PersuasionfortheLongRun3JamesBestyDanielQuigleyzDecember92016LastUpda - PDF document

1 PersuasionfortheLong-RunJamesBestyDa
PersuasionfortheLong-RunJamesBestyDanielQuigleyzDecember9,2016[LastUpdated:30thJanuary2017]AbstractWeexaminethelimitsofpersuasionwhencredibilitytodayissustainedbytheincen-tiveoffuturecredibility.Wemodelthisasalong-runsenderwithprivateinformationplayingacheaptalkgameagainstshort-runreceiverswherethereisanoisysignalattheendofeachperiodonthesender'sex-anteprivateinformation.Wecompareourmodeloflong-runpersuasiontothepersuasionbaselineofcommittedpersuasion,wherethesendercancommittostrategiesatthestagegame.Long-runpersuasioncanonlyachievetheoptimalcommittedpersuasionpayosiftheoptimalcommittedpersuasionstrategyishonest.Whentheoptimalcommittedstrategyisnothonesttheuseofeitheraweakcommunicationmechanismcalleda`CoinandCup'(CnC)orastandardcommunicationmechanism(amediator)expandstheParetofrontierofthegame.Forsucientlypatientsenders,aCnCmechanismreplicatescommittedpersuasionpay-oswhenthesender'sinformationisperfectlyobservedex-post,whereasamediatorcangetarbitrarilyclosewheneversystematicdeviationfromtruthtellingisasymptot-icallyidentied.TheadvantageoftheCnCoverthemediatoristhatitisrelativelyeasytomanufactureandimplement.Finally,weshowhow`emergentcommunicationmechanisms'arisewhentherearemanysimultaneousreceivers.1IntroductionWouldbepersuaders,suchasadvertisers,salesmen,mediaoutlets,politicians,centralbanks,nancialanalysts,creditratingagencies,lobbyists,think-tanks,charities,activists,employ- SpecialthankstoBillZameforhisinvaluablehelponTheorem1ofthispaper.WewouldalsoliketothankPeterEso,IanJewitt,MegMeyer,AdrienVigier,JoelShapiro,InaTanevaandPeytonYoungfortheircommentsandadvice.yDepartmentofEconomicsandNueldCollege,UniversityofOxford(Email:james.best@economics.ox.ac.uk)zDepartmentofEconomicsandNueldCollege,UniversityofOxford(Email:daniel.quigley@economics.ox.ac.uk)1 mentagencies,managers,andmanyothers,relyontheirrecordofhonestytopersuadeotherstoday.Itfollowsthen,thatthedesiretopersuadeinthefuturecangeneratecredibility,andhencepersuasion,today.Allthisrequiresthatthereissomerecordedhistoryofhonesty,anddishonesty.Tothisend,manyinstitutionshavearisentokeeprecordsofhonesty:advertisersfundombudsmen;centralbankshav

2 ebegunpublicisingtheminutesofmeetingspas
ebegunpublicisingtheminutesofmeetingspast;theNationalAssociationofSecurityDealerslobbiedforlegislationenforcingthedisclosureofhistoricalnancialad-vice;theDodd-Frankactrequirescreditratingsagenciestodisclosepastperformanceandmethodologies;andsoon.Further,theinternethascreatedcompaniesthatalsofulllthisrole.Forexample,Amazon,E-Bay,andAlibabaallusevariousaggregatorsofthesellers'claimsandbuyers'feedbackinamannerthatincreasesthepersuasivepowerofsomesell-ers.Anotherrelatedexample,isthecurrentmovebysocialmediasitessuchasFacebookandTwittertomeasureandrecordwhoproduces`fakenews'.Finally,theinternetnowallowsustouseblockchaintocreatedecentralisedpublicrecordsof`claims'and`feedback'that,inconjunctionwithmoderncryptographictechnologies,openupnewfrontiersinbothmeasuringandrecordinghonesty.Whiletheimprovedrecordsgeneratedbysuchinstitutionsandtechnologiesincreasetheabilityofpersuaderstobecredibletoday,itisnotobviousthatitwillmakethemmorehonesttoday.Howmuchthencanapersuadergainfrombeingsystematicallydishonest?Howdodierentinstitutionsandtechnologieshelppersuadersusedishonestymoreeectively?Whenaresuchinstitutionsandtechnologiessociallyoptimal?Toanswerthesequestionwedevelopagenerallong-runpersuasiongamewhere,contraKamenicaandGentzkow[2011],thereisnocommitmentatthestagegame.ApatientSender(`he')playsacheaptalkgamewithasequenceofshortlivedReceivers(each`she').EachperiodastatethatispayorelevanttotheReceiverisrealisedaccordingtoani.i.d.process.TheSendergetsanoisysignalofthestate;hethensendsamessagetotheReceiver,whothenchoosesanactionbasedonherbeliefaboutthestate;nally,thereisapublicnoisysignalaboutthestate.TheReceiver'sactiondeterminestheSender'spayo.EachnewReceiverobservesthejointhistoryofmessagesandpublicsignalswhichsheusestoformherbeliefsabouttheSender'sstrategy.TheSenderthen,mustalsoconsidertheeectoftodaysadviceonthebeliefsandactionsofallfutureReceivers.Hence,thevalueofbeingtrustedinthefutureyieldsapotentialsourceofcommitmentfortheSender'sreportingstrategytoday.Ourrstmainresultisthat,evenwhentheSender'sprivateinformationisperfectlyobservedex-post,theSendercanonlyachievetheoptimalaveragepayoiftheoptimalcommittedpersuas

3 ionstrategyishonest.Whereastra
ionstrategyishonest.WhereastrategyishonestifandonlyiftheSender'sprivateinformationdeterminesexactlywhatmessagetheysend,i.e.,foreachsignal2 theysendonlyonemessage.1Theresultthenfollowsbecausedishoneststrategiesrequiremixingbetweenmultiplemessagesinsomestateoftheworld,andhenceindierenceacrossmessages.Itishelpfultothinkofthemessageyieldingtheloweststagepayoasthetruthandtheothermessagesaslies.Mixingisonlythencredibleifeachliereceivesanexpectedpunishmentthatwipesoutanygainatthestagefromlying.Thisonpathpunishmentpinstheaveragestagepayodowntothepayofromalwayssendingthetruthateachstageofthegame.Therefore,onlywhentheoptimalcommittedpersuasionstrategyincludesnolies,i.e.onlyonemessageforanysignal,canlong-runpersuasionachievethepayosavailableundercommittedpersuasion.Bycontrast,dishonestequilibriaarecharacterisedbytrustcyclesasReceiverspunishtheSenderbyignoringhimafterdetectinglies,harmingbothSenderandReceiversontheequilibriumpath.2Next,welookathowwecanusedierentcommunicationmechanismstoincreasetheeciencyofpersuasion.First,welookataveryweakformofcommunicationmechanismthatwecallaCoinandCup(CnC).ACnCisapayoirrelevantrandomvariablethatthesenderprivatelyobservesbeforesendinghisreporteachstage.ThisvariableisthenpubliclyrevealedonlyaftertheReceiverhastakenheraction.OursecondmainresultisthatwhenthereisnonoisethentheSendercanachievethesameaveragepayosasincommittedpersuasionthroughuseofaCnC.ThisfollowsbecausetheSenderneverneedstomixwithaCnC-insteadhecanconditionhismessagingstrategyonthevalueoftheCnCatthatstage.FromtheperspectiveoftheReceiveratagivenstagetheSenderisstillmixingmessagesacrossstateshowever,yetex-posteveryonecanverifythattheSenderstucktohispureCnCconditionedstrategyandthereisnoneedforpunishmentontheequilibriumpath.TherearethreefurtherthingstonoteabouttheCnC.First,thisdevicehasnoeectontheequilibriumoftheone-shotcheaptalkgame.Second,evenwithnoiseitstillexpandstheParetofrontierwhenoptimalcommittedpersuasionisdishonest.Finally,somethingliketheCnCalreadyexistsincryptography-calledacommitmentscheme-thatisusedforverydierentpurposes.Itwouldbeeasytoma

4 nufacturesuchamechanismwithacoinandacup,
nufacturesuchamechanismwithacoinandacup,oralargenumberofcoinsandatime-locksafe.However,thisiscumbersome,andwiththeadventoftheinternetitbecomesveryeasytoembedsuchacryptographicdeviceintoablockchainthatgeneratesapubliclyveriablerecordoftherandomvariable.ItisthissimplicityandeaseofpracticalimplementationthatmakestheCnCmechanismappealing. 1Thisnotionofhonestyallowsforcoarsemessagessolongasthemessagespaceisapartitionofthestatespace.Forexample,havingatotallyuninformativemessagestrategycanbehonesti.e.neversayinganything.Honestyalsoincludesthestandardinterpretationoftruthtelling:Sender'ssignalcanbeuniquelydeterminedfromhismessage.Typicallythepersuasionliteratureisinterestedincaseswheretheoptimalcommittedpersuasionstrategyisdishonest.2OnemighttentativelyventurethatthecurrentepisodeofvotersignoringexpertsregardingtheBrexitreferendumandtheUSpresidentialelectionsisanexampleofalowtrustphaseinsuchanequilibrium.3 Thethirdmainresultisthatastandardcommunicationmechanism,ormediator,cangetusarbitrarilyclosetothecommittedpersuasionpayossolongasthenoisesatisessomestandardlinearindependenceconditions.Themediatorusesareviewstrategytoswitchbetweenpersuasionphasesandpunishmentphases.ThereisareviewperiodwithalargeknownnumberofstagesandastartingpointknowntotheSenderandmediator,butnotthereceivers.ThemediatormapsthemessagesfromtheSenderintoadvicefortheReceiver.IftheSenderreportstruthfully,letthismappingyieldanexpectedstagepayoclosetothepayounderoptimalcommittedpersuasioninthepersuasionphase.Attheendofthereviewperiodthemediatorestimatestheprobabilityofobservingtheex-postnoisysignalsconditionalontheSenderalwaystellingtruth.Ifthisprobabilityislowenoughthemediatorwillswitchtoapunishmentphaseofbabblingforalargenitenumberofperiods.BychoosingalongenoughreviewperiodtheprobabilityofpunishingatruthfulSendercanbemadearbitrarilylow.Theproportionoflies,throughthismechanism,canbepushedclosetozero.TheReceiver'sobeythemediator'sadviceastheadvicegivesthemclosetothepayotheywouldreceiveunderoptimalcommittedpersuasion.However,theSenderwillstilllie,andimportantly,theywillfrontloadthoseliestowardsthebeginningofthereviewperiodduetoimpatience.Themediator'srolethe

5 n,istoaggregatehistoriessothatReceiversd
n,istoaggregatehistoriessothatReceiversdon'tknowwheretheyareineachreviewperiod-otherwise,earlyReceiversdon'tobeyandtheequilibriumunravels.AnaturalexamplewheresuchmediationstrategiesarepossiblewouldbeonlinesalesplatformssuchasE-bay,wheretheyusebandstoclassifythequalityofsellers-forexamplePowerSellers-basedonthefeedbackqualityoverxedwindowsofsales.Finally,weexposittwoexamplesofgameswhichhaveemergentcommunicationmecha-nisms.Therstisthecaseofmultipleidenticalreceiversactingsimultaneouslyeachperiodwithasinglestatevariable.Inthisgamethecontinuationequilibriumcanbeconditionedontheactionsofallagentsinthepriorperiod.ThesendercannowhaveastrategywherethereisabudgetforeachmessageconditionalontheSender'sprivateinformation,butthesendermixesoverwhoreceivesthemessage.ThisendsuphavingasimilarresulttoaCnCmechanism.WedemonstratethisinapublicgoodgamewhichalsoshowshowtheSender'sabilitytopersuadecanbewelfareenhancingforallplayers.Thesecondcasewelookatisonewherethesendersendssimultaneousmessagestomanyreceiversactingsimultaneouslyeachwiththeirownidiosyncraticpayorelevantstate.Inthisgamewecansimilarlyconditionthecontinuationequilibriumonthedistributionofactions,allowingasimilarbudgetingstrategyasinthepreviouscase.However,Receiver'shavealargersampletodetectdeviationsfromtheequilibriumstrategy,whichreducestheprobabilityofonpathpunishmentoccurringduetonoiseabouttheSender'sprivateinfor-4 mation.AsthenumberofReceiversbecomeslargetheprobabilityofonpathpunishmentgoestozero.Thissecondcaseissimilartothatofthecaseofamediator.Weshowhowthismapsintothecaseofnancialanalystsandbrokerages.Inparticular,weexaminehowalegislativechangein2002pushedbyanindustrybodymadeitpossibletoverifythehistoryofadvicegivenbybrokeragesandhowthisincreasedtheabilityofbrokeragestopersuadeclients.Therestofthissectionreviewstheliterature;section2providesasimpleexampletoillustratethemainresultsofthepaper;insection3wedescribethefullmodel;insection4weanalysetheequilibriawithnonoiseandtheCnCmechanism;insection5weanalysethegamewithnoiseandtheroleofamediator;insection6welookatemergentcommunicationsmechanisms;andinsection7weconclude.1.1LiteratureReviewOnecontributionofourpaperisitexamineshowandwhen

6 thedesireforcredibilityinadynamicsetting
thedesireforcredibilityinadynamicsettingcanmicrofoundthecommitmentassumptionpresentedinKamenicaandGentzkow[2011]andappliedinthesubsequentliterature(RayoandSegal,2010;Kolotilin,2015;Taneva,2015;Tamura,2016).WhiledynamicpersuasionhasalreadybeenanalysedinRayoandSegal[2010],Kremeretal.[2014],Ely[2015],Bizzottoetal.[2016]andelsewhere,theseanalysesallrelyoncommitmentatthestagegame.Hence,thesepapersdon'tprovidefoundationsforcommitmentanddon'tspeaktotheinteractionbetweencredibilitynowandcredibilityinthefuture.OurpaperrelatestoarecentandgrowingbodyofworkonrepeatedBayesiangameswithcommunicationandlong-runplayers:AtheyandBagwell[2001,2008],EscobarandToikka[2013],Renaultetal.[2013],MargariaandSmolin[2015],Hörneretal.[2015]andBarron[2016].InHörner,Takahashi,andVieille[2015],(HTVhereafter),theyreducethesegamestoone-shotgameswithex-posttransfers-similartoastaticBayesianmechanismdesignproblem-andprovidearesultanalogoustotherevelationprinciple:truthtellingdoesnotrestrictpayosetswhenplayersaresucientlypatient.However,unlikethecasesexaminedinHTV,non-truthfulequilibriaexpandthepayosetinlong-runpersuasiongamesbecauseex-posttransfersarerestrictedbytheimpatienceoftheReceivers.Thismakespersuasionameaningfulexerciseinlong-runpersuasiongames,whereasintheseotherpaperspersuasioncanbereplacedwithtransfers.Notethen,long-runpersuasionistothisliteraturealmostasBayesianpersuasionistomechanismdesign;amechanismdesignerwithunconstrainedex-posttransfersandnohiddenactionscan,asinHTV,doawaywithpersuasion.Therearethreeotherpapersweknowofthatdropthecommitmentassumptionof5 Bayesianpersuasioninoneformoranother:ChakrabortyandHarbaugh[2010],Perez-Richet[2014]andPiermont[2016].InChakrabortyandHarbaugh[2010]theyexamineastaticcheaptalkgame,asinCrawfordandSobel[1982],withamultidimensionalstateandasenderwithstateindependentpreferences.Theyshowthatthesendercanmakecred-iblecomparativestatementsthattradeotheincentivesforthesendertoexaggerateonmultipledimensions;ifsenderpreferencesarequasiconvextheycandobetterthroughsuchpersuasion.Unlikeourpapertheydonotmicrofoundcommitment,theirsendercanonlymakecomparativestatements,andourresultsdonotrelyonthesenderhavingquasi-convexpreferenc

7 es.InPerez-Richet[2014]andPiermont[2016]
es.InPerez-Richet[2014]andPiermont[2016]thesenderobservestheprobabilitydistribu-tionofstatesinthefutureandthenchoosesaninformationstructuretowhichtheyhavefullcommitment.Afterthatthesubgameiseectivelyplayedbynaturewiththereceiverasnaturedetermineswhatmessagesthereceivernowgetsbecausethesenderhascommittedtotheinformationstructureatallsubsequentstages.Inthesepapersthen,thereisonlyalackofcommitmentintheinitialchoiceoftheinformationstructure,onceitischosentheyhavecommitmentateverystageofthesubgame.Theinitialchoiceofthemechanismispinneddownbywhatthatchoicerevealsabouttheunderlyingdistributionofthestate.Thisstandsinstarkcontrasttoourpaperwherethereisnoexogenousabilitytocommittotheinformationstructureofthegameatanypoint.2ASimpleExampleTheheadofaconstructionFirmwantsaMayortogivethempermissionforalargecon-structionproject.TheMayorisuncertainwhethertheprojectwillbeanetbenetorlossforthecity.IftheprojectisrejectedboththeFirmandtheMayorgetapayoofzero.IfthemayorAcceptstheprojectshegetspayoonewhentheprojectisGoodandminusonewhenBad-thermgetszeroeitherway.Hence,theMayorwillonlyokaytheprojectifshebelievestheprojectwillbeGoodwithatleastaftypercentprobability.ThesolidlineinFigure1belowthengivestheFirm'spayos,v(:),asafunctionoftheMayor'sposteriorprobability,,thattheprojectwillbeGoodv()=8:1if0:50if0:5;theMayor'sexpectedpayoasafunctionofherposteriorbeliefisgivenbythediagonaldashedline,6 u()=8:2(�0:5)if0:50if0:5:ThepriorprobabilityoftheprojectbeingGoodis0=1=3.However,theFirmaccuratelylearnsthequalityoftheproject,GoodorBadFigure1 TheFirmthensendsareportof`Accept'or`Reject'.Werstconsiderthecommittedpersuasioncasewherethermcommitstoapolicyofsendingan`Accept'or`Reject'withaprojectcondi-tionalprobability,byconventiontheprobabilityofsendingAcceptisweaklygreaterforGoodprojectsthanBad.The(Bayesian)Mayor,afteranygivenreport,formspos-teriorbeliefsbasedontherelativeprobabilitiesofreceivingthatreportwhentheprojectisGoodversusBad.TheFirm'spolicycaninduceanypairofposteriorsthatsatisfythelawoftotalproba-bilities,Pr(Good)=Pr(GoodjAccept)Pr(Accept)+Pr(GoodjReject)(1�Pr(Accept)):(1)Hence,

8 wecanthinkoftheFirm'spolicyasachoiceofan
wecanthinkoftheFirm'spolicyasachoiceofanytwoposteriors,A=Pr(GoodjAccept)andR=Pr(GoodjReject),straddlingtheprior0;wheretheposte-riorspindownthefrequencyofsendinganAccept.WecanrestrictattentiontopoliciessuchthatA0:5astheFirmcanonlymakeprotswithpoliciesthatgiveincentivecompatibleadvicewheretheMayoracceptstheprojectifshereceivesanAccept.TheexpectedpayototheFirmisthenjusttheprobabilityofsendinganaccept,from(1)thisisPr(Accept)=0�R A�R;7 Figure2(a) (b) (c) whichisjusttheheightofthelineconnectingtheposteriorsRandAevaluatedattheprior0,asshowningure2abelow.FromtheMayor'sperspective,thebestpolicyisTruthTelling-theFirmsendsanAcceptifandonlyiftheprojectisGood,(R;A)=(0;1).ThispaysoutforGoodprojectsonly-ascanbeseenfromgure2b,thishasanexpectedpayoofonethirdtobothparties.FromtheFirm'sperspectivethisiswaste-ful,theFirmdoesn'tneedcertaintyafteranAc-cepttogetpermission,justA0:5.InsteadtheFirmprefersamaximallyPersuasivepolicywhereitreportsAccepthalfthetimewhentheprojectisbadandallthetimewhentheprojectisGood.UnderthispolicytheFirmisnowre-portingAccepttwothirdsofthetimeanditgivestheMayorposteriors(R;A)=(0;0:5)sotheMayorstillacceptsafterreceivingaposi-tivereport.WhilethisisthebestpolicyfortheFirmitistheworstfortheMayor:theMayorgetsexpectedsurplusofzeroirrespectiveofthereport.Wecanseefromgure2cthatthispol-icyplacesusontheconvexhulloftheFirm'spayofunction,asinKamenicaandGentzkow(2011).Commitmentatthestagegameisfrequentlyinfeasible,inwhichcasethereisnoequilibriuminwhichtheMayorfollowstheFirm'sadvice.TheMayoronlyacceptsiftheFirmhasastrat-egyofsendingAcceptnomorethanhalfofthetimewhentheprojectisBad.Butthiscan'tbeanequilibriumastheFirmwouldalwaysbreakitswordandreportAccept.Consequently,withoutcommitment,neitherTruthTellingnorPersuasionpoliciesarefeasible:alltheequi-libriaofthegamearepayoequivalenttoaBabblingequilibriumwheretheFirmrandomlysendsAcceptandRejectreportswhiletheMayorignoresthereports.However,inarepeated8 "!#$% setting,wheretheFirmcaresaboutfuturecredibilitythereisroomtogeneratecommitment.Figure3 Inthislight,considernowaFirmthatisl

9 onglivedandproposesasequenceofex-anteide
onglivedandproposesasequenceofex-anteidenticalprojectstoasequenceofonetermMayors.Asbefore,aMayorattermtacceptstheprojectonlyifshebelievesithasatleastaftypercentchanceofbeingGood:herposteriort0:5.EachMayorobservesalltheFirm'spriorreportsandtheoutcomesofallpriorprojects-sheformsherbeliefsabouttheFirm'sstrategyaccordingly.TheFirmdiscountsfuturepayosatratesothattheFirm'slifetimediscountedpayoisV0=1Xt=0tv(t):Wecomparethepayosoflong-termper-suasiontothoseofcommittedpersuasion,describedinFigure3.BisthepayofromBab-bling;TisthepayofromTruthTelling;andPisthepayofromoptimalcommittedpersuasion.Note,thelinePTistheParetofrontierandBistheworstpayopairofthestagegame.AsBabblingisanequilibriumoftheoneshotgameitisalsoanequilibriumoftherepeatedgame.TheTruthTellingequilibriumcanthenbesupportedbythethreatofaBabblingequilibriumiftheFirmissucientlypatient.Ontheequilibriumpath,MayorsbelievetheFirmtellstheTruthandaccepttheprojectifandonlyiftheFirmsendsanAcceptreport.IftheFirmsendsanAcceptwhentheprojectisBadthentheMayorsknowtheyareopathandhavethebeliefthatthermisbabbling.TheFirm'sonpathdiscountedpayofromTruthTellingatanystageisthen:VT=0 1�=1 3(1�):Ifatsometermt,theFirmlearnstheprojectisBadbutreportsanAccepttheFirmgainsapayoofonebutreplacestheTruthTellingcontinuationequilibriumwithBabbling,replacingVTbyzero.Hence,TruthTellingcanbesupportedasanequilibriumif 3(1�)�1:9 !"#$%$&'() Thisresultisnotsurprising.Perhapsmoresurprisingthough,isthattheFirmcandonobetterthantheTruthTellingequilibrium.ConsideranequilibriumwheretheexpectedstagepayoishigherthanTruthTellingatsomestage.Hence,atthisstagetheFirm'sstrategyistosometimesliewhentheprojectisBad:mixbetweenAcceptandRejectsothattA2[0:5;1).IftheFirmismixingthenitmustbeindierentbetweenthetworeports,asthestagepayofromAcceptishigherthanRejectitfollowsthatthelowercontinuationpayofromsendinganAcceptintheBadstatemustexactlyosetthehigherstagepayo.SupposethenaFirmhasaBadprojectinsometermanditismixing,butbysheerchanceitsendsaReject;thensupposethishappenseachtimeitgetsaBadprojectforw

10 hichit'sstrategyistomix,forever;inthisca
hichit'sstrategyistomix,forever;inthiscasetheFirmneversendsanincentivecompatibleAcceptreportforaBadproject.TheupperboundontheexpectedpayoconditionedonthisaccidentaloutcomeofneversendinganincentivecompatibleAcceptforaBadprojectmustthenbetheTruthTellingequilibrium.Now,theFirmhasnotlostoutbyneversendinganincentivecompatibleAcceptforaBadprojectbecausetheFirmwasalwaysindierentbetweensendingAcceptandRejectwheneveritwasmixing.AstheFirmhasnotlostoutitfollowsthattheupperboundontheexpectedpayototheFirmisgivenbytheTruthTellingequilibrium.WhiletheaboveargumentrulesouttheFirmgettingahigherpayothanTruthTellingitdoesnotruleoutequilibriainwhichtheFirmlies.Forexample,consideranequilibriumwheretheFirmfollowstheoptimalPersuasionstrategyin`normal'periodsandbabblingin'punishment'periods.PunishmentperiodsaretriggeredwheneveraFirmliesaboutaBadprojectandgoonlongenoughtomaketheFirmindierentbetweenlyingandtellingthetruth.InsuchanequilibriumtheMayorsgetzerosurplusandtheFirmgetthesamesurplusasfromTruthTelling.Ingeneral,wecananyoftheaveragepayosonthelineDTinequilibrium.WhileitpossibletosupportsomedegreeofpersuasioninequilibriumthisisParetodominatedbyTruthTelling.ThesolutiontotheFirm'sproblemisacoinandacup.AtthebeginningofeachtermtheFirmshakesacoininacup,placesitonthetableandpeeksunderthecuptoseewhetherthecoincameupheadsortails.TheMayorobservestheFirmdoallthis,butdoesnotseethecoin.Thecup,withthecoinstillunderit,isleftonthetable.Then,asbefore,theFirmlearnsthequalityoftheproject,sendsareport,andtheMayormakesherdecision.AfterthedecisiontheMayorliftsthecup,observesthecoin,andrecordswhetheritwasheadsortails.AllMayorsnowobservethehistoryofreports,projectqualities,andcoinips.Thisprocessisrepeatedeveryterm.ForsucientlypatientFirms,wecanreplicateoptimalcommittedpersuasion.TheFirmonlyeversendsaRejectiftheprojectisBadandthecoincomesuptails,otherwisethe10 FirmsendsAccept.OntheequilibriumpathMayortalwaysacceptsaftertheFirmsendsAccept,asshehasposteriorstR=0andtA=0:5.OtheequilibriumpathMayorsbelievetheFirmisbabbling.TheopaththreatofBabblingisenoughtostopapatientFirmdeviating.Recall,thatwithoutaCnC,mixingbetweenthetruthandlyingiscr

11 edibleonlywhenthereareonpathpunishments.
edibleonlywhenthereareonpathpunishments.Now,byintroducingthecoinandcup,theFirm'spurestrategyisstillex-antestochasticfortheMayormakingthedecision-itisasiftheFirmmixesovermessageswhenprojectsareBad-yetitisalsoex-postdeterministic.3ThisimpliespunishmentsareonlyrequirediftheFirmdeviatesfromitspurestrategyandneverontheequilibriumpath.Hence,thissimplemechanismachievesthemaximallypersuasiveequilibriumwithnoneedforthirdpartyverication,contracts,ortransfers.3TheModelASender(`he')andapopulationofReceivers(each`she')playthefollowinginnitelyre-peatedpersuasiongame.3.1StageGameEachperiod,aReceiverRtmusttakeanactionatfromacompactsetA.Herpayosfromactionatdependonanunknownstateoftheworld,(t;!t)2 =1;2;:::;N !1;!2;:::;!N! .HerpayosaregivenbytheutilityfunctionuR(at;t;!t).Eachtisdrawnindependentlyacrosstime,fromapriordistributionrepresentedbythevector02
.4Conditionalont,!tfollowsthedistributiong(!tjt)�0,where!tisalsoindependentof,!,6=t.5However,notethatweallowfortand!ttobecontempora-neouslycorrelated.Ateacht,aReceiverRtarrivesexanteuninformedabout(t;!t)andleavesthemodelattheendoftheperiod.Atthebeginningofeachperiod,aninnitely-livedSenderSprivatelyobservesthere-alization,t.BeforeRttakesanaction,Ssendsamessagemtfromsomeset,M.Within 3TheCnCmayappearsimilartoasunspot,howeverasunspotisnosubstituteasasunspotcontingentstrategywouldbedeterministicex-antedeterministicfromtheMayor'sperspective:theywouldseethevalueofthesunspotandknowwhethertheFirmwasgoingtolieornotforbadprojects.TheessentialfeatureoftheCnCisthattherevelationofit'svalueisstaggered,privatetotheFirmatthepointoftheMayor'sdecision,andonlyrevealedpubliclyex-post.4Followingstandardnotation,weuse
XtodenotethesimplexoversetX.5Noteinparticularthatweassumetheconditionaldistributionof!thasfullsupport,forallt.Whilethisassumptionisnotstrictlynecessaryforthemainresults,itsimpliestheexpositionofourresultsrelatedtothereviewmechanismswediscussbelowsignicantly.11 aperiod,theSenderonlycaresabouttheactiontakenbyagentRtandhasstageutilityuS(at).Withinperiod,thetimingofthisstati

12 ccheaptalkgameisasfollows:1.t,!tare
ccheaptalkgameisasfollows:1.t,!taredrawnrespectivelyfromdistributions0,g(!tjt).Sprivatelyobservestherealizationoft.2.Ssendsamessagemt2M(possiblyrandom)toRt.3.Afterobservationofmt,Rtchoosesanactionat2A.4.Aftertakingactionat,!tisobservedbyallplayers.WeinterprettasSender'sprivateinformationrelevantforReceiver'sdecisionproblem,and!tasexpostfeedbackthattheReceiverlearnsonlyaftertakingaction.Writtenthisway,themodelisexibleenoughtodescribeinteractionswhere(i)SenderisbetterinformedaboutReceiver'spreferences,uR(a;);6(ii)Receiver'snalinformationofherownpreferencesaftertakingactionwillbebetterthanSender's,uR(a;!);7and(iii)neitherSendernorReceiverinformationcanberankedassuperiorfordecision-making,uR(a;;!).8Afterreceivingmessagemt,theReceiverRtformsherposteriorbelieftandchoosesheractionat(t)tomaximizeE[uR(at;t;!t)jmt].Hence,weoftenreferdirectlytotheSender'sequilibriumperiod-tstagepayo,asafunctionoftheReceiver'sposterior:v(t):=uS(a(t)):AsinKamenica&Gentzkow(2011),weassumethatwheneverRt'sposteriorbeliefleavesherindierentbetweentwoormoreactions,weassumeshechoosestheoneSprefers.Thisensuresthatv(t)isanuppersemi-continuousfunction.Werefertothisstagegameby�t.Asthestagegameisastandardcheaptalkgame,therealwaysexistsababblingequilibrium.9Wecontrastthisstagegametoastaticinformationdesignproblem,inwhichScancommitinadvancetoa(mixed)reportingstrategybeforelearningt.Inthepersuasiongame,thetimingandavailableactionsareasfollows:1a.Schoosesanexperiment:amessagespaceM,andarandommapping^s:!
M.2a.tisprivatelydrawnfromdistribution0.Conditionalont,mt2Misdrawnfroms0. 6Forinstance,whereuncertaintyisoverthequalityofagoodforsale7Thismodelismostappropriateforexperiencegoods,whereconsumershaveidiosyncraticpreferencesforgoods.8Thissettingismostappropriateforenvironmentssuchasassetand/ormatchingmarkets.9AswediscussfurtherinSection4,thisisoftennottheonlyequilibriumofthestagegame.12 3a.Rtobservesmtandchoosesanactionat2A.Inthestaticinformationdesignproblem,Scommits(beforeobservingt)toanexperiment(amessagespaceM,andagarbling^soft).Thekeydistinguishingfeatureofanexperime

13 ntisthatScancommittoastochasticpolicy.Rt
ntisthatScancommittoastochasticpolicy.Rthenobservesadrawmtfromtheexperimentandusesthisinformationtochooseanoptimalaction.NoticeinparticularthatweassumethatSendercanonlyconductexperimentsbasedonhisowninformation,t.Thisaidscomparisonwiththecheap-talkvariantofthegameandhasthesimpleinterpretationthatthereisnowayfortheSendertopredictReceiver'slikelyfeedback,beyondhisownprivateinformation.Ofcourse,Scandoatleastaswellusinginformationdesignashecaninanyequilibriumofthestaticcheaptalkgame.Dene^v()asthesmallestconcavefunctionthatiseverywhereweaklygreaterthanv().Thatis,^v():=supf:2co(v)gwhereco(v)denotestheconvexhullofthegraphofv.Kamenica&Gentzkow(2011)showthatS'soptimalpayoviainformationdesignisexactly^v(0),whichwerefertoasOptimalPersuasion.Bydenition,^v(0)v(0).If^v(0)=v(0),thenS'soptimalpayocanbeachievedbysendingnoinformationtoRt,orbyagarblingequilibriumofthecheaptalkgame.ToensurethatpersuasionisausefultoolforSender,weassumeintherestofthepaperthatv,0aresuchthat^v(0)�v(0)3.2TheRepeatedGameThestagegame�tisrepeatedeachperiodt=0;1;2;:::,adinnitum-werefertothisinnitelyrepeatedgameby�1.Ateachperiodtandpublichistoryt=(mt;at;!t)t�1=0,theSenderandReceiverRobservet(theSenderalsoobservestheprivatehistoryt=()t=0)andplaygame�t.TheSender'sdiscountedpayofromasequenceofReceiveractionsa=(a1;a2;:::)is1Xt=0tuS(at):Letthesetofallperiod-thistoriesbet.Atperiodt,letthemapst:tt!
MexpressahistoryandstatedependentprobabilitydistributionovertheSender'smessages.AstrategyfortheSenderisacollections=(st)1t=0.Similarly,letamixedstrategyforReceiver13 Rtbeamapt:tM!
A.WeusethetermequilibriumtorefertoPerfectBayesianequilibriaoftheabovegame.Anequilibriumspecies:astrategysfortheSender;strategies=(t)1t=0foreachRt;andposteriorbeliefsftg1t=0,wheret2
isanN-dimensionalvector,suchthat:1.GiventheReceivers'strategiesandhistory(t;t),smaximizestheSender'sexpecteddiscountedpayoE"1X=tuS(at)jt;t;#:2.GiventheSender'

14 sstrategy,tmaximizesRt'sexpectedpay
sstrategy,tmaximizesRt'sexpectedpayoE[uR(a;t)jmt]=NXi=1it
uR�a;it
:3.Wherepossible,theReceiver'sposteriorbeliefst=�1t;:::;Nt
satisfyit=Pr�t=itjt;mt;s
Inparticular,notethatbeliefssatisfythestandardnosignallingwhatyoudon'tknowrestriction(Fudenberg&Tirole,(1991)).Regardlessofplayinroundst,condition3.aboveensuresmessagesinthesupportofSender'sstrategyatanyhistorytmustbeconsistentwiththepriorbeliefovert.However,foro-pathmessageschosenattimet,equilibriumplacesnorestrictionsonReceiverRt'sbeliefs.3.3ADirectEquilibriumBuildingonaninsightfromKamenicaandGentzkow[2011],thenotionofequilibriuminourinnitelyrepeatedgamecanbecastentirelyintermsofhistory-dependentlotteriesoverbeliefs,t.Denethestagegame^�tasfollows:^�tspeciestheSender'sfeasiblemessagespaceasthesetofpossibleposteriorbeliefsthatRtmayhold,
,andiselsewherethesameas�t.Theinnitelyrepeatedgame,^�1,isanalogouslydened.Insuchanenvironment,historiesarenowvectorsoftheformht=(~;a;!)t�1=0,thesetofallperiod-thistoriesHt,and(behavioural)strategiesfunctionsoftheform=((h;))1=0,whereeacht:Htt!
M,andt:tM!
AforS,Rtrespectively.WedenotethesetofallstrategiesforSby.Wedeneadirectequilibriumofthisrepeatedgameasfollows:14 1.(Bestresponses)GiventheReceivers'belieffunctionst(ht;~t),~t2[t2supp(t(ht;t))maximizestheSender'sexpecteddiscountedpayoVt(ht;t)=v(t(ht;~t))+E[Vt+1((ht;~t;t);t+1)](2)whereVtisSender'scontinuationpayoathistory(ht;t).2.(Obedientbeliefs)TheReceiverbelievesanyequilibriummessage,~t2[t2supp(t(ht;t))t(ht;~t)=~t3.(Bayesplausibility)02co([t2supp(t(ht;t))).Thefunctiont(ht;~t)speciesRt'sbeliefs,givenobservationofhistoryhtandmessage~tsentbySinperiodt.Giventhesebeliefs,theoptimalbehaviouroftheReceiverisimplicitinthefunctionv(t(ht;~t)),whichdenesS'sstagepayofromthisbehaviour.Vti

15 ssimplythesumofS'sdiscountedpayofro
ssimplythesumofS'sdiscountedpayofromequilibriumplay,fromhistory(ht;t)onwards.Inanyequilibrium,Smustmaximize(2)atallhistoriesofthegametree,given(h;~),t.Moreover,adirectequilibriumrequiresthat(i)Rt'sbeliefsconformtotherecommendationmadebyS,forany~tontheequilibriumpath,(ii)atanyhistory,S'smixedstrategyovermessagescanbe`averagedback'thetheReceiver'sprior.Whilethesetwoconditionsappearstrongerthanrequiredforanyequilibrium,thefollowingLemmaestablishesthatitiswithoutlosstorestrictattentiontosuchdirectequilibriaofgame�1:Lemma1.Foranyequilibriumofgame�1,thereisadirectequilibriumofgame^�1thatinducesthesamedistributionoverReceivers'actions,foreachstatetandhistoryhtontheequilibriumpath.Lemma1extendstheinsightofKamenica&Gentzkow(2011)toequilibriaofrepeatedgames,inwhichSenderisunabletocommittohissignallingstrategyatanyhistory.Asforcondition2,sinceallthatmattersaboutS'sstrategyistheeectithasonRt'sbeliefs,wecanreplaceS'smessageswithrecommendationsofthebeliefsthatRtshouldhold.MoreovertheserecommendationsmustbeoptimalforS,sincetheywereingame�1(condition1).10Finally,condition3followsfromtheobservationthatanyequilibriumstrategymustinduceposteriorsthatsatisfytheLawofTotalProbability,(??).Wesometimeswriteaninducedlotteryofposteriors,athistoryht,as2(0)
(
),where(0)denotesthesetoflotteriessatisfyingcondition3.Inotherwords,ifatsomehistoryht,inducesalottery 10Formessagesnotinthesupportofacontinuationequilibriumatht,Rt'sbeliefsarenotconstrainedbyBayes'ruleandthusitiseasytopreventSenderfromplaying`opath'choicesof~.15 ofMposteriors(t;1;t;2;:::;t;M)withj=Pr(=1;j),j=1;2;:::;Mand2(0),thenitsatises0=MXj=1jt;j(3)4TheValueofRepetitionforPersuasionInthisSection,weareprimarilyinterestedinunderstandingwhentheopportunityforre-peatedinteractioncanallowStoachievehisoptimaldiscountedaveragepayosunderpersuasion,despiteonlybeingabletomakecheaptalkstatements.First,weestablishthatthereisgenerallyaneedforrepeatedinteractiontoimprovethepossiblepayosthatSendercanachieveinequilibrium.Interestingly,it

16 ispossibletondpreferencesv()fo
ispossibletondpreferencesv()forwhichoptimalpersuasioncanbeachievedasanequilibriumofastaticcheaptalkgamewithnon-trivialcommunication(seeFigure4).However,ourrstmainresultestablishesthatthesekindoffunctionsarenottypicalandthereforestaticcheaptalkdoesnotusuallyallowSendertodoaswellashewouldundercommitmenttooptimalpersuasion:Theorem1.Foranyprior0,optimalpersuasionisgenericallynotpossibleinastaticcheap-talkgame.11Whilethereexistnon-convexfunctionsv()forwhichtheconcavication^v()involvesatleasttwopointsx,ysuchthatv(x)=(y)and^v(x;y),8x;y=x+(1�)y,2[0;1].Insuchcases,Sisindierentbetweensendingmessagesxandyandmoreoverthesemessagesarefeasibleinadirectequilibriumif0=x;yforsome2[0;1].Thus,thereexistsanequilibriumofthestaticcheaptalkgamewhichachieves^v()(andtherefore,S'soptimalstagepayoundercommitment12)withouttheneedforrepeatedplay.TheproofofTheorem1showsthatsuchfunctionsareinfactnon-generic.Sincesuchcasesarerare,wefocusintherestofthepaperonfunctionsvandpriors0forwhichnocheaptalkgamecanachieveS'soptimalpayosundercommitmenttopersuasion.Theorem1tellsusthatcasesinwhichSendercanachieveoptimalpersuasionusingcheaptalkwithouttheneedforrepeatedinteractionarerare.Moreoverweareinterestedstudyingtheroleofrepetitioninpersuasion.WethereforefocusourattentiononthegenericcasesofSenderpayofunctionforwhichstaticcheaptalkcannotbeusedtosustainoptimalpersuasion.ForanysubsetP,dene
P:=f2
:i2P()i=0gasthesetofposteriorswhichputpositiveprobabilityonastateifandonlyifitisinP. 11SpecialthankstoBillZameforadviceonthisproof.12ThisfollowsimmediatelyfromCorollary2,Kamenica&Gentzkow(2011).16 Figure4:A(non-generic)exampleofoptimalpersuasionattainableviacheaptalkAssumption1.ForallP,optimalpersuasioncannotbeachievedbyinformative,staticcheaptalk,conditionalon2P.IntheAppendix,maketheformalstatementofAssumption1.Assumption1ensuresthatthereisnosubsetoftypesforwhichScoulduseone-shot,informativecommunicationtoachievetheoptimalcommitmentpayo^vP,conditionalonRalsoknowingthat2P.Consideringr

17 stP=,Theorem1assuresusthatfunctionsv
stP=,Theorem1assuresusthatfunctionsviolatingAssumption1inthiscasearenon-generic.Inotherwords,suchfunctionsarerare.SinceTheorem1appliestoanynitestatespace,,wecansimilarlyapplythelogictothepayofunctionv,denedoverthesubspace
P.Inthisway,Theorem1alsoassuresusthatvfunctionsforwhichcheaptalkcouldachievetheoptimalcommitmentoutcomeforSonanysubsetPofthestatespacearealsorare.Sincethesecasesarenon-generic,weomittheiranalysisfromthemainresultsinordertoaidexpositionofthetypicalpersuasionproblem.ByrulingoutonlyinformativecheaptalkasamethodofachievingtheKG(2011)solutiononanyP,Assumption1allowsforsituationsinwhichSpreferstocommunicatenomoreinformationifheknowsthatRbelieves2P.Thisisimportant,aswedonotwishtoruleoutcasesinwhichSneverwishestoconcealinformation-indeed,strategicallyconcealinginformationisattheheartofpersuasion.Ourassumptionallowsforthis.Infact,aswehaveemphasizeditallowsforalmostanycombinationofinteriorandboundarybeliefsaspartoftheoptimalcommitmentsignal.ItonlyrequiresthatsuchsignalsdonotallleaveSindierentwheninducedinthestagegame.Beforeweanalysethepropertiesoftherepeatedgameindetail,wepresentauseful17 preliminaryresult:Lemma2.Inanyequilibrium,SendercandonobetterthanastrategywhichatanyhistoryhtinducesatmostNpossibleposteriorbeliefs,t(ht).Lemma2establishesthatfromtheperspectiveofS'spayositiswithoutlosstorestrictattentiontodirectequilibriainwhichatanyhistory,S'sstrategyinducesnomorethananN-pointdistributionoverposteriorbeliefs(recallthatN=jj).Theresultsimpliesthesearchforoptimalequilibriasignicantly.Mostimportantly,itensuresusthatweonlyneedconsiderstrategiesthatinduceanitedistributionofposteriorsforanyReceiver,Rt.WeusesomekeypropertiesofdirectequilibriaforS'spayosandofconvexsetstoestablishthatifN0�Nsignalswereeverbeingsentatsomehistoryht,oneofthesesignalswouldberedundantforS'scontinuationpayoatthathistory(andforfeasibilityofinducedposteriorsatthathistory).RemovingsuchanalternativefromS'sstrategyathtisfeasibleatht,andsinceitdoesnotaectpayosatht,itdoesnotaectS'sincentivesatearlierhistoriesorindeedhisexpecteddiscountedpayofromthegame,E[V0

18 (0)].Interestingly,thepropertiesofe
(0)].Interestingly,thepropertiesofequilibriumallowustoreducethecardinalityofthesignalspacebymorethanunderstandardpersuasion,whichcanreducethesearchoversignalstoN+1-pointdistributions.134.1RepeatedPersuasionwithoutNoiseInthissection,weprovidearstanalysisforthecasewhereSender'sprivateinformationisaperfectpredictionReceiver'snallypreferences:thatis,weassume =andg(tjt)=1.Intuitively,becauseSender'stypecanbeveriedattheendofeachround,thisenvironmentrepresentstheverybestcasescenariofortheuseofreputationasadeviceforallowingSenderstoconvincinglypersuadeapopulationofReceivers.Somewhatsurprisingly,however,weshowthatrepeatedinteractioncannottypicallyachievetheoptimalcommitmentpayofrompersuasionevenwhenthereisnonoiseexpostbetweentheinformationofSendersandReceivers.WeextendourresultstoallowforSender'sinformationtobeonlyanoisysignalofReceiver'snalutilityinthenextsection.SupposenowShastheopportunitytointeractsequentiallywitha(potentiallyinnite)setofshort-runReceivers.Asapreliminaryresult,weshowthatrepeatedplayofthecheap-talkgamecansustaintruth-tellingbytheSenderasanequilibrium. 13Theproofdealswithtwocomplicationsascomparedwithstandardpersuasionarguments.First,itdealswiththefactthatSdoesnotcommittohisstrategy.Andsecond,wemustensurethattheequilibriumdynamicsarenotviolated.18 Proposition1.Thereexists 1suchthattruth-tellingisanequilibriumoftherepeatedgame8 1,iSender'struth-tellingpayoexceedshisworststagegameequilibriumpayo.Tosustainon-pathtruth-tellingineveryperiodtheequilibriumemploysatriggerstrat-egy,movingtotheworstcheap-talkequilibriumforeverifadeviationisdetected14.Whentcanbeobservedattheendofeachround,deviationsfromtruth-tellingareeasilyde-tectabletoReceivers.Therefore,solongasSenderissucientlypatientandtheworstcheaptalkequilibriumyieldsalowerexpectedSenderstagepayothandoestruth-telling,suchastrategyenforcestruthfulequilibria.WenowaskhowthepotentialforusingrepetitionasacommitmentdeviceaectsS'sabilitytoearnrentsfrompersuasionintheequilibriumofsomerepeatedcheaptalkgame.Inaskingthisquestion,wemovetoafocusonSender-preferredequilibria.Inparticular,considerthepro

19 blemofmaximizingS'speriod-0discounteduti
blemofmaximizingS'speriod-0discountedutility,acrossallpossibleequilibriaoftherepeatedcheaptalkgame:max2E[V0(0)](4)s.t.Vt(ht;t)=v(t)+E[Vt+1((ht;t;t);t+1)]v(0t)+E[Vt+1((ht;0t;t);t+1)];8ht2Ht;t2supp(t(ht;t)),0t2[t2supp(t(ht;t)),and02co([t2supp(t(ht;t)));8ht2Ht.Problem(4)involveschoosingastrategyprole=(1;2(h2);:::)forSthatmax-imizeshispresentdiscountedutility,subjectto:(i)eachchoiceoft2supp(t(ht;t))involvesa(weakly)higherpresentdiscountvalueforSathistoryhtthananyalternative0tthatisplayedbySwithsomepositiveprobabilityatht;(ii)satisfyingBayesplausibil-ityateachhistory.Thereisasubtledierencebetweenproblem(4)andthedescriptionofequilibrium.Inequilibrium,Sneedonlymaximizehischoiceof~tateachhistoryht,subjecttoRtbelievingthatthesechoicesbeconsistentwiththeequilibriumstrategy,.Inparticular,non-equilibriumchoicesof~canberuledoutbecausewecanchooseRt'sbeliefs 14SuchanequilibriumcaningeneralbeworseforSenderthanbabbling.19 tobeskepticalaftersuchreports.15Inproblem(4),whenwechooseastrategy0,wearealsoabletovaryRt'sbeliefsfollowinganymessagesent,solongastheyconformtoequilibriumrestrictions.Inaddition,thechoiceofstrategymustbeoptimalforS,giventheReceivers'beliefs.Next,wecharacterizethesolutiontothevaluefunctionforproblem(4).First,weintroducesomenotation.Letv i():=minfv():2supp();i�0gbetheminimumpayotoSamongallposteriorsthat(i)areinthesupportofN0-pointdistribution2
(
),forN0N,and(ii)occurwithstrictlypositiveprobabilityconditionalonstateit(under).Thenwehave:Proposition2.Sender'sdiscountedaveragecontinuationvaluefromanyrepeatedcheaptalkgameisboundedaboveby(1�)E[V0(0)]max2(0)Xi0v i()(5)Thereexists suchthat81� ,thisupperboundcanbeattainedatsomeequilibrium.Proposition2establishesanupperboundonthepayosthatSendercanachieveinanyequilibriumoftherepeatedgame.Moreover,itshowsthatthisupperboundisattainableinsomeequilibrium,solon

20 gasSenderissucientlypatient.Theboun
gasSenderissucientlypatient.Theboundinequation(5)statesthatSender'sbestdiscountedaveragepayomustbenogreaterthanthebestexpectedstatewise-minimalpayof,amongalllotteriesofposteriors2(0).Theimportanceofthestatewise-minimalpayoisasfollows:inanyequilibrium,iftherearemultiplemessagesinthesupportofSender'sstrategyconditionalonsome,thenSendermustbeheldindierentacrossthesemessages.Thus,messageswhichleadtomorepreferablecurrentactionsmustalsobeassociatedwithlargerfuturepunishments.InaSender-preferredequilibrium,theleast-preferredcurrentactionisneverassociatedwithfuturepunishment,pinningdowntheupperboundonSender'sequilibriumdiscountedaveragepayos.IntheAppendix,weconstructastrategyprolewhichachievesthisboundusingnitepunishmentperiodsforsendingmessagesotherthantheSender'sleast-preferredoneatsomet,followedbyreversiontotheoriginalstrategythereafter.Sincetheworstcheaptalkequilibriumyieldsstatewiseminimalpayos,itisnecessarily(weakly)worsethantheboundin(5),andthusmayalwaysbeusedasapunishment(oritselfachievestheSender'slargestpayo).Incomparingtherepeatedcheaptalkgametothecommitmentbenchmark,aparticularclassofexperimentsisimportantforunderstandingwhentheboundonS'spayo,(5), 15Tosupporttheequilibrium,weadditionallyspecifythecontinuationplayaftersending~asequaltotheworston-pathmessageatht,thereafter.20 istight,relativetothecommitmentpayo^v(0).Weintroducethenotionofanhonestexperiment,asfollows:Denition1.Anexperiment(^s;M)(resp.behaviouralstrategy,(^t;M))ishonestifthereexistsapartitionPofsuchthat^s(resp.^t)canbeexpressedasabijection^s(^t):P!M.Underanhonestreportingstrategythereisno2forwhichSmightrandomizeoversendingtwoormoremessages.Honestexperiments(reportingstrategies)arethereforeonesforwhicheachmessageconvincestheReceiverthatliesinadierent,disjointsubsetof.Inotherwords,assumingSplaysaccordingto^tatroundt,RtreceivesatruthfulreportabouttheelementofthepartitionPinwhichlives.Whilethesereportsaretruthfulaboutthepartitioninwhichlies,theyarenonethelesscoarsesignals.Inparticular,ourdenitionofhonestyallowsforcompletelyuninfo

21 rmative`babbling'experiments(reports)and
rmative`babbling'experiments(reports)andperfecttruth-telling.Withthisdenitioninhand,wecanestablishthefollowingTheorem:Theorem2.SupposeAssumption1holds.Optimalpersuasionisattainablebyrepeatedcheap-talkifandonlyiftheoptimalexperimentishonest,for .Theorem2tellsusthatS'soptimalpersuasionpayo^v(0)canbeattainedintheequilibriumoftherepeatedcheap-talkgameifandonlyiftheoptimalexperimentundercommitmentinvolvessendingmessageswhicharehonest.Theintuitionforthisresultrestsonasimpleobservation:ReceiverscanverifyexpostwhetherShasdeviatedfrommakingreportsthatareconsistentwithhonestexperiments.Therefore,honestreportingstrategiescanbesustainedinequilibriumoftherepeatedgamebyusingonlythethreatofo-pathpunishments.Iftheoptimalexperimentundercommitmenthappenstobehonest,thenthereisanequilibriumofthegameinwhichSmakesreportsthatmimicthisexperimenton-path,sustainedbyo-pathpunishmentswhosecostsareneverrealized(forlargeenough).Bycontrastiftheoptimalcommitmentexperimentonlyinvolvesrandomizationbetweenmessagesatsome,S'sstrategycannevermimicsuchanexperimentatanyhistorywithoutleavingReceiversindoubtaboutwhetherShasdeviatedfromtheexperimentatsomet.Inordertoensureincentivecompatibility,ReceiversthusneedtopunishSontheequilibriumpathfollowingsomemessages.4.1.1RecoveringSenderOptimality:`CoinandCup'MechanismsAswehaveseeninSection4,therepeatedopportunityforcheaptalkdoesnottypicallyallowSendertoachievehisoptimalcommitmentpayo^v(0).However,Theorem2alsoprovideshopethattheremightbewaystodesigninstitutionssuchthatScanachieve^v(0)21 insomeequilibrium.Aswepointedoutabove,akeyfeatureofhonestexperimentsisthatineachperiodReceiverscanverifyexpostwhetherShasdeviatedfrommakingreportsthatareconsistentwithsuchexperiments.Importantly,thisallowedforon-pathequilibriumstrategiestobesustainedusingonlyo-pathpunishments.Thus,ifwecanndmechanismswhichallowforthisfeaturewithoutalsoinsistingonstricthonesty(withrespectto)thenwemightbeabletorecoveroptimallong-runequilibriainthesegamestoo.Inthissection,weintroduceasimple`CoinandCup'(CnC)mechanismwhichcanindeedbeusedtoensureScanachieve^v(0),withoutviolatingincentivecompatibil

22 ity.TheCnCmechanismaugmentstherepeatedga
ity.TheCnCmechanismaugmentstherepeatedgameinapayoirrelevantwayforallplayersbutnonethelessintroducesequilibriathatattain^v(0)forS.InadditionthesemechanismsarerobusttochangesinthepayosofSenderandReceiver.Finallyweidentifyseveralexamplesofreal-worldapplicationsinwhichinstitutionsappeartousenaturallyoccurringversionsofCnCmechanismsandshowhowtheseinstitutionscanbeusedtomaximizeSender'spayos.A`CoinandCup'mechanismintroducesapayo-irrelevantstatevariable,ct2[0;1],totherepeatedgameinSection3.Toeaseexposition,wesupposethatctisi.i.d.overtimectU[0;1].TheCnCmechanismisacombinationofsequenceofrandomvariablesctandrepeatedplayofthefollowingadaptedstagegame:1a.t,ctaredrawnindependentlyfromtheirrespectivedistributions.Sprivatelyobservesbothrealizations.2a.Ssendsamessagemt2MttoRt3a.Afterobservationofmt,Rtchoosesanactionat2At.4a.Aftertakingactionat,ctandthestatetareobservedbyallReceivers.TheCnCmechanismrequiresthatwecanndapayo-irrelevantrandomvariableeachperiodsuchthat(i)SenderisabletoprivatelyobservectbeforeRt;(ii)Sendercannotmanipulatetherealizationofct;(iii)Rtisabletoobservect.Inthisway,onecanthinkofctasastaggeredsunspot.AsthenextPropositionshows,theCnCmechanismalwaysadmitsequilibriainwhichSenderachieveshisfullcommitmentpayo:Theorem3.SupposeSendercanachievepayo?viacommitmenttosomeexperiment.?isattainableinanequilibriumofsomeCnCmechanism,if?exceedsS'sworststagegamepayo.22 Intuitively,theCnCmechanismimprovesonpurerepetitionbecauseitallowspunish-mentstobeconditionedontherealizationofamuchlarger`augmented'state,(t;ct),wherectdoesnotinvolveanynewpayoconsiderationsforSenderorReceiver.Onthisexpandedstatespace,wecanusectasawaytoassigna`budget'forreportssentgiveneacht.Thus,iftheoptimalexperimenteverproscribesmixingbetweentwomessages1,2withprob-abilities1,2atsomestate0t,wecansimplychoosetheequilibriumstrategiestoallowmessage1tobereportedwithoutsubsequentpunishmentifct2[a;a+1],a+11,and2tobereportedifct2[b;b+1];b+11,fordisjointintervals[a;a+1],[b;b+1].16Importantly,thismeansthateachroundSendercancrediblymim

23 ictheoptimalexperiment,knowingthatifhere
ictheoptimalexperiment,knowingthatifhereportsaccordingtohis`budget'(given0t),hewillfacenopunishment.Otherwise,hewillfacea`GrimTrigger'-stylepunishmentinwhichfutureReceiversallreverttothebabblingequilibrium.Importantly,fortheCnCmechanismtoimproveequilibriumoutcomes,Sendermustknowtherealizationofctbeforereporting(toknowwhichmessagetosendwithoutpunishment),whileReceivermustonlyobservectaftertakingactionat.IfReceiverlearnedcttooearly,thiswoulddestroySender'sabilitytoeectivelypersuadewithoutfacingon-pathpunishments.Inotherwords,theenvironmentwouldreverttotherepeatedgamessettingofSection3.Thus,theabilitytodelayobservationofcttoReceiversisacrucialelementofthedesignofCnCmechanisms.Suchstaggeredsunspotsctareimplementableanddon'trequireanyspecialistknowledgeofthedecisionproblemorforinstancetherealizationof.17Asthesimplestpossibleexam-ple,wecancreatethemusingnomorethansomecoinsandacup.SolongastheReceiverobservesthecoinsbeingtossedintothecup,Sendercanprivatelypeerinandcheckifeachisa`Heads'ora`Tails',beforesendinghismessagestoReceiver.18Theproportionof`Heads'acrossthecupscanthenplaytheroleofct.Asamorerealisticexample,Blockchaintech-nologies(suchasthatunderpinningBitcoin)supportdecentralizedrecordingandupdatingofinformationamongpeersusingcryptographicmethods.19Thesetechnologiescanbeusedtoshareinformationinawaythatcannotsubsequentlybetamperedwith,andallowforinformationtobewithheldfromsomeparticipantsuntilpre-speciedtimes.20Programmed 16Sincemixingprobabilitiessumto1and!tisuniform,itiseasytocharacterizedisjointsetsintervalson thatsupportsuchastrategy.Asimilarlogicgoesthroughformoregeneraldistributionsof!,solongasthedistributionateachtisatomless.17NoticethatwedonotrequireStobeabletocommittoaspecicexperimentalprocedureforgeneratingaparticulardistribution,!t.18Senderisnotallowedtotouchthecup.19OtherrecentusesofBlockchaintechnologyinclude:`smartcontracts'forverifyingtheperformanceofobligations,reducingmanipulationofexperimentaldesigninmedicaltrials,andcreatingtrustworthydigitalaccountsofpropertyownership.20http://www.economist.com/news/science-and-technology/21699099-blockchain-technology-could-23 witharandomnumbergeneratortoupdatetheBlockc

24 hainwithnewmessagesct,thesetechnologiesc
hainwithnewmessagesct,thesetechnologiescouldbeusedasthebasisofaCnCmechanism.Interestingly,theavailabilityof`trust'technologiessuchasBlockchaincanthereforeintroduceequilibriainwhichthepayostopersuasionareimproved.5RepeatedPersuasionwithNoiseWenowreturntothemoregeneralsettingofSection3,inwhichReceiverscanneververifytheinformationonwhichSenderbasedhisrecommendations.Inthissetting,itisnowtyp-icallynotpossibletoverifyexpostwhetheraSenderhasplayedaccordingtoaproscribedstrategy,eveniftheoptimalsignalundercommitmentishonest(orifthecoinandcuptech-nologyisavailable.Moreover,givenarealizedt,itisusuallynotevenpossibletodeterminewhetherSenderhasmadethe`worst'reportintheconditionalsupportoft(ht;t).21ToensuretheSenderisindierentbetweenmessagesinthesupportofhisequilibriumstrategythereforerequiresevengreaterpunishmentsontheequilibriumpath.Inparticular,attimetpunishmentsmustnowbeassignedinexpectationacrossdierentrealizationsof!.Asaresult,the`worst'messageassociatedwithsomestate(e.g.`Accept'whent=GoodintheMayor-Firmequilibrium,P)willtypicallyhavetosometimesbeproscribedan`accidental'punishment,meantforthatmessageinanotherstate(e.g.`Accept'whent=Badinequi-librium,P).Accordingly,notonlydoesequation(5)continuestoprovideanupperboundonpayosbuttheboundisnolongeralwaysachievableintherepeatedgamewithnoise.TotheextentthatpunishmentperiodsmayalsobecostlyforReceivers(asistruewhenreversiontoperiodsofbabblingactsasthepunishment)thecostsassociatedwith`acciden-tal'punishmentscanalsofallonReceivers.22Thisdiscussionissummarizedinthefollowingremark:Remark1.Inthepresenceofnoise,Sender'spayofromrepeatedpersuasionisnobetter,andcanbestrictlyworse,thanwithoutnoise.ThekeyproblemintroducedbyidiosyncrasiesinSenderandReceivers'typesisthatexpostvericationisnolongerpossible.Asaresult,somepunishmentisrequiredtodisciplineSendersfrom`over-sending'messagesthatleadtoattractiveactions.Inthisenvironment,wherethecostsofsustainingpunishmentsarelarge,itisarguablyevenmoreimportant improve-reliability-medical-trials-better21ExceptwhereSender'srecommendationinduceshisleastfavouredactionamongallpossiblemessagesin[i2f1;:::;Ngsupp(t�ht;it

25 ).22However,thisisnotguaranteedtobecostl
).22However,thisisnotguaranteedtobecostlyforReceivers-asdiscussedabove,therearetypicallymultipleNashequilibriaofthestagegame,someofwhichmayoerReceiversahigherpayothantheyachieve`onthepath'.24 tondcommunicationdeviceslikethe`CoinandCup'thatcanrestorecrediblepersuasiveequilibriawithouttheneedforpunishmentphases.Asournextremarkpointsout,whiletheCnCmechanismcannottypicallycompletelyovercometheproblemofexpostunveriablereportingandrecoverthefullcommitmentpayofortheSender,itdoesstillwidenthesetofavailablepayopairsavailabletoSenderandReceivers.Remark2.TheCnCmechanismallowsustostrictlyexpandtheParetofrontierwheneveranequilibriuminducingatleastoneinteriorposteriort2int(
)liesontheParetofrontieroftherepeatedgame.Ofcourse,theCnCcannevermakethesetofequilibriumpayossmallerforSenderandReceivers,sinceitcanalwaysbeignored.Asanillustrativeexample,consideravariantoftheMayor-FirmgameinwhichPr(!=goodjGood)=Pr(!=badjBad)=p�0:5.SupposethatintheSender-optimalequilibrium,theFirm'sequilibriumstrategyproscribesreporting`Accept'withprobability1ift=Good,and`Accept'withprobabilityq0:5ift=Bad.TokeepSenderindierentbetweenreportswhent=Bad,suchastrategyrequires`accidental'punishments(following!=bad)evenwhenSendersays`Accept'intheGoodstate.TheCoinandCupcanhelpreducepunishmentshere,withoutaectingreportingincentives.TheequilibriumsimplyallowsSendertoreport`Accept'withoutpunishment(regardlessof!)for0ctq.Otherwise,continuationplayisasbefore.SinceSenderprivatelyobservesctupfront,thischangedoesnotaecthisincentivesforct�qandhecanreport`Accept'ifandonlyift=Good.However,for0ctq,healwayspreferstosay`Accept'(thereforematchingtheoveralldistributionofreportsfromtherepeatedgame)inwhichcaseheavoidspunishmentregardlessoft.However,ingeneraltheCnCmechanismwillnotallowustoretrievethefullsetofpayosavailableundercommitment.ThisraisesthequestionofwhetheratrustedmediatorcanimprovethescopeofcommunicationpossibilitiesbetweentheSenderandReceiversbycarefullycontrollingtheinformationthatReceiversobserveabouttheSender'sreputation.Suchmediatedtrustmechanismsarecommonintherealworld

26 ,mostnotablyinthevarietyofonlineplatform
,mostnotablyinthevarietyofonlineplatformssuchaseBay,Alibaba,Amazon,AirBnBandothers.Onthesewebsites,independentsellersaregivenratingsbytheplatformbasedonthenoisyaggregationofbuyers'feedbackaboutseveralaspectsoftheirexperiencewiththeseller,includingthequalityofthegoodorservicepurchased,theaccuracyoftheseller'sclaimsetc.Inordertoaddressthisquestion,wenowdescribeaparticularmediatedcommunicationmechanism,whichwecallaReviewMechanism,sinceitisbasedonRadner[1985]'sreviewstrategies.AformaldescriptionofthemechanismispostponedtoAppendixBandherewesettleforanintuitivedescription.IntheReviewMechanism,theMediatorcontrolstheowofinformationavailabletoReceivers.Themechanismrunsiterativelyinphases.Attime0,25 themechanismstartsina`Good'phase,whichlastsforTperiods.Ateachperiodwithinthisphase,areceiverisonlytoldthattheSender'spastplaymeansthathisreputationis`Good',G.Importantly,Receiversobservenothingelse.Inparticular,weassumethattheMediatorcanrandomlypermutetheorderingoftherstTReceiverssothattheydonotevenknowtheirpositionintheline,andcannotinferthisfromtheirindext.DuringphaseG,theMediatorprivatelyasksSenderforreportsofhistype,andcommitstogarblethesereportsusingamessagefunctionm(;G):!
whichoutputsrecommendedbeliefs,~,fortheReceiverwhoisassignedtoplayattimet.SenderobservesnothingelseaboutthemechanismotherthanthephaseGandhishistoryofreportsandinducedactions.Inparticular,hedoesnotobserve(!)t=0.AttheendoftherstTperiods,themediatorconductsastatisticalreviewofSender'sreports,~tT�1t=0,giventhesequenceoffeedback(!t)T�1t=0.ThetestcheckswhethertheSenderissucientlylikelytohavereportedtruthfullyonaverageacrosstheG-phase.Ifhepasses,themechanismmovesontoanotherG-phasefromperiodTonwards.Otherwise,mediatorswitchestoannouncingtothenextT2NReceiversthatthetheSender'sreputationisB.Inthiscase,theMediatorcommitstosenda`babbling'messagetoeachReceiver,regardlessofSender'sprivatereportsintheB-phase.AttheendoftheB-phase,mediatorrevertstoannouncingaG-phase.Toestablishthemainresultofthissection,wemakeacommonassumptionontheformofrelationshipbetween!tandt.Denotetothejointprobabilityof�it;!jt
,i;j2f1;

27 2;:::;Ngbyf(i;!j):=i0g�!jtj
2;:::;Ngbyf(i;!j):=i0g�!jtjit
.Then:Assumption2.Thereexistsasubset 0:=!1;:::;!N  suchthat�f�it;!1t

Ni=1;:::;�f�it;!Nt

Ni=1arelinearlyindependent.Assumption2requiresthatwhiletheReceivercannottypicallyinferSender'ssignaltfromobservingherown!t,eachpossiblesignalprovidesstatisticallydistinctposteriordistributionsj!iinthesensethatsuchaposteriorcouldbeformedonlyby!iandnotsomelinearcombinationofsignals,f!jgj6=i.Tounderstandthisconditionbetter,considerthefollowingexample:Example1.(SymmetricBinarySignals)SupposeSender'ssignalattimetisbinary,t2 ;  ,distributedaccordingtoPr�t= 
=02(0;1).ReceiverRtobserves!t2f! ; !g,wherePr�!t= !j t
=Pr(!t=! j t)=p.ItiseasytoverifythatAssumption2issatisedip6=1 2.Inotherwords,Assumption2issatisedsolongas!tisaninformativesignaloft,inthatPr� tj!t
6=0.26 Inaddition,weneedthefollowingregularityassumptiononSender'sreduced-formpref-erences:Assumption3.Forall02(0),thereexistsanopensetX(0)(0)suchthatPNi=1iv(i)iscontinuousinonX(0)and02cl(X(0)).Assumption3ensuresthattheSender'sreduced-formutilityfunctiondoesnotinvolvediscontinuousjumpsinSender'spayoswhichareonlyattainableonalower-dimensionalsubsetofbeliefsin
.SinceuSiscontinuousinactionsinourmodel,such`jumps'canonlyeveroccuriftheReceiver'sbeliefsjustinducehimtochangehisactionsinadiscontinuousway,ashisbeliefsentersucha`knifeedge'regioninthespaceofposteriorbeliefs.Weruleoutthiskindofknife-edgepreferenceoftheReceivertotakediscontinuouslyhighactionsfortworelatedreasons.First,suchutilityfunctionsareclearlynotrobusttosmallperturbationsofReceiver'spreferences.Second,suchutilityfunctionsarenotrobusttosmallamountsofnoiseinReceiver'sposteriorbeliefs.Finally,toavoidsometechnicalissuesthatdonotaddanyinsighttothemainresult,weassumeintherestofthissectionthatAisacountableset.WiththecorrectchoiceofReviewMechanismandapatientenoughSender,itturnsoutwecanrecoverpayostotheSenderarbitrarilyclosetohisoptimalcommitmentpayo,evenwhenSender'sprivateinformationisn

28 otveriableexpost:Theorem4.SupposeAs
otveriableexpost:Theorem4.SupposeAssumptions2and3hold.Forany"�0,thereexistparameters�Rand 1suchthatforall1&#x]TJ/;༕ ;.9;Ւ ;&#xTf 1;.6 ;� Td;&#x [00; thereisanequilibriumofreviewmechanism�RinwhichSender'snormalizeddiscountedaverageutilityisatleast^v(0)�":(1�)E"1X=tuS(at)j~?;?;�R#^v(0)�":whereEh
j~?;?;�Ridenotesexpectationstakenwithrespecttoequilibriumplayof�R.Theorem4establishesthattrustmechanisms,suchasthemediatedreviewsystemsonsalesplatformslikeeBay,areavaluabletoolforpersuasionpurposes.TheycanallowSenderstoextractalmostalltheavailablegainsfrompersuasionevenwhentheSendercannotcommittosignals,andwhererepeatedplayaloneboundsSender'spayoby(5),whichisoftenstrictlybelow^v(0).ToensureanequilibriuminwhichtheSenderachieves`closeto'optimalpayos,theproofconstructsaReviewMechanismwiththefeaturesthat(i)themediatorpromisestouseagarblingfunctionm(
;G)`closeto'theoptimalexperimentundercommitment,whererecommendations~iareeachatcontinuitypointsofv(guaranteedbyAssumption3);(ii)27 thetestisstrictenoughthatSender'soptimalstrategycanbebounded`closeenough'totruthtellingbytheexpectedcostsoffallingintoaB-phase;(iii)ifSenderdidadoptatruthtellingstrategy,theexpectedcostoffallingintoaB-phaseisalmost0.AnimportantfeatureofthetrustmechanismisthatitkeepsReceiversuncertainaboutwhentheSenderisnextvulnerabletoareviewchange.ToseewhythisaidstheSender'sabilitytoearnrents,considertheFirm'soptimalstrategy(fromSection2)inaReviewMechanism.Attime0,theFirmknowsthatitwillhavetheveracityofitsreportstestedstatisticallyTperiodslater.However,whenTislargeenoughfortheLawofLargeNumberstomakethesetestsreliable,themarginaleectofasinglemisreportonthechancesoffailingareinsignicant.23Thus,theFirmhasanincentivetomakesomemisreports,andduetodiscounting,itwillprefertooverreport`Good'att=0.But,iftherstMayorknowssheisrstinline,shewillbeawareoftheFirm'sincentiveandtreatitasbabbling,underminingitsincentiveto`overreport'intherstplace.Thisincentivetofront-loadmisreportscausestheFirm'sabilitytoextractpersuasiverentstounravel

29 whenMayorsknowtheirpositionintheline.How
whenMayorsknowtheirpositionintheline.However,whenaggregationsystemsleaveMayorsuncertainaboutwhentheFirmisupforitsnextreview,thisuncertaintyconsiderablyrestrictstheMayor'sskepticismonseeingan`Accept'recommendation.Then,despitetheincentiveoftheFirmtofront-loaditslies,theMayorcandonobetterthantofollowtherecommendationssincetheReviewMechanismisdesignedtokeeptheSender`closeto'truthfulonaverageacrosstime.6EmergentCommunicationMechanismsTheroleoftheCnCmechanismistoallowReceiverstoallowtheSendertomixfromtheperspectiveoftheReceiverbutallowstheReceivertosubsequentlyverifythattheSenderwasmixingaccordingtosomeparticularrule.ThisallowstheReceivertoonlypunishtheSenderopath.Therearetwoclassesofgamewherethesepropertiesendogenouslyemergeinequilibrium:1)onestatemaysimultaneousactions;and2)manystatesmanysimultaneousactions.WewillillustratethisrstcasebyadaptingtherepeatedpublicgoodsgamedevelopedinHermalin[2007],thissimultaneouslyillustratesasettingwheresustaininggreaterlevelsofpersuasioncanbesociallyoptimal.Inthesecondexampleweuseanancecasewhichrelatestosomerealworldattemptstoimprovepersuasionandalsospeakstohowwecandobetterwhenthereisnoisyfeedbackaboutwhatthesenderobserves. 23TheeectofasinglemisreportontheaverageisoftheorderofT�1.Bycontrast,toensuretruth-tellingpassesthetestwithahighprobability,thetest's`errorbounds'areoftheorderT�1 2.28 6.1OneStateManyAgents:ARepeatedPublicGoodsGameConsideranevennumberedteamofL�3myopicworkers(receivers)andonepatientbenev-olentboss(sender).Atperiodteachworkerexertscostlyeortelt2f0;1gonanewproject.Thetotalstatedependentoutputfromaprojectatstagetis:t=L 2tXLeltWheret2f1;3gwherePr(t=3)=0=1 3andwewillrefertothehighpayoprojectasgoodandthelowpayoasbad.Allworkersshareequallyinthetotaloutputandtheireortisnotmeasured.Thestagepayoofworkerlisul= L�el:Thebossesstagepayowilljusttheaveragepayooftheworkers,i.e.,theboss,liketheworkers,wantseveryonetoexerteortindependentofthestate.Asinthestandardcase,thebossobservesthestateandthenmakesareporttotheworkerswhothensimultaneouslychoosetheireortlevel.Atstagetworkersobservethehistoryofoutputsf0;

30 1;:::;t�1g.Noterstthatit
1;:::;t�1g.NoterstthatitisParetooptimalforallworkerstoexerteortateverystageindependentofthestate.However,itisalsoeasytoseethatitisonlyindividuallyoptimalfortheeachworkertoexerteortiftheirposteriorprobabilityoftheprojectbeinggoodisPr(t=3)=t0:5.Further,becausetheworkersaremyopictheycannotenforcehigheortthroughpunishmentstrategies.IfthebossisonlyabletosendasinglemessagetoallworkersthenthegameissimilartotheMayorandFirmcase.Thereisatruthtellingequilibriumwherethebossreportsthetruestateandallworkersexerteortifandonlyifthebosssaystheprojectisgood.Iftheoutputislow,L2=2,afterthebosssentamessagestatingtheprojectwasgoodtheworkersknowthebossliedandwepermanentlymovetoababblingequilibrium.Wecanalsosupportnon-truthfulequilibriainthesamewayaswiththemayorandrm,butinthiscasebothbossandworkerwillbeworseoastheboss'sincentivesarealignedwiththeaverageworker.However,ifthebossisabletosendavectorofindividualizedmessagestoworkerswecandomuchbetter.Inthegoodstatethebossalwaystellsworkerstoexerteort;inthebadstateherandomlyselects(equiprobably)halftheworkersandtellsthemtheprojectisgood.ConditionalonbeingtoldtheprojectisgoodeachworkerhasposteriorPr(t=3jGood)0:5andexertseortifandonlyifthebosssaystheprojectisgood.Ontheequilibriumpath29 weonlyobserve2L2 4;3L2 2,ifworkersobserveanyotheroutputtheyknowthebosshasdeviatedfromtheequilibriumstrategyandthegamemovestoababblingequilibriumforever.ThisequilibriumParetodominatestruthtelling.24Notethatachievinghigherlevelsofpersuasiondoesnotrelyonthisbeingapublicgood.Forexample,iftheirwerenopublicgoodproblemandtheworkersabovejustearned�elwhilethebossgetsapayoofPelthesameequilibriumasdescribedaboveisstillfeasible.Ingeneral,allthatisnecessaryisthattherearemanysimultaneousactionsandsomeformoffeedbackmechanismaboutthehistoryofstatesandmessagevectors.Solongasthesender'schoiceoftowhomtosendamessageisex-antestochasticatthestagegamefromthereceiversperspectivewecangetobedientbehaviourfromthereceiverswhilestillbeingabletoverifythatthesenderisstickingtotheequilibriumstrategyex-post-justaswiththeCnCmechanism.25Aconsiderationinthesecasesistheincentivesan

31 dcapabilitiesofReceiverstosharetheirmess
dcapabilitiesofReceiverstosharetheirmessageswithoneanotheratthestagegame.Inthepublicgoodgameexaminedabovetheworkerswouldneverwanttodiscourageothersfromworking,andsoitisplausiblethatevenifabletheywouldnothavemeaningfulcommunicationwithoneanother.However,ifwetakethevariantwherethereisnopublicgoodtheworkershaveweakincentivestocommunicatewithoneanotheratthestagegame;inthiscase,andsimilar,thesenderwouldnditdesirabletopreventcommunicationbetweenreceivers,suchasChinesewallsordepartmentalsegregation.6.2ManyStatesandManyAgents:FinancialAdvice&DisclosureRulesWewillillustratethisusinganexampleofbrokerages(sender)givinginvestmentadvicetotheirclients(receivers).Thereareseveralpapersshowingthatadvisersandbrokerageshaveincentivestooversellproductstotheirclientsandarefartoooptimisticintheirrecommen-dations[DugarandNathan,1995,LinandMcNichols,1998,MichaelyandWomack,1999,Krigmanetal.,2001,HongandKubik,2003].Thiscouldbemodeledusinganassetpricingmodelwithcostlytransactionsandinformationacquisition,butthiswouldbeapaperinitself.Soforexpositionalpurposesweuseasimplebinaryactionmodelwhereabrokerageadvisesclientsonproductsthatitalwayswantsthemtobuy. 24NotethatbyallowingthebosstosendindividualizedmessageswecanachievebetterpayosviacheaptalkthaninHermalin[2007].25Ifthesenderwerenotindierentthiswouldbeokaysolongasthesender'sdidnotknowwhothesenderhad30 ThereareLinvestors,eachinvestorlismakingadecisionaboutanidiosyncraticportfoliothat,giventheirpreferences,isovervaluedorundervaluedlt2fLow;Highg.ThePr(lt=High)=0=1=3forallt.26Theportfoliovaluesareindependentacrossinvestors,E[ltjk6=lt]=E[lt].27Letinvestorpayosbeidenticaltothemayorandrmcasesothattheinvestorbuysiflt0:5;Thebrokeragealwayswantstheirclientstobuy.Thebrokerageobservesltforalllandsendsavectorofmessagest=(1t;::::;Lt)toinvestorswhothenchoosetobuyorsell.IfinvestorlonlyobservesthehistoryoftheirownportfoliolandtherespectiveadvicegiventothemthenwejusthaveLreplicationsofthestandardcaseandtheequilibriumpayosetisidenticaltothatofthemayorandrmexample.However,ifinvestorscanobservesomemeasureofthehistoryofallmessagesthenwecangetarbitraril

32 yclosetothesetofpaymentsavailableunderfu
yclosetothesetofpaymentsavailableunderfullcommitment.Forconcreteness,supposeinvestorsobservetheproportionof`buy'recommendationssentbythebrokerageinthepastinadditiontotheirownadvice.NowtheproportionofbuyrecommendationscanbeusedinafashionthatissimilartotheCnC.Thebrokerageisgivenaperperiodbudgetof`buy'oerstoallocateacrossinvestorsandtellsinvestorstobuywhenlt=Highandthengivestheremainingbuyrecommendationsatrandomtotheremaininginvestors.Ifthebudgetislessthanorequaltotwothirdsthenitisincentivecompatiblefortheinvestorstofollowtheadvice.Thisissustained,ofcourse,bythethreatofmovingtoababblingequilibrium.28Movingfromaworldinwhichinvestorsonlyobservetheirownhistoryofmessagestoobservingtheaggregateofmessagesmakesafarmoreprotableequilibriumforthebrokerage.Itsohappens,thatin2002,theNationalAssociationofSecurityDealers,anancialindustryself-regulatingbody,imposedrulesthatrequirebrokeragestodisclosetheaggregatedistributionoftheirrecommendationstoclients.Barberetal.[2006]andKadanetal.[2009]analysetheadvicegivenbybrokeragesaswellasthepricereactiontothatadvice.Priortotheintroductionoftheserules,analystsgave'buy'recommendations60%ofthetime.OnintroductionofthesenewrulesBuyrecommendationsdroppedalmostimmediatelyto51%andbythefollowingyearBuyrecommendationsweredownto42%.Thiswasnotadropfromananomaloushigh,sincethebeginningofthedatatheyanalyse(1996)Buy 26Wecouldrelaxthisassumption.Itissucientjusttohaveex-postrealisationoftheoverallsuccessratesacrossassets.27ThisassumptionissubstitutingforamodelwheretheinvestorgivesadviceonLassetclasses,therearemanyinvestorseachwithanidiosyncraticcomponenttotheirportfolio,andinvestorsarenotcontemporane-ouslymonitoringtheadvicegivenonallassets.Suchamodelcaneasilybegroundedinrationalinattentionorcostlyinformationacquisition.Further,weknowasanempiricalmatter,thatdierentinvestorsdoinfactfocusondierentassetclasses.28Thelackofcompetitionmaybeofconcern,weknowhoweverfromGentzkowandKamenica[2016]thatthereisstillaroleforpersuasionincompetitiveenvironments.But,weareassumingthatsomethingakintothisresultpersistsinadynamicenvironmentwithoutcommitment.31 recommendationshadalwaysexceeded60%.Alongwiththedropinbuyrecommend

33 ationsKadanetal.[2009]showthatpricesbeca
ationsKadanetal.[2009]showthatpricesbecamemoreresponsivetobuyrecommendations,suggestinginvestorsfoundthenewreportsmorepersuasive.Thisiscompatiblewithourmodel.Supposethatpriortothechangethebrokerageswererecommendingbuywhentheassetswereundervaluedandsomeproportionofthetimewhentheassetswereovervalued.Onpaththeywouldbepunishedgoingthroughbabblingperiodswithinvestorlwherethermalwayssaysacceptandtheinvestorignoresthebrokerage.29Usinganadaptationofourmodel,wecanstudyhowtheintroductionofthesedisclosurerulesaectedtheabilityofbrokeragermstopersuadeclientstotradewiththem.Intheabsenceoftheserules,analystsfacedrepeatedinteractionwithclients.GiventheresultsofSection4,wewouldexpectanyinformativecommunicationinthissettingtobesustainedbylongperiodsofbabblingasaclientspunishoverlyoptimisticrecommendations.Indeed,theaggregatereportingdatabearthisoutsomewhat-reportswereskewedheavilytowards`Buy'calls,andyetsimultaneouslyhadlittleeectonmarketbehaviour.Nonetheless,thenatureofrepeatedinteractionmayhaveprovidedenoughdisciplinetopreventanalystsfromcompletelybabbling.Oneeectoftherulechangewastoallowinvestorstoseethedistributionofrecom-mendationsmadebytheinvestmentbank'sanalystsacrossassets,providingcontextfortherecommendationmadeinanindividualreport.Indoingthis,theNASDruleseectivelypro-videdbankswithacommitmentdevicetoenactoptimalpersuasion.Indeed,weshowthattheintroductionofaggregatereportingstandardsintroducesequilibriainwhichthebankscanachievetheoptimalpayosfrompersuasion.Inequilibrium,theaggregatereportingstandardcanbeusedasapayo-irrelevantdiscipliningdevice,aroundwhichinvestorscan`punish'thebankwithbabblingifitsaggregatereportsbecomeexcessivelyskewedtoward`Buy'(whencomparedtoindividualoutcomes,expost).Givensuchadeviceandalargeenoughassetspace,anequilibriumcanbesustainedinwhichthebankcrediblyreportsaccordingtotheoptimalpersuasivestrategyandinwhichReceiversneverneedtoemployon-pathpunishments,sincetheaggregatestatisticscanbeusedasadiscipliningdevice.Noticethat,fromtheperspectiveofasingleasset,wehaveintroducedapayo-irrelevant`state'variablewhichisseenrstbythebankandonlylaterbyinvestors.-thisissimplythevectorofperformanceou

34 tcomesacrosstheotherassets. 29Thisformof
tcomesacrosstheotherassets. 29Thisformofbabblingisevenmorenaturalinaworldwithnaiveinvestorswhotakeadviceatfacevalue.32 7ConclusionTherearethreemainresultsinthispaper.Therst,inlinewithpreviousthinking,isthatmeaningfulcommunicationtodaycanbesustainedoutofthedesireformeaningfulcommunicationinthefuture.Thesecond,isthatwhilerepeatedinteractioncangeneratecommitmenttodishonestreportingstrategiesitcannotgenerallyachievetheoptimalout-comefortheSender.ThethirdisthattherearevariousalterationstothegamethatachievetheoptimaloutcomefortheSender.Theseresultshaveimportantimplicationsfortherealworld,andtheliteratureonper-suasion.PerhapsthemostimportantimplicationisthattheuseofasimplemechanismsuchasaCnCcangenerateParetoimprovementsforSendersandReceivers.Further,mediatorscandrasticallyimproveoutcomesbyservingtoaggregatereportsandreducethenecessityofonthepathpunishmentsinthepresenceofnoise.However,italsofollowsthattheintroduc-tionofsuchmechanismsinsomecircumstancescanincreasetheextenttowhichdishonestpersuasionispossible,andgiveSendersanincentivetopushthegametowardslesshonestequilibria.Itseemsthatsomeofthesecommunicationmechanismshavealreadybeensetuptosomeextenteitherthroughtheuseoflaws,thirdpartiesoronlinetechnologies.Although,itisnotcleartheextenttowhichthey'vebeendesignedtobeoptimalfromthepointofviewoftheSenders,Receivers,orsociety.Finally,itseemsthatnotalltoolsarecurrentlyinuse,andthatitmightbepossibletobenetboththeSendersandReceiversoftheworldbyidentifyingthosecaseswheretheycanbefruitfullyimplemented.ReferencesS.AtheyandK.Bagwell.Optimalcollusionwithprivateinformation.RANDJournalofEconomics,32(3),2001.S.AtheyandK.Bagwell.Collusionwithpersistentcostshocks.Econometrica,pages493540,2008.B.M.Barber,R.Lehavy,M.McNichols,andB.Trueman.Buys,holds,andsells:Thedistributionofinvestmentbanks'stockratingsandtheimplicationsfortheprotabilityofanalysts'recommendations.JournalofAccountingandEconomics,41(1-2):87117,2006.D.Barron.Attainingeciencywithimperfectpublicmonitoringandone-sidedmarkovadverseselection.TheoreticalEconomics,2016.33 J.Bizzotto,J.Rüdiger,andA.Vigier.Delayedpersuasion.Technicalreport,UniversityofOxfordWorkingPaper,2016.A.Chakrabort

35 yandR.Harbaugh.Persuasionbycheaptalk.Ame
yandR.Harbaugh.Persuasionbycheaptalk.AmericanEconomicReview,100(5):23612382,2010.V.CrawfordandJ.Sobel.Strategicinformationtransmission.Econometrica,50(6):14311451,1982.A.DugarandS.Nathan.Theeectofinvestmentbankingrelationshipsonnancialanalysts'earningsforecastsandinvestmentrecommendations.ContemporaryAccountingResearch,12(1):131160,1995.J.C.Ely.Beeps.WorkingPaper.,2015.J.F.EscobarandJ.Toikka.Eciencyingameswithmarkovianprivateinformation.Econo-metrica,81(5):18871934,2013.M.GentzkowandE.Kamenica.Competitioninpersuasion.TheReviewofEconomicStudies,(September):136,2016.URLhttp://faculty.chicagobooth.edu/matthew.gentzkow/research/BayesianComp.pdf{%}5Cnhttp://www.nber.org/papers/w17436.B.E.Hermalin.Leadingforthelongterm.JournalofEconomicBehaviorandOrganization,62(1):119,2007.H.HongandJ.D.Kubik.Analyzingtheanalysts:Careerconcernsandbiasedearningsforecasts.TheJournalofFinance,58(1):313351,2003.J.Hörner,S.Takahashi,andN.Vieille.Truthfulequilibriaindynamicbayesiangames.Econometrica,83(5):17951848,2015.O.Kadan,L.Madureira,R.Wang,andT.Zach.Conictsofinterestandstockrecommen-dations:Theeectsoftheglobalsettlementandrelatedregulations.ReviewofFinancialStudies,22(10):41894217,2009.E.KamenicaandM.Gentzkow.Bayesianpersuasion.AmericanEconomicReview,101(6):25902615,2011.A.Kolotilin.Experimentaldesigntopersuade.GamesandEconomicBehavior,90:215226,2015.I.Kremer,Y.Mansour,andM.Perry.ImplementingtheWisdomoftheCrowd.JournalofPoliticalEconomy,122(5):9881012,2014.34 L.Krigman,W.H.Shaw,andK.L.Womack.Whydormsswitchunderwriters?JournalofFinancialEconomics,60(2):245284,2001.H.-w.LinandM.F.McNichols.Underwritingrelationships,analysts'earningsforecastsandinvestmentrecommendations.JournalofAccountingandEconomics,25(1):101127,1998.C.MargariaandA.Smolin.DynamicCommunicationwithBiasedSenders.2015.R.MichaelyandK.L.Womack.Conictofinterestandthecredibilityofunderwriteranalystrecommendations.Reviewofnancialstudies,12(4):653686,1999.E.Perez-Richet.InterimBayesianpersuasion:Firststeps.AmericanEconomicReview,104(5):469474,2014.E.Piermont.DistributionalUncertaintyandLong-Ru

36 nUncertainty.2016.R.Radner.RepeatedPrinc
nUncertainty.2016.R.Radner.RepeatedPrincipal-AgentGameswithDiscounting.Econometrica,53(5):11731198,1985.L.RayoandI.Segal.Optimalinformationdisclosure.JournalofpoliticalEconomy,118(5):949987,2010.J.Renault,E.Solan,andN.Vieille.Dynamicsenderreceivergames.JournalofEconomicTheory,148(2):502534,2013.W.Tamura.OptimalMonetaryPolicyandTransparencyunderInformationalFrictions.JournalofMoney,CreditandBanking,48(6):12931314,2016.I.A.Taneva.Informationdesign.WorkingPaper,2015.35 AppendixABelowweprovidetheformalstatementunderlyingAssumption1inthemaintext:Assumption.Forany1;:::;n2
Psuchthatv(1)=v(2)=


=v(n),wherei6=jforatleasttwoi;j2f1;2;:::;ng,n�1,theconcavicationofvon
P,^vP(),satises^vP nXi=1ii!�nXi=1iv(i)ProofofLemma1Proof.Takeanyequilibriumofgame�1,anddenotethecorrespondingstrategiesandbeliefsrespectivelyby?=(?t(t;t))1t=0,?t(t;mt),t=0;1;2;:::,forS,Rt,and?t(t;mt),t=0;1;2;:::.Sincethesestrategiesandbeliefsformanequilibriumof�1,theyobeyconditions1.-3.insection3.2.Weconstructstrategiesandbeliefswithingame^�1which(i)inducethesameconditionaldistributionsofReceiver'sactions,givenappropriatelydenedhistories,and(ii)formadirectequilibriumof^�1;thatis,theysatisfyconditions1.-3.insection3.3.Weassumeforeaseofexpositionthatthereisamessagem 2Mwhichisneverplayedontheequilibriumpath.30Foreachpublichistoryt=(m;a;)t=0,denethefunction~t(t;mt):=?t(t;mt)andacorrespondinghistoryofgame^�1,ht:=(~t(t;mt);at;t).LetthesetofallsuchhistoriesbeH.Foranyhistoryhtofgame^�1suchthatht=2H,recursivelydene0tasahistoryof�1inwhichm=m foranysuchthat~t6=~t(0;m)forsomemt2M.Considernowstrategies^,^t,t=0;1;:::,andbeliefst(ht;~t)ingame^�1,where:^t(ht;t):=8:?t(t;t);ifht2H?t(0t;t);otherwise:where?t2
(
)satisesPr(^=~t(0;m)jt;t)=?t(m;t;t)and?

37 t(m;t;t):=Pr(m=mj
t(m;t;t):=Pr(m=mjt;t;?t(t;t))isthemeasureoverMinducedbylottery?t(t;t),^t(ht;^t)=8&#x]TJ ;� -2;
.52;&#x Td ;&#x[000;&#x]TJ ;� -2;
.52;&#x Td ;&#x[000;&#x]TJ ;� -2;
.52;&#x Td ;&#x[000;&#x]TJ ;� -2;
.51; Td;&#x [00;&#x]TJ ;� -2;
.51; Td;&#x [00;&#x]TJ ;� -2;
.51; Td;&#x [00;:?(t;mt);ifht2H;^t=~t(0;m);forsomem2M?(0t;mt);ifht=2H;^t=~t(0;m);forsomem2Ma ;otherwise. 30Lemma3.3doesnotrequiresuchanassumption(proofavailableonrequest),butitsignicantlyreducesthenotationrequiredtodescribestrategiesfully.36 andt(ht;^t)=8���&#x]TJ ;� -2;
.52;&#x Td ;&#x[000;&#x]TJ ;� -2;
.52;&#x Td ;&#x[000;&#x]TJ ;� -2;
.52;&#x Td ;&#x[000;:t(t;mt);ifht2H;^t=~t(0;m);forsomem2Mt(0t;mt);ifht=2H;^t=~t(0;m);forsomem2Ma ;otherwise.whereat=a 2AistheactionwhichminimizesS'sstagepayo.Atanyhistoryht2H1,strategy^assignsprobability?t(m;t;t)tomessage^t=t(t;mt).Moreover,givenmessage^t,^tht;mtinducesthesamelotteryovera2Athat?(t;mt)does,conditionalon(t;mt).Thus,strategyprole(^;^)inducesthesamedistributionoverReceiveractionsathistoryhtingame^�1asdoes(?;?)doesin�1.Moreover,thetransitionprobabilitiesbetweenhtandht+1=ht;~t(t;mt);at;tareclearlyidenticaltothosebetweentandt+1=t;;mt;at;t.Thus,(^;^)inducesthesameexantedistributionoveratas(?;?)aswell.GivenS'sstrategy,amessage^t=~t(0;m)inducesRt'sbeliefsatanyhistoryht2HtobePr�t=itj^t
=Pr^tjht;iti0 PjPr^tjht;jtj0=Pr(mtjt;it)i0 PjPr(mtjt;it)j0=~t(0;m)Thus,condition2.ofdirectequilibriumissatisedforsuchstrategies.Mor

38 eover,since~t(0;m)ar
eover,since~t(0;m)areposteriorprobabilities,theynaturallyintegratebacktothepriorasinKa-menicaandGentzkow[2011].Thus,condition3.issatised.Finally,wearguethatcondition1.ofadirectequilibriumalsoholds.Trivially,^t(ht;^t)isabestresponseforRttosignal^t=~t(0;m)sentbyS,since?(t;mt)wasoptimalforhimingame�1giventhesesamebeliefs.Moreover,^ispreferredforSatanyhistoryhtthananystrategy^0inwhich^t=~t(0;m),forsome,m2M,sincebothsuchstrategiesarerelabelingsof?andsomealternativefeasiblestrategy0ingame�,respectively(andsimilarlyfor^,?).Since?isoptimalingame�1,^ispreferredto^0.Finallyforanyotherdeviation,wehavespeciedthestrategiesforSenderandReceiverinsuchawaythatthestagepayofromSenderdeviationcouldonlybelowerthanthatofadeviationtosomealternativestrategy^0inwhich^t=~t(0;m),forsome,m2M,andmoreover,thecontinuationpayowouldbethesameasunder^0. 37 ProofofTheorem1Proof.Fix0.LetUbethesetofallreal-valuedupper-semicontinuousfunctionson
,withtypicalmemberv2U,andconsiderthemetricspace(U;jj
jj)endowedwiththesupnorm.Asinthetext,wedenotetheconcavicationofvby^v,andanelementof
by.WeshowthatthesetU?,denedasU?(0)=(v2U:^v(0)�Xiiv(i);8figN+1i=1;s:t:0=Xiii;v(i)=v(j);i6=j;i;j=1;2;:::;N+1)isopenanddense.31ToestablishdensityofU?,considerafunctionv02U=U?.Weshowthatthereexistarbitrarilysmallperturbationsofv0underthesupnormsuchthattheperturbedfunction,v,livesinU?.Theconcavicationofv0canbeexpressed^v0=sup2~(0)N+1Xi=1iv(i)where~(0)isthesubsetofN+1-pointdistributionsin(0).32Sincev0isuppersemi-continuousand~(0)iscompactin,thefunctionPN+1i=1iv(i)attainsitsmaximumin~(0).33Moreovertheset~(0)isbothupperandlowerhemicontinuousin0.Toestab-lishupperhemicontinuity,takeasequencefn0g!0,andanycorrespondingconvergentsequence,fng!,n2(n0).Weshowthat2(0).S

39 upposenot.Then,forany�0,ther
upposenot.Then,forany�0,therealwaysexistasubsequencefn0g2NsuchthatN+1Xi=1n0in0i�0�8n0.Butsincefn0g!0,andn0i1,8i,n0,forsome00 N+1,theremustexistanmsuchthatforalln0m,wehaveN+1Xi=1n0in0i�0N+1Xi=1n0in0i�0-acontradiction.Toshowlowerhemicontinuity,takeanysequencefn0g!0,andany 31ByCaratheodory'sTheorem,itiswithoutlossofgeneralitytodeneU?withregardtonitesets,figN+1i=1.32ByCaratheodory'sTheorem,thisiswithoutlossfornding^v0.33Indeed,itiseasytoseethat~(0)isacompactsubsetofRN(N+2).38 2(0).Weshowthatthereexistsasubsequencen000 ,suchthat9n002�n000
satisfyingn00!.Fixingsome�0,wecanndverticesp()=(1();:::;N())forwhichall~0satisfyingj~0�0jcanbeexpressedasconvexcombinationsoftheverticesinp()(Rockafeller,Theorem20.4).Moreoverforan 2k-ball,k2N,k1,wecanencloseallpointsinj~0�0j 2kbythesimplexgeneratedbypk()=2k�1 2k0+1 2ki()Ni=1.Notingthatwecanwrite0=Pii0,eachvertexofpk()canbewrittenPii0+1 2k(i()�0).Thus,foranysequencen000!0andm2N,wecanndksuchthatalln00canbeexpressedasaconvexcombinationofpointspk,whichgetarbitrarilyclosetoPii0.Thisestablishesthelimitingsequencen00!.ApplyingBerge'sTheoremoftheMaximum,thevaluefunction,^v0,iscontinuouson
.Consequently,thesubgraphof^v0,sub(^v0)=f(;):^v0();2
g,isaclosedconvexset.Boundsub(^v0)belowbysomeB 2R,suchthatB min2
^v0()anddenethebounded,closedconvexset,H(^v0):=sub(^v0)Tf(;v):vB g.NotethatH(^v0+)hasthesamepropertiesandcontainsH(^v0+2).Wearenowabletondan-perturbationofv0suchthatthenewfunctionvsatisesv2U?.Partition
intotwosets:C=f:v0()=^v0(&#

40 22;)gand C:=
=C.Wenowconstructapo
22;)gand C:=
=C.WenowconstructapolyhedralconvexsetPforwhichint(sub(^v))Pint(sub(^v+))andalltheverticesofP,f(?i;?i)gMi=1,forwhichi�^v0()satisfyproj
(vi)2C.Foranyx2H(^v0),wecanchooseasimplexSxsuchthatx2SxandSx2int(D).BecausePN+1i=1iv(i)attainsitsmaximumon
,wecaninfactchooseSxsuchthatitsvertices(s1;s2;:::;sN+1)satisfysi=(0i;^v0(i)+)and0i2C,ifsi=2int(H(^v)).FromthisunionofsimplicesSfSxg,wecanndanitesubsetofsimpliceswhoseconvexhullalsocoversH(^v0)-thisisthepolyhedronP(Rockafeller,Theorem20.4).Moreover,byconstruction,novertexofPthatliesaboveH(^v0)has?i2 C.Finally,perturb^v0tosome~vbyaddingto^v0ateachvertexf(?i;?i)gMi=1ofP.~visclearlystilluppersemi-continuous.Moreover,theconcavicationof~visP.On~v,itsucestocheckthat?i6=?jfori6=j2f1;:::Mg.Iftwosuchi,jcanbefound,wecanndaperturbationof?ibysome~satisfying0~&#x]TJ/;༕ ;.9;Ւ ;&#xTf 1;.51; 0 ;&#xTd [;,suchthat?i6=?k,k2f1;2;:::;Mg=figandthenewpolyhedronP0stillcontainsCeverywhere.Thisnewperturbedfunctionvisuppersemi-continuous,satisesv2U?andjv�^v0j(M+1),whichcanbechosenarbitrarilycloseto0.WenowshowthatU?isopeninv,forallpriorscloseto0,forany0satisfyingv(0)^v(0).Specically,weshowthatforanyfunctionv2U?,thereexist1,2&#x]TJ/;༕ ;.9;Ւ ;&#xTf 1;.86; 0 ;&#xTd [;0s.t.forall~vsatisfyingjj~v�vjj1,~0satisfyingj~0�0j2,wehave~v2Sj~0�0j2U?(0).Takesomev2U?.Wearguethat,forsome1&#x]TJ/;ད ;.97;v T; 10;&#x.01 ;&#x-1.1; T; [0;0,thereexists~1,~2&#x]TJ/;ད ;.97;v T; 10;&#x.01 ;&#x-1.1; T; [0;0suchthat8j~0�0j~1,ifj^v(~0)�Piiv(i)j1,forsome2(~0)thenjv(i)�v(j)j~239 forsomei6=j,i;j2f1;2;:::;N+1g.34Supposethiswerenotthecase.Then,forany,~1,~2�0,wecouldndsome~0and2(~0)suchthat(i)j~0�0j~1,(ii)jv(i)�v(j)j~

41 ;2,(iii)j^v(~0)�Piiv(i
;2,(iii)j^v(~0)�Piiv(i)j1.Nowconsideranysequence(n;~n1;~n2)1n=1satisfyinglimn!1(n;~n1;~n2)=0.Thus,wecanndacorrespondingsequence((~n0;n0))1n=1inwhicheach(~n0;n0)satises(i)-(iii)evaluatedat=n,~1=~n1and~2=~n2.Butsince(0;)2RNRN+1and(0)iscompactin(0;),theBolzano-WeierstrassTheoremimpliesthatwecanndaconvergentsubsequence��~n00;n00

!(0;)forsome?2(0).35Moreover,uppersemi-continuityofvimpliesthatatthislimit,wemusteitherhave(i)^v(0)=Pi?iv(?i),andv(i)=^v(j),8i;j2f1;2;:::;N+1g;(ii)Piiv(i)&#x-566;^v(0),or(iii)^v(0)=Piiv(i),v(i)6=v(j),foratleasttwoi,j2f1;2;:::;N+1g.36Sincev2U?,case(i)yieldsacontradiction.Bydenitionof^v,case(ii)alsoimpliesacontradiction.Finally,weruleoutcase(iii).Sincetherethediscreteupwardjumpat?i,forsomei,mustalsocauseadiscontinuityat^v(0)onthepathn0!0-acontradiction,tothecontinuityof^v,whichweprovedabove.Finally,foranyperturbedfunctionv0suchthatjjv0�vjjmin 3;2 3 ,wemustalsohave8j~0�0j~1,ifj^v0(~0)�Piiv0(i)j01,forsome2(~0)thenjv0(i)�v0(j)j~02forsomei6=j,i;j2f1;2;:::;N+1g,forsome~02,01&#x-282;0. ProofofLemma2Proof.SupposeforacontradictionthatforsomeequilibriumpayoE[V(h;)]oftheSenderandsomehistoryh,theminimumnumberofmessagesintheSender'sstrategycompatiblewithobtainingE[V(h;)]inequilibriumisjM0j=N0�N,whereM0=[2tsupp((h;)).ThisstrategyinducesaN0-pointdistribution2
�
N0
ofpos-teriorbeliefsf(m)gm2M0overandacorrespondingdistributionoverReceiverR'sactions,a((m)),wherea((m))2argmaxa2AE[uR(a;)jt]=NXi=1i
uR�a;i
Forthistobeanequilibrium,itmustbethatforallm2supp((h;))andanym2M,V(h;):=v((m))+E[V((h+1;+1))]

42 1;v((~m))+EhV~h
1;v((~m))+EhV~h+1;+1i 34Again,byCaratheodory'sTheoremitiswithoutlosstorestrictattentiontoN+1-pointdistributions,2~(~0).35Avariationonourargumentthat(0)iscompactincanbeusedtoestablishcompactnessin(0;).36Since^viscontinuous,wecannothaveatthelimit^v(0)�Piiv(i).40 whereh+1=(h;m;a;)and~h+1=(h;~m;~a;).Inparticular,givenanystateandmessagesm;~m2supp((h;)),wemusthavev((m))+E[V((h+1;+1))]=v((~m))+EhV~h+1;+1iGivenanyhistory,wedeneanequilibriummessagem2M0tobeuniquelyproscribedatstateifsupp((h;))=m .Thesetofallmessagesthatareuniquelyproscribedatsomestate2isdenotedM.Wedividethesetofequilibriummessagessentathistoryhintotwomutuallyexclusiveandexhaustivesub-groups:thosethatareuniquelyproscribed,m2M,andthosethatarenot,m2M0=M.SinceN0�N,thereexistsan~m2M0andcorresponding(~m)2f(m)gm2M0thatcanberemovedfromthesupportsuchthatremainingposteriorsstillsatisfyBayes'plausibilityXm2M0=f~mgm(m)=0(6)forsomeweightsmsuchthatm0,Pm=1(followsfromCaratheodory'sTheoremappliedtotheconvexset,
�
N0
).ByProposition1inKamenica&Gentzkow(2011),theposteriors(~m)2f(m)gm2M0=f~mgcanbesustainedbyafeasiblesignalstructurewithN0�1distinctmessages.Moreover,themessage~mcannotbeuniquelyproscribedinanystate2.Otherwise,therewouldexistsomeiforwhichi(m)=0,8m2M0=f~mg,whilei0�0,violating(6).Therefore,~m2M0=MandforeverystateinwhichproscribesPr(m=~mjh;)�0,thereexistsanothermessagem0sentwithpositiveprobabilityinstate.Constructanewstrategy?whichinducesthedistribution(m)m2M0=f~mgovertheposteriorsf(m)gm2M0athistoryh,andplaysaccordingtootherwise(thisisfeasible,byProposition1ofKamenica&Gentzkow(2011

43 )).Foranym2M0=f~mg,thestrategycontinuest
)).Foranym2M0=f~mg,thestrategycontinuestoinducebelief(m)athistoryhandleavescontinuationpayosunchangedatV(h;)thereafter(forany2).Moreover,thiscontinuationpayoiswelldenedforeachmsince~mwasneveruniquelyproscribed.Therefore,strategy?achievesthesamepayosfortheSenderfromhistoryh,leavespayosotherwiseunchangedatotherhistories,andinvolvesonlyN0�1messagessentathistoryh.Therefore,italsodoesnotaectincentivecompatibilityofequilibriumplayatanypriorhistory,ht,fort.Ittriviallydoesnotaecttheincentivecompatibilityofanyhistoryht,fort&#x-277;.ButthisisacontradictiontoN0astheminimumnumberofmessagesinanystrategyconsistentwithE[V(h;)]. 41 ProofofProposition1Proof.LetvBbetheworstexpectedstagepayotothesenderfromanyequilibriumofthestagegame.DeneBtastheequilibriumstrategyofthestagegamethatgivetheSendervB,giventheReceiver'sequilibriumbeliefs.Atanystaget,Bt+forallconstitutesacontinuationequilibrium.DeneTtasthestagegamestrategyofreportingthetruestateoftheworld~it=1ifandonlyift=itand0otherwise.LetvTbetheexpectedstagepayototheSenderfromastagegameconditionalonthesenderfollowingthestrategyTt.LetHTtHtbethesetofhistoriesattimetconsistentwiththesenderfollowingstrategyTatallt.DenethefollowingstrategyTsuchthatforallt:t=8:Ttifht2HTtBtifht=2HTt:GivensuchastrategyandtheobedientbeliefsoftheReceiver,thegapinthecontinuationpayosfromplayingthestrategyTtatahistoryht2HTtis:(vT�vB) 1�:Atanyhistoryht2HTtthestagepayofromdeviatingisboundedabovebyv1,astheSendersstagepayoisuppersemi-continuousandthemessagespaceiscompact.Hence,ifvT&#x]TJ/;༤ ;.9;Ւ ;&#xTf 2;.21; 0 ;&#xTd [;vB;thereexists1suchthat(vT�vB) 1��vforall�:TconstitutesanequilibriumstrategywithobedientBayesplausiblebeliefsfor�: ProofofProposition2Proof.Inanyequilibrium,Smustbeindierentatanyhistory(ht;it)betweenallmessages~2t(ht;it).Since&

44 #22; i(ht):=argminv i(t(ht;it)
#22; i(ht):=argminv i(t(ht;it))isbydenitioninthesupportoft(ht;),wemusthavethatpayosfromanyequilibriummessageatthishistoryareVt(ht;t)=v i(ht)+E[Vt+1((ht;t;t);t+1)]:Considerthefollowingproblem:sup2E[Vt(h t;0)](7)42 s.t.Vt+�h t+;t+
=v t++EhVt++1h t+; t+;t+;t++1i;02co([t2supp(t(ht;t)));8h ts.t.~t= iatallsubsequencesh 0,00,ofh tatwhichSacts.Werefertothesetofcontinuationpayosthatsatisfyallconstraintsin(7),byV.NoticeVisnon-empty.37Att=0(whereh0=;),problem(7)isarelaxedversionofproblem(4):itonlyretainsconstraintsforhistoriesinwhichShasalwaysreportedthe`worst'currentmessage i0amongallthoseavailableinthesupportofhisstrategyatprevioushistories,h0,0.Allotherconstraintsfrom(4)aredropped.Thus,theoptimalvalueof(7)providesanupperboundon(4).LetV?t(h t;t)bethesupremumachievedinproblem(7)athistoryh t.Fromtherstconstraint,wemusthaveEV?t+�h t+;t+
=supt(ht;);Vt++1Ehv it++Vt++1h t+;i t+;t+;t++1i(8)wherethesupremumistakenoverfeasiblelotteriest(ht;)2(0)andfeasiblepayosfromthecontinuationequilibrium,Vt++12V.38Noticethat,foranyt+,history�h t+;t+
andcorrespondingt+�h t+;t+
,Vt+�h t+;t+
ismaximizedbychoosingthehighestfeasibleexpectedcontinuation,EhV?t++1h t+; t+;t+;t++1i.Moreover,sincethecontinuationgamesathistories�h t+
andh t+; t+;t+areiden-tical,theexpectedcontinuationvaluesmustbeequal:EV?t+�h t+;t+
=EhV?t++1h t+; t+;t+;t++1iSubstitutinginto(8)yields,onrearrangement:(1�)E[V?0(0)]=max2(0)Xi0

45 v i()(9)Sinceanyequilibriumvalueisb
v i()(9)Sinceanyequilibriumvalueisboundedbythissupremum,therstpartofourresultholds.Sincev i()istheminimumofnitelymanyuppersemicontinuousfunctions(fromLemma2,weneedonlychoosefromNdistinctposteriors,,andvisuppersemicontinuous),itis 37Thediscountedpayofromrepeatedplayofthestaticbabblingequilibriumateachhistory,v(0) 1�,isfeasible.38Focusingonexpectedcontinuations(ratherthanvaluesconditionalon)ensuresthatwedonotviolatetheconstraint02co([t2supp(t(ht;t))).43 uppersemi-continuous.Moreover,theset(0)isclearlycompact.Therefore,bytheExtremeValueTheorem,themaximumexists.Letthelotterythatachievesthisoptimumbe?2(0),withassociatedsupportf?1;?2;:::;?N0g,whereforconvenienceweindexsuchthatv(?1)v(?2)


v(?N0).FollowingProposition1,vBistheworstexpectedstagepayotothesenderfromanyequilibriumofthestagegame;BtistheequilibriumstrategythatinducesvBatanystaget;Bt+isafeasiblecontinuationequilibriumforall10andany.Nowdenev :=PNi=1i0v (),thisistheputativeupperboundontheaveragestagepayodescribedinProposition2.Nowdenetasthestagegamestrategyofinducingthelottery.Considerthefollowingstrategysuchthatforallt:t=8:tifht2H tBtifht2HtSHBt:WhereHBtisthesetofhistoriesinconsistentwithatt.Hence,ifht2HBtthenht+2HBt+forall�0.GivenstrategyweonlyplayathistoriesinH torHt.Atthebeginningofthegameh02H 0.Ifht2Htwesaythatweareinapunishmentperiod.Apunishmentperiodcommencesif~t6= iatht2H tforsome~t2supp(?)andlastsforKi;+1(bt+1i;)periods.Ki;andi;2[0;1]arethevaluesofkandthatsolvethefollowingequation:v()�v( i)=k+1(v �vB)+kX=1(v �vB):(10)First,notethatifv =vBthenthispayocanbesustainedasanaveragepayoinanequilibriumofthestagegameandhencealsointherepeatedgame,leavingustofocusonthecasewherev �vB.Whenv �vB�0thenlet 1bedenedbythefollowingequat

46 ion:v= 1� (v �v
ion:v= 1� (v �vB):Hence,for� therealwaysexistsakandsolvingequation10asv()�v( i)v1.Now,toseethatisanequilibriumfor&#x]TJ/;༤ ;.9;Ւ ;&#xTf 2;
.83; 0 ;&#xTd [;,rstnotethatBtisanequilibriumofthestagegameandsocanbeanequilibriumatanyhistory;second,thereisasolutiontoequation(10)andsotheSenderismadeindierentbetweenchoosing~tand iforalliandall~t2supp(?);andnally,deviatingfromtforsomeht2H tresultsinanexpectedutilitylossof1X=1(v �vB)�v:44 ProofofTheorem2Proof.(If)Clearly,theoptimaldiscountedaveragepayoachievableviainformationdesignoneachReceiverRtweaklyexceedstheoptimalpayofromanyrepeatedgame(sincethisproblemissimilarto(2),butwithoutincentiveconstraints).Supposethatatprior0,theoptimalpayounderinformationdesign,^v(0),canbeimplementedbyabijection^sPbetweensomepartitionPoftoM:=fm1;m2;:::;mN0g,whereN0N.Thus,foreachi2,^s(i)=m(i),forsomeuniquem2M.Moreover,wecandeneaninversefunctionm�1(mj):=f:m()=mjg,withthepropertythatm�1(mj)\m�1(mk),8j;k2f1;2;:::;N0g,j6=kand[j2f1;:::;N0gm�1(mj)=.Undersuchastrategy,aReceiver'sposteriorbelief,conditionalonobservingamessagemj2Misavector(mj),wherethejthentryofisi(mj)=Pr�j2m�1()
S'spayofromexperiment(M;^sP)isXi2f1;:::;Ngi0v((mj()))Now,considertherepeatedcheaptalkgameandthefollowinglottery,P,whosesupportisf(mj)gj2f1;2;:::;N0g.UnderlotteryP,Pr(=(mj))=Xi2m�1(mj)i0LotteryPreplicatestheinduceddistributionofposteriorsunder^sP:therefore,itisclearlyfeasible,P2(0).Moreover,sinceeachiinducesoneandonlyonemessageunderP,v i(P)=v((mj())).Therefore,Xi0v i(P)=Xi2f1;:::;Ngi0v((mj()))Sincetheoptimalpayofrominformationdesignisanupperboundonthatunderrepeatedpersuasion,Pmustachievethemaximumvalueof(5).Finally,byProposition2thereexistsa 1suchthatwecanobtainthispayoasanequilibriumoftherepeated

47 gameforall 1-establishing
gameforall 1-establishingnecessity.(Onlyif)Supposethatthediscountedaveragepayofromoptimalsignaldesignoneach45 Receiver,^v(0),cannotbeobtainedbyanypartitionstrategy.Takeanyoptimalexperiment(M;s??)thatdoesachieve^v(0),anddenotethelotteryoverposteriorsinducedbytheexperimentby??2
(
).Letthesupportofthisdistributionbef??1;??2;:::;??N0g,andlettheprobabilityofposterior??junder??be??j.Then,theexpectedpayounderlottery??isXj2f1;2;:::;N0g??jv�??j
=Xii0 Xj??ji;??j i0v�??j
!wherei;??j:=Pr�ij=??j
istheithcomponentofvector??j,andPj??ji;??j i0=1.However,bydenitionofv i(??)wehavev i(??)Xj??ji;??j i0v�??j
(11)Wenowarguethat,generically,theremustexisti2f1;2;:::;Ngsuchthat(9)holdswithstrictinequality.Supposenot.Thenexperiments??mustinvolveapartitionofintoasetofsubsetsfP1;P2;:::;PN00gandcorrespondingpartitionsofMintofM1;:::;MN00gsuchthatPr(m2Mjj2Pk)8:&#x]TJ ;� -2;
.52;&#x Td ;&#x[000;0;ifj=k=0;otherwise:j;k2f1;2;:::;N00gandv((m))=v((m0))8m;m02Mj,j=1;2;:::;N00.Sincebyassumption,thissignalisnothonest,theremustex-istatleastonej2f1;2;:::;N00gandmessagesm;m02Mjsuchthatv((m))=v((m0))but(m)6=(m0).However,noticethatthesemessagesareonlyeversentinthesubsetofPj.Therefore,fors??tomaximizeE[v()]on(0),itmustalsobemaximizingE[v()]amongalllotterieson 2
Pj,subjecttotherestrictionthatXl l^^l=(Pj)(:=Pr(j2Pj))forsomebeliefs^^linthesupportof
Pj.39Refertothisfeasiblesetas�Pj((Pj)).However,byTheorem1,thisgenericallycannotbetrue:thereexistarbitrarilysmallperturbationsofvonthesubset
Pjsuchthatv((m))6=v((m0))attheoptimalvalueofE[v()]ontheset�Pj((Pj)). 39Notethat^^il=0fori=2Pj.46 Thus,genericallytheremustexistsomestatei2forwhichv i(??)Xj??ji;??j i0v�??j
andthereforeXi0v i(??)Xj2f1;2;:::;N0g??jv�??j
=^v(0)(12)Asimilarargumentestablishesthatthepayofromany

48 experimentinducingarbitrarylottery0
experimentinducingarbitrarylottery0attainsaweaklyhigherpayothanPi0v i(0)evaluatedat0.Therefore,Xi0v i(?)^v(0)where?solves(9).Theinequalityisgenericallystrictsinceeither:(i)?isnotanhoneststrategy(inwhichcase(12)holdsgenerically),or(ii)?ishonest,inwhichcasePj2f1;2;:::;N0g?jv�?j
^v(0)byourassumptionthat^v(0)cannotbeimplementedbyanhonestexperiment. ProofofTheorem3Proof.Werstprovetheresultfor?=^v(0).Deningthestatevariableas(t;ct)2[0;1],itiseasytoshowthattheoptimalexperimentundercommitmentcanbewrittenashonestwithrespectto(t;ct).TheTheoremthenfollowsasaCorollaryofTheorem2.Finally,foranyv (0)?^v(0),wherev (0)isSender'sworststagegamepayo,wecanachieve?withapublicrandomizationdevice,whichrandomizesbetweenplayoftheoptimalexperimentandthecheaptalkequilibrium.Forlargeenough,thiscontinuestobeanequilibriumsupportedbyGrimTriggertotheworststageequilibriumforever. ProofofTheorem4Proof.InarbitraryreviewphaseGj,denoterealizedsequencesofSenderreports,typesandReceiversignalsrespectivelyby~()G(j)+T�1jG(j)=~G(j);~G(j)+1;:::;~G(j)+T�1,()G(j)+T�1jG(j)=�G(j);G(j)+1;:::;G(j)+T�1
and(!)G(j)+T�1jG(j)=�!G(j);!G(j)+1;:::;!G(j)+T�1
.Similarly,inpunishmentphasesBjdene~()G(j)+(1+)TjG(j)+T,()G(j)+(1+)TjG(j)+T,(!)G(j)+(1+)TjG(j)+T.Duetotherecursivenatureoftheequilibriumweestablishbelow,thesameargumentswillholdacrossallGj,Bj,j=1;2;:::,respectively,andthereforeitissucienttoestablishargumentsforj=1.Toeasenotation,werefertotheabovesequencesas~()T�1G:=~()G(1)+T�1jG(1),()T�1G:=()G(1)+T�1jG(1)and(!)T�1G:=(!)G(1)+T�1jG(1)inphaseG0and47 ~()(1+)TB,()(1+)TB,(!)(1+)TBinB0.Forany(i;!j)2 ,wedenetheempiricalfre-quencyofthejointobservation�it;!jt
givensequences()T�1Gand(!)T�1GasFT(i;!j):=PT�1t=01(it;!jt) T.Similarly,given~T�1G,(!)T�1G,lettheempiricalfrequencyofjointobserva-tion~t=

49 i;!jtbe~FT~=i;!
i;!jtbe~FT~=i;!j:=PT�1t=01(~t=i;!jt) T.Finally,givensequences~T�1G,()T�1G,letb~=ijk,i;k2f1;2;:::;Ng,bethefrequencyofreports~=iwhenSobservesk,b~=ijk:=8:PT�1t=01(~t=i;kt) PT�1t=01(kt);ifPTt=11�k
�00;otherwise.ConsiderreviewphaseG0andxT,T�0.Werstshowthatthereexist^,^Tsuchthat(i)ifSenderadoptsatruth-tellingstrategywithinreviewperiodG0,hepassesthereviewwithprobabilityatleastPr\i;j2f1;2;:::;Ngn~FT~=i;!j�f(i;!j)o1� T;(ii)Pr�\i;j2f1;2;:::;NgfjFT(i)�i0jg
1� T;(iii)foranysequences~()T�1G,()T�1GsuchthatjFT(i)�i0j,i=1;2;:::;N,ifb~=kjiNk=1�ei�T,thenSfailsthereviewwithprobabilityatleast1� T,whereei=(0;0:::;1;:::;0)isanN1vectorwhoseithrowis1,andallothersare0.Toestablish(i),choose(T)= Ty, �0,0y1 2,forsupposethatSusesthetruthfulstrategy^t~()t�1;()t�1;t=t.Underthetruthfulstrategy,~;!isasequenceofindependentBernoullitrials.ByChebyshev'sinequality,Pr^FT^=i;!j�f�i;!j
(T)1� �2 4T1�2yLettingAij:=~;;!T�1G:^FT^=i;!j�f(i;!j)(T),wecanwritePr�\i;j2f1;2;:::;NgAij
1�Xi;j2f1;2;:::;NgPr� Aij
=1�N2 �2 4T1�2yChoosingT0,(T0)foranyT0 T,whereN2 �2 4 T1�2y T,establishespart(i).Similarly,claim(ii)canbeshowntofollowfromChebyshev'sinequalityforanyT0,(T0)suchthatT0 T.Toestablishclaim(iii),xT0 Tandsequences~()T�1G,()T�1GsuchthatFT�k
�k0(T0),8k2f1;2;:::;Ng.Considertheempiricalfrequency~FT0~=

50 8;i;!j,conditionalon~()T0�1
8;i;!j,conditionalon~()T0�1G,()T0�1G.Ateacht2f1;2;:::;T0g,theevent~t=i;!jtisaBernoullitrialwithprobabilityof48 success,conditionalon()T0�1G8:f�!jjk
;if~=i;t=k;k2f1;2;:::;Ng0;otherwise:Moreover,since(givent)!tisconditionallyindependentofall,!,6=t,wehaveasequenceofindependentBernoullitrials.Calculatingtheexpectationof~FT0~=i;!j,given~()T0�1G,()T0�1G,wehaveE~FT0~=i;!jj~;T0�1G=Xk2b~=ijkFT0�k
f�!jjk
Sincen~t=i;!jtoT0t=0isasequenceofindependentBernoullitrials,itfollowsfromCheby-shev'sinequalitythatPr~FT0~=i;!j�E~FT0~=i;!jj~;T�1G(T0)j~;T�1G1� �2 4(T0)1�2y(13)Now,sinceFT0�k
�k0(T0),andbydenition0Pk2b~=ijkf�!jjk
Pk2b~=ijkN,wecanboundE~FT0~=i;!jj~;T0�1G�Xk2b~=ijkf�k;!j
N(T0);whichfollowsafterrecallingthatf�k;!j
=i0f�!jjk
,8k.Usingthetriangleinequal-ityand(13),wecanboundPr~Ai;jj~;T0�1G1� �2 4(T0)1�2y,where~Aij:=(~;;!T0�1G:~FT0~=i;!j�Xk2b~=ijkf�k;!j
(N+1)(T0));ItiseasytoseethatthisimpliesPr[i;j2(1;2;:::;N)~Aijj~;T0�1G1�T0:Theevents~Aij,i;j2f1;2;:::;Ng,canbewrittenusingthesystemofinequalitiesF;!
bi�(N+1)(T0)
1~FT0F;!
bi+(N+1)(T0)
1(14)49 whereF;!=0BB@f(1;!1)


f�N;!1
.........f�

51 N;!1



f�N;!N&
N;!1



f�N;!N
1CCA,~FT0=0BB@~FT0(i;!1)...~FT0�i;!N
1CCA,bi=0BB@b(ij1)...b�ijN
1CCAand1=0BB@1...11CCA.Assumption2impliesthatF;!isinvertible.Thus,thelinearfunctionF;!:RN!RNiscontinuousandinjective.Itfollowsthatforany�0,wecanndsuchthatifjbi�eij�,thenjF;!
bi�F;!
eij=jF;!
bi�F;!
eij�,where!0as!0.Giventhis,wecanselectTsuchthat2(N+1)(T).Thus,foratleastonei2f1;2;:::;Ng,wehave�(N+1)(T)�0~FT~=i;!j�f�i;!j
Fi;!
bi�Fi;!
ei�~FT0~=i;!j�Fi;!
bi�(N+1)(T)(N+1)(T)ApplyingChebsyshev'sinequalityagain,wehavefoundaTsuchthatforanyjbi�eij�,thereisi2NsuchthatPr~FT0�i;!j
�f�i;!j
�(T)Pr~FT0�i;!j
�Fi;!
bi(N+1)(T)1�T N2Selecting^T=max T;T ,^=^Testablishesclaim(iii).Next,x^,^T,andsomeMediatorreportfunctionrsuchthatr(G;)=:!
,withsupport~2f1;:::;Ng,andr(B;)= :!
withsupport~2n 1;:::; No,where isSender'sreportingstrategyintheworstequilibriumofthestagegameinSectionXXXX.SupposefurtherthatReceiverposteriorbeliefssatisfy,forallt=0;1;:::;T�1,^(~t;P)=8:^i;ifP=G;~=i i;ifP=B;~=i50 where^2
.40LetQtdenotethesetofallpairsofsequencesqt=~()t�1;()t,andQ=[t2f0;1;:::;T�1gQt.Withsomenotationalabuse,let~QdenotethesetoffeasiblereportingplanswithinphaseG,~Q:=~t:t�1t!T�1t=0.Givenareportingplan~t(qt)andreportfunction,wewritetheinducedlotteryoverf1;:::;

52 Ng,conditionalon~t&#
Ng,conditionalon~t�1,()t�1asi(qt):=Pr~=ij~t�1;t�1.Associatedwiththislottery,denetheinducedlotteryoverReceiverposteriorsas^,where^i:=Pr^=^ij~t�1;t�1.NotethatinanyPBE,^2(0).Clearly,inanyBphase,truthfulreportingbySendercanbesustainedaspartofaPBE,sinceMediator'sreports(conditionalon~)andReceiverbeliefsarespeciedasthoseinthePBEoftheworststagegame.Forconvenience,wenormalizePNi=1 jv j=0.ConsiderSender'sbestresponseinreviewphase,G.Letting~'denotetheprobabilitythatSfailsthereviewattheendofperiodT�1whenfollowingreportingplan~Q,wecanwriteSender'spayosatthebeginningofphaseGrecursivelyasVG=max~Q2QT�1Xt=0tE"NXi=1^i~t�qt
v(^t;i)#+T(1�~')+~'TVG(15)(15)isastandarddynamicprogrammingproblemwithanitesetofstates-therefore,astationaryoptimalstrategyforSenderexists.Lettingthebestresponsefunctionbe~t= t,wecanequivalentlywrite(15)as41(1�)VG=(1�)PT�1t=0tEhPNi=1^i t(qt)v(^t;i)i 1�(1�~')T�~'(1+)T(16)or(1�)VG=(1�) 1�T"T�1Xt=0tE"NXi=1^i t�qt
v(^t;i)#�T�1�T
~'(1�)VG#(17)Wenowarguethat,forany' 2(0;1)and"�0and�0,thereexist , T,r, , T,suchthatifReceiverposteriorsgivenMediatormessage~isatisfyj^i�~ijforalli2f1;2;:::;Ng,then(i)Sender'soptimalstrategyinvolves~'' ,and(ii)(1�)VG^v(0)�",forall T.FixaMediatorreportfunctionrG;~t=0:!
,withsupportf01;:::;0Ngand 40Weshowbelowthatinequilibrium,ReceiverbeliefsareindeedinvarianttothecalendartimetoftheReceiveratdate,conditionalonreportphaseG.41Wesuppressexplicitdependenceof on^,^T,r,^,,fornotationalease.51 inducedlottery0,where0satises02X(??)forsomeoptimalexperimentundercommit-ment,??2(0),andPNi=1&#

53 21;0i(t)v(0i)�^v(0)

21;0i(t)v(0i)�^v(0)" 4.SuchareportfunctionexistsunderAssumption3.SupposefurtherthatReceiverposteriorbeliefs,^0(~),satisfyj^0i�~ij0,foralli2f1;2;:::;Ng,where0issmallenoughthatPNi=10i(t)v(0i)�Pj0i(t)v(^i)" 4forall^0i2N0(~i),i=1;:::;N.Supposethat(1�)VG^v(0)�3" 4forall1(veriedbelow).Takinglimitsofequation(17)as!1foranyreportingplan~t(qt)inphaseGthatinvolvesaprobabilityoffailingthereviewofatleast~',wecanwriteanupperboundSender'spayoaslim!1(1�)VG v�~'lim!1(1�)VG v�~'^v(0)�3" 4where v:=maxfv(i)gNi=1.Clearly,givena '�0,wecannda 2N�0suchthat v�  '�^v(0)�3" 4
^v(0)�3" 4-acontradictiontothelowerbound,(1�)VG^v(0)�3" 4.Thus,forany ,itcannotbeoptimal(inthelimit)forSendertouseareportingplan~t(qt)inphaseGthat~'� '.Wenowestablishthatsuchalowerboundcanbeimposed,given .Fixing �0,notethatwecanboundSender'slimitpayo,lim!1(1�)VG,as!1ineachG-phasebySender'slimitpayofromtruthfulreporting.Fromequation(16),lim!1(1�)VGT
EhPNi=1^i(t)v(^t;i)i T+T T^v(0)�" 2 1+TBychoosing Tsucientlyhigh,and = Twecanset Tsmallenoughthat^v(0)�" 2 1+T^v(0)�3" 4.Denotingby~VGthediscountedpayofromanarbitraryrecursivestrategyofusingreportingplan~t(qt)inphaseG,and~t=tinphaseB.Notethatareareonlynitelymanysuchchoicesofrecursivestrategy,becauseeachphaseshasnitelymanynodesandmessages.Given T,wecanthereforeselect T1suchthat(1�)~VG�lim(1�)~VG" 4uniformlyacrossallrecursivestrategies.Inparticular,sincetheoptimalstrategyisrecursive,thisimpliesthat(1�)VGv(^0)�",forall T-asrequired.GiventhatAiscountable,Assumption3ensuresthereexistsan&

54 #17;00�0suchthata?(^0i)=a?(~
#17;00�0suchthata?(^0i)=a?(~i),forallj^0i�~ij00.Thus,tocompletetheproofofexistenceofanequilibriumwithdesiredproperties,weneedonlyshowthereexists 'suchthatReceiverbeliefsareguaranteedtosatisfyj^0i�~ij00inequilibrium-Sender'sandReceivers'optimalstrategiesarethenmutualbestresponses,andbeliefsarecorrect.52 Forany '2(0;1),wehaveshownthereexist , T,r, , Tsuchthat~'' ,and(ii)(1�)VG^v(0)�",forall T.Additionally,foranyT? T,T?�0wecanchoose(T?),T?,whereT?=maxn T;^To,toensurethatsequences()T�1GsatisfyingjFT(i)�i0j(T?),foralli=1;2;:::;N,occurwithprobabilitygreaterthan1�T?inreviewperiodG(claim(i));conditionalonanysuchsequence,areportingstrategythatinvolvesjbi�eij�Tfailsthereviewwithprobabilityatleast1� T(claim(iii)).Combiningtheseobservations,wecanboundtheprobabilitythatSender'soptimalre-portingplan� t(qt)
T�1t=0involvesjbi�eij�atsomehistory� (qt)
T�1,()T�1,forsomei=1;2;:::;N,asfollows.BydenitionPr(jbi�eij�T?)=Pr�\iFT?()�i0? ;jbi�eij�T?
+Pr�[iFT?()�i0�? ;jbi�eij�T?
foranyi=1;2;:::;N.Forallsuchi,wecanboundeachtermontherighthand-sideoftheinequalityrespectivelybyPr�\iFT?()�i0? ;jbi�eij�T?
(1�T?)Pr(SfailsreviewatT?�1) 'andPr�[iFT?()�i0�? ;jbi�eij�T?
Pr�[iFT?()�i0�?
T?Thus,Pr(jbi�eij�T?) ' 1�T?+T?!0,T?!0asT?!1.LetT22f0;1;:::;T?�1gdenoteanarbitrarysubsetoff0;1;:::;T?�1gandforanypossiblehistory~()T�1G,()T�1GdenebT(i;j)=Pt2T1(~t=i)
1(t=j) Pt2T1(t=j).Similarly,letbTi:=�bT(i;j)
Nj=1.

55 Onanyhistory~()T�1G,()T�
Onanyhistory~()T�1G,()T�1GdeneTLasthesolutiontoTL():=maxT22f0;1;:::;T?�1gjTj(18)s.t.bTi�ei1 2wherejTjdenotesthecardinalityofsetT.Sinceproblem(18)isanitechoiceproblem,ithasawell-denedsolution.Itiseasytoshowarithmeticallythatforanyhistory~()T�1G,()T�1Gsuchthatjbi�eij,wehaveTL() T1 2.Now,considerReceiverRt'sinferenceproblem,givenobservations(G;^).GivenuniformpermutationofReceiversinanyphaseofthereviewmechanism,Rt'sbeliefsdonotdepend53 onherindext.Letting(jB�Ej):=~(qt);T�1G:jbi�eij;8i2f1;:::;Ng,wecanwriteposteriorbeliefsgivenmessage~asPr�~(t)=ijG;~
=Pr~(t)=i;jB�Ej1 2;~(t)=2TLjG;~(19)+Pr~(t)=i;jB�Ej1 2;~(t)2TLjG;~+Pr~(t)=i;jB�Ej�1 2jG;~ThenaltwotermscanbeboundedrespectivelybyPr~(t)=i;jB�Ej1 2;~(t)2TLjG;~PrjB�Ej1 2;~(t)2TLjG;~Pr~(t)2TLjjB�Ej1 2;G;~Pr~(t)=i;jB�Ej�1 2jG;~PrjB�Ej�1 2jG;~As,T!0,wehaveshownabovethat42Pr~(t)2TLjjB�Ej1 2;G;PrjB�Ej�1 2jG!0:FurtheritisstraightforwardtoshowthatPr�~~(t)jG
!~(~(t(qt)))as,T!0,wheret(qt)isthetruthfulreportingstrategy,and~(~)istheinducedfrequencyofmessage~whenMediatorusesmessagerule~~;G.Sincethereareonlynitelymanymessagessentin~(LemmaXXXX),itiswithoutlossthat~(~)�0forany~2supp(~).Thus,usingBayes'rule,PrjB�Ej�1 2jG;~=PrjB�Ej�1 2jGPr~jjB�Ej�1 2;G Pr(~jG)PrjB�Ej�1 2jG Pr(~jG)!0.Similarly,Pr~(t)2TLjjB�E

56 j1 2;G;~!0.Thus,as&#
j1 2;G;~!0.Thus,as,T!0,(19)impliesthatPr�~(t)=ijG;~
!Pr~(t)=ijG;~;jB�Ej1 2;~(t)=2TLPrjB�Ej1 2;~(t)=2TLjG;~Butforanytimeperiod~(t)inwhichPr�~(t)=~
�0,PrjB�Ej1 2;~(t)=2TLjG;~;~(t)!1sinceotherwise,1�PrjB�Ej1 2;~(t)=2TLjG;~wouldbeboundedbelowbysomePr�~(t)=~
x,x�0-acontradictiontolim;T!0PrjB�Ej1 2;~(t)=2TLjG;~= 42NoticethatPr~(t)2TLjjB�Ej1 2;G:=EhTL() TjjB�Ej1 2;Gi.54 1.Finally,byBayes'Rule,Pr~(t)=ijG;~;jB�Ej1 2;~(t)=2TL=Pr~(t)=ijG;jB�Ej1 2;~(t)=2TLPr~j~(t)=i;G;jB�Ej1 2;~(t)=2TL Pr~jG;jB�Ej1 2;~(t)=2TLUsinganidenticalargumenttothatabove,Pr~(t)=ijG;jB�Ej1 2;~(t)2TL!Pr�~(t)=i
.Furthermore,foranyi,!0impliesb^Ti!ei,for^T=f0;1;:::;T�1g=TL.Bycontinuity,lim;T!0Pr~j~(t)=i;G;jB�Ej1 2;~(t)=2TL=~(~ji)(20)where~(~ji)istheprobabilityMediatorsends~,giventruthfulreportingbySender.Finally,(20)impliesthatwecannd,Tsmallenoughthatj^�~j00,forany00&#x-278;0.SelectingT?,(T?)largeenoughensuressuch,Tcanbefound.Thiscanbesustainedasshownabovebyaboundonthediscountrate,T?forsomeT?1.55 AppendixB:DenitionofReviewMechanismsWeintroduceaclassofreviewmechanismsasfollows.ThesemechanismsareimplementedbyaMediator,whocommitsexantetoplayapre-speciedroleaswedescribebelow.LetG(j),B(j),j=1;2;3;:::bedenedinductivelyby:B(j)2f0;Tg(21)G(j+1)=G(j)+T+B(n)(22)foreachj2N�0,where2R�0,T2N�0,areparameterssuchthatT2N�

57 ;0,andG(1)=0(23)Wedenethe(T-period)
;0,andG(1)=0(23)Wedenethe(T-period)reviewphaseGjasthesetoftimeperiodsGj=fG(j);G(j)+1;:::;G(j)+T�1g:IfB(j)�0,thenwesaythatreviewphasejisfollowedbyanassociatedpunishmentphaseBj:=fG(j)+T;G(j)+T+1;:::;G(j)+T+B(j)g;otherwise(B(j)=0),reviewphasejispunishment-freeandwewriteBj:=f;g.Letthesetofhistoriesofphases(Gj;Bj)j=1satisfying(21)-(23),=1;2;:::,andt2G[BbeAt.Finally,letj:Gj!Gjbeanarbitrarypermutationfunction(bijection)onGjanddenotethesetofallsuchpermutationfunctionsby(Gj).TheMediatorisabletodesignconstraintsonthehistoryofplaythatReceiverscanobserve.Inparticular,atthebeginningofanyreviewphaseGj,theMediatorrandomlyselects~2(Gj)fromtheuniformdistribution,wherePr(j=~)=1 T!.Given~,theMediatorcommitstopermutetheorderingofReceiversRG(j);:::;RG(j)+T�1byassigningReceiverRtoplaythemechanismatperiod~(),for2Gj.Attime,heprivatelyinformsReceiverR~�1()ofonlythecurrentphaseofthemechanism,Gj.Receiversdonotlearntherealizationof~andareunabletoseetheirpositionintheline,~(),directly.43Similarly,theMediatoruniformlypermutestheorderingofReceiversinpunishmentphases. 43ItisconvenientbutnotcrucialforourmainresultsthatReceiversobservenothingaboutthemechanismoutsidetheirownphase.Itiscrucial,however,thatReceiverscannotinferanythingabouttheirpositionwithinthecurrentphasefromanyotherinformationtheymayhave.ForrealrecommendationplatformssuchaseBay,uncertaintyabouthow/whentheplatformupdatesitsreviewsand/orthefrequencyofcustomerinteractionhelptokeepbuyersuninformedinthisway.56 Ateachtimet2f0;1;:::g,thereviewmechanismasksSendertomakeaprivatereport,~t2,totheMediator.Ateachtimetatwhichheisaskedtoreport,theSenderobservesthehistoryofhisownsignals,()t=(1;:::;t),reports,~t�1=~1;:::;~t�1,andthehistoryofphasesannouncedbytheMediator,(Gj;Bj)j=1forsomesuchthatt2G[B.Givenareportof~tinperiodt,theMediatorthenprivatelysendsamessagem2MtoReceiverR~�1(t)accordingtoa(possiblyrandom)messagefunctionr:G[B!
M,wherewenoteinparticularthemessagefunctiondependsonwh

58 etherthemechanismisinareviewphaseorapuni
etherthemechanismisinareviewphaseorapunishmentphaseatperiodt.Finally,ReceiverR~�1(t)observesmessagemandchoosesanactionafromthesetA.Tobeclear,inareviewmechanismtheSender'sstrategyisacollectionof(possiblymixed)reportingfunctions~t:tt�1At�1Pt!
,fort=0;1;2;:::wherePt:=[:t2G[BG[B.Inotherwords,ateachtime,theSendercanconditionhisreportsontheentirehistoryofplayhehasobservedinthemechanism,includinghisownpasttypes,reportsandtheMediator'sannouncementsofpastandpresentphases.AstrategyforReceiverRtisafunction:PtM!A,inwhichshechoosesanactionasafunctionofthecurrentlyannouncedphaseandthemessagemsenttoherbytheMediator.Toclosethedescriptionofourreviewmechanisms,weneedtodenethecircumstancesunderwhichreviewphaseGjisfollowedbyapunishment,B(j)�0,forj=1;2;:::.Todothis,considersomereviewphaseGj.GivenanyhistoryofReceiversignals(!)G(j)+T�1=�!1;!2;:::;!G(j)+T�1
,reports~G(j)+T�1andany!k2 0,denethesubsequenceoftimeperiodsinwhich!kwasrealizedinreviewphaseGjas�!k;(!)t;Gj
:=:2Gjs:t:!t=!k ,andthecorrespondingempiricalfrequencyofreports,~i2,as:F~ij~G(j)+T�1;!k;Gj=P2(!k;Gj)1:~=~i j(!k;Gj)jWithsomenotationalabuse,wewilloftensuppressthedependenceofFon~G(j)+T�1,GjandsimplywriteF~ij!kwhereclear.GivenF,wecancalculate,foreachi2,!k2 ,thedierenced�i;!k
betweentheempiricalfrequencyandtheoreticalprobabilityofobservingigiven!k,asd�i;!k
=F�ij!k
�f�ij!k
ConditionalonbeinginreviewphaseGj,theMediatorperformsthefollowingtestsinperiodG(j)+T�1.Atthistime,theMediatorcalculatestherealizedvaluesofd�i;!k
57 foralli2,andall!k2 .TheSenderpassesthereviewphaseGjifd�i;!k
;foralli2,andall!k2 satisfying�!k;Gj
&#x-278; ,forsomexedparameter 2N�0,where�0isaxedparameterofthemechanism.Otherwise,hefailsthereview.IftheSenderpasses,th