A Formalism for Quantum Games and an Application

TL;DR

This paper introduces a new mathematical formalism for quantum games, clarifying foundational issues and demonstrating the existence of quantum equilibria with better payoffs than classical ones in specific games.

Contribution

It provides the first comprehensive formalism for quantum games and addresses key controversies, including the existence of superior quantum equilibria.

Findings

Quantum equilibria can outperform classical correlated equilibria in Prisoner's Dilemma.

The formalism resolves ambiguities in the definition of quantum games.

Existence of quantum equilibria not corresponding to classical ones is demonstrated.

Abstract

This paper presents a new mathematical formalism that describes the quantization of games. The study of so-called quantum games is quite new, arising from a seminal paper of D. Meyer \cite{Meyer} published in Physics Review Letters in 1999. The ensuing near decade has seen an explosion of contributions and controversy over what exactly a quantized game really is and if there is indeed anything new for game theory. What has clouded many of the issues is the lack of a mathematical formalism for the subject in which these various issues can be clearly and precisely expressed, and which provides a context in which to present their resolution. Such a formalism is presented here, along with proposed resolutions to some of the issues discussed in the literature. One in particular, the question of whether there can exist equilibria in a quantized version of a game that do not correspond to classical correlated equilibria of that game and also deliver better payoffs than the classical correlated equilibria is answered in the affirmative for the Prisoner's Dilemma and Simplified Poker.