Strategy Iteration Is Strongly Polynomial for 2-Player Turn-Based Stochastic Games with a Constant Discount Factor

Thomas Dueholm Hansen, Peter Bro Miltersen, Uri Zwick
2013 Journal of the ACM  
Ye showed recently that the simplex method with Dantzig pivoting rule, as well as Howard's policy iteration algorithm, solve discounted Markov decision processes (MDPs), with a constant discount factor, in strongly polynomial time. More precisely, Ye showed that both algorithms terminate after at most O mn 1−γ log n 1−γ iterations, where n is the number of states, m is the total number of actions in the MDP, and 0 < γ < 1 is the discount factor. We improve Ye's analysis in two respects. First,
more » ... e improve the bound given by Ye and show that Howard's policy iteration algorithm actually terminates after at most O m 1−γ log n 1−γ iterations. Second, and more importantly, we show that the same bound applies to the number of iterations performed by the strategy iteration (or strategy improvement) algorithm, a generalization of Howard's policy iteration algorithm used for solving 2-player turn-based stochastic games with discounted zero-sum rewards. This provides the first strongly polynomial algorithm for solving these games, resolving a long standing open problem.
doi:10.1145/2432622.2432623 fatcat:ntouvh6v4jd25cn32bugrefwy4