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 parameterM. 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, a 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, a 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.