Defensive Plays in Repeated Games

In obtaining the classical folk theorems for repeated games, the players must collectively coordinate their actions, even after someone deviates. In this paper, we explore the impacts of defensive plays on the set of equilibrium payoffs when all players have sufficiently high discount factor. Once some player deviates, everyone else will play the continuation game defensively by refusing to coordinate with the deviator and hence always choose their best responses in all subsequent periods. We show that this line of reasoning can significantly limit what can be supported by equilibrium in certain classes of repeated games. We characterize the set of limiting equilibrium payoffs under defensive plays.