ABSTRACT
Title of thesis: INTERACTION BETWEEN A SERVICE
PROVIDER AND CONTENT PROVIDERS
AND SOCIAL EFFICIENCY
Ramy ElDelgawy, Master of Science, 2014
Thesis directed by: Professor Richard J. La
Department of Electrical and Computer Engineering
University of Maryland, College Park
It is unclear how the introduction of “fast lanes” would affect the evolution
of the Internet. While this is with no doubt an important issue, we do not aim to
address this question. Instead, in this work we study the interaction between an
internet service provider and one or more content providers when such fast lanes are
allowed. Each entity is considered a player in a game, and it is assumed that these
players are both rational and selfish, i.e., they aim to maximize their own gains.
Clearly, their objectives are different, and their bargaining powers may vary. We
examine how they may reach an agreement and how their interaction affects the
resulting quality-of-service on the fast lanes. To get a better understanding of the
kind of contract these entities may be willing to sign we look at two cases: (i) a
monopoly market that has one internet service provider and one content provider
and (ii) one where an extra content provider enters the market.
Our findings show that, in the case of a monopoly market, an efficient contract
can be reached in Stackelberg games, which maximizes the aggregate payoff of both
the providers. Similarly, when another content provider enters the market, the
entity with the strongest bargaining power chooses a contract that maximizes the
aggregate payoff of all providers.
INTERACTION BETWEEN A SERVICE PROVIDER AND
CONTENT PROVIDERS AND SOCIAL EFFICIENCY
by
Ramy ElDelgawy
Thesis submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Master of Science
2014
Advisory Committee:
Professor Richard J. La, Chair/Advisor
Professor Mark A. Shayman
Professor Michael C. Rotkowitz
c© Copyright by
Ramy ElDelgawy
2014
“The God of heaven, he will prosper us; therefore we his servants will arise and build”
—Nehemiah 2:20
ii
Acknowledgments
It would be hard to appropriately thank all the beloved people who stood by
me and helped me through my journey.
First and foremost I’d like to thank my advisor, Professor Richard La for
giving me an invaluable opportunity to work with him. He mentored me through a
topic that I had very few experience in, he was both kind and patient during that
course. His passion to research is admirable and I definitely learned a lot from him
professionally and, most importantly, personally.
I would like to thank Professor Michael Rotkowitz and Professor Mark Shay-
man for serving in my M.Sc. thesis committee. I am especially thankful for Professor
Mark Shayman for giving me the opportunity to work with him as a grader for his
course. I am also thankful to Professor Robert Newcomb, whom I assisted in teach-
ing two courses.
I would also like to thank my professors back in Alexandria University in Egypt
and Professor Kosuke Sato in Osaka University, Japan for without them I would not
be having this opportunity of studying at UMD. Finally, I would like to thank Mrs.
Melanie Prange for her continuous help during my stay here at UMD.
I owe myself and everything that I am, to my beloved family who provide
me – endlessly and unconditionally – with love, energy and prayers. Words cannot
express the gratitude I owe them. Thank you for always believing in me even when
I did not believe in myself.
I owe a special and a personal thank you message to all the people who en-
iii
couraged me through these past two years. Abanoub Gad, Mary Gad, Christina
Fahmi, Karim Labib, Peter Mansour, Sandra Shaker and all the Coptic Orthodox
Christian Association members, as well as the congregation of the St. Mary’s Coptic
Orthodox Church, thank you for being my family away from home.
iv
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Internet Neutrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Forms of Traffic Management . . . . . . . . . . . . . . . . . . . . . . 4
2. Existing literature on net neutrality . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Discrimination by traffic origin: Termination Fees . . . . . . . . . . . 7
2.2 Discrimination by traffic content: Various lane speeds . . . . . . . . . 10
2.2.1 ISPs offering a range of qualities . . . . . . . . . . . . . . . . . 10
2.2.2 Vertical Foreclosure . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Other Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3. Interaction Between a Single SP and a Single CP . . . . . . . . . . . . . . 13
3.1 Setup and System Model . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Stackelberg Game Models . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1 Stackelberg Game with the SP as a Leader . . . . . . . . . . . 20
3.2.2 Stackelberg Game with the CP as a Leader . . . . . . . . . . . 21
3.3 Bargaining Problem and Social Efficiency . . . . . . . . . . . . . . . . 22
3.3.1 Nash Bargaining Solution . . . . . . . . . . . . . . . . . . . . 22
3.3.2 Social Optimum . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.1 Proofs of Main Results . . . . . . . . . . . . . . . . . . . . . . 28
4. Interaction Between a Single SP and Two CPs . . . . . . . . . . . . . . . . 34
4.1 Stackelberg Game Model . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.1 The Shapley value . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1.2 Stackelberg Game with the SP as a Leader . . . . . . . . . . . 42
4.2 Social Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3.1 Proofs of Main Results . . . . . . . . . . . . . . . . . . . . . . 46
5. Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 49
v
Chapter 1: Introduction
1.1 Internet Neutrality
Network neutrality, a term coined by Columbia media law professor Tim Wu in
2003 [18], has been a hot topic for the past decade [11,12,19]. Regulators, legislators
and courts have been involved, and wide media and advocacy campaigns occurred.
Moreover, scholarly interest from different disciplines emerged to study its effects.
The regulatory and judicial contention among network users and access providers
has been centered on the Internet service provider’s (ISP) existing or potential au-
thorities, which include discriminatory traffic management, i.e., traffic generated by
varying Internet applications receives different treatment for the purpose of man-
aging the network. Proponents of net neutrality have advocated for some form of
regulatory intervention as they believe that any form of traffic discrimination gives
network operators the final say on which applications would succeed or fail. At the
other end of the spectrum, some prefer an open market where competition dictates
the behavior of network operators and argue that any form of regulation would
constrain the future development of broadband.
While the Internet is global, the debate over net neutrality is, however, specific
to each country and also to each state. The debate started in the U.S, and has
1
extended to other countries in Europe and Asia but with distinct differences. On
May 15, 2014, the Federal Communications Commission (FCC) in the U.S. decided
to consider two options regarding Internet services: first, permit the presence of
different broadband lane speeds, which in its essence contradicts net neutrality; and
second, reclassify broadband as a telecommunications service, and thus preserving
net neutrality.
In early 2014, Comcast and Netflix signed an agreement that allows Netflix
to stream their content directly through the Comcast network for an undisclosed
fee. The key idea behind this agreement between an Internet service provider (ISP),
namely Comcast, and a content provider (CP), namely Netflix, is to bypass the
“congested” Internet and to improve the quality-of-service (QoS) experienced by the
Netflix streaming users. Soon afterward, another ISP, Verizon, signed a contract
with Netflix as well though it is unclear what the terms of this contract are and
how similar it is to the one signed with Comcast. The idea of fast lanes could be
exemplified in Figure 1.1.
The debate between whether to regulate the ISPs or leave competition take
its course and the legal and economic policies to be taken by the regulatory body
were triggers to the net neutrality discussions in the academia and among policy
makers. In reality though, due to the lack of consistent and coherent messages –
and sometimes unrealistic assumptions – found in academia, these findings have not
found an impact on the decisions of network operators and regulators. If fast lanes
are to be allowed in the future, it is important to understand how the ISPs and
other content/service providers would behave and how it would shape the ensuing
2
Fig. 1.1: The express lane on Southbound I-60 is a good example of what is
proposed by the FCC. Cars could avoid traffic by going on an express lane in
return for a fee.
3
economic and social benefits of the Internet.
The rest of this chapter discusses the concepts and terminologies of traffic
management that would aid us in the rest of this study. In Chapter 2 we review
related literature. Chapters 3 and 4 deal with a market where a single ISP interacts
with a single CP and two CPs, respectively. We conclude in Chapter 5.
1.2 Forms of Traffic Management
Performance or quality-of-service (QoS) is often the key word used in the net
neutrality debate. At the network level, the QoS may refer to the congestion level,
delay, or delay jitter that network traffic experiences. At the user level, the QoS
may be measured by the user’s satisfaction level from the service. For example, the
user’s satisfaction level can be determined by streaming video speed and delays, the
clarity of voice calls made over the internet, and file download speed. We use the
QoS as one of parameters stipulated in the contract signed between an ISP and a
CP, which is required to be provided by the ISP.
The ISP can offer varying QoS using different means. These include packet
filtering (i.e., packets from a certain source are not delivered), rate limiting (i.e.,
outgoing rate of traffic is artificially limited), and differentiation (i.e., certain traffic
is given a “fast lane”, while other traffic is sent on the “regular lane”). In this thesis,
we focus on the differentiation via fast lanes on the basis of contracts signed between
an ISP and a CP.
The network is comprised of physical links (wires, cables, and fibers) and the
4
network devices that connect them (e.g., switches and routers). Increasing network
capacity requires investments for (i) replacing existing links with faster links or
deploying additional ones, (ii) upgrading network equipment to handle higher link
speeds, and/or (iii) installing additional storage devices, e.g., caches and content
storage space. Storage devices, such as caches, can be used in a network to reduce
the delays experienced by end users by (temporarily) storing popular objects at
the edge of the network closer to end users, thereby improving performance. We
examine how the ISP’s investments in its network are influenced by the relative
bargaining positions of involved parties at an equilibrium and the agreed contract
duration in different settings we consider in this thesis.
5
Chapter 2: Existing literature on net neutrality
The Internet was made available for commercial use in the late 1980s. During
that time, the Internet was categorized as an information service. However, by
the end of the 1990s and early 2000s, new internet devices and services began to
emerge, making the Internet more than just an information service. Since then,
the U.S. regulators, independent ISPs, CPs and advocate groups have been in a
constant battle over the issue of net neutrality.
Loosely speaking, net neutrality refers to the network operation where all data
packets in the network are treated equally without discrimination. There are dif-
ferent interpretations of net neutrality; one interpretation is that network operators
should not distinguish packets on the basis of their origins. Under this interpretation,
the network operator would not be able to charge the content provider a termination
fee without violating net neutrality. Another interpretation is that traffic belonging
to different applications should not be differentiated or given priority over others.
Different possible interpretations of net neutrality and an increasing call for
allowing “fast lanes” sparked academic interests on this topic, which started off as
studies of its legal and economical implications by legal and economics scholars.
However, later on it also attracted researchers from other disciplines as well.
6
Despite the growing research interest in net neutrality, there has not been a
formal economic model adopted by researchers. In spite of recent progresses, it is still
hard to rely on simple theoretical models to draw meaningful concrete conclusions
on this dynamic and complex issue. In many of the existing studies, results are often
ambiguous and model parameters are unrealistic.
In this thesis, we adopt the second interpretation of net neutrality mentioned
earlier. The recent FCC’s decision to consider various broadband lane speeds in
May 2014. However, we briefly summarize the literature pertaining to either of the
aforementioned interpretations.
2.1 Discrimination by traffic origin: Termination Fees
The Internet can be viewed as a two-sided market where ISPs provide a plat-
form of interaction between CPs and consumers. Consumers and CPs pay their
respective ISPs for access to the internet. However, ISPs do not collect extra fees
from CPs for delivering content to the consumers, called termination fees [17]. Two
central questions arise in this kind of market: What type of pricing mechanism is
more likely to be adopted by ISPs to maximize their own profits? And, how does
the introduction of this pricing mechanism by ISPs affect the interaction between
CPs and consumers and how does it affect the overall social welfare?
The pricing imposed on both sides of the market affects the number of users
subscribing to the broadband service on one side and the number of content providers
and applications on the other side. In [20], users on the same side of the market
7
could have different membership values. But, their interaction with users from the
other side is the same. In [21], there are no membership values. However, usuage
fees varies according to the interactions between agents on different sides of the
platform.
Economides and T˚ag [22] build on the two-sided market model employed in [20]
to examine the effect of departing from net neutrality by allowing the ISPs to charge
CPs a fee in addition to the price they charge their subscribers. They study the case
of a monopoly and duopoly ISP market and suggest that imposing net neutrality
always enhances social welfare when the market is fully covered, i.e., everyone has
Internet access, under the condition that the CPs’ valuation of an extra consumer
connecting to the Internet is greater than the consumers’ valuation of an extra CP
connecting to the Internet.
Instead of charging a fee, an ISP could also subsidize CPs to increase its value
to consumers. In this case, the model discussed above would decrease the total
surplus. Lee and Wu [17] argue that one possible explanation for the fact that ISPs,
at this day and time, do not charge CPs with termination fees is that it would
lead to lower content variety over the Internet which causes consumer discontent.
Therefore, ISPs do not find it in their interest to charge these kind of fees.
Musacchio et al. [23] study the interaction between multiple CPs and multiple
ISPs connecting the CPs to consumers. The CPs collect profits from advertising
revenues arising from consumer clicks. Their finding indicates that ISPs overcharge
CPs as compared to the social optimum. An ISP charging a termination fee to CPs
have a negative effect on the overall market as consumers are less willing to pay
8
due to reduced availability of Internet content. This effect is more prominent as the
number of ISPs in the market increases. They also make these suggestions for the
following two cases:
Case 1: In the case that the CP’s advertising revenue is low, it might be desirable to
offer negative termination fees by the ISP to the CP to increase the CP’s incentive
to invest in its content;
Case 2: If the number of the ISP subscribers is low or the CP’s advertising revenue
is high, the ISP charges the CP with positive termination fees, so that the ISP would
have an incentive to invest in its own network.
A related problem in which the CPs offer sponsored contents, e.g., advertise-
ments, is studied by Andrews et al. [1] where the interaction between CPs and SPs
is modelled as a Stackelberg game. The service provider offers a pricing schedule
and the CP responds with how much content it wants to sponsor. They suggest
that CPs and service providers are better off coordinating together to maximize the
total system profit, assuming that the additional profit from cooperation can be
split between both entities in an arbitrary manner.
Liang and Wang [15] study a similar problem but with a market that has
heterogeneous CPs, which interact with a monopolistic ISP. Their finding indicates
that none of the CPs may be willing to sponsor their data if the ISP discriminates
the price of sponsored data between CPs according the CPs’ net worth. Then, they
consider a competition between one small CP and one large (or rich) CP. Their
result shows that, on the short run, the smaller CP would benefit from sponsoring
its data unlike the larger CP. In the long run, however, the larger CP will gain more
9
market share if it chooses to sponsor its content.
2.2 Discrimination by traffic content: Various lane speeds
As discussed before, the second interpretation for net neutrality implies that
the traffic generated by various applications would be treated equally. On the other
hand, traffic discrimination would mean that either ISPs would provide CPs with
varying QoS for different prices or traffic is managed by the ISP in a certain manner
without giving CPs a say on how their traffic is to be handled.
2.2.1 ISPs offering a range of qualities
Hermalin and Katz [8] analyze a model where a monopolistic SP provides a
continuum of QoS. The CPs choose the QoS they would like to receive and pay a
corresponding price. They compare this scenario to the case where a single level
of QoS is provided by the SP, as proposed by the net neutrality regulation. Not
surprisingly, their findings indicate that CPs that would select lower QoS (resp.
higher QoS) are almost always harmed because they would be driven out of the
market (resp. utilize less efficient, lower quality services), which argue against the
net neutrality. On the other hand, CPs in the middle of the market benefit from a
single QoS level. They also study the duopoly case and suggest that the same results
apply as in the monopoly case. If the ISP is required to provide the highest possible
quality, the charge it would require would be unattractive for content providers that
prefer lower QoS. Thus, the content available to consumers would be reduced. On
10
the other hand, not allowing the ISP to collect fees from CPs would force ISPs to
provide a single level of QoS that is inefficient.
Cheng et al. [5], Choi and Kim [6] and Kra¨mer and Wiewiorra [10] study the
effects of network congestion. They use the concepts of queuing theory in operations
research to study the effects of traffic discrimination in the market. In Choi and
Kim [6], “fast lanes” are sold to the CP with the highest bid, ensuring that its traffic
would receive higher QoS over that of other CPs in the market. Their results show
that the absence of discrimination leads to a higher social welfare. However, they
were not able to confirm similar findings in the long run. Cheng et al. [5] consider a
similar approach to the problem; a key difference is that they provide the premium
lane service to all CPs at the same price. Clearly, if all CPs pay for the priority
service, no CP enjoys better QoS over the other CPs. Their results suggest that,
contrary to ISPs’ claim, net discrimination leads to little incentive for ISPs to invest
in their network capacity. In short, net discrimination benefits the ISPs, but hurts
the CPs.
In Kra¨mer and Wiewiorra [10], CPs do not compete with each other. But,
their advertised content is sensitive to waiting times. Their model also assumes
that there is minimum QoS provided to those CPs who choose not to get a priority
service. Under these assumptions, their results show that overall welfare is improved
in the short run as well as in the long run.
11
2.2.2 Vertical Foreclosure
Many ISPs have their own content to provide which may compete with that of
other CPs offering similar service. In that case, ISPs may purposely degrade other
CPs’ services in order to promote their own service. Chen and Nalebuff [24] study
the case with two firms in the market; one provides an essential service and the
other offers a non-essential service. Their finding suggests that if the firm providing
the essential service decides to enter the other firm’s market and compete, it has no
incentive to degrade the other firm’s service as it collects most of its revenue from its
essential product. However, their results seem a bit too optimistic as they are based
on a critical assumption that the non-essential good has a low value. Empirical
data in real life show a different result. This can be seen in cable operators and
the programs that they air, and in mobile companies blocking Viber and Skype
services on their 3G network, for example, as in some Middle Eastern and European
countries.
2.3 Other Practices
Kocsis and de Bijl [25] and Economides and T˚ag (2012) [22] express their con-
cern over ISPs striking exclusive deals with CPs. They suspect that these exclusive
deals would eventually lead to the monopolization of the market. They argue that
a regulatory body should play an important role in such situations.
12
Chapter 3: Interaction Between a Single SP and a Single CP
The problem of interest to us can be viewed as one of task delegation [14]. The
task delegation problem is often studied in the context of contract design [3]. One
issue that attracted much attention in task delegation is asymmetry of information,
which arises when one party has private information that the other party cannot
observe. While this could happen even in our problem, here we assume that there is
no information asymmetry and we can model the problem as a complete-information
game. For an interested reader, we refer to [13] for a study that examines the
consequence of information asymmetry in a related setting.
3.1 Setup and System Model
We focus on a scenario with only a single (internet) SP, e.g., Comcast or
Verizon, and a single CP, e.g., Netflix or Amazon Prime. Subscribers to the SP (resp.
the CP) pay a subscription fee fSP (resp. fCP ). We assume that the subscribers to
the CP must also have internet service as well, hence subscribe to the SP.
We assume that, in the absence of any other incentives, the SP only provides
certain minimum QoS, which we denote by Qmin. Typically, the performance of
streaming service deteriorates during the peak hours when the Internet is congested.
13
For this reason, the CP may be interested in convincing the SP to improve the QoS
experienced by streaming traffic during congestion.
One possible means of achieving this goal is via a contract; it stipulates the
QoS that the SP is required to deliver to the streaming traffic, and the SP collects a
payment from the CP in return for the higher QoS enjoyed by the streaming service.
Here, we consider a class of contracts given by a pair (Q, p), where Q is the QoS
to be provided to the streaming traffic of the CP, and p is the price that the CP
will pay per CP subscriber to the SP.1 Throughout this chapter, we assume that,
even when the SP and the CP agree to a contract, the SP continues to provide the
minimum QoS Qmin to other non-streaming traffic. Thus, only the streaming traffic
of the CP receives (higher) QoS stipulated in the contract.
In order to capture the payoffs of both the SP and the CP, we introduce the
following notation. Let Q denote the QoS offered to the streaming service by the
SP. We assume that Q ∈ Q := [Qmin, Qmax] for some maximum QoS Qmax satisfying
Qmin < Qmax < ∞. Let NCP (t;Q) and NSP (t;Q) be the number of subscribers for
CP and SP, respectively, at time t ∈ IR+ := [0,∞) when the streaming service QoS
is Q. The duration of the contract, if any is signed, is denoted by M ∈ (0,∞).
i. Revenues: The payoff of the CP and the SP from their revenue over the
interval [0, M ], i.e., the contract duration, when the SP offers the streaming service
1 While we assume that p is a price per CP subscriber for convenience, we can also consider a
price per unit bandwidth, assuming that CP subscribers are homogeneous.
14
QoS Q is given by2
UCP,R(Q) =
∫ M
0
NCP (t;Q) fCP e
−τt dt and (3.1)
USP,R(Q) =
∫ M
0
NSP (t;Q) fSP e
−τt dt, (3.2)
where τ > 0 is the discount rate that captures the time value of money, i.e., a dollar
today is worth more than a dollar tomorrow [4].3
ii. SP investment costs: When the SP wishes to provide a certain level
of QoS Q ∈ Q, it has to invest in the network capacity and/or infrastructure and
the required investment depends on (i) the QoS Q and (ii) the increase in streaming
traffic. We model this investment using the cost function c : Q×[0,M ]→ IR+, where
c(Q, t) denotes the additional investment necessary to serve a new CP subscriber
joining at time t with QoS Q (as opposed to QoS Qmin).
The investment needed to maintain the minimum QoS Qmin will be made by
the SP whether a contract with the CP is signed or not. For this reason, we do
not explicitly model this investment and only focus on the additional investment
needed to provide the agreed QoS Q in the contract for the streaming traffic. For
this reason, we assume that c(Qmin, t) = 0 for all t ∈ [0,M ].
The total cost of the SP for this investment over the contract duration is given
by
CSP (Q) = c(Q, 0) NCP (0;Q) +
∫ M
0
c(Q, t)
∂
∂t
NCP (t;Q) dt, (3.3)
2 To facilitate our analysis, we assume that the subscription fees fCP and fSP do not change
during the duration of the contract, and leave the case where the CP can change the subscription
fee as a function of QoS Q for a future study.
3 Throughout the paper, we assume that all fees and costs are adjusted for inflation over time.
15
where ∂NCP (t;Q)/∂t is the rate at which the number of CP subscribers increases
at time t when QoS Q is provided to its subscribers.
Note that the investment in (3.3) consists of two parts; the first is the up-front
investment needed to improve the QoS for existing CP subscribers from Qmin to Q.
The second accounts for the additional investment required to handle new CP sub-
scribers over the duration of the contract. Since the initial number of CP subscribers
NCP (0;Q) does not depend on Q, we denote it by NCP (0) when convenient.
iii. Payment by the CP to the SP: When the CP and the SP sign a
contract Γ = (Q, p) with Q > Qmin, the CP must pay the SP for the improved QoS
provided to the streaming traffic by the SP according to the contract Γ. The payoff
of the CP and the SP from this payment is given by
UCP,P (Γ) =−
∫ M
0
p NCP (t;Q) e
−τt dt and (3.4)
USP,P (Γ) =
∫ M
0
p NCP (t;Q) e
−τt dt (3.5)
=−UCP,P (Γ),
where τ is the same discount rate introduced in (3.1) and (3.2) to model the time
value of money.
Given a contract Γ = (Q, p) the providers agree to, we define their payoffs
using (3.1) through (3.5) as follows:
UCP (Γ) = UCP,R(Q) + UCP,P (Γ) and (3.6)
USP (Γ) = USP,R(Q) + USP,P (Γ)− CSP (Q). (3.7)
The aggregate payoff of the providers, which we denote by Uag, is equal to the sum
16
of the payoffs of the providers, i.e.,
Uag(Γ) = UCP (Γ) + USP (Γ)
= UCP,R(Q) + USP,R(Q)− CSP (Q). (3.8)
Note that the aggregate payoff depends only on the QoS Q, but not on the price
p because UCP,P (Γ) = −USP,P (Γ) and they cancel each other out in the sum. For
this reason, with a little abuse of notation, we write the aggregate payoff as Uag(Q)
when there is no confusion.
In order to facilitate our analysis, we introduce the following simplifying as-
sumption.
Assumption 3.1. (a) The rate at which the number of CP subscribers increases,
i.e., ∂NCP (t;Q)/∂t, is given by a concave and strictly increasing mapping ζ : Q → IR.
In other words, for fixed QoS Q for streaming service, ∂NCP (t;Q)/∂t = ζ(Q).
(b) Similarly, the rate at which the number of SP subscribers increases is given by a
concave and strictly increasing function λ : Q → IR, e.g., λ(Q) = ζ(Q) + δ for some
δ ≥ 0.
The interpretation of Assumption 3.1(b) is that, in addition to the new sub-
scribers for streaming service, there may be a separate group of new subscribers to
the SP solely for access to the Internet without subscribing to streaming service.
In general, it is expected that the cost of providing a certain level of QoS
will diminish over time due to advances in technologies. The following assumption
captures this.
17
Assumption 3.2. For each fixed QoS Q ∈ Q, the cost function c(Q, t) is strictly
decreasing in t. In particular, we assume that c(Q, t) = c(Q, 0) · e−ηt, where 0 < τ <
η. In addition, the cost c(Q, 0) is strictly convex and strictly increasing in Q.
When there is no confusion, we denote c(Q, 0) simply by c(Q). Here, we
assume that η > τ because η reflects both (i) the decrease in investment costs due
to advances in technologies and (ii) the time value of money.
3.2 Stackelberg Game Models
Clearly, the price that the CP will be willing to pay will depend on the QoS
delivered by the SP. The terms of the contract will likely be dependent on the
bargaining powers of these two parties. Furthermore, they are private entities most
likely interested only in their own payoffs.
In this section, we consider the scenarios in which one of the providers is in a
much better position to shape the terms of the contract to be offered. We examine
two separate cases. In the first (resp. second) case, we assume that the SP (resp.
the CP) has more bargaining power. We model each case as a Stackelberg game
with the party with stronger bargaining power as a leader and the other as a follower
and examine their NEs [7].
First, we note that both the CP and the SP can ensure a minimum payoff,
which is the payoff they will receive without any contract between them. These
minimum payoffs are equal to
UminCP := UCP,R(Qmin) and U
min
SP := USP,R(Qmin). (3.9)
18
Therefore, a contract Γ is feasible in the sense that the CP and the SP may agree to
it, only if the aggregate payoff is greater than or equal to UminCP +U
min
SP =: U
min
ag . The
reason for this is that if the aggregate payoff is strictly less than Uminag , either the
CP or the SP will have to accept a payoff strictly smaller than its minimum payoff.
Since a rational player will not agree to a payoff less than its minimum payoff, such
a contract will never be signed by rational CP and SP. We denote the set of feasible
contracts by Γfc.
In a Stackelberg game, the leader first offers a contract Γ = (Q, p) to the
follower and then the follower decides whether to accept the contract or not. There-
fore, the action space of the leader is the set of all contracts (of the form under
consideration)4 and that of the follower is {accept, reject}. If the follower accepts
the contract Γ, the payoffs of the CP and the SP are given by the payoff vector
(UCP , USP ) = (UCP (Γ), USP (Γ)). Otherwise, their payoff vector is (UCP , USP ) =
(UminCP , U
min
SP ). Throughout the rest of this section and following sections, we assume
that the set of feasible contracts is nonempty, i.e., Γfc 6= ∅.
Second, at an NE of the Stackelberg game, the leader will be able to extract all
the increase in the aggregate payoff and the follower will receive its minimum payoff.
The reason is that, assuming that a contract Γ = (Q, p) is feasible, the leader can
choose a price p such that the payoff to the follower is equal to its minimum payoff.
Recall that at an NE, no player can increase its payoff via unilateral deviation, Thus,
if the follower receives a payoff larger than its minimum payoff at an NE, the leader
4 In practice, however, the action space of the leader can be assumed to be the set of feasible
contracts for the reason explained earlier.
19
can change the price so that it can increase its own payoff, thereby contradicting
that it is an NE.
3.2.1 Stackelberg Game with the SP as a Leader
In the Stackelberg game in which the SP is the leader, the following procedure
can be used to further reduce the set of contracts that will be considered by the
SP at an NE. Given the QoS Q, compute the maximum price pCPmax(Q) that the CP
would be willing to pay by setting UCP (Q, pCPmax(Q)) = U
min
CP . Clearly, there exists a
unique maximum price.
After a little algebra, we obtain
pCPmax(Q) =
(A1 + A2)fCP (ζ(Q)− ζ(Qmin))
NCP (0)τA1 + ζ(Q)(A1 + A2)
, (3.10)
where A1 := 1− e−τM and A2 := −τ M e−τM . Note that A1 + A2 = 1− e−τM(1 +
τM) > 0 because log(1) = 0 > −τM + log(1 + τM).
The goal of the SP is to maximize its payoff by searching over all feasible
contracts of the form Γ = (Q, pCPmax(Q)), i.e.,
max
Q∈Q:(Q, pCPmax(Q))∈Γfc
USP ((Q, p
CP
max(Q))). (3.11)
Suppose that Q?SP is a solution to (3.11). Then, an NE of the Stackelberg game is
given by ((Q?SP , p
CP
max(Q
?
SP )), accept).
5
5 There may exist other NEs of the Stackelberg game where no contract is signed by the
providers.
20
3.2.2 Stackelberg Game with the CP as a Leader
In the second Stackelberg game model, the roles of the providers are reversed;
the CP is the leader and the SP is the follower. Similarly as in the first Stackelberg
game outlined in the previous subsection, when the CP is the leader, we can use the
following procedure to reduce the set of contracts that will be considered by the CP
at an NE: for any fixed Q ∈ Q, calculate the minimum price pSPmin(Q) that the SP
would be willing to accept for providing the QoS Q by setting USP (Q, pSPmin(Q)) =
UminSP .
From (3.7) and (3.9), we get
pSPmin(Q) =
Num1(Q) +Num2(Q)
τ NCP (0) A1 + ζ(Q)(A1 + A2)
, (3.12)
where A3 := 1− e−ηM ,
Num1(Q) = τ
2(NCP (0)(c(Q)− c(Qmin)) + (ζ(Q)− ζ(Qmin))A3/η)
Num2(Q) =−fSP (A1 + A2)(λ(Q)− λ(Qmin))
The goal of the CP is to maximize its payoff by searching over all feasible
contracts of the form Γ = (Q, pSPmin(Q)), i.e.,
max
Q∈Q:(Q, pSPmin(Q))∈Γfc
UCP ((Q, p
SP
min(Q))). (3.13)
LetQ†CP be a solution to (3.13). An NE of the Stackelberg game is then ((Q
†
CP , p
SP
min(Q
†
CP )),
accept).
21
3.3 Bargaining Problem and Social Efficiency
Another possible scenario that may emerge in practice is that neither the CP
nor the SP has much stronger bargaining power than the other. In this case, the
interaction between the two providers may be modeled as a two-person bargaining
problem [16, 26]. A two-person bargaining problem tells us how two entities (with
comparable bargaining power) should cooperate in order to mitigate the well-known
inefficiency of Nash equilibria of non-cooperative games.
In this section, we model the problem of contract design/selection as a two-
person bargaining problem between the two providers and examine the NBS. In the
process, we establish the relation between the NBS and the Pareto optimum that
maximizes the aggregate payoff of both providers.
3.3.1 Nash Bargaining Solution
Let us denote the set of feasible payoff vectors by P , i.e.,
P = {(UCP (Γ), USP (Γ)) | Γ ∈ Γfc}.
The NBS is a payoff vector (U?CP , U
?
SP ) for the two providers and is defined via an
axiomatic approach [26].
1. (Individual rationality): (U?CP , U
?
SP ) ≥ (U
min
CP , U
min
SP );
2. (Feasibility): (U?CP , U
?
SP ) ∈ P ;
3. (Pareto optimality): if there exists another feasible payoff vector (U ′CP , U
′
SP ) ≥
(U?CP , U
?
SP ), then (U
′
CP , U
′
SP ) = (U
?
CP , U
?
SP );
22
4. (Independence of irrelevant alternatives): Suppose that another bargaining
problem has (i) a feasible payoff vector set P∗ ⊂ P such that (U?CP , U
?
SP ) ∈ P
∗
and (ii) the same minimum payoffs. Then, the NBS of the new bargaining
problem is (U?CP , U
?
SP );
5. (Independence of linear transformations): Consider another bargaining prob-
lem in which the payoffs of the two parties are given by linear transformations
of those in the original bargaining problem. Then, the NBS of the new bargain-
ing problem is given by the linear transformations of the NBS of the original
bargaining problem;
6. (Symmetry): Suppose that P is symmetric, i.e., if v = (v1, v2) ∈ P , then
(v2, v1) ∈ P , and UminCP = U
min
SP . Then, U
?
CP = U
?
SP .
Moreover, it is well known that, if there exists another feasible payoff vector
(U ′CP , U
′
SP ) > (U
min
CP , U
min
SP ), then there exists a unique payoff vector that maximizes
the product (UCP −UminCP )× (USP −U
min
SP ) among all feasible payoff vectors, and the
unique maximizer is the NBS.
Our problem, however, has a special structure that simplifies the analysis of the
NBS. Recall from Section 3.1 that, for fixed QoS Q, any feasible contract Γ = (Q, p)
achieves the same aggregate payoff because the aggregate payoff depends only on
the QoS Q of the contract from (3.8). Therefore, the set of feasible payoff vectors
achieved by contracts with fixed Q ∈ Q (assuming Uag(Q) > Uminag ) is given by the
line segment between (Uag(Q) − UminSP , U
min
SP ) and (U
min
CP , Uag(Q) − U
min
CP ). This is
illustrated in Fig. 3.1.
23
SG2
UCP
UISP
UCP
min
UISP
min
NBS
set of payoff vectors achievable
with QoS E
Pareto frontier
NE
NE
SG1
Fig. 3.1: Example of a set of achievable payoff vectors with a fixed QoS Q ∈
Q and Pareto frontier. NESG1 (resp. NESG2) = Nash equilibrium of the
Stackelberg game with the SP (resp. the CP) as the leader, and NBS = Nash
bargaining solution.
24
For the reason explained above, the Pareto frontier is given by the set of
feasible payoff vectors that can be achieved with Q? ∈ arg maxQ∈Q Uag(Q). Let us
denote maxQ∈Q Uag(Q) by Umaxag and the difference ∆Uag := U
max
ag −U
min
ag . Then, the
NBS is given by
NBS =
(
0.5∆Uag + U
min
CP , 0.5∆Uag + U
min
SP
)
=
(
Umaxag + U
min
CP − U
min
SP
2
,
Umaxag + U
min
SP − U
min
CP
2
)
.
This is shown in Fig. 3.1.
3.3.2 Social Optimum
It is clear from the above discussion that the NBS is socially optimal in that it
maximizes the aggregate payoff of the two providers. A related interesting question
is how efficient the NEs of the Stackelberg games (discussed in the previous section)
are. In order to answer this question, we need to first compute arg maxQ∈Q Uag(Q).
From (3.1) through (3.3), we obtain
Uag(Q) = fCP
[
NCP (0) A1
τ
+
ζ(Q)(A1 + A2)
τ 2
]
+ fSP
[
NSP (0) A1
τ
+
λ(Q)(A1 + A2)
τ 2
]
− c(Q) NCP (0)−
c(Q)ζ(Q)A3
η
.
After removing the terms that do not depend on the QoS Q, the aggregate payoff
maximization problem is equivalent to
max
Q∈Q
(
1
τ 2
((fCP · ζ(Q) + fSP · λ(Q))(A1 + A2))
−c(Q)NCP (0)−
c(Q)ζ(Q)A3
η
)
. (3.14)
25
From Assumptions 3.1 and 3.2, one can show that there exists a unique optimal
solution to the above maximization problem, i.e., |arg maxQ∈Q Uag(Q)|= 1.
3.4 Main Results
One interesting question that arises naturally in the Stackelberg game models
discussed in Section 3.2 is which Stackelberg game yields a more efficient NE when
a contract is signed. Let us denote the NE of the Stackelberg game with the SP
(resp. the CP) as the leader by NESG1 (resp. NESG2). The proofs of our main
results are provided in the following subsection.
Theorem 1. The QoS chosen at the NEs NESG1 and NESG2 is the same, i.e.,
Q?SP = Q
†
CP . Hence, the aggregate payoff achieved by both NEs NESG1 and NESG2
is identical.
A next question of interest to us is where these NEs lie in relation to the Pareto
frontier, in particular the NBS. The following theorem answers this question.
Theorem 2. The NBS achieves the maximum aggregate payoff. Furthermore, the
aggregate payoff of the NBS is equal to that of NESG1 and NESG2.
An important implication of this finding is that when one of the providers
has a much stronger bargaining position than the other provider and can dictate
the terms of the contract, then regardless of which provider is the leader, there is
no loss of efficiency due to the selfish nature of the providers in the sense that the
aggregate payoff of the providers is maximized. Another way of stating this is that
the price of stability is one [9].
26
Before we present the proofs of these two theorems in the subsequent subsec-
tion, we first offer some intuition behind them. Recall from Section 3.2 that, in a
Stackelberg game, the leader offers a contract that yields the minimum payoff to
the follower at the NE NESG1 or NESG2. Hence, in both maximization problems
in (3.11) and (3.13), the leader essentially tries to maximize the aggregate payoff,
knowing that it will be able to keep all of the aggregate payoff except for the min-
imum payoff of the follower. For this reason, it is in the interest of the leader to
maximize the aggregate payoff, thereby achieving a Pareto optimal payoff vector at
the NEs.
Another question we are interested in is how the contract duration M affects
the QoS at the Pareto optimum that maximizes the aggregate payoff of the two
providers. It is not immediately clear from (3.11) and (3.13) how the equilibrium
QoS Q?SP or Q
†
CP would behave as the contract duration M is varied.
It turns out that there is no general trend that holds in all cases. However,
under certain mild conditions, in particular when the contract duration is large in
relation to τ−1, increasing contract duration tends to encourage the providers to
improve the equilibrium QoS. This is stated formally in the following theorem: Let
Q?(M) denote the QoS at the Pareto frontier with varying contract duration M .
Recall that the QoS at the NEs of Stackelberg games is identical to the QoS at the
Pareto optimum that maximizes the aggregate payoff.
Theorem 3. Suppose that c, ζ and λ are twice differentiable. Fix a contract duration
M∗ and assume that the following hold:
27
c1. Q?(M∗) ∈ (Qmin, Qmax);
c2. τ ·M∗ ≥ 1; and
c3. c(Q) ddQζ(Q) + ζ(Q)
d
dQc(Q) is strictly increasing in Q over [Q
?(M∗), Qmax].
Then, Q?(M) is nondecreasing in M over [M∗,∞). In fact, it is strictly increasing
when Q?(M) < Qmax.
Note that the condition c3 in the theorem is satisfied if ζ is an affine function
by Assumption 3.2.
Theorem 3 tells us that if the contract duration is long relative to the discount
rate modeling the time value of money, i.e., τ , then the Pareto optimal QoS, which
is the same as the QoS at the NEs of Stackelberg games, increases with the contract
duration. Hence, the content service subscribers will likely enjoy better service when
the providers sign a long-term contract.
3.4.1 Proofs of Main Results
Proof of Theorem 1: Note that in both (3.11) and (3.13), the leader searches over
the set of feasible contracts in order to find a feasible contract that (i) yields the
minimum payoff to the follower and (ii) maximizes the leader’s payoff. Therefore,
the maximization problem in (3.11) is equivalent to the following:
max
Q∈Q
(
USP ((Q, p
CP
max(Q))) + U
min
CP
)
(3.15)
This is because adding a constant to the objective function does not change the so-
lution. Clearly, the new objective function in (3.15) is the aggregate payoff achieved
28
in (14)
UCP
UISP
UCP
min
UISP
min
NBS
NE
NE
SG1
SG2
Pareto frontier
se
ar
ch
 di
rec
tio
n i
n (
15)
search direction in (16)
search area
Fig. 3.2: Search directions and area in maximization problems (3.14), (3.15) and (3.16).
by the contract. Therefore, in order to solve (3.15), the SP simply searches along
the dotted vertical line connecting the payoff vector (UminCP , U
min
SP ) and NESG1 as
indicated by the vertical purple arrow in Fig. 3.2.
Similarly, (3.13) is equivalent to
max
Q∈Q
(
UCP ((Q, p
SP
min(Q))) + U
min
SP
)
. (3.16)
By an analogous argument, in order to solve (3.16), the CP looks for a feasible
contract that maximizes the aggregate payoff along the horizontal line connecting
29
the payoff vector (UminCP , U
min
SP ) and NESG2 as shown by the horizontal purple arrow
in Fig. 3.2.
This observation can be made more formal as follows. First, by substituting
(3.10) in (3.7) and removing the terms that do not depend on Q, we can show that
(3.11) is equivalent to the following:
max
(Q, pCPmax(Q))∈Γfc
(
(fCP ζ(Q) + fSP λ(Q))(A1 + A2)
τ 2
−c(Q)NCP (0)−
c(Q)ζ(Q)A3
η
)
. (3.17)
Similarly, plugging (3.12) in (3.6) and deleting the terms that are independent of
Q, we find the following equivalent maximization problem for (3.13):
max
(Q, pSPmin(Q))∈Γfc
(
(fCP ζ(Q) + fSP λ(Q))(A1 + A2)
τ 2
−c(Q)NCP (0)−
c(Q)ζ(Q)A3
η
)
. (3.18)
First, note that the objective functions in (3.17) and (3.18) are identical. Since
both of these optimizations are over the QoS Q (with the price being given by
either (3.10) or (3.12)), they yield the same optimal value and optimal solution Q?.
Because the aggregate payoff depends only on the QoS Q, this completes the proof
of Theorem 1.
Proof of Theorem 2: The proof of the theorem follows directly from the proof of
Theorem 1 and (3.14). Note that the aggregate payoff maximization problem in
(3.14) has the same objective function as (3.17) (or (3.18)). Therefore, the only
difference between (3.14) and (3.15) (or (3.16)) is that, rather than limiting the
search to the line segment between the payoff vector (UminCP , U
min
SP ) and NESG1 (or
30
NESG2), (3.14) searches over the set of feasible contracts, i.e., the yellow area in
Fig. 3.2, and the maximum aggregate payoff is achieved on the Pareto frontier shown
as the red line in Fig. 3.2. Since the NBS, NESG1 and NESG2 all lie on the Pareto
frontier, they achieve the same aggregate payoff.
Proof of Theorem 3: For notational simplicity, we denote Q?(M∗) by Q∗. In order
to make the dependence on contract duration M explicit, we denote the aggregate
payoff Uag(Q) by Uag(Q;M). From (3.14) and condition c1 in the theorem, the
first-order necessary condition [2] tells us
∂
∂Q
Uag(Q;M
∗)
∣
∣
∣
∣
Q=Q∗
=
1
τ 2
(fCP · ζ
′(Q∗) + fSP · λ
′(Q∗))(A1(M
∗) + A2(M
∗))
−c′(Q∗)NCP (0)
−
1
η
A3(M
∗)(c′(Q∗)ζ(Q∗) + c(Q∗)ζ ′(Q∗))
= 0, (3.19)
where A1(M) = 1− e−τM , A2(M) = −τMe−τM and A3(M) = 1− e−ηM .
Define G : IR+ × (Qmin, Qmax)→ IR, where
G(M,Q) =
∂
∂Q
Uag(Q;M).
By its definition and from (3.19), we have G(M∗, Q∗) = 0. Thus, by the im-
plicit function theorem [27], there exist (i) open sets Q+ ⊂ Q and M+ ⊂ IR+
such that M∗ ∈ M+ and Q∗ ∈ Q+ and (ii) a continuously differentiable function
φ :M+ → Q+ such that G(M,φ(M)) = 0 for all M ∈ M+. Therefore, in order to
31
prove the theorem, it suffices to prove that dφ(M)/dM > 0 for all M ∈ [M∗,∞)
such that Q?(M) < Qmax.
The implicit function theorem also tells us
d
dM
φ(M) = −
(
∂
∂Q
G(M,φ(M))
)−1 ∂
∂M
G(M,φ(M)). (3.20)
Let us first consider ∂∂QG(M,φ(M)).
∂
∂Q
G(M,φ(M))
=
1
τ 2
(fCP · ζ
′′(φ(M)) + fSP · λ
′′(φ(M)))(A1 + A2)
−c′′(φ(M))NCP (0) (3.21)
−
1
η
A3(c
′′(φ(M))ζ(φ(M)) + 2c′(φ(M))ζ ′(φ(M))
+c(φ(M))ζ ′′(φ(M)))
The first term in (3.21) is nonpositive since both ζ and λ are assumed concave
by Assumption 3.1. The second term is nonpositive because the cost function c is
assumed convex. Finally, by condition c3 in the theorem, the last term is strictly
negative. Therefore, ∂G(M,φ(M))/∂Q < 0.
In light of this and (3.20), in order to prove dφ(M)/dM > 0, we only need to
prove ∂G(M,φ(M))/∂M > 0.
∂
∂M
G(M,φ(M)) = Ξ1(M)Me
−τM − Ξ2(M)e
−ηM , (3.22)
where Ξ1(M) := fCP · ζ ′(φ(M)) + fSP ·λ′(φ(M)) and Ξ2(M) := c′(φ(M))ζ(φ(M)) +
c(φ(M))ζ ′(φ(M)).
Recall that G(M,φ(M)) = 0 for all M ∈M+. Thus, from (3.19)
1
τ 2
Ξ1(M)(A1(M) + A2(M))≥
1
η
A3(M)Ξ2(M).
32
This implies that, for all M ∈M+,
Ξ1(M)
Ξ2(M)
≥
τ 2A3(M)
η(A1(M) + A2(M))
>
τ 2
η
, (3.23)
where the second inequality follows from the inequality
A3(M)
A1(M) + A2(M)
=
1− e−ηM
1− e−τM(1 + τM)
> 1
for all M > 0 because τ < η.
We now proceed to complete the proof of the theorem. From (3.22), proving
∂G(M,φ(M))/∂M > 0 is equivalent to showing
Ξ1(M)
Ξ2(M)
>
exp(−ηM)
M exp(−τM)
.
To prove this inequality, using (3.23), we will demonstrate
Ξ1(M∗)
Ξ2(M∗)
>
τ 2
η
≥
exp(−ηM)
M exp(−τM)
or, equivalently,
τ 2M exp(−τM) ≥ η exp(−ηM). (3.24)
Rewriting η as τ(1 + ) with  > 0 and dividing both sides by τ exp(−ηM),
(3.24) is equivalent to
τM exp(τM) ≥ (1 + )
For all M ≥M∗, from the assumption τM∗ ≥ 1,
τM exp(τM)≥ exp() > (1 + ).
This completes the proof of Theorem 3.
33
Chapter 4: Interaction Between a Single SP and Two CPs
In this chapter we extend our study to the case where there are one SP and
two competing CPs in the market. Subscribers to the SP (resp. CP1 and CP2) pay
a subscription fee fSP (resp. f 1CP and f
2
CP ). As in the previous chapter, we assume
that the subscribers to the CPs must have internet service and subscribe to the SP.
Moreover, the SP provides at least minimum QoS, Qmin, to all traffic.
We assume that the SP offers only a single fast lane with higher QoS. In other
words, when the SP signs a contract with both CPs, it offers the same QoS to
both CPs. Hence, the class of contracts considered by the SP is given by a triple
(Q, p1CP , p
2
CP ), where Q ∈ (Qmin, Qmax] is the common QoS to be provided to the
streaming traffic of the CPs that sign a contract with the SP, and p1CP (resp. p
2
CP ) is
the price that CP1 (resp. CP2) will pay per subscriber to the SP. From the earlier
assumption, the streaming traffic of a CP that does not sign a contract with the SP
continues to receive the minimum QoS Qmin.
In order to capture the payoffs of the SP, CP1 and CP2, we introduce the
following notation. Suppose Q = (Q1, Q2) and T = (T1, T2). Let N iCP (t; Q,T),
i = 1, 2, and NSP (t; Q,T) be the number of subscribers for CPi and SP, respectively,
at time t ∈ IR+ when the streaming service QoS is Qi for CPi, i = 1, 2, starting
34
at time t = Ti.1 The duration of the contract, if any is signed, is denoted by
M ∈ (0,∞) as before. If CPi signs a contract at time t = 0, then the other CP can
sign a contract only at time t = m, where 0 ≤ m ≤ M . Hence, m = 0 means that
both CPs sign the contract at the same time, and t = M implies that the second
CP never signs a contract.
i. Revenues: Analogous to the previous case, the payoff of the CPi, i = 1, 2,
and the SP from their revenue over the interval [0, M ] when the SP provides the
streaming service QoS Qi to CPi’s traffic is given by
U iCP,R(Q,T) =
∫ M
0
N iCP (t; Q,T) f
i
CP e
−τt dt and (4.1)
USP,R(Q,T) =
∫ M
0
NSP (t; Q,T) fSP e−τt dt, (4.2)
where τ > 0 is the discount rate that captures the time value of money.
ii. SP investment costs: The investment necessary for expanding network
capacity and improving network equipments to offer QoS Q ∈ Q to a CP subscriber
depends on the QoS Q. We model this investment using the cost function c :
Q× [0,M ]→ IR+, where c(Q, t) tells us the (additional) investment the ISP needs
to make at time t in order to serve a new CP subscriber joining at time t with the
QoS Q. We assume that this investment does not depend which CP the subscriber
belongs to.
Recall that the ISP is assumed to make necessary investments to maintain at
least the minimum QoS Qmin for all traffic, which we do not explicitly model. In
other words, c(Qmin, t) = 0 for all t ∈ [0,M ].
1 An implicit assumption is that, prior to Ti, the traffic from CPi receives QoS Qmin.
35
The total cost of the SP for this investment over the contract duration is then
given by
CSP (Q,T) =
2∑
i=1
(
c(Qi, Ti)N
i
CP (Ti; Q,T)
+
∫ M
Ti
c(Qi, t)
∂
∂t
N iCP (t; Q,T) dt
)
, (4.3)
where ∂N iCP (t; Q,T)/∂t is the rate at which the number of CPi subscribers increases
at time t when QoS Qi, i = 1, 2, is provided to subscribers of CPi, starting at time
Ti. Since the number of CPi’s subscribers, i.e., N iCP (0; Q,T), does not depend on
Qi, i = 1, 2, we denote it by N iCP (0) for notational ease.
As in the previous case with a single CP, the investment in (4.3) consists of two
parts – the up-front investment needed for existing subscribers at time t = Ti, i.e.,
when the contract takes effect, and the investment for newly arriving subscribers
over the contract duration.
iii. Payment by the CPi to the SP: When a CP and the SP sign a
contract Γ = (Q, p) with Q ∈ (Qmin, Qmax], starting at time t = T , the CP must pay
the SP according to the contract Γ. Suppose that CP1 signs the contract at t = 0
and CP2 signs the contract at t = m.2 We denote the contract signed between the
SP and CPi by Γi = (Qi, pi), i = 1, 2, the time at which contract Γi takes effect by
Ti, and (Γ1,Γ2) by Γ1,2.
When CPi does not sign a contract with the SP, we assume either (Qi, pi) =
(Qmin, 0) or Ti = M . Thus, we can interpret (Qmin, 0) as an implicit contract between
the SP and a CP in the absence of an explicit contract with Q > Qmin. Moreover, in
2 We will study the effect of who signs first shortly.
36
the rest of thip chapter, we impose the assumption that the SP offers common QoS
Q to both CPs if contracts are signed. Hence, when both the CPs sign a contract
with the SP, then Q1 = Q2 ∈ (Qmin, Qmax].
The payoff of the CPi and the SP from this payment from the CP to the SP
is given by
U iCP,P (Γ1,2,T) =−
∫ M
Ti
pi N
i
CP (t; Q,T) e
−τt dt and (4.4)
USP,P (Γ1,2,T) =
2∑
i=1
(∫ M
Ti
pi N
i
CP (t; Q,T) e
−τt dt
)
(4.5)
=−U1CP,P (Γ1,2,T)− U
2
CP,P (Γ1,2,T),
where τ is the same discount rate introduced in (4.1) and (4.2) to model the time
value of money.
From (4.1) through (4.5), the payoffs of the providers are defined to be
U iCP (Γ1,2,T) = U
i
CP,R(Q,T) + U
i
CP,P (Γ1,2,T), i = 1, 2, (4.6)
USP (Γ1,2,T) = USP,R(Q,T) + USP,P (Γ1,2,T)− CSP (Q,T). (4.7)
The aggregate payoff of the providers, denoted by Uag, is given by
Uag(Γ1,2,T) = USP (Γ1,2,T) +
2∑
i=1
U iCP (Γ1,2,T)
=
2∑
i=1
U iCP,R(Q,T) + USP,R(Q,T)− CSP (Q,T). (4.8)
Obviously, the aggregate payoff depends only on (Qi, Ti), i = 1, 2, but not on the
prices pi, i = 1, 2, because
∑2
i=1 U
i
CP,P (Γ1,2,T) and USP,P (Γ1,2,T) cancel each other
out. Therefore, we write the aggregate payoff as Uag(Q,T) when there is no confu-
sion.
37
To make progress, we introduce the following simplifying assumption.
Assumption 4.1. (a) The rate at which new content service users join CPi, i = 1, 2,
is given by a mapping ζi : Q → IR for all t ∈ [0,M ]. In other words, when CPi is
served with QoS Qi, i = 1, 2, at time t ∈ [0,M ], new content service subscribers
join CPi at the rate ζi(Qi). In addition, ζi is strictly increasing and concave. (b)
The rate at which the number of SP subscribers increases is given by a concave
function λ : Q×Q → IR. Moreover, λ is strictly increasing in each argument, e.g.,
λ(Q) = ζ1(Q) + ζ2(Q) + δ for some δ ≥ 0 and functions ζi, i = 1, 2, satisfying
Assumptions 4.1(a).
Note that we assume that the rate at which new content service subscribers
joint a CP depends only on the CP’s QoS, but is independent of that of the other
CP. While we make this assumption to facilitate our analysis, it may not hold in
practice.
In addition to the arrivals of new content service subscribers, we also model
so-called churn rates when subscribers change their CP from one to the other based
on, for instance, different QoS experienced by the CPs. Suppose that one of the
CPs, say CP1, signs a contract with the SP and CP2 does not. Then, some of
CP2’s subscribers may switch to CP1. We model this switching of subscribers using
a mapping θ : Q → IR+ as follows; the rate at which CP2’s subscribers leave CP2
for CP1 at time t is given by θ(Q1) times the number of CP2 subscribers at time t.
Assumption 4.2. The mapping θ : Q → IR+ is strictly increasing and concave with
θ(Qmin) = 0 and θ(Qmax) = θmax .
38
When both the CPs have a contract with the common QoS or neither has a
contract with the SP, we assume that there is no churning.
As argued in Chapter 3, the cost of providing a certain level of QoS is expected
to decline over time owing to advances in technologies which we capture via the
following assumption.
Assumption 4.3. For each fixed QoS Q ∈ Q, the cost function c(Q, t) is strictly
decreasing in t. In particular, we assume that c(Q, t) = c(Q, 0) · e−ηt, where 0 < τ <
η. In addition, the cost c(Q, 0) is strictly convex and strictly increasing in Q.
For notational simplicity, we denote c(Q, 0) by c(Q) in the remainder of this
chapter.
4.1 Stackelberg Game Model
The terms of the contract between the CPs and the SP will likely depend on
the bargaining powers of these parties. Unlike in the previous chapter, however,
we argue that the SP will be in a much stronger position to dictate the terms of
the contract to be offered to the CPs using a well known power index, namely
the Shapley value. The Shapley value, introduced by Shapley [26], is an axiomatic
approach to determining the importance or bargaining power of each player in an
n-player cooperative game.
39
4.1.1 The Shapley value
Let N := {1, 2, . . . , n} denote the set of n players. Suppose that a vector
function φ(v) = (φ1(v), . . . , φn(v)) defined on the set of characteristic functions of
n-person games satisfies the following axioms [26]:
1. (Efficiency) Suppose that a coalition S ′ satisfies that, for any coalition S, the
equality v(S) = v(S ∩ S ′) holds. Then,
∑
i∈S′ φi(v) = v(S
′);
2. (Symmetry) If pi is a permutation of the set N and, for any coalition S, the
equality v′(piS) = v(S) holds, then φpii(v′) = φi(v);
3. (Linearity) φi(v + u) = φi(v) + φi(u).
These axioms uniquely determine the mapping φ, which is given by [26]
φi(v) =
∑
S⊂N :i∈S
(|S|−1)! (n− |S|)!
n!
[v(S)− v(S \ {i})], i ∈ N . (4.9)
In our cooperative game v, there are three players – the SP, CP1 and CP2.
Hence, possible coalitions are {CP1}, {CP2}, {SP}, {CP1, CP2}, {CP1, SP}, {CP2,
SP} and {CP1, CP2, SP}. Note that the characteristic function v : 2N → IR assigns
to each coalition the largest aggregate payoff the coalition can achieve. First, note
that, as in Chapter 3, all three players can ensure a minimum payoff, which is the
payoff they will receive without any contract between them. These minimum payoffs
are equal to
U1,minCP := U
1
CP,R(Qmin, Qmax,M, 0), U
2,min
CP = U
2
CP,R(Qmax, Qmin, 0,M), (4.10)
and UminSP := USP,R(Qmin, Qmin,M,M). (4.11)
40
A contract Γ1,2 and effective times Ti, i = 1, 2, are feasible in the sense that the
players – CP1, CP2 and SP – may agree to it, only if the aggregate payoff is greater
than or equal to U1,minCP + U
2,min
CP + U
min
SP =: U
min
ag for the same reason explained in
Chapter 3. We denote the set of the pairs of feasible contracts and effective times
by Ξfeas.
We assume that there exist feasible contracts that increase the aggregate pay-
off. Put differently, for each a coalition S with SP ∈ S, the SP can find ξ ∈ Ξfeas
that yields the aggregate payoff strictly larger than the sum of the minimum payoffs
of all members in the coalition. This implies that, for any coalition S including the
SP, v(S) >
∑
p∈S v({p}). Since any contract requires the SP, a coalition without
the SP cannot increase the aggregate payoff.
Using this observation, we can show that the SP has the largest Shapley value
as follows. From (4.9), we obtain
φCP1(v) =
1
6
(
2v({CP1}) + 2(v({SP,CP1, CP2})− v({SP,CP2}))
+(v({CP1, CP2})− v({CP2})) + (v({SP,CP1})− v({SP}))
)
,
φCP2(v) =
1
6
(
2v({CP2}) + 2(v({SP,CP1, CP2})− v({SP,CP1}))
+(v({CP1, CP2})− v({CP1})) + (v({SP,CP2})− v({SP}))
)
,
and
φSP (v) =
1
6
(
2v({SP}) + 2(v({SP,CP1, CP2})− v({CP1, CP2}))
+(v({SP,CP1})− v({CP1})) + (v({SP,CP2})− v({CP2}))
)
.
41
Subtracting φCP1(v)− v({CP1}) from φSP (v)− v({SP}) yields
(φSP (v)− v({SP}))− (φCP1(v)− v({CP1}))
= 0.5
(
(v({SP,CP2})− v({SP}))− (v({CP1, CP2})− v({CP1}))
)
≥ 0.5
(
v({SP,CP2})− v({SP})− v({CP2})
)
> 0,
where the last strict inequality is the consequence of the assumption that, for any
coalition S that includes the SP, v(S) >
∑
p∈S v({p}). Similar calculation gives us
(φSP (v)− v({SP}))− (φCP2(v)− v({CP2})) > 0. This finding implies that the SP
will possess stronger bargaining power than the CPs.
4.1.2 Stackelberg Game with the SP as a Leader
In this section, we consider a Stackelberg game where the SP is the leader and
the two CPs are followers and examine its NEs. Recall from the previous chapter
that, at an NE of the Stackelberg game, the SP will be able to enjoy all the increase
in the aggregate payoff and both CPs will receive their minimum payoffs U i,minCP ,
i = 1, 2. This is because the SP can choose prices pi, i = 1, 2, such that the payoffs
to the CPs are equal to their minimum payoffs in (4.10). We denote by Ξconfeas the
set of the pairs of feasible contracts Γ1,2 and effective times T, which satisfy (i)
Q1 = Q2, (ii) Ti = 0 for some i, and (iii) U iCP (Γ1,2,T) ≥ U
i,min
CP , i = 1, 2.
In the Stackelberg game, the SP first offers a contract Γ = (Q, pi) at time
t = 0 to CPi, and CPi decides whether to accept the contract or not (with no
delay). Then, the SP offers a contract Γ = (Q, pj) at time t = m to CPj, and CPj
42
decides if it will accept the contract (with no delay). Therefore, an action of the SP
consists of (i) the offered QoS Q, (ii) prices charged to the CPs, (iii) time at which
the second CP is approached, i.e., m, and (iv) the first CP to be offered a contract.
Hence, aSP = (Γ1,2,T), where Γ1,2 = (Γ1,Γ2) and Γi = (Qi, pi), i = 1, 2. We require
Q1 = Q2 and interpret Ti to be the time at which the SP offers a contract to CPi3.
The action space of the CPs is {accept, reject}, i.e., aiCP ∈ {accept, reject}. We
denote the action profile chosen by the players by a = (a1CP , a
2
CP , aSP ). With a little
abuse of notation, we denote the payoff vector by (U1CP (a), U
2
CP (a), USP (a)).
In addition, the following procedure can be used to further reduce the set of
contracts that will be considered by the SP at an NE: Suppose that the SP offers
a contract to CP1 first and then to CP2. The other case can be handled in an
analogous manner. Given the QoS Q, compute the maximum price pCP1max (Q) that
CP1 would be willing to pay at t = 0 by setting U1CP (Γ1,2, 0,m) = U
1,min
CP and the
maximum price pCP2max (Q) that CP2 would be willing to pay at t = m by setting
U2CP (Γ1,2, 0,m) = U
2,min
CP , where Γ1,2 = ((Q, p
CP1
max (Q)), (Q, p
CP2
max (Q))). One can show
that there exists a unique maximum price that each CP is willing to pay and these
prices do not depend on each other for any fixed Q.
Let Q = (Q,Q) and T = (0,m). After some algebra, we obtain
pCP1max (Q) =
f 1CP
∫M
0 N
1
CP (t; Q,T) e−τt dt
[
A1ζ1(Q) + A2ζ2(Qmin)
+
(
N2CP (0)−
ζ2(Qmin)
θ
)(A3θ − A4τe−τM
θ + τ
)
+
(
N1CP (0)−
ζ1(Qmin)
θmax
)(A5θmax − A6τe−τM
θmax + τ
)]
, (4.12)
3 We assume that the contract takes effect immediately if the CP accepts it
43
pCP2max (Q) =
f 2CP
∫M
m N
2
CP (t; Q,T) e−τt dt
[
A7ζ2(Q) +
ζ2(Qmin)
θ
(A3θ − A4τe−τM
θ + τ
)
+N2CP (0)
(A8θe−θm + A9τ
θ + τ
)
−N2CP (0)
( A10
θmax + τ
)
−
ζ2(Qmin)
θmax
(A5θmax − A6τe−τM
θmax + τ
)]
, (4.13)
where
A1 := 1−e
−τM−τMe−τM
τ2 , A2 :=
1−e−τm−τme−τM
τ2 , A3 :=
1−e−τM−e−(τ+θ)m+e−(τM+θm)
τ ,
A4 := 1−e
−θm
τ , A5 :=
1−e−τM
τ , A6 :=
1−e−θmaxM
τ , A7 :=
e−τm−e−τM−τ(M−m)e−τM
τ2 ,
A8 := e
−τm−e−τM
τ , A9 :=
1−e−(τM+θm)
τ , and A10 := 1− e
(θmax+τ)M .
The goal of the SP is to maximize its payoff by searching over all feasible
contracts that ensures the payoffs of at least U i,minCP , i = 1, 2.
max
ξ=(Γ1,2,T)∈Ξconfeas
USP (Γ1 = (Q, p
CP1
max (Q)),Γ2 = (Q, p
CP2
max (Q)),T) (4.14)
The NE of interest to us is of the form (a?CP1, a
?
CP2, a
?
SP ), where a
?
CP1 = a
?
CP2 =
accept, and a?SP = ((Q
?, pCP1max (Q
?), (Q?, pCP2max (Q
?), (0,m?)) or a?SP = ((Q
?, pCP1max (Q
?),
(Q?, pCP2max (Q
?), (m?, 0)).
4.2 Social Efficiency
Consider a scenario where a social player dictates the actions to be adopted
by all the players. The aim of the social player is to maximize the aggregate payoff
of all players. Recall that the aggregate payoff depends only on the QoS Q provided
by the SP and the times at which the QoS Q will be provided to the CPs.
max(Q,T):Q∈Q Uag(Q = (Q,Q),T) (4.15)
44
subject to Ti = 0 for some i
We are interested in determining how efficient the NEs of the Stackelberg game,
which has the form mentioned at the end of the previous section, are in relation to
the social optimum.
4.3 Main Results
One interesting question that arises naturally in the Stackelberg game models
discussed in Section 4.1 is whether the Stackelberg game yields an efficient NE when
both contracts are signed or not. Let us denote the NE of the Stackelberg game with
the SP as the leader by NESG
Theorem 4. Both QoS Q? and time m? chosen at the NE are identical to those
chosen by the social player.
As stated in the previous chapter, an important implication of this finding is
that when one of the providers, which is the SP in this case, has a much stronger
bargaining position than the CPs and can dictate the terms of the contract, there is
no loss of efficiency due to the selfish nature of the SP in the sense that the aggregate
payoff of the providers is maximized. Hence, the price of stability is one.
Analogously to Theorem 2, the intuition behind this theorem is that the SP, as
a leader, offers a contract that yields the minimum payoff for both CPs. Therefore,
it is in the SP’s own interest to maximize the aggregate payoff, thereby achieving a
Pareto optimal payoff vector at the NEs.
45
Another question we are interested in is which CP the SP would offer a contract
to first. A general rule that holds for all cases is given in the following section. But,
for the special case where (a) f 1CP = f
2
CP and (b) ζ1 and ζ2 are identical, the following
holds:
Theorem 5. Suppose that fCP1 = fCP2 and ζ1(Q) = ζ2(Q) for all Q ∈ Q. Then,
the ISP is indifferent as to which CP to approach first.
For the general case, the answer to the above question depends on many
parameters the SP has to take into consideration.
4.3.1 Proofs of Main Results
Proof of Theorem 4: The proof follows directly from Theorems 1 and 2 in the
previous chapter. The difference between the aggregate utility and the SP’s payoff
is constant, namely U1,minCP + U
2,min
CP (under the contracts offered by the SP). Since
the presence of a constant does not affect the optimization problem, the resulting
optimal Q and m coincide at the social optimum and the NE of Stackelberg game.
Proof of Theorem 5: Substituting (4.12) and (4.13) in (4.7) for pCP1max and p
CP2
max ,
respectively, the payoff of the SP when approaching CP1 at t = 0 and CP2 at t = m
is given by
USP ((Q, p
CP1
max (Q)), (Q, p
CP2
max (Q)), 0,m)
= fSP
[
λ(Qmin, Q)A11 + λ(Q,Q)A12 −m(λ(Q,Q)− λ(Qmin, Q))A8 +NSP (0)A5
]
+f 1CP
[
A1ζ1(Q) + A2ζ2(Qmin) +
(
N2CP (0)−
ζ2(Qmin)
θ(Q)
)(A3θ(Q)− A4τe−τM
θ(Q) + τ
)
46
+
(
N1CP (0)−
ζ1(Qmin)
θmax
)(A5θmax − A6τe−τM
θmax + τ
)]
+f 2CP
[
A7ζ2(Q) +
ζ2(Qmin)
θ(Q)
(A3θ(Q)− A4τe−τM
θ(Q) + τ
)
+N2CP (0)
(A8θ(Q)e−θ(Q)m + A9τ
θ(Q) + τ
)
−N2CP (0)
( A10
θmax + τ
)
−
ζ2(Qmin)
θmax
(A5θmax − A6τe−τM
θmax + τ
)]
−
c(Q)
η
(1− e−ηm)ζ1(Q)−
c(Q)
η
(e−ηm − e−ηM)(ζ1(Q) + ζ2(Q)),
where A11 := 1−e
−τm−τme−τm
τ2 and A12 :=
e−τm−e−τM+τme−τm−τMe−τM
τ2 .
Due to the symmetry, when the SP approaches CP2 first at t = 0 and then
CP1 at t = m, the payoff of the SP is very similar to the payoff given above except
for the following differences: (1) the indices 1 and 2 are reversed, and (ii) λ(Qmin, Q)
becomes λ(Q,Qmin). It is reasonable to assume that λ(Qmin, Q) = λ(Q,Qmin).
The SP compares the payoffs it would receive, i.e., USP ( (Q, pCP1max (Q)), (Q, p
CP2
max (Q)),
0,m) and USP ( (Q, p˜CP1max (Q)), (Q, p˜
CP2
max (Q)), m, 0), where p˜
CPi
max(Q), i = 1, 2, are the
maximum prices the SP would charge if it approached CP2 first, and offers a con-
tract to CP1 first if the first is greater than the latter. After deleting the common
terms, this is equivalent to the following inequality.
f 1CP
[
A1ζ1(Q) + A2ζ2(Qmin) +
(
N2CP (0)−
ζ2(Qmin)
θ
)(A3θ − A4τe−τM
θ + τ
)
+
(
N1CP (0)−
ζ1(Qmin)
θmax
)(A5θmax − A6τe−τM
θmax + τ
)]
+f 2CP
[
A7ζ2(Q) +
ζ2(Qmin)
θ
(A3θ − A4τe−τM
θ + τ
)
+N2CP (0)
(A8θe−θm + A9τ
θ + τ
)
−N2CP (0)
( A10
θmax + τ
)
−
ζ2(Qmin)
θmax
(A5θmax − A6τe−τM
θmax + τ
)]
−
c(Q)
η
(1− e−ηm)ζ1(Q)
> f 2CP
[
A1ζ2(Q) + A2ζ1(Qmin) +
(
N1CP (0)−
ζ1(Qmin)
θ
)(A3θ − A4τe−τM
θ + τ
)
47
+
(
N2CP (0)−
ζ2(Qmin)
θmax
)(A5θmax − A6τe−τM
θmax + τ
)]
+f 1CP
[
A7ζ1(Q) +
ζ1(Qmin)
θ
(A3θ − A4τe−τM
θ + τ
)
+N1CP (0)
(A8θe−θm + A9τ
θ + τ
)
−N1CP (0)
( A10
θmax + τ
)
−
ζ1(Qmin)
θmax
(A5θmax − A6τe−τM
θmax + τ
)]
−
c(Q)
η
(1− e−ηm)ζ2(Q).
This gives us the general rule for the SP to decide which CP to approach
first. When f 1CP = f
2
CP and ζ1(Q) = ζ2(Q) for all Q ∈ Q, all the terms that are
independent of N1CP (0) and N
2
CP (0) will cancel each other out. After a little algebra,
we would be left with
(N1CP (0) +N
2
CP (0))
[
A5θmax−A6τe−τM
θmax+τ
]
on both sides. Therefore, the ISP receives the identical payoff whether it approaches
CP1 first or CP2 first. This proves Theorem 5.
48
Chapter 5: Conclusions and Future Work
We studied both the interactions between a single SP and a single CP, and a
single SP and two CPs and showed that the aggregate payoff and QoS at the NEs of
Stackelberg games do not degrade due to selfish nature of these providers when one of
them has a much stronger bargaining position and their interaction can be modeled
as a Stackelberg game. Furthermore, we demonstrated that the Pareto optimal
QoS and the NE QoS are the same. Finally, we demonstrated that the Pareto
optimal QoS improves as the contract duration increases under mild conditions for
the monopoly market case. For the market with two content providers we have
shown a strategy by which the SP determines which CP to approach first in order
to increase its payoff.
There are multiple ways to improve our model and make it more realistic.
In our study, we have assumed that none of the CPs increase their subscription fee
after it signs a contract with the SP, i.e., CP subscribers receive a higher QoS for the
same price. This led to the other CP being obligated to sign a contract as well with
the SP. However, it is unclear what would happen if the CP that signed a contract
would increase its subscription fee. Some of the users might find the subscription
fee too high for its budget and so would turn away to the CP with the lower QoS
49
(and lower subscription fee). Therefore, there might be a NE where one of the CPs
decides to not sign a contract.
Another improvement would be that each CP agrees on a different QoS with
the SP. In our model, the SP provides only one QoS to both CPs. This might not
be optimal as one of the CPs may need a higher QoS than the other CP. It is not
directly clear to say which CP is to be approached first.
50
Bibliography
[1] M. Andrews, U. Ozen, M.I. Reiman and Q. Wang, “Economic models of spon-
sored content in wireless networks with uncertain demand,” Proc. of IEEE
INFOCOM Computer Communications Workshops, pp. 345-350, Apr. 2013.
[2] D.P. Bertsekas, Nonlinear Programming, Athena Scientific, 1995.
[3] P. Bolton and M. Dewatripont, Contract Theory, The MIT Press, Cambridge
(MA), 2005.
[4] E.F. Brigham and P.R. Daves, Intermediate Financial Management, 10th edi-
tion, Cengage Learning, 2009.
[5] H. K. Cheng, S. Bandyopadhyay and H. Guo, “The debate on net neutrality:
a policy perspective,” Information Systems Research, 22(1):60-82, Mar. 2010.
[6] J. P. Choi and B.-C. Kim, “Net neutrality and investment incentives,” RAND
Journal of Economics, 41(3), Mar. 2010.
[7] D. Fudenberg and D.K. Levine, The Theory of Learning in Games, The MIT
Press, 1998.
[8] B. E. Hermalin and M. L. Katz, “The economics of product-line restrictions
with an application to the network neutrality debate,” Information Economics
and Policy, Elsevier, 19(2):215-248, Jun. 2007.
[9] E. Koutsoupias and C.H. Papadimitriou, “Worst-case equilibria,” Proc. of the
16th Annual Symposium on Theoretical Aspects of Computer Science (STACS),
pp. 404-413, 1999.
[10] J. Kra¨mer and L. Wiewiorra, “Network neutrality and congestion sensitive
content providers: implications for content variety, broadband investment, and
regulation,” Information Systems Research, 23(4):1303-1321, May. 2012.
[11] M. A. Lemley and L. Lessig, “The End of End-to-End: Preserving the Archi-
tecture of the Internet in the Broadband Era,” UCLA Law Review, Vol. 48, p.
925, 2001.
[12] J. B. Speta, “Handicapping the Race for the Last Mile?: A Critique of Open
Access Rules for Broadband Platforms,” Yale Journal On Regulation, 2000.
51
[13] R. J. La and J. Mo, “Interaction of service providers in task delegation under
simple payment rules,” Proc. of IEEE Conference on Decision and Control
(CDC), Dec. 2009.
[14] J.-J. Laffont and D. Martimort, The Theory of Incentives: The Principal-Agent
Model, Princeton University Press, Princeton (NJ), 2002.
[15] Z. Liang and D. Wang, “Sponsoring content: motivation and pitfalls for con-
tent service providers,” Proc. of IEEE INFOCOM Computer Communications
Workshops, pp. 577-582, Apr. 2014.
[16] J.F. Nash, “The bargaining problem,” Econometrica, 18:155-162, 1950.
[17] R. S. Lee and T. Wu, “Subsidizing Creativity through Network Design: Zero-
Pricing and Net Neutrality,” Journal of Economic Perspectives, Vol. 23(3), pp.
61-76, 2009.
[18] T. Wu, “Network Neutrality, Broadband Discrimination,” Journal of Telecom-
munications and High Technology Law, Vol. 2, p. 141, 2003.
[19] C. S. Yoo, “Would Mandating Network Neutrality Help or Hurt Broadband
Competition? A Comment on the End-to-End Debate,” Journal of Telecom-
munications and High Technology Law, Vol. 3, 2004.
[20] M. Armstrong,“ Competition in Two-Sided Markets,” RAND Journal of Eco-
nomics, V ol. 37(3), pp. 668-691, 2006.
[21] J-C. Rochet and J. Tirole, “Platform Competition in Two-Sided Markets,”
Journal of the European Economic Association, Vol. 1(4), pp. 990-1029, 2003.
[22] N. Economides and J. T˚ag, “Network Neutrality on the Internet: A Two-Sided
Market Analysis,” Information Economics and Policy, Vol. 24, 2012.
[23] J. Musacchio, G. Schwartz and J. Walrand “A T wo-Sided Market Analysis
of Provider Investment Incentives with an Application to the Net-Neutrality
Issue,” Review of Network Economics, Vol. 8(1), pp. 22-39, 2009.
[24] M. K. Chen and B. Nalebuff, “One-Way Essential Complements,” Cowles Foun-
dation Discussion Paper No. 1588, Yale School of Management, 2006.
[25] V. Kocsis and P. W. J. de Bijl, “Network Neutrality and the Nature of Com-
petition between Network Operators,” International Economics and Economic
Policy, Vol. 4(2), pp. 159-184.
[26] G. Owen, Game Theory, 3rd edition, Academic Press, 1995.
[27] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1976.
52