Determinacy and Decidability of Reachability Games with Partial Observation on Both Sides

We consider two-players stochastic reachability games with partial observation on both sides and finitely many states, signals and actions. We prove that in such games, either player 1 has a strategy for winning with probability 1, or player 2 has such a strategy, or both players have strategies that guarantee winning with non-zero probability (positively winning strategies). We give a fix-point algorithm for deciding which of the three cases holds, which can be decided in doubly-exponential time.