trading activity occurs. The spread between the highest bid price and the lowest ask price may be very large, or one or both queues may be completely empty, discouraging trading.2 One way to induce liquidity is to provide a market maker who is willing to accept a large number of buy and sell orders at particular prices. We call this mechanism a CDA with market maker (CDAwMM).3 Conceptually, the market maker is just like any other trader, but typically is willing to accept a much larger volume of trades. The market maker may be a person, or may be an automated algorithm. Adding a market maker to the system increases liquidity, but exposes the market maker to risk. Now, instead of only matching trades, the system actually takes on risk of its own, and depending on what happens in the future, may lose considerable amounts of money. 2.3 Wagering markets The typical Las Vegas bookmaker or oddsmaker functions much like a market maker in a CDA. In this case, the market institution (the book or house) sets the odds,4 initially according to expert opinion, and later in response to the relative level of betting on the various outcomes. Unlike in a pari-mutuel environment, whenever a wager is placed with a bookmaker, the odds or terms for that bet are fixed at the time of the bet. The bookmaker profits by offering different odds for the two sides of the bet, essentially defining a bidask spread. While odds may change in response to changing information, any bets made at previously set odds remain in effect according to the odds at the time of the bet; this is precisely in analogy to a CDAwMM. One difference between a bookmaker and a market maker is that the former usually operates in a take it or leave it mode: bettors cannot place their own limit orders on a common queue, they can in effect only place market orders at prices defined by the bookmaker. Still, the bookmaker certainly reacts to bettor demand. Like a market maker, the bookmaker exposes itself to significant risk. Sports betting markets have also been shown to provide high quality aggregate forecasts [4, 9, 23]. 2.4 Market scoring rule Hanson"s [11] market scoring rule (MSR) is a new mechanism for hedging and speculating that shares some properties in common with a DPM. Like a DPM, an MSR can be conceptualized as an automated market maker always willing to accept a trade on any event at some price. An MSR requires a patron to subsidize the market. The patron"s final loss is variable, and thus technically implies a degree of risk, though the maximum loss is bounded. An MSR maintains a probability distribution over all events. At any time any 2 Thin markets do occur often in practice, and can be seen in a variety of the less popular markets available on http://TradeSports.com, or in some financial options markets, for example. 3 A very clear example of a CDAwMM is the interactive betting market on http://WSEX.com. 4 Or, alternatively, the bookmaker sets the game line in order to provide even-money odds. 172 trader who believes the probabilities are wrong can change any part of the distribution by accepting a lottery ticket that pays off according to a scoring rule (e.g., the logarithmic scoring rule) [27], as long as that trader also agrees to pay off the most recent person to change the distribution. In the limit of a single trader, the mechanism behaves like a scoring rule, suitable for polling a single agent for its probability distribution. In the limit of many traders, it produces a combined estimate. Since the market essentially always has a complete set of posted prices for all possible outcomes, the mechanism avoids the problem of thin markets or illiquidity. An MSR is not pari-mutuel in nature, as the patron in general injects a variable amount of money into the system. An MSR provides a two-sided automated market maker, while a DPM provides a one-sided automated market maker. In an MSR, the vector of payoffs across outcomes is fixed at the time of the trade, while in a DPM, the vector of payoffs across outcomes depends both on the state of wagering at the time of the trade and the state of wagering at the market"s close. While the mechanisms are quite different-and so trader acceptance and incentives may strongly differ-the properties and motivations of DPMs and MSRs are quite similar. Hanson shows how MSRs are especially well suited for allowing bets on a combinatorial number of outcomes. The patron"s payment for subsidizing trading on all 2n possible combinations of n events is no larger than the sum of subsidizing the n event marginals independently. The mechanism was planned for use in the Policy Analysis Market (PAM), a futures market in Middle East related outcomes and funded by DARPA [20], until a media firestorm killed the project.5 As of this writing, the founders of PAM were considering reopening under private control.6 3. A DYNAMIC PARI-MUTUEL MARKET 3.1 High-level description In contrast to a standard pari-mutuel market, where each dollar always buys an equal share of the payoff, in a DPM each dollar buys a variable share in the payoff depending on the state of wagering at the time of purchase. So a wager on A at a time when most others are wagering on B offers a greater possible profit than a wager on A when most others are also wagering on A. A natural way to communicate the changing payoff of a bet is to say that, at any given time, a certain amount of money will buy a certain number of shares in one outcome the other. Purchasing a share entitles its owner to an equal stake in the winning pot should the chosen outcome occur. The payoff is variable, because when few people are betting on an outcome, shares will generally be cheaper than at a time when many people are betting that outcome. There is no pre-determined limit on the number of shares: new shares can be continually generated as trading proceeds. For simplicity, all analyses in this paper consider the binary outcome case; generalizing to multiple discrete outcomes should be straightforward. Denote the two outcomes A and B. The outcomes are mutually exclusive and ex5 See http://hanson.gmu.edu/policyanalysismarket.html for more information, or http://dpennock.com/pam.html for commentary. 6 http://www.policyanalysismarket.com/ haustive. Denote the instantaneous price per share of A as p1 and the price per share of B as p2. Denote the payoffs per share as P1 and P2, respectively. These four numbers, p1, p2, P1, P2 are the key numbers that traders must track and understand. Note that the price is set at the time of the wager; the payoff per share is finalized only after the event outcome is revealed. At any time, a trader can purchase an infinitesimal quantity of shares of A at price p1 (and similarly for B). However, since the price changes continuously as shares are purchased, the cost of buying n shares is computed as the integral of a price function from 0 to n. The use of continuous functions and integrals can be hidden from traders by aggregating the automated market maker"s sell orders into discrete lots of, say, 100 shares each. These ask orders can be automatically entered into the system by the market institution, so that traders interact with what looks like a more familiar CDA; we examine this interface issue in more detail below in Section 4.2. For our analysis, we introduce the following additional notation. Denote M1 as the total amount of money wagered on A, M2 as the total amount of money wagered on B, T = M1 + M2 as the total amount of money wagered on both sides, N1 as the total number of shares purchased of A, and N2 as the total number of shares purchased of B. There are many ways to formulate the price function. Several natural price functions are outlined below; each is motivated as the unique solution to a particular constraint on price dynamics. 3.2 Advantages and disadvantages To my knowledge, a DPM is the only known mechanism for hedging and speculating that exhibits all three of the following properties: (1) guaranteed liquidity, (2) no risk for the market institution, and (3) continuous incorporation of information. A standard pari-mutuel fails (3). A CDA fails (1). A CDAwMM, the bookmaker mechanism, and an MSR all fail (2). Even though technically an MSR exposes its patron to risk (i.e., a variable future payoff), the patron"s maximum loss is bounded, so the distinction between a DPM and an MSR in terms of these three properties is more technical than practical. DPM traders can cash out of the market early, just like stock market traders, to lock in a profit or limit a loss, an action that is simply not possible in a standard pari-mutuel. A DPM also has some drawbacks. The payoff for a wager depends both on the price at the time of the trade, and on the final payoff per share at the market"s close. This contrasts with the CDA variants, where the payoff vector across possible future outcomes is fixed at the time of the trade. So a trader"s strategic optimization problem is complicated by the need to predict the final values of P1 and P2. If P changes according to a random walk, then traders can take the current P as an unbiased estimate of the final P, greatly decreasing the complexity of their optimization. If P does not change according to a random walk, the mechanism still has utility as a mechanism for hedging and speculating, though optimization may be difficult, and determining a measure of the market"s aggregate opinion of the probabilities of A and B may be difficult. We discuss the implications of random walk behavior further below in Section 4.1 in the discussion surrounding Assumption 3. A second drawback of a DPM is its one-sided nature. 173 While an automated market maker always stands ready to accept buy orders, there is no corresponding market maker to accept sell orders. Traders must sell to each other using a standard CDA mechanism, for example by posting an ask order at a price at or below the market maker"s current ask price. Traders can also always hedge-sell by purchasing shares in the opposite outcome from the market maker, thereby hedging their bet if not fully liquidating it. 3.3 Redistribution rule In a standard pari-mutuel market, payoffs can be computed in either of two equivalent ways: (1) each winning $1 wager receives a refund of the initial $1 paid, plus an equal share of all losing wagers, or (2) each winning $1 wager receives an equal share of all wagers, winning or losing. Because each dollar always earns an equal share of the payoff, the two formulations are precisely the same: $1 + Mlose Mwin = Mwin + Mlose Mwin . In a dynamic pari-mutuel market, because each dollar is not equally weighted, the two formulations are distinct, and lead to significantly different price functions and mechanisms, each with different potentially desirable properties. We consider each case in turn. The next section analyzes case (1), where only losing money is redistributed. Section 5 examines case (2), where all money is redistributed. 4. DPM I: LOSING MONEY REDISTRIBUTED For the case where the initial payments on winning bets are refunded, and only losing money is redistributed, the respective payoffs per share are simply: P1 = M2 N1 P2 = M1 N2 . So, if A occurs, shareholders of A receive all of their initial payment back, plus P1 dollars per share owned, while shareholders of B lose all money wagered. Similarly, if B occurs, shareholders of B receive all of their initial payment back, plus P2 dollars per share owned, while shareholders of A lose all money wagered. Without loss of generality, I will analyze the market from the perspective of A, deriving prices and payoffs for A only. The equations for B are symmetric. The trader"s per-share expected value for purchasing an infinitesimal quantity of shares of A is E[ shares] = Pr(A) · E [P1|A] − (1 − Pr(A)) · p1 E[ shares] = Pr(A) · E » M2 N1 ˛ ˛ ˛ ˛ A − (1 − Pr(A)) · p1 where is an infinitesimal quantity of shares of A, Pr(A) is the trader"s belief in the probability of A, and p1 is the instantaneous price per share of A for an infinitesimal quantity of shares. E[P1|A] is the trader"s expectation of the payoff per share of A after the market closes and given that A occurs. This is a subtle point. The value of P1 does not matter if B occurs, since in this case shares of A are worthless, and the current value of P1 does not necessarily matter as this may change as trading continues. So, in order to determine the expected value of shares of A, the trader must estimate what he or she expects the payoff per share to be in the end (after the market closes) if A occurs. If E[ shares]/ > 0, a risk-neutral trader should purchase shares of A. How many shares? This depends on the price function determining p1. In general, p1 increases as more shares are purchased. The risk-neutral trader should continue purchasing shares until E[ shares]/ = 0. (A riskaverse trader will generally stop purchasing shares before driving E[ shares]/ all the way to zero.) Assuming riskneutrality, the trader"s optimization problem is to choose a number of shares n ≥ 0 of A to purchase, in order to maximize E[n shares] = Pr(A)·n·E [P1|A]−(1−Pr(A))· Z n 0 p1(n)dn. (1) It"s easy to see that the same value of n can be solved for by finding the number of shares required to drive E[ shares]/ to zero. That is, find n ≥ 0 satisfying 0 = Pr(A) · E [P1|A] − (1 − Pr(A)) · p1(n), if such a n exists, otherwise n = 0. 4.1 Market probability As traders who believe that E[ shares of A]/ > 0 purchase shares of A and traders who believe that E[ shares of B]/ > 0 purchase shares of B, the prices p1 and p2 change according to a price function, as prescribed below. The current prices in a sense reflect the market"s opinion as a whole of the relative probabilities of A and B. Assuming an efficient marketplace, the market as a whole considers E[ shares]/ = 0, since the mechanisms is a zero sum game. For example, if market participants in aggregate felt that E[ shares]/ > 0, then there would be net demand for A, driving up the price of A until E[ shares]/ = 0. Define MPr(A) to be the market probability of A, or the probability of A inferred by assuming that E[ shares]/ = 0. We can consider MPr(A) to be the aggregate probability of A as judged by the market as a whole. MPr(A) is the solution to 0 = MPr(A) · E[P1|A] − (1 − MPr(A)) · p1. Solving we get MPr(A) = p1 p1 + E[P1|A] . (2) At this point we make a critical assumption in order to greatly simplify the analysis; we assume that E[P1|A] = P1. (3) That is, we assume that the current value for the payoff per share of A is the same as the expected final value of the payoff per share of A given that A occurs. This is certainly true for the last (infinitesimal) wager before the market closes. It"s not obvious, however, that the assumption is true well before the market"s close. Basically, we are assuming that the value of P1 moves according to an unbiased random walk: the current value of P1 is the best expectation of its future value. I conjecture that there are reasonable market efficiency conditions under which assumption (3) is true, though I have not been able to prove that it arises naturally from rational trading. We examine scenarios below in which 174 assumption (3) seems especially plausible. Nonetheless, the assumption effects our analysis only. Regardless of whether (3) is true, each price function derived below implies a welldefined zero-sum game in which traders can play. If traders can assume that (3) is true, then their optimization problem (1) is greatly simplified; however, optimizing (1) does not depend on the assumption, and traders can still optimize by strategically projecting the final expected payoff in whatever complicated way they desire. So, the utility of DPM for hedging and speculating does not necessarily hinge on the truth of assumption (3). On the other hand, the ability to easily infer an aggregate market consensus probability from market prices does depend on (3). 4.2 Price functions A variety of price functions seem reasonable, each exhibiting various properties, and implying differing market probabilities. 4.2.1 Price function I: Price of A equals payoff of B One natural price function to consider is to set the price per share of A equal to the payoff per share of B, and set the price per share of B equal to the payoff per share of A. That is, p1 = P2 p2 = P1. (4) Enforcing this relationship reduces the dimensionality of the system from four to two, simplifying the interface: traders need only track two numbers instead of four. The relationship makes sense, since new information supporting A should encourage purchasing of shares A, driving up both the price of A and the payoff of B, and driving down the price of B and the payoff of A. In this setting, assumption (3) seems especially reasonable, since if an efficient market hypothesis leads prices to follow a random walk, than payoffs must also follow a random walk. The constraints (4) lead to the following derivation of the market probability: MPr(A)P1 = MPr(B)p1 MPr(A)P1 = MPr(B)P2 MPr(A) MPr(B) = P2 P1 MPr(A) MPr(B) = M1 N2 M2 N1 MPr(A) MPr(B) = M1N1 M2N2 MPr(A) = M1N1 M1N1 + M2N2 (5) The constraints (4) specify the instantaneous relationship between payoff and price. From this, we can derive how prices change when (non-infinitesimal) shares are purchased. Let n be the number of shares purchased and let m be the amount of money spent purchasing n shares. Note that p1 = dm/dn, the instantaneous price per share, and m = R n 0 p1(n)dn. Substituting into equation (4), we get: p1 = P2 dm dn = M1 + m N2 dm M1 + m = dn N2 Z dm M1 + m = Z dn N2 ln(M1 + m) = n N2 + C m = M1 h e n N2 − 1 i (6) Equation 6 gives the cost of purchasing n shares. The instantaneous price per share as a function of n is p1(n) = dm dn = M1 N2 e n N2 . (7) Note that p1(0) = M1/N2 = P2 as required. The derivation of the price function p2(n) for B is analogous and the results are symmetric. The notion of buying infinitesimal shares, or integrating costs over a continuous function, are probably foreign to most traders. A more standard interface can be implemented by discretizing the costs into round lots of shares, for example lots of 100 shares. Then ask orders of 100 shares each at the appropriate price can be automatically placed by the market institution. For example, the market institution can place an ask order for 100 shares at price m(100)/100, another ask order for 100 shares at price (m(200)−m(100))/100, a third ask for 100 shares at (m(300)− m(200))/100, etc. In this way, the market looks more familiar to traders, like a typical CDA with a number of ask orders at various prices automatically available. A trader buying less than 100 shares would pay a bit more than if the true cost were computed using (6), but the discretized interface would probably be more intuitive and transparent to the majority of traders. The above equations assume that all money that comes in is eventually returned or redistributed. In other words, the mechanism is a zero sum game, and the market institution takes no portion of the money. This could be generalized so that the market institution always takes a certain amount, or a certain percent, or a certain amount per transaction, or a certain percent per transaction, before money in returned or redistributed. Finally, note that the above price function is undefined when the amount bet or the number of shares are zero. So the system must begin with some positive amount on both sides, and some positive number of shares outstanding on both sides. These initial amounts can be arbitrarily small in principle, but the size of the initial subsidy may affect the incentives of traders to participate. Also, the smaller the initial amounts, the more each new dollar effects the prices. The initialization amounts could be funded as a subsidy from the market institution or a patron, which I"ll call a seed wager, or from a portion of the fees charged, which I"ll call an ante wager. 4.2.2 Price function II: Price of A proportional to money on A A second price function can be derived by requiring the ratio of prices to be equal to the ratio of money wagered. 175 That is, p1 p2 = M1 M2 . (8) In other words, the price of A is proportional to the amount of money wagered on A, and similarly for B. This seems like a particularly natural way to set the price, since the more money that is wagered on one side, the cheaper becomes a share on the other side, in exactly the same proportion. Using Equation 8, along with (2) and (3), we can derive the implied market probability: M1 M2 = p1 p2 = MPr(A) MPr(B) · M2 N1 MPr(B) MPr(A) · M1 N2 = (MPr(A))2 (MPr(B))2 · M2N2 M1N1 (MPr(A))2 (MPr(B))2 = (M1)2 N1 (M2)2N2 MPr(A) MPr(B) = M1 √ N1 M2 √ N2 MPr(A) = M1 √ N1 M1 √ N1 + M2 √ N2 (9) We can solve for the instantaneous price as follows: p1 = MPr(A) MPr(B) · P1 = M1 √ N1 M2 √ N2 · M2 N1 = M1 √ N1N2 (10) Working from the above instantaneous price, we can derive the implied cost function m as a function of the number n of shares purchased as follows: dm dn = M1 + m √ N1 + n √ N2 Z dm M1 + m = Z dn √ N1 + n √ N2 ln(M1 + m) = 2 N2 [(N1 + n)N2] 1 2 + C m = M1 " e 2 r N1+n N2 −2 r N1 N2 − 1 # . (11) From this we get the price function: p1(n) = dm dn = M1 p (N1 + n)N2 e 2 r N1+n N2 −2 r N1 N2 . (12) Note that, as required, p1(0) = M1/ √ N1N2, and p1(0)/p2(0) = M1/M2. If one uses the above price function, then the market dynamics will be such that the ratio of the (instantaneous) prices of A and B always equals the ratio of the amounts wagered on A and B, which seems fairly natural. Note that, as before, the mechanism can be modified to collect transaction fees of some kind. Also note that seed or ante wagers are required to initialize the system. 5. DPM II: ALL MONEY REDISTRIBUTED Above we examined the policy of refunding winning wagers and redistributing only losing wagers. In this section we consider the second policy mentioned in Section 3.3: all money from all wagers are collected and redistributed to winning wagers. For the case where all money is redistributed, the respective payoffs per share are: P1 = M1 + M2 N1 = T N1 P2 = M1 + M2 N2 = T N2 , where T = M1 + M2 is the total amount of money wagered on both sides. So, if A occurs, shareholders of A lose their initial price paid, but receive P1 dollars per share owned; shareholders of B simply lose all money wagered. Similarly, if B occurs, shareholders of B lose their initial price paid, but receive P2 dollars per share owned; shareholders of A lose all money wagered. In this case, the trader"s per-share expected value for purchasing an infinitesimal quantity of shares of A is E[ shares] = Pr(A) · E [P1|A] − p1. (13) A risk-neutral trader optimizes by choosing a number of shares n ≥ 0 of A to purchase, in order to maximize E[n shares] = Pr(A) · n · E [P1|A] − Z n 0 p1(n)dn = Pr(A) · n · E [P1|A] − m (14) The same value of n can be solved for by finding the number of shares required to drive E[ shares]/ to zero. That is, find n ≥ 0 satisfying 0 = Pr(A) · E [P1|A] − p1(n), if such a n exists, otherwise n = 0. 5.1 Market probability In this case MPr(A), the aggregate probability of A as judged by the market as a whole, is the solution to 0 = MPr(A) · E[P1|A] − p1. Solving we get MPr(A) = p1 E[P1|A] . (15) As before, we make the simplifying assumption (3) that the expected final payoff per share equals the current payoff per share. The assumption is critical for our analysis, but may not be required for a practical implementation. 5.2 Price functions For the case where all money is distributed, the constraints (4) that keep the price of A equal to the payoff of B, and vice versa, do not lead to the derivation of a coherent price function. A reasonable price function can be derived from the constraint (8) employed in Section 4.2.2, where we require that the ratio of prices to be equal to the ratio of money wagered. That is, p1/p2 = M1/M2. In other words, the price of A is proportional to the amount of money wagered on A, and similarly for B. 176 Using Equations 3, 8, and 15 we can derive the implied market probability: M1 M2 = p1 p2 = MPr(A) MPr(B) · T N1 · N2 T = MPr(A) MPr(B) · N2 N1 MPr(A) MPr(B) = M1N1 M2N2 MPr(A) = M1N1 M1N1 + M2N2 (16) Interestingly, this is the same market probability derived in Section 4.2.1 for the case of losing-money redistribution with the constraints that the price of A equal the payoff of B and vice versa. The instantaneous price per share for an infinitesimal quantity of shares is: p1 = (M1)2 + M1M2 M1N1 + M2N2 = M1 + M2 N1 + M2 M1 N2 Working from the above instantaneous price, we can derive the number of shares n that can be purchased for m dollars, as follows: dm dn = M1 + M2 + m N1 + n + M2 M1+m N2 dn dm = N1 + n + M2 M1+m N2 M1 + M2 + m (17) · · · n = m(N1 − N2) T + N2(T + m) M2 ln » T(M1 + m) M1(T + m) . Note that we solved for n(m) rather than m(n). I could not find a closed-form solution for m(n), as was derived for the two other cases above. Still, n(m) can be used to determine how many shares can be purchased for m dollars, and the inverse function can be approximated to any degree numerically. From n(m) we can also compute the price function: p1(m) = dm dn = (M1 + m)M2T denom , (18) where denom = (M1 + m)M2N1 + (M2 − m)M2N2 +T(M1 + m)N2 ln » T(M1 + m) M1(T + m) Note that, as required, p1(0)/p2(0) = M1/M2. If one uses the above price function, then the market dynamics will be such that the ratio of the (instantaneous) prices of A and B always equals the ratio of the amounts wagered on A and B. This price function has another desirable property: it acts such that the expected value of wagering $1 on A and simultaneously wagering $1 on B equals zero, assuming (3). That is, E[$1 of A + $1 of B] = 0. The derivation is omitted. 5.3 Comparing DPM I and II The main advantage of refunding winning wagers (DPM I) is that every bet on the winning outcome is guaranteed to at least break even. The main disadvantage of refunding winning wagers is that shares are not homogenous: each share of A, for example, is actually composed of two distinct parts: (1) the refund, or a lottery ticket that pays $p if A occurs, where p is the price paid per share, and (2) one share of the final payoff ($P1) if A occurs. This complicates the implementation of an aftermarket to cash out of the market early, which we will examine below in Section 7. When all money is redistributed (DPM II), shares are homogeneous: each share entitles its owner to an equal slice of the final payoff. Because shares are homogenous, the implementation of an aftermarket is straightforward, as we shall see in Section 7. On the other hand, because initial prices paid are not refunded for winning bets, there is a chance that, if prices swing wildly enough, a wager on the correct outcome might actually lose money. Traders must be aware that if they buy in at an excessively high price that later tumbles allowing many others to get in at a much lower price, they may lose money in the end regardless of the outcome. From informal experiments, I don"t believe this eventuality would be common, but nonetheless it requires care in communicating to traders the possible risks. One potential fix would be for the market maker to keep track of when the price is going too low, endangering an investor on the correct outcome. At this point, the market maker could artificially stop lowering the price. Sell orders in the aftermarket might still come in below the market maker"s price, but in this way the system could ensure that every wager on the correct outcome at least breaks even. 6. OTHER VARIATIONS A simple ascending price function would set p1 = αM1 and p2 = αM2, where α > 0. In this case, prices would only go up. For the case of all money being redistributed, this would eliminate the possibility of losing money on a wager on the correct outcome. Even though the market maker"s price only rises, the going price may fall well below the market maker"s price, as ask orders are placed in the aftermarket. I have derived price functions for several other cases, using the same methodology above. Each price function may have its own desirable properties, but it"s not clear which is best, or even that a single best method exists. Further analyses and, more importantly, empirical investigations are required to answer these questions. 7. AFTERMARKETS A key advantage of DPM over a standard pari-mutuel market is the ability to cash out of the market before it closes, in order to take a profit or limit a loss. This is accomplished by allowing traders to place ask orders on the same queue as the market maker. So traders can sell the shares that they purchased at or below the price set by the market maker. Or traders can place a limit sell order at any price. Buyers will purchase any existing shares for sale at the lower prices first, before purchasing new shares from the market maker. 7.1 Aftermarket for DPM II For the second main case explored above, where all money 177 is redistributed, allowing an aftermarket is simple. In fact, aftermarket may be a poor descriptor: buying and selling are both fully integrated into the same mechanism. Every share is worth precisely the same amount, so traders can simply place ask orders on the same queue as the market maker in order to sell their shares. New buyers will accept the lowest ask price, whether it comes from the market maker or another trader. In this way, traders can cash out early and walk away with their current profit or loss, assuming they can find a willing buyer. 7.2 Aftermarket for DPM I When winning wagers are refunded and only losing wagers are redistributed, each share is potentially worth a different amount, depending on how much was paid for it, so it is not as simple a matter to set up an aftermarket. However, an aftermarket is still possible. In fact, much of the complexity can be hidden from traders, so it looks nearly as simple as placing a sell order on the queue. In this case shares are not homogenous: each share of A is actually composed of two distinct parts: (1) the refund of p · 1A dollars, and (2) the payoff of P1 · 1A dollars, where p is the per-share price paid and 1A is the indicator function equalling 1 if A occurs, and 0 otherwise. One can imagine running two separate aftermarkets where people can sell these two respective components. However, it is possible to automate the two aftermarkets, by automatically bundling them together in the correct ratio and selling them in the central DPM. In this way, traders can cash out by placing sell orders on the same queue as the DPM market maker, effectively hiding the complexity of explicitly having two separate aftermarkets. The bundling mechanism works as follows. Suppose the current price for 1 share of A is p1. A buyer agrees to purchase the share at p1. The buyer pays p1 dollars and receives p1 · 1A + P1 · 1A dollars. If there is enough inventory in the aftermarkets, the buyer"s share is constructed by bundling together p1 ·1A from the first aftermarket, and P1 ·1A from the second aftermarket. The seller in the first aftermarket receives p1MPr(A) dollars, and the seller in the second aftermarket receives p1MPr(B) dollars. 7.3 Pseudo aftermarket for DPM I There is an alternative pseudo aftermarket that"s possible for the case of DPM I that does not require bundling. Consider a share of A purchased for $5. The share is composed of $5·1A and $P1 ·1A. Now suppose the current price has moved from $5 to $10 per share and the trader wants to cash out at a profit. The trader can sell 1/2 share at market price (1/2 share for $5), receiving all of the initial $5 investment back, and retaining 1/2 share of A. The 1/2 share is worth either some positive amount, or nothing, depending on the outcome and the final payoff. So the trader is left with shares worth a positive expected value and all of his or her initial investment. The trader has essentially cashed out and locked in his or her gains. Now suppose instead that the price moves downward, from $5 to $2 per share. The trader decides to limit his or her loss by selling the share for $2. The buyer gets the 1 share plus $2·1A (the buyer"s price refunded). The trader (seller) gets the $2 plus what remains of the original price refunded, or $3 · 1A. The trader"s loss is now limited to $3 at most instead of $5. If A occurs, the trader breaks even; if B occurs, the trader loses $3. Also note that-in either DPM formulation-traders can always hedge sell by buying the opposite outcome without the need for any type of aftermarket. 8. CONCLUSIONS I have presented a new market mechanism for wagering on, or hedging against, a future uncertain event, called a dynamic pari-mutuel market (DPM). The mechanism combines the infinite liquidity and risk-free nature of a parimutuel market with the dynamic nature of a CDA, making it suitable for continuous information aggregation. To my knowledge, all existing mechanisms-including the standard pari-mutuel market, the CDA, the CDAwMM, the bookie mechanism, and the MSR-exhibit at most two of the three properties. An MSR is the closest to a DPM in terms of these properties, if not in terms of mechanics. Given some natural constraints on price dynamics, I have derived in closed form the implied price functions, which encode how prices change continuously as shares are purchased. The interface for traders looks much like the familiar CDA, with the system acting as an automated market maker willing to accept an infinite number of buy orders at some price. I have explored two main variations of a DPM: one where only losing money is redistributed, and one where all money is redistributed. Each has its own pros and cons, and each supports several reasonable price functions. I have described the workings of an aftermarket, so that traders can cash out of the market early, like in a CDA, to lock in their gains or limit their losses, an operation that is not possible in a standard pari-mutuel setting. 9. FUTURE WORK This paper reports the results of an initial investigation of the concept of a dynamic pari-mutuel market. Many avenues for future work present themselves, including the following: • Random walk conjecture. The most important question mark in my mind is whether the random walk assumption (3) can be proven under reasonable market efficiency conditions and, if not, how severely it effects the practicality of the system. • Incentive analysis. Formally, what are the incentives for traders to act on new information and when? How does the level of initial subsidy effect trader incentives? • Laboratory experiments and field tests. This paper concentrated on the mathematics and algorithmics of the mechanism. However, the true test of the mechanism"s ability to serve as an instrument for hedging, wagering, or information aggregation is to test it with real traders in a realistic environment. In reality, how do people behave when faced with a DPM mechanism? • DPM call market. I have derived the price functions to react to wagers on one outcome at a time. The mechanism could be generalized to accept orders on both sides, then update the prices wholistically, rather than by assuming a particular sequence on the wagers. • Real-valued variables. I believe the mechanisms in this paper can easily be generalized to multiple discrete 178 outcomes, and multiple real-valued outcomes that always sum to some constant value (e.g., multiple percentage values that must sum to 100). However, the generalization to real-valued variables with arbitrary range is less clear, and open for future development. • Compound/combinatorial betting. I believe that DPM may be well suited for compound [8, 11] or combinatorial [2] betting, for many of the same reasons that market scoring rules [11] are well suited for the task. DPM may also have some computational advantages over MSR, though this remains to be seen. Acknowledgments I thank Dan Fain, Gary Flake, Lance Fortnow, and Robin Hanson. 10. REFERENCES [1] Mukhtar M. Ali. Probability and utility estimates for racetrack bettors. Journal of Political Economy, 85(4):803-816, 1977. [2] Peter Bossaerts, Leslie Fine, and John Ledyard. Inducing liquidity in thin financial markets through combined-value trading mechanisms. European Economic Review, 46:1671-1695, 2002. [3] Kay-Yut Chen, Leslie R. Fine, and Bernardo A. Huberman. Forecasting uncertain events with small groups. In Third ACM Conference on Electronic Commerce (EC"01), pages 58-64, 2001. [4] Sandip Debnath, David M. Pennock, C. Lee Giles, and Steve Lawrence. Information incorporation in online in-game sports betting markets. In Fourth ACM Conference on Electronic Commerce (EC"03), 2003. [5] Robert Forsythe and Russell Lundholm. Information aggregation in an experimental market. Econometrica, 58(2):309-347, 1990. [6] Robert Forsythe, Forrest Nelson, George R. Neumann, and Jack Wright. Anatomy of an experimental political stock market. American Economic Review, 82(5):1142-1161, 1992. [7] Robert Forsythe, Thomas A. Rietz, and Thomas W. Ross. Wishes, expectations, and actions: A survey on price formation in election stock markets. Journal of Economic Behavior and Organization, 39:83-110, 1999. [8] Lance Fortnow, Joe Kilian, David M. Pennock, and Michael P. Wellman. Betting boolean-style: A framework for trading in securities based on logical formulas. In Proceedings of the Fourth Annual ACM Conference on Electronic Commerce, pages 144-155, 2003. [9] John M. Gandar, William H. Dare, Craig R. Brown, and Richard A. Zuber. Informed traders and price variations in the betting market for professional basketball games. Journal of Finance, LIII(1):385-401, 1998. [10] Robin Hanson. Decision markets. IEEE Intelligent Systems, 14(3):16-19, 1999. [11] Robin Hanson. Combinatorial information market design. Information Systems Frontiers, 5(1), 2002. [12] Robin D. Hanson. Could gambling save science? Encouraging an honest consensus. Social Epistemology, 9(1):3-33, 1995. [13] Jens Carsten Jackwerth and Mark Rubinstein. Recovering probability distributions from options prices. Journal of Finance, 51(5):1611-1631, 1996. [14] Joseph B. Kadane and Robert L. Winkler. Separating probability elicitation from utilities. Journal of the American Statistical Association, 83(402):357-363, 1988. [15] David M. Pennock, Steve Lawrence, C. Lee Giles, and Finn ˚Arup Nielsen. The real power of artificial markets. Science, 291:987-988, February 9 2001. [16] David M. Pennock, Steve Lawrence, Finn ˚Arup Nielsen, and C. Lee Giles. Extracting collective probabilistic forecasts from web games. In Seventh International Conference on Knowledge Discovery and Data Mining, pages 174-183, 2001. [17] C. R. Plott, J. Wit, and W. C. Yang. Parimutuel betting markets as information aggregation devices: Experimental results. Technical Report Social Science Working Paper 986, California Institute of Technology, April 1997. [18] Charles R. Plott. Markets as information gathering tools. Southern Economic Journal, 67(1):1-15, 2000. [19] Charles R. Plott and Shyam Sunder. Rational expectations and the aggregation of diverse information in laboratory security markets. Econometrica, 56(5):1085-1118, 1988. [20] Charles Polk, Robin Hanson, John Ledyard, and Takashi Ishikida. Policy analysis market: An electronic commerce application of a combinatorial information market. In Proceedings of the Fourth Annual ACM Conference on Electronic Commerce, pages 272-273, 2003. [21] R. Roll. Orange juice and weather. American Economic Review, 74(5):861-880, 1984. [22] Richard N. Rosett. Gambling and rationality. Journal of Political Economy, 73(6):595-607, 1965. [23] Carsten Schmidt and Axel Werwatz. How accurate do markets predict the outcome of an event? The Euro 2000 soccer championships experiment. Technical Report 09-2002, Max Planck Institute for Research into Economic Systems, 2002. [24] Wayne W. Snyder. Horse racing: Testing the efficient markets model. Journal of Finance, 33(4):1109-1118, 1978. [25] Richard H. Thaler and William T. Ziemba. Anomalies: Parimutuel betting markets: Racetracks and lotteries. Journal of Economic Perspectives, 2(2):161-174, 1988. [26] Martin Weitzman. Utility analysis and group behavior: An empirical study. Journal of Political Economy, 73(1):18-26, 1965. [27] Robert L. Winkler and Allan H. Murphy. Good probability assessors. J. Applied Meteorology, 7:751-758, 1968. 179

trader interface;information speculation;bet;dynamic pari-mutuel market;pari-mutuel market;gamble;dpm;demand;double auction format;risk allocation;hedge;bid-ask queue;price;continuous double auction;combinatorial bet;selling;price function;market institution;loss;gain;event resolution;automated market maker;information aggregation;trade;payoff per share;hybrid;cda;speculate;automate market maker;compound security market;infinite buy-in liquidity;wager;zero risk

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Applying Learning Algorithms to Preference Elicitation

We consider the parallels between the preference elicitation problem in combinatorial auctions and the problem of learning an unknown function from learning theory. We show that learning algorithms can be used as a basis for preference elicitation algorithms. The resulting elicitation algorithms perform a polynomial number of queries. We also give conditions under which the resulting algorithms have polynomial communication. Our conversion procedure allows us to generate combinatorial auction protocols from learning algorithms for polynomials, monotone DNF, and linear-threshold functions. In particular, we obtain an algorithm that elicits XOR bids with polynomial communication.

1. INTRODUCTION In a combinatorial auction, agents may bid on bundles of goods rather than individual goods alone. Since there are an exponential number of bundles (in the number of goods), communicating values over these bundles can be problematic. Communicating valuations in a one-shot fashion can be prohibitively expensive if the number of goods is only moderately large. Furthermore, it might even be hard for agents to determine their valuations for single bundles [14]. It is in the interest of such agents to have auction protocols which require them to bid on as few bundles as possible. Even if agents can efficiently compute their valuations, they might still be reluctant to reveal them entirely in the course of an auction, because such information may be valuable to their competitors. These considerations motivate the need for auction protocols that minimize the communication and information revelation required to determine an optimal allocation of goods. There has been recent work exploring the links between the preference elicitation problem in combinatorial auctions and the problem of learning an unknown function from computational learning theory [5, 19]. In learning theory, the goal is to learn a function via various types of queries, such as What is the function"s value on these inputs? In preference elicitation, the goal is to elicit enough partial information about preferences to be able to compute an optimal allocation. Though the goals of learning and preference elicitation differ somewhat, it is clear that these problems share similar structure, and it should come as no surprise that techniques from one field should be relevant to the other. We show that any exact learning algorithm with membership and equivalence queries can be converted into a preference elicitation algorithm with value and demand queries. The resulting elicitation algorithm guarantees elicitation in a polynomial number of value and demand queries. Here we mean polynomial in the number of goods, agents, and the sizes of the agents" valuation functions in a given encoding scheme. Preference elicitation schemes have not traditionally considered this last parameter. We argue that complexity guarantees for elicitation schemes should allow dependence on this parameter. Introducing this parameter also allows us to guarantee polynomial worst-case communication, which usually cannot be achieved in the number of goods and agents alone. Finally, we use our conversion procedure to generate combinatorial auction protocols from learning algorithms for polynomials, monotone DNF, and linear-threshold functions. Of course, a one-shot combinatorial auction where agents provide their entire valuation functions at once would also have polynomial communication in the size of the agents" valuations, and only require one query. The advantage of our scheme is that agents can be viewed as black-boxes that provide incremental information about their valuations. There is no burden on the agents to formulate their valuations in an encoding scheme of the auctioneer"s choosing. We expect this to be an important consideration in practice. Also, with our scheme entire revelation only happens in the worst-case. 180 For now, we leave the issue of incentives aside when deriving elicitation algorithms. Our focus is on the time and communication complexity of preference elicitation regardless of incentive constraints, and on the relationship between the complexities of learning and preference elicitation. Related work. Zinkevich et al. [19] consider the problem of learning restricted classes of valuation functions which can be represented using read-once formulas and Toolbox DNF. Read-once formulas can represent certain substitutabilities, but no complementarities, whereas the opposite holds for Toolbox DNF. Since their work is also grounded in learning theory, they allow dependence on the size of the target valuation as we do (though read-once valuations can always be succinctly represented anyway). Their work only makes use of value queries, which are quite limited in power. Because we allow ourselves demand queries, we are able to derive an elicitation scheme for general valuation functions. Blum et al. [5] provide results relating the complexities of query learning and preference elicitation. They consider models with membership and equivalence queries in query learning, and value and demand queries in preference elicitation. They show that certain classes of functions can be efficiently learned yet not efficiently elicited, and vice-versa. In contrast, our work shows that given a more general (yet still quite standard) version of demand query than the type they consider, the complexity of preference elicitation is no greater than the complexity of learning. We will show that demand queries can simulate equivalence queries until we have enough information about valuations to imply a solution to the elicitation problem. Nisan and Segal [12] study the communication complexity of preference elicitation. They show that for many rich classes of valuations, the worst-case communication complexity of computing an optimal allocation is exponential. Their results apply to the black-box model of computational complexity. In this model algorithms are allowed to ask questions about agent valuations and receive honest responses, without any insight into how the agents internally compute their valuations. This is in fact the basic framework of learning theory. Our work also addresses the issue of communication complexity, and we are able to derive algorithms that provide significant communication guarantees despite Nisan and Segal"s negative results. Their work motivates the need to rely on the sizes of agents" valuation functions in stating worst-case results. 2. THE MODELS 2.1 Query Learning The query learning model we consider here is called exact learning from membership and equivalence queries, introduced by Angluin [2]. In this model the learning algorithm"s objective is to exactly identify an unknown target function f : X → Y via queries to an oracle. The target function is drawn from a function class C that is known to the algorithm. Typically the domain X is some subset of {0, 1}m , and the range Y is either {0, 1} or some subset of the real numbers Ê. As the algorithm progresses, it constructs a manifest hypothesis ˜f which is its current estimate of the target function. Upon termination, the manifest hypothesis of a correct learning algorithm satisfies ˜f(x) = f(x) for all x ∈ X. It is important to specify the representation that will be used to encode functions from C. For example, consider the following function from {0, 1}m to Ê: f(x) = 2 if x consists of m 1"s, and f(x) = 0 otherwise. This function may simply be represented as a list of 2m values. Or it may be encoded as the polynomial 2x1 · · · xm, which is much more succinct. The choice of encoding may thus have a significant impact on the time and space requirements of the learning algorithm. Let size(f) be the size of the encoding of f with respect to the given representation class. Most representation classes have a natural measure of encoding size. The size of a polynomial can be defined as the number of non-zero coefficients in the polynomial, for example. We will usually only refer to representation classes; the corresponding function classes will be implied. For example, the representation class of monotone DNF formulae implies the function class of monotone Boolean functions. Two types of queries are commonly used for exact learning: membership and equivalence queries. On a membership query, the learner presents some x ∈ X and the oracle replies with f(x). On an equivalence query, the learner presents its manifest hypothesis ˜f. The oracle either replies ‘YES" if ˜f = f, or returns a counterexample x such that ˜f(x) = f(x). An equivalence query is proper if size( ˜f) ≤ size(f) at the time the manifest hypothesis is presented. We are interested in efficient learning algorithms. The following definitions are adapted from Kearns and Vazirani [9]: Definition 1. The representation class C is polynomialquery exactly learnable from membership and equivalence queries if there is a fixed polynomial p(·, ·) and an algorithm L with access to membership and equivalence queries of an oracle such that for any target function f ∈ C, L outputs after at most p(size(f), m) queries a function ˜f ∈ C such that ˜f(x) = f(x) for all instances x. Similarly, the representation class C is efficiently exactly learnable from membership and equivalence queries if the algorithm L outputs a correct hypothesis in time p(size(f), m), for some fixed polynomial p(·, ·). Here m is the dimension of the domain. Since the target function must be reconstructed, we also necessarily allow polynomial dependence on size(f). 2.2 Preference Elicitation In a combinatorial auction, a set of goods M is to be allocated among a set of agents N so as to maximize the sum of the agents" valuations. Such an allocation is called efficient in the economics literature, but we will refer to it as optimal and reserve the term efficient to refer to computational efficiency. We let n = |N| and m = |M|. An allocation is a partition of the objects into bundles (S1, . . . , Sn), such that Si ∩ Sj = ∅ for all distinct i, j ∈ N. Let Γ be the set of possible allocations. Each agent i ∈ N has a valuation function vi : 2M → Ê over the space of possible bundles. Each valuation vi is drawn from a known class of valuations Vi. The valuation classes do not need to coincide. We will assume that all the valuations considered are normalized, meaning v(∅) = 0, and that there are no externalities, meaning vi(S1, ..., Sn) = vi(Si), for all agents i ∈ N, for any allocation (S1, ..., Sn) ∈ Γ (that is, an agent cares only about the bundle allocated to her). Valuations satisfying these conditions are called general valuations.1 We 1 Often general valuations are made to satisfy the additional 181 also assume that agents have quasi-linear utility functions, meaning that agents" utilities can be divided into monetary and non-monetary components. If an agent i is allocated bundle S at price p, it derives utility ui(S, p) = vi(S) − p. A valuation function may be viewed as a vector of 2m − 1 non-negative real-values. Of course there may also be more succinct representations for certain valuation classes, and there has been much research into concise bidding languages for various types of valuations [11]. A classic example which we will refer to again later is the XOR bidding language. In this language, the agent provides a list of atomic bids, which consist of a bundle together with its value. To determine the value of a bundle S given these bids, one searches for the bundle S of highest value listed in the atomic bids such that S ⊆ S. It is then the case that v(S) = v(S ). As in the learning theory setting, we will usually only refer to bidding languages rather than valuation classes, because the corresponding valuation classes will then be implied. For example, the XOR bidding language implies the class of valuations satisfying free-disposal, which is the condition that A ⊆ B ⇒ v(A) ≤ v(B). We let size(v1, . . . , vn) = Èn i=1 size(vi). That is, the size of a vector of valuations is the size of the concatenation of the valuations" representations in their respective encoding schemes (bidding languages). To make an analogy to computational learning theory, we assume that all representation classes considered are polynomially interpretable [11], meaning that the value of a bundle may be computed in polynomial time given the valuation function"s representation. More formally, a representation class (bidding language) C is polynomially interpretable if there exists an algorithm that given as input some v ∈ C and an instance x ∈ X computes the value v(x) in time q(size(v), m), for some fixed polynomial q(·, ·).2 In the intermediate rounds of an (iterative) auction, the auctioneer will have elicited information about the agents" valuation functions via various types of queries. She will thus have constructed a set of manifest valuations, denoted ˜v1, . . . , ˜vn.3 The values of these functions may correspond exactly to the true agent values, or they may for example be upper or lower bounds on the true values, depending on the types of queries made. They may also simply be default or random values if no information has been acquired about certain bundles. The goal in the preference elicitation problem is to construct a set of manifest valuations such that: arg max (S1,...,Sn)∈Γ i∈N ˜vi(Si) ⊆ arg max (S1,...,Sn)∈Γ i∈N vi(Si) That is, the manifest valuations provide enough information to compute an allocation that is optimal with respect to the true valuations. Note that we only require one such optimal allocation. condition of free-disposal (monotonicity), but we do not need it at this point. 2 This excludes OR∗ , assuming P = NP, because interpreting bids from this language is NP-hard by reduction from weighted set-packing, and there is no well-studied representation class in learning theory that is clearly analogous to OR∗ . 3 This view of iterative auctions is meant to parallel the learning setting. In many combinatorial auctions, manifest valuations are not explicitly maintained but rather simply implied by the history of bids. Two typical queries used in preference elicitation are value and demand queries. On a value query, the auctioneer presents a bundle S ⊆ M and the agent responds with her (exact) value for the bundle v(S) [8]. On a demand query, the auctioneer presents a vector of non-negative prices p ∈ Ê(2m ) over the bundles together with a bundle S. The agent responds ‘YES" if it is the case that S ∈ arg max S ⊆M v(S ) − p(S ) ¡ or otherwise presents a bundle S such that v(S ) − p(S ) > v(S) − p(S) That is, the agent either confirms that the presented bundle is most preferred at the quoted prices, or indicates a better one [15].4 Note that we include ∅ as a bundle, so the agent will only respond ‘YES" if its utility for the proposed bundle is non-negative. Note also that communicating nonlinear prices does not necessarily entail quoting a price for every possible bundle. There may be more succinct ways of communicating this vector, as we show in section 5. We make the following definitions to parallel the query learning setting and to simplify the statements of later results: Definition 2. The representation classes V1, . . . , Vn can be polynomial-query elicited from value and demand queries if there is a fixed polynomial p(·, ·) and an algorithm L with access to value and demand queries of the agents such that for any (v1, . . . , vn) ∈ V1 × . . . × Vn, L outputs after at most p(size(v1, . . . , vn), m) queries an allocation (S1, . . . , Sn) ∈ arg max(S1,...,Sn)∈Γ È vi(Si). Similarly, the representation class C can be efficiently elicited from value and demand queries if the algorithm L outputs an optimal allocation with communication p(size(v1, . . . , vn), m), for some fixed polynomial p(·, ·). There are some key differences here with the query learning definition. We have dropped the term exactly since the valuation functions need not be determined exactly in order to compute an optimal allocation. Also, an efficient elicitation algorithm is polynomial communication, rather than polynomial time. This reflects the fact that communication rather than runtime is the bottleneck in elicitation. Computing an optimal allocation of goods even when given the true valuations is NP-hard for a wide range of valuation classes. It is thus unreasonable to require polynomial time in the definition of an efficient preference elicitation algorithm. We are happy to focus on the communication complexity of elicitation because this problem is widely believed to be more significant in practice than that of winner determination [11].5 4 This differs slightly from the definition provided by Blum et al. [5] Their demand queries are restricted to linear prices over the goods, where the price of a bundle is the sum of the prices of its underlying goods. In contrast our demand queries allow for nonlinear prices, i.e. a distinct price for every possible bundle. This is why the lower bound in their Theorem 2 does not contradict our result that follows. 5 Though the winner determination problem is NP-hard for general valuations, there exist many algorithms that solve it efficiently in practice. These range from special purpose algorithms [7, 16] to approaches using off-the-shelf IP solvers [1]. 182 Since the valuations need not be elicited exactly it is initially less clear whether the polynomial dependence on size(v1, . . . , vn) is justified in this setting. Intuitively, this parameter is justified because we must learn valuations exactly when performing elicitation, in the worst-case. We address this in the next section. 3. PARALLELSBETWEEN EQUIVALENCE AND DEMAND QUERIES We have described the query learning and preference elicitation settings in a manner that highlights their similarities. Value and membership queries are clear analogs. Slightly less obvious is the fact that equivalence and demand queries are also analogs. To see this, we need the concept of Lindahl prices. Lindahl prices are nonlinear and non-anonymous prices over the bundles. They are nonlinear in the sense that each bundle is assigned a price, and this price is not necessarily the sum of prices over its underlying goods. They are non-anonymous in the sense that two agents may face different prices for the same bundle of goods. Thus Lindahl prices are of the form pi(S), for all S ⊆ M, for all i ∈ N. Lindahl prices are presented to the agents in demand queries. When agents have normalized quasi-linear utility functions, Bikhchandani and Ostroy [4] show that there always exist Lindahl prices such that (S1, . . . , Sn) is an optimal allocation if and only if Si ∈ arg max Si vi(Si) − pi(Si) ¡ ∀i ∈ N (1) (S1, . . . , Sn) ∈ arg max (S1,...,Sn)∈Γ i∈N pi(Si) (2) Condition (1) states that each agent is allocated a bundle that maximizes its utility at the given prices. Condition (2) states that the allocation maximizes the auctioneer"s revenue at the given prices. The scenario in which these conditions hold is called a Lindahl equilibrium, or often a competitive equilibrium. We say that the Lindahl prices support the optimal allocation. It is therefore sufficient to announce supporting Lindahl prices to verify an optimal allocation. Once we have found an allocation with supporting Lindahl prices, the elicitation problem is solved. The problem of finding an optimal allocation (with respect to the manifest valuations) can be formulated as a linear program whose solutions are guaranteed to be integral [4]. The dual variables to this linear program are supporting Lindahl prices for the resulting allocation. The objective function to the dual program is: min pi(S) πs + i∈N πi (3) with πi = max S⊆M (˜vi(S) − pi(S)) πs = max (S1,...,Sn)∈Γ i∈N pi(Si) The optimal values of πi and πs correspond to the maximal utility to agent i with respect to its manifest valuation and the maximal revenue to the seller. There is usually a range of possible Lindahl prices supporting a given optimal allocation. The agent"s manifest valuations are in fact valid Lindahl prices, and we refer to them as maximal Lindahl prices. Out of all possible vectors of Lindahl prices, maximal Lindahl prices maximize the utility of the auctioneer, in fact giving her the entire social welfare. Conversely, prices that maximize the È i∈N πi component of the objective (the sum of the agents" utilities) are minimal Lindahl prices. Any Lindahl prices will do for our results, but some may have better elicitation properties than others. Note that a demand query with maximal Lindahl prices is almost identical to an equivalence query, since in both cases we communicate the manifest valuation to the agent. We leave for future work the question of which Lindahl prices to choose to minimize preference elicitation. Considering now why demand and equivalence queries are direct analogs, first note that given the πi in some Lindahl equilibrium, setting pi(S) = max{0, ˜vi(S) − πi} (4) for all i ∈ N and S ⊆ M yields valid Lindahl prices. These prices leave every agent indifferent across all bundles with positive price, and satisfy condition (1). Thus demand queries can also implicitly communicate manifest valuations, since Lindahl prices will typically be an additive constant away from these by equality (4). In the following lemma we show how to obtain counterexamples to equivalence queries through demand queries. Lemma 1. Suppose an agent replies with a preferred bundle S when proposed a bundle S and supporting Lindahl prices p(S) (supporting with respect to the the agent"s manifest valuation). Then either ˜v(S) = v(S) or ˜v(S ) = v(S ). Proof. We have the following inequalities: ˜v(S) − p(S) ≥ ˜v(S ) − p(S ) ⇒ ˜v(S ) − ˜v(S) ≤ p(S ) − p(S) (5) v(S ) − p(S ) > v(S) − p(S) ⇒ v(S ) − v(S) > p(S ) − p(S) (6) Inequality (5) holds because the prices support the proposed allocation with respect to the manifest valuation. Inequality (6) holds because the agent in fact prefers S to S given the prices, according to its response to the demand query. If it were the case that ˜v(S) = v(S) and ˜v(S ) = v(S ), these inequalities would represent a contradiction. Thus at least one of S and S is a counterexample to the agent"s manifest valuation. Finally, we justify dependence on size(v1, . . . , vn) in elicitation problems. Nisan and Segal (Proposition 1, [12]) and Parkes (Theorem 1, [13]) show that supporting Lindahl prices must necessarily be revealed in the course of any preference elicitation protocol which terminates with an optimal allocation. Furthermore, Nisan and Segal (Lemma 1, [12]) state that in the worst-case agents" prices must coincide with their valuations (up to a constant), when the valuation class is rich enough to contain dual valuations (as will be the case with most interesting classes). Since revealing Lindahl prices is a necessary condition for establishing an optimal allocation, and since Lindahl prices contain the same information as valuation functions (in the worst-case), allowing for dependence on size(v1, . . . , vn) in elicitation problems is entirely natural. 183 4. FROM LEARNING TO PREFERENCE ELICITATION The key to converting a learning algorithm to an elicitation algorithm is to simulate equivalence queries with demand and value queries until an optimal allocation is found. Because of our Lindahl price construction, when all agents reply ‘YES" to a demand query, we have found an optimal allocation, analogous to the case where an agent replies ‘YES" to an equivalence query when the target function has been exactly learned. Otherwise, we can obtain a counterexample to an equivalence query given an agent"s response to a demand query. Theorem 1. The representation classes V1, . . . , Vn can be polynomial-query elicited from value and demand queries if they can each be polynomial-query exactly learned from membership and equivalence queries. Proof. Consider the elicitation algorithm in Figure 1. Each membership query in step 1 is simulated with a value query since these are in fact identical. Consider step 4. If all agents reply ‘YES", condition (1) holds. Condition (2) holds because the computed allocation is revenue-maximizing for the auctioneer, regardless of the agents" true valuations. Thus an optimal allocation has been found. Otherwise, at least one of Si or Si is a counterexample to ˜vi, by Lemma 1. We identify a counterexample by performing value queries on both these bundles, and provide it to Ai as a response to its equivalence query. This procedure will halt, since in the worst-case all agent valuations will be learned exactly, in which case the optimal allocation and Lindahl prices will be accepted by all agents. The procedure performs a polynomial number of queries, since A1, . . . , An are all polynomial-query learning algorithms. Note that the conversion procedure results in a preference elicitation algorithm, not a learning algorithm. That is, the resulting algorithm does not simply learn the valuations exactly, then compute an optimal allocation. Rather, it elicits partial information about the valuations through value queries, and periodically tests whether enough information has been gathered by proposing an allocation to the agents through demand queries. It is possible to generate a Lindahl equilibrium for valuations v1, . . . , vn using an allocation and prices derived using manifest valuations ˜v1, . . . , ˜vn, and finding an optimal allocation does not imply that the agents" valuations have been exactly learned. The use of demand queries to simulate equivalence queries enables this early halting. We would not obtain this property with equivalence queries based on manifest valuations. 5. COMMUNICATION COMPLEXITY In this section, we turn to the issue of the communication complexity of elicitation. Nisan and Segal [12] show that for a variety of rich valuation spaces (such as general and submodular valuations), the worst-case communication burden of determining Lindahl prices is exponential in the number of goods, m. The communication burden is measured in terms of the number of bits transmitted between agents and auctioneer in the case of discrete communication, or in terms of the number of real numbers transmitted in the case of continuous communication. Converting efficient learning algorithms to an elicitation algorithm produces an algorithm whose queries have sizes polynomial in the parameters m and size(v1, . . . , vn). Theorem 2. The representation classes V1, . . . , Vn can be efficiently elicited from value and demand queries if they can each be efficiently exactly learned from membership and equivalence queries. Proof. The size of any value query is O(m): the message consists solely of the queried bundle. To communicate Lindahl prices to agent i, it is sufficient to communicate the agent"s manifest valuation function and the value πi, by equality (4). Note that an efficient learning algorithm never builds up a manifest hypothesis of superpolynomial size, because the algorithm"s runtime would then also be superpolynomial, contradicting efficiency. Thus communicating the manifest valuation requires size at most p(size(vi), m), for some polynomial p that upper-bounds the runtime of the efficient learning algorithm. Representing the surplus πi to agent i cannot require space greater than q(size(˜vi), m) for some fixed polynomial q, because we assume that the chosen representation is polynomially interpretable, and thus any value generated will be of polynomial size. We must also communicate to i its allocated bundle, so the total message size for a demand query is at most p(size(vi), m) + q(p(size(vi), m), m)+O(m). Clearly, an agent"s response to a value or demand query has size at most q(size(vi), m) + O(m). Thus the value and demand queries, and the responses to these queries, are always of polynomial size. An efficient learning algorithm performs a polynomial number of queries, so the total communication of the resulting elicitation algorithm is polynomial in the relevant parameters. There will often be explicit bounds on the number of membership and equivalence queries performed by a learning algorithm, with constants that are not masked by big-O notation. These bounds can be translated to explicit bounds on the number of value and demand queries made by the resulting elicitation algorithm. We upper-bounded the size of the manifest hypothesis with the runtime of the learning algorithm in Theorem 2. We are likely to be able to do much better than this in practice. Recall that an equivalence query is proper if size( ˜f) ≤ size(f) at the time the query is made. If the learning algorithm"s equivalence queries are all proper, it may then also be possible to provide tight bounds on the communication requirements of the resulting elicitation algorithm. Theorem 2 show that elicitation algorithms that depend on the size(v1, . . . , vn) parameter sidestep Nisan and Segal"s [12] negative results on the worst-case communication complexity of efficient allocation problems. They provide guarantees with respect to the sizes of the instances of valuation functions faced at any run of the algorithm. These algorithms will fare well if the chosen representation class provides succinct representations for the simplest and most common of valuations, and thus the focus moves back to one of compact yet expressive bidding languages. We consider these issues below. 6. APPLICATIONS In this section, we demonstrate the application of our methods to particular representation classes for combinatorial valuations. We have shown that the preference elicitation problem for valuation classes V1, . . . , Vn can be reduced 184 Given: exact learning algorithms A1, . . . , An for valuations classes V1, . . . , Vn respectively. Loop until there is a signal to halt: 1. Run A1, . . . , An in parallel on their respective agents until each requires a response to an equivalence query, or has halted with the agent"s exact valuation. 2. Compute an optimal allocation (S1, . . . , Sn) and corresponding Lindahl prices with respect to the manifest valuations ˜v1, . . . , ˜vn determined so far. 3. Present the allocation and prices to the agents in the form of a demand query. 4. If they all reply ‘YES", output the allocation and halt. Otherwise there is some agent i that has replied with some preferred bundle Si. Perform value queries on Si and Si to find a counterexample to ˜vi, and provide it to Ai. Figure 1: Converting learning algorithms to an elicitation algorithm. to the problem of finding an efficient learning algorithm for each of these classes separately. This is significant because there already exist learning algorithms for a wealth of function classes, and because it may often be simpler to solve each learning subproblem separately than to attack the preference elicitation problem directly. We can develop an elicitation algorithm that is tailored to each agent"s valuation, with the underlying learning algorithms linked together at the demand query stages in an algorithm-independent way. We show that existing learning algorithms for polynomials, monotone DNF formulae, and linear-threshold functions can be converted into preference elicitation algorithms for general valuations, valuations with free-disposal, and valuations with substitutabilities, respectively. We focus on representations that are polynomially interpretable, because the computational learning theory literature places a heavy emphasis on computational tractability [18]. In interpreting the methods we emphasize the expressiveness and succinctness of each representation class. The representation class, which in combinatorial auction terms defines a bidding language, must necessarily be expressive enough to represent all possible valuations of interest, and should also succinctly represent the simplest and most common functions in the class. 6.1 Polynomial Representations Schapire and Sellie [17] give a learning algorithm for sparse multivariate polynomials that can be used as the basis for a combinatorial auction protocol. The equivalence queries made by this algorithm are all proper. Specifically, their algorithm learns the representation class of t-sparse multivariate polynomials over the real numbers, where the variables may take on values either 0 or 1. A t-sparse polynomial has at most t terms, where a term is a product of variables, e.g. x1x3x4. A polynomial over the real numbers has coefficients drawn from the real numbers. Polynomials are expressive: every valuation function v : 2M → Ê+ can be uniquely written as a polynomial [17]. To get an idea of the succinctness of polynomials as a bidding language, consider the additive and single-item valuations presented by Nisan [11]. In the additive valuation, the value of a bundle is the number of goods the bundle contains. In the single-item valuation, all bundles have value 1, except ∅ which has value 0 (i.e. the agent is satisfied as soon as it has acquired a single item). It is not hard to show that the single-item valuation requires polynomials of size 2m − 1, while polynomials of size m suffice for the additive valuation. Polynomials are thus appropriate for valuations that are mostly additive, with a few substitutabilities and complementarities that can be introduced by adjusting coefficients. The learning algorithm for polynomials makes at most mti +2 equivalence queries and at most (mti +1)(t2 i +3ti)/2 membership queries to an agent i, where ti is the sparcity of the polynomial representing vi [17]. We therefore obtain an algorithm that elicits general valuations with a polynomial number of queries and polynomial communication.6 6.2 XOR Representations The XOR bidding language is standard in the combinatorial auctions literature. Recall that an XOR bid is characterized by a set of bundles B ⊆ 2M and a value function w : B → Ê+ defined on those bundles, which induces the valuation function: v(B) = max {B ∈B | B ⊆B} w(B ) (7) XOR bids can represent valuations that satisfy free-disposal (and only such valuations), which again is the property that A ⊆ B ⇒ v(A) ≤ v(B). The XOR bidding language is slightly less expressive than polynomials, because polynomials can represent valuations that do not satisfy free-disposal. However, XOR is as expressive as required in most economic settings. Nisan [11] notes that XOR bids can represent the single-item valuation with m atomic bids, but 2m − 1 atomic bids are needed to represent the additive valuation. Since the opposite holds for polynomials, these two languages are incomparable in succinctness, and somewhat complementary for practical use. Blum et al. [5] note that monotone DNF formulae are the analogs of XOR bids in the learning theory literature. A monotone DNF formula is a disjunction of conjunctions in which the variables appear unnegated, for example x1x2 ∨ x3 ∨ x2x4x5. Note that such formulae can be represented as XOR bids where each atomic bid has value 1; thus XOR bids generalize monotone DNF formulae from Boolean to real-valued functions. These insights allow us to generalize a classic learning algorithm for monotone DNF ([3] Theorem 6 Note that Theorem 1 applies even if valuations do not satisfy free-disposal. 185 1, [18] Theorem B) to a learning algorithm for XOR bids.7 Lemma 2. An XOR bid containing t atomic bids can be exactly learned with t + 1 equivalence queries and at most tm membership queries. Proof. The algorithm will identify each atomic bid in the target XOR bid in turn. Initialize the manifest valuation ˜v to the bid that is identically zero on all bundles (this is an XOR bid containing 0 atomic bids). Present ˜v as an equivalence query. If the response is ‘YES", we are done. Otherwise we obtain a bundle S for which v(S) = ˜v(S). Create a bundle T as follows. First initialize T = S. For each item i in T, check via a membership query whether v(T) = v(T − {i}). If so set T = T − {i}. Otherwise leave T as is and proceed to the next item. We claim that (T, v(T)) is an atomic bid of the target XOR bid. For each item i in T, we have v(T) = v(T − {i}). To see this, note that at some point when generating T, we had a ¯T such that T ⊆ ¯T ⊆ S and v( ¯T) > v( ¯T − {i}), so that i was kept in ¯T. Note that v(S) = v( ¯T) = v(T) because the value of the bundle S is maintained throughout the process of deleting items. Now assume v(T) = v(T − {i}). Then v( ¯T) = v(T) = v(T − {i}) > v( ¯T − {i}) which contradicts free-disposal, since T − {i} ⊆ ¯T − {i}. Thus v(T) > v(T − {i}) for all items i in T. This implies that (T, v(T)) is an atomic bid of v. If this were not the case, T would take on the maximum value of its strict subsets, by the definition of an XOR bid, and we would have v(T) = max i∈T { max T ⊆T −{i} v(T )} = max i∈T {v(T − {i})} < v(T) which is a contradiction. We now show that v(T) = ˜v(T), which will imply that (T, v(T)) is not an atomic bid of our manifest hypothesis by induction. Assume that every atomic bid (R, ˜v(R)) identified so far is indeed an atomic bid of v (meaning R is indeed listed in an atomic bid of v as having value v(R) = ˜v(R)). This assumption holds vacuously when the manifest valuation is initialized. Using the notation from (7), let ( ˜B, ˜w) be our hypothesis, and (B, w) be the target function. We have ˜B ⊆ B, and ˜w(B) = w(B) for B ∈ ˜B by assumption. Thus, ˜v(S) = max {B∈ ˜B | B⊆S} ˜w(B) = max {B∈ ˜B | B⊆S} w(B) ≤ max {B∈B | B⊆S} w(B) = v(S) (8) Now assume v(T) = ˜v(T). Then, ˜v(T) = v(T) = v(S) = ˜v(S) (9) The second equality follows from the fact that the value remains constant when we derive T from S. The last inequality holds because S is a counterexample to the manifest valuation. From equation (9) and free-disposal, we 7 The cited algorithm was also used as the basis for Zinkevich et al."s [19] elicitation algorithm for Toolbox DNF. Recall that Toolbox DNF are polynomials with non-negative coefficients. For these representations, an equivalence query can be simulated with a value query on the bundle containing all goods. have ˜v(T) < ˜v(S). Then again from equation (9) it follows that v(S) < ˜v(S). This contradicts (8), so we in fact have v(T) = ˜v(T). Thus (T, v(T)) is not currently in our hypothesis as an atomic bid, or we would correctly have ˜v(T) = v(T) by the induction hypothesis. We add (T, v(T)) to our hypothesis and repeat the process above, performing additional equivalence queries until all atomic bids have been identified. After each equivalence query, an atomic bid is identified with at most m membership queries. Each counterexample leads to the discovery of a new atomic bid. Thus we make at most tm membership queries and exactly t + 1 equivalence queries. The number of time steps required by this algorithm is essentially the same as the number of queries performed, so the algorithm is efficient. Applying Theorem 2, we therefore obtain the following corollary: Theorem 3. The representation class of XOR bids can be efficiently elicited from value and demand queries. This contrasts with Blum et al."s negative results ([5], Theorem 2) stating that monotone DNF (and hence XOR bids) cannot be efficiently elicited when the demand queries are restricted to linear and anonymous prices over the goods. 6.3 Linear-Threshold Representations Polynomials, XOR bids, and all languages based on the OR bidding language (such as XOR-of-OR, OR-of-XOR, and OR∗ ) fail to succinctly represent the majority valuation [11]. In this valuation, bundles have value 1 if they contain at least m/2 items, and value 0 otherwise. More generally, consider the r-of-S family of valuations where bundles have value 1 if they contain at least r items from a specified set of items S ⊆ M, and value 0 otherwise. The majority valuation is a special case of the r-of-S valuation with r = m/2 and S = M. These valuations are appropriate for representing substitutabilities: once a required set of items has been obtained, no other items can add value. Letting k = |S|, such valuations are succinctly represented by r-of-k threshold functions. These functions take the form of linear inequalities: xi1 + . . . + xik ≥ r where the function has value 1 if the inequality holds, and 0 otherwise. Here i1, . . . , ik are the items in S. Littlestone"s WINNOW 2 algorithm can learn such functions using equivalence queries only, using at most 8r2 + 5k + 14kr ln m + 1 queries [10]. To provide this guarantee, r must be known to the algorithm, but S (and k) are unknown. The elicitation algorithm that results from WINNOW 2 uses demand queries only (value queries are not necessary here because the values of counterexamples are implied when there are only two possible values). Note that r-of-k threshold functions can always be succinctly represented in O(m) space. Thus we obtain an algorithm that can elicit such functions with a polynomial number of queries and polynomial communication, in the parameters n and m alone. 186 7. CONCLUSIONS AND FUTURE WORK We have shown that exact learning algorithms with membership and equivalence queries can be used as a basis for preference elicitation algorithms with value and demand queries. At the heart of this result is the fact that demand queries may be viewed as modified equivalence queries, specialized to the problem of preference elicitation. Our result allows us to apply the wealth of available learning algorithms to the problem of preference elicitation. A learning approach to elicitation also motivates a different approach to designing elicitation algorithms that decomposes neatly across agent types. If the designer knowns beforehand what types of preferences each agent is likely to exhibit (mostly additive, many substitutes, etc...), she can design learning algorithms tailored to each agents" valuations and integrate them into an elicitation scheme. The resulting elicitation algorithm makes a polynomial number of queries, and makes polynomial communication if the original learning algorithms are efficient. We do not require that agent valuations can be learned with value and demand queries. Equivalence queries can only be, and need only be, simulated up to the point where an optimal allocation has been computed. This is the preference elicitation problem. Theorem 1 implies that elicitation with value and demand queries is no harder than learning with membership and equivalence queries, but it does not provide any asymptotic improvements over the learning algorithms" complexity. It would be interesting to find examples of valuation classes for which elicitation is easier than learning. Blum et al. [5] provide such an example when considering membership/value queries only (Theorem 4). In future work we plan to address the issue of incentives when converting learning algorithms to elicitation algorithms. In the learning setting, we usually assume that oracles will provide honest responses to queries; in the elicitation setting, agents are usually selfish and will provide possibly dishonest responses so as to maximize their utility. We also plan to implement the algorithms for learning polynomials and XOR bids as elicitation algorithms, and test their performance against other established combinatorial auction protocols [6, 15]. An interesting question here is: which Lindahl prices in the maximal to minimal range are best to quote in order to minimize information revelation? 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Sandholm, S. Suri, A. Gilpin, and D. Levine. CABOB: A fast optimal algorithm for combinatorial auctions. In Proc. the 17th International Joint Conference on Artificial Intelligence (IJCAI), pages 1102-1108, 2001. [17] R. Schapire and L. Sellie. Learning sparse multivariate polynomials over a field with queries and counterexamples. In Proceedings of the Sixth Annual ACM Workshop on Computational Learning Theory, pages 17-26. ACM Press, 1993. 187 [18] L. Valiant. A theory of the learnable. Commun. ACM, 27(11):1134-1142, Nov. 1984. [19] M. Zinkevich, A. Blum, and T. Sandholm. On polynomial-time preference elicitation with value-queries. In Proc. 4th ACM Conference on Electronic Commerce (ACM-EC), San Diego, CA, June 2003. 188

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Competitive Algorithms for VWAP and Limit Order Trading

We introduce new online models for two important aspects of modern financial markets: Volume Weighted Average Price trading and limit order books. We provide an extensive study of competitive algorithms in these models and relate them to earlier online algorithms for stock trading.

1. INTRODUCTION While popular images of Wall Street often depict swashbuckling traders boldly making large gambles on just their market intuitions, the vast majority of trading is actually considerably more technical and constrained. The constraints often derive from a complex combination of business, regulatory and institutional issues, and result in certain kinds of standard trading strategies or criteria that invite algorithmic analysis. One of the most common activities in modern financial markets is known as Volume Weighted Average Price, or VWAP, trading. Informally, the VWAP of a stock over a specified market period is simply the average price paid per share during that period, so the price of each transaction in the market is weighted by its volume. In VWAP trading, one attempts to buy or sell a fixed number of shares at a price that closely tracks the VWAP. Very large institutional trades constitute one of the main motivations behind VWAP activity. A typical scenario goes as follows. Suppose a very large mutual fund holds 3% of the outstanding shares of a large, publicly traded company - a huge fraction of the shares - and that this fund"s manager decides he would like to reduce this holding to 2% over a 1-month period. (Such a decision might be forced by the fund"s own regulations or other considerations.) Typically, such a fund manager would be unqualified to sell such a large number of shares in the open market - it requires a professional broker to intelligently break the trade up over time, and possibly over multiple exchanges, in order to minimize the market impact of such a sizable transaction. Thus, the fund manager would approach brokerages for help in selling the 1%. The brokerage will typically alleviate the fund manager"s problem immediately by simply buying the shares directly from the fund manager, and then selling them off laterbut what price should the brokerage pay the fund manager? Paying the price on the day of the sale is too risky for the brokerage, as they need to sell the shares themselves over an extended period, and events beyond their control (such as wars) could cause the price to fall dramatically. The usual answer is that the brokerage offers to buy the shares from the fund manager at a per-share price tied to the VWAP over some future period - in our example, the brokerage might offer to buy the 1% at a per-share price of the coming month"s VWAP minus 1 cent. The brokerage now has a very clean challenge: by selling the shares themselves over the next month in a way that exactly matches the VWAP, a penny per share is earned in profits. If they can beat the VWAP by a penny, they make two cents per share. Such small-margin, high-volume profits can be extremely lucrative for a large brokerage. The importance of the VWAP has led to many automated VWAP trading algorithms - indeed, every major brokerage has at least one VWAP box, 189 Price Volume Model Order Book Model Macroscopic Distribution Model OWT Θ(log(R)) (From[3]) O(log(R) log(N)) 2E(Pbins maxprice ) 2(1 + )E(Pbins maxprice ) for -approx of Pbins maxprice Θ(log(Q)) (same as above plus...) VWAP Θ(log(R)) O(log(R) log(N)) (from above) 2E(Pbins vol ) Ω(Q) fixed schedule O(log(Q)) for large N (1 + )2E(Pbins vol ) for -approx. of Pbins vol 1 for volume in [N, QN] Figure 1: The table summarizes the results presented in this paper. The rows represent results for either the OWT or VWAP criterion. The columns represent which model we are working in. The entry in the table is the competitive ratio between our algorithm and an optimal algorithm, and the closer the ratio is to 1 the better. The parameter R represents a bound on the maximum to the minimum price fluctuation and the parameter Q represents a bound on the maximum to minimum volume fluctuation in the respective model. (See Section 4 for a description of the Macroscopic Distribution Model.) All the results for the OWT trading criterion (which is a stronger criterion) directly translate to the VWAP criterion. However, in the VWAP setting, considering a restriction on the maximum to the minimum volume fluctuation Q, leads to an additional class of results which depends on Q. and some small companies focus exclusively on proprietary VWAP trading technology. In this paper, we provide the first study of VWAP trading algorithms in an online, competitive ratio setting. We first formalize the VWAP trading problem in a basic online model we call the price-volume model, which can be viewed as a generalization of previous theoretical online trading models incorporating market volume information. In this model, we provide VWAP algorithms and competitive ratios, and compare this setting with the one-way trading (OWT) problem studied in [3]. Our most interesting results, however, examine the VWAP trading problem in a new online trading model capturing the important recent phenomenon of limit order books in financial markets. Briefly, a limit buy or sell order specifies both the number of shares and the desired price, and will only be executed if there is a matching party on the opposing side, according to a well-defined matching procedure used by all the major exchanges. While limit order books (the list of limit orders awaiting possible future execution) have existed since the dawn of equity exchanges, only very recently have these books become visible to traders in real time, thus opening the way to trading algorithms of all varieties that attempt to exploit this rich market microstructure data. Such data and algorithms are a topic of great current interest on Wall Street [4]. We thus introduce a new online trading model incorporating limit order books, and examine both the one-way and VWAP trading problems in it. Our results are summarized in Figure 1 (see the caption for a summary). 2. THEPRICE-VOLUMETRADINGMODEL We now present a trading model which includes both price and volume information about the sequence of trades. While this model is a generalization of previous formalisms for online trading, it makes an infinite liquidity assumption which fails to model the negative market impact that trading a large number of shares typically has. This will be addressed in the order book model studied in the next section. A note on terminology: throughout the paper (unless otherwise specified), we shall use the term market to describe all activity or orders other than those of the algorithm under consideration. The setting we consider can be viewed as a game between our algorithm and the market. 2.1 The Model In the price-volume trading model, we assume that the intraday trading activity in a given stock is summarized by a discrete sequence of price and volume pairs (pt, vt) for t = 1, . . . , T. Here t = 0 corresponds to the day"s market open, and t = T to the close. While there is nothing technically special about the time horizon of a single day, it is particularly consistent with limit order book trading on Wall Street. The pair (pt, vt) represents the fact that a total of vt shares were traded at an (average) price per share pt in the market between time t − 1 and t. Realistically, we should imagine the number of intervals T being reasonably large, so that it is sensible to assign a common approximate price to all shares traded within an interval. In the price-volume model, we shall make an infinite liquidity assumption for our trading algorithms. More precisely, in this online model, we see the price-volume sequence one pair at a time. Following the observation of (pt, vt), we are permitted to sell any (possibly fractional) number of shares nt at the price pt. Let us assume that our goal is to sell N shares over the course of the day. Hence, at each time, we must select a (possibly fractional) number of shares nt to sell at price pt, subject to the global constraint T t=1 nt = N. It is thus assumed that if we have left over shares to sell after time T − 1, we are forced to sell them at the closing price of the market - that is, nT = N − T −1 t=1 nt is sold at pT . In this way we are certain to sell exactly N shares over the course of the day; the only thing an algorithm must do is determine the schedule of selling based on the incoming market price-volume stream. Any algorithm which sells fractional volumes can be converted to a randomized algorithm which only sells integral volumes with the same expected number of shares sold. If we keep the hard constraint of selling exactly N shares, we might incur an additional slight loss in the conversion. (Note that we only allow fractional volumes in the price-volume model, where liquidity is not an issue. In the order book model to follow, we do not allow fractional volumes.) In VWAP trading, the goal of an online algorithm A which sells exactly N shares is not to maximize profits per se, but to track the market VWAP. The market VWAP for an intraday trading sequence S = (p1, v1), . . . , (pT , vT ) is simply the average price paid per share over the course of the trading 190 day, ie VWAPM (S) = T t=1 ptvt /V where V is the total daily volume, i.e., V = T t=1 vt. If on the sequence S, the algorithm A sells its N stocks using the volume sequence n1, . . . nT , then we analogously define the VWAP of A on market sequence S by VWAPA(S) = T t=1 ptnt /N . Note that the market VWAP does not include the shares that the algorithm sells. The VWAP competitive ratio of A with respect to a set of sequences Σ is then RVWAP(A) = max S∈Σ {VWAPM (S)/VWAPA(S)} In the case that A is randomized, we generalize the definition above by taking an expectation over VWAPA(S) inside the max. We note that unlike on Wall Street, our definition of VWAPM does not take our own trading into account. It is easy to see that this makes it a more challenging criterion to track. In contrast to the VWAP, another common measure of the performance of an online selling algorithm would be its one-way trading (OWT) competitive ratio [3] with respect to a set of sequences Σ: ROWT(A) = max S∈Σ max 1≤t≤T {pt/VWAPA(S)} where the algorithms performance is compared to the largest individual price appearing in the sequence S. In both VWAP and OWT, we are comparing the average price per share received by a selling algorithm to some measure of market performance. In the case of OWT, we compare to the rather ambitious benchmark of the high price of the day, ignoring volumes entirely. In VWAP trading, we have the more modest goal of comparing favorably to the overall market average of the day. As we shall see, there are some important commonalities and differences to these two approaches. For now we note one simple fact: on any specific sequence S, VWAPA(S) may be larger that VWAPM (S). However, RVWAP(A) cannot be smaller than 1, since on any sequence S in which all price pt are identical, it is impossible to get a better average share per price. Thus, for all algorithms A, both RVWAP(A) and ROWT(A) are larger than 1, and the closer to 1 they are, the better A is tracking its respective performance measure. 2.2 VWAP Results in the Price-Volume Model As in previous work on online trading, it is generally not possible to obtain finite bounds on competitive ratios with absolutely no assumptions on the set of sequences Σbounds on the maximum variation in price or volume are required, depending on the exact setting. We thus introduce the following two assumptions. 2.2.0.1 Volume Variability Assumption.. Let 0 < Vmin ≤ Vmax be known positive constants, and define Q = Vmax /Vmin . For all intraday trading sequences S ∈ Σ, the total daily volume V ∈ [Vmin , Vmax ]. 2.2.0.2 Price Variability Assumption.. Let 0 < pmin ≤ pmax be known positive constants, and define R = pmax/pmin. For all intraday trading sequences S ∈ Σ, the prices satisfy pt ∈ [pmin, pmax], for all t = 1, . . . , T. Competitive ratios are generally taken over all sets Σ consistent with at least one of these assumptions. To gain some intuition consider the two trivial cases of R = 1 and Q = 1. In the case of R = 1 (where there is no fluctuation in price), any schedule is optimal. In the case of Q = 1 (where the total volume V over the trading period is known), we can gain a competitive ratio of 1 by selling vt V N shares after each time period. For the OWT problem in the price-volume model, volumes are irrelevant for the performance criterion, but for the VWAP criterion they are central. For the OWT problem under the price variability assumption, the results of [3] established that the optimal competitive ratio was Θ(log(R)). Our first result establishes that the optimal competitive ratio for VWAP under the volume variability assumption is Θ(log(Q)) and is achieved by an algorithm that ignores the price data. Theorem 1. In the price-volume model under the volume variability assumption, there exists an online algorithm A for selling N shares achieving competitive ratio RVWAP(A) ≤ 2 log(Q). In addition, if only the volume variability (and not the price variability) assumption holds, any online algorithm A for selling N shares has RVWAP(A) = Ω(log(Q)). Proof. (Sketch) For the upper bound, the idea is similar to the price reservation algorithm of [3] for the OWT problem, and similar in spirit to the general technique of classify and select [1]. Consider algorithms which use a parameter ˆV , which is interpreted as an estimate for the total volume for the day. Then at each time t, if the market price and volume is (pt, vt), the algorithm sells a fraction vt/ ˆV of its shares. We consider a family of log(Q) such algorithms, where algorithm Ai uses ˆV = Vmin 2i−1 . Clearly, one of the Ai has a competitive ratio of 2. We can derive an O(log(Q)) VWAP competitive ratio by running these algorithms in parallel, and letting each algorithm sell N/ log(Q) shares. (Alternatively, we can randomly select one Ai and guarantee the same expected competitive ratio.) We now sketch the proof of the lower bound, which relates performance in the VWAP and OWT problems. Any algorithm that is c-competitive in the VWAP setting (under fixed Q) is 3c-competitive in the OWT setting with R = Q/2. To show this, we take any sequence S of prices for the OWT problem, and convert it into a price-volume sequence for the VWAP problem. The prices in the VWAP sequence are the same as in S. To construct the volumes in the VWAP sequence, we segment the prices in S into log(R) intervals [2i−1 pmin , 2i pmin ). Suppose pt ∈ [2i−1 pmin , 2i pmin ), and this is the first time in S that a price has fallen in this interval. Then in the VWAP sequence we set the volume vt = 2i−1 . If this is not the first visit to the interval containing pt, we set vt = 0. Assume that the maximum price in S is pmax . The VWAP of our sequence is at least pmax /3. Since we had a c competitive algorithm, its average sell is at least pmax /3c. The lower bound now follows using the lower bound in [3]. An alternative approach to VWAP is to ignore the volumes in favor of prices, and apply an algorithm for the OWT problem. Note that the lower bound in this theorem, unlike in the previous one, only assumes a price variation bound. 191 Theorem 2. In the price-volume model under the price variability assumption, there exists an online algorithm A for selling N shares achieving competitive ratio RVWAP(A) = O(log(R)). In addition, if only the price variability (and not the volume variability) assumption holds, any online A for selling N shares has RVWAP(A) = Ω(log(R)). Proof. (Sketch) Follows immediately from the results of [3] for OWT: the upper bound from the simple fact that for any sequence S, VWAPA(S) is less than max1≤t≤T {pt}, and the lower bound from a reduction to OWT. Theorems 1 and 2 demonstrate that one can achieve logarithmic VWAP competitive ratios under the assumption of either bounded variability of total volume or bounded variability of maximum price. If both assumptions hold, it is possible to give an algorithm accomplishing the minimum of log(Q) and log(R). This flexibility of approach derives from the fact that the VWAP is a quantity in which both prices and volumes matter, as opposed to OWT. 2.3 RelatedResultsinthePrice-VolumeModel All of the VWAP algorithms we have discussed so far make some use of the daily data (pt, vt) as it unfolds, using either the price or volume information. In contrast, a fixed schedule VWAP algorithm has a predetermined distribution {f1, f2, . . . fT }, and simply sells ftN shares at time t, independent of (pt, vt). Fixed schedule VWAP algorithms, or slight variants of them, are surprisingly common on Wall Street, and the schedule is usually derived from historical intraday volume data. Our next result demonstrates that such algorithms can perform considerably worse than dynamically adaptive algorithms in terms of the worst case competitive ratio. Theorem 3. In the price-volume model under both the volume and price variability assumptions, any fixed schedule VWAP algorithm A for selling N shares has sell VWAP competitive ratio RVWAP(A) = Ω(min(T, R)). The proofs of all the results in this subsection are in the Appendix. So far our emphasis has been on VWAP algorithms that must sell exactly N shares. In many realistic circumstances, however, there is actually some flexibility in the precise number of shares to be sold. For instance, this is true at large brokerages, where many separate VWAP trades may be pooled and executed by a common algorithm, and the firm would be quite willing to carry a small position of unsold shares overnight if it resulted in better execution prices. The following theorem (which interestingly has no analogue for the OWT problem) demonstrates that this trade-off in shares sold and performance can be realized dramatically in our model. It states that if we are willing to let the number of shares sold vary with Q, we can in fact achieve a VWAP competitive ratio of 1. Theorem 4. In the price-volume model under the volume variability assumption, there exists an algorithm A that always sells between N and QN shares and that the average price per sold share is exactly VWAPM (S). In many online problems, there is a clear distinction between benefit problems and cost problems [2]. In the VWAP setting, selling shares is a benefit problem, and buying shares is a cost problem. The definitions of the competitive ratios, Rbuy VWAP(A) and Rbuy OWT(A), for algorithms which Figure 2: Sample Island order books for MSFT. buy exactly N shares are maxS∈Σ{VWAPA(S)/VWAPM (S)} and maxS∈Σ maxt{VWAPA(S)/pt} respectively. Eventhough Theorem 4 also holds for buying, in general, the competitive ratio of the buy (cost) problem is much higher, as stated in the following theorem. Theorem 5. In the price-volume model under the volume and price variability assumptions, there exists an online algorithm A for buying N shares achieving buy VWAP competitive ratio Rbuy VWAP(A) = O(min{Q, √ R}). In addition any online algorithm A for buying N shares has buy VWAP competitive ratio Rbuy VWAP(A) = Ω(min{Q, √ R}). 3. A LIMIT ORDER BOOK TRADING MODEL Before we can define our online trading model based on limit order books, we give some necessary background on the detailed mechanics of financial markets, which are sometimes referred to as market microstructure. We then provide results and algorithms for both the OWT and VWAP problems. 192 3.1 Background on Limit Order Books and Market Microstructure A fundamental distinction in stock trading is that between a limit order and a market order. Suppose we wish to purchase 1000 shares of Microsoft (MSFT) stock. In a limit order, we specify not only the desired volume (1000 shares), but also the desired price. Suppose that MSFT is currently trading at roughly $24.07 a share (see Figure 2, which shows an actual snapshot of a recent MSFT order book on Island (www.island.com), a well-known electronic exchange for NASDAQ stocks), but we are only willing to buy the 1000 shares at $24.04 a share or lower. We can choose to submit a limit order with this specification, and our order will be placed in a queue called the buy order book, which is ordered by price, with the highest offered unexecuted buy price at the top (often referred to as the bid). If there are multiple limit orders at the same price, they are ordered by time of arrival (with older orders higher in the book). In the example provided by Figure 2, our order would be placed immediately after the extant order for 5,503 shares at $24.04; though we offer the same price, this order has arrived before ours. Similarly, a sell order book for sell limit orders (for instance, we might want to sell 500 shares of MSFT at $24.10 or higher) is maintained, this time with the lowest sell price offered (often referred to as the ask). Thus, the order books are sorted from the most competitive limit orders at the top (high buy prices and low sell prices) down to less competitive limit orders. The bid and ask prices (which again, are simply the prices in the limit orders at the top of the buy and sell books, respectively) together are sometimes referred to as the inside market, and the difference between them as the spread. By definition, the order books always consist exclusively of unexecuted orders - they are queues of orders hopefully waiting for the price to move in their direction. How then do orders get executed? There are two methods. First, any time a market order arrives, it is immediately matched with the most competitive limit orders on the opposing book. Thus, a market order to buy 2000 shares is matched with enough volume on the sell order book to fill the 2000 shares. For instance, in the example of Figure 2, such an order would be filled by the two limit sell orders for 500 shares at $24.069, the 500 shares at $24.07, the 200 shares at $24.08, and then 300 of the 1981 shares at $24.09. The remaining 1681 shares of this last limit order would remain as the new top of the sell limit order book. Second, if a buy (sell, respectively) limit order comes in above the ask (below the bid, respectively) price, then the order is matched with orders on the opposing books. It is important to note that the prices of executions are the prices specified in the limit orders already in the books, not the prices of the incoming order that is immediately executed. Every market or limit order arrives atomically and instantaneously - there is a strict temporal sequence in which orders arrive, and two orders can never arrive simultaneously. This gives rise to the definition of the last price of the exchange, which is simply the last price at which the exchange executed an order. It is this quantity that is usually meant when people casually refer to the (ticker) price of a stock. Note that a limit buy (sell, respectively) order with a price of infinity (0, respectively) is effectively a market order. We shall thus assume without loss of generality that all orders are placed as limit order. Although limit orders which are unexecuted may be removed by the party which placed them, for simplicity, we assume that limit orders are never removed from the books. We refer the reader to [4] for further discussion of modern electronic exchanges and market microstructure. 3.2 The Model The online order book trading model is intended to capture the realistic details of market microstructure just discussed in a competitive ratio setting. In this refined model, a day"s market activity is described by a sequence of limit orders (pt, vt, bt). Here bt is a bit indicating whether the order is a buy or sell order, while pt is the limit order price and vt the number of shares desired. Following the arrival of each such limit order, an online trading algorithm is permitted to place its own limit order. These two interleaved sources (market and algorithm) of limit orders are then simply operated on according to the matching process described in Section 3.1. Any limit order that is not immediately executable according to this process is placed in the appropriate (buy or sell) book for possible future execution; arriving orders that can be partially or fully executed are so executed, with any residual shares remaining on the respective book. The goal of a VWAP or OWT selling algorithm is essentially the same as in the price-volume model, but the context has changed in the following two fundamental ways. First, the assumption of infinite liquidity in the price-volume model is eliminated entirely. The number of shares available at any given price is restricted to the total volume of limit orders offering that price. Second, all incoming orders, and therefore the complete limit order books, are assumed to be visible to the algorithm. This is consistent with modern electronic financial exchanges, and indeed is the source of much current interest on Wall Street [4]. In general, the definition of competitive ratios in the order book model is complicated by the fact that now our algorithm"s activity influences the sequence of executed prices and volumes. We thus first define the execution sequence determined by a limit order sequence (placed by the market and our algorithm). Let S = (p1, v1, b1), . . . , (pT , vT , bT ) be a limit order sequence placed by the market, and let S = (p1, v1, b1), . . . , (pT , vT , bT ) be a limit order sequence placed by our algorithm (unless otherwise specified, all bt are of the sell type). Let merge(S, S ) be the merged sequence (p1, v1, b1), (p1, v1, b1), . . . , (pT , vT , bT ), (pT , vT , bT ), which is the time sequence of orders placed by the market and algorithm. Note that the algorithm has the option of not placing an order, which we can view as a zero volume order. If we conducted the order book maintenance and order execution process described in Section 3.1 on the sequence merge(S, S ), at irregular intervals a trade occurs for some number of shares and some price. In each executed trade, the selling party is either the market or the algorithm. Let execM (S, S ) = (q1, w1), . . . , (qT , wT ) be the sequence of executions where the market (that is, a party other than the algorithm) was the selling party, where the qt are the execution prices and wt the execution volumes. Similarly, we define execA(S, S ) = (r1, x1), . . . , (rT , xT ) to be the sequence of executions in which the algorithm was the selling party. Thus, execA(S, S ) ∪ execM (S, S ) is the set of all executions. We generally expect T to be (possibly much) smaller than T . The revenue of the algorithm and the market are defined 193 as: REVM (S, S ) ≡ T t=1 qtwt , REVA(S, S ) ≡ T t=1 rtxt Note that both these quantities are solely determined by the execution sequences execM (S, S ) and execA(S, S ), respectively. For an algorithm A which is constrained to sell exactly N shares, we define the OWT competitive ratio of A, ROWT(A), as the maximum ratio (under any S ∈ Σ) of the revenue obtained by A, as compared to the revenue obtained by an optimal offline algorithm A∗ . More formally, for A∗ which is constrained to sell exactly N shares, we define ROWT(A) = max S∈Σ max A∗ REVA∗ (S S∗ ) REVA(S, S ) where S∗ is the limit order sequence placed by A∗ on S. If the algorithm A is randomized then we take the appropriate expectation with respect to S ∼ A. We define the VWAP competitive ratio, RVWAP(A), as the maximum ratio (under any S ∈ Σ) between the market and algorithm VWAPs. More formally, define VWAPM (S, S ) as REVM (S, S )/ T t=1 wt, where the denominator is just the total executed volume of orders placed by the market. Similarly, we define VWAPA(S, S ) as REVA(S, S )/N, since we assume the algorithm sells no more than N shares (this definition implicitly assumes that A gets a 0 price for unsold shares). The VWAP competitive ratio of A is then: RVWAP(A) = max S∈Σ {VWAPM (S, S )/VWAPA(S, S )} where S is the online sequence of limit orders generated by A in response to the sequence S. 3.3 OWT Results in the Order Book Model For the OWT problem in the order book model, we introduce a more subtle version of the price variability assumption. This is due to the fact that our algorithm"s trading can impact the high and low prices of the day. For the assumption below, note that execM (S, ∅) is the sequence of executions without the interaction of our algorithm. 3.3.0.3 Order Book Price Variability Assumption.. Let 0 < pmin ≤ pmax be known positive constants, and define R = pmax/pmin. For all intraday trading sequences S ∈ Σ, the prices pt in the sequence execM (S, ∅) satisfy pt ∈ [pmin, pmax], for all t = 1, . . . , T. Note that this assumption does not imply that the ratios of high to low prices under the sequences execM (S, S ) or execA(S, S ) are bounded by R. In fact, the ratio in the sequence execA(S, S ) could be infinite if the algorithm ends up selling some stocks at a 0 price. Theorem 6. In the order book model under the order book price variability assumption, there exists an online algorithm A for selling N shares achieving sell OWT competitive ratio ROWT(A) = 2 log(R) log(N). Proof. The algorithm A works by guessing a price p in the set {pmin2i : 1 ≤ i ≤ log(R)} and placing a sell limit order for all N shares at the price p at the beginning of the day. (Alternatively, algorithm A can place log(R) sell limit orders, where the i-th one has price 2i pmin and volume N/ log(R).) By placing an order at the beginning of the day, the algorithm undercuts all sell orders that will be placed during the day for a price of p or higher, meaning the algorithm"s N shares must be filled first at this price. Hence, if there were k shares that would have been sold at price p or higher without our activity, then A would sell at least kp shares. We define {pj} to be the multiset of prices of individual shares that are either executed or are buy limit order shares that remained unexecuted, excluding the activity of our algorithm (that is, assuming our algorithm places no orders). Assume without loss of generality that p1 ≥ p2 ≥ . . .. Consider guessing the kth highest such price, pk. If an order for N shares is placed at the day"s start at price pk, then we are guaranteed to obtain a return of kpk. Let k∗ = argmaxk{kpk}. We can view our algorithm as attempting to guess pk∗ , and succeeding if the guess p satisfies p ∈ [pk∗ /2, pk∗ ]. Hence, we are 2 log(R) competitive with the quantity max1≤k≤N kpk. Note that ρ ≡ N i=1 pi = N i=1 1 i ipi ≤ max 1≤k≤N kpk N i=1 1 i ≤ log(N) max 1≤k≤N kpk where ρ is defined as the sum of the top N prices pi without A"s involvement. Similarly, let {pj} be the multiset of prices of individual executed shares, or the prices of unexecuted buy order shares, but now including the orders placed by some selling algorithm A . We now wish to show that for all algorithms A which sell N shares, REVA ≤ N i=1 pi ≤ ρ. Essentially, this inequality states the intuitive idea that a selling algorithm can only lower executed or unmatched buy order share prices. To prove this, we use induction to show that the removal of the activity of a selling algorithm causes these prices to increase. First, remove the last share in the last sell order placed by either A or the market on an arbitrary sequence merge(S, S ) - by this we mean, take the last sell order placed by A or the market and decrease its volume by one share. After this modification, the top N prices p1 . . . pN will not decrease. This is because either this sell order share was not executed, in which case the claim is trivially true, or, if it was executed, the removal of this sell order share leaves an additional unexecuted buy order share of equal or higher price. For induction, assume that if we remove a share from any sell order that was placed, by A or the market, at or after time t then the top N prices do not decrease. We now show that if we remove a share from the last sell order that was placed by A or the market before time t, then the top N prices do not decrease. If this sell order share was not executed, then the claim is trivially true. Else, if the sell order share was executed, then claim is true because by removing this executed share from the sell order either: i) the corresponding buy order share (of equal or higher value) is unmatched on the remainder of the sequence, in which case the claim is true; or ii) this buy 194 order matches some sell order share at an equal or higher price, which has the effect of removing a share from a sell order on the remainder of the sequence, and, by the inductive assumption, this can only increase prices. Hence, we have proven that for all A which sell N shares REVA ≤ ρ. We have now established that our revenue satisfies 2 log(R)ES ∼A[REVA(S, S )] ≥ max 1≤k≤N {kpk} ≥ ρ/ log(N) ≥ max A {REVA }/ log(N), where A performs an arbitrary sequence of N sell limit orders. 3.4 VWAP Results in the Order Book Model The OWT algorithm from Theorem 6 can be applied to obtain the following VWAP result: Corollary 7. In the order book model under the order book price variability assumption, there exists an online algorithm A for selling N shares achieving sell VWAP competitive ratio RVWAP(A) = O(log(R) log(N)). We now make a rather different assumption on the sequences S. 3.4.0.4 Bounded Order Volume and Max Price Assumption. The set of sequences Σ satisfies the following two properties. First, we assume that each order placed by the market is of volume less than γ, which we view as a mild assumption since typically single orders on the market are not of high volume (due to liquidity issues). This assumption allows our algorithm to place at least one limit order at a time interleaved with approximately γ market executions. Second, we assume that there is large volume in the sell order books below the price pmax , which means that no orders placed by the market will be executed above the price pmax . The simplest way to instantiate this latter assumption in the order book model is to assume that each sequence S ∈ Σ starts by placing a huge number of sell orders (more than Vmax ) at price pmax . Although this assumption has a maximum price parameter, it does not imply that the price ratio R is finite, since it does not imply any lower bound on the prices of buy or executed shares (aside from the trivial one of 0). Theorem 8. Consider the order book model under the bounded order volume and max price assumption. There exists an algorithm A in which after exactly γN market executions have occurred, then A has sold at most N shares and REVA(S, S ) N = VWAPA(S, S ) ≥ (1 − )VWAPM (S, S ) − pmax N where S is a sequence of N sell limit orders generated by A when observing S. Proof. The algorithm divides the trading day into volume intervals whose real-time duration may vary. For each period i in which γ shares have been executed in the market, the algorithm computes the market VWAP of only those shares traded in period i; let us denote this by VWAPi. Following this ith volume interval, the algorithm places a limit order to sell exactly one share at a price close to VWAPi. More precisely, the algorithm only places orders at the discrete prices (1− )pmax , (1− )2 pmax , . . .. Following volume interval i, the algorithm places a limit order to sell one share at the discretized price that is closest to VWAPi, but which is strictly smaller. For the analysis, we begin by noting that if all of the algorithm"s limit orders are executed during the day, the total revenue received by the algorithm would be at least (1 − )VWAPM (S, S )N. To see this, it suffices to note that VWAPM (S, S ) is a uniform mixture of the VWAPi (since by definition they each cover the same amount of market volume); and if all the algorithm"s limit orders were executed, they each received more than (1 − )VWAPi dollars for the interval i they followed. We now count the potential lost revenue of the algorithm due to unexecuted limit orders. By the assumption that individual orders are placed with volume less than γ, then our algorithm is able to place a limit order during every block of γ shares have been traded. Hence, after γN market orders have been executed, A has placed N orders in the market. Note that there can be at most one limit order (and thus, at most one share) left unexecuted at each level of the discretized price ladder defined above. This is because following interval i, the algorithm places its limit order strictly below VWAPi, so if VWAPj ≥ VWAPi for j > i, this limit order must have been executed. Thus unexecuted limit orders bound the VWAPs of the remainder of the day, resulting in at most one unexecuted order per price level. A bound on the lost revenue is thus the sum of the discretized prices: ∞ i=1(1 − )i pmax ≤ pmax / . Clearly our algorithm has sold at most N shares. Note that as N becomes large, VWAPA approaches 1 − times the market VWAP. If we knew that the final total volume of the market executions is V , then we can set γ = V/N, assuming that γ >> 1. If we have only an upper and lower bound on V we should be able to guess and incur a logarithmic loss. The following assumption tries to capture the market volume variability. 3.4.0.5 Order Book Volume Variability Assumption. We now assume that the total volume (which includes the shares executed by both our algorithm and the market) is variable within some known region and that the market volume will be greater than our algorithms volume. More formally, for all S ∈ Σ, assume that the total volume V of shares traded in execM (S, S ), for any sequence S of N sell limit orders, satisfies 2N ≤ Vmin ≤ V ≤ Vmax . Let Q = Vmax /Vmin . The following corollary is derived using a constant = 1/2 and observing that if we set γ such that V ≤ γN ≤ 2V then our algorithm will place between N and N/2 limit orders. Corollary 9. In the order book model, if the bounded order volume and max price assumption, and the order book volume variability assumption hold, there exists an online algorithm A for selling at most N shares such that VWAPA(S, S ) ≥ 1 4 log(Q) VWAPM (S, S ) − 2pmax N 195 0 2 4 6 8 x 10 7 0 20 40 60 80 100 QQQ: log(Q)=4.71, E=3.77 0 2 4 6 8 10 x 10 6 0 20 40 60 80 JNPR: log(Q)=5.66, E=3.97 0 0.5 1 1.5 2 x 10 6 0 10 20 30 40 50 60 70 MCHP: log(Q)=5.28, E=3.86 0 2 4 6 8 10 x 10 6 0 50 100 150 200 250 CHKP: log(Q)=6.56, E=4.50 Figure 3: Here we present bounds from Section 4 based on the empirical volume distributions for four real stocks: QQQ, MCHP, JNPR, and CHKP. The plots show histograms for the total daily volumes transacted on Island for these stocks, in the last year and a half, along with the corresponding values of log(Q) and E(Pbins vol ) (denoted by "E"). We assume that the minimum and maximum daily volumes in the data correspond to Vmin and Vmax , respectively. The worst-case competitive ratio bounds (which are twice log(Q)) of our algorithm for those stocks are 9.42, 10.56, 11.32, and 13.20, respectively. The corresponding bounds on the competitive ratio performance of our algorithm under the volume distribution model (which are twice E(Pbins vol )) are better: 7.54, 7.72, 7.94, and 9.00, respectively (a 20−40% relative improvement). Using a finer volume binning along with a slightly more refined bound on the competitive ratio, we can construct algorithms that, using the empirical volume distribution given as correct, guarantee even better competitive ratios of 2.76, 2.73, 2.75, and 3.17, respectively for those stocks (details omitted). 4. MACROSCOPIC DISTRIBUTION MODELS We conclude our results with a return to the price-volume model, where we shall introduce some refined methods of analysis for online trading algorithms. We leave the generalization of these methods to the order book model for future work. The competitive ratios defined so far measure performance relative to some baseline criterion in the worst case over all market sequences S ∈ Σ. It has been observed in many online settings that such worst-case metrics can yield pessimistic results, and various relaxations have been considered, such as permitting a probability distribution over the input sequence. We now consider distributional models that are considerably weaker than assuming a distribution over complete market sequences S ∈ Σ. In the volume distribution model, we assume only that there exists a distribution Pvol over the total volume V traded in the market for the day, and then examine the worst-case competitive ratio over sequences consistent with the randomly chosen volume. More precisely, we define RVWAP(A, Pvol ) = EV ∼Pvol max S∈seq(V ) VWAPM (S) VWAPA(S) . Here V ∼ Pvol denotes that V is chosen with respect to distribution Pvol , and seq(V ) ⊂ Σ is the set of all market sequences (p1, v1), . . . , (pT , vT ) satisfying T t=1 vt = V . Similarly, for OWT, we can define ROWT(A, Pmaxprice ) = Ep∼Pmaxprice max S∈seq(p) p VWAPA(S) . Here Pmaxprice is a distribution over just the maximum price of the day, and we then examine worst-case sequences consistent with this price (seq(p) ⊂ Σ is the set of all market sequences satisfying max1≤t≤T pt = p). Analogous buy-side definitions can be given. We emphasize that in these models, only the distribution of maximum volume and price is known to the algorithm. We also note that our probabilistic assumptions on S are considerably weaker than typical statistical finance models, which would posit a detailed stochastic model for the step-by-step evolution of (pt, vt). Here we instead permit only a distribution over crude, macroscopic measures of the entire day"s market activity, such as the total volume and high price, and analyze the worst-case performance consistent with these crude measures. For this reason, we refer to such settings as the macroscopic distribution model. The work of El-Yaniv et al. [3] examines distributional assumptions similar to ours, but they emphasize the worst196 case choices for the distributions as well, and show that this leads to results no better than the original worst-case analysis over all sequences. In contrast, we feel that the analysis of specific distributions Pvol and Pmaxprice is natural in many financial contexts and our preliminary experimental results show significant improvements when this rather crude distributional information is taken into account (see Figure 3). Our results in the VWAP setting examine the cases where these distributions are known exactly or only approximately. Similar results can be obtained for macroscopic distributions of maximum daily price for the one-way trading setting. 4.1 Results in the Macroscopic Distribution Model We begin by noting that the algorithms examined so far work by binning total volumes or maximum prices into bins of exponentially increasing size, and then guessing the index of the bin in which the actual quantity falls. It is thus natural that the macroscopic distribution model performance of such algorithms (which are common in competitive analysis) might depend on the distribution of the true bin index. In the remaining, we assume that Q is a power of 2 and the base of the logarithm is 2. Let Pvol denote the distribution of total daily market volume. We define the related distribution Pbins vol over bin indices i as follows: for all i = 1, . . . , log(Q) − 1, Pbins vol (i) is equal to the probability, under Pvol , that the daily volume falls in the interval [Vmin 2i−1 , Vmin 2i ), and Pbins vol (log(Q)) is for the last interval [Vmax /2, Vmax ] . We define E as E(Pbins vol ) ≡ Ei∼P bins vol 1/Pbins vol (i) 2 =   log(Q) i=1 Pbins vol (i)   2 . Since the support of Pbins vol has only log(Q) elements, E(Pbins vol ) can vary from 1 (for distributions Pvol that place all of their weight in only one of the log(Q) intervals between Vmin , Vmin 2, Vmin 4, . . . , Vmax ) to log(Q) (for distributions Pvol in which the total daily volume is equally likely to fall in any one of these intervals). Note that distributions Pvol of this latter type are far from uniform over the entire range [Vmin , Vmax ]. Theorem 10. In the volume distribution model under the volume variability assumption, there exists an online algorithm A for selling N shares that, using only knowledge of the total volume distribution Pvol , achieves RVWAP(A, Pvol ) ≤ 2E(Pbins vol ). All proofs in this section are provided in the appendix. As a concrete example, consider the case in which Pvol is the uniform distribution over [Vmin , Vmax ]. In that case, Pbins vol is exponentially increasing and peaks at the last bin, which, having the largest width, also has the largest weight. In this case E(Pbins vol ) is a constant (i.e., independent of Q), leading to a constant competitive ratio. On the other hand, if Pvol is exponential, then Pbins vol is uniform, leading to an O(log(Q)) competitive ratio, just as in the more adversarial price-volume setting discussed earlier. In Figure 3, we provide additional specific bounds obtained for empirical total daily volume distributions computed for some real stocks. We now examine the setting in which Pvol is unknown, but an approximation ˜Pvol is available. Let us define C(Pbins vol , ˜Pbins vol ) = log(Q) j=1 ˜Pbins vol (j) log(Q) i=1 P bins vol (i) √ ˜P bins vol (i) . C is minimized at C(Pbins vol , Pbins vol ) = E(Pbins vol ), and C may be infinite if ˜Pbins vol (i) is 0 when Pbins vol (i) > 0. Theorem 11. In the volume distribution model under the volume variability assumption, there exists an online algorithm A for selling N shares that using only knowledge of an approximation ˜Pvol of Pvol achieves RVWAP(A, Pvol ) ≤ 2C(Pbins vol , ˜Pbins vol ). As an example of this result, suppose our approximation obeys (1/α)Pbins vol (i) ≤ ˜Pbins vol (i) ≤ αPbins vol (i) for all i, for some α > 1. Thus our estimated bin index probabilities are all within a factor of α of the truth. Then it is easy to show that C(Pbins vol , ˜Pbins vol ) ≤ αE(Pbins vol ), so according to Theorems 10 and 11 our penalty for using the approximate distribution is a factor of α in competitive ratio. 5. REFERENCES [1] B. Awerbuch, Y. Bartal, A. Fiat, and A. Ros´en. Competitive non-preemptive call control. In Proc. 5"th ACM-SIAM Symp. on Discrete Algorithms, pages 312-320, 1994. [2] A. Borodin and R. El-Yaniv. Online Computation and Competitive Analysis. Cambridge University Press, 1998. [3] R. El-Yaniv, A. Fiat, R. M. Karp, and G. Turpin. Optimal search and one-way trading online algorithms. Algorithmica, 30:101-139, 2001. [4] M. Kearns and L. Ortiz. The Penn-Lehman automated trading project. IEEE Intelligent Systems, 2003. To appear. 6. APPENDIX 6.1 Proofs from Subsection 2.3 Proof. (Sketch of Theorem 3) W.l.o.g., assume that Q = 1 and the total volume is V . Consider the time t where the fixed schedule f sells the least, then ft ≤ N/T. Consider the sequences where at time t we have pt = pmax , vt = V , and for times t = t we have pt = pmin and vt = 0. The VWAP is pmax and the fixed schedule average is (N/T)pmax + (N − N/T)pmin . Proof. (Sketch of Theorem 4) The algorithm simply sells ut = (vt/Vmin )N shares at time t. The total number of shares sold U is clearly more than N and U = t ut = t (vt/Vmin )N = (V/Vmin )N ≤ QN The average price is V WAPA(S) = ( t ptut)/U = t pt(vt/V ) = V WAPM (S), where we used the fact that ut/U = vt/V . 197 Proof. (of Theorem 5) We start with the proof of the lower bound. Consider the following scenario. For the first T time units we have a price of √ Rpmin , and a total volume of Vmin . We observe how many shares the online algorithm has bought. If it has bought more than half of the shares, the remaining time steps have price pmin and volume Vmax − Vmin . Otherwise, the remaining time steps have price pmax and negligible volume. In the first case the online has paid at least √ Rpmin /2 while the VWAP is at most √ Rpmin /Q + pmin . Therefore, in this case the competitive ratio is Ω(Q). In the second case the online has to buy at least half of the shares at pmax , so its average cost is at least pmax /2. The market VWAP is√ Rpmin = pmax / √ R, hence the competitive ratio is Ω( √ R). For the upper bound, we can get a √ R competitive ratio, by buying all the shares once the price drops below √ Rpmin . The Q upper bound is derive by running an algorithm that assumes the volume is Vmin . The online pays a cost of p, while the VWAP will be at least p/Q. 6.2 Proofs from Section 4 Proof. (Sketch of Theorem 10) We use the idea of guessing the total volume from Theorem 1, but now allow for the possibility of an arbitrary (but known) distribution over the total volume. In particular, consider constructing a distribution Gbins vol over a set of volume values using Pvol and use it to guess the total volume V . Let the algorithm guess ˆV = Vmin 2i with probability Gbins vol (i). Then note that, for any price-volume sequence S, if V ∈ [Vmin 2i−1 , Vmin 2i ], VWAPA(S) ≥ Gbins vol (i)VWAPM (S)/2. This implies an upper bound on RVWAP(A, Pvol ) in terms of Gbins vol . We then get that Gbins vol (i) ∝ Pbins vol (i) minimizes the upper bound, which leads to the upper bound stated in the theorem. Proof. (Sketch of Theorem 11) Replace Pvol with ˜Pvol in the expression for Gbins vol in the proof sketch for the last result. 198

share;modern financial market;online algorithm;online trade;sequence of trade;trade sequence;competitive analysis;market order;volume weighted average price trading model;vwap;competitive algorithm;stock trading;online model;limit order book trading model

train_J-74

On Cheating in Sealed-Bid Auctions

Motivated by the rise of online auctions and their relative lack of security, this paper analyzes two forms of cheating in sealed-bid auctions. The first type of cheating we consider occurs when the seller spies on the bids of a second-price auction and then inserts a fake bid in order to increase the payment of the winning bidder. In the second type, a bidder cheats in a first-price auction by examining the competing bids before deciding on his own bid. In both cases, we derive equilibrium strategies when bidders are aware of the possibility of cheating. These results provide insights into sealedbid auctions even in the absence of cheating, including some counterintuitive results on the effects of overbidding in a first-price auction.

1. INTRODUCTION Among the types of auctions commonly used in practice, sealed-bid auctions are a good practical choice because they require little communication and can be completed almost instantly. Each bidder simply submits a bid, and the winner is immediately determined. However, sealed-bid auctions do require that the bids be kept private until the auction clears. The increasing popularity of online auctions only makes this disadvantage more troublesome. At an auction house, with all participants present, it is difficult to examine a bid that another bidder gave directly to the auctioneer. However, in an online auction the auctioneer is often little more than a server with questionable security; and, since all participants are in different locations, one can anonymously attempt to break into the server. In this paper, we present a game theoretic analysis of how bidders should behave when they are aware of the possibility of cheating that is based on knowledge of the bids. We investigate this type of cheating along two dimensions: whether it is the auctioneer or a bidder who cheats, and which variant (either first or second-price) of the sealed-bid auction is used. Note that two of these cases are trivial. In our setting, there is no incentive for the seller to submit a shill bid in a first price auction, because doing so would either cancel the auction or not affect the payment of the winning bidder. In a second-price auction, knowing the competing bids does not help a bidder because it is dominant strategy to bid truthfully. This leaves us with two cases that we examine in detail. A seller can profitably cheat in a second-price auction by looking at the bids before the auction clears and submitting an extra bid. This possibility was pointed out as early as the seminal paper [12] that introduced this type of auction. For example, if the bidders in an eBay auction each use a proxy bidder (essentially creating a second-price auction), then the seller may be able to break into eBay"s server, observe the maximum price that a bidder is willing to pay, and then extract this price by submitting a shill bid just below it using a false identity. We assume that there is no chance that the seller will be caught when it cheats. However, not all sellers are willing to use this power (or, not all sellers can successfully cheat). We assume that each bidder knows the probability with which the seller will cheat. Possible motivation for this knowledge could be a recently published expos´e on seller cheating in eBay auctions. In this setting, we derive an equilibrium bidding strategy for the case in which each bidder"s value for the good is independently drawn from a common distribution (with no further assumptions except for continuity and differentiability). This result shows how first and second-price auctions can be viewed as the endpoints of a spectrum of auctions. But why should the seller have all the fun? In a first-price auction, a bidder must bid below his value for the good (also called shaving his bid) in order to have positive utility if he 76 wins. To decide how much to shave his bid, he must trade off the probability of winning the auction against how much he will pay if he does win. Of course, if he could simply examine the other bids before submitting his own, then his problem is solved: bid the minimum necessary to win the auction. In this setting, our goal is to derive an equilibrium bidding strategy for a non-cheating bidder who is aware of the possibility that he is competing against cheating bidders. When bidder values are drawn from the commonly-analyzed uniform distribution, we show the counterintuitive result that the possibility of other bidders cheating has no effect on the equilibrium strategy of an honest bidder. This result is then extended to show the robustness of the equilibrium of a first-price auction without the possibility of cheating. We conclude this section by exploring other distributions, including some in which the presence of cheating bidders actually induces an honest bidder to lower its bid. The rest of the paper is structured as follows. In Section 2 we formalize the setting and present our results for the case of a seller cheating in a second price auction. Section 3 covers the case of bidders cheating in a first-price auction. In Section 4, we quantify the effects that the possibility of cheating has on an honest seller in the two settings. We discuss related work, including other forms of cheating in auctions, in Section 5, before concluding with Section 6. All proofs and derivations are found in the appendix. 2. SECOND-PRICE AUCTION, CHEATING SELLER In this section, we consider a second-price auction in which the seller may cheat by inserting a shill bid after observing all of the bids. The formulation for this section will be largely reused in the following section on bidders cheating in a first-price auction. While no prior knowledge of game theory or auction theory is assumed, good introductions can be found in [2] and [6], respectively. 2.1 Formulation The setting consists of N bidders, or agents, (indexed by i = 1, · · · , n) and a seller. Each agent has a type θi ∈ [0, 1], drawn from a continuous range, which represents the agent"s value for the good being auctioned.2 Each agent"s type is independently drawn from a cumulative distribution function (cdf ) F over [0, 1], where F(0) = 0 and F(1) = 1. We assume that F(·) is strictly increasing and differentiable over the interval [0, 1]. Call the probability density function (pdf ) f(θi) = F (θi), which is the derivative of the cdf. Each agent knows its own type θi, but only the distribution over the possible types of the other agents. A bidding strategy for an agent bi : [0, 1] → [0, 1] maps its type to its bid.3 Let θ = (θ1, · · · , θn) be the vector of types for all agents, and θ−i = (θ1, · · · , θi−1, θi+1, · · · θn) be the vector of all types except for that of agent i. We can then combine the vectors so that θ = (θi, θ−i). We also define the vector of bids as b(θ) = (b1(θ1), . . . , bn(θn)), and this vector without 2 We can restrict the types to the range [0, 1] without loss of generality because any distribution over a different range can be normalized to this range. 3 We thus limit agents to deterministic bidding strategies, but, because of our continuity assumption, there always exists a pure strategy equilibrium. the bid of agent i as b−i(θ−i). Let b[1](θ) be the value of the highest bid of the vector b(θ), with a corresponding definition for b[1](θ−i). An agent obviously wins the auction if its bid is greater than all other bids, but ties complicate the formulation. Fortunately, we can ignore the case of ties in this paper because our continuity assumption will make them a zero probability event in equilibrium. We assume that the seller does not set a reserve price.4 If the seller does not cheat, then the winning agent pays the highest bid by another agent. On the other hand, if the seller does cheat, then the winning agent will pay its bid, since we assume that a cheating seller would take full advantage of its power. Let the indicator variable µc be 1 if the seller cheats, and 0 otherwise. The probability that the seller cheats, Pc , is known by all agents.5 We can then write the payment of the winning agent as follows. pi(b(θ), µc ) = µc · bi(θi) − (1 − µc ) · b[1](θ−i) (1) Let µ(·) be an indicator function that takes an inequality as an argument and returns is 1 if it holds, and 0 otherwise. The utility for agent i is zero if it does not win the auction, and the difference between its valuation and its price if it does. ui(b(θ), µc , θi) = µ bi(θi) > b[1](θ−i) · θi − pi(b(θ), µc ) (2) We will be concerned with the expected utility of an agent, with the expectation taken over the types of the other agents and over whether or not the seller cheats. By pushing the expectation inward so that it is only over the price (conditioned on the agent winning the auction), we can write the expected utility as: Eθ−i,µc [ui(b(θ), µc , θi)] = Prob bi(θi) > b[1](θ−i) · θi − Eθ−i,µc pi(b(θ), µc ) | bi(θi) > b[1](θ−i) (3) We assume that all agents are rational, expected utility maximizers. Because of the uncertainty over the types of the other agents, we will be looking for a Bayes-Nash equilibrium. A vector of bidding strategies b∗ is a Bayes-Nash equilibrium if for each agent i and each possible type θi, agent i cannot increase its expected utility by using an alternate bidding strategy bi, holding the bidding strategies for all other agents fixed. Formally, b∗ is a Bayes-Nash equilibrium if: ∀i, θi, bi Eθ−i,µc ui b∗ i (θi), b∗ −i(θ−i) , µc , θi ≥ Eθ−i,µc ui bi(θi), b∗ −i(θ−i) , µc , θi (4) 2.2 Equilibrium We first present the Bayes-Nash equilibrium for an arbitrary distribution F(·). 4 This simplifies the analysis, but all of our results can be applied to the case in which the seller announces a reserve price before the auction begins. 5 Note that common knowledge is not necessary for the existence of an equilibrium. 77 Theorem 1. In a second-price auction in which the seller cheats with probability Pc , it is a Bayes-Nash equilibrium for each agent to bid according to the following strategy: bi(θi) = θi − θi 0 F( N−1 P c ) (x)dx F( N−1 P c ) (θi) (5) It is useful to consider the extreme points of Pc . Setting Pc = 1 yields the correct result for a first-price auction (see, e.g., [10]). In the case of Pc = 0, this solution is not defined. However, in the limit, bi(θi) approaches θi as Pc approaches 0, which is what we expect as the auction approaches a standard second-price auction. The position of Pc is perhaps surprising. For example, the linear combination bi(θi) = θi − Pc · θi 0 F (N−1) (x)dx F (N−1)(θi) of the equilibrium bidding strategies of first and second-price auctions would have also given us the correct bidding strategies for the cases of Pc = 0 and Pc = 1. 2.3 Continuum of Auctions An alternative perspective on the setting is as a continuum between first and second-price auctions. Consider a probabilistic sealed-bid auction in which the seller is honest, but the price paid by the winning agent is determined by a weighted coin flip: with probability Pc it is his bid, and with probability 1 − Pc it is the second-highest bid. By adjusting Pc , we can smoothly move between a first and second-price auction. Furthermore, the fact that this probabilistic auction satisfies the properties required for the Revenue Equivalence Theorem (see, e.g., [2]) provides a way to verify that the bidding strategy in Equation 5 is the symmetric equilibrium of this auction (see the alternative proof of Theorem 1 in the appendix). 2.4 Special Case: Uniform Distribution Another way to try to gain insight into Equation 5 is by instantiating the distribution of types. We now consider the often-studied uniform distribution: F(θi) = θi. Corollary 2. In a second-price auction in which the seller cheats with probability Pc , and F(θi) = θi, it is a Bayes-Nash equilibrium for each agent to bid according to the following strategy: bi(θi) = N − 1 N − 1 + Pc θi (6) This equilibrium bidding strategy, parameterized by Pc , can be viewed as an interpolation between two well-known results. When Pc = 0 the bidding strategy is now welldefined (each agent bids its true type), while when Pc = 1 we get the correct result for a first-price auction: each agent bids according to the strategy bi(θi) = N−1 N θi. 3. FIRST-PRICE AUCTION, CHEATING AGENTS We now consider the case in which the seller is honest, but there is a chance that agents will cheat and examine the other bids before submitting their own (or, alternatively, they will revise their bid before the auction clears). Since this type of cheating is pointless in a second-price auction, we only analyze the case of a first-price auction. After revising the formulation from the previous section, we present a fixed point equation for the equilibrium strategy for an arbitrary distribution F(·). This equation will be useful for the analysis the uniform distribution, in which we show that the possibility of cheating agents does not change the equilibrium strategy of honest agents. This result has implications for the robustness of the symmetric equilibrium to overbidding in a standard first-price auction. Furthermore, we find that for other distributions overbidding actually induces a competing agent to shave more off of its bid. 3.1 Formulation It is clear that if a single agent is cheating, he will bid (up to his valuation) the minimum amount necessary to win the auction. It is less obvious, though, what will happen if multiple agents cheat. One could imagine a scenario similar to an English auction, in which all cheating agents keep revising their bids until all but one cheater wants the good at the current winning bid. However, we are only concerned with how an honest agent should bid given that it is aware of the possibility of cheating. Thus, it suffices for an honest agent to know that it will win the auction if and only if its bid exceeds every other honest agent"s bid and every cheating agent"s type. This intuition can be formalized as the following discriminatory auction. In the first stage, each agent"s payment rule is determined. With probability Pa , the agent will pay the second highest bid if it wins the auction (essentially, he is a cheater), and otherwise it will have to pay its bid. These selections are recorded by a vector of indicator variables µa = (µa1 , . . . , µan ), where µai = 1 denotes that agent i pays the second highest bid. Each agent knows the probability Pa , but does not know the payment rule for all other agents. Otherwise, this auction is a standard, sealed-bid auction. It is thus a dominant strategy for a cheater to bid its true type, making this formulation strategically equivalent to the setting outlined in the previous paragraph. The expression for the utility of an honest agent in this discriminatory auction is as follows. ui(b(θ), µa , θi) = θi − bi(θ) · j=i µaj · µ bi(θi) > θj + (1 − µaj ) · µ bi(θi) > bj(θj) (7) 3.2 Equilibrium Our goal is to find the equilibrium in which all cheating agents use their dominant strategy of bidding truthfully and honest agents bid according to a symmetric bidding strategy. Since we have left F(·) unspecified, we cannot present a closed form solution for the honest agent"s bidding strategy, and instead give a fixed point equation for it. Theorem 3. In a first-price auction in which each agent cheats with probability Pa , it is a Bayes-Nash equilibrium for each non-cheating agent i to bid according to the strategy that is a fixed point of the following equation: bi(θi) = θi − θi 0 Pa · F(bi(x)) + (1 − Pa) · F(x) (N−1) dx Pa · F(bi(θi)) + (1 − Pa) · F(θi) (N−1) (8) 78 3.3 Special Case: Uniform Distribution Since we could not solve Equation 8 in the general case, we can only see how the possibility of cheating affects the equilibrium bidding strategy for particular instances of F(·). A natural place to start is uniform distribution: F(θi) = θi. Recall the logic behind the symmetric equilibrium strategy in a first-price auction without cheating: bi(θi) = N−1 N θi is the optimal tradeoff between increasing the probability of winning and decreasing the price paid upon winning, given that the other agents are bidding according to the same strategy. Since in the current setting the cheating agents do not shave their bid at all and thus decrease an honest agent"s probability of winning (while obviously not affecting the price that an honest agent pays if he wins), it is natural to expect that an honest agent should compensate by increasing his bid. The idea is that sacrificing some potential profit in order to regain some of the lost probability of winning would bring the two sides of the tradeoff back into balance. However, it turns out that the equilibrium bidding strategy is unchanged. Corollary 4. In a first-price auction in which each agent cheats with probability Pa , and F(θi) = θi, it is a BayesNash equilibrium for each non-cheating agent to bid according to the strategy bi(θi) = N−1 N θi. This result suggests that the equilibrium of a first-price auction is particularly robust when types are drawn from the uniform distribution, since the best response is unaffected by deviations of the other agents to the strategy of always bidding their type. In fact, as long as all other agents shave their bid by a fraction (which can differ across the agents) no greater than 1 N , it is still a best response for the remaining agent to bid according to the equilibrium strategy. Note that this result holds even if other agents are shaving their bid by a negative fraction, and are thus irrationally bidding above their type. Theorem 5. In a first-price auction where F(θi) = θi, if each agent j = i bids according a strategy bj(θj) = N−1+αj N θj, where αj ≥ 0, then it is a best response for the remaining agent i to bid according to the strategy bi(θi) = N−1 N θi. Obviously, these strategy profiles are not equilibria (unless each αj = 0), because each agent j has an incentive to set αj = 0. The point of this theorem is that a wide range of possible beliefs that an agent can hold about the strategies of the other agents will all lead him to play the equilibrium strategy. This is important because a common (and valid) criticism of equilibrium concepts such as Nash and BayesNash is that they are silent on how the agents converge on a strategy profile from which no one wants to deviate. However, if the equilibrium strategy is a best response to a large set of strategy profiles that are out of equilibrium, then it seems much more plausible that the agents will indeed converge on this equilibrium. It is important to note, though, that while this equilibrium is robust against arbitrary deviations to strategies that shave less, it is not robust to even a single agent shaving more off of its bid. In fact, if we take any strategy profile consistent with the conditions of Theorem 5 and change a single agent j"s strategy so that its corresponding αj is negative, then agent i"s best response is to shave more than 1 N off of its bid. 3.4 Effects of Overbidding for Other Distributions A natural question is whether the best response bidding strategy is similarly robust to overbidding by competing agents for other distributions. It turns out that Theorem 5 holds for all distributions of the form F(θi) = (θi)k , where k is some positive integer. However, taking a simple linear combination of two such distributions to produce F(θi) = θ2 i +θi 2 yields a distribution in which an agent should actually shave its bid more when other agents shave their bids less. In the example we present for this distribution (with the details in the appendix), there are only two players and the deviation by one agent is to bid his type. However, it can be generalized to a higher number of agents and to other deviations. Example 1. In a first-price auction where F(θi) = θ2 i +θi 2 and N = 2, if agent 2 always bids its type (b2(θ2) = θ2), then, for all θ1 > 0, agent 1"s best response bidding strategy is strictly less than the bidding strategy of the symmetric equilibrium. We also note that the same result holds for the normalized exponential distribution (F(θi) = eθi −1 e−1 ). It is certainly the case that distributions can be found that support the intuition given above that agents should shave their bid less when other agents are doing likewise. Examples include F(θi) = −1 2 θ2 i + 3 2 θi (the solution to the system of equations: F (θi) = −1, F(0) = 0, and F(1) = 1), and F(θi) = e−e(1−θi) e−1 . It would be useful to relate the direction of the change in the best response bidding strategy to a general condition on F(·). Unfortunately, we were not able to find such a condition, in part because the integral in the symmetric bidding strategy of a first-price auction cannot be solved without knowing F(·) (or at least some restrictions on it). We do note, however, that the sign of the second derivative of F(θi)/f(θi) is an accurate predictor for all of the distributions that we considered. 4. REVENUE LOSS FOR AN HONEST SELLER In both of the settings we covered, an honest seller suffers a loss in expected revenue due to the possibility of cheating. The equilibrium bidding strategies that we derived allow us to quantify this loss. Although this is as far as we will take the analysis, it could be applied to more general settings, in which the seller could, for example, choose the market in which he sells his good or pay a trusted third party to oversee the auction. In a second-price auction in which the seller may cheat, an honest seller suffers due the fact that the agents will shave their bids. For the case in which agent types are drawn from the uniform distribution, every agent will shave its bid by P c N−1+P c , which is thus also the fraction by which an honest seller"s revenue decreases due to the possibility of cheating. Analysis of the case of a first-price auction in which agents may cheat is not so straightforward. If Pa = 1 (each agent cheats with certainty), then we simply have a second-price auction, and the seller"s expected revenue will be unchanged. Again considering the uniform distribution for agent types, it is not surprising that Pa = 1 2 causes the seller to lose 79 the most revenue. However, even in this worst case, the percentage of expected revenue lost is significantly less than it is for the second-price auction in which Pc = 1 2 , as shown in Table 1.6 It turns out that setting Pc = 0.2 would make the expected loss of these two settings comparable. While this comparison between the settings is unlikely to be useful for a seller, it is interesting to note that agent suspicions of possible cheating by the seller are in some sense worse than agents actually cheating themselves. Percentage of Revenue lost for an Honest Seller Agents Second-Price Auction First-Price Auction (Pc = 0.5) (Pa = 0.5) 2 33 12 5 11 4.0 10 5.3 1.8 15 4.0 1.5 25 2.2 0.83 50 1.1 0.38 100 0.50 0.17 Table 1: The percentage of expected revenue lost by an honest seller due to the possibility of cheating in the two settings considered in this paper. Agent valuations are drawn from the uniform distribution. 5. RELATED WORK Existing work covers another dimension along which we could analyze cheating: altering the perceived value of N. In this paper, we have assumed that N is known by all of the bidders. However, in an online setting this assumption is rather tenuous. For example, a bidder"s only source of information about N could be a counter that the seller places on the auction webpage, or a statement by the seller about the number of potential bidders who have indicated that they will participate. In these cases, the seller could arbitrarily manipulate the perceived N. In a first-price auction, the seller obviously has an incentive to increase the perceived value of N in order to induce agents to bid closer to their true valuation. However, if agents are aware that the seller has this power, then any communication about N to the agents is cheap talk, and furthermore is not credible. Thus, in equilibrium the agents would ignore the declared value of N, and bid according to their own prior beliefs about the number of agents. If we make the natural assumption of a common prior, then the setting reduces to the one tackled by [5], which derived the equilibrium bidding strategies of a first-price auction when the number of bidders is drawn from a known distribution but not revealed to any of the bidders. Of course, instead of assuming that the seller can always exploit this power, we could assume that it can only do so with some probability that is known by the agents. The analysis would then proceed in a similar manner as that of our cheating seller model. The other interesting case of this form of cheating is by bidders in a first-price auction. Bidders would obviously want to decrease the perceived number of agents in order to induce their competition to lower their bids. While it is 6 Note that we have not considered the costs of the seller. Thus, the expected loss in profit could be much greater than the numbers that appear here. unreasonable for bidders to be able to alter the perceived N arbitrarily, collusion provides an opportunity to decrease the perceived N by having only one of a group of colluding agents participate in the auction. While the non-colluding agents would account for this possibility, as long as they are not certain of the collusion they will still be induced to shave more off of their bids than they would if the collusion did not take place. This issue is tackled in [7]. Other types of collusion are of course related to the general topic of cheating in auctions. Results on collusion in first and second-price auctions can be found in [8] and [3], respectively. The work most closely related to our first setting is [11], which also presents a model in which the seller may cheat in a second-price auction. In their setting, the seller is a participant in the Bayesian game who decides between running a first-price auction (where profitable cheating is never possible) or second-price auction. The seller makes this choice after observing his type, which is his probability of having the opportunity and willingness to cheat in a second-price auction. The bidders, who know the distribution from which the seller"s type is drawn, then place their bid. It is shown that, in equilibrium, only a seller with the maximum probability of cheating would ever choose to run a second-price auction. Our work differs in that we focus on the agents" strategies in a second-price auction for a given probability of cheating by the seller. An explicit derivation of the equilibrium strategies then allows us relate first and second-price auctions. An area of related work that can be seen as complementary to ours is that of secure auctions, which takes the point of view of an auction designer. The goals often extend well beyond simply preventing cheating, including properties such as anonymity of the bidders and nonrepudiation of bids. Cryptographic methods are the standard weapon of choice here (see [1, 4, 9]). 6. CONCLUSION In this paper we presented the equilibria of sealed-bid auctions in which cheating is possible. In addition to providing strategy profiles that are stable against deviations, these results give us with insights into both first and second-price auctions. The results for the case of a cheating seller in a second-price auction allow us to relate the two auctions as endpoints along a continuum. The case of agents cheating in a first-price auction showed the robustness of the first-price auction equilibrium when agent types are drawn from the uniform distribution. We also explored the effect of overbidding on the best response bidding strategy for other distributions, and showed that even for relatively simple distributions it can be positive, negative, or neutral. Finally, results from both of our settings allowed us to quantify the expected loss in revenue for a seller due to the possibility of cheating. 7. REFERENCES [1] M. Franklin and M. Reiter. The Design and Implementation of a Secure Auction Service. In Proc. IEEE Symp. on Security and Privacy, 1995. [2] D. Fudenberg and J. Tirole. Game Theory. MIT Press, 1991. 80 [3] D. Graham and R. Marshall. Collusive bidder behavior at single-object second-price and english auctions. Journal of Political Economy, 95:579-599, 1987. [4] M. Harkavy, J. D. Tygar, and H. Kikuchi. Electronic auctions with private bids. In Proceedings of the 3rd USENIX Workshop on Electronic Commerce, 1998. [5] R. Harstad, J. Kagel, and D. Levin. Equilibrium bid functions for auctions with an uncertain number of bidders. Economic Letters, 33:35-40, 1990. [6] P. Klemperer. Auction theory: A guide to the literature. Journal of Economic Surveys, 13(3):227-286, 1999. [7] K. Leyton-Brown, Y. Shoham, and M. Tennenholtz. Bidding clubs in first-price auctions. In AAAI-02. [8] R. McAfee and J. McMillan. Bidding rings. The American Economic Review, 71:579-599, 1992. [9] M. Naor, B. Pinkas, and R. Sumner. Privacy preserving auctions and mechanism design. In EC-99. [10] J. Riley and W. Samuelson. Optimal auctions. American Economic Review, 71(3):381-392, 1981. [11] M. Rothkopf and R. Harstad. Two models of bid-taker cheating in vickrey auctions. The Journal of Business, 68(2):257-267, 1995. [12] W. Vickrey. Counterspeculations, auctions, and competitive sealed tenders. Journal of Finance, 16:15-27, 1961. APPENDIX Theorem 1. In a second-price auction in which the seller cheats with probability Pc , it is a Bayes-Nash equilibrium for each agent to bid according to the following strategy: bi(θi) = θi − θi 0 F( N−1 P c ) (x)dx F( N−1 P c ) (θi) (5) Proof. To find an equilibrium, we start by guessing that there exists an equilibrium in which all agents bid according to the same function bi(θi), because the game is symmetric. Further, we guess that bi(θi) is strictly increasing and differentiable over the range [0, 1]. We can also assume that bi(0) = 0, because negative bids are not allowed and a positive bid is not rational when the agent"s valuation is 0. Note that these are not assumptions on the setting- they are merely limitations that we impose on our search. Let Φi : [0, bi(1)] → [0, 1] be the inverse function of bi(θi). That is, it takes a bid for agent i as input and returns the type θi that induced this bid. Recall Equation 3: Eθ−i,µc ui(b(θ), µc , θi) = Prob bi(θi) > b[1](θ−i) · θi − Eθ−i,µc pi(b(θ), µc ) | bi(θi) > b[1](θ−i) The probability that a single other bid is below that of agent i is equal to the cdf at the type that would induce a bid equal to that of agent i, which is formally written as F(Φi(bi(θi))). Since all agents are independent, the probability that all other bids are below agent i"s is simply this term raised the (N − 1)-th power. Thus, we can re-write the expected utility as: Eθ−i,µc ui(b(θ), µc , θi) = FN−1 (Φi(bi(θi)))· θi − Eθ−i,µc pi(b(θ), µc ) | bi(θi) > b[1](θ−i) (9) We now solve for the expected payment. Plugging Equation 1 (which gives the price for the winning agent) into the term for the expected price in Equation 9, and then simplifying the expectation yields: Eθ−i,µc pi(b(θ), µc ) | bi(θi) > b[1](θ−i) = Eθ−i,µc µc · bi(θi) + (1 − µc ) · b[1](θ−i) | bi(θi) > b[1](θ−i) = Pc · bi(θi) + (1 − Pc ) · Eθ−i b[1](θ−i) | bi(θi) > b[1](θ−i) = Pc · bi(θi) + (1 − Pc ) · bi(θi) 0 b[1](θ−i)· pdf b[1](θ−i) | bi(θi) > b[1](θ−i) db[1](θ−i) (10) Note that the integral on the last line is taken up to bi(θi) because we are conditioning on the fact that bi(θi) > b[1](θ−i). To derive the pdf of b[1](θ−i) given this condition, we start with the cdf. For a given value b[1](θ−i), the probability that any one agent"s bid is less than this value is equal to F(Φi(b[1](θ−i))). We then condition on the agent"s bid being below bi(θi) by dividing by F(Φi(bi(θi))). The cdf for the N − 1 agents is then this value raised to the (N − 1)-th power. cdf b[1](θ−i) | bi(θi) > b[1](θ−i) = FN−1 (Φi(b[1](θ−i))) FN−1(Φi(bi(θi))) The pdf is then the derivative of the cdf with respect to b[1](θ−i): pdf b[1](θ−i) | bi(θi) > b[1](θ−i) = N − 1 FN−1(Φi(bi(θi))) · FN−2 (Φi(b[1](θ−i)))· f(Φi(b[1](θ−i))) · Φi(b[1](θ−i)) Substituting the pdf into Equation 10 and pulling terms out of the integral that do not depend on b[1](θ−i) yields: Eθ−i,µc pi(b(θ), µc ) | bi(θi) > b[1](θ−i) = Pc · bi(θi)+ (1 − Pc ) · (N − 1) FN−1(Φi(bi(θi))) · bi(θi) 0 b[1](θ−i) · FN−2 (Φi(b[1](θ−i)))· f(Φi(b[1](θ−i))) · Φi(b[1](θ−i)) db[1](θ−i) 81 Plugging the expected price back into the expected utility equation (9), and distributing FN−1 (Φi(bi(θi))), yields: Eθ−i,µc ui(b(θ), µc , θi) = FN−1 (Φi(bi(θi))) · θi− FN−1 (Φi(bi(θi))) · Pc · bi(θi)− (1 − Pc ) · (N − 1) · bi(θi) 0 b[1](θ−i) · FN−2 (Φi(b[1](θ−i)))· f(Φi(b[1](θ−i))) · Φi(b[1](θ−i)) db[1](θ−i) We are now ready to optimize the expected utility by taking the derivative with respect to bi(θi) and setting it to 0. Note that we do not need to solve the integral, because it will disappear when the derivative is taken (by application of the Fundamental Theorem of Calculus). 0 = (N−1)·FN−2 (Φi(bi(θi)))·f(Φi(bi(θi)))·Φi(bi(θi))·θi− FN−1 (Φi(bi(θi))) · Pc − Pc ·(N−1)·FN−2 (Φi(bi(θi)))·f(Φi(bi(θi)))·Φi(bi(θi))·bi(θi)− (1−Pc )·(N−1)· bi(θi)·FN−2 (Φi(bi(θi)))·f(Φi(bi(θi)))·Φi(bi(θi)) Dividing through by FN−2 (Φi(bi(θi))) and combining like terms yields: 0 = θi − Pc · bi(θi) − (1 − Pc ) · bi(θi) · (N − 1) · f(Φi(bi(θi))) · Φi(bi(θi)) − Pc · F(Φi(bi(θi))) Simplifying the expression and rearranging terms produces: bi(θi) = θi − Pc · F(Φi(bi(θi))) (N − 1) · f(Φi(bi(θi))) · Φi(bi(θi)) To further simplify, we use the formula f (x) = 1 g (f(x)) , where g(x) is the inverse function of f(x). Plugging in function from our setting gives us: Φi(bi(θi)) = 1 bi(θi) . Applying both this equation and Φi(bi(θi)) = θi yields: bi(θi) = θi − Pc · F(θi) · bi(θi) (N − 1) · f(θi) (11) Attempts at a derivation of the solution from this point proved fruitless, but we are at a point now where a guessed solution can be quickly verified. We used the solution for the first-price auction (see, e.g., [10]) as our starting point to find the answer: bi(θi) = θi − θi 0 F( N−1 P c ) (x)dx F( N−1 P c ) (θi) (12) To verify the solution, we first take its derivative: bi(θi) = 1− F(2· N−1 P c ) (θi) − N−1 P c · F( N−1 P c −1) (θi) · f(θi) · θi 0 F( N−1 P c ) (x)dx F(2· N−1 P c ) (θi) This simplifies to: bi(θi) = N−1 P c · f(θi) · θi 0 F( N−1 P c ) (x)dx F( N−1 P c +1) (θi) We then plug this derivative into the equation we derived (11): bi(θi) = θi − Pc · F(θi) · N−1 P c · f(θi) · θi 0 F( N−1 P c ) (x)dx (N − 1) · f(θi) · F( N−1 P c +1) (θi) Cancelling terms yields Equation 12, verifying that our guessed solution is correct. Alternative Proof of Theorem 1: The following proof uses the Revenue Equivalence Theorem (RET) and the probabilistic auction given as an interpretation of our cheating seller setting. In a first-price auction without the possibility of cheating, the expected payment for an agent with type θi is simply the product of its bid and the probability that this bid is the highest. For the symmetric equilibrium, this is equal to: F(N−1) (θi) · θi − θi 0 F(N−1) (x)dx F(N−1)(θi) For our probabilistic auction, the expected payment of the winning agent is a weighted average of its bid and the second highest bid. For the bi(·) we found in the original interpretation of the setting, it can be written as follows. F(N−1) (θi) · Pc θi − θi 0 F( N−1 P c ) (x)dx F( N−1 P c ) (θi) + (1 − Pc ) 1 F(N−1)(θi) · θi 0 x− x 0 F( N−1 P c ) (y)dy F( N−1 P c ) (x) ·(N −1)·F(N−2) (x)·f(x)dx By the RET, the expected payments will be the same in the two auctions. Thus, we can verify our equilibrium bidding strategy by showing that the expected payment in the two auctions is equal. Since the expected payment is zero at θi = 0 for both functions, it suffices to verify that the derivatives of the expected payment functions with respect to θi are equal, for an arbitrary value θi. Thus, we need to verify the following equation: F(N−1) (θi) + (N − 1) · F(N−2) (θi) · f(θi) · θi − F(N−1) (θi) = Pc · F(N−1) (θi) · 1 − 1− (N−1 P c ) · F( N−1 P c −1) (θi) · f(θi) · θi 0 F( N−1 P c ) (x)dx F2( N−1 P c ) (θi) + (N − 1) · F(N−2) (θi) · f(θi) · θi − θi 0 F( N−1 P c ) (x)dx F( N−1 P c ) (θi) + (1−Pc ) θi− θi 0 F( N−1 P c ) (y)dy F( N−1 P c ) (θi) ·(N−1)·F(N−2) (θi)·f(θi) 82 This simplifies to: 0 = Pc · (N−1 P c ) · F(N−2) (θi) · f(θi) · θi 0 F( N−1 P c ) (x)dx F( N−1 P c ) (θi) + (N − 1) · F(N−2) (θi) · f(θi) · − θi 0 F( N−1 P c ) (x)dx F( N−1 P c ) (θi) + (1−Pc ) − θi 0 F( N−1 P c ) (y)dy F( N−1 P c ) (θi) ·(N−1)·F(N−2) (θi)·f(θi) After distributing Pc , the right-hand side of this equation cancels out, and we have verified our equilibrium bidding strategy. Corollary 2. In a second-price auction in which the seller cheats with probability Pc , and F(θi) = θi, it is a Bayes-Nash equilibrium for each agent to bid according to the following strategy: bi(θi) = N − 1 N − 1 + Pc θi (6) Proof. Plugging F(θi) = θi into Equation 5 (repeated as 12), we get: bi(θi) = θi − θi 0 x( N−1 P c ) dx θi ( N−1 P c ) = θi − P c N−1+P c θi ( N−1+P c P c ) θi ( N−1 P c ) = θi − Pc N − 1 + Pc · θi = N − 1 N − 1 + Pc · θi Theorem 3. In a first-price auction in which each agent cheats with probability Pa , it is a Bayes-Nash equilibrium for each non-cheating agent i to bid according to the strategy that is a fixed point of the following equation: bi(θi) = θi − θi 0 Pa · F(bi(x)) + (1 − Pa) · F(x) (N−1) dx Pa · F(bi(θi)) + (1 − Pa) · F(θi) (N−1) (8) Proof. We make the same guesses about the equilibrium strategy to aid our search as we did in the proof of Theorem 1. When simplifying the expectation of this setting"s utility equation (7), we use the fact that the probability that agent i will have a higher bid than another honest agent is still F(Φi(bi(θi))), while the probability is F(bi(θi)) if the other agent cheats. The probability that agent i beats a single other agent is then a weighted average of these two probabilities. Thus, we can write agent i"s expected utility as: Eθ−i,µa ui(b(θ), µa , θi) = θi − bi(θi) · Pa · F(bi(θi)) + (1 − Pa ) · F(Φi(bi(θi))) N−1 As before, to find the equilibrium bi(θi), we take the derivative and set it to zero: 0 = θi − bi(θi) · (N − 1)· Pa · F(bi(θi)) + (1 − Pa ) · F(Φi(bi(θi))) N−2 · Pa · f(bi(θi)) + (1 − Pa ) · f(Φi(bi(θi))) · Φi(bi(θi)) − Pa · F(bi(θi)) + (1 − Pa ) · F(Φi(bi(θi))) N−1 Applying the equations Φi(bi(θi)) = 1 bi(θi) and Φi(bi(θi)) = θi, and dividing through, produces: 0 = θi − bi(θi) · (N − 1)· Pa · f(bi(θi)) + (1 − Pa ) · f(θi) · 1 bi(θi) − Pa · F(bi(θi)) + (1 − Pa ) · F(θi) Rearranging terms yields: bi(θi) = θi − Pa · F(bi(θi)) + (1 − Pa) · F(θi) · bi(θi) (N − 1) · Pa · f(bi(θi)) · bi(θi) + (1 − Pa) · f(θi) (13) In this setting, because we leave F(·) unspecified, we cannot present a closed form solution. However, we can simplify the expression by removing its dependence on bi(θi). bi(θi) = θi − θi 0 Pa · F(bi(x)) + (1 − Pa) · F(x) (N−1) dx Pa · F(bi(θi)) + (1 − Pa) · F(θi) (N−1) (14) To verify Equation 14, first take its derivative: bi(θi) = 1 − 1− (N − 1) · Pa · F(bi(θi)) + (1 − Pa ) · F(θi) (N−2) · Pa · f(bi(θi)) · bi(θi)) + (1 − Pa ) · f(θi) · θi 0 Pa · F(bi(x)) + (1 − Pa ) · F(x) (N−1) dx Pa · F(bi(θi)) + (1 − Pa) · F(θi) 2(N−1) This equation simplifies to: bi(θi) = (N −1)· Pa ·f(bi(θi))·bi(θi))+(1−Pa )·f(θi) · θi 0 Pa · F(bi(x)) + (1 − Pa ) · F(x) (N−1) dx Pa · F(bi(θi)) + (1 − Pa) · F(θi) N Plugging this equation into the bi(θi) in the numerator of Equation 13 yields Equation 14, verifying the solution. 83 Corollary 4. In a first-price auction in which each agent cheats with probability Pa , and F(θi) = θi, it is a BayesNash equilibrium for each non-cheating agent to bid according to the strategy bi(θi) = N−1 N θi. Proof. Instantiating the fixed point equation (8, and repeated as 14) with F(θi) = θi yields: bi(θi) = θi − θi 0 Pa · bi(x) + (1 − Pa ) · x (N−1) dx Pa · bi(θi) + (1 − Pa) · θi (N−1) We can plug the strategy bi(θi) = N−1 N θi into this equation in order to verify that it is a fixed point. bi(θi) = θi − θi 0 Pa · N−1 N x + (1 − Pa ) · x (N−1) dx Pa · N−1 N θi + (1 − Pa) · θi (N−1) = θi − θi 0 x(N−1) dx θ (N−1) i = θi − 1 N θN i θ (N−1) i = N − 1 N θi Theorem 5. In a first-price auction where F(θi) = θi, if each agent j = i bids according a strategy bj(θj) = N−1+αj N θj, where αj ≥ 0, then it is a best response for the remaining agent i to bid according to the strategy bi(θi) = N−1 N θi. Proof. We again use Φj : [0, bj(1)] → [0, 1] as the inverse of bj(θj). For all j = i in this setting, Φj(x) = N N−1+αj x. The probability that agent i has a higher bid than a single agent j is F(Φj(bi(θi))) = N N−1+αj bi(θi). Note, however, that since Φj(·) is only defined over the range [0, bj(1)], it must be the case that bi(1) ≤ bj(1), which is why αj ≥ 0 is necessary, in addition to being sufficient. Assuming that bi(θi) = N−1 N θi, then indeed Φj(bi(θi)) is always welldefined. We will now show that this assumption is correct. The expected utility for agent i can then be written as: Eθ−i ui(b(θ), θi) = Πj=i N N − 1 + αj bi(θi) · θi −bi(θ) = Πj=i N N − 1 + αj · θi· bi(θi) (N−1) − bi(θi) N Taking the derivative with respect to bi(θi), setting it to zero, and dividing out Πj=i N N−1+αj yields: 0 = θi · (N − 1) · bi(θi) (N−2) − N · bi(θi) (N−1) This simplifies to the solution: bi(θi) = N−1 N θi. Full Version of Example 1: In a first-price auction where F(θi) = θ2 i +θi 2 and N = 2, if agent 2 always bids its type (b2(θ2) = θ2), then, for all θ1 > 0, agent 1"s best response bidding strategy is strictly less than the bidding strategy of the symmetric equilibrium. After calculating the symmetric equilibrium in which both agents shave their bid by the same amount, we find the best response to an agent who instead does not shave its bid. We then show that this best response is strictly less than the equilibrium strategy. To find the symmetric equilibrium bidding strategy, we instantiate N = 2 in the general formula the equation found in [10], plug in F(θi) = θ2 i +θi 2 , and simplify: bi(θi) = θi − θi 0 F(x)dx F(θi) = θi− 1 2 · θi 0 (x2 + x)dx 1 2 · (θ2 i + θi) = θi− 1 3 θ3 i + 1 2 θ2 i θ2 i + θi = 2 3 θ2 i + 1 2 θi θi + 1 We now derive the best response for agent 1 to the strategy b2(θ2) = θ2, denoting the best response strategy b∗ 1(θ1) to distinguish it from the symmetric case. The probability of agent 1 winning is F(b∗ 1(θ1)), which is the probability that agent 2"s type is less than agent 1"s bid. Thus, agent 1"s expected utility is: Eθ2 ui((b∗ 1(θ1), b2(θ2)), θ1) = F(b∗ 1(θ1)) · θ1 − b∗ 1(θ1) = (b∗ 1(θ1))2 + b∗ 1(θ1) 2 · θ1 − b∗ 1(θ1) Taking the derivative with respect to b∗ 1(θ1), setting it to zero, and then rearranging terms gives us: 0 = 1 2 · 2 · b∗ 1(θ1) · θ1 − 3 · (b∗ 1(θ1))2 + θ1 − 2 · b∗ 1(θ1) 0 = 3 · (b∗ 1(θ1))2 + (2 − 2 · θ1) · b∗ 1(θ1) − θ1 Of the two solutions of this equation, one always produces a negative bid. The other is: b∗ 1(θ1) = θ1 − 1 + θ2 1 + θ1 + 1 3 We now need to show that b1(θ1) > b∗ 1(θ1) holds for all θi > 0. Substituting in for both terms, and then simplifying the inequality gives us: 2 3 θ2 1 + 1 2 θ1 θ1 + 1 > θ1 − 1 + θ2 1 + θ1 + 1 3 θ2 1 + 3 2 θ1 + 1 > (θ1 + 1) θ2 1 + θ1 + 1 Since θ1 ≥ 0, we can square both sides of the inequality, which then allows us to verify the inequality for all θ1 > 0. θ4 1 + 3θ3 1 + 17 4 θ2 1 + 3θ1 + 1 > θ4 1 + 3θ3 1 + 4θ2 1 + 3θ1 + 1 1 4 θ2 1 > 0 84

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