Reputation Effects and Equilibrium Degeneracy in Continuous-Time Games

We study a class of continuous-time reputation games between a large player and a population of small players in which the actions of the large player are imperfectly observable. The large player is either a normal type, who behaves strategically, or a behavioral type, who is committed to playing a certain strategy. We provide a complete characterization of the set of sequential equilibrium payoffs of the large player using an ordinary differential equation. In addition, we identify a sufficient condition for the sequential equilibrium to be unique and Markovian in the small players' posterior belief. An implication of our characterization is that when the small players are certain that they are facing the normal type, intertemporal incentives are trivial: the set of equilibrium payoffs of the large player coincides with the convex hull of the set of static Nash equilibrium payoffs.