Abstract
In this paper, a Cournot game with two competing firms is studied. The two competing firms seek the optimality of their quantities by maximizing two different objective functions. The first firm wants to maximize an average of social welfare and profit, while the second firm wants to maximize their relative profit only. We assume that both firms are rational, adopting a bounded rationality mechanism for updating their production outputs. A two-dimensional discrete time map is introduced to analyze the evolution of the game. The map has four equilibrium points and their stability conditions are investigated. We prove the Nash equilibrium point can be destabilized through flip bifurcation only. The obtained results show that the manifold of the game’s map can be analyzed through a one-dimensional map whose analytical form is similar to the well-known logistic map. The critical curves investigations show that the phase plane of game’s map is divided into three zones and, therefore, the map is not invertible. Finally, the contact bifurcation phenomena are discussed using simulation.
Highlights
Mathematics 2021, 9, 3119. https://The duopoly market structure has been deeply studied and investigated as a significant aspect of economic dynamics and game theory
Some important dynamic characteristics such as stability, invariant manifold and global bifurcation for a duopoly game whose players adopt two different objective functions were analyzed in this manuscript
The obtained results showed that the Nash point can be destabilized through flip bifurcation only
Summary
The duopoly market structure has been deeply studied and investigated as a significant aspect of economic dynamics and game theory. Several investigations in the literature have analyzed the dynamic characteristics of such models based on several types of inverse and cost functions including linear and nonlinear ones. Other studies in the literature ([10,11,12,13,14,15,16,17,18]) have analyzed market structure and kinds of market competitions, including several types of inverse demand and cost functions in addition to different strategies used by competing firms for productions updating.The above studies and others in the literature have reported that such games possess periodic, quasi-periodic, and complex attractors with peculiar structures that route to chaos. The authors analyzed the game’s players that followed corporate social responsibility (CSR), which depended on managerial delegation.In [20], the weighted average between two objective functions, profit and social welfare, was maximized to seek the equilibrium points of Cournot–Bertrand model, whose productions are differentiated.
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