Dynamics of a delayed duopoly game with ... - Semantic Scholar
Mathematical and Computer Modelling 52 (2010) 1479–1489
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Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Dynamics of a delayed duopoly game with bounded rationality A.A. Elsadany ∗ Department of Mathematics, Faculty of Education in Al-Ijailat, Seventh of April University, Zawia, Libya
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info
Article history: Received 17 September 2009 Received in revised form 31 May 2010 Accepted 1 June 2010 Keywords: Delayed bounded rationality Duopoly game Chaotic Strange attractor
abstract A bounded rationality duopoly game with delay is formulated. Its dynamical evolution is analyzed. The existence of an economic equilibrium of the game is derived. The local stability analysis has been carried out. The analysis showed that firms using delayed bounded rationality have a higher chance of reaching a Nash equilibrium point. Numerical simulations were used to show bifurcation diagrams and phase portraits. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction An oligopoly is the case where the market is controlled by a small number of firms. Even the duopoly situation, an oligopoly of two producers, can be more complex than one might imagine since the duopolists have to take into account their actions and reactions when a decision is made. Oligopoly theory is one of the oldest branches of mathematical economics dating back to 1838 when its basic model was proposed by Cournot [1,2]. In the repeated oligopoly game all players maximize their profits. Recently, the dynamics of the duopoly game has been studied in [3–13]. Bischi and Naimzada [5] gave the general formula of the oligopoly model with a form of bounded rationality. They discussed the global and local stability of the duopoly game with a particular form of bounded rationality. They showed that the dynamics of the game can lead to complex behavior such as cycles and chaos. Ahmed et al. [13] used the Jury condition to discuss the stability of a modification of Puu’s model with bounded rationality. Agiza et al. examined the dynamical behavior of Bowley’s model with bonded rationality [12]. Agiza et al. [11] have been studied the complex dynamics of a bounded rationality duopoly game with a nonlinear demand function. The modification of the duopoly game depends on the strategy that the firms use, such as homogeneous and heterogeneous; and the expectations of the output the firms have to maximize, such as bounded rationality, naive expectation and adaptive expectation, see [10,9]. They developed duopoly game with heterogeneous players. The development of complex oligopoly dynamics theory has been reviewed in [14]. Other studies on the dynamics of oligopoly models with more firms and other modifications have been studied [15–17,6,18]. Also in the past decade, there has been a great deal of interest in chaos control of duopoly games because its complexity see [19,20] and its references. The present work aims to formulate a bounded rationality duopoly game with delay and studying its dynamical behaviors. In additions it is aimed to check if the delay case used (that is, considering markets with memory) is a more realistic assumption than the non-delay case and increases the stability of the system. This paper is organized as follows. In Section 2, the delayed duopoly game with bounded rationality is briefly described. In Section 3, we analyze the dynamics for a simple case of a delayed duopoly game with bounded rationality. Explicit parametric conditions of the existence, local stability of equilibrium points will be given. In Section 4, we present the numerical simulations, to verify our results which taken place by the theoretical analysis. Finally, some remarks are confined in Section 5.
∗
Tel.: +218 255620649; fax: +218 255620649. E-mail address: [email protected].
0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.06.011
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2. Delayed duopoly game with bounded rationality method We consider a Cournot duopoly game where qi denotes the quantity supplied by firm i, i = 1, 2. In addition let P (qi + qj ), i 6= j, denote to a twice differentiable and nonincreasing inverse demand function and let Ci (qi ) denote the twice differentiable increasing cost function. For firm i the profit resulting from the above Cournot game is given by
Πi (t ) = P (qi (t ) + qej (t + 1))qi (t ) − Ci (qi (t )).
(1)
The profit maximizing behavior of player i, taking the quantity supplied to the opponent j, i 6= j, as given, results in the well-known reaction function for firm i qi (t + 1) = ri (qej (t + 1)) = arg max[P (qi (t ) + qej (t + 1))qi (t ) − Ci (qi (t ))].
(2)
qi
Cournot assumed that the expected quantity in the next time step qej (t + 1) is given by qej (t + 1) = qj (t ).
(3)
Then the Cournot duopoly game defined as a discrete dynamical system has the form qi (t + 1) = ri (qj (t )),
i, j = 1, 2,
i 6= j.
(4)
However, as pointed out by Bischi and Naimzada [5] there is an unrealistic assumption in this approach. It is implicitly assumed that the duopolist knows the market’s demand function. A more realistic approach is to assume a bounded rationality i.e. each firm (say ith one) modifies its production according to its marginal profit Hence the dynamical system of a duopoly game with a bounded rationality is
qi (t + 1) = qi (t ) + αi (qi )
∂ Πi (qi (t ), qej (t + 1)) ∂ qi
i = 1, 2,
∂ Πi (qi (t ),qej (t +1)) ∂ qi
i 6= j
, i = 1, 2, i 6= j.
(5)
where αi (qi ) is the adjustment of the ith firm i = 1, 2. They assumed that qej (t + 1) = qj (t ) in the term of the bounded rationality and also assumed αi (qi ) = αi qi . Then the Bischi-Naimzada bounded rationality duopoly game has the form
qi (t + 1) = qi (t ) + αi qi
∂ Πi (qi (t ), qj (t )) i = 1, 2, ∂ qi
i 6= j .
(6)
This means that, if the marginal profit is positive/negative he increases /decreases its production qi in the next output period. Also they assumed that the expected product of a firm qe (t + 1) is equal to its previous quantity q(t ) in the bounded rationality term. However it may make more sense to use previous productions i.e. q(t − 1), q(t − 2), . . . , q(t − T ) with different weights. This point of view has been studied in [13,21–23] in a different context. Ahmed et al. and Agiza et al. have been examined this point in the monopoly case only. In [13,21], they assumed that delay was put in the full term of bounded rationality for all players in the game. In this paper, we think that a greater reality for this game is put the delay in the term of the bounded rationality for all players except the ith player. Here both realistic ideas of bounded rationality and delay are combined. It will be shown that delay increases the stability domain. The dynamical system will be qi (t + 1) = qi (t ) + αi qi
∂ Πi (qi , qD ) , ∂ qi
i = 1, 2
(7)
where qD = qej (t + 1) = l=0 qj (t − l)ωl , ωl ≥ 0, l=0 ωl = 1. The factors ωl , l = 0, 1, 2, . . . , T are the weights given to previous productions. From Eq. (7), it is clear that the delay was put in the bounded rationality term for all players except the player i. This argument is the basic difference between our paper and the other papers [13,21,24].
PT
PT
3. Dynamics of the simple delayed duopoly game with bonded rationality For simplicity set T = 1, and consider the duopoly case and the profit of i th firm is given by
Πi = qi (a − b(q1 + q2 )) − ci qi ,
i = 1, 2.
Under the above assumption, the delayed duopoly game with bounded rationality (7) is given by q1 (t + 1) = q1 (t ) + α1 q1 {a − c1 − 2bq1 (t ) − b[ω2 q2 (t ) + (1 − ω2 )q2 (t − 1)]} . q2 (t + 1) = q2 (t ) + α2 q2 {a − c2 − 2bq2 (t ) − b[ω1 q1 (t ) + (1 − ω1 )q1 (t − 1)]}
(8)
To study the stability of dynamical system (8), rewrite it as a fourth dimensional system in the form p1 (t + 1) = q1 (t ) p2 (t + 1) = q2 (t ) q1 (t + 1) = q1 (t ) + α1 q1 (t )(a − c1 − 2bq1 (t ) − b[ω2 q2 (t ) + (1 − ω2 )p2 (t )]) q2 (t + 1) = q2 (t ) + α2 q2 (a − c2 − 2bq2 (t ) − b[ω1 q1 (t ) + (1 − ω1 )p1 (t )]).
(9)
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3.1. Equilibrium points and local stability It is clear that the system (9) has four fixed points in the following form: E0 = (0, 0, 0, 0)
E1 =
2b
E2 =
a − c1
0,
, 0,
a − c2 2b
a − c1
,0 , a − c2
if c1 < a
2b
, 0, ,
2b
,
if c2 < a
E∗ = (q∗1 , q∗2 , q∗1 , q∗2 )
(10)
such that q∗1 =
a + c2 − 2c1
and
3b
q∗2 =
a + c1 − 2c2 3b
.
(11)
Obviously, E0 , E1 , E2 are boundary equilibrium points. The fixed point E∗ is a Nash equilibrium point and has economic meaning when
2c1 − c2 < a 2c2 − c1 < a.
(12)
To investigate the local stability of the equilibrium points E0 , E1 , E2 and E∗ we have find the Jacobian matrix of the system equations (9). The Jacobian matrix for the model system (9) at any point (p1 , p2 , q1 , q2 ) takes the form
0 0 J (p1 , p2 , q1 , q2 ) = 0 −α2 b(1 − ω1 )q2
0 0 −α1 b(1 − ω2 )q1 0
1 0 A α2 bω1 q2
0 1 −α1 bω2 q1 B
(13)
where A = 1 + α1 (a − c1 − 4bq1 − b[ω2 q2 + (1 − ω2 )p2 ]) and B = 1 + α2 (a − c2 − 4bq2 − b[ω1 q1 + (1 − ω1 )p1 ]). The stability of equilibrium points will be determined by the nature of the eigenvalues of the Jacobian matrix evaluated at the corresponding equilibrium points. Theorem 1. The trivial equilibrium point E0 of system (9) is an unstable equilibrium point. Proof. At the trivial equilibrium point E0 (0, 0, 0, 0) the Jacobian matrix given by (13) takes the form
0 0 J (E0 ) = 0 0
0 0 0 0
1 0 1 + α1 (a − c1 ) 0
0 1 . 0 1 + α2 (a − c2 )
The eigenvalues of J (E0 ) are given by λ1 = λ2 = 0, λ3 = 1 + α1 (a − c1 ) and λ 3 = 1 + α2 (a − c2 ). From the conditions that a, αi , ci (i = 1, 2) are positive parameters and ci < a, i = 1, 2. We have that λ3,4 > 1. Hence the trivial equilibrium point E0 is unstable Theorem 2. The equilibrium points E1 , E2 of system (9) are saddle points. Proof. At the boundary equilibrium point E1 (
0
0 J (E1 ) = 0 α (1 − ω )(a − c ) 2 2 2 − 2
with eigenvalues λ1 = λ2 = 0, λ3 = 1 +
0 0 0 0 α1 2
, 0, a−2bc1 , 0), the Jacobian matrix (13) takes the form of
a − c1 2b
1+
α1
−
1 0
(a − 2c1 + c2 ) α2 ω1 (a − c2 )
2
2
0 1
0 1 − α2 (a − c2 )
(a − 2c1 + c2 ) and λ4 = 1 − α2 (a − c2 ). Since αi , ci (i = 1, 2) and a are positive
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Fig. 1. Bifurcation diagrams of q1 with respect to α1 in two cases: non-delay (blue) and delay (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
parameters, ci < a, i = 1, 2 for E1 have economic meaning and when (12) are satisfied, then the eigenvalue λ3 is greater than 1 and λ4 is less than 1. Therefore E1 is a saddle point. Similarly we can prove that E2 is also a saddle point. Now we investigate the local stability of the Nash equilibrium point E∗ . The Jacobian matrix (13) at E∗ is
0 0 J (E∗ ) = 0 −α2 b(1 − ω1 )q∗2
0 0 −α1 b(1 − ω2 )q∗1 0
1 0 1 − 2α1 bq∗1 −α2 bω1 q∗2
0 1 . −α1 bω2 q∗1 ∗ 1 − 2α2 bq2
(14)
The Nash equilibrium point is given in (10) and stability conditions are that all roots of the equation P (λ) = 0 satisfy |λ| < 1,where P (λ) = λ4 + a1 λ3 + a2 λ2 + a3 λ + a4
(15)
such that a1 = −2 + 2α1 bq∗1 + 2α2 bq∗2 a2 = 1 − 2α1 bq∗1 − 2α2 bq∗2 + 4α1 α2 b2 q∗1 q∗2 − α1 α2 b2 ω1 ω2 q∗1 q∗2 a3 = −α1 α2 b2 ω1 q∗1 q∗2 + 2α1 α2 b2 ω1 ω2 q∗1 q∗2 − α1 α2 b2 ω2 q∗1 q∗2 a4 = −α1 α2 b2 q∗1 q∗2 + α1 α2 b2 ω1 q∗1 q∗2 + α1 α2 b2 ω2 q∗1 q∗2 − α1 α2 b2 ω1 ω2 q∗1 q∗2 . A necessary and sufficient condition for (14) to have only roots of absolute value less than one is the following (see [25]): a4 < 1
(16)
3 + 3a4 > a2
(17)
1 + a1 + a2 + a3 + a4 > 0
(18)
1 − a1 + a2 − a3 + a4 > 0
(1 − a4 )(1 −
a24
(19)
) − a2 (1 − a4 ) + (a1 − a3 )(a3 − a1 a4 ) > 0. 2
(20)
According the Jury criteria, the Nash equilibrium point is locally asymptotically stable if the condition equations (16)–(20) are satisfied. When ωi , i = 1, 2 are sufficiently small and for 0 < b < 1 the Nash equilibrium point is stable, see Figs. 10–14. Hence, we deduced from above analysis that delay has a stabilization effect for the Nash equilibrium point. 4. Numerical simulations To provide some numerical evidence for the dynamical behavior of model (9), we present various numerical results here to show that the delay has the effect of increasing the stability domain. In order to study the local stability properties of the equilibrium points it is convenient to take the parameters’ values as follows: a = 10, b = 0.5, c1 = 3, c2 = 5. Fig. 1 shows the bifurcation diagrams of q1 with respect to α1 in two cases: non-delay (ω1 = ω2 = 1, blue graph) and delay (ω1 = ω2 = 0.5, red graph). It also shows that the bifurcation diagram of q1 in the non-delay case ω1 = ω2 = 1
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Fig. 2. Bifurcation diagrams of q2 with respect to α1 in two cases: non-delay (blue) and delay (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Strange attractor for the duopoly game (9) in the non-delay case ω1 = ω2 = 1.
(blue graph) converges to the Nash equilibrium point as α1 < 0.27. As α1 > 0.27, the Nash equilibrium point becomes unstable. Period-doubling bifurcations appear and finally chaotic behaviors occur. Also the bifurcation diagram of q1 with respect to α1 in delay case ω1 = ω2 = 0.5 (red graph) is plotted. The Nash equilibrium in the delayed case is converges as α1 < 0.332. As α1 > 0.332, the Nash equilibrium point becomes unstable. Comparing the bifurcation diagrams (blue and red), it is observed that period-doubling bifurcations are delayed in the system, as expected. Fig. 2 shows the bifurcation diagrams of q2 with respect to α1 in two cases: non-delay (ω1 = ω2 = 1, blue graph) and delay (ω1 = ω2 = 0.5, red graph). Also, period-doubling bifurcations are delayed in system (9) for q2 . Fig. 3 shows the graph of the strange attractor of system (9) in the non-delay case. Phase portrait of system (9) in the delayed case ω1 = ω2 = 0.5 is plotted in Fig. 4.
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Fig. 4. Strange attractor for the delayed duopoly game (9) when ω1 = ω2 = 0.5.
Fig. 5. Bifurcation diagram of q1 with respect to the α1 delayed case when ω1 = ω2 = 0.1.
From Figs. 1–4, everyone can deduce that a delay has the effect of delaying a period-doubling appearance. So, the delay has a stabilization effect for the Nash equilibrium point. Figs. 5 and 6 show the bifurcation diagrams of q1 in two delayed cases: first when ω1 = ω2 = 0.1, other ω1 = ω2 = 0.7. It is clear that also, that when period-doubling occurs it is later under the delay effect than those observed in the non-delay case. Strange attractor of the delay case when ω1 = ω2 = 0.7 is plotted in Fig. 7. From above, it is clear that delay increases stability more than the case of non-delay. As an example compare the case ω1 = ω2 = 1 (non-delay) with ω1 = ω2 = 0.5 (delay). Fig. 8 shows the stability region of the Nash equilibrium point in the non-delay case. Also Fig. 9 shows the stability region of the Nash equilibrium point in the delay case when ω1 = ω2 = 0.5. Comparing between Figs. 8 and 9, one can see that the stability region in the delay case is larger than that in the non-delay case.
A.A. Elsadany / Mathematical and Computer Modelling 52 (2010) 1479–1489
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Fig. 6. Bifurcation diagram of q1 with respect to the α1 delayed case when ω1 = ω2 = 0.7.
Fig. 7. Strange attractor for the delayed duopoly game (9) when ω1 = ω2 = 0.7.
Consequently firms using delay have a higher chance of reaching a Nash equilibrium point than these not using delay. Figs. 10–13 show the bifurcation diagrams of q1 with respect to ω1 . The bifurcation diagram of q1 with respect to ω1 is polluted in Fig. 14. From Figs. 10–14, we deduce that the delay increases the stability domain.
5. Conclusion In this paper, a duopoly delayed bounded rationality game has been proposed and analyzed. The local stability of four equilibrium points is investigated in this game. Basic properties of the game have been analyzed by means of bifurcation diagrams, phase portraits and stability regions. From above analysis, everyone can see that the delay case increases the
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Fig. 8. Stability region of the Nash equilibrium point in the non-delay case.
Fig. 9. Stability region of the Nash equilibrium point in the delay case when ω1 = ω2 = 0.5.
domain of stability. Firms using a delayed bounded rationality method have a higher chance of reaching the Nash equilibrium point than those using bounded rationality without delay. It is important in studying oligopoly games to take the memory of the market into consideration, because, in an oligopoly it is very hard to sell all the quantity for each year in the next year. So, we must consider the oligopoly market with memory. In the near future, we will publish other papers containing more details about oligopoly games with memory.
Acknowledgements The author thanks the anonymous reviewers for providing some helpful comments which help to improve the style of this work. Also I wish to thank Professors E. Ahmed, Y. Moustafa for discussion and help.
A.A. Elsadany / Mathematical and Computer Modelling 52 (2010) 1479–1489
Fig. 10. Bifurcation diagram of q1 with respect to ω1 when ω2 = 0.
Fig. 11. Bifurcation diagram of q1 with respect to ω1 when ω2 = 0.2.
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Fig. 12. Bifurcation diagram of q1 with respect to ω1 when ω2 = 0.5.
Fig. 13. Bifurcation diagram of q1 with respect to ω1 when ω2 = 0.8.
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Fig. 14. Bifurcation diagram of q2 with respect to ω1 when ω2 = 0.5.
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A. Cournot, Researches into the Principles of the Theory of Wealth, Irwin Paper Back Classics in Economics, 1963, Engl. trans. (Chapter VII). J. Friedman, Oligopoly Theory, in: K.j. Arrow, M.d. Intriligator (Eds.), Handbook of Mathematical Economics, vol. II, North-Holland, 1982. T. Puu, Chaos in duopoly pricing, Chaos Solitons Fractals 1 (1991) 573–581. T. Puu, The chaotic duopolists revisited, J. Econ. Behav. Organ. 37 (1998) 385–394. G.I. Bischi, A. Naimzada, Global Analysis of a Dynamic Duopoly Game with Bounded Rationality, in: Advanced in Dynamic Games and Application, vol. 5, Birkhouser, 1999 (Chapter 20). G.I. Bischi, A.K. Naimzada, L. Sbragia, Oligopoly Games with Local Monopolistic Approximation, J. Econ. Behav. Organ. 62 (2007) 371–388. G.I. Bischi, L. Stefanini, L. Gardini, Synchronization intermittency and critical curves in a duopoly game, Math. Comput. Simulution 44 (1998) 559–585. H.N. Agiza, On the stability, bifurcations, chaos and chaos control of Kopel map, Chaos Solitons Fractals 11 (1999) 1909–1916. H.N. Agiza, A.A. Elsadany, Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Appl. Math. Comput. 149 (2004) 843–860. H.N. Agiza, A.A. Elsadany, Nonlinear dynamics in the Cournot duopoly game with heterogeneous players, Physica A 320 (2003) 512–524. H.N. Agiza, A.S. Hegazi, A.A. Elsadany, Complex dynamics and synchronization of duopoly game with bounded rationality, Math. Comput. Simulation 58 (2002) 133–146. H.N. Agiza, A.S. Hegazi, A.A. Elsadany, The dynamics of Bowley’s model with bounded rationality, Chaos Solitons Fractals 9 (2001) 1705–1717. E. Ahmed, H.N. Agiza, S.Z Hassan, On modifications of Puu’s dynamical duopoly, Chaos Solitons Fractals 7 (2000) 1025–1028. J.B. Rosser, The development of complex oligopoly dynamics theory, in: Text Book Oligopoly Dynamics : Models and Tools, Springer-Verleg, 2002. A. Naimzada, L. Sbragia, Oligopoly games with nonlinear demand and cost functions: Two boundedly rational adjustment processes, Chaos Solitons Fractals 2 (9) (2006) 707–722. H.N. Yao, F. Xu, Complex dynamics analysis for a duopoly advertising model with nonlinear cost, Appl. Math. Comput. 180 (2006) 134–145. J. Zhang, Q. Da, Y. Wang, Analysis of nonlinear duopoly game with hetergeneous players, Econ. Model. 24 (2007) 138–148. J. Du, T. Huang, New results on stable region of Nash equilibrium point of output game model, Appl. Math. Comput. 192 (2007) 12–19. J. Du, T. Huang, Z. Sheng, Anaylsis of decision-making in economic chaos control, Nonlinear anaylsis: Real World Applications 10 (2009) 2493–2501. Z. Sheng, T. Huang, J. Du, Q Mei, H. Haung, Study on self-adaptive proportional control method for a class of output models, Discrete Contin. Dyn. Syst. Ser. B 11 (2009) 459–477. F. Aicardi, S. Invernizzi, Memory effects in discrete dynamical systems, Internat. J. Bifur. Chaos 2 (4) (1992) 815–830. V. Bohm, J Wenzelburger, Cycles and local Stability in discrete dynamical systems with delay structure, Discussion Paper 320, Bielefeleld Univ., Department of Economics, 1996. A. Stiefenhofer, Stability and predictability of dynamical economic Systems with memory, Discussion Paper 331, Bielefeleld Univ., Department of Economics, 1997. M.T. Yassen, H.N. Agiza, Analysis of a duopoly game with delayed bounded rationality, Appl. Math. Comput. 138 (2003) 387–402. S.N. Elaydi, An Introduction to Difference Equations, Springer-Verlag Publishers, 1996.
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Dynamics of a delayed duopoly game with bounded rationality A.A. Elsadany ∗ Department of Mathematics, Faculty of Education in Al-Ijailat, Seventh of April University, Zawia, Libya
article
info
Article history: Received 17 September 2009 Received in revised form 31 May 2010 Accepted 1 June 2010 Keywords: Delayed bounded rationality Duopoly game Chaotic Strange attractor
abstract A bounded rationality duopoly game with delay is formulated. Its dynamical evolution is analyzed. The existence of an economic equilibrium of the game is derived. The local stability analysis has been carried out. The analysis showed that firms using delayed bounded rationality have a higher chance of reaching a Nash equilibrium point. Numerical simulations were used to show bifurcation diagrams and phase portraits. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction An oligopoly is the case where the market is controlled by a small number of firms. Even the duopoly situation, an oligopoly of two producers, can be more complex than one might imagine since the duopolists have to take into account their actions and reactions when a decision is made. Oligopoly theory is one of the oldest branches of mathematical economics dating back to 1838 when its basic model was proposed by Cournot [1,2]. In the repeated oligopoly game all players maximize their profits. Recently, the dynamics of the duopoly game has been studied in [3–13]. Bischi and Naimzada [5] gave the general formula of the oligopoly model with a form of bounded rationality. They discussed the global and local stability of the duopoly game with a particular form of bounded rationality. They showed that the dynamics of the game can lead to complex behavior such as cycles and chaos. Ahmed et al. [13] used the Jury condition to discuss the stability of a modification of Puu’s model with bounded rationality. Agiza et al. examined the dynamical behavior of Bowley’s model with bonded rationality [12]. Agiza et al. [11] have been studied the complex dynamics of a bounded rationality duopoly game with a nonlinear demand function. The modification of the duopoly game depends on the strategy that the firms use, such as homogeneous and heterogeneous; and the expectations of the output the firms have to maximize, such as bounded rationality, naive expectation and adaptive expectation, see [10,9]. They developed duopoly game with heterogeneous players. The development of complex oligopoly dynamics theory has been reviewed in [14]. Other studies on the dynamics of oligopoly models with more firms and other modifications have been studied [15–17,6,18]. Also in the past decade, there has been a great deal of interest in chaos control of duopoly games because its complexity see [19,20] and its references. The present work aims to formulate a bounded rationality duopoly game with delay and studying its dynamical behaviors. In additions it is aimed to check if the delay case used (that is, considering markets with memory) is a more realistic assumption than the non-delay case and increases the stability of the system. This paper is organized as follows. In Section 2, the delayed duopoly game with bounded rationality is briefly described. In Section 3, we analyze the dynamics for a simple case of a delayed duopoly game with bounded rationality. Explicit parametric conditions of the existence, local stability of equilibrium points will be given. In Section 4, we present the numerical simulations, to verify our results which taken place by the theoretical analysis. Finally, some remarks are confined in Section 5.
∗
Tel.: +218 255620649; fax: +218 255620649. E-mail address: [email protected].
0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.06.011
1480
A.A. Elsadany / Mathematical and Computer Modelling 52 (2010) 1479–1489
2. Delayed duopoly game with bounded rationality method We consider a Cournot duopoly game where qi denotes the quantity supplied by firm i, i = 1, 2. In addition let P (qi + qj ), i 6= j, denote to a twice differentiable and nonincreasing inverse demand function and let Ci (qi ) denote the twice differentiable increasing cost function. For firm i the profit resulting from the above Cournot game is given by
Πi (t ) = P (qi (t ) + qej (t + 1))qi (t ) − Ci (qi (t )).
(1)
The profit maximizing behavior of player i, taking the quantity supplied to the opponent j, i 6= j, as given, results in the well-known reaction function for firm i qi (t + 1) = ri (qej (t + 1)) = arg max[P (qi (t ) + qej (t + 1))qi (t ) − Ci (qi (t ))].
(2)
qi
Cournot assumed that the expected quantity in the next time step qej (t + 1) is given by qej (t + 1) = qj (t ).
(3)
Then the Cournot duopoly game defined as a discrete dynamical system has the form qi (t + 1) = ri (qj (t )),
i, j = 1, 2,
i 6= j.
(4)
However, as pointed out by Bischi and Naimzada [5] there is an unrealistic assumption in this approach. It is implicitly assumed that the duopolist knows the market’s demand function. A more realistic approach is to assume a bounded rationality i.e. each firm (say ith one) modifies its production according to its marginal profit Hence the dynamical system of a duopoly game with a bounded rationality is
qi (t + 1) = qi (t ) + αi (qi )
∂ Πi (qi (t ), qej (t + 1)) ∂ qi
i = 1, 2,
∂ Πi (qi (t ),qej (t +1)) ∂ qi
i 6= j
, i = 1, 2, i 6= j.
(5)
where αi (qi ) is the adjustment of the ith firm i = 1, 2. They assumed that qej (t + 1) = qj (t ) in the term of the bounded rationality and also assumed αi (qi ) = αi qi . Then the Bischi-Naimzada bounded rationality duopoly game has the form
qi (t + 1) = qi (t ) + αi qi
∂ Πi (qi (t ), qj (t )) i = 1, 2, ∂ qi
i 6= j .
(6)
This means that, if the marginal profit is positive/negative he increases /decreases its production qi in the next output period. Also they assumed that the expected product of a firm qe (t + 1) is equal to its previous quantity q(t ) in the bounded rationality term. However it may make more sense to use previous productions i.e. q(t − 1), q(t − 2), . . . , q(t − T ) with different weights. This point of view has been studied in [13,21–23] in a different context. Ahmed et al. and Agiza et al. have been examined this point in the monopoly case only. In [13,21], they assumed that delay was put in the full term of bounded rationality for all players in the game. In this paper, we think that a greater reality for this game is put the delay in the term of the bounded rationality for all players except the ith player. Here both realistic ideas of bounded rationality and delay are combined. It will be shown that delay increases the stability domain. The dynamical system will be qi (t + 1) = qi (t ) + αi qi
∂ Πi (qi , qD ) , ∂ qi
i = 1, 2
(7)
where qD = qej (t + 1) = l=0 qj (t − l)ωl , ωl ≥ 0, l=0 ωl = 1. The factors ωl , l = 0, 1, 2, . . . , T are the weights given to previous productions. From Eq. (7), it is clear that the delay was put in the bounded rationality term for all players except the player i. This argument is the basic difference between our paper and the other papers [13,21,24].
PT
PT
3. Dynamics of the simple delayed duopoly game with bonded rationality For simplicity set T = 1, and consider the duopoly case and the profit of i th firm is given by
Πi = qi (a − b(q1 + q2 )) − ci qi ,
i = 1, 2.
Under the above assumption, the delayed duopoly game with bounded rationality (7) is given by q1 (t + 1) = q1 (t ) + α1 q1 {a − c1 − 2bq1 (t ) − b[ω2 q2 (t ) + (1 − ω2 )q2 (t − 1)]} . q2 (t + 1) = q2 (t ) + α2 q2 {a − c2 − 2bq2 (t ) − b[ω1 q1 (t ) + (1 − ω1 )q1 (t − 1)]}
(8)
To study the stability of dynamical system (8), rewrite it as a fourth dimensional system in the form p1 (t + 1) = q1 (t ) p2 (t + 1) = q2 (t ) q1 (t + 1) = q1 (t ) + α1 q1 (t )(a − c1 − 2bq1 (t ) − b[ω2 q2 (t ) + (1 − ω2 )p2 (t )]) q2 (t + 1) = q2 (t ) + α2 q2 (a − c2 − 2bq2 (t ) − b[ω1 q1 (t ) + (1 − ω1 )p1 (t )]).
(9)
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3.1. Equilibrium points and local stability It is clear that the system (9) has four fixed points in the following form: E0 = (0, 0, 0, 0)
E1 =
2b
E2 =
a − c1
0,
, 0,
a − c2 2b
a − c1
,0 , a − c2
if c1 < a
2b
, 0, ,
2b
,
if c2 < a
E∗ = (q∗1 , q∗2 , q∗1 , q∗2 )
(10)
such that q∗1 =
a + c2 − 2c1
and
3b
q∗2 =
a + c1 − 2c2 3b
.
(11)
Obviously, E0 , E1 , E2 are boundary equilibrium points. The fixed point E∗ is a Nash equilibrium point and has economic meaning when
2c1 − c2 < a 2c2 − c1 < a.
(12)
To investigate the local stability of the equilibrium points E0 , E1 , E2 and E∗ we have find the Jacobian matrix of the system equations (9). The Jacobian matrix for the model system (9) at any point (p1 , p2 , q1 , q2 ) takes the form
0 0 J (p1 , p2 , q1 , q2 ) = 0 −α2 b(1 − ω1 )q2
0 0 −α1 b(1 − ω2 )q1 0
1 0 A α2 bω1 q2
0 1 −α1 bω2 q1 B
(13)
where A = 1 + α1 (a − c1 − 4bq1 − b[ω2 q2 + (1 − ω2 )p2 ]) and B = 1 + α2 (a − c2 − 4bq2 − b[ω1 q1 + (1 − ω1 )p1 ]). The stability of equilibrium points will be determined by the nature of the eigenvalues of the Jacobian matrix evaluated at the corresponding equilibrium points. Theorem 1. The trivial equilibrium point E0 of system (9) is an unstable equilibrium point. Proof. At the trivial equilibrium point E0 (0, 0, 0, 0) the Jacobian matrix given by (13) takes the form
0 0 J (E0 ) = 0 0
0 0 0 0
1 0 1 + α1 (a − c1 ) 0
0 1 . 0 1 + α2 (a − c2 )
The eigenvalues of J (E0 ) are given by λ1 = λ2 = 0, λ3 = 1 + α1 (a − c1 ) and λ 3 = 1 + α2 (a − c2 ). From the conditions that a, αi , ci (i = 1, 2) are positive parameters and ci < a, i = 1, 2. We have that λ3,4 > 1. Hence the trivial equilibrium point E0 is unstable Theorem 2. The equilibrium points E1 , E2 of system (9) are saddle points. Proof. At the boundary equilibrium point E1 (
0
0 J (E1 ) = 0 α (1 − ω )(a − c ) 2 2 2 − 2
with eigenvalues λ1 = λ2 = 0, λ3 = 1 +
0 0 0 0 α1 2
, 0, a−2bc1 , 0), the Jacobian matrix (13) takes the form of
a − c1 2b
1+
α1
−
1 0
(a − 2c1 + c2 ) α2 ω1 (a − c2 )
2
2
0 1
0 1 − α2 (a − c2 )
(a − 2c1 + c2 ) and λ4 = 1 − α2 (a − c2 ). Since αi , ci (i = 1, 2) and a are positive
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Fig. 1. Bifurcation diagrams of q1 with respect to α1 in two cases: non-delay (blue) and delay (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
parameters, ci < a, i = 1, 2 for E1 have economic meaning and when (12) are satisfied, then the eigenvalue λ3 is greater than 1 and λ4 is less than 1. Therefore E1 is a saddle point. Similarly we can prove that E2 is also a saddle point. Now we investigate the local stability of the Nash equilibrium point E∗ . The Jacobian matrix (13) at E∗ is
0 0 J (E∗ ) = 0 −α2 b(1 − ω1 )q∗2
0 0 −α1 b(1 − ω2 )q∗1 0
1 0 1 − 2α1 bq∗1 −α2 bω1 q∗2
0 1 . −α1 bω2 q∗1 ∗ 1 − 2α2 bq2
(14)
The Nash equilibrium point is given in (10) and stability conditions are that all roots of the equation P (λ) = 0 satisfy |λ| < 1,where P (λ) = λ4 + a1 λ3 + a2 λ2 + a3 λ + a4
(15)
such that a1 = −2 + 2α1 bq∗1 + 2α2 bq∗2 a2 = 1 − 2α1 bq∗1 − 2α2 bq∗2 + 4α1 α2 b2 q∗1 q∗2 − α1 α2 b2 ω1 ω2 q∗1 q∗2 a3 = −α1 α2 b2 ω1 q∗1 q∗2 + 2α1 α2 b2 ω1 ω2 q∗1 q∗2 − α1 α2 b2 ω2 q∗1 q∗2 a4 = −α1 α2 b2 q∗1 q∗2 + α1 α2 b2 ω1 q∗1 q∗2 + α1 α2 b2 ω2 q∗1 q∗2 − α1 α2 b2 ω1 ω2 q∗1 q∗2 . A necessary and sufficient condition for (14) to have only roots of absolute value less than one is the following (see [25]): a4 < 1
(16)
3 + 3a4 > a2
(17)
1 + a1 + a2 + a3 + a4 > 0
(18)
1 − a1 + a2 − a3 + a4 > 0
(1 − a4 )(1 −
a24
(19)
) − a2 (1 − a4 ) + (a1 − a3 )(a3 − a1 a4 ) > 0. 2
(20)
According the Jury criteria, the Nash equilibrium point is locally asymptotically stable if the condition equations (16)–(20) are satisfied. When ωi , i = 1, 2 are sufficiently small and for 0 < b < 1 the Nash equilibrium point is stable, see Figs. 10–14. Hence, we deduced from above analysis that delay has a stabilization effect for the Nash equilibrium point. 4. Numerical simulations To provide some numerical evidence for the dynamical behavior of model (9), we present various numerical results here to show that the delay has the effect of increasing the stability domain. In order to study the local stability properties of the equilibrium points it is convenient to take the parameters’ values as follows: a = 10, b = 0.5, c1 = 3, c2 = 5. Fig. 1 shows the bifurcation diagrams of q1 with respect to α1 in two cases: non-delay (ω1 = ω2 = 1, blue graph) and delay (ω1 = ω2 = 0.5, red graph). It also shows that the bifurcation diagram of q1 in the non-delay case ω1 = ω2 = 1
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Fig. 2. Bifurcation diagrams of q2 with respect to α1 in two cases: non-delay (blue) and delay (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Strange attractor for the duopoly game (9) in the non-delay case ω1 = ω2 = 1.
(blue graph) converges to the Nash equilibrium point as α1 < 0.27. As α1 > 0.27, the Nash equilibrium point becomes unstable. Period-doubling bifurcations appear and finally chaotic behaviors occur. Also the bifurcation diagram of q1 with respect to α1 in delay case ω1 = ω2 = 0.5 (red graph) is plotted. The Nash equilibrium in the delayed case is converges as α1 < 0.332. As α1 > 0.332, the Nash equilibrium point becomes unstable. Comparing the bifurcation diagrams (blue and red), it is observed that period-doubling bifurcations are delayed in the system, as expected. Fig. 2 shows the bifurcation diagrams of q2 with respect to α1 in two cases: non-delay (ω1 = ω2 = 1, blue graph) and delay (ω1 = ω2 = 0.5, red graph). Also, period-doubling bifurcations are delayed in system (9) for q2 . Fig. 3 shows the graph of the strange attractor of system (9) in the non-delay case. Phase portrait of system (9) in the delayed case ω1 = ω2 = 0.5 is plotted in Fig. 4.
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Fig. 4. Strange attractor for the delayed duopoly game (9) when ω1 = ω2 = 0.5.
Fig. 5. Bifurcation diagram of q1 with respect to the α1 delayed case when ω1 = ω2 = 0.1.
From Figs. 1–4, everyone can deduce that a delay has the effect of delaying a period-doubling appearance. So, the delay has a stabilization effect for the Nash equilibrium point. Figs. 5 and 6 show the bifurcation diagrams of q1 in two delayed cases: first when ω1 = ω2 = 0.1, other ω1 = ω2 = 0.7. It is clear that also, that when period-doubling occurs it is later under the delay effect than those observed in the non-delay case. Strange attractor of the delay case when ω1 = ω2 = 0.7 is plotted in Fig. 7. From above, it is clear that delay increases stability more than the case of non-delay. As an example compare the case ω1 = ω2 = 1 (non-delay) with ω1 = ω2 = 0.5 (delay). Fig. 8 shows the stability region of the Nash equilibrium point in the non-delay case. Also Fig. 9 shows the stability region of the Nash equilibrium point in the delay case when ω1 = ω2 = 0.5. Comparing between Figs. 8 and 9, one can see that the stability region in the delay case is larger than that in the non-delay case.
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Fig. 6. Bifurcation diagram of q1 with respect to the α1 delayed case when ω1 = ω2 = 0.7.
Fig. 7. Strange attractor for the delayed duopoly game (9) when ω1 = ω2 = 0.7.
Consequently firms using delay have a higher chance of reaching a Nash equilibrium point than these not using delay. Figs. 10–13 show the bifurcation diagrams of q1 with respect to ω1 . The bifurcation diagram of q1 with respect to ω1 is polluted in Fig. 14. From Figs. 10–14, we deduce that the delay increases the stability domain.
5. Conclusion In this paper, a duopoly delayed bounded rationality game has been proposed and analyzed. The local stability of four equilibrium points is investigated in this game. Basic properties of the game have been analyzed by means of bifurcation diagrams, phase portraits and stability regions. From above analysis, everyone can see that the delay case increases the
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Fig. 8. Stability region of the Nash equilibrium point in the non-delay case.
Fig. 9. Stability region of the Nash equilibrium point in the delay case when ω1 = ω2 = 0.5.
domain of stability. Firms using a delayed bounded rationality method have a higher chance of reaching the Nash equilibrium point than those using bounded rationality without delay. It is important in studying oligopoly games to take the memory of the market into consideration, because, in an oligopoly it is very hard to sell all the quantity for each year in the next year. So, we must consider the oligopoly market with memory. In the near future, we will publish other papers containing more details about oligopoly games with memory.
Acknowledgements The author thanks the anonymous reviewers for providing some helpful comments which help to improve the style of this work. Also I wish to thank Professors E. Ahmed, Y. Moustafa for discussion and help.
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Fig. 10. Bifurcation diagram of q1 with respect to ω1 when ω2 = 0.
Fig. 11. Bifurcation diagram of q1 with respect to ω1 when ω2 = 0.2.
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Fig. 12. Bifurcation diagram of q1 with respect to ω1 when ω2 = 0.5.
Fig. 13. Bifurcation diagram of q1 with respect to ω1 when ω2 = 0.8.
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Fig. 14. Bifurcation diagram of q2 with respect to ω1 when ω2 = 0.5.
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