Parrondo Games with Two-Dimensional Spatial Dependence

Abstract

Parrondo games with one-dimensional (1D) spatial dependence were introduced by Toral and extended to the two-dimensional (2D) setting by Mihailović and Rajković. [Formula: see text] players are arranged in an [Formula: see text] array. There are three games, the fair, spatially independent game [Formula: see text], the spatially dependent game [Formula: see text], and game [Formula: see text], which is a random mixture or non-random pattern of games [Formula: see text] and [Formula: see text]. Of interest is [Formula: see text] (or [Formula: see text]), the mean profit per turn at equilibrium to the set of [Formula: see text] players playing game [Formula: see text] (or game [Formula: see text]). Game [Formula: see text] is fair, so if [Formula: see text] and [Formula: see text], then we say the Parrondo effect is present. We obtain a strong law of large numbers (SLLN) and a central limit theorem (CLT) for the sequence of profits of the set of [Formula: see text] players playing game [Formula: see text] (or game [Formula: see text]). The mean and variance parameters are computable for small arrays and can be simulated otherwise. The SLLN justifies the use of simulation to estimate the mean. The CLT permits evaluation of the standard error of a simulated estimate. We investigate the presence of the Parrondo effect for both small arrays and large ones. One of the findings of Mihailović and Rajković was that “capital evolution depends to a large degree on the lattice size.” We provide evidence that this conclusion is partly incorrect. A paradoxical feature of the 2D game [Formula: see text] that does not appear in the 1D setting is that, for fixed [Formula: see text] and [Formula: see text], the mean function [Formula: see text] is not necessarily a monotone function of its parameters.