Stability, Optimality and Complexity of Network Games with Pricing and Player Dropouts

(1)

STABILITY, OPTIMALITYANDCOMPLEXITY OF NETWORKGAMESWITH PRICING

AND PLAYER DROPOUTS

ANDREW LOMONOSOV

∗

AND MEERA SITHARAM

†

Abstrat. Westudybasipropertiesofalassofnonooperativegameswhoseplayersareselsh,distributedusersofanetwork

andthegame'sbroadobjetiveistooptimizeQualityofServie(QoS)provision. Thislassofgameswaspreviouslyintrodued

bytheauthorsandisageneralizationofwell-studiednetworkongestiongames.

TheoverallgoalistodetermineaminimalsetofstatigamerulesbasedonpriingthatresultinstableandnearoptimalQoS

provision.

Weshowthefollowing. (i)Standardtehniquesforexhibiting stabilityorexisteneofNashequilibriafailforthesegames

speially,neitherarethe utilityfuntions onvex, nordoesageneralized potential funtion exist. (ii)Theproblemofnding

whetheraspeigameinstaneinthislasshasaNashequilibriumisNP-omplete.

Toosettheapparentinstabilityofthesegames,weshowpositiveresults.(iii)Fornaturalsublassesofthesegames,although

generalized potential funtions donot exist, approximateNash equilibriado existand are easyto ompute. (iv) These games

performwellintermsofprieofstabilityandprieofanarhy. I.e.,alloftheseapproximateNashequilibrianearlyoptimizea

ommunal(orsoial)welfarefuntion,andthereisatleastoneNashequilibriumthatisoptimal.

Finally,wegiveomputerexperimentsillustratingthebasidynamisofthesegameswhihindiatethatpriethresholdsould

speeduponvergenetoNashequilibria.

Keywords. Congestiongames,Selshrouting,Atomiunsplittablemodel,NashEquilbria,Networkpriing

1. Introdution. Reentlymuhresearhhasbeendoneinapplyinggame-theoretioneptsandgeneral

eonomistehniquestoanalysisofomputernetworktra[2,3,5,10,11,12,16,14,20,21,24℄. Forageneral

surveysee[1℄. StabilityingamesreferstowhetherthegamereahesaNash equilibrium,astatewherenoplayer

hasinentiveto move. Optimality is a measure of how losea Nash equilibrium is to optimizing a soial or

ommunalwelfare funtion,usually thesumoftheindividualplayers'utilityfuntions.

We onsider primarily atomi games, where thenumber of players(network users)is nite. The ase of

non-atomi gameswhere there is an innitenumberof innitesimally small players is easierto analyze. For

similarreasons,spittablegames,wherenetworkusersansplittheirvolumeontomanyservielassesareeasier

toanalyzeandhavemoreorderlybehaviorthanunsplittablegames,whereeahuserisforedtoplaealltheir

volumeontothesamelass.

Theatomisplittablenetworkgamemodelhasbeenstudied[20,12℄,withearlyresultsinthetransportation

literature. Eieny(or optimality)ofNashequilibria inatomi splittablenetwork gameswasstudied in[24℄

and[28℄.

Hereweonsiderprimarilytheunsplittableasethathasalsobeenstudiedforsometime,forexample[26℄.

Most of the researh dealswith ongestion gameswhere payo to a playerdepends only on the player's

strategy and on the number of players hoosing the same strategy. Thanks to [26℄ it is known that suh

gamesalwayshaveNashequilibrium. Twoommontehniquesthat areusedtodemonstrateexisteneofNash

Equilbriaare thefollowing. Whentheplayerutilityfuntionsare onvex,Kakutani'sxed pointtheorem[25℄

diretlyshowsexistene. Alsowhensuhonvexitypropertiesarenotpresent,potentialfuntions,[18℄,ertain

funtionsthatinreaseaftereverymove,areusedtoshowexistene. Thesehavealonghistory,forexample,as

Lyapunovstabilityfuntionslassiallyusedtodesribeequilibriain dynamialsystems.

The[23℄networkgameshaverealistifeaturesthatmakethemsomewhatdierentfrom ongestiongames:

in partiular,playershavenon-onvexutility funtionsaused byathresholdoftotaltravolumein servie

lassesthat theyarewilling totolerate. Inadditionin the[15℄ games, theplayersareallowedto refrainfrom

partiipation,ortodropout,iftheirtraqualitydemandsarenotsatised. HeneexisteneofNashequilibria

orpotentialfuntionsisnotguaranteedforthese lassesofgames. However,wewereabletoshowexisteneof

Nashequilibriaforsomeoftheselassesofgamesbyonstrutinggeneralizedpotentialfuntions. (Generalized)

potentialfuntionshavealsobeenusedbyotherstostudyversionsofongestionandothergamese.g.,[7,21,22℄.

Forthelassesofgamesin[15,16℄weadditionallyshowedthattheNashequilibriaestablishedvia

general-izedpotentialfuntionsareeasytoompute. Ingeneral,however,whilepotentialfuntionsguaranteeexistene

ofNashequilibrium,theproblemofatuallyndingsuhanequilibriumremainsomputationallyhallenging.

∗

UGSIn.,10824HopeStreet,Cypress,CA90630USA(lomonosougs.om).

†

(2)

It has been shown [7℄ that the problem of nding Nash Equilbrium in ongestion games is PLS-Complete,

whihintuitivelymeansashard toomputeasanyobjetwhoseexisteneis guaranteedbyapotential

fun-tion.

ConsiderableresearhhasgoneintotheprieofanarhyandprieofstabilityofNashequilibria[27℄. These

notionsdesribehowfarorhowloseNashequilbriaanbeto theSystemOptimumof agame, wheresystem

optimumisaonguration(not neessarilyaNashequilibrium)thathasgreatestommunalwelfare.

We showedthat for the lassesof games with Nash equilibria in [15, 16℄, the ommunal welfare at these

equilibria waspoor, i. e., theyare far from thesystem optimum. Toretify this, we further generalizedour

lasses of games by introduing priing inentives (not to beonfused with the word prie in theprevious

paragraph). Theeet ofpriingonongestiongameshasalsobeenstudied in[9,6, 8℄. Ouroriginalgoalwas

to modify ouroriginal lassof games sothat theNash equilibriawouldbelose to systemoptima. However,

thepriedgameswereshowntonothaveNashequilibria,ingeneral. Weinsteadshowedthatthereistrade-o

betweengamestability(existeneofNashEquilbria)andommunalwelfareahievedbysuhgames. I.e.,while

thepriedgamesdidnotalwayshaveNashequilibria,theNashequilibria,whentheyexisted, werelosetothe

systemoptima.

This trade-o has sine beenformalized by examining approximate Nash equilibria i. e. states where no

playeranimprovetheirindividualwelfarebymorethanaertainfator,andthevalueofommunalwelfareat

suhapproximate equilibria [4℄. Forexample, [2℄demonstratedatradeo betweenwelfareand stability when

ostsfuntionsaresemionvex.

Inthispaper,ouroverallgoalistoanalyzeourlassesofrealistinetworkongestiongameswithrespetto

thesestabilityandommunalwelfaremeasures;investigatemehanismsforgamesto optimizethese measures;

andtoposeformalquestionsaboutthestrutureofgamelassesimposedbysuh measures.

Morespeially,theoriginallassesofgamesintroduedin[15℄were: thelass

Q

whereplayersweresolely

motivatedbytheirtraqualitydemandsandlasses

PQ

whereplayerswerealsoinunedbypriesimposed

on tra. Stability of games in

Q

was demonstrated by means of general potentialfuntions, and onrete

examplesofinstabilityof

PQ

werethengiven.

Inthis paper, weestablishthe NP-ompletenessof determining existeneofNash equilibriaand for

om-puting Nash Equilbria in

PQ

. We further study stability and ommunal welfare of (a modied version of)

approximateNashequilibriain

PQ

,asomparedtolass

Q

(i.e. eetofpriingonstabilityandsoialwelfare

inourgames).

We also briey look at game dynamis, i. e. numberof steps that it atually takesto onverge to Nash

Equilbria forsomeof ourgamesand ondut omputerexperimentsto studytrade-o betweenwillingness to

payandspeedofonvergene.

Setion 2 presents preliminary denitions, Setion 3 presents previous results on the lass

Q

of games,

Setion4presentsthemainresultsofthispaperonerningthelass

PQ

,andSetion5onludesbytabulating

andomparingtheresultsofSetions 3and4,followedbyopenproblems.

2. Denitions. Agame (instane)

G

inthebaselassofQoSprovisionnetworkgamesisspeiedbythe

game parameters

G

=

hn, m

∈

N,

{λi

∈

R

+

_{: 1}

_≤

_i

_≤

_n},

_{bi,j

_∈

_R

+

_{: 1}

_≤

_i

_≤

_n

_{; 1}

_≤

_j

_≤

_m},

_{pj

_:

_R

+

_→

R,

1 ≤

j

≤

m}i

. Thebestwaytodene

G

isbyidentifyingitwithitsnitegameongurationgraph(formally

dened below)whih onsists of aset of feasible gameongurations (verties)and the valid orselsh game

moves(orientededges). Thegame

G

isplayedby

n

users or players eah wantingto sendatraof

λi

units

throughoneof

m

network servielassesand (foronvenieneof analysis) anoveroworDummy Class with

index 0,referredtoasDC. Eah player

i

additionallyhasavolume threshold

bi,j

(tobedesribedbelow)for

eahlass

j

. Apriefuntion

pj

()

foreahservielassisanoninreasingfuntionthatmapsthetotal(tra)

volumeinthelasstoaunitprie. (Unitprietypiallydereaseswithinreasingongestionortotalvolumein

anyservielass). TheprieforusingDCis0. Afeasibleonguration

Λ

of

G

isfullyspeiedbyanalloation

J

Λ

:

{

1 , . . . , n} → {

1 , . . . , m}

whihdesribeswhihservielass

J

Λ(

i

)

that theuserorplayer

i

hasdeided to

plaetheirhunk

λi

oftra. This alloation

J

Λ

resultsin atotaltravolume

q

Λ

,j

=

P

i

:1

≤

i

≤

n

∧

J

Λ

(

i

)=

j

λi

in eahlass

1 ≤

j

≤

m

at thegame onguration

Λ

. Theset offeasible gameongurations

F

form thevertex

set ofthegame onguration graph

Ω

. Individual utilityfuntion

Ui

(Λ)

isatypeofstepfuntion basedon

i

's

volumethresholdbeingmetat theonguration

Λ

,and ontheunit prieinurred bytheplayer

i

inits lass

j

=

J

Λ(

i

)

.

Ui

(Λ)

is:

(3)

• −ǫ

,forsmall

ǫ >

0

if

bi,j

< q

Λ

,j

(volumethresholdexeeded)

•

equalto

λi

(1

−

pj

q

Λ

,j

)

otherwise.

Itisassumedthatthepriefuntionsarealwaysappropriatelynormalizedsothatthisquantityisalwaysstritly

positive forall players

i

and theirlasses

J

Λ(

i

)

at anyonguration

Λ

. Atypialutilityfuntion is shown on

Figure2.1. Wesaythatuser

i

issatisedatonguration

Λ

if

Ui

(Λ)

6 = 0

,andnotsatisedotherwise. Wedene

b

_i

Utility

Volume

Price

Fig.2.1. Utilityasafuntionofvolume,volumethresholdandprie

afuntion

Sat

Λ(

i

) = 1

if

U

Λ(

i

)

6 = 0

,otherwise

Sat

Λ(

i

) = 0

. Aselsh movebyuser

i

at aonguration

Λ1

isa

realloationof

i

's volume

λi

from adeparture lass

j

1

(i.e

J

Λ

1 (

i

) =

j

1

),toaadestinationlass

j

2

resultingin

aonguration

Λ2

(i.e,

J

Λ

2 (

i

) =

j

2)

thatinreasesutility ofthis user, i.e,

Ui

(Λ1)

< Ui

(Λ2)

. Movesto DC by

auserwhose volumethresholdis exeededarealleduser dropouts. Note thatuserdropoutsqualify asselsh

movesaordingtoourdenition.

Eah selshmoveisanordered pair offeasible gameongurations(for example

(Λ1

,

Λ2)

∈

F

×

F

),and

representsanoriented edge of thegame ongurationgraph

Ω

. A generalizedpotential funtion is afuntion

dened on ongurations that inreases after every player move. A game play for

G

is a sequene of valid

selshmovesin

G

, i.e

(Λ1

,

Λ2)

,

(Λ2

,

Λ3)

, . . . ,

(Λ

k

−

1 ,

Λ

k

)

,orapath in thegameongurationgraph

Ω

. ANash EquilibriumorNEof agame

G

isaonguration

Λ

suh thatthere isnoselsh movepossibleforanyuser

i

. Nashequilibriaare exatlysinkvertiesofagameongurationgraph

Ω

that haveno outgoingedges toward

other verties. For our lasses of games, the ommunal welfare funtion for onguration

Λ

is dened as

C

(Λ) = Σ

iSat

Λ(

i

)

λi

. Thefeasible gameongurationthat hashighestvalue ofommunalwelfarefuntion is

alled the System Optimum or SO. Let

Λ

N

bea Nash Equilibrium that has the smallestvalue of ommunal welfarefuntion takenoverallNashEquilibriums,while

Λ

M

beaNashEquilibriumthat hasthelargestvalue. Asdenedinsay[27℄aprieofanarhyofagameisequalto

C

(Λ

N

)

/C

(Λ

∗

)

,where

Λ

∗

isSO.Aprieofstability isequalto

C

(Λ

M

)

/C

(Λ

∗

)

.

Class of games that do nothave priing, i. e.

pj

(

x

) = 0

for alllasses

j

and theirvolumes

x

is denoted

by

Q

. In suh gamesplayers are motivated onlyby their desireto satisfy their volume thresholds. Sublass

Q

E

⊂ Q

is a lass of games with nopriing where all playershave equalvolume. Class of games that have

onlyonepriingfuntion

p

(

x

)

foralllasses

j

andthisfuntion isstritlydereasing(

p

(

x

)

< p

(

y

)

↔

x > y

)is denoted by

PQ

. Sublass

PQ

E

⊂ PQ

is alassof games withsinglestritly dereasingpriefuntion where allplayershaveequalvolume. Here wewillgiveapitorialexample,Figure2.2,ofsomenotionsintroduedin

thissetion. Agameongurationgraph

Ω

andongurations

Λ

ofapartiular game

G

areshown. Columns

representlasses, retangles representusers, thesize of a retangle orresponds to volume of a user, volume

thresholdsofusersareindiatedontheright. Inthisexamplethegame

G

inlass

PQ

has2lasses,2users

A

and

B

thathaveequalvolumesand thevolumethresholdof

A

is greaterthanthatof

B

. Gameonguration

graph

Ω

has4verties. Thisgame

G

hasnoNashequilibrium.

Throughout this paper we assume wlog that every player

i

has the same volume threshold

bi

=

bi,

1 =

bi,

2 =

. . . bi,m

ineverylass

j

= 1

. . . m

. Wealsoassumethatplayersaresortedintheinreasingorderoftheir

thresholds, i.e

b

1 ≤

b

2 ≤

. . .

≤

bn

. (Theformer assumption ould be easily generalizedfor all resultsin this

paper,thelatterassumptionisrealistiandommonlymade[23℄).

In proofs when desribing a game onguration

Λ

, we will speify values of game parameters

n

and

m

,

provide alist of usersin theform User(Volume, Volume Threshold)(forexampleA(5,12)meansthat UserA

(4)

B

b

A

b

B

b

A

b

B

b

A

b

B

b

A

b

B

1

2 DC

Configuration II

II

_III

IV

I

Configuration

graph

DC

1

2 A

Configuration I

1

2 DC

Configuration III

1

2 DC

Configuration IV

A

B

A

B

A

Fig.2.2. Gameongurationgraph andindividualongurations

3. Previously known properties of

Q

. Welistrelevantpropertiesof thelass

Q

ofgamesestablished

in[15℄onerningexistene,optimalityandomplexityofomputingNashequilibria.

Theorem 3.1. Every game in

Q

hasageneralizedpotentialfuntion andthereforeevery suhgamehas a

Nash Equilbrium.

Theorem 3.2. For any

ǫ >

0

there is agame in

Q

that has prie of anarhy andprie of stabilityequal

to

ǫ

.

Theorem 3.3. A NashEquilibrium thatisalsoaSystemOptimum ofagame in

Q

E

anbefound intime linearinthe game parameters.

Theorem 3.4. AnyNash Equilibriumofanygame

G

∈ Q

E

has ommunalwelfareof atleastahalfof that of

G

'sSystemOptimum.

Theorem 3.5. For any initial onguration of every game in

Q

E

there is a sequene of selsh moves by players that will terminate at Nash Equilibrium after

O

(

n

2 ₎

steps. This sequene an be determined by

onsidering players indereasing orderoftheir volume thresholdsandlettingthemmake theirselsh hoies.

4. Newresults. Inthissetionweonsiderstabilityofgamesin lass

PQ

andvariouspropertiesoftheir

Nashequilibria. Resultswill beomparedtothoseof

Q

inTable5.

We begin by establishingthe following simpleresult about the priesof anarhyand stabilityof general

gamesinthelass

PQ,

showingthattheyarenotpartiularlywellbehaved.

Theorem 4.1. Forany

ǫ >

0

thereisagame in

PQ

thathasauniqueNash equilibrium, whoseommunal

welfare is

ǫ

,whilethe systemoptimum of this game has ommunalwelfare equal to1. Thisimpliesthat pries

of anarhyandstabilityof suhagameareequal to

ǫ

.

Proof. Consideragamewithonenon-DClass,andtwoplayers,

A

(

ǫ,

1+

ǫ

)

and

B

(1

,

1)

. Theonlyequilibrium

this gamehasis whenplayer

A

is in lass1and player

B

is in DC, asopposed tothe systemoptimumwhen

theirpositions arereversed.

4.1. ApproximateNashequilibria. AswehavenotedintheIntrodutionandFigure2.2,Nash

equilib-riadonotneessarilyexistingames

PQ

thatinvolvepriing. Oneapproahto examiningsuhgamesinvolves

α−

approximate Nash equilibria, dened inforexample[4℄. Aongurationissaidto be

α−

approximateNash

equilibriumifnoplayeranmoveanddereaseherostbymorethanan

α

multipliativefator.

Notethatsinepriingfuntionsof

PQ

arearbitrarydereasinglinearfuntions,wewillinsteaduseamore

appropriatenotionof

δ−

approximateNash equilibriuminstead,where

δ

is anadditive fator.

Let

PQ

E

bethesubsetof

PQ

whereallplayershavevolume

ǫ

=

δ

. Insuhagameaongurationwhere all playersare satisedand all lasses haveequal total volume would be a

ǫ−

approximate Nash equilibrium,

sinenoplayerwouldhaveaninentiveto move.

When

ǫ

goesto zeroand numberof playersgoesto innity,thelass

PQ

E

willbedenotedas

PQ

∞

. This lassofgameshassimilarbehaviortothelassofgameswhereplayersareallowedtosplittheirvolumebetween

severallasses.

Theorem 4.2. ANash equilibrium(

δ−

approximateNashequilibrium) thatisalsosystemoptimum anbe

onstrutedfor any game in

PQ

∞

(

PQ

E

) intimeof

O

(

n

)

.

(5)

n

−

2

inlass 1if

bn

−

2 ≥

3 ǫ

et. Theresultingongurationis asystemoptimumanda

δ−

approximateNash

equilibrium.

NotethatwhilethepreeedingtheoremguaranteesexisteneofanapproximateNashequilibriumforgames

PQ

E

,itdoesnotpromisethateverysequeneofselshmoveswill arriveatanapproximateNashequilibrium.

Considerthefollowingobservation, whih also disprovesexisteneofgeneralpotentialfuntions forall games

in

PQ

E

. Thisisalsotrueforgamesin

PQ

∞

.

Theorem 4.3. Thereisagame in

PQ

E

where thereisayleof selshmoves. Proof. Let

δ

= 1

. Consideragamewith2non-DClassesand12players:

A

1(1

,

9)

, A

2(1

,

9)

, A

3(1

,

9)

, B

1(1

,

6)

, B

2(1

,

6)

, B

3(1

,

6)

, C

1(1

,

3)

, . . . , C

6(1

,

3)

.

Initialonguration

Λ

: players

C

4 , C

5

and

C

6

areinlass2,allotherplayersareinlass1. Firstplayers

B

1 , B

2

and

B

3

movetolass2,afterthat players

C

1 , C

2 , C

3

movetoDC, thenplayers

A

1 , A

2

and

A

3

movetolass2

andnallyplayers

C

1 , C

2 , C

3

movefrom DCtolass1. Theresultingongurationisessentiallyisomorphito

Λ

,heneaylehasourred.

NowwewillexaminepropertiesoforrespondingNashequilibria.

Theorem 4.4. Prie of anarhy of games in

PQ

∞

is equal to 1/2. Prie of stability of suh games is equal 1.

Ifprieofanarhyandprieofstabilitywereredenedover

ǫ

-approximateNashequilibriainsteadofregular

Nash equilibria, thenitwouldhold that prie ofanarhy ofgamesin

PQ

E

isequalto1/2andprieof stability ofsuhgamesisequal1.

Proof. Prie ofstabilityfollowsfrom thefat that Nashequilibriaonstrutedin Theorem4.2 aresystem

optima.

Prieofanarhyanbedemonstratedbyfollowingargumentforgamesin

PQ

E

,andtheprooffor

PQ

∞

is similar. Let

Λ

beaNashequilibriumwhenallplayershavethesamevolume

ǫ

. Considertheunsatisedplayer

i

that has thelargest volume threshold

bi

. (Ifthere areno unsatised playersthen suh a Nashequilibrium

isasystemoptimum). Totaltra volume

qj

in everylass

j

is stritlygreater than

bi

−

ǫ

, heneommunal

welfare of

Λ

is greater thanor equalto

m

(

bi

−

ǫ

)

but ommunal welfare ofsystem optimumannot bemore

than

2(

m

(

bi

−

ǫ

))

.

4.2. Finding a Nash equilibrium. It wasshownin [16℄ that the problem of nding systemoptimum

of a game in lass

Q

is NP-Complete. It was also shown that the problem of nding a Nash equilibrium in

Q

anbesolvedin

O

(

n

2 ₎

time. Similarlytheproblem ofnding asystemoptimum ofagame in lass

PQ

is

NP-Complete. NowwewillexaminetheproblemofndingaNashequilibrium(ordeterminingthatitdoesnot

exists)forgamesin

PQ

.

Theorem 4.5. Problem ofnding Nash equilibriumfor gamesin

PQ

isNP-Complete.

Proof. ConsiderthefollowingversionofMAXIMUM SUBSETSUM problemgivenset

S

=

{s

1 , . . . , sn}

andtargets

t

1 , t

2

,nd

A

⊆

S

suhthat

t

1 ≤

P

i

∈

A

si

≤

t

2

. Thisproblemanbereduedto problemofnding aNashequilibriumas follows. Thereare

n

+ 1

playersandtwonon-DC lasses. Players

1 , . . . , n

allhavesame

threshold

b

1 =

b

2 =

. . . bn

=

t

2

, individualvolumes

λi

=

si

. Player

n

+ 1

hasvolume

λn

+1

=

t

2

andthreshold

bn

+1

=

t

1 +

t

2

. Thenthis gamewill haveaNashequilibrium ifandonly iftheoriginalMAXIMUMSUBSET

SUMproblemhadafeasiblesolution.

4.3. Prie thresholds. In [16℄it wasshownthat games in lass

Q

will terminatein

O

(

n

2 ₎

steps, given

ertain assumptions on order of player moves. Here we will desribe aomputer experiment that examined

speedofonvergeneofgameswheretherewasnosuhorderingofplayermoves.

Thisexperimentinvolvedafollowingnaturalassumption aboutplayersbehavior. Inpratie,there ould

bealimitonhowmuhauseriswillingtopay,andthisoneptanbeeasilyaddedtoourgames,resultingin

thenewlassesofgames. Thisonepthasadesirableeetonthedynamisofthegame,asexplainedbelow.

Formally, for players

i

we dene prie thresholds (in addition to theold volume thresholds)

ti

that have the

followingproperty. Ifthe priein alassexeeds player

i

'spriethreshold, thenplayer

i

is notsatised. We

assumethat

bi

≤

bj

ifandonlyif

ti

≥

tj

,i.euserswhodemandbetterqualityofservie(smallertravolume

intheirlass)arewillingtopaymore.

(6)

Totestthisonjetureweranaomputerprogramsimulatingagameinlass

PQ

. Laterweaddedpriing

thresholds to the game whih has onsiderably improved time lapsed before onvergene to Nash equilibria.

Game parameterswere hosensuh that Nash equilibrium would alwaysexist. Parameters of thegame were

M

=

numberoflasses,

M/T

=

numberoftypesofusersthathavethesamevolumeandvolumethreshold,

K

=

numberofusersofthesametypethat an tinonelasswithoutexeedingtheirvolumethreshold. Volumes

were in inrementsofone,i.ethere are

T

∗

K

usersthathavevolume 1andvolume threshold

K

,

T

∗

K

users

that havevolume2andthreshold

2 K

,

. . . , T

∗

K

usersthathavevolume

M/T

andthreshold

M

∗

K/T

. Thus

thereareatotalof

M

∗

K

users. Forexamplelet

K

= 10

, M

= 20

, T

= 5

. Thismeansthatthereare20lasses,

4typesofusersandatmost10usersofanyonetypeantintoonelass. Usersare

A

1(1

,

10)

, . . . , A

50(1

,

10)

, B

1(2

,

20)

, . . . , B

50(2

,

20)

, C

1(3

,

30)

, . . . , C

50(3

,

30)

,

D

1(4

,

40)

, . . . , D

50(4

,

40)

.

Initially allusers are in the dummy lass (DC). A game proeeds by piking oneof the

M

∗

K

users at

random and this user moves either to the largestlass where his threshold would not beexeeded or to the

DC.Evenifthis moveexeedsthevolumethresholdof someotherusersinthedestinationlassofthemoving

user, these unsatised usersannot moveuntilit is their turn to moveand turns are determined at random.

EventuallyaNashequilibrium wasalwaysreahed,where allusersofthersttypewerein

T

lasses,allusers

oftheseondtypewereintheseondsetof

T

lasseset. Resultsareshownintable4.1. Moves1 denotesthe

totalnumberofusermovesuntil Nashequilibrium wasreahed.

Laterasimulationofpriingthresholdswasaddedto theexperiment. Eetivelyitwouldprohibitauser

i

that hasvolumethreshold

bi

to moveintoanylass

j

suhthat

qj

+

λi

< bi

−

∆

where

∆

is someonstant.

Thereasonforthisisthat lass

j

istooexpensiveforthe

i

th

user.

Table4.1

K M T Moves1

∆

Moves2

5 20 1 161,000 5 7,000

10 20 1 17,077,000 10 9,000

20 20 2 1,354,000 20 25,000

50 20 1 56,000 50 35,000

100 20 1 49,000 100 46,000

100 20 10 3,000 100 5,000

1000 20 10 35,000 1000 49000

5 40 1 2,360,000 5 190,000

5 50 1 8,391,000 5 940000

When

∆ =

∞

thisisequivalenttotheoldexperimentwithoutpriingthresholds. Ingeneralintrodutionof

small

∆

signiantlyimprovednumberofmovesthatwasneededtoreahtheNashequilibrium. SeeMoves2

inthetable4.1.

5. Conlusions, Diretions. HerewesummarizeknownresultsaboutNashEquilibriafor various

sub-lassesof

Q

and

PQ

.

NE/GenPotentialalwaysexists Prieofanarhy Prieofstability ComplexityofndingNE

Q

Yes/Yes

ǫ

O

(

n

2 ₎

Q

E

Yes/Yes 1/2 1

O

(

n

)

P Q

No/No

ǫ

NP-Complete

P Q

E

Yes/No 1/2 1

O

(

n

)

P Q

∞

Yes/No 1/2 1

O

(

n

)

(7)

ofgeneralizedpotentialfuntions fortheselassesisshownin Theorem4.3. Priesof anarhyandstabilityof

Q

are shownin Theorem3.2, of

Q

E

in Theorem3.4, of

PQ

in Theorem 4.1(assumingthatNash Equilibrium

exists), of

PQ

E

and

PQ

∞

in Theorem 4.4. Complexity of ndingaNash Equilibrium in gamesof lass

Q

is showninTheorem3.5,aseof

Q

E

isTheorem3.3,forgamesin

PQ

thisproblemisNP-Complete(Theorem4.5), forgamesin

PQ

E

and

PQ

∞

resultfollowsfromTheorem4.2.

5.1. Open questions. Thelass

PQ

ontainsbothgamesthathaveNashequibriaandthosewhodonot.

Whatisthestrutureofgamesinlass

PQ

whereNashequilibriaorapproximateNashequilibria(additive

ormultipliative)areguaranteedtoexistbut theyarehardtoompute? Forexample,aretherePLS-omplete

games in the lass

PQ

? For thesublasses suh as

PQ

E

Nash equilibria existeneis easy to determine, and (approximate) Nash equilibria are easy to ompute. Formally state and prove the onjeture of Setion 4.3

onerningtheusageofpriethresholdsandspeedofonvergenetoNashequilibria.

REFERENCES

[1℄ E.Altman,T.Boulogne,R.El-Azouzi,T.JimenezandL.Wynter,Asurveyonnetworkinggamesin

teleommuni-ations,Comput.Oper.Res.,pages286311,2006.

[2℄ E.Anshelevih,A.Dasgupta,J.Kleinberg,E.Tardos,T.WexlerandT.Roughgarden,ThePrieofStability

forNetworkDesignwithFairCostAlloation,FOCS'04:Proeedingsofthe45thAnnualIEEESymposiumon

FoundationsofComputerSiene(FOCS'04),pages295304,2004.

[3℄ B. Awerbuh, Y. Azar and A. Epstein,LargeThe prie of routingunsplittable ow, STOC'05: Proeedingsof the

thirty-seventhannualACMsymposiumonTheoryofomputing,pages5766,2005.

[4℄ H. Chen and T. Roughgarden, Network design with weighted players, ACM Symposiumon parallel algorithms and

arhiteture,pages29-38,2006.

[5℄ R. Cohi andD. Estrin andS. Shenker,Priing in ComputerNetworks: Motivation,IEEE/ACMTransationson

Networking,Vol.1,No.6,Deember1993,pages614-627.

[6℄ R.Cole,Y.DodisandT.Roughgarden,Priingnetworkedgesforheterogeneousselshusers,STOC'03:Proeedings

ofthethirty-fthannualACMsymposiumonTheoryofomputing,pages521530,2003.

[7℄ A.Fabrikant,C.PapadimitriouandK.Talwar,TheomplexityofpureNashequilibria,STOC'04:Proeedingsofthe

thirty-sixthannualACMsymposiumonTheoryofomputing,pages604612,2004.

[8℄ L.Fleisher, K.Jain andM. Mahdian,Tolls forHeterogeneousSelsh Usersin Multiommodity Networksand

Gen-eralized CongestionGames,FOCS'04: Proeedingsofthe45thAnnualIEEESymposiumonFoundationsofComputer

Siene(FOCS'04),pages277285,2004.

[9℄ L.Fleisher,Lineartollssue:newboundsandalgorithmsfortollsinsinglesourenetworks,Theor.Comput.Si.,(348),

pages217225,2005.

[10℄ D.Fotakis,S. C.Kontogiannis,E.Koutsoupias, M.Mavroniolas andP.G. Spirakis,TheStrutureand

Com-plexityofNashEquilibriaforaSelshRoutingGame,ICALP'02:Proeedingsofthe29thInternationalColloquiumon

Automata,LanguagesandProgramming,pages123134,2002.

[11℄ D.Fotakis,S.Kontogiannis,P.Spirakis,Selshunsplittableows,Theor.Comput.Si.,(348),pages226239,2005.

[12℄ Y.Korilis,A.Lazar,andA.Orda,Arhitetingnonooperativenetworks,IEEEJournalofSeletedAreasin

Communi-ations,(13)pages12411251,1995.

[13℄ E.KoutsoupiasandC.H.Papadimitriou,Worst-aseequilibria,SymposiumonTheoretialAspetsofComputerSiene,

pages404413,1999.

[14℄ ,Optimaltransportstrategiesforbest-eorttraoverpriedonnetions,Tehnialreport,Tehnion,2000.

[15℄ A.Lomonosov,M.SitharamandK.Park,StabilityvsoptimalitytradeoingametheoretimehanismsforQoS

provi-sion,SAC'03:Proeedingsofthe2003ACMsymposiumonAppliedomputing,pages2832,2003.

[16℄ ,NetworkQoSgames: stabilityvsoptimalitytradeo,J.Comput.Syst.Si.,(69),pages281302,2004.

[17℄ I. Milhtaih,Congestiongameswithplayer-speipayofuntions,GamesandEonomiBehavior(13),pages111124,

1996.

[18℄ D.MondererandL.Shapley,PotentialGames,GamesandEonomiBehavior(14),pages124143,1996.

[19℄ N.NisanandA.Ronen,Algorithmimehanismdesign,Pro.31stACMSymp.onTheoryofComputing,pages129140,

1999.

[20℄ A.Orda,R.Rom,andN.Shimkin,Competitiveroutinginmultiuserommuniation networks,IEEE/ACMTransations

onNetworking,(1),pages510521,1993.

[21℄ ,Computingorrelated equilibria inmulti-player games,STOC '05: Proeedingsof the thirty-seventhannual ACM

symposiumonTheoryofomputing,pages4956,2005.

[22℄ C. Papadimitriou and T. Roughgarden, Computing equilibria in multi-player games, SODA '05: Proeedings of the

sixteenthannualACM-SIAMsymposiumonDisretealgorithms,pages8291,2005.

[23℄ K.Park,M.Sitharam,andS.Chen,Qualityofservieprovisioninnonooperativenetworks:Heterogeneouspreferenes,

ProeedingsoftheFirstInt.Conf.onInformationandComputationEonomisICE'98,1998.

[24℄ K.Park,M.Sitharam,andS.Chen,Qualityofservieprovisioninnonooperativenetworkswithdiverseuser

require-ments, DeisionSupport Systems,SpeialIssue on Informationand Computation Eonomies, vol.28, pages 101122,

2000.

(8)

[26℄ R.W.Rosenthal,AClassofgamespossessingpurestrategynashequilibria,InternationalJournalofGameTheory,pages

6567,1973.

[27℄ T.Roughgarden,SelshRoutingandthePrieofAnarhy,TheMITPress,2005.

[28℄ T. Roughgarden andE. Tardos, How bad isselsh routing? IEEE Symposiumon Foundations ofComputerSiene,

pages93102,2000.

[29℄ ,MehanismdesignandtheInternet,PresentationinDIMACSWorkshoponComputationalIssuesinGameTheory

andMehanismDesign,2001.

Editedby: MarinPaprzyki,NiranjanSuri

Reeived: Otober1,2006