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 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 believe that the large player could be a strategic normal type or a commitment type, who plays the same action at all times. With this perturbation, nontrivial intertemporal incentives arise. In this two-type setting, we characterize the set of sequential equilibrium payoffs of the large player using an ordinary differential equation. Unlike in discrete time, in a large class of games in continuous time the sequential equilibrium is unique and Markov in the small players ’ belief for any prior. 1 Introduction
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