Nash Equilibria in Quantum Games

TL;DR

This paper introduces a new quantum game model based on classical two-by-two games, classifies all Nash equilibria for a broad class of these quantum games, and reveals geometric structures underlying equilibrium strategies.

Contribution

It provides a complete classification of Nash equilibria in quantum versions of classical two-player games for generic cases, revealing geometric configurations of strategies.

Findings

Equilibrium strategies are supported on at most four points.

These points lie in specific geometric configurations.

In zero-sum games, equilibrium payoffs are averages of four possible payoffs.

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 a certain standard protocol originally introduced by Eisert and Wilkins [EW]. In the game $\G^Q$, 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 $\G^Q$ for a Zariski-dense set of games $\G$ 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.