On a Bertrand Duopoly Game With Heterogeneous Expectations and Emissions Costs

This study was based on the dynamics of a nonlinear Bertrand-type duopoly game with differentiated goods, linear demand and a cost function that included emissions costs. This duopoly game was modeled with a system of two difference equations. Existence and stability of equilibria of this system were studied. It was shown that the model gave more complex chaotic and unpredictable trajectories as a consequence of changes in the speed of adjustment parameter and horizontal product differentiation parameter. Numerical simulations showed that a higher value of the speed of adjustment and a higher or lower (negative) value of product differentiation (weaker or fiercer competition) can destabilize the economy. The chaotic features were justified numerically via computing Lyapunov numbers and sensitive dependence on initial conditions. Also, it was shown that in this case of a duopoly game, there were stable trajectories, and a higher (lower) degree of product differentiation did not tend to destabilize the economy. Keywords: Bertrand duopoly game, discrete dynamical system, heterogeneous expectations, stability, chaotic behavio

Complexity And Chaos Control of A Cournot Duopoly Game With Relative Profit Maximization And Heterogeneous Expectations

The paper considers a Cournot-type duopoly game, in which linear demand and cost functions are used. The two players produce differentiated goods and offer them at discrete times on a common market. In the cost functions of the players, in addition to the production cost, the cost of transporting the products is also included. Each firm does not care only about its profits but also about the percentage of its opponents’ profits, using a generalized relative profit function. In this model, the players follow different strategies. More specifically, the first player is characterized as a bounded rational player while the second player follows an adaptive mechanism. The existence of the Nash Equilibrium is proved, and its stability conditions are found. The complexity that appears for some values of the game’s parameters is shown. A mechanism by which the chaotic behavior of the discrete dynamical system is presented, importing a new parameter m. The algebraic results are verified, and the apparent complexity is shown by plotting bifurcation diagrams and strange attractors, computing the Lyapunov numbers, and checking the system’s sensitivity on its initial conditions. Keywords: Cournot duopoly game, discrete dynamical system, heterogeneous expectations, stability, chaotic behavior, chaos contro

Chaotic dynamics in Bertrand model with technological innovation

In this paper, the dynamics of a Bertrand duopoly game with technology innovation have been studied, which contains boundedly rational and naive players. They have been analyzed that the stability of the equilibrium point, the bifurcation and chaotic behavior of the dynamic system. It has been proved that technology innovation has played a very important role in the stability of Nash equilibrium point. Technology innovation can enlarge the stability region of the speed and control the chaos of the dynamic system effectively

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

Stability analysis in a Bertrand duopoly with different product quality and heterogeneous expectations

We study the local stability properties of a nonlinear Bertrand duopoly with vertical differentiation and heterogeneous players under the hypotheses of both covered and uncovered markets. In the former case, the unique pure strategy Nash equilibrium can undergo a flip bifurcation when the extent of consumer's heterogeneity increases. In the latter, the quality differential determines the stability of prices over time. Numerical evidence is provided to show the occurrence of endogenous fluctuations

On the Stability of a Cournot Dynamic Game Under the Influence of Information

In this paper we study the impact of information on the stability of a dynamic Cournot – type duopoly game. We suppose that one player searches for information about his rival before he makes his decision. We study how the amount of information acquired by player, influences the stability of Nash equilibrium. The game is modeled with a system of two difference equations. Existence and stability of equilibrium of this system are studied. We show numerically that the model gives chaotic and unpredictable trajectories as a consequence of change in the information parameter, but in our case there are also stable trajectories for each value of the information parameter. To provide some numerical evidence for the chaotic behavior of the system we present various numerical results including bifurcation diagrams, strange attractors, Lyapunov numbers and sensitive dependence on initial conditions.     Keywords: Cournot dynamic game, Nash equilibrium, Stability, Chaos

On a Cournot Game with Bounded Rational Players, Convex, Log-Linear Demand and Consumer Surplus

Based on the Cournot oligopoly game and the nonlinear dynamics theory, we study the behavior of semi-public enterprises by considering corporate social responsibility into their objectives. The model that is established is a dynamical Cournot-type duopoly model with bounded rationality containing the consumer surplus. We suppose quadratic cost function and a convex, log-linear demand function. The game is modeled with a system of two difference equations. Existence and stability of equilibriums of this system are studied. More complex chaotic and unpredictable trajectories are resulted studying this discrete dynamical system. The complex dynamics of the system are demonstrated numerically via computing Lyapunov numbers, sensitivity dependence on initial conditions, and bifurcation diagrams. Keywords: Cournot duopoly game; Discrete dynamical system; Homogeneous expectations; Stability; Chaotic Behavior; Consumer Surplus

Efficient bargaining versus right to manage: a stability analysis in a Cournot duopoly with trade unions.

The present study considers a unionised (nonlinear) duopoly with two different labour market institutions, i.e. efficient bargaining (EB) and right to manage (RTM), to analyse product market stability under quantity competition with trade unions. We show that when the preference of unions towards wages is small, (i) the parametric stability region under RTM is higher than under EB, and (ii) a rise in the union power in the Nash bargaining played between firms and unions monotonically increases (resp. reduces) the parametric stability region under RTM (resp. EB). In contrast, when the preference of unions becomes larger, an increase in the union's bargaining power acts: (1) as an economic stabiliser when the union power is small; (2) as an economic de-stabiliser when the union power is high. In addition to established results with regard to equilibrium outcomes, our findings shed some light on the effects of how the labour market regulation affects out-of-equilibrium behaviours in a Cournot duopoly

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