Perfect Bayesian Equilibria in Reputation Games with Nested Information Structure

We analyze reputation games where a strategic long-lived player acts in a repeated game against a collection of short-lived (myopic) Bayesian players. The running assumption in our model is that the information at the short-lived players is nested in that of the long-lived player in a causal fashion. Under this setup, we show that the set of perfect Bayesian equilibria coincide with a properly defined Markov perfect equilibria. Hence, a dynamic programming formulation is obtained for the computation of optimal strategies for the strategic long-lived player in any perfect Bayesian equilibrium. We also establish the existence of an optimal stationary strategy and, under further assumptions, the continuity of the equilibrium payoffs. The undiscounted average payoff setup is considered separately, this case is also used as a tool leading to an upper payoff bound for an arbitrarily patient long-lived player in the discounted average payoff setup. This achievable upper bound is identified with a stage game Stackelberg equilibrium payoff. Furthermore, by using measure concentration techniques, we obtain a lower payoff bound on the value of reputation. Our findings contribute to the reputations literature by obtaining structural and computational results on the equilibrium behavior in finite-horizon, infinite-horizon and undiscounted settings, as well as continuity results in the prior probabilities, and refined upper and lower bounds on the value of reputations. In particular, we exhibit that a control theoretic formulation can be utilized to characterize the equilibrium behavior.