A Structure Theorem for Rationalizability in Infinite-horizon Games

We show that in any game that is continuous at infinity, if a plan of action ai is rationalizable for a type ti, then there are perturbations of ti for which following ai for an arbitrarily long future is the only rationalizable plan. One can pick the perturbation from a finite type space with common prior. Furthermore, if ai is part of a Bayesian Nash equilibrium, the perturbation can be picked so that the unique rationalizable belief of the perturbed type regarding the play of the game is arbitrarily close to the equilibrium belief of ti. As an application to repeated games, we prove an unrefinable folk theorem: Any individually rational and feasible payoff is the unique rationalizable payoff vector for some perturbed type profile. This is true even if perturbed types are restricted to believe that the repeated-game payoff structure and the discount factor are common knowledge. JEL Numbers: C72, C73.