Expectation in Stochastic Games with Prefix-independent Objectives

Abstract: Stochastic two-player games model systems with an environment that is both adversarial and stochastic. In this paper, we study the expected value of quantitative prefix-independent objectives in stochastic games. We show a generic reduction from the expectation problem to linearly many instances of almost-sure satisfaction of threshold Boolean objectives. The result follows from partitioning the vertices of the game into so-called value classes where each class consists of vertices of the same value. Our procedure further entails that the memory required by both players to play optimally for the expectation problem is no more than the memory required by the players to play optimally for the almost-sure satisfaction problem for a corresponding threshold Boolean objective. We show the applicability of the framework to compute the expected window mean-payoff measure in stochastic games. The window mean-payoff measure strengthens the classical mean-payoff measure by computing the mean-payoff over a window of bounded length that slides along an infinite path. Two variants have been considered: in one variant, the maximum window length is fixed and given, while in the other, it is not fixed but is required to be bounded. For both variants, we show that the decision problem to check if the expected value is at least a given threshold is in UP $\cap$ coUP. The result follows from guessing the expected values of the vertices, partitioning them into value classes, and proving that a unique short certificate for the expected values exists. It also follows that the memory required by the players to play optimally is no more than that in non-stochastic two-player games with the corresponding window objectives.