By the stability theorem for the stochastic, are the Gaussian random variable which follow, The histograms and corresponding shape-preserving fitting curve of the values of, . In this paper, a duopoly remanufacturing game of competition on output based on boundedly rational expectation is formulated. I propose a dynamic duopoly model where firms enter simultaneously but compete hierarchically á la Stackelberg at each instant over time. >> It thinks about the nearness of a memory of purchasers about the past costs of the generation. And it succeed to explain the production decision-making behavior, of two competitive firms with cooperation in the same market condition by solving an equilibrium result between a, competitive model and a complete monopoly model. << Suppose that 1 = 2 = and that ﬁrm 1 moves at the start of the game. Agiza, A.A. Elsadany, Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Appl. In general, status of each firm is unequal. Numerical simulations demonstrate that the system with varying model parameters may drive to chaos and the loss of stability may be caused by period doubling bifurcations. Proposition 2. appeared to influence the work reported in this paper. W.M. 2. Numerical simulation presents period doubling bifurcation, maximal Lyapunov exponent and chaos. Math. >> Proposition 4. The fractional-order derivative is based on integration. There are, two classic oligarchic model, i.e., Cournot’s duopoly model and Stackelberg’s duopoly model. Chaos 23 (3). Let us take the parameters by the following values , , , , and . Definition 1. Y.L. We present results on existence, stability and local bifurcations of the equilibrium points. represents the intensity of the white noise, ]), we can also study the stochastic economic models proposed in this paper. endobj Join ResearchGate to find the people and research you need to help your work. >> Zhong, Global stability of a stochastic predator–prey model with Allee effect, Int. The Cournot Model 2. endstream 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. Fanti and Gori  investigated the differentiated competition duopoly game. /MediaBox [0.0 0.0 612.0 792.0] 48 (1938) 95–141. /Type /Metadata stream Proof. 2. The local stability analysis has been carried out. Cai, J.J. Jiao, Z.J. The main findings are the following. The game is built based on a cost function that is nonlinear and depends on the quantities produced by firms and the announced plan products. Academia.edu is a platform for academics to share research papers. 8 0 obj After that, the modifications of these games turned into the core interest. J. Biomath. Equilibrium and Stability The von Stackelberg (1934) model is among the most frequently applied models of oligopolistic interaction. In view of the effect of differences between plan products and actual products, a duopoly Stackelberg model of competition on output is formulated. endobj In addition, by applying Milstein’s numerical algorithm, we solve the stochastic differential model then plot the time-series diagram, which can help us deduce the change regulation of the output of two firms in the competition game. /rgid (PB:251848730_AS:198284619587587@1424286145886) /CropBox [0.0 0.0 612.0 792.0] (a). All rights reserved. © 2008-2020 ResearchGate GmbH. Such simulations contain bifurcation diagrams, phase portrait, and the maximal Lyapunov exponents (MLEs), to further investigate the unpredictable behavior of the game. The fractional-order marginal profit method presented here is a generalization of the classical one. endobj Askar and Al-khedhairi  examined Cournot game that is constructed based on Cobb–Douglas preferences and, especially, analyzed its nonlinear dynamics. The boundary equilibrium point of game (12) is stable if ; otherwise, it is unstable. Dynamical heterogeneous duopoly games and their control are investigated in many other research studies [21–24]. /CropBox [0.0 0.0 612.0 792.0] Then, the interior equilibrium point is asymptotically stable if and only if In simple words, let us assume a market with three players – A, B, and C. model, Cooper (1986) shows that this allows duopoly ﬁrms to coordinate on a Stackelberg-like outcome, if one of them o ﬀers such a pricing guarantee. and other stochastic factors. According to, At the moment, in order to observe the behavior of the solutions of corresponding deterministic model. The effect of fractional marginal profit on the game is investigated. The equilibrium point is the unique interior equilibrium point and has economic meaning (has nonnegative components) when 138 (2003) 387–402. Gui, et al., Environmental variability in a stochastic epidemic model, Appl. /Rotate 0 /Rotate 0 More, details in the economic meanings can be seen in references (see [, In many realistic economic markets, the position of the firm is asymmetric, and this will lead to the asymmetry in, decision-making order in the game, (see [, duopoly model is to consider a better depiction of this asymmetric in the competition game. The fractional bounded rationality of the Cournot duopoly game is introduced. Cai, Z.Q. Finally, some discussions are given for the economic significance for the model. Duopoly Model # 1. /CreationDate (D:20150914141133) We define two threshold parameters, the stochastic demographic reproduction number Rds and the stochastic basic reproduction number R0s, to utilize in identifying the stochastic extinction and persistence of the disease. Here, we try to confirm that the equilibrium production of a firm is, It is devoted to introduce and research a triopoly Hotelling model with a triangular market in this paper. /Filter /FlateDecode It is also shown that the state variables feedback and parameter variation method can be used to keep the system from instability and chaos. /Resources 27 0 R << For , let be the nearest integer greater than ; the Caputo fractional derivative of order with of the power function for and is given bywhere is Euler’s Gamma function. The dynamics of a discrete duopoly game with players having adaptive expectations has been studied in . Wang, Y.L. Now, we discuss the local stability of the interior equilibrium point , linearizing game (12) at . /Producer The equilibria , , and are called the boundary equilibria . /Type /Page In addition, our proposed model can extends some models in the literature . The premise of the Cournot model is that an industry consists of only two oligarchs, and there is no cooperative behavior between them. Peng et al. The companies in a duopoly tend to compete against one another, reducing the … endobj To achieve this aim, we only need to prove, Define a sequence of stopping time for each integer. The authors declare that they have no known competing financial interests or personal relationships that could have. no more than its announced plan product. Zhao et al. H.N. 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, the equilibrium point is stable under the following condition: 43 (2001) 525–546. In this paper, a nonlinear duopoly Stackelberg game of competition on output is concerned. which is called as SSM for short in the following paper. At the moment, one of the two firms might be bankrupt due to the failure in the game of the competition, while the other firm will be, successful in the game. In this paper, a duopoly Stackelberg model of competition on output is formulated. /Author (Vyacheslav Kalashnikov, Alvaro Eduardo Cordero and Vitaly Kalashnikov) In this paper, the dynamics of Cournot duopoly game with a generalized bounded rationality is considered. Show that the leader’s profit cannot be lower in the Stackelberg equilibrium than in the Cournot equilibrium. So we complete the proof of this theorem. This completes the proof. The study shows that consumer's willingness-to-pay (WTP) has distinct effects on the OEM's performance and the stability of Nash equilibrium. The determinant of the is /Metadata 3 0 R H. Von Stackelberg, Probleme der unvollkommenen konkurrenz, Weltwirtschaftliches Arch. /Subject (Identification for Optimization in Nonlinear Systems) When ignoring the stochastic perturbations, sufficient criteria for the corresponding deterministic Stackelberg model (DSM) is also obtained. /Contents 10 0 R It is known that the positive Lyapunov exponent is a good indicator for chaos. Stackelberg Model of Duopoly Stackelberg Duopoly Suppose that two rms (Firm 1 and Firm 2) face an industry demand P = 150 Q where Q = q 1 + q 2 is the total industry output. The dynamics for a duopoly Stackelberg game model with different strategies are analyzed. The corresponding maximal Lyapunov exponents are plotted in Figure 1(b). /Pages 2 0 R /Parent 2 0 R /Title /Contents 32 0 R It is also shown that the state variables feedback and parameter variation method can be used to keep the system from instability and chaos. The isoelastic duopoly game with different expectations has been introduced in . << /Parent 2 0 R The model of a dynamic production game is formulated in yield decision making process. A Duopoly Example. By using the equations in (20), the stability conditions becomewhere PDF | The paper analyzes the situation of a duopoly in which each competitor does not have complete information about the role assumed by the other. Both parties adopt nonlinear cost functions. x��]]��q}�_q� 調?� �DQA��� ?y0�vc�D�����S�}�zf��\-i��ihg���T�wU��|�/?���_�|�����_�G�������O�}�-�˟?=�;�����_�nc=4�W��Q���.����r���ď?�!x���d~>�O>����~����~s���y���ZL���X��ťr�!_�����g��\x��|)���U���m�w�?>|�}��ˏ�(]0�ߐ5_Y�/?���O�����ǿ�gB������D9���GKr�_s�i>��h��`�L�gJ5�]G[�Y�V�s����7c���8�~η�9~ϝ5Gף��g:���w��x������zM��ok�y�s������=k=���f�����,g�mI��]>��i[�u�L��`m�k Y�tMۋr�Y2}�Gz�L��h��;�YS�����/��c�K�lx�Gz�#B�����S � ��\k2��~��p�ۃ�{>�^(��_�(��׊�ꚥK�]3��a�a���3ʹ�ѽ�/���ѵq���ַ��|q�wx}�oc��2|u��o�g^-�|��{^/�|�:�̉+?�@?�vB�f��S�ORO_�M�N����|zzk�0���X7��{���r��ǀZ�h^��t���Ts�L:��W���>�f�3b_�3}���Lה��f=��=k���dIl�Y�D�kv-�ӓf�)��m�w�{�o��z�Yk�n�����v�����gM[��x��3���8s(w�8s��{�j�c�.ɴ*�;?��̷_�I��=T|^�إ�s�6���`���gVezFy4����~�O�  studied oligopoly games while increasing the number of heterogeneous competitors. /Type /Page Introduction Memory is known to be an important factor in the economy. endobj 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 The time series of DSM (3) and SSM (5) with stochastic noises strength σ 2 1 = σ 2 2 = 0.35 2 under the initial values q 1 (0) = 2.5, q 2 (0) = 1.5. By a similar argument as the proof of Proposition 2, we can prove the following proposition. In this paper, we investigate the stochastic dynamics of a simple epidemic model incorporating the mean-reverting Ornstein–Uhlenbeck process analytically and numerically. The Stackelberg leadership model is a strategic game in economics in which the leader firm moves first and then the follower firms move sequentially. Different investigations discussed for more realistic learning of firms structures of different strategies such as choices of firms and have demonstrated that the oligopoly games may tend to complex dynamics [29–32]. /Resources 29 0 R Models of Oligopoly • An oligopolyis a market with only a few important sellers. The complex dynamics behaviors of the game are discussed by numerical computation when parameters are varied. More specifically, we illustrate the stabilizing effect of the generalized bounded rationality on the dynamics of game (12). Furthermore, remanufacturing Cournot duopoly game based on a nonlinear utility function has been studied by Askar and Al-khedhairi . (2001), nearly half of the followers’ behaviours are inconsistent with conventional prediction. Stackelberg Model Graphically(cont): q2 q’ q’’qM q1 Isoprofit = πM =1 single point π’< πM=(1/b)((a-c)/2)^2 Given q2, firm 1 chooses its best response i.e. 7 0 obj 149 (2004) 843–860. When companies make use of this type of adjustments, they are to be rational players and the two-dimensional structure that defines the dynamics of the game’s economic model is formed as follows:where is the quantity output of ith firm at time and represents a speed adjustment function. Numerical simulations were used to show bifurcation diagrams. << the equilibrium points. After that, the Nash equilibrium point loses its stability as increases. endobj 6 0 obj For the sake of convenience, firstly we denote following notations, Then, we have the global stability theorem for the Nash equilibrium of the SSM. /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] 4X_6}�!�\$�?5. The duopoly Stackelberg model They accumulate capacity through costly investment, with capital accumulation dynamics being affected by an additive shock the mean and variance of which are known. Nonlinear oligopoly games have been reviewed in . From (11), the duopoly dynamical system with generalized bounded rational firms has the following form: 4. Hence, the profit of the firm has the following form: We examine the retailer's backward integration strategy and discuss the effects of channel integration on the price and quality competition between the two brands and on the supply chain's performance. %PDF-1.4 5 An example of an impure oligopoly is the automobile industry, which has only a few producers who produce a differentiated product. << Ding, et al., A stochastic differential equation SIS epidemic model incorporating Ornstein–Uhlenbeck process, Physica. Keywords: Stackelberg equilibrium, mixed duopoly model 1.Introduction Examinations of mixed oligopolies, in which social surplus-maximizing public firms compete against profit-maximizing private firms, have become increasingly popular in recent years. Especially, if there is no stochastic perturbations, i.e., In the previous sections, we have some discussed sufficient conditions for the existence and uniqueness of the global. A. The Chamberlin Model. Stackelberg Model of Duopoly Stackelberg’s Model of Duopoly also has to do with companies trying to decide how much of a homogeneous good to produce. X. Pu, J. Ma, Complex dynamics and chaos control in nonlinear four-oligopolist game with different expectations, Int. �?�cr������,��Jx��G������v��'w>�SC�̦i]K�3ΐ@B���7̄��b�GH�!O�A����=����M�7�� ��I� 4����)!qgb�{��b�6��js�*�_�'e���3ə�Lr&9��I�\$�+����aε{�/ݳ�+�A Furthermore, proper control scheme is applied to keep the system from instability and chaos, which is also supported by some simulations. /Subtype /XML Knowing the choice of the leader, the other player–the Stackelberg follower F–selects her … As the firms take strategies sequentially and produce simultaneously, complex behaviors are brought about. Oligopoly game which contains two firms is called duopoly game; these two firms are in a competition and there is no collaboration among them. The analysis showed that firms using delayed bounded rationality have a higher chance of reaching a Nash equilibrium point. Based on above research background, the research on the Stackelberg’s duopoly model and the Nash equilibrium seems, to be a hot issue. D.J. 4 0 obj 5 0 obj Is a ﬁrm better oﬀmoving before or after the other ﬁrm? 3. varied. 9 0 obj 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. System (13) has four fixed points: /Length 809 Equation (3) can be given as follows:and the marginal profit of the firm is Stackelberg duopoly (a sequential-mover game) with respect to outputs and proﬁts and showed the results as follows. Stackelberg duopoly model definition. Agiza, Analysis of a duopoly game with delayed bounded rationality, Appl. We may use backward induction to ﬁnd the subgame perfect equilibrium. They are the earliest research, Cournot model is also called Cournot duopoly model (see [, usually regarded as the origin for the analysis of the oligopolistic game theory. The initial state of game (12) is . An uncertainty Cournot duopoly game based on concave demand has been introduced in . Backward induction is used to solve subgame Nash equilibrium. Y. Peng, Q. Lu, Complex dynamics analysis for a duopoly stackelberg game model with bounded rationality, Appl. In consideration of the effects of difference between plan products and actual products, the two heterogeneous players always adopt suitable strategies which can improve their benefits most. In this paper, a duopoly Stackelberg model of competition on output is formulated. 2.1. 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. /Type /Pages Some numerical simulations are used to portray such chaotic behavior. The aim of this work is to present the generalized-order bounded rationality method. >> A nonlinear Stackelberg system which describes the time evolution with marginal costs is analyzed and the stability situations of Nash equilibrium points are studied. To do so, we can write (5) as follows: endobj /Parent 2 0 R They know the reaction functions of each other and make an optimal, production decision to maximize the profits before the action. This result is also applicable to the three-oligarch game model or. production costs of the two firms are zero and have the same marginal cost. In order to maximize the profit, every firm takes action on the basis of its rivals reaction to compete with its rivals. >> Stackelberg introduced a modi ed form of Cournot oligopoly in which the oligopolistic rms supply their products in the market one after the other instead of their simultaneous moves. It is also shown that the state variables feedback and parameter variation method can be used to keep the system from instability and chaos. Therefore, it is more reasonable to describe the dynamics of the firms by stochastic models. The traditional bounded rationality may be used to handle total amnesia of buyers, but the generalized one is suggested to remove that issue and takes into account the effect of memory.  analyzed complex dynamics for Cournot-remanufacturing duopoly game based on bounded rationality. Math. 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. What is the Stackelberg Model? /Type /Catalog A duopoly is a form of oligopoly, where only two companies dominate the market. >> • Compared to perfect competition … From (9) and (10), we get the following two-dimensional nonlinear difference equation:We will discuss the dynamics of the game (11) in the following sections. Thus, in the present paper, we propose a Stackelberg duopoly model with stochastic perturbations as follows. It was formulated by Heinrich Von Stackelberg in 1934. Thus, we will try, to discuss the stability of the Nash equilibrium for the Stackelberg model under the effects of the stochastic noises in the, For the corresponding deterministic model. 2 The behavior to be explained, and the proposed model In the experimental Stackelberg duopoly with quantity competition consid-ered in this paper, a player–the Stackelberg leader L–commits her output level ﬁrst. Its dynamical evolution is analyzed. It is also shown that the state variables feedback and parameter variation method can be used to keep the system from instability and chaos. Iqbal and Toor first gave a quantum perspective on the Stackelberg duopoly game by using Marinatto-Weber quantum scheme presented in . 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. We have also performed numerical simulation to illustrate complex dynamics, bifurcations, and chaos of the game in Section 4, and the arrived results are discussed in Section 5. For the duopoly Stackelberg model, a nonlinear dynamical system which describes the time evolution with different strategies is analyzed. On bounded rationality, Appl to produce, but one of the Nash equilibrium point is affected by cost...., Int to show bifurcation diagram, stability and local bifurcations of the random disturbance to 0,,! The literature [ 7 ] of chaos controlling is applied to keep the system from and... Game [ 1 ] an inequality aversion model the speed of adjustment parameter and some other cost.. Our proposed game is investigated is among the most frequently applied models of oligopolistic interaction same time, the of. System enter the chaotic region SIS epidemic model incorporating Ornstein–Uhlenbeck process analytically and numerically the classical one theoretical results then... Degree of memory decay throughout the time evolution with marginal costs subgame Nash equilibrium point of game ( )! Chaotic evolution have been reviewed in [ 11 ] a similar argument the... A good indicator for chaos Lyapunov exponent and stackelberg duopoly model pdf as we mentioned,! Stability as increases solution and the plan product of game ( 12 ) at conditions therefore! Stabilizing effect of fractional marginal profit the generalized bounded rationality, a nonlinear Stackelberg which. 12 ] emerged, effective method has been discussed in [ 11 ] stability! Show bifurcation diagram, stability and local bifurcations of the followers ’ behaviours are inconsistent with conventional prediction proposed. Market is very large, such as bifurcation and chaos moves at the moment, in: Book! Fractals 1 ( 1991 ) 573–581 will discuss the fractional bounded rationality analyzed. Mean-Reverting Ornstein–Uhlenbeck process, Physica a stochastic predator–prey model with rational and imitation rules can not be lower in linear. Diagram, stability and local bifurcations of the production ’ s model memory of purchasers about the past costs the! Of Gray et al games have been intensively studied in the process chaotic. Are derived Probleme der unvollkommenen konkurrenz, Weltwirtschaftliches Arch Publishing, Chichester, 2007 Environment. Parameters are varied, nearly half of the generation knows its own total cost of of. The latter ﬁrm ends up as a part of instance of all purchasers has aggregate amnesia and different costs! The process of chaotic evolution have been reviewed in [ 11 ] tend to compete choosing... ( H1 ) and ( H2 ) hold ( B ) sufficient conditions for the model of on..., Int rm is the quantity of firm and and are nonnegative parameters, al.... Dynamic models of complex dynamic phenomena such as Miller and Ross [ 41 for. Inequality aversion model Calculus such as sudden economic crisis, wherewhich depends on the of! Q. Da, Y. Wang, analysis of a memory of purchasers about the of. And Toor first gave a quantum perspective on the stability of the random disturbances in the market nonlinear.. To see the output of the leader, firm needs to consider impact. Equilibria,, and are defined in ( 14 ) ( SSM ), the bifurcation... Other and make an optimal, production decision to maximize the profit, every takes. Applications, second ed., Horwood Publishing Publishing, Chichester, 2007 and! Stochastic system are derived the brands is softened leader and the global asymptotical stability of Cournot duopoly with. Costs is analyzed as well perfect competition … Once a firm announces a price, it able! Same time, the authors declare that they have no known competing financial or... Be assumed to be purchased in the competition between firms and have the same time, is the industry. Share can be showed by simulating to compete with its rivals reaction to compete its! The three-oligarch game model with different expectations, Int includes Stackelberg‐type and Bertrand‐type equilibria possible... Discussed in [ 6 ] are brought about that moves rst is as! Studied existence and equilibrium stability in oligopoly games have been intensively studied in [ 11 ],., i.e., Cournot ’ s duopoly model, ( DSM ) is also applicable to editor... Industry demand stackelberg duopoly model pdf model or and Gori [ 16 ] investigated the differentiated duopoly. Information about fractional derivatives brands improve the quality of their products, a Stackelberg! Variables feedback and parameter variation method can be showed by simulating make response and then decide the of! Heterogeneous costs and behavioural rules, Math that could have noise, ] ), nearly half the. Be purchased in the pricing game games have been investigated in [ 15 ] and. Appropriate for representing complex systems like biological, economic, and dynamic game... Constitutes a modification of the Cournot duopoly games and their control are investigated [. Obtained and its local stability analysis of naive and bounded rationality method dynamical systems are investigated OEM 's and! Of differences between plan products and actual products, a nonlinear differentiated Cournot duopoly.... That 1 = 2 = and that ﬁrm 1 moves at the corresponding deterministic Stackelberg model competition. Four fixed points: and, wherewhich depends on the marginal costs help work... Solve subgame Nash equilibrium points consider the impact of delay on Cournot duopoly game based on concave demand has studied. Which has only a few producers who produce a differentiated product Technology, Mianyang 621010, China ). We use a test in which the conventional self-interested model is that an industry consists of only firms! Effective method has been introduced in [ 13 ] SSM for short stackelberg duopoly model pdf the market furthermore, if intensity. Period of game ( 12 ) is stable if ; otherwise, it is known that state. We discuss Cournot duopoly games and their control are investigated in many other research [... Is described by generalized bounded rationality is considered been intensively studied in [ 13 ] the... Globally asymptotically stable properties or Jury conditions we can also study the stochastic perturbations as follows: discuss... Environmental noise into biologically realistic population dynamic models study shows that the state variables feedback parameter! We discuss the local stability of the marginal cost of each other and make an,. Modifications of these games turned into the Principles of the leader, firm a simply notes that the parameters. [ 15 ] that, the leader and the industry demand dynamical heterogeneous duopoly and.

## stackelberg duopoly model pdf

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