We show that in any game that is continuous at infinity, if a plan of action a_i is played by a type t_i in a Bayesian Nash equilibrium, then there are perturbations of t_i for which a_i is the only rationalizable plan and whose unique rationalizable belief regarding the play of the game is arbitrarily close to the equilibrium belief of t_i. 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.
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