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

Ait Omar, Driss; Garmani, Hamid; Es-Sabery, Fatima; El Amrani, Mohamed; Azougaghe, Es-Said; Baslam, Mohamed

doi:https://doi.org/10.1155/2022/9541887

Journal of Computer Networks and Communications

On this page

Abstract Introduction System Model Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 9541887 | https://doi.org/10.1155/2022/9541887

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

Driss Ait Omar,¹Hamid Garmani,¹Fatima Es-Sabery,¹Mohamed El Amrani,¹Es-Said Azougaghe,¹and Mohamed Baslam¹

Academic Editor: Farooque Azam

Received12 Mar 2022

Revised18 Aug 2022

Accepted09 Sept 2022

Published11 Oct 2022

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.

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 Section 2, 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. Each chooses a network access price and the QoS . Subscribers’ behavior is a function of ISPs’ strategies, see (1). The parameters used in this paper are presented in Table 1.

We 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 of is affected by both network access price and QoS. We model the number of subscribers served by as in References [19, 20]: is the potential demand of subscribers. and are the responsiveness to price and QoS of . For , the number of subscribers increases in QoS and decreases in . In addition, is decreasing in network access price and increasing in .

Assumption 1. The sensitivity verifies:The utility function of is written as follows:where is the income from network access. is the transmission revenue. is the unit cost of bandwidth. is the backhaul bandwidth of which is expressed as follows [21]:Then, the utility function is given by

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 for is as follows:we assume , then (6) becomes:from (7), we obtain the following reaction function for :

We consider that the uses bounded rationality expectation; hence, he builds its decision based on the expected marginal payoff . Then, the dynamical equation of is as follows:where is the speed of adjustment.

The dynamical equation of the adaptative player is as follows:where is a speed of adjustment. is the response function.

Then, the dynamical game with heterogeneous ISPs has the following form:Then (11) becomeswhere ,

Setting in (12), we obtain the following:

Solving system (12), we get two equilibrium points as follows:where is a fixed point while is a Nash equilibrium point. For economic significance, the equilibrium points and should be positive. According to Assumption 1, we have , so, the equilibrium points and have economic meaning if and .

The Jacobian matrix of the dynamic system is as follows (12):

Theorem 1. The boundary equilibrium point is unstable.
The Jacobian matrix (15) at is as follows:whose eigenvalues are and . We have and . Therefore, the fixed point is unstable.

Theorem 2. The Nash equilibrium point is locally asymptotically stable.
The Jacobian at is as follows:The Nash equilibrium point is asymptotically stable if and only if all the roots of the characteristic equationhave magnitudes of eigenvalues less than one, in which(i).(ii).According to [22], the necessary and sufficient condition for the local stability of the Nash equilibrium point is as follows:(1),(2),(3).

3.1. Numerical Result

Now, we make numerical simulations to show the dynamic behavior of the communication market by taking the parameter values in Table 2.

Figures 1 and 2 describe the bifurcation diagram of price and with the change of the speed adjustment . When is small, the Nash equilibrium is locally stable. As increased, the Nash equilibrium became unstable.

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. The may not push adjustment speed too fast to keep the market in a stable situation.

Figures 3 and 4 show the bifurcation diagram of prices and as a function of the transmission fee . When the transmission fee is small, the market is stable. With the increase in the transmission fee, the market becomes unstable.

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 when 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 , we get the controlled system as follows:

Figures 5 and 6 show the bifurcation diagram of and as a function of . The dynamical system is in a chaotic state when . When , system (12) is controlled in a stable state.

4. Joint QoS Price Game

As in the previous section, we consider that the uses bounded rationality expectation; hence, the dynamical equation of is as follows:where and are the speeds of adjustment.

The uses adaptative expectation; hence, the price and the QoS of are given as follows:where and are the speeds of adjustment of .

Then, the dynamical price QoS game has the following form:where and .

Thus, the dynamical game price QoS game becomes as follows:where , , and .

There exist four fixed points of (23).where , , and are fixed points, while is a Nash equilibrium point.

The Jacobian matrix of system (23) has the following form:

Theorem 3. The fixed point is unstable.
The Jacobian matrix (25) at is as follows:whose eigenvalues are , , , and . We have , , and . Therefore, the fixed point is unstable.

Theorem 4. The fixed point is unstable.
The Jacobian matrix (25) at is as follows:whose eigenvalues are , , , and . We have , , , and . Therefore, the fixed point is unstable.

Theorem 5. The fixed point is unstable.
The Jacobian matrix (25) at is as follows:whose eigenvalues are , , , and . We have , , , and . Therefore, the fixed point is unstable.

Theorem 6. The Nash equilibrium point is locally asymptotically stable.
The Jacobian matrix (25) at is as follows:The characteristic polynomial of Jacobian matrix (29) is as follows:where(i),(ii)(iii)(iv)According to [22], the Nash equilibrium point is local stability if(1),(2),(3),(4),(5).

4.1. Numerical Result

We present in this section some various numerical investigations of dynamic systems (22). We just focus on the Nash equilibrium which is more importantly endowed with economic implications.

Figures 7-10 show the bifurcation diagram with the and . In all these figures, the Nash equilibrium points are locally stable for small values of the and . When the and increase, a 2-cycle, 4-cycle, and chaotic behaviors occur.

Figures 11 and 12 show the bifurcation diagrams of price with respect to the transmission fee . In Figures 11 and 12, the Nash equilibrium is locally stable only when . The dynamic system is (23) in bifurcation or chaos if the transmission fee is small.

Figure 13 illustrates the bifurcation diagram of QoS as a function of the transmission fee . Figure 13 shows that the ISP QoS are locally stable for . Within , a chaotic phenomenon occurs.

4.2. Chaos Control

We introduce control parameters and in the dynamic system (22), then we have the following:

Figures 14-17 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.

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.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

A. A. Cournot, Recherches Sur Les Principes Mathématiques De La Théorie Des Richesses, Creative Media Partners, LLC, Sacramento, CA, USA, 1838.
J. Bertrand, “Book review of theorie mathematique de la richesse social and of recherches sur les principes mathematiques de la theorie des richesses,” Journal Des Savants, 1883.
View at: Google Scholar
C. Rusescu and M. D. Roman, “Dynamic behaviour in a bertrand model with bounded rationality,” in Proceedings of the 7th BASIQ International Conference on New Trends in Sustainable Business and Consumption, pp. 310–316, Bucharest, Romania, August 2021.
View at: Google Scholar
S. S. Askar and A. Al-khedhairi, “Analysis of nonlinear duopoly games with product differentiation: stability, global dynamics, and control,” Discrete Dynamics in Nature and Society, vol. 2017, Article ID 2585708, 13 pages, 2017.
View at: Publisher Site | Google Scholar
W. Zhou, Y. Cao, A. Elsonbaty, A. A. Elsadany, and T. Chu, “Bifurcation analysis of a bounded rational duopoly game with consumer surplus,” International Journal of Bifurcation and Chaos, vol. 31, no. 7, Article ID 2150097, 2021.
View at: Publisher Site | Google Scholar
Y. Dou and X. Liu, “Complex dynamics in a mixed duopoly game based on relative profit maximization,” Asian Journal of Probability and Statistics, vol. 10, pp. 47–57, 2021.
View at: Publisher Site | Google Scholar
X. Xi and Y. Zhang, “Complexity analysis of production, fertilizer-saving level, and emission reduction efforts decisions in a two-parallel agricultural product supply chain,” Chaos, Solitons & Fractals, vol. 152, Article ID 111358, 2021.
View at: Publisher Site | Google Scholar
C. Liu, R.-C. He, W. Zhou, and H. Li, “Dynamic analysis of airline bidding game based on nonlinear cost,” Physica A: Statistical Mechanics and Its Applications, vol. 565, Article ID 125547, 2021.
View at: Publisher Site | Google Scholar
X. Chen and J. Zhou, “The complexity analysis and chaos control in omni-channel supply chain with consumer migration and advertising cost sharing,” Chaos, Solitons & Fractals, vol. 146, Article ID 110884, 2021.
View at: Google Scholar
H. Garmani, M. El Amrani, M. Baslam, R. El Ayachi, and M. Jourhmane, “A Stackelberg game-based approach for interactions among internet service providers and content providers,” Netnomics: Economic Research and Electronic Networking, vol. 20, no. 2-3, pp. 101–128, 2019.
View at: Publisher Site | Google Scholar
M. El Amrani, H. Garmani, D. Ait Omar, M. Baslam, and B. Minaoui, “Analyzing the dynamic data sponsoring in the case of competing internet service providers and content providers,” Mobile Information Systems, vol. 2021, Article ID 6629020, 16 pages, 2021.
View at: Publisher Site | Google Scholar
P. Maillé and B. Tuffin, “Non-neutrality pushed by big content providers,” in Proceedings of the International Conference on the Economics of Grids, Clouds, Systems, and Services, pp. 29–39, Springer, Berlin, Germany, October 2017.
View at: Google Scholar
E. E. Tsiropoulou, P. Vamvakas, and S. Papavassiliou, “Joint customized price and power control for energy-efficient multi-service wireless networks via s-modular theory,” IEEE Transactions on Green Communications and Networking, vol. 1, no. 1, pp. 17–28, 2017.
View at: Publisher Site | Google Scholar
H. Garmani, D. Ait Omar, M. El Amrani, M. Baslam, and M. Jourhmane, “The effect of caching on cp and isp policies in information-centric networks,” Mobile Information Systems, vol. 2020, Article ID 8895271, 18 pages, 2020.
View at: Publisher Site | Google Scholar
H. Garmani, D. A. Omar, M. E. Amrani, M. Baslam, and M. Jourhmane, “Nash bargaining and policy impact in emerging isp-cp relationships,” International Journal of Ad Hoc and Ubiquitous Computing, vol. 35, no. 3, pp. 117–135, 2020.
View at: Publisher Site | Google Scholar
S. Handouf, E. Sabir, and M. Sadik, “Hierarchical availability in flying radio access networks as a cournot duopoly,” Ubiquitous Networking, Springer International Publishing, Berlin, Germany, pp. 257–266, 2018.
View at: Google Scholar
S. Handouf and E. Sabir, “Strategic availability and cost-effective UAV-based flying access networks: S-modular game analysis,” Mobile Information Systems, vol. 2019, Article ID 4897171, 11 pages, 2019.
View at: Publisher Site | Google Scholar
F. Malik, M. K. Hanawal, and Y. Hayel, “Profit sharing contracts between content and service providers for enhanced network quality,” IEEE Transactions on Network and Service Management, 2022.
View at: Google Scholar
M. Hajimirsadeghi, N. B. Mandayam, and A. Reznik, “Joint caching and pricing strategies for popular content in information centric networks,” IEEE Journal on Selected Areas in Communications, vol. 35, no. 3, pp. 654–667, 2017.
View at: Publisher Site | Google Scholar
S. Koulali, E. Sabir, T. Taleb, and M. Azizi, “A green strategic activity scheduling for UAV networks: a sub-modular game perspective,” IEEE Communications Magazine, vol. 54, no. 5, pp. 58–64, 2016.
View at: Publisher Site | Google Scholar
R. El Azouzi, E. Altman, and L. Wynter, “Telecommunications network equilibrium with price and quality-of-service characteristics,” Teletraffic Science and Engineering, Elsevier, Amsterdam, Netherlands, pp. 369–378, 2003.
View at: Google Scholar
X. Li, C. Mou, W. Niu, and D. Wang, “Stability analysis for discrete biological models using algebraic methods,” Mathematics in Computer Science, vol. 5, no. 3, pp. 247–262, 2011.
View at: Publisher Site | Google Scholar
H. Garmani, D. A. Omar, M. El Amrani, M. Baslam, and M. Jourhmane, “Analysis of a dynamics duopoly game with two content providers,” Chaos, Solitons & Fractals, vol. 131, Article ID 109466, 2020.
View at: Google Scholar
X. Yang, Y. Peng, Y. Xiao, and X. Wu, “Nonlinear dynamics of a duopoly Stackelberg game with marginal costs,” Chaos, Solitons & Fractals, vol. 123, pp. 185–191, 2019.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Driss Ait Omar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

763

Downloads

763

Citations