H.N. Agiza

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.

Multistability in a dynamic Cournot game with three oligopolists

The time evolution of a dynamic oligopoly game with three competing firms is modeled by a discrete dynamical system obtained by the iteration of a three-dimensional non-invertible map. For the symmetric case of identical players a complete analytical study of the stability conditions for the fixed points, which are Nash equilibria of the game, is given. For the situation of several coexisting stable Nash equilibria a numerical study of their basins of attraction is provided. This gives, evidence of the occurrence of global bifurcations at which the basins are transformed from simply connected sets into non-connected sets, a basin structure which is peculiar of non-invertible maps. The presence of several coexisting attractors (or multistability) is observed even when complex attractors exist. Two different routes to complexity are presented: one related to the creation of more and more complex attractors; the other related to the creation of more and more complex structures of the basins. Starting from the benchmark case of identical players, the effects of heterogeneous behavior of the players, causing the loss of the symmetry properties of the dynamical system, are investigated through numerical explorations.

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.

Analysis of a duopoly game with delayed bounded rationality

A dynamic of Cournot duopoly game is analyzed, where players use different production methods and choose their quantities with bounded rationality. A dynamic of nonlinear Cournot duopoly game is analyzed, where players choose quantities with delayed bounded rationality and similar methods of production. The equilibria of the corresponding discrete dynamical systems are investigated. The stability conditions of Nash equilibrium under a local adjustment process are studied. The stability of Nash equilibrium, as some parameters of the model are varied, gives rise to complex dynamics such as cycles of higher order and chaos. We show that firms using delayed bounded rationality have a higher chance of reaching Nash equilibrium. Numerical simulations are used to show bifurcations diagrams, stability regions and chaos.

On modeling epidemics Including latency, incubation and variable susceptibility

A cellular automata model for an epidemic is given.It includes inhomogeneous mixing, latency, incubation andvariable suscepibility to the disease. These features are shown to be extremely difficult to be studied using differential equations.