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The Relationship Between Classical and Quantum Correlation in Games

Adam M. Brandenburger

Published Aug 3, 2008• 1 citation
OpenAlex

Abstract

Quantum games have been argued to extend classical games (Eisert-Wilkens-Lewenstein [8, 1999], Meyer [18, 1999], and subsequent papers). Levine [17, 2005] has questioned this assertion. We offer what we believe to be the simplest expression of the disagreement. In doing so, we also establish a general correspondence, at the probabilistic level, between the analyses of correlation in game theory and in quantum mechanics.

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Quantum games have been argued to extend classical games (Eisert-Wilkens-Lewenstein [8, 1999], Meyer [18, 1999], and subsequent papers). Levine [17, 2005] has questioned this assertion. We offer what we believe to be the simplest expression of the disagreement. In doing so, we also establish a general correspondence, at the probabilistic level, between the analyses of correlation in game theory and in quantum mechanics.
In these lecture notes we investigate the implications of the identification of strategies with quantum operations in game theory beyond the results presented in [J. Eisert, M. Wilkens, and M. Lewenstein, Phys. Rev. Lett. 83, 3077 (1999)]. After introducing a general framework, we study quantum games with a classical analogue in order to flesh out the peculiarities of game theoretical settings in the quantum domain. Special emphasis is given to a detailed investigation of different sets of quantum strategies.
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.
Quantum games have proposed a new point of view for the solution of the classical problems and dilemmas in game theory. Certain quantization relationships can be proposed with the objective that a game can be generalized into a quantum domain where the linear superposition of actions is allowed. This quantization let us describe and solution problems originated by conflicting or cooperative behaviors among the members of a system from the point of view of quantum mechanical interactions. This leads us to interesting aspects which only can be observed through the quantization of a game like the possibility of the entanglement between players, the definition of a socioeconomical temperature in a system, the collective welfare principle,the analysis of a game through elements of quantum information theory and the globalization process. The relationships between quantum mechanics and game theory are also studied and its shown how although both systems analyzed are described through two theories apparently different (quantum mechanics and game theory) both are analogous and thus exactly equivalents.