DurableCheapTalkEquilibriaSandeepBaligaMEDSDepartmentKelloggSchoolofMa - PDF document

1 DurableCheapTalkEquilibriaSandeepBaligaM
DurableCheapTalkEquilibriaSandeepBaligaM.E.D.S.Department,KelloggSchoolofManagementTomasSjöströmDepartmentofEconomics,RutgersUniversityJuly,2006Abstract WearegratefultoMattJacksonforencouragingustoworkonthisproblemandtoStephenMorrisforcomments. receiverallofthepowertorenegotiate.ItisalsointhespiritofDewatripontandMaskin[4],LaontandMartimort[9]andMaskinandTirole[10]whotaketheviewthatarenegotiationorcollusionoermadebyanuninformedpartyinthepresenceofasymmetricinformationdoesnotsuerfromthesignallingproblemofanoerfromaninformedparty.OurrecursivenotionisalsointhespiritofBernheim,PelegandWhinston’s[2]recursiveequilibriumconceptforgameswithcompleteinformation.Intheirtermi-nology,aNashequilibriumiscoalition-proofifnocoalitionofplayerscandeviateandbebetteroandthereisnosub-coalitionthatcaninturnprotablede-viatefromthedeviation.Also,anypotentialdeviationbythesub-coalitionmustbejudgedbythesamecriterionandsoon.Inourcontext,acheap-talkequilibriumconceptisdurableifnocoalitionof“types”

2 hasacredibleobjectionandthereisnosub-coa
hasacredibleobjectionandthereisnosub-coalitionoftypesthatcancrediblyobjecttotheobjection.Moreover,anycredibleobjectionbythesub-coalitionoftypesmustbejudgedbythesamecriterion,andsoon.InSection4,weformallyconsiderasequentialcheap-talkgamewherethesenderkeepssendingnewmessageseachperioduntilanactionistaken.Incontrasttotheone-shotgame,thesendermayrevealinformationslowlyovertime,andthenitionofdurabilitymustbemodied.Ifthereceiveronlylistenstoneologismsoftheform“mytypeisintheset”thenwemustallowmulti-stageobjectionsofthefollowingform.Onday1thereceiverclaimstobeinset,saya,bThereceiveranticipatesthatmorepreciseinformationwillbegivenonday2;onday2typesays“Iamandtypesays“Iam”(Ofcourse,therelevantincentivecompatibilityconstraintsmustbesatised).Thisleadstoadenitionofsequentialneologismproofequilibria.However,thepossibilityofrevealinginformationslowlyovertimeisspuriousifamulti-stagecredibleneologismcouldbecollapsedtoasinglemorecomplicatedannouncementonday1.Ratherthansaying“Iaminseta,b”

3 onday1,andthen“Iam”onday2,thesendercould
onday1,andthen“Iam”onday2,thesendercouldsayallatonce:“Iaminseta,bifIweretypeIwouldsayso,andifIweretypeIwouldsayso;actuallyIam”Thereceiverwhoissophisticatedenoughtounderstandthemultistagespeechesshouldbeabletounderstandthissingle(butmorecomplex)message.SuchcomplexmessagesarecalledannouncementsMatthews,Okuno-FujiwaraandPostlewaite[11].Consequently,thedenitionofsequentialneologismproofequilibriaisformallyequivalenttothedenitionofdurableannouncementproofequilibria.Onecanthinkofannouncementsasthecollapsingofasequenceofneologismsintoonemessage.Neologismshavetheadvantageofbeingsimplerspeeches.However,sequentialneologismsrequiresophisticatedforecastingabilityonthepartofthereceiver,whiledurableannouncementsdonot.nalpointshouldbemade.Farrell’soriginalargument(seeFarrellandRabin[6])wasthatreceiversshouldtakethemeaningofamessageinthenaturallanguageasastartingpoint,andthenask“whywouldhewantmetothinkthat?”.Inthisspirit,weassumethatifthesenderonMondayclaimshistypeiseither Example1Rec

4 eiverActionActionActionActionActionType2
eiverActionActionActionActionActionType2,50,00,00,03,14,2SenderType0,02,50,00,03,14,2Type0,00,02,50,03,10,-8Type0,00,00,02,50,08,-8Suppose)=1foralla,b,c,d.Thissender-receivergamehastwoperfectBayesianequilibrium(PBE)outcomes.IntherstPBEoutcome,typesa,bandpoolandreceivewhiletypereceivesThus,Thisoutcomeisneologism-proof.For,adeviationcanonlybeprotabletothesenderifitresultsintheactionHowever,thiswouldattracttypeandthenwouldnotbeanoptimalresponseforthereceiver.Ontheotherhand,ifthereceivercannotcommittostoptalking,thenthisequilibriumisnotdurablesequentiallycredible.Forinthisequilibrium,themessageconvincesthereceiverthatthetruestateisintheseta,b,c,eachstateinthissetbeingequallylikely.Supposeratherthantakingtheactionfollowingmessagethereceivergivesthesenderonemorechancetospeak(hecannotcommitnottodothis).Whatfollowsisacheaptalkgamewherethe-rowhasbeendeleted.Completepoolingofthethreetypesa,bandisimplausibleherebecause,usingthelogicofFGP,ifthesender’struetypeisthenheshouldsay“Iam

5 ”andfollowingthisspeechthereceivershould
”andfollowingthisspeechthereceivershouldassignprobabilityonehalfeachtoandandtakeactionTypedoesnotwanttomakethisspeechbecauseactionisbadforhim.Typelikesactionbutthistypehasbeenruledoutbyhypothesis:messagehasalreadyconvincedthereceiverthatthetypeisnot,and(wesuppose)thefactthatthereceiverhimselfreturnsformoreinformationdoesnotchangehisbeliefthatthesenderisnot.Moreover,afterthereceiverhasruledoutbothtypesandbutthinkstypesandareequallylikely,thereisnowaytoreturnyetanothertimetoseparatefrom:neithernorhasanyincentivetoconfesstohistruetype(becausebynotconfessingheexpectsTherefore,thereceiverwillbeabletoobtainmoreinformationbyfurtherinterrogationfollowingmessagesothisequilibriumisnotdurableneologism-prooforasequentialneologism-proof.But,isthesender’sspeechacrediblesequential InExamples1and2,thesetofneologismproofequilibriaisidenticaltothesetofequilibriathatareannouncement-proofinthesenseofMatthews,Okuno-FujiwaraandPostlewaite[11].Forsimplicity,wecouchthediscussionintermsofneolo

6 gismsratherthanannouncements. aftersomes
gismsratherthanannouncements. aftersomesequenceofmessages,thereceiverbecomesconvincedthatthetypesbelonga,b,cConsiderthetimewhenhe(accordingtohisequilibriumstrategy)isabouttotaketheactionNowsupposehedeviatesfromhisequilibriumstrategyandasksforanothermessage.Afterthisdeviationbyhimselfheremainsconvinced(wesuppose)thatthesender’stypeisnotThen,ifthesendersays“Iamitshouldbebelieved,sopoolingofthetypesa,bandisnotaplausibleoutcomeofthesequentialgame.Asimilarargumentappliestotheanalysisoftheobjections.Againstthesecond(fullyseparating)equilibrium,“Iama,b”shouldnotbeabelievableobjection,becausethisspeechconvincesthereceiverthatthetruetypeisina,b,cthenhecanextractfurtherinformation,whichleadstoacontradiction.ItisimportantfortheanalysisofExample1thatoncethereceiverisconvincedthatthetruetypeisina,bhecannotcomebackyetagaintoseparatefromsomepoint,furtherinterrogationbecomesimpossible,andthisisthepoint.Durableneologism-proofnessisdenedrecursively,usingthestabilityorinstabilityofsmalle

7 rsetsofpoolingtypestocheckthestabilityor
rsetsofpoolingtypestocheckthestabilityorinstabilityoflargersets.Therecursionisstartedobynoticingthataspeechwhichrevealstypescompletelycannotbefurtherdestabilized,sodurabilityistriviallysatisExample1showsthatourequilibriumconceptcanbeneitherweakernorstrongerthanneologism-proofness(orannouncement-proofness,cf.footnote3).Italsoshowsthatthethreattoreturnformoreinformationcanmakethereceiverbettero.ThatthisisnotalwaysthecaseisseeninExample2.Example2ReceiverActionActionActionType4,50,01,32,4SenderType0,04,51,32,4Type3,-33,-31,3-1,-1Suppose.ThissenderreceivergamehastwoPBEoutcomes.IntherstPBEoutcome,and.Thisequilibriumoutcomeisneologism-proof.Anydeviationbythesenderthatcausesthereceivertorespondwitheitherwouldattracttypebutthenneitherwouldbeanoptimalresponse.However,thisoutcomeisnotdurable.Forsupposethereceiverisabouttochooseconvincedthatthesenderiseither(butnot).Ifhedoesnottakeanyaction,butinsteadtellsthesender“nowIknowyouareeitherwhichoneisit?”,thentypeshouldcertainlysayandty

8 peshouldcertainlysaybecauseoncethe-rowis
peshouldcertainlysaybecauseoncethe-rowisdeletedfromthematrixanyreasonablecriterionpredictsseparationoffrom.Andofcourse,onceandhavebeenseparated,thereisnomoreinformationtobeobtained.Thisdestroystheoriginalequilibrium.Thepartiallypoolingoutcomeisnotdurable.Moreover,completeseparationisalsonotaneologism-proofoutcomeastypewouldthemimitatetype(ortypeTherealsoexistsatotallyuninformativepoolingequilibriumwhereandthereceiverrespondswithItisnotneologism-proof.For nextday”toobtainmoreinformation.Ofcourse,whetherornotcompletepoolinginisareasonableoutcomeofacheaptalkgameisitselfsomethingtobetestedusingourcriterion.Henceourdenitionisrecursive.Thesecondwayinwhichourdenitiondiersisthatitwillnotbeenoughforanobjectionofsizegreaterthanonetobecredibletobebelieved,ifitcanbefollowedbyfurtherinformationtransmission.Ifthereceiverbecomesconvincedbytheobjectionthatthesender’struetypebelongstobutcompletepoolingisnotareasonableoutcomeofthecheap-talkgamewhentypesarerestrictedtothesetthenthereceiv

9 ershouldbeabletoobtainmoreinformation.Th
ershouldbeabletoobtainmoreinformation.ThesetwodierencesareillustratedinExample1.Itcannotbeanequilibriumoutcomefora,b,ctopool,becausethereceiverwillbeabletogetmoreinformation.Forthesamereason,attheseparatingequilibriumthereceivercannotbelievetheobjection“Iama,b”Inbothcases,poolingofthetypesa,bandisnotdurable,becauseoncethereceiverisconvincedthestateisina,b,checanpressontodistinguishandfrom(butafterthistherecanbenomoreinformationtransmissionbecausepoolingofanddurable).Wenowgivetheformalrecursivedenitionofdurability.Acredibleneologismofsizeisalwaysdurable.SupposedurabilityhasbeendenedforcredibleneologismsofsizeatmostIfacredibleneologismisofsizethenitisdurableifthereisnocredibleanddurableneologismwithrespecttoHeremeansthepartitionofthathasonlyoneelement,thewholeofNoticethataneologismwithrespecttowouldbeofsizeatmostThiswaydurabilityisdeforneologismsofanysize.nition1AperfectBayesianequilibriumisadurableneologism-proofequilibriumif:(i)thereisnocredibleanddurableneologismwithr

10 especttoand(ii)forallthereisnocrediblean
especttoand(ii)forallthereisnocredibleanddurableneologismwithrespecttonition2Apartitionisadurableneologism-proofpartitionifthereexistsadurableneologism-proofequilibriumsuchthatConsiderExample2.Observethattheneologismiscrediblewithrespecttowherea,bThisisbecauseandandA,aD,awhileA,bD,bItistriviallyalsodurable.NowtherstPBEgeneratestheinformationpartition,wherea,band.Thereisnocredibleanddurableneologismwithrespectto,becauseanydeviationthatresultsinattracts.Thus,condition(i)ofDenition1issatised.Butcondition(ii)isviolated:thereexistsacredibleanddurableneologismwithrespecttoa,bThus,thisPBEisnotdurableneologismproof.InthesecondPBE,whichgeneratesthecompletelypoolinginformationpartitiona,b,cHereconditions(i)and(ii)bothamounttocheckingforobjectionswithrespecttoThereis Bydenition,acredibleannouncementofsizeisalwaysdurable.SupposedurabilityhasbeendenedforcredibleannouncementsofsizeatmostIfacredibleobjectionisofsizethenitisdurableifforallthereisnocredibleanddurableannouncementwithres

11 pectto(Anysuchannouncementwouldbeofsizea
pectto(Anysuchannouncementwouldbeofsizeatmost)Thiswaydurabilityisdeforannouncementsofanysize.nition3AperfectBayesianequilibriumisadurableannouncement-proofequilibriumif:(i)thereisnocredibleanddurableannouncementwithrespectand(ii)forallthereisnocredibleanddurableannouncementwithrespectto(4SequentialCheapTalkConsiderasequentialcheaptalkgameofthefollowingform.Thereareaninnitenumberof“days,”,....Eachdayisdividedintotwosubperiodscalledmorningandevening.Inthemorningofeachday,thesendersendsamessagetothereceiver.The(timeinvariant)messagespaceisagainsucientlyrichtoatleastincludeallthesubsetsofand“neologisms”tousetodeviatefromanymessageprole.Also,thereisa“null”messagewiththeinterpretationofnotsayinganything.Intheeveningofday,thereceivercaneitherdonothing()ortakeanactionWhenanactionistaken,thegameends,andthepayosareforthesenderandforthereceiver.Ifinsteadthesendersendsanewmessageinthenextperiod.Thiscontinuesuntilthereceiverhastakenanactionin.Thereisnodiscounting.bethesender’sstrat

12 egy.Thestrategyspecies:foreachtimeandeac
egy.Thestrategyspecies:foreachtimeandeachstringofmessages,...,mifthereceiverhasnottakenanactionbeforetimewhichmessagetosendattimeWeassumeforallandsothereceiverisbetterotakinganactionatsomepointratherthanpostponingitindenitely,whichweassumegiveshimzero.Giventhesender’sstrategy,thereceiver’sdecisionproblemissimple.Thereceiveroughttowaituntilthesenderhasrevealedalltheinformationhewilleverreveal,thentaketheoptimalaction.Giventhatthereisanitenumberoftypes,ifthesenderplaysaccordingtotherewillalwaysexistanitetimesuchthataftertimenomoreinformationwillberevealed.Thisgeneratesapartitionofdenoted,asfollows:belongtothesameelementofthepartition(say)ifandonlyifwhenthesenderplaysaccordingtothentypesandsendthesamestringofmessagesfromday1todayInotherwords,ifthesenderplaysthenthereceiverwillnotbeabletodistinguishbetween,evenifhewaitsuntiltime(thereisnoneedtowaitlonger).Letbethatsetinthepartitionwhichcontains,i.e. chooseactionifthesendersaidhewastypethatmorning,andotherwise.Thismakesboth

13 typesandbettero,ascomparedtotheequilibri
typesandbettero,ascomparedtotheequilibriumallocationthatwouldhaveresultedfrom.Moreover,thereisnoincentivefortypetopretendtobetypeorvice-versa.Finally,typehasnoincentivetomimicthedeviationasheprefersandThus,ifthetruestateisthenthesendercancrediblycommunicatehistypebysequentiallyannouncingsubsetsofstates.Noticethatcommunicationnecessarilyisslow(takestwodays).Iftheonlybelievablespeechesareannouncementsofsetsoftypes,thenbyannouncingsuccessivelysmallersetsoftypes,thesendercancrediblytransmitmoreinformationthanhecouldinaone-shotgame.Formally,letandlet,...,Tbeapartitionofneologismwithrespectto(isamessageconsistingofasubsetoftypes.Letdenotethesizeofthentheneologismissequentiallycredibleandonlyifitiscredible.Supposesequentiallycredibleneologismsofsizehavebeendened,andletbeofsizesequentiallycredibleifthereexistsapartition,..,Ssuchthatthefollowingconditionsholdforall(1)Ifthen(2)Ifthen(3)Ifthenforall(4)ThereisnosequentiallycredibleneologismwithrespecttoanyThisway,wedenesequentiall

14 ycredibleneologismsofanysize.Asequential
ycredibleneologismsofanysize.Asequentialneologism-proofequilibriumisaperfectBayesianequilibriumofthesequentialcheaptalkgamesuchthatthereisnosequentiallycredibleneologismwithrespectto,whereisthepartitioningofwhichisinducedbyAsequentiallycredibleneologismcanbethoughtofasamulti-stagespeechwherethesenderrstannounces,anduponhearingtheforwardlookingreceiverdoesnottakeanyaction.Thenonthenextdaythesenderannouncessomesubsetof.Theeventualoutcomeofthelongconversationsatisestherelevantincentiveconstraints,andthenalinformationpartitionisitselfsequentialcredible.Thisnotionimputesahighdegreeofrationalityonthepartofthesenderandthereceiver.Thereceiverforeseesthatthesubsetthatisreportedinoneperiodcanbesplitintosubsets,..,Swithouttherebeinganyincentiveforadeviatingsenderofonetypetopretendtobeanother(condition(3)),thatthereceiver’seventualresponsewillmakethesedeviatingtypesbettero(condition(1)),andthatthenon-deviatingtypesdonotwanttodeviategivenhiseventualresponses(condition(2)). 5Conclus

15 ionIfthereceiverdoesnotcommittolistentoo
ionIfthereceiverdoesnotcommittolistentoonlyonemessage,thelogicofFarrell[5]andGrossmanandPerry[7]canstillbeusedtoformalizenotionsofcrediblespeeches,buttheirprecisedenitionsnolongerapply.Ourdenitionofdurableneologism-proofequilibriumisamodicationofthestaticnotionofneologism-proofnesswhichgivesreasonablepredictions,asillustratedinExamples1and2.Butitdoesnotsucientlytakeintoaccountthepossibilitysendingmultistagespeeches.Therearetwowaysofhandlingthisproblem.Wecouldexplicitlyallowobjectionstobesequential,whichleadstothenotionofsequentialneologismproofness.Or,inthespiritofMatthews,Okuno-FujiwaraandPostlewaite[11],wecouldallowmorecomplexspeeches,wherethesendersummarizesallinformationthatcouldberevealedslowlyovertimeinonesinglemessage.Thisleadstothenotionofdurableannouncementproofness.Thesetwowaysofarguingleadtothesameconclusion,forsequentialneologismproofnessanddurableannouncementproofnessarejusttwodierentwaysofexpressingthesameidea.Wethinktheseargumentsprovideagoodfoundationfo

16 rarguinginfavorofsequentialneologismproo
rarguinginfavorofsequentialneologismproofness(or,equivalently,durableannouncementproofness)asasolutionconceptforsequentialcheaptalkgames.References[1]Baliga,S.CorchonL.andSjöströmT.(1997)“TheTheoryofImplementationwhenthePlannerisaPlayer,”JournalofEconomicTheory,forthcoming.[2]D.Bernheim,B.PelegandM.Whinston,“Coalition-ProofNashEquilibriaI:Concepts,”JournalofEconomicTheory,42(1987),1-12.[3]V.CrawfordandJ.Sobel,“StrategicInformationTransmission,”Econometrica50(1982),579-594.[4]M.DewatripontandE.Maskin,“ContractRenegotiationinModelswithAsym-metricInformation,”EuropeanEconomicReview34(1990),311-321.[5]J.Farrell,“MeaningandCredibilityinCheap-TalkGames,”GamesandEco-nomicBehavior,(1993)5:514-531.[6]J.FarrellandM.Rabin“CheapTalk,”JournalofEconomicPerspectives(1996)10:103-118.[7]S.GrossmanandM.Perry,“PerfectSequentialEquilibrium,”JournalofEco-nomicTheory39(1986),97-119.[8]B.HolmströmandR.B.Myerson,“EcientandDurableDecisionRuleswithIncompleteInformation,”Econometrica51(1983),179