Trust and Betrayals: Reputational Payoffs and Behaviors without Commitment

Abstract

I introduce a reputation model in which all types of the reputation-building player are facing lack-ofcommitment problems. I study a repeated trust game in which a patient player (e.g., a seller) wants to win the trust of some myopic opponents (e.g., buyers) but can strictly benefit from betraying them. Her benefit from betrayal is her persistent private information. I provide a tractable formula for the highest equilibrium payoff for every type of that patient player. Interestingly, incomplete information affects this payoff only through the lowest benefit in the support of the prior belief. In every equilibrium that attains this highest payoff, the patient player’s behavior depends nontrivially on past play. I establish bounds on her long-run action frequencies that apply to all of her equilibrium best replies. These features of her behavior are essential for her to extract information rent while preserving her informational advantage. I construct a class of such high-payoff equilibria in which the patient player’s reputation depends only on the number of times she has betrayed as well as the number of times she has been trustworthy in the past. The method I developed is useful for studying other repeated incomplete information games between a patient informed player and a sequence of myopic uninformed players.