Order of limits in reputations

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.