信息级联和阈值实现：理论及其在众筹中的应用 Information Cascades and Threshold Implementation：Theory and An Application to Crowdfunding

NBERWORKINGPAPERSERIESFORMATIONCASCADESANDTHRESHOLDIMPLEMENTATIONORYANDANAPPLICATIONTOCROWDFUNDINGinWilliamCongizhouXiaoWorkingPaper30820http//papers/w30820NATIONALBUREAUOFECONOMICRESEARCHCambridgeMA8January3CongthankstheEwingMarionKauffmanFoundationforresearchfunding,andXiaothanksthegrantfromtheResearchGrantsCounciloftheHongKongSpecialAdministrativeRegion,China(ProjectNo.CUHK24500417)TheviewsexpressedhereinarethoseoftheauthorsanddonotnecessarilyreflecttheviewsoftheNationalBureauofEconomicResearch.NBERworkingpapersarecirculatedfordiscussionandcommentpurposes.Theyhavenotbeenpeer-reviewedorbeensubjecttothereviewbytheNBERBoardofDirectorsthataccompaniesofficialNBERpublications.©2023byLinWilliamCongandYizhouXiao.Allrightsreserved.Shortsectionsoftext,nottoexceedtwoparagraphs,maybequotedwithoutexplicitpermissionprovidedthatfullcredit,including©notice,isgiventothesource.InformationCascadesandThresholdImplementationTheoryandAnApplicationtoCrowdfundingLinWilliamCongandYizhouXiaoNBERWorkingPaperNo.30820January3JELNoD,D83,G12,G14STRACTEconomicinteractions,suchascrowdfunding,ofteninvolvesequentialactions,observationallearning,andcontingentprojectimplementation.Weincorporateall-or-nothingthresholdsinacanonicalmodelofinformationcascades.Earlysupporterseffectivelydelegatetheirdecisionstoa"gatekeeper,"resultinginuni-directionalcascadeswithoutherdingonrejections.Projectproposersconsequentlycanchargehigherprices.Proposalfeasibility,projectselection,andinformationaggregationallimprove,evenwhenagentscanwait.Equilibriumoutcomesdependonthecrowdsize,andprojectimplementationandinformationaggregationachieveefficiencyinthelarge-crowdlimit.Ourkeyinsightsremainrobustunderthresholdsindollaramounts,alternativeequilibriumselection,amongothermodelextensions.LinWilliamCongSCJohnsonCollegeofBusinessCornellUniversitySageHallIthaca,NY14853andNBERwill.cong@izhouXiaoChineseUniversityofHongKongHongKongyizhou@.hkAdataappendixisavailableat/data-appendix/w3082011Introductionsinessactivitiesandgatheringsupportofteninvolvesequentialcontributorsectimplementationcontingentonachievingcertainthresholdhepastdecadecrowdbasedfundraisingwhichincludesequityandrewardcrowdfunding,peer-to-peerlending,andinitialcoinofferings,constitutesthemostgentsarepronetoinformationcascadesthatcauseincompleteinformationaggregationandsubopti-1992)focusonpureinformationalexternalitieswitheachagent’spayoffstructureindepen-dentofothers’actions.Weincorporateintoamodelofdynamiccontributiongamesthefactthatmanyprojectsorproposalsinpracticeareonlyimplementedwithasufficientlevelofsupport—an“all-or-nothing”(AoN)threshold.Weshowthatthresholdimplementationdrasticallyaltersinformationalenvironmentsandeconomicoutcomes,withimplicationsforfinancingprojectsandaggregatinginformation—arguablythetwomostimportantfunctionsofmodernfinancialmarkets.1Specifically,weintroducethresholdimplementationinastandardframeworkofinfor-mationcascade`alaBikhchandani,Hirshleifer,andWelch(1992).AprojectproposalischoosetosupportorrejectEachsupporterpaysapre-specifiedcontributionprice,andgetsaneventualpayoffnormalizedtooneiftheprojectisgood.Allagentsarerisk-neutralwithacommonpriorbeliefabouttheproject’squality.Theyeachreceiveaprivate,informativesignalandobservetheactionsofprecedingagentsbeforedecidingwhethertosupport.Deviatingfromtheliterature,supporterspaythepriceandreceivethepayoffifandonlyifthesupportlevelreachesanAoNthreshold,whichiseitherexogenouslygivenorendogenouslydeterminedjointlywiththepricebytheproposer.AoNthresholdsleadtouni-directionalcascadesinwhichagentsneverrationallyignorepositiveprivatesignalstorejecttheproject(i.e.,thereispracticallynoDOWNcascade,whichwedefinepreciselyinthemodel),butmayrationallyignorenegativeprivatesig-nalstosupporttheproject(i.e.,UPcascadesarepossible),makingtheagentsappearto1AoNthresholdispredominantoncrowdfundingplatformsandinventurefinancing.Moreover,super-majorityruleorq-ruleisacommonpracticeinmanyvotingprocedures;assurancecontractorcrowdactioninpublicgoodsprovisionischaracterizedbysequentialdecisionsandimplementationthresholds(e.g.,BagnoliandLipman,1989).Inasimilarspirit,charitableprojectssettargetlevelsoffundraisingtoproceed(e.g.,Andreoni,1998).2havefearsofmissingout.Informationaggregationalsobecomesmoreefficient,especiallywithalargecrowd.Withendogenousimplementationthresholdandprice,theproposernoproposalfeasibility(positiveprobabilityforimplementation),projectselection(goodprojectsbeingmorelikelyimplementedthanbadprojects),andinformationaggregation(publicsupportinghistoryrevealingprojectquality)allimprove.Inparticular,whenthenumberofagentsapproachesinfinity,equilibriumprojectimplementationandinformationaggregationbecomeefficient,eratureoninformationcascadesBanerjee1992;Lee,1993;Bikhchandani,Hirshleifer,andWelch,1998;AliandKartik,2012).Toderivetheseresults,wefirsttaketheAoNthresholdandpriceasgiveninthesub-gameofagentcontributionandlearning.Weshowthatbeforereachingthethreshold,theaggregationofprivateinformationonlystopsuponanUPcascade.TheintuitionisthattheAoNthresholdlinksagents’payoffstosubsequentagents’actions,makingthempartiallyin-ternalizetheinformationalexternalitiesoftheiraction.Suchforward-lookingconsiderationsleadtointerestingasymmetries:EvenbeforeanUPcascade,agentswithpositiveprivateagentwhosesupportingdecisionbringsthetotalsupporttothethreshold.This“delegation”hedgesagainstmistakenlysupportingabadprojectbecausethesubsequent“gate-keeper”makesthecontributiondecisionwithbetterinformationbyobservingalongersequenceofpreviousactions.DOWNcascadesarethereforealwaysinterruptedbyagentswithpositivesignalsbeforetheAoNthresholdisreached.Incontrast,anagentwithanegativesignalisreluctanttosupportaprojectbeforeitreachestheAoNthresholdoranUPcascade,forfearthattheirsupportingactions(whichwouldbeindistinguishablefromtheactionsofagentswithpositivesignalsnow)maymisleadsubsequentagentstopositivelyupdatethebeliefontheprojectqualityinspiteofthenegativesignaltheagentprivatelyobserves.Thisagent’ssupportingactionthenincreasesthelikelihoodofabadprojectbeingfunded,reducingherexpectedpayoff.Butoncetheagent’sbeliefontheprojectqualityissufficientlyhigh,antsbecausetheydonotpositivelyupdatesfromheractionanyway.old,aswellasthecontributionprice,tomaximizethelevelofsupport.AhigherAoNthresh-ed3earlier,aDOWNcascadecannothappenbeforetheAoNthresholdhasbeenreached.Theentrepreneur’soptimalAoNisthussettobejustsufficientsothatachievingitimpliesahighvaluationrelativetothecontributionpriceandessentiallyexcludesDOWNcascades.Meanwhile,theproposertradesoffincreasingtheproceedsfromsupporters(bychargingahigherpricewithloweringtheAoNandchargingacorrespondinglylowerpricesoastostilleffectivelyexcludeDOWNcascades)toboosttheprobabilityofimplementingtheproject.Ingeneral,alargercrowdmitigatestheconcernaboutimplementationfailureandgenerallypermitsahigheroptimalprice,makingpricesendogenouslydependentonthecrowdsize.AoNthresholdsanduni-directionalcascadeshavethreeimportantimplications.First,theyimproveprojectfeasibilitybyallowinggoodprojectswithhighproductioncoststobesupported.Standardinformationcascadetheoriessuggestthatforprojectswithhighproductioncosts,thecontributionpricetoatleastcoverthecostissohighthatthefirstagentwillrejectitevenwithapositiveprivatesignal,resultinginaDOWNcascadeandaguaranteedfundingfailure(Welch,1992).AoNthresholdsmitigatetheconcernoverDOWNcascades,makingitpossibletochargeahighpricetocovertheproductioncosts.Second,Nthresholdsimproveprojectimplementationeciencybecausechargingahighpriceimntheposteriorbeliefissucientlypositivewhichiscorrelatedwiththeproject’spositivequality.Third,AoNthresholdsfacilitateinformationaggregationbymitigatingDOWNcascadesanddelayingthearrivalofUPcascades.AproposerfacingalargenumberofpotentialsupporterscanutilizethresholdimplementationtoguardagainstDOWNcascadesandtochargeahighcontributionprice(whichdelaysUPcascades)forgreaterproceedsorsupportregardlessofwhetherthethresholdiseventuallyreached.Whileoutcomesinstandardmodelsofinformationcascadesareindependentofthesizeoftheagentbase,thecasewithAoNthresholdsdiffers:theerrorsofmis-supportingormis-rejectingdecreasewiththecrowdsize,andtheendogenouspriceconvergestothehighestlevelatwhichtheproposerextractsfullsurplus.Inthelimit,projectsareimplementedifandonlyiftheyareofhighquality.Publicknowledgeabouttheproject’struetypealsobecomesperfect.Wethereforeobtainsociallyefficientprojectimplementation(underprivatesignals)andfullinformationaggregationwithalargecrowd,hithertounachievableinmostmodelsevantintheageofdigitalplatformsandtheInternet,whichfeatureoutreachestoextremelylargecrowds.Finally,wedemonstratethatourkeyinsightsapplyevenwhenagentshavetheoptiontosubjecttotheusualcritiquesofexogenousactiontiming.Wealsoshowthatourfindingsarerobusttointroducinginvestorheterogeneityandthresholdsbasedondollaramounts(andtointroducingsmallcontributionfrictionsorlearningcosts,whichisdiscussedintheappendix).WefurtheranalyzeotherperfectBayesianNashequilibriaunderthesamemildtie-breakingconventionandtounderstandthestrategiccomplementarityintroducedbyAoNthresholds.Intermsofprojectimplementationandinformationaggregation,theequilibriumThetheoreticalinsightswederiveapplytomanysequentialcontributiongamessuchasventurefinancingorsyndicatedloans.Wehighlighttheapplicationtocrowdfundingforseveralreasons:First,crowdfundinghasquicklybecomeamainstreamsourceofcapitalforentrepreneurs,withitstotalvolumesurpassingthemarketsizeforangelfundsin2015andreachingawhopping35billionUSDgloballyin2017evenbeforetheexplosionofcrypto-tokenofferings.Second,itpresentsasettingwherethetechnologyallowsanoutreachtolargecrowds,whichrendersthelimitingresultsforlargecrowdsrelevantandimportant.Third,thesequentialnatureofcontributionsandthresholdimplementationaresalientincrowdfunding,makingitrepresentativeofgeneraldynamiceconomicinteractionswithob-servationallearningandthresholdimplementation,unlikeauctions.Moreover,otherformsofentrepreneurialorcorporatefinancealsofeatureinvestorsfre-quentlyinquiringaboutprecedinginvestmentsaswellasthresholdimplementationwrittenmoneybackguaranteesorprivateplacementmemoranda.2Therefore,theycanalsobeanalyzedthroughourcon-ceptuallens,furtherdemonstratingthepracticalimportanceofthresholdimplementationdesigninaconsiderablevarietyofeconomicinteractionsandfinancingsituations.Literature—Ourpaperaddstothetheoryofinformationalcascades,sequentialdeci-sions,andobservationallearning.Theinsightsfrompriordynamicinformationalmodelsprimarilyconcernsignalstructureandlearningbias(Banerjee,1992;Bikhchandani,Hirsh-2InanangelorAroundoffinancing,investorswhoareapproachedlaterinthefundraisingprocessoftenlearnwhichotherfinanciersindicatedtheirsupportfortheprojectandofferadditionalcontributionsontheconditionthatthefundraisingreachescertainthresholds(Halac,Kremer,andWinter,2020).InIPOprocesses,lateinvestorslearnfromobservingthebehaviorofearlyinvestors,andtheissuermaychoosetowithdrawtheofferingifthemarketreactionislukewarm(e.g.,RitterandWelch,2002).Infact,intheearly1980s,manytinyfirmsintheUnitedStatesconductedanIPOwithabesteffortscontractthatfrequentlyhadanAoNfeature.WethankJayRitterforprovidingthisexampleandSteveKaplanforshowingussampleproprietarydocumentsofprivateplacementmemoranda.45erandWelchWelchBikhchandaniHirshleiferandWelchChamley2004;Callander,2007).Traditionally,informationalcascadescanbeasymmetricorevenrinoHarmgart,andHuck,2011;HerreraandH¨orner,2013).Ourcontributionstothislitera-turearetwo-fold.First,weobtainasymmetricinformationalcascadesendogenouslyduetothresholdimplementationevenwithobservableactions.Second,weshowthatfulllearningimplementation.Importantly,weobtainperfectinformationaggregationinlarge-crowdlim-its,whichistypicallyunachievablewithinformationcascades(AliandKartik,2012).Ourhaviorbylargecrowdsandaddstotheunderstandingofhowthelatesttechnologies,suchastheInternetandblockchain,impactthesocialefficiencyininformationaggregationandfundraisinginfinancialmarkets.ingandmarketplacelending.Strausz(2017)andEllmanandHurkens(2015)findthatAoNiscrucialformitigatingmoralhazardsandpricediscrimination.ChemlaandTinn(2018)sharetheconcernformoralhazardasinStrausz(2017),butinadditionemphasizetherealoptionoflearningthroughcrowdfunding.Chang(2016)showsthatinsimultaneousmovegamesasinChemlaandTinn(2018),AoNalsogeneratesmoreprofitundercommon-valueassumptionsbymakingtheexpectedpaymentspositivelycorrelatedwithvalues.HakenesandSchlegel(2014)arguethatendogenousloanratesandAoNthresholdsencouragein-formationacquisitionbyindividualhouseholdsinlending-basedcrowdfunding.BrownandDavies(2020)focusonasimultaneous-actionsettinginwhichthresholdimplementation,whensetbyanentrepreneurafterobservingthetotalcontributioncreatesalosers’blessingthatdiscouragesinvestors’informationacquisitionandreducesfinancingefficiency.Insteadofintroducingmoralhazardorfinancialconstraint,weofferthefirstdynamicmodelofsequentialcontributionunderthresholdimplementation.Ouremphasisonobser-vationallearning,asalientfeatureofcrowdfundingandsupport-gatheringprocessesinreallife,distinguishesourpaperfromandcomplementstheexistingcrowdfundingliteratureand3Anaveragecrowdfundingcampaignlasts9weeksorlonger(/crowdfunding_statistics/).AsarticulatedbyCanal(2020),oneofthebestfeaturesofcrowdfundingplatformsisthat“userscanseethesuccessofacampaignasitprogresses,”nottomentiontheampleempiricalevidenceforagents’sequentialarrivals(e.g.,Vismara,2018).6ringcrowdfundingcampaignisendogenouslydeterminedbyboththetrueunderlyingqualityoftheprojectandthedynamiclearningun-derinformationalfrictions.WealsoconfirmthesuperiorityofAoNdesignsover“keep-it-all”designsinadynamicenvironmentandthevalueofcommittingtothresholdimplementa-tionforimprovingfinancingefficiency(forwhichBrownandDavies,2020,alsocontainsanexampleinasimultaneous-actionsetting)andinformationaggregation.Modelsofdynamiclearningbecomecomplicatedveryquickly.Regardingtheparticularapplicationofourtheory,wedonotclaimtocoverallaspectsofcrowdfunding,especiallyuranderandPerryOurpapershouldbeviewedasarststepinunderstandingtheconsequencesofintroducingthresholdimplementationsindynamiccon-tributiongameswithlargecrowds.InsteadofallowingtheentrepreneurtopossessprivateinformationaboutproductioncostasinStrausz(2017),weemphasizetheaggregationofinvestors’privatesignalsaboutprojectquality.WhereasBrownandDavies(2020)empha-sizesinvestors’informationacquisition,wefocusonentrepreneurs’ex-antecommitmenttoimplementationthresholdsinaffectinginformationaggregation,andwederivetheoptimalthresholdsinadynamicsetting.Therestofthepaperisorganizedasfollows:Section2setsupthemodel;Section3characterizestheequilibrium,startingwiththesubgameofcontributiontoillustratethemainmechanismbeforeendogenizingcontributionpricesandimplementationthresholds;Section4discussesmodelimplicationsonproposalfeasibility,projectselection,andinformationaggregationSectionextendsthemodeltoallowoptionstowait,budgetheterogeneityandthresholdsindollaramounts,andcharacterizationsofotherequilibria;Section6concludes.Theinternetappendicescontainalltheproofsanddetailsofvariousmodelextensions.2ADynamicModelofCrowd-basedSupport-gathering2.1ModelSetupConsideraprojectproposalpresentedtoagentsi=1,2,...,Nwhosequentiallytakeactionsaie{_1,1}toeithersupport(ai=1)orreject(ai=_1)it.4Incrowdfunding,4Weuse“support”and“invest”interchangeably,althoughourmodelcanbeappliedtosituationwherethecontributionisnon-pecuniary.Inpractice,crowdfunderstypicallyobserveboththetotalcapitalraised7supportingmeanscontributingfinancially;morebroadly,supportingcanbeinterpretedasadoptingoradvocatingforcertainbehaviorsbyincurringapersonalcost.Iftheproposalisimplemented,thentheproposercollectsfromeverysupportingagentapre-specified“con-tribution”p,andeachagentintheendreceivesaprojectpayoffV,whichiseither0or1.5Giventhatcrowdfundingoftenservesademanddiscoveryfunctioninmanycases(Strausz,2017),Vcanbeinterpretedasacrudetransformationoftheuncertainaggregatemarketdemand,whichcouldbehigh(V=1)orlow(V=0).Thresholdimplementation.Wedepartfromthepriorliteraturebyincorporating“all-or-nothing”(AoN)thresholdscommonlyobservedinpractice:theproposerreceives“all”contributionsifthecampaignreachesapre-specifiedthresholdlevelofsupport,or“noth-ing”otherwise.6Inotherwords,theprojectisimplementedifandonlyifatleastTagentssupportit,wherethethresholdTcouldbeexogenous,e.g.,drivenbytheneedtocoveraminimumscaleoftheprojectthatisoutsidetheentrepreneur’scontrol,oncethecontribu-tionpriceisspecified.Inmanycasesincludingcrowdfunding,however,Tisendogenouslysetbytheentrepreneur,whichisequivalenttosettingatotaldollaramountwhenagentsfacethesamecontributionprice.WediscussthresholdsindollaramountsunderinvestorheterogeneityinSection5.2.Notealsothatsupporterspayponlywhentheprojectisim-plemented.Thresholdimplementationsareansalientfeatureofcrowdfundingmarkets,andourcontributioncentersaroundprovidinginsightsontheirinformationaleffects,especiallyconcerningfinancingandinformationaggregationoutcomes.Agents’informationanddecision.Allagents(indexedbyi)andtheproposerarerational,risk-neutral,andsharethecommonpriorthattheprojectpaysV=0andV=1withequalprobability.Ourspecificationdescribesfittinglyequity-basedcrowdfundingandandthenumberofsupportersto-date(Vismara,2018),whosedistinctionisimmaterialinthebaselinemodel.Importantly,oursettingdiffersfromthatforvotingbecausenon-contributorsdonotbeartheriskyoutcomesoftheprojectwhereasnon-voterstypicallyfacetheconsequencesofavotingoutcome.5Aseparateliteratureallowspricetodynamicallychangeandfocusesonassetpricingimplications(AveryandZemsky,1998;Brunnermeier,2001;Vives,2010;ParkandSabourian,2011).Wefollowthestandardcascademodelstofixthepricefortakinganactionexante,whichcloselymatchesapplicationsincrowd-fundingandentrepreneurialfinance.Inotheractivitiessuchaspoliticalpetitions,pcanbeinterpretedasthesupportingeffortorreputationcostifthepetitiongoesthroughandbecomespublic.6TheJOBSActmandatesthatcrowdfundingplatformsadoptthresholdimplementation(Sec.4A.a.7.Seehttp:///bill/112th_congress/senate_bill/2190/text).TheAoNmechanism,al-ternativelyknownas“provisionpointmechanism,”hasalsobeenusedinRegulationDfilingssince1982(BagnoliandLipman,1989).AsinHakenesandSchlegel(2014);Chang(2016),weassumeanentrepreneurcancommitexantetoanimplementationthreshold.8peer-to-peerlending,whichconstitute80%oftheentirecrowdfundingmarketasof2020.preferences,thereisacommonvaluecorrespondingtothebasicqualityoftheproduct.Whileitdoesnotfullycapturecasessuchassalesofartpieceormusicwhereprivateprojectimplementationandinformationaggregationwithpriorstudies(e.g.,Fey,1996;Wit,Eachagentiobservesoneconditionallyindependentinformativeprivatesignalxie{1,_1}suchthat:Pr(xi=1|V=1)=Pr(xi=_1|V=0)=qe╱,1←.(1)Wedenotethesequenceofprivatesignalsbyx=(x1,...,xN)andthesetofallsuchsequencesbyX={1,_1}N.7esherdecision,sheobservesxiandthehistoryofactionsHi-1三(a1,a2...,ai-1)e{_1,1}i-1.Herstrategycanthusberepresentedasai(.,.):{1,_1}×{_1,1}i-1-∆({_1,1}),whichincludesmixedstrategiesintermsofprobabilitydistributionsoftheactionset{_1,1}.Tosimplifyexposition,wedefineAi==1aj✶(ai=1},for1<i<Nasthetotalnumberofsupportersuptoagenti.When1<i\<i<NandHi\hasthesamefirsti\elementsasHidoes,wesayHie{_1,1}inestsHi\e{_1,1}i\,aconceptweuseforequilibriumdefinitionlater.Agenti’soptimizationis:max✶(ai=1}了┌(V_p)✶(AN2T}|xi,Hi-1,ai=1┐,aie(-1,1}where,ANisthetotalnumberofsupportersamongallagents,and✶(AN2T}istheindicator7Thebinaryinformationandactionstructureherearestandardintheliterature(Bikhchandani,Hirsh-leifer,andWelch,1992).WeshowintheInternetAppendixC)thatthemainresultsandintuitionarerobustwithmultipleinvestmentamountsandwhensignalsareasymmetricallydistributed.8Whilerealworldexamplessuchascrowdfundingmayinvolveendogenousorderingsofagents,oursetupallowsacomparisonwiththelargeliteratureoninformationcascadeswhichtypicallyassumesexogenousordersofagents(Kremer,Mansour,andPerry,2014).Moreover,becauseagentsinpracticeupdatetheirbeliefsbasedonthepassageofcampaigntime(alsoseeninHerreraandH¨orner,2013)andusecontributioninformationalonetopredictfinalfundingoutcomes(HYPERLINK\l"_bookmark47