Games with additional winning strategies

In game theory, deciding whether a designed player wins a game corresponds to check whether he has a winning strategy. There are situations in which it is important to know whether some extra winning strategy also exists. In this paper we investigate this question over two-player finite games, under the reachability objective. We provide an automata-based technique that, given such a game, it allows to decide in linear time whether the game admits more than a winning strategy. We discuss along the paper some case studies and use them to show how to apply our solution methodology.