# The Effect of Caching on CP and ISP Policies in Information-Centric Networks

Authors: Hamid Garmani; Driss Ait Omar; Mohamed El Amrani; Mohamed Baslam; Mostafa Jourhmane
Journal: Mobile Information Systems
 (2020)

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## Abstract

Internet traffic volume is increasing, and this causes scalability issues in content delivery. Information-centric network has been introduced to support this increase in Internet traffic through caching. While collaborative caching in information-centric network is a crucial feature to improve network performance and reduce delivery costs in content distribution, the current pricing strategies on the Internet are not incentive compatible with information-centric network interconnection. In this paper, we focus on the economic incentive interactions in caching deployment between several types of information-centric network providers (content provider and Internet service provider). In particular, we develop game-theoretic models to study the interaction between providers in an information-centric network model where the providers are motivated to cache and share content. We use a generalized Zipf distribution to model content popularity. We formulate the interactions between the Internet service providers and between the content providers as a noncooperative game. We use a Stackelberg game model to capture the interactions between the content provider and Internet service providers. Through mathematical analysis, we prove the existence and uniqueness of the Nash equilibrium under some conditions. An iterative and distributed algorithm based on best response dynamics is proposed to achieve the equilibrium point. The numerical simulations illustrate that our proposed game models result in a win-win solution.


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## Body

## 1. Introduction

The growth of Internet traffic data is rapidly increasing due to the explosion of sites on the Internet such as YouTube, Dailymotion, and Netflix. Many technologies have been proposed to control the large volume of information, improve the spectrum efficiency, and increase network capacity. Besides traditional technologies, information-centric network (ICN) has drawn researchers’ attention in recent years, which is a new communication paradigm to increase the efficiency of content delivery. The main ideas in ICNs are the following: (1) content is located by name instead of by IP, and (2) every ICN node can cache and serve the requested content. The advantages motivating the ICN approach are scalable, persistent, security, mobility, etc [1].It is considered that network caching can get many advantages to end-users, the Internet service provider (ISP), and the content providers (CPs). For end-users, caching content in ISP cache can minimize the delay in getting content, which enhances the quality of experience. From the viewpoint of the ISPs, network caching can lower the traffic outgoing to the other ISPs or CPs, which reduce the backhaul bandwidth consumption. So, literature studies [2–4] suggest that ISPs should deploy a cache with capacity. Furthermore, for CPs, content requests can be satisfied from the ISP cache, which can lower the load on the CPs, economize backhaul bandwidth resources, and avert network congestions. More importantly, when the CP is temporarily offline, the ISP cache can continue to provide content to end-users, which improves the robustness of the network.Inspired by the above discussion, ICN has many technical benefits; however, ICN has so far stayed only in the research literature, compared to CDN or HTTP caches that have a massive deployment. The reason is that the vast providers (ISPs and CPs) need economic incentives in addition to benefits of technic to collaborate in the deployment of the ICN. This paper aims at designing a flexible economic model to prompt the deployment of ICN.In our previous work [5], we study the interaction among two ISPs in ICN, where each ISP controls the amount of content cached and adjusts their prices and its quality of service. In this paper, we extend our previous studies by considering a general framework for incentive CPs and ISPs to collaborate in caching and distributing content in ICN. We consider a hierarchical network model with multiple CPs and multiple ISPs. We use a game-theoretic approach to study the interaction between CPs and ISPs.Our main contributions are summarized as follows:(i)
We formulate the interactions among ISPs based on four market parameters, network access price, caching, cache access price, and quality of service. In addition, we formulate the competition among CPs as a function of two parameters, content access price and credibility of content. The interaction between providers is analyzed as a noncooperative game.(ii)
We use the Stackelberg game to analyze competition among CPs and ISPs, and we adopt the method backward induction to solve the proposed Stackelberg game.(iii)
We analytically prove the existence and uniqueness of the Nash equilibrium.(iv)
Through simulation, the interactions between model parameter and provider strategies are represented. Also, the impacts of model parameter on provider revenue are revealed. We finally carry out that the ISPs, the CPs, and end-users can receive benefit from caching investment.The rest of this paper is organized as follows.Section 2 discusses related work. In Section 3, we describe the system model. In Section 4, we formulate and analyze the noncooperative game. In Section 5, we model a Stackelberg game. Then, we present the numerical results in Section 6. Conclusions are provided in Section 7.


## 2. Related Work

The existing literature has studied several problems related to caching in the small cell network, cloud radio access networks, and ISPs. In [6], the authors have used the mean-field game to study the problem of distributed caching in a dense wireless small cell network. In [7], the authors proposed a joint network slicing and content caching for in-band wireless backhaul-based virtualized wireless networks, where requested contents can be retrieved from either the associated base stations or the close content cached at the edge of the core network. The authors in [8] propose an ICN-based caching approach for videos in 5G networks, which considers both the mobility of users and the popularity of videos. In [9], the authors designed a proactive caching policy for a HetNet scenario to jointly optimize the choice of which files to store at the mobile users and the small base stations. The authors in [10] studied distributed cooperative hybrid caching problems based on content-awareness in D2D networks. In [11], the authors proposed a dynamic Stackelberg–Cournot game in which multiple mobile network operators offer payable storage service to multiple CPs competing for caching space. The authors in [12] proposed an edge caching scheme for vehicular content networks to provide vehicular content delivery. An analytical model is developed, which takes account of vehicle velocity, road traffic, and content popularity for characterizing the content distribution at the edge of vehicular content networks. In [13], the authors have studied caching with asynchronous requests and presented a coded joint pushing and caching method to determine the packets to be pushed and cached. The authors in [14] proposed a new cooperative edge caching architecture for content-centric 5G networks, where mobile edge computing resources are utilized for enhancing edge caching capability. They show that the proposed scheme minimizes content access latency and improves caching resource utilization. The authors in [15] presented an interdomain cache-sharing market routing technique based on the bargaining game, which took full advantage of the cache resource to extend cache sharing among domains to Internet service providers.The second research line [3, 16–19] attracts more attention compared to the first one, as how to incentivize network players to collaborate in the deployment of the ICN. The authors in [20] explored the interdomain routing policies in ICN. They argued that the network caching will influence the business relationships between neighboring autonomous systems. The authors in [19] model the economic incentive independencies between ISPs and CP and propose a reward-based incentive caching model for content caching and sharing in ICN, where the CP decides the reward price of caching each content and ISPs decide which content to be cached. They used a Stackelberg game model to capture the interactions between ISP and CP. In [21], the authors formalized the profit of ISPs when implementing content-centric networking and investigate the likelihood of content-centric networking spreading through numerical evaluation. The authors in [22] analyzed the economic interactions in one ISP and proposed a policy to make ICN economically profitable for ISPs. In [18], the authors proposed a new incentive mechanism that improves both the ISPs’ and CPs’ profits via the use of auction theory. In the proposed model, the ISP earns profits from caching by alleviating traffic load on transit links and participating in content selling, where the contents are from multiple CPs. The authors in [23] proposed a new incentive mechanism for paid content caching that satisfies both ISP and CPs through the use of a reverse auction. The authors in [3] have modeled the economic interactions between the different actors of an information-centric network, using the framework of game theory. They show the uniqueness of the Nash equilibrium in the pricing caching game. The authors of this work show that caching is profitable. The authors in [24] proposed a pricing model to establish an economic incentive mechanism for caching and sharing content in ICN that consists of an ISP, a transit ISP, and a CP. In [25], the authors investigate a noncooperative game between an ISP and a CP, where the ISP changes its caching strategy, and the CP can control its pricing strategy. They considered that the polarity of the side-payment (from the ISP to the CP) in an ICN is different from that in the current Internet model (i.e., host-centric communication model). In [16], the authors modeled the content popularity by the generalized Zipf distribution and used the noncooperative game to decide caching and pricing strategies in information-centric network (ICN). In [26], the authors investigated an economical pricing approach to address caching resource management issues in the 5G wireless networks. They used Cournot, Stackelberg, and Bertrand models to study this model. The authors in [27] analyzed the impact of different factors on the incentives of related players in ICN, including the ISPs, CDNs, and CPs. The authors built a simple model to address the economic incentive problem of different ICN network players in a hierarchical caching infrastructure.All these related works mentioned above economics of service pricing in the ICN, which consists of an ISP and a CP. Their model restricts the network to one CP and few ISPs. Our work belongs to the second research line but differs from these existing works in the following five aspects:(1)
We developed an analytical framework for the distribution of popular content in ICN, which comprises multiple CPs, multiple ISPs, and a set of end-users.(2)
We model and analyze the interplay between ISPs as a function of four market parameters: network access price, quality of services, caching, and cache access price.(3)
We model and analyze the interplay between CPs as a function of four market parameters: content access price, the credibility of content, caching cost, and transmission price.(4)
We study a game between ISPs and among CPs as a noncooperative game. The analyzed economic interaction in our model is not involved by the existing works.(5)
We analytically prove the existence of equilibrium between CPs and between ISPs, and then we applied the best response algorithm for learning Nash equilibrium. In addition, we use the price of anarchy to evaluate the performance of the system at the Nash equilibrium.


## 3. Problem Modeling

The system model is illustrated in Figure1, with F ISPs, N CPs, and several end-users. These entities have diverse roles in ICN, the CPs produce content, the ISP provides access service or connects end-users, and end-users consume content. The strategies of each CP consists of four parts, the price for delivering content to end-users, the price for selling content to ISPs, the price for forwarding content trough ISP, and the credibility of content, which is a function of the quality of service and quality of content. Each ISP can cache the entire or portion of all content provided by CPs. Also, each ISP chooses five market parameters: network access price, the price to access to content in the cache, caching, the credibility of content in the cache, and the quality of service. A detailed summary of notations is presented in Table 1.Figure 1
Interaction between different providers in a simplified model of ICN.Table 1
Summary of notation.
NotationDescriptionNNumber of CPsHNumber of items that the CP sellsFNumber of ISPspsjNetwork access price ofISPjpccjPrice to access to the content in the cache ofISPjqsjQuality of service ofISPjcjCredibility of content ofISPjCihCaching cost of itemh provided by CPivjBackhaul bandwidth costpciPrice to access to content ofCPiciCredibility of content ofCPiptiTransmission fee ofCPiαjfSensitivity ofISPj to price psf of ISPfβjfSensitivity ofISPj to QoS qsf of ISPfρjfSensitivity ofISPj to price pccf of ISPfδinSensitivity ofCPi to price pcn of CPnσinSensitivity ofCPi to credibility of content cn of CPndijThe total potential demand of end-usersDijDemand ofCPi through ISPjBjBackhaul bandwidthhηThe rank of itemhηThe skewness of the popularity distributionλThe sensitivity of the credibility of content to the quality of serviceμThe sensitivity of the credibility of content to the quality of contentθiThe cost to produce a unit of credibilityci


### 3.1. Content Popularity

LetH be the number of items provided by CPs, and each item has a measure of popularity reflected by the probability of requests for it. We consider a model where the popularity of content is the same for all users. As in previous works (e.g., [28–30]), in this paper, the probability of requests follows generalized Zipf distribution function as follows:(1)ϕh=A−1h−η,where A=∑h=1Hh−η, hη is the rank of item h−th, and η is the skewness of the popularity distribution. The item ranked in the order of their popularity where the item h is the h most popular item, i.e., h=1 is the most popular item and h=H is the least popular item.


### 3.2. Demand Model

Dij is the average demand of end-users who connect to the CPi through the ISPj which depends on content access price pci, credibility ci, network access price psj, QoS qsj, and cache access price pccj (see [31–33]). The demand Dij depends also on prices pc−i, credibilities c−i, prices ps−j, prices pcc−j, and QoS set by the opponents. Eventually, Dij is decreasing with respect to pci, psj, and pccj and increasing with respect to pcn, n≠i, psf, pccf, and f≠j, whereas it is increasing with respect to ci and qsj and decreasing with respect to cn, n≠i, qsf, and f≠j. Then, the demand functions Dij can be written as follows:(2)Dij=dij−αjjpsj+βjjqsj−ρjjpccj−δiipci+σiici−∑f=1,f≠jFαjfpsf−βjfqsj+ρjfpccf+∑n=1,n≠iNδinpcn−σincn,where dij, αjf, βjf, ρif, δin, and σin are the positive constant. The parameter dij reflects the total potential demand of end-users. δin and σin denote the responsiveness of demand Dij to price pcn and credibility cn of CPn. αjf, βjf, and ρjf denote the responsiveness of demand Dij to price psf, price pccf, and QoS qsf of ISPf.Assumption 1.
The sensitivityα verifies:(3)αjj≥∑f=1,f≠jFαjf,∀j=1,…,F.
The sensitivityβ verifies:(4)βjj≥∑f=1,f≠jFβjf,∀j=1,…,F.
The sensitivityρ verifies:(5)ρjj≥∑f=1,f≠jFρjf,∀j=1,…,F.
The sensitivityδ verifies:(6)δii≥∑n=1,n≠iNδin,∀i=1,…,N.
The sensitivityσ verifies:(7)σii≥∑n=1,n≠iNσin,∀i=1,…,N.
Assumption1 will be needed to ensure the uniqueness of Nash equilibrium.


### 3.3. Utility Model of the ISP

The utility of theISPj is equal to revenue minus cost:(8)UISPj=psj∑i=1NDij+∑i=1N∑h=1HϕhτjikijhDijpccj−Cih−ptiDijci1−τjikijh−vjBj∑i=1NH−τij∑h=1Hkijh−θjcj2.psj∑i=1NDij is the revenue of network access. Cih is the caching cost of the item h provided by CPi. τji is the proportion of demand for content produced by CPi and served by ISPj cache, which can be expressed as τji=max1,cj/ci. τjiDij is the demand served by ISPj cache. ϕhτjikijhDijpccj−Cih is the revenue that results when the ISPj serves the requested demand ϕhτjikijhDij of the item h from its cache. In ICN models, the ISP pays to the CP the transmission fees ([25]). ϕhptiDijci1−τjikijh is the transmission fee that the ISPj pays to the CPi, when the CPi forwards the request demand ϕhDij1−τjikijh of the item h through the ISPj. θj is the cost to produce a unit of the credibility of content cj. θjcj2 is the cost to produce credibility of content cj. vj is the unit backhaul bandwidth cost paid by ISPj. Bj is the backhaul bandwidth needed to serve demand ∑i=1NDij. Bj∑i=1NH−τij∑h=1Hkijh is the backhaul bandwidth needed for serving the demand ∑i=1NDijH−τij∑h=1Hkijh. Bj is the backhaul bandwidth required by the ISPj. The quality of service qsj is defined as the expected delay (see [34]):(9)qsj=1Delay=Bj−∑i=1NDij,which means that(10)Bj=qsj2+∑i=1NDij.Then, the utility of theISPj is given by(11)UISPj=psj∑i=1NDij+∑i=1N∑h=1HϕhτjikijhDijpccj−Cih−ptiDijci1−τjikijh−vjqsj2+∑i=1NDijHN−∑i=1Nτij∑h=1Hkijh−θjcj2.kijh=1 means that the ISPj decides to cache the item h of the CPi, because the revenue of ISPj by serving the requested demand of the item h from their cache is larger than the cost for forwarding the request demand of the item h to the CPi. kijh=0 means that the ISPj does not cache the item h of CPi, because the cost for forwarding the request demand of the item h to the CPi is larger than the revenue of ISPj by serving the requested demand of the item h from their cache.


### 3.4. Utility Model of the CP

The utility ofCPi is equal to revenue minus cost:(12)UCPi=∑j=1F∑h=1Hϕhpci+pticiDij1−τjikijh+τjikijhCihDij−θici2.pciϕh∑jDij1−τjikijh is the revenue of CPi by serving the request demand ϕhDij1−τjikijh of item h. pti is the transmission fee that CPi charges to ISPs. ptiϕhDij1−τjikijh is the transmission revenue of the CPi, when CPi forwards the request demand of item h through ISPj. τjikijhCihDij is the profit of CPi when the demand τjikijhDij are satisfied from the ISPj cache. θici2 is the cost to produce the credibility of content ci. The credibility of content ci of CPi is a function of the quality of service qssi and quality content qci, which can be expressed as [33, 35–37](13)ci=λqssi+μqci,where λ and μ are the two positive constants, which represent, respectively, the sensitivity of the credibility of content to QoS and QoC. We define the QoS as the “expected delay,” which is computed by the Kleinrock function (see [34]). The quality of content QoC can be specified for a specific domain of content, e.g., quality of video streaming.Then, the utility ofCPi is given by(14)UCPi=∑j=1F∑h=1Hϕhpci+ptiλqssi+μqciDij1−τjikijh+τjikijhCihDij−θiλqssi+μqci2.


## 3.1. Content Popularity

LetH be the number of items provided by CPs, and each item has a measure of popularity reflected by the probability of requests for it. We consider a model where the popularity of content is the same for all users. As in previous works (e.g., [28–30]), in this paper, the probability of requests follows generalized Zipf distribution function as follows:(1)ϕh=A−1h−η,where A=∑h=1Hh−η, hη is the rank of item h−th, and η is the skewness of the popularity distribution. The item ranked in the order of their popularity where the item h is the h most popular item, i.e., h=1 is the most popular item and h=H is the least popular item.


## 3.2. Demand Model

Dij is the average demand of end-users who connect to the CPi through the ISPj which depends on content access price pci, credibility ci, network access price psj, QoS qsj, and cache access price pccj (see [31–33]). The demand Dij depends also on prices pc−i, credibilities c−i, prices ps−j, prices pcc−j, and QoS set by the opponents. Eventually, Dij is decreasing with respect to pci, psj, and pccj and increasing with respect to pcn, n≠i, psf, pccf, and f≠j, whereas it is increasing with respect to ci and qsj and decreasing with respect to cn, n≠i, qsf, and f≠j. Then, the demand functions Dij can be written as follows:(2)Dij=dij−αjjpsj+βjjqsj−ρjjpccj−δiipci+σiici−∑f=1,f≠jFαjfpsf−βjfqsj+ρjfpccf+∑n=1,n≠iNδinpcn−σincn,where dij, αjf, βjf, ρif, δin, and σin are the positive constant. The parameter dij reflects the total potential demand of end-users. δin and σin denote the responsiveness of demand Dij to price pcn and credibility cn of CPn. αjf, βjf, and ρjf denote the responsiveness of demand Dij to price psf, price pccf, and QoS qsf of ISPf.Assumption 1.
The sensitivityα verifies:(3)αjj≥∑f=1,f≠jFαjf,∀j=1,…,F.
The sensitivityβ verifies:(4)βjj≥∑f=1,f≠jFβjf,∀j=1,…,F.
The sensitivityρ verifies:(5)ρjj≥∑f=1,f≠jFρjf,∀j=1,…,F.
The sensitivityδ verifies:(6)δii≥∑n=1,n≠iNδin,∀i=1,…,N.
The sensitivityσ verifies:(7)σii≥∑n=1,n≠iNσin,∀i=1,…,N.
Assumption1 will be needed to ensure the uniqueness of Nash equilibrium.


## 3.3. Utility Model of the ISP

The utility of theISPj is equal to revenue minus cost:(8)UISPj=psj∑i=1NDij+∑i=1N∑h=1HϕhτjikijhDijpccj−Cih−ptiDijci1−τjikijh−vjBj∑i=1NH−τij∑h=1Hkijh−θjcj2.psj∑i=1NDij is the revenue of network access. Cih is the caching cost of the item h provided by CPi. τji is the proportion of demand for content produced by CPi and served by ISPj cache, which can be expressed as τji=max1,cj/ci. τjiDij is the demand served by ISPj cache. ϕhτjikijhDijpccj−Cih is the revenue that results when the ISPj serves the requested demand ϕhτjikijhDij of the item h from its cache. In ICN models, the ISP pays to the CP the transmission fees ([25]). ϕhptiDijci1−τjikijh is the transmission fee that the ISPj pays to the CPi, when the CPi forwards the request demand ϕhDij1−τjikijh of the item h through the ISPj. θj is the cost to produce a unit of the credibility of content cj. θjcj2 is the cost to produce credibility of content cj. vj is the unit backhaul bandwidth cost paid by ISPj. Bj is the backhaul bandwidth needed to serve demand ∑i=1NDij. Bj∑i=1NH−τij∑h=1Hkijh is the backhaul bandwidth needed for serving the demand ∑i=1NDijH−τij∑h=1Hkijh. Bj is the backhaul bandwidth required by the ISPj. The quality of service qsj is defined as the expected delay (see [34]):(9)qsj=1Delay=Bj−∑i=1NDij,which means that(10)Bj=qsj2+∑i=1NDij.Then, the utility of theISPj is given by(11)UISPj=psj∑i=1NDij+∑i=1N∑h=1HϕhτjikijhDijpccj−Cih−ptiDijci1−τjikijh−vjqsj2+∑i=1NDijHN−∑i=1Nτij∑h=1Hkijh−θjcj2.kijh=1 means that the ISPj decides to cache the item h of the CPi, because the revenue of ISPj by serving the requested demand of the item h from their cache is larger than the cost for forwarding the request demand of the item h to the CPi. kijh=0 means that the ISPj does not cache the item h of CPi, because the cost for forwarding the request demand of the item h to the CPi is larger than the revenue of ISPj by serving the requested demand of the item h from their cache.


## 3.4. Utility Model of the CP

The utility ofCPi is equal to revenue minus cost:(12)UCPi=∑j=1F∑h=1Hϕhpci+pticiDij1−τjikijh+τjikijhCihDij−θici2.pciϕh∑jDij1−τjikijh is the revenue of CPi by serving the request demand ϕhDij1−τjikijh of item h. pti is the transmission fee that CPi charges to ISPs. ptiϕhDij1−τjikijh is the transmission revenue of the CPi, when CPi forwards the request demand of item h through ISPj. τjikijhCihDij is the profit of CPi when the demand τjikijhDij are satisfied from the ISPj cache. θici2 is the cost to produce the credibility of content ci. The credibility of content ci of CPi is a function of the quality of service qssi and quality content qci, which can be expressed as [33, 35–37](13)ci=λqssi+μqci,where λ and μ are the two positive constants, which represent, respectively, the sensitivity of the credibility of content to QoS and QoC. We define the QoS as the “expected delay,” which is computed by the Kleinrock function (see [34]). The quality of content QoC can be specified for a specific domain of content, e.g., quality of video streaming.Then, the utility ofCPi is given by(14)UCPi=∑j=1F∑h=1Hϕhpci+ptiλqssi+μqciDij1−τjikijh+τjikijhCihDij−θiλqssi+μqci2.


## 4. Game Analysis

LetG1=F,Psj,Pccj,Qsj,UISPj. denote the noncooperative price QoS price game (NPQPG), where F=1,…,F is the set of ISPs, Psj is the strategy set of price, Pccj is the strategy set of price, Qsj is the strategy set of QoS, and UISPj. is the utility function of ISPj defined in equation (11). We assume that the strategy spaces Psj, Pccj, and Qsj of each ISPs are compact and convex sets with maximum and minimum constraints. Thus, for each ISPj, we consider as respective strategy spaces the closed intervals: Psj=p¯sj,p¯sj, Pccj=p¯ccj,p¯ccj, and Qsj=q¯sj,q¯sj. Let, the price vector ps=ps1,…,psFT∈PsF=Ps1×Ps2×⋯×PsF, the price vector pcc=pc1,…,pcFT∈PccF=Pc1×Pc2×⋯×PcF, and the QoS vector qs=qs1,…,qsFT∈QsF=Qs1×Qs2×⋯×QsF.LetG2=N,Pci,Qssi,Qci,UCPi. denote the noncooperative QoS price QoC game (NQPQG), where N=1,…,N is the set of CPs, Pci is the strategy set of price, Qssi is the strategy set of QoS, Qci is the strategy set of QoC, and UCPi. is the utility function of CPi defined in equation (14). We assume that the strategy spaces Pci, Qssi, and Qci of each CP are compact and convex sets with maximum and minimum constraints. Thus, for each CPi, we consider as respective strategy spaces the closed intervals: Pci=p¯ci,p¯ci, Qssi=q¯ssi,q¯ssi, and Qci=q¯ci,q¯ci. Let, the price vector pc=pc1,…,pcNT∈PcN=Pc1×Pc2×⋯×PcN, the QoS vector qss=qss1,…,qssNT∈QssN=Qss1×Qss2×⋯×QssN, and the QoC vector qc=qc1,…,qcNT∈QcN=Qc1×Qc2×⋯×QcN.


### 4.1. PricePs Game

A NPQPG in price is defined for fixedpcc∈Pcc, qs∈Qs as G1pcc,qs=FPsj,UISPj.,pcc,qs.Definition 1.
A price vectorps∗=ps1∗,…,psF∗ is a Nash equilibrium of the NPQPG G1pcc,qs if(15)∀j,psj∈F,Psj,UISPjpsj∗,ps−j∗,pcc,qs≥UISPjpsj,ps−j∗,pcc,qs.Theorem 1.
For eachpcc∈Pcc, qs∈Qs, the game F,Psj,UISPj.,pcc,qs admits a unique Nash equilibrium.Proof:.
The game G1(Pcc, qs) admits at least one equilibrium, if and only if its second derivative of the utility UISPj function with respect to price Psi is nonpositive. psj is nonpositive. Indeed, we first compute the first derivative of UISPj:(16)∂UISPj∂psj=−Nαjjpsj+∑i=1NDij+∑i=1N∑h=1Hϕh−αjjτjikijhpccj−Cih+αjjptici1−τjikijh+vjNαjjHN−τji∑h=1Hkijh.
Then, we calculate the second derivative:(17)∂2UISPj∂psj2=−2αjjN≤0.
The second derivative of the utility function is always nonpositive, then the utility function is concave, which ensures the existence of a Nash equilibrium.
We use the following proposition that holds for a concave game [38]: if a concave game satisfies the dominance solvability condition(18)−∂2UISPj∂psj2≥∑f=1,f≠jF∂2UISPj∂psj∂psf,then the game G1pcc,qs admits a unique Nash equilibrium point.
The mixed partial is written as(19)∂2UISPj∂psj∂psf=αjfN.
Then,(20)−∂2UISPj∂psj2−∑f=1,f≠jF∂2UISPj∂psj∂psf=N2αjj−∑f=1,f≠jFαjf≥0.
Thus, theG1pcc,qs admits a unique Nash equilibrium point.


### 4.2. PricePcc Game

A NPQPG in price is defined for fixedps∈Ps, qs∈Qs as G1ps,qs=F,Pccj,UISPjps,.,qs.Definition 2.
A price vectorpcc∗=pcc1∗,…,pccF∗ is a Nash equilibrium of the NPQPG G1ps,qs if(21)∀j,pccj∈F,Pccj,UISPjps,pccj∗,pcc−j∗,qs≥UISPjps,pccj,pcc−j∗,qs.Theorem 2.
For eachps∈Ps, qs∈Qs, the game F,Pccj,UISPjps,.,qs admits a unique Nash equilibrium.
The proof of the above theorem can be found in Appendix A.


### 4.3. QoSQs Game

An NPQPG in QoS is defined for fixedps∈Ps, pcc∈Pcc as G1ps,pcc=F,Qsj,UISPjps,pcc,..Definition 3.
A QoS vectorqs∗=qs1∗,…,qsF∗ is a Nash equilibrium of the NPQPG G1ps,pcc if(22)∀j,qsj∈F,Qsj,UISPjps,pcc,qsj∗,qs−j∗≥UISPjps,pcc,qsj,qs−j∗.Theorem 3.
For eachps∈Ps, pcc∈Pcc, the game F,Qsj,UISPjps,pcc,. admits a unique Nash equilibrium.
The proof of the above theorem can be found in Appendix B.


### 4.4. PricePc Game

An NQPQG in pricepc is defined for fixed qss∈Qss, qc∈Qc as G2qss,qc=N,Pci,UCPi.,qss,qc.Definition 4.
A price vectorpc∗=pc1∗,…,pcN∗ is a Nash equilibrium of the NQPQG G2qss,qc if(23)∀i,pci∈N,Pci,UCPipci∗,pc−i∗,qs,qc≥UCPipci,pc−i∗,qs,qc.Theorem 4.
For eachqss∈Qss, qc∈Qc, the game N,Pci,UCPi.,qss,qc admits a unique Nash equilibrium.
The proof of the above theorem can be found in Appendix C.


### 4.5. QoCQc Game

An NQPQG in QoC is defined for a fixedpc∈Pc, qss∈Qss as G2pc,qss=N,Qci,UCPipc,qss,..Definition 5.
A QoC vectorqc∗=qc1∗,…,qcN∗ is a Nash equilibrium of the NQPQG G2pc,qss if(24)∀i,qci∈N,Qci,UCPipc,qss,qci∗,qc−i∗≥UCPipc,qss,qci,qc−i∗.Theorem 5.
For eachpc∈Pc, qss∈Qss, the game N,Qci,UCPipc,qss,. admits a unique Nash equilibrium.
The proof of the above theorem can be found in Appendix D.


### 4.6. QoSQss Game

An NQPQG in QoS is defined for a fixedpc∈Pc, qc∈Qc as G2pc,qc=N,Qssi,UCPipc,.,qc.Definition 6.
A QoS vectorqss∗=qss1∗,…,qssN∗ is a Nash equilibrium of the NQPQG G2pc,qc if(25)∀i,qssi∈N,Qssi,UCPipc,qssi∗,qss−i∗,qc≥UCPipc,qssi,qss−i∗,qc.Theorem 6.
For eachpc∈Pc, qc∈Qc, the game N,Qssi,UCPipc,.,qc admits a unique Nash equilibrium.


### 4.7. Learning Nash Equilibrium

The idea of best response is useful when we are trying to find an approach to reach a steady-state of a game, i.e., a Nash equilibrium of a game. The best response dynamics scheme consists of a sequence of rounds, where in each round after the first, each provider observes the policy taken by its competitors in previous rounds and inputs them in its decision process to update its policy. In the first round, the choice of each provider is the best response based on its arbitrary belief about what the other players will choose.Algorithm1 summarizes the best response learning steps that each provider has to perform to find the Nash equilibrium.Algorithm 1: Best response algorithm.
(1)
Initialize vectorsx0=x10,…,xO0 randomly;(2)
For eachEo, o∈O at time instant t computes:(a)
xot+1=argmaxxo∈XoUEoxt.(3)
If∀o∈O, xjt+1−xjt<ε, then STOP.(4)
Else, t⟵t+1 and go to step (2)Such as(i)
E denotes CP or ISP(ii)
O refers to N or K(iii)
x refers to the vector price pc, vector price ps, vector price pcc, vector qs, vector qc or vector qss(iv)
Xo refers to the policy profile price, QoS or QoCThe proof of the above theorem can be found in Appendix E.


### 4.8. Price of Anarchy

The price of anarchy (PoA) is defined as the ratio between the performance measures of the worst equilibrium and the optimal outcome. A PoA close to 1 indicates that the equilibrium is approximately socially optimal, and thus, the consequences of selfish behavior are benign.In [39], we measure the loss of efficiency due to actors’ selfishness as the quotient between the social welfare obtained at the Nash equilibrium and the maximum value of the social welfare:(26)PoA=minps,pcc,qs,pc,qc,qssWNEps,pcc,qs,pc,qc,qssmaxps,pcc,qs,pc,qc,qssWps,pcc,qs,pc,qc,qss,where Wps,pcc,qs,pc,qc,qss=∑j=1MUips,pcc,qs,pc,qc,qss is the social welfare function and WNEps∗,pcc∗,qs∗,pc∗,qc∗,qss∗=∑j=1MUips,pcc,qs,pc,qc,qss is a sum of utilities of all actors at Nash equilibrium.


## 4.1. PricePs Game

A NPQPG in price is defined for fixedpcc∈Pcc, qs∈Qs as G1pcc,qs=FPsj,UISPj.,pcc,qs.Definition 1.
A price vectorps∗=ps1∗,…,psF∗ is a Nash equilibrium of the NPQPG G1pcc,qs if(15)∀j,psj∈F,Psj,UISPjpsj∗,ps−j∗,pcc,qs≥UISPjpsj,ps−j∗,pcc,qs.Theorem 1.
For eachpcc∈Pcc, qs∈Qs, the game F,Psj,UISPj.,pcc,qs admits a unique Nash equilibrium.Proof:.
The game G1(Pcc, qs) admits at least one equilibrium, if and only if its second derivative of the utility UISPj function with respect to price Psi is nonpositive. psj is nonpositive. Indeed, we first compute the first derivative of UISPj:(16)∂UISPj∂psj=−Nαjjpsj+∑i=1NDij+∑i=1N∑h=1Hϕh−αjjτjikijhpccj−Cih+αjjptici1−τjikijh+vjNαjjHN−τji∑h=1Hkijh.
Then, we calculate the second derivative:(17)∂2UISPj∂psj2=−2αjjN≤0.
The second derivative of the utility function is always nonpositive, then the utility function is concave, which ensures the existence of a Nash equilibrium.
We use the following proposition that holds for a concave game [38]: if a concave game satisfies the dominance solvability condition(18)−∂2UISPj∂psj2≥∑f=1,f≠jF∂2UISPj∂psj∂psf,then the game G1pcc,qs admits a unique Nash equilibrium point.
The mixed partial is written as(19)∂2UISPj∂psj∂psf=αjfN.
Then,(20)−∂2UISPj∂psj2−∑f=1,f≠jF∂2UISPj∂psj∂psf=N2αjj−∑f=1,f≠jFαjf≥0.
Thus, theG1pcc,qs admits a unique Nash equilibrium point.


## 4.2. PricePcc Game

A NPQPG in price is defined for fixedps∈Ps, qs∈Qs as G1ps,qs=F,Pccj,UISPjps,.,qs.Definition 2.
A price vectorpcc∗=pcc1∗,…,pccF∗ is a Nash equilibrium of the NPQPG G1ps,qs if(21)∀j,pccj∈F,Pccj,UISPjps,pccj∗,pcc−j∗,qs≥UISPjps,pccj,pcc−j∗,qs.Theorem 2.
For eachps∈Ps, qs∈Qs, the game F,Pccj,UISPjps,.,qs admits a unique Nash equilibrium.
The proof of the above theorem can be found in Appendix A.


## 4.3. QoSQs Game

An NPQPG in QoS is defined for fixedps∈Ps, pcc∈Pcc as G1ps,pcc=F,Qsj,UISPjps,pcc,..Definition 3.
A QoS vectorqs∗=qs1∗,…,qsF∗ is a Nash equilibrium of the NPQPG G1ps,pcc if(22)∀j,qsj∈F,Qsj,UISPjps,pcc,qsj∗,qs−j∗≥UISPjps,pcc,qsj,qs−j∗.Theorem 3.
For eachps∈Ps, pcc∈Pcc, the game F,Qsj,UISPjps,pcc,. admits a unique Nash equilibrium.
The proof of the above theorem can be found in Appendix B.


## 4.4. PricePc Game

An NQPQG in pricepc is defined for fixed qss∈Qss, qc∈Qc as G2qss,qc=N,Pci,UCPi.,qss,qc.Definition 4.
A price vectorpc∗=pc1∗,…,pcN∗ is a Nash equilibrium of the NQPQG G2qss,qc if(23)∀i,pci∈N,Pci,UCPipci∗,pc−i∗,qs,qc≥UCPipci,pc−i∗,qs,qc.Theorem 4.
For eachqss∈Qss, qc∈Qc, the game N,Pci,UCPi.,qss,qc admits a unique Nash equilibrium.
The proof of the above theorem can be found in Appendix C.


## 4.5. QoCQc Game

An NQPQG in QoC is defined for a fixedpc∈Pc, qss∈Qss as G2pc,qss=N,Qci,UCPipc,qss,..Definition 5.
A QoC vectorqc∗=qc1∗,…,qcN∗ is a Nash equilibrium of the NQPQG G2pc,qss if(24)∀i,qci∈N,Qci,UCPipc,qss,qci∗,qc−i∗≥UCPipc,qss,qci,qc−i∗.Theorem 5.
For eachpc∈Pc, qss∈Qss, the game N,Qci,UCPipc,qss,. admits a unique Nash equilibrium.
The proof of the above theorem can be found in Appendix D.


## 4.6. QoSQss Game

An NQPQG in QoS is defined for a fixedpc∈Pc, qc∈Qc as G2pc,qc=N,Qssi,UCPipc,.,qc.Definition 6.
A QoS vectorqss∗=qss1∗,…,qssN∗ is a Nash equilibrium of the NQPQG G2pc,qc if(25)∀i,qssi∈N,Qssi,UCPipc,qssi∗,qss−i∗,qc≥UCPipc,qssi,qss−i∗,qc.Theorem 6.
For eachpc∈Pc, qc∈Qc, the game N,Qssi,UCPipc,.,qc admits a unique Nash equilibrium.


## 4.7. Learning Nash Equilibrium

The idea of best response is useful when we are trying to find an approach to reach a steady-state of a game, i.e., a Nash equilibrium of a game. The best response dynamics scheme consists of a sequence of rounds, where in each round after the first, each provider observes the policy taken by its competitors in previous rounds and inputs them in its decision process to update its policy. In the first round, the choice of each provider is the best response based on its arbitrary belief about what the other players will choose.Algorithm1 summarizes the best response learning steps that each provider has to perform to find the Nash equilibrium.Algorithm 1: Best response algorithm.
(1)
Initialize vectorsx0=x10,…,xO0 randomly;(2)
For eachEo, o∈O at time instant t computes:(a)
xot+1=argmaxxo∈XoUEoxt.(3)
If∀o∈O, xjt+1−xjt<ε, then STOP.(4)
Else, t⟵t+1 and go to step (2)Such as(i)
E denotes CP or ISP(ii)
O refers to N or K(iii)
x refers to the vector price pc, vector price ps, vector price pcc, vector qs, vector qc or vector qss(iv)
Xo refers to the policy profile price, QoS or QoCThe proof of the above theorem can be found in Appendix E.


## 4.8. Price of Anarchy

The price of anarchy (PoA) is defined as the ratio between the performance measures of the worst equilibrium and the optimal outcome. A PoA close to 1 indicates that the equilibrium is approximately socially optimal, and thus, the consequences of selfish behavior are benign.In [39], we measure the loss of efficiency due to actors’ selfishness as the quotient between the social welfare obtained at the Nash equilibrium and the maximum value of the social welfare:(26)PoA=minps,pcc,qs,pc,qc,qssWNEps,pcc,qs,pc,qc,qssmaxps,pcc,qs,pc,qc,qssWps,pcc,qs,pc,qc,qss,where Wps,pcc,qs,pc,qc,qss=∑j=1MUips,pcc,qs,pc,qc,qss is the social welfare function and WNEps∗,pcc∗,qs∗,pc∗,qc∗,qss∗=∑j=1MUips,pcc,qs,pc,qc,qss is a sum of utilities of all actors at Nash equilibrium.


## 5. Stackelberg Game Formulation

The Stackelberg game is an extension of the noncooperative game with a bilevel hierarchy. The Stackelberg game models a game between two groups of players, namely, leaders and followers. The leaders have the privilege of acting the first while the followers work according to the leaders’ actions. Backward induction is always utilized to solve the Stackelberg game. In the backward induction method, the followers’ optimal strategies are solved firstly, given the leader strategies. Secondly, leader players’ optimal strategies are solved. At the first stage of the Stackelberg game, the CP sets its QoS and QoC. At the second stage of the game, given the QoS and QoC of the CP, the ISP sets the QoS and QoC of content in its cache. The proposed Stackelberg game modeling is shown in Figure2.Figure 2
Stackelberg game modeling.Firstly, we consider the second stage game. Given the QoS and QoC of the CP, the ISP needs to determine the optimal QoS and QoC to maximize this payoff.Proposition 1.
The optimal QoS and QoC of the ISPs is given as follows:(27)qcj=∑i=1N∑h=1HϕhkijhDijpccj−Cih2μθjci+ptiDijkijh2μθj+vjBj∑h=1Hkijh2μθjci−λqssjμ,qssj=∑i=1N∑h=1HϕhkijhDijpccj−Cih2λθjci+ptiDijkijh2λθj+vjBj∑h=1Hkijh2λθjci−μqcjλ.Proof:.
the optimal condition for QoS and QoC is to let the first-order differentiation equal to 0 as follows:(28)∂UISPj∂qcj=0,∂UISPj∂qssj=0.
On the other hand, we have(29)∂UISPj∂qcj=∑i=1N∑h=1HϕhμkijhDijpccj−Cihci+μptiDijkijh+μvjBj∑h=1Hkijhci−2μθjλqssj+μqcj,∂UISPj∂qssj=∑i=1N∑h=1HϕhλkijhDijpccj−Cihci+λptiDijkijh+λvjBj∑h=1Hkijhci−2λθjλqssj+μqcj.
Then, the QoS and QoC that maximize the ISPs’ profits are given as follows:(30)qcj=∑i=1N∑h=1HϕhkijhDijpccj−Cih2μθjci+ptiDijkijh2μθj+vjBj∑h=1Hkijh2μθjci−λqssjμ,qssj=∑i=1N∑h=1HϕhkijhDijpccj−Cih2λθjci+ptiDijkijh2λθj+vjBj∑h=1Hkijh2λθjci−μqcjλ.
Secondly, we solve the first stage of the Stackelberg game, where the CPs set QoS and QoC. We formulate the game between CPs as a noncooperative game. The noncooperative game is described in the above Sections4.5 and 4.6.


## 6. Numerical Investigation

In this section, we provide the simulation results to illustrate the competitive interactions in the ICN market. Additionally, we will present a detailed description of the impact of the system parameters on the provider policies. Furthermore, we will discuss the effect of the caching cost and the number of cached items on the price of anarchy. To simplify the complex market environment, we consider a network scenario that includes two ISPs and two CPs. Simulation settings are listed in Table2.Table 2
Parameter setting used for numerical examples.
α11=α22αjf,j≠fβ11=β22βjf,j≠fρ11=ρ22ρjf,j≠fd110.550.450.550.450.550.45250σ11=σ22σin,i≠nδ11=δ22δin,i≠npt1=pt2θi,i∈1,2d120.550.45100.5102250p¯c1=p¯c2p¯c1=p¯c2q¯ss1=q¯ss2q¯ss1=q¯ss2q¯c1=q¯c2q¯c1=q¯c2d21100011000110001200p¯s1=p¯s2p¯s1=p¯s2q¯s1=q¯s2q¯s1=q¯s2p¯cc1=p¯cc2p¯cc1=p¯cc2d22100011000110001250v1=v2HC1h=C2h,∀hAηθj,j∈1,2λ=μ104040100.5100.5Figures3–8 show the convergence toward Nash equilibrium price, Nash equilibrium QoS, and Nash equilibrium QoC of all providers. Figures 3–8 demonstrate the existence and uniqueness of a Nash equilibrium point at which no providers can profitably deviate given the strategies of another opponent.Figure 3
Nash equilibrium price under the best response dynamic.Figure 4
Nash equilibrium price under the best response dynamic.Figure 5
Nash equilibrium QoS under the best response dynamic.Figure 6
Nash equilibrium price under the best response dynamic.Figure 7
Nash equilibrium QoC under the best response dynamic.Figure 8
Nash equilibrium QoS under the best response dynamic.Figure9 shows the price pc of the CP under different cost C. From Figure 9, we can observe that price pc of the designed model is decreasing as cost C increases. The reason is that a higher cost generates a larger profit from caching, which makes CPs decrease their price pc to induce increased demand from the end-users.Figure 9
Nash equilibrium pricepc evolution with respect to caching cost C.Figure10 illustrates the Nash equilibrium price ps with caching cost C for two ISPs. Figure 10 shows that as the caching cost C increases, the ISPs set a greater price ps. The reason is that as the caching cost C becomes expensive, the ISPs do not have any motivation to cache content. Hence, the ISPs forward more requests of content to the CPs, which increases the transmission fee and the cost of backhaul bandwidth. Then, the ISPs are obliged to increase their price ps to compensate for the cost of transmission and the cost of backhaul bandwidth.Figure 10
Nash equilibrium priceps evolution with respect to caching cost C.Figure11 shows the price pcc under different content popularities. From Figure 11, we can observe that pcc is decreasing as content popularity increases. The reason is that a larger Zipf skewness parameter generates a larger proportion of content requests for the popular contents, which makes more requests directly satisfied by the ISP caches and then brings a higher profit.Figure 11
Nash equilibrium pricepcc evolution as a function of Zipf’s factor η.The effect of the number of cached itemsH on QoS and QoC is shown in Figures 12 and 13. QoS and QoC of the proposed model are growing as H increases because a more significant number of cached items can make more content cached and more requests of end-users satisfied by the ISP cache. As a result, the revenue of CPs increases because the revenue from caching increases. Therefore, the CPs increase their QoS and QoC to induce increased demand from end-users.Figure 12
Nash equilibrium QoCqc evolution with respect to H.Figure 13
Nash equilibrium QoSqc evolution as a function of H.Figure14 illustrates the Nash equilibrium price with the number of cached items H for two ISPs. As H increases, the number of cached item increases. When the number of cached item increases, the proportion of content requests of ISP cache increases, the transmission cost decreases, and the cost of backhaul bandwidth decreases, which brings a higher profit. Therefore, the ISPs decrease their price ps to induce increased demand from the end-users.Figure 14
Nash equilibrium priceps evolution as a function of H.Figure15 shows the ISP PoA variation curve as a function of caching cost C. In that figure, PoA increases with respect to C. When C is lower, the PoA is lower; then, the Nash equilibrium is not socially efficient, the ISPs are selfish, and each one seeks to maximize its profit individually. However, when C increases, the equilibrium becomes more and more socially efficient; this increase finds the simple intuition that when C increases, ISPs cooperate with each other for optimizing Nash equilibrium.Figure 15
Price of anarchy evolution with respect to caching costC.Figure16 shows the ISP PoA variation curve as a function of the number of items cached H. The price of anarchy decreases with respect to H. When H is lower, the price of anarchy is socially efficient; moreover, when H is lower, the income from caching decreases and ISPs cooperate for optimizing the Nash equilibrium. On the other hand, when H increases, the price of anarchy is lower; then, the Nash equilibrium is not socially efficient, ISPs are selfish, and each one seeks to maximize its profit individually.Figure 16
Price of anarchy evolution with respect toH.Figure17 shows the CP PoA variation curve as a function of H. The price of anarchy increases with respect to H. A special feature is that the Nash equilibrium performs well, and the loss of efficiency is only around 6%. This result indicates that the Nash equilibrium of this game is fair and socially efficient.Figure 17
Price of anarchy evolution with respect toH.


## 7. Conclusion

In this paper, we developed an analytical framework for the distribution of accessible content in an ICN that comprises multiple ISPs and multiple CPs. The interaction among ISPs and among CPs is investigated by using the noncooperative game. Additionally, we have used a Stackelberg game to study the interaction between CP and ISP. Each ISP can control the amount of content cached in the network and adjust its strategies (network access price, price to access to content in the cache, QoS, and credibility of content in the cache). The CP can adjust its strategy (content access price and credibility of content). We consider that the popularity of content follows a generalized Zipf distribution. We proved the existence and uniqueness of the Nash equilibrium in a competitive ICN market. This result is significant because it implies that a stable solution with suitable economic incentives in collaborative caching is feasible in the ICN paradigm. Next, we describe a learning mechanism that allows each ISP to discover accurately and rapidly its equilibrium policies. Also, we use the price of anarchy to evaluate the effectiveness of the Nash equilibrium. The simulation results show that caching investment is beneficial for the ISPs, the CPs, and end-users.



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*Source: 8895271-2020-10-30.xml*