Analyzing theDynamicData Sponsoring in the Case of Competing Internet Service Providers and Content Providers

With a sponsored content plan on the Internet market, a content provider (CP) negotiates with the Internet service providers (ISPs) on behalf of the end-users to remove the network subscription fees. In this work, we have studied the impact of data sponsoring plans on the decision-making strategies of the ISPs and the CPs in the telecommunications market. We develop gametheoretic models to study the interaction between providers (CPs and ISPs), where the CPs sponsor content. We formulate the interactions between the ISPs and between the CPs as a noncooperative game.We have shown the existence and uniqueness of the Nash equilibrium.We used the best response dynamic algorithm for learning the Nash equilibrium. Finally, extensive simulations show the convergence of a proposed schema to the Nash equilibrium and show the effect of the sponsoring content on providers’ policies.


Introduction
One of the principal trends on the Internet in the last few years is the explosion in demand for cellular data usage. erefore, one of the main challenges for CP is how to motivate end-users to access their content to achieve a higher profit. In addition, this increase in cellular data usage needs higher investments in wireless capacity. ISPs launched a new type of data pricing called data sponsoring to get additional revenue and to increase the capacity of their existing network architectures [1]. e critical idea of data sponsoring is to allow the CPs to pay the ISP on behalf of the end-users the network access fees. Data sponsoring plans benefit all entities in the network; the ISPs can generate more revenue with CP's subsidies, and end-users can enjoy free network access to the CPs, which increases the demand and attracts more traffic, resulting in higher income of the CP. As a real-world example, AT&T allows advertisers to sponsor video to attract more end-users to watch advertisements [2,3], and GS Shop (Korea TV) has cooperated with SK Telecom to sponsor the traffic of its application [4].
Sponsoring data have recently been subject to modeling and analysis in the literature. e authors in [5] proposed a novel joint optimization approach of a Stackelberg contract game to characterize the market-oriented model for sponsored content market and to capture the interactions among the ISPs, CPs, and end-users. ey have developed a Stackelberg game, where the ISP acts as the leader and the CPs and end-users act as the followers. In [6], the authors developed a new model to study the competition among CPs under sponsored data plans. e authors in [7] analyzed the interactions of three network entities, i.e., the end-users, the ISPs, and the CP, based on the game theory. e authors designed an effective data sponsoring control scheme using a novel dual-leader Stackelberg game model. e authors in [8,9] investigated joint sponsored and caching content under the noncooperative game. e interactions among ISP, CP, and end-users are modeled as a three-stage Stackelberg game. In [10], the authors studied the sponsored data as the noncooperative game among ISPs. e authors derived the best response of the CP and the ISP and analyzed their implications for the sponsoring strategy. e authors in [11,12] analyzed content sponsoring data from an economic point of view. ey examined the implications of sponsored data on the CPs and the end-users and identified how the sponsored data influence the CP inequality. In [13], the authors studied the sponsorship competition among CPs in the Internet content market and demonstrated that the competitions improve the welfare of the ISP and the CP. e authors in [3] considered a sponsored data market with one ISP, one CP, and a set of end-users. ey have modeled the interactions among three entities as a two-stage Stackelberg game, where the ISP and the CP act as the leaders determining the pricing and sponsoring strategies, respectively, in the first stage and the end-users act as the followers deciding on their data demand in the second stage. In [14], the authors analyzed the interaction among ISP and CPs and proposed a pricing mechanism for sponsored content that is truthful in CP's valuation. e sponsored content plan has been extensively investigated during the past few years; most papers focus on a simple model with a single ISP and a single CP interacting in a game-theoretic setting; few works study the completion between multiple CPs and multiple ISPs with content sponsoring plans. However, to the best of our knowledge, none of the current works includes the time constraint that makes sponsoring content more dynamic. e present paper moves toward this less explored direction.
e main objective of this paper is to study a sponsored content market consists of multiple CPs, multiple ISPs, and end-users, with the time constraint using game theory.
Game theory has been used to solve many problems in communication networks [15][16][17][18][19][20][21][22][23]. It has been used to propose new pricing strategies for Internet services [24,25]. Many other issues relating to wireless networks have been modeled and analyzed using game theory, such as resource allocation [26,27], power control [28,29], network routing [30], network caching [31], and security [32]. e contributions of this paper are as follows: (i) We present new features in the mathematical modeling that include sponsoring content, CPs revenues, and ISPs revenues with the time constraint. (ii) We model the interplay among ISPs as a function of two market parameters network access prices and quality of service; each ISP wants to maximize its utility. We formulate a competitive problem between ISP as a noncooperative game. (iii) We model the interplay between CPs as a function of three market parameters content access price, the credibility of content, and the number of sponsored content. e number of sponsored content is modeled as a function of time. We formulate a competitive problem between CPs as a noncooperative game.
(iv) We analytically prove the existence and uniqueness of the Nash equilibrium in the noncooperative game between ISPs and between CPs, which means that there exists a stable state where all providers do not have an incentive to change their strategies. e best response algorithm is used to find the Nash equilibrium point.
(v) Numerical analysis shows the effect of sponsoring content on providers' policies.
is paper is organized as follows. Section 2 discusses the system model with temporality constraint. We prove the existence and uniqueness of a Nash equilibrium point in Section 3.
en, we present a numerical investigation in Section 4, and we conclude this paper in Section 5.

Problem Modeling
In our setting, we consider a telecommunication network with three types of actors: ISPs, CPs, and end-users. e ISP provides the network infrastructure to the end-users. e CPs provide N content for end-users and sponsor a fraction of content on behalf of the end-users to lower the network access price. e ISP k sets two decision parameter network access price p s k and quality of service (QoS) q s k . Let p c f and c f , respectively, be the content access price and the credibility of content decided by CP f . End-users behavior is a function of CP and ISP policies (see (1)).

Demand
Model. D fk is the demand of end-users for the content provided by CP f and transferred by ISP k which is a function of content access price p c f , credibility of content c f , network access price p s k , and QoS q s k (see [33,34]).
is demand function is also a function of prices p c −f , credibilities c −f , prices p s −k , and QoSs q s −k set by the opponents. e demand D fk is decreasing with respect to p c f and p s k and increasing with respect to p c g , g ≠ f and p s j , j ≠ k, whereas it is increasing with respect to c f and q s k and decreasing with respect to c g , g ≠ f and q s j , j ≠ k. en, the demand functions D ij can be written as follows: e parameter d fk is the potential demand of endusers. σ g f and ς g f are two positive constants representing, respectively, the responsiveness of demand D fk to content access price and credibility of CP g . Moreover, τ j g and ϱ j g are two positive constants representing, respectively, the responsiveness of demand D fk to network access price and QoS of ISP j .

Assumption 1.
e sensitivity σ verifies the following: (2) e sensitivity ς verifies the following: e sensitivity τ verifies the following: Assumption 1 means that the effect of provider policies on its demand is greater than the effect of the policies of its opponent on its demand function [24,35]. Assumption 1 will be used to prove the uniqueness of the Nash equilibrium.

Utility Model of the CP.
e utility of CP f can be modeled as follows: (5) where S fk is the fraction of content sponsored by CP f for each ISP k subscriber. Let S fk � 1 if full sponsoring is decided and S fk � 0 if CP f decides not to sponsor any content. Recall that sponsoring could be an incentive to consume more CP f content.
(1 + χ f S fk )D fk is the new demand for contents provided by CP f and distributed by ISP k , which is a function of the proportion of sponsored content S fk . e quantity χ f S fk reflects the change in the demand for the contents of the CP. χ f is a nonnegative constant. θ i is the cost to produce a unit of the credibility of content c i . p t k is the transmission is transmission fee results when the CP f forwards to the end-users the demand with credibility of content c f . e fourth term θ f c f is the cost to produce the credibility of content c f .
Credibility of content c f of CP f is a linear function of the quality of service (QoS) q ss f and the quality content (QoC) q c f , which is written as follows [16,34,36,37]: where λ and μ are nonnegative constants. e QoS is defined as the expected delay (see [17,34]). e QoC can be specified for a specific domain of content (e.g., video streaming). en, the utility of CP f is expressed as follows: 2.3. Utility Model of the ISP. e utility function of ISP k is the difference between the revenue and the fee: is the revenue of ISP f by forwarding the amount of content requests to the end-users. e fourth term υ k B k is the investment of ISP k , where υ k is a cost per unit of requested bandwidth and B k is the backhaul bandwidth. e QoS q s f is defined as the expected delay computed by the Kleinrock function (see [38,39]): this means that en, the utility of ISP f is given as follows: 2.4. Adding Temporality to the Model. We study in this section the impact of time on the number of sponsored Mobile Information Systems 3 content. We model the proportion of sponsored content by attaching it with the time parameter. is proportion is expressed in the following form: where w fk represents the speed at which CP f sponsors content for each ISP k subscriber. We notice that when t � 0, S fk � 0 and when t � ∞, S fk � 1.
e temporal analysis of the number of sponsored content can be performed in the networks; we consider ξ as a discount factor, so that a monetary unit in t years is worth e − ξt monetary units of today. CP f with a profit U CP f at time t can predict this profit over a period ranging from [0, T] as the average of the discounted revenue in this period as follows: Similarly, we have

Game Analysis
} is the set of the ISPs, P s k is the price strategy set of ISP k , and Q s k is the QoS strategy set of ISP k . We assume that the strategy spaces P s k and Q s k of each ISP k are compact and convex sets with maximum and minimum constraints; for any given ISP k , we consider as strategy spaces the closed intervals P s k � [p s k , p s k ] and Q s k � [q s k , q s k ]. Let the price vector p s � (p s 1 , . . . , p s K ) T ∈ P K s � P s 1 × P s 2 × · · · × P s K and QoS vector q s � (q s 1 , . . . , q s K ) T ∈ Q K s � Q s 1 × Q s 2 × · · · × Q s K .
3.1. Price P s Game. A NPQG in network access price is defined for fixed q s ∈ Q s as G 2 (q s ) � [K, P s k , U ISP k (., q s ) ].
Theorem 1. For each q s ∈ Q s , the game [K, P s k , U ISP k (., q s ) ] admits a unique Nash equilibrium.
Appendix A gives a proof of the above theorem. Appendix B gives a proof of the above theorem.

Price P c Game.
A NQPQG in price p c is defined for fixed q ss ∈ Q ss and q c ∈ Q c as G 1 (q ss , q c ) � [F,

Definition 3. A price vector p
Theorem 3. For each q ss ∈ Q ss and q c ∈ Q c , the game [F, P c f , U CP f (., q ss , q c ) ] admits a unique Nash equilibrium.
Appendix C gives a proof of the above theorem.

QoC Q c Game. A NQPQG in QoC is defined for a fixed
p c ∈ P c and q ss ∈ Q ss as G 1 (p c , q ss ) � [F, Q c f , Theorem 4. For each p c ∈ P c and q ss ∈ Q ss , the game , q ss , .) ] admits a unique Nash equilibrium.
Appendix D gives a proof of the above theorem. Appendix E gives a proof of the above theorem.

Definition 5. A QoS vector q
3.6. Learning Nash Equilibrium. In this section, based on our previous analysis, we introduce two distributed and iterative learning processes that convergence toward the Nash equilibrium point. In this algorithm, each provider observes the policy taken by its competitors in previous rounds and inputs them in its decision process to update its policy. erefore, the best response and Nash seeking algorithms will converge to the unique equilibrium point. e best response (BR) algorithm is known to reach equilibria for S-modular games, by exploiting the monotonicity of the best response functions. Each player fixes its desirable strategies to maximize its profit. en, each player can observe the policy taken by its competitors in previous rounds and input them in its decision process to update its policy. en, it becomes natural to accept the Nash equilibrium as the attractive point of the game. Yet, the best response algorithm requires perfect rationality and complete Mobile Information Systems information, which is not practical for real-world applications and may increase the signaling load as well. erefore, we propose an adaptive distributed learning framework to discover equilibria for the activation game based on the "Nash seeking algorithm" (NSA) with stochastic state-dependent payoffs for continuous actions. e equilibrium-learning framework is an iterative process. At each iteration t, the player I chooses its policy and obtains from the environment the realization of its payoff. e improvement of the strategy is based on the current observation of the realized payoff and previously chosen strategies. Hence, we say players learn to play an equilibrium if after a given number of iterations; the strategy profile converges to an equilibrium strategy. e proposed learning framework has the following parameters: φ f is the perturbation phase, z f is the growth rate, b f is the perturbation amplitude, and Ω f is the perturbation frequency.
is procedure is repeated for the window T.
Algorithms 1 and 2 summarize the best response learning and Nash seeking algorithm steps that each player has to perform to discover its Nash equilibrium strategy.
such as (i) E denotes a CP or ISP (ii) L refers to F or K (iii) x refers to the vector price p c , vector price p s , vector q s , vector q c , or vector q ss (iv)X f refers to the policy profile price, QoS, or QoC

Numerical Investigations
In this section, we study the telecommunication network numerically as a noncooperative game while considering the expressions of the utility functions and using the best response algorithm. We consider a network scenario that includes two ISPs and two CPs. Figures 1-5 show the convergence toward Nash equilibrium price, Nash equilibrium QoS, and Nash equilibrium QoC of all providers. Figures 1-5 demonstrate the existence and uniqueness of a Nash equilibrium point at which no providers can profitably deviate given the strategies of another opponent. So, our model ensures the existence of an equilibrium for keeping the economy stable and achieving economic growth. Table 1 gives a comparison between the two algorithms proposed for the learning of the numerical results; we notice that the algorithm of best response gives the same results as the Nash seeking algorithm but in fewer iterations and in a very small time compared with a Nash seeking algorithm. e effect of the parameter χ on the QoS and the QoC is shown in Figures 6 and 7. e QoS and the QoC of the proposed model are growing as χ increases. When χ increases, the demand of end-users increases, and then, the revenue of CPs increases. As a result, the CPs increase their QoS and QoC to attract more end-users. Figures 8-10 represent the impact of sponsoring price p u on the content access price, the QoS, and the QoC of the two CPs. As the sponsoring price increases, the content access price p c increases, the QoC decreases, and the QoS decreases. When the sponsoring price is low, the CP invests (1) Initialize vectors x(0) � [x 1 (0), . . . , x F (0)] randomly; (2) For each E f , f ∈ L at time instant t compute: ( (4) Else, t ⟵ t + 1 and go to step (2). ALGORITHM 1: Best response algorithm.
2π]: perturbation phase; (ii) b f > 0: perturbation amplitude; (iii) Ω f : perturbation phase; (iv) z f : the growth rate; . . , τ * F (0)] randomly; (4) Learning Pattern. For each iteration t: (5) Observes the payoff U E f (x(t)) and estimates τ * (t + 1) using ALGORITHM 2: Nash seeking algorithm. 6 Mobile Information Systems     Mobile Information Systems more to offer better QoS, better QoC, and low content access price to induce increased demand from end-users. On the other hand, when the sponsoring price is high, the CP increases their content access price and decreases their QoS and QoC to compensate the increase in the sponsoring price. e impact of sponsoring price p u on the network access price and QoS of the two ISPs is illustrated in Figures 11 and   12. Figures 11 and 12 show that the network access price decreases and the QoS increases when sponsoring price increases. e reason is that as sponsoring price increases, the revenue of sponsoring increases, which leads to a rise in the income of the ISPs. erefore, the ISPs decrease their network access price and invest for more bandwidth to increase their QoS to attract more end-users.  Figures 13 and 14 show the influence of sponsoring content speed w, respectively, on network access price and QoS equilibrium. From the two figures, we notice that when the speed of sponsoring content increases, the number of sponsored content increases and then the revenue of ISPs increases. erefore, the ISP needs to lower the price and improve the QoS to induce increased demand from endusers.
We plot in Figures 15-17, respectively, the interplay of the speed of sponsoring content w on the content access price, the QoC, and the QoS at Nash equilibrium, for both CPs that we consider in this example. On the one hand, we note that the equilibrium content access price for both CPs is increasing with respect to the speed of sponsored content. On the other hand, we indicate that the equilibrium QoS and QoC for all CPs is decreasing with the speed of sponsored content. When the speed of sponsoring content is low, the CPs invest more to offer better QoS, QoC, and an attractive content access price. However, as the speed of sponsoring content increases, the CPs choose to raise their content access price and decrease their QoS and QoC to compensate the rise in the sponsoring price.

Conclusion
In this paper, we study the data sponsoring problem with time constraint in the Internet market with multiple ISPs, multiple CPs, and a set of end-users. e interaction among ISPs and among CPs is investigated by using the noncooperative game. en, we have proved the existence and uniqueness of the Nash equilibrium. is result is significant because it implies that a stable solution with suitable economic incentives in collaborative data sponsoring is feasible in the Internet paradigm. In addition, we describe a learning mechanism that allows each provider to discover accurately and rapidly its equilibrium policies. At last, we have presented a numerical investigation to validate the proposed approach, and we found that the sponsoring content has a negative effect on the strategies of CPs and positive one on the strategies of ISPs and to motivate the CPs to sponsor more content to reduce the cost of sponsorship in a long term. e second derivative of the utility function is negative, and then the utility function is thus concave, which ensures existence of a Nash equilibrium in the game G 2 (q s ).