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arXiv:0910.5275 · Journal of Physics A Mathematical and Theoretical

Uniqueness of Nash equilibria in a quantum Cournot duopoly game

Yohei Sekiguchi, Kiri Sakahara, Takashi Sato

Published Mar 18, 2010• 33 citations
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Abstract

A quantum Cournot game whose classical form game has multiple Nash equilibria is examined. Although the classical equilibria fail to be Pareto optimal, the quantum equilibrium exhibits the following two properties: (i) if the measurement of entanglement between strategic variables chosen by the competing firms is sufficiently large, the multiplicity of equilibria vanishes, and (ii) the more strongly the strategic variables are entangled, the more closely the unique equilibrium approaches to the optimal one.

DOI
10.1088/1751-8113/43/14/145303

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A quantum Cournot game whose classical form game has multiple Nash equilibria is examined. Although the classical equilibria fail to be Pareto optimal, the quantum equilibrium exhibits the following two properties: (i) if the measurement of entanglement between strategic variables chosen by the competing firms is sufficiently large, the multiplicity of equilibria vanishes, and (ii) the more strongly the strategic variables are entangled, the more closely the unique equilibrium approaches to the optimal one.
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Abstract In the Cournot duopoly market, each firm is not completely rational, one firm's possession of information is not symmetrical, and their Nash equilibria are also usually inferior to Pareto optimality. The intersection of quantum game theory, nonlinear dynamics and asymmetric information may be an effective way to solve this problem. In this paper, we set up a multi-period quantum Cournot duopoly game model with bounded rationality based on asymmetric information, investigate the effects of quantum entanglement and asymmetric information on the local stability of quantum Nash equilibrium and complex dynamic behavior of the system. The result shows that an increase in the quantum entanglement decreases the quantum Nash equilibrium local stability and accelerates the occurrence of the system’s chaotic behavior, while the asymmetric information stabilizes the quantum Nash equilibrium and suppresses the chaotic behavior.The contradictive impact on the local stability of quantum Nash equilibrium creates the interesting properties of the game. Numerical simulations demonstrate these complex dynamical features via bifurcation, strange attractors, fractal dimensions and sensitivity to initial conditions.
This paper investigates Nash equilibria in pure strategies for quantum approach to the Prisoner's Dilemma. The quantization process involves extending the classical game by introducing two additional unitary strategies. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. For each class, we identify and analyse all possible Nash equilibria. Our results reveal the complexity and diversity of strategic behaviour in the quantum setting, providing new insights into the dynamics of classical decision-making dilemmas. In the case of the standard Prisoner's Dilemma, the resulting Nash equilibria of quantum extensions are found to be closer to Pareto optimal solutions than those of the classical equilibrium.
A simulation scheme of quantum version of Cournot's duopoly is proposed, in which there is a new Nash equilibrium that may also be Pareto optimal without any entanglement involved. The unique property of this simulation scheme is decoherence-free against the symmetric photon loss. Furthermore, we analyze the effects of the asymmetric information on this simulation scheme and investigate the case of asymmetric game caused by asymmetric photon loss. A second-order phase transition-like behavior of the average profits of firms 1 and 2 in a Nash equilibrium can be observed with the change of the degree of asymmetry of the information or the degree of ‘virtual cooperation’. It is also found that asymmetric photon loss in this simulation scheme plays a similar role as that with the asymmetric entangled states in the quantum game.