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Quantum Information and Computation

Full characterization of quantum correlated equilibria

Zhaohui Wei, Shengyu Zhang

Published Sep 1, 2013• 3 citations
Publisher

Abstract

Quantum game theory aims to study interactions of people (or other agents) using quantum devices with possibly conflicting interests. Recently Zhang studied some quantitative questions in general quantum strategic games of growing sizes~\cite{Zha12}. However, a fundamental question not addressed there is the characterization of quantum correlated equilibria (QCE). In this paper, we answer this question by giving a sufficient and necessary condition for an arbitrary state $\rho$ being a QCE. In addition, when the condition fails to hold for some player $i$, we give an explicit positive-operator valued measurement (POVM) for that player to achieve a strictly positive gain of payoff. Finally, we give some upper bounds for the maximum gain by playing quantum strategies over classical ones, and the bounds are tight for some games.

DOI
10.26421/qic13.9-10-7

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Quantum game theory aims to study interactions of people (or other agents) using quantum devices with possibly conflicting interests. Recently Zhang studied some quantitative questions in general quantum strategic games of growing sizes~\cite{Zha12}. However, a fundamental question not addressed there is the characterization of quantum correlated equilibria (QCE). In this paper, we answer this question by giving a sufficient and necessary condition for an arbitrary state ρ\rhoρ being a QCE. In addition, when the condition fails to hold for some player iii, we give an explicit positive-operator valued measurement (POVM) for that player to achieve a strictly positive gain of payoff. Finally, we give some upper bounds for the maximum gain by playing quantum strategies over classical ones, and the bounds are tight for some games.
Quantum game theory lays a foundation for understanding the interaction of people using quantum computers with conflicting interests. Recently Zhang proposed a simple yet rich model to study quantum strategic games, and addressed some quantitative questions for general games of growing sizes \cite{Zha10}. However, one fundamental question that the paper did not consider is the characterization of quantum correlated equilibria (QCE). In this paper, we answer this question by giving a sufficient and necessary condition for an arbitrary state ρρρ being a QCE. In addition, when the condition fails to hold for some player iii, we give an explicit POVM for that player to achieve a strictly positive gain. Finally, we give some upper bounds for the maximum gain by playing quantum strategies over classical ones, and the bounds are tight for some games.
Players’ choices in quantum game schemes are often correlated by a quantum state. This enables players to obtain payoffs that may not be achievable when classical pure or mixed strategies are used. On the other hand, players’ choices can be correlated due to a classical probability distribution, and if no player benefits by a unilateral deviation from the vector of recommended strategies, the probability distribution is a correlated equilibrium. The aim of this paper is to investigate relation between correlated equilibria and Nash equilibria in the MW-type schemes for quantum games.
Quantum entanglement has been recently demonstrated as a useful resource in conflicting-interest games of incomplete information between two players, Alice and Bob [Pappa et al., Phys. Rev. Lett. 114, 020401 (2015)]. The general setting for such games is that of correlated strategies where the correlation between competing players is established through a trusted common adviser; however, players need not reveal their input to the adviser. So far, the quantum advantage in such games has been revealed in a restricted sense. Given a quantum correlated equilibrium strategy, one of the players can still receive a higher than quantum average payoff with some classically correlated equilibrium strategy. In this work, by considering a class of asymmetric Bayesian games, we show the existence of games with quantum correlated equilibrium where the average payoff of both the players exceeds the respective individual maximum for each player over all classically correlated equilibriums.
We give a concise and self-contained introduction to the theory of Quantum Games by reviewing the seminal works of Meyer, Eisert-Wilkens-Lewenstein, Marinatto-Weber and Landsburg, which initiated the study of this field. By generalizing this body of work, we formulate a protocol to Quantumize\textit{Quantumize}Quantumize any finite classical nnn-player game, and use a novel approach of describing such a Quantum Game in terms of commuting Payoff Operators. We describe what advantages can be gained by players by quantumizing such a game, particularly, what additional Nash Equilibria the players can achieve and the Pareto-Optimality of these additional equilibria.