Double route to chaos in an heterogeneous triopoly game

We move from a triopoly game with heterogeneous players (E.M.Elabassy et al., 2009. Analysis of nonlinear triopoly game with heterogeneous players. Computers and Mathematics with Applications 57, 488-499). We remove the nonlinearity from the cost function and introduce it in the demand function. We also introduce a different decisional mechanism for one of the three competitors. A double route to complex dynamics is shown to exist, together with the possibility of multistability of different attractors, requiring a global analysis of the dynamical system.Triopoly game; Heterogeneous players; Global analysis

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

Analysis of a duopoly game with heterogeneous players participating in carbon emission trading

In this paper, a price competition model with two heterogeneous players participating in carbon emission trading is formulated. The stable conditions of the equilibrium points of this system are discussed. Numerical simulations are used to show bifurcation diagrams, strange attractors, and sensitive dependence on initial conditions. We observe that the speed of adjustment of bounded rational player may change the stability of the Nash equilibrium and cause the system to behave chaotically. In addition, we find that the price of emission permits plays an important role in the duopoly game. The chaotic behavior of the system has been stabilized on the Nash equilibrium point by applying delay feedback control method

Effect of information asymmetry in Cournot duopoly game with bounded rationality

We investigate the effect of information asymmetry on a dynamic Cournot duopoly game with bounded rationality. Concretely, we study how one player’s possession of information about the other player’s behavior in a duopoly affects the stability of the Cournot-Nash equilibrium. We theoretically and numerically show that the information stabilizes the Cournot-Nash equilibrium and suppresses chaotic behavior in the duopoly

An evolutionary model with best response and imitative rules

We formulate an evolutionary oligopoly model where quantity setting players produce following either the static expectation best response or a performance-proportional imitation rule. The choice on how to behave is driven by an evolutionary selection mechanism according to which the rule that brought the highest performance attracts more followers. The model has a stationary state that represents a heterogeneous population where rational and imitative rules coexist and where players produce at the Cournot–Nash level. We find that the intensity of choice, a parameter representing the evolutionary propensity to switch to the most profitable rule, the cost of the best response implementation as well as the number of players have ambiguous roles in determining the stability property of the Cournot–Nash equilibrium. This marks important differences with most of the results from evolutionary models and oligopoly competitions. Such differences should be referred to the particular imitative behavior we consider in the present modeling setup. Moreover, the global analysis of the model reveals that the above-mentioned parameters introduce further elements of complexity, conditioning the convergence toward an inner attractor. In particular, even when the Cournot–Nash equilibrium loses its stability, outputs of players little differ from the Cournot–Nash level and most of the dynamics is due to wide variations of imitators’ relative fraction. This describes dynamic scenarios where shares of players produce more or less at the same level alternating their decision mechanisms

Complex Dynamics in Nonlinear Triopoly Market with Different Expectations

A dynamic triopoly game characterized by firms with different expectations is modeled by three-dimensional nonlinear difference equations, where the market has quadratic inverse demand function and the firm possesses cubic total cost function. The local stability of Nash equilibrium is studied. Numerical simulations are presented to show that the triopoly game model behaves chaotically with the variation of the parameters. We obtain the fractal dimension of the strange attractor, bifurcation diagrams, and Lyapunov exponents of the system