Econometrica: Sep 2007, Volume 75, Issue 5

Games with Imperfectly Observable Actions in Continuous Time

DOI: 10.1111/j.1468-0262.2007.00795.x
p. 1285-1329

Yuliy Sannikov

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() of payoff pairs achievable by all public perfect equilibria of the continuous‐time game, where 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(). These public perfect equilibria are based on a pair of continuation values as a state variable, which moves along the boundary of ℰ() 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().

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Supplemental Material

Supplementary material for Games With Imperfectly Observable Actions in Continuous Time

The following supplement was uploaded after the first appearance of the article on December 18, 2007. The following files numerically implement the algorithm of Section 8 to find the set of equilibrium payoffs for the examples of the noisy partnership (scriptpd.m) and duopoly (script_duopoly.m).  Differential equations are solved using the 4th-order Runge Kutta method.  Figures 11, 12 and 13 in the paper are based on the outputs of these files.

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