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arXiv:1101.2388 · Journal of Physics A Mathematical and Theoretical

Simulation of continuous variable quantum games without entanglement

Shang-Bin Li

Published Jun 20, 2011• 12 citations
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Abstract

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.

DOI
10.1088/1751-8113/44/29/295302

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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.
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.
The aim of this paper is to investigate Cournot-type competition in the quantum domain with the use of the Li-Du-Massar scheme for continuous-variable quantum games. We derive a formula which, in a simple way, determines a unique Nash equilibrium. The result concerns a large class of Cournot duopoly problems including the competition, where the demand and cost functions are not necessary linear. Further, we show that the Nash equilibrium converges to a Pareto-optimal strategy profile as the quantum correlation increases. In addition to illustrating how the formula works, we provide the readers with two examples.
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.
Based upon a modification of Li et al. 's "minimal" quantization rules ( Phys. Lett. A 306 (2002) 73), we investigate the quantum version of the Cournot and Bertrand oligopoly. In the Cournot oligopoly, the profit of each of the N firms at the Nash equilibrium point rises monotonically with the measure of the quantum entanglement. Only at maximal entanglement, however, does the Nash equilibrium point coincide with the Pareto optimal point. In the Bertrand case, the Bertrand Paradox remains for finite entanglement ( i.e. , the perfectly competitive stage is reached for any N ⩾2), whereas with maximal entanglement each of the N firms will still have a non-zero shared profit. Hence, the Bertrand Paradox is completely resolved. Furthermore, a perfectly competitive market is reached asymptotically for N → ∞ in both the Cournot and Bertrand oligopoly.
We present a quantum model of Bertrand duopoly and study the entanglement behavior on the profit functions of the firms. Using the concept of optimal response of each firm to the price of the opponent, we find only one Nash equilibirum point for the maximally entangled initial state. The presence of quantum entanglement in the initial state gives payoffs higher to the firms than the classical payoffs at the Nash equilibrium. As a result, the dilemma-like situation in the classical game is resolved.
Playing the Cournot duopoly in the quantum domain can lead to the optimal strategy profile in the case of maximally correlated actions of the players. However, that result can be obtained if the fact that the players play the quantum game is common knowledge among the players. Our purpose is to determine reasonable game outcomes when players’ perceptions about what game is actually played are limited. To this end, we consider a collection consisting of the classical and quantum games that specifies how each player views the game and how each player views the other players’ perceptions of the game. We show that a slight change in how the players perceive the game may considerably affect the result of the game and, in the case of maximally correlated strategies, may vary from the inefficient Nash equilibrium outcome in the classical Cournot duopoly to the Pareto optimal outcome. We complete our work by investigating in the same way the Bertrand duopoly model.