Spatial Parrondo games with spatially dependent game $A$

TL;DR

This paper explores a modified spatial Parrondo game where a new game A' replaces game A, investigates conditions for the Parrondo effect to occur, and compares the effects in different models.

Contribution

It introduces a new spatially dependent game A' and analyzes the Parrondo effect in this context, extending previous models and providing conditions for the effect's occurrence.

Findings

Identification of parameter regions where the Parrondo effect occurs.

Conditions for the convergence of mean profit as the number of players grows.

Comparison of Parrondo regions between different spatial models.

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. Noting that game $A'$ is fair, we say that the \textit{Parrondo effect} occurs if game $B$ is losing or fair and game $C'$, determined by a random or periodic sequence of games $A'$ and $B$, is winning. We investigate numerically the region in which the Parrondo effect appears. We give sufficient conditions for the mean profit in game $C'$ to converge as $N\to\infty$. Finally, we compare the Parrondo region in the model of Xie et al.\ with that in the model of Toral.