行为-社会网络中的贝叶斯学习(纽约大学艾伦和盖尔金融经济学讲义)_

BayesianLearninginSocialNetworks1DouglasGale(CorrespondingAuthor)DepartmentofEconomics,NewYorkUniversity269MercerSt.,7thFloor,NewYork,NY,10003-6687.E-mail:[e-mail]::[phone]Fax:[phone]andShacharKarivDepartmentofEconomics,NewYorkUniversity269MercerSt.,7thFloor,NewYork,NY,10003-6687.E-mail:[e-mail]:~sk510Version:March13,2003.Weextendthestandardmodelofsociallearningintwoways.First,weintroduceasocialnetworkandassumethatagentscanonlyobservetheactionsofagentstowhomtheyareconnectedbythisnetwork.Secondly,weallowagentstochooseadifferentactionateachdate.Ifthenetworksatisfiesaconnectednessassumption,theinitialdiversityresultingfromdiverseprivateinformationiseventuallyreplacedbyuniformityofactions,thoughnotnecessarilyofbeliefs,infinitetimewithprobabilityone.Welookatparticularnetworkstoillustratetheimpactofnetworkarchitectureonspeedofconvergenceandtheoptimalityofabsorbingstates.Convergenceisremarkablyrapid,sothatasymptoticresultsareagoodapproximationeveninthemediumrun.JournalofEconomicLiteratureClassificationNumbers:D82,D83KeyWords:Networks,Sociallearning,Herdbehavior,Informationalcascades.RunningTitle:BayesianLearninginSocialNetworks.1OneofusdiscussedthisproblemwithBobRosenthalseveralyearsago,whenwewerebothatBostonUniversity.Atthattime,wefoundtheproblemoflearninginnetworksfascinatingbutmadenoprogressandwereeventuallydivertedintoworkingonboundedlyrationallearning,whichledtoourpaperonimitationandexperimentation.WethankseminarparticipantsatNYU,DELTA,INSEAD,Cergy,CornellandIowafortheircomments.ThefinancialsupportoftheNationalScienceFoundationthroughGrantNo.SES-0095109isgratefullyacknowledged.11.INTRODUCTIONThecanonicalmodelofsociallearningcomprisesasetofagentsI,afinitesetofactionsA,asetofstatesofnatureΩ,andacommonpayofffunctionU(a,ω),whereaistheactionchosenandωisthestateofnature.Eachagentireceivesaprivatesignalσi(ω),afunctionofthestateofnatureω,andusesthisprivateinformationtoidentifyapayoff-maximizingaction.Thissetupprovidesanexampleofapureinformationexternality.Eachagent’spayoffdependsonhisownactionandonthestateofnature.Itdoesnotdependdirectlyontheactionsofotheragents.However,eachagent’sactionrevealssomethingabouthisprivatesignal,soanagentcangenerallyimprovehisdecisionbyobservingwhatothersdobeforechoosinghisownaction.Insocialsettings,whereagentscanobserveoneanother’sactions,itisrationalforthemtolearnfromoneanother.ThiskindofsociallearningwasfirststudiedbyBanerjee(1992)andBikhchandani,HirshleiferandWelch(1992).TheirworkwasextendedbySmithandSørensen(2000).Thesemodelsofsociallearningassumeasim-plesequentialstructure,inwhichtheorderofplayisfixedandexogenous.Theyalsoassumethattheactionsofallagentsarepublicinformation.Thus,atdate1,agent1choosesanactiona1,basedonhisprivatein-formation;atdate2,agent2observestheactionchosenbyagent1andchoosesanactiona2basedonhisprivateinformationandtheinformationrevealedbyagent1’saction;atdate3,agent3observestheactionschosenbyagents1and2andchoosesanactiona3...;andsoon.Inwhatfollowswerefertothisstructureasthesequentialsocial-learningmodel(SSLM).Onegoalofthesociallearningliteratureistoexplainthestrikinguni-formityofsocialbehaviorthatoccursinfashion,fads,“mobpsychology”,andsoforth.InthecontextoftheSSLM,thisuniformitytakestheformofherdbehavior.2SmithandSørensen(2000)haveshownthat,intheSSLM,herdbehaviorarisesinfinitetimewithprobabilityone.Oncetheproportionofagentschoosingaparticularactionislargeenough,thepub-licinformationinfavorofthisactionoutweighstheprivateinformationofanysingleagent.Soeachsubsequentagent“ignores”hisownsignaland“followstheherd”.Thisisanimportantresultandithelpsusunderstandthebasisforuniformityofsocialbehavior.3Atthesametime,theSSLMhasseveral2Aherdoccursif,aftersomefinitedatet,everyagentchoosesthesameaction.Aninformationalcascadeoccursif,aftersomefinitedatet,everyagentfindsitoptimaltochoosethesameactionregardlessofthevalueofhisprivatesignal.Aninformationalcascadeimpliesherdbehavior,butaherdcanarisewithoutacascade.3ThemostinterestingpropertyofthemodelsofBikhchandani,HirshleiferandWelch(1992)andBanerjee(1992)isthatinformationalcascadesariseveryrapidly,beforemuchinformationhasbeenrevealed.Forexample,inthesemodelsifthefirsttwoagentsmakethesamechoice,allsubsequentagentswillignoretheirinformationandimitatethefirsttwo.Thebehaviorofapotentialinfinityofagentsisdeterminedbythebehaviorofthefirsttwo.ThisisbothinformationallyinefficientandParetoinefficient.2specialfeaturesthatdeservefurtherexamination:(i)eachagentmakesasingle,irreversibledecision;(ii)thetimingoftheagent’sdecision(hispositioninthedecision-makingqueue)isfixedandexogenous;(iii)agentsobservetheactionsofalltheirpredecessors;and(iv)thenumberofsignals,likethenumberofagents,isinfinite,soonceacascadebeginstheamountofinformationlostislarge.ThesefeaturessimplifytheanalysisoftheSSLM,buttheyarequiterestrictive.Inthispaper,westudytheuniformityofbehaviorinaframeworkthatallowsforaricherpatternofsociallearning.WedepartfromtheSSLMintwoways.First,wedroptheassumptionthatactionsarepu