Let game B be Toral's cooperative Parrondo game with (one-dimensional) spatial dependence, parameterized by N ≥ 3 and p0, p1, p2, p3 ∈ [0, 1], and let game A be the special case p0 = p1 = p2 = p3 = 1/2. In previous work we investigated μB and μ(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 3 ≤ N ≤ 19, at least, and appeared to converge as N → ∞, suggesting that the Parrondo region (i.e., the region in which μB ≤ 0 and μ(1/2, 1/2) > 0) 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 Ar Bs, where r and s are positive integers. We show that μ[r, s], the mean profit per turn to the ensemble of N players repeatedly playing the pattern Ar Bs, is computable for 3 ≤ N ≤ 18 and r + s ≤ 4, at least, and appears to converge as N → ∞, albeit more slowly than in the random-mixture case. Again this suggests that the Parrondo region (μB ≤ 0 and μ[r, s] > 0) has nonzero volume in the limit. Moreover, we can prove this convergence under certain conditions and identify the limits.
PARRONDO GAMES WITH SPATIAL DEPENDENCE II