Spatial Parrondo games and an interacting particle system

TL;DR

This paper studies a spatially dependent Parrondo game model, replacing the original game A with a new variant, and establishes conditions for the mean profit convergence using ergodicity of an associated interacting particle system.

Contribution

It introduces a new spatially dependent game A' and provides sufficient conditions for the convergence of mean profit in the model, extending prior Parrondo game analyses.

Findings

Sufficient conditions for convergence as N→∞.

Ergodicity criteria for the associated particle system.

Extension of Parrondo games with spatial dependence.

Abstract

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 $p_0,p_1,p_2\in[0,1]$, a randomly chosen player, player $x$ say, wins or loses one unit according to the toss of a $p_m$-coin, where $m\in\{0,1,2\}$ 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 $N\to\infty$ 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.