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Phase Transitions in Quantum Games

Jiangfeng Du, Hui Li, Xiaodong Xu, Xianyi Zhou, Rongdian Han

Published Nov 26, 2001• 0 citations
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Abstract

The discontinuous dependence of the properties of a quantum game on its entanglement has been shown up as phase transitions [Phys. Rev. Lett, 88, 137902]. In this paper we explore this kind of phase transitions in quantum games and reveal the underlying physical foundations, using a method which is appropriate for the task here. For the generalized version of Prisoners' Dilemma, we find that, with different settings of the numerical values in the payoff table, even though the classical game behaves the same, the quantum game behaves greatly differently and exhibits novel features in view of phase transitions.

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The discontinuous dependence of the properties of a quantum game on its entanglement has been shown up as phase transitions [Phys. Rev. Lett, 88, 137902]. In this paper we explore this kind of phase transitions in quantum games and reveal the underlying physical foundations, using a method which is appropriate for the task here. For the generalized version of Prisoners' Dilemma, we find that, with different settings of the numerical values in the payoff table, even though the classical game behaves the same, the quantum game behaves greatly differently and exhibits novel features in view of phase transitions.
The discontinuous dependence of the properties of a quantum game on its entanglement has been shown up to be very much like phase transitions viewed in the entanglement-payoff diagram [J. Du et al., Phys. Rev. Lett, 88, 137902 (2002)]. In this paper we investigate such phase-transition-like behavior of quantum games, by suggesting a method which would help to illuminate the origin of such kind of behavior. For the particular case of the generalized Prisoners' Dilemma, we find that, for different settings of the numerical values in the payoff table, even though the classical game behaves the same, the quantum game exhibits different and interesting phase-transition-like behavior.
The discontinuous dependence of the properties of a quantum game on its entanglement has been shown to be very much like phase transitions viewed in the entanglement-payoff diagram (J Du et al 2002 Phys. Rev. Lett. 88 137902). In this paper we investigate such phase-transition-like behaviour of quantum games, by suggesting a method which would help to illuminate the origin of such a kind of behaviour. For the particular case of the generalized Prisoners' Dilemma, we find that, for different settings of the numerical values in the payoff table, even though the classical game behaves the same, the quantum game exhibits different and interesting phase-transition-like behaviour.
In one of our previous works we investigated the correlation between a quantum game and its entanglement. For the particular case of the Prisoners' Dilemma, the result shows that the game's property depends discontinuously on the amount of the entanglement. This discontinuity can be considered as phase changes with respect to the varying of the game's entanglement. In this paper, we reveal the correlation between the phase changes and the numerical values in the payoff matrix of the game. We find that if these numerical values satisfy some certain condition, the phase in which the game has two asymmetric Nash equilibria will disappear. Furthermore, the phase changes exhibits interesting variation with respect to the numerical values in the payoff matrix.
This paper presents a new mathematical formalism that describes the quantization of games. The study of so-called quantum games is quite new, arising from a seminal paper of D. Meyer \cite{Meyer} published in Physics Review Letters in 1999. The ensuing near decade has seen an explosion of contributions and controversy over what exactly a quantized game really is and if there is indeed anything new for game theory. What has clouded many of the issues is the lack of a mathematical formalism for the subject in which these various issues can be clearly and precisely expressed, and which provides a context in which to present their resolution. Such a formalism is presented here, along with proposed resolutions to some of the issues discussed in the literature. One in particular, the question of whether there can exist equilibria in a quantized version of a game that do not correspond to classical correlated equilibria of that game and also deliver better payoffs than the classical correlated equilibria is answered in the affirmative for the Prisoner's Dilemma and Simplified Poker.