Learning from private information in noisy repeated games

We study the perfect type-contingently public ex-post equilibrium (PTXE) of repeated games where players observe imperfect public signals of the actions played, and both the payoff functions and the map from actions to signal distributions depend on an unknown state. The PTXE payoffs when players are patient are determined by the solutions to a family of linear programming problems. Using this characterization, we develop conditions under which play can be as if the players have learned the state. We provide a sufficient condition for the folk theorem, and a characterization of the PTXE payoffs in games with a known monitoring structure.