No fixed-point guarantee of Nash equilibrium in quantum games

TL;DR

This paper investigates the stability of Nash equilibria in quantum games, revealing that traditional fixed-point guarantees do not hold universally and identifying conditions where equilibrium can be assured.

Contribution

It demonstrates the lack of fixed-point guarantees in quantum games and specifies conditions under which Nash equilibrium can be established.

Findings

Glickberg's fixed-point theorem does not apply to pure quantum games with observable payoffs.

Nash equilibrium can be guaranteed when payoffs are based on state preparation.

Fixed-point stability is not inherently assured in quantum game settings.

Abstract

The theory of quantum games permits players to choose strategies that prepare and measure quantum states. Whereas conventional game theory provides guarantees for fixed-point stability in non-cooperative games, so-called Nash equilibria, we find this guarantee is not provided for quantum games. In particular, we show the conditions for Glickberg's fixed-point theorem do not apply to pure quantum games when the payoff is a physical observable. We further show that Nash equilibrium can be guaranteed when the payoff is defined with respect to state preparation.