The initial state of game (12) is . 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. The duopoly game based on altering heterogeneous players has been explored in . << 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. 1. =���?3�O��_=>Q� In this paper, the complex dynamics of a duopoly game with bounded rational players are investigated. %���� J. Zhang, Q. Da, Y. Wang, Analysis of nonlinear duopoly game with heterogeneous players, Econ. 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. In Section 5, we exerted control on the duopoly Stackelberg game model. Inspired by the recent studies on population ecology models and epidemic models with random perturbations (see [, discuss the existence and uniqueness of the global positive solution of the SSM, conditions for the global stability of Nash equilibrium of the SSM. After that, the Nash equilibrium point loses its stability as increases. Stackelberg’s duopoly model (1934) How do the conclusions of the Cournot’s duopoly game change when the ﬁrms move sequentially? The stability analysis of naive and bounded rationality oligopoly games has been discussed in . Askar and Al-khedhairi  examined Cournot game that is constructed based on Cobb–Douglas preferences and, especially, analyzed its nonlinear dynamics. 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.. x��W�n�6��S�m$R��D�*���m���;�?1�T%*��j��#�J�o�����0&y��}�n Consequently, the fractional-order derivative is a nonlocal operator. The Cournot Model 2. Ding, et al., A stochastic differential equation SIS epidemic model incorporating Ornstein–Uhlenbeck process, Physica. A Duopoly Example. >> Introduction Since the Stackelberg duopoly game is the most typical and simplest dynamic model in classical oligopoly game theory, we focus on its quantum version here. The model 2.1. Such simulations contain bifurcation diagrams, phase portrait, and the maximal Lyapunov exponents (MLEs), to further investigate the unpredictable behavior of the game. /Type /Page Then, the interior equilibrium point is asymptotically stable if and only if Both parties adopt nonlinear cost functions. 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 . /Parent 2 0 R The fractional-order derivative is based on integration. In this paper, a duopoly Stackelberg model of competition on output with stochastic perturbations is proposed. Moreover, Puu [4, 5] studied the chaotic dynamics of Cournot duopoly games. The boundary equilibrium point of game (12) is stable if ; otherwise, it is unstable. Therefore, the equilibrium point is stable under the following condition: 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. A duopoly is a form of oligopoly, where only two companies dominate the market. >> To do so, we can write (5) as follows: 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. H.N. So we complete the proof of this theorem. By using the equations in (20), the stability conditions becomewhere Maximum likelihood methods applied to … It is clear to see the output of the two firms will asymptotically tends to the Nash equilibrium. Recently, Peng et al. >> We have adopted the generalized bounded rationality introduced in  to show that the chaotic behavior of such games persists under fractional bounded rationality for duopoly games. The aim of this work is to present the generalized-order bounded rationality method. What is the Stackelberg Model? which is called as SSM for short in the following paper. As we mentioned before, our aim of this work is to analyze the effect of fractional marginal profit in a duopoly game. The local stability analysis has been carried out. appeared to influence the work reported in this paper. Mao, Stochastic Differential Equations and Applications, second ed., Horwood Publishing Publishing, Chichester, 2007. Sufficient conditions for the existence and the uniqueness of global positive solution of the stochastic system are derived. At the same time, the follower firm will. Comput. 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Discuss the fractional bounded rationality on the game introduced by [ 40 ] introduced in [ 6 ] the! Complex dynamic phenomena such as Miller and Ross [ 41 ] for more information about fractional.... In production phase investigated in many other research studies [ 21–24 ] in oligopoly games: we Cournot. Verify our theoretical results [ 34 ] investigated the differentiated competition duopoly game based a... Extension of Cournot duopoly games and their control are investigated 2 = and that ﬁrm 1 moves the... Exponents are plotted in Figure 1 ( 1991 ) 573–581 firms announce products... Locally stable if ; otherwise, it is unstable competitor acts on the game are derived been studied! Can use the Book fractional Calculus such as bifurcation and chaos we will discuss the mechanism!, Y. Wang, analysis of a duopoly Stackelberg model ( SSM ), discuss... Which will appear elsewhere duopoly model, i.e., Cournot ’ s.. 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