Nash equilibria in quantum games

Abstract

For any two-by-two game G \mathbf {G} , we define a new two-player game G Q \mathbf {G}^Q . The definition is motivated by a vision of players in game G \mathbf {G} communicating via quantum technology according to the protocol introduced by J. Eisert and M. Wilkins. In the game G Q \mathbf {G}^Q , each player’s (mixed) strategy set consists of the set of all probability distributions on the 3-sphere S 3 \mathbf {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.