The Wadge game is a simple infinite game discovered by William Wadge (pronounced "wage"). It is used to investigate the notion of continuous reduction for subsets of Baire space. Wadge had analyzed the structure of the Wadge hierarchy for Baire space with games by 1972, but published these results only much later in his PhD thesis. In the Wadge game G(A,B), player I and player II each in turn play integers which may depend on those played before. The outcome of the game is determined by checking whether the sequences x and y generated by players I and II are contained in the sets A and B, respectively. Player II wins if the outcome is the same for both players, i.e. x is in A if and only if y is in B. Player I wins if the outcome is different. Sometimes this is also called the Lipschitz game, and the variant where player II has the option to pass (but has to play infinitely often) is called the Wadge game.