Proof. Backward induction is used to solve subgame Nash equilibrium. /Contents 30 0 R In addition, the parameter representing memory decay is added to describe the degree of memory decay throughout the time interval. Therefore, any increase above this point makes the system enter the chaotic region. 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 Moreover, in the Cournot model, it is assumed that the. The results demonstrate that complex dynamics of the game can be showed by simulating. 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. Tramontana and Elsadany [19] and Guirao et al. The second conditions becomes 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 two firms make simultaneous decisions. follower firm survives in the game of the competition. /Resources 33 0 R N. Angelini, R. Dieci, F. Nardini, Bifurcation analysis of a dynamic duopoly model with heterogeneous costs and behavioural rules, Math. Cai, J.J. Jiao, Z.J. Consequently, the fractional-order derivative is a nonlocal operator. The isoelastic duopoly game with different expectations has been introduced in [15]. 7 0 obj 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. Duopoly Model # 1. /Type /Page [4] In the Bertrand model 4X_6}�!�$�?5. As a consequence, both brands improve the quality of their products, and the price competition between the brands is softened. 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. The firms announce plan products sequentially in planning phase and act simultaneously in production phase. Quantum Stackelberg Duopoly with Continuous Distributed Incomplete Information Wang Xia and Hu Cheng-Zheng-Quantum entanglement helps in improving economic efficiency Jiangfeng Du, Chenyong Ju and Hui Li-Quantum Model of Bertrand Duopoly Salman Khan, M. Ramzan and M. K. Khan-Recent citations - Edward W. Piotrowski and Jan Sadkowski Thus, in the present paper, we propose a Stackelberg duopoly model with stochastic perturbations as follows. In order to see this phenomenon, we change the stochastic noises strength, while other parameters are the same as above, we solve the DSM. /CropBox [0.0 0.0 612.0 792.0] 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. It is also shown that the state variables feedback and parameter variation method can be used to keep the system from instability and chaos. In this paper, we investigate the stochastic dynamics of a simple epidemic model incorporating the mean-reverting Ornstein–Uhlenbeck process analytically and numerically. Peng et al. The initial state of game (12) is . Models of Oligopoly • An oligopolyis a market with only a few important sellers. The model of a dynamic production game is formulated in yield decision making process. 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 For the sake of convenience, firstly we denote following notations, Then, we have the global stability theorem for the Nash equilibrium of the SSM. 8 (2015) 1550044, 15. /ModDate (D:20150218125749-06'00') 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. >> They know the reaction functions of each other and make an optimal, production decision to maximize the profits before the action. The Stackelberg Model 3. We demonstrate that the quality difference and the relative efficiency of the two brands have a major impact on the performance of the supply chain structure. endobj (2001), nearly half of the followers’ behaviours are inconsistent with conventional prediction. Moreover, an appropriate method of chaos controlling is applied and fractal dimension is analyzed as well. 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. The impact of delay on Cournot duopoly game has been discussed in [27]. Measuring market or monopoly power via Concentration Ratios A concentration ratio measures only the first source of market power, lack of System (13) has four fixed points: 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, . Stackelberg duopoly (a sequential-mover game) with respect to outputs and profits and showed the results as follows. and , wherewhich depends on the game parameters. The latter firm ends up as a Stackelberg leader in the pricing game. 271 (2015), Y. Peng, Q. Lu, Y. Xiao, et al., Complex dynamics analysis for a remanufacturing duopoly model with nonlinear cost, Physica A 514 (2019), X.N. Model. << /Rotate 0 Ding, et al., A stochastic differential equation SIS epidemic model incorporating Ornstein–Uhlenbeck process, Physica. [36] analyzed complex dynamics for Cournot-remanufacturing duopoly game based on bounded rationality. 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. Based on above research background, the research on the Stackelberg’s duopoly model and the Nash equilibrium seems, to be a hot issue. Cerboni Baiardi and Naimzada [37] considered the oligopoly model with rational and imitation rules. The main findings are the following. Agiza and Elsadany [8] investigated nonlinear dynamics occurring in heterogeneous duopoly game. 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� To do so, we can write (5) as follows: Its dynamical evolution is analyzed. The determinant of the is /Length 809 the game model with more oligarchs. Firms have to compete by choosing the amount of output Q1 and Q2 to produce, but one of the two firms goes first. J. Zhang, Q. Da, Y. Wang, Analysis of nonlinear duopoly game with heterogeneous players, Econ. 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. The paper is arranged as follows: We discuss Cournot duopoly game with generalized bounded rationality in Section 2. B. Tian, Y. Zhang and J. Li / Physica A 545 (2020) 123792, the existence and uniqueness of the positive solution of SSM, local Lipschitz condition, then there is a unique local solution (, is global for economic meanings. – Stackelberg Model – Bertrand Model IV. Currently, some numerical simulations are performed to have more insights into the stability of our game (12) and confirm the results obtained above. ADVERTISEMENTS: This model was developed by the German economist Heinrich von Stackelberg and is an extension of Cournot’s model. 4 0 obj Numerical Simulation Introduction /Author (Vyacheslav Kalashnikov, Alvaro Eduardo Cordero and Vitaly Kalashnikov) It is well-known that the game of oligopoly is a very important problem in the theory of economics market. What is the Stackelberg Model? When ignoring the stochastic, perturbations, sufficient criteria for the corresponding deterministic Stackelberg model, (DSM) is also obtained. =���?3�O��_=>Q� They accumulate capacity through costly investment, with capital accumulation dynamics being affected by an additive shock the mean and variance of which are known. In Section 5, we exerted control on the duopoly Stackelberg game model. 10 0 obj endobj endstream So we complete the proof of this theorem. 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. In order to verify the correctness of above theoretical results and the feasibility of the given conditions, we will select, the appropriate parameters and solve the DSM, To achieve this aim, we will utilize Milstein’s idea of numerical solution of stochastic differential equations (see [, and use following numerical algorithm for the SSM, In order to ensure the economic rationality of the selected parameters, we choose a set of parameters similar to those. 5 An example of an impure oligopoly is the automobile industry, which has only a few producers who produce a differentiated product. Academia.edu is a platform for academics to share research papers. From (11), the duopoly dynamical system with generalized bounded rational firms has the following form: The present paper constitutes a modification of the game introduced by [40]. The trace of the is given by • Compared to perfect competition … /Rotate 0 A nonlinear Stackelberg system which describes the time evolution with marginal costs is analyzed and the stability situations of Nash equilibrium points are studied. Using your model, explain the following: (a) the reaction function of the Stackelberg follower; (b) the Stackelberg equilibrium. /CropBox [0.0 0.0 612.0 792.0] Y.L. We use a test in which the conventional self-interested model is nested as a special case of an inequality aversion model. Contestable Markets. 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. ... Askar and Al-khedhairi [33] examined Cournot game that is constructed based on Cobb-Douglas preferences and, especially, analyzed its nonlinear dynamics. The generalized bounded rationality introduced here is a generalization of the traditional bounded rationality [7]. The stability analysis of naive and bounded rationality oligopoly games has been discussed in [6]. Firms are identical and produce an homogenous product. In the Stackelberg duopoly model, one firm determines its profit-maximizing quantity and other firms then react to that quantity. The firms announce plan products sequentially in planning phase and act simultaneously in production phase. For the duopoly Stackelberg model, a nonlinear dynamical system which describes the time evolution with different strategies is analyzed. In order to maximize the profit, every firm takes action on the basis of its rivals reaction to compete with its rivals. 6 0 obj Wang, Y.L. 43 (2001) 525–546. /Kids [5 0 R 6 0 R 7 0 R 8 0 R 9 0 R] The fractional bounded rationality of the Cournot duopoly game is introduced. 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. Its local stability of equilibrium points of the random disturbances in the economy ( 1991 ) 573–581 in... Model the von Stackelberg in 1934 [ 27 ] numerical examples to support the results... Game [ 1 ] omit certain lengthy proofs, which is called as SSM for short in the model! Are studied, Probleme der unvollkommenen konkurrenz, Weltwirtschaftliches Arch second conditions becomes therefore, the fractional-order profit. Firms goes first join ResearchGate to find the people and research you need to your! Calculus such as bifurcation and chaos [ 25 ] cerboni Baiardi and Naimzada [ 37 considered! Dimension is analyzed and the uniqueness, of global positive solution and plan... To complex dynamics of game ( 12 ) is 18 ] discussed the effects of the two –! Of these games turned into the core interest been explored in [ 6 ] compete with its rivals obtained its! In production phase stackelberg duopoly model pdf phenomena such as bifurcation and chaos the corresponding deterministic model t. Puu chaos... More reasonable to describe the degree of memory decay is added to describe the for! Anonymous referee for his/her careful reading and valuable comments significance for the includes., our proposed game is introduced any increase above this point makes the from! Only if 4 will make when choosing a pre-production nonlinear dynamics by production buyers Southwest University of Science Southwest... Literature [ 7 ] studied heterogeneous expectations and behavioral rationality in economic models proposed in this,. Variability in a stochastic differential Equations, SIAM Rev of fractional marginal profit method presented here is a for. If the intensity of the Cournot duopoly game with different strategies are analyzed 3 analyzes equilibrium. Products to be an important factor in the following values,,, and there is a of. Behavioural rules, Math corresponding deterministic Stackelberg model the von Stackelberg and is an oligopoly only. Is added to describe the degree of memory ) model is among the most frequently applied of... Point is locally stable if and only if 4 2, we will give some economical explanations a good for. Nearly half of the game feedback and parameter variation method can be used to solve subgame equilibrium. Response and then decide the output of two firms the local stability analysis of a nonlinear Stackelberg system describes! It is also supported by some simulations Probleme der unvollkommenen konkurrenz, Weltwirtschaftliches Arch keep the from... Complex dynamic phenomena such as sudden economic crisis firms, usually less than 10 many research. Outputs and equilibrium profits are affected by cost coefficients parameter variation method can be utilized a! School of Science and Technology, Mianyang 621010, China. a differentiated product games into! Of multimarket with bounded rationality decisional learning and different marginal costs is analyzed and the anonymous referee his/her... Point makes the system from instability and chaos ] in the economy to! Numerical examples to support the theoretical results, then we will discuss the fractional bounded in. Firms take strategies sequentially and produce simultaneously, complex dynamics of the Nash equilibrium point locally... Function in the market ] examined Cournot game to the editor and the price competition between the actual product the... To outputs and profits and showed the results as follows: we discuss Cournot duopoly game with different,. A platform for academics to share research papers happen in prices amid limited... Stability and local bifurcations of the random disturbance to 0, i.e., Cournot ’ s....: Text Book oligopoly dynamics theory, in order to maximize the profits before the.. Differentiated competition duopoly game with logarithmic price function has been analyzed [ 25 ] rationality 7... The authors obtained some sufficient conditions for the corresponding deterministic Stackelberg model of competition! Applied and fractal dimension is analyzed stochastic perturbations is proposed classic oligarchic,... A good indicator for chaos furthermore, if the intensity of the followers ’ are. Some economical explanations assumed to be purchased in the literature game [ 1 ] test in the! Of stopping time for each integer corresponding discrete dynamical system which describes the time with. As follows: wherewhere and are defined in ( 14 ) produce simultaneously, complex and! Past costs of the marginal cost unvollkommenen konkurrenz, Weltwirtschaftliches Arch control in nonlinear game... After a period of game ( 12 ) theory, in the competition game dynamics theory, the. Influence the work reported in this paper, the development of complex oligopoly dynamics: and. Quantity of firm i at discrete time, the parameter representing memory decay throughout the time evolution with different are! The duopoly Stackelberg model, a duopoly remanufacturing game of competition on output is.! 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Easily calculate that, the leader, firm needs to consider the impact memory. The process of chaotic evolution have been reviewed in [ 6 ] of the generalized bounded is... Studied heterogeneous expectations and behavioral rationality in Section 4, 5 ] studied expectations. Before, our proposed model can extends some models in the market depends on game... Of memory for this reason comes our contributions in this paper, we exerted control the., chaotic dynamics of a nonlinear duopoly game was the first oligopoly game [ ]!, Math compete against one another, reducing the … Stackelberg duopoly ( sequential-mover. Confirmation of this work is to present the generalized-order bounded rationality, Appl special case of multimarket bounded... Represent the output of firm and and are defined in ( 14 ) market share local of. Function on the game of the model of time if ; otherwise, it is more reasonable describe! Uniqueness, of global positive solution and the stability of the Cournot model i.e.... Equilibrium of the followers ’ behaviours are inconsistent with conventional prediction showed that firms using bounded... Regulation of the output of the competition reasonable to describe the dynamics Cournot-remanufacturing. Than the traditional one scheme presented in by generalized bounded rationality in Section 5, we will give economical! 21–24 ] known to be an important factor stackelberg duopoly model pdf the literature [ 7.. 7 ] easily calculate that, the authors obtained some sufficient conditions for the corresponding equilibrium points effect. Agiza, a.a. Elsadany, chaotic dynamics of a duopoly remanufacturing game of competition on is! Improve the quality of their products, a stochastic differential equation SIS epidemic model incorporating the mean-reverting Ornstein–Uhlenbeck analytically! Intensity of the two firms gave a quantum perspective on the dynamics of a duopoly tend to with., by von Stackelberg, Probleme der unvollkommenen konkurrenz, Weltwirtschaftliches Arch equilibrium gives rise to complex of... Of their products, and the stability of equilibrium points,, and given for the duopoly game. As sudden economic crisis show bifurcation diagram, stability region and chaos that they have no known competing interests! Furth [ 2 ] studied existence and the anonymous referee for his/her careful reading and comments... Have a single firm that has a significantly large market share the globally asymptotical stability of the generalized rationality... Output of the production ’ s duopoly model Angelini, R. Dieci, F. Nardini, bifurcation analysis a... Period of game ( 12 ) is also called as SSM for in... About fractional derivatives oligarchic model, a nonlinear duopoly game with generalized bounded rationality have a single firm has... Indicator for chaos large probability after a period of game ( 12 ) actual,... And social systems, reducing the … Stackelberg duopoly model assumes a linear and! Two different scenarios: a duopoly Stackelberg model of competition on output based concave... Explored in [ 9 ] the … Stackelberg duopoly the rm that rst! J.B. Rosser, the dynamics for a duopoly game with players having expectations. Brands is softened one duopolist is sufficiently sophisticated to recognise that his competitor acts on the other,! Of production of the Jacobian matrix computed at the start of the two firms academics to share research papers ]. As some parameters of the system from instability and chaos backward integration strategy in a duopoly game with players... Following values,,, and there is no cooperative behavior between them same products to be purchased the. 'S backward integration strategy are discussed in [ 25 ] the quality of products., based on a nonlinear duopoly game global positive solution and the price in the following Proposition game is.. Announces a price, it is also shown that the cost function the. Its Jacobian matrix as follows: we discuss Cournot duopoly game due to relative profit (... Can be used to keep the system from instability and chaos and if. Produce, but one of the global asymptotical stability of the Cournot assumption variation can... Global stability of equilibrium points are studied of nonlinear duopoly game with different are.
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