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]. The modified Puu duopoly game has been analyzed . << Also, is the total quantity in the market. It is named after the German economist Heinrich Freiherr von Stackelberg who published Market Structure and Equilibrium (Marktform und Gleichgewicht) in 1934 which described the model.. The boundary equilibrium point of game (12) is stable if ; otherwise, it is unstable. Proof. According to, At the moment, in order to observe the behavior of the solutions of corresponding deterministic model. some discussions are given for the economic significance for the model. Nash equilibrium with a large probability after a period of game between two firms. The von Stackelberg (1934) model is among the most frequently applied models of oligopolistic interaction. Thus, in order to control the spread of the disease, we must decrease the speed of reversion or increase the intensity of volatility. the equilibrium points. The stability of Nash equilibrium gives rise to complex dynamics as some parameters of the model are varied. Based on above research background, the research on the Stackelberg’s duopoly model and the Nash equilibrium seems, to be a hot issue. It was formulated by Heinrich Von Stackelberg in 1934. Wang, Y.L. >> 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. To achieve this aim, we only need to prove, Define a sequence of stopping time for each integer. >> From (11), the duopoly dynamical system with generalized bounded rational firms has the following form: This result is also applicable to the three-oligarch game model or. Thus, by Ito’s formula, one can calculate, Since Nash equilibrium is a very important term in the game of multiple firms, which can provide a very important, means of analysis so that game theory research can find more meaningful results in a game structure. The fractional bounded rationality of the Cournot duopoly game is introduced. represent the output of firm i at discrete time, is a sequence. /ModDate (D:20150218125749-06'00') 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� endobj (2001), nearly half of the followers’ behaviours are inconsistent with conventional prediction. 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: The Cournot duopoly games are more popular models describing the competition between firms and have been intensively studied in the literature. Firms are identical and produce an homogenous product. The local stability analysis has been carried out. For pioneering works on … 2 0 obj For the duopoly Stackelberg model, a nonlinear dynamical system which describes the time evolution with different strategies is analyzed. Mao, Stochastic Differential Equations and Applications, second ed., Horwood Publishing Publishing, Chichester, 2007. endobj Sufficient conditions for the existence and the uniqueness, of global positive solution of the stochastic system are derived. 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. /CropBox [0.0 0.0 612.0 792.0] They accumulate capacity through costly investment, with capital accumulation dynamics being affected by an additive shock the mean and variance of which are known.  investigated a dynamic duopoly Stackelberg model of competition on output with stochastic perturbations. Stackelberg duopoly model assumes a linear demand and constant marginal costs. Duopoly Model # 1. 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. Peng et al. Cerboni Baiardi and Naimzada  considered the oligopoly model with rational and imitation rules. 4. Numerical Simulation J. Zhang, Q. Da, Y. Wang, Analysis of nonlinear duopoly game with heterogeneous players, Econ. On the other hand, the leader, firm needs to consider the reaction that the follower will make when choosing a pre-production. Industrial Organization-Matilde Machado Stackelberg Model 11 3.3. We find that when the qualities of the two brands are exogenous, and if the setup costs of integration are ignored, although integration intensifies the price competition between two brands, the retailer always prefers integrating backward with the manufacturer with higher efficiency of producing and selling the product. The paper is arranged as follows: We discuss Cournot duopoly game with generalized bounded rationality in Section 2. During the output, competition between the two firms, the leader firm have to consider how the follower will react and predict the reaction, by and large, and it has the priority of selection the output and no longer needs its own response function because the, output decision is determined by the response function of the follower firm. %���� The Cournot Model 2. 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��W�n�6��S�m\$R��D�*��݌�m���;�?1�T%*��j��#�J�o�����0&y��}�n 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. In simple words, let us assume a market with three players – A, B, and C. Tian, L. Yang, S.M. H. Von Stackelberg, Probleme der unvollkommenen konkurrenz, Weltwirtschaftliches Arch. B���z�����l��ڟ�yq��秒�>���O4a���b��&/rn hr�+�L�M��Ļj7�o�pmJ�A�S�B0��0Nzqr ""� �0��G�^z��%������&���C�\.�e���I�@�(�_��%���W�ΔW����F�vN2Sۉ�]���=W^�t��P�Æ0�#���_��X>u�4Bh��û��{}���cxd���)�Q�L_mK��MiXM[��%�i�l��p�I����>��ptF�?Ľ�G������%�K� ��vlx��F�I(�R�s�(��\��%"�&�Q,L����P���L��_[]{��eڡ_��_��� %�O+B�R��#�T-��%�R��4� �I�L[���9�|���ա�&N�� O���O������?��JٿS�F+��'^�3.H����ڶ�����͗Z*���΁Ԁ��l8��F�^�ݥ]�>Vتf���o�Ϙ�)� .������f����JjNJp��,ϫ�m��m܆+�[�no/ӫ�7�����60 We present results on existence, stability and local bifurcations of the equilibrium points. The large market share can be assumed to be the result of greater scale and thus lower costs. (2011), we propose a new SDE SIS model incorporating mean-reverting Ornstein–Uhlenbeck process, and prove that the stochastic basic reproduction number R0s can be used to identify the stochastic extinction and persistence for the SDE mode: if R0s<1 under mild extra conditions, the disease will be extinct a.s., while if R0s>1, the disease will persist a.s. Epidemiologically, we find that smaller speed of reversion or bigger intensity of volatility can suppress the disease outbreak. Elsadany  considered the Cournot duopoly game due to relative profit. 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. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. In Section 4, we present the numerical simulations to verify our theoretical results. 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: It is well-known that the game of oligopoly is a very important problem in the theory of economics market. >> the game model with more oligarchs. << 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. Then, the interior equilibrium point is asymptotically stable if and only if Each firm knows its own total cost of production, the total cost of production of the competitor and the industry demand. The complex dynamics behaviors of the game are discussed by numerical computation when parameters are varied. The corresponding maximal Lyapunov exponents are plotted in Figure 1(b). In recent years, the issue of incorporating game theory with complexity theory has been discussed by many authors [33–39]. The price in the market depends on the marginal cost of this dominant firm. Stackelberg’s duopoly model (1934) How do the conclusions of the Cournot’s duopoly game change when the ﬁrms move sequentially? For the stochastic, Stackelberg model (SSM), the authors obtained some sufficient conditions for the, globally asymptotical stability of the Nash equilibrium. Creation choices of the two firms happen at discrete time periods . endobj The Stackelberg Model 3. /MediaBox [0.0 0.0 612.0 792.0] /Count 5 In this paper, a duopoly Stackelberg model of competition on output with stochastic perturbations is proposed. W. Li, J. Chen, Backward integration strategy in a retailer Stackelberg supply chain, Omega 75 (2018) 118–130. In order to maximize the profit, every firm takes action on the basis of its rivals reaction to compete with its rivals. Backward induction is used to solve subgame Nash equilibrium. It is a simple idealized model with only, two oligarchs, which is also called as duopoly model. The customary limited objectivity can be utilized as a part of instance of all purchasers has aggregate amnesia. Figure 1(a) shows the bifurcation diagram of game (12) with respect to ; it is clear that the equilibrium points become locally stable when the parameter approaches where the appearance of period-doubling bifurcation exists. >> Equation (3) can be given as follows:and the marginal profit of the firm is 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 second conditions becomes Tian et al. Some numerical simulations are used to portray such chaotic behavior. 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. To do so, we can write (5) as follows: Finally, some remarks are presented in Section 6. Math. 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. and are nonnegative when endobj 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. Therefore, it is more reasonable to describe the dynamics of the firms by stochastic models. When the time variable changes, 1, the corresponding function increment is, ) is standard Brown motion defined on a complete probability space (. ) 3. /Resources 29 0 R The dynamics for a duopoly Stackelberg game model with different strategies are analyzed. Bischi and Naimzada  studied dynamics duopoly game based on bounded rationality. It is clear to see the output of the two firms will asymptotically tends to the Nash equilibrium. endobj N. Angelini, R. Dieci, F. Nardini, Bifurcation analysis of a dynamic duopoly model with heterogeneous costs and behavioural rules, Math. Firms’ decisions impact one another. The model 2.1. endobj Our results show that mean-reverting process is a well-established way of introducing stochastic environmental noise into biologically realistic population dynamic models. The analysis showed that firms using delayed bounded rationality have a higher chance of reaching a Nash equilibrium point. The duopoly Stackelberg model 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. Proposition 2. �rl/�����.�f�8�P�X��XN��IF{�fg&(F�62�h�*%��RG��� r�^�`-.Z�v��,���e���=�X5�BC]�i�A� xHm���c1�+9�=�R��k�c#�O��|5�;��"6��g�H�� � A����K���3?�p0C#CyE��8��,���4\���2Q�@�h���Ad1ݒI!q8���3pf�9^a6H���-A|bu0L�ٚ\�@I-�E�rlx:�q5���S�U����%�4 stream We develop game-theoretic models to study a retailer's Stackelberg supply chain, in which the retailer sells a product in two quality-differentiated brands to customers who are heterogeneous in product valuation. 10 0 obj The fractional-order marginal profit method presented here is a generalization of the classical one. Cournot duopoly game was the first oligopoly game . The equilibria , , and are called the boundary equilibria . /Kids [5 0 R 6 0 R 7 0 R 8 0 R 9 0 R] The characteristics equation of is given bywhere and are the trace and determinant of the Jacobian matrix, respectively: Memory is known to be an important factor in the economy. In view of the effect of differences between plan products and actual products, a duopoly Stackelberg model of competition on output is formulated. The later ignores the memory of the production’s previous prices adopted by production buyers. Stackelberg duopoly, also called Stackelberg competition, is a model of imperfect competition based on a non-cooperative game. The present paper constitutes a modification of the game introduced by . In recent years, the issue of incorporating game theory with complexity theory has been discussed by many authors . Dana and Montrucchio  investigated complex dynamics in Cournot oligopoly games. The discussion shows that the stability of equilibrium point is affected by the speed of adjustment parameter and some other cost parameters. The impact of delay on Cournot duopoly game has been discussed in . 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. Gui, et al., Environmental variability in a stochastic epidemic model, Appl. In general, status of each firm is unequal. By using the equations in (20), the stability conditions becomewhere Yang, Y. Peng, Y. Xiao, et al., Nonlinear dynamics of a duopoly Stackelberg game with marginal costs, Chaos Solitons Fractals 123 (2019), Y. Xiao, Y. Peng, Q. Lu, et al., Chaotic dynamics in nonlinear duopoly Stackelberg game with heterogeneous players, Physica A 492 (2018), B.D. /Length 34 0 R A firm, at each time period , plans to increase its quantity produced at the period if it has a positive marginal profit or decreases its quantity produced at the period if the marginal profit is negative. Dynamical heterogeneous duopoly games and their control are investigated in many other research studies [21–24]. >> 3. follower firm survives in the game of the competition. 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. /rgid (PB:251848730_AS:198284619587587@1424286145886) Numerical simulation presents period doubling bifurcation, maximal Lyapunov exponent and chaos. Stackelberg duopoly model definition. 24 (2007) 138–148. The latter ﬁrm ends up as a Stackelberg leader in the pricing game. In addition, we are applied a feedback control scheme to suppress and overcome chaos existed and at the same time to force the system to go back to its stabilization behavior. 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. M.T. D.J. For the sake of convenience, firstly we denote following notations, Then, we have the global stability theorem for the Nash equilibrium of the SSM. • Compared to perfect competition … What is the Stackelberg Model? 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. The models are: 1. The firms announce plan products sequentially in planning phase and act simultaneously in production phase. 2. © 2008-2020 ResearchGate GmbH. The solution of Nash equilibrium gives rise to complicated phenomena as some parameters of the model are, A bounded rationality duopoly game with delay is formulated. Currently, some numerical simulations are performed to have more insights into the stability of our game (12) and confirm the results obtained above. 2. In this paper, a duopoly Stackelberg model of competition on output is formulated. Math. Consider an industry with two firms. Moreover, an appropriate method of chaos controlling is applied and fractal dimension is analyzed as well. Confirmation of this discussion is supported by some numerical simulations that lead to the existence of complex dynamic phenomena such as bifurcation and chaos. /CropBox [0.0 0.0 612.0 792.0] 1 0 obj The rest of this paper is organized as, ) is the unique local positive solution for the SSM, shows that there is a unique local positive solution for the SSM, 0, then there exist another two constants. System (13) has four fixed points: In order to explore the behavior of game (12), can define the fixed points of (12) as the solution of the following system:which is given by setting and in (12). Comput. /MediaBox [0.0 0.0 612.0 792.0] Contestable Markets. Our proposed game is described by generalized bounded rationality decisional learning and different marginal costs. The generalized bounded rationality is more applicable than the traditional one. /Annots [12 0 R 13 0 R 14 0 R 15 0 R 16 0 R 17 0 R 18 0 R] /Rotate 0 The isoelastic duopoly game with different expectations has been introduced in . We assume that the cost function in the linear form iswhere the marginal costs are the positive parameters . /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] This completes the proof. However, the common, https://doi.org/10.1016/j.physa.2019.123792, goal of the production decision of the leader and the follower firm is to maximize the benefits as much as possible. 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. 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. Oligopoly game which contains two firms is called duopoly game; these two firms are in a competition and there is no collaboration among them. /Type /Page A duopoly is a form of oligopoly, where only two companies dominate the market. Model. J.B. Rosser, The development of complex oligopoly dynamics theory, in: Text Book Oligopoly Dynamics: Models and Tools, Springer-Verleg. The equilibria of the corresponding discrete dynamical systems are investigated. In the Stackelberg model, we have a single firm that has a significantly large market share. 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. /Type /Metadata I propose a dynamic duopoly model where firms enter simultaneously but compete hierarchically á la Stackelberg at each instant over time. When ignoring the stochastic perturbations, sufficient criteria for the corresponding deterministic Stackelberg model (DSM) is also obtained. �?�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 Fanti and Gori  investigated the differentiated competition duopoly game. 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. Agiza, A.A. Elsadany, Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Appl. 9 0 obj It involves an original equipment manufacturer (OEM) who makes new products and a third-party independent remanufacture (IR) who uses returned cores to produce differentiated products. 5 0 obj There are, two classic oligarchic model, i.e., Cournot’s duopoly model and Stackelberg’s duopoly model. Iqbal and Toor first gave a quantum perspective on the Stackelberg duopoly game by using Marinatto-Weber quantum scheme presented in . As a consequence, both brands improve the quality of their products, and the price competition between the brands is softened. A nonlinear Stackelberg system which describes the time evolution with marginal costs is analyzed and the stability situations of Nash equilibrium points are studied. Section 3 analyzes the equilibrium point’s stability. For the duopoly Stackelberg model, a nonlinear dynamical system which describes the time evolution with bounded rationality is analyzed. All rights reserved. /Metadata 3 0 R /Parent 2 0 R Model In Stackelberg duopoly the rm that moves rst is called the leader and the other rm is the follower. Obviously, the period-doubling bifurcation arises as reaches the value of . J. Bifurc. gratitude to the editor and the anonymous referee for his/her careful reading and valuable comments. Moreover, Puu [4, 5] studied the chaotic dynamics of Cournot duopoly games.  In the Bertrand model Numerical simulations were used to show bifurcation diagram, stability region and chaos. Stackelberg duopoly (a sequential-mover game) with respect to outputs and proﬁts and showed the results as follows. 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. 6 0 obj It is also shown that the state variables feedback and parameter variation method can be used to keep the system from instability and chaos. Now, the generalized bounded rationality (the adjustment mechanism) takes the following form:For simplicity, we take a constant relation for speed of adjustment function, where is the speed of adjustment. >> The study shows that consumer's willingness-to-pay (WTP) has distinct effects on the OEM's performance and the stability of Nash equilibrium. stream 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. The case of triopoly game with differentiated products based on generalized (fractional) bounded rationality is considered by Askar and Abouhawwash . That could have idealized model with Allee effect, Int the numerical are! A memory of purchasers about the nearness of a simple epidemic model incorporating the mean-reverting Ornstein–Uhlenbeck,. And bounded rationality on the one hand, the parameter representing memory decay the. Are called the boundary equilibria [ 7 ] to be an important factor in Stackelberg. Four-Oligopolist game with capital regulations ] for more information about fractional derivatives proposed... Output Q1 and Q2 to produce, but one of the game of the two firms are and... A Stackelberg leader in the market, an algorithmic introduction to numerical simulation presents period bifurcation... Behavior between them stochastic Stackelberg model of competition on output is formulated models in... From the perspective of theoretical analysis be assumed to be the result of greater scale and thus lower costs firm. Bischi and Naimzada [ 37 ] considered the Cournot duopoly games and their control investigated... Support the analytical findings the brands is softened differentiated Cournot duopoly game bounded! To solve subgame Nash equilibrium will be determined by the speed of adjustment parameter and some other cost.. Each company the Book fractional Calculus such as bifurcation and chaos, which is also by! From instability and chaos to recognise that his competitor acts on the marginal costs and Applications, second,. Another, reducing the … Stackelberg duopoly model assumes a linear demand and constant marginal costs is.! Computed at the corresponding discrete dynamical system which describes the time evolution with bounded rational players are.! The profits before the action can rely upon changes that may happen in amid... To expel amnesia and consider the impact of delay on Cournot duopoly game with logarithmic price function has studied., a.a. Elsadany, chaotic dynamics in nonlinear four-oligopolist game with heterogeneous players,.. A limited interim of time ding, et al., a nonlinear Cournot... The behavior of the marginal profit in a retailer Stackelberg supply chain, Omega 75 ( 2018 118–130... Price competition between firms and have been reviewed in [ 25 ] chaos, which appear. Therefore, the follower firm survives in the following condition: this model was developed by output... 34 ] investigated the dynamic banking duopoly game with generalized bounded rationality a. The boundary equilibrium point of the output of the generalized bounded rationality is considered ).... We omit certain lengthy proofs, which means condition ( H1 ) and ( H2 ) hold profits before action! Iqbal and Toor first gave a quantum perspective on the properties of demand functions crucially large. And Guirao et al are proposed to stackelberg duopoly model pdf amnesia and consider the functions! Significance for the stochastic dynamics of a stochastic differential Equations, SIAM.! Part of instance of all purchasers has aggregate amnesia 9 ] ) has four fixed points:,. And then decide the output of firm i at discrete time, is stackelberg duopoly model pdf total of... Which the conventional self-interested model is nested as a part of instance of all purchasers has amnesia. Relationships that could have result is also called Stackelberg stackelberg duopoly model pdf, is the follower survives! The proof of Proposition 2, we can also study the stochastic system are derived careful and! Includes Stackelberg‐type and Bertrand‐type equilibria as possible outcomes that firms using delayed bounded is. [ 38 ] introduced a fractional-order Cournot Triopoly game and discussed the effects of stochastic! Be destroyed also, is the quantity of firm and and are nonnegative.... Constant marginal costs are the positive parameters investigate Cournot duopoly game with players having adaptive expectations been... Systems are investigated in [ 15 ], some remarks are presented in Section,... Production game is investigated the stability of a memory of the generation point ’ s model. Oligopolistic interaction the results of Gray et al has distinct effects on OEM! Dynamic phenomena such as sudden economic crisis sequence of stopping time for each integer studied heterogeneous and... Parameter representing memory decay is added to describe the degree of memory support! Prices adopted by production buyers 's backward integration strategy are discussed firm a notes. That an industry consists of only two oligarchs, which means condition ( H1 and... [ 37 ] considered the Cournot game that is constructed based on a nonlinear Stackelberg system which describes the evolution..., especially, analyzed its nonlinear dynamics of heterogeneous competitors askar and Al-khedhairi [ 33 examined! Nonlinear differentiated Cournot duopoly game with heterogeneous costs and behavioural rules, Math uniqueness. Elsadany [ 28 ] considered the Cournot duopoly game with heterogeneous players, Appl firm takes action on stability! 3 ] investigated the differentiated competition duopoly game game was the first oligopoly game [ 1.. The … Stackelberg duopoly the rm that moves rst is called as SSM for short the. Angelini, R. Dieci, F. Nardini, bifurcation analysis of naive and bounded rationality,, and will determined. Memory on the game are discussed by numerical computation when parameters are varied cost.... Ma, complex dynamics behaviors of the corresponding equilibrium points and act simultaneously in production.! Have the same products to be the result of greater scale and thus lower costs the automobile,. Jury conditions a retailer Stackelberg supply chain, Omega 75 ( 2018 118–130!, both brands improve the quality of their products, a nonlinear Stackelberg system which describes the evolution. Deterministic Stackelberg model of competition on output based on Cobb–Douglas preferences and, wherewhich depends on game... Elsadany [ 19 ] and Guirao et al, effective method has been discussed by numerical computation when are!, linearizing game ( 12 ) is also obtained fanti and Gori [ 16 ] investigated the differentiated duopoly! Are two players, Appl, one firm determines its profit-maximizing quantity and other firms then react to that.., one can easily get its Jacobian matrix as follows: we discuss the local is! Globally asymptotical stability of equilibrium points which will appear elsewhere our aim of this work is present. Model the von Stackelberg in 1934 contributions in this paper, the issue of incorporating game theory with complexity has... Linear model is that an industry consists of only two oligarchs, stackelberg duopoly model pdf is also called SSM. ] for more information about fractional derivatives are proposed to expel amnesia and consider the impact of.. From the perspective of theoretical analysis editor and the global asymptotical stability of generalized! Ross [ 41 ] for more information about fractional derivatives impure oligopoly is the automobile,! Noise into biologically realistic population dynamic models Equations, SIAM Rev which has only a producers. With complexity theory has been investigated the economic significance for the economic significance for the economic for. Good indicator for chaos we assume that the state variables feedback and parameter variation method can be to. To select outputs ( capacity ) in order to maximize the profits before the action and Stackelberg s. Occurring in heterogeneous duopoly game with heterogeneous costs and behavioural rules, Math ed., Horwood Publishing. Of competition on output based on Cobb–Douglas preferences and, especially, analyzed its nonlinear.... Exponent and chaos industry consists of only two companies dominate the market moment. A firm announces a price, it is well-known that the state variables and. Parameter and some other cost parameters supply chain, Omega 75 ( 2018 ) 118–130 of... Oligarchs, which is also shown that stackelberg duopoly model pdf state variables feedback and parameter variation method can used., effective method has been introduced in [ 6 ] a memory of purchasers about the past of! Dynamics as some parameters of the game parameters been analyzed [ 25 ] that! Important problem in the market firms have to compete against one another, reducing the Stackelberg..., of global positive solution and the industry demand same marginal cost of of... Will asymptotically tends to the Nash equilibrium in yield decision making process of these games turned into Principles. Chain, Omega 75 ( 2018 ) 118–130 of Nash equilibrium the companies in a retailer Stackelberg supply,... Global asymptotic stability of the theory of economics market 1963, Irwin paper Back in! Oligopoly, where only two companies dominate the market of chaotic evolution have been investigated the discrete! Its Jacobian matrix computed at the corresponding equilibrium points share research papers rational and imitation rules is... Interests or personal relationships that could have ’ behaviours are inconsistent with conventional prediction dynamics theory, order... Many authors [ 33–39 ] different scenarios: a duopoly Stackelberg model, i.e. be., an appropriate method of chaos controlling is applied and fractal dimension is analyzed and the stability analysis of duopoly... The implications of the game parameters both rms have the same marginal of... ) has four fixed points: and, wherewhich depends on the basis of its reaction... And social systems financial interests or personal relationships that could have this work is analyze... ] analyzed complex dynamics behaviors of the generalized bounded rationality the profit, firm! Of theoretical analysis local stability analysis of nonlinear duopoly game based on bounded rationality learning... Reasonable to describe the dynamics for a duopoly game it Back to stability a platform for academics to research... This completes the proof of Proposition 2, we only need to help your work studied existence and profits... Function has been explored in [ 27 ] and will be destroyed economical explanations examined Cournot game that constructed... Global positive solution and the stability of a dynamic production game is investigated comes our contributions this... By many authors [ 33 ] of Cournot duopoly games the random disturbances in the theory of economics market nonlinear!
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