Revenue Sharing in the Internet: A Moral Hazard Approach and a Net-neutrality Perspective
Fehmina Malik, Manjesh K.Hanawal, Yezekael Hayel, Jayakrishnan Nair
11 Revenue Sharing in the Internet: A Moral HazardApproach and a Net-neutrality Perspective
Fehmina Malik, Manjesh K. Hanawal, Yezekael Hayel and Jayakrishnan Nair
Abstract —Revenue sharing contracts between ContentProviders (CPs) and Internet Service Providers (ISPs) can actas leverage for enhancing the infrastructure of the Internet.ISPs can be incentivised to make investments in networkinfrastructure that improve Quality of Service (QoS) for usersif attractive contracts are negotiated between them and CPs.The idea here is that part of the net profit gained by CPsare given to ISPs to invest in the network. The Moral Hazardeconomic framework is used to model such an interaction, inwhich a principal determines a contract, and an agent reactsby adapting her effort. In our setting, several competitive CPsinteract through one common ISP. Two cases are studied: (i) theISP differentiates between the CPs and makes a (potentially)different investment to improve the QoS of each CP, and (ii) theISP does not differentiate between CPs and makes a commoninvestment for both. The last scenario can be viewed as networkneutral behavior on the part of the ISP. We analyse the optimalcontracts and show that the CP that can better monetize itsdemand always prefers the non-neutral regime. Interestingly,ISP revenue, as well as social utility, are also found to be higherunder the non-neutral regime.
Index Terms —revenue sharing, net neutrality, moral hazard
I. I
NTRODUCTION
The rapid growth of data-intensive services has resulted inan explosion of the Internet traffic, and it is expected to in-crease at an even faster rate in the future [1]. To accommodatethis increase in traffic, and to provide better Quality of Service(QoS) for end users, Internet service providers (ISPs) needto upgrade their network infrastructure and expand capacity.This development follows the deployment of next generationnetworks that will induce more business interactions betweenservice providers and content providers. For example, cachingtechnologies has recently received increased attention fromindustry and academia to be a key solution for next generationnetworks [2]. As it is specified in this special issue, theeconomics of caching will be one of important aspect indeciding the monetary interactions between the ISPs and CPs.In terms of return on investments, the ISPs, especially theones providing last-mile connectivity, feel that the revenuefrom end-users (mainly access charges) are often not enoughto recoup their investment costs and they propose that the CPsshare the risk by sharing part of their revenues. The CPs mayalso have the incentive to contribute to ISP capacity expansion,as increased capacity and better QoS trigger higher demandfor content and help them earn even higher revenues (mainly
Fehmina Malik and Manjesh K. Hanawal are with IEOR, IIT Bombay,India. E-mail: {fehminam, mhanawal, }@iitb.ac.in. Yezekael Hayel is withLIA/CERI, University of Avignon, France. E-mail: [email protected]. Jayakrishnan Nair is with Department of Electrical Engineering,IIT Bombay, E-mail: [email protected] from subscriptions and advertisements) [3]. For example, in[4], the authors propose a model in which a Mobile NetworkOperator leases its edge caches to a Content Provider. Thisis an increasingly relevant scenario and follows proposalsfor deploying edge storage resources at mobile 5G networks[5]. A recent announcement by Comcast that it will bundleNetflix subscription in its package is another example of anISP helping a CP to increase its demand.If a CP enters into a contract with an ISP to share its revenuein return for ISP putting efforts to improve demand for itscontent, the CP may want to monitor the efforts level of ISPso that the contract is honored. However such monitoring ofefforts levels may not be always feasible in the Internet as thereis an inherent asymmetry of information between CPs and ISPs– CPs cannot observe the exact investment (caching effort)made by the ISP, but can only observe the resulting increasein user demand for its content, which is a (random) functionof the ISP’s investment. In this situation, an obvious questionarises: how can this imperfect information about the ISP’sinvestment be used to formulate an optimal revenue sharingcontract? This is a situation where a privately taken action(investment) by the ISP influences the probability distributionof the outcome (demand) for the CP.The role of this information asymmetry between the CPand ISP is critical under such agreements, where ISP makesan investment which ultimately benefits the CP, and should beconsidered in the structure of the contract. Therefore, moralhazard can be applied to propose a contract in which the ISP(agent) knows that CP (principle) will pay to cover its risks,which in turn gives the ISP the incentive to make the (risky)investment. Also, the moral hazard model is proven to induceproper incentives for taking appropriate action [6], and thismay provide fair revenue sharing between CPs and ISPs. Inour work, we propose an incentivizing mechanism using the moral hazard approach in which CPs share a part of theirrevenues with an ISP expecting a better QoS for their contentand higher revenue in return from the ISP efforts.The classical moral hazard problem deals with a singleprinciple and a single agent, whereas we are faced withpossibility of multiple principles interacting with multipleagents. In this work, we focus on a monopolistic ISP con-necting end users to multiple CPs, i.e., multiple principlesand a single agent. Our interest is in determining optimalsharing mechanisms/contracts between each CP and the ISP.We distinguish two cases for the effort (investment) madeby the ISP. In one case, we allow the ISP to make differentamounts of effort for each CP, and in the other case, the ISPis constrained to put an equal amount of effort for all CPs. a r X i v : . [ ec on . GN ] A ug The former case corresponds to a ‘non-neutral’ regime wherethe ISP is allowed to differentiate between CPs, and the lattercase corresponds to a ‘neutral’ regime where the ISP cannotdifferentiate between CPs. We compare the revenue of eachplayers and social utilities under both the regimes and analyzewhich regime is preferred by the players.We consider competition between multiple CPs that providecontent of a particular type, for example, video, and aim toearn higher revenue by entering into a revenue sharing contractwith the ISP. Each CP separately negotiates with the ISP know-ing that the other CPs can also enter into similar negotiationswith the ISP. We consider linear contracts and analyze theequilibrium sharing contracts in both the neutral and non-neutral regime. We first consider the symmetric case where allthe CPs earn same revenue per unit demand. This correspondsto the case where ability of the CPs to monitize their demandis the same. We then study the asymetric case where the CPscapability to monetize their demand could be different. Ourcontributions and observations can be summarized as follows: • We model the competitive revenue sharing of CPs withan ISP in return for improved QoS in the moral hazardframework with multiple principles and a single agent. • We analyze the equilibrium contracts in a regime wherethe ISP can put a different level of effort for each CP(non-neutral) and in a regime where it is constrained toput equal efforts for all CPs (neutral). • In the symmetric case we show that all the players preferthe non-neutral regimes as their utilities are higher • In the asymmetric case we show that the CP which canbetter monetize the demand for its traffic always prefersthe non-neutral regime whereas the CP with weakermonetization power can prefer the neutral or non-neutralregime depending on its relative monetization power. • The ISP always prefers the non-neutral regime. Moreover,social utility (defined as the sum of the average earningof all players) is also higher in the non-neutral regime.This paper is organized as follows. In Section II we discussthe problem setup and define contracts under the neutraland non-neutral regime. We study the equilibrium contractsunder neutral and non-neutral regime under the symmetriccase in in Section III and under asymetric case in Section IV.Conclusions and future extensions are discussed in Section V.Proofs of all stated results can be found in the appendix.
A. Related works
Several works [7], [8], [9], [10], [11] study the possibility ofcontent charges by ISPs to recover investment costs. In [7] and[8], the authors investigated the feasibility of ISPs charging acontent charge to CPs, and evaluate its effect by modeling theStackelberg game between CPs and ISPs. In [9] and [10], arevenue-sharing scheme is proposed when the ISP provides acontent piracy monitoring service to CPs for increasing thedemand for their content. This work is extended to two ISPscompeting with each other in [11] where only one of themprovides the content piracy monitoring service. Several studiesconsidered cooperative settlement between service providers for profit sharing [12], [13], [14] where the mechanisms arederived using the Shapely value concept.A moral hazard framework is applied in [15] to studyinterconnection contracts between ISP and end users in themarket for network transport services. However, the contractdesign problem between ISPs and CPs remained unaddressedin this work. In the present paper, we apply the moralhazard approach to study revenue sharing between an ISPand multiple CPs. The CPs act as the principles that enterinto a contract with the ISP (agent) to improve QoS for theircontent. We thus end up with a multiple principle, single agentproblem.A moral hazard framework where two principals offer acontract to the same agent is studied in [16] where theprincipals can only observe correlated noisy signals of thecommon one-dimensional action taken by the agent, wherethe principals’ output is linearly increasing with the agent’saction. Our work is different from them as we consider thateven though the agent (ISP) is common, it chooses differentactions for each principal (CP). We also compare it with thesetting when the ISP is forced to choose the same actionfor both CPs. Also, we consider the demands of CPs to belogarithmic in the efforts of the agent, unlike the linear caseconsidered in [16]. Such a demand function comes from thefact that the delay experienced by end users is exponential inthe cache and not linear as assumed in [14] and [17].A moral hazard setup is also used in [18] for motivating endusers to participate in crowd-sourced services in one principaland several agent problem.II. P
ROBLEM D ESCRIPTION
We consider multiple Content Providers (CPs) and a singleInternet Service Provider (ISP) that connects end users to thecontent of the CPs. Each CP can enter into a contract withthe ISP under which the ISP agrees to offer a better quality ofservice on the contents of the CPs to the end users by investingin the network infrastructure [19] and in turn, each CP agreesto share a part of its revenue with the ISP. One example of ISPinvesting in network infrastructure to improve QoS is caching,where better caching efforts by the ISP for a CP’s contentresults in higher revenue for the CP. However, such cachingdecisions (effort or action) of ISP may not be directly visibleto the CPs, but each CP can observe the QoS experiencedby end users through demand for their contents. Thus, higherthe effort (caching) by the ISP, higher will be the revenuefor CP because of the price paid for their content or fromadvertisements (from the click-through rate) [20]. However,the ISP’s profit maximization strategy may not be aligned withthe interests of the CPs, and moreover, the ISP effort (thatinfluence CP revenue) is not directly observable by CPs. Thisscenario gives rise to the Moral Hazard problem.Let n denote the number of CPs and N = { , , . . . , n } theset of CPs. For each i ∈ N we denote i -th CP as CP i . Therevenue of CP i from its content is random and denoted as X i ∈ R + with probability density function F i . The amount ofefforts put by the ISP to improve the demand for the contentof CP i is quantified by a positive number denoted as a i ∈ R + . Fig. 1. Revenue flow between CPs, ISP and End Users. CP i shares part of his revenue with the ISP to incentivizeISP investments. This share is determined by the outcome assharing function s i : R + → R + , which is called contract oragreement in the moral hazard framework. Specifically, if x i is the realized revenue in a month for CP i , it gives s i ( x i ) to the ISP. Then, the net revenue of CP i is x i − s i ( x i ) . Ourgoal in this work is first to design optimal sharing functions s i (·) , i ∈ N that maximize the expected net revenues of eachCPs taking into account the rational behavior of the ISP andits participation constraints. We assume that the CPs offer asimilar type of content and compete with each other to attractmore demand. Our model with multiple CPs and a single ISPis depicted in Figure 1. In the terminology of moral hazard,the CPs are the principles, and the ISP is the agent. We thushave multiple principles and a single agent. Demand and Revenue:
The increment in demand for contentof CP i depends on the effort by the ISP to improve QoSfor CP i content and could be random. Let D i denote thisincrement in demand. We assume that the mean of D i growslogarithmically in a i (law of diminishing gains) and is givenby D i : = log ( a i + ) + (cid:15) i , i ∈ N , where (cid:15) i is the random variationin the demand for CP i .The revenue generated for CP i is proportional to the demandand given by X i = r i D i , where r i is a constant that captureshow each unit of demand translates to earnings. For example, r i could be revenue per click for CP i when D i is interpreted astotal number of clicks on CP i ’s content. The expected revenuefor CP i is then E [ X i ] = r i log ( r i + ) . A. Utilities and Objective
For a given contract s i (·) , i ∈ N the net revenue for CP i is X i − s i ( X i ) . Let V i (·) ∈ R denote the utility function of CP i andis typically assumed to an increasing concave function of the CP ’s net revenue. We assume the utility of CP i is linear in itsnet revenue and set V i ( X i − s i ( X i )) = X i − s i ( X i ) . The utility ofthe ISP depends on revenue-share it receives from both the CPsand also the cost involved in the efforts it puts for the CPs. Theearning for the ISP from the CPs is (cid:205) i s i ( X i ) while it incurs Our model generalizes trivially to the case where D i : = d i log ( a i + ) + (cid:15) i , i.e., the average demand scales with a CP-specific multiplicative constant; thisconstant can simply be absorbed into r i . a total cost of c (cid:205) i a i , where c is a positive constant. The netearnings for the ISP is then W ( s ( X ) . . . , s n ( X n ) , a . . . , a n ) = (cid:205) i s i ( X i ) − c (cid:205) i a i . We assume the ISP is risk-averse and setits utility as H ( W ) = − exp {− zW } , where z is a risk-averseparameter. This utility is referred to as Constant Absolute RiskAversion (CARA) in the literature [21]. The ISP enters intoagreement with the CPs only if its expected utility is largerthan a certain threshold denoted as H .The CPs compete against each other and aim to maximizetheir utility. The objective of CP i , i ∈ N taking into participa-tion constraints of the ISP is given as follows: max a ,..., a n , s i (·) U CP i : = E X i [ V i ( X i − s i ( X i ))] subjected to E X ,..., X n [ H ( s ( X ) , ..., s n ( X n ) , a , ..., a n )] ≥ H (1) max a ,..., a E X ,..., X n [ H ( s ( X ) , ..., s n ( X n ) , a , ..., a n )] . (2)Constraint (1) guarantees the ISP minimum expected utility H and is called as individual rationality (IR) or the participationconstraint. The second constraint (2) is the ISP’s optimizationproblem and is called an incentive compatibility constraint(IC). It also captures the fact that the CPs cannot observethe effort level of the ISP. Note that the Constraint in (2) mayhave multiple optima hence objective of each CP is optimizedover all of these possibilities. The structure of the optimizationproblem is hierarchical and can be studied considering aStackelberg solution concept where the principals (here eachCP) can be seen as leaders and the agent (here the ISP) as thefollower who plays after observing the action of the principals.In our setting, there is another level of complexity as there aremultiple strategic principals. This gives rise to a static gamebetween CPs with shared constraints. B. Linear Contracts
In the following we consider a specific type of contractwhere CPs share a fraction of their revenue to ISP. Thesecontracts are of the form s i ( X i ) = β i X i and are referred toas linear contracts, where β i ∈ [ , ] for all i ∈ N . TheISP chooses whether to accept or reject the contract. Linearcontracts are shown to be optimal in [22] particularly whenagent has a risk averse utility which is also the case in oursetting. Henceforth we denote a linear contract between theISP and CP i by its parameter β i . The expected utility of CP i is then: U CP i = E [ X i ( − β i )] = ( − β i ) r i log ( a i + ) , and the expected utility of the ISP is: U ISP = E (cid:34) H (cid:32)(cid:213) i ( s i ( X i ) − ca i ) (cid:33)(cid:35) = E (cid:34) H (cid:32)(cid:213) i ( β i r i ( log ( a i + ) + (cid:15) i ) − ca i ) (cid:33)(cid:35) = − exp (cid:40) − z (cid:213) i ( β i r i log ( a i + ) − ca i ) (cid:41) E (cid:34) exp (cid:40) − z (cid:213) i β i r i (cid:15) i (cid:41)(cid:35) . The ISP’s optimization problem is to find an effort level a i for each CP i which maximizes its expected utility. Notice thatmaximizing expected utility (IC constraint) is equivalent tomaximizing (cid:205) i ( β i r i log ( a i + ) − ca i ) over ( a , . . . , a n ) . C. Neutral vs Non-Neutral regime
We distinguish two scenarios based on differentiation in theefforts by the ISP for each CP. We say that the network is neutral if ISP puts the same amount of efforts for both theCPs irrespective of the revenue-share it can get from them,i.e., ISP always sets a = a ... = a n . We say that the networkis non-neutral if the ISP can put different amount of effortfor each CP, i.e., a (cid:44) a ... (cid:44) a n is permitted. Hence in theneutral regime the ISP treats each CP identically, whereas itcan differentiate between them in the non-neutral regime.Under the neutral regime and linear contracts, the ICconstraint of the ISP, i.e., max a E [ H ( s ( X ) , ..., s n ( X n ) , a )] ,simplifies to max a (cid:205) i ( β i r i log ( a + ) − ca ) . The optimal effortlevel for a given contracts ( β i , i ∈ N ) is given by: a ∗ = max (cid:18) (cid:205) i β i r i nc − , (cid:19) . (3)The objective function of CP i , i ∈ N in the neutral regime canthen be expressed as follows: Neutral: max β i ∈[ , ] ( − β i ) log ( a + ) r i subjected to (cid:213) i z ( β i r i log ( a + ) − ca )− log E (cid:34) exp (cid:40) − z (cid:213) i β i r i (cid:15) i (cid:41)(cid:35) ≥ − log (− H ) and a = max (cid:18) (cid:205) i β i r i nc − , (cid:19) . (4)Under the non-neutral regime, the IC constraint of the ISP,i.e., max a ,..., a n E [ H ( s ( X ) , ..., s n ( X n ) , a , ..., a n )] , simplifiesto max a ,..., a n (cid:205) i ( β i r i log ( a i + ) − ca i ) . The optimial effortslevel for a given contracts ( β i , i ∈ N ) are: a ∗ i = max (cid:18) β i r i c − , (cid:19) for i ∈ N . Simplified optimization problem for each CP i can then beexpressed as the following bi-level optimization problem: NonNeutral: max β i ∈[ , ] ( − β i ) log ( a i + ) r i subjected to (cid:213) i z ( β i r i log ( a i + ) − ca i )− log E (cid:34) exp (cid:40) − z (cid:213) i β i r i (cid:15) i (cid:41)(cid:35) ≥ − log (− H ) and a i = max (cid:18) β i r i c − , (cid:19) for i ∈ N . (5)Notice that the ISP has incentive to enter into contract withthe CPs only when H ≥ − , otherwise its net earning from theCPs is negative For any value of H ∈ (− , ] , the IR constraintmake the strategies of the players coupled and the game canhave continuum of equilibria as it is the case with generalcoupled constrained games [23]. However, with continuum ofequilibria we will be faced with equilibrium selection problemand a systematic comparison of CP utilities under the tworegime is not possible. We thus set H = − under which the IC constraint ensures that the IR constraint always holds andhence the objective of the CPs are no more jointly constrained.As we will see in the subsequent sections, this avoids thecontinuum of equilibrium. We note that even after relaxingthe IR constraint the problem is still challenging to analyze,but makes it possible to compare equilibrium utilities of allplayers under both the regimes.We say that a contract profile ( β i , i ∈ N ) is an equilibriumif no CP has an incentive for unilateral deviation from its con-tract. In the following we superscript the quantities computedat equilibrium with NN and N when they are associated withnon-neutral regime and neutral regime respectively.III. S YMMETRIC C ASE
In this section we consider the symmetric case whererevenue per unit demand for all the CPs is the same, i.e., r = r , . . . , = r n : = r . In other words, the CPs are symmetricwith regards to the ability to monetize their content. In thissetting, we analyze the equilibrium contracts arising in theneutral as well as non-neutral regime, and the resulting surplusof the CPs and the ISP. Our results highlight, surprisingly, thateven when the CPs are symmetric, the imposition of neutralityactually shrinks the surplus of all parties involved. Moreover,this ‘loss of surplus’ becomes more pronounced as the numberof CPs n grows. A. Non-neutral regime
In the non-neutral regime, it is easy to see that when H = − , the interactions between each CP and the ISP aredecoupled. The optimization problem in (5) for CP i , i ∈ N , after substituting the optimal effort simplifies to: max β i ∈[ , ] ( − β i ) r log (cid:18) max (cid:18) β i rc , (cid:19)(cid:19) . Moreover, we note that it is only interesting to consider thecase r > c . Indeed, since the monetization resulting from ISPeffort a i for CP i equals r log ( + a i ) , the marginal monetizationis at most r . Thus, if r ≤ c , it is not worthwhile for CPs tomake any investments to grow the demand.The following result characterizes the equilibrium contractsbetween each CP and the ISP. The contracts are expressedin terms of the LambertW function computed on its principlebranch, denoted as W (·) (see [24]). Theorem 1: If r > c , the equilibrium contract betweenCP i , i ∈ N and the ISP is given by β N N : = β N Ni = W (cid:0) rc e (cid:1) . (6)Since W (·) in strictly increasing and W ( e ) = , it follows that β N N ∈ ( , ) when r / c > . Moreover, note that equilibriumfraction β N N of CP revenue that is shared with the ISP isa strictly decreasing function of the ratio r / c , as might beexpected.Using Theorem 1, one can easily characterize the equilib-rium effort of the ISP as well as the surplus of each agent. Corollary 1:
Assume r > c . The equilibrium effort for eachCP i , i ∈ N put by the ISP is given by a N N : = a N Ni = r β N N c − > . (7)The equilibrium surplus of CP i , i ∈ N is given by U N N CP i = ( − β N N ) r log ( a N N + ) = ( − β N N ) β N N r > . (8)Finally, the equilibrium surplus of the ISP is given by U N N
ISP = nr + nc − n β N N r > . (9)Note that so long as r > c , the equilibrium contracts awardeach CP and the ISP a positive surplus. B. Neutral regime
We now consider the neutral regime. The CPs are stillassumed to be symmetric, only the ISP is now constrained to make the same investment decision for all CPs, i.e., a = . . . = a n : = a . The surplus of CP i in this case, aftersubstituting the optimal ISP effort simplifies to: ( − β i ) r log (cid:32) max (cid:32) (cid:205) nj = β j rnc , (cid:33)(cid:33) . Since the surplus of each CP in the neutral regime depends onthe actions of all CPs, we seek contract profiles ( β Ni , i ∈ N ) that constitute a Nash equilibrium between CPs. These equi-libria are characterized completely in the following theorem.As before, the only scenario of interest is r > c . Theorem 2:
Consider the neutral regime with r > c . In thiscase, only symmetric Nash equilibria exist. When < r / c ≤ n , there are two Nash equilibria, , and ( β Ni , i ∈ N ) , where β Ni = β N : = nW (cid:0) rnc e / n (cid:1) . (10)When r / c > n , ( β Ni , i ∈ N ) is the only Nash equilibrium,where β Ni is given by (10).Note that when < r / c ≤ n , unlike in the non-neutralregime, making no contributions to the ISP, resulting in zerosuplus for all parties, is an equilibrium between the CPs.The other equilibrium, given by (10), results in a positivesurplus for all parties (as is shown in the following corollary).In the remainder of this section, we will refer to this latterequilibrium as the non-zero equilibrium. Corollary 2:
Consider the neutral regime, with r > c . Underthe non-zero equilibrium: • The effort for CP i , i ∈ N , put by the ISP is given by a N : = a Ni ( n ) = max (cid:18) β N rc − , (cid:19) . (11) • The surplus of CP i , i ∈ N is given by U N CP i = ( − β N ) r log ( a N + ) = (cid:0) − β N (cid:1) n β N r > . (12) • The surplus of ISP is given by U N ISP = r + nc − ( n + ) β N r > . (13) C. Neutral regime v/s Non-neutral regime
Having now characterized the equilibrium contracts and thesurplus of each CP and the ISP under the neutral and thenon-neutral regime, we are now in a position to compare thetwo. As the following result shows, the non-neutral regimeis actually better for all parties as compared to the neutralregime.
Theorem 3:
Suppose r > c , and n ≥ . In the symmetriccase, at equilibrium, the following statements hold.1) CPs share a higher fraction of their revenue with the ISPin the non-neutral regime, i.e., β N N > β N .
2) The effort by the ISP for each CP is higher in the non-neutral regime, i.e., a N N > a N
3) The surplus of each CP is higher in the non-neutralregime, i.e., U N N CP i > U N CP i for all i ∈ N
4) The surplus of the ISP is higher in the non-neutralregime, i.e., U N N
ISP > U N ISP .The above result highlights that, surprisingly, constraining theISP to be neutral is actually sub-optimal for all parties, evenwhen the CPs are symmetric. In other words, the non-neutralregime is actually preferable to the ISP as well as the CPs.Intuitively, the reason for this tragedy of the commons is thatthe imposition of neutrality skews the payoff landscape foreach CP, such that the ‘benefit’ of any additional investmentit makes gets ‘shared’ across all CPs. This induces the CPs tocommit smaller fractions of their revenues to the ISP, which inturn results in a lower ISP effort, and a lower demand growthfor all CPs. Indeed, as we show below, this effect gets furthermagnified with an increase in the number of CPs.
D. The effect of number of CPs
In the non-neutral regime, the interactions between thedifferent CPs and the ISP are decoupled, implying that theimpact of scaling n is trivial. Thus, we now study the impactof scaling n in the neutral regime, on the equilibrium ISPeffort, and the surplus of each agent. Note that when n = , the neutral and the non-neutral regime coincide. Our mainresult is the following. Theorem 4:
Suppose that r > c . In the neutral regime, thenon-zero equilibrium satisfies the following properties.1) β N is a strictly decreasing function of n .
2) The effort by the ISP for each CP ( a N ) is a strictlydecreasing function of n , even though the total effort( na N ) by the ISP is a strictly increasing function of n .
3) The surplus of each CP is a strictly decreasing functionof n , and lim n →∞ U NCP i ( n ) = .
4) The surplus of the ISP is eventually strictly decreasingin n , and lim n →∞ U NISP ( n ) = . Theorem 4 highlights that an increase in the number of CPsfurther exacerbates the sub-optimality of the neutral regimefor the CPs as well as the ISP. As before, the explanationfor this is that with increasing n , the surplus resulting froman additional contribution by any CP gets ‘split’ further, thusdisincentivising the CPs from offering a significant fractionof their revenues to the ISP. The variation of ISP utility as afunction of n is depicted in Figure 2 for different values of r / c . In all cases, the utility first increase for some n and thedecreases thereafter. Fig. 2. ISP utility in the neutral regime as n varies for different r / c . IV. A
SYMMETRIC CASE
In this section we study the asymmetric case where mone-tizing power of all the CPs need not be the same, i.e., r i (cid:44) r j for i (cid:44) j . Our interest in this section is to understand howdisparity in the monetizing power influences preference of theplayers for the neutral and non-neutral regimes. We focus onthe case with two CPs ( n = ) and without loss of generalityassume that monetizing power of CP is more than that ofCP , i.e., r > r . We refer to CP as dominant and CP asnon-dominant.Recall that the objective of the CP i , i ∈ N in the non-neutralregime can be expressed as max β i ∈[ , ] ( − β i ) r i log (cid:18) max (cid:18) β i r i c , (cid:19)(cid:19) , and in the neutral regime it can be expressed as max β i ∈[ , ] ( − β i ) r i log (cid:18) max (cid:18) (cid:205) i β i r i nc , (cid:19)(cid:19) . As discussed in Section III, the case r i / c ≤ for all i ∈ N isnot interesting as none of the CPs would have the incentive tocontribute towards ISP effort. Thus, in this section, we restrictourselves to the case where r i / c > for at least one i ∈ N .The following results characterize the equilibrium contractsfor the neutral and the non-neutral regime. A. Equilibrium contracts
In the non-neutral regime, the interactions between eachCP and the ISP remain decoupled, and thus the equilibriumcontracts follow easily from Theorem 1.
Corollary 3:
In the non-neutral regime, the equilibriumcontract ( β N N , β N N ) is as follows: β N Ni = (cid:40) if r i c ≤ , W ( ric e ) if r i c > . Note that when r i c ≤ , the equilibrium contract between CP i and the ISP is not uniquely defined, since any β i ∈ [ , ] wouldresult in zero surplus to CP i . Next, we characterize equilibrium contract in the neutralregime.
Theorem 5:
Consider the neutral regime, with r > r . If r / c ≤ then ( β N , β N ) = ( , ) . If r / c > , then the The characterization of equilibrium contracts can actually be done for any n ; see Appendix E. equilibrium contract is given by: ( β N , β N ) = ( β , β ) if r + r r − r > W (cid:0) r + r c √ e (cid:1) , (cid:18) W ( r c e ) , (cid:19) otherwise , (14)where β = r + r r W (cid:0) r + r c √ e (cid:1) − r − r r , β = r + r r W (cid:0) r + r c √ e (cid:1) − r − r r . When r / c ≤ , the equilibrium contract is not unique,though the outcome is that ISP effort equals zero. When r / c > , the equilibrium contract is unique, and at least oneCP (specifically, CP ) is guaranteed to contribute a constantfraction of her revenue to the ISP. Note that when r / c ∈ ( , ] , there is no CP contribution in the neutral regime, even thoughthere is in the non-neutral regime.To interpret the equilibrium when r / c > , let r ∗ : = r ∗ ( r ) denote the value of r that satisfies the following relation fora given r r + r r − r = W (cid:16) r + r c √ e (cid:17) . For r ≤ r ∗ the condition in (14) holds where revenue sharedby both the CPs is strictly positive, i.e., β Ni > for all i ∈ N .For r > r ∗ the condition in (14) fails in which only the CP ’sshare is strictly positive and CP does not share anything, i.e., β N > and β N = . Further, it is easy to verify that r ∗ ismonotonically increasing in r and r ∗ > r . B. Comparison between Neutral and Non-neutral regimes
Having characterized the equilibrium contracts in bothregimes, we compare and contrast the neutral and non-neutralregimes in the remainder of this section. We begin by com-paring the equilibrium contracts, followed by CP/ISP utility,social utility, and finally ISP effort. Contracts : The following proposition provides a com-parison of the equilibrium contracts in both the regimes.
Proposition 1:
Fix r > . We have • For r > r , β N N ≥ β N . Moreover, β N decreases in r . • For r ≥ r ∗ , β N > β N N . Moreover, β N decreases in r for r ≥ r ∗ .The conclusions of Proposition 1 are summarised in the scatterplot in Fig. 3a. Note that the non-dominant CP always con-tributes a smaller fraction of its revenue in the neutral regime.With the dominant CP, the contribution factor is larger in theneutral regime when the revenue rates are highly asymmetric(see the green region in Figure 3a, and larger in the non-neutralregime when the revenue rates are symmetric (see the redregion in Figure 3a). A sufficient condition for the former is r ≥ r ∗ ( r ) . The latter observation is of course consistent withTheorem 3, which dealt with the case of perfect symmetry.Proposition 1 also establishes monotonicity properties of thesharing contracts of CP in the neutral regime in r for a fixed r . While β N descreasess in r , β N eventually decreasing in r ; see Figs. 3b and 3c. Note that β N can actually be increasingwith respect to r when the revenue rates are nearly symmetric,in contrast with the non-neutral setting. (a) (b) (c)Fig. 3. Fig. 3a gives scatter-plot of β s. Figs. 3b and 3c shows variation of equilibrium β vs r under neutral and non-neutral regime.(a) (b) (c)Fig. 4. Fig. 4a shows scatterplot for the CP utilities at equilibrium. Figs. (4b) & (4c) compare CP utility in both regimes as r varies. Utility of CPs : The following proposition characterizespreference of the CPs for the neutral and non-neutral regime.
Proposition 2:
Fix an r . We have • For all r > r , CP prefers the non-neutral regime. • For all r ≥ r ∗ , CP prefers the neutral regime.Figure 4a shows the scatter plot utilities of the players in boththe regimes. Note that the dominant CP has higher utility inthe non-neutral regime as can be observed from the red andmagenta regions. This is because in the neutral regime, thedominant CP is ‘forced’ to pay for capacity investments thatalso benefit the non-dominant CP. Indeed, note that in theregion r ≥ r ∗ , the dominant CP shares a smaller fraction ofits revenue in the non-neutral regime, but still ends-up witha higher utility. Interestingly, the non-dominant CP obtainsa higher utility in the neutral regime when the asymmetricrevenue rates are too separated (see the pink region in Fig. 4a)A sufficient condition for this is r ≥ r ∗ . This is of course dueto the ‘subsidization’ it receives from the dominant CP. Onthe other hand, when the revenue rates are nearly symmetric,even the non-dominant CP prefers the non-neutral regime,once again consistent with Theorem 3. The above observationsfurther illustrated in Figs. 4b and 4c. ISP utility : We next compare utility of the ISP in thenon-neutral and neutral regime. For simplicity, we take ISPutility to be the expected revenue given as U ISP = E [ (cid:205) i s ( X i ) − ca i ] (ignoring the risk-sensitive utility defined before). Itsvalue in the non-neutral regime is given by: U N N
ISP = ( − β N N ) r + ( − β N N ) r + c , and in the neutral regime for all r ≥ r ∗ is given by: U N ISP = ( − β N ) r + c . The utility for r < r ∗ in the neutral regime is cumbersomeand we skip its expression. The following lemma demonstratesthe ISPs earnings are higher in the non-neutral regime whenmonetization power of the dominant CP is much larger thanthe other, i.e., r is much larger than r . Lemma 1:
There exist r b > r ∗ , such that for all r > r b theISP’s utility is higher in the non-neutral regime.A general comparison of ISP utility in the two regimes isnot analytically tractable. We give a numerical illustration inFigure 5. As seen in the first figure, utility of ISP in the non-neutral regime is higher than in the neutral regime for all r for a given r and c . Scatter plot in the second figure showsthat this observation extends over the entire parameter range. Fig. 5. The first figure compares ISP utility in neutral and non-neutral with c = r = . The second figure gives a scatter plot. Social Utility : The social utility in the non-neutral andneutral regimes are given, respectively, as follows: SU N N = U N NCP + U N NCP + U N NISP = r log (cid:32) β N N r c (cid:33) + r log (cid:32) β N N r c (cid:33) − ( β N N r + β N N r ) + c and SU N = U NCP + U NCP + U NISP = ( r + r ) log (cid:32) β N r + β N r c (cid:33) − ( β N r + β N r ) + c . As it is not easy to compare the social utilities analytically,
Fig. 6. Comparison of social utility between neutral and non-neutral regimefor c = , r = and scatter plot. we resort to numerical comparison of the utilities in Figure6. As seen social utility in the non-neutral regime dominatesthat in the neutral regime for all values of r for a given r and c . The scatter plot in the second part of the figure showsthat the observation continue to hold for all parameters. Total Effort by ISP : Finally we compare the total effortby ISP for CPs in the non-neutral and neutral regime given,respectively, as follows A N N = a N N + a N N = β N N r c − + β N N r c − = r c W ( r c e ) + r c W ( r c e ) − ∀ r > r ∗ , and A N = a N = (cid:32) β N r c − (cid:33) = r c W ( r c e ) − ∀ r > r ∗ Lemma 2:
There exist a threshold r a > r ∗ such that totaleffort by ISP is higher in neutral regime than in the non-neutral. The threshold satisfies the following:The threshold r a is given by following equation: r a (cid:169)(cid:173)(cid:171) W ( r a c e ) − W ( r a c e ) (cid:170)(cid:174)(cid:172) = r W ( r c e ) It can be seen from above equation that r a is monotonicallyincreasing in r .The above lemma implies that total effort by ISP in theneutral regime becomes higher when there is high asymmetrybetween CP’s revenue per click rates.V. C ONCLUSIONS AND R EGULATORY I SSUES
We studied the problem of revenue sharing between multipleCPs and an ISP on the Internet using the moral hazard frame-work with multiple principles and a single agent. We comparedthe revenues of each player and the social utility in a regime
Fig. 7. Scatter plot for comparison between total effort (investment) by ISPin non-neutral and neutral regime over different range of r i / c where the ISP is forced to put equal effort for all the CPs(neutral) with a regime where there are no such restrictions(non-neutral) on the ISP. Our key take-away is that every oneis better off and social utility is higher in non-neutral regimewhen the CPs ability to montize their demand are ‘nearly’ thesame. When the there is a large disparity in the monetizationpower of the CPs, for the case of two CPs we showed thatnon-neutral regime is preferable from the standpoint of thedominant CP (with higher monetizing power), the ISP, andfrom the standpoint of social utility. On the other hand, thenon-dominant CP is benefited by a neutrality stipulation sinceit gets to ‘free-ride’ on the contribution made by the dominantCP, the very reason that makes this regime less preferred bythe dominat CP.Our analysis throws up an intriguing dilemma for a regulator–enforcing neutrality brings in parity in the way ISP treat theCPs, but it worsens the social utility and pay-off of all theplayers compared to the neutral regime if the players act non-cooperatively. It is then interesting to study mechanisms thatthe regulator can use to induce cooperation among the theplayers so that the social utility and players pay-off are noworse than in the non-neutral regime.A PPENDIX
A. Proof of Theorem 1
From CP i optimization problem, it can be observed that for r / c < U CP i = for all i . Henc e no CP has an incentiveto share a fraction of their revenue with the ISP and β i = ∀ i ∈ N is the equilibirum. Now assume r / c ≥ . For thiscase the optimial value of β i will be such that r β i / c ≥ andthe optimization problem of CP i reduces to max β i ∈[ , ] ( − β i ) r log (cid:18) β i rc (cid:19) . The first order optimality condition ∂ U CP i / ∂ β i = then gives: log (cid:18) β i rc (cid:19) = − β i β i ∀ i = , , ..., n . Solving the first order conditions for each CP i , we get: − β i β i = log (cid:18) β i rc (cid:19) = ⇒ β i = log (cid:18) β i rec (cid:19) = ⇒ e β i = β i rec = ⇒ β i e β i = rec Using the definition of the LamebertW funtion we get β i = W (cid:16) rc e (cid:17) = ⇒ β i = W (cid:0) rc e (cid:1) Hence we get equilibrium contract given in (6).
B. Proof of Theorem 2
Recall the objective of CP i , i ∈ N max β i ∈[ , ] ( − β i ) r log (cid:32) max (cid:32) (cid:205) nj = β j rnc , (cid:33)(cid:33) . First assume that r / c < . In this case for any given ( β , β , . . . , β i − , β i + , . . . , β n ) , best response of CP i is to set β i = . Thus β i = ∀ i ∈ N is an equilibrium.Next consider the case ≤ r / c < n . Fix an i ∈ N and assume β j = for all j (cid:44) i . Then the object of CP i simplifies to max β i ∈[ , ] ( − β i ) r log (cid:18) max (cid:18) β i rnc , (cid:19)(cid:19) , and the best response of CP i is to set β i = . Hence β i = for all i ∈ N is an equilibrium. We next look for a non-zeroequilibrium. By symmetry, it must be such that β = β . . . = β n ∈ ( , ] . Further, at equilibrium it must be the case that (cid:205) ni = β j r / nc ≥ , otherwise CPs have incentive to deviate tomake their share zero. Writing the first order condition for theoptimaization problem of CP i , i ∈ N , i.e., max β i ∈[ , ] ( − β i ) r log (cid:18) (cid:205) j β j rnc (cid:19) , we get log (cid:32) (cid:205) nj = β j rnc (cid:33) = − β i (cid:205) nj = β j . Setting β = β , . . . , = β n = β we have log (cid:18) β rc (cid:19) = − β n β . Simplyfying the above as earlier in the format of LambertWfunction we get β = nW ( rnc e / n ) .For the case r / c ≥ n , β i = , ∀ i ∈ N at equilibrium is notarise, however the equilibrium β = nW ( rnc e / n ) still holds. Thiscompletes the proof. C. Proof of Theorem 3Part 1:
When r / c ≤ , β N N = β N = and the relation β N N ≥ β N holds trivially. In the range < r / c ≤ n , two equilibriaare possible in the neutral regime, β N = or nW ( r / ce / n ) . If β N = is the equilibrium, again the relation holds trivially.Consider the case when β N = nW ( r / ce / n ) is the equilibriumfor < r / c . Define b : = r / c and f ( b ) = β N N β N . lim b → f ( b ) = lim b → nW ( bn e n ) W ( e ) = nW ( n e n ) W ( e ) = n . n = ( using x = W ( xe x )) Fig. 8. Utility of CP vs β The limit holds as the equlibrium definition holds for all b > and W is continuous at b = . Also, f ( b ) is monotonicallyincreasing in b forall b > as ∂ f ( b ) ∂ b = nW ( be n / n ) bW ( be ) (cid:34) W ( be ) − W ( be n / n )( + W ( be ))( + W ( be n / n )) (cid:35) > ∀ b > Hence β N N > β N . It holds similarly for the case r / c > n . Part 2:
Since investment decision by ISP is monotonicallyincreasing in the share in gets fromt the CPs (from Eqns. (7)and (11), by Part it is clear that ISP make more investmentin non-neutral regime as compared to neutral regime. Part 3:
In both non-neutral and neutral regime equilibriumeffort, a for given β is a + = max (cid:16) β rc , (cid:17) .Now, in both non-neutral and neutral regimes, each CP’sutility at equilibrium is the same function given by ( − β ) r log (cid:16) max (cid:16) β rc , (cid:17)(cid:17) which is concave in β ∈ ( c / r , ) andfrom Part 1 we have that β N ≤ β N N . This implies that U NCP ≤ U N NCP (seen Fig. 8)
Part 4: U
N NISP = [ n β N N r log ( a N N + ) − nc ( a N N )] Substituting the value of a N N ( β N N ) = β N N rc − in the secondterm of above expression, we get: U N NISP = { n β N N r (cid:2) log ( a N N + ) − (cid:3) + nc } (15)Similarly, U NISP = [ n β N r (cid:2) log ( a N + ) − (cid:3) + nc ] (16)Now, from Part and , we know that β N N ≥ β N and a N N ≥ a N , respectively. Thus, by comparing Eqns. (15) and (16), wehave that U N NISP ≥ U NISP . D. Proof of Theorem 4Part 1:
Considering n to be continuous variable, ∂ β N ( n ) ∂ n = n W (cid:0) rnc e / n (cid:1) (cid:34) − + + n + W (cid:0) rnc e / n (cid:1) (cid:35) Now, β N ( n ) decreases with n iff ∂β N ∂ n < ⇐⇒ (cid:34) − + + n + W (cid:0) rnc e / n (cid:1) (cid:35) < ⇐⇒ (cid:34) + n + W (cid:0) rnc e / n (cid:1) (cid:35) < ⇐⇒ + n < + W (cid:16) rnc e / n (cid:17) ⇐⇒ < nW (cid:16) rnc e / n (cid:17) it holds as β N ( n ) = nW ( rnc e / n ) < = ⇒ nW (cid:16) rnc e / n (cid:17) > . Part 2:
Effort of ISP for each CP is decreasing followingdirectly as β N is decreasing in n . The total effort of ISP is A N ( n ) = na N ( n ) = n (cid:16) β N rc − (cid:17) . In the following we showthat A N ( n + ) > A N ( n ) for any n . We have A N ( n + ) > A N ( n ) ⇐⇒ ( n + ) β N ( n + )− n β N ( n ) > cr (17)We prove that the above inequality holds in two part. Part (i):
We first prove that g ( n ) = n β N ( n ) is concave in n ,which implies the difference ( n + ) β N ( n + ) − n β N ( n ) shrinksas n increases. Now, g ( n ) = n β N ( n ) = W (cid:16) bn e n (cid:17) ; where b = rc It is clear that g ( n ) is increasing in n . Treating n as continuousvariable, we have ∂ f ( n ) ∂ n = n + n W (cid:16) bn e n (cid:17) (cid:16) + W (cid:16) bn e n (cid:17)(cid:17) > ∂ f ( n ) ∂ n = n W (cid:16) bn e n (cid:17) (cid:16) + W (cid:16) bn e n (cid:17)(cid:17) × ( n + ) (cid:16) + W (cid:16) bn e n (cid:17)(cid:17)(cid:16) + W (cid:16) bn e n (cid:17)(cid:17) − n ( n + ) (cid:18) + W (cid:18) bn e n (cid:19)(cid:19) Now, f ( n ) is strictly concave in n iff ∂ f ( n ) ∂ n < ⇐⇒ ( n + ) n ( n + ) < (cid:16) + W (cid:16) bn e n (cid:17)(cid:17) (cid:16) + W (cid:16) bn e n (cid:17)(cid:17) After cross multiplying and expanding, we get + W (cid:16) bn e n (cid:17) nW (cid:16) bn e n (cid:17) + W (cid:16) bn e n (cid:17) < nW (cid:18) bn e n (cid:19) We know that / β N = nW (cid:16) an e n (cid:17) > at equilibrium,thereforeLHS < >
1. Thus, the above inequality holds .
Part (ii):
Now, we show that ( n + ) β N ( n + ) − n β N ( n ) → c / r as n → ∞ . Consider asymptotic expansion of LambertWfunction, W ( x ) = x − x + o ( x ) = x ( − x + o ( x )) W ( x ) = x ( − x + o ( x )) = x · ( + x + o ( x )) = x + + o ( ) second equality comes by using − x = + x + o ( x ) .Now, f ( n ) = n β N ( n ) = W (cid:16) rnc e n (cid:17) = ncre n + + o ( ) = ncr (cid:18) − n + o (cid:18) n (cid:19)(cid:19) + + o ( ) = ncr − cr + + o ( ) (18)third equality comes by using e − x = − x + o ( x ) .Therefore, f ( n + ) − f ( n ) = ( n + ) cr − ncr → cr as n → ∞ Part 3:
For any n each CP’s utility for the given a atequilibrium, is given by ( − β ) r log (cid:16) max (cid:16) β rc , (cid:17)(cid:17) .We know that the above function is concave in β ∈ ( c / r , ] and from Part , as n increases β N ( n ) decreases and approaches c / r when n → ∞ (cid:169)(cid:173)(cid:171) nW (cid:16) rnc e n (cid:17) = n (cid:18) rnce / n − (cid:16) rnce / n (cid:17) − o (cid:16) n (cid:17)(cid:19) → cr , as n → ∞ (cid:170)(cid:174)(cid:172) .This implies that U NCP ( n ) decreases as n increases (see Fig.8). Also it can be seen that as n → ∞ , U NCP → . Part 4: U
NISP = n ( β N r log ( a N + ) − ca N ) = n (cid:18) β N r log (cid:18) β N rc (cid:19) − c (cid:18) β N rc − (cid:19)(cid:19) = r + nc − ( n + ) β N r Utility of ISP from each CP is U NISP n = β N r log ( a N + ) − ca N (19)Substituting the value of a N = β N rc − in the second term,we get U NISP n = β N r (cid:0) log ( a N + ) − (cid:1) + c . Since β N ans a N decreases with increase in n , it is clear from above expressionthat utility of ISP from each CP also decreases with increasein number of CPs.Now, U ISP ( n ) N increases with increase in n iffU ISP ( n ) N ∂ n > (considering n to be continuous) c − n W (cid:16) rnc e n (cid:17) (cid:169)(cid:173)(cid:173)(cid:171) n + n + − nW (cid:16) rnc e n (cid:17) n (cid:16) + W (cid:16) rnc e n (cid:17)(cid:17) (cid:170)(cid:174)(cid:174)(cid:172) r > n W (cid:16) rnc e n (cid:17) (cid:169)(cid:173)(cid:173)(cid:171) n + n + − nW (cid:16) rnc e n (cid:17) n (cid:16) + W (cid:16) rnc e n (cid:17)(cid:17) (cid:170)(cid:174)(cid:174)(cid:172) < cr Since, c / r > , we have n W (cid:16) rnc e n (cid:17) (cid:169)(cid:173)(cid:173)(cid:171) n + n + − nW (cid:16) rnc e n (cid:17) n (cid:16) + W (cid:16) rnc e n (cid:17)(cid:17) (cid:170)(cid:174)(cid:174)(cid:172) < ⇐⇒ n + n + − nW (cid:16) rnc e n (cid:17) < . E. Asymmetric case: Equilibrium Contracts for n > Theorem 6:
In the non-neutral regime equilibrium contractfor CP i is given by β N Ni = (cid:40) if r i c < , W ( ric e ) if r i c ≥ , (20)Further, the effort levels of the ISP for are given by a N Ni = max (cid:32) β N Ni r i c − , (cid:33) ∀ i = , , · · · n . (21) Theorem 7:
In the neutral regime, each CP i , i = , , · · · , n shares a positive fraction of the revenue at equilibrium with theISP only if r i / c > ∀ i = , , · · · , n and r , r , ..., r n are closeenough to each other. Specifically, the equilibrium contract isas follows β Ni = (cid:205) nj = r j n r i W (cid:16) (cid:205) nj = r j n c e n (cid:17) − (cid:205) nj = , j (cid:44) i r j − ( n − ) r i nr i ; ∀ i = , , ..., n . (22)and the equilibrium effort is a N = (cid:16) (cid:205) nj = β j r j nc − (cid:17) .When r >> r , only CP shares positive fraction at equilib-rium, and the equilibrium contract is given as follows: β N = W (cid:0) r nc e (cid:1) , & β ∗ i = , ∀ i = , , ..., n (23)and the equilibrium effort is a N = (cid:16) β r nc − (cid:17) Proof: Substituting the best action of ISP determined inCP i ’s optimization problem, we get: max β i ( − β i ) r i log (cid:32) max (cid:32) (cid:205) nj = β j r j nc , (cid:33)(cid:33) First order necessary condition for CP i gives ( − β i ) r i (cid:205) nj = β j r j − log (cid:32) (cid:205) nj = β j r j nc (cid:33) = ∀ i = , , ..., n (24)Comparing these set of eqns, we get, ( − β ) r = ( − β ) r = ... = ( − β n ) r n = ⇒ β i r i = β j r j + r i − r j ; ∀ i = , , ..., n ; i (cid:44) j ∴ n (cid:213) j = β j r j = n β i r i + n (cid:213) j = , j (cid:44) i r j − ( n − ) r i ∀ i = , , ..., n Substituting, we get ( − β i ) r i n β i r i + (cid:205) nj = , j (cid:44) i r j − ( n − ) r i = log (cid:32) n β i r i + (cid:205) nj = , j (cid:44) i r j − ( n − ) r i nc (cid:33) (25)Adding / n to both the sides of eqn.(25), we get (cid:205) nj = r j n (cid:16) n β i r i + (cid:205) nj = , j (cid:44) i r j − ( n − ) r i (cid:17) = log (cid:32) n β i r i + (cid:205) nj = , j (cid:44) i r j − ( n − ) r i nc (cid:33) + log e n Rearranging and solving, we get β i = (cid:205) nj = r j n r i W (cid:16) (cid:205) nj = r j n c e n (cid:17) − (cid:205) nj = , j (cid:44) i r j − ( n − ) r i nr i ; i = , , ..., n Since, r > r , β > , however, β , β , ..., β n in above expres-sion can tale negative value. therefore, the above solution holdsonly if r i ’s are sufficiently close s.t. above solution is positive ∀ i = , , ..., n . Else, β , β , ..., β n = , and β is obtained from ( − β ) β − log (cid:16) β r nc (cid:17) = . Solution of which is β = W ( r nc e ) . F. Proof of Proposition 1Part 1:
We know that there exist some r < r ∗ , for which β N N is positive given by β N = r + r r W (cid:0) r + r c e . (cid:1) − r − r r Now, differentiating β N w.r.t r , we get: ∂ β N ∂ r = (cid:0) + W (cid:0) r r + r c e . (cid:1)(cid:1) − ≤ ∀ r ≥ r which implies decreases β N with increase in r .And for r > r ∗ , β N = . And β N N = W ( r c e . ) > which remain unchanged with increase in r . Also at r = r (symmetric case), β N N ≥ β N . Thus, β N N ≥ β N ∀ r ≥ r . Part 2:
It is clear from the expression of β N N that it isdecreasing in r Now, there exist some r > r ∗ for which β N is β N = W ( r c e ) ,which also decreases with increase in r .Also, for r > r ∗ , β N > β N N . Now, consider the case when r ≤ r ∗ where β N = r + r r W (cid:0) r + r c e . (cid:1) − r − r r Now, differentiating β N w.r.t r , we get: ∂ β N ∂ r = (cid:0) + W (cid:0) r + r c e . (cid:1)(cid:1) (cid:0) W (cid:0) r + r c e . (cid:1) − (cid:1) − r r W (cid:0) + W (cid:0) r + r c e . (cid:1)(cid:1) And it can take both negative and positive values dependingupon value of r . Therefore, it if not apparent that whether β N increases or decreases when r < r ∗ . G. Proof of Proposition 2Part 1:
Utility of CP in non-neutral regime is given by U N NCP = ( − β N N ) r log (cid:32) β N N r c (cid:33) = ( − β N N ) β N N r (Using first order condition ( − β N N ) β N N = log (cid:16) β N N r c (cid:17) )Utility of CP in neutral regime is given by for all r > r ∗ U NCP = ( − β N ) r log (cid:32) β N r c (cid:33) = ( − β N ) β N r (Using first order condition ( − β N ) β N = log (cid:16) β N r c (cid:17) )We know for all r > r ∗ , β N > β N N . = ⇒ ( − β N ) < ( − β N N ) and β N < β N N = ⇒ ( − β N ) β N r < ( − β N N ) β N N r , thus, U NCP < U N NCP Part 2:
Utility of CP in non-neutral regime is given by U N NCP = ( − β N N ) r log (cid:32) β N N r c (cid:33) = ( − β N N ) β N N r (Using first order condition ( − β N ) β N = log (cid:16) β N r c (cid:17) )Utility of CP in non-neutral regime is given by for r ≥ r ∗ U NCP = ( − β N ) r log (cid:32) β N r + β N r c (cid:33) = ( − β N ) β N r (Using first order condition ( − β N ) β N = log (cid:16) β N r c (cid:17) )Now, U N NCP ≤ U NCP iff ( − β N N ) β N N ≤ ( − β N ) β N We know that when r > r ∗ , β N decreases with increase in r .Thus, RHS of above inequality is increasing in r . However, β N N remain unchanged with increase in r , implying that LHSof the above inequality is constant. Therefore, there exist some r , beyond which the above inequality holds. We know that U N NCP remain constant with increase in r . H. Proof of lemma 1
Utility of
ISP in non-neutral regime is given by U N NISP = β N N r log (cid:32) β N N r c (cid:33) + β N N r log (cid:32) β N N r c (cid:33) − c (cid:32) β N N r c + β N N r c − (cid:33) = ( − β N N ) r + ( − β N N ) r + c (Using first order condition ( − β Ni ) β Ni = log (cid:16) β Ni r i c (cid:17) ; i = , )Utility of ISP in neutral regime is given by for r ≥ r ∗ U NISP = β N r log (cid:32) β N r c (cid:33) − c (cid:32) β N r c − (cid:33) = ( − β N ) r + c (Using first order condition ( − β N ) β N = log (cid:16) β N r c (cid:17) )Now, U N NISP ≥ U NISP iff ( − β N N ) r + ( − β N N ) r ≥ ( − β N ) r ( β N − β N N ) r ≥ −( − β N N ) r r (cid:18) W ( r c e ) − W ( r c e ) (cid:19) ≥ (cid:18) W ( r c e ) − (cid:19) r LHS in increasing in r , however RHS remain unchanged.Therefore, there must exist some r > r ∗ say r b s.t. for all r > r b the above inequality holds. Also, plot shows that ISPis always better off in non-neutral regime. I. Proof of Lemma 2A N ≥ A N N ⇐⇒ r c W ( r c e ) − ≥ r c W ( r c e ) + r c W ( r c e ) − ⇐⇒ r (cid:18) W ( r c e ) − W ( r c e ) (cid:19) ≥ r W ( r c e ) LHS of above inequality is increasing in r and RHS remainsunchanged. With increase in r for fixed r , the above inequal-ity will start holding for some large value of r . Thus, theabove inequality will hold for some large enough r a >> r ∗ . R EFERENCES[1] H. Jung, “Cisco visual networking index: global mobile data trafficforecast update 2010–2015,” Technical report, Cisco Systems Inc, Tech.Rep., 2011.[2] G. Paschos, G. Iosifidis, M. Tao, D. Towsley, and G. Caire, “The role ofcaching in future communication systems and networks,”
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