On Strategy Improvement Algorithms for Simple Stochastic Games

Abstract

The study of simple stochastic games (SSGs) was initiated by Condon for analyzing the computational power of randomized space-bounded alternating Turing machines. The game is played by two players, MAX and MIN, on a directed multigraph, and when the play terminates at a sink s, MAX wins from MIN a payoff p(s) ∈ [0,1]. Condon showed that the SSG value problem, which given a SSG asks whether the expected payoff won by MAX exceeds 1/2 when both players use their optimal strategies, is in NP ∩ coNP. However, the exact complexity of this problem remains open as it is not known whether the problem is in P or is hard for some natural complexity class. In this paper, we study the computational complexity of a strategy improvement algorithm by Hoffman and Karp for this problem. The Hoffman-Karp algorithm converges to optimal strategies of a given SSG, but no nontrivial bounds were previously known on its running time. We show a bound of O(2ⁿ/n) on the convergence time of this algorithm, and a bound of O(2^{0.78 n}) on a randomized variant. These are the first non-trivial upper bounds on the convergence time of these strategy improvement algorithms.