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(a), (b). The feasible set D⊂Q is depicted in white; the gray points are initial conditions producing unfeasible trajectories. (a) α=1.5, d=-0.1, w=0.5, and b1=b2=0.2. (b) α=1.5, d=-0.2, w=0.5, and b1=b2=0.2. (c) Critical curves of rank-0, LC-1, for system T and the parameter values as in (b). (d) Critical curves of rank-1, LC=T(LC-1), for the same parameter values as in panel (c). These curves separate the plane into the regions Z4, Z2, and Z0, whose points have a different number of preimages.
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This paper tackles the issue of local and global analyses of a duopoly game with price competition and market share delegation. The dynamics of the economy is characterised by a differentiable two-dimensional discrete time system. The paper stresses the importance of complementarity between products as a source of synchronisation in the long term,...
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Citations
... Bischi [20,21] studied multiple steadystate and path dependence through the complex structure of attracting basin. Fanti [22] discussed the phenomenon of multiple attractor coexistence. Agiza et al. [23] established a three-oligarch Cournot dynamic model and studied its multiattractors coexistence and several stable Nash equilibrium coexistence situations. ...
A two-stage duopoly Cournot game model with nonlinear inverse demand function and R&D spillover is established in this paper, and the local stability of the equilibrium points is further analyzed. The complex dynamical behaviors of the built model under different parameters are simulated numerically. The results show that when the speed of adjustment speeds or the degree of R&D spillover exceed a certain range, the Nash equilibrium point will lose stability and the system will enter into a chaotic state. In addition, the global dynamics are analyzed using the basin of attraction, it is found that the system presents path dependency, which implies that the final state of the system is related to its initial state.
... In Ref. [5], Cui et al. studied the bidding game model of duopoly airlines with nonlinear variable cost, analyzed the local and global characteristics of the system, and analyzed the influence of various parameters on the nonlinear system by numerical simulation. Fanti et al. [6] also established Bertrand duopoly model based on market share and carried out global dynamic analysis. The result shows that the stronger the complementarity of products, the more conducive to the stable behavior of both sides. ...
In this paper, we propose a dynamic Cournot duopoly game model based on isoelastic demand and strategic delegation. The local stability conditions of each equilibrium point of the system are discussed. We also investigate the influence of parameters on system stability, and we find that with smaller speed of adjustment, relatively large price elasticity, relatively small profit weight coefficient and marginal cost, the duopoly is more stable. Then the problem of coexistence of attractors is studied through the basin of attraction, and it is found that the jump phenomenon in the one-dimensional bifurcation diagram is due to coexistence of attractors. Finally, the structure of the basins of attraction and the global dynamics of the system are analyzed by using critical curves. It is found that when a critical curve or an attractor contact with the boundary of the basin of attraction, global bifurcations occur in the system.
... On the basis of them, various scholars have studied the oligopoly game under management delegation. L. Fanti et al. [27] studied the Bertrand duopoly competition model based on market share, and studied the multistability and synchronization phenomenon of the model. Y. Nakamura [28] concluded that in a management mixed duopoly game composed of a public company with a welfare maximization owner and a private company with a profit maximization owner, the relevant authorities did not have to regulate the free determination of both the timing of setting the incentive parameter and the content of strategic contracts based on the individual incentive of the enterprises. ...
... After that, A.K. Naimzada and L. Sbragia [43], G.I. Bischi et al. [44] and others analyzed and discussed the gradient adjustment mechanism and the local monopolistic approximation. A large number of scholars [12,27,31,45,46] have also adopted adjustment mechanism to make the game process dynamic and ameliorate the static game process. ...
This paper considers a mixed duopoly competition model, in which two enterprises produce heterogeneous commodities and compete for output in the same market. One enterprise separates the ownership from the management rights, and the enterprise owner entrusts a manager to manage the enterprise. The other enterprise considers consumer surplus. Through stability theory and numerical simulation, the local stability of each equilibrium point and the way the system performs bifurcation are analyzed. By using the basin of attraction, the global stability of the system is studied. The multistability is explained and the properties and some phenomena of the basin of attraction and the chaotic attractor are analyzed by using the knowledge of noninvertible map and critical curve. Finally, the influence of parameters on enterprise profit and objective is also analyzed, which is helpful for enterprises to make decisions in the process of game.
... A notable and useful tool to study the noninvertible map T S is the critical curves [16][17][18]. The rank-1 critical curve is denoted as LC, which is combined by the locus of points that may have two, or more, coincident rank preimages. ...
... When v = 0.8118, only one chaotic attractor A exists in the basin of attraction, as shown in Fig. 9(d). The value of the transverse Lyapunov exponent when v = 0.8118 is calculated as Λ nat ⊥ = 0.5266 > 0. With the help of critical curves and the method of "trial and error" [16][17][18], we take two red segments LC a −1 and LC b −1 as starting segment, which can be seen in Fig. 9(e). Then the boundary of this chaotic attractor A can be obtained by ∂A = ∪ 4 k=1 T S(k) (η), where η = A ∩ LC −1 and LC −1 = LC a −1 ∪ LC b −1 . ...
On the basis of constant conjectural variation, a static Cournot duopoly model is built first in this paper, where the effects of knowledge spillover and product differentiation have been parameterized. Through the first order conditions, we can obtain the static conjectural variation equilibrium. By using the gradient adjustment mechanism, a dynamic conjectural variation model, represented by a two-dimensional difference equation, is established, in which two firms have bounded rationality. Then, the local stability of the Nash equilibrium is analyzed. From further studies on the effects of conjectural variation to the dynamics of the established model, we conclude that the more intense the market competition, the bigger the size of the stability region. In general case, the two axes are invariant sets, while the diagonal line is also an invariant set in symmetric case. Through the method of numerical simulation, we can deeply understand the complex characteristics of dynamic behaviors and the means for economic activities. Finally, synchronized behaviors are also studied by the method of critical curves and basins of attraction.
... Suppose that A is an attractor of the map T, where A may be a Nash equilibrium, a cycle or other complex attractor. The basin of attraction of A contains all points that generate bounded trajectories converging to A, which can be given as [38]). We denote the basin of infinity as Bð1Þ, which is the set of points having unbounded trajectories. ...
In this paper, taking the factor of product service provided by the manufacturers into consideration, a static Bertrand duopoly game with service factor is studied first, in which these two oligarchs produce differentiated products. A dynamic Bertrand duopoly game with bounded rationality is established by using the gradient mechanism. Keeping the adjustment speeds in a relatively small range may help the long-term stable operation of the market. It is found that there is another 2-cycle, different from the one flip bifurcated from the fixed point, which may appear through a saddle-node bifurcation. The unstable set of the saddle cycle, connecting the saddle to the node, gives a closed invariant curve. In addition, the emergence of intermittent chaos implies that the established economic system has the capability of self-regulating, where PM-I intermittency and crisis-induced intermittency have been studied. With the help of the critical curves, the qualitative changes on the basin of attraction are investigated.
... The above are some examples of scholars studying the Cournot duopoly game model. Many scholars [15][16][17][18][19][20][21][22][23][24] have also studied the Bertrand duopoly game with price as the decision variable. Zhang [15] studied a Bertrand oligopoly model based on the level of product differentiation with bounded rationality, and concluded that the adjustment speed would affect the stability of the internal equilibrium point, leading to bifurcation and chaos. ...
In this paper, we consider the Cournot-Bertrand duopoly mixed competition model, which is characterized by different decision-making variables and different objective functions of the two enterprises. This form of competition is more consistent with the complex actual economic market. The general stability conditions of the four equilibria are given and analyzed with eigenvalues and Jury criterion, so that enterprise decision makers could choose appropriate parameters to determine the development of the enterprise. Under the specific parameter conditions, the stability conditions are analyzed and demonstrated in detail by using two-dimensional bifurcation diagrams, stable region diagrams and bifurcation curves. Many fractal Arnold's tongues are found in two-dimensional bifurcation diagrams. These tongues are associated with the Neimark-Sacker bifurcation of the fixed point and are arranged in accordance with the organization law of the periodic tree of the Stern-Brocot tree, which is helpful for us to analyze the transitions of system between period and chaos. With the help of the basin of attraction, the coexistence of attractors is analyzed, on which the decision maker can choose the initial conditions that are beneficial to the development of the enterprise. We also analyze the topology structure of basin of attraction and the formation mechanism of holes in the basin of attraction by using the theory of critical curve and noninvertible map.
... As we all know, competition among players is a classical multi-player game. In order to further analyze the complicated reasons and regularities of dynamical evolution, many scholars have introduced nonlinear dynamics to explain and describe the intricate dynamical behaviors, such as stability [1], bifurcation [2,3], chaos [4,5], initial value sensitivity [6], multi-steady [7,8], etc. On the one hand, in terms of price competition in the market: For financial market research, Naimzada [9][10][11][12], et al. analyzed the nonlinear dynamics and multi-stability in a duopoly game model. ...
... The point E * could be regarded as an internal fixed point or known as unique Nash equilibrium point. Since system (7) describes the problem about dynamic price game of airlines, it has practical significance only when the equilibrium point is non-negative. According to the known conditions that α > 0, β > 0, γ > 0, and θ > 0, it can be determined that these three boundary equilibrium points all satisfy the non-negative. ...
In this paper, a duopoly airlines bidding game model based on nonlinear variable cost is built, then local and global dynamic characteristics of this bidding game model are extensively researched, with the purpose of discussing the influences of price adjustment speed and of price elasticity coefficient on the evolving regularity of bidding competition of airline. We study the model from both theoretical and numerical simulation aspects. The local and global properties of the model in two-parameter space are analyzed. Moreover, we systematically research the regularity of dynamic evolution about attractors and topological structures of attractors in symmetric and asymmetric information. The simulation results show that there are local steady and multi-steady in this nonlinear model. The adjustment speed and price elasticity coefficient do have significant effects on the model, specifically, once the adjustment speed and price elasticity coefficient exceed the threshold, the game model will irreversibly enter into chaotic state after multiple bifurcations. Furthermore, the speed of price adjustment and the coefficient of price sensitivity have a significant impact on the nonlinear game model. Even subtle changes can cause the system enter bifurcation phenomenon so far as to chaos state. Therefore, the airlines need prudently adjust the parameters of game model in order to keep market competition in steady state in the long run.
... Bischi et al. [8,12,13] used the method of critical curve to analyse the global dynamic behaviour of the Cournot oligopoly model. Fanti et al. [19][20][21] established a Bertrand oligopoly model of heterogeneous products and studied synchronization, intermittency and global dynamics of the built model. Agliari et al. [1] established a Cournot model of product differentiation and analysed the changes of attractors and basins of attraction in their model. ...
In this paper, the local and global dynamics of a two-stage Bertrand dynamic duopoly model with R&D spillover are studied. This paper aims to study the complex dynamics of the built model by changing the speed of adjustment and other parameters. Combining analysis tools and numerical techniques, we discuss the stability of the system and calculate the stability conditions at first. Then we infer the dynamic phenomena of the system with different sets of parameters and get some interesting results. If two firms operate under different market environment, the system will exhibit two kinds of chaotic intermittent phenomena. While if they operate in same market environment, the system will exhibit the synchronization and on–off intermittency. Finally, through the study of complex basins and multistability, we see how the number of attractors and the size or shape of the basin will change with the change of parameters.
... Gori et al. [2015] studied a delayed continuous Cournot oligopoly model, and they drew a conclusion that there would be no synchronization phenomenon according to the degree of time delay and inertia even in homogeneous firms. Fanti et al. [2015b] discussed the occurrence of synchronization in the Bertrand duopoly model by considering market share authorization. In [Bischi et al., 1999;Bischi & Lamantia, 2002;Bischi et al., 2015], Bischi offered a method by which the critical curve is able to study the chaotic synchronization of discrete dynamical system. ...
... As we know that an attractor A of T is asymptotically stable if and only if all the trajectories belonging to attractor A are transversely attractive, for an investigation on the stability of the attractor, its transverse Lyapunov exponent should be calculated as follows (see [Fanti et al., 2015b;Fanti et al., 2013]) ...
... In this case, if ∧ ⊥ < 0 holds, then the period-k cycle is transversely stable. The transverse Lyapunov exponent ∧ ⊥ is the natural transverse Lyapunov exponent ∧ nat ⊥ when the initial condition q(0) belongs to a generic aperiodic trajectory inlay in the chaotic attractors (see [Fanti (a) (b) al., 2015b;Fanti et al., 2013]). Therefore, a spectrum of transverse Lyapunov exponents of many unstable cycles, embedded in the chaotic attractor A along the diagonal, can be determined by the inequality (see [Fanti et al., 2013]). ...
Based on the oligopoly game theory, a dynamic duopoly Cournot model with bounded rationality and consumer surplus is established. On the one hand, the type and the stability of the boundary equilibrium points and the stability conditions of the Nash equilibrium point are discussed in detail. On the other hand, the potential complex dynamics of the system is demonstrated by a set of 2D bifurcation diagrams. It is found that the bifurcation diagrams have beautiful fractal structures when the adjustment speed of production is taken as the bifurcation parameter. And it is verified that the area with scattered points in the 2D bifurcation diagrams is caused by the coexistence of multiple attractors. It is also found that there may be two, three or four coexisting attractors. It is even found the coexistence of Milnor attractor and other attractors. Moreover, the topological structure of the attracting basin and global dynamics of the system are investigated by the noninvertible map theory, using the critical curve and the transverse Lyapunov exponent. It is concluded that two different types of global bifurcations may occur. Because of the symmetry of the system, it can be concluded that the diagonal of the system is an invariant one-dimensional submanifold. And it is controlled by a one-dimensional map which is equivalent to the classical Logistic map. The bifurcation curve of the system on the adjustment speed and the weight of the consumer surplus is obtained based on the properties of the Logistic map. And the synchronization phenomenon along the invariant diagonal is discussed at the end of the paper.
... The present article represents an extension in a research agenda on nonlinear duopolies with price competition, horizontal product differentiation, and managerial firms. In particular, it extends some results obtained in Fanti et al. (2014Fanti et al. ( , 2015 by considering the case in which managers are paid according to relative profit delegation contracts. The existence of a delegation variable in a context with limited information is responsible for rich dynamic phenomena, such as complex dynamics, multistability, and complex basins, emerging when products are complements or substitutes. ...
... A. The structure of the feasible set As it has been stressed, we are interested in the dynamics produced by system (7) for any initial condition belonging to Q. Obviously, a trajectory is economically meaningful only if, at any time t, the two prices x and y belong to Q. We pursue the following definition (see also Fanti et al. (2014Fanti et al. ( , 2015): ...
... If A D is an attracting set of / then in order to study the stability of A for T b we have to consider the transverse stability, as in Fanti et al. (2014Fanti et al. ( , 2015 for the case of market share bonus, so that we now underline the main results related to the present model in comparison with the previous ones, and we refer to them for further details. ...
In this article, we investigate the local and global dynamics of a nonlinear duopoly model with price-setting firms and managerial delegation contracts (relative profits). Our study aims at clarifying the effects of the interaction between the degree of product differentiation and the weight of manager's bonus on long-term outcomes in two different states: managers behave more aggressively with the rival (competition) under product complementarity and less aggressively with the rival (cooperation) under product substitutability. We combine analytical tools and numerical techniques to reach interesting results such as synchronisation and on-off intermittency of the state variables (in the case of homogeneous attitude of managers) and the existence of chaotic attractors, complex basins of attraction, and multistability (in the case of heterogeneous attitudes of managers). We also give policy insights.
