General quantum two-players games, their gate operators and Nash equilibria

TL;DR

This paper develops a general framework for quantized two-player games with multiple strategies, using group theory to analyze gate operators and Nash equilibria, revealing limitations in maximally entangled games.

Contribution

It introduces a general form of gate operators for multi-strategy quantum games and analyzes Nash equilibria using group theoretical methods, highlighting the role of the initial state's stability group.

Findings

Maximally entangled games lack nontrivial pure Nash strategies.

Explicit computations provided for three-strategy games.

Group theoretical approach clarifies structure of quantum game equilibria.

Abstract

The two-players $N$ strategies games quantized according to the Eisert-Lewenstein-Wilkens scheme (Phys. Rev. Lett. 83 (1999), 3077) are considered. Group theoretical methods are applied to the problem of finding a general form of gate operators (entanglers) under the assumption that the set of classical pure strategies is contained in the set of pure quantum ones. The role of the stability group of the initial state of the game is stressed. As an example, it is shown that the maximally entangled games do not admit nontrivial pure Nash strategies. The general arguments are supported by explicit computations performed in the three strategies case.