Discriminatory Congestion Pricing of Network Services: A Game Theoretic Approach Using Adverse...

Discriminatory Congestion Pricing of Network Services:A Game Theoretic Approach Using Adverse Selection

Sangkyu Rho*Seoul National University

Seoul, Korea

Jungnam An**Electronics and Telecommunications Research Institute

Daejeon, Korea

Song Chong***Korea Advanced Institute of Science and Technology

Daejeon, Korea

Abstract

The quality of network services deteriorates as network utilizationincreases beyond a certain point, i.e., congestion externalities. Manyresearchers have proposed congestion pricing models which internalizecongestion externalities. However, most studies are based on a unitpricing which cannot reflect user’s different congestion sensitivities. Thepurpose of this paper is to propose a discriminatory congestion pricingmodel using adverse selection. Our pricing mechanism provides a

Seoul Journal of BusinessVolume 13, Number 1 (June 2007)

* Main author, Professor of Management Information Systems, GraduateSchool of Business, Seoul National University ([email protected]).

** Coauthor, Researcher, IT Technology Strategy Research Division, Electronicsand Telecommunications Research Institute ([email protected]).

*** Coauthor, Associate Professor of Department of Electrical Engineering andComputer Science, Korea Advanced Institute of Science and Technology([email protected]).

This research was funded in part by the Korea Science and EngineeringFoundation under Grant R01-2001-000-00317-0, but does not necessarily reflectthe views of the KSEF.

congestion-sensitive user with a high quality service for a higher priceand a congestion-tolerant user with a low quality service for a lowerprice. Our model allows service providers to better control congestionswhile maximizing their profits.

Keywords: congestion pricing, congestion externalities, marketsegmentation, adverse selection

INTRODUCTION

The quality of network services such as Internet access andvideo on demand (VOD) services deteriorates as networkutilization increases beyond a certain point. In such situations,additional (marginal) user causes delay or loss in the service (i.e.,cost) for the rest of the service users. MacKie-Mason and Varian(1995a, 1995b) refer this state as a congestion externality. As theexplosive usage growth in network services makes congestionexternality problems very severe, prior research suggests that flatpricing should be replaced with usage pricing. However, usagepricing cannot control congestion since it does not reflectdynamically-changing network utilization. As a result, a usermay get the quality he/she has paid for in low networkutilization but he/she may not get the quality he/she has paidfor in high network utilization. This is not an acceptablesituation for service providers as well as for users (Rho and An2003). Service providers may lose their money in the long runbecause dissatisfied users are not likely to return for theirservices. Therefore, service providers must control congestions.

One way to control congestions is to block new users frompurchasing services when the number of current users is abovecertain thresholds (Bohn, et. al. 1993). Although this may be aneffective and easy way to control congestion, it is not likely tomaximize the profits of service providers (MacKie-Mason andVarian 1995a). Furthermore, it is not very user-friendly becausesome users may want to purchase services despite problems inthe service quality. A better way to control congestion problemsis to internalize congestion externalities by charging a higherunit price as congestion is getting worse (MacKie-Mason andVarian 1995a, 1995b). This congestion pricing mechanism ismore user-friendly as users can either pay more for the better

78 Seoul Journal of Business

service quality or delay their purchase when congestion occurs. Although most congestion pricing mechanisms internalize

congestion externalities, they do not reflect users’ differentcongestion sensitivities: some users are congestion-sensitive butothers are congestion-tolerant. To solve this problem, we useadverse selection to design differentiated menus: a higher pricefor a high quality service to a congestion-sensitive user versus alower price for a low quality service to a congestion-tolerant user.The purpose of this paper is to propose a discriminatorycongestion pricing mechanism for a better congestion control aswell as more profits for service providers. In the next section, wereview the prior congestion pricing research. In the followingsection, we present a discriminatory congestion pricingmechanism. We then analytically compare our model with otherpricing models and present a numerical example. Finally, wediscuss implications of our results and directions for futureresearch.

CONGESTION PRICING RESEARCH

Many researchers in information systems and computerscience have studied the allocation problems to controlcongestion in network services. Since early 1990s, economics-based research has started to investigate network resourceallocation problems (Lin, et. al. 2003). MacKie-Mason and Varian(1995a) show that a pricing mechanism can solve congestionexternality problems. Since then, many papers have suggestedthat congestion pricing models should internalize congestionexternalities by charging a congestion cost as well as a price forusage.

Varian (1995b) introduces a congestion pricing mechanismbased on auction. Simiu and Srikant (1999) solve congestionexternality problems by applying economic pricing mechanismsin network. Kelly (2000, 2001) proposes a congestion pricingmechanism to maximize social welfare. Basar and Srikant (2002)use a Stackelberg game to show a game-theoretic congestionpricing mechanism. Recently, Lin, et al. (2003) propose a pricingmodel to allocate computing resources such as networkbandwidth and storage quantity so that total benefits of the

Discriminatory Congestion Pricing of Network Services 79

service providers and the service consumers are optimized.Mathew, et al. (2004) suggest congestion pricing models basedon congestion variations by showing the difference between Webservice providers and application service providers (ASPs). Allthese pricing models design the same quality service for thesame price to all users. Therefore, we can name such a pricingmodel as a unit congestion pricing model.

In reality, however, there are different types of users. Some arecongestion-sensitive and the others are congestion-tolerant eventhough service providers do not know the information of users’private demand characteristics, i.e., which user is sensitive.

Gupta et. al. (2000) point out that most congestion pricingapproaches do not consider such a realistic problem. Theyaddress the issue of estimating unknown delay costs based onusers’ choices in order to implement incentive-compatiblenetwork externality pricing. Based on the research of Gupta et al.(2000), Lin, et al. (2002) also study the issues in implementingcongestion pricing mechanisms. Both studies implement morerealistic congestion pricing mechanisms in network byconsidering asymmetric information situation but do not showhow to segment a market for congestion-sensitive users andcongestion-tolerant users. To design differentiated menus forboth types of users, we suggest a game-theoretic discriminatorycongestion pricing mechanism using adverse selection. Weexpect that a discriminatory congestion pricing helps serviceproviders segment their markets and control congestion forhigher profits and a more efficient resource allocation.

DISCRIMINATORY CONGESTION PRICING MECHANISM

We model our pricing mechanism using an adverse selectionmodel assuming a monopolistic market with one service providerand two types of users: congestion-tolerant (type 1) users andcongestion-sensitive (type 2) users. Table 1 summarizes thenotations used in our model.

With notations in Table 1, we assume

1) qi units of network resource are needed to provide the


service quality of qi. 2) C and K are given.3) θ1 and θ2 are users’ private information.

Therefore, q1 and q2 measure the qualities that type 1 and type2 users get, respectively. Since we consider a short-term period,network utilization can be represented as Y= Q/C (i.e., the degreeof congestion).

Since the capacity cost is a sunk cost in the short run, weconsider only variable costs. According to MacKie-Mason andVarian (1995b), most costs of providing additional networkservices are more or less independent of the system utilization.As a result, we can model a service provider’s profit function withonly revenue from users since we can normalize variable cost tozero as in Varian (2000). Not knowing the congestion sensitivityof each user, a service provider can maximize its profit as statedin Salanie (1998).

maxt1,t2,q1,q2

n1t1 + n2t2 (1)

Most congestion pricing models such as Basar and Srikant(2002), MacKie-Mason and Varian (1995a), and Simiu andSrikant (1999) internalize congestion externalities with user’sutility function consisting of three terms: the utility of usage, the


Table 1. Notations

Symbol Explanation

n1 and n2 Numbers of type 1 users and type 2 users N = n1 + n2 Number of total usersq1 and q2 Quantities necessary to provide a low quality service

and a high quality service, respectivelyTotal quantity needed to provide services to usersincluding new users

t1 and t2 Prices of a low quality service and a high quality service,respectively

K Available system capacity at the time of new users’service requests

C System capacity in the short runθ1 and θ2 Congestion sensitivities (preferences) of type 1 users and

type 2 users, respectively, where θ1 ≤ θ2

disutility of loss of quality due to congestion, and the price forquantities.

Ui = ui (qi; θi) – νi (qi; qi–1, K) – ti (2)

Following their approach, we assume user ’s utility function asfollows:

(3)

where qi ≥ 1 and K > n1q1 + n2q2

The first term is the utility of usage which is a strictly concavefunction satisfying Spence-Mirrlees condition (u(q; θ2) isincreasing in for all θ2 > θ1). The second term is the disutilityfrom quality loss which is a strictly convex function as shown inSimiu and Srikant (1999). The last term is price. For simplicity,we normalize reservation price to zero. Then, a service providerlets each user participate in a transaction with the following twoindividual rationality (IR) constraints (i.e., participationconstraints).

(4)

At the same time, a service provider has to design an incentivewhich makes each user to prefer the menu designed for him/her:type 1 users choose a low quality service for a lower price whiletype 2 users choose a high quality service for a higher price. Aservice provider can design such an incentive with the followingtwo incentive compatibility (IC) constraints.

(5) ( ) log – log

– ––IC q

N

K

K n q n qt1 1 1

1 1 2 21

1θ

( ) log – log

– ––IR q

N

K

K n q n qt2 2 2

1 1 2 22

10θ

≥

( ) log – log

– ––IR q

N

K

K n q n qt1 1 1

1 1 2 21

10θ

≥

U qN

K

K n q n qtI I I I=

θ log – log – –

–1

1 1 2 2


Following Salanie (1998)’s approach, our model can beextended from two groups to n groups in sensitivity level with nincentive compatibility constraints. With (1), (4), and (5), aservice provider designs differentiated menus for type 1 and type2 users to maximize its profit while controlling congestion asfollows:

subject to

(6)

where K > n1q1 + n2q2

In model (6), we can neglect (IC1) and (IR2) since (IR1) andt (IC2) are

log – log – –

–qN

K

K n q n qt2 1

1 1 2 21

1θ≥

( ) log – log – –

–IC qN

K

K n q n qt2 2 2

1 1 2 2

1θ

log – log– –

– qN

K

K n q n qt1 2

1 1 2 22

1θ≥

( ) log – log – –

–IC qN

K

K n q n qt1 1 1

1 1 2 21

1θ

( ) log – log

– ––IR q

N

K

K n q n qt2 2 2

1 1 2 22

10θ

≥

( ) log – log – –

–IR qN

K

K n q n qt1 1 1

1 1 2 21

10θ

≥

max , , ,

n t n tt t q q

1 1 2 21 2 1 2

+

≥

θ2 11 1 2 2

11

log – log – –

–qN

K

K n q n qt

( ) log – log

– ––IC q

N

K

K n q n qt2 2 2

1 1 2 22

1θ

≥

θ1 21 1 2 2

21

log – log– –

– qN

K

K n q n qt


active and q2 ≥ q1 (See Appendix for a proof). Then, we can formulatea discriminatory congestion pricing as shown in model (7).

Discriminatory congestion pricing model

subject to (7)

t2 = t1 + θ2 (log q2 – log q1) where K > n1q1 + n2q2

ANALYSIS

In this section, we analytically compare our pricing model withunit congestion pricing (unable to segment markets) anddiscriminatory usage pricing (unable to control congestion). Wemodify our discriminatory congestion pricing into the followingmodels (8) and (9). (8) is a unit congestion pricing model whichdoes not consider market segmentation whereas (9) is adiscriminatory usage pricing model which does not considercongestion control.

Unit congestion pricing model

subject to (8)

where K > Nq

Discriminatory usage pricing model

subject to t1 = θ1 log q1 (9)where K > n1q1 + n2q2

max , , ,

n t n tt t q q

1 1 2 21 2 1 2

+

t q

K

K Nq=

θ log – log–

max ,

Ntt q

t qN

K

K n q n q1 1 11 1 2 2

1=

θ log – log– –

max , , ,

n t n tt t q q

1 1 2 21 2 1 2

+


The Necessity of Discriminatory Congestion Pricing Model

We first show why discriminatory congestion pricing isnecessary using propositions 1 and 2. Proposition 1 comparestotal allocated quantities Q*, Q**, and Q*** of model (7), (8), and(9), showing the necessity of internalizing congestion externalities(See appendix for proofs).

Proposition 1: Q* ≤ Q** ≤ Q*** = K

Total allocated quantity of model (9) equals to the availablesystem capacity. That is, a service provider allocates all availablesystem resources under a discriminatory usage pricing model (9).When the network capacity is fully utilized, users experiencesevere congestions. In this case, users’ disutility proportional tonetwork utilization may be too significant for service providers tokeep their users satisfied.

Total allocated quantities of congestion pricing models (7) and(8) are less than the available system capacity. Service providerscan limit total allocated quantities to a certain threshold,preventing their users from experiencing severe congestions.Furthermore, allocated quantity of model (7) is less than or equalto that of model (8), which implies that the former is better atcontrolling congestion than the latter. In conclusion, marketsegmentation in addition to congestion pricing could further helpservice providers control congestion.

In proposition 2, we compare a discriminatory congestionpricing model (7) with a unit congestion pricing model (8) interms of profits.

Proposition 2: Π* > Π**

Proposition 2 states that profit of model (7) is greater than thatof model (8). Service providers employing model (7) can makemore profits than that those employing model (8) even thoughresources allocated in model (7) are less than or equal to those inmodel (8). From propositions 1 and 2, we can infer that marketsegmentation help service providers make more profits as well asbetter control congestion.


Analysis of Discriminatory Congestion Pricing Model

Our model (7) designs differentiated menus for type 1 and 2users as shown in proposition 3.

Proposition 3:

1)

2)

3)

where

The equilibrium quantities and prices in model (7) aredetermined by the congestion preference and the number of eachtype user and total available system capacity. One of theinteresting result is that model (7) allocates the availableresources only to type 2 users when an additional usage makesthe degree of congestion severe

What this result implies is that our method allows serviceproviders to block a certain type of users from accessing networkservices when the network congestion is severe.

In proposition 4, we compare two differentiated menus interms of unit prices.

Proposition 4: 1)

2) p p q q q q p2 1 2 2 1 2 1 10* * * * * * * (log – log ) ( – )< = >if θ

p p q q q q p2 1 2 2 1 2 1 10* * * * * * * (log – log ) ( – )< = <if θ

(( – ) –

– ( ) .θθ

θθ

θ θ2

1 1 1

22

1

2

1

11

11

11 1< ≥ +

+n N n

n

N

n

n

n KN Nor

N

n

n

n KN N

21

2

1

11

11

1 1 – ( )≥ ++θ

θ θ

q tn N n

n1 1 21 1 1

220

1* * ( – ) –

= = <if orθθ

θ

q q t t2 1 2 1* * * *, > >

t qN

N t t q q1 1 1 1 2 1 2 2 11

1* * * * * *log – log ( ), (log – log )= + = +θ θ θ

q

N n

n NK q

N11 2 2

1 12

2

11 1* *–

( ), =

+=

+θ θ

θθ

θ


where

Proposition 4 states that the unit price of a high quality servicemay be lower or higher than that of a low quality servicedepending on their congestion preferences and networkutilization. If a type 2 user’s utility increment by switching menufrom a type 1’s to his/her own is less than the payment change(i.e. quantity change times the price for the type 1), he/she doesnot have any incentive to choose his/her own menu. In this case,a service provider has to discount a unit price to induce the type2 user to select his/her own option (i.e., quantity discount). As aresult, type 2 users get the quantity discount by revealing theirprivate information when the network utilization level is low.

If a type 2 user’s utility increment is more than the paymentchange, he/she has incentive to choose his/her own menu. Inthis case, a service provider can charge a premium to the type 2user (i.e., quantity premium). Type 2 users must pay thequantity premium for causing more congestion externalitieswhen the network utilization level is high.

Our adverse selection model is different from a traditionaladverse selection model (model (9) which does not considercongestion externalities) in that type 2 users may experience apenalty (quantity premium) as well as a benefit (quantitydiscount) depending on their congestion preferences and networkutilization. Type 2 users in most adverse selection studies get thequantity discount for revealing private information. However,type 2 users in this study can pay the quantity premium forcausing more congestion externalities in high network utilization.

NUMERICAL ILLUSTRATIONS

In this section, we show the difference between models (7) and(8) using the following numerical example: C = 100, n1 = n1 = 1,θ2 = 1.5θ1 = 1.5. Table 2 shows the equilibrium quantities,prices, total quantities, and profits from models (7) and (8)depending on the level of network utilization.

pt

qp

t

q11

12

2

2

**

**

*

*, .= =


In Table 2, network utilization is measured by the ratio of thetotal quantity used and the system capacity, (C – K)/C. Forexample, 80% of network utilization implies the available systemcapacity at the time of new user’s service request, K, is 20. Thenotation ‘-’ represents a situation when a service provider doesnot provide the service.

In this example, model (7) provides each user with adifferentiated menu until the network utilization is 80%.However, it blocks type 1 users from getting services when thenetwork utilization is over 80%. On the other hand, model (8)provides one menu for both type of users until the utilization is90%. The total quantities allocated in model (7) are less than orequal to those in model (8). However, model (7) provides a serviceprovider with higher profits than model (8) does.

Figure 1 shows the unit prices of models (7) and (8) dependingon the network utilization. In model (7), the unit prices generallyincrease as the network gets congested. The unit price of type 2users is less than that of type 1 users until Point B in Figure 1shown in Proposition 3. What this implies is that until a certainthreshold of congestion, a service provider should providequantity discount to type 2 users to induce the usage. When thenetwork gets congested, the unit price of type 2 users increases


Table 2. The Equilibrium Quantities, Prices, Total Quantities, andProfits

Discriminatory Congestion Unit CongestionUtilization

Pricing Model (7) Pricing Model (8)

q*1 q*2 t*1 t*2 Q* Π* q** t** Q** P**

0% 16.67 50.00 0.98 1.70 66.67 2.68 33.33 1.28 66.67 2.5710% 15.00 45.00 0.94 1.65 60.00 2.59 30.00 1.24 60.00 2.4820% 13.33 40.00 0.89 1.60 53.33 2.49 26.67 1.19 53.33 2.3730% 11.67 35.00 0.83 1.54 46.67 2.37 23.33 1.13 46.67 2.2640% 10.00 30.00 0.76 1.48 40.00 2.24 20.00 1.06 40.00 2.1250% 8.33 25.00 0.68 1.40 33.33 2.08 16.67 0.98 33.33 1.9760% 6.67 20.00 0.59 1.30 26.67 1.89 13.33 0.89 26.67 1.7770% 5.00 15.00 0.46 1.18 20.00 1.64 10.00 0.76 20.00 1.5280% 3.33 10.00 0.28 1.00 13.33 1.28 6.67 0.59 13.33 1.1790% – 5.00 – 0.72 5.00 0.72 3.33 0.28 6.67 0.57100% – – – – – – – – – –

rapidly. This implies that a service provider should chargepremium to type 2 users to control congestions while maximizingprofits. Unlike the unit price of type 2 users, the unit price oftype 1 users decreases when the network is congested (over PointA). Point A is a point when type 1 users are no longer willing topay more for each unit because the disutility due to congestiondecreases the values of the service.

CONCLUSIONS

This paper has proposed a discriminatory congestion pricingmechanism for network services using adverse selection in gametheory. Equilibrium quantities and prices in our model aredetermined by total available system capacity, the congestionpreferences, and the proportion of each type users. Therefore,service providers can offer differentiated menus for users withdifferent congestion sensitivities depending on dynamically-changing network utilization.

There are several implications for managers of network


Figure 1. Unit Prices

services. First, varying prices of network services depending onnetwork utilization is an effective way of controlling congestions.Services subject to congestions such as VOD and networkstorage services require efficient and effective congestion controlmethods. However, most service providers solve congestionproblems by increasing capacity, which is an expensive option.Our model provides an inexpensive alternative to increasingcapacity.

Second, segmenting markets can not only maximize profits butalso better control congestions. In most cases, marketsegmentation is used to increase profits. However, our modelshows that market segmentation in congestion pricing allowsbetter congestion control. Our model allocates less resources byblocking low quality service users from getting services andcharging more for high quality service users.

Third, quantity premium as well as quantity discount for highquality services are necessary depending on network utilization.High quality service users in most cases get quantity discount forrevealing their private information. However, high quality serviceusers (i.e., type 2 users) in this study may pay the quantitypremium for causing more congestion externalities in highnetwork utilization. .

Our research has a few limitations, each of which is a directionfor future research. First, we will extensively compare our modelwith other models using various sets of data. We will investigatethe effects of various factors such as utilization, user sensitivity,proportion of each type users, etc. Second, we will extend ourmodel in several directions. We will consider n types of users,because in real world, consumers’ sensitivity levels would bemuch more heterogeneous than two levels assumed in thispaper. We can also consider other factors such as administrationcost and initial cost which are assumed to be zero in this paper.Finally, we will address the implementation issues of congestionpricing models such as ours. They include how to estimate thecognitive costs of users and overcome users’ resistance, etc.


Appendix

Simplifying model (6) into model (7)

① Since θ2 ≥ θ1 and q1 ≥ 0, from(IC2),

If (IR1), is inactive, i.e., θ1 log q1

then (IR2) is inactive, i.e. θ2 log q2

In this case, if the a service provider raises price from t1 to t2,then it can increase its profit.

So, (IR1) is active at the optimum

② Since θ2 ≥ θ1, q1 ≥ 0, and (IR1) is active, if (IC2) is inactive,then

In this case, if the a service provider raises price t2 subject to(IC2), then it can increase its profit.

So, (IC2) is active, i.e. t2 – t1 = θ2(log q2 – log q1).

t qN

K

K n q n q1 2 11 1 2 2

1=

θ log – log – –

.

– log – –

–1

01 1 2 2

2N

K

K n q n qt

>

– log – –

–1

01 1 2 2

1N

K

K n q n qt

>

≥

θ1 11 1 2 2

11

log – log – –

–qN

K

K n q n qt

θ2 11 1 2 2

11

log – log – –

–qN

K

K n q n qt≥

θ2 2

1 1 2 22

1log – log

– ––q

N

K

K n q n qt


As a result, we get t2 = t1 + θ2(log q2 – log q1).

③ Sum (IC1) and (IC2), then

That is, θ2 log q2 – θ2 log q1 ≥ θ1 log q2 – θ1 log q1 > 0, since θ2 ≥θ1.

Therefore, θ2 (log q2 – log q1) > 0. Consequently, q2 ≥ q1.

④ From ②, t2 – t1 = θ2 (log q2 – log q1)

And from ③, θ2 log q2 – θ2 log q1 ≥ θ1 log q2 – θ1 log q1 > 0.Therefore, we get t2 – t1 = θ2 (log q2 – log q1) ≥ θ1 (log q2 – log q1) > 0.

+

>θ1 11 1 2 2

10– log log

– – q

N

K

K n q n q

≥

θ1 21 1 2 2

1log – log

– – q

N

K

K n q n q

qK

K n q n q2 11 1 2 2

– log – –

θ

t t q

N

K

K n q n q2 1 2 21 1 2 2

1– log – log

– –=

θ

+

θ2 1

1 1 2 21

1 log – log

– – –q

N

K

K n q n qt

≥

θ1 2

1 1 2 2

1log – log

– –q

N

K

K n q n q

θ2 21 1 2 2

21

log – log – –

–qN

K

K n q n qt+

θ1 11 1 2 2

11

log – log – –

– qN

K

K n q n qt


So,

Therefore, we can neglect (IC1).

From ①,

So,

Therefore, we can neglect (IC2).

Proposition 1

Model (7) can be rewritten as max

From FOC

we get qn

N nq

n

N n21 2

1 2 21

1 2

1 2 22 11= ≥ ≥

θθ θ

θθ θ

θ θ–

, –

, sin .where ce

∂∂

θΠq

n

K n q n q

n

q2

1

1 1 2 2

2 2

2

0= + = –– –

,and

∂∂

θ θΠq

N

q

n

K n q n q

n

q1

1

1

1

1 1 2 2

2 2

1

0= =–– –

–

N qK

K n q n qn q qθ θ1 1

1 1 2 22 2 2 2log – log

– – (log – log .

+

θ2 21 1 2 2

21

0log – log – –

– .qN

K

K n q n qt

≥

>

=θ1 11 1 2 2

21

0log – log – –

–qN

K

K n q n qt

θ2 21 1 2 2

11

log log – –

–qN

K

K n q n qt≥ +

θ2 21 1 2 2

21

log – log – –

–qN

K

K n q n qt

1 1 2 22

10log

– – –

N

K

K n q n qt+

>

θ θ1 21 1 2 2

1 1 21

log – log – –

– log qN

K

K n q n qt q

≥ +


Thus,

Since

As a result,

Since

Therefore, we get

Since a service provider should set

From

Consequently, we get

Therefore,

N

n

n

n KN N– ( ) ,< +

+1

2

1

11

11

1 1θ

θ θ

Q n q n q

N

NK

n N n

n* ,

( – ) –* *= + =+

≤1 1 2 21

1

1 1 1

221

1θθ

θθif

( – ) –– ( )

111 1 1

22

1

2

1

21

11

1n N n

n

N

n

n

n KN Nθ

θθ

θ θ≤ < ++

N

n

n

n KN N

21

2

1

11

11

1 1 – ( ) .< ++

θθ

θ θ

qN n

n NK q N N

11 2 2

1 11 1

1

11 1* *–

( ) ( ) , =

+> +

θ θθ

θ θand

t qN

N q N N1 2 1 1 1 1

11

1 0 1 1* * *log – log ( ) , ( ) .= + > > +θ θ θ θ

p

N n

n NK p

NK1

1 2 2

1 12

2

11 1* *–

( ) , .=

+=

+θ θ

θθ

θ

( )–( – )

, ( – ) –

.1 11

01

11

2

2 1

2

1 1 1

22

1

2

+ = + ≥ ≤ ≤NN

n

n N

n

n N n

n

N

nθ

θ θ θθ

θ

p

NK

NN2

2

1

2

12 11 1

1 1* . , .=+ +

≤ ≤ +θ

θθ

θθ θSince

N n

n NN n

n N n

n

N

n

θ θθ

θ θθ

θθ1 2 2

1 11 2 2

1 1 1

22

1

211 0

1–

( ), – ,

( – ) –

+≤ ≥ ≤ ≤we get

qN n

n NK1

1 2 2

1 11* –

( ) .=

+θ θ

θ


Similarly, from model (8),

From model (9),

Since

Proposition 2

Since

Then,

which leads to

Therefore, we can show

= + ≥N q q n q qθ θ1 2 2 2 2 1 0(log – log ) (log – log ) .* * * *

**– log – log( )+{ }N q Nθ θ1 11

Π Π* ** * * *– log – log( ) (log – log ) –= + +{ }N q N n q qθ θ θ1 1 1 2 2 2 11

N q q n q qθ θ1 2 2 2 2 1 0(log – log ) (log – log ) .* ** * *+ ≥

θθ

2 2 11

21(log – log ) (log – log ). * * ** *q q

N

nq q≥

log log log ,*

*

**

*

**

*

q

q

N

n

q

q

q

q2

1

1

2 2 1 1

≥

≥

θθ

q q

q

q

q

q12

1 1

* ****

*

**

*, . < >we get

N

NQ Q Q

θθ1

111

+≤ ≤ ≤ , * * * * * *.we get

qN n

n

K

Nq

K

NQ K1

1 2 2

1 12

2

1

*** *** –

, , * **=

=

=θ θ

θθθ

and

q

NK Q

N

NK* * * *=

+=

+θ

θθ

θ2

1

1

11 1and

N

n

n

n KN N – ( )≥ +

+

21

2

1

21

11

1 1θθ

θ θor

Q n qn

NK

n N n

n* ,

( – ) – *= =

+<2 2

2 2

12

1 1 1

21

1θθ

θθ

if


That is, Π* ≥ Π**.

Proposition 3

From the proof of proposition 1, we get the optimal quantitiesand prices as following:

where

Therefore, we get

where log q*2 – log q*1 > since q*2 > q*1 as shown in ③. As a result, we can show t*2 > t*1 .

Proposition 4

From the proof of proposition 3, we can get

where θ2 2 1 2 11

1

(log – log )– ( – ) * * * **

*q q q qt

q

=

1

22 2 1 2 1

1

1qq q q q

t

q** * * *

*

*(log – log )– ( – ) ,θ

= +{ }[ ]1

1 21 1 2 2 1 1 1q q

q t q q q t* ** * * * * *(log – log ) –θ

p p

t

q

t

q q qq t q t2 1

2

2

1

1 1 21 2 2 1

1* **

*

*

* * ** * * *– – ( – )= =

t t q q2 1 2 2 1* * * *– (log – log ), = θ

N

n

n

n KN N

21

2

1

21

11

1 1– ( )≥ ++

θθ

θ θ

q t in N n

n1 1 21 1

2

01* * ( – ) –

= = <f orθθ

t q

NN t t q q1 1 1 1 2 1 2 2 1

11* * * * * *log – log( ), (log – log )= + = +θ θ θ

qN n

n NK q

NK1

1 2 2

1 12

2

11 1* *

–

( ), =

+=

+θ θ

θθ

θ


Therefore,

REFERENCES

Basar, T., and R. Srikant (2002), “Revenue-Maximizing Pricing andCapacity Expansion in a Many-users Regime,” IEEE INFOCOM.

Bohn, R., et al. (1993), “Mitigating the Coming Internet Crunch: MultipleService Levels via Precedence,” Technical report, UCSD, San DiegoSupercomputer Center, and NSF.

Gupta, A., et al. (2000), “Extracting Consumers’ Private Information forImplementing Incentive-compatible Internet Traffic Pricing,” Journalof Management Information Systems, Summer 17(4), 9-29.

Kelly, K.P. (2000), “Models for a Self-Managed Internet,” PhilosophicalTransactions of the Royal Society A, 358, 2335-2348.

________ (2001), “Mathematical Modeling of the Internet,” in MathematicsUnlimited, and Beyond, B. Engquist and W. Schmid, eds. Berlin:Springer-Verlag, 685-702.

Laffont, J., and D. Martimort (2002), The Theory of Incentives: ThePrincipal-Agent Model. Princeton University Press, 39-43.

Lin, Z., Zhao H., and S. Ramanathan (2003), “Pricing Network Servicesfor Optimizing Resource Allocation: an Implementation Scheme,”Web2003, WA. December, 13-14.

Lin, Z., et al. (2002), “Exploring Traffic Pricing for the Virtual PrivateNetwork,” Information Technology and Management, October 3, 301-327.

MacKie-Mason, J.K, and H. R. Varian (1995a), “Pricing CongestibleNetwork Resources,” IEEE JSAC, 13(8), 1141-1149.

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Salanie, B. (1998), The Economics of Contracts. Cambridge,Massachusetts: The MIT Press.

p p q q q q p2 1 2 2 1 2 1 1* * * * * * * (log – log ) ( – log ) .> >if θ

p p q q q q p2 1 2 2 1 2 1 1* * * * * * * (log – log ) ( – log ) .< <if θ

θ2 2 1 2 1 1 (log – log )– ( – ) .* * * * *q q q q pmeans


Simiu, E., and R. Srikant (1999), “Adaptive Pricing in Networks withNon-cooperative Users,” CiteSeer. IST, available from http://citeseer.ist.psu.edu/simiu99adaptive.html.

Varian, H. R. (2000), “Versioning Information Goods,” in InternetPublishing and Beyond: The Economics of Digital Information andIntellectual Property, Brain Kahin and Hal Varian, eds. Cambridge:MIT Press.


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