  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. Game $A'$ is fair,
so we say that the Parrondo effect occurs if game $B$ is losing or fair and the
game $C'$, determined by a random or periodic sequence of games $A'$ and $B$,
is winning. Here we give sufficient conditions for convergence as $N\to\infty$
of the mean profit per game played from game $C'$. This requires ergodicity of
an associated interacting particle system (not necessarily a spin system), for
which sufficient conditions are found using the basic inequality.
