Nash Equilibria in Quantum Games

For any two-by-two game $\text{G}$, we define a new two-player game ${\text{G}}^Q$. The definition is motivated by a vision of players in game $\text{G}$ communicating via quantum technology according to a certain standard protocol originally introduced by Eisert and Wilkins [EW]. In the game ${\text{G}}^Q$, each players’ strategy set consists of the set of all probability distributions on the 3-sphere $S^3$. Nash equilibria in this game can be difficult to compute. Our main theorems classify all possible equilibria in ${\text{G}}^Q$ for a Zariski-dense set of games $\text{G}$ that we call {\it generic games}. 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 geometric configurations. One easy consequence is that for zero-sum games, the payoff to either player in a mixed strategy quantum equilibrium must equal the average of that player’s four possible payoffs.