Abstract
In this paper, some important dynamic characteristics such as multistability and synchronization phenomena are investigated for a game of an economic Cournot duopoly whose time evolution is received by the iteration of a noninvertible map in the plane. In the asymmetric case, the equilibrium points of game’s map are calculated, and their stability conditions are obtained. The obtained results show that the Nash equilibrium point loses its stability through flip bifurcation. Under some restrictions, the map’s coordinate axes form an invariant manifold, and hence their dynamics are studied based on a one-dimensional discrete dynamic map. In the symmetric case where both firms are identical, the map has the property of symmetry, and this implies that the diagonal q 1 = q 2 forms an invariant manifold and therefore synchronization phenomena occur. Global analysis of the behavior of the noninvertible map is carried out through studying critical manifolds of the map that categorize it as Z 4 − Z 2 − Z 0 type. Furthermore, global bifurcation of the basins of attraction is confirmed through contact between the critical curves and the boundaries of escaping domain.
Highlights
The market possesses only two competing firms whose target is seeking the optimal quantities in the case of Cournot by maximizing their profits, or seeking the optimal prices in the case of Bertrand
Based on a linear inverse demand functions and adopting the bounded rationality mechanism, a Cournot duopoly game with players seeking the maximization of their average profits is introduced and analyzed in this manuscript
Due to the above discussion, we have introduced a general case with different objectives. e game is described by a nonlinear discrete dynamic map that possesses four equilibrium points, three of which are boundary points, and the fourth is an interior point and coincides with Nash equilibrium point
Summary
Dynamic economic games whose evolutions has been depended on discrete time maps have attracted many researchers due to the interesting dynamic behaviors and bifurcations types occurring in such maps. E first rule among those rules which has been extensively adopted is known as bounded rationality It has been deeply adopted in many studies for process of modeling discrete time maps representing the evolutions of these games. It is described as a gradient rule because it depends on firms’ marginal profits and requires firms to carry out an estimation of their marginal profits to determine whether they increase or decrease, so they can update their outputs period of time. Based on a linear inverse demand functions and adopting the bounded rationality mechanism, a Cournot duopoly game with players seeking the maximization of their average profits is introduced and analyzed in this manuscript.
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