Efficiency of Classical and Quantum Games Equilibria.

Marek Szopa

doi:10.3390/e23050506

Abstract

Nash equilibria and correlated equilibria of classical and quantum games are investigated in the context of their Pareto efficiency. The examples of the prisoner’s dilemma, battle of the sexes and the game of chicken are studied. Correlated equilibria usually improve Nash equilibria of games but require a trusted correlation device susceptible to manipulation. The quantum extension of these games in the Eisert–Wilkens–Lewenstein formalism and the Frąckiewicz–Pykacz parameterization is analyzed. It is shown that the Nash equilibria of these games in quantum mixed Pauli strategies are closer to Pareto optimal results than their classical counter-parts. The relationship of mixed Pauli strategies equilibria and correlated equilibria is also studied.

Highlights

We were looking for game solutions that would be closer to the Paretoefficient results than classical game solutions
We took into account: the prisoner’s dilemma game, battle of the sexes and two versions of the chicken game
For most of these games, correlated equilibria are better than Nash equilibria

Summary

Introduction

This, leads to an undesirable dependence of the equilibria on the selected parameterization [27] To resolve this dilemma, we propose using the criterion of quantum game invariance under isomorphic transformations of the input classic game introduced by Frackiewicz [28]. Thanks to the use of mixed quantum strategies, we obtain both: non-trivial Nash equilibria and that they are closer to Pareto-efficient solutions than classical equilibria. The third section, presents four classical games, discuss their Nash equilibria and analyzes their Paretooptimality We discuss their correlated equilibria, which thanks to the use of additional mechanisms of correlation of players’ behavior, allow for better Pareto optimization of the results of these games. In the last part we discuss the applicability of both correlation mechanisms and the perspective of physical implementation of quantum games

Game Theory Preliminaries

The Efficiency of Selected Classical Games

EWL Quantization Protocol in Frackiewicz–Pykacz

Efficiency of Quantum Games Equilibria

Conclusions