# Learning, Mutation, and Long Run Equilibria in Games

George J. Mailath, Michihiro Kandori, Rafael Rob

We analyze an evolutionary model with a finite number of players and with noise or mutations. The expansion and contraction of strategies is linked--as usual--to their current relative success, but mutations--which perturb the system away from its deterministic evolution--are present as well. Mutations can occur in every period, so the focus is on the implications of ongoing mutations, not a one-shot mutation. The effect of these mutations is to drastically reduce the set of equilibria to what we term "long-run equilibria." For $2 \times 2$ symmetric games with two symmetric strict Nash equilibria the equilibrium selected satisfies (for large populations) Harsanyi and Selten's (1988) criterion of risk-dominance. In particular, if both strategies have equal security levels, the Pareto dominant Nash equilibrium is selected, even though there is another strict Nash equilibrium.