Abstract:
This paper focuses on the role that production subsidies play in a Bertrand mixed duopoly. The paper examines four regimes: mixed and private duopoly, each with and without subsidies. The results of this study are compared with the findings of the existing Cournot mixed market literature. As a result, the paper shows that that the introduction of production subsidies into the analyses of Bertrand and Cournot mixed markets can improve social welfare.

Abstract:
The aim of the paper is to study the Bertrand duopoly example in the quantum domain. We use two ways to write the game in terms of quantum theory. The first one adapts the Li-Du-Massar scheme for the Cournot duopoly. The second one is a simplified model that exploits a two qubit entangled state. In both cases we focus on finding Nash equilibria in the resulting games.

Abstract:
Interesting explanation of an adjustment process in a Cournot duopoly game is shown in Varian[1]’s textbook, where players adjust their strategies sequentially. It is generally assumed that players adjust simultaneously. We investigate a game that determines whether to adjust or not. First, Nash principle is assumed and as the solution whether sequential or simultaneous is derived. The result depends on the initial state of the output of both firms. In some regions firms adjust simultaneously, and in other regions sequentially. The game under maximin principle is also examined and we compare the results with that of the Nash game.

Abstract:
We present the quantum model of Bertrand duopoly and study the entanglement behavior on the profit functions of the firms. Using the concept of optimal response of each firm to the price of the opponent, we found only one Nash equilibirum point for maximally entangled initial state. The very presence of quantum entanglement in the initial state gives payoffs higher to the firms than the classical payoffs at the Nash equilibrium. As a result the dilemma like situation in the classical game is resolved.

Abstract:
Cournot (1838) anticipated Nash's definition of equilibrium by overa century, but only in the context of a particular model of duopoly.Not surprisingly, Cournot's work is one of the classics of gametheory; it is also one of the cornerstones of the theory of industrialorganization. We consider a modification of the Cournot's modelwith an uncertainty in the demand. We find the Bayesian Nashequilibrium of the game.

Abstract:
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

Abstract:
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

Abstract:
This paper considers domestic (resp. international) Bertrand mixed duopoly competition in which a state-owned welfare-maximizing public firm and a domestic (resp. foreign) profit-maximizing private firm produce complementary goods. The main purpose of the paper is to present and to compare the equilibrium outcomes of the two mixed duopoly models.

Abstract:
We construct a Cournot duopoly model with production externality in which reaction functions are unimodal. We consider the case of a Cournot model which has a stable equilibrium point. Then we show the existence of analytic solutions of the model. Moreover, we seek general solutions of the model in the form of nonlinear second-order difference equation.

Abstract:
We analyze a nonlinear discrete-time Cournot duopoly game, where players have heterogeneous expectations. Two types of players are considered: boundedly rational and naive expectations. In this study we show that the dynamics of the duopoly game with players whose beliefs are heterogeneous, may become complicated. The model gives more complex chaotic and unpredictable trajectories as a consequence of increasing the speed of adjustment of boundedly rational player. The equilibrium points and local stability of the duopoly game are investigated. As some parameters of the model are varied, the stability of the Nash equilibrium point is lost and the complex (periodic or chaotic) behavior occurs. Numerical simulations is presented to show that players with heterogeneous beliefs make the duopoly game behaves chaotically. Also we get the fractal dimension of the chaotic attractor of our map which is equivalent to the dimension of Henon map.