Parrondo Game with Finite Memory and Flashing Ratchet

Abstract

The Parrondo paradox refers to the situation that for two games A and B, the random mixture of the game is winning but the individual game is losing. And in the original Parrondo game, game A is a biased coin tossing game with winning probability $p = 1/2 − \epsilon$, where $\epsilon$ is a small number; game B has two coins, a bad coin with winning probability $p_g = 1/10 − \epsilon$ if the player’s capital is divisible by 3, and a good coin with winning probability $p_b = 3/4 − \epsilon$ if the capital is not divisible by 3. We introduced a simple type of game illustrating the Parrondo Paradox and the return function can be exactly solved by the recursive equation. For the original Parrondo game, we reviewed several common switching scheme such as the random switching and the periodic switching, we further extended the switching scheme into a memory-based switching. The return for short-term memory and long-term memory has been analyzed and discussed. For the short-term memory switching, we found the optimal switching rate is independent with the winning probability. Also, the similarity between the Parrondo game and the Flashing ratchet has been discussed, and the numerical simulation of the flashing ratchet under different switching strategy has been performed. We found that there is a mapping of the switching in the Parrondo game and Flashing ratchet. At the end we have introduced a noise-based Parrondo game, and we have showed that there is a direct relation between the pay-off and the information entropy.