A Recursive Algorithm for Solving Simple Stochastic Games

Abstract

We present two recursive strategy improvement algorithms for solving simple stochastic games. First we present an algorithm for solving SSGs of degree $d$ that uses at most $O\left(\left\lfloor(d+1)^2/2\right\rfloor^{n/2}\right)$ iterations, with $n$ the number of MAX vertices. Then, we focus on binary SSG and propose an algorithm that has complexity $O\left(\varphi^nPoly(N)\right)$ where $\varphi = (1 + \sqrt{5})/2$ is the golden ratio. To the best of our knowledge, this is the first deterministic strategy improvement algorithm that visits $2^{cn}$ strategies with $c < 1$.