  The best-response dynamics is an example of an evolutionary game where
players update their strategy in order to maximize their payoff. The main
objective of this paper is to study a stochastic spatial version of this game
based on the framework of interacting particle systems in which players are
located on an infinite square lattice. In the presence of two strategies, and
calling a strategy selfish or altruistic depending on a certain ordering of the
coefficients of the underlying payoff matrix, a simple analysis of the
non-spatial mean-field approximation of the spatial model shows that a strategy
is evolutionary stable if and only if it is selfish, making the system bistable
when both strategies are selfish. The spatial and non-spatial models agree when
at least one strategy is altruistic. In contrast, we prove that, in the
presence of two selfish strategies and in any spatial dimensions, only the most
selfish strategy remains evolutionary stable. The main ingredients of the proof
are monotonicity results and a coupling between the best-response dynamics
properly rescaled in space with bootstrap percolation to compare the infinite
time limits of both systems.
