Nonlinear dynamics and chaos control of the quantum Cournot-Bertrand duopoly game model

This paper examines a duopoly market comprising two firms, where the first firm optimizes its output and the second firm optimizes its price. We establish a quantum Cournot-Bertrand dynamical system from the perspective of quantum game theory to study the existence and stability conditions of equilibrium points and conduct bifurcation analysis on equilibrium points. A controlled quantum Cournot-Bertrand dynamical system is also established, which delays, advances, or eliminates the generation of bifurcations and chaos by appropriately selecting model parameter values. Finally, numerical simulation and analysis are conducted to verify the theoretical correctness of the research model and explore the influence of model parameter values on quantum game strategies. We also provide relevant economic explanations in the process. The research results indicate that the potential market size increases will lead to the instability of quantum strategy. The impact of the two firms' cost control parameters on their respective quantum strategy is reversed. The second firm's control parameter of cost continuously decreasing will lead to its quantum strategy entering bifurcations and chaos. The quantum strategy's sensitivity to the second firm's marginal cost, strategy adjustment speed, and quantum entanglement degree is higher than that of the first firm's marginal cost, strategy adjustment speed, and quantum entanglement degree, respectively. The continuous increase of other parameter values will lead to the second firm's quantum strategy entering bifurcation and chaos, followed by the first firm's quantum strategy entering bifurcation and chaos.