Pricing and Revenue Sharing Strategies for Internet Service Providers


 Jocelyn Powers
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1 Pricing and Revenue Sharing Strategies for Internet Service Providers Linhai He and Jean Warand Department of Eectrica Engineering and Computer Sciences University of Caifornia at Berkeey Abstract One of the chaenges facing the networking industry today is to increase the profitabiity of Internet services. This cas for economic mechanisms that can enabe providers to charge more for better services and coect a fair share of the increased revenues. In this paper, we present a generic pricing mode for Internet services jointy offered by a group of providers. We show that noncooperative pricing strategies between providers may ead to unfair distribution of profit and may even discourage future upgrades to the network. As an aternative, we propose a fair revenuesharing poicy based on the weighted proportiona fairness criterion. We show that this fair aocation poicy encourages coaboration among providers and hence can produce higher profits for a providers. Based on the anaysis, we suggest a scaabe agorithm for providers to impement this poicy in a distributed way and study its convergence property. Keywords: Internet services, pricing, game theory I. INTRODUCTION Today s Internet is a confederation of thousands of service providers. As no provider presenty has the compete endtoend coverage, they have to work together, i.e. forward each other s traffic, to offer Internet services. This interdependency makes pricing of Internet services a very compex issue. For exampe, because service providers are usuay owned by different organizations and have their individua objectives, they woud price their services in ways that woud maximize their own profits. If a service is jointy provided by a group of providers, then each of them woud charge a share of the tota price as arge as possibe. This sefcentered competition for more revenues may infate the price charged to the end users and reduce the potentia demand for the service. Consequenty, providers are very ikey to be worse off than if they had been more coaborative in choosing the prices. This research is supported by DARPA under Grant No. BAA0018 and by NSF under Grant ANI
2 Even when the providers decide to coaborate, there are issues such as how to measure a provider s contribution to the service, and how that may be converted into a criterion for fair aocation of the tota revenues. These issues can be compicated further when different groups of providers have to share network resources in serving their customers. In that case, a fair aocation concerns not ony the providers within a group, but those across different groups as we. There are a number of recent papers that have expored pricing issues in networking. For exampe, Key et a [2], Kunniyur and Srikant [3], and Low and Lapsey [4] propose pricing mechanisms that can be used for congestion contro in the Internet. However, in many of such studies, prices are used mainy as a vehice for passing state or contro information in distributed agorithms. Those prices are often fictitious instead of refecting the actua vaue of the consumed network resources. In addition, those studies assume that networks pay the roe of a sociawefare maximizer for end users and do not have their own interest. This assumption ceary does not match with the rea situation in today s Internet, as most service providers are in the business for making profit and are ony interested in their own wefare. In this paper we are interested in anayzing how providers individua interests and their strategic interactions may affect the price of an Internet service, especiay when it is jointy offered by a group of service providers. In addition to the anaysis, we are aso interested in designing pricing schemes that coud improve providers wefare. We beieve that a good pricing scheme shoud be fair for a providers invoved and encourage future upgrades to their networks. We aso beieve that a good pricing scheme shoud provide right incentives for the providers to foow the protoco and not to cheat for their own advantage. Finay, it shoud be scaabe, i.e. suitabe for argescae depoyment. Our paper is organized as foows. In Section 2, we present a generic pricing mode for Internet services provided by a group providers. In the subsequent two sections, we first study the case in which providers adopt noncooperative pricing strategies. Through simpe exampes, we show
3 that such strategies may resut in undesirabe outcomes. We then propose a fair revenuesharing poicy based on weighted proportiona fairness criterion. We show that this poicy is abe to yied a better equiibrium, reachabe through a distributed protoco. Finay, we concude the paper with discussions on future research. II. BASIC MODEL We consider a generic Internet service offered by a group of interconnected service providers. This service has certain performance requirements, which coud be deay, packet oss probabiity, etc. For simpicity, we assume that those performance requirements coud be transated into oca capacity constraints. For instance, the maximum utiization of each ink by the traffic of premium service is imited to be ess than, say, 40% to ensure that a packets of that service cass experience ony sma deay when going through that ink. We assume that this service is offered on a set of routes, R. Each route is defined as an endtoend path which traverses a sequence of service providers. Figure 1 iustrates such an exampe, in which route 1 originates at Provider 1 and terminates at Provider 2. Each provider on a route charges a price for its share of the service. The endtoend price of that route is then defined as the sum of these prices, or in short, the price of a route. End users, each of which may have different utiity from using the service, decide if they woud use the service based on its price. In this paper, we do not expicity mode how this decision is made by individua users. Instead, we assume that in aggregate effect, the price of a route r, denoted by p r, contros the number of users on that route. This reationship is abstracty modeed by a demand function d r (p r ), which is stricty decreasing and differentiabe, for a r R. To maintain the performance requirements of the service, providers dynamicay adjust their prices to reguate the demand on their inks. There are different possibe approaches to impement such a scheme, for appications with either fixed or eastic bandwidth requirements (see [5]
4 for one of such schemes). Our mode, however, does not depend on the specific aspects of an impementation or the nature of appications. We simpy assume that when a provider sets its price, its objective is to maximize its own profit, whie maintaining the performance requirements for the service. The profit of a provider is its revenue from providing the service subtracted by the associated costs. Those costs, in abstraction, are proportiona to bandwidth and may be modeed by a costperunitbandwidth parameter, s. This choice of inear revenue and cost modes is to approximate the generay compex reationships between price, cost and bandwidth in the region of interest. Therefore, in the case of ony one provider offering the service, its choice of the optima price can be soved from by the foowing constrained optimization program: max p 0 s.t. J = (p s) d(p) (1) d(p) C where C is the capacity of the provider s botteneck ink, p is the price it charges for its service, and d(p) is the demand at price p. The firstorder condition for the optima soution is p = s + µ d(p )/d (p ), where µ 0 is some constant that satisfies µ(d(p ) C) = 0. A sufficient condition for a unique soution to exist is that d/d is a decreasing function of price p. In that case, the soution is aso a maximizer. So in the rest of the paper, we consider ony demand functions that satisfy this property. And for ater use, we define g r (p) d r (p)/d r(p). Note that our assumption on g r (p) impies the demand is reativey ineastic when price is ow, but it becomes more eastic when price increases. We beieve this assumption is not unreaistic, given the nature of Internet services: when price is ow, demand is dominated by users need to communicate, which woud not increase significanty even if price drops; but once price goes beyond a certain threshod, cost becomes a much more important factor infuencing users decision in whether to use the service or not.
5 Before proceeding to the anaysis of mutipeprovider mode, we make some additiona assumptions, for both simpicity and ease of presentation. First, we assume that capacity bottenecks are on inks between providers ony. This is because a provider s egress inks are often purchased from its downstream providers and are ikey to become saturated before its interna inks do. Under this assumption, providers may be viewed as ogica nodes connected by capacitated inks between them, and a route is a sequence of interprovider inks that it traverses. In the ater part of the paper we wi see that this simpicity can be easiy removed. It can be done by adding additiona constraints on the capacities of a provider s interna inks. The basic structure and properties of the optima soution wi not change. We aso assume that a route between a sourcedestination pair is fixed and does not change with prices. We make this assumption because in today s Internet, routing between providers is often performed based on a set of businessoriented poicies instead of shortterm costs or performance measures. Moreover, the prices in our schemes fuctuate with traffic demand much faster than the time scae on which the providers change their interdomain routing poicies. So for the time scae of interest, routing between service providers may be decouped from the dynamics of pricing. III. NONCOOPERATIVE PRICING STRATEGIES In this section, we assume a group of providers jointy offer services on a set of routes. Each provider chooses its price independenty to maximize its own profit. A. Formuation When the providers behave strategicay, there is often confict of interest, which may easiy ead to inefficient and/or unfair outcomes. For instance, in the scenario shown in Figure 1, if the demand on route 2 is much stronger than that on route 1, then it is in Provider 2 s interest to charge a very high price on route 1 and use most of its capacity to carry the more profitabe traffic on route 2. However, if route 1 is the ony traffic that Provider 1 carries and it has penty
6 of spare capacity, then a high price (hence ight oad) on route 1 may not fit Provider 1 s interest. Through interaction, they may have to sette on a comprise acceptabe to both of them. We mode this kind of strategic interaction between the providers by strategic games, with each provider as a strategic payer [6]. Under different assumptions about what strategic information is avaiabe to the providers, different types of formuation, e.g. Nash, Stackeberg, etc, may be appicabe. However, we argue that ony Nash games mode cosey how providers woud interact in rea situations. This is because in a argescae network with compex topoogy (e.g. the Internet), not much information about the game, or the goba states, is avaiabe to individua providers. At best, they coud optimize ocay their strategies by observing how their profit change with their own actions. Such behaviors fit naturay into the bestresponse framework of Nash games. In our formuation, a provider s payoff and strategies are modeed by the foowing program: max pr 0 J i = E i r R (p r s ) d r (p r ) s.t. r R d r (p r ) C, E i, where E i is the set of egress inks owned by provider i, R is the set of routes going through ink, s is the cost per unit bandwidth in forwarding traffic on ink, and C is the capacity constraint on ink. As for the prices, p r is the price charged by provider i for its service (i.e. providing ink ) on route r. Define L r as the set of inks that route r traverses, and p r = i L r p ir as the endtoend price for route r. For ater use, we define p r k L r \ p kr = p r p r, and L as the set of a the inks. Note that in this formuation, every route on the same ink may be charged with a different price. We make this assumption mainy because its generaity. For pricing schemes with different granuarity, e.g. uniform price on perink instead of perroute basis, the mode may be modified by adding additiona constraints. As we wi show ater in the paper, even with perroute pricing, there are schemes in which providers do not need to keep state information for every route on their inks. (2)
7 By definition, Nash equiibrium is a strategy profie from which no payer woud uniateray deviate [1]. For our game specified in (2), the Nash equiibrium, if it exists, is the set of prices, {p r, L, r R}, that sove (2) simutaneousy for a the providers. Equivaenty, it is the soution to the foowing system of equations, where {µ, L} are the Lagrangian mutipiers: p r = s + µ + g r (p r + p r ), L, r R, µ ( r R d r (p r ) C ) = 0, L. Due to noncooperative behaviors of payers, equiibria in Nash games are often inefficient or even have undesirabe properties. In the next section, we study a simpe exampe and show that noncooperative pricing strategy may ead to unfair distribution of revenues among providers. Moreover, a provider with botteneck inks may not have incentives to upgrade its capacity. B. Exampe Consider two providers connected by a singe ink in tandem, and there is ony one route going through them. The demand on that route is d(p 1 + p 2 ), where p i is the price charged by provider i(i = 1, 2). Without oss of generaity, we assume C 1 > C 2, so that provider 2 is aways the botteneck. In addition, the cost of carrying traffic, s, is the same for both providers. The resuting Nash game payed by the two providers is: Provider 1: max p1 0 (p 1 s) d(p 1 + p 2 ) Provider 2: (3) max p2 0 (p 2 s) d(p 1 + p 2 ) s.t. d(p 1 + p 2 ) C 2 It is easy to show that this game has a unique Nash equiibrium. We first consider the case when the capacity constraint in (3) is not active at equiibrium. By symmetry, the prices charged by the two providers at equiibrium must be the same. This price
8 may be soved from the firstorder optimaity condition of either program in (3) as: p = s + g(2p ) (4) The corresponding demand at this price then is d(2p ) X. For ater use, define K = 2p = d 1 (X ), where d 1 ( ) is the inverse function of d(p). Note that (4) can then be expressed in terms of K as K = 2s + 2g(K ). Now consider the case where C 2 X, i.e. the capacity constraint of Provider 2 is active at equiibrium. We first show that in this case Provider 2 aways charges a higher price than Provider 1 does, hence coects more revenue. First, at the equiibrium, due to the capacity constraint, we have d(p 1 + p 2 ) = C 2, or p 1 + p 2 = d 1 (C 2 ) K. (5) From the optimaity condition for provider 1, we get p 1 = s + g(p 1 + p 2 ) = s + g(k). (6) Since d( ) is a decreasing function, K is a decreasing function of C 2. So when C 2 < X, we have K > K and hence 2s + 2g(K) < 2s + 2g(K ) = K < K. By (5) and (6), the above inequaities impy that 2p 1 < p 1 + p 2, or p 1 < p 2. This cacuation shows that the provider with the smaer capacity aways makes more profit, which we beieve is unfair. Note that the ratio between the prices is p 2 /p 1 = K/(s + g(k)) 1. So the smaer C 2 is, the arger K is, hence the higher the ratio is. If C 2 is fixed, then the more eastic the demand is, the faster g(k) decays with K, and the higher the ratio is. Next we consider the providers incentive in upgrading their inks. When there is ony one provider, as described by (1), the profit J aways increases with the capacity C, as ong as the capacity constraint is active. This is because from optimization theory [8], we know that the
9 Lagrangian mutipier µ, which is positive in that case, indicates the sensitivity of the profit J w.r.t to the capacity C. However, this resut may no onger hod with two providers appying noncooperative pricing strategy. Consider the sensitivity of Provider 2 s profit w.r.t. its capacity, J2 / C 2 (equiibrium vaue of a variabes are superscripted with *). We are especiay interested in whether the equation J 2 C 2 = (p 2 s) C 2 C 2 = p 2 C 2 C 2 + p 2 s = 0 (7) has a soution. By differentiating both sides of d(p 1 + p 2) = C 2 w.r.t. C 2, we get ( ) p d (p 1 + p 1 p 2) = 1. (8) p 2 C 2 Simiary, by differentiating both sides of g(p 1 + p 2) = p 1 w.r.t. p 2, we get ( ) = g (p 1 + p p 1 2) + 1 p 2 p 1 p 2 p 1 p = [1 g (p 1 + p 2)] 1 > 0, by the fact that g(p) is a decreasing function of p. Since the demand function is aso a decreasing function of price, the first term in (8) is negative. Since s > 0, this impies that p 2/ C 2 must be negative as we. This resut suggests that there might exist a soution for (7) and hence a possibe maximum of J 2. If such a maximum does exist, then the botteneck provider may stop upgrading its ink after that maximizer, even before the demand is fuy met. It does not require very specia types of demand functions for this maximum does exist. For instance, it can be shown that if demand function is in the form of d(p) = A exp( Bp α ), α > 1, then a maximum exits. Figure 2 shows J 2 as a function of C 2, with the demand function chosen to be d(p) = 10 exp( p 2 ), and cost s = 0.1. It can be ceary seen that a maximum is achieved before C 2 moves into the unconstrained region. In summary, the resuts in this section have significant practica impications. They te us that if providers appy noncooperative pricing strategy, those with botteneck inks have an unfair advantage in getting more profits than their peers. Consequenty, to keep benefiting from this
10 unfair advantage, they may not have an incentive to upgrade their inks. This obviousy woud imit the evoution of the entire network. IV. REVENUE SHARING POLICY Given the undesirabe properties of noncooperative pricing strategy, it is then natura to ask if better pricing schemes coud be designed to overcome those drawbacks and yet sti be compatibe with providers interest (i.e. they have no incentive to cheat). Game theory itsef provides many usefu theories and soution concepts for such types of design probems. Possibe approaches may incude mechanism design and cooperative game theory ([6] [7]). However, those theories and concepts often are too difficut, if not impossibe, to compute and impement in a decentraized way. As an aternative, we adopt the weighted proportiona fairness criterion and propose a fair revenuesharing poicy for providers to improve their profits. We first study how providers woud behave under this poicy, and then suggest a scaabe agorithm for providers to reach that equiibrium. A. Fair Aocation of Revenue When providers reaize the undesirabe outcomes produced by noncooperative pricing strategy, they understand that they have incentives to coaborate and improve their own benefits together. Possiby, they woud reach some agreement on how to distribute revenues among themseves, instead of competing against each other for revenues. In that case, the foremost question to be answered is what agreement, among a the feasibe ways of aocating the revenues, woud providers reach among themseves. For an aocation acceptabe to a providers, we beieve that ideay it shoud possess at east the foowing properties. First, it shoud be Pareto efficient, i.e. there is no other aocation that can offer better payoff for every provider invoved. Second, this aocation shoud not depend on the scae by which the providers profits are measured, nor the order of the providers indices.
11 One fairness criterion that meets the above requirements is the socaed weighted proportionay fair aocation [9], which is a generaization of Nash s bargaining soution [10]. At this aocation, a providers make equa (weighted) proportiona compromise in their payoffs (hence its name). Mathematicay, it is the soution at which i w i J i /J i < 0, where J i is any feasibe deviation in provider i s payoff J i and w i is its associated weight (i.e. its bargaining power). In other words, under any feasibe deviation from this aocation, there must be at east one provider whose percentage change in payoff has to be sacrificed for some others gains. This deviation hence woud not be acceptabe, as it woud vioate the efficiency property. By this criterion, the soution to the weighted proportionay fair aocation may be found by maximizing the foowing objective function (in its genera form): i w iog(j i ). However, in the context of our mode, a direct appication of the weighted proportiona fairness criterion may not aways yied sensibe soutions. The scenario depicted in Figure 3 is such an exampe. In this case, one backbone provider is connected to N access providers. There are N routes, each of which originates from an access provider and terminates at the egress ink of the backbone provider. Suppose the demand functions are d(p) on a these routes, and the costs are s for a the providers. The egress ink of the backbone provider has a capacity of C and is the ony botteneck. Suppose the weights are the same for a the providers. Then by a symmetry argument, a routes shoud have the same endtoend price and the same aocation of revenue between the access and backbone providers. Define p a and p b as the corresponding price of the service by the access and backbone providers, and J i = (p i s) d(p a + p b ), for i = a, b, as the profit of the access and backbone providers, respectivey. Then the proportionay fair aocation is the soution to the foowing maximization program: max pa,p b N og(j a ) + og(j b ) s.t. N d(p a + p b ) C.
12 It is easy to verify that, under the assumption g(p) is decreasing, this program has a unique soution: p a = s + N g(p N+1 a + p b ), p b = s + 1 N+1 g(p a + p b ). What this indicates is that, on each route, the access provider gets N times more revenue than the backbone provider does. Moreover, the more routes the backbone provider serves, the ess share of the revenues per route it is abe to get. Obviousy, this is not a fair agreement that the backbone provider woud accept. The reason for this insensibe aocation is that, in our mode, the negotiation is dictated by two kinds of comprise. First, on a route traversing through a sequence of providers, these providers negotiate how to share the revenues coected from this route, according to their respective contributions. Second, for a provider carrying traffic on mutipe routes, because of its capacity constraint, it needs to negotiate with others on how to aocate its capacity among different routes, or equivaenty, the endtoend price of the routes it serves. For exampe, in the scenario described in Section III.A, Provider 1 and 2 not ony negotiate on how to spit p 1 d(p 1 ), but aso a pair of p 1 and p 2 that are acceptabe to both of them. In the rest of this section, we propose a fair aocation scheme that takes both intra and interroute negotiations into consideration. We first show that endtoend prices do not affect the negotiation on individua routes. We then propose a scaabe agorithm for finding fair endtoend prices. Suppose {p r, r R} is a set of endtoend prices that the providers woud agree on. Then consider the negotiation between the providers on route r. We assume that in the negotiation, the higher cost a provider has in forwarding traffic, the more bargaining power it has. We mode this assumption by assigning s i as the weight w i for the providers. With profit as a provider s payoff, the weighted proportionay fair aocation on route r can be found as the soution to the foowing
13 program: max pr 0 s.t. L r s og((p r s ) d r (p r )) L r p r = p r. The soution is unique and can be expressed in terms of profittocost ratio as the foowing: p r s s = p mr s m s m,, m L r. (9) This resut first impies that under weighted proportiona fair aocation, each provider s profit is equay proportiona to its cost. Since the ratios in (9) can aso be interpreted as return on investment rate, we beieve this aocation is more pragmatic and more ikey to be accepted by the service providers. Secondy, it impies that on any route, endtoend prices do not infuence providers reative share of revenues. This fact thus aows us to fix the aocation ratio on each route when computing fair endtoend prices. We assume that providers negotiate the endtoend prices based on their tota profit. Moreover, each route is assumed to have the same significance in the negotiation. Then proportionay fair endtoend prices may be found from soving the foowing program: max pr i J i = i s.t. ( P s p r E i r R j Lr s j r R d r (p r ) C, L. s ) d r (p r ) (10) Unfortunatey, this program in genera cannot be separated in its decision variabes and hence is difficut to sove by distributed agorithms. To get around this difficuty, we propose to trade efficiency for scaabiity. More specificay, instead of finding the fair endtoend prices by a centraized program as in (10), we propose to have the providers reach an agreement through ocaized earning. On each route, providers agree to share the revenue according the rue in (9). For the endtoend prices, each provider independenty chooses its oca price (i.e. p r ) in a way that when combined with those of others, the resuting endtoend prices woud maximize its own tota profit. In gametheoretic terms, this
14 agreement is the outcome of a Nash game payed between the providers, with each provider s aocated profit as its payoff and their oca prices as strategies. Note that the difference between this game and that described by (2) is that in this game, a provider s oca price does not directy determines its revenue and profit. Instead, they are decided by other providers choices as we, through the aocation rue specified in (9). In the next section, we present a formuation of this game and then show that it has an equiibrium (hence an agreement can be reached). In addition, we show that this equiibrium can be reached via a distributed earning agorithm. B. Equiibrium and Its Properties Mathematicay, on a route r, after the providers on that route have chosen their oca prices, the resuting tota revenues are distributed to each provider in proportion to its cost, according to the rue in (9). For provider i, the profit generated from route r on its ink, denoted J r, hence is J r (p r ; p r ) = ( ) s (p r + p r ) s d r (p r + p r ). m L r s m Again, p r here is defined as p r k L r \ p kr = p r p r. Provider i s strategy is to choose a oca price p r that soves the foowing program: max pr >0 J i = E i r R J r (p r ; p r ) s.t. r R d r (p r + p r ) C, E i. (11) First of a, we are interested in whether an equiibrium exists in this game, i.e. whether the providers woud reach a revenuesharing agreement under this new poicy. Theorem 4.1: Nash equiibrium exists for the game specified in (11). Proof: The proof is carried out in a few steps, by a sequence of emmas. We first show that
15 Lemma 4.1: For any given strategy profie of other providers, {p r, E i, r R }, a unique maximizer for (11) exists. Proof: Note that because there is no active capacity constraint on interna inks, routes existing through different egress inks do not interfere with each other at a. So a provider can optimize over each egress ink independenty. Consider the Lagrangian function for ink : M = r R J r (p r ; p r ) + µ [C r R d r (p r + p r )], where µ 0 is the Lagrangian mutipier for ink. Appying the firstorder optimaity condition to (11), we get p r = max{0, µ + s s m L r s m + g r (p r + p r) p r }. (12) Now define t r through the foowing fixedpoint equation: t r µ + s s m L r s m + g r (t r ). Intuitivey, t r is the optima endtoend price for route r, preferred by provider i, regardess of the prices chosen by other providers on that route. So (12) can be rewritten as p r = max{0, t r p r }. (13) So for any given p r, provider i has a unique bestresponse in its p r through (13). Since g( ) is continuous and decreasing, t r is an increasing continuous function of µ. So is p r by (13). Therefore, by the Intermediate Vaue theorem, we can concude that there exists either a unique µ > 0 which satisfies r R d r (p r + p r ) = C, or µ = 0. By duaity theory (see Luenberger [8]), this pair of (p r, µ ) hence is the optima soution to (11). This resut suggests that we ony need to consider the dua variabes µ {µ, L} when soving the equiibrium of the game, because µ uniquey determine p r s through (12). Foowing this idea, we then show that
16 Lemma 4.2: For any feasibe µ, on any route r, the soution to the foowing system of equations p r = max{0, t r p r }, L r, is unique and is given by t r, if t r > t r, L r \ ; p r = 0, otherwise. In other words, ony the ink with the argest t r sets a nonzero price, which is aso the endtoend price for that route. Proof: We first cassify inks with different vaues of t r into different sets, denoted by (14) A j, j = 1,, J. We then define the vaue of these sets, denoted by S j, by the corresponding t r of its members. For any two inks, say, m and n in the same set A j, p mr = max{0, S δ p nr } and p nr = max{0, S δ p mr }, where S δ S j k m,n p kr. By symmetry, the soution to these two equations is either p mr + p nr = S δ, if S δ > 0; or p mr = p nr = 0, if S δ 0. This resut can be extended to incude a members in the same set A j, i.e. either the sum of a p kr, k A j, equas some positive number, or they a equa zero. Now define y j A j p r, j = 1,, J, and consider the foowing system of equations y j = max{0, S j k j y k }, j = 1,, J. (15) Without oss of generaity, we assume that the sets are abeed so that S 1 < S 2 < < S J. We first argue that y 1 must be zero. Suppose it is not. Then by (15), y 1 = S 1 j=2 y j, or J j=1 y j = S 1. Now for y 2, y 2 = max{0, S 2 j 2 y j} = max{0, S 2 S 1 + y 2 } = S 2 S 1 + y 2 = S 2 j 2 y j
17 So we have J j=1 y j = S 2, which ceary is a contradiction, since S 2 S 1. With y 1 = 0, (15) can be reduced to y j = max{0, S j J k=2 y k}, j 2. By appying the same argument to y 2, we get y 2 = 0. This procedure is repeated unti j = J 1 to get y J 1 = 0 and y J = S J. This resut impies that price on a inks except those in A J shoud be set zero. If there is ony one ink in A J, then the price for that ink shoud be S J. Otherwise, to avoid ambiguity, we fix a tiebreaking rue that ony the most upstream ink in A J sets its price to S J, whie the rest of the inks in A J set zero price. Remark. By the definition of t r, if on two inks m and n, the ratio µ m /s m > µ n /s n, then t mr > t nr. Here the ratio µ /s is the normaized sensitivity of provider i s profit on route r w.r.t. the capacity of ink. Hence it aso indicates how we the demands are served on this ink, or in more intuitive terms, how congested this ink is. Therefore, this emma impies that ony the most congested ink can set the tota price for a route. For comparison, consider the case where a the inks are owned by a singe provider. The optima endtoend price for route r in that case is p r = L r (λ + s ) + g r (p r ), where λ s are the Lagrangian mutipiers associated with the capacity constraints. So a inks set a margina price based on its own degree of congestion, and it is the sum of a these prices, not the maximum of them, that determines the optima endtoend price p r. Another impication from this emma is that given any feasibe µ, the set of endtoend prices are competey determined. In this sense, one may view µ as the ony strategy payed in the game. So next we use this argument to prove the existence of equiibrium. Consider the mapping f : µ µ, i.e. the bestresponse of ink in µ given the set of Lagrangian mutipiers on other inks, µ. By the previous two emmas, this mapping is oneto
18 one. Moreover, it is easy to verify that f (µ) is bounded in [0, f (0)]. The Nash equiibrium, if exists, is the soution to the foowing system of fixedpoint equations: µ = f (µ), L. (16) To show the existence of this soution, by Brouwer s fixedpoint theorem, we need to show the mapping defined through (16) is continuous. Lemma 4.3: The mapping defined in (16) is continuous. Proof: Pease see Appendix. This emma concudes the proof for the theorem. The proof for the existence of equiibrium can aso be used to show that under this fair revenueaocation agreement, providers aways have incentives to upgrade their inks, as ong as there is unserved demand (i.e. their inks capacity constraints are active). Consider any provider with a constrained ink, say, indexed by. Then define R,1 as the set of routes whose endtoend prices are set by ink, and R,2 as the rest of routes traversing through ink, i.e. R,2 = R \ R,1. Because the tota price for a route is determined by the maximum of µ k /s k, k L r, routes in R,2 are not affected by any increase in C at a, because µ / C < 0. Moreover, r R,1, since µ /s > µ k /s k, k L r \, any infinitesima increase in C does not change the members in R,1. Therefore, Ji / C = µ > 0, i.e. increase in C wi increase provider i s profit as ong as µ > 0. Since providers aways have incentive to upgrade their inks under fairaocation agreement, eventuay the network wi move into the capacity region in which none of the inks is constrained. In that case, pricing of the routes is no onger couped through capacity constraints. As a resut, the fair revenue aocation probem reduces to that for a singeroute case: each provider coects its share of revenue as specified by (9), and the optima endtoend price for a route is determined from the profit maximization for that route, i.e. p r = arg max p 0 {(p L r s ) d r (p)}. So our
19 Nashgame based scheme woud produce the same aocation as the genera rue (11) woud. By the Pareto efficiency of proportionay fair aocation, the profit for each provider in this case Pareto dominates (i.e. is higher than or at east equa to) that produced by noncooperative pricing strategy. C. Impementation The proof on the existence of equiibrium aso suggests a distributed agorithm for reaching it. First, Lemma 14 suggests that the optima endtoend price for a route is determined by the ink with the argest scaed Lagrangian mutipier (i.e. µ/s), among those it traverses. Lemma 12 shows that the duaity gap for the oca optimization program is zero, so these Lagrangian mutipiers can be computed iterativey based on the traffic oad on the inks. Based on these observations, we propose the foowing agorithm: Each provider maintains a state variabe µ for each ink, which is updated periodicay according to the foowing rue: µ := max{0, µ + ω (X C )}, where ω > 0 is a sma constant and X r R d r (p r ) is the tota traffic oad on ink. Contro packets, or packets invoved in the pricing procedure (depend on the actua impementation), have two dedicated fieds in their headers. These fieds are used to carry information about µ/s and m s m, and are initiated to zero when the originating host sends such a packet. As the packet passes through a ink on its route to destination, the router on that ink updates the first fied ony if the oca ink has a arger scaed Lagrangian mutipier, i.e. µ/s := max{µ/s, µ /s }. It updates the second fied by m s m := m s m + s. After the packet reaches its destination, the vaues recorded in these two fieds are returned to the sending host via either an ACK packet or some specia contro packet.
20 We assume that a firsthop provider is abe to keep some estimates of the demand on each route that initiates from its network. When it receives an ACK or contro packet returned from a destination, it first computes the new optima price by soving p = m s m+µ/s+g(p ), and then updates the price for the corresponding route accordingy. Subsequent data packets in the estabished connection are stamped with the current endtoend price p and the tota cost s m of the route. In actua impementation, for appications with fixedbandwidth requirement, these variabes coud be the same as those initiay posted to the users when the connection is estabished; for appications with eastic bandwidth requirement, these variabes may change from packet to packet to refect the instantaneous demands for the service. As these data packets pass through a sequence of providers on the route, each provider records its share of the revenue p s / m s m for forwarding them. We assume that there is some system (e.g. a cearing house) estabished for the providers to coect or distribute the tota revenues, presumaby on a time scae much onger than that of the dynamics of the traffic. Note that cost parameter s generay is a piece of information private to individua providers. To prevent cheating, parameter s used in this agorithm may not need to be its actua vaue. Instead, it coud be a vaue negotiated in advance among the providers and hence indicates the bargaining power of different providers. In this agorithm, ony the firsthop providers need to keep state information for each of its routes, and if necessary, the ongoing price charged for each fow. This is feasibe because at edge of the Internet, the number of active fows and routes is reativey sma, and providers have to maintain that information for charging purpose anyway. On the other hand, athough transit providers may aggregate fows from the edge and carry more oad, they do not need to keep any
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