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Analysis of a duopoly game with delayed bounded rationality
Applied Mathematics and Computation 138 (2003) 387–402 www.elsevier.com/locate/amc
Analysis of a duopoly game with delayed bounded rationality M.T. Yassen *, H.N. Agiza Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Abstract A dynamic of Cournot duopoly game is analyzed, where players use different production methods and choose their quantities with bounded rationality. A dynamic of nonlinear Cournot duopoly game is analyzed, where players choose quantities with delayed bounded rationality and similar methods of production. The equilibria of the corresponding discrete dynamical systems are investigated. The stability conditions of Nash equilibrium under a local adjustment process are studied. The stability of Nash equilibrium, as some parameters of the model are varied, gives rise to complex dynamics such as cycles of higher order and chaos. We show that firms using delayed bounded rationality have a higher chance of reaching Nash equilibrium. Numerical simulations are used to show bifurcations diagrams, stability regions and chaos. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Duopoly game; Delayed bounded rationality; Nash equilibrium; Stability; Bifurcation; Chaos
1. Introduction Recently the complex dynamics of bounded rationality duopoly game of BowleyÕs model have been studied [1]. In the BowleyÕs model [2] the demand function has the linear form p ¼ f ðQÞ ¼ a bQ *
Corresponding author. E-mail addresses: [email protected] (M.T. Yassen), [email protected] (H.N. Agiza). 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 2 ) 0 0 1 4 3 - 1
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where a and b are the demand parameters. The constant a is the maximum price in the market and Q is the total quantity in the market. The dynamics of the duopoly model when firms use similar production methods was investigated in [1]. In this work we study the complex dynamics of duopoly BowleyÕs model when firms use different production method and the cost function is proposed in the nonlinear form Ci ðqi Þ ¼ ci q2i where ci , i ¼ 1; 2 is a positive shift parameter of the cost function. Generally the complex dynamics of duoploy models was studied with different assumptions to the demand function (see for examples [3–6]). In this paper, we study duopoly game which describes a market with two-players producing homogeneous goods, updates their production strategies in order to maximize their profits. Each player think with bounded rationality, adjust his output according to the expected marginal profit, therefore the decision of each player depends on local information about his output. In this Cournot game each player tries to maximize his profit according to local informations of his input. The plan of the paper is as follows. In Section 2, the description of the duopoly BowleyÕs model with bounded rationality. The existence and local stability of the equilibrium points are studied. The stability region of Nash equilibrium is determined in the plane of speeds adjustment (v1 v2 -plane) and also complex behavior such as cycles of higher order and chaos is studied. Numerical simulation is presented to show such behaviour. In Section 3, duopoly BowleyÕs model with delayed bounded rationality is studied when firms use similar production methods. The existence and local stability of the boundary equilibria and Nash equilibrium are investigated. The stability region of Nash equilibrium is determined in the plane of speeds adjustment (v1 v2 -plane). Numerical simulations are presented. Section 4 is the conclusion.
2. Duopoly model We consider two firms, labelled by i ¼ 1; 2 producing the same good for sale in the market. Production decisions of both firms occur at discrete time periods t ¼ 0; 1; 2 . . . Let qi ðtÞ represent the output of ith firm during period t, with a production cost function Ci ðqi Þ. The price prevailing in period t is determined by the total supply QðtÞ ¼ q1 ðtÞ þ q2 ðtÞ through a demand function p ¼ f ðQÞ In this model the demand function is assumed linear [7], which has the form f ðQÞ ¼ a bQ
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where a and b are positive constants. Also we assume that the cost function has the nonlinear form Ci ðqi Þ ¼ ci q2i ;
i ¼ 1; 2
ð1Þ
where the positive parameters ci , i ¼ 1; 2 are positive shift parameters to the cost function of the firm i, i ¼ 1; 2 respectively. With these assumptions the single-period profit of the ith firm is given by Pi ðq1 ; q2 Þ ¼ qi ða bQÞ ci q2i ;
i ¼ 1; 2
ð2Þ
and the marginal profit of the ith firm at the point (q1 , q2 ) of the strategy space is Ui ¼
oPi ¼ a 2ðb þ ci Þqi bqj ; oqi
i ¼ 1; 2; i 6¼ j
ð3Þ
The duopoly method with bounded rational players [6] for our assumptions has the following two dimensional nonlinear map T ðq1 ; q2 Þ ! ðq01 ; q02 Þ which is defined by oPi ; i ¼ 1; 2 T : q0i ¼ qi þ ai ðqi Þ oqi where ai ðqi Þ is a positive function which gives the extend of the production variation of the ith firm according to his computed profit signal Ui and prime (0 ) denotes the unit-time advancement operator, that is, if the right-hand side variables are productions of period t then the left-hand side ones represent productions of period (t þ 1). We take the function ai ðqi Þ in the linear form ai ðqi Þ ¼ vi qi . From (3) the two-dimensional nonlinear map T ðq1 ; q2 Þ ! ðq01 ; q02 Þ is defined by the following two nonlinear difference equations: q1 ðt þ 1Þ ¼ q1 ðtÞf1 þ v1 ½a 2ðb þ c1 Þq1 ðtÞ bq2 ðtÞ g q2 ðt þ 1Þ ¼ q2 ðtÞf1 þ v2 ½a 2ðb þ c2 Þq2 ðtÞ bq1 ðtÞ g
ð4Þ
where vi , i ¼ 1; 2 is a positive constant which is called the speed of adjustment of ith firm [6]. 2.1. Equilibrium points and local stability In order to study the qualitative behavior of the solutions of the nonlinear difference Eq. (4), we define the equilibrium points of the dynamic duopoly game as a nonnegative fixed points of the map (4), i.e. the solution of the nonlinear algebraic system q1 ða 2ðb þ c1 Þq1 bq2 Þ ¼ 0 ð5Þ q2 ða bq1 2ðb þ c2 Þq2 Þ ¼ 0 which is obtained by setting qi ðt þ 1Þ ¼ qi ðtÞ, i ¼ 1; 2 in (4). The algebraic system (5) as four equilibria:
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E0 ¼ ð0; 0Þ a ;0 E1 ¼ 2ðb þ c1 Þ a E2 ¼ 0; 2ðb þ c2 Þ
ð6Þ
E ¼ ðq 1 ; q 2 Þ where E0 , E1 and E2 are called the boundary equilibria [6], and the equilibrium, E is the unique Nash equilibrium with components aðb þ 2c2 Þ 3b2 þ 4bðc1 þ c2 Þ þ 4c1 c2 aðb þ 2c1 Þ q 2 ¼ 2 3b þ 4bðc1 þ c2 Þ þ 4c1 c2
q 1 ¼
ð7Þ
The study of the local stability of the fixed points of the two-dimensional system (4) depends on the eigenvalues of the Jacobian matrix of (4). The Jacobian matrix J at the solution ðq1 ; q2 Þ has the form Jðq1 ; q2 Þ ¼
1 þ v1 ða 4ðb þ c1 Þq1 bq2 Þ bv2 q2
bv1 q1 1 þ v2 ða bq1 4ðb þ c2 Þq2 Þ
Lemma 1. The boundary equilibria E0 , E1 and E2 of the system (4) are unstable equilibrium points. Proof. In order to prove this result, we find the eigenvalues of the Jacobian matrix J at each boundary equilibria E0 , E1 and E2 . At E0 the Jacobian matrix J is the diagonal matrix 0 1 þ av1 J ð0; 0Þ ¼ 0 1 þ av2 whose eigenvalues are k1 ¼ 1 þ av1 and k2 ¼ 1 þ av2 . Since v1 , v2 and a are positive constants then ki > 1, i ¼ 1; 2. Hence the equilibrium point E0 is unstable node. Also at E1 the Jacobian matrix J becomes a triangular matrix 2 3 v1 ab 1 2v a 1 6 7 a 2ðb þ c1 Þ 7 J ;0 ¼ 6 4 5 b þ 2c1 2ðb þ c1 Þ 0 1 þ v2 a 2ðb þ c1 Þ
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whose eigenvalues are given by the diagonal entries. They are b þ 2c1 k1 ¼ 1 2v1 a and k2 ¼ 1 þ v2 a 2ðb þ c1 Þ Since a, b, c1 , v1 and v2 are positive, then the eigenvalue k2 is greater than 1 and k1 is less than 1. Therefore E1 is a saddle point (unstable). Similarly we can prove that E2 is also a saddle point. Hence the proof is completed. 2.1.1. Local stability of Nash equilibrium point In order to study the local stability of Nash equilibrium E ¼ ðq 1 ; q 2 Þ of the system (4) where aðb þ 2c2 Þ 3b2 þ 4bðc1 þ c2 Þ þ 4c1 c2 aðb þ 2c1 Þ q 2 ¼ 2 3b þ 4bðc1 þ c2 Þ þ 4c1 c2 q 1 ¼
We estimate the Jacobian matrix J at E , which is 1 2v1 ðb þ c1 Þq 1 v1 bq 1
J ðq1 ; q2 Þ ¼ v2 bq 2 1 2v2 ðb þ c2 Þq 2 The characteristic equation is pðkÞ ¼ k2 Tr k þ Det ¼ 0 where Tr is the trace and Det is the determinant of the Jacobian matrix J ðq 1 ; q 2 Þ. Tr ¼ 2 2v1 ðb þ c1 Þq 1 2v2 ðb þ c2 Þq 2 Det ¼ 1 2v1 ðb þ c1 Þq 1 2v2 ðb þ c2 Þq 2 þ v1 v2 ð3b2 þ 8bðc1 þ c2 Þ þ 4c1 c2 Þq 1 q 2 Since 2
Tr2 4Det ¼ ½2v1 ðb þ c1 Þq 1 2v2 ðb þ c2 Þq 2 þ 4v1 v2 q 1 q 2 i:e: Tr2 4Det > 0 This means that the eigenvalues of Nash equilibrium are real. The local stability conditions of Nash equilibrium are given by recalling JuryÕs conditions [8] which are the necessary and sufficient conditions for jki j < 1, i ¼ 1; 2 ii(i) Det ¼ 1 2v1 ðb þ c1 Þq 1 2v2 ðb þ c2 Þq 2 þ v1 v2 ð3b2 þ 8bðc1 þ c2 Þ þ 4c1 c2 Þ q 1 q 2 < 1 This condition becomes
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v1 v2 ð3b2 þ 8bðc1 þ c2 Þ þ 4c1 c2 Þq 1 q 2 < 2v1 ðb þ c1 Þq 1 þ 2v2 ðb þ c2 Þq 2 i(ii) 1 Tr þ Det ¼ v1 v2 ð3b2 þ 8bðc1 þ c2 Þþ 4c1 c2 Þq 1 q 2 > 0 (iii) 1 þ Tr þ Det ¼ 4 4v1 ðb þ c1 Þq 1 4v2 ðb þ c2 Þq 2 þ v1 v2 ð3b2 þ 8bðc1 þ c2 Þ þ 4c1 c2 Þq 1 q 2 > 0: This condition becomes 4v1 ðb þ c1 Þq 1 þ 4v2 ðb þ c2 Þq 2 v1 v2 ð3b2 þ 8bðc1 þ c2 Þ þ 4c1 c2 Þq 1 q 2 4 < 0 ð8Þ
The inequalities (i), (ii) and (iii) define a region of stability in the plane of speeds of adjustment ðv1 ; v2 Þ this stability region is bounded by the portion of hyperbola with positive v1 and v2 , whose equation is given by the vanishing of the left-hand side of (8) i.e. 4v1 ðb þ c1 Þq 1 þ 4v2 ðb þ c2 Þq 2 v1 v2 ð3b2 þ 8bðc1 þ c2 Þ þ 4c1 c2 Þq 1 q 2 4 ¼ 0
For the values of ðv1 ; v2 Þ inside the stability region (see Fig. 1), the Nash equilibrium E is stable node and looses its stability through a period doubling bifurcation. This bifurcation curve intersects the axes v1 and v2 in the points A1 and A2 respectively whose coordinates are given by 1 1 ;0 and A2 ¼ 0; A1 ¼ ðb þ c1 Þq 1 ðb þ c2 Þq 2 Since q 1 ¼ q 2 ¼
aðb þ 2c2 Þ 3b2 þ 4bðc1 þ c2 Þ þ 4c1 c2 3b2
aðb þ 2c1 Þ þ 4bðc1 þ c2 Þ þ 4c1 c2
then
3b2 þ 4bðc1 þ c2 Þ þ 4c1 c2 ;0 aðb þ c1 Þðb þ 2c2 Þ 3b2 þ 4bðc1 þ c2 Þ þ 4c1 c2 A2 ¼ 0; aðb þ c2 Þðb þ 2c1 Þ
A1 ¼
and
The stability of E depends on the system parameters. For Example, an increase of the speeds of adjustment with the other parameters held fixed, has a destabilizing effect. In fact, an increase of v1 and v2 starting from a set of parameters which ensures the local stability of Nash equilibrium, can bring the point ðv1 ; v2 Þ out the region of stability, crossing the flip bifurcation curve. Similar arguments apply if the parameters v1 , v2 , c1 and c2 are fixed parameters and the parameter a, which represents the maximum price of the good produced, is increased. In this case the region of stability becomes small, and this
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Fig. 1. The region of stability of the Nash equilibrium of duopoly game (4) in v1 v2 -plane. The values of the parameters are a ¼ 7, b ¼ 0:5, c1 ¼ 1 and c2 ¼ 2.
can cause a loss of stability of E . An increase of the cost parameter c1 and c2 held fixed, causes a displacement of the point A1 to the right and of A2 downwards. Indeed, an increase of c2 with c1 held fixed, causes a displacement of the point A1 to the left and of A2 upwards. It is clear that using similar production methods increase stability of Nash equilibrium than the case of using different production methods. Consequently firms using similar production methods have a higher chance of reaching Nash equilibrium than these not using different production methods. Monopoly case: The following one-dimensional map is obtained from (4) with qi ðtÞ ¼ 0 qj ðt þ 1Þ ¼ qj ðtÞf1 þ vj ½a 2ðb þ cj Þqj ðtÞ g;
j 6¼ i; j ¼ 1; 2
ð9Þ
This map is conjugate to the standard logistic map X 0 ¼ lX ð1 X Þ
ð10Þ
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where 0 denotes the unit-time advancement operator. This map (10) is obtained from (9) by the linear transformation 1 þ vj a X ð11Þ qj ¼ 2vj ðb þ cj Þ from which we obtain the relation l ¼ 1 þ vj a This means that the dynamics of (9) can be obtained from the well-known dynamics of (10) [7,9,10]. 2.2. Numerical simulations Numerical experiments are simulated to show the stability and period doubling bifurcation route to chaos for the two-dimensional system (4). In all simulations here, we take a ¼ 7, b ¼ 0:5, c1 ¼ 1 and c2 ¼ 2. If we fix the system parameters and vary one for example v2 , bifurcations and chaos occur, which is detected by using Lyapunov exponents. Fig. 1 shows the region of stability of the Nash equilibrium for the system (4). Crossing the bifurcation curve we obtain period doupling bifurcation and finally chaotic behaviour (see Fig. 2).
Fig. 2. The bifurcation diagram of the solutions q1 and q2 of the system (4) vs v2 and down the maximum Lyapunov exponent Lyp vs v2 (a ¼ 7, b ¼ 0:5, v1 ¼ 0:31, c1 ¼ 1 and c2 ¼ 2Þ.
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Fig. 3. The strange attractor for the two-dimensional system (4) for the values of parameters a ¼ 7, b ¼ 0:5, c ¼ 1, c2 ¼ 2, v1 ¼ 0:3 and v2 ¼ 0:42:
Lyapunov exponents are plotted in Fig. 2 to show bifurcations and chaos were at bifurcation point the maximum Lyapunov exponents is zero while is positive when chaos exist. Fig. 3 shows the strange attractor of the two-dimensional system (4) for v1 ¼ 0:3 and v2 ¼ 0:42. 3. Duopoly Bowley’s model with delayed bounded rationality Discrete time dynamical systems defined by so-called delay equations arise in economic models [11]. The primary reason for the occurrence of such a lagged structure in economic models is that 1. Decisions made by economic agents at time t depend on past observed variables by means of a prediction feedback, and 2. the functional relationships describing the dynamics of the model may not only depend on the current state of the economy but also in a nontrivial manner on past states [12]. It is customary [13] to assume that the expected product of a firm qe ðt þ 1Þ is equal to its previous quantity qðtÞ (niave expectations). However it may make
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more sense to use previous productions i.e. qðt 1Þ; qðt 2Þ; . . . ; qðt T Þ with different weights this point of view has been studied [11,14] in different context. Here both the realistic ideas of bounded rationality and delay are combined. It will be shown that delay increases stability domain. The dynamical system will be qi ðt þ 1Þ ¼ qi ðtÞ þ ai ðqi ðtÞÞ
oPi ðqD Þ oqi
ð12Þ
D D where i ¼ 1; 2; . . . ; n, qD ¼ ðqD 1 ; q2 ; . . . ; qn Þ and
qD i ¼
T X
qi ðt lÞxl
ð13Þ
l¼0
where xl P 0;
T X
xl ¼ 1
l¼0
where the constants xl , l ¼ 0; 1; 2; . . . ; T are the weights given to previous productions. For simplicity we set T ¼ 1, and we consider the case of duopoly (n ¼ 2) in BowleyÕs model where the profit of the ith firm Pi is given in the form Pi ¼ qi ða bQÞ cq2i ;
i ¼ 1; 2
ð14Þ
where a and b are positive constants and QðtÞ ¼ q1 ðtÞ þ q2 ðtÞ. In this case the bounded rationality dynamical model (4) with one step (T ¼ 1) delay is given by q1 ðt þ 1Þ ¼ q1 ðtÞ þ v1 q1 ðtÞfa 2ðb þ cÞ½xq1 ðtÞ þ ð1 xÞq1 ðt 1Þ bðxq2 ðtÞ þ ð1 wÞq2 ðt 1ÞÞg q2 ðt þ 1Þ ¼ q2 ðtÞ þ v2 q2 ðtÞfa 2ðb þ cÞ½xq2 ðtÞ þ ð1 xÞq2 ðt 1Þ bðxq1 ðtÞ þ ð1 wÞq1 ðt 1ÞÞg
ð15Þ
where 0 < x < 1, a, b, v1 , v2 and c are positive constants. To study the stability of (15), we rewrite it as a fourth dimensional system in the form p1 ðt þ 1Þ ¼ q1 ðtÞ p2 ðt þ 1Þ ¼ q2 ðtÞ q1 ðt þ 1Þ ¼ q1 ðtÞ þ v1 q1 ðtÞfa 2ðb þ cÞ½xq1 ðtÞ þ ð1 xÞp1 ðtÞ bðxq2 ðtÞ þ ð1 xÞp2 ðtÞÞg q2 ðt þ 1Þ ¼ q2 ðtÞ þ v2 q2 ðtÞfa 2ðb þ cÞ½xq2 ðtÞ þ ð1 xÞp2 ðtÞ bðxq1 ðtÞ þ ð1 xÞp1 ðtÞÞg
ð16Þ
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The system (16) has four fixed points in the following form: E0 ¼ ð0; 0; 0; 0Þ a a ; 0; E1 ¼ 0; 2ðb þ cÞ 2ðb þ cÞ a a ; 0; ;0 E3 ¼ 2ðb þ cÞ 2ðb þ cÞ a a a a
E ¼ ; ; ; 3b þ 2c 3b þ 2c 3b þ 2c 3b þ 2c where E is the Nash equilibrium of the system (16) and the equilibria E0 , E1 and E3 are the boundary equilibria. 3.1. Stability of boundary equilibria In order to study the stability of the equilibria of the nonlinear system (16) we calculate the Jacobian matrix of the system (16) at the state (p1 , p2 , q1 , q2 ) 2 3 0 0 1 0 6 7 0 0 0 1 7 J ¼6 4 2v1 q1 ðb þ cÞð1 xÞ bð1 xÞv1 q1 a11 bxv1 q1 5 bð1 xÞv2 q2 2ðb þ cÞð1 xÞv2 q2 bxv2 q2 a22 where a11 ¼ 1 þ v1 fa 2ðb þ cÞ½xq1 þ ð1 xÞp1 bðxq2 þ ð1 xÞp2 Þg 2ðb þ cÞxv1 q1 and a22 ¼ 1 þ v2 fa 2ðb þ cÞ½xq2 þ ð1 xÞp2 bðxq1 þ ð1 xÞp1 Þg 2ðb þ cÞxv2 q2 Lemma 2. The zero solution E0 ¼ ð0; 0; 0; 0Þ of the system (16) is unstable node. Proof. The Jacobian matrix of the system (16) at E0 ¼ ð0; 0; 0; 0Þ is given by 2 3 0 0 1 0 60 0 0 1 7 7 J ¼6 4 0 0 1 þ a1 a 0 5 0 0 0 1 þ a2 a There exists two eigenvalues: 1 þ a1 a, and 1 þ a2 a, which are greater than 1. Since the parameters a1 , a2 , and a are positive. Hence the zero solution is unstable node.
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a a Lemma 3. The boundary equilibrium E1 ¼ 0; 2ðbþcÞ ; 0; 2ðbþcÞ of the system (16) is unstable. Proof. The Jacobian matrix J of the system (16) at the boundary equilibrium a a ; 0; 2ðbþcÞ E1 ¼ 0; 2ðbþcÞ is given by 2 6 6 J ¼6 4
0 0
0 0
0
0
bð1 xÞv2 q
2ðb þ cÞð1 xÞv2 q
3 1 0 7 1 0 7 2c þ b 7 0 1 þ a1 a 5 2ðb þ cÞ bxv2 q 1 v2 ax
2cþb The Jacobian matrix J has the eigenvalue k ¼ 1 þ a1 fa 2ðbþcÞ g > 1 which is sufficient for unstability of the boundary equilibrium E1 . Similarly we can prove that the boundary equilibrium E2 is unstable and this completes the proof of the lemma.
Fig. 4. The region of stability of the Nash equilibrium of duopoly BowleyÕs model (15) with nondelayed bounded rationality. The values of the parameters are a ¼ 7, b ¼ 0:5, c ¼ 1 and x ¼ 1.
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Fig. 5. The region of stability of Nash equilibrium of duopoly BowleyÕs model (15) with delayed bounded rationality. The values of the parameters are a ¼ 7, b ¼ 0:5, c ¼ 1 and x ¼ 0:7.
3.2. Stability of Nash equilibrium E The coefficients a11 and a22 of the Jacobian matrix J at Nash equilibrium of the system (16) are given by a11 ¼ 1 2ðb þ cÞxa1 q1
and
a22 ¼ 1 2ðb þ cÞxa2 q2
Then the Jacobian matrix J at Nash equilibrium E is given by 2
0 6 0 6 J ¼4 2ðb þ cÞð1 xÞv1 q bð1 xÞv2 q
0 0 bð1 xÞv1 q 2ðb þ cÞð1 xÞv2 q
1 0 1 2ðb þ cÞxv1 q bxv2 q
3 0 7 1 7 5 bxv1 q 1 2ðb þ cÞxv2 q
where q ¼ a=ð3b þ 2cÞ. The characteristic polynomial pðkÞ of the Jacobian matrix J has the form
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Fig. 6. The bifurcation diagram of the states q1 and q2 of the system (16) vs. v2 and down the maximum Lyapunov exponent Lyp vs. v2 (the values of the parameters are a ¼ 7, b ¼ 0:5, c ¼ 1 and v1 ¼ 0:33 and x ¼ 1).
pðkÞ ¼ k4 þ l1 k3 þ l2 k2 þ l3 k þ l4 ¼ 0 where l1 ¼ xðb þ aÞ 2, l2 ¼ 1 þ ðb þ aÞð1 2xÞ þ x2 ðab dcÞ, l3 ¼ 2ðx2 2 xÞðdc abÞ ð1 xÞða þ bÞ, and l4 ¼ ðx 1Þ ðab dcÞ such that a ¼ 2v1 qðb þ cÞ, b ¼ 2v2 qðb þ cÞ, d ¼ v1 qb and c ¼ v2 qb. If b1 ¼ 1 l24 , b2 ¼ l1 l4 l3 , b3 ¼ l2 l4 l2 , b4 ¼ l3 l4 l1 , c1 ¼ b24 b21 , c2 ¼ b4 b3 b1 b2 and c3 ¼ b4 b2 b1 b3 , then recalling Jury test [8], for stability of Nash equilibrium, we get the necessary and sufficient conditions for jki j < 1, i ¼ 1; 2; 3; 4 ii(i) i(ii) (iii) (iv) (v)
1 þ l1 þ l2 þ l3 þ l4 > 0 1 l1 l2 l3 þ l4 > 0 jl4 j < 1 jb4 j < jb1 j, and jc1 j < jc3 j
Else period doubling bifurcation to chaos occurs [15].
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Fig. 7. The bifurcation diagram of the states q1 and q2 of the system (16) vs. v2 and down Lyp vs. v2 at the values a ¼ 7, b ¼ 0:5, c ¼ 1 and v1 ¼ 0:33 and x ¼ 0:7.
It is clear that delay (0 < x < 1) increases stability than the case of nondelay (x ¼ 1) see Figs. 4–7. Consequently firms using delay have a higher chance of reaching Nash equilibrium than these not using delay. 3.2.1. Numerical simulation Numerical simulations are carried to show the different two cases of weight factor. In these numerical experiments the parameters a, b and c are fixed (a ¼ 7, b ¼ 0:5, and c ¼ 1). The first case we take x ¼ 1, the region of stability of the Nash equilibrium of the system (15) is shown in Fig. 4. Fig. 5 shows the stability region of the second case when delay is introduced to the duopoly model x ¼ 0:7. It is clear from Figs. 4 and 5 that the stability region is increased. In order to detect chaos and bifurcations, bifurcation diagrams and Lyapunov exponents are drawn in Figs. 6 and 7. We see that duopoly BowleyÕs model with bounded rationality is stable for v2 < 0:08 and v1 ¼ 0:33, but the BowleyÕs model with delayed bounded rationality (x ¼ 0:7) is stable for v2 < 0:4 and v ¼ 0:33 (see Figs. 6 and 7).
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4. Conclusion The dynamic of duopoly BowleyÕs model with bounded rationality is analyzed, where players use different production methods. Also the dynamic of nonlinear duopoly BowleyÕs model with delayed bounded rationality is analyzed, where players use similar methods of production. This study shows that the stability region of Nash equilibrium of duopoly BowleyÕs model with bounded rationality is decreased when players use different production methods. The stability region of Nash equilibrium of duopoly BowleyÕs model with delayed bounded rationality is larger than the case when players use nondelayed bounded rationality. Thus we see that firms using delay have a higher chance of reaching Nash equilibrium than those not using delay.
References [1] H.N. Agiza, A.S. Hegazi, A.A. Elsadany, The dynamics of BowleyÕs Model with bounded rationality, Chaos, Solitons & Fractals 12 (2001) 1705–1717. [2] J. Fredman, Oligopoly theory, in: K.J. Arrow, M. Intriligator (Eds.), Handbook of Mathematical Economics, vol. 11, North-Holland, Amsterdam, 1982. [3] T. Puu, Chaos in duopoly pricing, Chaos, Soliton & Fractals 1 (1991) 573–581. [4] T. Puu, Complex dynamics with three oligopolists, Chaos, Soliton & Fractals 7 (1996) 2075– 2081. [5] M. Kopel, Simple and complex adjustment dynamics in Cournot duopoly model, Chaos, Soliton & Fractals 7 (12) (1996) 2031–2045. [6] G.I. Bischi, A. Naimzada, Global analysis of a dynamic duopoly game with bounded rationality, in: Advanced in Dynamics Games and Application, vol. 5, Birkhauser, Basel, 1999, Chapter 20. [7] G.I. Bischi, L. Stefanini, L. Gardini, Synchronization intermittency and critical curves in a duopoly game, Mathematical and Computer Simulation 44 (1998) 559–585. [8] L. Edelstein-Kashet, Mathematical Models in Biology, Random House, New York, 1992. [9] R.L. Deveney, An Introduction to Chaotic Dynamical Systems, Addison-Wesely, Reading MA, 1989. [10] D. Glulick, Encounters with Chaos, McGraw-Hill, New York, 1992. [11] A. Stiefenhofer, Stability and predictability of dynamical economic systems with memory, discussion paper 331, Department of Economics, Bielefeled University, 1997. [12] V. Bohm, J. Wenzelbarger, Cycles and local stability in discrete dynamical systems with delay structure, discussion paper 320, Department of Economics, Bielefeled University, 1996. [13] R. Gibbons, A Primer in Game Theory, Simon & Schuster, New York, 1992. [14] E. Ahmed, H.N. Agiza, S.Z Hassan, On modification of PuuÕs dynamical duopoly, Chaos, Solitons & Fractals 11 (7) (2000) 1025–1028. [15] H.-w. Lorenz, Nonlinear dynamical economics and chaotic motion, in: Lectures Notes in Economics and Mathematical Systems, vol. 334, Springer-Verlag, Berlin, 1993.
Analysis of a duopoly game with delayed bounded rationality M.T. Yassen *, H.N. Agiza Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Abstract A dynamic of Cournot duopoly game is analyzed, where players use different production methods and choose their quantities with bounded rationality. A dynamic of nonlinear Cournot duopoly game is analyzed, where players choose quantities with delayed bounded rationality and similar methods of production. The equilibria of the corresponding discrete dynamical systems are investigated. The stability conditions of Nash equilibrium under a local adjustment process are studied. The stability of Nash equilibrium, as some parameters of the model are varied, gives rise to complex dynamics such as cycles of higher order and chaos. We show that firms using delayed bounded rationality have a higher chance of reaching Nash equilibrium. Numerical simulations are used to show bifurcations diagrams, stability regions and chaos. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Duopoly game; Delayed bounded rationality; Nash equilibrium; Stability; Bifurcation; Chaos
1. Introduction Recently the complex dynamics of bounded rationality duopoly game of BowleyÕs model have been studied [1]. In the BowleyÕs model [2] the demand function has the linear form p ¼ f ðQÞ ¼ a bQ *
Corresponding author. E-mail addresses: [email protected] (M.T. Yassen), [email protected] (H.N. Agiza). 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 2 ) 0 0 1 4 3 - 1
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where a and b are the demand parameters. The constant a is the maximum price in the market and Q is the total quantity in the market. The dynamics of the duopoly model when firms use similar production methods was investigated in [1]. In this work we study the complex dynamics of duopoly BowleyÕs model when firms use different production method and the cost function is proposed in the nonlinear form Ci ðqi Þ ¼ ci q2i where ci , i ¼ 1; 2 is a positive shift parameter of the cost function. Generally the complex dynamics of duoploy models was studied with different assumptions to the demand function (see for examples [3–6]). In this paper, we study duopoly game which describes a market with two-players producing homogeneous goods, updates their production strategies in order to maximize their profits. Each player think with bounded rationality, adjust his output according to the expected marginal profit, therefore the decision of each player depends on local information about his output. In this Cournot game each player tries to maximize his profit according to local informations of his input. The plan of the paper is as follows. In Section 2, the description of the duopoly BowleyÕs model with bounded rationality. The existence and local stability of the equilibrium points are studied. The stability region of Nash equilibrium is determined in the plane of speeds adjustment (v1 v2 -plane) and also complex behavior such as cycles of higher order and chaos is studied. Numerical simulation is presented to show such behaviour. In Section 3, duopoly BowleyÕs model with delayed bounded rationality is studied when firms use similar production methods. The existence and local stability of the boundary equilibria and Nash equilibrium are investigated. The stability region of Nash equilibrium is determined in the plane of speeds adjustment (v1 v2 -plane). Numerical simulations are presented. Section 4 is the conclusion.
2. Duopoly model We consider two firms, labelled by i ¼ 1; 2 producing the same good for sale in the market. Production decisions of both firms occur at discrete time periods t ¼ 0; 1; 2 . . . Let qi ðtÞ represent the output of ith firm during period t, with a production cost function Ci ðqi Þ. The price prevailing in period t is determined by the total supply QðtÞ ¼ q1 ðtÞ þ q2 ðtÞ through a demand function p ¼ f ðQÞ In this model the demand function is assumed linear [7], which has the form f ðQÞ ¼ a bQ
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where a and b are positive constants. Also we assume that the cost function has the nonlinear form Ci ðqi Þ ¼ ci q2i ;
i ¼ 1; 2
ð1Þ
where the positive parameters ci , i ¼ 1; 2 are positive shift parameters to the cost function of the firm i, i ¼ 1; 2 respectively. With these assumptions the single-period profit of the ith firm is given by Pi ðq1 ; q2 Þ ¼ qi ða bQÞ ci q2i ;
i ¼ 1; 2
ð2Þ
and the marginal profit of the ith firm at the point (q1 , q2 ) of the strategy space is Ui ¼
oPi ¼ a 2ðb þ ci Þqi bqj ; oqi
i ¼ 1; 2; i 6¼ j
ð3Þ
The duopoly method with bounded rational players [6] for our assumptions has the following two dimensional nonlinear map T ðq1 ; q2 Þ ! ðq01 ; q02 Þ which is defined by oPi ; i ¼ 1; 2 T : q0i ¼ qi þ ai ðqi Þ oqi where ai ðqi Þ is a positive function which gives the extend of the production variation of the ith firm according to his computed profit signal Ui and prime (0 ) denotes the unit-time advancement operator, that is, if the right-hand side variables are productions of period t then the left-hand side ones represent productions of period (t þ 1). We take the function ai ðqi Þ in the linear form ai ðqi Þ ¼ vi qi . From (3) the two-dimensional nonlinear map T ðq1 ; q2 Þ ! ðq01 ; q02 Þ is defined by the following two nonlinear difference equations: q1 ðt þ 1Þ ¼ q1 ðtÞf1 þ v1 ½a 2ðb þ c1 Þq1 ðtÞ bq2 ðtÞ g q2 ðt þ 1Þ ¼ q2 ðtÞf1 þ v2 ½a 2ðb þ c2 Þq2 ðtÞ bq1 ðtÞ g
ð4Þ
where vi , i ¼ 1; 2 is a positive constant which is called the speed of adjustment of ith firm [6]. 2.1. Equilibrium points and local stability In order to study the qualitative behavior of the solutions of the nonlinear difference Eq. (4), we define the equilibrium points of the dynamic duopoly game as a nonnegative fixed points of the map (4), i.e. the solution of the nonlinear algebraic system q1 ða 2ðb þ c1 Þq1 bq2 Þ ¼ 0 ð5Þ q2 ða bq1 2ðb þ c2 Þq2 Þ ¼ 0 which is obtained by setting qi ðt þ 1Þ ¼ qi ðtÞ, i ¼ 1; 2 in (4). The algebraic system (5) as four equilibria:
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E0 ¼ ð0; 0Þ a ;0 E1 ¼ 2ðb þ c1 Þ a E2 ¼ 0; 2ðb þ c2 Þ
ð6Þ
E ¼ ðq 1 ; q 2 Þ where E0 , E1 and E2 are called the boundary equilibria [6], and the equilibrium, E is the unique Nash equilibrium with components aðb þ 2c2 Þ 3b2 þ 4bðc1 þ c2 Þ þ 4c1 c2 aðb þ 2c1 Þ q 2 ¼ 2 3b þ 4bðc1 þ c2 Þ þ 4c1 c2
q 1 ¼
ð7Þ
The study of the local stability of the fixed points of the two-dimensional system (4) depends on the eigenvalues of the Jacobian matrix of (4). The Jacobian matrix J at the solution ðq1 ; q2 Þ has the form Jðq1 ; q2 Þ ¼
1 þ v1 ða 4ðb þ c1 Þq1 bq2 Þ bv2 q2
bv1 q1 1 þ v2 ða bq1 4ðb þ c2 Þq2 Þ
Lemma 1. The boundary equilibria E0 , E1 and E2 of the system (4) are unstable equilibrium points. Proof. In order to prove this result, we find the eigenvalues of the Jacobian matrix J at each boundary equilibria E0 , E1 and E2 . At E0 the Jacobian matrix J is the diagonal matrix 0 1 þ av1 J ð0; 0Þ ¼ 0 1 þ av2 whose eigenvalues are k1 ¼ 1 þ av1 and k2 ¼ 1 þ av2 . Since v1 , v2 and a are positive constants then ki > 1, i ¼ 1; 2. Hence the equilibrium point E0 is unstable node. Also at E1 the Jacobian matrix J becomes a triangular matrix 2 3 v1 ab 1 2v a 1 6 7 a 2ðb þ c1 Þ 7 J ;0 ¼ 6 4 5 b þ 2c1 2ðb þ c1 Þ 0 1 þ v2 a 2ðb þ c1 Þ
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whose eigenvalues are given by the diagonal entries. They are b þ 2c1 k1 ¼ 1 2v1 a and k2 ¼ 1 þ v2 a 2ðb þ c1 Þ Since a, b, c1 , v1 and v2 are positive, then the eigenvalue k2 is greater than 1 and k1 is less than 1. Therefore E1 is a saddle point (unstable). Similarly we can prove that E2 is also a saddle point. Hence the proof is completed. 2.1.1. Local stability of Nash equilibrium point In order to study the local stability of Nash equilibrium E ¼ ðq 1 ; q 2 Þ of the system (4) where aðb þ 2c2 Þ 3b2 þ 4bðc1 þ c2 Þ þ 4c1 c2 aðb þ 2c1 Þ q 2 ¼ 2 3b þ 4bðc1 þ c2 Þ þ 4c1 c2 q 1 ¼
We estimate the Jacobian matrix J at E , which is 1 2v1 ðb þ c1 Þq 1 v1 bq 1
J ðq1 ; q2 Þ ¼ v2 bq 2 1 2v2 ðb þ c2 Þq 2 The characteristic equation is pðkÞ ¼ k2 Tr k þ Det ¼ 0 where Tr is the trace and Det is the determinant of the Jacobian matrix J ðq 1 ; q 2 Þ. Tr ¼ 2 2v1 ðb þ c1 Þq 1 2v2 ðb þ c2 Þq 2 Det ¼ 1 2v1 ðb þ c1 Þq 1 2v2 ðb þ c2 Þq 2 þ v1 v2 ð3b2 þ 8bðc1 þ c2 Þ þ 4c1 c2 Þq 1 q 2 Since 2
Tr2 4Det ¼ ½2v1 ðb þ c1 Þq 1 2v2 ðb þ c2 Þq 2 þ 4v1 v2 q 1 q 2 i:e: Tr2 4Det > 0 This means that the eigenvalues of Nash equilibrium are real. The local stability conditions of Nash equilibrium are given by recalling JuryÕs conditions [8] which are the necessary and sufficient conditions for jki j < 1, i ¼ 1; 2 ii(i) Det ¼ 1 2v1 ðb þ c1 Þq 1 2v2 ðb þ c2 Þq 2 þ v1 v2 ð3b2 þ 8bðc1 þ c2 Þ þ 4c1 c2 Þ q 1 q 2 < 1 This condition becomes
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v1 v2 ð3b2 þ 8bðc1 þ c2 Þ þ 4c1 c2 Þq 1 q 2 < 2v1 ðb þ c1 Þq 1 þ 2v2 ðb þ c2 Þq 2 i(ii) 1 Tr þ Det ¼ v1 v2 ð3b2 þ 8bðc1 þ c2 Þþ 4c1 c2 Þq 1 q 2 > 0 (iii) 1 þ Tr þ Det ¼ 4 4v1 ðb þ c1 Þq 1 4v2 ðb þ c2 Þq 2 þ v1 v2 ð3b2 þ 8bðc1 þ c2 Þ þ 4c1 c2 Þq 1 q 2 > 0: This condition becomes 4v1 ðb þ c1 Þq 1 þ 4v2 ðb þ c2 Þq 2 v1 v2 ð3b2 þ 8bðc1 þ c2 Þ þ 4c1 c2 Þq 1 q 2 4 < 0 ð8Þ
The inequalities (i), (ii) and (iii) define a region of stability in the plane of speeds of adjustment ðv1 ; v2 Þ this stability region is bounded by the portion of hyperbola with positive v1 and v2 , whose equation is given by the vanishing of the left-hand side of (8) i.e. 4v1 ðb þ c1 Þq 1 þ 4v2 ðb þ c2 Þq 2 v1 v2 ð3b2 þ 8bðc1 þ c2 Þ þ 4c1 c2 Þq 1 q 2 4 ¼ 0
For the values of ðv1 ; v2 Þ inside the stability region (see Fig. 1), the Nash equilibrium E is stable node and looses its stability through a period doubling bifurcation. This bifurcation curve intersects the axes v1 and v2 in the points A1 and A2 respectively whose coordinates are given by 1 1 ;0 and A2 ¼ 0; A1 ¼ ðb þ c1 Þq 1 ðb þ c2 Þq 2 Since q 1 ¼ q 2 ¼
aðb þ 2c2 Þ 3b2 þ 4bðc1 þ c2 Þ þ 4c1 c2 3b2
aðb þ 2c1 Þ þ 4bðc1 þ c2 Þ þ 4c1 c2
then
3b2 þ 4bðc1 þ c2 Þ þ 4c1 c2 ;0 aðb þ c1 Þðb þ 2c2 Þ 3b2 þ 4bðc1 þ c2 Þ þ 4c1 c2 A2 ¼ 0; aðb þ c2 Þðb þ 2c1 Þ
A1 ¼
and
The stability of E depends on the system parameters. For Example, an increase of the speeds of adjustment with the other parameters held fixed, has a destabilizing effect. In fact, an increase of v1 and v2 starting from a set of parameters which ensures the local stability of Nash equilibrium, can bring the point ðv1 ; v2 Þ out the region of stability, crossing the flip bifurcation curve. Similar arguments apply if the parameters v1 , v2 , c1 and c2 are fixed parameters and the parameter a, which represents the maximum price of the good produced, is increased. In this case the region of stability becomes small, and this
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Fig. 1. The region of stability of the Nash equilibrium of duopoly game (4) in v1 v2 -plane. The values of the parameters are a ¼ 7, b ¼ 0:5, c1 ¼ 1 and c2 ¼ 2.
can cause a loss of stability of E . An increase of the cost parameter c1 and c2 held fixed, causes a displacement of the point A1 to the right and of A2 downwards. Indeed, an increase of c2 with c1 held fixed, causes a displacement of the point A1 to the left and of A2 upwards. It is clear that using similar production methods increase stability of Nash equilibrium than the case of using different production methods. Consequently firms using similar production methods have a higher chance of reaching Nash equilibrium than these not using different production methods. Monopoly case: The following one-dimensional map is obtained from (4) with qi ðtÞ ¼ 0 qj ðt þ 1Þ ¼ qj ðtÞf1 þ vj ½a 2ðb þ cj Þqj ðtÞ g;
j 6¼ i; j ¼ 1; 2
ð9Þ
This map is conjugate to the standard logistic map X 0 ¼ lX ð1 X Þ
ð10Þ
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where 0 denotes the unit-time advancement operator. This map (10) is obtained from (9) by the linear transformation 1 þ vj a X ð11Þ qj ¼ 2vj ðb þ cj Þ from which we obtain the relation l ¼ 1 þ vj a This means that the dynamics of (9) can be obtained from the well-known dynamics of (10) [7,9,10]. 2.2. Numerical simulations Numerical experiments are simulated to show the stability and period doubling bifurcation route to chaos for the two-dimensional system (4). In all simulations here, we take a ¼ 7, b ¼ 0:5, c1 ¼ 1 and c2 ¼ 2. If we fix the system parameters and vary one for example v2 , bifurcations and chaos occur, which is detected by using Lyapunov exponents. Fig. 1 shows the region of stability of the Nash equilibrium for the system (4). Crossing the bifurcation curve we obtain period doupling bifurcation and finally chaotic behaviour (see Fig. 2).
Fig. 2. The bifurcation diagram of the solutions q1 and q2 of the system (4) vs v2 and down the maximum Lyapunov exponent Lyp vs v2 (a ¼ 7, b ¼ 0:5, v1 ¼ 0:31, c1 ¼ 1 and c2 ¼ 2Þ.
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Fig. 3. The strange attractor for the two-dimensional system (4) for the values of parameters a ¼ 7, b ¼ 0:5, c ¼ 1, c2 ¼ 2, v1 ¼ 0:3 and v2 ¼ 0:42:
Lyapunov exponents are plotted in Fig. 2 to show bifurcations and chaos were at bifurcation point the maximum Lyapunov exponents is zero while is positive when chaos exist. Fig. 3 shows the strange attractor of the two-dimensional system (4) for v1 ¼ 0:3 and v2 ¼ 0:42. 3. Duopoly Bowley’s model with delayed bounded rationality Discrete time dynamical systems defined by so-called delay equations arise in economic models [11]. The primary reason for the occurrence of such a lagged structure in economic models is that 1. Decisions made by economic agents at time t depend on past observed variables by means of a prediction feedback, and 2. the functional relationships describing the dynamics of the model may not only depend on the current state of the economy but also in a nontrivial manner on past states [12]. It is customary [13] to assume that the expected product of a firm qe ðt þ 1Þ is equal to its previous quantity qðtÞ (niave expectations). However it may make
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more sense to use previous productions i.e. qðt 1Þ; qðt 2Þ; . . . ; qðt T Þ with different weights this point of view has been studied [11,14] in different context. Here both the realistic ideas of bounded rationality and delay are combined. It will be shown that delay increases stability domain. The dynamical system will be qi ðt þ 1Þ ¼ qi ðtÞ þ ai ðqi ðtÞÞ
oPi ðqD Þ oqi
ð12Þ
D D where i ¼ 1; 2; . . . ; n, qD ¼ ðqD 1 ; q2 ; . . . ; qn Þ and
qD i ¼
T X
qi ðt lÞxl
ð13Þ
l¼0
where xl P 0;
T X
xl ¼ 1
l¼0
where the constants xl , l ¼ 0; 1; 2; . . . ; T are the weights given to previous productions. For simplicity we set T ¼ 1, and we consider the case of duopoly (n ¼ 2) in BowleyÕs model where the profit of the ith firm Pi is given in the form Pi ¼ qi ða bQÞ cq2i ;
i ¼ 1; 2
ð14Þ
where a and b are positive constants and QðtÞ ¼ q1 ðtÞ þ q2 ðtÞ. In this case the bounded rationality dynamical model (4) with one step (T ¼ 1) delay is given by q1 ðt þ 1Þ ¼ q1 ðtÞ þ v1 q1 ðtÞfa 2ðb þ cÞ½xq1 ðtÞ þ ð1 xÞq1 ðt 1Þ bðxq2 ðtÞ þ ð1 wÞq2 ðt 1ÞÞg q2 ðt þ 1Þ ¼ q2 ðtÞ þ v2 q2 ðtÞfa 2ðb þ cÞ½xq2 ðtÞ þ ð1 xÞq2 ðt 1Þ bðxq1 ðtÞ þ ð1 wÞq1 ðt 1ÞÞg
ð15Þ
where 0 < x < 1, a, b, v1 , v2 and c are positive constants. To study the stability of (15), we rewrite it as a fourth dimensional system in the form p1 ðt þ 1Þ ¼ q1 ðtÞ p2 ðt þ 1Þ ¼ q2 ðtÞ q1 ðt þ 1Þ ¼ q1 ðtÞ þ v1 q1 ðtÞfa 2ðb þ cÞ½xq1 ðtÞ þ ð1 xÞp1 ðtÞ bðxq2 ðtÞ þ ð1 xÞp2 ðtÞÞg q2 ðt þ 1Þ ¼ q2 ðtÞ þ v2 q2 ðtÞfa 2ðb þ cÞ½xq2 ðtÞ þ ð1 xÞp2 ðtÞ bðxq1 ðtÞ þ ð1 xÞp1 ðtÞÞg
ð16Þ
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The system (16) has four fixed points in the following form: E0 ¼ ð0; 0; 0; 0Þ a a ; 0; E1 ¼ 0; 2ðb þ cÞ 2ðb þ cÞ a a ; 0; ;0 E3 ¼ 2ðb þ cÞ 2ðb þ cÞ a a a a
E ¼ ; ; ; 3b þ 2c 3b þ 2c 3b þ 2c 3b þ 2c where E is the Nash equilibrium of the system (16) and the equilibria E0 , E1 and E3 are the boundary equilibria. 3.1. Stability of boundary equilibria In order to study the stability of the equilibria of the nonlinear system (16) we calculate the Jacobian matrix of the system (16) at the state (p1 , p2 , q1 , q2 ) 2 3 0 0 1 0 6 7 0 0 0 1 7 J ¼6 4 2v1 q1 ðb þ cÞð1 xÞ bð1 xÞv1 q1 a11 bxv1 q1 5 bð1 xÞv2 q2 2ðb þ cÞð1 xÞv2 q2 bxv2 q2 a22 where a11 ¼ 1 þ v1 fa 2ðb þ cÞ½xq1 þ ð1 xÞp1 bðxq2 þ ð1 xÞp2 Þg 2ðb þ cÞxv1 q1 and a22 ¼ 1 þ v2 fa 2ðb þ cÞ½xq2 þ ð1 xÞp2 bðxq1 þ ð1 xÞp1 Þg 2ðb þ cÞxv2 q2 Lemma 2. The zero solution E0 ¼ ð0; 0; 0; 0Þ of the system (16) is unstable node. Proof. The Jacobian matrix of the system (16) at E0 ¼ ð0; 0; 0; 0Þ is given by 2 3 0 0 1 0 60 0 0 1 7 7 J ¼6 4 0 0 1 þ a1 a 0 5 0 0 0 1 þ a2 a There exists two eigenvalues: 1 þ a1 a, and 1 þ a2 a, which are greater than 1. Since the parameters a1 , a2 , and a are positive. Hence the zero solution is unstable node.
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a a Lemma 3. The boundary equilibrium E1 ¼ 0; 2ðbþcÞ ; 0; 2ðbþcÞ of the system (16) is unstable. Proof. The Jacobian matrix J of the system (16) at the boundary equilibrium a a ; 0; 2ðbþcÞ E1 ¼ 0; 2ðbþcÞ is given by 2 6 6 J ¼6 4
0 0
0 0
0
0
bð1 xÞv2 q
2ðb þ cÞð1 xÞv2 q
3 1 0 7 1 0 7 2c þ b 7 0 1 þ a1 a 5 2ðb þ cÞ bxv2 q 1 v2 ax
2cþb The Jacobian matrix J has the eigenvalue k ¼ 1 þ a1 fa 2ðbþcÞ g > 1 which is sufficient for unstability of the boundary equilibrium E1 . Similarly we can prove that the boundary equilibrium E2 is unstable and this completes the proof of the lemma.
Fig. 4. The region of stability of the Nash equilibrium of duopoly BowleyÕs model (15) with nondelayed bounded rationality. The values of the parameters are a ¼ 7, b ¼ 0:5, c ¼ 1 and x ¼ 1.
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399
Fig. 5. The region of stability of Nash equilibrium of duopoly BowleyÕs model (15) with delayed bounded rationality. The values of the parameters are a ¼ 7, b ¼ 0:5, c ¼ 1 and x ¼ 0:7.
3.2. Stability of Nash equilibrium E The coefficients a11 and a22 of the Jacobian matrix J at Nash equilibrium of the system (16) are given by a11 ¼ 1 2ðb þ cÞxa1 q1
and
a22 ¼ 1 2ðb þ cÞxa2 q2
Then the Jacobian matrix J at Nash equilibrium E is given by 2
0 6 0 6 J ¼4 2ðb þ cÞð1 xÞv1 q bð1 xÞv2 q
0 0 bð1 xÞv1 q 2ðb þ cÞð1 xÞv2 q
1 0 1 2ðb þ cÞxv1 q bxv2 q
3 0 7 1 7 5 bxv1 q 1 2ðb þ cÞxv2 q
where q ¼ a=ð3b þ 2cÞ. The characteristic polynomial pðkÞ of the Jacobian matrix J has the form
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Fig. 6. The bifurcation diagram of the states q1 and q2 of the system (16) vs. v2 and down the maximum Lyapunov exponent Lyp vs. v2 (the values of the parameters are a ¼ 7, b ¼ 0:5, c ¼ 1 and v1 ¼ 0:33 and x ¼ 1).
pðkÞ ¼ k4 þ l1 k3 þ l2 k2 þ l3 k þ l4 ¼ 0 where l1 ¼ xðb þ aÞ 2, l2 ¼ 1 þ ðb þ aÞð1 2xÞ þ x2 ðab dcÞ, l3 ¼ 2ðx2 2 xÞðdc abÞ ð1 xÞða þ bÞ, and l4 ¼ ðx 1Þ ðab dcÞ such that a ¼ 2v1 qðb þ cÞ, b ¼ 2v2 qðb þ cÞ, d ¼ v1 qb and c ¼ v2 qb. If b1 ¼ 1 l24 , b2 ¼ l1 l4 l3 , b3 ¼ l2 l4 l2 , b4 ¼ l3 l4 l1 , c1 ¼ b24 b21 , c2 ¼ b4 b3 b1 b2 and c3 ¼ b4 b2 b1 b3 , then recalling Jury test [8], for stability of Nash equilibrium, we get the necessary and sufficient conditions for jki j < 1, i ¼ 1; 2; 3; 4 ii(i) i(ii) (iii) (iv) (v)
1 þ l1 þ l2 þ l3 þ l4 > 0 1 l1 l2 l3 þ l4 > 0 jl4 j < 1 jb4 j < jb1 j, and jc1 j < jc3 j
Else period doubling bifurcation to chaos occurs [15].
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Fig. 7. The bifurcation diagram of the states q1 and q2 of the system (16) vs. v2 and down Lyp vs. v2 at the values a ¼ 7, b ¼ 0:5, c ¼ 1 and v1 ¼ 0:33 and x ¼ 0:7.
It is clear that delay (0 < x < 1) increases stability than the case of nondelay (x ¼ 1) see Figs. 4–7. Consequently firms using delay have a higher chance of reaching Nash equilibrium than these not using delay. 3.2.1. Numerical simulation Numerical simulations are carried to show the different two cases of weight factor. In these numerical experiments the parameters a, b and c are fixed (a ¼ 7, b ¼ 0:5, and c ¼ 1). The first case we take x ¼ 1, the region of stability of the Nash equilibrium of the system (15) is shown in Fig. 4. Fig. 5 shows the stability region of the second case when delay is introduced to the duopoly model x ¼ 0:7. It is clear from Figs. 4 and 5 that the stability region is increased. In order to detect chaos and bifurcations, bifurcation diagrams and Lyapunov exponents are drawn in Figs. 6 and 7. We see that duopoly BowleyÕs model with bounded rationality is stable for v2 < 0:08 and v1 ¼ 0:33, but the BowleyÕs model with delayed bounded rationality (x ¼ 0:7) is stable for v2 < 0:4 and v ¼ 0:33 (see Figs. 6 and 7).
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4. Conclusion The dynamic of duopoly BowleyÕs model with bounded rationality is analyzed, where players use different production methods. Also the dynamic of nonlinear duopoly BowleyÕs model with delayed bounded rationality is analyzed, where players use similar methods of production. This study shows that the stability region of Nash equilibrium of duopoly BowleyÕs model with bounded rationality is decreased when players use different production methods. The stability region of Nash equilibrium of duopoly BowleyÕs model with delayed bounded rationality is larger than the case when players use nondelayed bounded rationality. Thus we see that firms using delay have a higher chance of reaching Nash equilibrium than those not using delay.
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