Reputation in Dynamic Games

We consider an infinite dynamic game played by one large player and a large number of small players. State variables are allowed, and public histories include only the play of the large player, the aggregate play of the small players and the aggregate state variable. We use a reputational argument that restricts the set of equilibria to profiles that give the large player almost what he could get by committing to an optimal strategy as his discount factor approaches 1. Furthermore we identify a class of dynamic games where this result holds even if the small players' discount factor also approaches 1.