Analysis of the dynamics of Cournot team-game with heterogeneous players

Applied Mathematics and Computation 215 (2009) 1098–1105

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Analysis of the dynamics of Cournot team-game with heterogeneous players Zhanwen Ding *, Qinglan Hang, Lixin Tian Faculty of Science, Jiangsu University, Zhenjiang 212013, PR China

a r t i c l e

i n f o

Keywords: Cournot game Team-game Heterogeneous players Dynamics Complexity

a b s t r a c t In this paper, we study a dynamical system of a two-team Cournot game played by a team consisting of two firms with bounded rationality and a team consisting of one firm with naive expectation. The equilibrium solutions and the conditions of their locally asymptotic stability are studied. It is demonstrated that, as some parameters in the model are varied, the stability of the equilibrium will get lost and many such complex behaviors as the period bifurcation, chaotic phenomenon, periodic windows, strange attractor and unpredictable trajectories will occur. The great influence of the model parameters on the speed of convergence to the equilibrium is also shown with numerical analysis. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction Competition and cooperation often coexist such that a real system may consist of many competing teams and in each team there are a number of cooperative members with compatible objectives. This type of system is called multi-team system [1]. Recently, multi-team Cournot game with bounded rationality is studied by some authors [2–5]. In the model given by Ahmed and Hegazi [2], each producer in the same team considers the allocation of the team’s total output. Ahmeda et al. [3], Elettreby and Hassan [4] (Model I) take the weighted profit of the total team as the objective function of every producer in the same team. In these articles, players are assumed to be homogeneous and the producers adjust their output by the marginal profit method. The equilibrium solutions and the stability are studied in [2–4]. Asker [5] considered a dynamical multi-team Cournot game describing the exploitation of renewable resources. If each team consists of only one player, it becomes a classical Cournot game. Much work has been done to Cournot game with heterogeneous expectations [6–8]. Agiza and Elsadany [6] investigated a nonlinear duopoly game played by heterogeneous players with bounded rationality and adaptive expectation. The authors [7,8] used similar method to analyze the Nash equilibrium in non-linear duopoly game with heterogeneous players, boundedly rational player and naive player. It is shown in these articles that the model parameters, such as the adjustment rate of boundedly rational players, have great influence on the stability region of Nash equilibrium and may give rise to complex dynamics, such as bifurcations, chaos, strange attractors and so on. The speed of convergence to Nash equilibrium in the Cournot game with bounded rationality is discussed by Du and Huang [9]. In a multi-team game, the players who are in the same team may have a tendency to cooperate and take the team’s weighted profit as their common objective function [3–5]. Together with the parameters in the classical Cournot game, the weighting coefficients are included in the Cournot team-game. The influence of all the parameters on the complex dynamical behaviors of the dynamics of Cournot team-game is worth further analysis. * Corresponding author. E-mail address: [email protected] (Z. Ding). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.046

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In this paper, we investigate a Cournot team-game with heterogeneous players and look for the stability conditions for the equilibriums. Numerical simulations are made to show what influence such model parameters as the adjustment rate and profit weights will have on the dynamic features of the dynamics of this team-game and on the speed of convergence to the equilibriums. This paper is organized as the following. Section 2 formulates the dynamical multi-team Cournot game and introduces two heterogeneous expectations. In Section 3, the dynamics of a two-team game with boundedly rational players and naive player is proposed. In Section 4, the equilibrium points and local stability of this team-game are analyzed. In Section 5, numerical analysis is used to illustrate that the complex dynamical behaviors will occur at some parameters and to analyze the speed of convergence to the equilibrium. 2. Multi-team Cournot game, boundedly rationality and naive expectation In the real market, the players having common benefit are willing to cooperate and join a cooperative team and the players having incompatible objectives join different competitive teams. It is reasonable to take into consideration a multi-team system where several competing teams control the market and each team contains some cooperative players. While making decision, the players in the same team care for not only their own benefits but also the team’s total profit, for instance, the weighted profit in the team [3–5]. In the multi-team Cournot model, the players are firms producing homogeneous goods and are divided into N teams, of which team i consists of mi cooperative firms, m1 + m2 +    + mN = n. The profit of a player is decided by the price determined by the supply quantity of all the players in the market and its own quantity. Let P(Q) denote the inverse demand function, P whereQ ¼ ni¼1 qi ; qi is the supply of i, and let Ci(qi) denote i’s cost function, i = 1, 2, . . . , n. Then the profit of player i is given by

Pi ðq1 ; q2 ; . . . ; qn Þ ¼ PðQ Þqi  C i ðqi Þ:

ð1Þ

If the profit of i belonging to team X is denoted by PXi , the weighted profit in team X will be:

X

PteamX ðq1 ; q2 ; . . . ; qn Þ ¼

xXi PXi ðq1 ; q2 ; . . . ; qn Þ;

ð2Þ

i2teamX

where xXi is the weighting coefficient of firm i in team X. Taking the team’s weighted profit as objective function, each producer will adjust its current strategy and decide the production for the next period. The players can employ the same adjusting strategy (homogeneous players) or use different adjusting strategies (heterogeneous players). In this work, two types of players are assumed to play the game: a type of naı¨ve players and a type of boundedly rational players. In each period, some players may make expectations of their rivals’ output in the next period in order to determine the corresponding profit-maximizing output for period t + 1. If we denote by qti the output of such firm i of team X in period t, its for the period t + 1 may be decided by solving the maximization problem production qtþ1 i

  ; qtþ1 ¼ arg max PteamX qi ; Q e;tþ1 i i

ð3Þ

qi

e;tþ1 e;tþ1 e;tþ1 e;tþ1 where Q e;tþ1 ¼ ðqe;tþ1 is the expectation of firm i about the production of firm j in period i i;1 ; . . . ; qi;i1 ; qi;iþ1 ; . . . ; qi;n Þ, and qi;j

t + 1. Cournot [10] assumed that qe;tþ1 ¼ qtj hence Q e;tþ1 ¼ Q ti ðQ ti represents the production vector of all the other players exi i;j cept i in period t), i.e. firm i expects that the production of other firms will remain the same as in the current period (such of such firm i can be chosen to satisfy the following equation: expectation is called naive one). Now the optimal strategy qtþ1 i

 @ PteamX ðqi ; Q ti Þ   @qi

¼ 0:

ð4Þ

qi ¼qtþ1 i

Information in the market is usually incomplete, and some players may use more complicated techniques, such as bounded rationality method. The boundedly rational firms do not have complete understanding of the market, they hence   make their production decision from a local estimation of the marginal profit @ Pi qti ; Q ti =@qti . The boundedly rational firm increases (decreases) its production for period t + 1 if the marginal profit in period t is positive (negative) [2–9,11,12]. For this case, the adjustment strategy of such producer in team X has the form

qtþ1 i

¼

qti

t i qi

þa

@

QteamX 

qti ; Q ti

@qti

 ;

t ¼ 0; 1; 2; . . . ;

ð5Þ

where ai is the positive adjustment rate. In the next section, we are going to apply these two techniques to a two-team game model with linear demand function and nonlinear cost function.

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3. Two-team Cournot game with heterogeneous players We consider a model of two teams. Team 1 is cooperative and contains two firms, firm 1 and firm 2. Team 2 contains only one firm, firm 3. We assume that firm 1 and firm 2 are boundedly rational players and firm 3 is a naive player. Assume the inverse demand function is linear, that is

P ¼ PðQÞ ¼ a  Q : where Q ¼

P3

i¼1 qi .

C i ðqi Þ ¼

ð6Þ

And the production cost function of firm i takes a nonlinear form

ci q2i ;

i ¼ 1; 2; 3;

ð7Þ

with ci > 0. We can assume c1 = c2, since firms 1 and 2 are in the same cooperative team and may sometimes share the technology of production. The weighting coefficient of firm 1and firm 2 in team 1 is x and 1-x, respectively. Now the objective Q Q function team1 of team 1 and the one team2 of team 2 are as follows:

Yteam1 Yteam2

¼ xP1 þ ð1  xÞP2 ¼ xq1 ða  Q  c1 q1 Þ þ ð1  xÞq2 ða  Q  c1 q2 Þ;

ð8Þ

¼ q3 ða  Q  c3 q3 Þ;

where 0 < x < 1; Q ¼

P3

i¼1 qi .

ð9Þ

From (5): qtþ1 ¼ qti þ ai qti i

@

Qteam1

ðqt1 ;qt2 ;qt3 Þ

@qti

ði ¼ 1; 2Þ and from (4):

@

Qteam2

ðqt1 ;qt2 ;qtþ1 3 Þ

@qtþ1 3

¼ 0, we get the

dynamical systems of q1, q2 and q3 as follows:

  8 tþ1 t t t t t > < q1 ¼ q1 þ a1 q1 xa  2xð1 þ c1 Þq1  q2  xq3 ;  qtþ1 ¼ qt2 þ a2 qt2 ð1  xÞa  qt1  2ð1  xÞð1 þ c1 Þqt2  ð1  xÞqt3 ; 2 >   : tþ1 q3 ¼ a  qt1  qt2 =2ð1 þ c3 Þ:

ð10Þ

4. Equilibrium points and local stability ¼ qti ¼ qi in (10), it is easy to work out four equilibrium points: By setting qtþ1 i

E0 ¼ 0; 0; E1 ¼ 0;

a ; 2ð1 þ c3 Þ

ð1 þ 2c3 Þa ð1 þ 2c1 Þa ; ; 4ð1 þ c1 Þð1 þ c3 Þ  1 4ð1 þ c1 Þð1 þ c3 Þ  1

ð1 þ 2c3 Þa ð1 þ 2c1 Þa E2 ¼ ; 0; ; 4ð1 þ c1 Þð1 þ c3 Þ  1 4ð1 þ c1 Þð1 þ c3 Þ  1   E ¼ q1 ; q2 ; q3 ; where q1 ¼ DD1 ; q2 ¼ DD2 ; q3 ¼ DD3 , and

D ¼ 8xð1  xÞð1 þ c1 Þ2 ð1 þ c3 Þ  4xð1  xÞð1 þ c1 Þ  2ð1 þ c3 Þ þ 1; D1 ¼ 4xð1  xÞð1 þ c1 Þð1 þ c3 Þa  2ð1  xÞð1 þ c3 Þa þ ð1  xÞa  2xð1  xÞð1 þ c1 Þa; D2 ¼ 4xð1  xÞð1 þ c1 Þð1 þ c3 Þa  2xð1 þ c3 Þa þ xa  2xð1  xÞð1 þ c1 Þa; D3 ¼ 4xð1  xÞð1 þ c1 Þ2 a  4xð1  xÞð1 þ c1 Þa: E0, E1 and E2 are on the boundary of the decision set S = {(q1, q2, q3)jq1 P 0, q2 P 0, q3 P 0}, and are hence boundary equilibriums. E* is the unique interior equilibrium provided that q1 > 0; q2 > 0 and q3 > 0. The stability of these equilibriums is based on the eigenvalues of the Jacobian matrix of system (10) 0

1  a1 q 1 xa1 q1 1 þ a1 ½xa  4xð1 þ c1 Þq1  q2  xq3  B  a2 q 2 1 þ a2 ½ð1  xÞa  q1  4ð1  xÞð1 þ c1 Þq2  ð1  xÞq3  ð1  xÞa2 q2 C Jðq1 ;q2 ;q3 Þ ¼ @ A: 1  2ð1þc 3Þ

1  2ð1þc 3Þ

0

ð11Þ

The equilibrium solution will be stable if the eigenvalues ki, i = 1, 2, 3 of the above Jacobian matrix satisfy the inequalities jkij < 1, i = 1, 2, 3. Now, we apply the condition of stability to all these equilibrium points.

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(i) The point E0 = (0, 0, q3)(q3 = a/2(1 + c3)) is an unstable fixed point. Because the Jacobian matrix

0 B Jð0; 0; q3 Þ ¼ @

1 þ a 1 ½ xa  xq 3  0 1  2ð1þc 3Þ

0

0

1

1 þ a2 ½ð1  xÞa  ð1  xÞq3  0 C A; 1  2ð1þc 3Þ

0

has two eigenvalues k1 = 1 + a1[xa  xq3] = 1 + a1xa(2c3 + 1)/(2(1 + c3)) and k2 = 1 + a2 [(1  x)a  (1  x)q3] = 1 + a2(1  x)a(2c3 + 1)/(2(1 + c3)), both of which are greater than 1. In a similar way, we have (ii) E1 is asymptotically stable if

1 < 1 þ a1 ðxa  q2  xq3 Þ < 1;

1  2a2 ð1  xÞð1 þ c1 Þq2 > 2;

(iii) E2is asymptotically stable if

1  2a1 xð1 þ c1 Þq1 > 2;

1 < 1 þ a2 ½ð1  xÞa  q1  ð1  xÞq3  < 1:

  (iv) E ¼ q1 ; q2 ; q3 is somewhat complex. The Jacobian matrix at E* takes the form: 

0



 

B J q1 ; q2 ; q3 ¼ @

1  2a1 xð1 þ c1 Þq1

a1 q1

a2 q2 1  2ð1þc 3Þ

1  2a2 ð1  xÞð1 þ c1 Þq2 1  2ð1þc 3Þ

a1 xq1

1

a2 ð1  xÞq2 C A: 0

  By calculation, we get the characteristic polynomial p(k) of the matrix J q1 ; q2 ; q3 as following:

pðkÞ ¼ k3 þ u1 k2 þ u2 k þ u3 ;   where u1 ¼ 2ð1 þ c1 Þ a1 xq1 þ a2 ð1  xÞq2  2,

  1 Þ a1 xq1 þ a2 ð1  xÞq2 þ a1 a2 ð4xð1  xÞð1 þ c1 Þ2  1Þq1 q2 ; 2ð1 þ c3 Þ   1 1 a1 xq1 þ a2 ð1  xÞq2  a1 a2 ð4xð1  xÞð1 þ c1 Þ  1Þq1 q2 : u3 ¼ 2ð1 þ c3 Þ 2ð1 þ c3 Þ u2 ¼ 1  ð2ð1 þ c1 Þ þ

Fig. 1. The bifurcation diagram of system (10) with respect to a1.

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Z. Ding et al. / Applied Mathematics and Computation 215 (2009) 1098–1105

Fig. 2. The bifurcation diagram of system (10) with respect to x.

A

q3

1 0.95 0.9 0.85 1.5

q2

1

0

0.5

1

q1

C

B 1.05 1.3

1

1.1

q3

q2

1.2

0.95

1 0.9

0.9

0.8 0

0.5

q1

1

0.8

Fig. 3. The strange attractor of system (10).

1

q2

1.2

1.4

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Z. Ding et al. / Applied Mathematics and Computation 215 (2009) 1098–1105

From Jury conditions, the necessary and sufficient conditions for jkij < 1, i = 1, 2, 3 are:

8 1 þ u1 þ u2 þ u3 > 0 > > > <1  u þ u  u > 0 1 2 3 : > ju 3j < 1 > > : ju2  u1 u3 j < 1  u23

ð12Þ

So the equilibrium point E* of the system (10) is stable, if the conditions in (12) are all satisfied. The result that E0 = (0, 0, a/2(1 + c3)) is not stable implies that in the long run the market cannot clear out every one of the cooperative team 1. But under some conditions E1 or E2 is asymptotically stable, which means in the long run one firm of team 1 will go out of the market so that this team just acts as a single player. However, it is very interesting that E* will be stable if the parameters satisfy (12), since at this equilibrium every one will stay in the market. 5. Numerical analysis The main purpose of this section is to show the complicated dynamic features of the dynamics of this two-team game and demonstrate the great influence of parameters on the speed of convergence to the equilibrium. To provide some numerical evidence for complicated behavior of system (10), we show some numerical results such as bifurcation diagrams, periodic windows, strange attractors and sensitive dependence on initial conditions. Fig. 1 shows the bifurcation diagram with respect to a1 while other parameters are fixed as a = 15, x = 0.3, a2 = 0.2, c1 = 5 and c3 = 6. In Fig. 1A, the equilibrium E* = (0.8756, 1.0680, 0.9326) is locally stable for small values of the parameter a1. If a1 increases, the equilibrium point E* becomes unstable, period doubling bifurcations occurs and the system becomes chaotic. In Fig. 1B, profit i represents the profit of firm i, and there is the same phenomenon as in Fig. 1A. We can also see many periodic windows in Fig. 1. Fig. 2 shows the bifurcation diagram for system (10) with respect to the weight x, where we take a = 15, a1 = 0.2, a2 = 0.2, c1 = 5 and c3 = 6. From Fig. 2, we can see that if x is small, player 1 will depart from the competition and actually, only 2 and

q1

1

0.5

0 0

20

40

60

80

100

120

80

100

120

t

q2

1.2

1

0.8

0

20

40

60

t

q3

1

0.95

0.9

0

20

40

60

80

100

t Fig. 4. Senstitive dependence on initial conditions for system (10).

120

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1000 800

t

600 400 200 0 0.3 0.25 0.2

α

0.15

2

0.1 0.05

0.1

0.2

0.3

0.5

0.4

0.6

0.7

α

1

Fig. 5. The influence of different adjustment rate on the speed of convergenceto the equilibrium point E*.

1000 800 600 400 200 0 1 0.8 0.6

ω

0.4 0.2

0

0.1

0.2

0.4

0.3

0.5

α

1

Fig. 6. The influence of different values of (a1, x) on the speed of convergence to the equilibrium point E*.

3 play the duopoly game in which their outputs are chaotic. To the opposite, if x is very large, player 2 will depart from the team and the team-game is equal to a duopoly game played by 1 and 3 whose outputs are also chaotic. We can see the inverse period doubling bifurcation in the interval of small weight and the period doubling bifurcation in the interval of large weight. There are also some periodic windows in Fig. 2. It is somewhat interesting that when the weight is small or is large, one player in team 1 will provide zero output rather than negative output, although negative output may occur mathematically. Fig. 3 shows the strange attractors for system (10) when (a, w, a1, a2, c1, c3) = (15, 0.3, 0.788, 0.2, 5, 6). Fig. 3A shows the three-dimension strange attractor, Fig. 3B and Fig. 3C show the strange attractors from two-dimension. In order to demonstrate the sensitivity to initial conditions of system (10), when (a, w, a1, a2, c1, c3) = (15, 0.3, 0.788,  0 0 0 0.2, 5, 6), we simulated two orbits  0 0  (in Fig. 4). The blue curve starts from the initial points q1 ; q2 ; q3 ¼ ð0:4; 0:5; 0:6Þ and 0 1 red one from q1 ; q2 þ 0:0001; q3 . It shows that although they are indistinguishable, after a number of iterations, the difference between them builds up rapidly. 1

For interpretation of color in Fig. 4, the reader is referred to the web version of this article.

Z. Ding et al. / Applied Mathematics and Computation 215 (2009) 1098–1105

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From the numerical analysis above, we can see that both the adjustment rate and profit weight may change the stability of the equilibrium and cause a market structure to behave chaotically. Different adjustment rate has also influence on the speed of convergence to the equilibrium. Fig. 5 shows the relationship between the convergence speed and adjustment rate, where x = 0.3,  a = 15, c1 = 5 and c3 = 6. The coordinate t represents the iterative time that it will take the outputs, from the initial output q01 ; q02 ; q03 ¼ ð0:4; 0:5; 0:6Þ, to be close to the equilibrium point E* (the error is set to be within 0.00001). When the iterative time t exceeds 1000, we do not portray it, and if this happens then either the iterative time for the output to reach the equilibrium is too long or the system becomes unstable. Fig. 5 shows that a too low or too high adjustment rate will cause too long time for the system to converge to the equilibrium point. Other parameters, such as x, c1, c2, also have great influence on the speed of convergence to the equilibrium. For example, Fig. 6 shows the influence of the model parameters (a1, x) on the convergence speed. It is much similar to Fig. 5 in the sense that, only when the value of parameter is medium, can the output be close to the equilibrium point E* soon. 6. Conclusion In this paper, we have studied a two-team Cournot game played by a team of bounded rationality players who keep in mind the team’s weighted profit and a team of only one naive player. The equilibrium solutions are obtained and their stability is studied. The conditions of local asymptotic stability of the equilibrium points are given. The numerical simulation shows that the complicated behaviors in the output dynamics may be caused by the main parameters such as the adjustment rate of boundedly rational players and the profit weighting coefficients in the cooperative team. The great influence of these parameters on the speed of convergence to the equilibrium is also illustrated with numerical results. Acknowledgements This work is financially supported by the National Natural Science Foundation of China (No. 90610031), the Top Talents Foundation of Jiangsu University (No. 07JDG023) and the Philosophy Social Science Foundation for Higher Education Institutions from Jiangsu Education Department (No. 1291190024). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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