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Journal of Physics A Mathematical and General

Quantum Bertrand duopoly of incomplete information

Gan Qin, Xi Chen, Min Sun, Jiangfeng Du

Published Apr 25, 2005• 29 citations
Publisher

Abstract

We study Bertrand's duopoly of incomplete information. It is found that the effect of quantum entanglement on the outcome of the game is dramatically changed by the uncertainty of information. In contrast with the case of complete information where the outcome increases with entanglement, when information is incomplete the outcome is maximized at some finite entanglement. As a consequence, information and entanglement are both crucial factors that determine the properties of a quantum oligopoly.

DOI
10.1088/0305-4470/38/19/013

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We study Bertrand's duopoly of incomplete information. It is found that the effect of quantum entanglement on the outcome of the game is dramatically changed by the uncertainty of information. In contrast with the case of complete information where the outcome increases with entanglement, when information is incomplete the outcome is maximized at some finite entanglement. As a consequence, information and entanglement are both crucial factors that determine the properties of a quantum oligopoly.
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.
A general model of the quantum Stackelberg duopoly is constructed by introducing the "minimal" quantum structure into the Stackelberg duopoly with continuous distributed incomplete information, where both players only know the continuous distribution of the competitor's unit cost. In this model, the cases with complete information, discrete distributed incomplete information, and continuous distributed asymmetric information are all involved. Because of different roles played by the total information uncertainty and the information asymmetry, the game exhibits some new interesting features, such as the total information uncertainty can counteract or improve the first-mover advantage according to the value of the quantum entanglement. What's more, this general model will be helpful for the government to reduce the abuses of oligopolistic competition and to improve the economic efficiency.
The minimal quantization structure is employed to investigate the quantum version of the Stackelberg duopoly with continuous distributed asymmetric information, i.e. the first mover has incomplete information that obeys a continuous distribution while the second mover has complete information. It is found that the effects of the positive quantum entanglement on the outcomes exhibit many interesting features due to the information asymmetry. Moreover, although the first-mover advantage is counteracted by the information asymmetry, the positive quantum entanglement still enhances the first-mover advantage and improves the first-mover tolerance of the information asymmetry beyond the classical limit.
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.