Reputation for Playing Mixed Actions: A Characterization Theorem

TL;DR

This paper characterizes the conditions under which a patient player can secure his commitment payoff in all equilibria despite the complexities of mixed actions and interdependent values, highlighting the fragility of equilibrium guarantees.

Contribution

It provides a characterization theorem for environments where a patient player can guarantee his commitment payoff across all equilibria, considering interdependent values and mixed strategies.

Findings

Patient player can guarantee commitment payoff in certain environments.

Small changes in pure commitment actions can greatly affect guaranteed payoffs.

Interdependent values prevent imitation of mixed commitment types.

Abstract

A patient player privately observes a persistent state that directly affects his myopic 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 interdependent values, the patient player cannot guarantee his mixed commitment payoff by imitating the mixed-strategy commitment type, and small perturbations to a pure commitment action can significantly reduce the patient player's guaranteed equilibrium payoff.