Partial-Observation Stochastic Games: How to Win when Belief Fails

In two-player finite-state stochastic games of partial observation on graphs, in every state of the graph, the players simultaneously choose an action, and their joint actions determine a probability distribution over the successor states. We consider reachability objectives where player 1 tries to ensure a target state to be visited almost-surely or positively. On the basis of information, the game can be one-sided with either (a)player 1 or (b)player 2 having partial observation, or two-sided with both players having partial observation. On the basis of randomization (a)players may not be allowed to use randomization (pure strategies), or (b)may choose a probability distribution over actions but the actual random choice is not visible (actions invisible), or (c)may use full randomization. Our results for pure strategies are as follows: (1)For one-sided games with player 2 perfect observation we show that belief-based strategies are not sufficient, and present an exponential upper bound on memory both for almost-sure and positive winning strategies; we show that the problem of deciding the existence of almost-sure and positive winning strategies for player 1 is EXPTIME-complete and present symbolic algorithms that avoid the explicit exponential construction. (2)For one-sided games with player 1 perfect observation we show that non-elementary memory is both necessary and sufficient for both almost-sure and positive winning strategies. (3)We show that for the two-sided case finite memory strategies are sufficient for both positive and almost-sure winning. We establish the equivalence of the almost-sure winning problem for pure strategies with randomized strategies with actions invisible. Our equivalence result exhibit serious flaws in previous results in the literature: we show a non-elementary memory lower bound for almost-sure winning whereas an exponential upper bound was claimed.