Games with Incomplete Information Played by Risk-Revising Players

Abstract: This paper introduces risk-revising players to a class of games with incomplete information. These players enter the game with ex ante risk preferences represented by coherent risk measures and develop time-consistent interim revisions of them contingent on their private information. The standard Nash equilibrium at ex ante stage and Bayesian Nash equilibrium at interim stage are extended to their risk-averse counterparts. Risk-revising Bayesian Nash equilibrium is proposed to capture behavioral outcomes resulting from interim plays based on revised risk preferences. We discuss existence results of these equilibrium concepts. When players' risk revisions correspond to their ex ante equilibrium play, connections are established between equilibria at the ex ante and interim stages. The effect of risk-aversion is analyzed using comparative statics. With the help of the dual representation of risk measures, we illustrate the role of risk-aversion in representing inconsistent beliefs. A numerical example is presented to illustrate the proposed equilibrium concepts under a specific choice of risk preference.