©CopyrightJASSS

FranciscoJoséLeónMedina,FranciscoJoséMiguelQuesadaandVanessaAlcaideLozano(2014)

TheProductionofStep-LevelPublicGoodsinStructuredSocialNetworks:AnAgent-BasedSimulation

JournalofArtificialSocietiesandSocialSimulation 17(1)4<http://jasss.soc.surrey.ac.uk/17/1/4.html>

Received:26-May-2012Accepted:03-Nov-2013Published:31-Jan-2014

Abstract

Thispaperpresentsamulti-agentsimulationoftheproductionofstep-levelpublicgoodsinsocialnetworks.Inpreviouspublicgoodsexperimentalresearchthedesignofthesequenceorderingofdecisionshavebeenlimitedbecauseofthenecessityofsimplicitytakingpriorityoverrealism,whichmeanstheyneveraccuratelyreproducethesocialstructurethatconstrainstheavailableinformation.Multi-agentsimulationcanhelpustoovercomethislimitation.Inourmodel,agentsareplacedin230differentnetworksandeachnetworks’successratesareanalyzed.Wefindthatsomenetworkattributes-densityandglobaldegreecentralityandheterogeneity-,someinitialparametersofthestrategicsituation-theprovisionpoint-andsomeagents’attributes-beliefsabouttheprobabilitythatotherswillcooperate-,allhaveasignificantimpactonthesuccessrate.Ourpaperisthefirstapproachtoanexplanationforthescalarvariantofproductionofpublicgoodsinanetworkusingcomputationalsimulationmethodology,anditoutlinesthreemainfindings.(1)Alessdemandingcollectiveeffortleveldoesnotentailmoresuccess:theeffortshouldneitherbeashighastodiscourageothers,norsolowastobelettoothers.(2)Moreinformedindividualsdonotalwaysproduceabettersocialoutcome:acertaindegreeofignoranceaboutotheragents’previousdecisionsandtheirprobabilityofcooperatingaresociallyusefulaslongasitcanleadtocontributionsthatwouldnothaveoccurredotherwise.(3)Densehorizontalgroupsaremorelikelytosucceedintheproductionofstep-levelpublicgoods:socialtiesprovideinformationabouttherelevanceofeachagent’sindividualcontribution.Thissimulationdemonstratestheexplanatorypowerofthestructuralpropertiesofasocialsystembecauseagentswiththesamedecisionalgorithmproducedifferentoutcomesdependingonthepropertiesoftheirsocialnetwork.

Keywords:PublicGoods,CollectiveBehaviour,DecisionMaking,SocialNetworks

Introduction

1.1 Sinceindividualrationalitycanleadtoasuboptimaloutcome,acollectiveactionproblememergesintheproductionprocessofpublicgoods(Taylor1987).Severalcharacteristicsofpublicgoodsexplainwhytheirproductiontakestheformofa"socialdilemma"(Hardin1982;Olson1965;Taylor1987):a)thegoodisjointlyproducedbythecontributionsoftheindividualgroupmembers,buttypicallynoteverysinglemember'scontributionisrequired;b)contributionsarecostlyevenwhenthevaluethatisobtainedfromthepublicgoodishigherthantheindividualcostofthecontribution;c)oncethegoodisproduced,itwillbeavailabletoallmembersofthegroupsinceexcludingnon-contributingmembersfromitsenjoymentisdifficultorcostly;d)amember'susageofthegooddoesnotdiminishitsavailabilitytoothermembers(non-rivalness).Thislastcharacteristicisveryinfrequent,andherewewillconsiderthatthegoodprovidesanequalbenefittoallmembersofthegroup.

1.2 Theproductionofpublicgoodshasbeenstudiedextensivelythroughthefollowingexperimentaldesign:a)agroupofindividualsreceiveanendowmentof(e)unitsandhavetodecide,anonymously,howmuch(x)theywishtocontributetoacommonpool;b)theshareoftheendowmentthatisnotcontributedtothecommonpool(e-x)retainsitsoriginalvalue,whilethesharethatiscontributedismultipliedbyafactorc;c)thecommonpool(∑x*c)isdividedinequalpartsamongallthemembersofthegroup;d)factorcisdeterminedsuchthatitwouldbebetterforeachplayertokeepratherthancontributingaunit,butitwouldbeworseforeachplayerifnoonecontributesratherthanifeverybodycontributesalltheirendowments.

1.3 Thisexperimentaldesignallowsforcountlessvariations:subjectscanhaveequalorunequalendowments,decisionsonhowmuchtocontributetothecommonpoolcanbebinary–tocontributeallornottocontribute–ordiscrete–tocontributeafractionofe–,etc.Thevariantweareinterestedinspecifiesadifferentproductionfunction;productionfunctionsmakethelevelofproductionofagooddependentonthecontributions.Themaindistinctionisbetweencontinuousand

step-levelfunctions(León2010)[1].Ineconomicterms,wecanstipulatethateitherthegoodhasapricebyunit(continuousgood)orithasauniquepricethathastobereached(step-levelgood).

1.4 Intheexperimentaldesignpresentedabove,theproductionfunctioniscontinuous.Eachcontributionaddstotheproductionofthegood,andallcontributionsaddthesame(itisalinearfunction).However,inthisarticlewefocusonpublicgoodswithastep-levelproductionfunction.These"step-levelgoods"(SLG)or"lumpygoods"arecharacterisedbytheexistenceofaprovisionpoint:aminimumofcontributionsisneededtoproducethepublicgood.Iftheprovisionpointisreached,thegoodisproduced;ifthepointisnotreached,thegoodisnotproducedatall.ThebasicSLGmodelstatesthatoncetheprovisionpointisreached

additionalcontributionsdonotgenerateahigherlevelofproduction.[2]

1.5 Intheearly1980s,MarwellandAmes(1980),Hardin(1982),Taylor(1987)andTaylorandWard(1982)highlightedthedistinctionbetweencontinuousandstep-levelgoods.Itisnowwidelyacceptedthatpublicgoodsanalysisisnotpossiblewithoutexplicitlystatingthekindofgoodwearereferringto(Kollock1998:189;KomoritaandParks1995;Ledyard1995;MarwellandOliver1993:24).

1.6 ThevariationoftheproductionfunctionweintroduceinourMABexperimentaldesignisoneofthemostinterestingpossiblevariations,mainlybecausetheinclusionofaprovisionpointcompletelychangesthelogicofthesituation.Intruth,thevariationofotherdesignelements,suchastheendowment–equitable/inequitable–,thetypeofdistribution–equitable/inequitable–,thetypeofdecision–binary/discrete–canhaveasignificantimpactontheresults,butdonotmakeapublicgoodsgameanydifferenttoann-personprisoner'sdilemma.Theintroductionofaprovisionpointdoesachievethedifference:sincethereis

aprovisionpoint,defectionisnolongeradominantstrategyinaone-shotgame.[3]

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1.7 Collectiveactionintheformofproducingastep-levelpublicgoodisratherfrequent.Forexample,thecasepresentedbyTaylor(1987)regardinganelectionwithtwooptions,AandB,onacommitteeoravolunteerassistanceparliamentinwhichtherearetwoparties:amajorityandaminority.ThelargestpartyprefersoptionAtoB.Whichmajoritygroupmemberswillattendthesession?Totheextentthatattendanceisvoluntaryandcostly,eachmemberfromthemajoritygroupwouldpreferthatothermembersofhisgroupattendsandhelpwiththeirvotetoelectoptionA.Moreover,inthissituation,agreaternumberofmajoritymembersthanminoritymembersattendingisenough,butbeyondthatpointfurtherattendeesofthemajoritygrouparenotrequiredtoproducethe"good":winningtheelection.NotethatinthiscaseitisnotstrictlynecessaryforAtobeapublicgood,butthelogicofthesituationisexactlywhatwemodelinthispaper.Anotherrealexampleofscalarpublicgoodswouldbethefollowing:insomeuniversitiesandcolleges,thedepartment'sbudgetpartiallydependsontheproductivityoftheirmembers,measuredasthetotalnumberofjournalpapersproducedbythedepartment.Ifthereisaminimumthresholdofarticlestoobtaintheadditionalbudget,thismaybeconsideredascalarpublicgood.Apartfromanyotherpersonalbenefitofpublication,eachmemberofthedepartmenthastoestablishitscontributionatalevelofeffort,knowingthatonlyifthecumulativeeffortsreachtherequiredlevelthedepartmentwillgetanadditionalamountintheannualbudget.

1.8 AnexperimentaldesignforstudyingSLGproductioncouldstartfromthesimplemodeloutlinedabove,andthenmodifythethirdcharacteristicbyaddingaminimumamountor"provisionpoint"requiredforthegoodtobeproduced.ThefirstexperimentswiththistypeofdesignwereconductedbyvandeKragtetal.(1983)andRapoport(1985,1987,1988,1993).Inthispaperweaimtocomputationallyreplicateavariantoftheseexperiments.Thepaperisorganisedasfollows.First,wepresentdifferenttypesofsequentialrunningorderingoftheagents'decisionsandourproposalofanetwork-dependentsequenceformalisation.Second,webrieflyreviewtheliteratureonsocialnetworksandexperimentationinpublicgoods.Third,wepresentanagent-basedmodelthatreplicatestheexperimentalsetting(Miguel2011).Fourth,wepresenttheresultsofdifferentsimulationsandthediscussion.

ANetwork-DependentSequenceOrderingRulefortheProductionofPublicGoods

2.1 TheliteratureonthesubjectsuggeststhatthedecisionsofexperimentalsubjectsinSLGexperimentsareverysensitivetothesequenceofthedecisions(AbeleandEhrhart2005;ErevandRapoport1990)becausedifferentsequencingimpliesdifferentinformationavailableatthedecisionmoment.Fourdifferentrulesfordeterminingtheorderofagents'decisionsequencingcanbefoundinexperimentalresearch(Budescuetal.1997):(1)the"simultaneousprotocol",wheresubjectshavenoinformationabouttheoneanother'sdecisions;(2)the"sequentialprotocol",wheresubjectshavecompleteinformationaboutallpreviousdecisions;(3)the"positionalprotocol",whereasubjecthasinformationaboutherpositioninthesequencebutnotaboutpreviousdecisions;and(4)the"cumulativeprotocol",wherethesubjectknowsthequantityofthecontributionsmadebeforeher,butnotherownpositioninthesequence.

2.2 Inlaboratoryexperiments,allthesesequencing–orrunningorder–ruleshavebeenlimitedintheirdesignbythenecessityofsimplicitytakingpriorityoverrealism,andcanthereforeresultinunrealisticassumptionsabouttheavailableinformationwhenpeoplemaketheirchoices.Thissimplificationimpliesahighercapacitytocontrolforvariables,generatingcumulativeknowledgeaboutfactorsthatinfluencebehaviour.Thelackofrealisminherentinthesesequencingrulesdoesnotnecessarilyimplyalackofvalueorheuristicutility.Ourproposalaimstoelaborateandapplyanewruletocomplementtheexistingones.

2.3 Amorerealisticsequenceorderingruleshouldrepresentadecisionalsituationwhereindividualsareembeddedinarelationalstructurethatconstrainstheavailableinformation.Inanon-artificialsituationofpublicgoodsproduction,individualstypicallyhavecertainlocalknowledgeaboutthedimensionsoftheirgroup,buttheyarenotindirectcontactwithallmembers.Inotherwords,itisthenetworkstructurethatdeterminesavailableinformationintermsofwhatothersthink,sayordo.Thatiswhy,withtheexceptionofexperimentsbySuriandWatts(2001),itiscommonlyacceptedinexistingliteraturethatasocialnetwork'stopology,ormorespecifically,thestructuralpropertiesofasocialnetworkhaveaneffectonthelevelofcontributiontocollectiveaction.Co-operativebehaviourcanspreadinanetworkduetosuchmechanismsasmimicry(FowlerandChristakis2010)orsocialcomparison(Zschache2012),butinthispaperwefocusontheroleplayedbytheuseoflocalinformationonthedecisionsofadjacentnodes.

2.4 Anexperimentalset-upaimedatreproducingthissituationwouldbetoocomplextobeexecutedinalaboratory,soweinsteadtestthisnewsequenceorderingrulewithinacomputationalmulti-agentsimulationmodel.Thissolutionallowustoavoidthetrade-offbetweensimplicityandrealismtypicalforlaboratoryexperiments,andsubstituteitforatrade-offbetweenrealisminsomeaspects–agentswillbeembeddedinanetwork–andrealisminothers–decisionswillbemadebysoftwareagentsnotbyrealpeople.Inadditiontotheaimtocomputationallyreplicatesomeexperimentalset-upsofthereviewedliterature,otherconsiderationssuchastherelativeadvantageintermsofcosts,resourcesandtime,orthefactthatthesimulationcanpredictwhentherearedifferencesintheparametersthatdescribeagivenenvironment,playinfavouroftheuseofABMresearchmethodology.

TheFormalModelDescription

3.1 Inthissectionweoutlinetheformalmodeltobecomputationallyimplemented:

1. AgroupofagentsN={1,…,n}participatesinapublicgoodsgame.2. Agentsknowthesizeofthegroup–thenetwork–theybelongto.3. Eachplayeri(i∈N)receivesanendowmentofeiunits(ei>0).Thevalueofallendowmentsisequal.4. Eachagenti∈Ndecidesaboutthequantity(xi)fromthesetX={0,ei}thatshewishestocontributetotheproductionofthepublicgood.Agentimust

makeabinarydecision:toeithercontributeallherendowment(xi=ei)orcontributenothing(xi=0).[4]

5. Thepositionofeachagentinthesequenceofdecisionsisrandomlydetermined.6. Agentsareplacedinanetwork,g,wheretheyonlyinteractwithsomemembersofN.Networkgisconstitutedbysymmetricallinks:gij=gji.7. Eachagenti∈NfacesasetofadjacentnodesorneighboursNi⊂Nwithwhichsheisconnected.Ni={j∈N|gij=1}.Thenumberofi'sneighboursiski–i's

nodedegreecentrality–.8. Ifgij=1agentjwillknowxiifintherandomlydecidedsequence(seepoint5)xihastakenplacebeforexj,andviceversa.9. Agentsknowtheirki,butnotthatoftheirneighbours.TheyalsodonotknowtheirpositioninthesequenceofdecisionsofN,althoughtheyknowtheir

positioninthesequenceofdecisionsofNi.10. Atthedecisionmomentt,eachagenthasthreepiecesofinformationaboutotheragents'decisions:a)howmanyneighbourshavemadedecisions

previously–B,for"Behaviours"–;b)∑ j∈Nixj,t-1,thatis,howmanyadjacentnodeshavecontributedtheirendowmenttotheproductionofthepublicgood

int-1–C,for"Cooperators"–;c)howmanyagentsparticipateinthenetworkaboutwhomshehasnoinformation,n-ki.11. Thereexistsaprovisionpoint(m)knownbyallagents,0<m<∑i∈Nei.12. If∑i∈Nxi≥m,eachagentreceivesapayoffr.If∑i∈Nxi<m,thegoodisnotproducedandallthecontributionsarelost.13. Thepayoffireceivesifthegroupreachestheprovisionpointisr=mc/n,wherecisafactorknowntoallagents,suchthatei<r.14. Thefinalpayoff(πi)thatagentireceivesisdeterminedbythefollowingpayofffunction:

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(1)

3.2 Inourmodel,called"NetCommons",peoplearelocatedinasocio-matrixthatconstrainstheavailableinformation,conditioningthedecisionprocess.Letusillustratetwodifferentdecisionsequencingordersinthisnetwork–thesequenceofdecisionsinthematrixfollowsanalphabeticalordering–.

Diagram1.AtypicaldecisionorderinginaN=7network

3.3 AgentBdecidesknowingthedecisionofAandknowingthatGhasnotdecidedyet.SincesheknowsherpositioninthesequenceofdecisionsofNBbutshedoesnotknowherpositioninthesequenceofN–seepoint9–,agentBcannotknowwhetherC,D,E,orFhavealreadydecided.Similarly,agentCdecidesknowingthedecisionofAandknowingthatEandDhavenotdecidedyet.SincesheknowsherpositioninthesequenceofdecisionsofNCbutshedoesnotknowherpositioninthesequenceofN,agentCcannotknowwhetherB,F,orGhasalreadydecided.Eachindividualhasaccesstoinformationaboutpartofthedecisionsalreadytaken,butfacesuncertaintywithrespecttotherestofthepastandfuturedecisions.

NetworksandExperimentationonPublicGoods

4.1 Moreno'swork(1951)isoftenreferredtoasthestartingpointofsocialnetworkanalysis.Onlyinthelast20yearshasnetworkanalysisbeenappliedinexperimentaleconomics–forareview,seeKosfeld(2004).Thesestudieshavefocusedmostlyoncoordinationnetworks,buyer-sellernetworksandnetworkformation.

4.2 Concerningexperimentalresearchonpublicgoods,labexperimentsconsideringtheroleofnetworksarescarce–someexceptionsareBonacich1990;CárdenasandJaramillo2009;Sonnemansetal.2006;FowlerandChristakis2010–andavailableresearchusuallyconsidersnetworkstobetheoutcomeofagenerativeprocessratherthanapreconditionforinteraction.KniggeandBuskens(2010),forexample,presentedanexperimentinwhichsubjectsseektoestablishrelationshipswithothersthatshareaccesstothegoodproduced,sothatthenetworkemergesfromtheinteractions.Thispaucityofnetworkanalysisinthecontextoflaboratoryresearchonpublicgoodsissurprisingastheconsensusisthatovercomingthefree-riderproblemrequiresestablishingsomeformofinterdependencebetweendecisions(Marwelletal.1988).Therelevanceofsocialtiesindeterminingparticipationincollectiveactionwaspointedoutalongtimeago(Tilly1978;Oberschall1973).Theexplanationforthisismostlikelythetechnicaldifficultiesofreproducingcountlessspecificconfigurationsofnetworksinalaboratorycontext.

4.3 Consistentwiththis,themaincontributionstounderstandingtherelationbetweennetworksandpublicgoodshavebeenrealisedoutsidethelab,throughmathematicalmodelsandsimulations.NovakandMay(1992)werethefirsttostudythedynamicsoftheprisoner'sdilemmawhenagentsareplacedinatwodimensionalspacethatconstrainsthemtointeractwiththeirneighbours.Sincethen,thesociallocationoftheagentshasbeenconsideredakeyelementtobemodelledintheoriesofcollectiveaction,thusopeningthedoortothestudyoftherelationshipbetweennetworksandpublicgoods.BramoulléandKranton(Bramoullé2007)werethefirsttopresentanetworkmodelofpublicgoods,whileotherauthorsuseanevolutionaryapproachtopublicgoodswithoutpayingattentiontothesocialstructure(e.g.,Yeetal.2011).Inthefieldofmathematicalandcomputationalmodelling,asinexperimentalresearch,theanalysisoftherelationshipbetweennetworksandpublicgoodshasemphasisedtheanalysisofreticulardynamics,i.e.theprocessesofnetworksemergency.Thus,forexample,Takácsetal.(2008)andSkyrmsandPemantle(2000)presentedmodelsinwhichtheparticipantsincollectiveactioncanstrategicallyreviewtheirrelationships,thusgeneratingreticulardynamics,andBravoetal.(2012)claimedthattheendogenousformationofthenetworkisevenmoreimportantforcooperationthanthenetworktopology.Ingeneral,thesemodelsarepartofabroadertradition,studyingtheemergencyofcomplexnetworksfromlocalinteractions,suchasmodelsinSutcliffeetal.(2012)andPujoletal.(2005).

4.4 Table1summarisesthedifferencesbetweenoursimulationmodelandtheleadingmodelsonnetworksandcollectiveactioninliterature.Ascanbeseen,NetCommonsisthefirstsimulationmodelofSLGproduction.Totheextentthatthelogicofthesituationiscompletelydifferentwhenthegoodhasaprovisionpoint,weunderstandthattheoriginalcontributionofthisarticleisthatitwillbethefirsttoexploretheroleofreticulartopologyonthesuccessofcollectiveactionwhichaimstoproduceastep-levelpublicgood.

Table1:NetCommonscomparedwithleadingmodelsofnetworksandpublicgoods

Marwell(1988) Gould(1993) Chwe(1999) NetCommons

Method Simulationandmathematicalmodel

Mathematicalmodel Mathematicalmodel Simulation

Agents Motivationanddecisionalmechanism

Expectedutilitymaximisation

Rationalityandjusticenorms

Probabilityofreachingthethreshold

Expectedutilitymaximisation

(criticality)Decisioncost Considered Irrelevant Notconsidered ConsideredEndowment Differentforeachagent Notconsidered Notconsidered Equalforallagents

Reward Differentforeachagent Notconsidered Notconsidered EqualforallagentsUnconditionalcontribution

Impossible Possible Impossible Impossible

Threshold Nothreshold Nothreshold Endogenousand Exogenousand

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precondition objectiveIndividualefficacy Dependentonthe

decisionofthegroupDependenton

previousdecisionsNotDependentonpreviousdecisions

Dependentonpreviousdecisions

Networks Links Asymmetricalanddirect Symmetricalanddirect

Asymmetrical,directandindirect

Symmetricalanddirect

Useofthelinks Costly Notcostly Notcostly NotcostlyAgent'sknowledgeof

thenetworkAgentsknowthe

structureAgentsknowtheir

neighboursAgentsknowthe

structureAgentsknowtheirneighboursandn

Informationobtainedinthenetwork

Perfectinformation Behavioursandpredispositions

Others'personalthresholds

Others'behaviours

Output Mainresult Sumofcontributions Sumofcontributions Sumofcontributions Successrate

TheSimulationModelDescription

5.1 ThesimulationwasconductedthroughtheNetlogoplatform.Thecodeisavailableat"CoMSESComputationalModelLibrary",thepublicrepositoryofOpenABMConsortium,http://www.openabm.org/model/2522/version/1/view(Miguel2011).Thismodelimplementstheformalnetwork-dependentdecisionsequenceorderingoutlinedintheprevioussection.However,sincethemodelintendstocapturetheessentialtraitsofSLGproductions,wecompleteherethespecificationoftheformalmodelbyincludingfourelements:(1)theagents'decisionprocess;(2)theinitialparameters;(3)thenetworkproperties;and(4)thedynamicsoftheprocessanditsoutput.

Howagentsdecideinthemodel

5.2 IntheNetCommonsmodel,agentsareassumedtobeself-interestedutilitymaximisers.Experimentshaveshownthatsubjectswhorepeatedlyfaceapublicgoodsgametendtoactinaccordancewiththepredictionsofrationalchoicetheory(Ledyard1995).Theoreticalpredictionsfromoursimulationcanbeconsideredaswell-groundedhypothesesofwhatwereasonablyexpecttofindinrealcases(includinglaboratoryexperiments).Moreover,ourmodelcanoffersomepracticaladviceonthemosteffectivewayofproducingapublicgood.Wecouldconsiderprosocialagents,butconstructingamodelbasedonthemostunfavourableassumptionsregardingtheproductionofpublicgoods–withself-interestedutilitymaximiseragents–isthebeststrategy.Prosocialagentswouldmerelyraisesuccessrates–theirprobabilityofcooperationishigher–.

5.3 Tomodeldecisionsweshallusetheconceptofself-efficacyorcriticality,anditsformalisationbasedonprobabilitytheory.InthecontextoftheproductionofSLG,criticalityreferstotheconditionwherebyadecisionisnecessaryandsufficienttoreachtheprovisionpoint.InNetCommons,agentshavethecognitivecapacitytoestimatetheirprobabilityofbeingcritical.

5.4 Whenanagentdecideswhethertocontributetothecommonpool,shewillconsidertheprovisionpoint(m),therewardthatsheexpectstoobtainifthegoodisproduced(r),thecostofhercontribution(x),thenumberofneighboursthathavepreviouslydecided(B),andthenumberofneighboursthathavealreadycooperated(C).Moreover,thedecisionwilldependonherbeliefthatanexactnumberofm-1willcontribute,makinghercontributioncritical,sothedecisionultimatelydependsontheprobabilityofcooperationagentsattributetoeachother(p).

5.5 AnobviousimplicationfortheNetCommonsmodelisthatanagentcannotrationallyforeseethedecisionsoftheotheragentsatthemomentofherdecision.Shedoesnotknowtheinformationthatotherswillhaveatthemomentoftheirdecision,sinceshedoesnotknowwhichotheragentsareneighboursorwhatinformationtheyreceivefromtheiradjacentlinks.Inthissituationofuncertaintyregardingothers'decisions,therationaloptionwouldbetoattributetoothersacertainprobabilityofcooperation,0≤p≤1.Inourmodel,weintegrateahomogeneityassumption(Rapoport1985),wherebyallagentsattributethesame

probabilitytoallotheragents[5].

5.6 Takingallthesefactorsintoaccount,agentswillfindthemselvesinoneofthefollowingsixexhaustiveandmutuallyexclusivesituations:

Situation1:Thenumberofneighbourswhohavealreadycooperatedisequalorhigherthantheprovisionpoint.Inthiscasethepublicgoodwillbeproducedandtheagent'scontributionisnotstrictlyrequired.Arationalagentwillnotcooperateinthissituation,forcontributionsabovetheprovisionpointhavenoeffectandconstituteapurelossfortheagent.

Firstsituation:IfC≥m,thenDONOTcooperate.

Situation2:Thenumberofneighboursthathavealreadycooperatedisequaltotheprovisionpointminusone,andeachagentbelievesthatsheisnotthelastonetodecide.Aslongastherewardishigherthanthecostofcontributing,weexpecttheagent'scooperationtoensuretheprovisionpointisreached.However,theagentalsotakesintoaccountthepossibilitythatotherswillbearthecostofcontribution,inwhichcaseshecanretaintheendowmentandreceive

ashareoftherewardwithoutexpendingthecost.Inthissituation,anagent'scooperationdependsonherbeliefthatnobodyelsewillco-operate,thatis(1-p)n-B-

1,becauseonlyinthiscaseishercontributioncritical.

Secondsituation:IfC=m-1andn-B>1,then:cooperateifr[1-(1-p)n-B-1]>eDONOTco-operateifr[1-(1-p)n-B-1]≤e

Situation3:Thenumberofneighbourswhohavealreadyco-operatedislowerthantheprovisionpointminusone,andthenumberofpendingdecisionsisequalorhigherthanthenumberofcontributionsneededtoreachtheprovisionpoint.Inthiscase,theagenthastocalculateherprobabilityofbeingcriticalby

consideringbothhowmanyneighbourshavenotdecidedyetandhowmanystrangersareinthenetwork.[6]BuildingontheprobabilisticmodelofRapoport(1985),whichhasbeenwidelyusedasareferencemodelforcalculating"objective"criticality(Kerr1989),wecalculatetheprobabilityofbeingcritical,Pm-C-1,asfollows:

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Thatis,theprobabilitythatineachcombinationofn-B-1players,exactlym-C-1willco-operateandtherest(n-B-m)willnot.Thiswouldmakehercontributioncritical.

Thirdsituation:IfC<m-1andm-C≤n-B,then:co-operateifrPm-C-1>e.DONOTco-operateifrPm-C-1≤e

Situation4:Theagent'scontributioniscriticalonlyifitisaddedtothefuturecooperationofalltheagentsaboutwhomshehasnoinformation–becausethey

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havenotdecidedorbecausetheyarenotplacedinadjacentnodes.Thisrequiresabeliefthatsheisnotthelastonedeciding.

Fourthsituation:If1<n-B=m-C,then:co-operateifrpn-B>eDONOTco-operateifrpn-B≤e

Situation5:Anagentfacesasituationinwhichthenumberofinstancesofcooperationneededtoreachmishigherthaneitherthenumberofagentsthathavenotyetdecidedorthoseaboutwhomshehasnoinformation.Inthiscase,arationalagentwillnotcooperate,sincethegoalisunattainableandhercontributionwouldbelost.

Fifthsituation:Ifm-C>n-B,thenDONOTcooperate.

Situation6:Anagentfindsherselfinasituationwhereshebelievesthatsheisthelastonetodecide,andonlyonemorecontributionisneededtoreachm.

Givenri>ei,arationalagentwillalwaysco-operate.[7]

Sixthsituation:If1=n-B=m-C,thencooperate.

Theinitialparametersofthesimulation

5.7 TheinitialparametersinNetCommonsarealmostidenticaltothoseusedinthepioneeringSLGexperimentsofvandeKragtetal.(1983):

m=3,n=7,p=0.5,e=5,r=15

InthevandeKragtdesign,therewasnopparameter,andourreward(r)isslightlyhigherthantheirsinresponsetosomeearlytestingofNetCommons.However,thequantityofrisnotarbitrary.Ithasbeenestablishinsidethemarginsthatleadstoafailureintheproductionofthegoodwhendecisionsaresimultaneous(Rapoport1985:151).Toillustrateit,inasimultaneousprotocolwiththeparametersm=3,n=7,p=0.5,e=5,rationalutilitymaximisersevaluatetheirprobabilityofbeingcriticalasPm-1=0,234.IftheirdecisionistocooperateonlywhenrPm-1>e,onlyr=22couldleadthemtocooperate.Thesameisalsotrueforacumulativeprotocol.Ifthenetwork-dependentdecisionsequenceorderingleadstosuccesswherethesimultaneousandthecumulativesequenceorderingsfail,orifatleastdoessoundercertainwell-specifiedconditions,wecouldconcludethatworkinginanetwork,oratleastincertainnetworks,facilitatestheachievementofthecommongoal.Underequalconditionsofincentivesfortheproductionofthegood,thenetwork-dependentsequenceordering

wouldnotonlybemorerealistic,butalsomoreeffectivethanthesimultaneousandthecumulativeorderingrules.[8]

Inputnetworksproperties

5.8 Multi-agentsimulationnotonlyallowsustoplaceagentsinanetworkstructureandobservetheresults,butalsoallowsustocomparetheresultsobtainedwhentheseagentsareplacedindifferentkindsofsocialnetworks.Forthispurpose,wehavebuilt–withUcinet6.0andinDLformat–acatalogueofsymmetricalnetworkscomposedby7nodesandwithdifferentdensitiesandglobaldegreecentralityandheterogeneity.

5.9 Densityisdefinedastheratioofthenumberoftiesoverthemaximumpossiblenumberofties(Scott2005).

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wherelisthenumberoftiesinthenetwork.Thevalueofdensityoscillatesbetween0,whennonodeislinked,and1,whenallthenodesareconnectedwitheachother.Fornetworkswithn=7andonlyonecomponent,densityrangesbetween0.29–foranetworkwiththeshapeofaline–and1.Densityisoneofthestructuralpropertiesofnetworksthatexistingliteraturehasidentifiedasrelevantduetoitsimpactonthesuccessofcollectiveaction.Thisimpact,however,isfarfromclear(Bravo2008).AuthorslikeTakácsetal.(2008),forexample,havenotedthatdensenetworksinvolvemoreinformationandgreatercapacityforpunishment,whichfosterscooperation.However,atthesametime,highdensitycanprovidetoolstoresistpressureandevenformutualaffirmationofdefectingbehaviours.Punishmentfordefectionmaybereducedpreciselybecausesomeneighboursanticipatefutureinteractionsthattheydonotwanttocompromise(Gould2003).Also,ifthedecisiontocontributedependsonobservedbehaviourinthelocalneighbourhood,thenobservingnon-co-operativebehaviourcanstimulatedefection(Gould1993).

5.10 Givenacertainlevelofdensity,itispossibletobuildnetworkswithdifferentglobaldegreecentralityofheterogeneity.Forexample,anetworkwithadensityof0.52canbecomposedofasetofnodeswithdegrees5,5,4,3,2,2,1orwithdegrees5,4,3,3,3,3,1(amongmanyotherpossiblecombinations).WeusedBlau'sindextomeasureheterogeneity.Insymmetricalnetworkstheindexisdefinedasfollows:

(4)

5.11 Thenetworksthatweusehavejustonecomponent;allnodesaredirectlyorindirectlyconnectedformingasinglestructure.Thisisfortwomainreasons.First,asamatterofrealism:insmallgroups(n=7)thelikelihoodofmorethanonecomponentisquitelow,andinthecaseofpublicgoodsthepopulationcouldbeseenasbelongingtocommunitiesthat,generally,aremorecohesivethesmallertheyare.Second,becauseourultimateinterestisinanalysingtheimpactofreticularpropertiessuchasdensityandheterogeneityinthesuccessorfailureofcollectiveaction.Thenumberofnetworkcomponentsaffectstheseindicatorsandthereforemustbeexperimentallycontrolledasseparatefromthesetofanalysedvariables.

5.12 Giventhesecharacteristics,thecatalogueiscomposedof230differentnetworks.[9]Thissetisexhaustive:itincludesallthepossiblenetworkswithn=7,symmetricaltiesandonlyonecomponent.Thismeansthatthecataloguecontainsverydifferentnetworktopologies:small-worldnetworks,scale-freenetworks,randomnetworks,etc.However,theaimofthisarticleisnottocomparethedifferentialsuccessofthesespecialtypesofnetworksbuttoexploretheimpactofreticularpropertiesonthesuccessorfailureoftheproductionofSLGs.

Dynamicsoftheprocessandoutputs

5.13 Thepreviousformalmodeldescriptionsectionspecifiesthedynamicsoftheproductionofthepublicgood,andcorrespondstoasingle"turn"intheNetCommonssimulationmodel,whereallmembersofthegroupdecideoneafteranotherresultingeitherintheprovisionpointbeingreached(success)ornot(failure).

5.14 Thetop-levelalgorithmforthemodelis:

Create initial network (options: random network OR import-DLfile)Unless maximum number of time steps (5000) has been reached, do:

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Reset all agents' attributes Unless all agents have decided, eavh agent will: Scan neighborhood for counting deciders (B) and cooperators (C) Estimate the decision situation Make a decision about cooperation behaviour Record micro-decisional outcomes in a buffer-matrix Next agent Compute the turn macro-indicators from the buffer-matrix Update outputs and plotting Next turnStop simulation run

5.15 Thefinalresultofaturndependsonthesequenceofdecisions,which,asstatedbefore,israndomlydecidedineachturn.Forn=7,thenumberofallpossiblesequenceorderingis7!=5,040foreachdifferentnetwork.Thismeansanetworkhas5,040differentpossibleresults.Inouranalysiswefocusonthesuccessrates(SR)ofeachnetwork,definedasthenumberofsuccessfulturns–wheretheprovisionpointisreached–overthetotalnumberofrealizedturns.Inthenextsectionwepresenttheresultsafterasimulationof5,000turnsforeachnetwork.

ResultsandDiscussion

6.1 Inthissectionwediscusstwodifferenttypesofresults.Thefirsttypeconcernshowthechangeincertaininitialparametersimpactsonthesuccessrate(SR).Forthisanalysiswefocusonasinglenetwork("18_16.dl")comprising18tiesandaheterogeneityof16%.Thereasonforselectingthisparticularnetworkwasitsaveragesuccessrate(0.63)whichallowedustobetteranalysetheimpactofchangesintheinitialparameters.

6.2 Thesecondtypeofresultsrelatestothevariabilityinsuccessratesofdifferentnetworksasafunctionoftheirstructuralproperties.Forthisanalysisweusedthefullcatalogueof230networks.

Thestabilityofthesuccessrate

6.3 Ifthereexistsafinitenumberofpossiblesequenceorderingsforanetwork,andiftheresultsofaparticularturndonotaffectthesubsequentturns,asuccessratethatisupdatedaftereachturncannotshowanunstablepattern.Asthesimulationisrunningturnafterturn,thenetworkwouldasymptoticallyapproachtheaveragesuccessrate(SR)producedby–hypothetically–calculatingtheresultofthe5,040possiblesequencesofthenetwork.WecanseeinFigure1,withthisstabilisationevident,theupdatedSRaftereachturnconcludes.

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Figure1.Successrateofthenetwork"18_16"(5000rounds)

6.4 Tworemarksshouldbemadehere.First,thefactthateachnetworkor"socialunit"hasitsownsuccessrateisunsurprisinginsociologicalanalysis.Lesscommonistheabilitytoidentifythesituationalmechanisms–responsibleforthemacro-microlink–,theaction-formationmechanisms–responsibleforthemicro-microlink–andthetransformationalmechanisms–responsibleforthemicro-macrolink–thatexplainhowindividualactionsjointlyproducethatoutcome(HedstromandSwedberg1996,1998).Inourcase,thesimulationallowsustogenerateandanalysethe35,000individualdecisionsthatcausethisrate,andmoreoverspecifyhowthishappens.

6.5 Second,weestimatethatthe"18_16"networkneeds212roundstostabiliseataratewithina±3%marginoftherateobtainedafter5,000rounds.Ifweuseamarginof±5%,ittakesthesimulation100roundstoachievestability.Inbothcases,itisanexcessivenumberofroundstobereproducedinalaboratoryexperimentwithhumansubjects.Thisiswhytheanalysisofthenetwork-dependentsequenceorderingcanonlybeproducedthroughamulti-agentsimulationoramathematicalmodel.

Theeffectsofthechangeintheparameters

6.6 InthefollowingsectionswepresentsomeresultsofanalysingtheeffectsontheSRofdifferentrelevantparameters.Wewillexplorethreespacesofparametersinturn:theprovisionpointofthepublicgood(m),theprobabilityofcooperationattributedtootheragents(p),andtwostructuralpropertiesofthenetworks(density,andglobaldegreeheterogeneity).Inallthreecasestheanalyticalstrategywillbefirsttopresentsomeaggregateormacro-results,andthenanalysethedetailedrecordstoestablishthemicro-foundationsexplainingtheseresults.

Changesintheprovisionpoint(m)

Macroresults

6.7 After5000rounds,thesuccessrateofthenetwork"18_16",whichcomprisesdifferentprovisionpointswhilekeepingtherestoftheparametersconstant,presentsthefollowingdistribution.

Figure2.Successratesofthenetwork"18_16"withdifferentprovisionpoints

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6.8 Inthisanalysiswedonotconsiderthepossibilityofm=1:thisparticularsituationdoesnotconstituteacollectiveactionproblem.Inaddition,ourformalisation0<m<∑i∈Neiallowsustodismissm=7aswell,becausethissituationexcludesthepossibilityofreceivingabenefitwithoutcost.

6.9 AsshownbyFigure2,highprovisionpointsgenerateafailureof100%.Thelowerprovisionpointm=2ismoresuccessful,butthisoneinturnisoutperformedbytheintermediateprovisionpointm=3.Toacertaindegree(m=3inourcase),higherSRsareachievedthemorepeoplecollaborate.

Microfoundations

6.10 Table2allowsustoidentifytheexplicativemicro-foundationsoftheSRsobtainedwithdifferentprovisionpoints.

Table2:Numberofdecisionsdependingontheprovisionpoint,theorderinthesequence,thesituationthattheagentfaces,andthetypeofdecisionsmade(network"18_16",5000roundsforeachprovisionpoint)

Situation1C>=m

Situation2C=m-1&n-B>1

Situation3C<m-1&m-C=<n-B

Situation4C=m-n+B

Situation5m-C>n-B

m Order Def. Coop. Def. Coop. Def. Coop. Def. Coop. Def. Coop.

2 1 50002 50003 50004 4602 3985 236 3902 8626 21 838 3181 852 1087 166 1393 2328 414 699

3 1 50002 50003 4163 8374 3171 18295 2279 2476 2456 25 1155 3239 463 1187 65 4251 222 462

4 1 50002 50003 50004 4590 4105 4143 624 2336 3966 370 6647 3644 665 691

Note:Anynetworkconsistingof7membersrepresentsatotalof7decisionseachround,sothat5000roundsinvolve35000choices.Eachofthesedecisions(a)hasbeenproducedinasequenceorderwithintheround(fromfirsttoseventh,thelast),(b)hasoccurredinoneofthe5decisionsituations,and(c)hasbeenwhethertoco-operateornot.Thetableabovedisplaystheabsolutefrequencycross-distributionof35,000decisions,accountingforthethreevariablesandcontrollingby3levelsofprovisionpoint(m).

6.11 Table2showsthatfirstchoicesarealwaysproducedinSituation3andalwaysconsistofadefection,independentofthevalueofm.TheexplanationofSRrequiresclarificationofhow,whenandwhyagentsstarttocooperate.InSituation3,PmC1indicatesthat(whileC=0)withalowBnoagentcooperates,butfromacertainthresholdinBvaluesonwardsagentsstartcooperating.Toexplainwhythishappenswehavetoexplorethethreecomponentsofthecriticalitycalculus(Formulae2).Thefirstcomponent,thebinomialcoefficientC(n-B-1,m-C-1),islowerthehigherBis.Inthesefirstroundsinwhichnobodycooperates,

thesecondcomponent(pm-C-1)isconstantandcanthereforebeignored.Thelastcomponent((1-p)n-B-m)providesthekeytounderstandingtheunderlyingmechanisms,becausehereweobservethatthehigherBis,thehigherthepossibilitythatn-mwillnotcooperate,whichinturnincreasestheprobabilityoftheagent'scriticality.

6.12 Thisisasituationalmechanism(HedströmandSwedberg1998),sinceitexplainsatwhichmomentandwhyapioneeringcooperativebehaviour–cooperationthatisbelievedtobepioneeringbytheagent–istriggered.Wecallthismechanismthepioneeringcooperationtrigger.Oncethismechanismisactivated,twosituationscanoccur.Insomecases,pioneeringcontributionsestablishtheconditionsforotheragentstoestimatetheirprobabilityofbeingcriticalashigh,andthereforetheydecidetoco-operate.Wecallthismechanismthespeculativecooperationtrigger.Inothercases,havingrelevantinformationconcerningpreviouscooperationdirectlyaffectsanagent'sbeliefthathercontributionmeanstheprovisionpointisreached(althoughshecannotbesurethathercontributioniscritical,sinceotheragentscouldbearthecostofthecontribution).Wecallthismechanismtheeffectivecooperationtrigger.

6.13 Theonlytransformationalmechanismthatexplainswhyaturnconcludessuccessfullyisthemechanismoftheaggregationofindividualchoices.However,thismechanismcantakedifferentformsdependingonthewayinwhichsituationalmechanismsthattriggerpioneering,speculativeandeffectivecooperationareactivated.DifferentcumulativeprocessesthatleadtosuccessexplainwhyeachcasehasitsSR,andwhythatrateisdifferentineachcase.

6.14 Further,differentcumulativeprocessesexplainwhyanetworkwithm=2achievesaspecificSR,andwhythisSRislowerthantheSRproducedwhenm=3.ThekeytounderstandingthisdifferenceisthefactthatthethresholdofBvalueswhichactivatethepioneeringcooperationtriggerishigherinnetworkswithlowerprovisionpoints.KeepingC=0,m=2requiresB=3toco-operate,whilem=3onlyrequiresB=2.

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6.15 ThisoutcomeisexplainedastheresultofachangeinthefirstcomponentofPm-C-1.KeepingBandCconstant,goodswithahighermproduceahighervalueforthebinomialcoefficientC(n-B-1,m-C-1),whichinturnimpliesthatthecriticalityestimationfavourscooperation.Whenp=0.5,theprobabilityofasituationwith1cooperationand5defectionsisequaltotheprobabilityofsituationswith2cooperationsand4defections.However,thelatterismorelikelythantheformerbecausetherearemorecombinationsofthisoutcome.Inotherwords,anagentismorelikelytobecriticalwhenthenumberofcontributionsneededishigher.

6.16 Whenm=2,startingfromthefourthplayeritispossiblethatsomeonemayhaveknowledgeofB=3andthendecidetocooperate.Whenm=3,thesameprocessstartsearlier,butnowitonlytakesB=2fortheagenttoco-operate.Thismeansthatwhenm=3pioneeringcooperationistriggeredearlierthanwhenm=2,whichinturninducestheearliertriggeringofspeculativeandeffectivecooperation.Inthecaseofm=3,previouscooperationleadseithertospeculativecooperationortotheagentsfindingthemselvesinSituation4characterisedbynocooperation.However,sincethebalanceispositive,theincreaseinspeculativecooperationisresponsiblefortheincreasingsuccessrate.Thecaseofm=3impliesanearlytriggeringofpioneeringcooperation,whichallowsasubsequenttriggeringofspeculativecooperationsufficientlyhighfortheprovisionpointtobereached.Whenm=2,pioneeringcooperationistriggeredatalaterpoint,whichmeansspeculativeandeffectivecooperationcanemergetoolatefortheprovisionpointtobereached.

6.17 SRgrowsfromm=2tom=3,butfallsfromm=3tom=4.Thisisexplainedbecause,whilewiththeseparameters B≤3leadstoadefection,incontrasttowhenm=3thepioneeringcooperationtriggerisnotactivated.Inthissituation,agentswithki≤3nevercooperate,whilethosewithki≥3eitherfaceSituation3,withacriticalityestimationthatdiscouragescooperation,orelsefaceSituation4,wherethecriticalitycalculusalwaysleadstodefection.Theconditionsthatleadtocooperativebehaviourarenevermet.

6.18 Inshort,theexplanationofthedifferentsuccessratesforeachvalueofmresemblesaninstanceofpath-dependence,becauseofthewayinwhichthecumulativeprocessofcontributionstakesplace.Wheretheconditionsforanearlytriggeringofpioneeringcooperationaremet,thesubsequenttriggeringofspeculativeandeffectivecooperationmaytakeplaceearlyenoughfortheprovisionpointtobereached.High(m=3)provisionpointstriggerpioneeringcooperationearlier,thusgeneratinganearliercascadeofdecisionsthatleadtotheproductionofthepublicgood.However,provisionpointsthataretoohigh(m>3)havetheoppositeeffect,sincetheaccumulationofdefectionsreducesthepossibilityofreachingaverydemandinggoal.Ingeneral,agentsdonotexpecttobecriticalwhenthegoaliseasy,becausetheybelieveotheragentswilllikelybearthecost,norwhenthegoalisverydemanding,becausetheythinktherewillnotbeenoughotheragentscontributing.

Changesintheprobabilityofcooperationattributedtootheragents(p)

6.19 Aswehavediscussedandargued,westartfromtheassumptionofhomogeneityasraisedbyRapoport(1985),i.e.allagentsattributetoothersthesameprobabilityofcooperation(p).Whenwefocusonanalysingtheroleofthestructuralpropertiesofthenetwork,wemustsetthisparametertoaspecificvaluetorunsimulations.Thisposesaproblem,becauseforarationalandmaximising-utilityagent,thereisnoreasonwhatsoevertosetthisparametertoanyvalue.Therefore,ifallpossibilitiesareequallylikely–andtheagenthasnoreasontobelieveotherwise–,thePrincipleofInsufficientReasonsuggeststhatwesettheparametertop=0.5.Thereareotherpossibilities,suchasfixingthevalueofpasafunctionofobservedbehavioursbytheagentinhisneighbourhood,buttoattributetothosewhohavenotyetdecidedacooperativeprobabilitydependentonthebehaviourofthosewhohavealreadydecidedseemsanexercisemoreunreasonablethanattributingtoallagentsanequallikelihood.However,ABMallowsustoexplorethebreadthofpossibilitiesbytestingforpossiblevaluesofp,andthatispreciselywhatwedointhenextsection.

Macroresults

6.20 Figure3showsthatthebeliefinahighprobabilitythatotheragentswillcooperatehasthesameeffectasalowprobability(theybothgeneratealowSR),whileintermediatelevelsoftrustordistrustinotheragents'cooperationproduceahigherSR.

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Figure3.Successrateofnetwork"18_16"dependingonbeliefsabouttheprobabilitythatotheragentswillco-operate(5000roundsforeachvalueofp)

Microfoundations

6.21 Table3helpsustoidentifytheexplanatorymechanismsoftheseresults.

Table3:Numberofdecisionsdependingontheprobabilityofcooperationattributedtootheragents,theorderinthesequence,thesituationthattheagentfaces,andthetypeofdecisionsmade(network"18_16",5000roundforeachvalueofp)

Situation1C>=m

Situation2C=m-1&n-B>1

Situation3C<m-1&m-C=<n-B

Situation4C=m-n+B

Situation5m-C>n-B

P Order Def. Coop. Def. Coop. Def. Coop. Def. Coop. Def. Coop.

0.3 1 50002 50003 50004 50005 4765 2356 4275 610 1157 4274 726

0.4 1 50002 2835 21653 1579 34214 113 899 39885 15 575 302 41086 106 1343 35517 506 1875 2619

0.5 1 50002 50003 4131 8694 3287 17135 2307 2432 2616 29 1183 3194 480 1147 72 4210 247 471

0.6 1 50002 50003 4200 8004 3164 18365 2303 2508 1896 19 1182 3186 485 1287 69 4208 252 471

0.7 1 50002 50003 50004 4617 3835 4148 626 226

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6 3927 367 598 1087 3551 742 707

Note:Anynetworkconsistingof7membersrepresentsatotalof7decisionseachround,sothat5000roundsinvolve35000choices.Eachofthesedecisions(a)hasbeenproducedinasequenceorderwithintheround(fromfirsttolast,theseventh),(b)hasoccurredinoneofthe5decisionsituations,and(c)hasbeentoco-operateornotco-operate.Thetableabovedisplaystheabsolutefrequencycross-distributionof35000decisions,accountingforthethreevariablesandcontrollingby5levelsofcooperationprobabilityattributedtoothers(p).

6.22 Again,thekeytoexplainingthedifferencesintheSRliesintheconditionsthatactivatethetriggermechanismsofpioneeringcooperation,thenumberofpreviousdecisions(B)necessaryforanagenttoevaluateherownprobabilityofbeingcriticalassufficientlyhightocooperate.

6.23 Forthisexplanation,webuildonFigure4.ThisfigurepresentsthemathematicalrelationbetweenB(keepingC=0)andtheprobabilityofcontributingtothecollectiveaction.OrdinateaxisrepresentsthevalueofD(p)foreachofthepossiblevaluesofp,whereD(p)=rPm-C-1−edenotesthedifferencebetweentheexpectedbenefitandtheexpectedcost.WhenD(p)hasapositivevaluetheagentdecidestocooperate,whileshedecidestodefectwhenD(p)isnegative.

Figure4.Probabilityofbeingcriticaldependingontheprobabilityofcooperationattributedtootheragentsandthenumberofpreviousdecisions

6.24 Figure4showstwoimportantpoints.First,thecriticalitycalculusnecessarilyhasaninvertedU-shapebecause,whilethefirstcomponentofPm-C-1isconstant,

theremainingtwoarenegativelycorrelated,suchthattheincreaseofpm-C-1necessarilyimpliesadecreaseof(1-p)n-B-m.Second,Figure4showsthatthevaluesofBthatleadtocooperationaredependentonthevaluesofp.Inotherwords,Figure4holdsthekeytoidentifyingthetransformationalmechanismsthatexplaintheresultsofFigure3.

6.25 Withp=0.3,thereisnopossibilityforpioneeringcooperationtobetriggered.Withp=0.4,however,cooperationbecomespossibleevenwhenB=1andB=2.Oncethesecooperationsareproduced,speculativeandeffectivecooperationcanbetriggered.Withp=0.5andp=0.6thesameprocessstartsatalaterpoint,whensomeagentsknowthatB=2.Thisimpliesthatinsomecasestheroundfailsbecausethetriggeringofspeculativecooperationtakesplacetoolate(notenoughcontributionsareproducedtoreachtheprovisionpoint).

6.26 Withp=0.7theSRfallsto0.0becausehereaminimumofB=3isnecessaryforthetriggeringofpioneeringcooperation,soonlyafractionofthosewhodecideinthefourthorahigherplacehavetheoptiontocooperate.Moreover,whenthishappenssomeoftheagentsfaceSituation4,whichagainimpliesthattherewillnotbeenoughcontributionsfortheprovisionpointtobereached.

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6.27 Inshort,ontheonehand,excessiveconfidenceaboutotheragents'cooperationincreasesthebeliefthatitisimprobablethatenoughdefectionswillbeproducedfortheagent'scontributiontobecritical.Ontheotherhand,excessivelackofconfidenceaboutotheragents'cooperationincreasesthebeliefthatitisimprobablethatenoughcontributionswillbeproducedfortheagent'scontributiontobecritical.Withintermediatevaluesofp,bycontrast,thenumberofpreviousdefectionsfortriggeringpioneeringcooperationislow,andthecascadeofcooperationstartssufficientlyearlyinaroundforthenecessarycontributionstobeaccumulated.

Successratesandthestructuralpropertiesofthenetworks

6.28 Inthissectionwepresenttheresultsobtainedforthe230networksofourcatalogueandwetrytoidentifysomemicro-foundationsofthoseresults.

Macroresults

6.29 InTable4wepresentOLSregressionmodelstotesttheinfluenceoftwostructuralpropertiesofthenetworksontheSR.Sincetherelationsbetweentheindependentvariablesandthedependentvariablehaveanexponentialcharacter,alltheindependentvariablesaretransformedintotheirbase-10logarithm.

Figure5.Networkdensityandsuccessrate

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Figure6.Networkglobaldegreeheterogeneityandsuccessrate

6.30 Table4showsthatmodelB,whichusesthebase-10logarithmofthedensityandheterogeneityasindependentvariables,explains84%oftheSRvariability.DensitypositivelycorrelateswithSRanditisthevariablewiththehighestpredictivevalue.GlobaldegreeheterogeneitynegativelycorrelateswithSRand,althoughitspredictivepowerislower,itsinclusionresultsinamodelwithabetterfit.

Table4:Successrate.OLSregressionmodels(non-standardisedcoefficientsandstandarderrors)

(A) (B)Non-standardisedcoefficientsandstandarderrors

BetaNon-standardisedcoefficientsandstandarderrors

Beta

Constant 1.271(.022)***

3.781(.260)***

LOG_DENSITY 2.114(.076)***

.881 1.505(.090)***

.627

LOG_HETEROGENEITY -2.195(.227)***

-.361

N 227 227AdjustedR-square .775 .841

F 780.461*** 598.051***R-squarechange -- .066

Fchange -- 93.787***

***Significancelowerorequalto0.001

6.31 TsukamotoandShirayama(2010)presentedamodelontheevolutionofcooperationincomplexnetworks,andfoundthatintermediatevaluesofheterogeneitygeneratemaximumcooperation.Inthesamevein,Fuetal.(2007)presentedanevolutionaryprisoner'sdilemmaonanetwork,andalsofoundthatthefrequencyofcooperationishighestatintermediatelevelsofheterogeneity,andYangetal.(2012)reachedthesameconclusionanalysingscale-freenetworks.Itseems,therefore,thatmuchresearchpointstothesameconclusion:intermediatevaluesofheterogeneityoptimisethelevelsofcooperation.However,Fuetal.(2007)studytheeffectofheterogeneitystartingfromanetworkwithagivenhomogeneityandthenaddinglinkstosomeselectednodes.Thus,itisnotonlytheheterogeneitywhichincreases,butalsothedensity.RegressionmodelB,presentedinTable4,allowsustoassessthenetimpactforeachofthetwoindependentvariables.Inourcase,unlikeFuetal.(2007),TsukamotoandShirayama(2010),andYangetal.(2012),wefindthat,whilecontrollingforthedensity,heterogeneityhasanegativerelationshipwiththerateofsuccess.Thedifferencebetweenthecontinuousproductionofagood,suchasthatmodelledbyFuetal.,TsukamotoandShirayama,andYangetal.,andascalarvariant–likeours–impliesadifferenceintheimpactoftheindependentvariables.

Microfoundations

6.32 Todiscoverthemicro-foundationsthatexplainthemacro-resultsoutlinedinTable4,weanalyseagents'decisionsinrelationtotheirnodaldegree.Afterall,networksaredistinguishedintermsoftheirrelationalstructures,resultingfromthedifferenceinthenodaldegreeoftheagents.Moreover,agentswiththesamenodaldegreebehaveinthesameway,independentofthenetworkinwhichtheyaresituated.

6.33 Forsimplicity,weuseasubsetof8networksrepresentativesofdifferentcombinationsofdensityandheterogeneity.Figure7showsthepercentageofcooperationanddefectionsofagentsdependingontheirnodaldegree.

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Figure7.Agents'decisionsdependingontheirnodedegreefor8selectednetworks(5000roundsforeachnetwork,280000decisionsintotal)

6.34 TherelationbetweenthenodedegreeandtheprobabilityofcooperationhasaninvertedU-shape,althoughtheprobabilityofcooperationgenerallyincreasesforhighernodedegrees.Thehigherthedensityofanetwork,thehighertheprobabilitythatasufficientnumberofagentscontributetotheproductionofthepublicgood.

6.35 Atthesametime,thehighertheglobaldegreeofheterogeneity,thehigherthedispersioninthenodaldegreesoftheagents.Inthiscasewecanfindmoreagentswithextremelyhighorextremelylownodaldegreesandthosearepreciselythenodaldegreesthatcorrespondwithalowerprobabilityofcooperation.Theeffectofheterogeneityisnotasclearastheeffectofdensity:whileahighheterogeneityleadstofewercontributions,alowheterogeneityleadseithertofewerormorecontributionsdependingonthemeannodaldegreeofthenetwork.

6.36 TheseeffectsofdensityandheterogeneityarerepresentedinFigures5and6above.Giventhattheinfluenceofdensityoutweighsthatofheterogeneity,thereductioninSRthatisproducedbyanincreaseinheterogeneitywillbelowerwhendensityishigh;thisexplainsthedecelerativeshapeofthepointcloudinFigure5.

6.37 Theinfluenceofthestructuralpropertiesofanetworkcanbeexplainedbythedifferentprobabilitiesofcooperationofagentswithdifferentnodaldegrees.However,westillneedtomakeexplicitwhyagentswithdifferentnodaldegreeshavedifferentprobabilitiesofcooperation.

Table5:Numberofdecisionsdependingonagents'nodedegree(k)andthesituationatthemomentofdecision,for8selectednetworks.

k Situation Defect Cooperate Total%row %col. %row %col. %row %col.

1 Situation3 100% 100% 100% 100%Total 100% 100% 100% 100%

2 Situation2 100% 8.7% 100% 2.9%Situation3 68.7% 100% 31.3% 91.3% 100% 97.1%

Total 66.7% 100% 33.3% 100% 100% 100%

3 Situation1 100% 1.9% 100% 100%Situation2 100% 17.6% 100% 8.6%Situation3 55.3% 98.1% 44.7% 82.4% 100% 90.4%

Total 50.9% 100% 49.1% 100% 100% 100%

4 Situation1 100% 4.8% 100% 2.1%Situation2 100% 17.7% 100% 9.9%Situation3 46.5% 90.1% 53.5% 82.3% 100% 85.7%

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Situation4 100% 5.1% 100% 2.3%Total 44.3% 100% 55.7% 100% 100% 100%

5 Situation1 100% 15.5% 100% 7.2%Situation2 100% 25.6% 100% 13.7%Situation3 45.6% 71.6% 54.4% 74.4% 100% 73.1%Situation4 100% 100% 100% 4.7%Situation5 100% 2.8% 100% 1.3%

Total 46.6% 100% 53.4% 100% 100% 100%

6 Situation1 100% 6.6% 100% 3.9%Situation2 100% 10.3% 100% 4.2%Situation3 46.1% 48.8% 53.9% 82.3% 100% 62.5%Situation4 81.1% 21.9% 18.9% 7.4% 100% 16,0%Situation5 100% 22.7% 100% 13.4%

Total 59.1% 100% 40.9% 100% 100% 100%

Note:Asetof8selectednetworksconsistingof7memberswithonedecisioneachperroundand5000roundsinvolvesatotalof280000choices.Eachofthesedecisions(a)hasoccurredinoneofthe5decisionsituationsand(b)hasbeentoco-operateornottoco-operate.Thetabledisplaysboththerowandcolumnpercentagecross-distributionof280000decisions,accountingforthetwovariablesandcontrollingby6levelsofagents'nodedegree(k).

6.38 Table5presentsbehaviouraldifferencesasafunctionofagents'nodedegree(k).Inthecaseofagentswithki=1,theeffectivecooperationtriggerisnotactivatedbecausem=3.Pioneeringandspeculativecooperationwillnotbetriggeredeither,sincefortheseagentsBandConlyhavevalues0or1andnocombinationofthesevaluesleadstoacriticalitycalculusfavourabletocooperation.Table5showsthatagentswithki=1arealwaysinthesamesituationanddecideinfavourofdefection.Giventhelargenumberofagentsaboutwhomtheyhavenoinformation,agentsconsidertheprobabilityoftheircontributionbeingcriticaltobeverylow.

6.39 Foragentswithki>1therangeofpossiblesituationsisbroader,whichfavourscooperationinsomecases.Forexample,agentswithki=2whofaceSituation3donotalwaysdefect,asagentswithki=1do.Infact,thecombinationsB=2&C=1andB=2&C=0favourcooperation,whilecombinationsB=1&C=1,B=1&C=0andB=0&C=0leadtodefection.Thisexplainswhyagentswithki=2inSituation3co-operateonlyinsomecases(Table5).However,agentswiththisnodaldegreemayalsofaceSituation2,inwhichtheagentbelievesthatonlyonecontributionisneededtoreachtheprovisionpointandcooperationistherationaldecision.Inotherwords,agentswithki=2faceasituationwherebothspeculativecooperation–whenthecombinationsofBandChold–andeffectivecooperationcanbetriggered–whenthenumberofpreviouscooperationsis2,thatism-1.Anincreaseinthenodaldegreeproducesabroaderrangeofpossiblesituationsthattheagentfaces,someofwhichfavourcooperation.

6.40 However,thecaseofki=6isanexception.Figure7showsthattheprobabilityofcooperationdecreasesfortheseagents.Table5suggeststhatthisreductionisexplainedbytherelationbetweenahigherkiandanincreasedproportionofoccasionsinwhichagentsfaceSituation4(C=m-n+B).WithourparametersthissituationisproducedwhenC=B-4.Thisoccursinthreesituations:withC=0&B=4,withC=1&B=5,andwithC=2&B=6.Anagentwithki=4facesonlyoneofthesesituations(C=0&B=4).Agentswithki=5andki=6facesituationsC=0&B=4andC=1&B=5,butthelatterhasahigherprobabilityofencounteringacombinationC=1&B=5becauseshehasmoreneighbours.DefectionischoseninallthesecombinationsofCandB.AslongasthemajorityofpossiblecombinationsofCandBinSituation4leadtodefection,ahigherkiincreasestherelativeweightofthesecombinationsinthesesituations.

6.41 Inshort,agentsthatarelinkedinawaythatallowsthemtoobtainsignificantinformationaboutotheragents'behaviourareawareofthelowprobabilitythattheprovisionpointwillbereached,andthereforedecidenottocontribute.Ontheotherhand,thelackofinformationthatcharacterisesthelessconnectednetworksfavoursspeculativecontributions.Whenki=6isreachedthepossibilityfortriggeringspeculativecooperationdecreases.Paradoxically,inthiscasewefoundthatconstrainingthelevelofinformationgenerateslargersocialbenefits–asElsterclaims(1979,2000)–,andthesemechanismscouldbeexaminedatadetailedlevelbymeansofcomputationalsimulation.

ConclusionsandFurtherWork

7.1 Weconcludethispaperbyhighlightingseveralresultsthatweconsiderespeciallyrelevant.Oursimulationbearswitnesstotheexplanatorypowerofthestructuralpropertiesofasocialsystem.Rationalagentswithanidenticaldecisionalgorithmcangenerateverydifferentsocialoutcomesdependingontherelationalstructureinwhichtheyareembedded.

7.2 Thetopologyorstructureofasocialnetworkisnottheonlyrelevantfactor;thedecisionsequenceorderingisalsoessential.Animportantaspectofthedecisionisthequantityofinformationavailabletotheagentatthemomentofdecision,andinasocialnetworkthisvariabledependsonthenodaldegreeoftheagentandonthesequencepositionofherdecision.Theseissuescanbeaddressedthroughtheanalysisofdynamicnetworks,throughobservationorsimulation.

7.3 Wetendtothinkthatthelowertherequiredeffortlevel,thehighertheprobabilityforapublicgoodtobeproduced.However,ourresearchsuggeststhat,insomecases,institutionaldesignersshouldbalancetheeffortlevel;nottoohightodiscourage,nortoolowtostimulateadefectiongroundedontheconfidencethatotherswillbearthecost.

7.4 Theresultsofoursimulationrunsalsosuggestthatbothhighandlowconfidenceaboutotheragents'cooperationcanhaveanegativeeffectontheprobabilityofsuccessofacollectiveaction.Intermediatelevelsofconfidenceseemtostimulatemorecooperationbecausetheysupporttheagent'sbeliefthathercontributionisrelevant.

7.5 Socialgroupswithadensestructurehaveahigherprobabilityofsuccessintheproductionofstep-levelpublicgoodsbecauseneighboursprovideusefulinformationabouttherelevanceoftheagent'scontribution.However,ifthelinksarecostly,thehighestnumberoflinksdoesnotnecessarilymakeamorecohesivegroup:fromacertaindegreeofdensityonwards,theprobabilitiesofsuccessincollectiveactiondonotchangesignificantly.

7.6 Groupswithahorizontalstructurehaveahigherprobabilityofsuccessintheproductionofstep-levelpublicgoods.Poorlyconnectedagentshavelittleusefulinformationabouttherelevanceoftheirdecision,whilewell-connectedagentsmayhaveanexcessofinformationsuchthattheymaybeawareofdifficultiesthat

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cannotbeperceivedfromotherpositions.Alimitationofinformationstimulatesagentstoco-operatebecausetheyignorecertaindifficulties,thereforegeneratingasociallyoptimalaggregateeffect.Inotherwords,moreinformationforanindividualdoesnotalwaysgenerateabetterresultforthegroup.Thisconclusionisinlinewithrecentfindingsaccordingtowhichlimitedinformation,cognitiveorsocial,canfavourtheevolutionofcooperation;forexample,Horváthetal.(2012)haveshownhowthenumberofroundsrememberedbyaniteratedgameagentdoesnotmonotonicallyincreasethelikelihoodofreachingaco-operativestate.

7.7 Thesameagentsthatfailtoproducethepublicgoodinasimultaneousorcumulativedecisionsequenceorderingcouldsucceedwhentheyaresituatedincertainnetworks,withrelevantimplicationsforinstitutionaldesignandformulationofmoreeffectivesocialpolicyalternatives.

7.8 Oncewehaveproceededwiththecomputationalreplicationofpreviouslaboratoryexperiments,ournextgoalistoextendthemodelandthestudywithfurtherABMpossibilities.Forinstance,infutureversionsofNetCommonsweplantoincludeanewagentheterogeneityassumptionregardingexpectationsofothers'behaviour;toovercomethehypothesisofhomogeneitywefocusoncertainnetworktopologytypesandweintroducenetworksizeasaparameterunderscrutiny.

Acknowledgements

ThisworkhasbenefitedfromaMICINNR&Dprojectgrant(referencenumberCSO2009-09890),fromaMINECOR&Dprojectgrant(referencenumberCSO2012-31401)andfromaCONSOLIDER-INGENIO2010projectgrant(referencenumberCSD2010-00034).PartofthisresearchhasbeencarriedoutduringaresearchstayofF.León'sattheEqualityStudiesCentre,UniversityCollegeDublin.

Notes

1Amongthecontinuousfunctionstherearealsoimportantdifferencesbetweenaccelerating,deceleratingandlinearfunctions(Heckathorn1996;Linares2007).

2Taylor(1987)writesthatwhenweareinterestedintheconstructionofabridge,moremoneydoesnotimplymorebridges.However,extracontributionscanbededicatedtomakingabetterbridge(useofsuperiormaterialsorbetterdesigns)Inthiscase,wewouldhavea"mixed"formpublicgood,inwhichthefirstparttakestheformofaSLGfollowedbyacontinuousfunctiononcetheprovisionpointisreached.Inexperimentalterms,thishasbeenmadeconcreteindifferent"rebaterules"(MarksandCroson1998).

3Taylorhasargued(1987:46)thatlumpygoodsareoftenbestmodelledbytheChickenGame:dependingontheexpectationsregardingthecontributionsofothergamers,cooperationmaybethepreferredoptionforarationalmaximizingutilityagent.

4Somemodelshavedemonstratedthepositiveeffectoncooperationofheterogeneitywithregardtodecisionsabouthowmuchtoinvest(see,e.g.,Caoetal.2010;Santosetal.2008).Inourmodel,however,andforreasonsofoutcometractability,allagentshavethesameendowmentandfaceabinarydecision(fullycontributeitornottothecommonpool).

5Accordingtotheformalisationofourmodel,thesamedistributionofvaluesofpbetweenagentsofagivennetworkwouldentaildifferentoutcomesdependingonwhichvaluesofpareassociatedwithagentshavingdifferentnodaldegrees.Incorporatingtheassumptionofheterogeneitywillexponentiallyincreasethenumberofcombinationsnecessarytoobtainarealisticassessmentofitsimpactonthesuccessrate.Consideringalsothatourultimategoalistotesttherelevanceofthestructuralpropertiesofthenetwork,andthatasdiscussedbelowwewillworkwithacatalogueof230networks,itseemsreasonabletochoosethehypothesisofhomogeneitytoassuretheanalyticaltractabilityofthemodeloutputs.

6Regardingthelatter,thissituationisstrategicallyequivalenttoasimultaneousdecision.

7InNetCommonsthissixthsituationhasbeensubsumedinSituation4,becauseindecisionaltermstheresultisthesame.

8Comparingthe"sequentialprotocol"orderingandournetwork-dependentsequenceorderingisbeyondthescopeofthispaper.Insequentialprotocolorderingthereisnonecessityforapparameter,becauseagentsknowwhichinformationothershaveatthemomentoftheirdecision.

9Togainaccesstothefullcatalogueofnetworksusedasinput,pleasecontacttheauthorsdirectly.

References

ABELE,S.&Ehrhart,K.M.(2005).Thetimingeffectinpublicgoodsgames.JournalofExperimentalSocialPsychology,41,470–481.[doi:10.1016/j.jesp.2004.09.004]

BONACICH,P.(1990).CommunicationDilemmasinSocialNetworks:AnExperimentalStudy.AmericanSociologicalReview,55(3),448–459.[doi:10.2307/2095768]

BRAMOULLÉ,Y.&Kranton,R.(2007).Publicgoodsinnetworks.JournalofEconomicTheory,135,478–494.[doi:10.1016/j.jet.2006.06.006]

BRAVO,G.(2008).ImitationandcooperationinDifferentHelpingGames.JournalofArtificialSocietiesandSocialSimulation11(1),8http://jasss.soc.surrey.ac.uk/11/1/8.html.

BRAVO,G.,Squazzoni,F.andBoero,R.(2012).Trustandpartnerselectioninsocialnetworks:anexperimentallygroundedmodel.SocialNetworks,34,481–492.[doi:10.1016/j.socnet.2012.03.001]

BUDESCU,D.V.,WingTung,A.andChen,X.P.(1997).Effectsofprotocolofplayandsocialorientationonbehaviorinsequentialresourcedilemmas.Organisationalbehaviorandhumandecisionprocesses,69(3),179–193.[doi:10.1006/obhd.1997.2684]

CAO,X.,Du,W.andRong,Z.(2010).Theevolutionarypublicgoodsgameonscale-freenetworkswithheterogeneousinvestment,PhysicaA,389,1273–1280.[doi:10.1016/j.physa.2009.11.044]

CÁRDENAS,J.C.&Jaramillo,C.R.(2009).cooperationinlargenetworks:anexperimentalapproach.http://wwwprof.uniandes.edu.co/~chjarami/cardenas_jaramillo_cooperation_in_large_networks_CEDE200706_20070430.pdf.Archivedathttp://www.webcitation.org/67w5FhNia.

CHWE,M.S.Y.(1999).Structureandstrategyincollectiveaction.AmericanJournalofSociology,105(1),128–156.[doi:10.1086/210269]

http://jasss.soc.surrey.ac.uk/17/1/4.html 16 16/10/2015

ELSTER,J.(1979).UlyssesandtheSirens:Studiesinrationalityandirrationality.Cambridge[Eng.];NewYork:CambridgeUniversityPress.

ELSTER,J.(2000).Ulyssesunbound:Studiesinrationality,precommitment,andconstraints.Cambridge;NewYork:CambridgeUniversityPress.[doi:10.1017/CBO9780511625008]

EREV,I.&Rapoport,A.(1990).Provisionofstep-levelpublicgoods.Thesequentialcontributionmechanism.JournalofConflictResolution,34(3),401–425.[doi:10.1177/0022002790034003002]

FOWLER,J.H.&Christakis,N.(2010).co-operativebehaviorcascadesinhumansocialnetworks,PNAS,107(12),5334–5338.[doi:10.1073/pnas.0913149107]

FU,F.,Liu,L.-H.andWang,L.(2007).EvolutionaryPrisoner'sDilemmaonheterogeneousNewman-Wattssmall-worldnetwork,Eur.Phys.J.B,56,367–372.[doi:10.1140/epjb/e2007-00124-5]

GOULD,R.V.(1993)CollectiveActionandNetworkStructure.AmericanSociologicalReview,58(2),182–196.[doi:10.2307/2095965]

GOULD,R.V.,(2003).Whydonetworksmatter?Rationalistandstructuralistinterpretations.InM.Diani&D.McAdam(Eds.),SocialMovementsandNetworks(pp.233-257).OxfordUniversityPress.[doi:10.1093/0199251789.003.0010]

HARDIN,R.(1982).Collectiveaction.Baltimore:JohnsHopkinsUniversityPress.

HECKATHORN,D.D.(1996).Thedynamicsanddilemmasofcollectiveaction.AmericanSociologicalReview,61(2),250–277.[doi:10.2307/2096334]

HEDSTRÖM,P.&Swedberg,R.(1996).SocialMechanisms,ActaSociologica,39,281–308.[doi:10.1177/000169939603900302]

HEDSTRÖM,P.&Swedberg,R.(eds.)(1998).SocialMechanisms:AnAnalyticalApproachtoSocialTheory.Cambridge,CambridgeUniversityPress.[doi:10.1017/cbo9780511663901]

HORVATH,G.,Kovárík,J.andMengel,F.(2012).Limitedmemorycanbebeneficialfortheevolutionofcooperation.JournalofTheoreticalBiology,300,93–205.

KERR,N.(1989).Illusionsofefficacy:theeffectsofgroupsizeonperceivedefficacyinsocialdilemmas.Journalofexperimentalsocialpsychology,25(4),287–313.[doi:10.1016/0022-1031(89)90024-3]

KNIGGE,A.&Buskens,V.(2010).Coordinationandcooperationproblemsinnetworkgoodproduction.Games,1,357–380.[doi:10.3390/g1040357]

KOLLOCK,P.(1998).Socialdilemmas:theanatomyofcooperation.Annualreviewofsociology,24,183–214.[doi:10.1146/annurev.soc.24.1.183]

KOMORITA,S.S.&Parks,C.D.(1995).Interpersonalrelations:mixed-motiveinteraction.Annualreviewofpsychology,46,183–207.[doi:10.1146/annurev.ps.46.020195.001151]

KOSFELDM.(2004).EconomicNetworksintheLaboratory:ASurvey.ReviewofNetworkEconomics,3(1),BerkeleyElectronicPress.[doi:10.2202/1446-9022.1041]

LEDYARD,J.O.(1995).Publicgoods:asurveyofexperimentalresearch.InJ.Kagel&A.E.Roth(Eds.).Thehandbookofexperimentaleconomics.Princeton:PrincetonUniversityPress.

LEÓN,F.J.(2010).Lalógicadelaaccióncolectivaenlaproduccióndebienespúblicosescalares.Papers.Revistadesociología,95(2),363–387.

LINARES,F.(2007).Elproblemadelaemergenciadenormassocialesenlaaccióncolectiva:Unaaproximaciónanalítica.RevistaInternacionaldeSociología,46,131–160.

MARKS,M.&Croson,R.(1998).Alternativerebaterulesintheprovisionofathresholdpublicgood:Anexperimentalinvestigation.JournalofPublicEconomics,67,195–220.[doi:10.1016/S0047-2727(97)00067-4]

MARWELL,G.&Ames,R.E.(1980).Experimentsontheprovisionofpublicgoods.II.Provisionpoints,stakes,experience,andthefree-riderproblem.Americanjournalofsociology,85(4),926–937.[doi:10.1086/227094]

MARWELL,G.&Oliver,P.(1993).Thecriticalmassincollectiveaction.Amicro-socialtheory.Cambridge:CambridgeUniversityPress.[doi:10.1017/CBO9780511663765]

MARWELL,G.,Pamela,E.andPrahl,R.(1988)Socialnetworksandcollectiveaction:atheoryofthecriticalmass.AmericanJournalofSociology,94,502–534.[doi:10.1086/229028]

MORENO,J.L.(1951).Sociometry,ExperimentalMethodandtheScienceofSociety.AnApproachtoaNewPoliticalOrientation.BeaconHouse,Beacon,NewYork.

MIGUEL,F.J.(2011).NetCommons(code).Archivedat:http://www.webcitation.org/67w5ftehr.

OBERSCHALL,A.(1973).Socialconflictandsocialmovements.EnglewoodCliffs,Prentice-Hall.

OLSON,M.(1965).Thelogicofcollectiveaction.Cambridge,MA:HarvardUniversityPress.

PUJOL,J.M.,Flache,A.,Delgado,J.andSangüesa,R.(2005).Howcansocialnetworkseverbecomecomplex?Modellingtheemergenceofcomplexnetworksfromlocalsocialexchanges.JournalofArtificialSocietiesandSocialSimulation,8(4),http://jasss.soc.surrey.ac.uk/8/4/12.html.

RAPOPORT,A.(1985).ProvisionofpublicgoodsandtheMCSexperimentalparadigm.AmericanPoliticalScienceReview,79(1),148–155.[doi:10.2307/1956124]

RAPOPORT,A.(1987).Researchparadigms,andexpectedutilitymodelsfortheprovisionofstep-levelpublicgoods.PsychologicalReview,94,74–83.[doi:10.1037/0033-295X.94.1.74]

RAPOPORT,A.(1988).Provisionofstep-levelpublicgoods:Effectsofinequalityinresources.JournalofPersonalityandSocialPsychology,45,432–440.[doi:10.1037/0022-3514.54.3.432]

RAPOPORT,A.(1993).Incrementalcontributioninstep-levelpublicgoodsgameswithasymmetricplayers.Organisationalbehaviorandhumandecisionprocesses,55,171–194.[doi:10.1006/obhd.1993.1029]

SANTOS,F.C.,Santos,M.andPacheco,J.M.(2008).Socialdiversitypromotestheemergenceofco-operativebehavior.Nature,454,213-216.[doi:10.1038/nature06940]

http://jasss.soc.surrey.ac.uk/17/1/4.html 17 16/10/2015

SCOTT,J.(2005).Socialnetworkanalysis:Ahandbook.London:Sage.

SKYRMS,B.&Pemantle,R.(2000).Adynamicmodelofsocialnetworkformation.PNAS,97(16),9340–9346.[doi:10.1073/pnas.97.16.9340]

SONNEMANS,J.,vanDijk,F.andvanWinden.F.(2006).Onthedynamicsofsocialtiesstructuresingroups.Journalofeconomicpsychology,27,187–204.[doi:10.1016/j.joep.2005.08.004]

SURI,S.&Watts,D.J.(2011).cooperationandcontagioninweb-based,networkedpublicgoodsexperiments,PLoSONE,6(3),1–18.[doi:10.1371/journal.pone.0016836]

SUTCLIFFE,A.,Wang,D.andDunbar,R.(2012).Socialrelationshipsandtheemergenceofsocialnetworks.JournalofArtificialSocietiesandSocialSimulation,15(4),3,http://jasss.soc.surrey.ac.uk/15/4/3.html.

TAKÁCS,K.,Janky,B.andFlache,A.(2008).Collectiveactionandnetworkchange,SocialNetworks,30,177–189.[doi:10.1016/j.socnet.2008.02.003]

TAYLOR,M.(1987).Thepossibilityofcooperation.Cambridge:CambridgeUniversityPress.

TAYLOR,M.&Ward,H.(1982).Chickens,whalesandlumpygoods:alternativemodelsofpublic-goodsprovision.Politicalstudies,30,350–370.[doi:10.1111/j.1467-9248.1982.tb00545.x]

TILLY,Ch.(1978).Frommobilisationtorevolution.Reading:Addison-Wesley.

TSUKAMOTO,E.&Shirayama,S.(2010).Influenceofthevarianceofdegreedistributionsontheevolutionofcooperationincomplexnetworks,PhysicaA,389,577–586.[doi:10.1016/j.physa.2009.10.002]

VANDEKRAGT,A.J.C.,Orbell,J.M.andDawes,R.M.(1983).TheMinimalContributingSetasasolutiontopublicgoodsproblems.TheAmericanpoliticalsciencereview,77(1),112–122.[doi:10.2307/1956014]

YANG,H.,Wu,Z.andDu,W.(2012).Evolutionarygamesonscale-freenetworkswithtunabledegreedistribution.EPL,99,10006.[doi:10.1209/0295-5075/99/10006]

YE,H.,Tan,F.,Ding,M.,Jia,Y.andChen,Y.(2011).SympathyandPunishment:EvolutionofcooperationinPublicGoodsGame.JournalofArtificialSocietiesandSocialSimulation,14(4),20<http://jasss.soc.surrey.ac.uk/14/4/20.html>.

ZSCHACHE,J.(2012).Producingpublicgoodsinnetworks:Someeffectsofsocialcomparisonandendogenousnetworkchange.SocialNetworks,34,539–548.[doi:10.1016/j.socnet.2012.05.003]

http://jasss.soc.surrey.ac.uk/17/1/4.html 18 16/10/2015