Socially-Aware Network Design Games

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Jocelyne Elias, Fabio Martignon, Konstantin Avrachenkov, Giovanni Neglia

To cite this version:

Jocelyne Elias, Fabio Martignon, Konstantin Avrachenkov, Giovanni Neglia. Socially-Aware Network

Design Games. [Research Report] RR-7141, INRIA. 2009, pp.21. �inria-00439687�

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Thème COM

Socially-Aware Network Design Games

J. Elias — F. Martignon — K. Avrachenkov — G. Neglia

N° 7141

J. Elias

∗

,F. Martignon

†

, K.Avra henkov

‡

,G. Neglia

§

ThèmeCOMSystèmes ommuni ants ProjetEquipeMaestro

Rapport dere her he n°7141De embre200921pages Abstra t:

Inmanys enariosnetworkdesignisnotenfor edbya entralauthority,butarisesfrom the intera tions of several self-interested agents. This is the ase of the Internet, where onne tivity is due to Autonomous Systems' hoi es, but also of overlaynetworks, where ea h user lient an de idethesetof onne tionstoestablish.

Re entworkshaveusedgametheory,andinparti ularthe on eptofNashEquilibrium, to hara terize stable networks reated by a set of selsh agents. The majority of these worksassumethatusersare ompletelynon- ooperative,leading,inmost ases,toine ient equilibria.

Toimprovee ien y, inthispaperweproposetwonovelso ially-awarenetworkdesign games. Intherstgamewein orporateaso ially-aware omponentintheusers'utility fun -tions,whileinthese ondgameweuseadditionallyaSta kelberg(leader-follower)approa h, where aleader (e.g., the networkadministrator) ar hite ts thedesirednetwork buying an appropriatesubsetofnetwork'slinks,drivinginthiswaytheuserstooveralle ientNash equilibria.

We provide bounds on the Pri e of Anar hy and other e ien y measures, and study the performan e of the proposed s hemes in several network s enarios, in luding realisti topologieswhereplayersbuildanoverlayontopofrealInternetServi eProvidernetworks. Numeri al results demonstrate that (1) introdu ing some in entives to make users more so ially-awareisanee tivesolutiontoa hievestableande ientnetworksinadistributed way, and (2)the proposed Sta kelbergapproa hpermitsto a hievedramati performan e improvements,designingalmostalwaystheso iallyoptimalnetwork.

Key-words: NetworkDesign,So ialAwareness,Game Theory,NashEquilibrium, Sta k-elbergGame

∗

Polite ni odiMilano,Italy,eliaselet.polimi.it

†

UniversityofBergamo,Italy,fabio.martignonunibg.it

‡

INRIASophiaAntipolis,Fran e,K.Avra henkovsophia.inria.fr

§

Résumé : Dans plusieurs s énarios le déploiement d'un réseau n'est pas imposé par une autorité entrale, mais résulte des intera tions entre plusieursagents égoïstes. C'est le as, par exemple, de l'Internet, où la onne tivité est due aux dé isions des systèmes autonomes,maisaussidesréseauxoverlay,où haqueutilisateurpeutdé iderl'ensembledes onnexionsàétablir. Destravauxré entsontutilisélathéoriedesjeux,etenparti ulierle on eptde l'équilibre deNash, pour ara tériserdes réseauxstables qui sont rééspar un ensemble d'agentségoïstes. Lamajoritéde estravauxsupposent quelesutilisateurssont omplètementnon oopératifs,menant,danslaplupart des as,àdeséquilibresine a es. Pouraméliorerl'e a ité,dans etarti lenousproposonsdeuxjeuxoriginauxpartiellement so iauxpour laplani ation des réseaux. Dans le premier jeu, nous avons in orporé une omposanteso iale(quitient omptedu outso ialouglobalduréseau)danslesfon tions d'utilité des utilisateurs, alors que dans le deuxième jeu nous avons utilisé en plus une appro heSta kelberg(leader-follower),oùleleader(parexemple,l'administrateurderéseau) ar hite teleréseaudésiréena hetantunsous-ensembleappropriédesliensderéseau,menant ommeçalesutilisateursàdeséquilibresdeNashe a es.Nousfournissonsdeslimitespour lePri e ofAnar hyet d'autresmesuresd'e a ité, et étudionsles performan es des jeux proposésdansplusieurss énarios,y omprisdestopologiesréellesoùlesjoueurs onstruisent unréseauoverlaysurdevraisréseauxdefournisseursd'Internet. Lesrésultatsnumériques montrentque(1)l'introdu tiondesin itationspourrendrelesutilisateursplusso iauxest unesolutione a epourobtenirdesréseauxstables et e a esd'unemanièredistribuée, et(2)l'appro heSta kelbergpeutproduireuneaméliorationimportantedesperforman es, en réantpresquetoujoursdesréseauxoptimauxd'unpointdevueglobal.

Mots- lés : Plani ation desRéseaux, So ialité,Théorie des Jeux,Equilibresde Nash, JeuxdeSta kelberg

1 Introdu tion

Networkdesignwithselshusershasbeenthefo usofseveralre entworks[1,2,3,4,5,6℄, whi h havemodeled howindependentselsh agents anbuild ormaintainalargenetwork bypayingforpossibleedges. Ea huser'sgoalisto onne tagivensetofterminalswiththe minimumpossible ost. Gametheoryisthenaturalframeworktoaddresstheintera tionof su hself-interestedusers(orplayers). ANashEquilibrium(NE)isasetofusers hoi es,su h thatnoneofthemhasanin entivetodeviateunilaterally. Forthisreasonthe orresponding networksaresaidto bestable.

However,Nashequilibriainnetworkdesigngames anbemu hmoreexpensivethanthe optimal, entralizedsolution. Thisismainlydueto thela kof ooperationamongnetwork users,whi hleadstodesign ostlynetworks.

A tually,themajorityofexistingworksassumethatusersare ompletelynon- ooperative. However,thisassumption ould benotentirelyrealisti , forexamplewhennetworkdesign involveslong-term de isions(e.g., in the aseof AutonomousSystems peering relations)

1 . Moreover,in entives ould beintrodu edby someexternal authority(e.g., theoverlay ad-ministrator)inorderto in reasetheusers' ooperationlevel.

Inthisworkweover omethislimitationbyrstproposinganovelnetworkdesigngame, the So ially-Aware Network Design (SAND) game, where users are hara terized by an obje tivefun tionthat ombinesbothindividual andso ial on ernsinauniedandexible manner. Morespe i ally,the ostfun tionofea huserisa ombinationofitsownpath ost (theselsh omponent)andtheoverallnetwork ost,whi hrepresentstheso ial omponent. A parameter (

α

) weights therelative importan e of the network ost with respe t to the userpath ost. Changingthevalueof

α

permitstotakeintoa ountdierentlevelsofso ial awarenessoruser ooperation.

We investigate systemati allythe impa tof ooperationamong network agentsonthe systemperforman e,throughthedeterminationofboundsonthePri eof Anar hy (

P oA

), thePri e of Stability (

P oS

)andthe Rea hable Pri e of Anar hy (

RP oA

) oftheproposed game. Theyallquantifythelossofe ien yastheratiobetweenthe ostofaspe i stable networkandthe ostoftheoptimalnetwork,whi h ouldbedesignedbya entralauthority. Inparti ularthe

P oA

,rstintrodu edin[7℄, onsiderstheworststablenetwork(thatwith thehighest ost),whilethe

P oS

[1℄ onsidersthebest stablenetwork(thatwiththelowest ost);nally,the

RP oA

onsidersonlyNashequilibriarea hableviabestresponsedynami s from the empty solution [6℄. Hen e,

P oA

and

RP oA

indi ate the maximum degradation dueto distributed usersde isions(anar hy),whilethe

P oS

indi ates theminimum ostto paytohaveasolutionrobusttounilateraldeviations. Ouranalyti alresultsshowthatas

α

in reases,i.e.,whenusersaremoresensitivetotheso ial ost,the

P oS

onvergesto

1

,i.e., thebeststablenetworkismoree ient,asexpe ted. Surprisingly,anoppositeresultholds fortheworst ase. Indeed,forlarge

α

values(highlyso ially-awareusers)theworststable network anbemu hmoreexpensivethanthenetworksdesignedbypurelyselshusers.

1

Thisobservationmotivatesin[4℄thestudyofstrongNE, onsidering oalitionsthat ouldtakede isions bene ialtoallthemembersofthegroup.

Forthisreason,wefurtherproposeaSta kelbergapproa h,theNetwork Administrator-Driven SAND game (NAD-SAND), whi h enablesverye ient Nash equilibria, avoiding worst- ases enarios: aleader(e.g., thenetwork administrator)buys anappropriatesubset of the network links (i.e., those belonging to the minimum ost generalized Steiner tree overingall sour e/destination pairs), indu ingthe followers(the network users)to rea h ane ientNashequilibrium.

Wemeasuredtheperforman eoftheproposed gamesin severalnetworktopologies, in- luding realisti s enarios where players build an overlay on top of real Internet Servi e Provider networks,and weobserved that so ially-aware usersalwaysgenerate better net-works. Furthermore,weobservedthattheproposedSta kelbergapproa ha hievesdramati performan eimprovementsin allthe onsidered s enarios,evenforsmall

α

values,sin eit leadsmostofthetimestotheoptimal(least ost)network.Hen e,introdu ingsome in en-tivestomakeusersmoreso ially-aware ouldbeanee tivesolutiontoa hievestableand e ientnetworksinadistributedway.

Insummary,themain ontributionsofthispaperarethefollowing:

ˆ the proposition of the So ially-Aware Network Design game, whi h ombines both individualandso ial on ernsin auniedandexiblemanner.

ˆ Thedetermination ofbounds onthePri e ofAnar hy,the Pri eof Stabilityandthe Rea hablePri e ofAnar hyoftheproposedgame.

ˆ ThepropositionofaSta kelberggamewherethenetworkadministratorleadstheusers toasystem-widee ientequilibriumbybuyinganappropriatesubsetofthenetwork links.

ˆ A thorough numeri al evaluation of the proposed games in several realisti network s enarios,in ludingrealISPtopologies.

Thepaperisorganizedasfollows: Se tion2dis ussesrelatedwork. Se tion3introdu es theproposed So ially-AwareNetwork Designgame. Se tion4proposesagreedyalgorithm that implements best response dynami s, whi h permits to rea h a Nash equilibrium in theproposed game. Se tion 5providespre ise bounds onthe Pri e ofAnar hy, the Pri e of Stability and theRea hablePri e of Anar hy for theSAND game. Se tion 6des ribes the proposed Sta kelberg game (the Network Administrator-Driven SAND game), whi h enablesverye ientNashequilibria. Se tion7presentsnumeri alresultsthatdemonstrate theee tivenessoftheSANDandNAD-SANDgamesinseveralrealisti networks enarios. Finally,Se tion8 on ludes thispaper.

2 Related Work

Severalre entworksfo usedonnetworkdesignwithselshusers[1,2,3,4,5,6℄.

Theso- alledShapley networkdesign game isproposedin[1℄. Inthisgame,ea hplayer hooses a path from its sour e to its destination, and the overall network ost is shared

amongtheplayersinthefollowingway: ea hplayerpaysforea hedgeaproportionalshare oftheedge ost,i.e.,theedge ostdividedbythenumberofplayersthatpassthroughsu h edge.

Of all the ways to share the so ial ost among the players, this proportional sharing methodenjoysseveraldesirableproperties. First,itisbudgetbalan ed,inthatitpartitions theso ial ostamongtheplayers. Se ond,it anbederivedfromtheShapleyvalue,andas a onsequen eistheunique ost-sharingmethodsatisfying ertainfairnessaxioms.Third,it admitspurestrategyNEs. Spe i ally,Anshelevi hetal. showedin[1℄thatapure-strategy Nashequilibriumalwaysexists,that

P oA = k

and

P oS = H

k

=

P

k

i=1

1/i = O(ln(k))

. Thenetworkdesignmodelpresentedin[2℄byAnshelevi hetal. isgeneralanddoesnot admitpure Nashequilibria, evenforverysimplenetworkinstan es. Briey, in su h model ea h player

i

hasaset ofterminalnodesthathemust onne t. A strategyofaplayeris a paymentfun tion

p

i

,where

p

i

(e)

ishowmu hplayer

i

is oeringto ontributetothe ost ofedge

e

. Any edge

e

su h that

P

i

p

i

(e) ≥ c(e)

is onsidered

bought

(

c(e)

beingthe link ost). Ea hplayer

i

triesto minimizeitstotalpayments,

P

e∈E

p

i

(e)

.

The authors in [3℄ haveextended the network design model in [1℄, in luding weighted players. But while easy to dene, this weighted network design game is hallenging to analyze. In parti ular, it is shown in [3℄ that: (1) pure-strategy Nash equilibria exist in all weightedShapley network designgames with

two

players, and (2) there are no larger lassesof weightedShapley networkdesign games that alwayspossesspure-strategy Nash equilibria.

The works in [4, 5℄ studythe existen eof strongNash equilibria(i.e., equilibria where no oalition animprovethe ostof ea h of itsmembers) in network design games under dierent ostsharingme hanisms. StrongNashequilibriaensurestabilityagainstdeviations byevery on eivable oalitionofagents. Morespe i ally,theauthorsin[4℄showthatthere aregraphsthat donotadmit strongNashequilibria, andthengivesu ient onditionson theexisten eofapproximatestrongNashequilibria.

Furthermore, the problem of designing a proto ol that optimizes the equilibrium be-havior of the indu ed network game is investigated in [6℄. The authors study the design of optimal ost-sharingproto ols forundire ted and dire ted graphs,single-sink and mul-ti ommoditynetworks,dierent lassesof ost-sharingmethods, and dierentmeasures of theine ien y of equilibria. Moreover,theyprovideupperand lowerbounds on thebest possibleperforman eofnon-uniform ost-sharingproto ols.

Few works have onsidered individual and so ial on ernsof user agents while dealing withdierenttypesofnetworkingproblems[8,9℄.

Apropositionthattakesintoa ountindividualandso ialbenetshasbeen onsidered in[8℄inthegeneral ontextofmulti-agentsystems,whereindividualandso ial on erns an oni t, leadingto ine ient system performan e. Toaddress su h problem, the authors haveproposedaformalde isionmakingframework,basedonso ialwelfarefun tions,that ombinesbothso ialandindividualperspe tives.

An experimental investigation of the impa t of ooperation in the ontext of routing games is ondu ted in [9℄. The game is studied onsidering parti ular network topologies

(i.e., parallellinks and loadbalan ing networks), sharedbyseveralusers. Ea h userseeks to optimizeeitherits ownperforman eorsome ombinationbetweenitsownperforman e andthatofotherusers,by ontrollingtheroutingofitsgivenowdemand.

However, unlikeour work,noneof the abovepapersapplies these on eptsto the net-workdesigngame,norprovidesatheoreti alanalysisofthee ien y ofthea hievedNash equilibria.

3 The So ially-Aware Network Design Game

In this Se tion we illustrate the proposed So ially-Aware Network Design (SAND) game, motivatingthereasontointrodu esu hmodel.

The SAND game o urs in a dire ted graph

G = (V, E)

, where ea h edge

e

has a nonnegative ost

c

e

,andea hplayer

i ∈ I = {1, 2, . . . k}

isidentiedwithasour e-sinkpair

(s

i

, t

i

)

. Everyplayer

i

pi ks apath

S

i

from itssour e to its destination, thereby reating thenetwork

(V, ∪

i

S

i

)

withtotal ostequalto

P

e∈∪

i

S

i

c

e

,whi hwillbereferredtoasso ial ost. Wewillrefertothepath

S

i

alsoasthestrategy hosenbyplayer

i

.

This so ial ostisassumedto besharedamongtheplayersin thefollowingway: let

x

e

denotethenumberofoverlaypathsthatgothroughoverlaylink

e

;ifedge

e

liesin

x

e

ofthe hosenpaths,thenea h player hoosingsu h anedge paysaproportionalshare

π

e

=

c

e

x

e

of the ost.

Theobje tivefun tion

J

i

thatuser

i

wantstominimizeisthereforegivenby:

J

i

₌

X

e∈S

i

π

e

+ α

X

e∈∪

j

S

j

c

e

(1)

Thersttermtakesintoa ounttheselsh natureofea hplayer,sin eitisthe ostfor user

i

tobuytheedgesbelongingtothe hosenpath,

S

i

;ontheotherhand,these ondterm representsthetotalnetwork ost(i.e.,theso ial ost),

α

beingaparameterthatpermitsto givemoreweighttoone omponentwithrespe tto theother.

Analternativeinterpretationofobje tivefun tion(1)isalsopossible: we anthinkthat usersare ompletelyselsh,but theso ial-awaretermin ostfun tion (1),

α

P

e∈∪

j

S

j

c

e

,is imposedbythenetworkoperator.Theadvantageofsu happroa histhattheoperatordoes notneedtosolvealarge-s aleIntegerLinearProgrammingprobleminordertooptimizethe routing hoi es forevery hangein the network (eitherin thetopology, links ost, players lo ationet ...), but anletthe userssolvethe problemin a distributed way, onverging toagoodandstablesolution.

Note that for

α = 0

obje tivefun tion (1) orrespondstothat of theShapleyNetwork Design Game proposed in [1℄, whi h represents a parti ular ase of our proposed game. Furthermore,inourgame,usersneedonlyalimitedamountofinformation,whi hisexa tly equaltothat oftheShapleynetwork designgame.

WeobservethattheSANDgameis apotentialgame withoutbeingat thesametime a ongestion game onthegivengraph

G

[10℄. Infa t,the ostin urredbyplayer

i

,

J

i

simplythe sumofthe ostsin urred bysu h playerfrom ea h link in the hosenpath

S

i

, butitin ludes afurtherterm,thetotalnetwork ost(multipliedbytheparameter

α

.)

It is easy to verify that the SAND game is hara terized by the following potential fun tion:

Φ(S) =

X

e∈E

x

e

X

x=1

c

e

x

+ α

X

e∈∪

j

S

j

c

e

(2)

S = (S

1

, S

2

. . . S

k

)

beingtheve torofplayers'strategies.

Asa onsequen e,su hgamehasatleastonepureNashequilibrium,namelythestrategy

S

that minimizes

Φ(S)

[11℄. Furthermore,in su h agame,best response dynami salways onvergetoaNashequilibrium.

Sin ewehaveintrodu edatermproportionaltothenetwork ostintheobje tive fun -tionofea hplayer,weexpe tthatplayersshould designbetternetworks,eventhough Se -tion5showsthatthisisnotne essarilythe ase. Beforeaddressingthisissue,wepresentin thefollowingSe tionapossibleasyn hronoususersoperationthatimplementsbestresponse dynami sandhen e onvergesto aNashequilibrium.

4 Best Response Algorithm

Wenowdes ribeasimplealgorithmthatimplementsbestresponsedynami s,allowingea h user to improve its ost fun tion in the proposed SAND game. Su h algorithm, detailed in thefollowing,is thebest response strategyforauserminimizingobje tivefun tion (1), assumingotherusersarenot hangingtheirstrategies.

Westill onsideradire tedgraph

G = (V, E)

,whereea hedge

e

hasanonnegative ost

c

e

.

Letus onsiderave torofplayers'strategies

S = (S

1

, S

2

. . . S

k

)

,where

S

i

⊂ E

isaset of edgesthat onne t thesour e-sink pair

(s

i

, t

i

)

. Letus denote by

S

−i

_{= ∪}

j6=i

S

j

the set ofedgesused byallplayersex eptplayer

i

; asa onsequen e,theset

E − S

−i

ontainsall linksthat are

not

usedbythesetofallplayersex ept

i

. Finally,let

k

e

denote thenumber oftimesthat edge

e

liesin thepaths hosenbysu hplayers(i.e., allplayersex ept player

i

).

Algorithm 1allows user

i

to determinethe best response to all other users'strategies. Therationalebehindsu halgorithmisverysimple: thelinkweightsaresetequaltothe ost in urredbyuser

i

to hoosethem, a ordingto obje tivefun tion (1). Then, theshortest path orresponds to the hoi e that minimizes the user's ost, and it representstherefore itsbest response. Notethat,ifmultipleshortestpathsexistwith thesame ost,theplayer hoosesoneofthemrandomly.

Algorithm1Pseudo- odespe i ationforthebestresponsedynami sfortheSANDgame 1: if

e ∈ S

−i

then 2: Set

Cost(e)

=

c

e

k

e

+1

3: else 4: Set

Cost(e)

=

c

e

· (1 + α)

5: endif

6: Computetheleast- ostpath(usingforexampletheDijkstraalgorithm)withlink osts

Cost(e)

7: Set

S

i

equalto theset oflinks ontainedintheleast- ostpath

5 Bounds on the Pri e of Anar hy, Pri e of Stability and Rea hable Pri e of Anar hy for the SAND game

InthisSe tion,wederiveboundsonthePri eofAnar hy(

P oA

),thePri eofStability(

P oS

) andtheso- alledRea hablePri eofAnar hy(

RP oA

)fortheSo ially-AwareNetworkDesign game,and omparethemwith theresultspresentedin [1℄ fortheShapleyNetworkDesign game. This allows us to determinethe worst and best ase performan e of ourproposed game.

5.1 Bound on the Pri e of Anar hy

We now establish a lower bound on the Pri e of Anar hy for the SAND game, whi h is dened as the ratio between the ost of the worst stable network (that with the highest ost),andthe ostoftheoptimalnetwork.

Proposition 1 Inthe SANDgame, alower boundonthePri eofAnar hy(

P oA

)isgiven bythe followingexpression:

P oA ≥ k(1 + α).

(3)

Proof: Letus onsiderthesimplenetworks enarioofFigure1withtwoparallel links, oneof ost equal to 1, the other with anarbitrarily high ost equalto

C

. Ea h of the

k

playersmust onne tthe ommonsour enode

s

tothe ommondestinationnode

t

. Inthe optimalout ome,ea hplayer

i

hoosesthelowerlink,with ost1,andthe ostoftheformed networkisobviously1.

However,thisisnottheonlyNashequilibriumfortheSANDgame. Letussupposethat theinitialnetwork ongurationseesall

k

usersroutedovertheupperlink,with ost

C

. It iseasytoshowthatif

α ≥

C

k

− 1

,nouserhasagaintodeviateand hoosethelinkwith ost equalto 1. Then,the ostofthenetworkis

C

. Nowif

C = k(1 + α)

,theaboveinequality issatised,and weobtainthatthePri eofAnar hyisat least

C

1

= k(1 + α)

.

t

s

1

C

Figure1: Two-linknetworktopology:

k

playersmust onne tthe ommonsour enode

s

to destinationnode

t

.

5.2 Bound on the Pri e of Stability

Inthefollowingwe omputeanupperboundonthePri e ofStabilityfortheSANDgame, whi h is dened asthe ratio between the ost of the best stable network (that with the lowest ost),andthe ostoftheoptimalnetwork.

Proposition 2 In the SAND game, the Pri e of Stability (

P oS

) is upper bounded by the following expression:

P oS ≤

H

k

+ α

1 + α

.

(4)

Proof: It is shown in [1, 11℄ that the fun tion

Ψ(S) =

P

e∈E

P

x

e

x=1

c

e

x

is an exa t potentialfun tion fortheShapleynetwork designgame revisedinSe tion II.Furthermore, itisshownin [11℄thatsu hfun tionsatisesthefollowinginequalities:

ost

(S) ≤ Ψ(S) ≤ H

k

ost

(S)

(5)

where ost

(S) =

P

e∈∪

j

S

j

c

e

and

H

k

=

P

k

i=1

1/i

isthe

k

thharmoni number.

Thepotentialfun tionoftheSANDgame,

Φ(S)

,isdire tlyrelatedtothatoftheShapley networkdesigngame,

Ψ(S)

. Infa t,from expression(2)it anbeeasilyseenthat

Φ(S) =

Ψ(S) + α ·

ost

(S)

. Hen e, if we repla e

Ψ(S) = Φ(S) − α · cost(S)

in expression (5), it followsthat:

(1 + α)

ost

(S) ≤ Φ(S) ≤ (H

k

+ α)

ost

(S).

(6) Finally, Theorem 19.13 in [11℄ states that if a potentialgame with potentialfun tion

Φ(S)

satisesthefollowinginequality:

ost

(S)

A

≤ Φ(S) ≤ B

ost

(S)

(7)

with

A

and

B

positive onstants,thenthePri eofStability(

P oS

)isatmost

AB

. Therefore,inourproblem,weobtainthatthePri eof Stabilityisatmost

H

k

+α

1+α

.

5.3 Bound on the Rea hable Pri e of Anar hy

Wenow derivea bound onthe so- alledRea hable Pri e of Anar hy (

RP oA

), aquantity dened in [6℄. The denominator of this ratio is the ost of the so ially optimal network, whi h wedenoteby

C

opt

;thenumeratoris thelargest ostofanequilibriumrea hablevia thefollowingpro ess: the

k

playersenter thegame one-by-one in anarbitrary order,and ea hpi ksapathofminimum ost(a ordingtoobje tivefun tion(1)),giventhe hoi esof previousplayers. Afterallplayershaveentered,thegamepro eedsexa tlyasfortheSAND game,withea hplayerre-optimizingitspath,giventhe urrentstrategiesofallotherplayers (usingforexamplethe best response algorithmillustratedin thepreviousSe tion). When thepro essrea hesaNashequilibrium(asitmust, sin eitisapotentialgame),itstops. Proposition 3 In the SAND game, the Rea hable Pri e of Anar hy (

RP oA

) is upper boundedbythe followingexpression:

RP oA < k

α + 1

α +

1

k

.

(8) Proof: Let

S

opt

i

denethepathfromsour enode

s

i

todestinationnode

t

i

forplayer

i

in theoptimalout ome(theleast ostnetwork),andlet

S

i

denotethepath hosenbyplayer

i

in itsrstmovein the game. Let

C(S

opt

i

) =

P

e∈S

_i

opt

c

e

and

C(S

i

) =

P

e∈S

i

c

e

denotethe ostsofsu htwopaths.

Fortherstplayerthatentersthenetwork,thefollowinginequalityholds:

X

e∈S

1

π

e

+ α

X

e∈S

1

c

e

= (1 + α)C(S

1

) < (1 + α)C(S

opt

1

).

Forthese ond player,thefollowinginequalityholds:

X

e∈S

2

π

e

+ αC(S

1

∪ S

2

) <

X

e∈S

opt

₂

π

e

+ αC(S

1

∪ S

2

opt

)

whi h analsobeexpressedasfollows,observingthat

C(S

1

∪ S

opt

2

) < C(S

1

) + C(S

2

opt

)

,and that

P

e∈S

opt

2

π

e

≤ C(S

opt

2

)

X

e∈S

2

π

e

+ αC(S

1

∪ S

2

) < (1 + α)C(S

opt

2

) + αC(S

opt

1

).

It anbefurtherobservedthat:

X

e∈S

i

π

e

≥

1

k

C(S

i

),

sin einthebestout ome,ea hofthelinksbelongingtothepath hosenbyplayer

i

isshared byall

k

players.

(α +

1

k

)C(

k

[

i=1

S

i

) < (1 + α)

k

X

i=1

C(S

i

opt

)

whi h anberewrittenasfollows,observingthat

P

k

i=1

C(S

opt

i

) ≤ kC

opt

:

(α +

1

k

)C(

k

[

i=1

S

i

) < (1 + α)kC

opt

.

Thisallowsus toobtainthefollowingresult:

C(

S

k

_i=1

S

i

)

C

opt

< k

α + 1

α +

1

k

.

(9)

Hen e, the ost of the network rea hed after the rst move of ea h of the

k

players,

C(

S

k

_i=1

S

i

)

, is no more than

k

α+1

α+

1

k

times the optimal ost,

C

opt

. Sin e subsequent best response movesperformed byplayersde rease thepotentialfun tion value

Φ(S)

givenby expression(2)(simulatingalo alsear honit),theplayerswill onvergetoaNash equilib-riumwhose ostis stillwithin

k

α+1

α+

1

k

timestheoptimum.

Therefore,we an on ludethattheRea hablePri e ofAnar hyisat most

k

α+1

α+

1

k

.

2

5.4 Comments

For

α = 0

the SAND game is equivalent to the original Shapley network design game. Indeed,expressions(3) and(4) onrmtheresultsalreadydemonstratedin [1℄:

P oA = k

and

P oS = H

k

. IntheSANDgame,for in reasing

α

values,theupperbound onthe

P oS

de reases, and tends to 1for

α → ∞

(then also

P oS → 1

). This anbeeasily explained sin e for

α → ∞

the so ial omponent of obje tive fun tion (1) is predominant. In this limiting behavior, all users share the same utility fun tion, whi h is the se ond term of expression(1); in su hasituation,theso ialnetwork optimum(i.e., thenetwork withthe minimum total ost)is obviously also aNash equilibrium, sin e the obje tivefun tion of ea h singleplayer oin ides withtheso ialnetwork ost(multipliedby

α

).

On the other hand, our lower bound on the

P oA

in reases when

α

in reases. This orresponds to the (quite ounter-intuitive) fa t that, in some ases, more so ially-aware users andesignlesse ientnetworks. Thishappensforexamplein thesimpleinstan eof Fig.1. Su h ine ien yisdueto themyopi de ision riterion: ea hplayeronly onsiders theee t ofitsown hoi e (i.e., hangingthesele tedpath),without onsidering eventual futurede isionsfromtheotherplayers.

However, it an be observed from expression (8) that the

RP oA

of the SAND game stri tly de reases for in reasing

α

values. Forlarge

α

values (notably, for

α → ∞

), the

Finally, oursimulationsshow thatthe Nashequilibria rea hed in theSANDgame are, inaverage, onsistentlybetterthanthosea hievedbytheShapleygame,aswewillshowin Se tion7.

6 The NetworkAdministrator-DrivenSo ially-Aware Net-work Design game

WenowillustrateavariationoftheSANDgame,namedtheNetworkAdministrator-Driven SAND(NAD-SAND) game,where anetwork administratorplaysbefore theusers,and his aimistodrivethemtothebest Nashequilibriumpossible.

Sin e omputingtheoptimalSta kelbergstrategyisNP-hard, wepresentin thispaper a simple strategy that a hieves onsistent performan e improvements. Su h approa h is implementedviathefollowingheuristi :

1. Given the network topology, the network administrator solvesa generalizedSteiner Treeproblem[12℄,determining theminimum- ostsubnetworksu h that thesour e/ destinationnodesofea hplayerare onne tedbyapath. Let

E

opt

bethesetofedges belongingtosu hoptimalsubnetwork.

2. Thenetworkadministrator hoosesalllinks belongingto

E

opt

,thus oeringto share eventuallytheir ostwiththeotherplayers. Therefore,usingthenotationintrodu ed inSe tionIII,afterthisstepwehave

x

e

= 1, ∀e ∈ E

opt

(thatis,thenetwork adminis-tratorhasalready hosenalllinks thatareoptimal fromaso ialpointofview). 3. At this point, all the

k

users play the SAND game, ea h trying to optimizeits own

obje tivefun tion,whi histhesameofexpression(1).

The rationale behind the proposed NAD-SAND game is the following: the network administratortriestomotivateallplayerstousethelinksthatbelongtotheso iallyoptimal solutionby sharing their ost with network users. Wewill show in the nextSe tion that su h heuristi isveryee tive, andpermitstoobtaindramati performan e improvements withrespe ttotheSANDgame.

WeobservethattherststepoftheNAD-SANDgameinvolvessolvinganNP-Complete problem. However,severale ientheuristi sandapproximationalgorithmshavebeen pro-posedtosolvesu hprobleminareasonable omputationtime[12,13,14℄. Inthenumeri al resultspresented in the nextSe tion we were able to ompute exa tly theminimum ost generalizedSteinertreeusingasimpleILPformulation,andsolvingitwiththeCPLEX 11 solver[15℄.

Finally,weobservethatas

α → ∞

,theNAD-SANDgamealwaysrea hestheminimum ostnetworksin eforea hplayerthe ostof hoosing anylinkthatdoesnotbelongtothe minimum- ostsubnetwork (i.e., to

E

opt

)hasanex eedingly large ost. As a onsequen e,

P oA → 1

as

α → ∞

.

Proposition 4 IntheNAD-SANDgame,alower boundonthe Pri eofAnar hy(

P oA

)is given bythe followingexpression:

P oA ≥

k

2(1 + α)

(10)

for

α ≤

k

2

− 1

.

Proof: Letus onsiderthenetworks enarioillustratedin Figure2,whereea hofthe

k

playersmust onne tthesour enode

s

i

tothe ommondestinationnode

t

. Link

ν → t

has a ostequalto

C

, with

2 ≤ C ≤ k

.

The minimum ost generalizedSteiner Tree is in this aseformed by alllinks

s

i

→ ν

(with ost zero) and by link

ν → t

, so that its ost is equal to

C

. Hen e, the network administratorwill hoosealltheselinks,whi h onstitutethe

E

opt

set.

Intheoptimalout ome,ea hplayer

i

hoosesthe

s

i

→ ν → t

path,andthe ostofthe formednetworkis

C

.

1

1

1

1

1

s

1

s

2

s

3

s

k−1

s

k

0

0

0

0

0

. . . .

. . . .

. . . .

C

v

t

Figure2: Parallellinktopology: bound onthePri eofAnar hyfortheNAD-SANDgame. TheNAD-SANDgame,however,admitsanotherNashequilibrium. Letussupposethat initiallyall

k

usersareroutedoverthedire tpath,

s

i

→ t

,withtotal ost

k

. It anbeeasily seenthatif

α ≤

C

2

− 1

,nouserhasagaintodeviateand hoosethe

s

i

→ ν → t

path,with ostequalto

C

.

Then,the ostofthenetworkis

k

. Nowif

C = 2(1 + α)

,theaboveinequalityissatised, andweobtainthat thePri e ofAnar hyisatleast

k

C

=

k

2(1+α)

.

7 Numeri al Results

In this Se tion, we report the results obtained by the proposed So ially-Aware Network Design(SAND)andNetworkAdministrator-DrivenSAND(NAD-SAND)gamesinrandom networktopologiesaswellasin real ISP topologies, omparing them both to theShapley networkdesigngame[1℄andtotheso ialoptimum. Tothisend,we onsiderbothrandomly generatednetworkinstan esandrealISPtopologiesmappedbytheRo ketfueltool[16,17℄. Randomnetworksareobtainedusinga ustomgeneratoraswellasadegree-basedgenerator (BRITE[18,19℄)to reatetopologieswithapowerlawdistribution ofnodedegrees.

Inall ases,westartfromtheemptynetworkandweapplyiterativelythebestresponse algorithmillustratedin Se tion4,untilaNashequilibrium isrea hed.

For ea h s enario we report the total network ost obtained by the proposed games for dierent

α

values, as well as the optimal network ost (the ILP olumn), obtained formulatingthegeneralizedSteinerTreeproblem[12℄withanIntegerLinearProgram(ILP), usingAMPL[20℄,andsolvingitwiththeCPLEXsolver[15℄. Solvingsu hproblemprovides theleast- ostnetworktopologythat onne tsallsour e/destinationpairs,thusrepresenting atermof omparisonforthee ien yoftheequilibriarea hedbyourproposedgames.

Finally,weevaluatednumeri allythePri eofAnar hyandthePri eofStabilityinallour simulationsettings,andwefoundthattheyare,respe tively,alwayslessthan1.78and1.56, hen e onsistentlylowerthantheboundsprovidedinSe tion V.

Full-Mesh Topologies

Werst onsiderfull-meshnetworktopologieswith50nodesrandomlydistributedona1000

×

1000squareareaand20players(sour e/destinationpairs). The ostofea hlinkisequal toitslength,andthenumeri alresults,averagedover20randomextra tions,arereported inTableI.

The SAND game permits to obtain network equilibria with a ost signi antly lower (approximately13%) than that obtained with the Shapleynetwork design game (i.e., the SANDrowwith

α = 0

),eventhoughthereisstillroomforimprovements,asdemonstrated bythegap existingbetweentheSANDgameandtheoptimal ost(theILP olumn).

This gap is lled bythe NAD-SAND game, whi h a hieves onsistently heaperNash equilibriaforall

α

values(uptomorethan30%),in ludingthe

α = 0

ase,thusrepresenting averyee tive approa h also when applied to the originalShapley network design game. Furthermore,when

α

issu ientlylarge(

≥ 10

),theNAD-SANDgamerea hesexa tlythe optimum(i.e.,thegeneralizedSteinerTree ost).

Random Topologies

To generate randomnetwork instan es, wehaveimplemented a topology generatorwhi h onsidersasquare areawith edgeequalto 1000, andrandomly extra tstheposition of

N

nodes,uniformlydistributedonthesquarearea.

Table1: Full-Meshtopologywith50nodesrandomlydistributedona1000

×

1000areaand 20players: average network ostsfortheSANDand theNAD-SANDgames. Theoptimal network ostisalso reported. The ostofea hlinkisset equaltoitslength.

Game

α = 0

α = 1

α = 10

α = 50

α = 100

α = 1000

ILP

SAND 6956.17 6124.62 6037.53 6043.82 6050.94 6046.66

4214.63 NAD-SAND 5718.86 4707.48 4214.63 4214.63 4214.63 4214.63

Asforthenetworklinks,whi h anbeboughtbyplayersto onne ttheirendpoints,we onsidertwoalternative hoi es:

ˆ Randomgeometri graphs: links existbetweenanytwonodeslo atedwithin arange

R

. Thelink ostisset equaltoitslength.

ˆ Randomnetwork modelUniform: agivennumberoflinks,

L

, isgeneratedbetween randomlyextra ted ouplesofnodes. The ostofea hlinkisuniformlydistributedin the0to

C

range(

C

beingthemaximumlink ost).

Givenarandomnetwork,20randomsele tionsof

k = 20

sour e/destination ouplesare onsidered.

Figure 3 shows thenetworks resulting at the Nash equilibria with the SAND game in arandom geometri graph s enariowith 50 nodes, range

R = 500

, 20 sour e/destination pairs,andtwodierent

α

values(viz.,

α = 0

and

α = 50

).

Weobservethatfor

α = 50

thetopologyismu h losertoatree-liketopologythanthat obtainedbytheShapleynetworkdesigngame(

α = 0

). Thisisree tedinthetotalnetwork ost,whi hisequalto7046.3for

α = 0

andto5336.2for

α = 50

,thusresultinginagainof morethan24%.

TableII illustratestheresultsobtainedinthesamerandomnetworks enarioofFigure 3, with 50 nodes, range

R = 500

and 20 sour e/destination pairs (players). The results areaveragedon20sour e/destinationrandomsele tions,andalsoon5randomtopologies. The Table reports the total ost of the network planned by the SAND and NAD-SAND games,aswellastheoptimalnetwork ost. Alsointhiss enario,theSANDgamea hieves improved equilibriawith respe t to theShapley network designgame (up to 13.2%). The NAD-SAND game outperforms the SAND game, further de reasingthe planned network ost,andrea hes,for

α ≥ 10

,theso ialoptimum.

Table III shows the numeri al results obtained using Random network model Uni-form in networks with 100 nodes,

L = 2000

links, maximum link ost

C = 100

and 20 sour e/destinationpairs. Thetotalnetwork ostsareillustratedin the Tablefor boththe SAND and NAD-SAND games, while the ILP olumn reports the optimal network ost. Althoughthere isstillroomforimprovement,inthis s enariothe SANDgameapproa hes theILPbound,and onsistentlyimprovesthequalityofnetworkequilibriawithrespe tto

Table 2: Random geometri graphs; random networks with 50 nodes,

R = 500

and 20 players: average network osts for the SAND and the NAD-SAND games. The optimal network ostisalso reported.

Game

α = 0

α = 1

α = 10

α = 50

α = 100

α = 1000

ILP

SAND 6567.73 6074.57 5708.95 5724.18 5736.17 5706.09

4213.82 NAD-SAND 5645.44 4675.13 4213.82 4213.82 4213.82 4213.82

Alsointhis ase,theNAD-SANDgameperformsbetterthantheSANDgame,lowering theoverallnetwork ostofmorethan

11%

,rea hingtheso iallyoptimalout omefor

α ≥ 50

.

Table3: Randomnetwork modelUniform; random networks with 100nodes,2000 links, maximum link ost

C = 100

, 20 players: average network osts for the SAND and the NAD-SANDgames. Theoptimalnetwork ostisalsoreported.

Game

α = 0

α = 1

α = 10

α = 50

α = 100

α = 1000

ILP SAND 314.69 294.18 296.38 295.09 295.57 296.48

263.25 NAD-SAND 295.85 274.38 263.70 263.25 263.25 263.25

Power Law topologies

Wefurther onsideredBRITE,adegree-basedtopologygenerator[18,19℄,whi hwasusedto generatepowerlawtopologies;theBarabási

−

Albert model[21℄wasadopted withdefault parametersprovidedbyBRITE.

Wegeneratedtwodierentpower-lawnetworks enarios:onewith100nodesandaverage node degree equal to 3, whi h ontained about 300 links; the other with 300 nodes and averagenodedegreeequalto5,whi h ontainedabout1500links. The ostofea hlinkwas setequaltoitslength.

TableIVreportsthe orrespondingnumeri alresults,in ludingtheoptimalnetwork ost obtainedsolvingtheILPmodel.

Inthis ase,theNashequilibriarea hedbytheShapleynetworkdesigngamearealready losetotheoptimum. Thisismainlyduetothefa tthatpowerlawtopologies,whi hresult in short hara teristi path lengthsand heavy lustering, arethe result ofso ial on erns thateventuallybringe ien ytothenetworkdesign(eventhoughnotexpli itlyexpressed intheunderlyingroutingproto olsdesign).

The SAND andNAD-SAND games, however, permit to improvethe e ien y ofsu h equilibria,nallyobtainingtheminimum ostnetworkprovidedbytheILPmodel.

Table4: PowerlawtopologiesgeneratedusingBRITE:averagenetwork ostsfortheSAND andtheNAD-SANDgames. Theoptimalnetwork ostisalsoreported.

100nodes Game

α = 0

α = 1

α = 10

α = 50

α = 100

α = 1000

ILP SAND 15980.20 15684.28 15596.31 15593.69 15593.43 15602.21 15314.14 NAD-SAND 15479.98 15314.14 15314.14 15314.14 15314.14 15314.14 300nodes Game

α = 0

α = 1

α = 10

α = 50

α = 100

α = 1000

ILP SAND 22580.82 22312.27 22224.65 22210.97 22210.55 22210.55 21927.41 NAD-SAND 22052.06 21987.68 21927.41 21927.41 21927.41 21927.41

Real ISP topologies

Finally, we onsider three real ISP topologiesmapped using Ro ketfuel [16, 17℄, listed in TableV,withanin reasingnumberofnodesandlinks. TableVIshowstheresultsobtained in su h topologies, that is, the total network osts as well as the optimal network osts obtainedsolvingtheILP model. Thelink ostsarethose providedby Ro ketfuel, and we performed20randomsele tionsof20sour e/destinationpairs.

Table5: Ro ketfuel-inferredISPtopologies: numberof networknodesand links. Network Lo ation Nodes Links

Telstra AU 108 306 Sprintlink US 141 748 Abovenet US 315 1944

In thesmall-size Telstra topology, the equilibria found by the SAND game (as well as those of the Shapley network formation game) are very lose to the optimum, whi h is rea hed bytheNAD-SAND gamefor

α ≥ 50

. As forthe otherISP topologies,theSAND gamediminishesthetotalnetwork osts,andapproa hestheoptimalILPsolution,whilethe NAD-SANDgame alwaysplans heapernetworks, a hieving theoptimal out omealready forrelativelysmall

α

values.

A nalinterestingpointthat wewould liketo mentionistheresilien eof theproposed network design game solutions to link failures. In parti ular, for the three ISP s enarios des ribedaboveweindividuatedthemost ongested link (i.e., theoneusedby thelargest numberof players), dropped it and re omputed newequilibria. It turns outthat thenew network ostsinallthe asesareatmost2%higherthanthe ostsoftheoriginalnetworks.

Table 6: Ro ketfuel topologies,20 sour e/destinationpairs: average network osts forthe SANDandtheNAD-SANDgames. Theoptimalnetwork ostisalso reported.

Network Game

α = 0

α = 1

α = 10

α = 50

α = 100

α = 1000

ILP Telstra SAND 165.55 164.55 163.15 163.10 163.10 163.10 157.95 NAD-SAND 163.25 160.70 158.55 157.95 157.95 157.95 Sprintlink SAND 286.25 282.60 280.90 280.30 280.30 280.30 255.60 NAD-SAND 270.30 263.35 256.85 255.65 255.60 255.60 Abovenet SAND 301.30 292.65 287.45 287.45 287.05 287.05 261.55 NAD-SAND 282.95 272.50 262.10 261.55 261.55 261.55 8 Con lusion

InthispaperweproposedtheSo ially-AwareNetworkDesigngame,anovelnetwork forma-tiongamewhereusersare hara terizedbyanobje tivefun tion that ombinesbothso ial andindividual on ernsinauniedandexiblemanner.

Westudiedthee ien yoftheNashequilibriaa hievedbytheproposedgame,providing boundsonthePri eofAnar hy,thePri eofStabilityandtheRea hablePri eofAnar hy. Ouranalyti alresultsshow thatwhen usersaremoresensitivetothe so ial ost, thebest stablenetworkismoree ient. However,anoppositeresultholdsfortheworst ase,sin e highly so ially-awareusers andesignstable networks that aremu h moreexpensivethan thenetworksdesignedbypurelyselshusers.

Forthis reason, wefurther proposed the Network Administrator-Driven SAND game, where the network administrator implements alink building strategy that drivesusers to thebest Nash equilibrium in termsofsystemperforman e, thus ar hite ting thedesired network.

We measuredtheperforman e ofour proposed games in several network s enarios, in- ludingrealISPtopologies,andweshowedhowtheyoutperform lassi alnetworkformation games(liketheShapleynetworkdesigngame),oftenobtainingtheso iallyoptimalout ome. Su hresultsdemonstratethatintrodu ingin entivestomakeusersmoreso ially-aware anbeaveryee tivesolutiontoa hievestableande ientnetworksinadistributedway. Furthermore, if in entives are deployed, the intervention of a network administrator an leadtodramati performan eimprovements,asdemonstratedbyourproposedSta kelberg game.

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[21℄ A.L.BarabásiandR.Albert.Emergen eofs alinginrandomnetworks.S ien e,pages 509512,vol.286,no.5439,1999.

0

100

200

300

400

500

600

700

800

900

1000

0

100

200

300

400

500

600

700

800

900

1000

(a)

α

= 0

0

100

200

300

400

500

600

700

800

900

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0

100

200

300

400

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600

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(b)

α

= 50

Figure3: NashequilibriaobtainedbytheSANDgameinarandomgeometri networkwith 50nodes,

R = 500

,20sour e/destinationpairs,anddierent

α

values(

α = 0

and

α = 50

).

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