Additional Winning Strategies in Reachability Games

In game theory, deciding whether a designed player wins a game amounts to check whether he has a winning strategy. However, there are several game settings in which knowing whether he has more than a winning strategy is also important. For example, this is crucial in deciding whether a game admits a unique Nash Equilibrium, or in planning a rescue as this would provide a backup plan. In this paper we study the problem of checking whether, in a two-player reachability game, a designed player has more than a winning strategy. We investigate this question both under perfect and imperfect information about the moves performed by the players. We provide an automata-based solution that results, in the perfect information setting, in a linear-time procedure; in the imperfect information setting, instead, it shows an exponential-time upper bound. In both cases, the results are tight.