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On Strategy Improvement Algorithms for Simple Stochastic Games

By Rahul Tripathi, Elena Valkanova and V. S. Anil Kumar


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 the Hoffman-Karp algorithm, and a bound of O(20.78n) on a randomized variant. These are the first non-trivial upper bounds on the convergence time of these strategy improvement algorithms

Topics: algorithms, computational complexity, stochastic games, strategy improvement algorithms, optimal strategies
Year: 2010
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