Spatial Evolutionary Games with Weak Selection
Mridu Nanda and Rick Durrett
Abstract. Recently a rigorous mathematical theory has been developed for spatial
games with weak selection, i.e., when the payoff differences between
strategies are small. The key to the analysis is that when
space and time are suitably rescaled the spatial model converges
to the solution of a partial differential equation (PDE). This approach
can be used to analyze all 2×2 games, but there are a number of 3×3
games for which the behavior of the limiting PDE is not known. In
this paper we give rules for determining the behavior of a large class
of 3 × 3 games and check their validity using simulation. In words,
the effect of space is equivalent to making changes in the payoff matrix,
and once this is done, the behavior of the spatial game can be
predicted from the behavior the replicator equation for the modified
game. We say predicted here because in some cases the behavior of
the spatial game is different from that of the replicator equation for
transformed game. For example, if a rock-paper-scissors game has a
replicator equation that spirals out to the boundary, space stabilizes
the system and produces an equilibrium.
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