Multistability and cyclic attractors in duopoly games

This study with the help of non-linear Cournot Duopoly model shows that duopoly market can be chaotic because of non-linearity. It establishes conditions for the stability of chaotic market. The study considers the conditions for generating chaos and controlling chaos from the perspectives of both the firms. It explains how adaptive expectations can be used due to inconsistency of naïve expectations with the help of autocorrelation coefficient and then explains the method of controlling chaos.

KnE Social Sciences

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...

Applied Mathematics and Computation, 2015

In this paper, dynamic duopolistic Cournot models are investigated with discrete time scales under the assumption of unknown inverse demand function and linear cost. With this motivation, we consider different types of models: bounded rational duopoly, Puu's duopoly, bounded rational duopoly with delay, and bounded rational multi-team model. In these models, the firms use two important adjustment mechanism, the bounded rationality and Puu's approach, to update their quantity in each period. The locally asymptotic stability of the fixed point of each model is investigated and complex dynamic characteristics including period doubling bifurcation, strang attractors and chaotic phenomena are also discussed. Numerical simulations are carried out to show such complex behavior of the four models and to point out the impact of the models' parameters on the stability of the fixed points.

The state trajectories of a special class of discrete dynamic economic systems will be considered. The control of their long-term behavior is a major research issue of dynamic markets. An N-firm production game known as oligopoly will be examined with isoelastic price function and linear costs. After the reaction functions of the firms are determined, dynamic systems with adaptive expectations and with adaptive adjustments will be introduced and the equivalence of their local asymptotic behavior will be verified. Then two special cases will be investigated in details, with two and three groups of identical firms, in which the dynamics is two and three dimensional, respectively. Stability conditions will be derived and the global behavior of the equilibria will be illustrated including chaos control.

Discrete Dynamics in Nature and Society, 2011

AnN-firm production game known as oligopoly will be examined with isoelastic price function and linear cost under al Cournot competition. After the best responses of the firms are determined, a dynamic system with adaptive expectations is introduced. It is first shown that the local asymptotic behavior of the system is identical with that of the adaptive adjustment process in which the firms cautiously determine their outputs. Dynamic analysis is confined to two special cases, one in whichNis divided into two groups and the other in whichNis divided into three groups. Then stability conditions will be derived and the global behavior of the equilibria will be illustrated including chaos control. Lastly the two- and three-group models are compared with two-firm (duopoly) and three-firm (triopoly) models to shed light on roles of the number of the firms.

Journal of Engineering Science and Technology Review, 2015

In this study we investigate the dynamics of a nonlinear discrete-time duopoly game, where the players have heterogeneous expectations linear demand and 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 marginal costs of the players. The chaotic features are justified numerically via computing Lyapunov numbers, sensitive dependence on initial conditions and the box dimension of the chaotic attractor.

Journal of Computational and Applied Mathematics, 2008

We consider a discrete map proposed by M. Kopel that models a nonlinear Cournot duopoly consisting of a market structure between the two opposite cases of monopoly and competition. The stability of the fixed points of the discrete dynamical system is analyzed. Synchronization of two dynamics parameters of the Cournot duopoly is considered in the computation of stability boundaries formed by parts of codim-1 bifurcation curves. We discover more on the dynamics of the map by computing numerically the critical normal form coefficients of all codim-1 and codim-2 bifurcation points and computing the associated two-parameter codim-1 curves rooted in some codim-2 points. It enables us to compute the stability domains of the low-order iterates of the map. We concentrate in particular on the second, third and fourth iterates and their relation to the period doubling, 1:3 and 1:4 resonant Neimark-Sacker points.

Chaos, Solitons & Fractals, 2000

This paper reconsiders the Cournot oligopoly (noncooperative) game with iso-elastic demand and constant marginal costs, one of the rare cases where the reaction functions can be derived in closed form. It focuses the case of three competitors, and so also extends the critical line method for non-invertible maps to the study of critical surfaces in 3D. By this method the various bifurcations of the attractors and their basins are studied. As a special case the restriction of the map to an invariant plane when two of the three ®rms are identical is focused. Ó

Mathematics, 2021

In this paper, a Cournot game with two competing firms is studied. The two competing firms seek the optimality of their quantities by maximizing two different objective functions. The first firm wants to maximize an average of social welfare and profit, while the second firm wants to maximize their relative profit only. We assume that both firms are rational, adopting a bounded rationality mechanism for updating their production outputs. A two-dimensional discrete time map is introduced to analyze the evolution of the game. The map has four equilibrium points and their stability conditions are investigated. We prove the Nash equilibrium point can be destabilized through flip bifurcation only. The obtained results show that the manifold of the game’s map can be analyzed through a one-dimensional map whose analytical form is similar to the well-known logistic map. The critical curves investigations show that the phase plane of game’s map is divided into three zones and, therefore, the...

2011

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.

Nonlinear Dynamics, 2015

In this paper we propose and compare three heterogeneous Cournotian duopolies, in which players adopt best response mechanisms based on different degrees of rationality. The economic setting we assume is described by an isoelastic demand function with constant marginal costs. In particular, we study the effect of the rationality degree on stability and convergence speed to the equilibrium output. We study conditions required to converge to the Nash equilibrium and the possible route to destabilization when such conditions are violated, showing that a more elevated degree of rationality of a single player does not always guarantee an improved stability. We show that the considered duopolies exhibit either a flip or a Neimark-Sacker bifurcation. In particular, in heterogeneous oligopolies models, the Neimark-Sacker bifurcation usually arises in the presence of a player adopting gradient-like decisional mechanisms, and not best response heuristic, as shown in the present case. Moreover, we show that the cost ratio crucially influences not only the size of the stability region, but also the speed of convergence toward the equilibrium.

International Game Theory Review, 2010

This study reconsiders a duopoly model with advertisement introduced earlier by Ahmed, Agiza and Hassan [1999]. It demonstrates three major findings. The first is that the model can be destabilized via either flip bifurcation or Hopf bifurcation. The second is that a half-pitchfork bifurcation of the output occurs when the advertisement dynamics is periodic and the nonlinearity of the output dynamics becomes stronger. Finally the third is that the existence of attractor and the coexistence of attracting sets are the main features of the model when it is locally unstable.

From a two-agent, two-strategy congestion game where both agents apply the multiplicative weights update algorithm, we obtain a two-parameter family of maps of the unit square to itself. Interesting dynamics arise on the invariant diagonal, on which a two-parameter family of bimodal interval maps exhibits periodic orbits and chaos. While the fixed point b corresponding to a Nash equilibrium of such map f is usually repelling, it is globally Cesàro attracting on the diagonal, that is, lim n→∞ 1 n n−1 k=0 f k (x) = b for every x in the minimal invariant interval. This solves a known open question whether there exists a nontrivial smooth map other than x → axe −x with centers of mass of all periodic orbits coinciding. We also study the dependence of the dynamics on the two parameters.

Communications in Nonlinear Science and Numerical Simulation, 2015

We study a heterogeneous duopolistic Cournotian game, in which the firms, producing a homogeneous good, have reduced rationality and respectively adopt a "Local Monopolistic Approximation" (LMA) and a gradient-based approach with endogenous reactivity, in an economy characterized by isoelastic demand function and linear total costs. We give conditions on reactivity and marginal costs under which the solution converges to the Cournot-Nash equilibrium. Moreover, we compare the stability regions of the proposed oligopoly to a similar one, in which the LMA firm is replaced by a best response firm, which is more rational than the LMA firm. We show that, depending on costs ratio, the equilibrium can lose its stability in two different ways, through both a flip and a Neimark-Sacker bifurcation. We show that the nonlinear, noninvertible map describing the model can give rise to several coexisting stable attractors (multistability). We analytically investigate the shape of the basins of attractions, in particular proving the existence of regions known in the literature as lobes.

International Journal of Non-Linear Mechanics, 2013

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.