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arXiv:quant-ph/0506219

An Introduction to Quantum Game Theory

J. Orlin Grabbe

Published Jun 27, 2005• 21 citations
arXivHTMLPDF

Abstract

This essay gives a self-contained introduction to quantum game theory, and is primarily oriented to economists with little or no acquaintance with quantum mechanics. It assumes little more than a basic knowledge of vector algebra. Quantum mechanical notation and results are introduced as needed. It is also shown that some fundamental problems of quantum mechanics can be formulated as games.

DOI
10.48550/arxiv.quant-ph/0506219

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This essay gives a self-contained introduction to quantum game theory, and is primarily oriented to economists with little or no acquaintance with quantum mechanics. It assumes little more than a basic knowledge of vector algebra. Quantum mechanical notation and results are introduced as needed. It is also shown that some fundamental problems of quantum mechanics can be formulated as games.
Main papers on quantum games are written by physicists for physicists, and the inevitable exploitation of physics jargon may create difficulties for mathematicians or economists. Our goal here is to make clear the physical content and to stress the new features of the games that may be revealed in their quantum versions. We first introduce from scratch the most fundamental facts of finite-dimensional quantum mechanics. In the main sections the foundations of quantum games are built via the basic examples. We omit sometimes the lengthy calculations (referring to the original papers) once the physical part is sorted out and the problem is reformulated as pure game-theoretic problem of calculating the Nash or dominated equilibria. Finally we touch upon general theory of finite quantum static games and provide further links and references.
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.
We give a self contained introduction to a few quantum game protocols, starting with the quantum version of the two-player two-choice game of Prisoners dilemma, followed by a n-player generalization trough the quantum minority games, and finishing with a contribution towards a n-player m-choice generalization with a quantum version of a three-player Kolkata restaurant problem. We have omitted some technical details accompanying these protocols, and instead laid the focus on presenting some general aspects of the field as a whole. This review contains an introduction to the formalism of quantum information theory, as well as to important game theoretical concepts, and is aimed to work as an introduction suiting economists and game theorists with limited knowledge of quantum physics as well as to physicists with limited knowledge of game theory.
With utilizing some quantum mechanism into the game theory, quantum game theory provides us with the ability to achieve results not otherwise possible. In this paper, a brief overview of classical game theory and some game-theoretic terms are depicted. Then, an introduction to quantum theory is presented as well as some correlative works in this field. Finally, some well-known quantum games are analyzed and discussed, which shows its predominant effect on some games problems (the Prisoners' Dilemma, Parrondo's Games, etc.).
Quantum computing offers the possibility of massively parallel computing that scales to large problems in a way not possible for classical computers. It also may make possible rapid and secure forms of communication that are not possible with classical devices. As an offshoot of the quantum computing literature, a small set of papers has started to examine quantum games. The question naturally arises: what if anything does quantum game theory have for economists? This brief note attempt to summarize the sometimes impenetrable notation used in quantum physics. I argue that quantum games fall within the existing framework of correlated equilibrium, cheap-talk equilibrium and mechanism design theory, where the correlation and/or communication devices are limited in a way not terribly relevant to economic theory. The notation is taken from Cleve et al [2004], and interpreted using the Wikipedia. A quantum mechanical system has a state � , which is simply a � -dimensional