Complex dynamics of a quantum Cournot duopoly game with two different objectives

In this work, a dynamic quantum Cournot duopoly game with two different objectives is proposed by applying the Li-Du-Massar quantization scheme. In this game, the first player is a (semi-) public firm adopting bounded rationality and aims to maximize the weighted sum of its own profit and social welfare, while the second player focus on the objective to maximize its own profit as a private firm with naïve expectation. The local stability conditions for the quantum Nash equilibrium of the system are analyzed. Numerical simulations are presented to display the dynamic behaviors including stability region, bifurcation and chaos diagrams, and sensitive dependence on initial conditions. The results of theoretical and numerical analysis show that a larger weight on profit or adjustment speed parameter can enhance the stability of the quantum Cournot duopoly system. Differently, a higher entanglement level will hasten the local instability of the quantum Nash equilibrium point. It is also shown that a larger quantum entanglement weakens the sensitivity to initial conditions. Moreover, the equilibrium of the system can loose stability via flip bifurcation while varying the value of the adjustment speed, and time-delayed feedback control method can be applied to stabilize the chaotic behaviors.