Reputation for playing mixed actions: A characterization theorem

A patient player privately observes a persistent state that directly affects his opponents' payoffs, and can be one of the several commitment types that plays the same mixed action in every period. I characterize the set of environments under which the patient player obtains at least his commitment payoff in all equilibria regardless of his stage-game payoff function. Due to the presence of interdependent values, the patient player cannot guarantee his mixed commitment payoff by imitating the mixed commitment type, and small perturbations to a pure commitment action can significantly reduce the patient player's lowest equilibrium payoff.