  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 $\frac 12\le p\le
1$). H\"orner, Rosenberg, Solan and Vieille identified an optimal strategy,
$\sigma^*$ for the informed player for $p$ in the range $[\frac 12,\frac 23]$.
We extend the range on which $\sigma^*$ is proved to be optimal to about
$[\frac 12,0.719]$ 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.
