Reputation and Equilibrium Selection in Games with a Patient Player

A single, long-run player plays a simultaneous-move stage game against a sequence of opponents who only play once, but observe all previous play. If there is a positive prior probability that the long-run player will always play the pure strategy he would most like to commit himself to (his Stackleberg strategy), then his payoff in any Nash equilibrium exceeds a bound that converges to the Stackleberg payoff as his discount factor approaches one. When the stage game is not simultaneous move, this result must be modified to account for the possibility that distinct strategies of the long-run player are observationally equivalent.