Complexity Of Nash Equilibrium In Quantum Bertrand Game Model Under Different Rational Expectations

Quantum game theory is an interdisciplinary subject that studies game theory with quantum information theory as a tool.Because quantum game can solve many problems that can not be solved in classical game theory,it has been paid more and more attention by scholars at home and abroad.While oligopoly is the most common market structure in the real society,Bertrand game model is one of the most basic models in oligopoly,which helps enterprises avoid profit loss caused by price war.But the core problem of Nash equilibrium has always puzzled scholars.In this paper,a quantum Bertrand nonlinear dynamic model based on different rational expected behaviors is constructed by using the relevant knowledge of quantum game theory and nonlinear dynamics theory.The effects of quantum entanglement on the local stability conditions of Nash equilibrium and the complex dynamics of the system are analyzed,which provides useful enlightenment for the application of the Bertrand game model in real life.The main content of this thesis has the following two parts:(1)Aiming at the complexity of the Bertrand oligopoly quantum Nash equilibrium,quantum entanglement is implanted into the classical Bertrand game model on the theoretical basis of the integration of quantum game theory and nonlinear dynamics theory under the condition of homogenous expectation of participants.The paper establishes a quantum Bertrand nonlinear dynamic model of participant bounded rationality anticipatory behavior.We analyzed the effect quantum entanglement degree and price adjustment speed on stability of the equilibrium points and complex dynamics behaviors.The results show that the speed of price adjustment affects the stability of the Nash equilibrium solution.When the price adjustment speed is not in the stable region,the system will have bifurcation or chaos phenomenon.Quantum entanglement degree can enhance the stability of the system and provide an effective measure and method for controlling the emergence of chaos in the system.(2)According to the participants' different cognition of oligopoly market,their rational expectation behaviors are also different.The paper establishes a dynamic quantum Bertrand duopoly game by applying quantum game theory and nonlinear dynamic system theory.Two types of players are considered: bounded rationality and naive expectation.We analyze the influence of quantum entanglement degree on stability of the equilibrium points and dynamics behavior of the system.Through theoretical analysis and numerical simulation,the result shows that the quantum entanglement degree can enhance the stability of the system.When the adjustment speeds of the two firms' output reaches a certain level,it will lead to complex chaotic characteristics of the system,and the entanglement degree can control the system's chaos effectively.Finally,the accuracy of the theory is verified by numerical simulation.Compared with the assumption of homogeneous expectations,the heterogeneous expectations are more likely to cause the complexity of the equilibrium point and bring it into the chaotic state,and the control factors can be selected to control the chaos of the system.