Reputational Bargaining with an Omniscient Type

This paper investigates the role of second-order beliefs in a reputational bargaining model involving two agents, A and B. Both agents can be either irrational (refusing to concede and sticking to their initial offer) or rational. B can take one of two rational forms: omniscient, who is certain of A’s rationality, or ignorant, who is uncertain. In typical reputational bargaining, agents make an offer at the beginning and adhere to it throughout the negotiation. However, we allow B to propose a ’fair’ 50-50 split of the surplus, which reveals B’s rationality and serves as a potential signal for the omniscient type. Using a hybrid discrete-continuous time framework proposed by Abreu and Pearce (2007), we examine how reputation effects can arise even when one agent (omniscient) is fully aware of the other’s true nature and decides whether to reveal or withhold this information. Our analysis reveals multiple equilibria, including scenarios where no fair offers are made, as rational players strategically avoid disclosing their rationality to preserve their advantage. If B’s irrational demand exceeds a fair division of the surplus, this scenario is the unique equilibrium. Conversely, when the demand is less than 50%, an equilibrium with a fair offer can occur. Every equilibrium of this type is characterized by a period t in which the fair deal is offered with positive probability exclusively at t.