Long Run Equilibrium in Discounted Stochastic Fictitious Play

In this paper we develop methods to analyze the long run behavior of models with multiple equilibria, and we apply them to a well known model of learning in games. Our methods apply to discrete-time continuous-state stochastic models, and as a particular application in we study a model of stochastic fictitious play. We focus on a variant of this model in which agents’ payoffs are subject to random shocks and they discount past observations exponentially. We analyze the behavior of agents’ beliefs as the discount rate on past information becomes small but the payoff shock variance remains fixed. We show that agents tend to be drawn toward an equilibrium, but occasionally the stochastic shocks lead agents to endogenously shift between equilibria. We then calculate the invariant distribution of players’ beliefs, and use it to determine the most likely outcome observed in long run. Our application shows that by making some slight changes to a standard learning model, we can derive an equilibrium selection criterion similar to stochastic evolutionary models but with some important differences.