Learning and equilibrium transitions: Stochastic stability in discounted stochastic fictitious play

In this paper I study how adaptive learning leads to switches between multiple stable equilibria, and I develop tools to characterize the rate of transition. While the methods I develop are relatively general, I focus on a two-player model of stochastic fictitious play, where agents’ payoffs are subject to random shocks. Each player forecasts his opponent’s future play as a discounted average of past play. I analyze the behavior of agents’ beliefs as the discount rate on past information becomes small, but the payoff shock variance remains fixed. I show that agents tend to be drawn toward an equilibrium, but occasionally the stochastic shocks lead to endogenous shifts between equilibria. I then calculate the limiting transition rates and the invariant distribution of players’ beliefs, and use it to determine the most likely outcome observed in long run. I show this stochastically stable equilibrium satisfies an important continuity condition which allows for relatively direct characterization.