Nash Bargaining Game Theory

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IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 8, NO. 5, OCTOBER 2000 667

A Game Theoretic Framework for BandwidthAllocation and Pricing in Broadband Networks

Hakel Yache, Ravi R. Mazumdar, Senior Member, IEEE, and Catherine Rosenberg, Senior Member, IEEE

AbstractIn this paper, we present a game theoretic frameworkfor bandwidth allocation for elastic services in high-speed net-works. The framework is based on the idea of the Nash bargainingsolution from cooperative game theory, which not only providesthe rate settings of users that are Pareto optimal from the point ofview of the whole system, but are also consistent with the fairnessaxioms of game theory. We first consider the centralized problemand then show that this procedure can be decentralized so thatgreedy optimization by users yields the system optimal bandwidthallocations. We propose a distributed algorithm for implementingthe optimal and fair bandwidth allocation and provide conditionsfor its convergence. The paper concludes with the pricing of elasticconnections based on users bandwidth requirements and users

budget. We show that the above bargaining framework can beused to characterize a rate allocation and a pricing policy whichtakes into account users budget in a fair way and such that thetotal network revenue is maximized.

IndexTermsBandwidthallocation, elastic traffic, game theory,Nash bargaining solution, pricing.

I. INTRODUCTION

CURRENT 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 cantolerate variations in transfer delays. These performance char-acteristics mean that the sources can be made to modify theirdata transfer rates according to network conditions. These ser-vices are referred to as elastic services. Their source rates areadjusted according to the network conditions so the networkcan carry a variable number of bursty connections in an effi-cient manner. Typical services, which share these properties, areTCP/IP based services, ATM available bit rate (ABR) services,or services using bandwidth-on-demand on a multiple accesssystem.

These applications are expected to ride on top of (at leastpartially since some minimum bandwidth may be reserved)bandwidth-guaranteed connections and utilize any residualbandwidth. 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 May8, 2000; approved by IEEE/ACM TRANSACTIONSON NETWORKINGEditorS. H.Low. This work was supported by a contract from the Centre National dEtudesdes Tlcommunications (CNET), France Telecom, through the ConsultationsThmatiques program.

H. Yache is with the Department of Electrical Engineering and ComputerScience, Ecole Polytechnique de Montral, Montral H3C3A7, Canada(e-mail:[email protected]).

R. R. Mazumdar andC. Rosenbergare with theSchoolof Electricaland Com-puter Engineering, Purdue University, West Lafayette, IN 47907-1285 USA(e-mail: [email protected]; [email protected]).

Publisher Item Identifier S 1063-6692(00)09116-0.

services being carried, the incoming elastic sources will haveto continually change their rates based on some notification bythe network on the available bandwidth. Thus the notion ofratecontrolof sources arises.

Since potentially there are many sources distributed in thenetwork which will be competing for the use of the availablebandwidth, there are several issues which arise and must bedealt with. These are: 1) efficient bandwidth allocation to thedifferent sources taking into account their different needs andperformance 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) theissue of pricing the bandwidth in such a way that the networkrevenue will be maximized if the users are allocated bandwidthaccording to 1) and 2) above.

In this paper, we propose a game theoretic framework, whichis very powerful, to address the above issues. In particular, bydrawing 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, andtherefore a precise framework in which different solutions canbe compared.

The idea of using the Nash bargaining solution (NBS) in thecontext of telecommunication networks is not new. This wasfirst presented in the context of packet-switched (data) networksby Mazumdar et al.[22]. The properties of Pareto optimalityas well as the development of local optimization procedureswhich lead to Pareto-optimal solutions (the local proceduresbeing greedy schemes) were studied in a series of papers byDouligeris and Mazumdar [10], [8], [9] in the context of datanetworks. This paper is thus an extension of those ideas as wellas a new approach in the context of elastic services in broadbandnetworks. 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

10636692/00$10.00 2000 IEEE
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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 shownthat the solution obtained has the property ofproportional fair-nessif 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 pricebased on user bandwidth demands. The combination of flowcontrol and pricing has also been addressed in [6] and [26].

The utility function approach used in [17] and [21] suffersfrom the point of view that user utilities or preferences are onlyknown 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 utilityfunctions. In the context of elastic services, one important mea-

sure is allocated rate. We propose a game theoretic frameworkbased on choosing this measure. We demonstrate that not onlyis it possible to address the issues of fairness and efficiency, butthe framework also allows us to put the solution in proper con-text.

Using the Nash bargaining framework from cooperativegame theory [24], [25] we show that proportional fairness(as introduced in [17]) is in fact an NBS. The bargainingframework allows us to address the bandwidth allocationproblem with nonzero minimum bandwidth guarantees [knownas minimum cell rate (MCR) in the ABR context] while alsoaccounting for peak-rate requirements of sources [referred toas peak cell rate (PCR) in the ABR context]. We then provide adistributed algorithm implemented at network links (or nodes),which achieves the desired bandwidth allocations that arePareto optimal and fair. This algorithm is based on the gradientof the dual of the basic optimization problem which resultswhen computing the NBS [2]. The algorithm proposed in [21]is also based on the dual of the social optimum problem withsecond-order differentiability or assumptions on the userutility functions. The performance functions we consider arenot in , and hence we provide a proof of the convergence ofour algorithm to the desired allocations.

We then address the issue of pricing and its relation tobandwidth allocation. It is shown that based on a users

budget or willingness-to-pay and its bandwidth demands, abargaining framework can be developed to allocate the networkbandwidths to the users in a way which is optimal in the Paretosense and is fair to the users. Furthermore, based on this,we can develop a pricing scheme based on the congestion inthe network for which network revenue is maximized whenthe network operates at the allocations corresponding to thebargaining solution. This pricing scheme has the followingproperty: a user is never charged more than its declared budgetbut could be charged less than its budget if the amount ofcongestion in the network links used by its connection is low.

The outline of this paper is as follows: In Section I, we presentthe salient facts about the NBS which is the base for our frame-

work. Section II considers the optimal and fair rate allocationproblem for elastic connections which have both minimum andpeak rate constraints. We discuss both the centralized (systemoptimality) as well as the user-based contexts. In Section III,we propose a distributed algorithm to implement the solutionand 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 andallocating bandwidth resources based on user budgets or will-ingness-to-pay. Technical proofs are deferred to the Appendix.

II. BASICFRAMEWORK

In this section, we present the salient concepts and resultsfrom cooperative game theory and the Nash bargaining (or arbi-trated) solutions (NBS) which are used in the sequel. For details,we refer the reader to the book by Muthoo [24] and the paperby Nash [25].

The basic setting of the problem is as follows: There areusers (connections) which compete for the use of a fixed re-source (bandwidth). Each user ( ) has a perfor-mance function and a desired initial performance whichis the minimal performance required by the user without anycooperation in order to enter the game. Each performance func-tion is defined on a subset of termed , which is the set ofgame strategies of the users. In a context of network resourceallocation, could represent the space of allocated rate vec-tors. The initial performance of each user represents a minimumguarantee that the network must provide the user. Therefore, wewill assume throughout our framework that each user involvedin the game can achieve its initial performance. In other words,there exists at least a vector in for which the performancevector is superior or equal to the initial per-

formance vector .Let be a nonempty convex closed andupper-bounded set. In our context, the set denotes theset of achievable performance. Let such that

. Here denotes the initialagreement point. Let denote the setof achievable performance with respect to the initial agreementpoint.

We first define the notion of Pareto optimality in the contextof multiple-criteria objectives which occurs in the typical gamesetting with multiple players.

Definition 2.1: The point is said to be Pareto optimalif for each , , then .

The interpretation of a Pareto optimum is that it is impos-sible to find another point which leads to strictlysuperior perfor-mance for all the players simultaneously. In general, in a gamewith players (or equivalently for a set of objectives), thePareto-optimal points form an dimensional hypersurface,which implies that there are an infinite number of points whichare Pareto optimal. From the definition of Pareto optimality, itis clear that an optimal network operating point should be aPareto-optimal point. The question that arises is at which of the(infinitely many) Pareto-optimal points should we operate thesystem?

One way in which we can define suitable Pareto-optimalpoints for operation is by introducing further criteria. From the
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This follows from the fact that every Pareto point can be ob-tained as the solution to the maximization of the sum of theweighted objectives (see [1]).

III. OPTIMAL ANDFAIRBANDWIDTHALLOCATION FORELASTICCONNECTIONS

It is natural to adopt a game theory approach to model and ad-dress the issue of network resource allocation. In the context offlow control in packet-switched networks, many schemes werebased on the use of game theory and gave a characterization forsome candidate points. Some of them considered Nash equi-librium points [4], [10] and others considered Pareto-optimalpoints [8], [9]. In [22], the Nash bargaining point was proposedas a suitable solution for the design of an optimal and fair flowcontrol.

As in [22], we consider the Nash bargaining point as the de-sired point for the operation of the network. This is due to thePareto 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 Paretoinefficient [11], [1]. Nash equilibria are important in that theyarise in the context of greedy optimization.

The definition of a Nash bargaining point is highly dependenton the consideration of an initial performance point (termedin the previous section). It represents a minimum performancethat a user wants to achieve and the user will not enter the gameif it is not possible. In the context of elastic services, for eachconnection (user) the initial performance can be 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 arethe available link capacities and each con-nection aims at maximizing its allocated rate beyond its min-imum desired rate. Giventhat there are many users who allsharethe same objective, the network performs an allocation which isfair to all the users while at the same time efficient from thepoint of view of the network. As argued above, this correspondsto finding the NBS for the allocation problem.

Then, we show that the NBS from the point of viewof the net-work can be achieved by solving a user-level greedy optimiza-tion problem by suitable modification of theuser objectives.Therequired modification comes in the form of implied costs asso-ciated with the global problem and these in turn play a role innetwork revenue maximization.

A. Network Optimal Rate Allocations

We consider a static model for the centralized (network)problem in which connections demand use of the networkand are identified by the routes (or paths) they take. We assumethere are links or nodes within the network. Each connectionis assumed to be elastic with a peak rate (PR) and an MR to beguaranteed by the network. Connections compete for availablebandwidth resources within the network. These resources arenetwork link available capacities and they are assumed to befixed (nontime-varying). With respect to the abstract frame-work already presented, the admissible rate vector space is

determined by network capacity constraints and the minimumand 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 peakrates o f the connections, a nd is a n incidencematrix, i.e., is equal to 1 if the link belongs to the pathand 0 otherwise.

In the context of elastic services, it is natural to assume thateach connection aims to obtain an allocated bandwidth greaterthan its minimum rate and as close to its peak bandwidth re-quirement as possible. Therefore, with respect to the frameworkdescribed above, the performance function for a user issimply defined as . Moreover, MR represents the initial (orminimum) performance desired by user .

For simplicity and without loss of generality, we assume thaton each link the spare capacity is strictly superior to the sum ofthe 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 thecorresponding minimum rate. One can show that this assump-tion ensures that has a nonempty interior.

Withrespect to theframework describedin Section I, theNBSof the centralized model is an optimal and fair rate allocation ofnetwork available capacities to the connections. From The-orem 1.1, the NBS is the solution of the following convex globaloptimization 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.
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Fig. 1. RM packet structure.

Fig. 2. Link updating and measurement process.

We propose a scheme using explicit-rate type of notificationmodeled after ABR schemes.

We assume that elastic sources regularly send forward re-source management (RM) packets in order to get feedback from

the network about the congestion state or resource availability.The information necessary for the operation of the control

scheme is conveyed by the RM packets, which are of twokinds: forward RM packets which are created by sources andconveyed along their corresponding paths, and backward oneswhich are created by destinations that turn around the forwardRM packets. The fields of an RM packet (Fig. 1) relevant to thedescription of the control scheme are DIR (direction: forwardor backward), MR (connection minimum rate), PR (connectionpeak rate), CP (congestion price), and ER (explicit rate). CPis used by the network nodes to communicate the value of theprice variables ( for link ) they control. ER stands for themaximal rate at which a given connection can transmit data.

There is a set of parameters associated with the controlscheme: a constant step-size used to update the price vari-ables, feedback intervals (FI in Fig. 2), and some measurementintervals (MI in Fig. 2). Each network link has its own feedbackinterval and measurement interval. A link price is updated atthe beginning of each feedback interval and the total link inputrate is measured during the measurement interval, as shown inFig. 2.

If we interpret as the current data rate of connectionand as a function of the current network link price vector, thenin (10) the sum can be interpreted as thecurrent total input rate at link . It is important to note that the

new price for a link is computed when the information aboutcurrent total input rate (the above sum) is available at the link.This helps determine the right values for the feedback and mea-surement intervals associated with network links.

In the following, we describe the local procedures associatedwith the allocation scheme.

Source Procedure:

A source sends a forward RM packet and inserts the MRand the PR in the corresponding fields. Then, it sends thepacket to the destination.

At the reception of a backward RM packet, a source ad-justs its transmissiondata rate according to the explicit ratenotification (ER) contained in the RM packet. We consider

that a source has a variable called allowed rate (AR) whichis updated as follows: AR ER. AR is the maximal rateat which a source is allowed to transmit.

Destination Procedure:

Upon the reception of a forward RM packet, a destinationcreates a backward RM packet, puts zero in the CP field,and sends it back to its corresponding source.

Network Node Procedure: For a particular output link: At the beginning of each feedback interval (Fig. 2), the

node updates the link priceusing the input rate measuredduring the previous measurement interval, a constantstep-size , and the link available capacity . The fol-lowing illustrates the price updating:

Upon the reception of a backward RM packet, ER and CPare modified using the current link price, the MR, and thePR. The modifications aredone as follows (ERis modifiedusing the new value of CP):

CP CP

ERPR if CP

PR MR

MRCP

if CPPR MR

Once the modifications are completed, the backward RMpacket is relayed back to the source.

A node, at regular intervals, measures (Fig. 2) the totalinput rate at the link.

It can be readily see that the ER contained in a backward RMpacket does not increase when going through network nodesin the backward direction. In addition, the implementation of

the scheme does not differentiate between network access nodesand the other nodes as far as the update of ER is concerned.

For the good operation of the control scheme, it is importantto dimension for each link the feedback and measurement in-tervals. Indeed, the feedback interval should be large enough toallow the sources traversing a particular link to react to the newprice (after update) conveyed by the backward RM packets andfor a link to experience the result of the sources reaction. Thetotal input rate at a link should be measured during that period,i.e., when the response of sources to the new price has reachedthe link.

The rate of convergence is governed by which depends onthe knowledge of defined earlier. This is the only quantitywhich needs to be broadcast to all nodes.

V. PRICINGFRAMEWORK FORELASTICSERVICES

We now address the issue of rate allocation together with thepricing issue in the context of elastic-rate connections consid-ering users bandwidth requirements and users budgets (will-ingness-to-pay) for bandwidth above their guaranteed minimumcell rates. As already shown in Proposition 2.2, if the networkcharges according to the user-implied costs, the network rev-enue is maximized when allocated rates are according to theNBS. This key property will allow us to formulate a pricingframework for the network to charge the users.


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[24] A. Muthoo,Bargaining Theory with Applications. Cambridge, U.K.:Cambridge Univ. Press, 1999.

[25] J. Nash, Thebargaining problem,Econometrica, vol.18, pp.155162,1950.

[26] S. Shenker, Fundamental design issues for the future Internet,IEEE J.Select. Areas Commun., vol. 13, pp. 11761188, Sept. 1995.

[27] A. Stefanescu and M. W. Stefanescu, The arbitrated solution for multi-objective convex programming,Rev. Roum. Math. Pure Applicat., vol.29, pp. 593598, 1984.

[28] E. Van Damme, Stability and Perfection of Nash Equilibria. NewYork, NY: Springer Verlag, 1991.[29] H. Yache, R. R. Mazumdar, and C. Rosenberg, A game theoretic

framework for rate allocation and charging of available bit rate (ABR)connections in ATM networks, in Broadband Communications98, P.Kuehn and R. Ulrich, Eds, pp. 222233.

[30] H. Yache, R. R. Mazumdar, and C. Rosenberg, Distributed algorithmsfor fair bandwidth allocation to elastic services, in Proc. INFOCOM,Tel Aviv, Israel, Mar. 2000.

Hakel Yachewas born in Sfax, Tunisia. He received the Engineering degreein computer science from the Ecole Nationale Superieure dElectro-technique,dElectronique, dInformatique et dHydraulique de Toulouse, France. He iscurrently working toward the Ph.D. degree at the Ecole Polytechnique de Mon-tral, Canada.

His research interests include congestion control in broadband networks.

Ravi R. Mazumdar (SM94)was bornin Bangalore,India. He received the B.Tech. degree in electricalengineering from the Indian Institute of Technology,Bombay, India, in 1977, theM.Sc.DIC degreein con-trol systems from Imperial College, London, U.K., in1978, and the Ph.D. degree in systems science fromthe University of California, Los Angeles (UCLA),in 1983.

He is currently Professor of electrical and com-puter engineering at Purdue University, Lafayette,IN. Prior to joining Purdue, he was Professor of

mathematics at the University of Essex, Colchester, U.K., from 1996 to1999. From 1988 to 1996, he was Professor at INRS-Tlcommunications, agraduate research institute of the Universit du Qubec, Canada, and an Invited

Professor of electrical engineering at McGill University, Montreal, Canada.During 19851988, he was an Assistant Professor of electrical engineering atColumbia University, New York, NY. He has held visiting positions at UCLA,the University of Twente, The Netherlands, the Indian Institute of Science,Bangalore, India, and the Ecole Nationale Suprieure des Tlcommunications,Paris, France. His research interests are in game theory, applied probability andstochastic analysis focusing on applications in telecommunication networks,statistical signal processing, and mathematical finance.

Dr. Mazumdar is a Fellow of the Royal Statistical Society.

Catherine Rosenberg (SM95) received theDipl.Ing. from the Ecole Nationale Suprieure desTlcommunications de Bretagne, Brest, France,in 1983, the M.S. degree in computer science fromthe University of California, Los Angeles (UCLA),in 1984, and the Ph.D. degree in computer sciencefrom Universit de Paris, Orsay, France, in 1986.

She is currently an Associate Professor of elec-trical and computer engineering at Purdue Univer-

sity, Lafayette, IN. Prior to joining Purdue, she wasthe Head of the Department of Broadband SatelliteNetworking at Nortel Networks, Harlow, U.K., and a Visiting Professor in theDepartment of Electrical Engineering at Imperial College, London, U.K., from1996 to 1999. From 1988 to 1996, she was on the faculty of the Ecole Polytech-nique de Montral, Canada. She was a Member of Technical Staff at AT&T BellLaboratories, Holmdel, NJ, from 1987 to 1988, and was with Alcatel, Lannion,France, from 1984 to 1987. She has held visiting appointments at the Universitde Paris, Jussieu, France, and the Indian Institute of Science.

Dr. Rosenberg is an Associate Editor for IEEE Communications Magazine,IEEE Communications Surveys and Telecommunications Systems. Herresearch interests are in broadband networks, IP, ATM, broadband satellitenetworks (GEO or LEO based), traffic engineering, and wireless networks.

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Nash Bargaining Game Theory

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