Dynamic Cournot Duopoly Game with Delay

The delay Cournot duopoly game is studied. Dynamical behaviors of the game are studied. Equilibrium points and their stability are studied. The results show that the delayed system has the same Nash equilibrium point and the delay can increase the local stability region. Dedicated to Professor H. N. Agiza on the occasion of his 60th birthday 1. Introduction It is well known that the duopoly game is one of the fundamental oligopoly games [1, 2]. Even the duopoly situation is an oligopoly of two producers that can be more complex than one might imagine since the duopolists have to take into account their actions and reactions when decisions are made [3]. Oligopoly theory is one of the oldest branches of mathematical economics dating back to 1838 when its basic model was proposed by Bischi et al. [4]. In repeated duopoly game all players maximize their profits. Recently, the dynamics of duopoly game has been studied [5–11]. Bischi and Naimzada [8] gave the general formula of duopoly game with a form of bounded rationality. Agiza et al. [10] examined the dynamical behavior of Bowley’s model with bounded rationality. They also have studied the complex dynamics of bounded rationality duopoly game with nonlinear demand function [11]. In general, a player, in order to adjust his output, can choose his strategy rule among many available techniques. Na？ve, adaptive, and boundedly rational strategies are only a few examples. When literature deals with duopoly games, most papers focus on games with homogeneous players. Another branch of literature is interested in games with heterogeneous players. In this type of literatures, the assumption of players adopting heterogeneous rules to decide their production is, in our opinion, more realistic than the opposite case; see [12]. Agiza and Elsadany [13, 14] were of the first authors who studied games with heterogeneous players, and in particular they analyze the dynamic behaviors emerging in this kind of games. Recently, Zhang et al. [15] and Dubiel-Teleszynski [16] used the same technique to analyze a duopoly game with heterogeneous players and nonlinear cost function. In addition, Angelini et al. [17] and Tramontana [18] studied a duopoly game with heterogeneous players with isoelastic demand function. Other studies on the dynamics of oligopoly models with more firms and other modifications have been studied [19–21]. Also in the past decade, there has been a great deal of interest in chaos control of duopoly games because of its complexity [22–25]. Recently Askar [26] has shown complex dynamics such as bifurcation and