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Chinese Physics Letters

Quantum Stackelberg Duopoly of Continuous Distributed Asymmetric Information

Xia Wang, Xiaohua Yang, Miao Chu Lin, Zhou Xiang, Hu Cheng-zheng

Published Oct 17, 2007• 15 citations
Publisher

Abstract

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.

DOI
10.1088/0256-307x/24/11/007

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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.
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.
A more efficient model of the quantum Stackelberg duopoly is proposed by using the asymmetric quantization scheme of Qin et al. In this model, two entanglement factors α and γ are introduced, which greatly expands the functions of the previously reported symmetric one. By choosing proper values of α and γ, one can better manage the market, such as suppressing the first-mover advantage and enhancing the second-mover profit to avoid abuse of oligopolistic competition, and optimizing the total quantity of the products, so to overcome the deficiencies of "first mover always wins" and "positive quantum entanglement always reduces the total quantity" in the symmetric model. The proposed model here is believed to be a good tool for the government and the department to improve the economic efficiency and develop the market.
In this paper, we investigate the influence of entanglement and decoherence on the quantum Stackelberg duopoly (QSD) game. It is shown that the first-mover advantage can be weakened or enhanced due to the existence of entanglement for the QSD game without decoherence. The influence of decoherence induced by the amplitude damping and the phase damping are explicitly studied in the formalism of Kraus operator representations. We show that the amplitude damping drastically changes the Nash equilibrium of the QSD game and the profits of the two players while the phase damping does not affect the Nash equilibrium and the profits of the two players. It is found that under certain conditions there exists a 'critical point' of the damping parameter for the amplitude damping environment. At the 'critical point' the two players have the same moves and payoffs. The QSD game can change from the first-mover advantage game into the follower-mover advantage game when the damping parameter varies from the left-hand-side regime of the 'critical point' to the right-hand-side regime.
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.
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.
In search of a way to securely transfer information, the scientific world uses entanglement with several qubits. However, the experimental feasibility of implementing communication protocols remains unclear. In this work, we indicate a one-to-one correspondence between a quantum dialogue and the quantum Stackelberg duopoly game. If we consider a controller-dependent communication protocol to be a sequential quantum game, the two players, Alice and Bob, can be compared to two firms, [Formula: see text] and [Formula: see text]. Moreover, the controller Charlie can be assumed to be the arbiter. First mover advantage is a significant feature of the Stackelberg duopoly game, which is explained and proved in the protocol. Further, we propose a controller-independent bidirectional quantum dialogue using four-qubit maximally entangled states. The efficiency of the protocol is determined to be 50%.