Quantum game theory for 2 2 games: a mathematical framework

TL;DR

This paper develops a rigorous mathematical framework for quantum game theory applied to 2x2 games, extending classical concepts to include quantum strategies and proving the existence of Nash equilibria.

Contribution

It introduces a formal framework for quantum 2x2 games, including mixed strategies over SU(2), and generalizes Nash equilibrium existence to the quantum setting.

Findings

Established a fixed-point proof for Nash equilibria in quantum mixed strategies.

Extended classical game theory concepts to the quantum domain.

Presented the Eisert-Wilkens-Lewenstein protocol as a standard quantum game implementation.

Abstract

We develop a rigorous mathematical framework for quantum game theory applied to static 2x2 games, extending classical concepts to the quantum setting where players may employ arbitrary unitary operations (pure strategies) or probability measures over the continuous group SU(2) (mixed strategies). The Eisert-Wilkens-Lewenstein protocol is introduced as the standard implementation of quantum 2x2 games. We prove the existence of Nash equilibria for continuous quantum mixed strategies via a fixed-point argument, generalising the classical Nash theorem to the quantum case.