Equilibrium Degeneracy and Reputation Effects in Continuous-Time Games

We study a continuous-time reputation game between a large player and a population of small players in which the actions of the large player are imperfectly observable. We explore two versions of the game. In the complete information game, in which it is common knowledge that the large player is a strategic normal type, we show that \emph{intertemporal incentives collapse}: irrespective of players' patience and signal informativeness, the set of equilibrium payoffs of the large player coincides with (the convex hull of) the set of static Nash equilibrium payoffs. In the incomplete information game, the small players assign positive probability to the large player being a \emph{commitment type}, who plays the same action at all times, irrespective of the past history of play, or a \emph{normal type} (with payoffs from the complete information game). With this perturbation, nontrivial intertemporal incentives arise. In this two-type setting, we characterize the set of Nash equilibrium payoffs of the large player by means of a pair of ordinary differential equations. Furthermore, we show that for any prior probability on the large player's types, the large player's maximum Nash equilibrium payoff is attained by a Markov equilibrium, i.e., by a sequential equilibrium in strategies that depend only on the small players' common posterior belief. We illustrate our results with examples of applications in Industrial Organization, Corporate Finance and Macroeconomics.