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Extended Parrondo's Game and Brownian Ratchets: Strong and Weak Parrondo Effect

Inspired by the flashing ratchet, Parrondo's game presents an apparently paradoxical situation. Parrondo's game consists of two individual games, game A and game B. Game A is a slightly losing coin-tossing game. Game B has two coins, with an integer parameter $M$. If the current cumulative capital (in discrete unit) is a multiple of $M$, an unfavorable coin $p_b$ is used, otherwise a favorable $p_g$ coin is used. Paradoxically, combination of game A and game B could lead to a winning game, which is the Parrondo effect. We extend the original Parrondo's game to include the possibility of $M$ being either $M_1$ or $M_2$. Also, we distinguish between strong Parrondo effect, i.e. two losing games combine to form a winning game, and weak Parrondo effect, i.e. two games combine to form a better-performing game. We find that when $M_2$ is not a multiple of $M_1$, the combination of $B(M_1)$ and $B(M_2)$ has strong and weak Parrondo effect for some subsets in the parameter space $(p_b,p_g)$, while there is neither strong nor weak effect when $M_2$ is a multiple of $M_1$. Furthermore, when $M_2$ is not a multiple of $M_1$, stochastic mixture of game A may cancel the strong and weak Parrondo effect. Following a discretization scheme in the literature of Parrondo's game, we establish a link between our extended Parrondo's game with the analysis of discrete Brownian ratchet. We find a relation between the Parrondo effect of our extended model to the macroscopic bias in a discrete ratchet. The slope of a ratchet potential can be mapped to the fair game condition in the extended model, so that under some conditions, the macroscopic bias in a discrete ratchet can provide a good predictor for the game performance of the extended model. On the other hand, our extended model suggests a design of a ratchet in which the potential is a mixture of two periodic potentials.