Pricing and Revenue Sharing Strategies for Internet Service Providers

Transcription

1 Pricing and Revenue Sharing Strategies for Internet Service Providers Linhai He and Jean Warand Dept. of EECS, U.C. Berkeey 1 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 mode for pricing Internet services that are jointy offered by a group of providers. We show that non-cooperative 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 revenue-sharing 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 One of the chaenges facing the networking industry today is to increase the profitabiity of Internet services. For historica reasons, the current Internet protoco stack acks support for impementing efficient market mechanisms. Consequenty, service providers have imited economic incentives to invest in technoogy for new services that users may vaue more. This eads to a stagnant industry and imits the future evoution of the Internet. To correct this state of affairs, it is essentia to impement economic mechanisms that woud enabe service providers to charge more for better services and coect a fair share of the increased revenues. 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 This research is supported by DARPA under Grant No. BAA00-18 and by NSF under Grant ANI 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 socia-wefare 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. Therefore, in this paper, we are interested in anayzing how providers individua interest 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 arge-scae depoyment. Our paper is organized as foows. In Section 2, we propose a packet-marking based pricing scheme for networks with mutipe service providers and present a generic mode for the providers and the service they offer. In the subsequent two sections, we first study the case in which providers adopt non-cooperative pricing strategies. Through simpe exampes, we show that such strategies may resut in undesirabe outcomes. We then propose a fair revenue-sharing 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 work. II. BASIC MODEL In today s Internet, there are many service providers, which may be cassified into different categories based on the roe they pay in the Internet hierarchy. There are oca and regiona providers, which provide access service

2 2 end host request ACK $3 Provider1 request $1 end host Provider2 request $1 $2 end host Fig. 1. An exampe with two service providers and two routes. to the end users, and transit providers, which do not connect end users directy but provide connectivity among oca and regiona service providers. Because no singe service provider has compete end-to-end coverage, a service providers have to work together, i.e. forward each other s traffic, when offering an Internet service. In this paper, 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 end-to-end path which traverse 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 end-toend 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. For exampe, pricing for Voice over IP (VoIP) service coud be impemented based on the foowing packet-marking idea. When a new VoIP connection is to be estabished, a contro packet is sent from the originating host to inform the intended destination host. As this packet traverses a sequence of providers to the destination, each provider aong the way marks the price it woud ike to charge in the header of the contro packet. After this packet reaches the destination, the price information accumuated in the header is carried back to the originating host by an acknowedgement packet. The end user who initiated the connection then decides if she woud proceed with the proposed end-to-end price. If she woud, then the same price wi be charged for the entire duration of the connection. For the providers, we assume certain mechanisms, e.g. some kind of cearing-house system, exist for them to coect revenues based on the amount of traffic they have forwarded and the prices they have set for those traffic. Our mode, however, does not depend on the specific aspects of an impementation or the nature of appications. We simpy mode 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 cost-per-unit-bandwidth parameter, s. 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) 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 first-order 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. It is easy to show that a unique soution exists if d/d is a decreasing function of price p 1. 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 g r (p) is the reciproca of the easticity of the demand function on route r. Before proceeding to the anaysis of mutipe-provider mode, we make some additiona assumptions, for both (1) 1 This is a sufficient condition, not the ony condition, for the caim to hod

3 3 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. Therefore, under this assumption, providers may be viewed as ogica nodes connected by capacitated inks between them, and a route is a sequence of inter-provider inks that it traverses. 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 business-oriented poicies instead of short-term costs or performance measures. In addition, the prices in our schemes presumaby fuctuate with traffic demand much faster than the time scae on which the providers change the routes. III. NON-COOPERATIVE PRICING STRATEGIES In this section, we assume there are a group of providers jointy offering the service 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 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 [5]. 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 arge-scae 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 best-response 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 end-to-end price for route r. For ater use, we define p r k L p r\ 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 per-ink instead of per-route basis, the mode may be modified by adding additiona constraints. As we wi show ater in the paper, even with per-route pricing, schemes do exist so that providers do not need to keep state information for every route on their inks. 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: { pr = s + µ + g r (p r + p r ), L, r R, µ ( r R d r (p r ) C ) = 0, L. Due to non-cooperative 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 non-cooperative pricing strategy may ead to unfair distribution of revenue among providers. Moreover, a botteneck provider may not have an incentive 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, (2)

4 4 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: max p2 0 (p 2 s) d(p 1 + p 2 ) (3) provider 2 provider 1 d(p) = 10 exp( 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 may be soved from the first-order 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 C 2 Fig. 2. Revenues coected by two providers at the equiibrium, when demand function is d(p) = 10 exp( p 2 ). ong as if the capacity constraint is active. This is because from optimization theory [7], we know that the 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 non-cooperative pricing strategy. First consider the sensitivity of Provider 2 s profit w.r.t. its capacity, J 2 / 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) C 2 p 2 Simiary, by differentiating both sides of g(p 1 + p 2 ) = p 1 w.r.t. p 2, we get ( ) p 1 p = g (p 1 + p 2 ) p 1 2 p p 1 p + 1 = [1 g (p 1 + p 2 )] 1 > 0, 2 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 J2. 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 is not hard to find a demand function with which this maximum does exist. For instance, it can be shown that d(p) = A exp( Bp α ), α > 1, is one cass of such functions. Figure 2 shows J2 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.

5 5 In summary, the resuts in this section have significant practica impications. They te us that if providers appy non-cooperative pricing strategy, those with botteneck inks have an unfair advantage in getting more profits than their peers. Consequenty, to keep benefiting from this unfair advantage, they may not have an incentive to upgrade their inks. This obviousy woud imit the evoution of the entire network. d(p) 1 C backbone d(p) N access Fig. 3. An exampe where a direction appication of proportiona fairness criterion may not be sensibe. IV. REVENUE SHARING POLICY Given the undesirabe properties of non-cooperative pricing strategy, it is then natura to ask if better pricing schemes coud be designed to overcome those drawbacks and yet 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 (see Fudenberg and Tiroe [5] and Owen [6]). 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 revenue-sharing 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 non-cooperative pricing strategy, they understand that they have an incentive to coaborate and improve their mutua benefits. 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. One fairness criterion that meets the above requirements is the so-caed weighted proportionay fair aocation 2, which is a generaization of Nash s bargain- 2 Pease visit frank/pf/ for a coec- ing soution [8]. 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 one of such exampes. 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 end-to-end 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. It is easy to verify that, under the assumption g(p) is detion of papers on its appication in networking.

6 6 creasing, this program has a unique soution: p a = s + N N+1 g(p 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 aspects of the 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 end-to-end 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 inter-route negotiations into consideration. We first show that end-to-end prices do not affect the negotiation on individua routes. We then propose a scaabe agorithm for finding fair end-to-end prices. Suppose {p r, r R} is a set of end-to-end 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 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 profit-to-cost 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 their cost. Since the ratios in (9) can aso be interpreted as return of 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, end-to-end 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 end-toend prices. We assume that providers negotiate the end-to-end prices based on their tota profit. Moreover, each route is assumed to have the same significance in the negotiation. Then proportionay fair end-to-end prices may be found from soving the foowing program: max pr i J i = ( s p r i E i r R j Lr s s j r R d r (p r ) C, L. ) d r (p r ) s.t. (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 end-to-end prices by a centraized program as in (10), we propose to have the providers reach an agreement through ocaized earning. On each route, the providers agree to share the revenue according the rue in (9). For the end-to-end prices, each provider independenty chooses its oca price (i.e. p r ) in a way that when combined with those of others, the resuting end-to-end prices woud maximize its own tota profit. In game-theoretic terms, this agreement is the outcome of a 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 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 ) m L r s m s ) d r (p r + p r ).

7 7 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 revenue-sharing agreement under this new poicy. Theorem IV.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 Lemma IV.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 first-order optimaity condition to (11), we get p r = max{0, µ + s s m + g r (p r + p r s ) p r}. m L r (12) Now define t r through the foowing fixed-point equation: t r µ + s s m L r s m + g r (t r ). Intuitivey, t r is the optima end-to-end 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 [7]), 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 Lemma IV.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 { p r = t r, if t r > t r, L r \ ; 0, otherwise. (14) In other words, ony the ink with the argest t r sets a nonzero price, which is aso the end-to-end price for that route. Proof: We first cassify inks with different vaues of t r into different sets, denoted by 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 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

8 8 ambiguity, we fix a tie-breaking 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 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 a singe provider. The optima end-to-end 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 end-to-end price p r. Another impication from this emma is that given any feasibe µ, the set of end-to-end 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 one-to-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 fixed-point 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 IV.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 revenue-aocation agreement, providers aways have incentive 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 end-to-end 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, J i / 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 fair-aocation 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 singe-route case: each provider coects its share of revenue as specified by (9), and the optima end-to-end 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 Nash-game 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 non-cooperative 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 end-to-end 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

9 9 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. We assume that a first-hop 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 end-to-end price p and the tota cost s m of the route. In actua impementation, for appications with fixed-bandwidth 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. In this agorithm, ony the first-hop providers need to keep state information for each of its routes, and if necessary, the on-going 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 per-fow nor per-route state. The ony states they need to maintain are the µ parameters for each of their egress inks. Moreover, they do not even need to estimate the demand on different routes or inks. Therefore, we beieve this agorithm is quite scaabe. Next we show this agorithm converges. Theorem IV.2: The agorithm described above converges to the equiibrium for (11). Proof: We take a continuous-time approximation to the adaptation process of µ. Consider the foowing function V (µ) = µ (C x r ( µ r )) r R,2 r µr 0 x r (t)dt, r 2 r 1 r 3 r 4 C 1 =2 C 2 =5 C 3 =3 Fig. 4. A scenario used for the numerica study. where variabe µ r and function x r are defined in the proof of Lemma IV.3 (see Appendix). It is easy to show that V is continuous in µ. At the equiibrium, if µ = 0 on ink, then a terms in V (µ) that are function of µ are zero. Hence V/ µ = 0. For those inks with non-zero Lagrangian mutipier at the equiibrium, V/ µ = C r R,2 x r ( µ r ) r R,1 x r (µ ) = C r R x r ( µ r ) = C X = 0. (17) The ast equaity is by the fact that capacity constraint is active when µ > 0. Therefore, V (µ) achieves its minimum at the equiibrium. During adaptation, from the first two steps of (17), we have V/ µ = C X. And under continuous-time approximation, dµ /dt = ω (X C ). Hence we have V t = V µ dµ dt = ω (C X ) 2 < 0, or V is a Lyapunov function for the agorithm. Therefore, the agorithm converges. Figure 5 and 6 show the resuts of a numerica simuation of the above agorithm for a scenario depicted in Figure 4. In this study, there are four routes. Two of them are singe-hop oca traffic and the other two traverse across two providers. The demand functions on a these routes are 10e p2. The ink capacities of the providers are 2, 5, 3, respectivey. The costs on those inks are a 0.1. We can see that in the pots the end-to-end price p and Lagrangian mutipiers µ adapt over time and converge to their steadystate vaues (equiibrium). V. CONCLUSION In this paper, we have presented a generic mode for pricing Internet services with mutipe service providers. With a game-theoretic formuation, we have shown that non-cooperative pricing strategy not ony can ead to unfair distribution of revenue among the providers, but aso may discourage future upgrades by botteneck providers. As an aternative, we have proposed a fair revenuesharing scheme based on the concept of weighted proportiona fairness. By encouraging coaboration among the

10 p p 3 1 ink p 1 ink p ink time step Fig. 5. Adaptation of end-to-end price p over time in the numerica study time step Fig. 6. Adaptation of dua variabes µ over time in the numerica study. providers, it avoids the aforementioned drawbacks of noncooperative pricing and can eventuay ead to higher profit for a providers. We aso have proposed an agorithm for impementing this scheme in a distributed way and have studied its convergence property. We are interested in exporing further the mode and approaches deveoped in this paper. First, in our current fair-aocation scheme, we have compromised efficiency for scaabiity by setting on an suboptima aocation produced by a Nash game. We are interested in investigating how much oss in efficiency this compromise costs, especiay if there exists a bound on this oss. Second, in our current mode, the routes between sources and destinations are fixed and independent from the prices. We are interested in extending the mode to incude routing as part of providers strategies. In other words, a provider may choose which downstream provider to forward its traffic, possiby base on the prices those providers offer. This new strategy woud introduce new issues and chaenges, such as competition, into the anaysis. Lasty, the impementation protoco we have proposed in this paper is rather preiminary. We intend to compete those unspecified detais and perfect its procedures, so that it can be ready for actua impementation. REFERENCES [1] J. F. Nash. Equiibrium Points in N-Person Games. Proceedings of Nationa Academy of Sciences, 36: 48-49, [2] Key, F. P., Mauoo, A. K., and Tan, D. H. K. The Rate Contro for Communication Networks: Shadow Prices, Proportiona Fairness and Stabiity, Journa of the Operationa Research Society, pp , vo.49, [3] Kunniyur, S. and Srikant, R. Anaysis and Design of an Adaptive Virtua Queue Agorithm for Active Queue Management, Proc. ACM Sigcomm, [4] Low, S. H. and Lapsey, D. E. Optimization Fow Contro, I: Basic Agorithm and Convergence,, IEEE/ACM Transactions on Networking, 7(6):861-75, Dec [5] Fudenberg, D. and Tiroe, J.(1991). Game Theory. Cambridge, Mass.: MIT Press. [6] Owen, G. (1995). Game Theory. San Diego: Academic Press. [7] Luenberger, D. G. (1984). Linear and Noninear Programming. Addison-Wesey, Reading, Mass. [8] Nash, J. F. Two-Person Cooperative Games, Econometrica, vo. 21, pp , APPENDIX Before we proceed to the proof for Lemma IV.3, we first prove a short emma that wi be used ater in the proof. Lemma.1: Suppose f(x) is a stricty decreasing, continuous function over a finite interva. Then ɛ > 0, if f(x 1 ) f(x 2 ) < ɛ, then δ ɛ > 0 s.t. x 1 x 2 < δ ɛ. Proof: Since f(x) is continuous and is a one-toone mapping (stricty decreasing) over a finite interva, its inverse function f 1 exists and is continuous. The concusion then foows. (Note: x r is one of such functions.) Now we start the proof for Lemma IV.3. Proof: For r R, define µ r max {µ /s : L r }. Then given this µ r, the corresponding end-to-end optima price for this route, p r, may be soved from the foowing equation: p r = L r s + µ r + g r ( p r ), whie aso defines an impicit function h r : µ r p r. Since g r is continuous and decreasing, one can show that so is h r. In addition, through h r, we may express the corresponding demand on route r under price p r, denoted by x r, in terms of µ r, i.e. x r ( µ r ) d r ( p r ) = d r h r ( µ r ). Ceary, x r is stricty decreasing and continuous in µ r. Now consider two feasibe sets of Lagrangian mutipiers, µ (i), i = 1, 2, which satisfy µ (1) µ (2) < ɛ by some norm. Without oss of generaity, we assume µ (1) µ (2) < ɛ,, and hence µ (1) r µ (2) r < ɛ as we. Denote µ (i) as the best-response of ink under µ (i). To prove that the mapping f : µ µ is continuous for a, we need to show that ɛ, δ > 0 such that µ (1) µ (2) < δ,. Note that µ (1) r and µ (2) r may correspond to Lagrangian mutipiers on two different inks on

11 11 route r. So we need to study µ (1) µ (2) under different possibe scenarios. Case 1. The constraint at ink is active under both µ (1) and µ (2). Define R,1 {r R µ r = µ }, and R,2 R\R,1. In other words, R,1 is the set of routes whose endto-end price is set by ink. Next we break the anaysis for this case into a few subcases, based on how R,1 and R,2 change under µ (1) and µ (2). Case 1.1 Suppose R,1 and R,2 do not change under µ (1) and µ (2). By the continuity of x r, ɛ > 0, if µ (1) r µ (2) r < ɛ, δ ɛ > 0, s.t. x r ( µ (1) r ) x r ( µ (2) ) < δ ɛ. Since we have C = r R,1 x r ( µ (1) ) + r R,2 x r ( µ (1) r ) = r R,1 x r ( µ (2) ) + r R,2 x r ( µ (2) r ), r R,1 x r ( µ (1) ) r R,1 x r ( µ (2) ) = r R,2 x r ( µ (1) r ) r R,2 x r ( µ (2) r ) < R,2 δ ɛ R δ ɛ. So by Lemma.1, γ ɛ > 0, s.t. µ (1) µ (2) < γ ɛ. Case 1.2. Suppose R,1 and R,2 change with µ (1) and µ (2). Define R,1 R,2 {r R(1),1 under µ(1) and r R (2),2 under µ(2) }, {r R(2),2 under µ(1) and r R (1),1 under µ(2) }. We break the anaysis of this case into three smaer subcases. Case 1.2.a. Suppose neither R,1 nor R,2 is empty. Choose any r 1 from R,1 and any r 2 from R,2. Under µ (1), µ (1) 1 max{µ (1) k : k L r1 \ } < µ (1), and µ (1) 2 max{µ (1) k : k L r2 \ } > µ (1). Under µ (2), µ (2) 1 max{µ (2) k : k L r1 \ } < µ (2), and µ (2) 2 max{µ (2) k : k L r2 \ } > µ (2). We then have µ (1) µ (2) < max{ µ (1) 2, µ(2) 1 } max{ µ(1) 1, µ(2) 2 } 2ɛ. Case 1.2.b. Suppose R,1 φ but R,2 = φ. If µ(1) µ (2), then we may choose any r 1 R,1, and by the fact r that µ (1) 1 = max{µ (1) k : k L r1 \ } < µ (1) µ (2) < µ (2) 1 = max{µ (2) k : k L r1 \ } µ (1) µ (2) < ɛ For the case that µ (1) > µ (2), consider the constraints under these two µ (i) x r ( µ (1) ) + x r ( µ (1) r ) = C (18) r R (1),1 r R (1),2 r R (2),1 Expand (19) as r R (2),1 \R,1 x r ( µ (2) ) + x r ( µ (2) r ) = C (19) r R (2),2 x r ( µ (2) ) + r R x r ( µ (1) ) +,1 r R x r ( µ (2) r ),1 r R x r ( µ (1) ) +,1 r R (2) x r ( µ (2) r ) = C,,2 (20) and then subtract (18) from (20), we get 0 < r R (2),1 \R,1 = r R (1),2 r R (1),2 x r ( µ (2) ) x r ( µ (1) r ) + r R,1 r R (2),2 r R (1),1 \R,1 x r ( µ (2) r ) x r ( µ (1) r ) x r ( µ (1) r ) + r R,1 r R (2),2 r R,1 x r ( µ (2) r ) x r ( µ (1) ) x r ( µ (2) ) x r ( µ (1) r ) x r ( µ (2) ) By continuity of x r, δ ɛ,1, δ ɛ,2 > 0, such that the two terms in the ast step re bounded by δ ɛ,1 and δ ɛ,2, respectivey. Then by Lemma.1, γ ɛ > 0, s.t. µ (2) µ (1) < γ ɛ. Case 1.2.c. Suppose R,1 = φ but R,2 φ. The proof for this case is omitted as it uses exacty the same idea as that in the previous case. Case 2. µ (1) = 0, but µ (2) > 0. Define x r (µ : L r ) d r h r (max{µ : L r }). Then we have r R x r (0, µ (1),r ) < C, and r R x r (0, µ (2),r ) > C.

12 12 By continuity of x r, δ ɛ > 0, s.t. δ ɛ > r R x r (0, µ (2),r ) r R x r (0, µ (1),r ) > r R x r (0, µ (2),r ) C = r R x r (0, µ (2),r ) r R x r ( µ (2), µ (2),r ) By Lemma.1, γ ɛ > 0 s.t. 0 µ (2) = µ (1) µ (2) < γ ɛ. The opposite case, i.e. µ (1) > 0, but µ (2) = 0, can be proved in the same way. Its proof is omitted here.