Information transmission

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StrategicInformationTransmissionModelsPreliminaryLectureNotesHongbinCaiandXiWengDepartmentofAppliedEconomics,GuanghuaSchoolofManagementPekingUniversityOctober2011Contents1CheapTalk21.1ModelSetting...................................21.2AMotivatingExample..............................31.3ContinuousStateSpace.............................71.4OptimalCommunicationMechanism......................102DisclosureGames:VeriableTalk142.1AMotivatingExample..............................152.2SkepticismandUnraveling............................171 1CheapTalkIngametheory,cheaptalkiscommunicationbetweenplayerswhichdoesnotdirectlyaectthepayosofthegame.ThisisincontrasttoSpencer'ssignalingmodelinwhichsendingcertainmessagesmaybecostlyforthesenderdependingonthestateoftheworld.Theclassicexampleisofanexperttryingtoexplainthestateoftheworldtoanuninformeddecisionmaker.Thedecisionmaker,afterhearingthereportfromtheexpert,mustthenmakeadecisionwhichaectsthepayosofbothplayers.TheclassicalmodelofcheaptalkisintroducedbyCrawfordandSobel(1982).Realexamplesofcheaptalkinclude:1.Monetarymystique":Acentralbankisunwillingtomakeprecisestatementsaboutitspolicyobjectives.2.Securityanalystrecommendations.3.RatingAgency.1.1ModelSettingTherearetwoplayers,aSender(S)andaReceiver(R)ofinformation.Sholdssomeprivateinformationaboutapayo-relevantstate.ThetimingofthegameisspeciedinFigure1.2 012SprivatelySsendsaRtakesanactionobservesmessagemאMaאAthestateoftoRtheworldsאSFigure1:TimelineoftheCheapTalkGamePayos:PayosareUS(a;s)andUR(a;s).Inparticular,wewillusequadraticutilityfunctions:S2R2U(a;s)=[a(s+b)]andU(a;s)=[as];whereb>0measureshownearlytheS'sinterestscoincidewiththeR's.Noticethatthesignalmisirrelevanttothepayofunctions(talkischeap").AlsothemessagespaceMisindependentofthestates.Givenstates,thesender'smostlypreferredactionisaS(s)=b+sandthereceiver'smostlypreferredactionisaR(s)=s.BothaSandaRareincreasingins:bothplayers'interestsarealigned.However,therearecon ictsaswellsinceaS>aR.1.2AMotivatingExampleStatespaceS=f0;1g,messagespaceM=f0;1g,actionspaceA=[0;1].Thepriorissuchthateachstatehappenswithequalprobability(1).23 Figure2:IllustrationoftheUtilityFunctions4 Strategies:Astrategyisaplanofactioncoveringeverycontingencythatmightarise.ForS,astrategyisafunctionfromtypestoactions.Letq(mjs)betheprobabilitythatSsendsmessagemwhenthetruestateiss.ForR,astrategymustspecifyanactiona(m)2Aforeachmessagem2M.BeliefsareupdatedbyBayes'rule.IfRconjecturesthatSchoosesmaccordingtothestrategyq(mjs),thenafterreceivingmessage,R'sposteriorbeliefaboutstatesisderivedfromBayes'rule:1q(mjs)p(sjm)=2:(1)1(q(mjs)+q(mj1s))2Denitionofequilibrium:Denition1Thestrategiesfq(mjs);a(m)gformaperfectBayesianequilibriumif:P1.foreachs2f0;1g,q(mjs)=1andifm?2Misinthesupportofq(
js),thenm=0;1?Sm2argmaxmU(a(m);s);2.foreachm2M,XRa(m)2argmaxaU(a;s)p(sjm);swherep(
jm)isgivenbyequation1ifq(mj0)+q(mj1)>0.Dependingonthevaluesofb,therearedierenttypesofequilibria.1.Ifb2(1;1),therearemultipleequilibria:425 (a)(babblingequilibrium)Inababblingequilibrium,noinformationisconveyedfromthesendertothereceiver.Therearemanywaystoconstructababblingequilib-rium:1)q(mjs)=1;8m;sanda(m)=1;8mor2)q(1js)=1;8sanda(1)=1,222a(0)=0;(b)(fullyseparatingequilibrium)q(mjs)=1ifm=s,=0otherwiseanda(m)=m;8m;(c)(partialseparatingequilibrium)q(1j1)=1;q(1j0)=12banda(0)=0;a(1)=2b.2bComparisonoftheexpectedutilities:FortheR:fullyseparating(0)>partiallyseparating(1(12b))>babbling(1):24FortheS:fullyseparating(b2)>partiallyseparating(1(1b)21b2)22>babbling(1(1b)21(1+b)2):2222Whichequilibriumisrealizedcruciallydependsonthereceiver'sbeliefs(e.g.,there-ceiverthinkshowtrustworthythesenderis).Thisisalsointerpretedassocialnorms.2.Ifb1,onlybabblingandfullyseparatingcanbeequilibrium.43.Ifb>1,theunique"equilibriumisthebabblingequilibrium.26 Remark1Fromthissimpleexample,wecanseethattherearethreekindsofindetermi-nacyincheap-talkmodels:multipleo-the-pathresponses,multiplemeaningsofmessages,andmultipleequilibriumassociationsbetweentypesandactions.Therstindeterminacyistheresultofdierentpossiblespecicationsofbehaviorothepathofequilibrium(i.e.,thereceivermayhavedierentresponseswhenthesendersendsamessagewhichisNOTsupposedtoappearinequilibrium).Thesecondkindofindeterminacy,multiplemeaningsofmessages,arisesinanynon-babblingoutcome.Onecantakeanyequilibriumoutcomeandformanewequilibriumoutcomebypermutingthemessages.Usually,wedonotcareabouttheactualchoiceofmessagesbutonlyaboutthepayo-relevantassociationofactionstotypes.Therefore,thiskindofmultiplicityofequilibriumisnotaproblem.Thethirdkindofindeterminacyisthefocusofourattention.Sincebabblingequilibriaalwaysexist,iftherealsoexistsanequilibriumwithmeaningfulcommunication,theremustbemultipleequilibriumtype-actiondistributions.1.3ContinuousStateSpaceStatespaceS=[0;1],actionspaceA=[0;1],messagespaceMisnitebuthasasucientlylargenumberofelements.1ThepriorissuchthatsU[0;1].Similarly,letq(mjs)betheprobabilitythatSsendsmessagemwhenthetruestateissanda(m)2AbetheactionchosenbytheRfacingmessagem2M.BeliefisupdatedbyBayes'rule:1ThenitenessofthemessagespaceisNOTanimportantassumption.TheresultdoesnotchangeifweassumeM=[0;1].7 q(mjs)f(s)q(mjs)f(sjm)=R=R:(2)11q(mjt)f(t)dtq(mjt)dt00Denition2Thestrategiesfq(mjs);a(m)gformaperfectBayesianequilibriumif:P1.foreachs2[0;1],q(mjs)=1andifm?2Misinthesupportofq(
js),thenm2M?Sm2argmaxmU(a(m);s);2.foreachm2M,Z1Ra(m)2argmaxaU(a;s)f(sjm)ds;0wheref(
jm)isgivenbyequation2ifm2Missentwithstrictlypositiveprobability.Obviously,therealwaysexistsababblingequilibrium:a(m)=1;8mandSsendseach2messagewithequalprobabilityq(mjs)=1.WeareinterestedinaninformativeequilibriumjMjwhereatleasttwodierentactionsatinducesaandnot0tsendsmessagem=1andt0s+b.Sincey(
)iscontinuousandstrictlyincreasing,thereexistss0>>>>>s1+bifss1;>>>s+bifs1>>>>>:s2+bifss2:Thereceiverchoosess10ands22[s1;1]tomaximize:Zs1Zs2Z1222W(s1;s2)=(s1+bs)dsbds(s2+bs)ds:0s1s2ItisshownthatZs1@W=2(s1+bs)ds<0;@s1012 Figure3:OptimalCommunicationMechanismwhichimpliesthats?=0.Similarly,1Z1@W=2(s2+bs)ds:@s2s2@W=0ats=s?=12bors=1.However,thesecondderivativeisnegativeats?but@s22222positiveat1.Therefore,Wismaximizedats?=12b.2Theorem2Theoptimalcommunicationmechanismtakesthefollowingform:8>>>>>:1bifs12b:Theoptimalcommunicationmechanismcanbeimplementedbydelegation.Thereceiver13 Figure4:Cheaptalkequilibriumwithb=110delegatestheactionchoicetothesenderwitharestrictiononthesetofactions.Thesenderisfreetochoosehismostlypreferredactionaslongasy1b.Sometimes,delegationissubjecttocommitmentproblem:exante,thereceivercannotcommittoletthesenderchoosehismostlypreferredactionexpost.Inthissituation,theoptimalcommunicationmechanismcanalsobeimplementedbyarbitration.Underarbitration,thesendercommu-nicatesprivatelyorpubliclywithaneutraltrustworthyarbitrator(e.g.,theUnitedNations),whothenchoosesanaldecision.Weassumethearbitratorhasthesamepreferenceasthereceiverbuthasthecommitmentpowertoenforceabindingdecision.Comparisontothecheaptalkequilibria(b=1):10Whenthesender'smostlypreferredactionisalwayschosen(y(s)=s+b),EUR=b2=1>37.Whentheoptimalcommunicationisimplemented,EUR=4b3b2>1.100120031002DisclosureGames:VeriableTalkInthecheaptalkgame,thesenderisfreetosendanyunveriablesignal.However,inthereality,informationcanbeveriable"eitherbecausethereceiverscandirectlycheckitsaccuracyorbecausethereareinstitutionsinplacethateectivelydeterfalseclaimsbythesenders.Inotherwords,themessagespaceMdependsonthestates.Itturnsoutthatsomeresultsinthecheaptalkgamesdonotholdanylonger.14 2.1AMotivatingExampleStatespaceS=f0;1g,messagespaceM=f0;1g,actionspaceA=[0;1].Thepriorissuchthateachstatehappenswithequalprobability(1).2PayosareUS(a;s)andUR(a;s).Inparticular,wewillusequadraticutilityfunctions:S2R2U(a;s)=[a(s+b)]andU(a;s)=[as];whereb>0measureshownearlytheS'sinterestscoincidewiththeR's.Inthecheaptalkgame,therealwaysexistsababblingequilibriumandifb>1,the2unique"equilibriumisthebabblingequilibrium.Disclosuregame:WeallowthesendertosendasubsetmofM.Themessagemhastobeveriableinthesensethatitincludesthetruestates.Forexample,ifthetruestateis0,therearetwopossiblemessagestobesent:f0gandf0;1g.Ifthetruestateis1,therearealsotwopossiblemessagestobesent:f1gandf0;1g.ForS,astrategyq(m=f0;1gjs)istheprobabilitythatSsendsmessagem=f0;1gwhenthetruestateiss.ForR,astrategymustspecifyanactiona(m)2AforeachmessagemM.Also,weuse(sjm)todenotethereceiver'sbeliefthatthestateisswhenhereceivesmessagem.Denition3Atriplefq;;agisasequentialequilibriumif:1.ForeverypossiblemessagemM,a(m)maximizesE[(sa)2j(
jm)];2.Foreverys2S,ifmisinthesupportofq(
js),thenmmaximizes(sa(m)b)2;15 3.Beliefsatisesthestructuralconsistencyrequirement:Supp(
jm)m,andisupdatedbyBayes'rule.Noticethatthebabblingstrategies(thesendersendsf0;1girrespectiveofthestateandthereceiveralwayschooses1)isstillaNashequilibrium.ButitisNOTasequentialequilib-2rium.Nashequilibriumonlyrequiresthatformessagesappearedontheequilibriumpath,thereceiverchoosestheoptimalactionbasedonbelief.However,sequentialequilibriumrequiresthatforanypossiblemessage,thereceiverisobligedtoformaconjectureconsistentwiththatmessage,andbasehischoiceuponthatconjecture.Sincem=f1gcanbeonlysentatstate1,(1jm=f1g)=1andhencea(f1g)=1.Asaresult,thebabblingstrategiesisnotasequentialequilibriumbecauseatstate1,thesenderalwayswouldliketodeviatetosendmessagef1g.Wesayastrategyqisfullrevealingifthesupportsats=0ands=1havenointersection(q(m=f0;1gjs=0)q(m=f0;1gjs=1)=0).Proposition1Foranyb>0,ateverysequentialequilibrium,thesenderusesfull-revealingstrategy.Proof.Inasequentialequilibrium,(1jm=f1g)=1andthereforea(f1g)=1forsure.Ifq(m=f0;1gjs=0)q(m=f0;1gjs=1)>0,a(f0;1g)<1andthesenderisindierentbetweensendingf1gandf0;1gatstate1.Thisleadstoacontradiction.Inparticular,ifb>1,thesenderwillalwayssendf1gatstate1andthebeliefat2m=f0;1gisskeptical:(s=0jf0;1g)=1.16 2.2SkepticismandUnravelingThefollowinganalysisisbasedonMilgrom(1981)andMilgrom(2008).Therearetwoplayers:aninformedsellerandanuninformedbuyer.Thesellerhasprivateinformationaboutthestateoftheworld,whichbelongstoanitesetf1;


;Ng.Highervaluesofrepresentbetterquality.Theseller'sonlymoveinthegameistomakeareportabouttotheotherwiseuninformedbuyer,whothenmakesapurchasedecision.Thereporthastobeveriableinthesensethat2m().Thebuyerchoosesthequantityofpurchaseqbasedonthereport.Weassumethattheseller'spayoisgivenbyv(q)wherevisstrictlyincreasinginqandthebuyer'spayoisgivenbyu(;q),whereuisstrictlyconcaveanddierentiableinq.Forexample,wemayassumethatv(q)=Pqandu(;q)=u(q)Pq.Thereisauniqueinteriorsolutionq?()maximizingu(;q)andq?()isstrictlyincreasingin.Thisissatisedwhenu(;q)=u(q)PqifweimposetheInadaconditiononu(
).Theseller'sreportingstrategyisdenotedas(mj)2[0;1]wheremf1;


;Ngandthebuyer'spurchasingstrategyisq(m).Thebelieffunctionis(jm)foreach2m.Theequilibrium(m;q;)isafully-revealingequilibriumif00(mj)>0=)(mj)=0;86=:Wewillfocusonsequentialequilibriumwhere:1.foreveryminthesupportof(
j),mmaximizesq(m);17 2.foreverymf1;


;Ng,q(m)2argmaxqE[u(;q)j(
jm);3.beliefsatisesthestructuralconsistencyrequirement:Supp(
jm)m,andisupdatedbyBayes'rulewheneverthatapplies.DenoteL(m)=minfj2mg.Thebelieffunctionisskepticalif(L(m)jm)=1foreverynon-emptym.Theorem3Everysequentialequilibriummustbefullyrevealingandthebeliefineverysequentialequilibriumisskeptical.Theargumentusedtoprovetheabovestatementiscommonlycalledtheunravelingar-gument.Theusualpresentationrstshowsthatthehighestqualitysellersalwaysmakereportsofqualitythatdistinguishtheirproductsfromallothers,andthentheremainingsellersfaceasimilargame.Thenexthighestqualitysellersthereforereportqualitylevelstodistinguishthemselvesfromlower-qualitytypes,andtheprocessrepeatsitself.Forthisargumenttowork,itmustbecommonknowledgethatasellercandistinguishitsproductfromlower-qualityproductsandsellersmustbenetbydoingso.Thelatterconditionisguaranteedsinceq?()isstrictlyincreasingin.Giventheequilibriumisfullyrevealing,itisstraightforwardtoseethatthebeliefisskeptical.Ifnot,thereexistssomemsuchthatE(jm)>L(m).Thenasellerwith=L(m)hasincentivetoreportmbutthisequilibriumisnotfullyrevealing.18 ReferencesCrawford,V.,andJ.Sobel(1982):StrategicInformationTransmission,"Economet-rica,52(6),1431{1451.Golstsman,M.,J.Horner,G.Pavlov,•andF.Squintani(2009):Mediation,arbi-trationandnegotiation,"JournalofEconomicTheory,144(4),1397{1420.Holmstrom,B.•(1984):OntheTheoryofDelegation,"inBayesianModelsinEconomicTheory.ElsevierSciencePublishersB.V.,Amsterdam.Kovac,E.,andT.Mylovanov(2009):Stochasticmechanismsinsettingswithoutmonetarytransfers:Theregularcase,"JournalofEconomicTheory,144,1373{1395.Melumad,N.,andT.Shibano(1991):CommunicationinSettingswithNoTransfers,"TheRANDJournalofEconomics,22,173{198.Milgrom,P.(1981):GoodNewsandBadNews:RepresentationTheoremsandApplica-tions,"BellJournalofEconomics,12,380{391.(2008):WhattheSellerWon'tTellYou:PersuasionandDisclosureinMarkets,"JournalofEconomicPerspectives,22,115{132.19