A.A. Elsadany

Nonlinear dynamics in the Cournot duopoly game with heterogeneous players

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 are presented to show that players with heterogeneous beliefs make the duopoly game behave chaotically. Also, we get the fractal dimension of the chaotic attractor of our map which is equivalent to the dimension of Henon map.

Chaotic dynamics in nonlinear duopoly game with heterogeneous players

In this study we investigate the dynamics of a nonlinear discrete-time duopoly game, where the players have heterogeneous expectations. Two players with different expectations are considered; one is boundedly rational and the other thinks with adaptive expectations. The stability conditions of the equilibria are discussed. We show how the dynamics of the game depend on the model parameters. We demonstrate that as some parameters of the game are varied, the stability of Nash equilibrium is lost through period doubling bifurcation. The chaotic features are justified numerically via computing Lyapunov exponents, sensitive dependence on initial conditions and the fractal dimension.

Complex dynamics and synchronization of a duopoly game with bounded rationality

A dynamic Cournot game characterized by players with bounded rationality is modeled by two non-linear difference equations. The stability of the equilibria of the discrete dynamical system is analyzed. As some parameters of the model are varied, the stability of Nash equilibrium is lost and the complex chaotic behavior occurs. Synchronization of two dynamic Cournot duopoly games are considered. In the case of identical players, such dynamical system becomes symmetric, and this implies that synchronized dynamics can be obtained by a simpler one-dimensional model whose dynamics summarizes the common behavior of the two identical players.

The dynamics of Bowley's model with bounded rationality

Abstract A nonlinear dynamical system which describe the time evolution of n-competitors in a Cournot game (Bowleys model) with bounded rationality is analyzed. The existence and stability of the equilibria of this system is studied. The stability conditions of the steady states for two and three players are explicitly computed. Complex behavior such as cycles and chaotic behavior are observed by numerical simulation. Delayed Bowleys with bounded rationality in monopoly is studied. We show that firms using bounded rationality with delay has a higher chance of reaching Nash equilibrium.

Analysis of nonlinear triopoly game with heterogeneous players

A nonlinear triopoly game with heterogeneous players is presented. We consider three types of players; boundedly rational, adaptive, and naive. A triopoly game is modelled by a three dimensional discrete dynamical system. The stability conditions of the equilibrium points are analyzed. Numerical simulations are used to show bifurcation diagrams, phase portraits, sensitive dependence on initial conditions and fractal dimension. The chaotic behavior of the model has been stabilized on the Nash equilibrium point, by the use of the Pyragas delay feedback control method.

Complex dynamics and chaos control of heterogeneous quadropoly game

The dynamical system of four heterogeneous firms is derived. Existence and stability conditions of the fixed points are investigated and also complex dynamics is studied. Numerical simulations are used to illustrate the complex behaviors of the proposed dynamic game. The chaotic behavior of the game has been controlled by using feedback control method.

Bifurcation analysis and chaos in a discrete reduced Lorenz system

In this paper, the discrete reduced Lorenz system is considered. The dynamical behavior of the system is investigated. The existence and stability of the fixed points of this system are derived. The conditions for existence of a pitchfork bifurcation, flip bifurcation and Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. The complex dynamics, bifurcations and chaos are displayed by numerical simulations.

Dynamics of a Cournot duopoly game with bounded rationality based on relative profit maximization

The dynamics of a Cournot duopoly with relative profits maximizations and costs function with externalities is considered. Results concerning the equilibria of the economic model and their stability are presented and the occurrence of bifurcations is stated. A double route to chaotic dynamics, via flip bifurcations and via Neimark-Sacker bifurcations for game is studied. Numerical experiments are presented.

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.

Nonlinear Cournot and Bertrand-type dynamic triopoly with differentiated products and heterogeneous expectations

In a differentiated triopoly model with heterogeneous firms, the local stability of the Nash equilibrium under both quantity and price competition is analyzed. We find that the presence of a firm following a gradient rule based on marginal profits, and a player with adaptive expectations, determines the local stability of the Nash equilibrium, regardless the competition type, while the effects of the degree of product differentiation on the stability depend on the nature of products. Moreover, the Nash equilibrium is more stable under quantity competition than under price competition.