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Complex dynamics and chaos control of heterogeneous quadropoly game
Applied Mathematics and Computation 219 (2013) 11110–11118
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Complex dynamics and chaos control of heterogeneous quadropoly game A.A. Elsadany a,⇑, H.N. Agiza b, E.M. Elabbasy b a b
Department of Basic Science, Faculty of Computers and Informatics, Suez Canal University, Ismailia 41522, Egypt Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
a r t i c l e
i n f o
Keywords: Quadropoly Nash equilibrium Heterogeneous firms Chaos Feedback control method
a b s t r a c t The dynamical system of four heterogeneous firms is derived. Existence and stability conditions of the fixed points are investigated and also complex dynamics is studied. Numerical simulations are used to illustrate the complex behaviors of the proposed dynamic game. The chaotic behavior of the game has been controlled by using feedback control method. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Oligopoly theory is one of the oldest branches of mathematical economics dated back to 1838 when its basic model was proposed by Cournot in [1]. Oligopoly is the case where the market is controlled by a few number of firms (players) producing similar products. Each player in the game aims to maximize his expected profit, and profits maximized when marginal revenue equals marginal cost [2]. Game theory is an important approach to study oligopoly models. Game theory has attracted extensive attention of people and is widely studied in the fields such as economics, applied mathematics and biology (see [3]). Expectations play an important role in game theory. A player can choose his expectations rules of many available strategies to adjust his production outputs. In this work, we assume that the players can choose their strategy from three different expectations: naive, bounded rational and adaptive [4,5]. Recently, the dynamics of the oligopoly game have been studied [6–16]. For instance, references [6–11] investigated the chaotic dynamics of duopoly or triopoly game model with homogeneous players. Zhang et al. [12] and Tramontana and Naimzada [16] studied the complex dynamics of duopoly and triopoly game model with heterogeneous players respectively. In oligopoly model, all players maximize their profits while quadropoly game is an oligopoly market with four players. Zhang and Ma [17] studied the chaos phenomenon and control of quadropoly price game with heterogenous players. Recently Snyder et al. [18] have introduced model of continuous-time dynamic Cournot adjustment game for n players and the existence and uniqueness of a positive Cournot equilibrium are discussed. Also, they have showed for any n, the response functions are bounded. Furthermore, they have discussed the modification of the model to account for non-constant costs. Studying economic system with a large number of players is more realistic moreover the expected dynamical behavior will be much more complicated than with small number of players. Oligopolies are common in airline, banking, music, soft drinks and telecommunication companies. In the future, the Egyptian telecommunication market will be the oligopoly structure with four main companies. According to the experience of other countries, competition not only drives an improved sector performance, but it also energizes organizational reform of the incumbent and contributes to consolidating and legitimating the regulatory process. When chaos occurs, the market is irregular, and it is impossible for operators to build a
⇑ Corresponding author. E-mail addresses: [email protected] (A.A. Elsadany), [email protected] (H.N. Agiza), [email protected] (E.M. Elabbasy). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.05.029
A.A. Elsadany et al. / Applied Mathematics and Computation 219 (2013) 11110–11118
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strategy with confidence. Chaos in oligopoly game means that if one player changes its output even slightly then, in the long run, large unpredictable changes will occur in the outputs of all players. Thus, people hope to find some methods to control chaos of economic system. A player can use a feedback of his decision-making variable to control the adjustment magnitude. The research on the game model with more than three players and its complexity is still poor. In this paper we consider a oligopoly game with four players. Our model generalizes the heterogeneous triopoly game which was introduced in [13] and recently studied in [15,16]. The paper is organized as follows. In Section 2 we introduce the model and the nonlinear system describing the dynamics of the production of the players. Also, in Section 2 we study the stability conditions of two equilibrium points. In Section 3, we study the strange attractor, bifurcation and Lyapunov exponent by numerical simulations. The delay feedback control method will be used to control the chaos of the system in Section 4. Finally, a conclusion is drawn in Section 5. 2. The model In this article, We study heterogenous quadropoly game where four players producing perfect substitute goods. The players need to decide the production decision which maximize their profits every time period. Let qi ðtÞ; i ¼ 1; . . . ; 4 represent P the output of the ith player during period t. The price P during period t is determined from the total supply Q ¼ 4i¼1 qi through a linear demand function:
P ¼ a bQ ¼ a bðq1 þ q2 þ q3 þ q4 Þ:
ð1Þ
The cost function is assumed to be linear in the form:
C i ðqi Þ ¼ ci qi ;
i ¼ 1; . . . ; 4:
ð2Þ
So the marginal cost of the ith player is constant and equal to ci > 0; i ¼ 1; . . . ; 4. As a consequence of the assumptions concerning demand function and cost function, the profit of the ith player is given by
Pi ðqi Þ ¼ qi ða bQÞ ci qi ;
i ¼ 1; 2; 3; 4:
ð3Þ
The marginal profit of the ith player is
/i ðqi ; Q i Þ ¼
@ Pi ðqi ; Q i Þ ¼ a 2bqi bQ i ci ; @qi
ð4Þ
P where Q i ¼ 4j¼1;i–j qj . We assume the first player and second player are naive players. The naive player knows the shape of the demand function but it has to conjecture the choices of the other three players. We assume that the second player is also is a naive player, that is it uses static expectations like the first player and then it maximize the expected profit Pei which is given by:
qi ðt þ 1Þ ¼ arg maxqi ðtþ1Þ Pei ðt þ 1Þ ¼ arg maxqi ðtþ1Þ ½qi ðt þ 1Þða bQ ðtÞ ci qi ðt þ 1Þ;
i ¼ 1; 2;
ð5Þ
that permits to derive the dynamic equation:
qi ðt þ 1Þ ¼
a ci bQ i ; 2b
i ¼ 1; 2:
ð6Þ
We consider our third player to be an adaptive player. At each time t, player 3 changes its output proportionally to the difference between the previous output q3 ðtÞ and the naïve expectations value. The proportion is given by a parameter b 2 ½0; 1 that regulates how reluctant the third player is to change its previous period production. Then the dynamic equation of third player given by:
a c3 bðq1 ðtÞ þ q2 ðtÞ þ q4 ðtÞÞ : q3 ðt þ 1Þ ¼ ð1 bÞq3 ðtÞ þ b 2b
ð7Þ
The fourth player uses bounded rational expectation and adopts the so-called myopic adjustment mechanism (see [19]), that is:
q4 ðt þ 1Þ ¼ q4 ðtÞ þ aq4 ðtÞ/4 ðQ ðtÞÞ;
ð8Þ
where /4 ðQ ðtÞÞ is the marginal profit of the fourth player, that is:
/4 ðQ ðtÞÞ ¼ /4 ðq1 ðtÞ þ q2 ðtÞ þ q3 ðtÞ þ q4 ðtÞÞ ¼
@ P4 ðq1 ðtÞ þ q2 ðtÞ þ q3 ðtÞ þ q4 ðtÞÞ ¼ a bQ bq4 c4 ; @q4
ð9Þ
In other words, the fourth player increases/decreases its output according to the information given by the marginal profit of the last period. The positive parameter a represents the speed of adjustment. By substituting (9) in (8) we finally obtain the dynamic equation:
q4 ðt þ 1Þ ¼ q4 ðtÞ þ aq4 ðtÞ½a c4 bQ bq4 :
ð10Þ
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If we use x; y; z; w instead of q1 ; q2 ; q3 ; q4 in (6), (7) and (10) we have that the dynamics of the players outputs are given by the following discrete time dynamical system ðx0 ; y0 ; z0 ; w0 Þ ¼ Tðx; y; z; wÞ:
8 0 ac bðyþzþwÞ x ¼ 1 2b ; > > > > < 0 ac2 bðxþzþwÞ y ¼ ; 2b T: > z0 ¼ ð1 bÞz þ bðac3 bðxþyþwÞÞ; > > 2b > : 0 w ¼ w þ aw½a c4 2bw bðx þ y þ zÞ;
ð11Þ
where 0 denotes the unit-time advancement operator, and we are interested only in positive trajectories. 2.1. Equilibrium points and local stability Setting the fixed point conditions x0 ¼ x ¼ x ; y0 ¼ y ¼ y; z0 ¼ z ¼ z and w0 ¼ w ¼ w in the map (11) we obtain the following nonlinear algebraic system:
8 ac bðy þz þw Þ 1 x ¼ 0; > > 2b > > > < ac2 bðx þz þw Þ y ¼ 0; 2b > > bz þ b ac3 bðx2bþy þw Þ ¼ 0; > > > : aw ½a c4 2bw bðx þ y þ z Þ ¼ 0; which is solved by the two fixed points:
E1 ¼
a þ c2 þ c3 3c1 a þ c1 þ c3 3c2 a þ c1 þ c2 3c3 ; ; ;0 ; 4b 4b 4b
E2 ¼
a þ c2 þ c3 þ c3 4c1 a þ c1 þ c3 þ c4 4c2 a þ c1 þ c2 þ c4 4c3 a þ c1 þ c2 þ c3 4c4 ; ; ; ; 5b 5b 5b 5b
E1 has non-negative coordinates provided that
8 > < a þ c2 þ c3 > 3c1 a þ c1 þ c3 > 3c2 > : a þ c1 þ c2 > 3c3
ð12Þ
and E2 has positive coordinates provided that
8 a þ c2 þ c3 þ c 4 > > > a þ c þ c þ c 1 2 4 > > : a þ c1 þ c2 þ c 3
> 4c1 > 4c2
ð13Þ
> 4c3 > 4c4
The fixed point E2 , is called the Nash equilibrium of the dynamic game [8]. A fixed point E for T : R4 ! R4 is called hyperbolic if DTðEÞ has no eigenvalues on the unite circle, where DT is the Jacobian matrix of T at the point E. Such a hyperbolic point E is: 1. a sink fixed point if all eigenvalues of DTðEÞ are less than one in absolute value, 2. a source fixed point if all eigenvalues of DTðEÞ are greater than one in absolute value, 3. a saddle fixed point otherwise, i.e., if some eigenvalues of DTðEÞ are less and some larger than one in absolute value. In order to investigate the local stability of the equilibrium points, we estimate the Jacobian matrix of the map T which is given by the following matrix:
2 6 6 J¼6 6 4
0
1 2
1 2 b 2
0
1 2 1 2
b 2
1b
1 2 1 2 b 2
abw abw abw 1 þ aða c4 3bw bðx þ y þ z þ wÞÞ
3 7 7 7 7 5
By applying the stability condition to the equilibrium E1 we have the following result: Theorem 1. If the Nash equilibrium point is strictly positive, then the equilibrium point E1 is locally unstable.
ð14Þ
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Proof. The Jacobian matrix evaluated at the equilibrium point E1 , becomes
2
1 2
0
6 1 6 2 JðE1 Þ ¼ 6 6 b 4 2 0
0
1 2 1 2
b 2
1b
0
0
3
1þa
1 2 7 1 7 2 7 b 7 5 2 ðaþc1 þc2 þc3 4c4 Þ 4
one eigenvalues of this matrix is k1 ¼ 1 þ a ðaþc1 þc24þc3 4c4 Þ. If the Nash equilibrium point has positive coordinates, then a þ c1 þ c2 þ c3 > 4c4 , hence jk1 j > 1. Then E1 is locally unstable. h 2.2. Stability of the Nash equilibrium point E2 The local stability of Nash equilibrium of market dynamics system can be analyzed by judging the eigenvalues of the Jacobian matrix. The Jacobian matrix evaluated at the Nash equilibrium point E2 becomes:
2 6 6 JðE2 Þ ¼ 6 6 4
0
1 2
1 2 b 2
0
abw
1 2 1 2
b 2
abw
1b abw
3
1 2 1 2 b 2
1 2 abw
7 7 7 7 5
ð15Þ
where w ¼ aþc1 þc25bþc3 4c4 . whose characteristic polynomial takes the form:
k4 þ d1 k3 þ d2 k2 þ d3 k þ d4 ¼ 0;
ð16Þ
where
8 d1 ¼ 2abw þ b 2; > > > > < d2 ¼ 3abw ðb 1Þ þ 36b ; 2 4 > ; d3 ¼ abw ð1 bÞ þ 1þb > 2 > > : abbw2 d4 ¼ 8 :
Fig. 1. The stability region of Nash equilibrium point in the plane (a,b).
ð17Þ
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According to Jury conditions [20], the sufficient and necessary conditions that Nash equilibrium point is locally stable are
8 1 þ d1 þ d2 þ d3 þ d4 > 0; > > > > > > < 1 d1 þ d2 d3 þ d4 > 0;
1 d4 > 0; > > > 3 þ 3d4 d2 > 0; > > > : ð1 d4 Þð1 d24 Þ d2 ð1 d4 Þ2 þ ðd1 d3 Þðd3 d1 d4 Þ > 0;
ð18Þ
Computing the above inequalities, we can get local stable region of Nash equilibrium point. The stability region of the Nash equilibrium point for the values of parameters a ¼ 10; b ¼ 0:5; c1 ¼ 0:1; c2 ¼ 0:3; c3 ¼ 0:3 and c4 ¼ 0:4 in the plane of adjustment ða; bÞ is given in Fig. 1. For the values of ða; bÞ inside the stable region, the Nash equilibrium is stable. Economic meaning of the stable region is that whatever initial quantity is chosen by four companies in local stable region, they will eventually achieve Nash equilibrium point after finite games. 3. Numerical simulations In this section, we will perform numerical simulations to show the complicated behavior of the game. We show some numerical results such as period cycles, bifurcation diagrams, phase portraits, strange attractors and largest Lyapunov exponents. Fig. 2 shows the bifurcation diagram with respect to the speed of adjustment of boundedly rational player a while other parameters are fixed as a ¼ 10; b ¼ 0:5; c1 ¼ 0:1; c2 ¼ 0:3; c3 ¼ 0:3; c4 ¼ 0:4 and b ¼ 0:2. From Fig. 2 we can see that the Nash equilibrium point is locally stable for small value of parameter a. This figure shows that the Nash equilibrium point is locally stable for a < 0:24, after this value it bifurcates to period two, and finally the system became chaotic. The largest Lyapunov exponents corresponding to Fig. 2 are calculated and plotted in Fig. 3. In the range 0 < a < 0:24 the Lyapunov exponents are negative, corresponding to a stable coexistence of the system. When 0:24 < a < 0:62 most Lyapunov exponentsare non-negative, and few are negative. This means that there exist stable fixed points or periodic windows in the chaotic band. Also Fig. 4 shows the bifurcation diagram with respect to the speed of adjustment of adaptive player b while other parameters are fixed as a ¼ 10; b ¼ 0:5; c1 ¼ 0:1; c2 ¼ 0:3; c3 ¼ 0:3; c4 ¼ 0:4 and a ¼ 0:2. Figs. 5 and 6 show the strange attractor for system (12). Figs. 5 and 6 show the three-dimension strange attractors. The double chaotic attractor for the model system (12) in xzw dimension is presented for the values ða; b; c1 ; c2 ; c3 ; c4 ; a; bÞ ¼ ð10; 0:5; 0:1; 0:2; 0:3; 0:4; 0:61; 0:2Þ. we conclude from the numerical experiments, that the adjustment speeds may change the stability of the equilibrium and cause a market structure to behave chaotically.
Fig. 2. The bifurcation diagram of system (12) with respect to a.
A.A. Elsadany et al. / Applied Mathematics and Computation 219 (2013) 11110–11118
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Fig. 3. Largest Lyapunov exponent corresponding to Fig. 2.
Fig. 4. The bifurcation diagram of system (12) with respect to b.
4. Chaos control The appearance of chaos in the economic systems is not desired expected and even is harmful. Thus, people hope to find some methods to control the chaos of economic system. A wide variety of methods have been proposed for controlling chaos in oligopoly models, for example, chaos control with OGY method in the Kopel duopoly game model was applied in [21], chaos control with modified straight-line stabilization method in an output duopoly competing evolution model have been studied by Du et al. [22], Holyst and Urbanowicz [23] had studied chaos control with time-delayed feedback method in an
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Fig. 5. Strange attractor for system (12) in xyw dimension.
Fig. 6. Strange attractor for system (12) in xzw dimension.
economical model, chaos control with pole placement method in a nonlinear Cournot duopoly model achieved in [24], The dynamics and adaptive control of a duopoly advertising model based on heterogeneous expectations is presented in Ding et al. [25], and so on. Chaos in Cournot game means that if one player changes its output even slightly then, in the long run, large unpredictable changes will occur in the outputs of all players. A producer can use a feedback of his decision-making variable to control the adjustment magnitude. Elabbasy et al. [13] have considered such a feedback control in their triopoly game. Also Ding et al. [25] have applied feedback control on multi-team Bertrand model. In this section, we modify the fourth equation of system (12) in the same way as in Elabbasy et al. [13]. The controlled system is given by:
8 > x0 ¼ ac1 bðyþzþwÞ ; > 2b > > > ac bðxþzþwÞ 0 2 > ; > z ¼ ð1 bÞz þ b 2b > > : 0 w ¼ w þ aw½a c4 2bw bðx þ y þ zÞ kðw0 wÞ; where k is the controlling factor. The equivalent system of (19) is
ð19Þ
A.A. Elsadany et al. / Applied Mathematics and Computation 219 (2013) 11110–11118
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Fig. 7. Bifurcation diagram of w with respect to the controlling factor k.
Fig. 8. Stabilization of Nash equilibrium with k = 1.7.
8 0 ac bðyþzþwÞ x ¼ 1 2b > > > > > < y0 ¼ ac2 bðxþzþwÞ 2b
> z0 ¼ ð1 bÞz þ b ac3 bðxþyþwÞ > > 2b > > : 0 aw ½a c 2bw bðx þ y þ zÞ w ¼ w þ 1þk 4 and the Jacobian matrix of (20) is given by
ð20Þ
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2 6 6 J¼6 6 4
0
1 2
1 2 b 2 bw a1þk
b 2 bw a1þk
0
1 2 1 2
1b
1 2 1 2 b 2
bw a1þk
a ða c 3bw bQÞ 1 þ 1þk 4
3 7 7 7: 7 5
ð21Þ
As we know, the system (4) is chaotic if ða; b; c1 ; c2 ; c3 ; c4 ; a; bÞ ¼ ð10; 0:5; 0:1; 0:2; 0:3; 0:4; 0:61; 0:2Þ. But now the Jacobian matrix (21) has the form
3 1 1 1 0 2 2 2 6 1 1 1 7 7 6 2 0 2 2 7 J¼6 6 0:1 0:1 0:9 0:1 7 5 4 10:98 10:98 k9:98 10:98 1þk 1þk 1þk 1þk 2
and the absolute value of its eigenvalue is less than one when k > 1:58. Hence when k > 1:58, the controlled system (20) is stable around the Nash equilibrium point. From Fig. 7 we can see that with the control factor k increasing, the system gets rid of chaotic behaviors and is controlled to periodic states when k is large and to reach equilibrium state when k > 1:58. Fig. 8 shows the stable behavior of the controlled system when k ¼ 1:7. Such a feedback control method is able to control chaos. If the bounded rational player adopt this adjustment method, the quantity game can switch from a chaotic trajectory to a regular periodic orbit or equilibrium state. 5. Conclusion This study analyzed the dynamics of four heterogeneous player in quadropoly game with linear demand function and linear cost function. The boundary equilibrium point is locally unstable under some conditions. The stability of Nash equilibrium strongly depends on the speed of adjustment of boundedly rational player. We reveal that the player of adaptive expectations has a stabilizing effect on the game. The stabilization of the chaotic behavior can be obtained by applying a delayed feedback control method. Acknowledgments The author thank the reviewers for their careful reading and efforts and for providing some helpful suggestions. Also we thank Professor E. Ahmed for discussion. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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. R. Yu, R. Xiao, Quantum Stackelberg duopoly with isoelastic demand function, J. Comput. Inf. Syst. 8 (9) (2012) 3643–3650. H.N. Agiza, A.A. Elsadany, Nonlinear dynamics in the Cournot duopoly game with heterogenous players, Physica A 320 (2003) 512–524. H.N. Agiza, A.A. Elsadany, Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Appl. Math. Comput. 149 (2004) 843–860. T. Puu, Chaos in duopoly pricing, Chaos Solitons Fract. 1 (1991) 573–581. M. Kopel, Simple and complex adjustment dynamics in Cournot duopoly models, Chaos Solitons Fract. 12 (1996) 2031–2048. G.I. Bischi, M. Galletgatti, A. Naimazada, Symmetry-breaking bifurcations and representative firm in dynamic duopoly games, Ann. Oper. Res. 89 (1999) 253–272. T. Puu, The complex dynamics with three oligopolists, Chaos Solitons Fract. 7 (1996) 2075–2081. F. Chen, J. Ma, X. Chen, The study of dynamic process of the triopoly games in Chinese 3G telecommunication market, Chaos Solitons Fract. 42 (2009) 1542–1551. A.A. Elsadany, Competition analysis of a triopoly game with bounded rationality, Chaos Solitons Fract. 45 (2012) 1343–1348. J. Zhang, Q. Da, Y. Wang, Analysis of nonlinear duopoly game with heterogeneous players, Econ. Model. 24 (2007) 138–148. E.M. Elabbasy, H.N. Agiza, A.A. Elsadany, Analysis of nonlinear triopoly game with heterogeneous players, Comput. Math. Appl. 57 (2009) 488–499. Z. Ding, Q. Hang, L. Tian, Analysis of the dynamics of Cournot team-game with heterogeneous players, Appl. Math. Comput. 215 (2009) 1098–1105. F. Tramontana, A.A. Elsadany, Heterogeneous triopoly game with isoelastic demand function, Nonlinear Dyn. 68 (2012) 187–193. F. Tramontana. A.K. Naimzada, Double route to chaos in an heterogeneous triopoly game, submitted. J. Zhang, J. Ma, Research on the price game model for four oligarchs with different decision rules and its chaos control, Nonlinear Dyn. 70 (2012) 323– 334. B.C. Snyder, R.A. Van Gorder, K. Vajravelu, Continuous-time dynamic games for the Cournot adjustment process for competing oligopolists, Appl. Math. Comput. 219 (2013) 6400–6409. A. Dixit, Comparative statics for oligopoly, Int. Econ. Rev. 27 (1986) 107–122. S.N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, 2005. H.N. Agiza, On the stability, bifurcations, chaos and chaos control of Kopel map, Chaos Solitons Fract. 11 (1999) 1909–1916. J. Du, T. Huang, Z. Sheng, Analysis of decision-making in economic chaos control, Nonlinear Anal. Real World Appl. 10 (2009) 2493–2501. J. Holyst, K. Urbanowicz, Chaos control in economical model by time-delayed feedback method, Physica A: Stat. Mech. Appl. 287 (2000) 587–598. A. Matsumoto, Controlling the Cournot–Nash chaos, J. Optim. Theory Appl. 28 (2006) 379–392. J. Ding, Q. Mei, H. Yao, Dynamics and adaptive control of a duopoly advertising model based on heterogeneous expectations, Nonlinear Dyn. 67 (2012) 129–138.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Complex dynamics and chaos control of heterogeneous quadropoly game A.A. Elsadany a,⇑, H.N. Agiza b, E.M. Elabbasy b a b
Department of Basic Science, Faculty of Computers and Informatics, Suez Canal University, Ismailia 41522, Egypt Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
a r t i c l e
i n f o
Keywords: Quadropoly Nash equilibrium Heterogeneous firms Chaos Feedback control method
a b s t r a c t The dynamical system of four heterogeneous firms is derived. Existence and stability conditions of the fixed points are investigated and also complex dynamics is studied. Numerical simulations are used to illustrate the complex behaviors of the proposed dynamic game. The chaotic behavior of the game has been controlled by using feedback control method. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Oligopoly theory is one of the oldest branches of mathematical economics dated back to 1838 when its basic model was proposed by Cournot in [1]. Oligopoly is the case where the market is controlled by a few number of firms (players) producing similar products. Each player in the game aims to maximize his expected profit, and profits maximized when marginal revenue equals marginal cost [2]. Game theory is an important approach to study oligopoly models. Game theory has attracted extensive attention of people and is widely studied in the fields such as economics, applied mathematics and biology (see [3]). Expectations play an important role in game theory. A player can choose his expectations rules of many available strategies to adjust his production outputs. In this work, we assume that the players can choose their strategy from three different expectations: naive, bounded rational and adaptive [4,5]. Recently, the dynamics of the oligopoly game have been studied [6–16]. For instance, references [6–11] investigated the chaotic dynamics of duopoly or triopoly game model with homogeneous players. Zhang et al. [12] and Tramontana and Naimzada [16] studied the complex dynamics of duopoly and triopoly game model with heterogeneous players respectively. In oligopoly model, all players maximize their profits while quadropoly game is an oligopoly market with four players. Zhang and Ma [17] studied the chaos phenomenon and control of quadropoly price game with heterogenous players. Recently Snyder et al. [18] have introduced model of continuous-time dynamic Cournot adjustment game for n players and the existence and uniqueness of a positive Cournot equilibrium are discussed. Also, they have showed for any n, the response functions are bounded. Furthermore, they have discussed the modification of the model to account for non-constant costs. Studying economic system with a large number of players is more realistic moreover the expected dynamical behavior will be much more complicated than with small number of players. Oligopolies are common in airline, banking, music, soft drinks and telecommunication companies. In the future, the Egyptian telecommunication market will be the oligopoly structure with four main companies. According to the experience of other countries, competition not only drives an improved sector performance, but it also energizes organizational reform of the incumbent and contributes to consolidating and legitimating the regulatory process. When chaos occurs, the market is irregular, and it is impossible for operators to build a
⇑ Corresponding author. E-mail addresses: [email protected] (A.A. Elsadany), [email protected] (H.N. Agiza), [email protected] (E.M. Elabbasy). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.05.029
A.A. Elsadany et al. / Applied Mathematics and Computation 219 (2013) 11110–11118
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strategy with confidence. Chaos in oligopoly game means that if one player changes its output even slightly then, in the long run, large unpredictable changes will occur in the outputs of all players. Thus, people hope to find some methods to control chaos of economic system. A player can use a feedback of his decision-making variable to control the adjustment magnitude. The research on the game model with more than three players and its complexity is still poor. In this paper we consider a oligopoly game with four players. Our model generalizes the heterogeneous triopoly game which was introduced in [13] and recently studied in [15,16]. The paper is organized as follows. In Section 2 we introduce the model and the nonlinear system describing the dynamics of the production of the players. Also, in Section 2 we study the stability conditions of two equilibrium points. In Section 3, we study the strange attractor, bifurcation and Lyapunov exponent by numerical simulations. The delay feedback control method will be used to control the chaos of the system in Section 4. Finally, a conclusion is drawn in Section 5. 2. The model In this article, We study heterogenous quadropoly game where four players producing perfect substitute goods. The players need to decide the production decision which maximize their profits every time period. Let qi ðtÞ; i ¼ 1; . . . ; 4 represent P the output of the ith player during period t. The price P during period t is determined from the total supply Q ¼ 4i¼1 qi through a linear demand function:
P ¼ a bQ ¼ a bðq1 þ q2 þ q3 þ q4 Þ:
ð1Þ
The cost function is assumed to be linear in the form:
C i ðqi Þ ¼ ci qi ;
i ¼ 1; . . . ; 4:
ð2Þ
So the marginal cost of the ith player is constant and equal to ci > 0; i ¼ 1; . . . ; 4. As a consequence of the assumptions concerning demand function and cost function, the profit of the ith player is given by
Pi ðqi Þ ¼ qi ða bQÞ ci qi ;
i ¼ 1; 2; 3; 4:
ð3Þ
The marginal profit of the ith player is
/i ðqi ; Q i Þ ¼
@ Pi ðqi ; Q i Þ ¼ a 2bqi bQ i ci ; @qi
ð4Þ
P where Q i ¼ 4j¼1;i–j qj . We assume the first player and second player are naive players. The naive player knows the shape of the demand function but it has to conjecture the choices of the other three players. We assume that the second player is also is a naive player, that is it uses static expectations like the first player and then it maximize the expected profit Pei which is given by:
qi ðt þ 1Þ ¼ arg maxqi ðtþ1Þ Pei ðt þ 1Þ ¼ arg maxqi ðtþ1Þ ½qi ðt þ 1Þða bQ ðtÞ ci qi ðt þ 1Þ;
i ¼ 1; 2;
ð5Þ
that permits to derive the dynamic equation:
qi ðt þ 1Þ ¼
a ci bQ i ; 2b
i ¼ 1; 2:
ð6Þ
We consider our third player to be an adaptive player. At each time t, player 3 changes its output proportionally to the difference between the previous output q3 ðtÞ and the naïve expectations value. The proportion is given by a parameter b 2 ½0; 1 that regulates how reluctant the third player is to change its previous period production. Then the dynamic equation of third player given by:
a c3 bðq1 ðtÞ þ q2 ðtÞ þ q4 ðtÞÞ : q3 ðt þ 1Þ ¼ ð1 bÞq3 ðtÞ þ b 2b
ð7Þ
The fourth player uses bounded rational expectation and adopts the so-called myopic adjustment mechanism (see [19]), that is:
q4 ðt þ 1Þ ¼ q4 ðtÞ þ aq4 ðtÞ/4 ðQ ðtÞÞ;
ð8Þ
where /4 ðQ ðtÞÞ is the marginal profit of the fourth player, that is:
/4 ðQ ðtÞÞ ¼ /4 ðq1 ðtÞ þ q2 ðtÞ þ q3 ðtÞ þ q4 ðtÞÞ ¼
@ P4 ðq1 ðtÞ þ q2 ðtÞ þ q3 ðtÞ þ q4 ðtÞÞ ¼ a bQ bq4 c4 ; @q4
ð9Þ
In other words, the fourth player increases/decreases its output according to the information given by the marginal profit of the last period. The positive parameter a represents the speed of adjustment. By substituting (9) in (8) we finally obtain the dynamic equation:
q4 ðt þ 1Þ ¼ q4 ðtÞ þ aq4 ðtÞ½a c4 bQ bq4 :
ð10Þ
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If we use x; y; z; w instead of q1 ; q2 ; q3 ; q4 in (6), (7) and (10) we have that the dynamics of the players outputs are given by the following discrete time dynamical system ðx0 ; y0 ; z0 ; w0 Þ ¼ Tðx; y; z; wÞ:
8 0 ac bðyþzþwÞ x ¼ 1 2b ; > > > > < 0 ac2 bðxþzþwÞ y ¼ ; 2b T: > z0 ¼ ð1 bÞz þ bðac3 bðxþyþwÞÞ; > > 2b > : 0 w ¼ w þ aw½a c4 2bw bðx þ y þ zÞ;
ð11Þ
where 0 denotes the unit-time advancement operator, and we are interested only in positive trajectories. 2.1. Equilibrium points and local stability Setting the fixed point conditions x0 ¼ x ¼ x ; y0 ¼ y ¼ y; z0 ¼ z ¼ z and w0 ¼ w ¼ w in the map (11) we obtain the following nonlinear algebraic system:
8 ac bðy þz þw Þ 1 x ¼ 0; > > 2b > > > < ac2 bðx þz þw Þ y ¼ 0; 2b > > bz þ b ac3 bðx2bþy þw Þ ¼ 0; > > > : aw ½a c4 2bw bðx þ y þ z Þ ¼ 0; which is solved by the two fixed points:
E1 ¼
a þ c2 þ c3 3c1 a þ c1 þ c3 3c2 a þ c1 þ c2 3c3 ; ; ;0 ; 4b 4b 4b
E2 ¼
a þ c2 þ c3 þ c3 4c1 a þ c1 þ c3 þ c4 4c2 a þ c1 þ c2 þ c4 4c3 a þ c1 þ c2 þ c3 4c4 ; ; ; ; 5b 5b 5b 5b
E1 has non-negative coordinates provided that
8 > < a þ c2 þ c3 > 3c1 a þ c1 þ c3 > 3c2 > : a þ c1 þ c2 > 3c3
ð12Þ
and E2 has positive coordinates provided that
8 a þ c2 þ c3 þ c 4 > > > a þ c þ c þ c 1 2 4 > > : a þ c1 þ c2 þ c 3
> 4c1 > 4c2
ð13Þ
> 4c3 > 4c4
The fixed point E2 , is called the Nash equilibrium of the dynamic game [8]. A fixed point E for T : R4 ! R4 is called hyperbolic if DTðEÞ has no eigenvalues on the unite circle, where DT is the Jacobian matrix of T at the point E. Such a hyperbolic point E is: 1. a sink fixed point if all eigenvalues of DTðEÞ are less than one in absolute value, 2. a source fixed point if all eigenvalues of DTðEÞ are greater than one in absolute value, 3. a saddle fixed point otherwise, i.e., if some eigenvalues of DTðEÞ are less and some larger than one in absolute value. In order to investigate the local stability of the equilibrium points, we estimate the Jacobian matrix of the map T which is given by the following matrix:
2 6 6 J¼6 6 4
0
1 2
1 2 b 2
0
1 2 1 2
b 2
1b
1 2 1 2 b 2
abw abw abw 1 þ aða c4 3bw bðx þ y þ z þ wÞÞ
3 7 7 7 7 5
By applying the stability condition to the equilibrium E1 we have the following result: Theorem 1. If the Nash equilibrium point is strictly positive, then the equilibrium point E1 is locally unstable.
ð14Þ
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11113
Proof. The Jacobian matrix evaluated at the equilibrium point E1 , becomes
2
1 2
0
6 1 6 2 JðE1 Þ ¼ 6 6 b 4 2 0
0
1 2 1 2
b 2
1b
0
0
3
1þa
1 2 7 1 7 2 7 b 7 5 2 ðaþc1 þc2 þc3 4c4 Þ 4
one eigenvalues of this matrix is k1 ¼ 1 þ a ðaþc1 þc24þc3 4c4 Þ. If the Nash equilibrium point has positive coordinates, then a þ c1 þ c2 þ c3 > 4c4 , hence jk1 j > 1. Then E1 is locally unstable. h 2.2. Stability of the Nash equilibrium point E2 The local stability of Nash equilibrium of market dynamics system can be analyzed by judging the eigenvalues of the Jacobian matrix. The Jacobian matrix evaluated at the Nash equilibrium point E2 becomes:
2 6 6 JðE2 Þ ¼ 6 6 4
0
1 2
1 2 b 2
0
abw
1 2 1 2
b 2
abw
1b abw
3
1 2 1 2 b 2
1 2 abw
7 7 7 7 5
ð15Þ
where w ¼ aþc1 þc25bþc3 4c4 . whose characteristic polynomial takes the form:
k4 þ d1 k3 þ d2 k2 þ d3 k þ d4 ¼ 0;
ð16Þ
where
8 d1 ¼ 2abw þ b 2; > > > > < d2 ¼ 3abw ðb 1Þ þ 36b ; 2 4 > ; d3 ¼ abw ð1 bÞ þ 1þb > 2 > > : abbw2 d4 ¼ 8 :
Fig. 1. The stability region of Nash equilibrium point in the plane (a,b).
ð17Þ
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According to Jury conditions [20], the sufficient and necessary conditions that Nash equilibrium point is locally stable are
8 1 þ d1 þ d2 þ d3 þ d4 > 0; > > > > > > < 1 d1 þ d2 d3 þ d4 > 0;
1 d4 > 0; > > > 3 þ 3d4 d2 > 0; > > > : ð1 d4 Þð1 d24 Þ d2 ð1 d4 Þ2 þ ðd1 d3 Þðd3 d1 d4 Þ > 0;
ð18Þ
Computing the above inequalities, we can get local stable region of Nash equilibrium point. The stability region of the Nash equilibrium point for the values of parameters a ¼ 10; b ¼ 0:5; c1 ¼ 0:1; c2 ¼ 0:3; c3 ¼ 0:3 and c4 ¼ 0:4 in the plane of adjustment ða; bÞ is given in Fig. 1. For the values of ða; bÞ inside the stable region, the Nash equilibrium is stable. Economic meaning of the stable region is that whatever initial quantity is chosen by four companies in local stable region, they will eventually achieve Nash equilibrium point after finite games. 3. Numerical simulations In this section, we will perform numerical simulations to show the complicated behavior of the game. We show some numerical results such as period cycles, bifurcation diagrams, phase portraits, strange attractors and largest Lyapunov exponents. Fig. 2 shows the bifurcation diagram with respect to the speed of adjustment of boundedly rational player a while other parameters are fixed as a ¼ 10; b ¼ 0:5; c1 ¼ 0:1; c2 ¼ 0:3; c3 ¼ 0:3; c4 ¼ 0:4 and b ¼ 0:2. From Fig. 2 we can see that the Nash equilibrium point is locally stable for small value of parameter a. This figure shows that the Nash equilibrium point is locally stable for a < 0:24, after this value it bifurcates to period two, and finally the system became chaotic. The largest Lyapunov exponents corresponding to Fig. 2 are calculated and plotted in Fig. 3. In the range 0 < a < 0:24 the Lyapunov exponents are negative, corresponding to a stable coexistence of the system. When 0:24 < a < 0:62 most Lyapunov exponentsare non-negative, and few are negative. This means that there exist stable fixed points or periodic windows in the chaotic band. Also Fig. 4 shows the bifurcation diagram with respect to the speed of adjustment of adaptive player b while other parameters are fixed as a ¼ 10; b ¼ 0:5; c1 ¼ 0:1; c2 ¼ 0:3; c3 ¼ 0:3; c4 ¼ 0:4 and a ¼ 0:2. Figs. 5 and 6 show the strange attractor for system (12). Figs. 5 and 6 show the three-dimension strange attractors. The double chaotic attractor for the model system (12) in xzw dimension is presented for the values ða; b; c1 ; c2 ; c3 ; c4 ; a; bÞ ¼ ð10; 0:5; 0:1; 0:2; 0:3; 0:4; 0:61; 0:2Þ. we conclude from the numerical experiments, that the adjustment speeds may change the stability of the equilibrium and cause a market structure to behave chaotically.
Fig. 2. The bifurcation diagram of system (12) with respect to a.
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11115
Fig. 3. Largest Lyapunov exponent corresponding to Fig. 2.
Fig. 4. The bifurcation diagram of system (12) with respect to b.
4. Chaos control The appearance of chaos in the economic systems is not desired expected and even is harmful. Thus, people hope to find some methods to control the chaos of economic system. A wide variety of methods have been proposed for controlling chaos in oligopoly models, for example, chaos control with OGY method in the Kopel duopoly game model was applied in [21], chaos control with modified straight-line stabilization method in an output duopoly competing evolution model have been studied by Du et al. [22], Holyst and Urbanowicz [23] had studied chaos control with time-delayed feedback method in an
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Fig. 5. Strange attractor for system (12) in xyw dimension.
Fig. 6. Strange attractor for system (12) in xzw dimension.
economical model, chaos control with pole placement method in a nonlinear Cournot duopoly model achieved in [24], The dynamics and adaptive control of a duopoly advertising model based on heterogeneous expectations is presented in Ding et al. [25], and so on. Chaos in Cournot game means that if one player changes its output even slightly then, in the long run, large unpredictable changes will occur in the outputs of all players. A producer can use a feedback of his decision-making variable to control the adjustment magnitude. Elabbasy et al. [13] have considered such a feedback control in their triopoly game. Also Ding et al. [25] have applied feedback control on multi-team Bertrand model. In this section, we modify the fourth equation of system (12) in the same way as in Elabbasy et al. [13]. The controlled system is given by:
8 > x0 ¼ ac1 bðyþzþwÞ ; > 2b > > > ac bðxþzþwÞ 0 2
ð19Þ
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11117
Fig. 7. Bifurcation diagram of w with respect to the controlling factor k.
Fig. 8. Stabilization of Nash equilibrium with k = 1.7.
8 0 ac bðyþzþwÞ x ¼ 1 2b > > > > > < y0 ¼ ac2 bðxþzþwÞ 2b
> z0 ¼ ð1 bÞz þ b ac3 bðxþyþwÞ > > 2b > > : 0 aw ½a c 2bw bðx þ y þ zÞ w ¼ w þ 1þk 4 and the Jacobian matrix of (20) is given by
ð20Þ
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2 6 6 J¼6 6 4
0
1 2
1 2 b 2 bw a1þk
b 2 bw a1þk
0
1 2 1 2
1b
1 2 1 2 b 2
bw a1þk
a ða c 3bw bQÞ 1 þ 1þk 4
3 7 7 7: 7 5
ð21Þ
As we know, the system (4) is chaotic if ða; b; c1 ; c2 ; c3 ; c4 ; a; bÞ ¼ ð10; 0:5; 0:1; 0:2; 0:3; 0:4; 0:61; 0:2Þ. But now the Jacobian matrix (21) has the form
3 1 1 1 0 2 2 2 6 1 1 1 7 7 6 2 0 2 2 7 J¼6 6 0:1 0:1 0:9 0:1 7 5 4 10:98 10:98 k9:98 10:98 1þk 1þk 1þk 1þk 2
and the absolute value of its eigenvalue is less than one when k > 1:58. Hence when k > 1:58, the controlled system (20) is stable around the Nash equilibrium point. From Fig. 7 we can see that with the control factor k increasing, the system gets rid of chaotic behaviors and is controlled to periodic states when k is large and to reach equilibrium state when k > 1:58. Fig. 8 shows the stable behavior of the controlled system when k ¼ 1:7. Such a feedback control method is able to control chaos. If the bounded rational player adopt this adjustment method, the quantity game can switch from a chaotic trajectory to a regular periodic orbit or equilibrium state. 5. Conclusion This study analyzed the dynamics of four heterogeneous player in quadropoly game with linear demand function and linear cost function. The boundary equilibrium point is locally unstable under some conditions. The stability of Nash equilibrium strongly depends on the speed of adjustment of boundedly rational player. We reveal that the player of adaptive expectations has a stabilizing effect on the game. The stabilization of the chaotic behavior can be obtained by applying a delayed feedback control method. Acknowledgments The author thank the reviewers for their careful reading and efforts and for providing some helpful suggestions. Also we thank Professor E. Ahmed for discussion. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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