Best reply structure and equilibrium convergence in generic games

Game theory is widely used as a behavioral model for strategic interactions in biology and social science. It is common practice to assume that players quickly converge to an equilibrium, e.g. a Nash equilibrium, but in some situations convergence fails. Existing research studies the problem of equilibrium convergence in classes of games with special properties. Here we take a different approach, conventional in ecology and in other natural sciences: we generate payoff matrices at random to calculate how typical convergence is over ensembles of two-player normal-form games. We introduce a formalism based on best reply dynamics, in which each player myopically uses the best response to her opponent’s last action. We show that the presence of best reply cycles predicts non-convergence of six well-known learning algorithms that are used in biology or have support from experiments with human players. We find that best reply cycles become dominant as games get more complicated and more competitive, indicating that in this case equilibrium is typically an unrealistic assumption. Alternatively, if for some reason games that describe real applications have special constraints and do not possess cycles, we raise the interesting question of why this should be so, and introduce a method to study their robustness.