International Journal of Computer Science & Emerging Technologies (E-ISSN: 2044-6004) Volume 1, Issue 4, December 2010

254

Analysis of a Cournot Duopoly Model’s Stability Hong-xing Yao， Jia-xiu Zu （Faculty of Science，Jiangsu University , Zhenjiang 212013 , China)

Abstract: In this paper, the feedback control methods are

cannot be realized by personal interest’s maximization. The

applied to a duopoly model based on heterogeneous

main question which the prisoners’ dilemma poses is whether

expectations. This is the time-delayed feedback control of the

a cooperative behaviour can emerge among rational and

production system. This control aims to bring this system into

self-interested players whenever there is no formal agreement

instability equilibrium by using delay of state variables．The

[10]. In real economical markets we truly can observe that

validity of the control method is proved through theoretical analysis and numerical simulations ．Moreover ，scope of convergent condition is given．The production model can quickly reach Nash equilibrium after control, providing theoretical reference and production conditions to enterprises．

competitors are often able to achieve the cooperation. Although the duopoly game with output competition (Cournot game) is faced with prisoners’ dilemma (Nash equilibrium is not Pareto optimal), it cannot be studied in standard game model with prisoners’ dilemma. Because the collection of strategies in this model is a finite set, and in the

Key words: dynamical Cournot model, delayed feedback

output competition it is an infinite set. Cafagna [10] has built

control, the Pareto optimal, Nash equilibrium, game, bounded

a strategy with output adjustment (the ‘good’ strategy), and

rationality

makes the firms reach a cooperative equilibrium finally. The prisoners’ dilemma can be explained based on that. However,

1

Introduction

the ‘good’ strategy is based on the premise that producers

Oligopolistic market is a universal market mechanism, in

completely know about their competitors’ output and profit.

which a trade is completely controlled by several firms. The

In fact, the producers with mutual competition, or even the

firms manufacture the same or homogeneous products and

producers who have achieved certain cooperation, keep the

they must consider not only the demand of marker, but also

output, the profit and the related things as the business secrets

the actions of their competitors [1]. Game theory has been

for their own benefit. So the supposition of incomplete

widely applied to oligopolistic markets thank to its ability to

information is more rational. For example, in the model with

consider strategic interactions among firms. Oligopolist is

two producers, as long as one producer does not know the

competitive, and the basic solution which refers to

other’s cost of production, it is impossible for the first

competitive equilibrium in Cournot game is Nash equilibrium

producer to know about the other’s profit under different

or Cournot equilibrium. The adjust dynamics to get the Nash

combination of bilateral outputs. That is to say, the first

equilibrium and the stability are studied by many works [2–9].

producer cannot have complete information. Then under the

But just as what Nash equilibrium reveals, Nash equilibrium

premise that each producer incompletely knows about the

reflects individual rationality, but it violates collective

competitor’s information (output, profit and so on), is there a

rationality – Nash equilibrium of the duopoly game is not

strategy of output adjustment for the producers to use to

Pareto optimal. The prisoners’ dilemma shows that, there is a

achieved a cooperative equilibrium?

contradiction between individual rationality and collective

In this paper, we study that how firms get bigger profits

rationality, and the correct choice based on individual

by adjusting their own outputs. It is different from the paper

rationality will reduce everybody’s welfare. In other words,

[10-12] that the producers do not know about the market

Pareto improvement cannot be carried on and Pareto optimal

information of the competitor’s output and profit, and the cooperative behaviour in duopoly competition is considered

International Journal of Computer Science & Emerging Technologies (E-ISSN: 2044-6004) Volume 1, Issue 4, December 2010

with the ‘‘tit-for-tat” conduct.

255

cooperative profit

p c - p i ,t < 0 , then his own profit is more;

he extrapolates that the competitor is cooperative, then he will

2. The model

properly reduce his output to continue the cooperation as a

There are two firms produce a homogeneous good in a

‘‘reward”1; Otherwise, if

p c - p i ,t > 0 , the firm i cannot

market .Taking production decisions at discrete time periods

t = 1,2,3,... . Denoting the quantity of output by each firm at time

t is qi ,t (i = 1, 2). We have that cost function has the

linear form:

competitor is not cooperative, then he will increase his output as ‘‘penalty”.2 For this case ,we get the dynamical systems of

q1 , and q2 as follows: ci ,t = ci qi ,t

Let

realize the cooperative profit, and extrapolates that the

(1)

p(Q) denote the inverse demand function:

qi ,t + 1 = qi ,t + ui (p c - p i ,t )= qi ,t + ui éêp c - a - b Q - ci ùú (4) ë û

(

where

pt = a - b Q Where a, b >

ui (i = 1, 2) is a adjusting parameter, and

(2)

0 , a > ci and Qt =

Then the profit of player

)

å

ui > 0 .In this model ,Since the firms do not need know the

q i i ,t

competitor’s related information, it is an adjusting strategy

i at time t is given by:

with incomplete information. Although its form is simple, it is based on the thoughts of ‘‘tit-for-tat” strategy in prisoners’

(

)

p i ,t = a - b Q - ci qi ,t

(3)

This paper is about cooperation under the incomplete information, and the following models are based on the assumption that the firms compare their own profits with the cooperative profit. The solving of the cooperative profit has been introduced in duopoly game theory. The cooperative profit means the profit which is solved by maximizing the sum of all firms’ profit. We consider the symmetrical case:

c1 = c2 = c ,then can get the cooperative profit, 3

pc =

2 (a - c) 27b 2

2(a - c)2 ,and the cooperative output, qc = 9b 2

dilemma game. Now the question is that whether the firms can achieve a cooperative Pareto optimality. With above assumptions, the duopoly game with heterogeneous players is described

by

a

two-dimensional

nonlinear

map

T (q1 (t ), q2 (t )) ® (q1 (t + 1), q2 (t + 1)) defined as : íï q t + 1 = q t + u ép - a - c q t + bq t q t + q t ù ) 1( ) 1 ê c ( ) 1 ( ) 1 ( ) 1 ( ) 2 ( )ú (5) ïï 1 ( ë û T : ïì ïï q t + 1 = q t + u ép - a - c q t + bq t q t + q t ù ) 2( ) 2 ê c ( ) 2( ) 2( ) 1( ) 2 ( )ú ïïî 2 ( ë û

Where

qi (t ) denotes productions of period

t ,

qi (t + 1) represent productions of period t + 1 In the paper, we are considering an economic model where only nonnegative equilibrium points are meaningful.

3. The tit-for-tat dynamic strategy The tit-for-tat strategy is the best behaviour allowing the achievement of cooperation in repeated games [10]. Its characteristic is that every player consists in doing what the opponent did in the previous move. In the paper, we study the Cournot model with the tit-for-tat conduct. And the dynamic equations are based on the incomplete information. Each producer cannot obtain the competitor’s complete information, but he completely knows about his own output and profit. The firm

i can compare his profit p i ,t at time t with the

cooperative profit

p c which is Pareto optimal. If the

So we only study the nonnegative fixed points of the map (5), i.e. the solution of the nonlinear algebraic system as:

íï p - (a - c)q + bq q + q = 0 1 1 1 2 ï c ì ïï p - (a - c)q + bq q + q = 0 2 2 1 2 ïî c

(6)

By setting qi (t + 1)= qi (t ), i = 1, 2 in system (5), we obtained (6). Then it is easy to work out an unique fixed point of system (6): E = (q1* , q2* ) ,where 2

* 1

* 2

q = q = qc =

2 (a - c) 9b2

International Journal of Computer Science & Emerging Technologies (E-ISSN: 2044-6004) Volume 1, Issue 4, December 2010

256

The stability of these equilibriums is based on the 16

eigenvalues of the Jacobian matrix of system (5)

B 15.5

æ ö÷ æ u1b b çç ÷÷ö ççc - a + ÷÷ 1 + u ÷ çç 1 çç ÷÷ ÷ ÷ 2 q + q 2 q + q è çç 1 2ø 1 2 ÷÷ * * J (q1 , q2 ) = ç ÷ çç æ ö÷÷ u2 b b ÷÷ ç çç ÷÷ 1 + u2 ççc - a + ÷÷ çç çè ÷øø÷ 2 q1 + q2 2 q1 + q2 ÷÷ è

15 profit 1 14.5 14 13.5 profit 2 13

We compute the Jacobian matrix J at E then get

12.5

æ u1 (a - c ) ö u1 (a - c ) ÷ çç1÷ ÷ ç ÷ 6 6 ÷ J (q1* , q2* ) = ççç ÷ çç u2 (a - c) ÷ u2 (a - c )÷ ÷ ÷ 1çç ÷ è ø 6 6

12 11.5

By calculation, we get the characteristic polynomial P (l )of the matrix J (q1* , q2* ) as following:

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Fig. 1b. The profit of the system (4) is stable experiments show that its stability is sensitive to the parameter

p (l )= l 2 - Trl + Det = 0

Tr is the trace and Det is the determinant of

Where

the Jacobian matrix

13

J (q1* , q2* ) .

A 12

Tr = 2 -

u1 (a - c) u2 (a - c) 6 6

Det = 1-

u1 (a - c) u2 (a - c) 6 6

11

10

9

(

* 1

* 2

Then we have two eigenvalues of matrix J q , q

l 1 = 1 and l 2 = 1-

(u1 + u2 )(a - c) . 6

),

8 q1

If it holds that

7 q2

ui (i = 1, 2) is very small, we have l 2 < 1. Since l 1 = 1 is a critical condition we cannot know the stability of the

6 0

200

400

600

800

1000

1200

1400

1600

1800

2000

Fig. 2a The output t of the system (4) is unstable

system (5). But the following numerical 16 B 8

15 A

7.8

Profit 1

14

7.6

13 7.4

Profit 2

q1

12

7.2

11

7

10

6.8 q2

6.6

9

6.4

8

6.2 6

7 0

200

400

600

800

1000

1200

1400

1600

1800

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Fig. 2b. The profit of the system (4) is unstable Fig. 1a. The output of the system (4) is stable

2000

International Journal of Computer Science & Emerging Technologies (E-ISSN: 2044-6004) Volume 1, Issue 4, December 2010

We

take a = 8, c = 2, u1 = u2 = 0.0082,

and

257

æ 3b2 ö ÷ ç ÷ Where B1 = - k - 2 + (u1 + u2 )ça - c ÷ ÷ çè 4(a - c) ø

the

initial value q1,0 = 8, q2,0 = 6 . If we fix other parameters and

æ æ 3b 2 ö÷ö÷ ç ç ÷÷÷ B2 = 2k + 1 + ççu1 + u2 + u2 k + u1u2 ç- a + c + çè 4(a - c) ÷÷÷ øø èç

vary one, for instance b, the stability of system changes . Fig. 1 shows that it is stable, but if a parameter changes slightly, it is the contrary (Fig. 2). And Fig. 2 shows that the output and

æ 3b2 ö÷ 9u1u2b 4 ç ÷*ç- a + c + çè 4(a - c) ÷÷ø 16(a - c)2

the profit not only cannot achieve the Pareto optimality, but also appears the phenomenon of malignant competition – the

æ 3b2 ö ÷ ÷ B3 = - k + u2 k çça - c ÷ ÷ çè 4(a - c) ø

outputs of both sides increase infinitely (Fig. 2A), while the profits approach to zero (Fig. 2B). That is to say, the firms in

From Jury conditions, the necessary and sufficient

Cournot game cannot achieve the Pareto optimal equilibrium under the adjustment Eq. (4)

conditions for

ïíï 1 + ïï ïï 1ì ïï 1ïï ïïî B3

4. Delayed feedback control of the production system

4.1

By adding a time-delayed feedback control, we

consider a new strategy:

íï q = q + u p - (a - c)q + b q + q + k (q - q ) ïï 1,t+1 1,t 1 c 1,t 1,t 2,t 1,t 1,t - 1 (7) ì ïï q = q + u p - (a - c)q + b q + q 2,t 1,t 2,t ïî 2,t+1 2,t 2 c

( (

Where

) )

B1 + B2 - B3 > 0

(9)

B32 > B2 - B1 B3 0 , k (q1,t - q1,t- 1 ) is the delayed feedback

q1

8 q

control of the system. (7) equivalent to the following

B1 + B2 + B3 > 0

10

8.5

and

l i < 1, i = 1, 2,3 are:

7.5

three-dimensional equations: 7

íï q = q + u p - (a - c)q + b q + q + k (q - q ) ïï 1,t+1 1,t 1 c 1,t 1,t 2,t 1,t 1,t - 1 ïï ïì q = q + u p - (a - c)q + b q + q (8) 2,t 1,t 2,t ïï 2,t+1 2,t 2 c ïï q = q ïïî 3,t+1 1,t

( (

The Jacobian matrix at

) )

E* = (q1* , q2* ) takes the form:

æ ö÷ æ ö÷ çç u1b b ç ÷ ÷ - k÷ ÷ ççk + 1 + u1 çççc - a + ÷ * *÷ * * ÷ çè çç 2 q1 + q2 ø÷ 2 q1 + q2 ÷ ÷ çç ÷ ÷ ÷ çç æ ö÷ ÷ u b b ç ÷ 2 ç ÷ ç ÷ J (E *) = çç 1 + u2 çc - a + 0 ÷ ÷ * * * *÷ ÷ ç çç ÷ 2 q1 + q2 2 q1 + q2 ø÷ èç ÷ ÷ çç ÷ çç 1 0 0÷ ÷ ÷ çç ÷ ÷ çç ÷ ÷ çè ø÷

By calculation, we get the characteristic polynomial

f (l ) of the matrix J (q1* , q2* ) as following: f (l )= l 3 + B1l 2 + B2l + B3 = 0

6.5 q2

6 5.5

0

50

100

150 t

Fig3a

200

250

International Journal of Computer Science & Emerging Technologies (E-ISSN: 2044-6004) Volume 1, Issue 4, December 2010

258

17

16

15

p1

proft

14

13

12 p2 11

10 0

50

100

150

200

250

t

Fig3b So the equilibrium point

E * of the system (7) is

stable, if the conditions in (9) are all satisfied. We

reconsider

the

unstable

situation

( a = 8, b = 1.09, c = 2, u1 = u2 = 0.0082, q1,0 = 8, q2,0 = 6 ) in Section 3. Let

íï q = q + u (p - (a - c)q + bq q + q ) + k (q - q ) 1,t 1,t 1,t 2,t 1 1,t 3,t ïï 1,t + 1 1,t 1 c ïï ï q2,t + 1 = q2,t + u2 (p c - (a - c)q2,t + bq2,t q1,t + q2,t ) + k2 (q2,t - q4,t ) ì ïï q = q ïï 3,t + 1 1,t ïï q = q ïî 4,t + 1 2,t (11)

k = 0.4 ，now the output and profit system

E* = (q1* , q2* ) takes the form :

(7) become stable, as showed in Fig. 3(blue line) . In

The Jacobian matrix at

Fig.3a,the

æ ö bq1* ç ÷ M1 - k1 0 ÷ ç ÷ ç ÷ ç 2 q1* + q2* ÷ ç ÷ ç ÷ ç ÷ * ÷ ç bq ÷ 2 ç ÷ J (E) = ç M 0 k 2 2 ÷ ç * * ÷ ç 2 q + q ÷ 1 2 ç ÷ ç ÷ ÷ ç ÷ ç 1 0 0 0 ÷ ç ÷ ç ÷ ç ÷ 0 1 0 0 ø è By calculation, we get the characteristic polynomial

blue

point

shows

that

the

changes

of

Productions1,2,when adds a time-delayed feedback control strategy. In Fig.3b,the blue point shows the proft.

4.2

By adding two time-delayed feedback control, we

consider a new strategy:

íï q = q + u (p - (a - c)q + bq q + q ) + k (q - q ) 1,t 1,t 1,t 2,t 1 1,t 1,t - 1 ïï 1,t+ 1 1,t 1 c ì ïï q = q + u (p - (a - c)q + bq q + q ) + k (q - q ) 2,t 2,t 1,t 2,t 2 2,t 2,t - 1 ïî 2,t+ 1 2,t 2 c (10) Where

ui (i = 1, 2) is an adjustment parameter , and

ui > 0 , ki (q1,t - q1,t- 1 ) is the delayed feedback control of the system. (10) equivalent to the following

f (l ) of the matrix J (q1* , q2* ) as following: f (l ) = l 4 + ( M 1 + M 2 )l 3 + ( M 1M 2 + k2 )l

2

é b 2 q1*q2* ù úl + (k1k2 - k1M 2 ) + êk1 - k2 M 1 ê 4(q1* + q2* ) ú ë û Where

M1 = k1 + 1- u1 (a - c) + u1b(

3q1* + 2q2* * 1

* 2

2 q +q

four-dimensional equations:

M 2 = k 2 + 1- u2 ( a - c ) + u 2 b (

+ q1* )

2q1* + 3q2* * 1

* 2

2 q + q

+ q2* )

International Journal of Computer Science & Emerging Technologies (E-ISSN: 2044-6004) Volume 1, Issue 4, December 2010

strategy with time-delayed feedback. In conclusion, the

2(a - c) 2 9b 2

q1* = q2* =

cooperation may be achieved under the tit-for-tat strategy.

From Jury conditions, the necessary and sufficient conditions for

Where

By introducing the feedback control to the cooperation B1 + B2 + B3 + B4 > 0 B1 - B2 - B3 + B4 > 0

intention of the players, the firms’ cooperation can be （12）