New Results on Simple Stochastic Games

Abstract

We study the problem of solving simple stochastic games, and give both an interesting new algorithm and a hardness result. We show a reduction from fine approximation of simple stochastic games to coarse approximation of a polynomial sized game, which can be viewed as an evidence showing the hardness to approximate the value of simple stochastic games. We also present a randomized algorithm that runs in ${\tilde{O}}(\sqrt{|V_{\mbox{R}}|!})$ time, where $|V_{\mbox{R}}|$ is the number of RANDOM vertices and ${\tilde{O}}$ ignores polynomial terms. This algorithm is the fastest known algorithm when $|V_{\mbox{R}}| = \omega(\log n)$ and $|V_{\mbox{R}}| = o(\sqrt{\min{|V_{\mbox{min}}|, |V_{\mbox{max}}|}})$ and it works for general (non-stopping) simple stochastic games.

Supported by the National Natural Science Foundation of China Grant 60553001 and the National Basic Research Program of China Grant 2007CB807900, 2007CB807901.