Winning Concurrent Reachability Games Requires Doubly-Exponential Patience

We exhibit a deterministic concurrent reachability game PURGATORY_n with n non-terminal positions and a binary choice for both players in every position so that any positional strategy for Player 1 achieving the value of the game within given isin < 1/2 must use non-zero behavior probabilities that are less than (isin²/(1 - isin))^2n-2 . Also, even to achieve the value within say 1 - 2^-n/2, doubly exponentially small behavior probabilities in the number of positions must be used. This behavior is close to worst case: We show that for any such game and 0 < isin < 1/2, there is an isin-optimal strategy with all non-zero behavior probabilities being 20(n) at least isin2^O(n). As a corollary to our results, we conclude that any (deterministic or nondeterministic) algorithm that given a concurrent reachability game explicitly manipulates isin-optimal strategies for Player 1 represented in several standard ways (e.g., with binary representation of probabilities or as the uniform distribution over a multiset) must use at least exponential space in the worst case.