Spatial Parrondo games with spatially dependent game $A$

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.