Winning in Sequential Parrondo Games by Players with Short-Term Memory

The original Parrondo game, denoted as AB3, contains two independent games: A and B. The winning or losing of A and B game is defined by the change of one unit of capital. Game A is a losing game if played continuously, with winning probability $p=0.5-\epsilon$, where $\epsilon=0.003$. Game B is also losing and it has two coins: a good coin with winning probability $p_g=0.75-\epsilon$ is used if the player`s capital is not divisible by $3$, otherwise a bad coin with winning probability $p_b=0.1-\epsilon$ is used. Parrondo paradox refers to the situation that the mixture of A and B game in a sequence leads to winning in the long run. The paradox can be resolved using Markov chain analysis. We extend this setting of 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 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 AB mod($M$) Parrondo game for other integer $M$ has been made for the general domain of parameters $p_b<p=0.5=p_A <p_g$. (please read the PDF file for full abstract)