Abstract. For any two-by-two game G, we define a new two-player game G Q . The definition is motivated by a vision of players in game G communicating via quantum technology according to the protocol introduced by J. Eisert and M. Wilkins.In the game G Q , each player's (mixed) strategy set consists of the set of all probability distributions on the 3-sphere S 3 . Nash equilibria in the game can be difficult to compute.Our main theorems classify all possible mixed-strategy equilibria. First, we show that up to a suitable definition of equivalence, any strategy that arises in equilibrium is supported on at most four points; then we show that those four points must lie in one of a small number of allowable geometric configurations.

An n-person game consists of 1. n sets S i (i = 1, . . . , n).2. n real-valued functions

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