Nonlinear Duopoly Game Research Articles

Local and global analysis of a nonlinear duopoly game with heterogeneous firms

In this paper, we introduce a nonlinear duopoly game whose players are heterogeneous and their inverse demand functions are derived from a more general isoelastic demand. The game is modeled by a discrete time dynamic system whose Nash equilibrium point is unique. The conditions of local stability of Nash point are calculated. It becomes unstable via two types of bifurcations: flip and Neimark–Sacker. Some local and global numerical investigations are performed to show the dynamic behavior of game’s system. We show that the system is noninvertible and belongs to Z_{2}-Z_{0} type. We also show some multistability aspects of the system including basins of attraction and regions known as lobes.

On complex dynamic investigations of a piecewise smooth nonlinear duopoly game

Abstract In this paper, we introduce a nonlinear duopoly game which is modelled by a piecewise smooth map. The inverse demand functions of the game are derived from a proposed nonlinear utility function. In the game, we consider that the second firm remanufactures returned products and then sell them again in the market. This means that the second firm receives returned products after the first firm presents its products in the market. A discrete-time piecewise smooth map is used to describe this game based on bounded rationality mechanism. The phase space of the map is divided into two regions separated by a border curve. The map’s fixed points in both regions are obtained. Their conditions of stability are discussed. We carry out numerical experiments to confirm the local stability of these fixed points and identify types of bifurcations by which they become unstable. Through simulation we highlight on periodic cycles and chaotic attractors that jump from region to other or cross the border curve in the phase space.

Dynamic investigations in a duopoly game with price competition based on relative profit and profit maximization

This paper introduces an attempt to investigate the complex dynamics of a nonlinear duopoly game of players (firms) adopting prices as their decision variables. Two different models are introduced and studied in this paper. The models are built by following the traditional role in which players have limited information. The first model describes a competition between two firms whose main interest is to maximize their relative profits while the second is the classic Bertrand one with profit maximization. The paper shows several local and global analysis about the equilibrium points of each model. We show that classical Bertrand game studied in the second model gives better results than those obtained for the first one however the equilibrium in both models loses its stability via flip bifurcation only.

Chaotic Effect of Linear Marginal Cost 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 linear demand and quadratic cost functions. Two players with different expectations are considered; one is boundedly rational and the other thinks with adaptive expectations. We show that the model gives more complex chaotic and unpredictable trajectories as a consequence of change in the slope of the marginal costs. The chaotic features are justified numerically via computing Lyapunov numbers, sensitive dependence on initial conditions and the box dimension of the chaotic attractor.

Chaotic effect of constant marginal cost in nonlinear duopoly game with heterogeneous players

Chaotic effect of constant marginal cost in nonlinear duopoly game with heterogeneous players

Analysis of Nonlinear Duopoly Game: A Cooperative Case

We make further attempts to investigate equilibrium stability of a nonlinear Cournot duopoly game. Our studies in this paper focus on the cooperation that may be obtained among duopolistic firms. Discrete time scales under the assumption of unknown inverse demand function and linear cost are used to build our models in the proposed games. We introduce and study here an adjustment dynamic strategy beside the so-called tit-for-tat strategy. For each model, the stability analysis of the fixed point is analyzed. Numerical simulations are carried out to show the complex behavior of the proposed models and to point out the impact of the models’ parameters on the cooperation.

On the dynamics of a non-linear Duopoly game model

The Cournot duopoly game modeled by Kopel, with adaptive expectations, is generalized by introducing the self-diffusion and cross-diffusion terms. General properties, such as boundedness and uniqueness, are obtained. Non-linear stability results are reached by the analysis of the stability of a ODE system.

The dynamics of a differentiated duopoly with quantity competition

We analyse the dynamics of a Cournot duopoly with heterogeneous players to investigate the effects of micro-founded differentiated products demand on stability. The present study, which indeed modifies and extends Zhang et al. (2007) (Zhang, J., Da, Q., Wang, Y., 2007. Analysis of nonlinear duopoly game with heterogeneous players. Economic Modelling 24, 138–148) and Tramontana, F., (2010) (Tramontana, F., 2010. Heterogeneous duopoly with isoelastic demand function. Economic Modelling 27, 350–357), reveals that a higher degree of product differentiation may destabilise the market equilibrium. Moreover, we show that a cascade of flip bifurcations may lead to periodic cycles and ultimately chaotic behaviours. Since a higher degree of product differentiation implies weaker competition, then a theoretical implication of our findings, that also constitute a policy warning, is that a fiercer (weaker) competition tends to stabilise (destabilise) the unique positive Cournot–Nash equilibrium.

Heterogeneous duopoly with isoelastic demand function

In this paper we analyze a duopolistic market with heterogeneous firms when the demand function is isoelastic (Puu, T., 1991. Chaos in duopoly pricing. Chaos, Solitons and Fractals 1, 573–581.). We consider the same heterogeneous firms of Zhang et al. (Zhang, J., Da, Q., Wang, Y., 2007. Analysis of nonlinear duopoly game with heterogeneous players. Economic Modelling 24, 138–148.) introducing a nonlinearity in the demand function instead of the cost function. Stability conditions of the Nash equilibrium and complex dynamics are studied. In particular we show two different routes to complicated dynamics: a cascade of flip bifurcations leading to periodic cycles (and chaos) and the Neimark-Sacker bifurcation which originates an attractive invariant closed curve. Comparisons with respect to the Puu model and the model of Zhang et al. are performed.

RETRACTED: Analysis of nonlinear duopoly game with heterogeneous players

This article has been retracted: please see Elsevier Policy on Article Withdrawal ( http://www.elsevier.com/locate/withdrawalpolicy ). This paper has been removed on the grounds of plagiarism. This case was investigated by the REPEC plagiarism committee and plagiarism was confirmed. The complete plagiarism case is described at http://plagiarism.repec.org/zhang-da-wang.html Original article details: Physica A: Statistical Mechanics and its Applications, Volume 320, 15 March 2003, pages 512–524, DOI: http://dx.doi.org/10.1016/S0378-4371(02)01648-5

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.

Equilibrium selection in a nonlinear duopoly game with adaptive expectations

We analyze a nonlinear discrete time Cournot duopoly game, where players have adaptive expectations. The evolution of expected outputs over time is generated by the iteration of a noninvertible two-dimensional map. The long-run behavior is characterized by multistability, that is, the presence of coexisting stable consistent beliefs, which correspond to Nash equilibria in the quantity space. Hence, a problem of equilibrium selection arises and the long run outcome strongly depends on the choice of the players’ initial beliefs. We analyze the basins of attraction and their qualitative changes as the model parameters vary. We illustrate that the basins might be nonconnected sets and reveal the mechanism which is responsible for this often-neglected kind of complexity. The analysis of the global bifurcations which cause qualitative changes in the topological structure of the basins is carried out by the method of critical curves.