1IntroductionThereisempiricalevidencesuggestingthatpeereectsandthest - PDF document

1IntroductionThereisempiricalevidencesuggestingthatpeere¤ectsandthestructureofsocialinteractionsmatterstronglyinexplaininganindividual’sowncriminalordelinquentbehavior.1Acriminal’splaceinthenetworkandtheknow-howonthecrimebusinessofhispartnersdeterminehiscriminalopportunitiesandconstraints,aswellashisinformationabouttheseopportunitiesandconstraints.Itisthereforecrucialtounderstandhowsuchcriminalnetworksareformedandstructured,andhowtheyevolveandperform.Di¤erentwaysofcharacterizingwhichnetworkstructuresarestablehavebeenproposedintheliteraturedependingonwhether(andhowfar)agentsanticipatethattheiractionmayalsoinduceotherstochangethenetworkrelationstheymaintain.2Thenotionofpairwisestablenetwork,introducedbyJacksonandWolinsky(1996),assumesthatagentsareabletomodifythenetworkonelinkatatime,andchoosetochangethenetworkiftheresultingnetworkimplieshigherpayo¤sforthedeviatingagents.A

ssuch,pairwisestabilityinvolvesfullymyopicagentsinthesensethattheydonotanticipatethatothersmightreacttotheiractions.Attheotherextremeendofthespectrum,anumberofsolutionconceptsinvolveperfectlyfarsightedagents,i.e.,agentsthatfullyanticipatethecompletesequenceofreactionsthatresultsfromtheirownactionsinthenetwork.However,thisassumptionofperfectfarsightedness,especiallywhenthenumberofagentsbecomeslarge,requiresaveryhighlevelofforesightonbehalfoftheagents.Kirchsteiger,Mantovani,MauleonandVannetelbosch(2016)provideexperimentalevidencesuggestingthatsubjectsareconsistentwithanintermediateruleofbehavior,whichcanbeinterpretedasaformoflimitedfarsightedness.Agentsonlyanticipatealimitednumberofreactionsbytheotheragentstotheactionstheytakethemselves.3Inthispaper,westudythecriminalnetworksthatagentsformwhencriminalsareneitherfullymyopicnorcompletelyfarsightedbuthavesomelimiteddegreeoffarsightedness

.Inotherwords,weshowhowthepredictionsaboutstablecriminalnetworksrelatetothedegreeoffarsightedness. 1SeePatacchiniandZenou(2008)amongothers.2MauleonandVannetelbosch(2016)provideacomprehensiveoverviewofthesolutionconceptsforsolvingnetworkformationgames.3SimilarexperimentalevidenceforlimitedfarsightednessisfoundinvanDolderandBuskens(2014). 1 other,istheuniquepairwisefarsightedlystableset.Moreover,theyshowthatthecompletenetworkisapairwisefarsightedlystablesetforanynumberofplayers.Whicharethecriminalnetworksthatwillemergeinthelongrunwhencriminalshavealimiteddegreeoffarsightedness?Weadoptthehorizon-KfarsightedsetofHerings,MauleonandVannetelbosch(2019)toanswerthisquestion.TheconceptencompassesboththepairwisefarsightedlystablesetandthepairwisemyopicallystablesetintroducedbyHerings,MauleonandVannetelbosch(2009).7AsetofnetworksGKisahorizon-Kfarsightedsetifthreeconditionsaresatis…ed.First

,deviationsoutsidethesetshouldbehorizon-Kdeterred.Second,horizon-Kexternalstabilityisrequired.Thatis,fromanynetworkoutsideofGKthereisasequenceoffarsightedimprovingpathsoflengthsmallerthanorequaltoKleadingtosomenetworkinGK.Third,aminimalityconditionisrequired.Thatis,thereisnopropersubsetofGKsatisfyingthe…rsttwoconditions.Herings,MauleonandVannetelbosch(2019)showthatahorizon-Kfarsightedsetalwaysexistsandprovideeasytoverifyconditionsforasetofnetworkstobeahorizon-Kfarsightedset.Inthispaper,we…ndthatincriminalnetworkswithncriminals,thesetcon-sistingofthecompletenetworkisahorizon-Kfarsightedsetwheneverthedegreeoffarsightednessofthecriminalsislargerorequalthan(n�1).Moreover,thecompletenetworkistheuniquehorizon-(n�1)farsightedset.Hence,weobtainaverysharppredictionforintermediatedegreesoffarsightedness(i.e.,adegreeoffarsightednessequalton�1),andshowthatalimiteddegreeoffars

ightedness(i.e.,atleastn�1)issu¢cienttorecoverthepredictionsobtainedincaseofcompletelyfarsightedcriminals.Knowledgeaboutthedegreeoffarsightednessofcriminalsisthereforeimportanttodeterminewhichcriminalnetworksarelikelytoemergeinthelongrunandtoimplementadequatedelinquency-reducingpolicies.Thepaperisorganizedasfollows.InSection2weintroducesomenotationsandbasicpropertiesofcriminalnetworks.InSection3wede…nethenotionofahorizon-Kfarsightedset.InSection4weidentifythehorizon-Kfarsightedsetofcriminalnetworks.Finally,inSection5weconclude. 7ThemyopicstablesetofDemuynck,Herings,Saulle,andSeel(2019)generalizesthepairwisemyopicallystablesettoalargeclassofsocialenvironmentsandshowshowituni…esthemostimportantconceptsofnon-cooperativegametheorylikeNashequilibriumandcooperativegametheorylikethecore. 3 probabilityofwinningthelootisgivenbyS(g)=#S=n.Thenetworkarchitecturedetermineshowt

helootissharedamongthecriminalsinthegroup.Considersomecriminali2NandletS2P(g)bethecriminalgroupibelongsto.Letdi(g)denotethedegreeofcriminaliing;i.e.,thenumberoflinkscriminalihasing.Wede…neci(g)=maxj2Sdj(g)asthemaximumdegreeinthiscriminalgroup.AcriminaliwhoispartofagroupS2P(g)expectsasharei(g)ofthelootgivenbyi(g)=(1 #fj2Sjdj(g)=cj(g)g,ifdi(g)=ci(g),0,otherwise.Thatis,withineachcriminalgroup,thecriminalthathasthehighestnumberoflinksgetstheloot.Iftwoormorecriminalshavethehighestnumberoflinks,thentheysharethelootequallyamongthem.Criminalihasaprobabilityqi(g)ofbeingcaught,inwhichcasehisrewardsarepunishedatarate�0.Itisassumedthatthehigherthenumberoflinksacriminalhas,thelowerhisindividualprobabilityofbeingcaught.Weassumethattheprobabilityofbeingcaughtissimplygivenbyqi(g)=n�1�di(g) n.Thetotalpayo¤sofcriminalibelongingtocriminalgroupS2P(g)arethereforeequalto

Yi(g)=S(g)i(g)(1�qi(g))B (1) =(#S n1 1#fj2Sjdj(g)=ci(g)g(1�n�1�di(g) n)B,ifdi(g)=ci(g),0,otherwise.Werequiren=(n�1)toguaranteethatpayo¤sarenon-negativeandpositiveforacriminalwiththehighestdegreeinhisgroup.3Horizon-KFarsightedSetWeproposethenotionofhorizon-KfarsightedsetintroducedbyHerings,MauleonandVannetelbosch(2019)todeterminethecriminalnetworksthatemergeinthelongrunwhencriminalsareneitherfullymyopicnorcompletelyfarsightedbuthavesomelimiteddegreeoffarsightedness. 5 Thesetf2K(g)=fK(fK(g))=fg002Gj9g02fK(g)suchthatg002fK(g0)gconsistsofthosenetworksthatcanbereachedbyacompositionoftwofarsightedimprovingpathsoflengthatmostKfromg.Weextendthisde…nitionand,form2N,wede…nefmK(g)asthosenetworksthatcanbereachedfromgbymeansofmcompositionsoffarsightedimprovingpathsoflengthatmostK.Letf1Kdenotethesetofnetworksthatcanbereachedfromgbymeansofanynumbe

rofcompositionsoffarsightedimprovingpathsoflengthatmostK.Lemma2inHerings,MauleonandVannetelbosch(2019)showsthatforeveryK1,foreveryg2G,itholdsthatf1K(g)f1K+1(g),andthatforKn0�1,foreveryg2G,itholdsthatf1K(g)=f1K+1(g)=f11(g).JacksonandWatts(2002)havede…nedthenotionofaclosedcycle.AsetofnetworksCisacycleifforanyg02Candg2Cnfg0g,thereexistsasequenceofimprovingpathsoflength1connectinggtog0,i.e.g02f11(g).AcycleCisamaximalcycleifitisnotapropersubsetofacycle.AcycleCisaclosedcycleiff11(C)=C,sothereisnosequenceofimprovingpathsoflength1startingatsomenetworkinCandleadingtoanetworkthatisnotinC.Aclosedcycleisnecessarilyamaximalcycle.Foreverypairwisestablenetworkg2P1,thesetfggisaclosedcycle.Thesetofnetworksbelongingtoaclosedcycleisnon-empty.Thenotionofahorizon-Kfarsightedsetisbasedontwomainrequirements:horizon-Kdeterrenceofexternaldeviationsandhorizon-Kexternalstability.Asetofn

etworksGsatis…eshorizon-Kdeterrenceofexternaldeviationsifallpossibledeviationsfromanynetworkg2GtoanetworkoutsideGaredeterredbyathreatofendingworseo¤orequallywello¤.11 De…nition1. ForK1,asetofnetworksGGsatis…eshorizon-Kdeterrenceofexternaldeviationsifforeveryg2G; (a) 8ij=2gsuchthatg+ij=2G,9g02[fK�2(g+ij)\G][[fK�1(g+ij)nfK�2(g+ij)]suchthat(Yi(g0);Yj(g0))=(Yi(g);Yj(g))orYi(g0)Yi(g)orYj(g0)Yj(g), (b) 8ij2gsuchthatg�ij=2G,9g0;g002[fK�2(g�ij)\G][[fK�1(g�ij)nfK�2(g�ij)]suchthatYi(g0)Yi(g)andYj(g00)Yj(g). 11Weusethenotationalconventionthatf�1(g)=;foreveryg2G. 7 pairwisefarsightedlystablesetG1de…nedbyHerings,MauleonandVannetelbosch(2009),thereisasetG0G1suchthatG0isalevel-(n0+1)farsightedset.13ThefollowingtheoremofHerings,MauleonandVannetelbosch(2019)willbeusedinthenextsectiontoidentifythehorizon-Kfarsightedsetofc

riminalnetworks. Theorem1(Herings,MauleonandVannetelbosch(2019)). ConsidersomeK2.Ifg2DJforsomeJKandforeveryg02Gnfggitholdsthatg2f1K(g0),thenfggisahorizon-Kfarsightedset.If,moreover,g2PK,thenfggistheuniquehorizon-Kfarsightedset.Theorem1requiresthatg2DJforsomeJK,sowehavetoshowthatg2fJ(g0)forallg0adjacenttog.ThehigherJ,theweakerthisrequirement,sowecouldreplacetherequirementg2DJforsomeJKbyg2DK�1.Toshowthatg2f1K(g0)forallg06=g,wehaveto…ndasequenceoffarsightedimprovingpathsoflengthatmostKthatconnectg0tog.Veryoftentheanalysisoffarsightedimprovingpathsofsmalllengthsisalreadysu¢cient.ThehigherK,theeasieritistosatisfytheconditionsofTheorem1andto…ndasingletonhorizon-Kfarsightedset.Finally,toshowthatg2PKrequiresthatfK(g)=;.Thisrequirementismoredi¢culttosatisfyforincreasingvaluesofK.4Horizon-KFarsightedSetofCriminalNetworksThroughoutthissection,weassumen3.Figure1presents

thepayo¤sfor3-playercriminalnetworkswithB=9and=1inexpression(1).Table1showsthefarsightedimprovingpathsforthedi¤erentpossiblevaluesofK.Itcanbeveri…edthatthefarsightedimprovingpathsforthe3-playercasedonotdependonthespeci…cchoicesforBand.Forthethree-playercase,wecomputetheclosedcyclesanduseTheorem3inHer-ings,MauleonandVannetelbosch(2019)toconcludethatG1=P1=fg1;g2;g3;g7gisthehorizon-1farsightedset,soG1consistsofallpairwisestablenetworks.Therearemanynetworksthatarestablewhenplayersaremyopic. 13Herings,MauleonandVannetelbosch(2009)de…neapairwisefarsightedlystablesetasasetG1ofnetworkssatisfyinghorizon-1deterrenceofexternaldeviationsandminimality,butwithhorizon-1externalstabilityreplacedbytherequirementthatforeveryg02GnG1,f1(g0)\G16=;. 9 criminalcasetothen-criminalcasewouldbeanynetworkconsistingofcompletecomponents,wherenotwocomponentshavethesamedegree.Butalsoanyn

etworkwithasinglecomponentwhereallplayershaveadegreeatleastequaltotwoandoneplayerhasadegreethatisatleasttwotimeshigherthanthedegreeofanyotherplayerispairwisestable.WewillarguenextthatfgNgisahorizon-KfarsightedsetwheneverKn�1.Weshow…rstthatthecompletenetworkispairwisedominant. Lemma1. ForcriminalnetworksitholdsthatgN2D1. Proof. ConsiderthenetworkgN�ijforsomeij.Itholdsthatdi(gN�ij)=dj(gN�ij)ci(gN�ij)=cj(gN�ij),soYi(gN�ij)=Yj(gN�ij)=0Yi(gN)=Yj(gN),andgN2f1(gN�ij).WehaveshownthatgN2D1. Weshownextthatthecompletenetworkcanbereachedfromanystartingnetworkbyrepeatedapplicationofatmostn�1degreesoffarsightedness. Lemma2. Forcriminalnetworksitholdsforeveryg2GnfgNgthatgN2f1n�1(g). Proof. Step1.Ifghasacomponentwhichisnotcomplete,thenthereisg02fn�1(g)suchthatg(g0.LetS2P(g)beacriminalgroupsuchthatsomeinternallinksaremissing,gjS6=gS.Ifforeveryi2Sitholdst

hatdi(g)=ci(g),soallplayersinShavethesamedegree,thenanytwounlinkedplayersiandjinScreatealinktoformthenetworkg+ijandimprovetheirpayo¤ssincetheincreaseintheirdegreeincreasestheshareinthelootandlowerstheprobabilityofbeingcaughtforbothplayers,i(g+ij)�i(g),j(g+ij)�j(g),qi(g+ij)qi(g),andqj(g+ij)qj(g),soYi(g+ij)&#x-373;Yi(g)andYj(g+ij)&#x-373;Yj(g).Wehavethatg!1g+ij,soclearlyg+ij2fn�1(g).IftheplayersinSdonotallhavethesamedegree,leti2Sbeaplayerwithdi(g)=ci(g):Ifci(g)#S�1;thenPlayerilinkswithanyPlayerjsuchthatij=2gtoformthenetworkg+ij:ItholdsthatYi(g+ij)&#x]TJ/;༕ ;.9;U T; 13;&#x.868;&#x 0 T; [00;Yi(g)&#x]TJ/;༕ ;.9;U T; 13;&#x.868;&#x 0 T; [00;0since 11 Fork=0,wehaveYi(g0)=1 n(1�qi(g))BYi(gK),Yj0(g0)=1 n(1�qj0(g))BYj0(gK),whereweuseqi(g0)&#x-277;qi(gK)andqj0(g0)&#x-277;qj0(gK)togetthestrictinequ

alities.Fork=1;:::;K�1,itholdsthatPlayeriisconnectedtoPlayerj0,sodi(gk)dj0(gk)=ci(gk),soi(gk)=0and0=Yi(gk)Yi(gK).Similarly,itholdsthatPlayerjkisconnectedtoPlayerj0,sodjk(gk)dj0(gk)=cjk(gk),sojk(gk)=0and0=Yjk(gk)Yjk(gK).Step3.Foreveryg2GnfgNg,itholdsthatgN2f1n�1(g).BycombiningtheresultsofStep1andStep2,wehavethatforeveryg2GnfgNg,thereisg02fn�1(g)withstrictlymorelinksthang.SincethecompletenetworkgNhasn(n�1)=2links,we…ndthatgN2fn(n�1)=2n�1(g)f1n�1(g). UsingTheorem1,weprovenowthatthecompletenetworkfgNgisahorizon-KfarsightedsetforeveryKn�1.14NoticethattheleveloffarsightednessneededtosustainthecompletenetworkfgNgisquitesmallwhencomparedtothenumberofpotentialnetworksandthemaximumlengthofpaths.15 Theorem2. ForcriminalnetworksitholdsthatfgNgisahorizon-KfarsightedsetforeveryKn�1. Proof. ByLemma1wehavethatgN2D1.ByLemma2wehavethatforeveryg

02GnfgNgitholdsthatgN2f1n�1(g0)f1K(g0),wheretheinclusionfollowsfromLemma2inHerings,MauleonandVannetelbosch(2019).WearenowinapositiontoapplyTheorem1andconcludethatfgNgisahorizon-Kfarsightedset. HowabouttheuniquenessoffgNgasahorizon-Kfarsightedset?ItistemptingtousetheapproachofTheorem1andshowsucharesultbyprovingthatgN2PK.However,considerthecasewith6playersandletg0=gN�16�26�35�45.ForanyvalueofBand,16weclaimthatg02f12(gN),sogN=2P12.Sincethenetwork 14Herings,MauleonandVannetelbosch(2009)showthatintheexampleofcriminalnetworkswithnplayers,thecompletenetworkfgNgisapairwisefarsightedlystableset.15Oncethenetworkconnectingdelinquentsisendogenous,Calvo-ArmengolandZenou(2004)…ndthatallcompletenetworks,whereallplayersinthepoolofcriminalsarelinkedtoeachother,arepairwisestable.Noticethatthesizeofthepoolofcriminalsdependsonthewageonthelabormarket.16Wemaintaintheassumpti

onthatn=(n�1): 13 Proof. Supposeg0isanelementoffn�1(gN).Letg0;:::;gKwithg0=gNandgK=g0beafarsightedimprovingpathoflengthKn�1.ByLemma3itholdsthatci(g0)isindependentfromi,sowedenoteitbyc.LetMNbesuchthati2Mifandonlyifdi(g0)=canddenotethecardinalityofMbym.Itcannotbethatm=n,sincethenallplayershavelowerpayo¤sing0thaningNbecausetheprobabilityofbeingcaughtishighering0thaningN.SincebyLemma3g0isconnected,itfollowsthatYj(g0)=0forallj2NnM.Aplayerj2NnMwillthereforenotseveralinkatanynetworkinthefarsightedimprovingpathg0;:::;gK.ItfollowsthatXi2M(n�1�di(g0))Xj2NnM(n�1�dj(g0)).Sincedi(g0)�dj(g0)wheneveri2Mandj2NnM,wehavethatm�n=2.Sinceatleastonelinkijwithi2Mandj2Nismissinging0,itfollowsthatthemaximumdegreeing0satis…escn�2.ThenumberKisequaltothenumberoftimesalinkijisseveredwithi2Mandj2NnMplusthenumberoftimesalinkijiscutwithi;j2Mplust

henumberoflinkadditions.Wearguenextthatlowerboundsforthesethreenumbersaregivenby2(n�m),2m�n�1,and1,respectively.SinceallplayersinNnMexperiencedtheseveranceofatleasttwolinks,andanysuchlinkiscutbyaplayerinM,alowerboundforthe…rstnumberis2(n�m).Fork=0;:::;K,letL(gk)=fi2Njdi(gk)=n�1gbethesetofplayerswithdegreen�1andlet`(gk)=#L(gk)beitscardinality.Clearly,itholdsthat`(gN)=nand`(g0)=0.Letk0bethelowestvalueofksuchthat`(gk)mforallkk0.Since`(gk)�`(gk+1)2,we…ndthat`(gk0)=mor`(gk0)=m�1.Thesumofthecardinality`(gk0)ofL(gk0)andthecardinalitymofMisthereforeatleast2m�1.Sincethereareonlynplayers,itfollowsthat#(L(gk0)\M),thecardinalityofthesetofplayersinL(gk0)thatbelongtoM,isatleast2m�n�1.Forallkk0,foralli2L(gk),itholdsthatYi(gk)�Yi(g0),sincetheloothastobesharedwithlessorthesamenumberofcriminalsandtheprobabilityofbeingcaughtisstrict

lylesswhencomparinggktog0.Suchaplayeriwillthereforeneverchoosetoseveralinkhimself,sowheneveralinkinvolvingplayeri2L(gk)isseveredwhengoingfromgktogk+1,itmustbebyaplayerinMnL(gk).Itfollowsthat`(gk)�`(gk+1)1.Since#(L(gk0)\M)2m�n�1,we…ndthatgoingfromgk0tog0involvesthedeletionofatleast2m�n�1linksijwithi;j2M. 15 offarsightedness.Weadoptthehorizon-KfarsightedsetofHerings,MauleonandVannetelbosch(2019)toshowhowthepredictionsaboutstablecriminalnetworksrelatetothedegreeoffarsightedness.Ahorizon-Kfarsightedsetalwaysexists.We…ndthatincriminalnetworkswithncriminals,thesetconsistingofthecompletenetworkisahorizon-Kfarsightedsetwheneverthedegreeoffarsightednessofthecriminalsislargerthanorequalto(n�1).Moreover,thecompletenetworkistheuniquehorizon-(n�1)farsightedset.Hence,alimiteddegreeoffarsightednessissu¢cienttorecoverthepredictionsobtainedincaseofcompletely

farsightedcriminals.AcknowledgmentsAnaMauleonandVincentVannetelboschare,respectively,ResearchDirectorandSeniorResearchAssociateoftheNationalFundforScienti…cResearch(FNRS).FinancialsupportfromtheMSCAITNExpectationsandSocialIn‡uenceDynamicsinEconomics(ExSIDE)GrantNo721846(1/9/2017-31/8/2020),fromtheBelgianFrenchspeakingcommunityARCproject15/20-072ofSaint-LouisUniversity-Brussels,andfromtheFondsdelaRechercheScienti…que-FNRSresearchgrantT.0143.18isgratefullyacknowledged.References [1] Ballester,C.,A.Calvo-ArmengolandY.Zenou,2010.Delinquentnetworks.JournaloftheEuropeanEconomicAssociation8,34-61. [2] Bezin,E.,T.VerdierandY.Zenou,2021.Crime,brokenfamiliesandpunish-ment.AmericanEconomicJournal:Microeconomicsforthcoming. [3] Calvo-Armengol,A.andY.Zenou,2004.Socialnetworksandcrimedecisions:theroleofsocialstructureinfacilitatingdelinquentbehavior.InternationalEconomicReview45,93

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