Classification of Quantum Symmetric Non-zero Sum 2x2 Games in the Eisert Scheme

TL;DR

This paper develops a new classification scheme for symmetric non-zero-sum 2x2 quantum games in the Eisert framework, based on Nash equilibria and Pareto-optimality, considering both unitary and nonunitary strategies.

Contribution

It introduces a comprehensive classification method for quantum 2x2 games that accounts for strategy restrictions and entanglement effects, extending classical game classifications.

Findings

Reproduces known features like multiple equilibria

Identifies changes in equilibrium character with entanglement

Classifies strategies including convex mixtures of unitaries

Abstract

A quantum game in the Eisert scheme is defined by the payoff matrix, plus some quantum entanglement parameters. In the symmetric nonzero-sum 2x2 games, the relevant features of the game are given by two parameters in the payoff matrix, and only one extra entanglement parameter is introduced by quantizing it in the Eisert scheme. The criteria adopted in this work to classify quantum games are the amount and characteristics of Nash equilibria and Pareto-optimal positions. A new classification based on them is developed for symmetric nonzero-sum classical 2x2 games, as well as classifications for quantum games with different restricted subsets of the total strategy set. Finally, a classification is presented taking the whole set of strategies into account, both unitary strategies and nonunitary strategies studied as convex mixures of unitary strategies. The classification reproduces features which have been previously found in other works, like appearance of multiple equilibria, changes in the character of equilibria, and entanglement regime transitions.