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Pricing Differentiated Internet Services Linhai He and Jean Walrand Dept. of EECS, U.C. Berkeley {linhai,wlr}@eecs.berkeley.edu

Abstract— One of the critical challenges facing the networking industry today is to increase the profitability of Internet services. One well-known method in economics for increasing the revenues of a service is to segment its market through differentiation. However, special characteristics of Internet services, such as congestion externality, may complicate the design and provisioning of such offerings. In this paper, we study how a provider should price its services differentially based on their characteristics. By using a game-theoretic approach, we show that even with a simple two-class differentiated service model, if prices are not properly matched with service qualities, then the system may settle into an undesirable equilibrium similar to that in the classical “Prisoner’s Dilemma” game. In addition, there may not even be a stable equilibrium under certain conditions. We then show that dynamic pricing approaches, in which prices are chosen according to users’ relative preferences over different service classes, may be used to avoid such types of problems. Index Terms— Pricing, differentiated services, market segmentation, game theory.

I. I NTRODUCTION For historical reasons, most of the Internet service providers (ISP) today offer only one type of service, namely, the best-effort service to the Internet users. All networking applications at present are designed based on this service model, which does not provide any guarantee on its quality. On the other hand, as new applications, such as Voice over IP and streaming video, are becoming popular, users may benefit from Internet services with higher quality and hence may be willing to pay extra money for them. Therefore, if they become available, differentiated Internet services (DIS) would benefit both the ISPs and the Internet users. Internet service providers do seem to be moving in that direction. Recently, some ISPs have started marketing basic versions of DIS. One example is NetZero, which is “a nationwide ISP offering consumers a choice of three This research is supported by DARPA under Grant No. BAA00-18 and by NSF under Grant ANI-0331659.

dial-up Internet access options to meet their budget and personal Internet access needs” [3]. Those access options are priced differently according to the download speeds they provide. Past work has studied how an provider should provision its resources to implement different classes of services. In this paper, we focus on the other dimension of the problem — the pricing issue, i.e. how should an ISP price its service classes differently based on their respective quality? It may seem trivial at first, as an ISP’s primary objective is to maximize its revenue, which is the product of price and demand. However, as we will demonstrate in this paper, because of unique characteristics of Internet service [2], such as congestion externality and individual users’ strategic choices, demand for different service classes may change with their prices in complicated ways. Consequently, pricing schemes need to be carefully designed to avoid undesirable outcomes for the ISPs. In the rest of the paper, we will first describe a DIS model and our assumptions on users’ behaviors in Section II. We then introduce two simple examples to demonstrate possible instability and inefficiency issues in providing DIS, when they are improperly provisioned. In Section IV, we propose two dynamic pricing schemes to address those issues. In Section V, we discuss ISPs’ economic incentives in providing DIS. Finally, we conclude the paper with a summary and discussions on future research. II. M ODELS Consider an ISP offering two classes of service to its users. These two classes share a common set of resources, such as link capacity and buffer spaces, through some work-conserving scheduling policy. Those two classes are priced at p1 and p2 , respectively (without loss of generality, assume p1 > p2 always). Each user chooses independently which service class to use, based on factors such as the prices and the levels of service quality that they experience. Note that in our model the service provider does not provision any explicit level of service quality for the two classes. Instead, they are determined by the aggregation of users’ individual

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choices. When the prices are fixed, a user chooses the class with the performance that best suits her need. But when too many users choose the same class and cause the service quality to degrade, some users in that class may decide to leave, for the decreased net benefit this class provides. Consequently, the deflection of these users then helps stabilize the degradation in service quality. The main motivation behind this self-regulated service model is its flexibility. Today, there are many types of applications running over the Internet, each of which may have different requirements in service quality. But with this service model, an ISP does not need to know much about users’ preferences to provision its services. In fact, their actual performance is adaptively determined by users’ own choices, i.e. the “market demand.” More specifically, if users can reach an equilibrium in their choices, i.e. a state from which no one would unilaterally deviate, then the class with higher price must have better service quality, because otherwise users in that class would not pay more for it. Therefore, this service model allows ISPs to adapt quickly to any change in users’ preference and willingness to pay, and the types of users they serve, without actively monitoring and learning about them. This minimalist approach shares the same philosophy with the so-called Paris-Metro model for Internet services [4]. In that model, service providers deploy two identical but physically separate networks and charge them differently to achieve service quality differentiation. We believe that for an Internet service, the performance measure that matters most to the end users is delay. This is because at the application level, what ultimately determines a user’s perception about the quality of a service is how soon information can be sent and received, e.g. how long it takes to fetch a web page, or how much a voice-over-IP packet is delayed when sent across the Internet. Therefore, for the purpose of analysis, we assume a user’s decision is made based on the maximization of her net benefit, defined by the following quasi-linear objective function: max Ju = ωu fu (Ti ) − pi . i=1,2

(1)

Here Ti is the delay experienced by user u if she chooses to use service class i. The function fu is a bounded, differentiable, decreasing function of Ti . In addition, for the convenience of analysis, we assume fu (t) → 0 as t → ∞, but fu (t) 6= 0 for any t ∈ R+ . The scaling parameter, ωu ∈ R+ , is unique to each user and indicates each user’s sensitivity to the price pi . In other words, ωu models a user’s willingness to pay (WTP). So the product, ωu fu (Ti ), indicates user u’s utility from using

B A

Class 1

Class 2

Class 1

fa(T0) – p1 fb(T0) – p1

fa(T1) – p1

Class 2

fa(T2) – p2 fb(T1) – p1

fa(T0) – p2 fb(T0) – p2

fb(T2) – p2

Fig. 1. User a and b’s net benefits corresponding to different combinations of their choices.

service class i when its delay is Ti . In this paper, we assume each user generates a fixed amount of traffic (i.e. no flow control is applied), so that with appropriate scaling, the cost of using service class i for a user equals the price of that class. In addition, users do not randomize in which class they use, nor split their traffic between two service classes. Lastly, we assume the service provider is also a strategic player. Its main objective is to maximize its own interest. III. M OTIVATING E XAMPLES In this section, we illustrate some of the issues may arise in pricing DIS, through two simple cases. In these two cases, there are two users (u = 1, 2) with the same WTP parameters, ωu = 1. The traffic they generate is exchangeable and does not depend on which service class they choose. When both users choose the same class, the resulting delay for them is the same and is denoted by T0 . When they choose differently, the user in the higher-price service class experiences a smaller delay (denoted by T1 ) than the other does (denoted by T2 ). Since the scheduling policy is work conserving, we have T1 < T0 < T2 and T1 + T2 = 2T0 . We can see that one user’s choice affects the amount of delay that the other may experience. This interdependency between each other’s choice leads to a strategic game [5] played between the users. The table in Figure 1 lists both users’ net benefits corresponding to different combinations of their choices. In each entry of the table, the first row corresponds to user 1’s net benefit, and the second row is for user 2. We are interested in finding what equilibrium [5] this game may have. We first show that no pure-strategy Nash equilibrium would exit if two users have different preferences in whether to share the same service class or not. Without loss of generality, assume user 1 does not prefer to share the same service class with user 2, whereas user 2 prefers the opposite. This happens only if f1 (T0 ) − f1 (T2 ) < p1 − p2 < f1 (T1 ) − f1 (T0 ), f2 (T1 ) − f2 (T0 ) < p1 − p2 < f2 (T0 ) − f2 (T2 ).

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Or equivalently, fi s should have the following properties:

IV. P RICING S CHEMES A. Many-User Model

f1 (T1 ) + f1 (T2 ) > 2f1 (T0 ), and f2 (T1 ) + f2 (T2 ) < 2f2 (T0 ).

Since T1 + T2 = 2T0 , a sufficient condition for the above to hold is that one of them is convex and the other is concave. As an example, user 1 may be running data applications whose throughput decreases quickly as delay increases. Whereas user 2 may be running realtime applications so that her utility decreases slowly with small amount of delay but diminishes quickly after the delay exceeds certain threshold. The non-existence of an equilibrium implies that neither user would be able to settle in her choice of classes. Both users would keep switching between classes for better net benefit for themselves. It is very likely that eventually both users are unhappy about the uncertainty in the service quality they experience and decide to leave, causing a loss of revenue for the provider. We next show that even if a pure-strategy Nash equilibrium exists, it may not be desirable for both users. Consider a case in which the prices satisfy the following condition fi (T0 ) − fi (T2 ) > p1 − p2 ,

for i = 1, 2.

(2)

It is straightforward to verify that this game has a unique Nash equilibrium. At this equilibrium, two users share the same service class, but they both choose the more expensive class 1 instead of class 2! Clearly, both of them could have been better off if they chose to use class 2. In that case, they could have the same delay T0 but pay a lower price p2 , thus achieving higher net benefits for both of them. This outcome closely resembles that of the classical “Prisoners’ Dilemma” game [6], in which both players end up in an undesirable equilibrium because of their selfish moves. We believe this dilemma happens because the degree of differentiation in the service qualities mismatch with that in the prices. When class 1 offers better quality than class 2 does, but p1 is not much higher than p2 , both users have a strong incentive to use class 1, consequently causing degradation in the service quality of that class. As the above two simple examples have shown, it is important for the service provider to match its prices appropriately with the service qualities. In the next section, we extend the above results to many-user cases and investigate how to design pricing schemes to achieve stable and efficient equilibrium.

In this model, we assume that there are infinite number of atomic users, each of which generates infinitesimal amount of traffic. Their objective functions are the same as the one described in (1), except that all users have the same f (·) function, but different WTP parameter ω . We assume there is a density function, ρ(ω), associated with ω , so that a group of users with a continuum of WTP parameterRover the range (ω1 , ω2 ) generate a total ω traffic load of ω12 ρ(ω)dω . The shape of the function ρ(ω) hence indicates the distribution of WTP among the users. Without losing too much generality, we assume that ρ(ω) is non-negative, differentiable and has support R over R+ . In addition, we assume ωρ(ω) dω is finite, i.e. a mean value of ω exists. With this model, we say a Nash equilibrium is reached if no single user has an incentive to unilaterally change her choice [5]. Proposition 4.1: Nash equilibrium exists for the game described above. Proof: If an equilibrium exists, define the load at each class by xi , and their corresponding delays by Ti . Note that Ti is a function of both x1 and x2 . By the definition of Nash equilibrium, this equilibrium should have the following properties: 1) If any user choose to use class 1, then her WTP parameter must satisfy the following condition, ωf (T1 ) − p1 > ωf (T2 ) − p2 ,

for class 1 must provide higher net benefit for her. Since p1 > p2 , and f (·) is a decreasing function, we then have T1 < T2 . This confirms our assertion earlier that if equilibrium exists, the class with higher price would offer better service quality. 2) For any user a and b with ωa < ωb , if user a chooses to use class 1, user b would choose to use class 1 as well. This is because if user a chooses to use class 1 at equilibrium, the following conditions must be satisfied: ωa f (T1 ) − p1 > ωa f (T2 ) − p2 , or ωa (f (T1 ) − f (T2 )) > p1 − p2 , So by ωa < ωb , ωb (f (T1 ) − f (T2 )) > p1 − p2 ,

or ωb f (T1 ) − p1 > ωb f (T2 ) − p2 . User b prefers to use class 1 as well, because it offers higher net benefit than class 2 does.

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3) By continuity assumption on ρ(ω), the above argument then implies that there must exist a marginal user with WTP parameter ω1 that satisfies the following condition: ω1 f (T1 ) − p1 = ω1 f (T2 ) − p2 ,

(3)

so that all users with WTP parameter ω > ω1 choose to use class 1 at the equilibrium. 4) By similar arguments, for any pair of users a and b with ωa > ωb , if user a decides not to join the system at all because she cannot find a class that offers non-negative net benefit, then user b would make the same choice. Moreover, there must exist a marginal user with WTP parameter ω2 that satisfies the same following condition: ω2 f (T2 ) = p2 ,

(4)

so that all users with WTP parameter ω < ω2 choose not to join the system. 5) Finally, for all users with WTP parameter ω ∈ (ω2 , ω1 ), they choose to use class 2, because by the definition of ω1 and ω2 , ωf (T1 ) − p1 < ωf (T2 ) − p2 , and ωf (T2 ) > p2 .

Based on the above properties, the existence of Nash equilibrium then depends on if there exists a solution {ω1 , ω2 } to the following system of fixed-point equations: ( ω1 (f (T1 ) − f (T2 )) = p1 − p2 , ω2 f (T2 ) = p2 . Since f is bounded and differentiable, the mapping between {ω1 , ω2 } defined by the above mapping is continuous and compact. Therefore, by Brower’s fixedpoint theorem, there exists at least one solution to the above equations. However, for any generic f and Ti functions, this proposition does not provide additional information about the Nash equilibrium other than its existence. Next we show that for certain combinations of those functions, there may be multiple equilibria and some of those may not be stable or efficient. As we will see, they closely resemble those equilibrium presented in Section III. First, let us consider a small perturbation applied to the Nash equilibrium. Without loss of generality, suppose some users with ω s in a small neighborhood ² of ω1 have switched from class 2 (their choices at equilibrium) to class 1. The change in their net benefits after the move

is the following:

´ n ³ ∂T1 1 − 4 = ² ω1 ρ(ω1 ) f 0 (T1 ) ∂T ∂x1 ∂x2 − ³ ´o ∂T2 2 f 0 (T2 ) ∂T − ∂x1 ∂x2

(5)

If 4 > 0, then the net benefits of this group of users actually increase after they switch the classes. Consequently, these users have incentive to stay in class 1, rather than going back to class 2. For the same reason, it can be shown that there are also a group of users who choose to use class 1 at Nash equilibrium but have incentive to switch to class 2 when the equilibrium is perturbed. This deviation from the equilibrium under perturbation suggests that this Nash equilibrium is not stable. In real operations, users experiment with the system in a randomized way. Therefore, one may expect that the exact Nash equilibrium could never be reached if it is not stable. What would happen instead is that a group of users whose ω s fall within some neighborhood around ω1 would switch back and forth between two classes. In the long run, such an outcome clearly is not very desirable for both users and the ISP. This instability condition (i.e. 4 > 0) in fact is not difficult to meet. One example is the following. Suppose f is a concave function of Ti . Then because T1 < T2 , f 0 (T1 ) > f 0 (T2 ). A sufficient condition for ∆ > 0 is to have ∂T2 2 ∂T ∂x1 > ∂x2 (6) ∂T1 + ∂T2 > ∂T1 + ∂T2 ∂x2 ∂x1 ∂x1 ∂x2 What these conditions suggest is that if under a scheduling policy, a user exerts more congestion effect on users in the other class than those in her own class, then the resulting Nash equilibrium may not be stable. It is easy to verify that for a wide range of values of xi s, strict-priority scheduling is one of such scheduling policies. In addition to stability, scheduling policy together with the function f also have effect in the multiplicity of the Nash equilibrium. A necessary condition is that d(ω1 ) , ω1 (f (T1 ) − f (T2 ))

is not a monotonic function of ω1 , with w2 as an implicit function of ω1 defined through condition (4). Figure IVA shows one of such examples. Here the classes are served by strict-priority policy, the utility function is p f (T ) = 2.5 1 − (T /10)2 , and ρ(ω) is uniform over [0, 20] with a total load of 0.75. The prices for the two classes are p1 = 4, and p2 = 1. Their difference (i.e. p1 − p2 ) is shown in the plot by the horizontal line. So

5 3.5

U(ω0)

defined in (5) thus becomes ¶ µ ∂T2 ∂T1 + f 0 (T2 ) < 0. 4 = ω1 ρ(ω1 ) f 0 (T1 ) ∂x1 ∂x2

p −p 1 2

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Nash Equilibrium (unstable)

Nash Equilibrium

2.5

p1 = 4, p2 =1 x(ω) ~ unif(0,20) f(T) = 2.5*[1−(T/10)2]0.5

2

0

ω

c

2

4

6

8

10

12

14

16

18

20

ω0

Fig. 2. Multiple equilibriums of a game when strict-priority scheduling is used.

its two intersections with d(ω1 ) correspond to the two Nash equilibria of the game. The slope of d(ω1 ) at these two intersections indicates the stability of those two equilibria. For instance, the downward slope at the right one indicates that when ω1 is perturbed slightly to the left, i.e. some users move into class 1 from class 2, then d(ω1 ) will increase, which corresponds to a positive 4. Hence this equilibrium is unstable. To the contrast, the upward slope at the left one indicates that when ω1 is perturbed slightly to the left, i.e. some users move into class 1 from class 2, then d(ω1 ) will decrease, which corresponds to a negative 4. Hence this equilibrium is stable. Although the equilibrium at the left is stable, we can see that a majority of the users choose to use more expensive class 1. This results in a longer average delay for all users, when compared to the one given by the other equilibrium, if it can be reached. Therefore, this outcome can be viewed as a generalization of that in the “Prisoner’s Dilemma” game described in Section III. This kind of outcomes clearly should be avoided. In the rest of this section, we investigate how to design pricing or scheduling schemes to ensure stable and efficient equilibrium. B. Stable Pricing Schemes We first investigate how to design the system so that it can lead to a stable equilibrium. First, one may conjecture from the previous analysis that the stability of the Nash equilibrium is directly related to the degree of coupling between two service classes. In this regard, strict-priority scheduling and the Paris-Metro model obviously represent two opposite extremes among all scheduling policies. Under strictpriority scheduling, the high-priority class has the most impact on the other class, whereas Paris-Metro model provides the most isolation because two service classes are physically separated. In fact, under the Paris-Metro model, [4], both cross partial derivatives are zero. 4

Therefore, Paris-Metro model always has a stable equilibrium. However, it is questionable if any ISP would adopt the Paris-Metro model to provide differentiated services. First, it is inflexible, because the resources allocated to the two classes are fixed and cannot be optimized when ρ(ω), i.e. the mix of different types of users, changes. Second, because resources allocated to the two classes cannot be shared, one may expect there may be some loss in efficiency, hence loss in revenue, for the ISP. Other work-conserving scheduling policies, such as weighted-fair queueing, may not lead to unstable equilibrium as easily as the strict-priority policy does, as long as they do not have strong cross-class congestion effect. However, they still need to be carefully provisioned to match well with the prices. For instance, for a weighted fair queueing based system with a small difference in the prices, if the weights assigned to the two classes are far apart, then the system would behave similarly to the one with strict-priority policy and has a high risk in getting into the instability problem. On the other hand, if the weights are set too close for the sake of avoiding instability, then there is little differentiation in service quality. Therefore, how to choose the right amount of differentiation in price determines all other issues in service provisioning. In the following, we investigate how to design pricing schemes to achieve stable equilibrium for any type of scheduling policy. As we have argued earlier, unstable equilibrium is caused by a strong cross-class congestion effect. Since scheduling policy is fixed, we may use the prices to regulate the traffic load distributed between classes, as a means to prevent the system from moving into the unstable region. Based on this idea, we keep p2 constant as before, but make price p1 a function of x1 . More specifically, we set p1 = p2 + g(x1 ), where g(x1 ) is an increasing, differentiable function of x1 over [0, 1], and g(0) = 0. Next, we show that Nash equilibrium still exists with this new pricing scheme, then derive conditions on g(x1 ) that would ensure stable Nash equilibrium. Proposition 4.2: The game with the above modified pricing function has a Nash equilibrium. Proof: Omitted. All the arguments in Proposition 1 still apply, with necessary modifications to accommodate g(x1 ). Now consider a user who chooses to use class 2 at

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equilibrium. Her ω must satisfy the following condition δ = ω(f (T1 ) − f (T2 )) − g(x1 ) < 0.

To ensure stability, the net benefit of a user should always decrease when she switches away from the equilibrium. Therefore, a sufficient condition for that is ∂δ < 0, or dg ∂T1 ∂T1 ∂T2 ∂T2 > ωρ(ω)[f 0 (T1 )( − )−f 0 (T2 )( − )] dx1 ∂x1 ∂x2 ∂x1 ∂x2

It can be verified that the same condition applies for users in class 1 switching to class 2. Here we have assumed that users who choose to stay in the system is not affected by a single user’s move, because the amount of traffic she generates is infinitesimal. Since both ωρ(ω) and |f 0 (Ti )| are bounded, a sufficient condition for the above inequality to hold is ¶ µ dg ∂T1 ∂T2 ∂T1 ∂T2 , >K + − − dx1 ∂x2 ∂x1 ∂x1 ∂x2 where K is a bound on ωρ(ω)|f 0 (T )|. To eliminate x2 from the above condition, we may take sup of its righthand side over [0, ρa ], which is the feasible range of x2 , then obtain the final sufficient condition for g(x1 ): µ ¶ dg ∂T1 ∂T2 ∂T1 ∂T2 > K sup + − − . dx1 ∂x1 ∂x1 ∂x2 x2 ∈[0,ρ] ∂x2 Please note that the terms in the parenthesis are the difference between the marginal congestion costs across service classes and that within classes. So if a scheduling policy has small marginal congestion cost across its service classes, e.g. the Paris-Metro model, then the above difference is negative and hence fixed prices are sufficient to ensure stable equilibrium. Otherwise, p1 , the price for “premium quality” class, should be adjusted according to the load level in class 1, at a rate at least K times faster than that of the difference in the congestion costs. C. Efficient Pricing Scheme Next we show how pricing scheme may be designed to avoid Prisoner’s Dilemma type of outcomes demonstrated in Section III. In this scheme, an ISP operates in a slightly different way as before. Instead of having fixed prices and unprescribed levels of service quality, on a short time scale, the ISP sets bounds for the maximum delay (denoted by Di , i = 1, 2.) permitted in each service class. Without loss of generality, we assume D1 < D2 . For a given scheduling policy, these delay bounds may be translated into the maximum traffic loads permitted for each service

class, which are denoted by λi 1 . The prices are adjusted to regulate the demand for each class, so that the delays do not exceed Di s. More specifically, p2 (x1 , x2 ) = pe2 I{x2 ≥ λ2 }, p1 (x1 , x2 ) = p2 (x1 , x2 )I {x1 < λ1 } + pe1 I{x1 ≥ λ1 }, (7) where I{·} is the indicator function, and pei s are a pair of prices through which the ISP is able to regulate xi at the target level, λi . After an equilibrium is reached under these prices, on a longer time scale, the ISP may experiment with different values of Ti s to achieve its objectives, such as revenue maximization. The users still choose the service class that maximizes their net benefits. Since the prices now in turn depend on the aggregation of users’ choices, we need to know if Nash equilibrium exists for this scheme. If it does, then what the prices pei are at the equilibrium.

Proposition 4.3: Nash equilibrium exists in the game with the pricing function described in (7). Proof: If there exists a pure-strategy Nash equilibrium, the delays in each service class should be Di , i = 1, 2; otherwise, the prices pei s would not be at equilibrium. Then for any two users a and b with WTP parameter ωa and ωb , if user a chooses to use class 1 at equilibrium, so must be user b, because ωb (f (D1 ) − f (D2 )) > ωa (f (D1 ) − f (D2 )) > pe1 − pe2 . (8) Therefore, all users in class 1 at equilibrium must be those with WTP parameters in the range of [ω1 , ∞), where ω1 satisfies the following condition, Z ∞ ρ(ω) dω = λ1 . (9) ω1

In addition, the prices should satisfy the condition that for any ω < ω1 , ωf (D1 ) − pe1 < ωf (D2 ) − pe2 ,

(10)

i.e. those users who are less willing to pay than ω1 would not have incentive to stay in class 1. Combining (8) and (10), we can conclude that pe1 − pe2 = ω1 (f (D1 ) − f (D2 )).

(11)

This condition sets the difference between the prices of two service classes. By a similar argument, if user a with ωa chooses to stay in class 2 at equilibrium, then user b with ωb > ωa would choose to stay in class 2 as well, provided that ωb < ω1 . Therefore, at equilibrium, all users in class 2 1 We assume λ1 + λ2 < ρa , so that users with small ωs are left out of the system

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are those with ω ∈ [ω2 , ω1 ], where ω2 is determined by the following constraint: Z ω1 ρ(ω) dω = λ2 . (12) ω2 pe2

Accordingly, price needs to be set sufficiently high, so that ∀ω < ω2 , ωf (D2 ) < pe2 , i.e. those users who are less willing to pay than ω2 would not have incentive to join class 2. In addition, by the fact that ∀ω < ω2 , < = = =

ωf (D1 ) − pe1 ω2 f (D1 ) − pe1 ω2 f (D1 ) − pe2 − ω1 (f (D1 ) − f (D2 )) (ω2 − ω1 )f (D1 ) + (ω1 − ω2 )f (D2 ) (ω1 − ω2 )(f (D2 ) − f (D1 )) < 0,

where the first equality is due to (11), those users would not choose to use class 1 either, for it offers negative net benefit to them. Combining the above arguments, we can completely specify the prices pei at equilibrium by the following: pe2 = ω2 f (D2 ), pe1 = ω1 (f (D1 ) − f (D2 )) + pe2 = ω1 f (D1 ) − (ω1 − ω2 )f (D2 ).

(13)

We now argue that the Nash equilibrium exists and in fact is unique. For any pair of λi , according to (9) and (12), there are a unique pair of ωi , which uniquely determine a pair of pei , according to (13). These ωi s and pei s together completely specify the equilibrium. Next we show that this Nash equilibrium in fact is socially efficient, i.e. it maximizes the sum of all users’ utilities, by comparing it with to the outcome of a Vickrey auction [7] designed according to the VCG mechanism [8]. In this Vickrey auction, users are asked to submit their WTP parameters to the ISP. Based on the collected information, the ISP then assigns the users to the service classes in a socially efficient way. The users are charged according to which class they are assigned to. These charges are carefully designed, so that it is in a user’s best interest to submit the true value of her WTP parameter. Similar schemes have been proposed for congestion pricing in the Internet [9] [10]. Proposition 4.4: The equilibrium specified in (9), (12), and (13) is the same as the outcome generated by the Vickrey auction described above. Proof: Since users have the same preference function f , the optimal assignment rule for maximizing the social welfare is to sort the users by their WTP parameters in descending order and then assign them

first to class 1 until the maximum load λ1 is reached, and then to class 2, until λ2 is reached. This assignment is socially efficient because a swapping between the assigned classes of any two users would not affect others, due to the assumption that all users’ traffic is exchangeable, yet would decrease the sum of these two users’ utilities. According to the VCG mechanism, to incentivize a user into revealing the true value of her WTP parameter, the price charged her should equal her “externality” to others, i.e. the difference in the total social welfare between the case in which she participates in the mechanism and the case in which she does not. Therefore, in our model, all users in class 2 should be charged with a price of ω2 f (D2 ). This is because if any of them is removed from the system, the user who presently is not assigned to either classes but now can move into class 2 is the one whose WTP parameter is “right before” ω2 . With other users unaffected, the change in the total social welfare equals her improved utility, which is ω2 f (D2 ). For any user u in class 1, her absence can affect two users. One is the user whose WTP parameter is “right after” ω1 . She is assigned to class 2 if user u is present, but can be moved up to class 1 when user u is removed from the system. This move can increase her utility by ω1 (f (D1 ) − f (D2 )). Moreover, her move to class 1 creates an open space in class 2, so that the user whose ω is “right after” ω2 now can join class 2, instead of being kept out of the system. This user’s utility consequently increases by ω2 f (D2 ). Therefore, the change in the total social welfare is the sum of the utilities of these two affected users, which is ω1 (f (D1 ) − f (D2 )) + ω2 f (D2 ). This gives the price ought to be charged to user u. Comparing with the equilibrium generated by our dynamic pricing scheme, we can see that the socially efficient Vickrey auction uses the same assignment rule and charges the same prices for the service classes. Yet our pricing scheme is simpler and more practical to implement, because it requires little communication between the ISP and the users. V. I NCENTIVES FOR M ARKET S EGMENTATION In this section, we investigate an ISP’s incentive in providing differentiated services in two cases. In the first case, users have homogeneous utility function; but in the second one, there is non-zero variance in users’ WTP paramters. A. Homogeneous Users Proposition 5.1: If all users have the exact same utility functions, then an ISP can generate no more revenue

8

than it could with a single service class. Proof: Consider a group of users generating a total load of ρa . For convenience, assume their WTP parameters are all equal to one. For the time being, assume that the provider is required to keep all users in the system. This constraint will be relaxed later in the proof. If the provider offers only one class of service, then the highest price it can charge under the aforementioned load constraint is p0 = f (T0 ), where T0 is the delay of the corresponding FIFO queue under a load of ρa . In this case, the provider collects a revenue of ρa f (T0 ). Now suppose that the provider offers two classes of services, priced at p1 and p2 , respectively. Denote the load in class 1 by α, and the delay of class i by Ti (α, ρ − α), i = 1, 2. With a slight abuse of notation, define fi (α, ρa − α) , f (Ti (α, ρa − α)). Depend on the prices and the delay function Ti s , there are three possible equilibria: 1) If f1 (α, ρa − α) − p1 > f2 (α, ρa − α) − p2 , ∀α ∈ [0, ρa ], then the only Nash equilibrium for the game is α = ρa , i.e. all users choose to use class 1. To keep all users in the system, p1 needs to satisfy p1 ≤ f1 (ρa , 0) = f (T0 ), i.e. all users should have non-negative net benefit. Therefore, the maximum possible revenue in this case is R∗ = ρa f (T0 ). 2) If f1 (α, ρa − α) − p1 < f2 (α, ρa − α) − p2 , ∀α ∈ [0, ρa ], then the only Nash equilibrium for the game is α = 0, i.e. all users choose to use class 2. To keep all users in the system, p2 needs to satisfy p2 ≤ f2 (0, ρa ) = f (T0 ). Therefore, the maximum possible revenue in this case is R∗ = ρa f (T0 ).

The above arguments show that for any given carried load, the provider gains nothing more from offering multiple classes of service. Using this result, we then show that at their respective optimal loads, the same conclusion still holds. Suppose that the multi-service and the single-service systems carry different loads, ρ∗f if o and ρ∗multi , at their respective optimum, Rf∗ if o and ∗ ∗ Rmulti . First, suppose that Rf∗ if o < Rmulti . Then by the arguments earlier, a single-service system carrying the same amount of load as ρ∗multi would yield the ∗ same revenue as Rmulti . So ρ∗f if o cannot be the optimal load for the single-service system, and its corresponding revenue cannot be the maximum revenue possible. This implies a contradiction. The same argument can be used ∗ to show that Rf∗ if o cannot be larger than Rmulti either. ∗ ∗ Hence we conclude that Rf if o must equal Rmulti . B. Heterogeneous Users In this model, we assume that ρ(ω) has a non-zero variance and all the properties described earlier. Proposition 5.2: By offering multiple service classes, an ISP is able to collect more revenues than, or at least the same amount as, that with a single service class. Proof: As in the proof for the previous case, first consider the constraint that a total load of ρa is to be carried by both multi-service and single-service systems. The total revenue generated by a multi-service system is Rmulti = x1 p1 + x2 p2 . As proved in Section IV, at equilibrium, p2 = ω2 f (T2 ), and hence p1 = ω1 (f (T1 ) − f (T2 )) + ω2 f (T2 ).

As a result, Rmulti = x1 ω1 (f (T1 ) − f (T2 )) + (x1 + x2 )ω2 f (T2 )

3) If ∃ α0 ∈ (0, ρa ) such that

= x1 ω1 (f (T1 ) − f (T2 )) + ρa ω2 f (T2 ).

f1 (α0 , ρa − α0 ) − p1 = f2 (α0 , ρa − α0 ) − p2 ,

then this is a Nash equilibrium for the game, because this distribution of the users makes the two service classes indifferent in term of the net benefit they provide. The revenue at this equilibrium thus is

Comparing it with the revenue generated in a singleservice system, Rf if o = ρa ω2 f (T0 ), we can see that Rmulti > Rf if o only if x1 ω1 (f (T1 ) − f (T2 )) > ρa ω2 (f (T0 ) − f (T2 )).

(14)

Note that f (T1 )−f (T2 ) > f (T0 )−f (T2 ) always, as long R = α0 p1 +(ρa −α0 )p2 = ρa p1 −(ρa −α0 )(p1 −p2 ). as p1 > p2 . Moreover, ρa ω2 is a constant with respect Since α0 increases as p1 − p2 decreases, revenue to p1 , because of the constraint to keep all of the given R is a decreasing function of p − p . This implies load ρa . There are two possible cases to consider: 1

2

that to increase the revenue, the provider should decrease the difference between p1 and p2 until α0 reaches ρa . At that point, p1 is set to f1 (ρa , 0) = f (T0 ) in order to keep all users in the system. Consequently, the maximum possible revenue in this case is R∗ = ρa p∗1 = ρa f1 (ρa , 0) = ρa f (T0 ).

Case 1. The condition is easily satisfied if x1 ω1 is an increasing function of ω1 , because then there must 2 be a large R ∞ enough ω1 to satisfy the inequality . Since x1 = ω1 ρ(ω) dω , x1 ω1 is increasing only if the tail of 2

Note that we can use p1 to lead the equilibrium to reach this ω1

9

ρ(ω) decays slowly enough. In other words, there is a significant percentage of high-paying users.

Case 2. If x1 ω1 is non-increasing, it is uncertain whether there always exists a ω1 to satisfy (14). However, the worst the provider can do is to choose its price so that ω2 = ω1 , i.e. to have all users choose to use class 1. This is essentially equivalent to offering a single class of service, which let the provider collect the same amount of revenue as Rf if o 3 . Requiring a multi-service system to carry the same total load as a single-service system obviously is suboptimal. But even in this case, the above results have shown that a provider should be able to generate at least the same amount of revenue by going from single service class to multiple service classes. So at the actual optimum, we can expect that the provider can do better with multiple service classes, especially when there is enough demand from the high-paying users. VI. S UMMARY AND F UTURE R ESEARCH In this paper, we have studied issues in pricing differentiated Internet services. We have shown that when prices for different service classes are improperly chosen to reflect the differentiation in service qualities, the resulting system may have inefficient and even unstable operating equilibria. These undesirable outcomes may affect users’ degree of satisfaction about the offered services and cause possible loss of revenue for the service provider. To avoid these problems, we have proposed two dynamic pricing schemes. In one scheme, we have derived a sufficient condition on the pricing function to ensure a stable operating equilibrium; in the other, we have demonstrated how prices can be used to regular traffic loads in a way to achieve socially efficient equilibrium. We also have studied the incentives of the ISPs in offering differentiated services, in the presence of homogeneous and heterogeneous users. We have shown that by offering multiple service classes, providers are able to collect more revenues than that with a single service class in most practical cases. There are more related research issues worth investigating. First, we would like to extend our results to more general models. For example, the user model could be expanded to include multiple types of utility functions, representing the case where users running different types of applications have different preference over delay. Second, we are also interested in exploring pricing issues with multiple service providers, such as competition and peering. Gibbens, Mason and Steinberg [11] have 3

It is straightforward to verify that p1 = ω2 f (T0 ) in this case.

shown in their paper that when two providers are in competition, they do not have economic incentives to offer multiple classes of services. However, this result is obtained based on the Paris-Metro service model and is verified by numerical studies only. It would be interesting to investigate if their findings still hold for more general models and scheduling policies. R EFERENCES [1] J. J. Gabszewicz, A. Shaked, J. Sutton and J. F. Thisse. “Segmenting the Market: the Monopolist’s Optimal Product Mix”, Journal of Economic Theory, 39(2),273-89, 1986. [2] P. Chander and L. Leruth. “The Optimal Product Mix for a Monopolist in the Presence of Congestion Effects: A Model and Some Results,” International Journal of Industrial Organization, 7(4), 437-49, 1989. [3] Netzero, http://www.netzero.com. [4] A. M. Odlyzko. “Paris Metro Pricing for the Internet,” Conf. on Electronic Commerce, 1999. [5] J. F. Nash. “Equilibrium Points in N-Person Games”, Proceedings of National Academy of Science, 1950. [6] D. Fudenberg and J. Tirole. “Game Theory,” MIT press, Cambridge, MA, 1991. [7] W. Vickery. “Counterspeculation, Auctions, and Competitive Sealed Tenders,” Journal of Finance, 16(1):8-37, Mar.1961 [8] A. Mas-Colell, M. D. Whinston and Jerry R. Green. “Microeconomic Theory,” Oxford University Press,New York, 1995. [9] J. Mackie-Mason and H. Varian. “Pricing the Internet,” Public Access to the Internet, B. Kahin and J. Keller, editors, PrenticeHall, Englewood Cliffs, NJ, 1995. [10] J. Shu and P. Varaiya. “Pricing Network Services,” Infocom, 2003. [11] R. Gibbens, R. Mason and R. Steinberg. “Internet Service Classes under Competition,” IEEE Journal on Selected Areas in Communications, 18(12), December 2000, pp. 2490-2498.

Pricing Differentiated Internet Services Linhai He and Jean Walrand Dept. of EECS, U.C. Berkeley {linhai,wlr}@eecs.berkeley.edu

Abstract— One of the critical challenges facing the networking industry today is to increase the profitability of Internet services. One well-known method in economics for increasing the revenues of a service is to segment its market through differentiation. However, special characteristics of Internet services, such as congestion externality, may complicate the design and provisioning of such offerings. In this paper, we study how a provider should price its services differentially based on their characteristics. By using a game-theoretic approach, we show that even with a simple two-class differentiated service model, if prices are not properly matched with service qualities, then the system may settle into an undesirable equilibrium similar to that in the classical “Prisoner’s Dilemma” game. In addition, there may not even be a stable equilibrium under certain conditions. We then show that dynamic pricing approaches, in which prices are chosen according to users’ relative preferences over different service classes, may be used to avoid such types of problems. Index Terms— Pricing, differentiated services, market segmentation, game theory.

I. I NTRODUCTION For historical reasons, most of the Internet service providers (ISP) today offer only one type of service, namely, the best-effort service to the Internet users. All networking applications at present are designed based on this service model, which does not provide any guarantee on its quality. On the other hand, as new applications, such as Voice over IP and streaming video, are becoming popular, users may benefit from Internet services with higher quality and hence may be willing to pay extra money for them. Therefore, if they become available, differentiated Internet services (DIS) would benefit both the ISPs and the Internet users. Internet service providers do seem to be moving in that direction. Recently, some ISPs have started marketing basic versions of DIS. One example is NetZero, which is “a nationwide ISP offering consumers a choice of three This research is supported by DARPA under Grant No. BAA00-18 and by NSF under Grant ANI-0331659.

dial-up Internet access options to meet their budget and personal Internet access needs” [3]. Those access options are priced differently according to the download speeds they provide. Past work has studied how an provider should provision its resources to implement different classes of services. In this paper, we focus on the other dimension of the problem — the pricing issue, i.e. how should an ISP price its service classes differently based on their respective quality? It may seem trivial at first, as an ISP’s primary objective is to maximize its revenue, which is the product of price and demand. However, as we will demonstrate in this paper, because of unique characteristics of Internet service [2], such as congestion externality and individual users’ strategic choices, demand for different service classes may change with their prices in complicated ways. Consequently, pricing schemes need to be carefully designed to avoid undesirable outcomes for the ISPs. In the rest of the paper, we will first describe a DIS model and our assumptions on users’ behaviors in Section II. We then introduce two simple examples to demonstrate possible instability and inefficiency issues in providing DIS, when they are improperly provisioned. In Section IV, we propose two dynamic pricing schemes to address those issues. In Section V, we discuss ISPs’ economic incentives in providing DIS. Finally, we conclude the paper with a summary and discussions on future research. II. M ODELS Consider an ISP offering two classes of service to its users. These two classes share a common set of resources, such as link capacity and buffer spaces, through some work-conserving scheduling policy. Those two classes are priced at p1 and p2 , respectively (without loss of generality, assume p1 > p2 always). Each user chooses independently which service class to use, based on factors such as the prices and the levels of service quality that they experience. Note that in our model the service provider does not provision any explicit level of service quality for the two classes. Instead, they are determined by the aggregation of users’ individual

2

choices. When the prices are fixed, a user chooses the class with the performance that best suits her need. But when too many users choose the same class and cause the service quality to degrade, some users in that class may decide to leave, for the decreased net benefit this class provides. Consequently, the deflection of these users then helps stabilize the degradation in service quality. The main motivation behind this self-regulated service model is its flexibility. Today, there are many types of applications running over the Internet, each of which may have different requirements in service quality. But with this service model, an ISP does not need to know much about users’ preferences to provision its services. In fact, their actual performance is adaptively determined by users’ own choices, i.e. the “market demand.” More specifically, if users can reach an equilibrium in their choices, i.e. a state from which no one would unilaterally deviate, then the class with higher price must have better service quality, because otherwise users in that class would not pay more for it. Therefore, this service model allows ISPs to adapt quickly to any change in users’ preference and willingness to pay, and the types of users they serve, without actively monitoring and learning about them. This minimalist approach shares the same philosophy with the so-called Paris-Metro model for Internet services [4]. In that model, service providers deploy two identical but physically separate networks and charge them differently to achieve service quality differentiation. We believe that for an Internet service, the performance measure that matters most to the end users is delay. This is because at the application level, what ultimately determines a user’s perception about the quality of a service is how soon information can be sent and received, e.g. how long it takes to fetch a web page, or how much a voice-over-IP packet is delayed when sent across the Internet. Therefore, for the purpose of analysis, we assume a user’s decision is made based on the maximization of her net benefit, defined by the following quasi-linear objective function: max Ju = ωu fu (Ti ) − pi . i=1,2

(1)

Here Ti is the delay experienced by user u if she chooses to use service class i. The function fu is a bounded, differentiable, decreasing function of Ti . In addition, for the convenience of analysis, we assume fu (t) → 0 as t → ∞, but fu (t) 6= 0 for any t ∈ R+ . The scaling parameter, ωu ∈ R+ , is unique to each user and indicates each user’s sensitivity to the price pi . In other words, ωu models a user’s willingness to pay (WTP). So the product, ωu fu (Ti ), indicates user u’s utility from using

B A

Class 1

Class 2

Class 1

fa(T0) – p1 fb(T0) – p1

fa(T1) – p1

Class 2

fa(T2) – p2 fb(T1) – p1

fa(T0) – p2 fb(T0) – p2

fb(T2) – p2

Fig. 1. User a and b’s net benefits corresponding to different combinations of their choices.

service class i when its delay is Ti . In this paper, we assume each user generates a fixed amount of traffic (i.e. no flow control is applied), so that with appropriate scaling, the cost of using service class i for a user equals the price of that class. In addition, users do not randomize in which class they use, nor split their traffic between two service classes. Lastly, we assume the service provider is also a strategic player. Its main objective is to maximize its own interest. III. M OTIVATING E XAMPLES In this section, we illustrate some of the issues may arise in pricing DIS, through two simple cases. In these two cases, there are two users (u = 1, 2) with the same WTP parameters, ωu = 1. The traffic they generate is exchangeable and does not depend on which service class they choose. When both users choose the same class, the resulting delay for them is the same and is denoted by T0 . When they choose differently, the user in the higher-price service class experiences a smaller delay (denoted by T1 ) than the other does (denoted by T2 ). Since the scheduling policy is work conserving, we have T1 < T0 < T2 and T1 + T2 = 2T0 . We can see that one user’s choice affects the amount of delay that the other may experience. This interdependency between each other’s choice leads to a strategic game [5] played between the users. The table in Figure 1 lists both users’ net benefits corresponding to different combinations of their choices. In each entry of the table, the first row corresponds to user 1’s net benefit, and the second row is for user 2. We are interested in finding what equilibrium [5] this game may have. We first show that no pure-strategy Nash equilibrium would exit if two users have different preferences in whether to share the same service class or not. Without loss of generality, assume user 1 does not prefer to share the same service class with user 2, whereas user 2 prefers the opposite. This happens only if f1 (T0 ) − f1 (T2 ) < p1 − p2 < f1 (T1 ) − f1 (T0 ), f2 (T1 ) − f2 (T0 ) < p1 − p2 < f2 (T0 ) − f2 (T2 ).

3

Or equivalently, fi s should have the following properties:

IV. P RICING S CHEMES A. Many-User Model

f1 (T1 ) + f1 (T2 ) > 2f1 (T0 ), and f2 (T1 ) + f2 (T2 ) < 2f2 (T0 ).

Since T1 + T2 = 2T0 , a sufficient condition for the above to hold is that one of them is convex and the other is concave. As an example, user 1 may be running data applications whose throughput decreases quickly as delay increases. Whereas user 2 may be running realtime applications so that her utility decreases slowly with small amount of delay but diminishes quickly after the delay exceeds certain threshold. The non-existence of an equilibrium implies that neither user would be able to settle in her choice of classes. Both users would keep switching between classes for better net benefit for themselves. It is very likely that eventually both users are unhappy about the uncertainty in the service quality they experience and decide to leave, causing a loss of revenue for the provider. We next show that even if a pure-strategy Nash equilibrium exists, it may not be desirable for both users. Consider a case in which the prices satisfy the following condition fi (T0 ) − fi (T2 ) > p1 − p2 ,

for i = 1, 2.

(2)

It is straightforward to verify that this game has a unique Nash equilibrium. At this equilibrium, two users share the same service class, but they both choose the more expensive class 1 instead of class 2! Clearly, both of them could have been better off if they chose to use class 2. In that case, they could have the same delay T0 but pay a lower price p2 , thus achieving higher net benefits for both of them. This outcome closely resembles that of the classical “Prisoners’ Dilemma” game [6], in which both players end up in an undesirable equilibrium because of their selfish moves. We believe this dilemma happens because the degree of differentiation in the service qualities mismatch with that in the prices. When class 1 offers better quality than class 2 does, but p1 is not much higher than p2 , both users have a strong incentive to use class 1, consequently causing degradation in the service quality of that class. As the above two simple examples have shown, it is important for the service provider to match its prices appropriately with the service qualities. In the next section, we extend the above results to many-user cases and investigate how to design pricing schemes to achieve stable and efficient equilibrium.

In this model, we assume that there are infinite number of atomic users, each of which generates infinitesimal amount of traffic. Their objective functions are the same as the one described in (1), except that all users have the same f (·) function, but different WTP parameter ω . We assume there is a density function, ρ(ω), associated with ω , so that a group of users with a continuum of WTP parameterRover the range (ω1 , ω2 ) generate a total ω traffic load of ω12 ρ(ω)dω . The shape of the function ρ(ω) hence indicates the distribution of WTP among the users. Without losing too much generality, we assume that ρ(ω) is non-negative, differentiable and has support R over R+ . In addition, we assume ωρ(ω) dω is finite, i.e. a mean value of ω exists. With this model, we say a Nash equilibrium is reached if no single user has an incentive to unilaterally change her choice [5]. Proposition 4.1: Nash equilibrium exists for the game described above. Proof: If an equilibrium exists, define the load at each class by xi , and their corresponding delays by Ti . Note that Ti is a function of both x1 and x2 . By the definition of Nash equilibrium, this equilibrium should have the following properties: 1) If any user choose to use class 1, then her WTP parameter must satisfy the following condition, ωf (T1 ) − p1 > ωf (T2 ) − p2 ,

for class 1 must provide higher net benefit for her. Since p1 > p2 , and f (·) is a decreasing function, we then have T1 < T2 . This confirms our assertion earlier that if equilibrium exists, the class with higher price would offer better service quality. 2) For any user a and b with ωa < ωb , if user a chooses to use class 1, user b would choose to use class 1 as well. This is because if user a chooses to use class 1 at equilibrium, the following conditions must be satisfied: ωa f (T1 ) − p1 > ωa f (T2 ) − p2 , or ωa (f (T1 ) − f (T2 )) > p1 − p2 , So by ωa < ωb , ωb (f (T1 ) − f (T2 )) > p1 − p2 ,

or ωb f (T1 ) − p1 > ωb f (T2 ) − p2 . User b prefers to use class 1 as well, because it offers higher net benefit than class 2 does.

4

3) By continuity assumption on ρ(ω), the above argument then implies that there must exist a marginal user with WTP parameter ω1 that satisfies the following condition: ω1 f (T1 ) − p1 = ω1 f (T2 ) − p2 ,

(3)

so that all users with WTP parameter ω > ω1 choose to use class 1 at the equilibrium. 4) By similar arguments, for any pair of users a and b with ωa > ωb , if user a decides not to join the system at all because she cannot find a class that offers non-negative net benefit, then user b would make the same choice. Moreover, there must exist a marginal user with WTP parameter ω2 that satisfies the same following condition: ω2 f (T2 ) = p2 ,

(4)

so that all users with WTP parameter ω < ω2 choose not to join the system. 5) Finally, for all users with WTP parameter ω ∈ (ω2 , ω1 ), they choose to use class 2, because by the definition of ω1 and ω2 , ωf (T1 ) − p1 < ωf (T2 ) − p2 , and ωf (T2 ) > p2 .

Based on the above properties, the existence of Nash equilibrium then depends on if there exists a solution {ω1 , ω2 } to the following system of fixed-point equations: ( ω1 (f (T1 ) − f (T2 )) = p1 − p2 , ω2 f (T2 ) = p2 . Since f is bounded and differentiable, the mapping between {ω1 , ω2 } defined by the above mapping is continuous and compact. Therefore, by Brower’s fixedpoint theorem, there exists at least one solution to the above equations. However, for any generic f and Ti functions, this proposition does not provide additional information about the Nash equilibrium other than its existence. Next we show that for certain combinations of those functions, there may be multiple equilibria and some of those may not be stable or efficient. As we will see, they closely resemble those equilibrium presented in Section III. First, let us consider a small perturbation applied to the Nash equilibrium. Without loss of generality, suppose some users with ω s in a small neighborhood ² of ω1 have switched from class 2 (their choices at equilibrium) to class 1. The change in their net benefits after the move

is the following:

´ n ³ ∂T1 1 − 4 = ² ω1 ρ(ω1 ) f 0 (T1 ) ∂T ∂x1 ∂x2 − ³ ´o ∂T2 2 f 0 (T2 ) ∂T − ∂x1 ∂x2

(5)

If 4 > 0, then the net benefits of this group of users actually increase after they switch the classes. Consequently, these users have incentive to stay in class 1, rather than going back to class 2. For the same reason, it can be shown that there are also a group of users who choose to use class 1 at Nash equilibrium but have incentive to switch to class 2 when the equilibrium is perturbed. This deviation from the equilibrium under perturbation suggests that this Nash equilibrium is not stable. In real operations, users experiment with the system in a randomized way. Therefore, one may expect that the exact Nash equilibrium could never be reached if it is not stable. What would happen instead is that a group of users whose ω s fall within some neighborhood around ω1 would switch back and forth between two classes. In the long run, such an outcome clearly is not very desirable for both users and the ISP. This instability condition (i.e. 4 > 0) in fact is not difficult to meet. One example is the following. Suppose f is a concave function of Ti . Then because T1 < T2 , f 0 (T1 ) > f 0 (T2 ). A sufficient condition for ∆ > 0 is to have ∂T2 2 ∂T ∂x1 > ∂x2 (6) ∂T1 + ∂T2 > ∂T1 + ∂T2 ∂x2 ∂x1 ∂x1 ∂x2 What these conditions suggest is that if under a scheduling policy, a user exerts more congestion effect on users in the other class than those in her own class, then the resulting Nash equilibrium may not be stable. It is easy to verify that for a wide range of values of xi s, strict-priority scheduling is one of such scheduling policies. In addition to stability, scheduling policy together with the function f also have effect in the multiplicity of the Nash equilibrium. A necessary condition is that d(ω1 ) , ω1 (f (T1 ) − f (T2 ))

is not a monotonic function of ω1 , with w2 as an implicit function of ω1 defined through condition (4). Figure IVA shows one of such examples. Here the classes are served by strict-priority policy, the utility function is p f (T ) = 2.5 1 − (T /10)2 , and ρ(ω) is uniform over [0, 20] with a total load of 0.75. The prices for the two classes are p1 = 4, and p2 = 1. Their difference (i.e. p1 − p2 ) is shown in the plot by the horizontal line. So

5 3.5

U(ω0)

defined in (5) thus becomes ¶ µ ∂T2 ∂T1 + f 0 (T2 ) < 0. 4 = ω1 ρ(ω1 ) f 0 (T1 ) ∂x1 ∂x2

p −p 1 2

3

Nash Equilibrium (unstable)

Nash Equilibrium

2.5

p1 = 4, p2 =1 x(ω) ~ unif(0,20) f(T) = 2.5*[1−(T/10)2]0.5

2

0

ω

c

2

4

6

8

10

12

14

16

18

20

ω0

Fig. 2. Multiple equilibriums of a game when strict-priority scheduling is used.

its two intersections with d(ω1 ) correspond to the two Nash equilibria of the game. The slope of d(ω1 ) at these two intersections indicates the stability of those two equilibria. For instance, the downward slope at the right one indicates that when ω1 is perturbed slightly to the left, i.e. some users move into class 1 from class 2, then d(ω1 ) will increase, which corresponds to a positive 4. Hence this equilibrium is unstable. To the contrast, the upward slope at the left one indicates that when ω1 is perturbed slightly to the left, i.e. some users move into class 1 from class 2, then d(ω1 ) will decrease, which corresponds to a negative 4. Hence this equilibrium is stable. Although the equilibrium at the left is stable, we can see that a majority of the users choose to use more expensive class 1. This results in a longer average delay for all users, when compared to the one given by the other equilibrium, if it can be reached. Therefore, this outcome can be viewed as a generalization of that in the “Prisoner’s Dilemma” game described in Section III. This kind of outcomes clearly should be avoided. In the rest of this section, we investigate how to design pricing or scheduling schemes to ensure stable and efficient equilibrium. B. Stable Pricing Schemes We first investigate how to design the system so that it can lead to a stable equilibrium. First, one may conjecture from the previous analysis that the stability of the Nash equilibrium is directly related to the degree of coupling between two service classes. In this regard, strict-priority scheduling and the Paris-Metro model obviously represent two opposite extremes among all scheduling policies. Under strictpriority scheduling, the high-priority class has the most impact on the other class, whereas Paris-Metro model provides the most isolation because two service classes are physically separated. In fact, under the Paris-Metro model, [4], both cross partial derivatives are zero. 4

Therefore, Paris-Metro model always has a stable equilibrium. However, it is questionable if any ISP would adopt the Paris-Metro model to provide differentiated services. First, it is inflexible, because the resources allocated to the two classes are fixed and cannot be optimized when ρ(ω), i.e. the mix of different types of users, changes. Second, because resources allocated to the two classes cannot be shared, one may expect there may be some loss in efficiency, hence loss in revenue, for the ISP. Other work-conserving scheduling policies, such as weighted-fair queueing, may not lead to unstable equilibrium as easily as the strict-priority policy does, as long as they do not have strong cross-class congestion effect. However, they still need to be carefully provisioned to match well with the prices. For instance, for a weighted fair queueing based system with a small difference in the prices, if the weights assigned to the two classes are far apart, then the system would behave similarly to the one with strict-priority policy and has a high risk in getting into the instability problem. On the other hand, if the weights are set too close for the sake of avoiding instability, then there is little differentiation in service quality. Therefore, how to choose the right amount of differentiation in price determines all other issues in service provisioning. In the following, we investigate how to design pricing schemes to achieve stable equilibrium for any type of scheduling policy. As we have argued earlier, unstable equilibrium is caused by a strong cross-class congestion effect. Since scheduling policy is fixed, we may use the prices to regulate the traffic load distributed between classes, as a means to prevent the system from moving into the unstable region. Based on this idea, we keep p2 constant as before, but make price p1 a function of x1 . More specifically, we set p1 = p2 + g(x1 ), where g(x1 ) is an increasing, differentiable function of x1 over [0, 1], and g(0) = 0. Next, we show that Nash equilibrium still exists with this new pricing scheme, then derive conditions on g(x1 ) that would ensure stable Nash equilibrium. Proposition 4.2: The game with the above modified pricing function has a Nash equilibrium. Proof: Omitted. All the arguments in Proposition 1 still apply, with necessary modifications to accommodate g(x1 ). Now consider a user who chooses to use class 2 at

6

equilibrium. Her ω must satisfy the following condition δ = ω(f (T1 ) − f (T2 )) − g(x1 ) < 0.

To ensure stability, the net benefit of a user should always decrease when she switches away from the equilibrium. Therefore, a sufficient condition for that is ∂δ < 0, or dg ∂T1 ∂T1 ∂T2 ∂T2 > ωρ(ω)[f 0 (T1 )( − )−f 0 (T2 )( − )] dx1 ∂x1 ∂x2 ∂x1 ∂x2

It can be verified that the same condition applies for users in class 1 switching to class 2. Here we have assumed that users who choose to stay in the system is not affected by a single user’s move, because the amount of traffic she generates is infinitesimal. Since both ωρ(ω) and |f 0 (Ti )| are bounded, a sufficient condition for the above inequality to hold is ¶ µ dg ∂T1 ∂T2 ∂T1 ∂T2 , >K + − − dx1 ∂x2 ∂x1 ∂x1 ∂x2 where K is a bound on ωρ(ω)|f 0 (T )|. To eliminate x2 from the above condition, we may take sup of its righthand side over [0, ρa ], which is the feasible range of x2 , then obtain the final sufficient condition for g(x1 ): µ ¶ dg ∂T1 ∂T2 ∂T1 ∂T2 > K sup + − − . dx1 ∂x1 ∂x1 ∂x2 x2 ∈[0,ρ] ∂x2 Please note that the terms in the parenthesis are the difference between the marginal congestion costs across service classes and that within classes. So if a scheduling policy has small marginal congestion cost across its service classes, e.g. the Paris-Metro model, then the above difference is negative and hence fixed prices are sufficient to ensure stable equilibrium. Otherwise, p1 , the price for “premium quality” class, should be adjusted according to the load level in class 1, at a rate at least K times faster than that of the difference in the congestion costs. C. Efficient Pricing Scheme Next we show how pricing scheme may be designed to avoid Prisoner’s Dilemma type of outcomes demonstrated in Section III. In this scheme, an ISP operates in a slightly different way as before. Instead of having fixed prices and unprescribed levels of service quality, on a short time scale, the ISP sets bounds for the maximum delay (denoted by Di , i = 1, 2.) permitted in each service class. Without loss of generality, we assume D1 < D2 . For a given scheduling policy, these delay bounds may be translated into the maximum traffic loads permitted for each service

class, which are denoted by λi 1 . The prices are adjusted to regulate the demand for each class, so that the delays do not exceed Di s. More specifically, p2 (x1 , x2 ) = pe2 I{x2 ≥ λ2 }, p1 (x1 , x2 ) = p2 (x1 , x2 )I {x1 < λ1 } + pe1 I{x1 ≥ λ1 }, (7) where I{·} is the indicator function, and pei s are a pair of prices through which the ISP is able to regulate xi at the target level, λi . After an equilibrium is reached under these prices, on a longer time scale, the ISP may experiment with different values of Ti s to achieve its objectives, such as revenue maximization. The users still choose the service class that maximizes their net benefits. Since the prices now in turn depend on the aggregation of users’ choices, we need to know if Nash equilibrium exists for this scheme. If it does, then what the prices pei are at the equilibrium.

Proposition 4.3: Nash equilibrium exists in the game with the pricing function described in (7). Proof: If there exists a pure-strategy Nash equilibrium, the delays in each service class should be Di , i = 1, 2; otherwise, the prices pei s would not be at equilibrium. Then for any two users a and b with WTP parameter ωa and ωb , if user a chooses to use class 1 at equilibrium, so must be user b, because ωb (f (D1 ) − f (D2 )) > ωa (f (D1 ) − f (D2 )) > pe1 − pe2 . (8) Therefore, all users in class 1 at equilibrium must be those with WTP parameters in the range of [ω1 , ∞), where ω1 satisfies the following condition, Z ∞ ρ(ω) dω = λ1 . (9) ω1

In addition, the prices should satisfy the condition that for any ω < ω1 , ωf (D1 ) − pe1 < ωf (D2 ) − pe2 ,

(10)

i.e. those users who are less willing to pay than ω1 would not have incentive to stay in class 1. Combining (8) and (10), we can conclude that pe1 − pe2 = ω1 (f (D1 ) − f (D2 )).

(11)

This condition sets the difference between the prices of two service classes. By a similar argument, if user a with ωa chooses to stay in class 2 at equilibrium, then user b with ωb > ωa would choose to stay in class 2 as well, provided that ωb < ω1 . Therefore, at equilibrium, all users in class 2 1 We assume λ1 + λ2 < ρa , so that users with small ωs are left out of the system

7

are those with ω ∈ [ω2 , ω1 ], where ω2 is determined by the following constraint: Z ω1 ρ(ω) dω = λ2 . (12) ω2 pe2

Accordingly, price needs to be set sufficiently high, so that ∀ω < ω2 , ωf (D2 ) < pe2 , i.e. those users who are less willing to pay than ω2 would not have incentive to join class 2. In addition, by the fact that ∀ω < ω2 , < = = =

ωf (D1 ) − pe1 ω2 f (D1 ) − pe1 ω2 f (D1 ) − pe2 − ω1 (f (D1 ) − f (D2 )) (ω2 − ω1 )f (D1 ) + (ω1 − ω2 )f (D2 ) (ω1 − ω2 )(f (D2 ) − f (D1 )) < 0,

where the first equality is due to (11), those users would not choose to use class 1 either, for it offers negative net benefit to them. Combining the above arguments, we can completely specify the prices pei at equilibrium by the following: pe2 = ω2 f (D2 ), pe1 = ω1 (f (D1 ) − f (D2 )) + pe2 = ω1 f (D1 ) − (ω1 − ω2 )f (D2 ).

(13)

We now argue that the Nash equilibrium exists and in fact is unique. For any pair of λi , according to (9) and (12), there are a unique pair of ωi , which uniquely determine a pair of pei , according to (13). These ωi s and pei s together completely specify the equilibrium. Next we show that this Nash equilibrium in fact is socially efficient, i.e. it maximizes the sum of all users’ utilities, by comparing it with to the outcome of a Vickrey auction [7] designed according to the VCG mechanism [8]. In this Vickrey auction, users are asked to submit their WTP parameters to the ISP. Based on the collected information, the ISP then assigns the users to the service classes in a socially efficient way. The users are charged according to which class they are assigned to. These charges are carefully designed, so that it is in a user’s best interest to submit the true value of her WTP parameter. Similar schemes have been proposed for congestion pricing in the Internet [9] [10]. Proposition 4.4: The equilibrium specified in (9), (12), and (13) is the same as the outcome generated by the Vickrey auction described above. Proof: Since users have the same preference function f , the optimal assignment rule for maximizing the social welfare is to sort the users by their WTP parameters in descending order and then assign them

first to class 1 until the maximum load λ1 is reached, and then to class 2, until λ2 is reached. This assignment is socially efficient because a swapping between the assigned classes of any two users would not affect others, due to the assumption that all users’ traffic is exchangeable, yet would decrease the sum of these two users’ utilities. According to the VCG mechanism, to incentivize a user into revealing the true value of her WTP parameter, the price charged her should equal her “externality” to others, i.e. the difference in the total social welfare between the case in which she participates in the mechanism and the case in which she does not. Therefore, in our model, all users in class 2 should be charged with a price of ω2 f (D2 ). This is because if any of them is removed from the system, the user who presently is not assigned to either classes but now can move into class 2 is the one whose WTP parameter is “right before” ω2 . With other users unaffected, the change in the total social welfare equals her improved utility, which is ω2 f (D2 ). For any user u in class 1, her absence can affect two users. One is the user whose WTP parameter is “right after” ω1 . She is assigned to class 2 if user u is present, but can be moved up to class 1 when user u is removed from the system. This move can increase her utility by ω1 (f (D1 ) − f (D2 )). Moreover, her move to class 1 creates an open space in class 2, so that the user whose ω is “right after” ω2 now can join class 2, instead of being kept out of the system. This user’s utility consequently increases by ω2 f (D2 ). Therefore, the change in the total social welfare is the sum of the utilities of these two affected users, which is ω1 (f (D1 ) − f (D2 )) + ω2 f (D2 ). This gives the price ought to be charged to user u. Comparing with the equilibrium generated by our dynamic pricing scheme, we can see that the socially efficient Vickrey auction uses the same assignment rule and charges the same prices for the service classes. Yet our pricing scheme is simpler and more practical to implement, because it requires little communication between the ISP and the users. V. I NCENTIVES FOR M ARKET S EGMENTATION In this section, we investigate an ISP’s incentive in providing differentiated services in two cases. In the first case, users have homogeneous utility function; but in the second one, there is non-zero variance in users’ WTP paramters. A. Homogeneous Users Proposition 5.1: If all users have the exact same utility functions, then an ISP can generate no more revenue

8

than it could with a single service class. Proof: Consider a group of users generating a total load of ρa . For convenience, assume their WTP parameters are all equal to one. For the time being, assume that the provider is required to keep all users in the system. This constraint will be relaxed later in the proof. If the provider offers only one class of service, then the highest price it can charge under the aforementioned load constraint is p0 = f (T0 ), where T0 is the delay of the corresponding FIFO queue under a load of ρa . In this case, the provider collects a revenue of ρa f (T0 ). Now suppose that the provider offers two classes of services, priced at p1 and p2 , respectively. Denote the load in class 1 by α, and the delay of class i by Ti (α, ρ − α), i = 1, 2. With a slight abuse of notation, define fi (α, ρa − α) , f (Ti (α, ρa − α)). Depend on the prices and the delay function Ti s , there are three possible equilibria: 1) If f1 (α, ρa − α) − p1 > f2 (α, ρa − α) − p2 , ∀α ∈ [0, ρa ], then the only Nash equilibrium for the game is α = ρa , i.e. all users choose to use class 1. To keep all users in the system, p1 needs to satisfy p1 ≤ f1 (ρa , 0) = f (T0 ), i.e. all users should have non-negative net benefit. Therefore, the maximum possible revenue in this case is R∗ = ρa f (T0 ). 2) If f1 (α, ρa − α) − p1 < f2 (α, ρa − α) − p2 , ∀α ∈ [0, ρa ], then the only Nash equilibrium for the game is α = 0, i.e. all users choose to use class 2. To keep all users in the system, p2 needs to satisfy p2 ≤ f2 (0, ρa ) = f (T0 ). Therefore, the maximum possible revenue in this case is R∗ = ρa f (T0 ).

The above arguments show that for any given carried load, the provider gains nothing more from offering multiple classes of service. Using this result, we then show that at their respective optimal loads, the same conclusion still holds. Suppose that the multi-service and the single-service systems carry different loads, ρ∗f if o and ρ∗multi , at their respective optimum, Rf∗ if o and ∗ ∗ Rmulti . First, suppose that Rf∗ if o < Rmulti . Then by the arguments earlier, a single-service system carrying the same amount of load as ρ∗multi would yield the ∗ same revenue as Rmulti . So ρ∗f if o cannot be the optimal load for the single-service system, and its corresponding revenue cannot be the maximum revenue possible. This implies a contradiction. The same argument can be used ∗ to show that Rf∗ if o cannot be larger than Rmulti either. ∗ ∗ Hence we conclude that Rf if o must equal Rmulti . B. Heterogeneous Users In this model, we assume that ρ(ω) has a non-zero variance and all the properties described earlier. Proposition 5.2: By offering multiple service classes, an ISP is able to collect more revenues than, or at least the same amount as, that with a single service class. Proof: As in the proof for the previous case, first consider the constraint that a total load of ρa is to be carried by both multi-service and single-service systems. The total revenue generated by a multi-service system is Rmulti = x1 p1 + x2 p2 . As proved in Section IV, at equilibrium, p2 = ω2 f (T2 ), and hence p1 = ω1 (f (T1 ) − f (T2 )) + ω2 f (T2 ).

As a result, Rmulti = x1 ω1 (f (T1 ) − f (T2 )) + (x1 + x2 )ω2 f (T2 )

3) If ∃ α0 ∈ (0, ρa ) such that

= x1 ω1 (f (T1 ) − f (T2 )) + ρa ω2 f (T2 ).

f1 (α0 , ρa − α0 ) − p1 = f2 (α0 , ρa − α0 ) − p2 ,

then this is a Nash equilibrium for the game, because this distribution of the users makes the two service classes indifferent in term of the net benefit they provide. The revenue at this equilibrium thus is

Comparing it with the revenue generated in a singleservice system, Rf if o = ρa ω2 f (T0 ), we can see that Rmulti > Rf if o only if x1 ω1 (f (T1 ) − f (T2 )) > ρa ω2 (f (T0 ) − f (T2 )).

(14)

Note that f (T1 )−f (T2 ) > f (T0 )−f (T2 ) always, as long R = α0 p1 +(ρa −α0 )p2 = ρa p1 −(ρa −α0 )(p1 −p2 ). as p1 > p2 . Moreover, ρa ω2 is a constant with respect Since α0 increases as p1 − p2 decreases, revenue to p1 , because of the constraint to keep all of the given R is a decreasing function of p − p . This implies load ρa . There are two possible cases to consider: 1

2

that to increase the revenue, the provider should decrease the difference between p1 and p2 until α0 reaches ρa . At that point, p1 is set to f1 (ρa , 0) = f (T0 ) in order to keep all users in the system. Consequently, the maximum possible revenue in this case is R∗ = ρa p∗1 = ρa f1 (ρa , 0) = ρa f (T0 ).

Case 1. The condition is easily satisfied if x1 ω1 is an increasing function of ω1 , because then there must 2 be a large R ∞ enough ω1 to satisfy the inequality . Since x1 = ω1 ρ(ω) dω , x1 ω1 is increasing only if the tail of 2

Note that we can use p1 to lead the equilibrium to reach this ω1

9

ρ(ω) decays slowly enough. In other words, there is a significant percentage of high-paying users.

Case 2. If x1 ω1 is non-increasing, it is uncertain whether there always exists a ω1 to satisfy (14). However, the worst the provider can do is to choose its price so that ω2 = ω1 , i.e. to have all users choose to use class 1. This is essentially equivalent to offering a single class of service, which let the provider collect the same amount of revenue as Rf if o 3 . Requiring a multi-service system to carry the same total load as a single-service system obviously is suboptimal. But even in this case, the above results have shown that a provider should be able to generate at least the same amount of revenue by going from single service class to multiple service classes. So at the actual optimum, we can expect that the provider can do better with multiple service classes, especially when there is enough demand from the high-paying users. VI. S UMMARY AND F UTURE R ESEARCH In this paper, we have studied issues in pricing differentiated Internet services. We have shown that when prices for different service classes are improperly chosen to reflect the differentiation in service qualities, the resulting system may have inefficient and even unstable operating equilibria. These undesirable outcomes may affect users’ degree of satisfaction about the offered services and cause possible loss of revenue for the service provider. To avoid these problems, we have proposed two dynamic pricing schemes. In one scheme, we have derived a sufficient condition on the pricing function to ensure a stable operating equilibrium; in the other, we have demonstrated how prices can be used to regular traffic loads in a way to achieve socially efficient equilibrium. We also have studied the incentives of the ISPs in offering differentiated services, in the presence of homogeneous and heterogeneous users. We have shown that by offering multiple service classes, providers are able to collect more revenues than that with a single service class in most practical cases. There are more related research issues worth investigating. First, we would like to extend our results to more general models. For example, the user model could be expanded to include multiple types of utility functions, representing the case where users running different types of applications have different preference over delay. Second, we are also interested in exploring pricing issues with multiple service providers, such as competition and peering. Gibbens, Mason and Steinberg [11] have 3

It is straightforward to verify that p1 = ω2 f (T0 ) in this case.

shown in their paper that when two providers are in competition, they do not have economic incentives to offer multiple classes of services. However, this result is obtained based on the Paris-Metro service model and is verified by numerical studies only. It would be interesting to investigate if their findings still hold for more general models and scheduling policies. R EFERENCES [1] J. J. Gabszewicz, A. Shaked, J. Sutton and J. F. Thisse. “Segmenting the Market: the Monopolist’s Optimal Product Mix”, Journal of Economic Theory, 39(2),273-89, 1986. [2] P. Chander and L. Leruth. “The Optimal Product Mix for a Monopolist in the Presence of Congestion Effects: A Model and Some Results,” International Journal of Industrial Organization, 7(4), 437-49, 1989. [3] Netzero, http://www.netzero.com. [4] A. M. Odlyzko. “Paris Metro Pricing for the Internet,” Conf. on Electronic Commerce, 1999. [5] J. F. Nash. “Equilibrium Points in N-Person Games”, Proceedings of National Academy of Science, 1950. [6] D. Fudenberg and J. Tirole. “Game Theory,” MIT press, Cambridge, MA, 1991. [7] W. Vickery. “Counterspeculation, Auctions, and Competitive Sealed Tenders,” Journal of Finance, 16(1):8-37, Mar.1961 [8] A. Mas-Colell, M. D. Whinston and Jerry R. Green. “Microeconomic Theory,” Oxford University Press,New York, 1995. [9] J. Mackie-Mason and H. Varian. “Pricing the Internet,” Public Access to the Internet, B. Kahin and J. Keller, editors, PrenticeHall, Englewood Cliffs, NJ, 1995. [10] J. Shu and P. Varaiya. “Pricing Network Services,” Infocom, 2003. [11] R. Gibbens, R. Mason and R. Steinberg. “Internet Service Classes under Competition,” IEEE Journal on Selected Areas in Communications, 18(12), December 2000, pp. 2490-2498.