A Recursive Algorithm for Solving Simple Stochastic Games

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 $O\left({\lfloor (d+1{)}^2\mathrm{/}2\rfloor }^{n\mathrm{/}2}\right)$ iterations, with $n$ the number of MAX vertices. Then, we focus on binary SSG and propose an algorithm that has complexity $O({\phi }^{n}Poly(N))$ where $\phi =(1+\sqrt{5})\mathrm{/}2$ is the golden ratio. To the best of our knowledge, this is the first deterministic strategy improvement algorithm that visits $2^{cn}$ strategies with $c<1$.