New Algorithms for Solving Simple Stochastic Games

Abstract

We present new algorithms for determining optimal strategies for two-player games with proba- bilistic moves and reachability winning conditions. Such games, known as simple stochastic games, were extensively studied by A.Condon [Anne Condon. The complexity of stochastic games. Information and Computation, 96(2):203–224, 1992, Anne Condon. On algorithms for simple stochastic games. In Jin-Yi Cai, editor, Advances in Computational Complexity Theory, volume 13 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 51–73. AMS, 1993]. Many interesting problems, including parity games and hence also mu-calculus model checking, can be reduced to simple stochastic games. It is an open problem, whether simple stochastic games can be solved in polynomial time. Our algorithms determine the optimal expected payoffs in the game. We use geometric interpre- tation of the search space as a subset of the hyper-cube [0,1]N. The main idea is to divide this set into convex subregions in which linear optimization methods can be used. We show how one can proceed from one subregion to the other so that, eventually, a region containing the optinal payoffs will be found. The total number of subregions is exponential in the size of the game but, in practice, the algorithms need to visit only few of them to find a solution. We believe that our new algorithms could provide new insights into the difficult problem of deter- mining algorithmic complexity of simple stochastic games and other, equivallent problems.