Nash equilibria in quantum games

Abstract:

For any two-by-two game $\mathbf {G}$, we define a new two-player game $\mathbf {G}^Q$. The definition is motivated by a vision of players in game $\mathbf {G}$ communicating via quantum technology according to the protocol introduced by J. Eisert and M. Wilkins.

In the game $\mathbf {G}^Q$, each player’s (mixed) strategy set consists of the set of all probability distributions on the 3-sphere $\mathbf {S}^3$. 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.