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

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.


Introduction
Oligopoly is a market mechanism, with a few players producing homogeneous products. ese players consider the market demand and the decisions of their opponents. e 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 e orts 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. e 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. ey show that the system goes into chaos throughout ip bifurcation. e authors in Reference [6] studied the complex behavior of a duopoly game with two parameters: price and quantity. e authors considered that each player maximizes its expected pro t with bounded rationality and adaptive expectation. In Reference [7], the authors investigated the dynamic game of agricultural product supply chain with bounded rationality. e 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. e 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 pro t by providing many services for the subscribers (X SMS, Y min Voice, Z Mo Data,. . .). e authors in Reference [10] analyzed the interaction between the ISPs in the communication market, where each ISP chooses price and QoS. e authors in Reference [11] analyzed the e ect of content sponsoring on the price and QoS of ISPs in the communication market.
e 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. e 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. e 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. e 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. e 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. e 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. e 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. rough 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. e 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.

System Model
We consider a communication market with several subscribers and two ISPs using adaptative expectation and bounded rationality expectation. Each ISP f chooses a network access price p f and the QoS q f . Subscribers' behavior is a function of ISPs' strategies, see (1). e 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. e number of subscribers in service of ISP f is affected by both network access price and QoS. We model the number of subscribers served by ISP f as in References [19,20]: (1) d f is the potential demand of subscribers. σ g f and ς g f are the responsiveness to price p g and QoS q g of ISP g . For ISP f , the number of subscribers D f increases in QoS q s f and decreases in q s g . In addition, D f is decreasing in network access price p f and increasing in p g .

Assumption 1.
e sensitivity ς verifies: e utility function of ISP f is written as follows: where p f D f is the income from network access. p t f D f is the transmission revenue. υ f is the unit cost of bandwidth. B f is the backhaul bandwidth of ISP f which is expressed as follows [21]: en, the utility function is given by

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 Table 1: Summary of notation. Notation Speed of adjustment of the price of the adaptive player.
where E 0 is a fixed point while E 1 is a Nash equilibrium point. For economic significance, the equilibrium points E 0 and E 1 should be positive. According to Assumption 1, we have 4σ 1 1 σ 2 2 − σ 2 1 σ 1 2 > 0, so, the equilibrium points E 0 and E * have economic meaning if c 1 > 0 and c 2 > 0.
e Nash equilibrium point E 1 is locally asymptotically stable.
e Jacobian at E 1 is as follows: The Nash equilibrium point E 1 is asymptotically stable if and only if all the roots of the characteristic equation have magnitudes of eigenvalues less than one, in which . According to [22], the necessary and sufficient condition for the local stability of the Nash equilibrium point E 1 is as follows:

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 p 1 and p 2 with the change of the speed adjustment α. When α is small, the Nash equilibrium E 1 is locally stable. As α 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. e ISP 1 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 p 1 and p 2 as a function of the transmission fee p t . When the transmission fee is small, the market is stable. With the increase in the transmission fee, the market becomes unstable. e flip bifurcation describes the communication market from a stable state to chaos. According to the previous results, the transmission fee plays an essential role. e communication market is more stable when p t is less. If the ISPs increase the transmission fee without     limit, the stability of the communication market will be destroyed.

Chaos Control.
e 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: Figures 5 and 6 show the bifurcation diagram of p 1 and p 2 as a function of κ. e dynamical system is in a chaotic state when κ < 0.2. When κ > 0.45, system (12) is controlled in a stable state.

Joint QoS Price Game
As in the previous section, we consider that the ISP 1 uses bounded rationality expectation; hence, the dynamical equation of ISP 1 is as follows: where α and β are the speeds of adjustment. e ISP 2 uses adaptative expectation; hence, the price and the QoS of ISP 1 are given as follows: where ε and τ are the speeds of adjustment of ISP 2 .
us, the dynamical game price QoS game becomes as follows:

Journal of Computer Networks and Communications
ere exist four fixed points of (23).
where E 2 , E 3 , and E 4 are fixed points, while E 5 is a Nash equilibrium point.
erefore, the fixed point E 3 is unstable.

Theorem 5.
e fixed point E 4 is unstable. e Jacobian matrix (25) at E 4 is as follows: 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. Figures 11 and 12 show the bifurcation diagrams of price with respect to the transmission fee p t . In Figures 11 and 12, the Nash equilibrium E 5 is locally stable only when p t > 8.5.
e 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 p t . Figure 13 shows that the ISP QoS are locally stable for p t ≤ 3. Within pt > 3, a chaotic phenomenon occurs.

Chaos Control.
We introduce control parameters Υ and Φ in the dynamic system (22), then we have the following:       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.
e 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.

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. e 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. e 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. is paper presents guidance for ISPs to choose their policies.

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