The Complexity of Solving Stochastic Games on Graphs

We consider some well-known families of two-player zero-sum perfect-information stochastic games played on finite directed graphs. Generalizing and unifying results of Liggett and Lippman, Zwick and Paterson, and Chatterjee and Henzinger, we show that the following tasks are polynomial-time (Turing) equivalent. – Solving stochastic parity games, – Solving simple stochastic games, – Solving stochastic terminal-payoff games with payoffs and probabilities given in unary, – Solving stochastic terminal-payoff games with payoffs and probabilities given in binary, – Solving stochastic mean-payoff games with rewards and probabilities given in unary, – Solving stochastic mean-payoff games with rewards and probabilities given in binary, – Solving stochastic discounted-payoff games with discount factor, rewards and probabilities given in binary. It is unknown whether these tasks can be performed in polynomial time. In the above list, “solving” may mean either quantitatively solving a game (computing the values of its positions) or strategically solving a game (computing an optimal strategy for each player). In particular, these two tasks are polynomial-time equivalent for all the games listed above. We also consider a more refined notion of equivalence between quantitatively and strategically solving a game. We exhibit a linear time algorithm that given a simple stochastic game or a terminal-payoff game and the values of all positions of that game, computes a pair of optimal strategies. Consequently, for any restriction one may put on the simple stochastic game model, quantitatively solving is polynomial-time equivalent to strategically solving the resulting class of games.