Pricing Internet Services With Multiple Providers


 Derek Sutton
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From this document you will learn the answers to the following questions:
 In which section do we first study the case in which providers adopt non  cooperative pricing strategies?
 What type of revenue do providers have to coect?
 What does r go through?
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1 Pricing Internet Services With Mutipe Providers Linhai He and Jean Warand Dept. of Eectrica Engineering and Computer Science University of Caifornia at Berkeey Berkeey, CA inhai, Abstract One of the chaenges facing the networking industry today is to increase the profitabiity of service providers. This cas for economic mechanisms that can enabe providers to charge more for better services and coect a fair share of the resuting increased revenue. In this paper we present a generic mode for pricing Internet services in a mutiprovider network. We show that noncooperative pricing is unfair and may discourage future upgrades of the network. As an aternative, we propose a simpe revenuesharing poicy and show that it is more efficient and encourages providers to coaborate without cheating. We aso suggest a scaabe agorithm for providers to impement this poicy in a distributed way and study its convergence properties. Introduction For historica reasons, the current architecture of the Internet acks the support for impementing efficient market mechanisms. Consequenty, service providers have imited economic incentives to invest in technoogy for new services. This situation 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 resuting increased revenue. In this paper we investigate how to design pricing schemes that coud meet these criteria. The idea of using economic mechanisms in network design is not new. For exampe, [] [2] [3] propose pricing mechanisms that can be used for congestion contro in the Internet. However, in these schemes the network acts as a sociawefare maximizer with no sefinterest. This assumption does not refect the situation in today s Internet, as most network service providers are in the business for making profit and are primariy interested in maximizing their own benefits [4]. Our pricing schemes try to incude these facts into the modes. We beieve that a good pricing scheme shoud provide the right incentives for providers to foow the protoco and not to cheat. In addition, it shoud be fair for a providers invoved and encourage upgrades to the network. In other words, a provider shoud be abe to coect more revenue by increasing the capacity of its network. Finay, a pricing scheme shoud be scaabe, i.e., feasibe for argescae depoyment. This research is supported in part by DARPA Grant No. BAA008.
2 Our paper is organized as foows. In Section 2, we describe the basic modes for the providers and the services that they offer. In the foowing two sections, we first study the case in which providers adopt noncooperative pricing strategies. Through simpe exampes, we show that such strategies woud resut in undesirabe equiibria. We then suggest a revenue sharing poicy as an aternative and show that it woud ead to a better equiibrium. In addition, it coud be reached through a distributed agorithm. We concude the paper with discussions on future work. 2 Basic Mode We consider a group of providers offering services with a certain eve of QoS guarantee. For simpicity, we assume that those QoS requirements coud be transated into oca capacity constraints. For instance, the maximum utiization on a ink may be imited to be ess than, say, 25% to ensure a packets experience ony sma deay going through that ink. We assume that there exists a set of routes across the network. On any of these routes each provider charges a price for its share of the service. The providers may adjust their prices dynamicay and signa them to end users to contro the demand for the services. There are many possibe approaches for impementing such a pricing scheme, for appications with either fixed or eastic bandwidth requirements. However, for the purpose of modeing, we do not specify detais of impementation in this paper. We simpy mode that when a price p is posted for a route r, the resuting traffic oad on that route is given by a function d r (p), which is stricty decreasing and differentiabe. Moreover, mechanisms exist for providers to coect revenues based on the amount of traffic that they have forwarded and the prices that they set. We assume that when a provider sets its price, its objective is to maximize its own revenues, whie maintaining the QoS for the service that it offers by respecting its oca capacity constraint. Therefore, in the case of ony one provider offering the service, the optima price can be determined by soving the foowing constrained optimization program max p 0 J = p d(p) () s.t. d(p) C where C is the capacity constraint. The firstorder condition for the soution is p = 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(p)/d (p) is an increasing function of 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. For ater use, we define g(p) d(p)/d (p). Notice that g(p) indicates the easticity of the demand function. To simpify anaysis, we assume that a providers have sufficient capacity on their interna inks. Capacity may be imited ony on the inks between providers. Loca QoS requirements by each provider are fixed and not affected by prices. A routes between sources and destinations are aso fixed. We choose this assumption because in today s Internet routing between providers is often performed based on a set of provisioned poicies instead of shortterm costs or performance measures.
3 3 NonCooperative Pricing Strategies In this section we try to understand how providers woud set their prices when they have to work together to offer a service. We assume that each provider acts in its own interest. In addition, each provider keeps its own capacity constraint as private information, but it may be possibe for each provider to observe prices marked by others (depending on the impementation). A these assumptions suggest a gametheoretic formuation of the probem in which each provider is a strategic payer. Under different assumptions on what strategic information is avaiabe to the providers, different types of formuation, such as Nash, Stackeberg, etc., are possibe. However, we argue that ony a Nash game modes cosey how providers woud interact in rea situations. In a argescae network with compex topoogy such as the Internet, not much information about the game is avaiabe. Providers generay do not know much about the goba state of the network. A provider s best strategy probaby is to optimize ocay based on its observations on how its payoff changes as it changes its prices. This mode fits naturay the bestresponse mode of a Nash game. Therefore, we mode the game between providers as foows: max pr 0 J i = ( ) E i r R p r d r k L p r\ kr + p r ( ) s.t. r R d r k L p (2) r\ kr + p r C, E i, where E i is the set of egress inks owned by provider i, L r is the set of inks that route r goes through, R is the set of routes going through ink, p r is the price charged for route r on ink, and C is the capacity constraint on ink. It is known that the equiibria in Nash games are often inefficient and may have undesirabe properties, due to the noncooperative nature of the games. The game described in (2) is not an exception. In the foowing we use a simpe exampe to show that noncooperative pricing coud ead to unfair distribution of revenues among providers, and that a botteneck provider may not have an incentive to update its capacity. Consider two providers connected in series with ony one route going through them. The demand on that route is d(p + p 2 ), where p i is the price charged by provider i for i =, 2. Without oss of generaity, we assume C > C 2, so that provider 2 is aways the botteneck. The resuting Nash game payed by the two providers is as foows: Provider : max p 0 p d(p + p 2 ) Provider 2: max p2 0 p 2 d(p + p 2 ) s.t. d(p + p 2 ) C 2 (3) It is easy to show that this game has a unique Nash equiibrium. We are interested in the case in which the capacity constraint of provider 2 is active at equiibrium. We first show that in that case provider 2 aways charges a higher price than provider, thus obtaining more revenues. The capacity constraint C 2 is active if C 2 X, where X is the traffic oad at Nash equiibrium when there is no capacity constraint for either provider. By appying a symmetry argument to (3) with the capacity constraint removed, X can be found from the foowing equations: { X = d(2p ) p = g(2p ), or equivaenty, { p = d (X )/2 K /2 p = g(2p ) = g(k ).
4 Here K is defined as the tota price charged to the users at equiibrium. The above equations impies that K = 2g(K ). When C 2 < X, define K d (C 2 ). From the optimaity condition for provider, p = g(p + p 2 ), we get p = g(k). Since the constraint is active, d(p + p 2 ) = C 2, or p + p 2 = K. So p 2 = K p = K g(k). Since d( ) is a decreasing function, K is a decreasing function of C 2. Therefore, when C 2 < X, K > K. Moreover, since g( ) is a decreasing function, 2g(K) < 2g(K ) = K < K. As a resut, p 2 = K g(k) > g(k) = p. What this means is that when potentia demand exceeds network capacity, botteneck provider aways charges higher price, thus obtain a arger share of the tota revenue, than the unconstrained one. This is certainy very unfair. In addition, note that the ratio between the prices is p 2 /p = K/g(K). So the smaer C 2 is, the arger K is, and the higher the ratio is. For fixed C 2, the more eastic the demand is, the faster g(k) decays with K, and the higher the ratio is. Next we show that provider 2 may not have incentive to upgrade its ink. In other words, increasing C 2 may not aways increase provider 2 s revenues! To estimate how the equiibrium changes with C 2, we are interested in the properties of J2 / C 2 (We designate the equiibrium vaue of a variabes by the superscript *). If the soution to the equation J2 = (p 2C 2 ) = p 2 C 2 + p 2 = 0 (4) C 2 C 2 C 2 exists, then it determines the capacity C 2 that yieds the argest revenues for provider 2. By differentiating both sides of d(p + p 2) = C 2 w.r.t. C 2, we get ( p d (p + p 2) p 2 ) p + 2 =. (5) C 2 Simiary, by differentiating both sides of g(p + p 2) = p w.r.t p 2, we get ( ) p p = g (p p + p 2) + = p + = 2 p 2 p 2 g (p + p 2) > 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 (5) is negative. This impies that p 2/ C 2 must be negative as we. This resut suggests that there might exist a soution for (4) 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 capacity, even before the demand is fuy met. We find that it is not hard to find a demand function with which this maximum exists. For instance, it can be shown that d(p) = A exp( Bp α ), α >, is one cass of such functions. Figure shows J 2 as a function of C 2, with the demand function chosen to be d(p) = 0 exp( p 2 ). It can be ceary seen that a maximum is achieved before C 2 moves into the unconstrained region. 4 RevenueSharing Poicy Given the undesirabe properties of noncooperative pricing strategy, it is natura to ask if a pricing scheme coud be designed to overcome those drawbacks and yet be compatibe with the providers interest by giving them no incentive to cheat. Game theory itsef provides many usefu concepts and toos for such a design probem, such as mechanism
5 provider 2.2 provider d(p) = 0 exp( p 2 ) C 2 Figure : Revenues coected by two providers at the equiibrium, when demand function is d(p) = 0 exp( p 2 ). design, cooperative game theory, and so on [5] [6]. However, we have found that these concepts are either too compex to compute and very difficut (if not impossibe) to impement in a scaabe way. As an aternative, we propose a straightforward revenue sharing poicy that has the aforementioned properties. We first study how providers woud behave under this poicy, and then suggest a scaabe agorithm for providers to reach that equiibrium. 4. Equiibrium In our revenue sharing poicy, a providers agree to eveny spit the tota revenue coected on a route. Nevertheess, they are sti aowed to choose prices based on their best interest. Mathematicay, a provider finds its optima prices by soving the foowing program: max pr 0 J i = ( ) ( ) E i r R N r k L p r\ kr + p r d r k L p r\ kr + p r ( ) s.t. r R d r k L p (6) r\ kr + p r C, E i, where N r is the number of providers on route r. As we wi see next, this change in the objective function competey aters providers strategy. First we show the existence of an equiibrium. Theorem 4.. A Nash equiibrium exists for the game described in (6). Proof. (To save space we ony give an outine of the proof here.) We first show that for any given strategy profie of other providers, a unique maximizer for (6) exists. Note that because there is no capacity constraint on interna inks, routes existing through different egress inks of a provider do not affect each other at a. So a provider can optimize over each egress ink independenty. By appying the firstorder optimaity condition, we get p r = max{0, N r µ + g r ( p kr + p r) p kr }, (7) k L r\ k L r\ where µ 0 is the Lagrangian mutipier for ink. Define t r N r µ +g r (t r ). Intuitivey, t r is the ocay optima tota price for route r, preferred by provider i on ink. Since g( ) is continuous and decreasing, t r, and thus p r, are nondecreasing, continuous functions of µ. Then by Intermediate Vaue theorem, we can concude that either there exists a unique µ > 0 which satisfies r R d r ( k L p r\ kr + p r ) = C, or µ = 0. By duaity
6 theory [7], this set of (p r, µ ) is the optima soution to (6). This resut aso suggests that we ony need to ook at the dua variabes when soving the equiibrium of the game. Foowing this idea, we first show the foowing Lemma. Lemma 4.2. On any route r, for any given set of µ 0, the soution to the foowing system of equations: p r = max{0, t r k L r\ p kr}, L r, is { p r = t r, if t r = max{t r : L r }; 0, otherwise. In other words, ony the ink with the argest t r sets nonzero price. Proof. (We ony give a brief outine of the proof here. Aso the subscript r is omitted for carity.) Cassify inks into different sets according to the vaue of their t s. Denote those sets by A j, j =,, J. Define the vaue of these sets, S j, by the corresponding t of its members. It can be shown that for inks beong to the same set A j, either j A j p j equas some positive number, or p j = 0, j A j. Based on this resut, define y j A j p, and consider the foowing system of equations: y j = max{0, S j k j y k}, j =,, J. By constructing a contradiction, one can show that y must be zero. The same procedure is repeated unti j = J to get y J = 0 and y J = S J. If there are more than one inks in A J, there are a set of prices that they coud choose, as ong as the sum of the prices equas S J. To avoid possibe ambiguity, we fix a rue that ony the most upstream ink in A J sets its price to S J, whie the rest of the inks a set zero price. Remark. Since t r = N r µ + g r (t r ), if µ m > µ n, then t mr > t nr. Roughy speaking, Lagrangian mutipier µ at the optimum indicates how congested a ink is. So this emma impies that on any route ony the most congested ink sets its tota price. Moreover, since µ fuy determines the optima price, we may view µ as the actua strategy payed in the game. We wi use this argument to prove the existence of the equiibrium. Consider the mapping f : µ {µ k, k} µ, the bestresponse of ink to µ on other inks. It is easy to see that f (µ) is bounded in [0, f (0)],. So to show the existence of the equiibrium, we ony need to show that the mapping f is continuous and then appy Brouwer s fixedpoint theorem. Define µ r max {µ : L r }, and the corresponding tota price for that route by p r h r ( µ r ), where h r is the impicit function defined through (7). So now the demand on route r can be expressed in terms of µ r, i.e. d r ( p r ) = d r (h r ( µ r )) = d r h r (max{µ : L r }). For use ater, define x r ( µ r ) d r h r ( µ r ). Ceary, x r is stricty decreasing and continuous. Consider µ (i), i =, 2, and suppose that µ () µ (2) < ɛ in some norm. Without oss of generaity, we may assume that µ () µ (2) < ɛ,. It can be shown that µ () r µ (2) r < ɛ, r, as we. Denote µ (i) as the bestresponse of ink under µ (i). Notice that µ () r and µ (2) r may be associated with different inks, so we have to study µ () µ (2) under different possibe scenarios. Before we proceed, we introduce a sma emma (we omit the proof). Lemma 4.3. Suppose f(x) is a stricty decreasing, continuous function over a finite interva. Then ɛ > 0, if f(x ) f(x 2 ) < ɛ, then δ ɛ > 0 s.t. x x 2 < δ ɛ.
7 Case. The constraint on ink is active under both µ () and µ (2). Define R, {r R µ r = µ }, and R,2 R \ R,. In other words, R, is the set of routes whose tota price is set by ink. Case. Suppose R, and R,2 do not change under µ () and µ (2). It can be shown that ɛ > 0, if µ () r µ (2) r < ɛ, δ ɛ > 0, s.t. r R, x r ( µ () ) r R, x r ( µ (2) ) < δ ɛ. Then by Lemma 4.3, γ ɛ > 0, s.t. µ () µ (2) < γ ɛ. Case.2. Suppose R, and R,2 change with µ () and µ (2). Define R,i as the set of routes that beong to R (),i under µ () but switch to R (2),j under µ (2), for i, j =, 2, and i j. There are two possibiities: Case.2.a. Suppose neither R, nor R,2 max{ µ () 2, µ (2) } max{ µ (), µ (2) 2 } 2ɛ. Case.2.b. Suppose R, φ but R,2 r R,, and by the fact that is empty. One can show that µ() µ (2) < = φ. If µ() µ (2), then we may choose any µ () r max{µ () k : k L r \ } < µ () µ (2) < µ (2) r max{µ (2) k : k L r \ } to concude µ () µ (2) < ɛ. For the case that µ () > µ (2), it can be shown that δ ɛ > 0, 0 < r R (2), \R, Then by Lemma 4.3, γ ɛ > 0, s.t. x r ( µ (2) ) µ (2) r R (), \R, µ () < γ ɛ. x r ( µ () ) < δ ɛ. The opposite case, i.e. R, = φ but R,2 φ, can be proved using the same idea. Case 2. µ () = 0, but µ (2) > 0. Define x r (µ : L r ) d r h r (max{µ : L r }). 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 (),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 ), because r R x r (0, µ (),r ) < C < r R x r (0, µ (2),r ). By Lemma (4.3), γ ɛ > 0 s.t. 0 µ (2) = µ () µ (2) < γ ɛ. The opposite case, i.e. µ () > 0, but µ (2) = 0, can be proved in the same way. Remark. If the decision on the distribution of revenues is made by a centra agent, the optima tota price for route r is p r = L r µ + g r (p r ), where µ s are the corresponding Lagrangian mutipiers which satisfy capacity constraints on a inks. In our sharing poicy, the tota price at equiibrium for the same route is p r = N r max{µ : L r } + g r (p r ). Therefore, one may consider this as a tradeoff between system efficiency and fairness for individua providers. This proof can aso be used to show that providers under revenuesharing poicy aways have incentive to upgrade their inks. Consider any provider with a constrained ink. Denote R, as the set of routes whose tota prices are set by ink and R,2 = R \ R, = {r R µ r µ }. Because the tota price for a route is determined by the maximum of µ k, over a k L r, routes in R,2 are not affected by increase in C at a,
8 because µ / C < 0. Moreover, on any route r R,, since µ > µ k for any k L r \, any infinitesima increase in C wi not change R,. Therefore, J / C = µ > 0. Since providers aways have incentive to upgrade their inks under sharing poicy, eventuay the network wi move into the capacity region in which none of the inks is constrained. For that case, we show next that revenue coected by each provider under sharing poicy stricty dominates that with noncooperative pricing. Because there is no capacity constraint, we ony need to prove the resut for the case of a singe route. Consider a route r connected by N providers and has associated demand d. Under revenuesharing poicy, by Lemma 4.2 in the proof, providers woud agree on a singe tota price, p s = arg max{p d(p)}, and then equay spit the tota revenue p s d(p s ). So in this case revenuesharing poicy is equivaent to centraized aocation, which resuts in a revenue of J s = p s d(p s )/N for each provider. Note that p s is soved from the optimaity condition p s = g(p s ). Under noncooperative pricing, suppose that the price set by individua provider i is p n,i. By symmetry, p n,i must be equa for a providers at the equiibrium. So the revenue coected by each provider is J n = p n d(np n ) = Np n d(np n )/N, where p n is the soution to the oca optimaity condition p n = g(np n ). Because g( ) is a decreasing function, it is straightforward to show that Np n > p s for any N >. Since p s is the ony maximizer to the unconstrained optimization program max p 0 p d(p), we can concude that p s d(p s ) > Np n d(np n ), i.e. J s > J n. 4.2 Impementation The proof of the existence of equiibrium aso suggests an agorithm for providers to compute it. Lemma 4.2 suggests that the optima tota price for a route is determined by the ink with the argest Lagrangian mutipier. Because the duaity gap for the oca optimization program is zero, these Lagrangian mutipiers can be computed iterativey based on the oca traffic oad. Theorem 4. then guarantees that an equiibrium exists even if each provider performs updates using oca information ony. Based on these resuts, we suggest 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 is the tota traffic oad on ink. Each packet has two dedicated fieds in its header, denoted by µ and N. Both fieds are initiaized to zero when a packet enters the network. As a packet passes through a ink on its route to destination, the router on that ink increments N by one and updates µ by the rue: µ := max{ µ, µ }, i.e. the router updates µ ony if its own µ is arger. After the packet reaches its destination, the vaues recorded in µ and N are returned to the sending host via either ACK or some specia contro packets. We assume that a provider is abe to keep some estimates of demand on each route that initiates from its network. When it receives an ACK or contro packet returned from a destination, it updates the price for the corresponding route with the soution from p = N µ + g(p).
9 r 2 r 4 r r 3 C =2 C 2 =5 C 3 = p p 3 ink p ink p ink time step time step Figure 2: Adaptation of p and µ over time in numerica simuation. In this impementation, ony firsthop providers need to keep states for each of their routes, and possiby the ongoing price charged for each fow. This is feasibe because at the edge of the network the numbers of active fows and routes are reativey sma, and the providers have to maintain those information for charging purpose anyway. Transit providers do not need to keep any perfow nor perroute state, and they do not even need to estimate any demand function in order to maximize its revenue. So this agorithm is quite scaabe for impementation. Next we show the foowing convergence resut. Theorem 4.4. The agorithm described above converges to the equiibrium described in Theorem 4.. Proof. We take a continuoustime approximation to the adaptation process of µ, i.e. dµ /dt = max{0, ω (X C )}. Consider the foowing function V (µ) = µ (C x r ( µ r )) r R,2 r µr 0 x r (t)dt. It is easy to show that V is continuous and achieves its minimum at the equiibrium. Then we have V µ = C r R,2 x r ( µ r ) r x r( µ r )I{µ = µ r } = C r R,2 x r ( µ r ) r R, x r (µ ) = C X V t = i.e. V is a Lyapunov function for the agorithm. V µ dµ dt = max{0, ω (X C )}(C X ) 0 Figure 2 shows the resuts of a numerica simuation of the above agorithm. In the pots p and µ adapt over time and converge to their equiibrium vaues. 5 Concusion In this paper we have presented a generic mode for pricing Internet services in a mutiprovider network. We have showed that noncooperative pricing is unfair and may discourage future upgrades of the networks. As an aternative, we have proposed a simpe revenue sharing poicy and have shown that it is fair, more efficient, and encourages
10 providers to coaborate without cheating. We aso have suggested a scaabe agorithm for providers to impement this poicy in a distributed way and studied its convergence property. The mode that we have presented in this paper is probaby more suited for users who prefer predictabe service quaity and are not very sensitive to fuctuations in price. Certainy there are users whose preference is the other way around. So it woud be usefu to extend our pricing schemes to support both service types in a fexibe way. Aso it woud be interesting to understand how optima pricing strategies woud change accordingy to refect different natures of these two services. References [] F. P. Key, A. K. Mauoo, and D. H. K. Tan, The Rate Contro for Communication Networks: Shadow Prices, Proportiona Fairness and Stabiity, Journa of the Operationa Research Society, pp , vo.49, 998. [2] S. Kunniyur and R. Srikant, Anaysis and Design of an Adaptive Virtua Queue Agorithm for Active Queue Management, Proc. ACM Sigcomm, 200. [3] S. H. Low and D. E. Lapsey, Optimization Fow Contro, I: Basic Agorithm and Convergence,, IEEE/ACM Transactions on Networking, 7(6):8675, Dec [4] G. Huston, Interconnection, peering, and settements, Proc. INET, June 999. [5] D. Fudenberg and J. Tiroe, Game Theory, MIT Press, Cambridge, Mass., 99. [6] G. Owen, Game Theory, Academic Press, San Diego, 995. [7] D. G. Luenberger, Linear and Noninear Programming, AddisonWesey, Reading, Mass., 984.
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