The relationship between quantum and classical correlation in games

Şahin Kaya Özdemir, Junichi Shimamura, Fumiaki Morikoshi +1 more

Adam M. Brandenburger

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.

M. Szopa

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.

Piotr Frąckiewicz

Ramón Alonso‐Sanz

Esteban Guevara Hidalgo

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.