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The Complexion of Multi-period Quantum Cournot Duopoly Game with Bounded Rationality Based on Asymmetric Information

Xinli Zhang, Jiahe Zhang

Published Oct 5, 2022• 0 citations
Publisher

Abstract

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.

DOI
10.21203/rs.3.rs-2114399/v1

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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.
On the basis of the bounded rationality and adaptive expectations, we investigate the dynamics of quantum Cournot duopoly game with heterogeneous players by using the Li et al. scheme. Further, we discuss the effect of quantum entanglement on local stability of the system. The results show that: (i) The special form of quantum Cournot duopoly game is the classical Cournot duopoly game, and the quantum Cournot duopoly game with naïve expectation is also a special form of quantum Cournot duopoly game with adaptive expectation. (ii) The increase or decrease of the stable region is dependent on the degree of quantum entanglement. It is related to the difference of relative marginal cost. (iii) The numerical simulation reveals the complexity characteristics of quantum dynamic Cournot game system through the analysis of stability region, bifurcation, strange attractor and sensitivity to initial conditions, and shows the difference from the classical dynamic Cournot model.
The development of quantum information theory based on quantum entangled states is advancing rapidly and has promoted the emergence of quantum game theory. Quantum game theory is quantum information processing applied to classical games, which solves problems that classical games cannot solve. This paper explores the quantum Cournot-Bertrand game model of two firms in a duopoly market with bounded rationality and product substitutability. The first and second firms choose production output and sales price as their decision variables, respectively. We first define the quantum Cournot-Bertrand dynamical system and study the existence and locally stable conditions of equilibrium points. Then we explore bifurcation conditions and bifurcation types. Finally, we do the bifurcation analysis and explore the impacts of the model parameters as the potential market size of products, the two firms' linear marginal cost, the players' speed of strategy adjustment, the product substitution index, and the quantum entanglement degree on the Nash equilibrium stability and bifurcation using numerical analysis and simulation, and provide economic explanations for the results.
Abstract Building on the existing symmetric quantization model of the dynamic Cournot duopoly game (CDG) with asymmetric information, we extend it to an asymmetric quantization model and study the stability of the quantum Bayesian Nash equilibrium (QBNE) under heterogeneous expectations. We analyze the influence of various parameters on the stability of QBNE, with a particular focus on the impact of the parameter α on system stability. The results show that when α < 1, under the same parameters, the quantum strategy of the asymmetric quantization model is more favorable for stabilizing the market. However, when α > 1, the quantum strategy of the symmetric quantization model is more conducive to stabilizing the market.
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.
The current trend in economics research is to incorporate quantum mechanical concepts to increase the security of business models. This interdisciplinary field of study represents real-world market dynamics more closely than do its classical counterparts. In this paper, we shed light on the significance of the two-qubit entangling operators in controlling chaos. We introduce a modified Eisert-Wilkens-Lewenstein scheme in a nonlinear Cournot duopoly game with complete and incomplete information. By doing so, the following interesting results are obtained: To begin, monopoly in a duopoly game can be avoided with the use of special perfect entanglers. Also, the stability analysis shows that there exists a class of entangling operators which can stabilize an unstable system and vice versa. Second, numerical analysis highlights the two-qubit entangling operators which can stabilize a chaotic system or at least delay chaos. Finally, we show that with an appropriate choice of initial state and speed of adjustments, entangling operators can decrease the sensitivity of the system. In short, while we know the importance of entangling operators in quantum game theory, in this paper we indicate the significance of operators in the context of a chaotic system.
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.