Games with Imperfectly Observable Actions in Continuous Time |
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| Authors: | Yuliy Sannikov |
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| Institution: | 1. Dept. of Economics, University of California, Berkeley, 593 Evans Hall 3880, Berkeley, CA 94720‐3880, U.S.A.;2. sannikov@gmail.com;3. I would like to thank Bob Wilson, Andy Skrzypacz, Peter DeMarzo, Paul Milgrom, Dilip Abreu, David Ahn, Manuel Amador, Anthony Chung, Darrell Duffie, Eduardo Faingold, Kyna Fong, Willie Fuchs, Drew Fudenberg, Mike Harrison, Vitali Kalesnik, Ayca Kaya, Patricia Lassus, Eddie Lazear, Deishin Lee, George Mailath, Day Manoli, Gustavo Manso, David Miller, William Minozzi, Dan Quint, Korok Ray, Ennio Stacchetti, Ivan Werning, Ruth Williams, Alexei Tchistyi, and all seminar participants at Stanford, Berkeley, Harvard, Princeton, Northwestern, NYU, MIT, the University of Chicago, Yale, the University of Minnesota, UCSD, Humboldt, Oxford, the Minnesota Workshop in Macroeconomic Theory, Rochester, the University of Pennsylvania, and the University of Michigan for valuable feedback on this paper. Also, I would like to thank a co‐editor and two anonymous referees for very thoughtful comments. |
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| Abstract: | This paper investigates a new class of two‐player games in continuous time, in which the players' observations of each other's actions are distorted by Brownian motions. These games are analogous to repeated games with imperfect monitoring in which the players take actions frequently. Using a differential equation, we find the set ℰ(r) of payoff pairs achievable by all public perfect equilibria of the continuous‐time game, where r is the discount rate. The same differential equation allows us to find public perfect equilibria that achieve any value pair on the boundary of the set ℰ(r). These public perfect equilibria are based on a pair of continuation values as a state variable, which moves along the boundary of ℰ(r) during the course of the game. In order to give players incentives to take actions that are not static best responses, the pair of continuation values is stochastically driven by the players' observations of each other's actions along the boundary of the set ℰ(r). |
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| Keywords: | Repeated games continuous time incentives optimal equilibria Brownian motion collusion computation |
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