Inspired by the flashing Brownian ratchet, Parrondo’s game presents an apparently paradoxical situation. Parrondo’s game consists of two individual games, namely game A and game B. Game A is a simple and 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 is used, otherwise a favorable coin is used. The paradoxical situation is that the combination of game A and game B that are slightly losing combine, could lead to a winning game, which is the Parrondian effect. We extended the original Parrondo’s game to include the possibility of M taking one of the two possible values M₁,M₂ randomly with M₂ M₁. In other words, we combined two game B with different...[ Read more ]

Inspired by the flashing Brownian ratchet, Parrondo’s game presents an apparently paradoxical situation. Parrondo’s game consists of two individual games, namely game A and game B. Game A is a simple and 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 is used, otherwise a favorable coin is used. The paradoxical situation is that the combination of game A and game B that are slightly losing combine, could lead to a winning game, which is the Parrondian effect. We extended the original Parrondo’s game to include the possibility of M taking one of the two possible values M₁,M₂ randomly with M₂ > M₁. In other words, we combined two game B with different values of M in the same way we combined game A and game B. We found that when M₂ is not a multiple of M₁, the combination of B (M₁) and B (M₂) could lead to a game that performs better than either of the individual games, i.e. there exists Parrondian effect. When M₂ is not a multiple of M₁, the further inclusion of game A may actually decrease or eliminate the Parrondian effect. Deterministic switching sequence of B (M₁) and B (M₂), which are losing when played individually, could also lead to a winning game. Parrondo’s game can be regarded as a discrete ratchet. Following the discretization scheme in the literature, we discretized the Fokker-Planck equation, and established the link between the discretized Fokker-Planck equation and our extended Parrondo’s game. The characteristics of the ratchet potential resulted from the discrete FP equation corresponding to the extended game have been shown and discussed.