Reputation in Continuous-Time Games

Authors

  • Eduardo Faingold,

    1. Dept. of Economics, Yale University, Box 208281, New Haven, CT 06520-8281, U.S.A.; eduardo.faingold@yale.edu
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  • Yuliy Sannikov

    1. Dept. of Economics, Princeton University, Princeton, NJ 08544-1021, U.S.A.; sannikov@princeton.edu
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    • We are grateful to a co-editor and two anonymous referees for exceptionally helpful comments and suggestions. We also thank Daron Acemoglu, Martin Cripps, Kyna Fong, Drew Fudenberg, George J. Mailath, Eric Maskin, Stephen Morris, Bernard Salanie, Paolo Siconolfi, Andrzej Skrzypacz, Lones Smith, and seminar audiences at Bocconi, Columbia, Duke, Georgetown, the Institute for Advanced Studies at Princeton, UCLA, UPenn, UNC at Chapel Hill, Washington University in St. Louis, Yale, the Stanford Institute of Theoretical Economics, the Meetings of the Society for Economic Dynamics in Vancouver, and the 18th International Conference in Game Theory at Stony Brook for many insightful comments.


Abstract

We study reputation dynamics in continuous-time games in which a large player (e.g., government) faces a population of small players (e.g., households) and the large player's actions are imperfectly observable. The major part of our analysis examines the case in which public signals about the large player's actions are distorted by a Brownian motion and the large player is either a normal type, who plays strategically, or a behavioral type, who is committed to playing a stationary strategy. We obtain a clean characterization of sequential equilibria using ordinary differential equations and identify general conditions for the sequential equilibrium to be unique and Markovian in the small players' posterior belief. We find that a rich equilibrium dynamics arises when the small players assign positive prior probability to the behavioral type. By contrast, when it is common knowledge that the large player is the normal type, every public equilibrium of the continuous-time game is payoff-equivalent to one in which a static Nash equilibrium is played after every history. Finally, we examine variations of the model with Poisson signals and multiple behavioral types.

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