Dynamical Analysis of a Repeated Game with Incomplete Information

Abstract

We study a two player repeated zero-sum game with asymmetric information introduced by Renault in which the underlying state of the game undergoes Markov evolution (parameterized by a transition probability $\frac{1}{2}\le p\le 1$). Hörner, Rosenberg, Solan and Vieille identified an optimal strategy, ${\sigma }^{\ast }$ for the informed player for $p$ in the range $[\frac{1}{2},\frac{2}{3}]$. We extend the range on which ${\sigma }^{\ast }$ is proved to be optimal to about $[\frac{1}{2},0.719]$ and prove that it fails to be optimal at a value around 0.7328. Our techniques make use of tools from dynamical systems, specifically the notion of pressure, introduced by D. Ruelle.

The title has been changed and many changes have been made following suggestions from the referees. The paper is now to appear in Mathematics of Operations Research