Games with Imperfectly Observable Actions in Continuous Time

Abstract

This paper investigates a new class of 2-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).1 I would like to thank especially Bob Wilson, Andy Skrzypacz, Peter DeMarzo, Paul Milgrom, Dilip Abreu, Manuel Amador, Darrell Duffie, Drew Fudenberg, Mike Harrison, Eddie Lazear, George Mailath, Ennio Stacchetti, Ivan Werning, Ruth Williams, David Ahn, Anthony Chung, Willie Fuchs, Patricia Lassus, Deishin Lee, Day Manoli, Gustavo Manso, David Miller, William Minozzi, Dan Quint, Korok Ray, 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 very valuable feedback on this paper. Also, I would like to thank the editor, Andrew Postlewaite, and two anonymous referees for very thoughtful comments.

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@inproceedings{SannikovGamesWI, title={Games with Imperfectly Observable Actions in Continuous Time}, author={Yuliy Sannikov and Mike Harrison and Eddie Lazear and George Mailath and Ennio Stacchetti and Ivan Werning and Ruth Williams and David Ahn and Anthony Chung and Willie Fuchs and Gustavo Manso and David Miller and William Minozzi and Dan Quint and Korok Ray and Alexei Tchistyi} }