Parrondo games with spatial dependence, II

TL;DR

This paper extends the analysis of Parrondo's paradox with spatial dependence by examining nonrandom periodic patterns, demonstrating convergence of mean profits and suggesting the Parrondo region remains significant as the number of players grows.

Contribution

It introduces analysis of nonrandom periodic patterns in spatial Parrondo games and proves convergence of mean profits, expanding understanding beyond random mixtures.

Findings

Mean profits for periodic patterns are computable for N=3 to 18.

Convergence of mean profits as N approaches infinity is observed.

The Parrondo region likely has nonzero volume in the limit.

Abstract

Let game B be Toral's cooperative Parrondo game with (one-dimensional) spatial dependence, parameterized by N (3 or more) and p_0, p_1, p_2, p_3 in [0,1], and let game A be the special case p_0=p_1=p_2=p_3=1/2. In previous work we investigated mu_B and mu_(1/2,1/2), the mean profits per turn to the ensemble of N players always playing game B and always playing the randomly mixed game (1/2)(A+B). These means were computable for N=3,4,5,...,19, at least, and appeared to converge as N approaches infinity, suggesting that the Parrondo region (i.e., the region in which mu_B is nonpositive and mu_(1/2,1/2) is positive) has nonzero volume in the limit. The convergence was established under certain conditions, and the limits were expressed in terms of a parameterized spin system on the one-dimensional integer lattice. In this paper we replace the random mixture with the nonrandom periodic pattern A^r B^s, where r and s are positive integers. We show that mu_[r,s], the mean profit per turn to the ensemble of N players repeatedly playing the pattern A^r B^s, is computable for N=3,4,5,...,18 and r+s=2,3,4, at least, and appears to converge as N approaches infinity, albeit more slowly than in the random-mixture case. Again this suggests that the Parrondo region (mu_B is nonpositive and mu_[r,s] is positive) has nonzero volume in the limit. Moreover, we can prove this convergence under certain conditions and identify the limits.