The Price of Anarchy for Network Formation in an Adversary Model

31 Pages·2011·0.4 MB·English

by  Lasse Kliemann

Games2011,2,302-332;doi:10.3390/g2030302
OPENACCESS
games
ISSN2073-4336
www.mdpi.com/journal/games
Article
The Price of Anarchy for Network Formation in an
Adversary Model
LasseKliemann
DepartmentofComputerScience,Christian-Albrechts-Universita¨tzuKiel,Christian-Albrechts-Platz4,
Kiel24098,Germany;E-Mail:[email protected];Tel.:[phone];
Fax:[phone]
Received: 27May2011;inrevisedform: 18July2011/Accepted: 8August2011/
Published: 23August2011
Abstract: We study network formation with n players and link cost α > 0. After the
network is built, an adversary randomly deletes one link according to a certain probability
distribution. Cost for player v incorporates the expected number of players to which v
will become disconnected. We focus on unilateral link formation and Nash equilibrium.
We show existence of Nash equilibria and a price of stability of 1 + o(1) under moderate
assumptions on the adversary and n ≥ 9. We prove bounds on the price of anarchy for
two special adversaries: one removes a link chosen uniformly at random, while the other
removes a link that causes a maximum number of player pairs to be separated. We show
an O(1) bound on the price of anarchy for both adversaries, the constant being bounded by
15+o(1)and9+o(1),respectively.
Keywords: network formation; equilibrium; price of anarchy; unilateral link formation;
adversarymodel;networkrobustness
1. NetworkFormation
In network formation, a multitude of individuals, called players, form a network in such a way that
each player decides for herself to which other players she would like to connect. So players can be
consideredvertices[1]ofa(to-be-built)network. Anyoutcomeofthis,i.e.,anynetwork,canbeevaluated
from the point of view of each player via an individual cost. Individual cost comprises building cost,
proportionaltothenumberoflinks[2]builtbytheplayer,andindirectcost,whichexpressespropertiesof
Games2011,2 303
thenetwork. Socialcost isthesumofindividualcostoverallplayers. Therearetwoparameters: nisthe
number of players and α > 0 is the cost of a link. Another crucial feature is how links can be formed:
unilaterallyorbilaterally. Underunilaterallinkformationaplayercanconnecttoanyotherplayerandis
charged theamountofα foreach link. Under bilateral linkformation, theconsentof bothendpointsis
requiredand iftheyboth agree,thenthey payα each. Whenthenetwork issothat noplayersees away
toimproveherindividualcost,wespeakofanequilibrium. Thefinerfacetsoftheequilibriumconcept
havetobechosenaccordingtothelinkformationrule: Nashequilibriumiswellsuitedforunilaterallink
formation, whereas for bilaterallink formation, pairwise Nash equilibriumor pairwise stabilityare better
suited. Whenthesocialcostisminimal,wespeakofanoptimum.[3]Thepriceofanarchy[4,5]measures
overallperformancelossduetodistributedoperation,comparedtowhenacentralauthoritywouldenforce
an optimum: the price of anarchy is the worst-case ratio of the social cost of an equilibrium to that of
anoptimum. Oneisinterestedinboundsonthepriceofanarchy,especiallyupperbounds. Thepriceof
anarchyisastaticmeasureinthesensethatitdoesnottrytoassesshowanetworkmightevolveovertime.
It instead builds upon the assumption that equilibrium networks will emerge from evolutionary processes.
Arelated conceptispriceof stability,i.e.,thebest-caseratioof thesocial costofan equilibriumto that
ofanoptimum. Thiswork’sfocusisonupperboundsonthepriceofanarchy,althoughweprovetight
boundsonthepriceofstabilityandsomestructuralresultsalongtheway.
OurContribution. Westudythepriceofanarchyinanadversarymodel. Afterthenetworkisbuilt,an
adversarydeletesexactlyonelinkfromit. Theadversaryismodeledbyarandomexperiment;hencein
generalthereisanuncertaintywhichlinkwillbedeleted,butplayersknowtheprobabilitydistribution
accordingtowhichtheadversarychoosesthelinktodestroy. Indirectcostofaplayerv isdefinedasthe
expected number of players to which v will lose connection when the adversary strikes. Formally, an
adversaryisamappingfromnetworkstoprobabilitydistributionsontheedgesoftheparticularnetwork.
Although it appears limiting that the adversary can only destroy one link, this model already is
challengingtoanalyze. Itisacontributiontotheunderstandingofhownetworksareformedwhenitis
importantthateveryvertexcanreacheveryothervertex,forexamplefordatatransmissionordelivery
ofgoods.
Afterpreparationsanddiscussionofrelatedwork(Sections2to5)westartoutwithasimpleO(1+ n)
α
bound on the price of anarchy for any adversary and independent of the link formation rule and the
equilibriumconcept,butundertheassumptionthatequilibriaonlyhavealinear(inn)numberofedges
(Section 6). This assumption will later be shown to be valid for the two special adversaries under
considerationandunilaterallinkformation. Inthethreeremainingsections(Sections7to9),weconsider
unilaterallinkformation. WeconstructivelyshowexistenceofNashequilibriaforalladversaries(under
somemoderateadditionalassumptions),includinga co-existenceof twoverydifferenttopologies forthe
samerangeofparameters. A1+o(1)boundonthepriceofstabilityfollowsfromourexistenceresults.
Thenfor twospecificadversaries weproveanO(1)boundon theprice ofanarchy forallranges ofα and
n ≥ 9. Thesetwoadversariesarechosentorepresentextremecases: thefirstone,calledsimple-minded,
choosesonelinkuniformlyatrandom. Thesecondone,calledsmart,choosesuniformlyatrandomfrom
thesetofthoselinkswhoseremovalcausesamaximumnumberofvertexpairstobeseparated. Theproof
techniquesforthesimple-mindedadversaryareroughlysimilartowhathasbeenusedforothermodels
Games2011,2 304
before,e.g., [6],namely werelatetothediameter ofequilibria. Forthe smartadversary,anewapproach
hastobetaken;itworksbyanappropriatedecompositionofthegraph.
OpenProblemsandFutureWork. Tightboundsonthepriceofanarchyforotheradversaries,orfora
generaladversaryareleftforfuturework. Asoneofthemostintriguingopenproblems,weleavethecase
ofanadversaryremovingmorethanonelink. Sinceourproofsrelyheavilyontherestrictionofonlyone
linkbeingremoved,thisisexpectedtobeanewchallenge.
This work’sfocus is onunilateral linkformation and Nashequilibrium. Bilateral linkformation with
its appropriate equilibrium concepts is planned to be studied in a separate publication. Currently, for
(cid:112)
bilaterallinkformationwecanshowaboundofO(1+ n/α)forthesimple-mindedadversary,ifα > 1.
2
However,itisnotknownwhetherthisboundistight. Forthesmartadversary,wecanshowatightΘ(n)
bound,ifα > 2consideredconstantandn ≥ 10.
2. ModelFramework
Wegivearigorousdescriptionofthemodelframeworkthatwillbeusedinthefollowing. Letn ≥ 3
(we will later refine this to n ≥ 9) and V a set of n vertices, say V = [n] := {1,...,n}. Each vertex
representsanindividual,calledaplayer. Eachplayernamesalistofotherplayerstowhichshewouldlike
tobuildanedge. Thedecisionsofplayerv arecollectedinavectorS ∈ {0,1}n,withS = 1meaning
v vw
that v would like to have the edge {v,w} in the network. Such an S is called a strategy for player v.
v
A vector of strategies S = (S ,...,S ), one for each player, is called a strategy profile. A strategy
1 n
profilecanbewrittenasamatrix{0,1}n×n andinterpretedastheadjacencymatrixofadirectedgraph
(cid:126) (cid:126) (cid:126)
G(S) = (V,E(S)). Then(v,w) ∈ E(S)ifandonlyifplayerv wouldliketohavetheedge{v,w}. We
willoftenworkwiththisrepresentationofstrategyprofiles. DenoteS(n)thesetofallstrategyprofiles
fornplayers. WeusesetsF ⊆ V ×V todenotestrategychanges. DefineS +F andS −F bysetting
forallx,y ∈ V
 
1 if(x,y) ∈ F 0 if(x,y) ∈ F
(S +F) := and (S −F) :=
xy xy
S otherwise S otherwise
xy xy
IfF = {(v,w)},wewriteS +(v,w)andS −(v,w). Forinstance,S +(v,w)meansthatweaddtoS
therequestofplayerv fortheedge{v,w}.
ThegraphwhichisactuallybuiltiscalledthefinalgraphanddenotedG(S) = (V,E(S)),where
E(S) := {{v,w}; S = 1∨S = 1}
vw wv
Sothewishofoneendpoint,eitherv orw,isenoughtohave{v,w}inthefinalgraph. Wealsocallthis
wayofformingthefinalgraphunilaterallinkformation(ULF).
Forbilaterallink formation(BLF),E(S) := {{v,w}; S = 1∧S = 1}are theedges of thefinal
vw wv
graph. Sobothendpoints, v andw,havetoagreeonhaving{v,w}inthefinalgraph. Otherwise, itwill
notbebuilt. BLFwillbeconsideredinfuturework,andweonlyconsiderULFhere(savesomeresults
thatholdindependentlyofthelinkformationruleandtheequilibriumconcept,cf.Section6).
Wespeakofselling(ordeleting,removing)anedgeeifaplayerchangesherstrategysothateisno
longerpartofthefinalgraph. Wespeakofbuying(oradding,building)anedgeeifaplayerchangesher
strategysothateisthenpartofthefinalgraph.
Games2011,2 305
Fix parameters nandα > 0. Given astrategy profileS ∈ S(n)each playerexperiences acostC (S),
v
her individual cost. It is comprised of building cost plus indirect cost. Building cost is computed by
countingαforeachedgethatv requested. Indirectcostcanbedefinedinmanydifferentwaysandusually
capturespropertiesofthefinalgraph,wedenoteitI (G(S))andsometimesjustI (S)forastreamlined
v v
(cid:80)
notation. Denoting|S | := S ,wecanwriteouttheindividualcostC (S) := |S |α+I (G(S)).
v w∈V vw v v v
An equivalent concept found in the literature is payoff: properties of the final graph are expressed by
income,andpayoffisincomeminusbuildingcost.
TheindirectcostI (·)isaplaceholdertobefilledininordertohaveaconcretemodel. Forexample,
v
(cid:80)
themodelin[6]usesI (G) = dist (v,w),wherethedistancedist(v,w)isthelengthofashortest
v w∈V G
pathbetweenv andw andequals∞ifthereisnosuchpath. Wecallthisthesum-distancemodel. The
priceofanarchyinthesum-distancemodelisparticularlywell-studied. Wewillintroduceourdefinition
ofindirectcostinSection3.
We call indirect cost anonymous if for each final graph G = (V,E) and each graph automorphism
φ : V −→ V ofG,wehaveI (G) = I (G)forallv ∈ V. Inotherwords,anonymityofindirectcost
v φ(v)
meansthatI (G)doesnotdependonv’sidentity,butonlyonv’spositioninthefinalgraphG. Thisisof
v
importanceinparticularifGhassymmetry. Forinstance,ifGisacycle,thenallverticesexperiencethe
sameindirectcost. IfGisapath,thenbothendpointsexperiencethesameindirectcost.
(cid:80)
Thesocialcost ofS isC(S) := C (S). Whenwesumupthebuildingcostoverallplayers,we
v∈V v
also speak of total building cost; when we sum up the indirect cost over all players, we speak of total
indirectcost. Hencesocialcostistotalbuildingcostplustotalindirectcost. AstrategyprofileS∗ iscalled
optimalifC(S∗) = min C(S). Thisiswithrespecttofixedα;denoteOPT(n,α)thesocialcostof
S∈S(n)
anoptimumforgivennandα. AnundirectedgraphGiscalledoptimalifG = G(S∗)foranoptimalS∗.
AstrategyprofileS iscalledessential[7]ifforallv,w ∈ V thefollowingimplicationholds:
S = 1 =⇒ S = 0
vw wv
Inotherwords,anessentialstrategyprofiledoesnotcontainunnecessaryrequests. Inanessentialstrategy
profile, building cost deserves its name in the following sense: players pay only for edges that would
not be in the final graph without this payment. This means that each edge in the final graph is paid
for α by exactly one of its endpoints, namely the one who requested it. Hence if S is essential then
(cid:80)
C(S) = |E(S)|α+ I (G(S)). Socialcostthenonlydependsonthefinalgraph.
v∈V v
ForeachstrategyprofileS,droppingallunnecessaryrequests[8]resultsinanessentialstrategyprofile
S(cid:48) withthesamefinalgraphandwiththesameorasmallerindividualcostforeachplayer. Moreover,it
iseasytoseethatifS isaNashequilibrium(introducedbelow),thenS(cid:48) isaNashequilibrium. Itishence
reasonabletorestricttoessentialstrategyprofiles,andwewilldosointhefollowing.
Recall thatwe can specifystrategyprofiles as directed graphs. Furthermore, since thesocial cost is
fullydeterminedbythefinalgraph(sincewerestricttoessentialstrategyprofiles),itsufficestoconsider
thefinalgraph(whichisanundirectedgraph)inplaceswhereonlythesocialcostisrelevant.
Astrategyprofile S iscalleda Nashequilibrium(NE)ifno playercanstrictlyimproveherindividual
costbychangingherstrategygiventhestrategiesoftheotherplayers,i.e.,
C (S +A−D) ≥ C (S) ∀A,D ⊆ {v}×V ∀v ∈ V
v v
Games2011,2 306
Denote the set of all NE for given n and α by NE(n,α). An undirected graph G is called a NE if there
exists a NE S such that G = G(S). The price of anarchy (with respect to NE) is the social cost of a
worst-caseNEdividedbythesocialcostofanoptimum,i.e.,
max C(S)
S∈NE(n,α)
OPT(n,α)
When wereplace “max” for“min”, wespeak ofpriceof stability. Both notions are meantrelativeto a
givennandα.
We call a NE S a maximal Nash equilibrium (MaxNE) if C (S +(v,w )+...+(v,w )) > C (S)
v 1 k v
forall{v,w },...,{v,w } (cid:54)∈ E(S). Thatis,weexcludethepossibilitythataplayercanbuyadditional
1 k
linkssothatthegaininherindirectcostandtheadditionalbuildingcostnullifyeachother. Thisnotionis
usefultocarryoverresultsforULFtoBLF,andwetreatithereforfuturereference.
2.1Remark. ANES ismaximal ifindirectcostI (S)hasits minimum possiblevalue forallplayersv
v
(whichis0formostmodels). ANEisalsomaximal,ifthereexistsε > 0suchthatitisstillaNEforlink
costα−εinsteadofα. Hence,ifS isaNEforallα ≥ f(n),forsomefunctionf,thisimpliesthatS isa
MaxNEforallα > f(n).
We require some basic graph-theoretic notions. Let an undirected graph G = (V,E) be given,
that is, V is a finite set and E ⊆ (cid:0)V(cid:1). A walk of length (cid:96) is a sequence of vertices W = (v ,...,v )
2 0 (cid:96)
such that {v ,v } ∈ E for all i ∈ [(cid:96)]. Denote V(W) := {v ,...,v } its vertices and
i−1 i 0 (cid:96)
E(W) := {{v ,v }; i ∈ [(cid:96)]} its edges. The walk is called a path if all its vertices are distinct,
i−1 i
that is, if |V(W)| = (cid:96) + 1. The walk is called a cycle if all its vertices except the last are distinct
(i.e.,|{v ,...,v }| = (cid:96))andthewalkisclosed(i.e.,v = v ). Sometimesweuseanotationthatgives
0 (cid:96)−1 0 (cid:96)
namestotheedgesinthewalk,like(v ,e ,v ,e ,...,e ,v ). IfC isacycleande = {u,w}isanedge
0 1 1 2 (cid:96) (cid:96)
withu,w ∈ V(C)bute (cid:54)∈ E(C),wecalleachord. ForasubsetW ⊆ V denoteG[W] := (W, (cid:0)W(cid:1)∩E)
2
theinducedsubgraphofW,orthesubgraphinduced byW. IfGisagraph,thenV(G)denotesitssetof
verticesandE(G)itssetofedges;thisisusefulwhenGwasnotintroducedwriting“G = (V,E)”. More
graph-theoreticnotionswillbeintroducedalongthewayasweneedthem.
One might suggest using multigraphs instead of graphs, since in our adversary model, connectivity
underremovalofedgesisrelevant. However,noneofourresultsbecomesfalsewhenweallowmultigraphs.
Wherenotobvious,aremarkonthisismade. Sowecansticktothesimplernotionofgraphs.
Inorder tonothaveto introducenamesfor alloccurringconstants, weuse“O”and “Ω”notation. For
ourresults,weusethisnotationinthefollowingunderstanding(itdoesnotnecessarilyapplytoallcited
results). Wewrite“x = O(y)”ifthereexistsaconstantc > 0suchthatx ≤ cy. Theconstantmayonly
dependonotherconstantsandisinparticularindependentofthenon-constantquantitiesthatconstitute
xandy,e.g.,parametersnandα. Wedonotimplicitlyrequirethatsomequantities,e.g.,n,havetobe
large. Analogously,wewrite“x = Ω(y)”ifthereexistsaconstantc > 0suchthatx ≥ cy. Notethat“O”
indicatesanupperbound, makingnostatementaboutalowerbound;while“Ω”indicatesalowerbound,
makingnostatementaboutanupperbound. Wewritex = Θ(y)ifx = O(y)andx = Ω(y);theconstants
usedinthe“O”andthe“Ω”statementmaybedifferent,ofcourse.
The“o”notationisonlyusedinoneform,namelyo(1)substitutingaquantitythattendsto0whenn
tendstoinfinity,regardlesswhetherotherparametersarefixedornot. Wheneverwewrite“o(1)”inan
expression,itismeantasanupperbound,makingnostatementaboutalowerbound.
Games2011,2 307
3. AdversaryModel
AnadversaryAisamappingassigningtoeachgraphG = (V,E)aprobabilitymeasurePrA onthe
G
edges E of G. Given a connected graph G, the relevance of an edge e for a player v is the number of
vertices thatcan, startingatv,onlybe reached viae. Wedenote therelevance byrel (e,v)andthe sum
G
(cid:80)
of all relevances for a player by R (v) := rel (e,v). An edge of a connected graph is called a
G e∈E G
bridge if its removal destroys connectivity, or equivalently, if it is no part of any cycle. The relevance
rel (e,v) is 0 iff e is not a bridge. Given a strategy profile S where G(S) is connected, we define the
G
individualcost ofaplayerv by
(cid:88)
C (S) := |S |α+ rel (e,v) PrA ({e})
v v G(S) G(S)
e∈E(S)
The indirect cost is the expected number of vertices to which v will lose connection when exactly one
edgeisremovedfromG(S)randomlyandaccordingtotheprobabilitymeasuregivenbytheadversaryA.
Forthisindirectcost,weusethetermdisconnectioncost inthefollowinginsteadof“indirectcost”. We
definetheindirectcosttobe∞whenG(S)isnotconnected,sowecanconcentrateonconnectedgraphs
inourstudyofoptimaandequilibria. Weusuallyomitthe“G(S)”subscriptsandalsothe“A”superscript;
wealsowrite“E”insteadof“E(S)”and“m”forthenumberofedges,i.e.,m = |E(S)|.
Remark. Since∞isassignedtodisconnectedfinalgraphs,optimaandNEareconnected.
Proof. Itis clearforoptima. ForNEnotethat sinceaconnected graphhasfinite indirect cost,a player
wouldalwayschoosetobuildenoughlinksinordertomakethegraphconnected.
Theseparationofanedgee,denotedsep(e),isthenumberoforderedvertexpairsthatwillbeseparated
bythe removal ofe. For abridgee,denoteν(e)thenumber ofvertices inthe componentofG−ethat
hasaminimumnumberofvertices;wehaveν(e) ≤ (cid:98)n(cid:99). Ifeisnotabridge,wedefineν(e) := 0. Then
2
(cid:80)
sep(e) = 2ν(e)(n−ν(e)) and also sep(e) = rel(e,v). If e is a bridge, then sep(e) ≥ 2(n−1).
v∈V
Wecanexpressthesocialcostnow:
(cid:88) (cid:88)(cid:88) (cid:88)
C(S) := C (S) = mα+ rel(e,v)Pr({e}) = mα+ sep(e)Pr({e})
v
v∈V v∈V e∈E e∈E
We call an adversary symmetric if for a fixed graph, the probability of an edge only depends on its
separation,i.e.,sep(e) = sep(e(cid:48))impliesPr({e}) = Pr({e(cid:48)})foralle,e(cid:48). Thefollowingpropositionis
provedstraightforwardly.
3.1Proposition. Asymmetricadversaryinducesanonymousdisconnectioncost.
Proof. Let G = (V,E) be connected and φ : V −→ V a graph automorphism of G. If e = {v,w}
is a non-bridge, then φ(e) := {φ(v),φ(w)} is a non-bridge as well, and so sep(e) = 0 = sep(φ(e))
and rel(e,v) = 0 = rel(φ(e),φ(v)) for all v ∈ V. Let e = {v,w} be a bridge and G , G be the two
1 2
components of G−e. Then φ(e) is a bridge as well. Let G(cid:48), G(cid:48) be the two components of G−φ(e).
1 2
Then φ(V(G )) = V(G(cid:48)) for all i ∈ {1,2}, or φ(V(G )) = V(G(cid:48)) for all i,j ∈ {1,2}, i (cid:54)= j. In either
i i i j
Games2011,2 308
case, sep(e) = sep(φ(e)), and also rel(e,v) = rel(φ(e),φ(v)) for all v ∈ V. In total, we have for all
e ∈ E andallv ∈ V:
sep(e) = sep(φ(e)) (1)
rel(e,v) = rel(φ(e),φ(v)) (2)
Letv ∈ V. Then:
(cid:88)
I (G) = rel(e,φ(v))Pr({e})
φ(v)
e∈E
(cid:88)
= rel(φ(e),φ(v))Pr({φ(e)}) φisbijective
e∈E
(cid:88)
= rel(e,v)Pr({φ(e)}) by(2)
e∈E
(cid:88)
= rel(e,v)Pr({e}) by(1)andsymmetricadversary
e∈E
= I (G)
v
The converse of Proposition 3.1 does not hold, as shown in Figure 1 on this page.
Proposition 3.1 is useful since symmetry can be recognized directly from the definitions of the two
specialadversariesstudiedlater,andsoweknowthattheyinduceanonymousdisconnectioncost.
Figure 1. Let this be the final graph G = (V,E) and G the subgraph to the left starting
1
with u (i.e., the cycle on 5 vertices), and G the subgraph to the right starting with w. Let
2
the adversary assign probabilities, say Pr({{u,v}}) := 2/3 and Pr({{v,w}}) := 1/3 and 0
to all other edges. Then all players in G experience the same disconnection cost, and the
1
sameholdsforG2. (Precisely,wehaveIx(G) = 2/3·6+1/3·5 = 4+5/3forallx ∈ V(G1)
and Iy(G) = 1/3 · 6 + 2/3 · 5 = 2 + 10/3 for all y ∈ V(G2) and Iv(G) = 2/3 · 5 + 1/3 · 5 =
5.) The adversary is not symmetric, since sep({u,v}) = 5 · 6 = sep({v,w}). However,
disconnectioncostisanonymous,sinceanautomorphismcanonlypermuteplayerswithinG
1
andG ,respectively.
2
2/3 1/3
u v w
4. RelatedWork
Thereisa vastbodyof literatureongame-theoreticnetwork formation,byfar notlimitedtostudies
ofthepriceofanarchy. AgoodstartingpointisthesurveybyJackson[12]from2004. Weciteseveral
publicationsbelowwithabiastowardsstudiesofthepriceofanarchy. Inaseparatesubsectiononpage311,
wegiveadetailedcomparisonofourmodelwithworkbeingparticularlyrelatedtoit,namely[13–18].
Games2011,2 309
Bilaterallinkformation(BLF)followsaconceptgivenbyMyerson[19](p.228)inadifferentcontext.
Jackson and Wolinsky [14] in 1996 introduced the symmetric connections model, using BLF, and the
equilibrium concept of pairwise stability. The symmetric connections model is best described using
the notions of income and payoff. The income for player v is (cid:80)w∈V δdistG(S)(v,w), where δ ∈ (0,1) is
w(cid:54)=v
a parameter. Her payoff is income minus building cost. Note that we have an exponential dependence
on distance. This models to some extent that each link has a probability of 1−δ for failure. We will
elaborateonthislater.
Pairwisestability(PS)isanequilibriumconceptsuitedforBLF. Essentially,itintroducesaminimum
of cooperation between players: a link not being in the final graph requires the additional justification
that building the link would be an impairment for at least one of its endpoints. On the other hand, PS
isonlyconcernedwithsingle-linkdeviations. JacksonandWolinskydiscussedseveralvariationsofPS,
includingwhatwouldlaterbeknownaspairwiseNashequilibrium(PNE),astrengtheningofPS.
Watts[20]in2001studiedthesymmetricconnectionsmodelwithanextendedequilibriumconcept:
a graph is considered stable if no player wishes to sell any link and if no two players wish to establish
anadditionallinkwhiledeletinganynumberoftheirlinks. Calvo´-Armengoland˙Ilkilic¸ [21]andCorbo
and Parkes [22] in 2005 discussed different equilibrium concepts and their relations: PNE, PS, and
properequilibrium[19].
BlochandJackson[23]in2007introducedamodelwithtransfers: eachplayerv decideshowmuch
she is willing to pay for a link {v,w} or how much she would demand the other endpoint w to pay for
the link. If v offers at least as much as w demands, or vice versa, the link {v,w} is established in the
finalgraph. Appropriateequilibriumconceptswereintroducedanddiscussed. BlochandJacksonalso
comparedPS,PNE,andtheirtransfermodelinaseparatepublication[24].
Bala and Goyal [18] in 2000 and in a unilateral setting studied a model where players wish to be
connected by a path to as many other players as possible, but path lengths are unimportant. They also
considered a unilateral version of the symmetric connections model. In another publication [9] in the
same year, they extended the first model by allowing each link to fail with a probability 1−p. Haller
andSarangi [16,17]in 2003extended thismodelagain byallowing eachlink{v,w}tofail withits own
probability1−p . Wewillelaborateonthislater.
vw
Anshelevich,Dasgupta, Tardos, andWexler [25]in 2003studied theprice ofanarchy andalgorithmic
aspects of a model in which each player has a set of terminals and aims to construct a network which
connects her terminals. For a related model, Anshelevich, Dasgupta, Kleinberg, Tardos, Wexler, and
Roughgarden[26] in2004studied thepriceofstability. Also in2004,Christin andChuang[27] studied
a model for network formation with an extended cost function modeling peer-to-peer networks, and
Christin,Grossklags,andChuang[28]lookedatitundertheaspectofdifferentgame-theoreticprinciples.
Chun,Fonseca,Stoica,andKubiatowicz[13]in2004experimentallystudiedanextendedversionof
thesum-distancemodel.
Johari, Mannor, and Tsitsiklis [29] in 2006 studied a model in which each vertex wishes to send a
givenamountoftraffictosomeofthe othervertices, andonlycareswhetherthetrafficeventuallyarrives
atthedestination. Thereisahandlingcostateachvertex,whichisproportionaltotheamountoftraffic
throughthatvertex.
Games2011,2 310
The work of Fabrikant, Luthra, Maneva, Papadimitriou, and Shenker [6] from 2003 is to the best
of the author’s knowledge the first quantitative study of the price of anarchy in a model that fits into
the framework considered here, as per Section 2. They considered the unilateral sum-distance model
√
and proved a bound of max{1, O( α)} on the price of anarchy in general, and an O(1) bound for
α > (n−1)n. Theyconjectured thatforα = Ω(1),all non-transientNEweretrees—the TreeConjecture.
2
ANEiscalled transient whenthere exists asequenceof strategy changesinwhich each playerchanging
her strategy maintains her individual cost, and finally a strategy profile is reached which is not a NE
anymore. TheTreeConjecture wasbasedontheobservationthatallNEconstructedsofaratthattime,
forα > 2,weretreesortransientones(namelythePetersengraphforα ≤ 4). TheTreeConjecturewas
later,in2006,disprovedbyAlbers,Eilts,Even-Dar,Mansour,andRoditty[30]byshowingthatforeach
(cid:112)
n0,thereexistsanon-transientNEonn ≥ n0 verticescontainingcycles,forany1 < α ≤ n/2.
CorboandParkes[22]in2005consideredthebilateralversionofthesum-distancemodel. Theyshowed
√
anO( α)boundfor1 ≤ α < n2 onthepriceofanarchy. Asnoticedlaterin2007byDemaineetal.[31],
√
theproofinfactyieldsO(min{ α, n/√α}).
Albersetal.[30]in2006notonlydisprovedtheTreeConjecture,butalsoimprovedtheboundsonthe
√
priceofanarchyfor theunilateralsum-distancemodel: they gaveconstantupper boundsforα = O( n)
andα ≥ 12n(cid:100)logn(cid:101),aswellasanupperboundforanyα of
15 (1+(min{α2/n, n2/α})1/3)
√
AnO(1)upperboundforα = O( n)wasalsoindependentlyprovedbyLin[32]. Theseboundswere
again improved by Demaine, Hajiaghayi, Mahini, and Zadimoghaddam [31] in 2007. They showed a
√
bound of 2O( logn) for any α and a constant bound for α = O(n1−ε) for any constant ε > 0. For the
√
bilateral version, they proved the O(min{ α, n/√α}) bound of Corbo and Parkes tight. Recently, in
2010, Mihala´k and Schlegel [33] proved that for the unilateral sum-distance model and α ≥ 273n, all
equilibriaaretrees,whichimpliesaconstantboundonthepriceofanarchyinthatrangeofα.
Moscibroda,Schmid,andWattenhofer[34]in2006studiedthepriceofanarchyinavariationofthe
sum-distancemodelwherethedistancebetweentwoverticesisgeneralized,thatis,itmaybegivenby
anymetric. Thecostfunctionusesthestretch,thatistheactualdistanceintheconstructedgraphdivided
bythedistancethatadirectconnectionwouldprovide. HaleviandMansour[35]in2007studiedtheprice
ofanarchyinthesum-distancemodelunderthegeneralizationthateachplayerhasalistof“friends”,that
is,alistofotherverticesandsheisonlyinterestedinherdistancetothose. Demaineetal.in[31]in2007
also considered the max-distance model: indirect cost for v is max dist(v,w). Upper bounds were
w∈V
shownforULFandtightboundsforBLF. ForULF,improvedboundswererecentlyshownin[33].
Brandes, Hoefer, and Nick [36] in 2008 studied a variant of the sum-distance model assigning a
finite distance to pairs of disconnected players, allowing for disconnected equilibria. They proved
structuralproperties andboundsonthe priceofanarchy. Laoutaris,Poplawski, Rajaraman, Sundaram,
andTeng[37]in2008consideredboundedbudgetconnectiongames,avariantofthesum-distancemodel
withplayer-dependentlinkcosts,lengths,andpreferencesw(u,v)expressingtheimportanceforplayer
u of having a good connection to player v, and finally a budget for each player limiting the number of
linksthatthisplayercanbuild. Theyconsideredexistenceofequilibriaandprovedboundsonthepriceof
anarchyandstability. Animportantspecialcaseistheuniformversion,whichhaslinkcosts,linklengths,
Games2011,2 311
and preferencesall equal, and also allplayers havethe same limit ontheir budget. Recently, this uniform
versionwasalsostudiedbyDemaineandZadimoghaddam[38]. Theyprovedatightupperboundand,
moreimportantly,showedhowtoinduceequilibriawithsmallsocialcost. Theyusedatechniquecalled
publicserviceadvertising,previouslystudiedfordifferentgamesbyBalcan,Blum,andMansour[39].
BaumannandStiller[15]in2008consideredthepriceofanarchyinthesymmetricconnectionsmodel.
Demaine et al. [40] in 2009 studied the price of anarchy in a cooperative variant of the sum-distance
model. Theyalsolooked atthecase thatlinkscan onlybe formedforcertain pairsofvertices, thatis, the
underlying“host”graphneedsnottobeacompleteone.
4.1. ComparisonofOurModelwithRelatedWork
Our adversary model addresses robustness in a way that, to the best of the author’s knowledge,
has not been studied theoretically before. We compare our approach to previous work that also
addressesrobustness.
Chun, Fonseca, Stoica, and Kubiatowicz [13] experimentally studied an extended version of the
sum-distance model and considered robustness. To simulate failures, they removed some vertices
randomly. Tosimulateattacks,theyremovedverticesstartingwiththosehavinghighestdegree.
The symmetric connections model of Jackson and Wolinsky [14] can also be interpreted from a
robustnesspoint-of-view. Recallthatinthe symmetricconnectionsmodelthereisaparameter δ ∈ (0,1),
andpayoffπ (S)forplayerv understrategyprofileS isdefined
v
(cid:88)
π (S) := δdistG(S)(v,w) −|S |α
v v
w∈V
w(cid:54)=v
Aninterpretationisthatv receivesoneunitofincomefromeachothervertexw alongashortestpath
betweenv andw. However,eachlinkhasaprobability1−δ offailure,sotheexpectedincomefromw is
the probability that none of the dist (v,w) links fails, which is δdistG(S)(v,w) if we assume stochastic
G(S)
independenceoffailures. ForBLF,BaumannandStiller[15]gaveanexpressionfortheexactpriceof
anarchyforα ∈ (δ−δ2,δ−δ3),whichimpliesanO(1)bound(theconstantbeingboundedby 4 ). The
1+2δ
priceofanarchyis1forα < δ−δ2,followingfrom[14]. Thepriceofanarchyintherangeα > δ−δ3 is
notfullyunderstoodyet.
Thesymmetricconnectionsmodelisdifferentfromoursinmanyrespects:
– Alllinkshavethesameprobabilityoffailure. Inourmodel,linkscanhavedifferentprobabilities,and
thesemayevendependonthefinalgraph.[41]
– The failure of a link e and the failure of a link f are independent events for e (cid:54)= f, at least along the
concernedpaths. Inourmodel,thefailuresofeandf aremutuallyexclusiveevents.
– Alternativepathsarenotconsidered; itisassumedthatroutinghappensalongaspecificshortestpath
that isfixed beforethe random experimentthat models thelink failuresis conducted. Inour model,all
pathsareconsidered. However,wedonotconsiderpathlengths.
BalaandGoyal[9]studiedavariationofthesymmetricconnectionsmodel,whichisclosertoours. In
theirmodel,eachvertexreceivesanamountof1fromeachvertexitisconnectedtoviasomepath. Each