Dynamics of a Bertrand Game with Heterogeneous Players and Different Delay Structures. 1 Introduction

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.5(8) No., pp.9-8 Dynamics of a Bertrand Game with Heterogeneous Players and Different Delay Structures Huanhuan Liu, Zhanwen Ding Faculty of Science, Jiangsu Uniersity, Zhenjiang, Jiangsu,, PR China (Receied 5 December 7, accepted 5 March 8) Abstract: In this paper we consider a duopoly Bertrand game played by two heterogeneous players with different delayed bounded rationality. We suppose that one player adjusts the price strategy according to his own marginal profit with time delay, while another player makes the price strategy as a best response to an expectation that aerages the opponent s actions in the preious periods. The dynamics is built for this decision making process and the local stability of its equilibrium states are studied. Numerical simulations are done to display the influence of the model parameters on the system complexity and system stability. It is shown that the system may lose stability and take complex behaiors, and the stability loss may be due to either period-doubling bifurcation or Neimark-Sacker bifurcation. It is shown that the time delay for the marginal profit has a greater effect on the system stability than the time delay considered for the price ariable, and an intermediate leel of delay on the marginal profit can greatly improe the system stability. Keywords: Bertrand game; Time delay; Marginal profit; System stability; System complexity Introduction Cournot [] game is a kind of quantity competition game, where each firm takes the quantity of output as its decision ariable. It is assumed that two firms produce homogeneous goods for sale in a duopoly market. As market price is dependent on the total supply, each player has to take into account not only the demand function but also the reaction of the other player. Being different from Cournot game, when the firms product is perceied as heterogeneous in the perspectie of the consumers, any firm may take price instead of quantity as its decision ariable. The model based on this idea was first proposed by Bertrand []. The traditional oligopoly games were considered as the case with naïe expectation, by which a player supposes that the opponents strategies keep as the same leel as in the preious period [e.g., 7]. Howeer, in a real market it is not always credible that a rational player can remain a constant strategy. As a kind of more sophisticated rationality, the adaptie expectation or adaptie adjustment method was put forward to ealuate the strategy of the opponents [e.g., 8 ]. In recent years, many scholars hae paid attention to so-called bounded rationality, with which a player uses his marginal profits to adjust his strategy in the following period. That is, a firm raises its ouput (in a Cournot game) or its price (in a Betrand game) if the marginal profit in the last period is positie, otherwise makes a negatie adjustment [e.g., 8]. These are early models built for the games with homogeneous players. As players are not all homogeneous in a real market, the models with heterogeneous players are also considered by other authors [e.g., 9 5]. A dynamical Cournot duopoly game which contains a bounded rationality player and a naie player is studied in [9]. Agiza and Elsadany [] made work on the duopoly model with a bounded rationality player and an adaptie player. Linear cost function is replaced by a nonlinear one in the duopoly game with heterogeneous players []. Quadratic cost functions are considered in [], where a heterogeneous duopoly game with diseconomies of scale is discussed. Fan et al. [] analyzed the duopoly game with two heterogeneous players for the case that a naie player makes its output strategy based on the market price of the preious period, while another bounded rationality player adjusts his strategy by the marginal profit method. Ding et al. [4] analyzed a dynamics of a multi-team Bertrand game played by Corresponding author. address: dgzw@ujs.edu.cn Copyright c World Academic Press, World Academic Union IJNS.8.4.5/999

2 International Journal of Nonlinear Science, Vol.5(8), No., pp. 9-8 a team consisting of two boundedlly rational firms and another team consisting of one naie firm. A master-slae Bertrand game for upstream and downstream monopolies owned by different parties is discussed in [5], in which the upstream monopolist s quantity is used as the main factor of production by the downstream monopolist who is a small purchaser of the upstream monopolist s output. Time delay plays an important role in oligopoly game, which can improe the system stability and delay the occurrence of complex behaiors [e.g., 6 9]. In the duopoly model with linear cost function discussed in [6] and a similar one with nonlinear cost function in [7], each firm is assumed to consider time delay for the output ariable and estimate its marginal profit according to the preious-period quantity of all players. A bounded-rationality duopoly game with time delay on opponents output was considered in [9], where each firm makes its output strategy by aeraging the preiousperiod quantity of its opponents. Lu [8] considered the case of increasing marginal cost instead of a constant one in [9]. In the aboe models with time delay, a potential assumption is that the opponents output history is a kind of common information for any player. But it may not be true in a real market when opponents behaiors are not easily obsered. Howeer, each player easily knows his own output and profit information in eery period. So Ding et al. [] considered time delay for the marginal profit and studied a case that each player makes his strategy adjustment by a smoothed marginal profit method, which aerages his own preious marginal profits with different weights. On the other hand, the market price is also easily to be obsered by each player. Peng et al. [] studied a repeated Bertrand game with delayed bounded rationality, where all players maximize their profit according to the local information of the market price. In a real market een with limited information, each player is able to get the information about his own marginal profit and the information about the price as well. In a Cournot market the market price is a kind of common information for all players, and in a Bertrand market any player s price strategy is also open although it is an indiidual behaior. So a player may adjust his behaior by considering the history of either the marginal profit or the price ariable. In this paper, we consider a Bertrand game played by players with heterogeneous and delayed rationality. One player considers time delay for the marginal profit so that adjusts his price strategy according to the aeraged marginal profit in the preious periods. Another player considers time delay for the opponent s price ariable by making an expectation that the opponent adopts a smoothed price strategy which is also aeraged from the preious periods. The main purpose is to inestigate the different effects caused by the two different time delay structures on the system stability. The model We consider a Bertrand game focusing on price competition between two firms, which produce similar goods and choose suitable price to sell. Let p i denote firm i s price strategy and q i represent the market demand for its product. The two firms price p i and p j together determine firm i s market demand q i through a function Q i (p i, p j ): q i = Q i (p, p ). () To gie a full supply q i, firm i has a cost that is described by a function C i (q i ). We suppose that the two firms are symmetric, and hence both of their demand functions and cost functions are identical; we also suppose that the two functions take linear forms as in [5, 6, 8, 5], i.e. where a >, b > and c >. Then the profit of firm i is gien by Q i (p i, p j ) = a p i + bp j, () C i (q i ) = cq i, i =,, () π i = p i Q i (p i, p j ) C i (q i ) = (p i c)(a p i + bp j ), i =,. (4) Many authors hae paid attention to the Cournot game (with incomplete information), where a player may not get accurate information of the opponent s quantity and hence updates its output strategy in period t + according to its marginal profit in period t [e.g.,, 7, 9 ]. Some other authors also studied the Bertrand game by a similar marginalprofit method, that is, a firm will raise its price strategy if its marginal profit is positie, otherwise reduces the price [e.g., 4 6, 8, 4, 5]. In these models mentioned, they only consider the marginal profit in period t, and do not take into account the marginal profit information in the preious periods. In a recent work Ding et al. [] considered delayed information of the marginal profit in a Cournot game. It may be true that in a Cournot game the opponent s quantity IJNS for contribution: editor@nonlinearscience.org.uk

3 H. Liu and Z. Ding: Dynamics of a Bertrand Game with Heterogeneous Players and Different Delay Structures strategy can not be obsered; but in a Bertrand game, the price information of each firm is open on the market and hence a player may adjust its behaior according to the price history of the opponent. In this work about a Bertrand game, we assume that the two firms are two heterogeneous players and both consider time delay. Firm uses a similar method discussed in [] so that makes its strategy according to the marginal profit with one step time delay, and firm adjusts its price strategy according to firm s price information in the last two periods. That is to say, firm updates its price strategy at time t+ according to an aerage of its marginal profits π(p,p) p t, π (p,p ) p t with different weights: p (t + ) = p (t) + p (t)[w π p t + ( w) π p t ], (5) where w is a weight coefficient, is a positie parameter which represents a relatie adjustment rate (for a Cournot game, see e.g. []). Firm is supposed to hae a different rationality and consider a time delay structure built for the price strategy of its opponent. In the period t + firm is not able to know what price strategy firm will take, but it may make some expectation about the behaior of firm. We suppose that firm takes a naïe expectation that firm will adopt a smoothed leel of price strategy p E (t + ), which aerages firm s price in the preious two periods with different weights, i.e. p E (t + ) = αp (t) + ( α)p (t ), (6) where α. Then for its price strategy p (t + ) at time t +, firm has an expected profit π E (t + ) as follows: π E (t + ) = [p (t + ) c ][a p (t + ) + bp E (t + )]. (7) To maximize its expected profit, firm will take an action as a best reply to (7). That is to say, firm will make a price strategy p (t + ) according to the following formula: p (t + ) = arg max p (t+) πe (t + ). (8) So far, for the Bertrand game we hae described two different time delay structures by (5) and (8), which are together written as a system: { p (t + ) = p (t) + p (t)[w π p t + ( w) π p t ], p (t + ) = arg max p πe (t + ). (9) (t+) From (4), we get the marginal profit of firm as And from (6) and (7), we obtain firm s expected profit π E (t + ) as which has a maximization solution π p t = a p (t) + bp (t) + c. () π E (t + ) = (p (t + ) c )[a p (t + ) + b(αp (t) + ( α)p (t ))], () p (t + ) = arg max p πe (t + ) = a + c + b(αp (t) + ( α)p (t )). () (t+) All the equations (9), () and () build a discrete dynamic system with one step delay: p (t + ) = p (t) + p (t)[w(a p (t) + bp (t) + c ) + ( w)(a p (t ) + bp (t ) + c )], p (t + ) = a+c +b(αp (t)+( α)p (t )). By setting p (t + ) = p (t) and p 4 (t + ) = p (t), we rewrite system () as a four-dimensional system as follows: p (t + ) = p (t) + p (t)[w(a p (t) + bp (t) + c ) + ( w)(a p (t) + bp 4 (t) + c )], p (t + ) = a+c +b(αp (t)+( α)p (t)), (4) p (t + ) = p (t), p 4 (t + ) = p (t). () IJNS homepage:

4 International Journal of Nonlinear Science, Vol.5(8), No., pp. 9-8 And let p i (t + ) = p i (t)(i =,,, 4) in system (4), then we obtain a boundary equilibrium E and a nonnegatie interior equilibrium E : where E = (, a + c,, a + c ), E = (p, p, p, p 4), p = p = ( a + ab + c + bc 4 b ), p = p 4 = ( a + ab + c + bc 4 b ). Since we only need to consider a nonnegatie state, we suppose < b < in our model. Stability of the equilibriums The local stability of an equilibrium point (p, p, p, p 4, ) is determined by the eigenalues of the Jacobian matrix J, which takes the form as: a bwp ( w)p b( w)p bα b( α) J(p, p, p, p 4 ) =, (5) where a = + [w(a 4p + bp + c ) + ( w)(a p + bp 4 + c )]. If all the absolute alues of the eigenalues of the Jacobian matrix J(p, p, p, p 4 ) are less than, then the equilibrium (p, p, p, p 4, ) will be locally asymptotically stable. On the contrary, the equilibrium (p, p, p, p 4 ) will be unstable if there is an eigenalue such that its absolute alue is greater than. At the boundary equilibrium point E, the Jacobian matrix (5) takes a form as: J(E ) = + (a+ab+bc+c) bα b( α) It is obious that J(E ) has an eigenalue λ = + (a+ab+bc+c) >, so we conclude that the boundary equilibrium point E is unstable. Now we consider the asymptotic stability of E. At the interior equilibrium E, the condition w(a p + bp + c ) + ( w)(a p + bp + c ) = must holds. Then the Jacobian matrix at E is gien by: J(E ) =. wp bwp ( w)p b( w)p bα b( α) Let Y (λ) denotes the characteristic polynomial of J(E ), then it has a form as: Y (λ) = λ 4 + Y λ + Y λ + Y λ + Y 4,. where Y = + wp, Y = ( w)p b wαp, Y = b p (wα w α), Y 4 = b p (w )( α). IJNS for contribution: editor@nonlinearscience.org.uk

5 H. Liu and Z. Ding: Dynamics of a Bertrand Game with Heterogeneous Players and Different Delay Structures Figure : Bifurcation diagrams with respect to the adjustment rate for different leels of α According to the Schur-Cohn Criterion (see e.g. []), all the roots of the polynomial Y (λ) (i.e., the eigenalues of the Jacobian matrix J(E )) will lie inside the unit disk if the following inequalities hold: (i)y () = + Y + Y + Y + Y 4 > ; (ii)( ) 4 Y ( ) = Y + Y Y + Y 4 > ; (iii)the determinates of the matrices B ± and the matrices B± are all positie, where B + = ( + Y 4), B = ( Y 4), B ± = Y ± Y 4 Y 4 Y Y Y Y 4 Y Y. It is obious that the condition Y 4 > always hold in our model, and + Y + Y + Y + Y 4 = p ( b ) is also positie since < b < is supposed in our model. Thus, we conclude that the interior equilibrium point E of system (4) is locally asymptotically stable if it satisfies the following conditions: which can be reduced to a group of inequalities as: Y + Y Y + Y 4 >, + Y 4 >, Det(B + ) >, Det(B ) >, (a) + Y 4 >, (b) Y + Y Y + Y 4 >, (c) Y Y Y 4 + Y4 + Y Y4 + Y < Y + Y 4 Y Y + Y Y 4 Y4 < + Y Y Y 4 + Y Y 4 Y4 Y Y4 Y. The interior equilibrium E will be locally stable if the coefficients of characteristic polynomial satisfy the conditions (a), (b) and (c). In the following section, we will proide numerical eidences for the complex dynamical behaiors of IJNS homepage:

6 4 International Journal of Nonlinear Science, Vol.5(8), No., pp. 9-8 Figure : Bifurcation diagrams with respect to the adjustment rate for different leels of w system when the stability conditions are broken, and use these stability conditions to analyze the stability region in the parameter space by numerical simulations. 4 Numerical simulations In this section, some detailed numerical results including bifurcation diagrams, phase portraits and stable regions are used to show the complicated dynamic behaior of the system and the different effect on the system stability of the two different delay structures. The numerical work is done for the main model parameters such as the delay weights w and α and the adjustment speed. In all the numerical simulations, we set the other parameters a, b, c and c as: a =, b =.6, c =, andc =.5. Fig. and Fig. show the bifurcation diagrams with respect to the adjustment rate when the system loses stability. To show the influence of the delay weight α on the system stability loss, Fig. is done for three cases that α takes three different alues (α =.,.6 and ) while w is fixed as w =.. Comparing the three bifurcation diagrams in Fig., we obsere that there is a little difference in the alues of for the system to begin stability loss. That is to say, the delay weight α has little effect on the system stability. Similarly, Fig. is plotted for three cases with different alues of w(w =.,.6, ) and a fixed α(α =.). Comparing the three diagrams in Fig., it is obious that there is a big difference in the alues of for the system to lose stability and an intermediate leel of w can make the system keep stability in a large region. So we can say that the delay weight w has a stronger effect on the system stability. From both Fig. and Fig., we can also see that the system will be stable for small alues of the adjustment speed, but the system becomes unstable as increases. Furthermore, the stability conditions (a), (b) and (c) obtained in Section can also be used to show the different influence of the delay weights w and α on the system stability. Based on the numerical solution of the three the stability conditions, Fig. and Fig.4 plot some two-dimensional parameter regions, from which eery couple of parameters together with other fixed ones satisfy the conditions (a), (b) and (c) and hence make the system stable in its eolution. Fig. shows the (, w)-stability regions for four different alues of α(α =.,.6,.75, ), and Fig.4 plots the stability regions IJNS for contribution: editor@nonlinearscience.org.uk

7 H. Liu and Z. Ding: Dynamics of a Bertrand Game with Heterogeneous Players and Different Delay Structures α=. α= w w α= w w. α= Figure : Stability region in (, w)-plane for different leels of α w=. w= α α w= α α. w= Figure 4: Stability region in (, α)-plane for different leels of w IJNS homepage: 5

8 6 International Journal of Nonlinear Science, Vol.5(8), No., pp. 9-8 =.5 =.45 =.46 =.48 = =.5. =.45. =.46. = = Figure 5: Phase portrait for Fig.(a) and Fig.(c) with arious w, α and in the (, α)-plane with four different alues of w(w =.,.6,.75, ).From Fig. we can find that there is nearly no difference among the four stability regions that are plotted for different alues of α. Howeer, form Fig.4 that is associated with different alues of w we can see that the four stability regions are much more distinguishable. And Fig.4 shows that the stability region for the delay weight w =.75 is much larger than the one for a non-delay case (w = ) and the one for a excessie delay case (w =.). From these numerical simulations for stability region, we also get a conclusion that the delay weight w for the marginal profit has a stronger effect on the system stability while the delay weight α for the price has little effect, and an appropriate leel of marginal-profit delay weight w can improe the system stability. If the stability conditions (a), (b) and (c) are not satisfied, the system will lose stability and displays some complicated dynamical behaiors, as shown in Fig. and Fig.. From Fig. and Fig., we can see that the system can lose stability either through period-doubling bifurcations or Neimark-Sacker bifurcations. It is obious that in Fig.(a)(b)(c) and Fig.(a) the system loses stability through Neimark-sacker bifurcations, while in Fig.(c) the system stability loss is caused by a period doubling bifurcation. In order to specify the two different ways of the stability loss, we can consider the phase portraits for each case. For instance, we plot the two-dimensional phase portraits associated with Fig.(a) and Fig.(c) in Fig.5. The first line of phase portraits are done for Fig.(a) with different alues of, which obiously shows a Neimark-sacker bifurcation because closed inariant cures take place when the system loses stability, and which also show there are period windows in the process of Neimark-sacker bifurcation. The second line in Fig.5 plots the phase portraits that are associated with Fig.(c); it is eidently related to a period-doubling process, which leads the system from doubling periods to chaos in the end. 5 Conclusions In a Bertrand market een with limited information, each player is able to get the information not only about his own marginal profit but also the opponents price strategy that is open in the market. When considering time delay, there may be players paying attention to either the marginal profit history or the open price memory. So we consider a duopoly Bertrand game played by heterogeneous players with two different time-delay rationality. We suppose that one firm IJNS for contribution: editor@nonlinearscience.org.uk

9 H. Liu and Z. Ding: Dynamics of a Bertrand Game with Heterogeneous Players and Different Delay Structures 7 adjusts its price strategy according to a kind of smoothed information that aerage its marginal profit in the preious period and the one in a delayed period, while another firm makes its price strategy as a best response to an expectation that aerages the opponents actions in the preious periods with also a one-step delay. Built for this game model, the dynamic system has a boundary equilibrium and an interior equilibrium. The instability of the boundary equilibrium is proed, and the stability conditions of the interior equilibrium are obtained. Numerical simulations including bifurcation diagrams, stability regions and phase portraits hae been done to show the influence of the main model parameters on system eolution. We hae shown that when the stability conditions are broken the system may take complex behaiors, and the loss of the system stability may be caused by Neimark-Sacker bifurcation or period doubling bifurcation; the time delay for the marginal profit has a stronger effect on the system stability than the time delay for the price ariable; an intermediate leel of time delay on the marginal profit can greatly improe the system stability. Acknowledgments Financial support by the National Natural Science Foundation of China (No. 7798) is gratefully acknowledged. References [] A. Cournot. Researches into the mathematical principles of the theory of wealth. Hachette Paris. 88. [] J. Bertrand. Thorie mathcmatique de la richesse sociale. Journal Des Saants, 67(88): [] T. Puu. Complex dynamics with three oligopolists. Chaos, Solitons & Fractals, 7(996)():75 8. [4] A. K. Naimzada and F. Tramontana. Dynamic properties of a Cournot-Bertrand duopoly game with differentiated products. Economic Modelling, 9()(4): [5] E. Ahmed and H. N. Agiza. Dynamics of a Cournot game with n-competitors. Chaos, Solitons & Fractals, 9(998)(9):5 57. [6] H. N. Agiza. Explicit stability zones for Cournot game with and 4 competitors. Chaos, Solitons & Fractals, 9(998)(): [7] J. B. Rosser Jr. The deelopment of complex oligopoly dynamics theory. In T. Puu and I. Sushko, eds., Oligopoly Dynamics: Models and Tools, 5 9. Springer Berlin Heidelberg.. [8] K. Okuguchi. Adaptie expectations in an oligopoly model. Reiew of Economic Studies, 7(97)(): 7. [9] F. Szidaroszky and K. Okuguchi. A linear oligopoly model with adaptie expectations: Stability reconsidered. Journal of Economics, 48(988)():79 8. [] S. Rassenti et al. Adaptation and conergence of behaior in repeated experimental Cournot games. Journal of Economic Behaior & Organization, 4()():7 46. [] G. I. Bischi and M. Kopel. Equilibrium selection in a nonlinear duopoly game with adaptie expectations. Journal of Economic Behaior & Organization, 46()():7. [] H. N. Agiza, G. I. Bischi and M. Kopel. Multistability in a dynamic Cournot game with three oligopolists. Mathematics and Computers in Simulation, 5(999)(-):6 9. [] G. I. Bischi and A. Naimzada. Global analysis of a dynamic duopoly game with bounded rationality. Annals of the International Society of Dynamic Games, 5():6 85. [4] E. Ahmed, A. A. Elsadany and T. Puu. On Bertrand duopoly game with differentiated goods. Applied Mathematics & Computation, 5(5): [5] F. Chen, J. H. Ma and X. Q. Chen. The study of dynamic process of the triopoly games in chinese g telecommunication market. Chaos, Solitons & Fractals, 4(9)(): [6] J. X. Zhang, Q. L. Da and Y. H. Wang. The dynamics of Bertrand model with bounded rationality. Chaos, Solitons & Fractals, 9(9)(5): [7] H. N. Agiza, A. S. Hegazi and A. A. Elsadany. Complex dynamics and synchronization of a duopoly game with bounded rationality. Mathematics and Computers in Simulation, 58()(): 46. [8] J. H. Ma and Z. B. Guo. The influence of information on the stability of a dynamic Bertrand game. Communications in Nonlinear Science & Numerical Simulation, (6)(-): 44. [9] H. N. Agiza and A. A. Elsadany. Nonlinear dynamics in the Cournot duopoly game with heterogeneous players. Physica A: Statistical Mechanics and its Applications, ()():5 54. IJNS homepage:

10 8 International Journal of Nonlinear Science, Vol.5(8), No., pp. 9-8 [] H. N. Agiza and A. A. Elsadany. Chaotic dynamics in nonlinear duopoly game with heterogeneous players. Applied Mathematics and Computation, 49(4)(): [] J. X. Zhang, Q. L. Da and Y. H. Wang. Analysis of nonlinear duopoly game with heterogeneous players. Economic Modelling, 4(7)():8 48. [] T. Dubiel-Teleszynski. Nonlinear dynamics in a heterogeneous duopoly game with adjusting players and diseconomies of scale. Communications in Nonlinear Science and Numerical Simulation, 6()():96 8. [] Y. Q. Fan, T. Xie and J. G. Du. Complex dynamics of duopoly game with heterogeneous players: A further analysis of the output model. Applied Mathematics & Computation, 8()(5): [4] Z. W. Ding, Q. L. Hang and H. L. Yang. Analysis of the dynamics of multi-team Bertrand game with heterogeneous players. Applied Mathematics and Computation, 5(9)():98 5. [5] B. G. Xin and T. Chen. On a master-slae Bertrand game model. Economic Modelling, 8()(4): [6] S. Z. Hassan. On delayed dynamical duopoly. Applied Mathematics and Computation, 5(4)(): [7] M. T. Yassen and H. N. Agiza. Analysis of a duopoly game with delayed bounded rationality. Applied Mathematics and Computation, 8()(-):87 4. [8] Y. L. Lu. Dynamics of a delayed duopoly game with increasing marginal costs and bounded rationality strategy. Procedia Engineering, 5(): [9] A. A. Elsadany. Dynamics of a delayed duopoly game with bounded rationality. Mathematical and Computer Modelling, 5()(9C): [] Z. W. Ding, X. F. Zhu and S. M. Jiang. Dynamical Cournot game with bounded rationality and time delay for marginal profit. Mathematics & Computers in Simulation, (4):. [] J. Peng, Z. Miao and F. Peng. Study on a -dimensional game model with delayed bounded rationality. Applied Mathematics & Computation, 8()(5): [] S. Elaydi. An Introduction to Difference Equations. Springer New York, third edn. 5. IJNS for contribution: editor@nonlinearscience.org.uk