Parrondo games with two-dimensional spatial dependence

Parrondo games with one-dimensional spatial dependence were introduced by Toral and extended to the two-dimensional setting by Mihailovi\'c and Rajkovi\'c. $MN$ players are arranged in an $M\times N$ array. There are three games, the fair, spatially independent game $A$, the spatially dependent game $B$, and game $C$, which is a random mixture or nonrandom pattern of games $A$ and $B$. Of interest is $\mu_B$ (or $\mu_C$), the mean profit per turn at equilibrium to the set of $MN$ players playing game $B$ (or game $C$). Game $A$ is fair, so if $\mu_B\le0$ and $\mu_C>0$, then we say the Parrondo effect is present. We obtain a strong law of large numbers and a central limit theorem for the sequence of profits of the set of $MN$ players playing game $B$ (or game $C$). 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 Mihailovi\'c and Rajkovi\'c was that "capital evolution depends to a large degree on the lattice size." We provide evidence that this conclusion is incorrect. Part of the evidence is that, under certain conditions, the means $\mu_B$ and $\mu_C$ converge as $M,N\to\infty$. Proof requires that a related spin system on ${\bf Z}^2$ be ergodic. However, our sufficient conditions for ergodicity are rather restrictive.