Spatial Parrondo games and an interacting particle system

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.