Abstract
•A Cournot duopoly game whose players use gradient rule and want to maximize their profits and social welfare is introduced.•The interior equilibrium point of the game can be destabilized due flip bifurcation only.•Two types of global bifurcations are occurred.•Global analysis shows the game map is noninvertible and possesses quite complicated attraction basins containing holes.
Considering a nonlinear price function a duopoly game with quantities setting is introduced. The two competitors in this game seek maximization of two different objectives. The first competitor want to detect the optimum of his/her production by maximizing an average of social welfare and profit while the second competitor wants to maximize his/her profit only. Due to the lack of market information, each competitor behaves rationally and so the bounded rationality mechanism is adopted in order to build the model describing the game. Studying the evolution of the game requires to investigate the model in discrete time periods. So a two-dimensional map is introduced to analyze the game’s evolution. For this map, we calculate its equilibrium points and study their stability. Through local and global dynamic analysis we prove that the Nash equilibrium point loses its stability because of flip bifurcation only. Other dynamic characteristics for the map such as contact bifurcation and multi-stability are analyzed. The obtained results show that the manifold of game’s map can be investigated based on a one-dimensional map whose analytical form looks like the famous logistic map. Through the critical curves analysis we prove that the phase plane of game’s map is divided into three zones that are Zi,i=0,2,4 and hence the map is noninvertible. Furthermore, an analysis of two types of contact bifurcation are discussed through simulation.