Dynamical Analysis of a Repeated Game with Incomplete Information

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, p, in the range $\frac{1}{2}$ to 1). Hörner, Rosenberg, Solan and Vieille identified an optimal strategy, σ^* for the informed player for p in the range $\left[\frac{1}{2},\frac{2}{3}\right]$. We extend the range on which σ^* is proved to be optimal to about $\left[\frac{1}{2},0.719\right]$ 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.