Order of limits in reputations

Abstract

The fact that small departures from complete information might have large effects on the set of equilibrium payoffs draws interest in the adverse selection approach to study reputations in repeated games. It is well known that these large effects on the set of equilibrium payoffs rely on long-run players being arbitrarily patient. We study reputation games where a long-run player plays a fixed stage-game against an infinite sequence of short-run players under imperfect public monitoring. We show that in such games, introducing arbitrarily small incomplete information does not open the possibility of new equilibrium payoffs far from the complete information equilibrium payoff set. This holds true no matter how patient the long-run player is, as long as her discount factor is fixed. This result highlights the fact that the aforementioned large effects arise due to an order of limits argument, as anticipated. Keywords

Stochastic Control Approach to Reputation Games

“…This further implies that the following conjecture made by Cripps, Mailath, and Samuelson [9] is indeed true in our setup: There exists a particular equilibrium in the complete information game and a bound such that for any commitment type prior (of Player 2s) less than this bound, there exists an equilibrium of the incomplete information game where the strategic long-lived Player 1's payoff is arbitrarily close to his payoff from the particular equilibrium in the complete information game. 14 This is also in line with the findings of [11], which uses the methods of [1] to show a similar upper semi continuity result. 13 The dynamic program (13) appears in the proof of Lemma III.3 in the Appendix.…”

“…Thus, every optimal strategy should be such that if Player 2's belief has converged, then the equilibrium behaviour must be of Stackelberg type. Note also that, by the analysis in the 11 Even though there is a large literature on repeated games with incomplete information in the undiscounted setup, the only papers that we know of that study the reputation games explicitly in the this setup are [10] and [40]. As opposed to our model, [10] analyzes a two-person reputation game where both of the players are long-lived.…”

“…payoff bound for the arbitrarily patient strategic long-lived Player 1 in the standard discounted average payoff case -through an Abelian inequality. 11 The only difference from our original setup is that the strategic long-lived Player 1 now wishes to maximize lim inf N →∞…”