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Parrondo's paradox, a paradox in game theory, has been described as: A combination of losing strategies becomes a winning strategy. Winning strategies consisting of various combinations of losing strategies were explored in biology before Parrondo's paradox was published. show via simulation that if $M=3$ and $\epsilon = 0.005,$ Game B is an almost surely losing game as well. In fact, Game B is a Markov chain, and an analysis of its state transition matrix (again with M=3) shows that the steady state probability of using coin 2 is 0.3836, and that of using coin 3 is 0.6164. As coin 2 is selected nearly 40% of the time, it has a disproportionate influence on the payoff from Game B, and results in it being a losing game.

However, when these two losing games are played in some alternating sequence - e.g. two games of A followed by two games of B (AABBAABB...), the combination of the two games is, paradoxically, a winning game. Not all alternating sequences of A and B result in winning games. For example, one game of A followed by one game of B (ABABAB...) is a losing game, while one game of A followed by two games of B (ABBABB...) is a winning game. This coin-tossing example has become the canonical illustration of Parrondo's paradox – two games, both losing when played individually, become a winning game when played in a particular alternating sequence.

The apparent paradox has been explained using a number of sophisticated approaches, including Markov chains, flashing ratchets, simulated annealing, and information theory. One way to explain the apparent paradox is as follows:

* While Game B is a losing game under the probability distribution that results for $C_t $ modulo $M$ when it is played individually ($C_t $ modulo $M$ is the remainder when $C_t $ is divided by $M$), it can be a winning game under other distributions, as there is at least one state in which its expectation is positive.

* As the distribution of outcomes of Game B depend on the player's capital, the two games cannot be independent. If they were, playing them in any sequence would lose as well.

The role of $M$ now comes into sharp focus. It serves solely to induce a dependence between Games A and B, so that a player is more likely to enter states in which Game B has a positive expectation, allowing it to overcome the losses from Game A. With this understanding, the paradox resolves itself: The individual games are losing only under a distribution that differs from that which is actually encountered when playing the compound game. In summary, Parrondo's paradox is an example of how dependence can wreak havoc with probabilistic computations made under a naive assumption of independence. A more detailed exposition of this point, along with several related examples, can be found in Philips and Feldman.

For a simpler example of how and why the paradox works, again consider two games Game A and Game B, this time with the following rules:

# In Game A, you simply lose $1 every time you play.

# In Game B, you count how much money you have left ⁠ ⁠—  if it is an even number you win $3, otherwise you lose $5.

Say you begin with $100 in your pocket. If you start playing Game A exclusively, you will obviously lose all your money in 100 rounds. Similarly, if you decide to play Game B exclusively, you will also lose all your money in 100 rounds.

However, consider playing the games alternatively, starting with Game B, followed by A, then by B, and so on (BABABA...). It should be easy to see that you will steadily earn a total of $2 for every two games.

Thus, even though each game is a losing proposition if played alone, because the results of Game B are affected by Game A, the sequence in which the games are played can affect how often Game B earns you money, and subsequently the result is different from the case where either game is played by itself.

Parrondo's paradox is used extensively in game theory, and its application to engineering, population dynamics, financial risk, etc., are areas of active research. Parrondo's games are of little practical use such as for investing in stock markets as the original games require the payoff from at least one of the interacting games to depend on the player's capital. However, the games need not be restricted to their original form and work continues in generalizing the phenomenon. Similarities to volatility pumping and the two envelopes problem have been pointed out. Simple finance textbook models of security returns have been used to prove that individual investments with negative median long-term returns may be easily combined into diversified portfolios with positive median long-term returns. Similarly, a model that is often used to illustrate optimal betting rules has been used to prove that splitting bets between multiple games can turn a negative median long-term return into a positive one. In evolutionary biology, both bacterial random phase variation and the evolution of less accurate sensors have been modelled and explained in terms of the paradox. In ecology, the periodic alternation of certain organisms between nomadic and colonial behaviors has been suggested as a manifestation of the paradox. There has been an interesting application in modelling multicellular survival as a consequence of the paradox and some interesting discussion on the feasibility of it. Applications of Parrondo's paradox can also be found in reliability theory. Interested readers can refer to the three review papers which have been published over the years, with the most recent one examining the Parrondo effect across biology.

In the early literature on Parrondo's paradox, it was debated whether the word 'paradox' is an appropriate description given that the Parrondo effect can be understood in mathematical terms. The 'paradoxical' effect can be mathematically explained in terms of a convex linear combination.

However, Derek Abbott, a leading researcher on the topic, provides the following answer regarding the use of the word 'paradox' in this context: