Winning in sequential Parrondo games by players with short-term memory

Abstract

The original Parrondo game, denoted as AB3, contains two independent games: A and B. The winning or losing of games A and B is defined by the change of one unit of capital. Game A is a losing game if played continuously, with winning probability , where . Game B is also losing and has two coins: a good coin with winning probability is used if the player’s capital is not divisible by 3, otherwise a bad coin with winning probability is used. The Parrondo paradox refers to the situation where the mixture of games A and B in a sequence leads to winning in the long run. The paradox can be resolved using Markov chain analysis. We extend this setting of the Parrondo game to involve players with one-step memory. The player can win by switching his choice of A or B game in a Parrondo game sequence. If the player knows the identity of the game he plays and the state of his capital, then the player can win maximally. On the other hand, if the player does not know the nature of the game, then he is playing a (C, D) game, where either (C = A, D = B), or (C = B, D = A). For a player with one-step memory playing the AB3 game, he can achieve the highest expected gain with switching probability equal to 3/4 in the (C, D) game sequence. This result has been found first numerically and then proven analytically. Generalization to an AB mod(M) Parrondo game for other integers M has been made for the general domain of parameters . We find that for odd M the Parrondo effect does exist. However, for even M, there is no Parrondo effect for two cases: the initial game is A and the initial capital is even, or the initial game is B and the initial capital is odd. There is still a possibility of the Parrondo effect for the other two cases when M is even: the initial game is A and the initial capital is odd, or the initial game is B and the initial capital is even. These observations from numerical experiments can be understood as the factorization of the Markov chains into two distinct cycles. Discussion of these effects on games is also made in the context of feedback control of the Brownian ratchet.