Winning in Sequential Parrondo Games by Players with Short-Term Memory
Ka Wai Cheung, Ho Fai Ma, Degang Wu, Ga Ching Lui, Kwok Yip Szeto

TL;DR
This paper investigates how players with short-term memory can optimize their winning strategies in Parrondo games, demonstrating that strategic switching with a 3/4 probability maximizes expected gains, supported by numerical and analytical methods.
Contribution
It introduces a model of Parrondo games involving players with one-step memory and derives the optimal switching probability for maximizing gains.
Findings
Players with one-step memory can achieve maximum gains by switching with probability 3/4.
Numerical and analytical methods confirm the optimal switching strategy.
Generalization to AB mod(M) games extends applicability across parameters.
Abstract
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 , where . Game B is also losing and it has two coins: a good coin with winning probability is used if the player`s capital is not divisible by , otherwise a bad coin with winning probability 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…
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