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Parrondo's paradox
From Wikipedia, the free encyclopedia
Parrondo's paradox, a paradox in game theory, has been described as: A losing strategy that wins. It is named after its creator, Spanish physicist Juan Parrondo, who discovered the paradox in 1996. A more explanatory description is:
Given two games, each with a higher probability of losing than winning, it is possible to construct a winning strategy by playing the games alternately.
Parrondo devised the paradox in connection with his analysis of the Brownian ratchet, a thought experiment about a machine that can purportedly extract energy from random heat motions popularized by physicist Richard Feynman.
The coin-tossing example
A example of Parrondo's paradox is drawn from the field of gambling. Consider playing two games, Game A and Game B with following rules. For convenience, define Ct to be our capital at time t, immediately before we play a game.
Winning a game earns us $1 and losing requires us to surrender $1. It follows that Ct + 1 = Ct + 1 if we win at step t and Ct + 1 = Ct − 1 if we lose at step t.
In Game A, we toss a biased coin, Coin 1, with probability of winning P1 = (1 / 2) − ε. If ε > 0, this is clearly a losing game in the long run.
In Game B, we first determine if our capital is a multiple of some integer M. If it is, we toss a biased coin, Coin 2, with probability of winning P2 = (1 / 10) − ε. If it is not, we toss another biased coin, Coin 3, with probability of winning P3 = (3 / 4) − ε. The role of modulo M provides the periodicity as in the ratchet teeth.
It is clear that by playing Game A, we will almost surely lose in the long run. Harmer and Abbott[1] show via simulation that if M = 3 and ε = 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.[2] 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 Ct modulo M when it is played individually (Ct modulo M is the remainder when Ct is divided 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.
Application of Parrondo's paradox
Parrondo's paradox is used extensively in game theory, and its application in engineering, population dynamics, financial risk, etc, are also being looked into as demonstrated by the reading lists below. Parrondo's games are of little practical use such as for investing in stock markets[8] 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-envelope problem[9] 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.[10] 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.

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