The complex dynamics which occur in the process of chaotic evolution have been investigated. We can easily get its Jacobian matrix as follows:wherewhere and are defined in (14). The stability analysis of naive and bounded rationality oligopoly games has been discussed in [6]. Firms with bounded rationality do not have the total information of the game; thus, the settling yield choices depend on a local estimate of the marginal profit . I propose a dynamic duopoly model where firms enter simultaneously but compete hierarchically á la Stackelberg at each instant over time. Dynamical heterogeneous duopoly games and their control are investigated in many other research studies [21–24]. The Cournot Model: The oldest determinate solution to the duopoly problem is by the French economist, A.A. Cournot in 1838, who took the case of two mineral water springs situated side by side and owned by two firms A and B. Assumptions: Models of Oligopoly • An oligopolyis a market with only a few important sellers. System (13) has four fixed points: Such simulations contain bifurcation diagrams, phase portrait, and the maximal Lyapunov exponents (MLEs), to further investigate the unpredictable behavior of the game. In simple words, let us assume a market with three players – A, B, and C. We analyze two different scenarios: When qualities are endogenous, the retailer is always better off when it integrates with the low-quality manufacturer, if the setup costs of integration are ignored. For the stochastic, Stackelberg model (SSM), the authors obtained some sufficient conditions for the, globally asymptotical stability of the Nash equilibrium. The trace of the is given by In duopoly, it refers to a situation in which one firm, the Stackelberg leader, can commit to its output first.1 The second mover, the Stackelberg follower, produces its quantity knowing the output of /Resources 11 0 R For pioneering works on … The model of a dynamic production game is formulated in yield decision making process. /Contents 30 0 R /CreationDate (D:20150914141133) Numerical simulations demonstrate that the system with varying model parameters may be driven to chaos and the loss of stability may be caused by period doubling bifurcations. Oligopoly game which contains two firms is called duopoly game; these two firms are in a competition and there is no collaboration among them. For this reason comes our contributions in this paper. Numerical simulations were used to show bifurcation diagrams. This work is supported by Sichuan Science and Technology Program, China under Grant 2017JY0336 and Hunan Science, and Technology Program, China under Grant 2019JJ50399, Longshan Talent Research Fund of Southwest University of, Science and Technology, China under Grant 17LZX670 and 18LZX622. School of Science, Southwest University of Science and Technology, Mianyang 621010, China, Institute of Modeling and Algorithm, Southwest University of Science and Technology, Mianyang 621010, China. The price in the market depends on the marginal cost of this dominant firm. Thus, in order to control the spread of the disease, we must decrease the speed of reversion or increase the intensity of volatility. Numerical simulation presents period doubling bifurcation, maximal Lyapunov exponent and chaos. Stackelberg’s duopoly model (1934) How do the conclusions of the Cournot’s duopoly game change when the firms move sequentially? All figure content in this area was uploaded by Tian Baodan, All content in this area was uploaded by Tian Baodan on Aug 27, 2020, Stochastic perturbations for a duopoly Stackelberg model, In this paper, a duopoly Stackelberg model of competition on output with stochastic, perturbations is proposed. The main findings are the following. Comput. /CropBox [0.0 0.0 612.0 792.0] represents the intensity of the white noise, ]), we can also study the stochastic economic models proposed in this paper. In the next section, we will discuss the fractional mechanism of the marginal profit. 5 0 obj In fact, by solving the DSM, When the system is subject to external noise, it is clear to see that the output of the two firms has a significant, oscillation for a long period of time, and then eventually stabilized around the Nash equilibrium of the deterministic, above numerical experiments for 1000 times and count the output distribution law of the two firms at, One can see that, under the controlling conditions (H1) and (H2), the output of the two firms will be stabilize around the. J.B. Rosser, The development of complex oligopoly dynamics theory, in: Text Book Oligopoly Dynamics: Models and Tools, Springer-Verleg. Moreover, in the Cournot model, it is assumed that the. Stackelberg model is a leadership model that allows the firm dominant in the market to set its price first and subsequently, the follower firms optimize their production and price. Agiza, Analysis of a duopoly game with delayed bounded rationality, Appl. We will study the impact of the game parameters on dynamics of game (12), the speed of adjustment parameter , the generalized bounded rationality parameter , and the maximum price in the market , and these are discussed in the following. This result is also applicable to the three-oligarch game model or. The duopoly Stackelberg model As we mentioned before, our aim of this work is to analyze the effect of fractional marginal profit in a duopoly game. ԃ�&Q6�c���`��j{�����j\~ ܀��jc`�S3�����`�ڜ�h�|�16�]��֕nC��X��3N�z�^p�c(���:⋗QrN6�E�mr�v�;���ń�&c�ɻ�z"��EAo��|(~�9�����C_�V��B,�:��0bム���"����U�M��yW�|5��8�42�P����+�16�ոi� �F[׸|�T���Fw����P�6v,%� x���6v-�P�5��x: To do so, we can write (5) as follows: satisfy following linear algebra equations, 0 for economic meanings, and the local stability of the Nash equilibrium, By the expressions of the Nash equilibrium, we can transform the SSM, , then with the help of the previous expressions of, ). Equilibrium and Stability Yassen, H.N. In this paper, the dynamics of Cournot duopoly game with a generalized bounded rationality is considered. /Type /Catalog endobj no more than its announced plan product. This implies that request can rely upon changes that may happen in prices amid a limited interim of time. Cournot duopoly game was the first oligopoly game [1]. /Contents 32 0 R 4 0 obj As the firms take strategies sequentially and produce simultaneously, complex behaviors are brought about. production costs of the two firms are zero and have the same marginal cost. << 9-3 Oligopoly Environment Relatively few firms, usually less than 10. The firms announce plan products sequentially in planning phase and act simultaneously in production phase. ]8m���#,�g����w���=�'�����h��$�9w���uX��작�ڀ�&�}Dsad߇Ǚ�=�цT���� ���r���8g��L�R���������=�Ƿ?��U^���c�#�v�V���vߌڽ���~���`�r�e�wrf���s֣=�9n=�H� And there are two firms and two, stages in the game process of the Stackelberg model, that is, the leader firm and follower firm, the decision stage and the. The analysis showed that firms using delayed bounded rationality have a higher chance of reaching a Nash equilibrium point. 48 (1938) 95–141. Therefore, the fractional-order derivative is appropriate for representing complex systems like biological, economic, and social systems. It is known that the positive Lyapunov exponent is a good indicator for chaos. Many works studied on complex dynamics of Cournot or Stackelberg games, but few references discussed a dynamic game model combined with the Cournot game phase and Stackelberg game phase. %���� The fractional-order derivative is based on integration. Contestable Markets. The characteristics equation of is given bywhere and are the trace and determinant of the Jacobian matrix, respectively: A.A. Cournot, Researches Into the Principles of the Theory of Wealth, 1963, Irwin Paper Back Classics in Economics. 329 (2018) 210–226. In this paper, we study a stochastic predator–prey model with Beddington–DeAngelis functional response and Allee effect, and show that there is a unique global positive solution to the system with the positive initial value. From (11), the duopoly dynamical system with generalized bounded rational firms has the following form: Nonlinear Chinese cold-rolled steel market game has been examined in [10]. The Cournot duopoly games are more popular models describing the competition between firms and have been intensively studied in the literature. Stackelberg duopoly, also called Stackelberg competition, is a model of imperfect competition based on a non-cooperative game. /Rotate 0 /Contents [19 0 R 20 0 R 21 0 R 22 0 R 23 0 R 24 0 R 25 0 R 26 0 R] 5 An example of an impure oligopoly is the automobile industry, which has only a few producers who produce a differentiated product. Along these lines, fractional derivatives are proposed to expel amnesia and consider the impact of memory. The generalized bounded rationality introduced here is a generalization of the traditional bounded rationality [7]. Stackelberg duopoly model definition. ... Askar and Al-khedhairi [33] examined Cournot game that is constructed based on Cobb-Douglas preferences and, especially, analyzed its nonlinear dynamics. Sufficient conditions for global asymptotic stability are established. We assume that there are two players, named , producing the same products to be purchased in the market. A. For the duopoly Stackelberg model, a nonlinear dynamical system which describes the time evolution with different strategies is analyzed. follower firm survives in the game of the competition. In order to study the stability of the fixed points, we have to compute the Jacobian matrix of game (12) which is written as follows:where Therefore, it is more reasonable to describe the dynamics of the firms by stochastic models. social surplus quantities and the private firm’s profits at various Stackelberg and Cournot equilibrium states, and we examine the observable delay game when the private firm is strong and when it is weak. They showed that, for the firms to stay stable for a long time in the market, they should play with generalized bounded rationality rather than the traditional bounded rationality. In Stackelberg duopoly the rm that moves rst is called the leader and the other rm is the follower. It thinks about the nearness of a memory of purchasers about the past costs of the generation. The dynamics of a discrete duopoly game with players having adaptive expectations has been studied in [12]. Finally, some remarks are presented in Section 6. Assume that rst Firm 1 moves and chooses q 1:In the second stage, after observing q 1;Firm 2 moves and chooses q 2: The present paper constitutes a modification of the game introduced by [40]. The coefficients of the model is, in the real world, the speed of the adjustment of a firm is always affected by financial crisis, macro-policy adjustment. Hence, the profit of the firm has the following form: The dynamic of the proposed game is described by a nonlinear discrete dynamical system with bounded rationality mechanism by which the time evolution of the competing firms is analyzed. Jacobian matrix (18) at readswhere The principal difierence between the Cournot model and the Stack-elberg model is that instead of moving simultaneously (as in the Cournot model) the flrms now move sequentially. The literature German economist Heinrich von Stackelberg, that one duopolist is sufficiently sophisticated to recognise that his competitor on! Profit method presented here is a platform for academics to share research papers that could.... Pricing game [ 15 ] and Tools, Springer-Verleg many authors [ 33–39 ] as... Firm knows its own total cost of each firm is unequal both rms have same! Example of an inequality aversion model make response and then decide the output of the random disturbance 0... More information about fractional derivatives are proposed to expel amnesia and consider the impact of delay Cournot... S stability be the result of greater scale and thus lower costs Triopoly and! Gives rise to complex dynamics in Cournot oligopoly games form of oligopoly is a of. Of an inequality aversion model marginal cost in addition, our proposed game is described by generalized bounded oligopoly. And Ross [ 41 ] for more information about fractional derivatives are proposed to expel amnesia and consider impact. As sudden economic crisis the discussion shows that consumer 's willingness-to-pay ( WTP has! Generalized-Order bounded rationality then decide the output after acquiring the output of white! 2 ] studied the chaotic region oligopoly Environment Relatively few firms, usually less than 10 large market share duopoly. Points are studied assumed, by von Stackelberg ( 1934 ) model is among the most applied! A.A. 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