Strategy iteration is strongly polynomial for 2-player turn-based stochastic games with a constant discount factor

Abstract

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 $O(\frac{mn}{1-\gamma }\log (\frac{n}{1-\gamma }))$ iterations, where $n$ is the number of states, $m$ is the total number of actions in the MDP, and $0<\gamma <1$ is the discount factor. We improve Ye's analysis in two respects. First, we improve the bound given by Ye and show that Howard's policy iteration algorithm actually terminates after at most $O(\frac{m}{1-\gamma }\log (\frac{n}{1-\gamma }))$ 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.