# Chaotic Dynamics in Joint Price QoS Game with Heterogeneous Internet Service Providers

Authors: Driss Ait Omar; Hamid Garmani; Fatima Es-Sabery; Mohamed El Amrani; Es-Said Azougaghe; Mohamed Baslam
Journal: Journal of Computer Networks and Communications
 (2022)

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## Abstract

This paper tries to investigate the complex characteristics of the communication market where Internet service providers (ISP) lease network access services and compete to serve a large pool of subscribers. For this purpose, we analyze the dynamics of a mixed duopoly game with two decision parameters: price and quality of service (QoS). We calculate and discuss the stability of each equilibrium solution by using the nonlinear system. A numerical simulation is used to show the flip bifurcation to chaos by the decisions of ISPs with different statuses. We discovered that the Nash equilibrium loses stability when the speed of adjustment and transmission fee increase. We show that the system parameter changes the stability of the communication market. In addition, we use a control method to keep the communication market in a stable state.


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## Body

## 1. Introduction

Oligopoly is a market mechanism, with a few players producing homogeneous products. These players consider the market demand and the decisions of their opponents. The Cournot model [1] is a duopoly game between two players, where the player uses naïve expectations. Bertrand in Reference [2] presented the Bertrand model with two players, in which the players’ decisions compete with price competition.In recent years, many works studied the dynamics of games and tremendous efforts have been devoted to investigating the complex nonlinear dynamic systems with bounded rationality behaviors. Several expectations have been proposed such as naïve expectation, adaptive expectation, and bounded rationality. In Reference [3], the authors studied a repeated Bertrand duopoly model with bounded rational players. The authors in reference [4] used a bounded rationality mechanism to study a dynamic game with a dynamical map. In Reference [5], the authors considered the Cournot model of consumer surplus with bounded rationality. They show that the system goes into chaos throughout flip bifurcation. The authors in Reference [6] studied the complex behavior of a duopoly game with two parameters: price and quantity. The authors considered that each player maximizes its expected profit with bounded rationality and adaptive expectation. In Reference [7], the authors investigated the dynamic game of agricultural product supply chain with bounded rationality. The authors in Reference [8] used nonlinear dynamics and game theory to investigate the dynamical behavior of airline bidding games with bounded rationality. In Reference [9], the authors established a dynamic model of a supply chain. The authors used nonlinear dynamic theory to investigate the stability of dynamic models. In addition, the authors controlled the dynamic game process based on an adaptive control method.In the communication market, many ISPs aim to increase their profit by providing many services for the subscribers (X SMS, Y min Voice, Z Mo Data,…). The authors in Reference [10] analyzed the interaction between the ISPs in the communication market, where each ISP chooses price and QoS.The authors in Reference [11] analyzed the effect of content sponsoring on the price and QoS of ISPs in the communication market. The authors in Reference [12] investigated ISPs’ best strategies in terms of quality offered to a big CP in a competitive context. In Reference [13], the authors used S-modular theory to investigate the price and power control competition in wireless networks. The authors in Reference [14] investigated the impact caching in Information Centric Network by building an analytical framework with multiple ISPs for the distribution of popular content. The authors used game theory to study the interaction between ISPs. In Reference [15], the authors investigated a non-neutral communication market where ISPs charge content providers (CP) for content distribution. The authors study the competition among ISP in two cases: (1) competitive case, where the ISP charge CP; (2) cooperative case, where CP cooperates with ISP to optimize their strategies.The authors in Reference [16, 17] investigated the interactions in price and beaconing duration between the UAV in the communication market. In Reference [18], the authors studied profit-sharing contracts between CP and ISPs in the communication market.In the literature on the communication market, most papers focused on games between ISPs adopted naive expectations to update their policies. However, we can hardly find a few papers that investigate the Bertrand game with two heterogeneous ISPs in the communication market. The present paper studies the interaction among heterogeneous ISPs in the communication market, where one ISP is rational and the second ISP is adaptative.In this paper, we investigate a communication market competition and focus on this more realistic problem. The ISPs compete to serve subscribers in terms of QoS and pricing. Based on maximizing the profits, the bounded rationality expectation rule, and adaptative expectation, this paper builds a dynamic game. In addition, we calculate and investigate the stability of the Nash equilibrium points by mathematical analysis. Through numerical results, we explore the effects of system parameters on the stability of the Nash equilibrium point. At last, we use a control method to control the chaos.The paper is organized as follows: in Section2, we describe dynamic model. In Section 3, we have presented the analysis and numerical result of the price game with bounded rationality. In Section 4, we have presented the analysis and numerical result of the joint QoS price game. In Section 5, we have presented the conclusion.


## 2. System Model

We consider a communication market with several subscribers and two ISPs using adaptative expectation and bounded rationality expectation. EachISPf chooses a network access price pf and the QoS qf. Subscribers’ behavior is a function of ISPs’ strategies, see (1). The parameters used in this paper are presented in Table 1.Table 1
Summary of notation.
NotationDescriptionpfNetwork access price ofISPf.qfQuality of service ofISPf.υfBackhaul bandwidth cost.ptfTransmission fee ofISPf.σfgSensitivity to pricepg.ςfgSensitivity to QoSqg.dfThe total potential demand of subscribers.DfDemandISPf.BfBackhaul bandwidth.αfSpeed of adjustment of the pricepf.βfSpeed of adjustment of the QoSqf.εSpeed of adjustment of the price of the adaptive player.τSpeed of adjustment of the QoS of the adaptive player.κThe controlling parameterWe assume all subscribers need the same type of service, and they achieve their demand by subscribing to one of the ISPs. The number of subscribers in service ofISPf is affected by both network access price and QoS. We model the number of subscribers served by ISPf as in References [19, 20]:(1)Df=df−σffpf+ςffqf+∑g=1,f≠g2σfgpg−ςfgqg,df is the potential demand of subscribers. σfg and ςfg are the responsiveness to price pg and QoS qg of ISPg. For ISPf, the number of subscribers Df increases in QoS qsf and decreases in qsg. In addition, Df is decreasing in network access price pf and increasing in pg.Assumption 1.
The sensitivityς verifies:(2)σff≥ςfg,∀f≠g∈1,2.
The utility function ofISPf is written as follows:(3)Πf=pfDf+ptfDf−υfBf,where pfDf is the income from network access. ptfDf is the transmission revenue. υf is the unit cost of bandwidth. Bf is the backhaul bandwidth of ISPf which is expressed as follows [21]:(4)Bf=Df+qf2.
Then, the utility function is given by(5)Πf=pf+ptfDf−υfDf+qf2.


## 3. Price Game

In the communication market, for the sake of becoming entirely rational and having complete knowledge about the communication market, the ISPs need to invest a significant cost, whereas most ISPs are not willing to do this activity. Hence, in this situation, we consider that some ISPs have incomplete knowledge about the market. In other words, the ISPs are boundedly rational. But in the communication market, a few ISPs willing to invest to grasp the market knowledge. We assume that ISPs will adopt expective expectations. In this paper, we study an asymmetric scenario with two ISPs, one ISP adopts expective expectations while the other is boundedly rational. The first-order condition forISPf is as follows:(6)∂Πf∂pf=df−2σffpf+ςffqf+σfgpg−ςfgqg−σffptf+υfσff,we assume cf=df+ςffqf−ςfgqg−σffptf+υfσff, then (6) becomes:(7)∂Πf∂pf=cf−2σffpf+σfgpg,from (7), we obtain the following reaction function for ISPf:(8)pf=cf+σfgpg2σff.We consider that theISP1 uses bounded rationality expectation; hence, he builds its decision based on the expected marginal payoff ∂Π1/∂p1. Then, the dynamical equation of ISP1 is as follows:(9)p1t+1=p1t+α1p1t∂Πp1,p2∂p1,where α1 is the speed of adjustment.The dynamical equation of the adaptative playerISP2 is as follows:(10)p2t+1=1−εp2t+εr2p1t,where ε∈0,1 is a speed of adjustment. r2p1=c2+σ21p1/2σ22 is the response function.Then, the dynamical game with heterogeneous ISPs has the following form:(11)p1t+1=p1t+α1p1t∂Πp1,p2∂p1,p2t+1=1−εp2t+εr2p1t.Then (11) becomes(12)p1t+1=p1t+αp1tc1−2σ11p1t+σ12p2t,p2t+1=1−εp2t+ε2σ22c2+σ21p1t,where c1=d1+ς11q1−ς12q2−σ11pt1+υ1σ11, c2=d2+ς22q2−ς21q1−σ22pt2+υ2σ22Settingp1t+1=p1t in (12), we obtain the following:(13)p1tc1−2σ11p1t+σ12p2t=0,−2σ22p2t+c2+σ21p1t=0.Solving system (12), we get two equilibrium points as follows:(14)E0=0,c22σ22E∗=2c1σ22+c2σ124σ11σ22−σ12σ21,c1σ21+2c2σ224σ11σ22−σ12σ21,where E0 is a fixed point while E1 is a Nash equilibrium point. For economic significance, the equilibrium points E0 and E1 should be positive. According to Assumption 1, we have 4σ11σ22−σ12σ21>0, so, the equilibrium points E0 and E∗ have economic meaning if c1>0 and c2>0.The Jacobian matrix of the dynamic system is as follows (12):(15)Jp1,p2=1+αc1−4σ11p1+σ12p2ασ12p11−εεσ212σ22.Theorem 1.
The boundary equilibrium pointE0 is unstable.
The Jacobian matrix (15) at E0 is as follows:(16)JE0=1+αc1+σ12c22σ2201−εεσ212σ22,whose eigenvalues are μ1=1+αc1+σ12c2/2σ22 and μ2=εσ21/2σ22. We have μ1>1 and μ2<1. Therefore, the fixed point E0 is unstable.Theorem 2.
The Nash equilibrium pointE1 is locally asymptotically stable.
The Jacobian atE1 is as follows:(17)JE1=1+αc1−4σ11p1∗+σ12p2∗ασ12p1∗1−εεσ212σ22.
The Nash equilibrium pointE1 is asymptotically stable if and only if all the roots of the characteristic equation(18)Pμ=μ2−aμ+b=0,have magnitudes of eigenvalues less than one, in which(i)
a=1+αc1−4σ11p1∗+σ12p2∗+εσ21/2σ22.(ii)
b=εσ21/2σ22+αεσ21/2σ22c1+σ12p2∗−αp1∗/2σ224αεσ11σ21+2α1−εσ12σ22.
According to [22], the necessary and sufficient condition for the local stability of the Nash equilibrium point E1 is as follows:(1)
1−a+b>0,(2)
1+a+b>0,(3)
1−b>0.


### 3.1. Numerical Result

Now, we make numerical simulations to show the dynamic behavior of the communication market by taking the parameter values in Table2.Table 2
Setting used for numerical simulations.
σ11=ς22σfg=ςfg,f≠gd1d20.90.12020υ1=υ2pt1=pt2q1=q2ε0.512200.5Figures1 and 2 describe the bifurcation diagram of price p1 and p2 with the change of the speed adjustment α. When α is small, the Nash equilibrium E1 is locally stable. As α increased, the Nash equilibrium became unstable.Figure 1
Bifurcation diagrams ofp1 with respect to α.Figure 2
Bifurcation diagrams ofp2 with respect to α.So now we can make a short conclusion that the system or we can say the communication market can be in a stable Nash equilibrium, but as the adjustment speed increases, the system will go into chaos, and the price of two ISPs will be unstable, and the market will be in chaos. TheISP1 may not push adjustment speed too fast to keep the market in a stable situation.Figures3 and 4 show the bifurcation diagram of prices p1 and p2 as a function of the transmission fee pt. When the transmission fee is small, the market is stable. With the increase in the transmission fee, the market becomes unstable.Figure 3
Bifurcation diagrams ofp1 with respect to pt.Figure 4
Bifurcation diagrams ofp2 with respect to pt.The flip bifurcation describes the communication market from a stable state to chaos. According to the previous results, the transmission fee plays an essential role. The communication market is more stable whenpt is less. If the ISPs increase the transmission fee without limit, the stability of the communication market will be destroyed.


### 3.2. Chaos Control

The numerical results demonstrate that when the adjustment of speed and the transmission fee increases, a chaotic behavior occurs. In this section, we use a control method [23, 24] to control the chaos of system (12). By introducing a control parameter κ, we get the controlled system as follows:(19)p1t+1=p1t+αp1tκc1−2σ11p1t+σ12p2t,p2t+1=1−εp2t+ε2σ22c2+σ21p1t.Figures5 and 6 show the bifurcation diagram of p1 and p2 as a function of κ. The dynamical system is in a chaotic state when κ<0.2. When κ>0.45, system (12) is controlled in a stable state.Figure 5
Bifurcation diagrams ofp1 with respect to κ.Figure 6
Bifurcation diagrams ofp2 with respect to κ.


## 3.1. Numerical Result

Now, we make numerical simulations to show the dynamic behavior of the communication market by taking the parameter values in Table2.Table 2
Setting used for numerical simulations.
σ11=ς22σfg=ςfg,f≠gd1d20.90.12020υ1=υ2pt1=pt2q1=q2ε0.512200.5Figures1 and 2 describe the bifurcation diagram of price p1 and p2 with the change of the speed adjustment α. When α is small, the Nash equilibrium E1 is locally stable. As α increased, the Nash equilibrium became unstable.Figure 1
Bifurcation diagrams ofp1 with respect to α.Figure 2
Bifurcation diagrams ofp2 with respect to α.So now we can make a short conclusion that the system or we can say the communication market can be in a stable Nash equilibrium, but as the adjustment speed increases, the system will go into chaos, and the price of two ISPs will be unstable, and the market will be in chaos. TheISP1 may not push adjustment speed too fast to keep the market in a stable situation.Figures3 and 4 show the bifurcation diagram of prices p1 and p2 as a function of the transmission fee pt. When the transmission fee is small, the market is stable. With the increase in the transmission fee, the market becomes unstable.Figure 3
Bifurcation diagrams ofp1 with respect to pt.Figure 4
Bifurcation diagrams ofp2 with respect to pt.The flip bifurcation describes the communication market from a stable state to chaos. According to the previous results, the transmission fee plays an essential role. The communication market is more stable whenpt is less. If the ISPs increase the transmission fee without limit, the stability of the communication market will be destroyed.


## 3.2. Chaos Control

The numerical results demonstrate that when the adjustment of speed and the transmission fee increases, a chaotic behavior occurs. In this section, we use a control method [23, 24] to control the chaos of system (12). By introducing a control parameter κ, we get the controlled system as follows:(19)p1t+1=p1t+αp1tκc1−2σ11p1t+σ12p2t,p2t+1=1−εp2t+ε2σ22c2+σ21p1t.Figures5 and 6 show the bifurcation diagram of p1 and p2 as a function of κ. The dynamical system is in a chaotic state when κ<0.2. When κ>0.45, system (12) is controlled in a stable state.Figure 5
Bifurcation diagrams ofp1 with respect to κ.Figure 6
Bifurcation diagrams ofp2 with respect to κ.


## 4. Joint QoS Price Game

As in the previous section, we consider that theISP1 uses bounded rationality expectation; hence, the dynamical equation of ISP1 is as follows:(20)p1t+1=p1t+αp1t∂Πp1,p2∂p1q1t+1=q1t+βq1t∂Πp1,p2,q1,q2∂q1,where α and β are the speeds of adjustment.TheISP2 uses adaptative expectation; hence, the price and the QoS of ISP1 are given as follows:(21)p2t+1=1−εp2t+εr3p1t,q1t,q2tq2t+1=1−τq2t+τr4p1t,p2t,q1t,where ε and τ are the speeds of adjustment of ISP2.Then, the dynamical price QoS game has the following form:(22)p1t+1=p1t+αp1t∂Πp1,p2,q1,q2∂p1,p2t+1=1−εp2t+εr3p1t,q1t,q2t,q1t+1=q1t+βq1t∂Πp1,p2,q1,q2∂q1,q2t+1=1−τq2t+τr4p1t,p2t,q1t,where r3p1t,q1t,q2t=n2+σ21p1t+ς22q2t−ς21q1t/2σ22 and r4p1t,p2t,q1t=m2+σ22p2t/2υ2.Thus, the dynamical game price QoS game becomes as follows:(23)p1t+1=p1t+αp1tn1−2σ11p1t+σ12p2t+ς11q1t−ς12q2t,p2t+1=1−εp2t+ε2σ22c2+σ21p1t+ς22q2t−ς21q1t,q1t+1=q1t+βq1tm1+σ11p1t−2υ1q1t,q2t+1=1−τq2t+τ2υ2m2+σ22p2t,where n1=d1−σ11pt1+υ1σ11, n2=d2−σ22pt2+υ2σ22, m1=ς11pt1−ς11υ1 and m2=ς22pt2−ς22υ2.There exist four fixed points of (23).(24)E2=0,n2ς222σ22,0,n2+2m2+ς224υ2E3=0,2n2υ1υ2−m1υ2ς21+m2υ1ς22υ1σ224υ2−ς22,m12υ1,2n2υ1−m1ς21+4m2υ12υ14υ2−ς22E4=4n1υ2σ22−n1σ22ς22+2n2υ2σ12−n2σ22ς12+m2σ12ς22−2m2σ22ς128υ2σ11σ22−2υ2σ12σ21−2σ11σ22ς22+σ21σ22ς12,2n1υ2σ21+4n2υ2σ11+2m2σ11ς22−m2σ21ς128υ2σ11σ22−2υ2σ12σ21−2σ11σ22ς22+σ21σ22ς12,0,n1σ21σ22+2n2σ11σ22+4m2σ11σ22−m2σ12ς218υ2σ11σ22−2υ2σ12σ21−2σ11σ22ς22+σ21σ22ς12E5=−4m2υ1σ22ς11−2n2υ1σ22ς12−2n1σ22υ1ς22+8n1σ22υ1υ2+4n2υ1υ2σ12+2m2σ12υ1ς22−m1σ22ς11ς22+4m1σ22υ2ς11+m1σ22ς12ς21−2m1υ2σ12ς21σ11σ22ς11ς22−4υ2σ11σ22ς11−4υ1σ11σ22ς22+16υ1υ2σ11σ22−σ11σ22ς12ς21+2υ2σ11σ12ς21+2υ1σ21σ22ς12−4υ1υ2σ12σ21,2m1υ2σ21ς11−2n2υ2σ11ς11+8n2υ1υ2σ11−4υ2m1σ11ς21−2υ2n1σ11ς21+4υ2n1υ1σ21−m2σ11ς11ς22+4m2υ1σ11ς22+m2σ11ς12ς21−2m2υ1σ21ς12σ11σ22ς11ς22−4υ2σ11σ22ς11−4υ1σ11σ22ς22+16υ1υ2σ11σ22−σ11σ22ς12ς21+2υ2σ11σ12ς21+2υ1σ21σ22ς12−4υ1υ2σ12σ21,−2m2σ11σ22ς12−n2σ11σ22ς12−n1σ22σ11ς22+4n1σ22υ2σ11−2m1σ22σ11ς22+8m1σ22υ2σ11+m1σ22σ21ς12+2n2υ2σ11σ12−2m1υ2σ12σ21+m2σ12σ11ς22σ11σ22ς11ς22−4υ2σ11σ22ς11−4υ1σ11σ22ς22+16υ1υ2σ11σ22−σ11σ22ς12ς21+2υ2σ11σ12ς21+2υ1σ21σ22ς12−4υ1υ2σ12σ21,−2m2σ11σ22ς11+8m2υ1σ11σ22+m1σ21σ22ς11−n2σ11σ22ς11+4n2υ1σ11σ22+n2σ11σ12ς21−2m2υ1σ12σ21−2m1σ11σ22ς21−n1σ11σ22ς21+2n1υ1σ21σ22σ11σ22ς11ς22−4υ2σ11σ22ς11−4υ1σ11σ22ς22+16υ1υ2σ11σ22−σ11σ22ς12ς21+2υ2σ11σ12ς21+2υ1σ21σ22ς12−4υ1υ2σ12σ21,where E2, E3, and E4 are fixed points, while E5 is a Nash equilibrium point.The Jacobian matrix of system (23) has the following form:(25)Jp1,p2,q1,q2=1+αn1−4σ11p1+σ12p2+ς11q1−ς12q2ασ12p1ας11p1−ας12p1εσ212σ221−ε−ες212σ22ες222σ22βσ11q101+βm1−4υ1q1+σ11p100τσ222υ201−τ.Theorem 3.
The fixed pointE2 is unstable.
The Jacobian matrix (25) at E2 is as follows:(26)JE2=1+αn1+σ12p2−ς12q2000εσ212σ221−ε−ες212σ22ες222σ22001+βm100τσ222υ201−τ,whose eigenvalues are μ1=1+αn1+σ12p2−ς12q2, μ2=1−ε, μ3=1+βm1, and μ4=1−τ. We have μ1>1, μ2<1, μ3>1 and μ4<1. Therefore, the fixed point E2 is unstable.Theorem 4.
The fixed pointE3 is unstable.
The Jacobian matrix (25) at E3 is as follows:(27)JE3=1+αn1+σ12p2+ς11q1−ς12q2000εσ212σ221−ε−ες212σ22ες222σ22βσ11q101+βm1−4υ1q100τσ222υ201−τ,whose eigenvalues are μ1=1+αn1+σ12p2+ς11q1−ς12q2, μ2=1−ε, μ3=1+βm1−4υ1q1, and μ4=1−τ. We have μ1>1, μ2<1, μ3<1, and μ4<1. Therefore, the fixed point E3 is unstable.Theorem 5.
The fixed pointE4 is unstable.
The Jacobian matrix (25) at E4 is as follows:(28)JE4=1+αn1−4σ11p1+σ12p2−ς12q2ασ12p1ας11p1−ας12p1εσ212σ221−ε−ες212σ22ες222σ22001+βm1+σ11p100τσ222υ201−τ,whose eigenvalues are μ1=1+αn1−4σ11p1+σ12p2−ς12q2, μ2=1−ε, μ3=1+βm1+σ11p1, and μ4=1−τ. We have μ1>1, μ2<1, μ3>1, and μ4<1. Therefore, the fixed point E4 is unstable.Theorem 6.
The Nash equilibrium pointE5 is locally asymptotically stable.
The Jacobian matrix (25) at E5 is as follows:(29)JE5=1+αn1−4σ11p1∗+σ12p2∗+ς11q1∗−ς12q2∗ασ12p1∗ας11p1∗−ας12p1∗εσ212σ221−ε−ες212σ22ες222σ22βσ11q1∗01+βm1−4υ1q1∗+σ11p1∗00τσ222υ201−τ.
The characteristic polynomial of Jacobian matrix (29) is as follows:(30)Pμ=μ4−aμ3+bμ2−cμ+d=0,where(i)
a=4+αn1−4ασ11p1∗+ασ12p2∗+ας11q1∗−ας12q2∗−ε+βm1−4βυ1q1∗+βσ11p1∗−τ,(ii)
b=2+1−ε2+αn1+βm1+βm1+αn1+αβn1m1+2+αn1−ε+βm11−τ−τες22σ22/4υ2σ22−4α1−εσ11p1∗−εασ21σ12/2σ22p1∗+βσ11p1∗−4ασ11p1∗+αβn1σ11p1∗−4αβm1σ11p1∗+16αβυ1σ12q1∗p1∗−4αβσ112p1∗2+αβσ11σ12p2∗p1∗+αβς11σ11q1∗p1∗−4α1−τσ11p1∗+β1−εσ11p1∗+β1−τσ11p1∗−αβς12σ11q2∗p1∗+α1−εσ12p2∗+ασ12p2∗+αβm1σ12p2∗−4αβυ1σ12q1∗p2∗+α1−τσ12p2∗+α1−ες11q1∗−4βυ1q1∗+ασ11q1∗−4αβn1υ1q1∗+αβσ11m1q1∗−4αβυ1σ11q1∗+4αβυ1σ12q2∗q1∗+α1−τσ11q1∗−4β1−ευ1q1∗−4αβσ11q1∗2−α1−ες12q2∗−ασ12q2∗−m1αβσ12q2∗−α1−τσ12q2∗,(iii)
c=1+αn1−4σ11p1∗+σ12p2∗+ς11q1∗−ς12q2∗1−ε1+βm1−4υ1q1∗+σ11p1∗−εασ12σ21p1∗/2σ221+βm1−4υ1q1∗+σ11p1∗−βσ11q1∗εας21σ12p1∗+α2σ22ς11p1∗1−ε/2σ22+1+αn1−4σ11p1∗+σ12p2∗+ς11q1∗−ς12q2∗1−τ1+βm1−4υ1q1∗+σ11p1∗−βας11σ111−τq1∗p1∗+1−τ1−ε1+βm1−4υ1q1∗+σ11p1∗−τεσ22ς22/4υ2σ221+βm1−4υ1q1∗+σ11p1∗(iv)
d=1+αn1−4σ11p1∗+σ12p2∗+ς11q1∗−ς12q2∗1−τ1−ε1+βm1−4υ1q1∗+σ11p1∗−τεσ22ς22/4υ2σ221+βm1−4υ1q1∗+σ11p1∗−εσ21/2σ22ασ12p1∗1−τ1+βm1−4υ1q1∗+σ11p1∗+τασ22ς12p1∗/2υ21+βm1−4υ1q1∗+σ11p1∗+βσ11q1∗−ασ12p1∗1−τες21/2σ22−ας11p1∗1−ε1−τ+ταε/4υ2ς11ς22−ς21ς12p1∗.
According to [22], the Nash equilibrium point E5 is local stability if(1)
1+a+b+c+d>0,(2)
1−a+b−c+d>0,(3)
1−d1−d4−b1−d2+a−cc−ad>0,(4)
3+3d>b,(5)
d<1.


### 4.1. Numerical Result

We present in this section some various numerical investigations of dynamic systems (22). We just focus on the Nash equilibrium E5 which is more importantly endowed with economic implications.Figures7-10 show the bifurcation diagram with the α and β. In all these figures, the Nash equilibrium points E5 are locally stable for small values of the α and β. When the α and β increase, a 2-cycle, 4-cycle, and chaotic behaviors occur.Figure 7
Bifurcation diagrams ofp1 with respect to α.Figure 8
Bifurcation diagrams ofp2 with respect to α.Figure 9
Bifurcation diagrams ofq2 with respect to α.Figure 10
Bifurcation diagrams ofq1 with respect to β.Figures11 and 12 show the bifurcation diagrams of price with respect to the transmission fee pt. In Figures 11 and 12, the Nash equilibrium E5 is locally stable only when pt>8.5. The dynamic system is (23) in bifurcation or chaos if the transmission fee is small.Figure 11
Bifurcation diagrams ofp1 with respect to pt.Figure 12
Bifurcation diagrams ofp2 with respect to pt.Figure13 illustrates the bifurcation diagram of QoS as a function of the transmission fee pt. Figure 13 shows that the ISP QoS are locally stable for pt≤3. Within pt>3, a chaotic phenomenon occurs.Figure 13
Bifurcation diagrams ofq1 with respect to pt.


### 4.2. Chaos Control

We introduce control parametersΥ and Φ in the dynamic system (22), then we have the following:(31)p1t+1=p1t+αp1tΥn1−2σ11p1t+σ12p2t+ς11q1t−ς12q2t,p2t+1=1−εp2t+ε2σ22c2+σ21p1t+ς22q2t−ς21q1t,q1t+1=q1t+βq1tΦm1+σ11p1t−2υ1q1t,q2t+1=1−τq2t+τ2υ2m2+σ22p2t.Figures14-17 illustrate the bifurcation diagram of price as a function of control parameters. When the control parameters increase, the controlled systems (31) are controlled from the chaotic state to a stable state.Figure 14
Bifurcation diagrams ofp1 with respect to ϒ.Figure 15
Bifurcation diagrams ofp2 with respect to ϒ.Figure 16
Bifurcation diagrams ofq2 with respect to ϒ.Figure 17
Bifurcation diagrams ofq1 with respect to Φ.So now we can make a short conclusion that the communication market can be in a stable Nash equilibrium, but with the changing of the value system parameters, the system will go into chaos, the policies of the ISPs will be unstable, and the communication market will be in chaos. The ISPs need to choose a low value of system parameters or apply a control method to stabilize the chaotic behaviors to keep the communication market in a stable situation.


## 4.1. Numerical Result

We present in this section some various numerical investigations of dynamic systems (22). We just focus on the Nash equilibrium E5 which is more importantly endowed with economic implications.Figures7-10 show the bifurcation diagram with the α and β. In all these figures, the Nash equilibrium points E5 are locally stable for small values of the α and β. When the α and β increase, a 2-cycle, 4-cycle, and chaotic behaviors occur.Figure 7
Bifurcation diagrams ofp1 with respect to α.Figure 8
Bifurcation diagrams ofp2 with respect to α.Figure 9
Bifurcation diagrams ofq2 with respect to α.Figure 10
Bifurcation diagrams ofq1 with respect to β.Figures11 and 12 show the bifurcation diagrams of price with respect to the transmission fee pt. In Figures 11 and 12, the Nash equilibrium E5 is locally stable only when pt>8.5. The dynamic system is (23) in bifurcation or chaos if the transmission fee is small.Figure 11
Bifurcation diagrams ofp1 with respect to pt.Figure 12
Bifurcation diagrams ofp2 with respect to pt.Figure13 illustrates the bifurcation diagram of QoS as a function of the transmission fee pt. Figure 13 shows that the ISP QoS are locally stable for pt≤3. Within pt>3, a chaotic phenomenon occurs.Figure 13
Bifurcation diagrams ofq1 with respect to pt.


## 4.2. Chaos Control

We introduce control parametersΥ and Φ in the dynamic system (22), then we have the following:(31)p1t+1=p1t+αp1tΥn1−2σ11p1t+σ12p2t+ς11q1t−ς12q2t,p2t+1=1−εp2t+ε2σ22c2+σ21p1t+ς22q2t−ς21q1t,q1t+1=q1t+βq1tΦm1+σ11p1t−2υ1q1t,q2t+1=1−τq2t+τ2υ2m2+σ22p2t.Figures14-17 illustrate the bifurcation diagram of price as a function of control parameters. When the control parameters increase, the controlled systems (31) are controlled from the chaotic state to a stable state.Figure 14
Bifurcation diagrams ofp1 with respect to ϒ.Figure 15
Bifurcation diagrams ofp2 with respect to ϒ.Figure 16
Bifurcation diagrams ofq2 with respect to ϒ.Figure 17
Bifurcation diagrams ofq1 with respect to Φ.So now we can make a short conclusion that the communication market can be in a stable Nash equilibrium, but with the changing of the value system parameters, the system will go into chaos, the policies of the ISPs will be unstable, and the communication market will be in chaos. The ISPs need to choose a low value of system parameters or apply a control method to stabilize the chaotic behaviors to keep the communication market in a stable situation.


## 5. Conclusion

In this paper, a communication market that consists of two ISPs is studied. We analyzed the dynamics of a Bertrand duopoly game. Each ISP maximizes its utility by using bounded rationality expectation and adaptative expectation. The existence and the stability of the equilibrium point of this dynamic Bertrand duopoly game are investigated. We showed numerically that the model gives chaotic and unpredictable trajectories. The main result is that a high value of transmission fee and a high value of the speed of adjustment may destabilize the communication market. But we also showed that for lower values of transmission fee and speed of adjustment, the communication market is stable. In addition, a control method is used to control the system. This paper presents guidance for ISPs to choose their policies.



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*Source: 9541887-2022-10-11.xml*