Network games under strategic complementarities

(1)

HAL Id: hal-01474250

https://hal-amu.archives-ouvertes.fr/hal-01474250

Submitted on 29 Nov 2018

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Distributed under a Creative Commons Attribution| 4.0 International License

Network games under strategic complementarities

Mohamed Belhaj, Yann Bramoullé, Frédéric Deroïan

To cite this version:

Mohamed Belhaj, Yann Bramoullé, Frédéric Deroïan. Network games under strategic complementarities. Games and Economic Behavior, Elsevier, 2014, 88 (C), pp.310–319. �10.1016/j.geb.2014.10.009�.

�hal-01474250�

(2)

Network games under strategic complementarities

Mohamed Belhaj

^a

, Yann Bramoullé

^b

^,∗ , Frédéric Deroïan

^b

^,

¹

aCentraleMarseille(Aix-MarseilleSchoolofEconomics),CNRS&EHESS bAix-MarseilleUniversity(Aix-MarseilleSchoolofEconomics),CNRS&EHESS

a b s t r a c t

Keywords:

Networkgames

Strategiccomplementarities Centrality

UniquenessInterdependence

Westudynetwork gamesunder strategic complementarities.Agentsare embeddedina fixednetwork.Theychooseapositive,continuousactionandinteractwiththeirnetwork neighbors.Interactions arepositiveandactions areboundedfromabove.Wefirstderive newsufficientconditionsforuniqueness,coveringallconcaveaswellassomenon-concave bestresponses.We thenstudytherelationshipbetweenpositionand actionandidentify situationswhere amore central agent alwaysplaysa higheraction inequilibrium.We finallyanalyzecomparativestatics.We showthatashockmaynotpropagatethroughout theentirenetworkanduncoverageneralpatternofdecreasinginterdependence.

1. Introduction

Inthispaper,westudynetworkgamesunderstrategiccomplementarities.Agentsareembeddedinaﬁxednetworkand choose a positive, continuous action. We assume that actions are bounded fromabove. Agents play a game of strategic complementswiththeirnetworkneighbors.Anagent’sactionisstrictlyincreasinginherneighbors’actionsuntilitreaches its upperbound.Thismaycapture, forinstance,peers’inﬂuenceincriminalactivityorarms racebetweencountries.² We do notimpose furtherassumptions onthe shapeofthebest response.It couldbe linear,concave,convexorofchanging curvaturebelowtheupperbound.OurmainobjectiveistoanalyzetheNashequilibriaofthesenetworkgames.

Wehavethreemaincontributions.First,wederivenewsufficientconditionsforuniqueness.Ourresultsprovidethefirst applicationofageneralprincipleoflocalizedcontraction.Insupermodulargames,contractionoftheiteratedbestresponse for ordered profiles lying betweenthe extremal equilibriaguarantees uniqueness. We show how to apply thisprinciple undervariouscurvatureassumptionsonthebestresponses.Wefindthatlocalizedcontractionalwaysholdsunderconcavity.

UniquenessinthatcasefollowsfromtheresultsofKennan (2001).Ourapproachprovidesanalternativeproofbasedonvery different premises.Under convexity,we provethat localizedcontraction holdswhenthe bestresponseis nottooconvex.

Andweobtainasuﬃcientuniquenessconditionforarbitrarybestresponses.

Second,westudytherelationshipbetweennetworkpositionandactioninequilibrium.Weﬁrstshowthatmorecentral agentsmayplayaloweraction.Wethenidentifytwosituationswhereactionisunambiguouslyrelatedtoposition.Anagent

*

Correspondingauthor.

E-mailaddress:[email protected](Y. Bramoullé).

1 WethankFedericoEchenique,MattJackson,threeanonymousrefereesandparticipantsofthe2012CoalitionTheoryNetworkandthe2013Journées Louis-AndréGérard-Varetforhelpfulcommentsanddiscussions.

2 WediscussthesetwoapplicationsinmoredetailinSection2below.

(3)

who hasfewer neighbors thananother inthe sense of setinclusion necessarilyplays alower action inany equilibrium.

Whenthe graphissymmetricandstructuredaround ageometric center,liketheline, actionisalignedwithcentralityin extremalequilibria.Theseresultsholdforanycurvatureofthebestresponsebelowtheupperbound.

Third,welookatcomparativestatics.Classicalresultsfromthetheoryofsupermodulargames tellusthatactioninan extremalequilibriumis weaklyincreasing inactions inisolation, upperboundsandlinkstrengths. Wethen studyhowa shockpropagatesthroughoutthenetwork.Wecharacterizepreciselywhoendsupbeingaffectedbyapositiveshockonone agent.Thepresenceandpositionsofbridgesbetweencommunitiesturnouttobecritical.Whenbridgingagentsreachtheir upperbounds,theystoptransmittinginﬂuenceacrossdifferentpartsofthenetworksandinterdependencemaybeseverely reduced.

Ouranalysiscontributestotheliteratureongamesplayedonfixednetworks. Alargepartofthisliterature todatehas studied games withlinear bestresponses.³ Ballester et al. (2006) studynetwork games underlinear best responses and smallnetworkeffects.TheyfindthattheequilibriumisuniqueandthatactionisalignedwithBonacichcentrality.However, noequilibriumexistsunderstrategiccomplementsandlargenetworkeffects.Agents’actionsfeedbackintoeachotherinan explosivewayanddivergetoinfinity.Thisdivergenceisunrealisticincontextswhereactionsarenaturallybounded.Intheir empiricalimplementationonpeereffectsandacademicachievement,Calvó-Armengolet al. (2009)discussthispossibility:

“Letusboundthestrategyspaceinsuchagamerathernaturallybysimplyacknowledgingthefactthatstudentshavea timeconstraintandallocatetheir timebetweenleisureandschool work.Inthatcase,multipleequilibriawillcertainly emerge,whichisaplausibleoutcomeintheschoolsetting.”,Calvó-Armengolet al.(2009,p. 1254)

We disprove their conjecture forpositive linear interactions. We extendBallester et al. (2006)’s analysisto situations withlargepositive networkeffectsandboundedactions.Weshow thatuniqueness holds.Wealsoshow thatequilibrium propertiesdifferquitedeeplyforsmallandlargeinteractions. Undersmallinteractions, amorecentralagentalways plays a higheraction andashock onone agent ends upaffecting every other agent inconnected networks. Incontrastunder large interactions,a morecentral agentmayplay a loweraction. Andan individual shockmaynotpropagate throughout thenetwork.

Tworecentpapers studynon-linearbestresponses innetworkcontexts. Hiller (2012)andBaetz (forthcoming) analyze networkgameswithhomogeneous concavebestresponses.TheyestablishuniquenessbyapplyingKennan (2001)’sresults.

Both papers then look atnetwork formation. In contrast,we consider ﬁxed networks andheterogeneous best responses here. We alsoshow that Kennan (2001)’sresults apply.We providean alternative approach, coveringconcave aswell as importantcasesofnon-concavebestresponses.Wetheninvestigatestructuralpropertiesoftheequilibria.

Ourstudyalsocontributestotheliteratureonsupermodulargames.⁴ Weprovideoneoftheﬁrstcrossoversbetweenthe theoriesofnetworkgamesandofsupermodulargames.Galeottiet al. (2010)studynetworkgamesunderstrategiccomple- mentswhenagentshaveincompleteinformationonthenetwork.Weanalyzeagameofcompleteinformationhere.⁵ Belhaj and Deroïan (2010) studycommunication efforts under strategic complements and indirect network interactions, when agentsplayonspeciﬁcsymmetricnetworks.Incontrast,weanalyzedirectnetworkinteractionsandarbitrarynetworks.Our analysisnotablyappliestotwoimportantcontexts:criminalactivityandarms race.Wecombineclassicalresultsfromthe theoryofsupermodulargameswithnetworkargumentstounderstandstructuralfeaturesoftheequilibria.

The restofthe paperisorganized asfollows.We presentthemodelanddiscussapplications inSection 2.We derive new suﬃcient conditions for uniqueness in Section 3. We study the relation between network position and action and comparativestaticsinSection4.WeconcludeinSection5.

2. Themodel

We considerthe followingnetwork games ofstrategic complements.Societyis composed ofn agents embedded ina ﬁxednetwork,representedbyann

×

ⁿ^matrix^G.^The

(

i

,

j

)

thentrygi j isanon-negativerealnumberrepresentingthelink between i and j. We consider an arbitraryweighted network. Formally gi j

∈ R

+ and, by convention, gii

=

^0.^Agent ⁱ ^is directlyaffectedbyagent j ifg_{i j}

>

0 andg_{i j} thenmeasuresthestrengthoftheirrelationship.Wedonotimposesymmetry so gi j coulddifferfrom gji.

Agentschooseactions xi

∈ [

⁰

,

bi

]

^and^play^a^game^with^best^response

f_i

(

x₋_i

) =

^min

a_i

+ ϕ

i

_n

j=1

g_{i j}x_j

,

b_i

(1)

wherebi

>

0 denotestheupperboundoni’saction;aidenotesi’soptimalactionabsentsocialinteractionswith0

<

ai

<

bi; and

ϕ

iisacontinuouslydifferentiablefunctionfrom

R

+into

R

+suchthat

ϕ

i

(

0

) =

^{0 and}

ϕ

_i

>

0.Thefunction

ϕ

i captures

3 See,forinstance,Ballesteret al. (2006),BelhajandDeroïan (2013),BramoulléandKranton (2007),Bramoulléet al. (2014),Calvó-Armengolet al. (2009).

4 See,forinstance,MilgromandRoberts (1990)andVives (1990).

5 However,weobservethatanagentonlyneedstoknowhisneighbors’actionstobeabletoplayabestresponse.

(4)

theimpactofneighbors’actionsbelowtheupperbound.ANashequilibriumofthegameisaproﬁleofactionsxsuchthat

∀

ⁱ

,

x_i

=

^fi

(

x₋_i

)

.

Ourmodelhasthreedistinguishingfeatureswithrespecttotherecentliteratureonnetworkgames.First,we focuson strategiccomplements.Anagent’sbestresponseisweaklyincreasinginothers’actions.⁶ Second,actionsareboundedfrom above byexogenouslygivenupperbounds.Asshownbelow,thishelpsguaranteeequilibriumexistence.Third,interactions neednotbelinearorconcave.Thefunctions

ϕ

i couldbelocallyorgloballyconvex.

Wecanapplyclassicalresultsofthetheoryofsupermodulargamestostudygameswithbestresponses(1).Toseewhy, considerthegame

Γ

withstrategiesx_i

∈ [

⁰

,

b_i

]

^and^payoffs

u_i

(

x_i

,

x₋_i

) =

^aix_i

−

¹ 2x²_i

+

^xi

ϕ

_n

j=¹ g_{i j}x_j

(2)

Note that

[

⁰

,

b_i

]

îs â ^complete ^lattice ând

∂

²u_i

/∂

x_i

∂

x_j

=

^gi j

ϕ

(

n

j=¹g_{i j}x_j

) ≥

^0,

∀

ⁱ

=

^j. ^Therefore,

Γ

is supermodular (MilgromandRoberts,1990).Inparticular,

Γ

alwayshasasmallestandalargestNashequilibrium.AndsinceNashequilib- riaonlydependonthebestresponses,thispropertyholdsforanygamewithbestresponses(1).Tosumup,

Corollary1.(SeeMilgromandRoberts,1990.)Anygamewithbestresponse(1)hasasmallestandalargestNashequilibrium.

Suchpositivenetworkinteractionsmayemergeinavarietyofcontexts.Wenextdescribetwoexamplesinsome detail:

crimeandarmsrace.Researchershavedocumentedthepowerfulroleplayedbysocialinteractionsindeterminingcriminal activity (Glaeser et al., 1996; Bayer et al.,2009). Building onCalvó-Armengol andZenou (2004), Liuet al. (2014) model choicesofcriminaleffortsinadelinquentnetworkasNashequilibriaofagamewithpayofffunctions

vi

(

xi

,

x₋i

) =

^aixi

−

¹

2x²_i

+ δ

i

gi jxixj

whereeffortisassumedtobenon-negative,xi

∈ R

+.Hereai

>

0 includesindividualdeterminantsofcriminalactivitywhile

δ

ig_{i j}x_ix_jcapturestheinﬂuenceofpeers’behaviors.Inthiscase,individualbestresponseissimplylinear:

fi

(

x₋i

) =

^ai

+ δ

n

j=1

gi jxj (3)

Linear networkgames havebeenanalyzedinBallesteret al. (2006) andBallesterandCalvó-Armengol (2010).Observe thatifxisanequilibrium,wehavex

=

^a

+ δ

Gxandhence,throughrepeatedsubstitutions,

x

=

^a

+ δ

Ga

+ δ

²G²a

+ · · · + δ

^tG^ta

+ δ

^t⁺¹G^t⁺¹x

foranyt

∈ N

^.^Denote^by

λ

max

(

G

)

thelargesteigenvalueofmatrixG.Therearetwocases.If

δλ

max

(

G

) <

1,thepreviousseries convergesandthereisa uniqueequilibriumgivenby x

= (

I

− δ

G

)

⁻¹a.Underhomogeneity(

∀

ⁱ

,

a_i

=

â),âctions âreâligned withBonacichcentralities inthenetwork.⁷ Amorecentralagent alwaysplays ahigheraction.Incontrastif

δλ

max

(

G

) ≥

^1, theprevious seriesdivergesandthereisnoequilibrium.Facedwiththisexistence issue,researchershave,sofar, typically imposed the assumptionthat

δλ

max

(

G

) <

1. Thisisnot very satisfactory. Itmeans restrictingattentiontorelatively small networkeffects.Andthisinequalitymaynotholdempirically(Calvó-Armengolet al.,2009).

The assumption that actionis unboundedis unrealisticin mostempiricalcontexts, however.With crime, presumably, xireachessomenaturalupperboundbi forindividualswhohavebecomefull-timecriminalsandarenotengagedinlegal activities anylonger.IfwekeepthesamepayoffsasinLiuet al. (2014)andintroduceupperboundsb_i,thebestresponse becomesacensoredversionof(3):

fi

(

x₋i

) =

min

ai

+ δ

n

j=1

gi jxj

,

bi

(4)

Best response (4)isa particularcaseof (1)with

ϕ

i

(

x

) = δ

x,andhencean equilibriumnow existsforany

δ

.In addition, theassumptionthatthemarginalimpactofpeers’actionsonownmarginalutilityisconstantandthesameforeveryoneis quitespeciﬁc.Inreality,thismarginalimpactmaywell dependontheindividualandonthelevelofpeers’actions,which wouldleadtopayoffs(2)andbestresponses(1).Forinstance,anindividualmaybemoresusceptibletocriminalinﬂuence oncehisfriendsarereallycommittedtocriminalactivity(

ϕ

iconvex)eventhoughsuchsusceptibilitymaythentaperoffas friendsgetclosetofull-timecrime(

ϕ

iconcave).

6 SeeBramoulléandKranton (2007)andBramoulléet al. (2014)forananalysisofnetworkgameswithlinearbestresponsesandstrategicsubstitutes.

7 TheprofileofBonacichcentralitiesisdefinedasc=(I−δG)G1(Bonacich,1987),whichyieldsx=â(1+δc).

(5)

Armsraceprovidesanothercontextwherenetworkeffectsmatter.MilgromandRoberts (1990)modelarmsracebetween twocountriesastheoutcomeofagamewithstrategiesx_i

∈ [

⁰

,

b

]

^and^payoffs

π

i

(

x_i

,

x₋_i

) =

^B

(

x_i

−

^x−i

) −

^C

(

x_i

)

.Toextendthis modelto manycountries,relationships betweencountrieshavetobe takenintoaccount. Militaryalliancesandhistorical partnershipslikelyshapeuptheinteractioninarms’ expenditures.Acountryprobablyworriesmoreabouttheincrease in thearmament ofapotentialfoethanofalong-timepartnerandally.Thus,onewaytoextendthismodeltoncountriesis toassumethat

π

i

(

x_i

,

x₋_i

) =

^B

x_i

−

n

j=1

g_{i j}x_j

−

^C

(

x_i

)

whereg_{i j}

=

^{1 if}^country ^j îsâ^potential^foeôf^countryî.^Here,^the^network^G^representsantagonisticrelationships.Inwhat follows,we assume that B andC are twice continuously differentiable, B is strictly increasing andstrictly concave,C is strictlyincreasingandstrictlyconvex,B

(

0

) >

C

(∞)

^and^B

(∞) <

C

(

0

)

.

Denote by a the unique value such that B

(

a

) =

^C

(

a

)

where a represents optimalarmament in isolation. Denote by I

(

x

) =

^x^theîdentity^functionândîntroduce

ϕ ₌ (

I

− (

B

)

⁻¹

◦

^C

)

⁻¹

−

^a^the^function^such^that:⁸

∀

^x

,

B

ϕ (

x

) +

^a

−

^x

=

^C

ϕ (

x

) +

^a

Then, observe that the best response of the arms race game between n countries is given by f_i

(

x₋_i

) =

^min

(

a

+ ϕ (

n

j=¹gi jxj

),

b

)

. A country tends to increase his armament when his potential opponents increase theirs,and the im- pactofothers’armament levelsdependsonthebeneﬁtsandcosts.InadditionifbeneﬁtsB_iandcostsC_iareheterogeneous, we wouldhaveheterogeneous armaments in isolationai andimpactsofenemies’arms

ϕ

i andhencea particularcaseof bestresponse(1).

Ourmain objectiveinthe remainderof thepaperis toanalyzethe Nash equilibriaofgameswithbest responses (1).

We investigatethree issuesin particular.When is therea unique equilibrium?How doestheposition inthe network of interactionsaffectequilibriumaction?AndhowdoNashequilibriadependontheparametersofthegame?

3. Uniqueness

In this section, we identify a new class of games with strategic complements which have a unique equilibrium. We derive formal conditions under which the iterated best response is contracting over ordered proﬁles lying between the extremal equilibria. This localized contraction leads to uniqueness in thissubset and, since it contains all equilibria, to globaluniqueness.Thismayhappeneven withlarge networkeffectsandwhenthe bestresponseisnon-contractingover largeportionsoftheoriginalstrategyspace.

We ﬁrstderive ourmain technicalresult. Weprovide asuﬃcient condition foruniqueness involvingthesmallestand the largest equilibrium. We then build on thiscondition to prove uniqueness under various assumptions on the game’s exogenous parameters.Let xandX denote thesmallestandlargest equilibriumanddenoteby I

= {

ⁱ

:

^xi

<

bi

}

^the^set ^of agentswhoplayaninterioractioninthesmallestequilibrium.

Lemma1.If

∀

ⁱ

∈

^I

,

a_i

+ ϕ

i

(

_n

j=¹g_{i j}x_j

) >

K_i

(

_n

j=¹g_{i j}x_j

)

whereK_i

=

^sup

ϕ

_iover

[

_n

j=¹g_{i j}x_j

,

_n

j=¹g_{i j}X_j

]

^,^then^the^game^has^a uniqueequilibrium.

Proof. Weprovethisresultinfoursteps.

1. Sincexisanequilibrium,

∀

^i,^xi

=

^fi

(

x₋_i

)

andhence,

∀

ⁱ

∈

^I

,

x_i

=

^ai

+ ϕ

i

_n

j=1

g_{i j}x_j

SubtractKi

(

n

j=¹gi jxj

)

frombothsidesoftheequalityandapplytheLemma’sconditions:

∀

ⁱ

∈

^I

,

x_i

−

^Ki

n

j=¹

g_{i j}x_j

>

0

2. DenotebyHthematrixdeﬁnedover Isuchthat

∀

ⁱ

,

j

∈

^I

,

hi j

=

^Kigi j.Introduce zi

=

^xi

−

^Ki

n

j=1gi jxj,

∀

ⁱ

∈

^I.^Denote byxI therestrictionofproﬁlexto I.Throughsuccessivesubstitutions,wecanwrite:

x_I

=

^z

+

^HxI

=

^z

+

^Hz

+

^H²^xI

= · · · =

t

s=0

H^sz

+

^H^t⁺¹^xI

foranyintegert.Since

∀

ⁱ

∈

^I

,

z_i

>

0 andx_i

>

0,thisisonlypossibleifH^t convergesto0ast tendstoinﬁnity.

8 Thefunctionϕiswell-deﬁnedsinceI−(B)⁻¹◦^C^is^strictlyincreasing.

(6)

3. Let S

= {

^y

∈ R

ⁿ₊

:

^x

≤

^y

≤

^X

}

^be ^the ^setôf âction^profiles ^lying ^between^the ^smallestând ^largest^Nash equilibrium.

Observeﬁrstthatf

(

S

) ⊂

^S.^Since^fîs^weaklyîncreasingând^xând^Xâreequilibria,thebestresponseofanyprofileinSlies in S.Next,foranyy

,

y

∈

^S^such^that^y

≤

^y^,^we^have:

0

≤

^f

y

−

f

(

y

) ≤ ϕ

y

− ϕ (

y

) ≤

^H

y

−

^y

wheretheﬁrstinequalityfollowsfromthemonotonicity ofthebestresponse;thesecond inequalityfromtheexamination ofpossiblecases⁹ andthethirdinequalityfromthemeanvaluetheoremandthedeﬁnitionofKiandH.

4. Applythepreviousinequalitytof

(

y

)

andf

(

y

)

:0

≤

^f²

(

y

) −

^f²

(

y

) ≤

^H

(

f

(

y

) −

^f

(

y

)) ≤

^H²

(

y

−

^y

)

sinceHhasnon-negative entries. By induction, 0

≤

^f^t

(

y

) −

^f^t

(

y

) ≤

^H^t

(

y

−

^y

)

for anyt

≥

^1. ^Since ^H^t ^converges ^to ^0, ^there^exists ^C

<

1 such that

^f^t

(

y

) −

^f^t

(

y

) ≤

^C

^y

−

^y

^for^t^high^enough.^Apply^to^y

=

^x^and^y

=

^X.^Since^f^t

(

x

) =

^x^and^f^t

(

X

) =

^X,

^X

−

^x

≤

^C

^X

−

^x

andX

−

^x

=

^0.^The^game^has^a^unique^Nashequilibrium.

2

We now show how to apply Lemma 1 to prove uniqueness for various assumptions on the games’ exogenous parameters. Consider linear best responses ﬁrst:

ϕ

i

(

u

) = δ

iu. In that case, observe that K_i

= δ

i and a_i

+ ϕ

i

(

_n

j=¹g_{i j}x_j

) =

ai

+ δ

i

(

n

j=1gi jxj

) > δ

i

(

n

j=1gi jxj

)

since ai

>

0. Thus,the conditionsof Lemma 1always hold whenthe

ϕ

i’s are linear.

More generally,weshow nextthat theseconditionsnecessarilyholdwhenthe

ϕ

i’sareconcave.Underconcavity,Kennan (2001)’sclassicalresultsalsoapplyhere.Kennan (2001)’sanalysisandoursarebasedonverydifferentpremises,however.

Ourapproachthusprovidesanalternativewaytoshowuniquenessunderconcavity.

Proposition1.Supposethat

∀

ⁱ

, ϕ

iisconcave.Then,thegamehasauniqueNashequilibrium.

Proof. Wederivetwoalternativeproofsofthisresult.WeshowthatLemma 1andTheorem1ofKennan (2001)bothapply here.

1. Since

ϕ

iisconcaveand

ϕ

i

(

0

) =

^0,

∀

^u

≥

⁰

, ϕ

i

(

u

) ≥

^u

ϕ

_i

(

u

)

.Thisimpliesthat:

ai

+ ϕ

i

_n

j=1

gi jxj

> ϕ

i

_n

j=1

gi jxj

≥ ϕ

_i

_n

j=1

gi jxj

_n

j=1

gi jxj

=

^Ki

_n

j=1

gi jxj

where the equality comes from the fact that

ϕ

_i is weakly decreasing. Therefore, the condition of Lemma 1 holds and uniquenessfollows.

2. Introduce h

(

x

) =

^f

(

x

) −

^x. ^Since ^f ^is ^weakly increasing, h is “quasi-increasing”.¹⁰ We next show that h is“strictly radially quasiconcave”. Consider x suchthat h

(

x

) =

⁰ ^and

λ ∈ ]

⁰

,

1

[

^.^We ^need ^to^show ^that

∀

ⁱ

,

hi

(λ

x

) >

0.There are two cases.(1)Ifx_i

=

^ai

+ ϕ

i

(

jg_{i j}x_j

)

,thenh_i

(λ

x

) = (

1

− λ)

a_i

+ ϕ

i

(λ

jg_{i j}x_j

) − λ ϕ

i

(

jg_{i j}x_j

)

.Since

ϕ

iisconcaveand

ϕ

i

(

0

) =

0,

ϕ

i

(λ

u

) ≥ λ ϕ

i

(

u

) ∀

^u

>

0 and hi

(λ

x

) ≥ (

1

− λ)

ai

>

0.(2). If xi

=

^bi, then hi

(λ

x

) =

^fi

(λ

x₋i

) − λ

bi. If fi

(λ

x₋i

) =

^bi, then hi

(λ

x

) = (

1

− λ)

bi

>

0. Otherwise, fi

(λ

x₋i

) =

^ai

+ ϕ

i

(λ

jgi jxj

) ≥ (

1

− λ)

ai

+ λ(

ai

+ ϕ

i

(

jgi jxj

)) ≥ (

1

− λ)

ai

+ λ

bi and h_i

(λ

x

) ≥ (

1

− λ)

a_i

>

0.Therefore,Theorem1inKennan (2001)applies.

2

We consider convexfunctions next. We can see that strong convexity in the best responses easily leads to multiple equilibria.Foruniquenesstoholdunderconvexity,theextentofthecurvaturemustbesomehowbounded.Thisisprecisely whatweobtainwhenapplyingLemma 1toconvexfunctions.DeﬁneBi

=

jgi jbjasthelargestpossiblelevelofneighbors’

actions,fromagenti’spointofview.

Proposition2.Supposethat

∀

ⁱ

, ϕ

iisconvex.If

∀

ⁱ

,

a_i

+ ϕ

i

(

B_i

) > ϕ

_i

(

B_i

)

B_i,thenthegamehasauniqueNashequilibrium.

Proof. Since

ϕ

iisconvex,

ϕ

_iisweaklyincreasingandKi

= ϕ

_i

(

n

j=¹gi jXj

) ≤ ϕ

_i

(

Bi

)

.Considertheauxiliaryfunction

ψ

i

(

u

) =

a_i

+ ϕ

i

(

u

) − ϕ

_i

(

B_i

)

u. We have:

ψ

_i

(

u

) = ϕ

_i

(

u

) − ϕ

_i

(

B_i

) ≤

^{0 if} ^u

≤

^Bi. Therefore,

ψ

i is weakly decreasing over

[

⁰

,

B_i

]

^and

ψ

i

(

_n

j=1g_{i j}x_j

) ≥ ψ

i

(

B_i

)

.Notethat

ψ

i

(

B_i

) >

0 amountspreciselytotheconditionstatedintheProposition.Thismeansthat ai

+ ϕ

i

(

n

j=¹gi jxj

) > ϕ

_i

(

Bi

)(

n

j=¹gi jxj

) ≥

^Ki

(

n

j=¹gi jxj

)

andLemma 1’sconditionholds.

2

Toillustrate thisresult,considerfunctionswithconstantelasticity:

ϕ

i

(

u

) = δ

iuηi with

η

i

>

1.Proposition 2’s suﬃcient conditionbecomes:

a_i

> δ

_i

( η

_i

₋

1

)

B^η_iⁱ

whichismoreeasilysatisﬁedwhenaiislarge,

δ

i issmalland,ifBi

≥

^1,^when

η

iiscloserto1.

9 If fi(y₋i)=bi,then fi(y₋_i)=bi.If fi(y₋i)<bi,then fi(y₋_i)−fi(y₋i)=fi(y₋_i)−(ai+ϕi(

jgi jyj))≤ϕi(

jgi jy_j)−ϕi(

jgi jyj). 10 WerefertoKennan (2001)fordetailsandformaldeﬁnitions.

(7)

Finally,we derivea suﬃcientuniqueness condition forfunctionswitharbitrarycurvature.Deﬁne Ai

=

jgi jaj asthe smallestpossiblelevelofneighbors’actions.

Proposition3.If

∀

ⁱ

,

ai

+ ϕ

i

(

Bi

) >

K_iBiwithK_i

=

^sup

ϕ

_iover

[

^Ai

,

Bi

]

^,^then^the^game^has^a^uniqueequilibrium.

Proof. Letu

∈ [

^Ai

,

B_i

]

^.^Byassumption,a_i

+ ϕ

i

(

B_i

) >

K_iB_i.Subtracting

ϕ

i

(

u

)

yields

ϕ

i

(

B_i

) − ϕ

i

(

u

) >

K_iB_i

−

^ai

− ϕ

i

(

u

)

.Bythe meanvaluetheorem,

ϕ

i

(

Bi

) − ϕ

i

(

u

) ≤

^K_i

(

Bi

−

^u

)

.Combiningbothinequalitiesleadsto

a_i

+ ϕ

i

(

u

) >

K_iu

Since K_i

≥

^Ki,applyinginu

=

_n

j=¹g_{i j}x_jleadsto ai

+ ϕ

i

_n

j=1

gi jxj

>

Ki

_n

j=1

gi jxj

andLemma 1applies.

2

Therefore,uniquenessmayprevaileveninthepresenceofconvexitiesandlargenetworkeffects.Whenpositivefeedbacks are initiallyexplosive, some agentswill quicklyreachtheir upper bounds.This tendsto dampenthe interactionandthis dampening may be suﬃciently strong to yield uniqueness. Our results clarify the circumstances under which the best responsesatisﬁesacontractingpropertybetweentheextremalequilibria.Suchlocalizedcontractionalwaysholdsforlinear andconcavebest responses.It mayalsoholdunderconvexbestresponses (below theupperbound)whenthe convexity isnot too strong.Whilethe uniqueness resultsforconcavebest responses are notnew, they hadnotbeen connectedto localizedcontractionbeforeandtheresultsforconvexandarbitraryfunctionsarenew.

Even though simple analytical expressions for equilibrium actions are generallyout of reach, they can be computed easily withthe help of classical algorithms. Inparticular, we know that inthese games repeatedmyopic bestresponses convergerapidly andmonotonicallytowards the smallestequilibriumwhen starting fromthesmallest proﬁle(

∀

ⁱ

,

xi

=

⁰⁾ andtowardsthelargestequilibriumwhenstartingfromthelargestproﬁle(

∀

ⁱ

,

xi

=

^bi)(Vives,1990; Echenique,2007).Thus, bothdynamic processesconvergetowardstheequilibriumunderuniquenessandwemakeuseofthesealgorithmsinour examplesbelow.

Inthenext section, weanalyzethepropertiesofNash equilibriainsome detail.Weﬁrstlook athow positionsinthe networkaffectactions.Wethenfocusoncomparativestatics.Ourresultsapplytoanygamewithbestresponses(1).Under multiplicity,theyapplytoextremalequilibriaandinoneinstance,toallequilibria.Underuniqueness,theyhelpcharacterize theequilibrium’sstructuralproperties.

4. Equilibriumproperties 4.1. Networkpositionandaction

Inthissection,westudyhowanagent’spositioninthenetworkaffectshisactioninequilibrium.Tofocusontheeffect ofthestructure,weassumethroughoutthesectionthatagentscanonlydifferintheirnetworkpositionsandthatlinksare binary.Formally,

∀

ⁱ

,

a_i

=

^a,^bi

=

^b,

ϕ

i

= ϕ

and

∀

ⁱ

,

j

,

g_{i j}

∈ {

⁰

,

1

}

^.^Best^responses^are^then^given^by:

f_i

(

x₋_i

) =

^min

a

+ ϕ

_n

j=¹ g_{i j}x_j

,

b

(5)

Weestablish,ﬁrst,thattheupperboundhasastrongimpactontherelationshipbetweenpositionandaction.Consider linearbestresponses:

ϕ (

u

) = δ

u.Withnoupperboundandsmallnetworkeffects,equilibriumactionsarealignedwiththe Bonacichcentralitiesoftheagentsinthenetwork(Ballesteret al.,2006). Weshownowthatthisalignment maynot hold inthepresenceofanupperbound.

Consider the networkdepicted inFig. 1. Ithas eightnodes andtwo cliques:one composed ofagents1 to4 andthe triangle6-7-8.¹¹Inaddition,agent5inthemiddleisconnectedtoagents4and6.When

δ =

⁰

.

3 andactionsareunbounded, theequilibriumissuchthatx5

≈

⁷

.

9

>

x6

≈

⁵

.

7.¹²Eventhoughagent6hasonemoreneighborthanagent5,hisneighbors arenotverycentral.Incontrast,agent5isconnectedtoagent4whoisthemostcentralagentinthegraph.When

δ

isnot toolow,theeffectofindirectpathsdominate;agent5ismorecentralthanagent6andplaysahigheraction.Supposenext that actionsare boundedfromabove by L

=

^5.^The equilibriumisnow suchthat x5

≈

³

.

7

<

x6

≈

⁴

.

0.¹³ Agents1–4reach

11 Acliqueisacompletelyconnectedsubgraph.

12 Thewholeequilibriumproﬁleisx≈(15.46,15.46,15.46,17.28,7.89,5.69,3.87,3.87). 13 Thewholeequilibriumproﬁleisnowx≈(5,5,5,5,3.70,3.99,3.14,3.14).

(8)

Fig. 1.A network for which action and centrality may not be aligned.

Fig. 2.An example of hierarchical community.

the upperboundandthisreducesthe actionpremiumthatagent 5gets fromhis linkwithagent4.Agent 6,whoisless central,nowplaysahigheraction.

Wenextidentifyspeciﬁcsituationsleading toaclear-cutrelationshipbetweenpositionandaction.Saythatagenti has fewer neighborsthan agent j inthe sense ofsetinclusion if

∀

^k

=

ⁱ

,

j, g_ik

≤

^gjk.Agents i and j could be connected.For instanceinthegraphdepictedinFig. 1,agents7and8havefewerneighborsthanagent6andagents1,2and3havefewer neighbors thanagent 4 inthesense ofset inclusion.Agents i and j could alsobe disconnected.Forinstance,ifthe link between7and8isremoved inFig. 1,agents7andagents8nowhavefewer neighborsthanagent 5inthesense ofset inclusion.Inthesesituations,theactionsofthetwoagentscanalwaysbeordered.

Proposition4.LetxbeanyNashequilibriumofagamewithbestresponses(5).Ifagenti hasfewerneighborsthanagent j inthe senseofsetinclusion,thenx_i

≤

^xj.

Proof. Let x be a Nash equilibrium. Suppose ﬁrst that g_{i j}

=

^0. ^Then,

kg_ikx_k

≤

kg_jkx_k. Therefore,

ϕ (

kg_ikx_k

) ≤ ϕ (

kg_jkx_k

)

andmin

(

a

+ ϕ (

kg_ikx_k

),

b

) ≤

^min

(

a

+ ϕ (

kg_jkx_k

),

b

)

.Sincexisanequilibrium,thismeansthatx_i

≤

^xj.Sup- posenextthatg_{i j}

=

^1.^Suppose^that^xi

>

x_j.Thisimpliesthat

kg_ikx_k

>

kg_jkx_kandx_j

+

k=ⁱ,jg_ikx_k

>

x_i

+

k=ⁱ,jg_jkx_k. Since

k=i,jgikxk

≤

k=i,jgjkxk,thenxj

>

xiwhichestablishesacontradiction.Therefore,xi

≤

xj.

2

We emphasize that Proposition 4 holdsfor anystrictly increasing function

ϕ

. Thisresult thereforeuncoversa robust property ofnetworkgameswithstrategiccomplements.Aniceimplicationisthat itleads toafullranking oftheactions fornestedsplitgraphs.Agraphisnestedsplit ifagentscanbe orderedsothat g_kl

=

¹

⇒

^gi j

=

^{1 whenever}ⁱ

≤

^k ^and ^j

≤

^l.

Thesegraphsemerge,forinstance,asoutcomesofcentrality-basedmodelsofnetworkformation(Königetal.,2014) andas solutionsofnetworkdesignproblems(Belhajet al.,2013).Onnestedsplitgraphs,ifagent i hasalowerdegreethanagent jthenshehasfewerneighborsinthesenseofsetinclusion.ByProposition 4,wethenhavexi

≤

^xj.Ifagenti hasthesame degreeasagent j,therearetwopossibilities.Eithertheyareunconnectedandtheyhavethesameneighbors:

∀

^k

,

gik

=

^gjk. Ortheyareconnectedand

∀

^k

=

ⁱ

,

j, gik

=

^gjk.Ineithercase,Proposition 4impliesthatxi

=

^xj.

Corollary2.Onnestedsplitgraphs,actionisanincreasingfunctionofdegreeinanyNashequilibriumofagamewithbestresponse(5).

Anotherfeaturethat mayhelpconnect positionandactionisgeometric symmetry.Inparticular, theanalysisofBelhaj and Deroïan (2010) applies here. Theystudysupermodular networkgames played on the lineandon more general“hierarchical communities”. These networks are deﬁned by the following three features. (1) They are structured around a geometriccenter.(2)Degreesdecreasesweaklyasnodesgetfurtherawayfromthecenter.(3)Allagentsatagivendistance from thecenter havesymmetricpositions. An exampleofsuch graphs isgivenin Fig. 2; we refer toBelhajand Deroïan (2010) formore details andformal deﬁnitions. Inthese structures, eventhough neighborhoods are not nestedcentrality isunambiguous.An agentismorecentralwhen herdistancetothecenterislower.Theirmainresultstatesthatinthese graphs,actionisalignedwithcentralityinthesmallestandlargestNash equilibrium.Theiranalysisdirectlyappliestoour setup.

Corollary3.(SeeBelhajandDeroïan,2010.)Onthelineandonmoregeneralhierarchicalcommunities,actionisalignedwithcentrality intheextremalNashequilibria.

(9)

Fig. 3.Two communities connected by a bridge.

4.2. Comparativestaticsandbrokeninterdependence

Inthis section,we studycomparative statics ofgames withbestresponse (1).We look,inparticular, athow a shock propagatesinthenetwork.Weﬁrstderiveweakmonotonecomparativestaticsbyapplyingclassicalresultsfromthetheory ofsupermodulargames.

Corollary4(Weakmonotonecomparativestatics).Theactionofanyagentisweaklyincreasingina,bandGinthesmallestandin thelargestNashequilibrium.

Proof. Compute payoff cross-derivatives in the game

Γ

:

∂

²ui

/∂

xi

∂

ai

=

^1;

∂

²ui

/∂

xi

∂

aj

=

^{0 if} ^j

=

^i;

∂

²ui

/∂

xi

∂

gi j

=

xj

ϕ

(

kgikxk

)

;

∂

²ui

/∂

xi

∂

gkl

=

^{0 if}^kl

=

^{i j.}^All^thesederivativesaregreaterthanorequaltozero.ByTheorem6inMilgrom andRoberts (1990),actionx_i intheextremalequilibriaisthen weaklyincreasing inall theseparametersforanyi.Next, deﬁnethesequenceyⁿ

(

b

)

byy⁰

(

b

) =

⁰^and^yⁿ⁺¹

(

b

) =

^f

(

yⁿ

(

b

),

b

)

.Byinduction,weseethatb

≤

^b

⇒

^yⁿ

(

b

) ≤

^yⁿ

(

b

)

.Since yⁿ

(

b

)

converges to x

(

b

)

as n tends to inﬁnity, this means that x

(

b

) ≤

^x

(

b

)

. A similar argument applies to the largest equilibrium.

2

Under strategic complements,direct and indirectnetwork effects are fully aligned. Consider, forinstance, the impact of an increase ina_i,the action ofagent i inisolation. The direct effectis to induceagent i to increase his action.As a consequence, agent i’s neighborswill also tend to increase their actions. In turn, their neighbors tend to increase their ownactions.Theimpactoftheoriginalshockpropagatesthroughoutthenetworkandallthedirectandindirecteffectsare greaterthanorequaltozero.Intheend,theactionofeveryother agentincreasesweakly.Thisstandsinsharpcontrastto setupswithstrategicsubstitutes,wheredirectandindirecteffectsaregenerallynotalignedandcomparativestaticscanbe quitecomplicated(Bramoulléet al.,2014).

Corollary 4coverstwodifferentsituations,however.Anincreaseinai couldleadtoanincreaseinxjoritcouldleavexj unaffected.Underlinearbestresponsesandsmallnetworkeffects,thesecondpossibilityisexcluded.Interdependenceis,in asense,maximal.If

ϕ

i

(

u

) = δ

uand

δ

small,thenx

= (

I

− δ

G

)

⁻¹aand,ifGisconnected,

∂

x_i

/∂

a_j

>

0

∀

ⁱ

,

j.Thepresence of upperboundsaffectsthispropertyquitedeeply.When anagent reacheshisupperbound,hestopsbeingatransmitterof inﬂuences in the network.Depending on how agentsare placed, interdependence maythen be broken andwhole parts of thenetworkmaybeunaffectedbyindividualshocks.

Letusﬁrstillustratehowthiscanhappen.Considerlinearbestresponsesunderhomogeneity,binarylinksandthegraph depictedin Fig. 3.Society iscomposed of two communities ofequal size. In each community,every agent is connected to every other agent. In addition, one agent in the ﬁrst community is connected to one agent in the second. The two communitiesarebridgedthroughauniquelink.ByProposition 1,theequilibriumisunique.Wecanshowthatthereexists twothresholdlevelsa^∗₁anda^∗₂ suchthatthefollowingpropertieshold.Ifa

<

a^∗₁,

∂

x_i

/∂

a_j

>

0 foranytwoagentsiand j in thepopulation.Ifa^∗₁

<

a

<

a^∗₂,thenxi

=

^b^for^both^bridge^agents;

∂

xi

/∂

aj

>

0 foranytwonon-bridgeagentsi and j inthe samecommunitywhile

∂

xi

/∂

aj

=

^{0 if}ⁱ^liesⁱⁿône^communityând ^j^liesⁱⁿ^theôther.^Finally,îfâ

≥

^a^∗2,everyagentplaysthe upperbound.Whenactionsinisolationaresmall,thebridgeplaysacrucialrole:ittransmitsshocksfromonecommunity totheother.Asactionsinisolationincrease,bridgeagentstendtoreachtheupperboundﬁrstbecauseoftheirmorecentral position.¹⁴Whenthishappens,theybecomeunresponsivetofurtherincreasesandthisturnsoffthetransmissionchannel.

Wenowanalyzethisissuemoregenerally.Consideranextremalequilibrium.Denote by Pi

= {

^j

: (

_∂^∂_a^xⁱ

j

)

⁺

>

0

}

^the^set^of

agents jwhoaffectiindirectlyinthesensethatanincreaseinajleadstoanincreaseinxi.¹⁵Forinstance,Pi

=

^N

\{

ⁱ

}

^under linearbest responses,smallnetwork effectsanda connectedgraph.Incontrast, Pi

= ∅

^if^xi

=

^bi.This setcan besimply characterized asfollows.Apath connectingi to j in G isa sequenceofagents i₁

=

^i, ⁱ2,. . . , i_l

=

^j ^such ^that ^gisi_s+1

>

0,

∀

^s

∈ {

¹

,

2

, . . . ,

l

−

¹

}

^.

Proposition5.Inthesmallestandlargestequilibrium,agent j affectsagenti indirectlyifandonlyifthereisapathfromi to j inGin whichallagentsplayaninterioraction.

14 Observethatabridgeagentinonecommunityhasmoreneighborsthanothercommunitymembersinthesenseofsetinclusion.ByProposition 4,she alwaysplaysaweaklyhigheraction.

15 Thelowestandthehighestequilibriumarecontinuousfunctionsofa.Theleft- andright-derivativesofindividualactionxiwithrespecttoaj may differ,however.Wedenoteby(_∂^∂_a^xⁱ

j)⁺theright-derivative.Alternatively,wecouldconsiderthesetofagentsjsuchthatadecreaseinajleadstoadecrease inxiandtheanalysiswouldcarryover.

(10)

Proof. Supposethatxi

<

bi anddenoteby Ithesetofagentsplayinganinterioraction.Wehave:

∀

ⁱ

∈

^I

,

x_i

=

^ai

+ ϕ

_i

k∈^I

g_ikx_k

+

k∈/^I g_ikb_k

ThesetI doesnotchangefollowinganinﬁnitesimalincreaseinaj.Taketheright-derivativewithrespecttoaj.

∀

ⁱ

∈

^I

, ∂

x_i

∂

a_j +

= δ

_{i j}

+

k∈^I g_ik

∂

x_k

∂

a_j +

ϕ

_i

k

g_ikx_k

where

δ

_{i j}

=

^{1 if}ⁱ

=

^j ^and0 otherwise.IntroducethematrixHdeﬁnedover Isuchthath_{i j}

=

^gi j

ϕ

_i

(

kg_ikx_k

)

,e^jtheproﬁle suchthate_k^j

= δ

_jk.Inmatrixnotations,wehave:

∂

x_I

∂

a_j +

=

^e^j

+

^H

∂

x_I

∂

a_j +

ByCorollary 4,

(

^∂_∂_a^xⁱ

j

)

⁺

≥

^0.^Byânârgument^similar^to^the^second^stepôf^the^proofôf^{Lemma 1,}^we^must^have

λ

max

(

H

) <

1.

ThismeansthatI

−

^Hîsînvertibleând^hence

(

^∂_∂^x_a^I

j

)

⁺

= (

I

−

^H

)

⁻¹e^j.Thisyields:

∂

x_i

∂

a_j +

=

∞ t=⁰

H^t

i j

Therefore, j

∈

^Pi iffthereexistsanintegertsuchthat

(

H^t

)

i j

>

0.Thisisequivalenttothefactthatthereisapathfromi to jinGinwhichallagentsplayaninterioraction.

2

AdirectimplicationofProposition 5 isthattheextentofinterdependencenecessarilydecreasesasactionsinisolation increase.

Corollary5.Asactionsinisolationaincrease,thesetofagentswhoaffecti indirectly,Pi,inthesmallestandlargestequilibrium shrinksmonotonicallytowardstheemptyset.

Proof. ByCorollary 4, moreagents reach their upperboundsas aincreases. ByProposition 5, Pi shrinks. When a

=

^b, Pi

= ∅ ∀

ⁱ^.

2

Thewayinterdependencegetsbrokenclearlydependsontheshapeofthenetworkandontheexistenceandpositionsof bridges.Whenbridgingagentsarerelativelymorecentralwithintheirowncommunities,wecanexpectinterdependenceto bereducedquitequicklybecausetheywilltendtoreachtheirupperboundsﬁrst.However,therearealsosituationswhere agents whoare betterconnectedoutsideof theircommunities are lessconnected within.Inthesesituations, interdepen- dencemaybemorerobustasbridgingagentsmaytendtoreachtheirupperboundslater.

5. Conclusion

Inthispaper,weanalyzenetworkgameswithpositivebutboundedactionsandstrategiccomplementarities.Wederive newsufficientconditionsforuniquenessandstudythestructuralpropertiesoftheequilibria.Weprovidethefirstapplica- tionsofageneralprincipleoflocalizedcontractionandproveuniquenessforallconcaveaswellassomenon-concavebest responses. We find that theupper boundshavea strongimpact on the relationshipbetweenaction andcentrality. They mayleadmorecentralagentstoplayaloweraction.Still,anagent whohasmoreneighborsthananotherinthesense of setinclusionalwaysplaysahigheraction.Anincreaseinthegameparametersnecessarilyleadstoaweakincreaseinevery agent’saction.However,itmayleavesomeagentsunaffected.Wecharacterizepreciselywhenanagentisindirectlyaffected byashockonanotheragentanduncoverageneralpatternofdecreasinginterdependence.

Ouranalysiscould potentiallybe usefulforempiricalstudies ofpeereffects innetworks.Atypical econometricmodel aimedatstudyingwhethersomeoutcomexissubjecttopeereffectscanbewritten:

x_i

=

^ai

+ δ

j

g_{i j}x_j

+ ε

_i (6)

(11)

where

δ

isthe mainparameter ofinterestto beestimatedand

ε

i isan errorterm.¹⁶ Mostoutcomesofinterest,such as grades, criminalactivity, or time spent exercising, are naturally boundedfromabove but existing empirical studies have neglected thesebounds. This likely generates biasesin existing estimates. To account for upper boundsproperly in the estimation,anaturalsolutionistoestimateacensoredversionof(6):

xi

=

min

ai

+ δ

j

gi jxj

+ ε

i

,

bi (7)

Thisdeﬁnesa systemofsimultaneous equationswhichisformally equivalent totheequilibriumconditionsof agame withbestresponse(1).Ourresultsshowthatwhenoutcomesarepositive,thissystemhasauniquesolution.Theempirical analysisthus doesnot sufferfromthe complications arising fromequilibriummultiplicity (Tamer, 2003). Since theequi- libriumxisafunctionofa

, δ,

Gand

ε

we can,inprinciple andgivensomeassumptions ontheerrorterms,computethe likelihoodL

(

x

|

^a

, δ,

G

)

andestimate

δ

throughmaximumlikelihoodinastraightforwardmanner.Moreover,ourresultsshow that thisproperty carries over toheterogeneous andnon-linear variants ofmodel (6). Therefore,our analysisprovides a steppingstoneforempiricalstudiesofpeereffectsinnetworkswithcontinuousbutboundedoutcomes.

References

Baetz,Oliver,forthcoming.Socialactivityandnetworkformation.TheoreticalEcon.

Ballester,Coralio,Calvó-Armengol,Antoni,2010.Interactionswithhiddencomplementarities.Reg.Sci.UrbanEcon. 40(6),397–406.

Ballester,Coralio,Calvó-Armengol,Antoni,Zenou,Yves,2006.Who’swhoinnetworks.Wanted:thekeyplayer.Econometrica 74(5),1403–1417.

Bayer,Patrick,Hjalmarsson,Randi,Pozen,David,2009.Buildingcriminalcapitalbehindbars:peereffectsinjuvenilecorrections.Quart.J.Econ. 124(1), 105–147.

Belhaj,Mohamed,Bervoets,Sébastian,Deroïan,Frédéric,2013.Networkdesignunderlocalcomplementarities.Workingpaper,Aix-MarseilleUniversity.

Belhaj,Mohamed,Deroïan,Frédéric,2010.Endogenouseffortincommunicationnetworksunderstrategiccomplementarity.Int.J.GameTheory 39,391–408.

Belhaj,Mohamed,Deroïan,Frédéric,2013.Strategicinteractionandaggregateincentives.J.Math.Econ. 49(3),183–188.

Blume,Lawrence,Brock,William,Durlauf,Steven,Ioannides,Yannis,2010.Identiﬁcationofsocialinteractions.In:Benhabib,J.,Bisin,A.,Jackson,M.(Eds.), HandbookofSocialEconomics.NorthHolland,Amsterdam.

Bonacich,P.,1987.Powerandcentrality:afamilyofmeasures.Amer.J.Sociology 92(5),1170–1182.

Bramoullé,Yann,Kranton,Rachel,2007.Publicgoodsinnetworks.J.Econ.Theory 135(1),478–494.

Bramoullé,Yann,Djebbari,Habiba,Fortin,Bernard,2009.Identiﬁcationofpeereffectsthroughsocialnetworks.J.Econometrics 150,41–55.

Bramoullé,Yann,Kranton,Rachel,D’amours,Martin,2014.Strategicinteractionandnetworks.Amer.Econ.Rev. 104(3),898–930.

Calvó-Armengol,Antoni,Zenou,Yves,2004.Socialnetworksandcrimedecisions:theroleofsocialstructureinfacilitatingdelinquentbehavior.Int.Econ.

Rev. 45(3),939–958.

Calvó-Armengol,Antoni,Patacchini,Eleanora,Zenou,Yves,2009.Peereffectsandsocialnetworksineducation.Rev.Econ.Stud. 76(4),1239–1267.

DeGiorgi,Giacomo,Pellizzari,Michele,Redaelli,Silvia,2010.Identiﬁcationofsocialinteractionsthroughpartiallyoverlappingpeergroups.Amer.Econ.J.

Appl.Econ. 2(2),241–275.

Echenique,Federico,2007.Findingallequilibriaingamesofstrategiccomplements.J.Econ.Theory 135,514–532.

Galeotti,Andrea,Goyal,Sanjeev,Jackson,Matthew,Vega-Redondo,Fernando,Yariv,Leeat,2010.Networkgames.Rev.Econ.Stud. 77(1),218–244.

Glaeser,Edward,Sacerdote,Bruce,Scheinkman,José,1996.Crimeandsocialinteractions.Quart.J.Econ. 111(2),507–548.

Hiller,Timo,2012.Peereffectsinendogenousnetworks.Workingpaper,UniversityofBristol.

Kennan,John,2001.UniquenessofpositivefixedpointsforincreasingconcavefunctionsonRⁿ^:ânêlementary^result.^Rev.Êcon.^{Dynam. 4}^(4),^893–899.

König,Michael,Tessone,Claudio,Zenou,Yves,2014.Nestednessinnetworks:atheoreticalmodelandsomeapplications.TheoreticalEcon. 9,695–752.

Lin,Xu,2010.IdentifyingPeereffectsinstudentacademicachievementsbyspatialautoregressivemodelswithgroupunobservables.J.Lab.Econ. 28(4), 825–860.

Liu,Xiaodong,Pattachini,Eleanora,Zenou,Yves,Lee,Lung-Fei,2014.Criminalnetworks:whoisthekeyplayer?Workingpaper,StockholmUniversity.

Milgrom,Paul,Roberts,John,1990.Rationalizability,learningandequilibriumingameswithstrategiccomplementarities.Econometrica 58(6),1255–1277.

Tamer,E.T.,2003.Incompletebivariatediscreteresponsemodelwithmultipleequilibria.Rev.Econ.Stud. 70,147–167.

Vives,Xavier,1990.Nashequilibriumwithstrategiccomplementarities.J.Math.Econ. 19,305–321.

16 Therearevariouspossibleempiricalspeciﬁcationsforthea’s(includingindividualcovariatesand,possibly,contextualpeereffects),theg’s(linear-in- sumorlinear-in-means),andtheerrortermsε.See,forinstance,Blumeet al. (2010),Bramoulléet al. (2009),DeGiorgiet al. (2010),Lin (2010)andLiu et al. (2014).