# Complexity Analysis of a Master-Slave Oligopoly Model and Chaos Control

Authors: Junhai Ma; Fang Zhang; Yanyan He
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We establish a master-slave oligopoly game model with an upstream monopoly whose output is considered and two downstream oligopolies whose prices are considered. The existence and the local stable region of the Nash equilibrium point are investigated. The complex dynamic properties, such as bifurcation and chaos, are analyzed using bifurcation diagrams, the largest Lyapunov exponent diagrams, and the strange attractor graph. We further analyze the long-run average profit of the three firms and find that they are all optimal in the stable region. In addition, delay feedback control method and limiter control method are used in nondelayed model to control chaos. Furthermore, a delayed master-slave oligopoly game model is considered, and the three firms’ profit in various conditions is analyzed. We find that suitable delayed parameters are important for eliminating chaos and maximizing the profit of the players.


---

## Body

## 1. Introduction

Oligopoly is a market structure between monopoly and perfect competition. It is characterized by a domination of several firms, which completely control trade. These firms manufacture the same or homogeneous products. They have to consider both the market demand, that is, the behavior of consumers, and the strategies of their competitors; that is, they form expectations concerning how their rivals will act. The most widely used and simultaneously the first formal model of oligopoly market were proposed by Cournot. Cournot model assumed that each company adjusts its quantity of production to that of its rivals and there is no retaliation at all, so that in every step the player perceives the latest move made by the competitors to remain his last. Besides Cournot model, there is another important model: the Bertrand model. The former is under the assumption that producers in an oligopoly decide their policy assuming that other producers will maintain their output at its existing level, while Bertrand model is based on the assumption that producers act on the belief that competitors will maintain their price to maximize profits rather than their output.Works on Cournot or Bertrand model showed that it has an ample dynamical behavior under different expectations [1–6]. A large number of literatures have been published. Puu considered a triopoly Cournot competition model [7]. Agiza and Elsadany studied the dynamics of a Cournot duopoly with heterogeneous players [8]. Zhang and Ma analyzed Bertrand competition model of four oligopolists with heterogeneous expectations [9]. And there are some articles about Cournot-Bertrand competition [10, 11]. For instance, Ma and Pu studied a Cournot-Bertrand duopoly model with bounded rational expectations [12]. The master-slave Cournot or Bertrand model exists in realistic economy. Xin and Chen studied a master-slave duopoly Bertrand game model in the setting that the upstream firm might be the monopolistic supplier of fresh water and the downstream firm might be the monopolistic supplier of pure distilled water [13]. And the upstream monopolistic firm’s output is used as the main factor of production by the downstream monopolistic firm who is a negligible purchaser of the upstream monopolist’s output.In general situations, a system with nonlinear term will not always be stable and sometimes can even be chaotic. However, the appearance of chaos in the economic system is not expected and even is harmful. Thus, people hope to find some methods to control the chaos of economic system. By controlling the chaotic phenomenon occurring on the market, bifurcation and chaos are delayed or eliminated, and the system is stabilized to the Nash equilibrium; that is, the market goes back to orderly competition. In recent years, scholars put forward many control methods to different chaotic systems such as OGY control, adaptive control, and feedback control.As the game player makes decisions at timet that depend on past observed variables by means of a prediction feedback and the functional relationships describing the dynamics of the model may depend on both the current state and the past states, a delayed structure in economics models emerges. Yassen and Agiza considered a delayed duopoly game and got some important conclusions [14]. Since then, many experts and scholars also extensively studied delay oligopoly models, such as Peng et al. [15] and Ma and Wu [16]. These studies focus on the changes of stability domain of system in the case of delay or the bifurcation of system with the parameters changing, but the research to the players’ profits is less.Based on the research of experts and scholars on these models, this paper builds a master-slave oligopoly game model in which upstream monopolistic firm competes in output and two downstream oligopolistic firms compete in price. We use two control methods: the delayed feedback control method and the limiter control method to control chaos. In addition, we study the delayed game model. And we have given thorough discussion on profits of players.The paper is organized as follows. In Section2, a nondelayed oligopoly model with bounded rational expectations is presented. Equilibrium points and stability are analyzed. And numerical simulations which confirm analytical results are given. Two chaos control methods, the delayed feedback control method and the limiter control method, are shown in Section 3. Delayed system is investigated in Section 4. Section 5 gives the conclusion.


## 2. The Nondelayed Master-Slave Model

### 2.1. Model

In an area, there are three firms in the market and firmi produces goods x
i, i
=
1,2
,
3. Firm 1 represents the upstream monopolistic firm and firms 2 and 3 represent two downstream oligopolistic firms. The output and price of firm i are represented, respectively, as q
i and p
i.This model is based on the following assumptions.(1)
The upstream monopolist supplies output (e.g., fresh water) to two downstream firms that compete in a final goods market (e.g., distilled bottled water). The upstream market is monopoly (a single firm: firm 1), and the downstream market is duopoly (two firms: 2 and 3). In addition, the two downstream oligopolistic firms do not cooperate. Then we haveq
1
>
q
2
+
q
3, p
1
<
p
2 and p
1
<
p
3.
(2)
The cost of the two upstream monopolistic firms is a quadratic function. Because firm 1’s price will affect the cost of the downstream firms (assuming 1 unit of fresh water is used to make 1 unit spring water), we assume the two firms’ marginal cost isp
1.
(3)
In the vertically connected market, the three firms make decisions at the same time. The upstream monopolistic firm’s decision variable is its outputq
1 while the two downstream firms’ decision variables are their prices p
2 and p
3.Assume that the inverse demand function of firm 1 and demand functions of firms 2 and 3, respectively, are(1)
p
1
(
t
)
=
a
1
-
b
1
q
1
(
t
)
,
q
2
(
t
)
=
a
2
-
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
,
q
3
(
t
)
=
a
3
-
b
3
p
3
(
t
)
+
d
3
p
2
(
t
)
,
where a
i
,
b
i
>
0
,
i
=
1,2
,
3, and d
i
>
0
,
i
=
2,3. d
2 and d
3 mean the competition parameters between firms 2 and 3.The cost functions of the first firm has the following form [17]:
(2)
C
1
(
t
)
=
c
1
q
1
2
(
t
)
,
where 2
c
1 is marginal cost and c
1 is positive for any t
>
0. Firm 1’s price will affect the cost of the two downstream firms. That is, C
j
=
f
(
p
1
), j
=
2,3. We assume firm j’s cost function is
(3)
C
j
(
t
)
=
p
1
(
t
)
q
j
(
t
)
,
where j
=
2,3.The profit functions of the three firms are(4)
Π
1
(
t
)
=
q
1
(
t
)
p
1
(
t
)
-
C
1
(
t
)
=
q
1
(
t
)
(
a
1
-
b
1
q
1
(
t
)
)
-
c
1
q
1
2
(
t
)
=
-
b
1
q
1
2
(
t
)
-
c
1
q
1
2
(
t
)
+
a
1
q
1
(
t
)
,
Π
2
(
t
)
=
q
2
(
t
)
p
2
(
t
)
-
C
2
(
t
)
=
(
a
2
-
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
)
p
2
(
t
)
-
p
1
(
t
)
q
2
(
t
)
=
a
2
p
2
(
t
)
-
b
2
p
2
2
(
t
)
+
d
2
p
2
(
t
)
p
3
(
t
)
-
(
a
1
-
b
1
q
1
(
t
)
)
(
a
2
-
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
)
,
Π
3
(
t
)
=
q
3
(
t
)
p
3
(
t
)
-
C
3
(
t
)
=
(
a
3
-
b
3
p
3
(
t
)
+
d
3
p
2
(
t
)
)
p
3
(
t
)
-
p
1
(
t
)
q
3
(
t
)
=
a
3
p
3
(
t
)
-
b
3
p
3
2
(
t
)
+
d
3
p
2
(
t
)
p
3
(
t
)
-
(
a
1
-
b
1
q
1
(
t
)
)
(
a
3
-
b
3
p
3
(
t
)
+
d
3
p
2
(
t
)
)
.We assume that the three firms do not have a complete knowledge of the market and the other players. In games, players behave adaptively, following a bounded rational adjustment process and they build decisions on the basis of the expected marginal profit; that is, if the marginal profit is positive (negative), they increase (decrease) their production or price in the next period. The marginal profit functions of the three firms are as follows:(5)
∂
Π
1
(
t
)
∂
q
1
(
t
)
=
-
2
b
1
q
1
(
t
)
-
2
c
1
q
1
(
t
)
+
a
1
,
∂
Π
2
(
t
)
∂
p
2
(
t
)
=
a
2
-
2
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
+
b
2
(
a
1
-
b
1
q
1
(
t
)
)
=
a
2
+
a
1
b
2
-
b
1
b
2
q
1
(
t
)
-
2
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
,
∂
Π
3
(
t
)
∂
p
3
(
t
)
=
a
3
-
2
b
3
p
3
(
t
)
+
d
3
p
2
(
t
)
+
b
3
(
a
1
-
b
1
q
1
(
t
)
)
=
a
3
+
a
1
b
3
-
b
1
b
3
q
1
(
t
)
+
d
3
p
2
(
t
)
-
2
b
3
p
3
(
t
)
.
Under the above assumptions, the dynamic adjustment equation of the master-slave game is
(6)
q
1
(
t
+
1
)
=
q
1
(
t
)
+
α
q
1
(
t
)
∂
Π
1
(
t
)
∂
q
1
(
t
)
,
p
2
(
t
+
1
)
=
p
2
(
t
)
+
β
p
2
(
t
)
∂
Π
2
(
t
)
∂
p
2
(
t
)
,
p
3
(
t
+
1
)
=
p
3
(
t
)
+
γ
p
3
(
t
)
∂
Π
3
(
t
)
∂
p
3
(
t
)
,
where α
,
β
,
γ are the adjustment speeds of the three firms, respectively. And α
,
β
,
γ
>
0.Substituting (5) into (6), then the model has the following form:
(7)
q
1
(
t
+
1
)
=
q
1
(
t
)
+
α
q
1
(
t
)
×
(
-
2
b
1
q
1
(
t
)
-
2
c
1
q
1
(
t
)
+
a
1
)
,
p
2
(
t
+
1
)
=
p
2
(
t
)
+
β
p
2
(
t
)
×
[
a
2
+
a
1
b
2
-
b
1
b
2
q
1
(
t
)
-
2
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
]
,
p
3
(
t
+
1
)
=
p
3
(
t
)
+
γ
p
3
(
t
)
×
[
a
3
+
a
1
b
3
-
b
1
b
3
q
1
(
t
)
+
d
3
p
2
(
t
)
-
2
b
3
p
3
(
t
)
]
.


### 2.2. Equilibrium Points and Local Stability

The system (7) has eight equilibrium points:
(8)
E
0
=
(
0,0
,
0
)
,
E
1
=
(
0,0
,
(
a
3
+
a
1
b
3
)
2
b
3
)
,
E
2
=
(
0
,
(
a
2
+
a
1
b
2
)
2
b
2
,
0
)
,
E
3
=
(
a
1
2
(
b
1
+
c
1
)
,
0,0
)
E
4
=
(
0
,
(
2
a
2
b
3
+
2
a
1
b
2
b
3
+
a
3
d
2
+
a
1
b
3
d
2
)
(
4
b
2
b
3
-
d
2
d
3
)
,
h
h
h
(
2
a
3
b
2
+
2
a
1
b
2
b
3
+
a
2
d
3
+
a
1
b
2
d
3
)
(
4
b
2
b
3
-
d
2
d
3
)
)
,
E
5
=
(
a
1
2
(
b
1
+
c
1
)
,
0
,
(
2
a
2
b
1
+
2
a
3
c
1
+
a
1
b
1
b
3
+
2
a
1
b
3
c
1
)
4
b
3
(
b
1
+
c
1
)
)
,
E
6
=
(
a
1
2
(
b
1
+
c
1
)
,
(
2
a
2
b
1
+
2
a
2
c
1
+
a
1
b
1
b
2
+
2
a
1
b
2
c
1
)
4
b
2
(
b
1
+
c
1
)
,
0
)
,
E
*
=
(
a
1
2
(
b
1
+
c
1
)
,
h
2
g
,
h
3
g
)
=
(
q
1
*
,
p
2
*
,
p
3
*
)
,
where
(9)
h
2
=
4
a
2
b
1
b
3
+
4
a
2
b
3
c
1
+
2
a
1
b
1
b
2
b
3
+
4
a
1
b
2
b
3
c
1
+
2
a
3
b
1
d
2
+
2
a
3
c
1
d
2
+
a
1
b
1
b
3
d
2
+
2
a
1
b
3
c
1
d
2
,
h
3
=
4
a
3
b
1
b
2
+
4
a
3
b
2
c
1
+
2
a
2
b
1
d
3
+
2
a
2
c
1
d
3
+
2
a
1
b
1
b
2
b
3
+
4
a
1
b
2
b
3
c
1
+
a
1
b
1
b
2
d
3
+
2
a
1
b
2
c
1
d
3
,
g
=
2
(
b
1
+
c
1
)
(
4
b
2
b
3
-
d
2
d
3
)
.
Since all the equilibrium points should be nonnegative, the parameters satisfy d
1
d
2
<
4
b
2
b
3. Since q
1
*
>
q
2
*
+
q
3
*, we should have
(10)
2
[
2
b
2
b
3
(
a
2
+
a
3
)
+
(
a
2
b
3
d
3
+
a
3
b
2
d
3
)
]
(
b
1
+
c
1
)
+
a
1
b
2
b
3
[
(
d
2
+
d
3
)
(
b
1
+
2
c
1
)
-
2
]
<
a
1
[
1
+
(
b
1
+
2
c
1
)
(
b
2
+
b
3
)
]
(
2
b
2
b
3
-
d
2
d
3
)
.The local stability of equilibrium points can be determined by the nature of the eigenvalues of the Jacobian matrix evaluated at the corresponding equilibrium points. The Jacobian matrix of the system (7) corresponding to the state variables (
q
1
,
p
2
,
p
3
) is
(11)
J
(
E
)
=
(
w
11
0
0
w
21
w
22
w
23
w
31
w
32
w
33
)
,
where
(12)
w
11
=
1
+
α
(
-
4
b
1
q
1
+
a
1
-
4
c
1
q
1
)
,
w
21
=
-
b
1
b
2
β
p
2
,
w
22
=
1
+
β
(
a
2
+
a
1
b
2
-
b
1
b
2
q
1
-
4
b
2
p
2
+
d
2
p
3
)
,
w
23
=
d
2
β
p
2
,
w
31
=
-
b
1
b
3
γ
p
3
,
w
32
=
d
3
γ
p
3
,
w
33
=
1
+
γ
(
a
3
+
a
1
b
3
-
b
1
b
3
q
1
+
d
3
p
2
-
4
b
3
p
3
)
.Theorem 1.
All the boundary equilibrium pointsE
0
,
E
1
,
…
E
6 are unstable.Proof.
E
0, E
1, E
2, and E
4 all have the eigenvalue λ
1
=
1
+
a
1
α. Since a
1
,
α
>
0, then λ
1
>
1. Hence the equilibrium points E
0
,
E
1
,
E
2, and E
4 are unstable equilibrium points [18, 19]. Similarly, E
3 has one eigenvalue λ
=
1
+
a
2
β
+
a
1
b
2
β
(
b
1
+
2
c
1
)
/
2
(
b
1
+
c
1
)
>
1; λ
=
1
+
a
2
β
+
β
(
2
a
3
b
1
d
1
+
2
a
3
c
1
d
2
+
2
a
1
b
1
b
2
b
3
+
4
a
1
b
2
b
3
c
1
+
a
1
b
1
b
3
d
2
+
2
a
1
b
3
c
1
d
2
)
/
4
b
3
(
b
1
+
c
1
)
>
1 for E
5 and λ
=
1
+
a
3
γ
+
γ
(
2
a
2
b
1
d
3
+
2
a
3
c
1
d
3
+
2
a
2
b
1
b
2
b
3
+
4
a
1
b
2
b
3
c
1
+
a
1
b
1
b
2
d
3
+
2
a
1
b
2
c
1
d
3
)
/
4
b
2
(
b
1
+
c
1
)
>
1 for E
6. Then all the boundary equilibrium points are unstable.Now we investigate the local stability of Nash equilibrium pointE
*. The Jacobian matrix J
(
E
*
) is
(13)
J
(
E
*
)
=
(
w
11
*
0
0
w
21
*
w
22
*
w
23
*
w
31
*
w
32
*
w
33
*
)
,
where
(14)
w
11
*
=
1
+
α
(
-
4
b
1
q
1
*
+
a
1
-
4
c
1
q
1
*
)
,
w
21
*
=
-
b
1
b
2
β
p
2
*
,
w
22
*
=
1
+
β
(
a
2
+
a
1
b
2
-
b
1
b
2
q
1
*
-
4
b
2
p
2
*
+
d
2
p
3
*
)
,
w
23
*
=
d
2
β
p
2
*
,
w
31
*
=
-
b
1
b
3
γ
p
3
*
,
w
32
*
=
d
3
γ
p
3
*
,
w
33
*
=
1
+
γ
(
a
3
+
a
1
b
3
-
b
1
b
3
q
1
*
+
d
3
p
2
*
-
4
b
3
p
3
*
)
.
The characteristic polynomial of the Jacobian matrix J
(
E
*
) is
(15)
f
(
λ
)
=
λ
3
+
A
λ
2
+
B
λ
+
C
.
And its local stability is given by the Jury conditions [20]:
(16)
(
i
)
:
f
(
1
)
=
A
+
B
+
C
+
1
>
0
,
(
ii
)
:
f
(
-
1
)
=
A
-
B
+
C
-
1
<
0
,
(
iii
)
:
C
2
-
1
<
0
,
(
iv
)
:
(
1
-
C
2
)
2
-
(
B
-
A
C
)
2
>
0
.In order to analyze the stability of Nash equilibrium point, we perform some numerical simulations.


### 2.3. Numerical Simulations

In this section, we will show the complex behaviors of the system (7) including bifurcation and strange attractor. In order to further analyze long-run profit of the three firms with parameters changing, the long-run average profit figures are given. It is convenient to take the parameters values as follows: a
1
=
2, a
2
=
4, a
3
=
3, b
1
=
0.2, b
2
=
2, b
3
=
1.5, c
1
=
0.3, d
2
=
0.5, and d
3
=
0.6. The initial values are chosen as (
q
1
(
0
)
,
p
2
(
0
)
,
p
3
(
0
)
)
=
(
1.8,2.1,2.2
). Through (7), the Nash equilibrium point is (
2,2.0769,2.2154
). Then its Jacobian matrix is
(17)
J
(
E
*
)
=
(
1
-
2
α
0
0
-
0.8308
β
1
-
8.3077
β
1.0385
β
-
0.6646
γ
1.3292
γ
1
-
6.6462
γ
)
,
where A
=
2
α
+
8.3077
β + 6.6462
γ
-
3, B
=
(
6.6462
γ
-
1
)
(
2
α
+
8.3077
β
-
2
) − 1.3804
β
γ
+
(
2
α
-
1
)
(
8.3077
β
-
1
), and C
=
-
1.3804
β
γ
(
2
α + 8.3077
β
-
2
) + 1.3804
β
γ
(
8.3077
β
-
1
) + (
2
α
-
1
)
(
8.3077
β
-
1
)
(
6.6462
γ
-
1
).Figure1 gives the stable region of the Nash equilibrium point E
*. We can see that the stable region is α
<
1, β
<
0.24, γ
<
0.3 approximately. From the figure, we can conclude that the stability region is asymmetric and the higher adjustment speeds will push the system out of the stable region.Figure 1
The stable region of the Nash equilibrium pointE
*.Figure2 displays the bifurcation diagram and the largest Lyapunov exponent with respect to the parameter α which is the adjustment speed of the upstream monopoly when β
=
0.15 and γ
=
0.2. By comparing the largest Lyapunov exponent diagram, one can have a better understanding of the particular properties of the system. In Figure 2, the system (7) converges to the Nash equilibrium point for 0
<
α
<
0.99. If α increases, that is, α
>
0.99, the system turns unstable and complex dynamic behavior is observed. At α
=
0.99, a flip bifurcation arises, which is followed by further flips and the largest Lyapunov exponent increases to zero for the first time; hence the system enters a period doubling routes to chaos. When α
≥
1.29, the largest Lyapunov exponent is positive and chaos emerges.Figure 2
Bifurcation diagram and the Largest Lyapunov exponent of the discrete dynamic system (7) for β
=
0.15, γ
=
0.2, and α varying from 0 to 1.5.Figure3 is the bifurcation diagram with respect to the parameter β, when given α
=
0.5 and γ
=
0.2. In Figure 3, the output q
1
(
t
) of upstream firm is always stable which illustrates that the adjustment speed β has little effect on the output q
1
(
t
) of the upstream firm, while the prices p
2
(
t
), p
3
(
t
) of the two downstream firms generate bifurcation behaviors at β
=
0.2312. And when β
≥
0.306, the largest Lyapunov exponent is positive; then the system is in a state of chaos.Figure 3
Bifurcation diagram and the Largest Lyapunov exponent of the discrete dynamic system (7) for α
=
0.5, γ
=
0.2, and β varying from 0 to 0.34.Similar to Figure3, Figure 4 gives the bifurcation diagram with respect to the parameter γ, when given α
=
0.5 and β
=
0.15. In Figure 4, the output q
1
(
t
) is also stable. And when γ
>
0.2856 the system turns unstable and enters chaos when γ
≥
0.3822.Figure 4
Bifurcation diagram and the Largest Lyapunov exponent of the discrete dynamic system (7) for α
=
0.5, β
=
0.15, and γ varying from 0 to 0.44.From the above analysis, it can be seen that the adjustment parameterα has an important influence on system (7). That is to say the behavior of upstream monopolist has a decisive influence on the market in economics. And it is harmful for the development of the two downstream firms if the changes of adjustment parameters α
,
β
,
γ are too big.Figure5 represents the graph of a strange attractors of the dynamical system (7) for the adjustment parameter values β
=
0.3, γ
=
0.35, and α
=
1.35, which exhibits fractal structure of the system.Figure 5
Strange attractors forα
=
1.35, β
=
0.3, and γ
=
0.35.Then we analyse the long-run average profit of the three firms. The results are shown in Figures6, 7, and 8.Figure 6
The long-run average profits of the players withα for system (7).Figure 7
The long-run average profits of the players withβ for system (7).Figure 8
The long-run average profits of the players withγ for system (7).From these figures, we can see that the long-run average profit of firm 1 is larger than the other firms. Figure6 shows that the long-run average profit of the three firms fluctuates and is lower than that of the stable state with the production adjustment speed increasing when the system (7) enters into bifurcation and chaotic states. When α
≥
0.99, the profit of the three firms begins to decrease. From Figures 7 and 8, we can see that the average profit of the upstream firm is a constant value, so we deduce that the adjustment speeds β and γ have little effect on the profit of firm 1. In Figure 7, the average profit of firms 1 and 3 is always higher than firm 2 and the long-run average profit of the downstream firms decreases when the bifurcation begins. When β
=
0.255, the profit of firm 2 is negative. In Figure 8, the long-run average profit of the downstream firms decreases when the bifurcation begins and when γ
=
0.3024, they are the same. From then on, the profit of firm 2 is higher than firm 3 and when γ
=
0.336, the profit of firm 3 turns to negative.Our study finds that the overall profit of the system will decrease in bifurcation and chaotic states with adjustment speeds increasing.


## 2.1. Model

In an area, there are three firms in the market and firmi produces goods x
i, i
=
1,2
,
3. Firm 1 represents the upstream monopolistic firm and firms 2 and 3 represent two downstream oligopolistic firms. The output and price of firm i are represented, respectively, as q
i and p
i.This model is based on the following assumptions.(1)
The upstream monopolist supplies output (e.g., fresh water) to two downstream firms that compete in a final goods market (e.g., distilled bottled water). The upstream market is monopoly (a single firm: firm 1), and the downstream market is duopoly (two firms: 2 and 3). In addition, the two downstream oligopolistic firms do not cooperate. Then we haveq
1
>
q
2
+
q
3, p
1
<
p
2 and p
1
<
p
3.
(2)
The cost of the two upstream monopolistic firms is a quadratic function. Because firm 1’s price will affect the cost of the downstream firms (assuming 1 unit of fresh water is used to make 1 unit spring water), we assume the two firms’ marginal cost isp
1.
(3)
In the vertically connected market, the three firms make decisions at the same time. The upstream monopolistic firm’s decision variable is its outputq
1 while the two downstream firms’ decision variables are their prices p
2 and p
3.Assume that the inverse demand function of firm 1 and demand functions of firms 2 and 3, respectively, are(1)
p
1
(
t
)
=
a
1
-
b
1
q
1
(
t
)
,
q
2
(
t
)
=
a
2
-
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
,
q
3
(
t
)
=
a
3
-
b
3
p
3
(
t
)
+
d
3
p
2
(
t
)
,
where a
i
,
b
i
>
0
,
i
=
1,2
,
3, and d
i
>
0
,
i
=
2,3. d
2 and d
3 mean the competition parameters between firms 2 and 3.The cost functions of the first firm has the following form [17]:
(2)
C
1
(
t
)
=
c
1
q
1
2
(
t
)
,
where 2
c
1 is marginal cost and c
1 is positive for any t
>
0. Firm 1’s price will affect the cost of the two downstream firms. That is, C
j
=
f
(
p
1
), j
=
2,3. We assume firm j’s cost function is
(3)
C
j
(
t
)
=
p
1
(
t
)
q
j
(
t
)
,
where j
=
2,3.The profit functions of the three firms are(4)
Π
1
(
t
)
=
q
1
(
t
)
p
1
(
t
)
-
C
1
(
t
)
=
q
1
(
t
)
(
a
1
-
b
1
q
1
(
t
)
)
-
c
1
q
1
2
(
t
)
=
-
b
1
q
1
2
(
t
)
-
c
1
q
1
2
(
t
)
+
a
1
q
1
(
t
)
,
Π
2
(
t
)
=
q
2
(
t
)
p
2
(
t
)
-
C
2
(
t
)
=
(
a
2
-
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
)
p
2
(
t
)
-
p
1
(
t
)
q
2
(
t
)
=
a
2
p
2
(
t
)
-
b
2
p
2
2
(
t
)
+
d
2
p
2
(
t
)
p
3
(
t
)
-
(
a
1
-
b
1
q
1
(
t
)
)
(
a
2
-
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
)
,
Π
3
(
t
)
=
q
3
(
t
)
p
3
(
t
)
-
C
3
(
t
)
=
(
a
3
-
b
3
p
3
(
t
)
+
d
3
p
2
(
t
)
)
p
3
(
t
)
-
p
1
(
t
)
q
3
(
t
)
=
a
3
p
3
(
t
)
-
b
3
p
3
2
(
t
)
+
d
3
p
2
(
t
)
p
3
(
t
)
-
(
a
1
-
b
1
q
1
(
t
)
)
(
a
3
-
b
3
p
3
(
t
)
+
d
3
p
2
(
t
)
)
.We assume that the three firms do not have a complete knowledge of the market and the other players. In games, players behave adaptively, following a bounded rational adjustment process and they build decisions on the basis of the expected marginal profit; that is, if the marginal profit is positive (negative), they increase (decrease) their production or price in the next period. The marginal profit functions of the three firms are as follows:(5)
∂
Π
1
(
t
)
∂
q
1
(
t
)
=
-
2
b
1
q
1
(
t
)
-
2
c
1
q
1
(
t
)
+
a
1
,
∂
Π
2
(
t
)
∂
p
2
(
t
)
=
a
2
-
2
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
+
b
2
(
a
1
-
b
1
q
1
(
t
)
)
=
a
2
+
a
1
b
2
-
b
1
b
2
q
1
(
t
)
-
2
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
,
∂
Π
3
(
t
)
∂
p
3
(
t
)
=
a
3
-
2
b
3
p
3
(
t
)
+
d
3
p
2
(
t
)
+
b
3
(
a
1
-
b
1
q
1
(
t
)
)
=
a
3
+
a
1
b
3
-
b
1
b
3
q
1
(
t
)
+
d
3
p
2
(
t
)
-
2
b
3
p
3
(
t
)
.
Under the above assumptions, the dynamic adjustment equation of the master-slave game is
(6)
q
1
(
t
+
1
)
=
q
1
(
t
)
+
α
q
1
(
t
)
∂
Π
1
(
t
)
∂
q
1
(
t
)
,
p
2
(
t
+
1
)
=
p
2
(
t
)
+
β
p
2
(
t
)
∂
Π
2
(
t
)
∂
p
2
(
t
)
,
p
3
(
t
+
1
)
=
p
3
(
t
)
+
γ
p
3
(
t
)
∂
Π
3
(
t
)
∂
p
3
(
t
)
,
where α
,
β
,
γ are the adjustment speeds of the three firms, respectively. And α
,
β
,
γ
>
0.Substituting (5) into (6), then the model has the following form:
(7)
q
1
(
t
+
1
)
=
q
1
(
t
)
+
α
q
1
(
t
)
×
(
-
2
b
1
q
1
(
t
)
-
2
c
1
q
1
(
t
)
+
a
1
)
,
p
2
(
t
+
1
)
=
p
2
(
t
)
+
β
p
2
(
t
)
×
[
a
2
+
a
1
b
2
-
b
1
b
2
q
1
(
t
)
-
2
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
]
,
p
3
(
t
+
1
)
=
p
3
(
t
)
+
γ
p
3
(
t
)
×
[
a
3
+
a
1
b
3
-
b
1
b
3
q
1
(
t
)
+
d
3
p
2
(
t
)
-
2
b
3
p
3
(
t
)
]
.


## 2.2. Equilibrium Points and Local Stability

The system (7) has eight equilibrium points:
(8)
E
0
=
(
0,0
,
0
)
,
E
1
=
(
0,0
,
(
a
3
+
a
1
b
3
)
2
b
3
)
,
E
2
=
(
0
,
(
a
2
+
a
1
b
2
)
2
b
2
,
0
)
,
E
3
=
(
a
1
2
(
b
1
+
c
1
)
,
0,0
)
E
4
=
(
0
,
(
2
a
2
b
3
+
2
a
1
b
2
b
3
+
a
3
d
2
+
a
1
b
3
d
2
)
(
4
b
2
b
3
-
d
2
d
3
)
,
h
h
h
(
2
a
3
b
2
+
2
a
1
b
2
b
3
+
a
2
d
3
+
a
1
b
2
d
3
)
(
4
b
2
b
3
-
d
2
d
3
)
)
,
E
5
=
(
a
1
2
(
b
1
+
c
1
)
,
0
,
(
2
a
2
b
1
+
2
a
3
c
1
+
a
1
b
1
b
3
+
2
a
1
b
3
c
1
)
4
b
3
(
b
1
+
c
1
)
)
,
E
6
=
(
a
1
2
(
b
1
+
c
1
)
,
(
2
a
2
b
1
+
2
a
2
c
1
+
a
1
b
1
b
2
+
2
a
1
b
2
c
1
)
4
b
2
(
b
1
+
c
1
)
,
0
)
,
E
*
=
(
a
1
2
(
b
1
+
c
1
)
,
h
2
g
,
h
3
g
)
=
(
q
1
*
,
p
2
*
,
p
3
*
)
,
where
(9)
h
2
=
4
a
2
b
1
b
3
+
4
a
2
b
3
c
1
+
2
a
1
b
1
b
2
b
3
+
4
a
1
b
2
b
3
c
1
+
2
a
3
b
1
d
2
+
2
a
3
c
1
d
2
+
a
1
b
1
b
3
d
2
+
2
a
1
b
3
c
1
d
2
,
h
3
=
4
a
3
b
1
b
2
+
4
a
3
b
2
c
1
+
2
a
2
b
1
d
3
+
2
a
2
c
1
d
3
+
2
a
1
b
1
b
2
b
3
+
4
a
1
b
2
b
3
c
1
+
a
1
b
1
b
2
d
3
+
2
a
1
b
2
c
1
d
3
,
g
=
2
(
b
1
+
c
1
)
(
4
b
2
b
3
-
d
2
d
3
)
.
Since all the equilibrium points should be nonnegative, the parameters satisfy d
1
d
2
<
4
b
2
b
3. Since q
1
*
>
q
2
*
+
q
3
*, we should have
(10)
2
[
2
b
2
b
3
(
a
2
+
a
3
)
+
(
a
2
b
3
d
3
+
a
3
b
2
d
3
)
]
(
b
1
+
c
1
)
+
a
1
b
2
b
3
[
(
d
2
+
d
3
)
(
b
1
+
2
c
1
)
-
2
]
<
a
1
[
1
+
(
b
1
+
2
c
1
)
(
b
2
+
b
3
)
]
(
2
b
2
b
3
-
d
2
d
3
)
.The local stability of equilibrium points can be determined by the nature of the eigenvalues of the Jacobian matrix evaluated at the corresponding equilibrium points. The Jacobian matrix of the system (7) corresponding to the state variables (
q
1
,
p
2
,
p
3
) is
(11)
J
(
E
)
=
(
w
11
0
0
w
21
w
22
w
23
w
31
w
32
w
33
)
,
where
(12)
w
11
=
1
+
α
(
-
4
b
1
q
1
+
a
1
-
4
c
1
q
1
)
,
w
21
=
-
b
1
b
2
β
p
2
,
w
22
=
1
+
β
(
a
2
+
a
1
b
2
-
b
1
b
2
q
1
-
4
b
2
p
2
+
d
2
p
3
)
,
w
23
=
d
2
β
p
2
,
w
31
=
-
b
1
b
3
γ
p
3
,
w
32
=
d
3
γ
p
3
,
w
33
=
1
+
γ
(
a
3
+
a
1
b
3
-
b
1
b
3
q
1
+
d
3
p
2
-
4
b
3
p
3
)
.Theorem 1.
All the boundary equilibrium pointsE
0
,
E
1
,
…
E
6 are unstable.Proof.
E
0, E
1, E
2, and E
4 all have the eigenvalue λ
1
=
1
+
a
1
α. Since a
1
,
α
>
0, then λ
1
>
1. Hence the equilibrium points E
0
,
E
1
,
E
2, and E
4 are unstable equilibrium points [18, 19]. Similarly, E
3 has one eigenvalue λ
=
1
+
a
2
β
+
a
1
b
2
β
(
b
1
+
2
c
1
)
/
2
(
b
1
+
c
1
)
>
1; λ
=
1
+
a
2
β
+
β
(
2
a
3
b
1
d
1
+
2
a
3
c
1
d
2
+
2
a
1
b
1
b
2
b
3
+
4
a
1
b
2
b
3
c
1
+
a
1
b
1
b
3
d
2
+
2
a
1
b
3
c
1
d
2
)
/
4
b
3
(
b
1
+
c
1
)
>
1 for E
5 and λ
=
1
+
a
3
γ
+
γ
(
2
a
2
b
1
d
3
+
2
a
3
c
1
d
3
+
2
a
2
b
1
b
2
b
3
+
4
a
1
b
2
b
3
c
1
+
a
1
b
1
b
2
d
3
+
2
a
1
b
2
c
1
d
3
)
/
4
b
2
(
b
1
+
c
1
)
>
1 for E
6. Then all the boundary equilibrium points are unstable.Now we investigate the local stability of Nash equilibrium pointE
*. The Jacobian matrix J
(
E
*
) is
(13)
J
(
E
*
)
=
(
w
11
*
0
0
w
21
*
w
22
*
w
23
*
w
31
*
w
32
*
w
33
*
)
,
where
(14)
w
11
*
=
1
+
α
(
-
4
b
1
q
1
*
+
a
1
-
4
c
1
q
1
*
)
,
w
21
*
=
-
b
1
b
2
β
p
2
*
,
w
22
*
=
1
+
β
(
a
2
+
a
1
b
2
-
b
1
b
2
q
1
*
-
4
b
2
p
2
*
+
d
2
p
3
*
)
,
w
23
*
=
d
2
β
p
2
*
,
w
31
*
=
-
b
1
b
3
γ
p
3
*
,
w
32
*
=
d
3
γ
p
3
*
,
w
33
*
=
1
+
γ
(
a
3
+
a
1
b
3
-
b
1
b
3
q
1
*
+
d
3
p
2
*
-
4
b
3
p
3
*
)
.
The characteristic polynomial of the Jacobian matrix J
(
E
*
) is
(15)
f
(
λ
)
=
λ
3
+
A
λ
2
+
B
λ
+
C
.
And its local stability is given by the Jury conditions [20]:
(16)
(
i
)
:
f
(
1
)
=
A
+
B
+
C
+
1
>
0
,
(
ii
)
:
f
(
-
1
)
=
A
-
B
+
C
-
1
<
0
,
(
iii
)
:
C
2
-
1
<
0
,
(
iv
)
:
(
1
-
C
2
)
2
-
(
B
-
A
C
)
2
>
0
.In order to analyze the stability of Nash equilibrium point, we perform some numerical simulations.


## 2.3. Numerical Simulations

In this section, we will show the complex behaviors of the system (7) including bifurcation and strange attractor. In order to further analyze long-run profit of the three firms with parameters changing, the long-run average profit figures are given. It is convenient to take the parameters values as follows: a
1
=
2, a
2
=
4, a
3
=
3, b
1
=
0.2, b
2
=
2, b
3
=
1.5, c
1
=
0.3, d
2
=
0.5, and d
3
=
0.6. The initial values are chosen as (
q
1
(
0
)
,
p
2
(
0
)
,
p
3
(
0
)
)
=
(
1.8,2.1,2.2
). Through (7), the Nash equilibrium point is (
2,2.0769,2.2154
). Then its Jacobian matrix is
(17)
J
(
E
*
)
=
(
1
-
2
α
0
0
-
0.8308
β
1
-
8.3077
β
1.0385
β
-
0.6646
γ
1.3292
γ
1
-
6.6462
γ
)
,
where A
=
2
α
+
8.3077
β + 6.6462
γ
-
3, B
=
(
6.6462
γ
-
1
)
(
2
α
+
8.3077
β
-
2
) − 1.3804
β
γ
+
(
2
α
-
1
)
(
8.3077
β
-
1
), and C
=
-
1.3804
β
γ
(
2
α + 8.3077
β
-
2
) + 1.3804
β
γ
(
8.3077
β
-
1
) + (
2
α
-
1
)
(
8.3077
β
-
1
)
(
6.6462
γ
-
1
).Figure1 gives the stable region of the Nash equilibrium point E
*. We can see that the stable region is α
<
1, β
<
0.24, γ
<
0.3 approximately. From the figure, we can conclude that the stability region is asymmetric and the higher adjustment speeds will push the system out of the stable region.Figure 1
The stable region of the Nash equilibrium pointE
*.Figure2 displays the bifurcation diagram and the largest Lyapunov exponent with respect to the parameter α which is the adjustment speed of the upstream monopoly when β
=
0.15 and γ
=
0.2. By comparing the largest Lyapunov exponent diagram, one can have a better understanding of the particular properties of the system. In Figure 2, the system (7) converges to the Nash equilibrium point for 0
<
α
<
0.99. If α increases, that is, α
>
0.99, the system turns unstable and complex dynamic behavior is observed. At α
=
0.99, a flip bifurcation arises, which is followed by further flips and the largest Lyapunov exponent increases to zero for the first time; hence the system enters a period doubling routes to chaos. When α
≥
1.29, the largest Lyapunov exponent is positive and chaos emerges.Figure 2
Bifurcation diagram and the Largest Lyapunov exponent of the discrete dynamic system (7) for β
=
0.15, γ
=
0.2, and α varying from 0 to 1.5.Figure3 is the bifurcation diagram with respect to the parameter β, when given α
=
0.5 and γ
=
0.2. In Figure 3, the output q
1
(
t
) of upstream firm is always stable which illustrates that the adjustment speed β has little effect on the output q
1
(
t
) of the upstream firm, while the prices p
2
(
t
), p
3
(
t
) of the two downstream firms generate bifurcation behaviors at β
=
0.2312. And when β
≥
0.306, the largest Lyapunov exponent is positive; then the system is in a state of chaos.Figure 3
Bifurcation diagram and the Largest Lyapunov exponent of the discrete dynamic system (7) for α
=
0.5, γ
=
0.2, and β varying from 0 to 0.34.Similar to Figure3, Figure 4 gives the bifurcation diagram with respect to the parameter γ, when given α
=
0.5 and β
=
0.15. In Figure 4, the output q
1
(
t
) is also stable. And when γ
>
0.2856 the system turns unstable and enters chaos when γ
≥
0.3822.Figure 4
Bifurcation diagram and the Largest Lyapunov exponent of the discrete dynamic system (7) for α
=
0.5, β
=
0.15, and γ varying from 0 to 0.44.From the above analysis, it can be seen that the adjustment parameterα has an important influence on system (7). That is to say the behavior of upstream monopolist has a decisive influence on the market in economics. And it is harmful for the development of the two downstream firms if the changes of adjustment parameters α
,
β
,
γ are too big.Figure5 represents the graph of a strange attractors of the dynamical system (7) for the adjustment parameter values β
=
0.3, γ
=
0.35, and α
=
1.35, which exhibits fractal structure of the system.Figure 5
Strange attractors forα
=
1.35, β
=
0.3, and γ
=
0.35.Then we analyse the long-run average profit of the three firms. The results are shown in Figures6, 7, and 8.Figure 6
The long-run average profits of the players withα for system (7).Figure 7
The long-run average profits of the players withβ for system (7).Figure 8
The long-run average profits of the players withγ for system (7).From these figures, we can see that the long-run average profit of firm 1 is larger than the other firms. Figure6 shows that the long-run average profit of the three firms fluctuates and is lower than that of the stable state with the production adjustment speed increasing when the system (7) enters into bifurcation and chaotic states. When α
≥
0.99, the profit of the three firms begins to decrease. From Figures 7 and 8, we can see that the average profit of the upstream firm is a constant value, so we deduce that the adjustment speeds β and γ have little effect on the profit of firm 1. In Figure 7, the average profit of firms 1 and 3 is always higher than firm 2 and the long-run average profit of the downstream firms decreases when the bifurcation begins. When β
=
0.255, the profit of firm 2 is negative. In Figure 8, the long-run average profit of the downstream firms decreases when the bifurcation begins and when γ
=
0.3024, they are the same. From then on, the profit of firm 2 is higher than firm 3 and when γ
=
0.336, the profit of firm 3 turns to negative.Our study finds that the overall profit of the system will decrease in bifurcation and chaotic states with adjustment speeds increasing.


## 3. Chaos Control of Nondelayed Master-Slave Model

### 3.1. Delay Feedback Control Method

In this part, we first take the delay feedback control method [21, 22] to control chaos. The method is based on the difference between the T-time delayed state and the current state, where T denotes a period of the stabilized orbits. The controlled system is
(18)
x
(
t
+
1
)
=
f
(
x
(
t
)
,
u
(
t
)
)
,
where u
(
t
) is the input signal and x
(
t
) is the state. Consider
(19)
u
(
t
)
=
k
(
x
(
t
+
1
-
T
)
-
x
(
t
+
1
)
)
,
t
>
T
,
where T is the time delay and k is the controlling factor.Because the behavior of firm 1 has great effect on the system, we add the function in the first equation from the system (7). Then the controlled system is
(20)
q
1
(
t
+
1
)
=
q
1
(
t
)
+
α
q
1
(
t
)
(
-
2
b
1
q
1
(
t
)
-
2
c
1
q
1
(
t
)
+
a
1
)
+
k
(
q
1
(
t
+
1
-
T
)
-
q
1
(
t
+
1
)
)
,
p
2
(
t
+
1
)
=
p
2
(
t
)
+
β
p
2
(
t
)
×
[
a
2
+
a
1
b
2
-
b
1
b
2
q
1
(
t
)
-
2
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
]
,
p
3
(
t
+
1
)
=
p
3
(
t
)
+
γ
p
3
(
t
)
×
[
a
3
+
a
1
b
3
-
b
1
b
3
q
1
(
t
)
+
d
3
p
2
(
t
)
-
2
b
3
p
3
(
t
)
]
.By choosingT
=
1, the system becomes
(21)
q
1
(
t
+
1
)
=
q
1
(
t
)
+
α
q
1
(
t
)
(
-
2
b
1
q
1
(
t
)
-
2
c
1
q
1
(
t
)
+
a
1
)
(
1
+
k
)
p
2
(
t
+
1
)
=
p
2
(
t
)
+
β
p
2
(
t
)
×
[
a
2
+
a
1
b
2
-
b
1
b
2
q
1
(
t
)
-
2
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
]
,
p
3
(
t
+
1
)
=
p
3
(
t
)
+
γ
p
3
(
t
)
×
[
a
3
+
a
1
b
3
-
b
1
b
3
q
1
(
t
)
+
d
3
p
2
(
t
)
-
2
b
3
p
3
(
t
)
]
.It can be seen from Figure9, (α
=
1.35, β
=
0.15, γ
=
0.2), that the chaotic system was gradually controlled with controlling parameter k increasing. When k
=
0, it turns uncontrolled system (7) which is at chaotic state. And when k
≥
0.36, the system is controlled. Taking k
=
0.8, we can see that the stable region of α expands to 1.775 from Figure 10, which indicates that chaos is delayed or eliminated completely. Figure 11 gives the long-run average profit of the players for the controlled system (21). Comparing Figure 11 with Figure 6, we see that in Figure 6 the three firms’ profit starts to fall in α
=
0.99 while in Figure 11 this phenomenon does not happen.Figure 9
The trend of system withk increasing for α
=
1.35, β
=
0.15, and γ
=
0.2.Figure 10
The bifurcation diagram of the controlled system (21) for k
=
0.8, β
=
0.15, γ
=
0.2, and α varying from 0 to 2.5.Figure 11
The long-run average profits of the players withα for the controlled system (21).


### 3.2. Limiter Control Method

We use limiter control method [23, 24] which is better for firms 2 and 3. This method only requires the player who wants to improve his performance to take measures without the other players’ cooperation. We impose lower limiters on the price of firms 2 and 3, noted p
2
min
⁡, p
3
min
⁡, and it has no effect on the behavior of firm 1.The controlled system is(22)
q
1
(
t
+
1
)
=
q
1
(
t
)
+
α
q
1
(
t
)
(
-
2
b
1
q
1
(
t
)
-
2
c
1
q
1
(
t
)
+
a
1
)
,
p
2
(
t
+
1
)
=
Max
⁡
[
f
2
(
q
1
(
t
)
,
p
2
(
t
)
,
p
3
(
t
)
)
,
p
2
min
⁡
]
,
p
3
(
t
+
1
)
=
Max
⁡
[
f
3
(
q
1
(
t
)
,
p
2
(
t
)
,
p
3
(
t
)
)
,
p
3
min
⁡
]
,
where p
2
min
⁡
,
p
3
min
⁡
>
0, and
(23)
f
2
(
q
1
(
t
)
,
p
2
(
t
)
,
p
3
(
t
)
)
=
p
2
(
t
)
+
β
p
2
(
t
)
×
[
a
2
+
a
1
b
2
-
b
1
b
2
q
1
(
t
)
-
2
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
]
,
f
3
(
q
1
(
t
)
,
p
2
(
t
)
,
p
3
(
t
)
)
=
p
3
(
t
)
+
γ
p
3
(
t
)
×
[
a
3
+
a
1
b
3
-
b
1
b
3
q
1
(
t
)
+
d
3
p
2
(
t
)
-
2
b
3
p
3
(
t
)
]
.By choosingp
2
min
⁡
=
2.0907 and p
3
min
⁡
=
2.2264, Figures 12 and 13 give bifurcation diagram and long-run average profit of the three firms with the α changing. They show that the behaviors of firm 1 are the same with the original system (7). Figure 12 shows that the bifurcation and chaotic behaviors of firms 2 and 3 have been controlled. In Figure 13, the decreasing speed of π
2 and π
3 is under control.Figure 12
The bifurcation diagram of the controlled system (22) for β
=
0.15 and γ
=
0.2.Figure 13
The long-run average profits of the players withα for the controlled system (22).


## 3.1. Delay Feedback Control Method

In this part, we first take the delay feedback control method [21, 22] to control chaos. The method is based on the difference between the T-time delayed state and the current state, where T denotes a period of the stabilized orbits. The controlled system is
(18)
x
(
t
+
1
)
=
f
(
x
(
t
)
,
u
(
t
)
)
,
where u
(
t
) is the input signal and x
(
t
) is the state. Consider
(19)
u
(
t
)
=
k
(
x
(
t
+
1
-
T
)
-
x
(
t
+
1
)
)
,
t
>
T
,
where T is the time delay and k is the controlling factor.Because the behavior of firm 1 has great effect on the system, we add the function in the first equation from the system (7). Then the controlled system is
(20)
q
1
(
t
+
1
)
=
q
1
(
t
)
+
α
q
1
(
t
)
(
-
2
b
1
q
1
(
t
)
-
2
c
1
q
1
(
t
)
+
a
1
)
+
k
(
q
1
(
t
+
1
-
T
)
-
q
1
(
t
+
1
)
)
,
p
2
(
t
+
1
)
=
p
2
(
t
)
+
β
p
2
(
t
)
×
[
a
2
+
a
1
b
2
-
b
1
b
2
q
1
(
t
)
-
2
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
]
,
p
3
(
t
+
1
)
=
p
3
(
t
)
+
γ
p
3
(
t
)
×
[
a
3
+
a
1
b
3
-
b
1
b
3
q
1
(
t
)
+
d
3
p
2
(
t
)
-
2
b
3
p
3
(
t
)
]
.By choosingT
=
1, the system becomes
(21)
q
1
(
t
+
1
)
=
q
1
(
t
)
+
α
q
1
(
t
)
(
-
2
b
1
q
1
(
t
)
-
2
c
1
q
1
(
t
)
+
a
1
)
(
1
+
k
)
p
2
(
t
+
1
)
=
p
2
(
t
)
+
β
p
2
(
t
)
×
[
a
2
+
a
1
b
2
-
b
1
b
2
q
1
(
t
)
-
2
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
]
,
p
3
(
t
+
1
)
=
p
3
(
t
)
+
γ
p
3
(
t
)
×
[
a
3
+
a
1
b
3
-
b
1
b
3
q
1
(
t
)
+
d
3
p
2
(
t
)
-
2
b
3
p
3
(
t
)
]
.It can be seen from Figure9, (α
=
1.35, β
=
0.15, γ
=
0.2), that the chaotic system was gradually controlled with controlling parameter k increasing. When k
=
0, it turns uncontrolled system (7) which is at chaotic state. And when k
≥
0.36, the system is controlled. Taking k
=
0.8, we can see that the stable region of α expands to 1.775 from Figure 10, which indicates that chaos is delayed or eliminated completely. Figure 11 gives the long-run average profit of the players for the controlled system (21). Comparing Figure 11 with Figure 6, we see that in Figure 6 the three firms’ profit starts to fall in α
=
0.99 while in Figure 11 this phenomenon does not happen.Figure 9
The trend of system withk increasing for α
=
1.35, β
=
0.15, and γ
=
0.2.Figure 10
The bifurcation diagram of the controlled system (21) for k
=
0.8, β
=
0.15, γ
=
0.2, and α varying from 0 to 2.5.Figure 11
The long-run average profits of the players withα for the controlled system (21).


## 3.2. Limiter Control Method

We use limiter control method [23, 24] which is better for firms 2 and 3. This method only requires the player who wants to improve his performance to take measures without the other players’ cooperation. We impose lower limiters on the price of firms 2 and 3, noted p
2
min
⁡, p
3
min
⁡, and it has no effect on the behavior of firm 1.The controlled system is(22)
q
1
(
t
+
1
)
=
q
1
(
t
)
+
α
q
1
(
t
)
(
-
2
b
1
q
1
(
t
)
-
2
c
1
q
1
(
t
)
+
a
1
)
,
p
2
(
t
+
1
)
=
Max
⁡
[
f
2
(
q
1
(
t
)
,
p
2
(
t
)
,
p
3
(
t
)
)
,
p
2
min
⁡
]
,
p
3
(
t
+
1
)
=
Max
⁡
[
f
3
(
q
1
(
t
)
,
p
2
(
t
)
,
p
3
(
t
)
)
,
p
3
min
⁡
]
,
where p
2
min
⁡
,
p
3
min
⁡
>
0, and
(23)
f
2
(
q
1
(
t
)
,
p
2
(
t
)
,
p
3
(
t
)
)
=
p
2
(
t
)
+
β
p
2
(
t
)
×
[
a
2
+
a
1
b
2
-
b
1
b
2
q
1
(
t
)
-
2
b
2
p
2
(
t
)
+
d
2
p
3
(
t
)
]
,
f
3
(
q
1
(
t
)
,
p
2
(
t
)
,
p
3
(
t
)
)
=
p
3
(
t
)
+
γ
p
3
(
t
)
×
[
a
3
+
a
1
b
3
-
b
1
b
3
q
1
(
t
)
+
d
3
p
2
(
t
)
-
2
b
3
p
3
(
t
)
]
.By choosingp
2
min
⁡
=
2.0907 and p
3
min
⁡
=
2.2264, Figures 12 and 13 give bifurcation diagram and long-run average profit of the three firms with the α changing. They show that the behaviors of firm 1 are the same with the original system (7). Figure 12 shows that the bifurcation and chaotic behaviors of firms 2 and 3 have been controlled. In Figure 13, the decreasing speed of π
2 and π
3 is under control.Figure 12
The bifurcation diagram of the controlled system (22) for β
=
0.15 and γ
=
0.2.Figure 13
The long-run average profits of the players withα for the controlled system (22).


## 4. The Delayed Master-Slave Model

### 4.1. Model

The primary reason for the occurrence of such a delayed structure in economic models is that (a) decisions made by economic agents at timet depend on past observed variables by means of a prediction feedback and (b) the functional relationships describing the dynamics of the model may not only depend on the current state of the firm but also, in a nontrivial manner, on past states. Considering these reasons, we introduce the delayed model and compare the three firms’ profits in various cases.Then the bounded rationality dynamical model evolved from system (7) with one step delayed is given by
(24)
q
1
(
t
+
1
)
=
q
1
(
t
)
+
α
q
1
(
t
)
(
-
2
b
1
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
q
1
(
t
-
1
)
)
hhhhhhhhhhh
hh
-
2
c
1
(
(
1
-
w
1
)
q
1
(
t
)
+

w
1
q
1
(
t
-
1
)
)

+
a
1
-
2
b
1
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
q
1
(
t
-
1
)
)
)
,
p
2
(
t
+
1
)
=
p
2
(
t
)
+
β
p
2
(
t
)
[
+
d
2
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
p
3
(
t
-
1
)
)
a
2
+
a
1
b
2
-
b
1
b
2
gggggggggggggggi
×
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
q
1
(
t
-
1
)
)
gggggggggggggggi
-
2
b
2
(
(
1
-
w
2
)
p
2
(
t
)
+
w
2
p
2
(
t
-
1
)
)
hhhhhhhhhhhhhh
+
d
2
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
p
3
(
t
-
1
)
)
]
,
p
3
(
t
+
1
)
=
p
3
(
t
)
+
γ
p
3
(
t
)
[
-
2
b
3
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
p
3
(
t
-
1
)
)
a
3
+
a
1
b
3
-
b
1
b
3
ggggggggggggggg
×
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
q
1
(
t
-
1
)
)
ggggggggggggggg
+
d
3
(
(
1
-
w
2
)
p
2
(
t
)
+
w
2
p
2
(
t
-
1
)
)
ggggggggggggggg
-
2
b
3
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
p
3
(
t
-
1
)
)
]
.
w
i represent the weights given to previous production and prices, and 0
≤
w
1
,
w
2
,
w
3
≤
1.It is convenient to takex
(
t
)
=
q
1
(
t
-
1
), y
(
t
)
=
p
2
(
t
-
1
), and z
(
t
)
=
p
3
(
t
-
1
); then system (24) becomes
(25)
q
1
(
t
+
1
)
=
q
1
(
t
)
+
α
q
1
(
t
)
(
-
2
c
1
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
x
(
t
)
)
+
a
1
-
2
b
1
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
x
(
t
)
)
gggggggggggggig
-
2
c
1
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
x
(
t
)
)
+
a
1
-
2
c
1
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
x
(
t
)
)
+
a
1
-
2
b
1
(
(
1
-
w
1
)
q
1
(
t
)
)
,
p
2
(
t
+
1
)
=
p
2
(
t
)
+
β
p
2
(
t
)
[
+
d
2
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
z
(
t
)
)
a
2
+
a
1
b
2
-
b
1
b
2
hhhhhhhhhhhhihi
×
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
x
(
t
)
)
hhhhhhhhhhhhihi
-
2
b
2
(
(
1
-
w
2
)
p
2
(
t
)
+
w
2
y
(
t
)
)
hhhhhhhhhhhhihi
+
d
2
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
z
(
t
)
)
]
,
p
3
(
t
+
1
)
=
p
3
(
t
)
+
γ
p
3
(
t
)
[
-
2
b
3
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
z
(
t
)
)
a
3
+
a
1
b
3
-
b
1
b
3
bbbbbbbbbbbhbbi
×
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
x
(
t
)
)
bbbbbbbbbbbhbbi
+
d
3
(
(
1
-
w
2
)
p
2
(
t
)
+
w
2
y
(
t
)
)
bbbbbbbbbbhbbbi
-
2
b
3
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
z
(
t
)
)
]
,
x
(
t
+
1
)
=
q
1
(
t
)
,
y
(
t
+
1
)
=
p
2
(
t
)
,
z
(
t
+
1
)
=
p
3
(
t
)
.We can get eight equilibrium points, denoted byE
i
=
(
q
1
i
,
p
2
i
,
p
3
i
,
x
i
,
y
i
,
z
i
). Consider (
q
1
i
,
p
2
i
,
p
3
i
)
=
(
x
i
,
y
i
,
z
i
), and (
q
1
i
,
p
2
i
,
p
3
i
)  (
i
=
0,1
,
…
,
7
) have the same values as equilibrium points of system (7). The Jacobi matrix for the system (25) is
(26)
J
(
E
)
=
(
J
11
0
0
J
14
0
0
J
21
J
22
J
23
J
24
J
25
J
26
J
31
J
32
J
33
J
34
J
35
J
36
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
)
,
where J
11
=
w
11
+
2
α
q
1
w
1
(
b
1
+
c
1
), J
14
=
-
2
w
1
α
q
1
(
b
1
+
c
1
), J
21
=
w
31
(
1
-
w
1
), J
22
=
w
22
+
2
β
w
2
p
2
b
2, J
23
=
w
23
(
1
-
w
3
), J
24
=
w
1
w
21, J
25
=
-
2
b
2
w
2
β
p
2, J
26
=
w
3
w
23, J
31
=
w
31
(
1
-
w
1
), J
32
=
w
32
(
1
-
w
2
), J
33
=
w
33
+
2
γ
w
3
p
3
b
3, J
34
=
w
2
w
32, J
35
=
w
2
w
32, and J
36
=
-
2
b
3
γ
p
3
w
3.E
0
,
…
,
E
6 are boundary equilibrium points; E
7 is the unique Nash equilibrium point. The Jury conditions are
(27)
(
i
)
:
F
(
1
)
=
δ
0
+
δ
1
+
δ
2
+
δ
3
+
δ
4
+
δ
5
+
δ
6
>
0
,
(
ii
)
:
F
(
-
1
)
=
δ
0
-
δ
1
+
δ
2
-
δ
3
+
δ
4
-
δ
5
+
δ
6
>
0
,
(
iii
)
:
|
δ
0
|
<
δ
6
,
(
iv
)
:
|
f
1
(
δ
0
,
…
,
δ
6
)
|
>
|
φ
1
(
δ
0
,
…
,
δ
6
)
|
,
(
v
)
:
|
f
2
(
δ
0
,
…
,
δ
6
)
|
>
|
φ
2
(
δ
0
,
…
,
δ
6
)
|
,
(
vi
)
:
|
f
3
(
δ
0
,
…
,
δ
6
)
|
>
|
φ
3
(
δ
0
,
…
,
δ
6
)
|
,
(
vii
)
:
|
f
4
(
δ
0
,
…
,
δ
6
)
|
>
|
φ
4
(
δ
0
,
…
,
δ
6
)
|
,
where F
(
λ
)
=
δ
0
+
δ
1
λ
+
δ
2
λ
2
+
⋯
+
δ
6
λ
6 is the characteristic polynomial, and f
i, φ
i (i
=
1,2
,
3,4) are the functions of δ
i (i
=
1,2
,
3,4
,
5,6).


### 4.2. Numerical Simulations

In order to discuss the complexity of delayed system (25), we first take the α
,
β
,
γ as constant values and then analyze behaviors with w
1
,
w
2
,
w
3 changing.Figure14 gives the bifurcation diagrams with respect to the parameters w
i
,
(
i
=
1,2
,
3
) when α
=
1.36, β
=
0.15, γ
=
0.2 in (a), (b), and (c). In the nondelayed system, when α
,
β
,
γ take the values as above, as we see from Figure 2, system (7) is at the state of chaos while in Figure 14(a), when w
2
=
w
3
=
0, system (25) from chaotic state changes to stable state, then from stable state to chaotic state with the changing of w
1. And when w
1
=
0, w
i
=
0, as w
j (i
,
j
=
2,3
,
i
≠
j) changing, the system (25) is always in the chaotic state. We conclude that w
1 plays an important role in changing system (7) from chaos to stable state with the rest parameters fixed.The bifurcation diagram and the corresponding largest Lyapunov exponent of the delayed system (25).
(a)
w
2
=
0, w
3
=
0
(b)
w
1
=
0, w
3
=
0
(c)
w
1
=
0, w
2
=
0Similarly, takingα
=
0.5, β
=
0.32, and γ
=
0.2, we know from Figure 4 that the system is in chaotic state. In Figure 15, the bifurcation diagram (b) shows that when w
1
=
w
3
=
0, the system (25) is also from the chaotic state to stable state, then from stable state to unstable state with w
2 changing.The bifurcation diagram and the corresponding largest Lyapunov exponent of the delayed system (25).
(a)
w
2
=
0, w
3
=
0
(b)
w
1
=
0, w
3
=
0
(c)
w
1
=
0, w
2
=
0Figure16 shows the bifurcation diagrams and the largest Lyapunov exponent diagram taking α
=
0.5, β
=
0.15, γ
=
0.41. When w
1
=
w
2
=
0, system (25) also experiences the process from chaos to stability then again to the chaos as the changes of the w
3.The bifurcation diagram and the corresponding largest Lyapunov exponent of the delayed system (25).
(a)
w
2
=
0, w
3
=
0
(b)
w
1
=
0, w
3
=
0
(c)
w
1
=
0, w
2
=
0From the analysis of the Figures14–16, we conclude that the appropriate delay parameters play an important role for controlling the system from chaos to stability in system (7).Tables1, 2, and 3 show three firms’ profit and the total profit when the systems are nondelayed, delay feedback controlled, limiter controlled, and delayed. We conclude the following.(a)
The profit of upstream firm is apparently higher than the other two downstream firms’.
(b)
In Table1, although the value of α goes beyond the stable region of system (7), when w
1
=
0.2, the three firms achieve maximal profit: π
1
=
60, π
2
=
13.6473, π
3
=
17.0414, total profit ∑
‍
π
=
90.6888. In Table 2, when the value of β goes beyond the stable region, the three firms achieve maximal profit with w
2
=
0.2. While in Table 3, when the value of γ goes beyond the stable region, the three firms achieve maximal profit with w
3
=
0.2. It can be seen that the effect of selecting appropriate delay parameters is the same as applying a control on the system, which can make profit of the system maximization.
(c)
Taking the appropriate value ofk can reach the best control effect for this case where α goes beyond the stable region. This proves that the delay feedback control method is advantageous to the upstream firm.
(d)
According to the comparison of the three tables, limiter control method is also effective in controlling chaos. And it has great significance when requiring the player who wants to improve his performance to take measures without the other players’ cooperation.Table 1
Comparative analysis of oligarchs’ profits whenα
=
1.36, β
=
0.15, and γ
=
0.2.
Parameters
Nondelayed
Delay feedback control
Limiter control
Delayed
Delayed
Delayed
w
1
0
0
0
0.2
0
0
w
2
0
0
0
0
0.2
0
w
3
0
0
0
0
0
0.2
k
0
0.8
0
0
0
0
p
2
min
⁡
0
0
2.0842
0
0
0
p
3
min
⁡
0
0
2.2213
0
0
0
π
1
53.9601
60
53.9601
60
53.9601
53.9601
π
2
12.1514
13.6473
12.6415
13.6473
12.4463
12.7958
π
3
15.8208
17.0414
16.3254
17.0414
15.5874
15.3587
∑
‍
π
i
81.9322
90.6888
82.9271
90.6888
82.9938
82.1146Table 2
Comparative analysis of oligarchs’ profits whenα
=
0.5, β
=
0.32, and γ
=
0.2.
Parameters
Nondelayed
Limiter control
Delayed
Delayed
Delayed
w
1
0
0
0.2
0
0
w
2
0
0
0
0.2
0
w
3
0
0
0
0
0.2
p
2
min
⁡
0
2.0842
0
0
0
p
3
min
⁡
0
2.2213
0
0
0
π
1
60
60
60
60
60
π
2
-
16.7885
9.9633
-
15.6247
13.6473
-
13.0483
π
3
10.9234
16.5478
10.6893
17.0414
12.4387
∑
‍
π
i
54.1349
86.5111
55.0646
90.6888
59.3904Table 3
Comparative analysis of oligarchs’ profits whenα
=
0.5, β
=
0.15, and γ
=
0.41.
Parameters
Nondelayed
Limiter control
Delayed
Delayed
Delayed
w
1
0
0
0.2
0
0
w
2
0
0
0
0.2
0
w
3
0
0
0
0
0.2
p
2
min
⁡
0
2.0842
0
0
0
p
3
min
⁡
0
2.2213
0
0
0
π
1
60
60
60
60
60
π
2
9.4274
12.6782
8.8646
10.2317
13.6473
π
3
-
13.2909
13.9479
-
19.2747
-
10.3930
17.0414
∑
‍
π
i
56.1364
86.6260
49.5898
59.8387
90.6888


## 4.1. Model

The primary reason for the occurrence of such a delayed structure in economic models is that (a) decisions made by economic agents at timet depend on past observed variables by means of a prediction feedback and (b) the functional relationships describing the dynamics of the model may not only depend on the current state of the firm but also, in a nontrivial manner, on past states. Considering these reasons, we introduce the delayed model and compare the three firms’ profits in various cases.Then the bounded rationality dynamical model evolved from system (7) with one step delayed is given by
(24)
q
1
(
t
+
1
)
=
q
1
(
t
)
+
α
q
1
(
t
)
(
-
2
b
1
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
q
1
(
t
-
1
)
)
hhhhhhhhhhh
hh
-
2
c
1
(
(
1
-
w
1
)
q
1
(
t
)
+

w
1
q
1
(
t
-
1
)
)

+
a
1
-
2
b
1
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
q
1
(
t
-
1
)
)
)
,
p
2
(
t
+
1
)
=
p
2
(
t
)
+
β
p
2
(
t
)
[
+
d
2
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
p
3
(
t
-
1
)
)
a
2
+
a
1
b
2
-
b
1
b
2
gggggggggggggggi
×
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
q
1
(
t
-
1
)
)
gggggggggggggggi
-
2
b
2
(
(
1
-
w
2
)
p
2
(
t
)
+
w
2
p
2
(
t
-
1
)
)
hhhhhhhhhhhhhh
+
d
2
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
p
3
(
t
-
1
)
)
]
,
p
3
(
t
+
1
)
=
p
3
(
t
)
+
γ
p
3
(
t
)
[
-
2
b
3
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
p
3
(
t
-
1
)
)
a
3
+
a
1
b
3
-
b
1
b
3
ggggggggggggggg
×
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
q
1
(
t
-
1
)
)
ggggggggggggggg
+
d
3
(
(
1
-
w
2
)
p
2
(
t
)
+
w
2
p
2
(
t
-
1
)
)
ggggggggggggggg
-
2
b
3
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
p
3
(
t
-
1
)
)
]
.
w
i represent the weights given to previous production and prices, and 0
≤
w
1
,
w
2
,
w
3
≤
1.It is convenient to takex
(
t
)
=
q
1
(
t
-
1
), y
(
t
)
=
p
2
(
t
-
1
), and z
(
t
)
=
p
3
(
t
-
1
); then system (24) becomes
(25)
q
1
(
t
+
1
)
=
q
1
(
t
)
+
α
q
1
(
t
)
(
-
2
c
1
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
x
(
t
)
)
+
a
1
-
2
b
1
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
x
(
t
)
)
gggggggggggggig
-
2
c
1
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
x
(
t
)
)
+
a
1
-
2
c
1
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
x
(
t
)
)
+
a
1
-
2
b
1
(
(
1
-
w
1
)
q
1
(
t
)
)
,
p
2
(
t
+
1
)
=
p
2
(
t
)
+
β
p
2
(
t
)
[
+
d
2
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
z
(
t
)
)
a
2
+
a
1
b
2
-
b
1
b
2
hhhhhhhhhhhhihi
×
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
x
(
t
)
)
hhhhhhhhhhhhihi
-
2
b
2
(
(
1
-
w
2
)
p
2
(
t
)
+
w
2
y
(
t
)
)
hhhhhhhhhhhhihi
+
d
2
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
z
(
t
)
)
]
,
p
3
(
t
+
1
)
=
p
3
(
t
)
+
γ
p
3
(
t
)
[
-
2
b
3
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
z
(
t
)
)
a
3
+
a
1
b
3
-
b
1
b
3
bbbbbbbbbbbhbbi
×
(
(
1
-
w
1
)
q
1
(
t
)
+
w
1
x
(
t
)
)
bbbbbbbbbbbhbbi
+
d
3
(
(
1
-
w
2
)
p
2
(
t
)
+
w
2
y
(
t
)
)
bbbbbbbbbbhbbbi
-
2
b
3
(
(
1
-
w
3
)
p
3
(
t
)
+
w
3
z
(
t
)
)
]
,
x
(
t
+
1
)
=
q
1
(
t
)
,
y
(
t
+
1
)
=
p
2
(
t
)
,
z
(
t
+
1
)
=
p
3
(
t
)
.We can get eight equilibrium points, denoted byE
i
=
(
q
1
i
,
p
2
i
,
p
3
i
,
x
i
,
y
i
,
z
i
). Consider (
q
1
i
,
p
2
i
,
p
3
i
)
=
(
x
i
,
y
i
,
z
i
), and (
q
1
i
,
p
2
i
,
p
3
i
)  (
i
=
0,1
,
…
,
7
) have the same values as equilibrium points of system (7). The Jacobi matrix for the system (25) is
(26)
J
(
E
)
=
(
J
11
0
0
J
14
0
0
J
21
J
22
J
23
J
24
J
25
J
26
J
31
J
32
J
33
J
34
J
35
J
36
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
)
,
where J
11
=
w
11
+
2
α
q
1
w
1
(
b
1
+
c
1
), J
14
=
-
2
w
1
α
q
1
(
b
1
+
c
1
), J
21
=
w
31
(
1
-
w
1
), J
22
=
w
22
+
2
β
w
2
p
2
b
2, J
23
=
w
23
(
1
-
w
3
), J
24
=
w
1
w
21, J
25
=
-
2
b
2
w
2
β
p
2, J
26
=
w
3
w
23, J
31
=
w
31
(
1
-
w
1
), J
32
=
w
32
(
1
-
w
2
), J
33
=
w
33
+
2
γ
w
3
p
3
b
3, J
34
=
w
2
w
32, J
35
=
w
2
w
32, and J
36
=
-
2
b
3
γ
p
3
w
3.E
0
,
…
,
E
6 are boundary equilibrium points; E
7 is the unique Nash equilibrium point. The Jury conditions are
(27)
(
i
)
:
F
(
1
)
=
δ
0
+
δ
1
+
δ
2
+
δ
3
+
δ
4
+
δ
5
+
δ
6
>
0
,
(
ii
)
:
F
(
-
1
)
=
δ
0
-
δ
1
+
δ
2
-
δ
3
+
δ
4
-
δ
5
+
δ
6
>
0
,
(
iii
)
:
|
δ
0
|
<
δ
6
,
(
iv
)
:
|
f
1
(
δ
0
,
…
,
δ
6
)
|
>
|
φ
1
(
δ
0
,
…
,
δ
6
)
|
,
(
v
)
:
|
f
2
(
δ
0
,
…
,
δ
6
)
|
>
|
φ
2
(
δ
0
,
…
,
δ
6
)
|
,
(
vi
)
:
|
f
3
(
δ
0
,
…
,
δ
6
)
|
>
|
φ
3
(
δ
0
,
…
,
δ
6
)
|
,
(
vii
)
:
|
f
4
(
δ
0
,
…
,
δ
6
)
|
>
|
φ
4
(
δ
0
,
…
,
δ
6
)
|
,
where F
(
λ
)
=
δ
0
+
δ
1
λ
+
δ
2
λ
2
+
⋯
+
δ
6
λ
6 is the characteristic polynomial, and f
i, φ
i (i
=
1,2
,
3,4) are the functions of δ
i (i
=
1,2
,
3,4
,
5,6).


## 4.2. Numerical Simulations

In order to discuss the complexity of delayed system (25), we first take the α
,
β
,
γ as constant values and then analyze behaviors with w
1
,
w
2
,
w
3 changing.Figure14 gives the bifurcation diagrams with respect to the parameters w
i
,
(
i
=
1,2
,
3
) when α
=
1.36, β
=
0.15, γ
=
0.2 in (a), (b), and (c). In the nondelayed system, when α
,
β
,
γ take the values as above, as we see from Figure 2, system (7) is at the state of chaos while in Figure 14(a), when w
2
=
w
3
=
0, system (25) from chaotic state changes to stable state, then from stable state to chaotic state with the changing of w
1. And when w
1
=
0, w
i
=
0, as w
j (i
,
j
=
2,3
,
i
≠
j) changing, the system (25) is always in the chaotic state. We conclude that w
1 plays an important role in changing system (7) from chaos to stable state with the rest parameters fixed.The bifurcation diagram and the corresponding largest Lyapunov exponent of the delayed system (25).
(a)
w
2
=
0, w
3
=
0
(b)
w
1
=
0, w
3
=
0
(c)
w
1
=
0, w
2
=
0Similarly, takingα
=
0.5, β
=
0.32, and γ
=
0.2, we know from Figure 4 that the system is in chaotic state. In Figure 15, the bifurcation diagram (b) shows that when w
1
=
w
3
=
0, the system (25) is also from the chaotic state to stable state, then from stable state to unstable state with w
2 changing.The bifurcation diagram and the corresponding largest Lyapunov exponent of the delayed system (25).
(a)
w
2
=
0, w
3
=
0
(b)
w
1
=
0, w
3
=
0
(c)
w
1
=
0, w
2
=
0Figure16 shows the bifurcation diagrams and the largest Lyapunov exponent diagram taking α
=
0.5, β
=
0.15, γ
=
0.41. When w
1
=
w
2
=
0, system (25) also experiences the process from chaos to stability then again to the chaos as the changes of the w
3.The bifurcation diagram and the corresponding largest Lyapunov exponent of the delayed system (25).
(a)
w
2
=
0, w
3
=
0
(b)
w
1
=
0, w
3
=
0
(c)
w
1
=
0, w
2
=
0From the analysis of the Figures14–16, we conclude that the appropriate delay parameters play an important role for controlling the system from chaos to stability in system (7).Tables1, 2, and 3 show three firms’ profit and the total profit when the systems are nondelayed, delay feedback controlled, limiter controlled, and delayed. We conclude the following.(a)
The profit of upstream firm is apparently higher than the other two downstream firms’.
(b)
In Table1, although the value of α goes beyond the stable region of system (7), when w
1
=
0.2, the three firms achieve maximal profit: π
1
=
60, π
2
=
13.6473, π
3
=
17.0414, total profit ∑
‍
π
=
90.6888. In Table 2, when the value of β goes beyond the stable region, the three firms achieve maximal profit with w
2
=
0.2. While in Table 3, when the value of γ goes beyond the stable region, the three firms achieve maximal profit with w
3
=
0.2. It can be seen that the effect of selecting appropriate delay parameters is the same as applying a control on the system, which can make profit of the system maximization.
(c)
Taking the appropriate value ofk can reach the best control effect for this case where α goes beyond the stable region. This proves that the delay feedback control method is advantageous to the upstream firm.
(d)
According to the comparison of the three tables, limiter control method is also effective in controlling chaos. And it has great significance when requiring the player who wants to improve his performance to take measures without the other players’ cooperation.Table 1
Comparative analysis of oligarchs’ profits whenα
=
1.36, β
=
0.15, and γ
=
0.2.
Parameters
Nondelayed
Delay feedback control
Limiter control
Delayed
Delayed
Delayed
w
1
0
0
0
0.2
0
0
w
2
0
0
0
0
0.2
0
w
3
0
0
0
0
0
0.2
k
0
0.8
0
0
0
0
p
2
min
⁡
0
0
2.0842
0
0
0
p
3
min
⁡
0
0
2.2213
0
0
0
π
1
53.9601
60
53.9601
60
53.9601
53.9601
π
2
12.1514
13.6473
12.6415
13.6473
12.4463
12.7958
π
3
15.8208
17.0414
16.3254
17.0414
15.5874
15.3587
∑
‍
π
i
81.9322
90.6888
82.9271
90.6888
82.9938
82.1146Table 2
Comparative analysis of oligarchs’ profits whenα
=
0.5, β
=
0.32, and γ
=
0.2.
Parameters
Nondelayed
Limiter control
Delayed
Delayed
Delayed
w
1
0
0
0.2
0
0
w
2
0
0
0
0.2
0
w
3
0
0
0
0
0.2
p
2
min
⁡
0
2.0842
0
0
0
p
3
min
⁡
0
2.2213
0
0
0
π
1
60
60
60
60
60
π
2
-
16.7885
9.9633
-
15.6247
13.6473
-
13.0483
π
3
10.9234
16.5478
10.6893
17.0414
12.4387
∑
‍
π
i
54.1349
86.5111
55.0646
90.6888
59.3904Table 3
Comparative analysis of oligarchs’ profits whenα
=
0.5, β
=
0.15, and γ
=
0.41.
Parameters
Nondelayed
Limiter control
Delayed
Delayed
Delayed
w
1
0
0
0.2
0
0
w
2
0
0
0
0.2
0
w
3
0
0
0
0
0.2
p
2
min
⁡
0
2.0842
0
0
0
p
3
min
⁡
0
2.2213
0
0
0
π
1
60
60
60
60
60
π
2
9.4274
12.6782
8.8646
10.2317
13.6473
π
3
-
13.2909
13.9479
-
19.2747
-
10.3930
17.0414
∑
‍
π
i
56.1364
86.6260
49.5898
59.8387
90.6888


## 5. Conclusions

In this paper, we investigate the dynamics of a bounded rational oligopoly model which contains three players: one is the upstream monopoly who chooses his production to compete and the other two downstream oligopolies adjust their prices to maximize their expected profit. The existence and stability of equilibria, bifurcation, and chaotic behavior are analyzed in this game. In addition, the largest Lyapunov exponent and strange attractors are also applied to display the bifurcation and chaotic behavior of this system. We show that if the adjustment speeds of players are too high, then they will change the stability of equilibrium and cause a market structure to behave chaotically. Furthermore, we give long-run average profit of the three firms, which demonstrates that the equilibrium state is satisfactory to the three firms.We adopt two kinds of control methods and consider the delayed system. Then the following conclusions are obtained.(1)
Delay feedback control method and limiter control method both can control chaos and make profit increase.
(2)
From the perspective of profits, the delay feedback control method is advantageous to the upstream firm. The limiter control method is effective in preventing and controlling the two downstream firms’ profit decline.
(3)
The effect of selecting appropriate delay parameters is the same as applying a control on the system.



---

*Source: 970205-2014-08-13.xml*