IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 8, NO. 5, OCTOBER 2000 667
A Game Theoretic Framework for Bandwidth
Allocation and Pricing in Broadband Networks
Haïkel Yaïche, Ravi R. Mazumdar, Senior Member, IEEE, and Catherine Rosenberg, Senior Member, IEEE
Abstract—In thispaper, we present a game theoretic framework
for bandwidth allocation for elastic services in high-speed net-
works. The framework is based on the idea of the Nash bargaining
solution from cooperative game theory, which not only provides
the rate settings of users that are Pareto optimal from the point of
view of the whole system, but are also consistent with the fairness
axioms of game theory. We first consider the centralized problem
and then show that this procedure can be decentralized so that
greedy optimization by users yields the system optimal bandwidth
allocations. We propose a distributed algorithm for implementing
the optimal and fair bandwidth allocation and provide conditions
for its convergence. The paper concludes with the pricing of elastic
connections based on users’ bandwidth requirements and users’
budget. We show that the above bargaining framework can be
used to characterize a rate allocation and a pricing policy which
takes into account users’ budget in a fair way and such that the
total network revenue is maximized.
Index Terms—Bandwidthallocation, elastic traffic, game theory,
Nash bargaining solution, pricing.
I. INTRODUCTION
C
URRENT high-speed networks have to support applica-
tions which have no way of predicting their traffic require-
ments in advance, but have stringent loss requirements and can
tolerate variations in transfer delays. These performance char-
acteristics mean that the sources can be made to modify their
data transfer rates according to network conditions. These ser-
vices are referred to as elastic services. Their source rates are
adjusted according to the network conditions so the network
can carry a variable number of bursty connections in an effi-
cient manner. Typicalservices, which share these properties, are
TCP/IP based services, ATM available bit rate (ABR) services,
or services using bandwidth-on-demand on a multiple access
system.
These applications are expected to ride “on top of” (at least
partially since some minimum bandwidth may be reserved)
bandwidth-guaranteed connections and utilize any residual
bandwidth. Since the available bandwidth will change de-
pending on the amount of “background” bandwidth-guaranteed
Manuscript received January 9, 1998; revised November 26, 1998 and May
8, 2000; approved by IEEE/ACM T
RANSACTIONSON NETWORKING Editor S. H.
Low. This work was supported by a contract from the Centre National d’Etudes
des Télécommunications (CNET), France Telecom, through the Consultations
Thématiques program.
H. Yaïche is with the Department of Electrical Engineering and Computer
Science, EcolePolytechniquede Montréal, Montréal H3C3A7, Canada (e-mail:
yaiche@comm.polymtl.ca).
R. R. Mazumdar and C. Rosenbergare with the School of Electrical and Com-
puter Engineering, Purdue University, West Lafayette, IN 47907-1285 USA
(e-mail: mazum@ecn.purdue.edu; cath@ecn.purdue.edu).
Publisher Item Identifier S 1063-6692(00)09116-0.
services being carried, the incoming elastic sources will have
to continually change their rates based on some notification by
the network on the available bandwidth. Thus the notion of rate
control of sources arises.
Since potentially there are many sources distributed in the
network which will be competing for the use of the available
bandwidth, there are several issues which arise and must be
dealt with. These are: 1) efficient bandwidth allocation to the
different sources taking into account their different needs and
performance requirements; 2) the crucial notion of fairness; 3)
the ability to implement the allocation scheme in a distributed
manner with minimal communication overheads; and 4) the
issue of pricing the bandwidth in such a way that the network
revenue will be maximized if the users are allocated bandwidth
according to 1) and 2) above.
In this paper, we propose a game theoretic framework, which
is very powerful, to address the above issues. In particular, by
drawing upon the Nash bargaining framework from coopera-
tive game theory [24], [25], we show that one can obtain a uni-
fied framework in which we can address issues of network ef-
ficiency, fairness, revenue maximization, and pricing. The ad-
vantage of such a framework is that we have precise mathemat-
ical characterization of the solutions and their properties, and
therefore a precise framework in which different solutions can
be compared.
The idea of using the Nash bargaining solution (NBS) in the
context of telecommunication networks is not new. This was
first presented in the contextof packet-switched (data) networks
by Mazumdar et al. [22]. The properties of Pareto optimality
as well as the development of local optimization procedures
which lead to Pareto-optimal solutions (the local procedures
being greedy schemes) were studied in a series of papers by
Douligeris and Mazumdar [10], [8], [9] in the context of data
networks. This paper is thus an extension of those ideas as well
as a new approach in the context of elastic services in broadband
networks. Preliminary results have been presented in [29] and
[30].
The issue of rate control for elastic sources has been the focus
of much attention. In the ATM ABR context the primary con-
cern has been to develop algorithms which adapt quickly to con-
gestion while trying to be fair in a so-called
sense
[5], [13], [16]. This notion of fairness is different from the no-
tion of the
solutions in game theory. More recently,
[17], [18] and [21] have considered the problem of rate allo-
cation and charging based on knowledge of user utility func-
tions. All consider the issue of maximizing the social benefit,
which is the sum of the user utilities. In [17] it is also shown
1063–6692/00$10.00 © 2000 IEEE
668 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 8, NO. 5, OCTOBER 2000
that the socially optimizing solution can be obtained as the so-
lution to a user optimization problem. Furthermore, it is shown
that the solution obtained has the property of proportional fair-
ness if the utility functions are logarithmic functions of the al-
located bandwidth. Allocating bandwidth based on user will-
ingness-to-pay is considered. Both [18] and [21] provide dis-
tributed algorithms for achieving the socially optimal rate allo-
cations. The pricing issues the authors consider are different; in
[17] users state their prices and the network allocates the band-
width accordingly, while in [21] the network charges a price
based on user bandwidth demands. The combination of flow
control and pricing has also been addressed in [6] and [26].
The utility function approach used in [17] and [21] suffers
from the point of view that user utilities or preferences are only
known in some qualitative sense. Thus, although reasonable as-
sumptions can be made on the behavior of utility functions,
such an approach cannot be used to provide concrete numer-
ical answers. Hence, the approach we take is to consider mea-
surable performance characteristics rather than abstract utility
functions. In the context of elastic services, one important mea-
sure is allocated rate. We propose a game theoretic framework
based on choosing this measure. We demonstrate that not only
is it possible to address the issues of fairness and efficiency, but
the framework also allows us to put the solution in proper con-
text.
Using the Nash bargaining framework from cooperative
game theory [24], [25] we show that proportional fairness
(as introduced in [17]) is in fact an NBS. The bargaining
framework allows us to address the bandwidth allocation
problem with nonzero minimum bandwidth guarantees [known
as minimum cell rate (MCR) in the ABR context] while also
accounting for peak-rate requirements of sources [referred to
as peak cell rate (PCR) in the ABR context]. We then provide a
distributed algorithm implemented at network links (or nodes),
which achieves the desired bandwidth allocations that are
Pareto optimal and fair. This algorithm is based on the gradient
of the dual of the basic optimization problem which results
when computing the NBS [2]. The algorithm proposed in [21]
is also based on the dual of the social optimum problem with
second-order differentiability or
assumptions on the user
utility functions. The performance functions we consider are
not in
, and hence we provide a proof of the convergence of
our algorithm to the desired allocations.
We then address the issue of pricing and its relation to
bandwidth allocation. It is shown that based on a user’s
budget or willingness-to-pay and its bandwidth demands, a
bargaining framework can be developed to allocate the network
bandwidths to the users in a way which is optimal in the Pareto
sense and is fair to the users. Furthermore, based on this,
we can develop a pricing scheme based on the congestion in
the network for which network revenue is maximized when
the network operates at the allocations corresponding to the
bargaining solution. This pricing scheme has the following
property: a user is never charged more than its declared budget
but could be charged less than its budget if the amount of
congestion in the network links used by its connection is low.
The outlineof thispaper isas follows: In Section I, we present
the salient facts about the NBS which is the base for our frame-
work. Section II considers the optimal and fair rate allocation
problem for elastic connections which have both minimum and
peak rate constraints. We discuss both the centralized (system
optimality) as well as the user-based contexts. In Section III,
we propose a distributed algorithm to implement the solution
and analyze its behavior in terms of convergence. In Section IV,
we then show how the game theoretic framework we have in-
troduced leads to a very elegant framework for charging and
allocating bandwidth resources based on user budgets or will-
ingness-to-pay. Technical proofs are deferred to the Appendix.
II. B
ASIC FRAMEWORK
In this section, we present the salient concepts and results
from cooperative game theory and the Nash bargaining (or arbi-
trated) solutions (NBS) which are used inthe sequel. For details,
we refer the reader to the book by Muthoo [24] and the paper
by Nash [25].
The basic setting of the problem is as follows: There are
users (connections) which compete for the use of a fixed re-
source (bandwidth). Each user
( ) has a perfor-
mance function
and a desired initial performance which
is the minimal performance required by the user without any
cooperation in order to enter the game. Each performance func-
tion is defined on a subset of
termed , which is the set of
game strategies of the
users. In a context of network resource
allocation,
could represent the space of allocated rate vec-
tors. The initial performance of each user represents a minimum
guarantee that the network must provide the user. Therefore, we
will assume throughout our framework that each user involved
in the game can achieve its initial performance. In other words,
there exists at least a vector in
for which the performance
vector
is superior or equal to the initial per-
formance vector
.
Let
be a nonempty convex closed and
upper-bounded set. In our context, the set
denotes the
set of achievable performance. Let
such that
. Here denotes the initial
agreement point. Let
denote the set
of achievable performance with respect to the initial agreement
point.
We first define the notion of Pareto optimality in the context
of multiple-criteria objectives which occurs in the typical game
setting with multiple players.
Definition 2.1: The point
is said to be Pareto optimal
if for each
, , then .
The interpretation of a Pareto optimum is that it is impos-
sible to findanother pointwhich leadsto strictlysuperior perfor-
mance for all the players simultaneously. In general, in a game
with
players (or equivalently for a set of objectives), the
Pareto-optimal points form an
dimensional hypersurface,
which implies that there are an infinite number of points which
are Pareto optimal. From the definition of Pareto optimality, it
is clear that an optimal network operating point should be a
Pareto-optimal point. The question that arises is at which of the
(infinitely many) Pareto-optimal points should we operate the
system?
One way in which we can define suitable Pareto-optimal
points for operation is by introducing further criteria. From the
YAÏCHE et al.: GAME THEORETIC FRAMEWORK FOR BANDWIDTH ALLOCATION AND PRICING 669
perspective of resource sharing, one of the natural criteria is the
notion of fairness. This, in general, is a loose term and there are
many notions of fairness. One of the commonly used notions is
that of
fairness which penalizes large users. From
the definition of
fairness [3], it can be seen that it
corresponds to a Pareto optimum. However, it is not easy to
take into account the notions that users might have different
requirements within this framework. A much more satisfactory
approach is to use the fairness axioms from game theory as the
fairness criteria [25].
We now define the NBS, which encapsulates the above
requirements of yielding Pareto optima as well as being fair
in a precise sense. Except in trivial cases, it differs from the
solution.
Definition 2.2: A mapping
is said to be an NBS
if:
1)
.
2)
is Pareto optimal.
3)
satisfies the linearity axiom if ,
with , , then
.
4)
satisfies the irrelevant alternatives axiom if ,
, and then
.
5)
satisfies the symmetry axiom if is symmetric with
respect to a subset
of indices (i.e.,
and , then if then
for ).
Remark 2.1: The items 3, 4, and 5 above are the so-called
axioms of fairness. The linearity property of the solution im-
plies that the bargaining solution is scale invariant, i.e., the bar-
gaining solution is unchanged if the performance objectives are
affinely (i.e., of the form
) scaled. The irrelevant-alter-
natives axiom states that the bargaining point is not affected by
enlarging the domain if agreement can be found on a restricted
domain. The symmetry property states that the bargaining point
does not depend on the specific labels, i.e., users with the same
initial points and objectives will realize the same performance.
Having defined the NBS, we define the optimal point as fol-
lows:
Definition 2.3: Let
be given by . Then is the
(Nash) bargaining point and
is called the set of the
(Nash) bargaining solutions.
The following result, due to Stefanescu [27], provides for a
characterization of the Nash bargaining point and will form the
basis for the results in the sequel.
Theorem 2.1: Let
, be con-
cave upper-bounded functions defined on
which is a convex
and compact subset of
. Let .
Let
s.t. . Denote by
and the subset
of strategies that enable the users to achieve at least their initial
performances.
Then there exists a bargaining solution and a unique bar-
gaining point
. Moreover the set of the bargaining solutions
(
) is determined as follows:
Let
be the set of users able to achieve a performance strictly
superior to their initial performance, i.e.,
is defined as
. Each vector in the bar-
gaining solution set verifies
and solves the fol-
lowing maximization problem (
):
Hence, satisfies that for and
otherwise.
Remark 2.2: From the assumption that there exists a
nonempty set
of users who can achieve performance superior
to their initial performance, it implies that
. Note that
for each
, . The users in are not
considered in the optimization above.
It can be readily shown that if each function
( )is
injective on
, then the bargaining solution set is a singleton
and therefore there exists a unique NBS (in the space
).
We now state an equivalent optimization problem, which will
also result in the NBS. The proof can be found in the Appendix.
We first need the following result whose proof is given in the
Appendix.
Lemma 2.1: Let
be concave. Then
is concave. If is injective,
then
is strictly concave.
Using the above, we can now formulate an equivalent opti-
mization problem, which we will consider in the sequel.
Theorem 2.2: In addition to the assumptions in Theorem 2.1,
let
; be injective on .
Consider the two maximization problems (
) and ( ):
Then:
1) (
) has a unique solution; the bargaining solution set is
a singleton.
2) (
) is a convex program and has a unique solution.
3) (
) and ( ) are equivalent. Hence, the unique solution
of (
) is the bargaining solution.
Remark 2.3: In [17], it has been shown that if the user utility
functions are logarithmic, then the maximization of the sum of
the utility functionsleads to anallocation which hasbeen termed
as proportionally fair by Kelly. In light of Theorem 2.2, this
corresponds to a NBS. However, the definition of a NBS does
not require the user objectives to be logarithmic functions. In
general, all that can be said in the case when the sums of user
utilities are maximized as considered in [17], [21] is that the
allocation will be a Pareto optimum. This optimum is referred
to as a social optimum.
Remark 2.4: Since the NBS is Pareto optimal, it im-
plies that there exists a set of weights
such that
where
(1)
670 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 8, NO. 5, OCTOBER 2000
This follows from the fact that every Pareto point can be ob-
tained as the solution to the maximization of the sum of the
weighted objectives (see [1]).
III. O
PTIMAL AND FAIR BANDWIDTH ALLOCATION FOR
ELASTIC CONNECTIONS
It is natural to adopt a game theory approach to model and ad-
dress the issue of network resource allocation. In the context of
flow control in packet-switched networks, many schemes were
based on the use of game theory and gave a characterization for
some candidate points. Some of them considered Nash equi-
librium points [4], [10] and others considered Pareto-optimal
points [8], [9]. In [22], the Nash bargaining point was proposed
as a suitable solution for the design of an optimal and fair flow
control.
As in [22], we consider the Nash bargaining point as the de-
sired point for the operation of the network. This is due to the
Pareto optimality and fairness property associated with NBSs.
It is important to note that NBSs are not related to Nash equi-
libria which (except in the case of inessential games) are Pareto
inefficient [11], [1]. Nash equilibria are important in that they
arise in the context of greedy optimization.
The definition of a Nash bargaining point is highly dependent
on the consideration of an initial performance point (termed
in the previous section). It represents a minimum performance
that a user wants to achieve and the user will not enter the game
if it is not possible. In the context of elastic services, for each
connection (user) the initialperformance canbe viewed as a per-
formance achieved by the minimum rate (MR) they want guar-
anteed by the network.
First, we consider a centralized (or global) model in which
network resources are the availablelink capacities and each con-
nection aims at maximizing its allocated rate beyond its min-
imum desiredrate. Giventhat there are many users who allshare
the same objective, the network performs an allocation which is
fair to all the users while at the same time efficient from the
point of view of the network. As argued above, this corresponds
to finding the NBS for the allocation problem.
Then, we show that the NBS from the pointof view ofthe net-
work can be achieved by solving a user-level greedy optimiza-
tion problem by suitable modification of the user objectives. The
required modification comes in the form of implied costs asso-
ciated with the global problem and these in turn play a role in
network revenue maximization.
A. Network Optimal Rate Allocations
We consider a static model for the centralized (network)
problem in which
connections demand use of the network
and are identified by the routes (or paths) they take. We assume
there are
links or nodes within the network. Each connection
is assumed to be elastic with a peak rate (PR) and an MR to be
guaranteed by the network. Connections compete for available
bandwidth resources within the network. These resources are
network link available capacities and they are assumed to be
fixed (nontime-varying). With respect to the abstract frame-
work already presented, the admissible rate vector space
is
determined by network capacity constraints and the minimum
and peak rates of the connections. It is defined as follows:
MR PR and (2)
where
is the vector of link capacities, PR is the vector of peak
rates of the connections, and
is an incidence
matrix, i.e.,
is equal to 1 if the link belongs to the path
and 0 otherwise.
In the context of elastic services, it is natural to assume that
each connection aims to obtain an allocated bandwidth greater
than its minimum rate and as close to its peak bandwidth re-
quirement as possible. Therefore, with respect to the framework
described above, the performance function
for a user is
simply defined as
. Moreover, MR represents the initial (or
minimum) performance desired by user
.
For simplicity and without loss of generality, we assume that
on each link the spare capacity is strictly superior to the sum of
the MR
s of the connections crossing this link. If this assump-
tion is not valid, then our model and results are still valid for the
subset of connections to which we can allocate more than the
corresponding minimum rate. One can show that this assump-
tion ensures that
has a nonempty interior.
With respectto the frameworkdescribed in SectionI, the NBS
of the centralized model is an optimal and fair rate allocation of
network available capacities to the
connections. From The-
orem 1.1, the NBS is the solution of the following convex global
optimization problem
:
MR
MR
PR
Proposition 3.1: Under the hypothesis that MR
; , there is a unique NBS for the centralized
problem
which is characterized as follows:
There exist
( ) and ( )
such that:
• for each
MR PR MR (3)
•
PR ;
•
;
•
.
Proof: Now under the assumption that
MR
; , the set is nonempty, convex, and com-
pact.
Define
MR
then is strictly concave.
YAÏCHE et al.: GAME THEORETIC FRAMEWORK FOR BANDWIDTH ALLOCATION AND PRICING 671
Noting that the constraints are linear in and is ,
it implies that the first-order Kuhn–Tucker [23] conditions are
necessary and sufficient for optimality.
Let
denote the Lagrangian where
, ; , and ;
denote the Lagrange multipliers associated with the
MR, PR, and capacity constraints respectively.
Then
MR
PR
Then the first-order necessary and sufficient conditions are
given by
MR
and
MR
PR
Under the assumption MR , we see that the
constraints
MCR are nonactive and hence for
all
. Furthermore, if PR and
PR otherwise.
Hence, the result follows as stated.
Remark 3.1: The Lagrange multiplier has the interpreta-
tion as the implied cost associated with the network link
.It
represents the marginal cost of a rate unit allocated for any con-
nection crossing link
.
Having obtained the characterization of the optimal (in the
Nash bargaining sense) rates allocated in the centralized or net-
work framework we now address the issue of how we can de-
fine a local optimization problem (for each connection or user)
which yields the above allocations.
B. The User Problem
In the previous section, we formulated and solved the cen-
tralized network optimal rate allocation problem. In general,
this will involve centralized coordination amongst the connec-
tions. In anetwork distributedover a vast geographical area, this
will require much communication overheads. Thus, an impor-
tant issue is whether such a computation can be decentralized at
a user level in which the user tries to optimize its performance
greedily. In general, greedy procedures lead to Nash equilibria
[28] which, beingParetoinefficient, arenot NBSs. Thus,clearly,
users must use modified criteria if the greedy optimization is to
lead to the NBS for the network.
The answer to the above question is in the affirmative. This
is well known in the theory of nonlinear programming as the
concept of tolls or penalties. This is also the approach used by
Kelly [17]. The basic idea is that if we think of the implied costs
as the penalties to be paid by the users, then local optimization
of the net user “goodput,” i.e., the desired performance minus
the penalty to be paid, will yield a Pareto-optimal point. This
will be the optimal of the weighted sum of the original objec-
tive functions, the weights being the penalties. Such an idea has
also been discussed in the context of packet-switched networks
in the thesis of Douligeris [7], where the decentralized proce-
dure attempts to arrive at the centralized or Pareto-optimal flow
control settings via the imposition of penalties.
In the decentralized model, each connection can optimize
only its allocated rate. The rate for the connection is bounded
from below by the MR and from above by the PR. It is assumed
that each user optimizes its rate without regard to the other users
(i.e., local optimization over
for user ). However, offering
unrestricted access to each user or connection is not in the net-
work’s interest and thus the network penalizes or charges each
user for use of network resources. This is reflected in a penalty
in the user optimization criteria.
We introduce
positive network parameters, denoted by
, which represent the penalty or cost in-
curred per unit of bandwidth or capacity by the
users, given
that they share the resources. The
s also can be interpreted as
the penalty per bandwidth unit that the network imposes on user
for consuming bandwidth within the network. We show how
the
s are determined such that the corresponding rate alloca-
tions lead to the centralized NBS rate allocations.
The objective of each user is to maximize its net utility which
is, for a particular rate, the difference between the utility ob-
tained from theallocated bandwidth
and the cost of accessing
the network given by
.
Hence, let
denote the following convex problem associ-
ated with user
:
MR
MR
PR
The network aims to determine the optimal rate allocation to
users that maximizes its total “revenue” based upon “charging”
per unit of bandwidth to user . Hence, the network has to
solve the following convex problem (
):
MR
PR
The following proposition shows that by appropriate choice
of network costs, the
s, the NBS of the centralized model
maximizes each user’s net utility and the network total revenue.
Proposition 3.2: Let
be the unique
NBS of the centralized problem
. Let ,
where
denotes the implied cost associated with link ;
obtained from the solution to .
Then
is the solution to the user optimization problem
MR (4)
