Stochastic Window Mean-Payoff Games

Stochastic two-player games model systems with an environment that is both adversarial and stochastic. The environment is modeled by a player ( \(\text {Player}~{2}\) ) who tries to prevent the system ( \(\text {Player}~{1}\) ) from achieving its objective. We consider finitary versions of the traditional mean-payoff objective, replacing the long-run average of the payoffs by payoff average computed over a finite sliding window. 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 present complexity bounds and algorithmic solutions for computing strategies for \(\text {Player}~{1}\) to ensure that the objective is satisfied with positive probability, with probability 1, or with probability at least p, regardless of the strategy of \(\text {Player}~{2}\) . The solution crucially relies on a reduction to the special case of non-stochastic two-player games. We give a general characterization of prefix-independent objectives for which this reduction holds. The memory requirement for both players in stochastic games is also the same as in non-stochastic games by our reduction. Moreover, for non-stochastic games, we improve upon the upper bound for the memory requirement of \(\text {Player}~{1}\) and upon the lower bound for the memory requirement of \(\text {Player}~{2}\) .