On a New Cournot Duopoly Game

This paper presents a new Cournot duopoly game. The main advantage of this game is that the outputs are nonnegative for all times. We investigate the complexity of the corresponding dynamical behaviors of the game such as stability and bifurcations. Computer simulations will be used to confirm our theoretical results. It is found that the chaotic behavior of the game has been stabilized on the Nash equilibrium point by using delay feedback control method. 1. Introduction The classic model of oligopolies was proposed by the French mathematician A. Cournot [1]. He treated the case with nave expectations; at each time step players assume that the competitors produce the same quantity of goods already produced in the last period. The presence of complex dynamic phenomena in Cournot oligopoly models is well documented in the mathematical economics literature, starting from Rand [2] and Dana and Montrucchio [3]. The oligopoly market structure showing the action of only two companies is called duopoly. In duopoly game, each duopolist believes that he can calculate the quantity he should produce in order to maximize his profits. In fact, the properties of the best reply dynamics of Cournot duopoly games have been extensively studied by Puu [4, 5] who showed that trajectories may not converge to the Nash equilibrium and that complex trajectories are possible. Over the past decade, many researchers, such as Kopel [6], Bischi et al. [7], Ahmed and Agiza [8], and Agiza and Elsadany [9], have paid a great attention to the dynamics of games. The theoretical development of complex duopoly dynamics has been recently surveyed in [10, 11]. We consider a market consisting of a duopoly in which both firms, the domestic and the foreign firm, compete on quantities rather than price of production for a certain good. Let , , represent the quantity of th firm during the period and the selling prices. Suppose that the goods in a market are identical. The inverse demand functions of products come from the maximization by the representative consumer of the following fractional utility function: subject to the budget constraint Using Lagrange multiplier to maximize utility function (1) subject to the budget constraint (2), one gets In this paper, we present a new Cournot’s duopoly game by using inverse demand function which was deduced in (3). The dynamical behavior of this game and the stability conditions for the Nash equilibrium are given. Theoretical analysis and numerical simulations of the system are made in detail. Finally, we give a feedback control to control chaos and