On Stochastic Games with Multiple Objectives

We study two-player stochastic games, where the goal of one player is to satisfy a formula given as a boolean combination of expected total reward objectives and the behaviour of the second player is adversarial. Such games are important for modelling, synthesis and veri cation of open systems with stochastic behaviour. We show that nding a winning strategy is PSPACE-hard in general and undecidable for deterministic strategies. We also prove that optimal strategy, if such exists, may require in nite memory and randomisation. However, when restricted to disjunctions of objectives only, memoryless deterministic strategies suffice, and the problem of deciding whether a winning strategy exists is NP-complete. We also present algorithms to approximate the Pareto sets of achievable objectives for the class of stopping games