2021/01/05 by Sung Chan Choi, Choi, Sung Chan
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2101.01778
openalex publication_date 2021/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Parrondo games with spatial dependence were introduced by Toral (2001) and have been studied extensively. In Toral's model N players are arranged in a circle. The players play either game A or game B. In game A, a randomly chosen player wins or loses one unit according to the toss of a fair coin. In game B, which depends on parameters p0,p1,p2∈[0,1], a randomly chosen player, player x say, wins or loses one unit according to the toss of a pm-coin, where m∈\0,1,2\ is the number of nearest neighbors of player x who won their most recent game. In this paper, we replace game A by a spatially dependent game, which we call game A', introduced by Xie et al.~(2011). In game A', two nearest neighbors are chosen at random, and one pays one unit to the other based on the toss of a fair coin. Game A' is fair, so we say that the Parrondo effect occurs if game B is losing or fair and the game C', determined by a random or periodic sequence of games A' and B, is winning. Here we give sufficient conditions for convergence as N→∞ of the mean profit per game played from game C'. This requires ergodicity of an associated interacting particle system (not necessarily a spin system), for which sufficient conditions are found using the basic inequality.