Global Dynamics and Synchronization in a Duopoly Game with Bounded Rationality and Consumer Surplus

Abstract

Based on the oligopoly game theory, a dynamic duopoly Cournot model with bounded rationality and consumer surplus is established. On the one hand, the type and the stability of the boundary equilibrium points and the stability conditions of the Nash equilibrium point are discussed in detail. On the other hand, the potential complex dynamics of the system is demonstrated by a set of 2D bifurcation diagrams. It is found that the bifurcation diagrams have beautiful fractal structures when the adjustment speed of production is taken as the bifurcation parameter. And it is verified that the area with scattered points in the 2D bifurcation diagrams is caused by the coexistence of multiple attractors. It is also found that there may be two, three or four coexisting attractors. It is even found the coexistence of Milnor attractor and other attractors. Moreover, the topological structure of the attracting basin and global dynamics of the system are investigated by the noninvertible map theory, using the critical curve and the transverse Lyapunov exponent. It is concluded that two different types of global bifurcations may occur. Because of the symmetry of the system, it can be concluded that the diagonal of the system is an invariant one-dimensional submanifold. And it is controlled by a one-dimensional map which is equivalent to the classical Logistic map. The bifurcation curve of the system on the adjustment speed and the weight of the consumer surplus is obtained based on the properties of the Logistic map. And the synchronization phenomenon along the invariant diagonal is discussed at the end of the paper.

Dynamics of a Heterogeneous Constraint Profit Maximization Duopoly Model Based on an Isoelastic Demand

“…e required condition of the local asymptotic stability of Nash equilibrium point is that the eigenvalues of the corresponding Jacobian matrix are inside the unit circle. According to the Jury criterion [34], the local stability conditions of the Nash equilibrium point are as follows:…”

“…e literature review reported above has shown that the issue of profit maximization for firms has been discussed without constraints in dynamic situation. Recently, certain games models have been investigated and have given rise to complex dynamics with some constraints, such as global analysis and multistability of Cournot duopoly market game based on consumer surpluses [34,35]. e effect of model parameters on game dynamics, coexisting attractors, and global dynamics of constrained competition games by both corporate social responsibility (CSR) and social welfare (SW) has been studied in [36,37].…”

The Impacts of Fairness Concern and Different Business Objectives on the Complexity of Dual-Channel Value Chains

“…(1) e manufacturer and the retailer sell the same products from two different channels on the basis of price competition, and the marginal cost of the product is c. (2) e manufacturer and the retailer can only obtain part of market information and have limited rationality in decision-making [43]. (3) Both manufacturer and the retailer consider fairness concern in the market competition [25].…”

The coevolution of the firm and the product attribute space

“…However, models à la Cournot have also been used in evolutionary settings to study, for example, reinforcement learning effects [30], implications of social and individual learning [31], impact of competing behavioral rules [32], effects of firm entry and exit on industry dynamics [33,34], and the influence of product differentiation on equilibrium stability [35]. Much recent work takes Cournot competition in dynamic settings as the steppingstone to study nonlinear behavior and equilibrium stability (e.g., [36,37,38,39]).…”