Exploiting small uncertainties on the part of opponents, players in long, finitely repeated games can maintain false reputations that lead to a large variety of equilibrium outcomes. Even cooperation in a finitely repeated prisoners' dilemma is obtainable. Can such false reputations be maintined in a society if the same repeated game is played recurringly by many different groups and each group observes the play paths of the earlier groups? We argue that such false reputations must die out over time. To prove this in environments that allow for rich (uncountable) sets of types of players, we combine ideas of purification with recent results from the rational learning literature.