strategic inaccuracy in bargaining

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TSE‐541November2014“StrategicInaccuracyinBargaining”SinemHidir StrategicInaccuracyinBargainingSinemHidiryNovember2014AbstractThispaperstudiesabuyer-sellergamewithpre-tradecommunicationofprivatehorizontaltastefromthebuyerfollowedbyatakeitorleaveitoerbytheseller.Theamountofinformationtransmittedimprovesthegainsfromtrade,butalsodetermineshowthissurpluswillbesharedbetweenthetwo.Lackofcommitmenttoapricecreatesahold-upproblemandatradeobetweeneciencyandrentextraction.Inthissetting,coarseinformationarisesduetotheconcernsonthetermsofthetransaction.Asthepreferencesgetlessimportant,informationtransmissionbecomeslessprecise.Itisshownthatinthebuyeroptimalequilibriaofthestaticanddynamicgames,themessagessentarejustinformativeenoughtoensuretrade.Inthedynamicgame,thebuyerisalwaysbetterosendinginfor-mativemessagesonlyattherstperiod,implyingnogainsfromgradualrevelationofinformation.Keywords:information;cheap-talk;bargaining;buyer-sellerrelationJELclassication:C72;D83IamgratefultomyadvisorJacquesCremerforhistimeandadvice,alsoLucasMaestriandHarryDiPeiforusefulcomments.IhavealsobenetedfromdiscussionsSFBSeminarMannheim,NASMinMinnesotaandEEA-ESEMinToulouse.Mistakesremainmine.yToulouseSchoolofEconomics.email:sinem.hidir@tse-fr.eu1 1IntroductionCommunicationplaysanimportantroleinbilateralrelationsinthepresenceofprivateinformation.Insettingswherethereisroomfornegotiation,thepartieswanttobestrategicwhenrevealingtheirpreferences.Abuyermaynotwanttorevealherpreferenceoverdierentoptionswhenfacinganopportunisticseller,inordertoavoidlosinginformationrents.Amongthegoodsthathavethesameintrinsicvalue,thesellercouldproposeadierentpricedependingonhowmuchheestimatesthevaluationofthebuyeris.Byprovidingonlyanestimateofherpreferredoption,thebuyercouldavoidgivingupallbargainingpowertotheseller.Thispaperaddressestheissueofhowabuyercanstrategicallyrevealherpreferencestoanuncommittedsellerviapre-tradecommunication.Therearecaseswhenabuyerchooseshowmuchinformationtoreveal.Whengoingtoarealestateagencyinsearchofahouse,youcouldeitherprovideacertainneighborhoodyouaresearchingfor,orabroaderrangeofthecitywhereyouwouldbewillingtoreside.Whenyoudescribeaveryspecicarea,thesellingagentcanshowhousesthatareexpensiveoronwhichhehasahigherprot,knowingthatyouarerestrictedtothatneighborhood.Ontheotherhand,ifyoudonotrevealyourspecicpreferenceoverneighborhoods,heisbetteroshowingyouhousesofmorereasonablepricesfromdierentareasinordertoincreasetheprobabilitythatyouactuallyendupbuying.Beingveryspecicaboutpreferencescanmakeyougaintimebutloseinformationrents.Thispaperconsidersabuyersearchingforagoodorserviceandhasprivateinformationabouthertasteorneedandshouldcommunicatewiththeseller.Thesellerwillcomeupwithanoerasaresponsetothebuyer'smessage.However,asthesellerhaslackofcommitmenttoaprice,thiscreatesaholdupproblem:knowingwhatthebuyervalues,thesellercanbetterextracthersurplus.Thenthetradeofacedbythebuyeristheonebetweenndingabettertgoodandbeingchargedahigherprice.Inthissetting,thesharingofthegainsfromtradedependsontheextentofinformationrevelation.Thebuyercouldeitherwaitforthesellertomakeoersorprovideinformationabouthertaste.Ifthebuyerprovidesnoinformation,shemightendupnotgettingarelevantoer.Ontheotherhand,themorethebuyerispreciseaboutherpreferences,thehigherthepricethatthesellercanaskher.Tostudythissituation,Iintroduceinformationtransmissionbypre-tradecheaptalkintoabuyer-sellergameandshowthatcoarseinformationarisesduetothecon ictonthetermsofthetransaction.Bargaininghappensonthehorizontaldimension,amonggoodsthathavethesameintrinsicvalue.Thelackofcommit-mentbythesellertoapriceimpliesthatanyinformationonthebuyer'stasteservesasatoolforrentextraction.Thesellerisfreetomakeanyoerwhilethebuyerholdsprivateinformation,whichcanbeseenasawaytostudyhowmuch2 informationthebuyerwouldliketorevealinthepresenceofextremehold-up.Theamountofinformationprovidedimprovesthematchbetweenthegoodandthebuyer'stype,butprovidesahighershareofthesurplustotheseller.Hencethereisacleartradeobetweeneciencyandrentextraction.Themainresultisthecharacterizationofthebuyeroptimalequilibriumaftershowingtheexistenceofacontinuumofequilibria.Themultiplicitycomesfromthewiderangeofpossibilitiesofcommunication,whichisusualincheaptalkmodels.Aftershowingthatanyequilibriumshouldconsistofmonotonepartitions,meaningthebuyertypespoolinintervals,Isearchfortheequilibriawhicharethebestforthebuyerfromanex-antepointofview,andcallthesebuyer-optimal.Thebuyerdecideshowmuchinformationtoprovide,andalthoughthereexistsmoreinformativeequilibria,includingtheperfectrevelation,thebuyerisbetterounderequilibriainwhichinformationislesspreciseevenifthatmaydecreasethechancesoftrade.Iexplorebothoneperiodanddynamicsettings.Thebuyer-optimalequilibriumintheoneperiodgameistheonehavingthecoarsestinformationstructure(theleastnumberofintervals)thatcoversthemarket(tradetakesplaceforalltypes).Inthisequilibrium,thebuyer'smessagesarejustpreciseenoughtoguaranteetrade:partitionintervalsarethelargestsubjecttotheconstraintthatthesellerdoesn'texcludeanytype.Inotherwords,thebuyeralwaysbenetsfrombeingimpreciseaslongastheseller'soerisstillacceptable.Inthetwoperiodgame,thebuyer'soptimalstrategyistosendinformativemessagesonlyattherstperiodandbabbleincasenextperiodisreached,asthisinducesthesellertomakemorefavorableoersandreducetheexpecteddelayintrade.Hence,itisnotoptimaltograduallyrevealinformationfortheownerofinformationinthissetting.1Now,inapoolingintervalinperiod1,therearesometypesofthebuyerwhowillaccepttheoerandotherswhowillrefuseandmoveontothenextperiod.Oncethenextperiodisreached,theseller'sprioraboutthebuyer'stypeisupdated.Thesellercanaskforahigherpriceinperiod1comparedtothestaticgameandleaveoutsometypestobeservedthenextperiod,ashecangetmoreinformationanddierentiateamongthetypesovertime.Twoinformationalbenetsofsecondperiodforthesellerareidentied:therstoneisthattheintervalofpossibletypesofthebuyershrinkswhenperiod2isreached,andthesecondoneisthepossibilityofanotherinformativesignalinthecomingperiod.Whentheonlyinformativesignalsaresentatperiod1andperiod2messagesarebabbling,thesellernolongerenjoysthesecondtypeofinformationalbenet.Still,thetotalinformationprovidedisenoughtoensuretradeoverthetwoperiods.1ThisisincontrasttoHornerandSkrzypacz(2013),who,althoughinadierentsetting,ndthattheownerofveriableandvaluableinformationwillprefergradualrevelation.3 Inthissetting,thesellerandthebuyerarebothworseowhenthediscountfactorishighenoughandthereisexpecteddelay.Whenthediscountfactorislow,thebuyeroptimalequilibriumistheno-delayequilibriuminwhichtheperiod1intervalsareneenoughthatthesellerdoesn'texcludeanytypeandperiod2isneverreached.Thisisduetodelaybecomingsocostlythatthebuyerisbetterorevealingmoreinformationinordertoavoidit.Theno-delayequilibriumgiveshigherprotstothesellerandresultsinhigherwelfarecomparedtotheoneperiodgame.Toeliminateinecientequilibriainthedynamicsetting,asubdivisioncondi-tionisprovidedwhichistheseparationoftypeswhorejecttheoerinagivenperiodandareinitiallyconnectedinanintervalintotwoormoreintervalsbysendingseparatemessagesinthenextperiod.ItisshownthatanyequilibriumwithsubdivisionisParetodominated.Thedynamicmodelistractableasthethresholdtypesinoneperiodarealsothresholdtypesinthenextperiodincaseofrejectingthecurrentoer,hencetheirincentivecompatibilityconstraintissat-ised.Then,duetothemonotonicityofstrategies,theincentivecompatibilityofthetypesinsidetheintervalisalsosatised.Finally,whenthegameisextendedtoinnitehorizon,itisshownthatitisalwaysmoreoptimalthattheonlyinformativesignalsaresentatperiod1.Sendingbabblingmessagesatt>1decreasesexpecteddelay:theperiod1intervalsarenerwhenfuturemessagesarebabblingandthesellerexcludeslesstypesineachperiod.Foranyequilibriumwhichhasinformativesignalsinfutureperiods,thereisanotheronefoundbymodifyingthesignalstructuresuchthattheonlyinformativesignalsaresentatperiod1anddoesstrictlybetterintermsofbuyerandsellersurplus.Inthedynamicgame,thediscountfactoriswhatpreventsthesellerfromndingthegoodthatperfectlytsthebuyer'stasteandextractingallhersurplus.Thereasonfornongradualismininformationrevelationisthatifinformationiseventuallygoingtoberevealed,itisalwaysmoreecienttorevealitatperiod1insteadoflater,asthereisatimecostofdelayingtradeandthesellerwillleaveoutmoretypestodayifheexpectsmoreinformationtoarrivetomorrow.Whenfuturemessagesareuninformative,thesellerismoreeagertosellearlier,whichmeansalowerpriceandlessexpecteddelay.Theuncertaintyisnotontheextentoftheinformationthebuyeriswillingtoprovide,butontheinformationitself.Hence,thebuyerisbetteroresolvingthisuncertaintyinthebeginningandwaitingforthesellertomakeoersbutnotprovidingmoreinformationinthecomingperiods.Finally,theresultsofthedynamicgamesuggeststhattheownerofinformationdoesnotgainfromgraduallyrevealinginformationinthissetting.4 1.1RelatedLiteratureHornerandSkyrypacz(2013)studyadynamicproblembetweenarmandanagentwhohasvaluableandveriableinformation.Thelackofcommitmentleadstoaholdupproblem:thermhastopayfortheinformationbeforeknowingifitisvaluable,asoncetheinformationisprovidedthermwouldnotbewillingtopayatall.Hence,thesellerofinformationbenetsfromthegradualrevelationofhisinformation.Thereisarecentliteratureaboutthesellerincentivestodisclosehorizontalmatchinformation,suchasAndersonandRenault(2006),Sun(2011)andKoesslerandRenault(2012).Thedeparturefromthisliteratureisthatinthispaperthefocusisonthebuyer'sincentivetoprovideinformationinabargainingsetting.Thispaperisalsorelatedtotheliteratureonstrategicinformationtransmis-sion,pioneeredbyCrawfordandSobel(1982).However,thecon ictofinterestontheoutcomepresentintheirpaperandtheliteraturethatfollowsdoesnotarisehere,andinsteadthecon ictisduetothesharingofthesurplus.Golosovetal.(2013)studythedynamicversionoftheCrawfordandSobelmodelinwhichthesellertakesadecisioneveryperiod2andKrishnaandMorgan(2004)showthataddinganex-antecommunicationstageimprovestheoutcomeofthecommunica-tiongame.LevyandRazin(2007)showthatlinkingdecisionstogethermayreducecommunicationduetospillovereects,specicallythatthecon ictofinterestononeissuemayimpedecommunicationonanotherissue.3Finally,thepaperisrelatedtothebargainingliterature.Thesettingofthispapercanbeseenasanexantestageinwhichagoodtobebargainedisbeingpickedamongmany,usingtheinformationprovidedbythebuyer.Asthereisnoverticaluncertainty,thesamegoodisoeredonlyonce.ThereasonIruleoutverticaluncertaintyisinordertohighlightthepreferencerevelationaspect,andinpresenceofverticaldierentiationthebuyerwouldnothaveanincentivetorevealthatshehasahighvaluation.Intheextensionpart,Ishowthisbyconsideringasimpleoneperiodexamplebyintroducinguncertaintyontheverticalvaluation.Therestofthepaperisorganizedasfollows:Section2explainsthegeneralmodel,section3solvesfortheequilibriumoftheoneperiodgame,section4studiesthetwoperiodssetting,section5studiestheinnitegame,section6providesanextension,andsection7concludes.TheomittedproofsandsolutionscanbefoundintheAppendixattheendofthepaper.2Theyshowthatfullrevelationispossiblebyconstructinganon-monotonicpartitionequi-librium,inwhichfarawaytypespoolinitiallyandseparatelateron.Incontrast,inthispaperthereisaonetimedecisionanditiscostlytodelaytrade.Hence,althoughfullrevelationcouldbesupportedasanequilibrium,itisinferiorforthebuyerfromanex-antepointofview.3Thisrelatestothecurrentpaperastwodimensionsarebeingagreedupon,howevertheinformationtransmissionhappensonlyononedimensionandthereisabargainingstagethatfollowsthemessage.5 2TheModelTherearetwoplayers,abuyer(B)andaseller(S)whointeractinordertoagreeontheonetimeexchangeofagoodorservice.Thebuyerprivatelyobserveshertype,arandomvariablein[0;1]whichdeterminesherhorizontaltasteforthegood.Thesellercanoerfromacontinuumofgoods,y2[0;1].TheutilityofthebuyerfromagoodyisU(;y)=kf(jyj),wheref,thecostofmismatchbetweenthegoodandthebuyer'stype,isstrictlyconvexinjyj4,U>0(the12preferredgoodofthebuyerisincreasinginhistype),U22<0(thereisasinglegoodthatperfectlytseachtypeofthebuyer).Thesellerisassumedindierentamongdierentgoodsandhisvaluationisnormalizedtozero,whichwillimplythattradeisalwaysoptimal.Thesellerwouldliketondthebesttforthebuyerorderinordertogetthehighestprots.Outsideoptionsarezero.Verticalheterogeneityinbuyervaluationisstudiedasanextension,andthemainpartofthepaperfocusesonthehorizontaltasteparameterandtheseller'sincentivetochargedierentpricesforgoodshavingthesameintrinsicvalue.First,thebuyersendsacostlessmessage,m2Mtotheseller,whorespondsbyanoerconsistingofagoody(m)andatransfer(m).IfBacceptstheoer,thegameendswithrealizedpayosU(;y)and.Themaximumutility,k,iswhatthebuyercangetfromagoodthatperfectlymatcheshertaste.Itissmallenoughthatintheabsenceofinformationthemarketwillnotbecovered,inotherwordstheseller'soerwillexcludesometypesofthebuyer.Thetotalsurplusfromtradeismaximizedwheny=,incaseeverytypegetstheirmostpreferredgood.Thecrucialassumptionisthatthesellercannotcommittoapricescheduleex-ante.Ifhecould,thenhewouldextractthemaximumrent,kthroughatruthfulrevelationmechanism.2.1EquilibriumAnalysisThesolutionconceptthatwillbeusedisPerfectBayesianEquilibrium.Given,(mj)isthesignalingstrategyofthebuyer.Theseller'sstrategyisanoer(y(m);(m)),towhichthebuyerrespondsby(y;)20;1)5.StrategiesthatconstituteaPBEsatisfy:forany2(0;1),if(mj)>0,thenU(;y(m))(m)U(;y(m0))(m0)forallm02M.(B'smessagemaximizesherutilityamongfeasiblemessagesgivenS'soptimalresponse.)4theresultscontinuetoholdforanyfaslongasitisnottooconcave,showninthelemma35duetotheassumptionthatwhenthebuyerisindierentbetweenacceptingtheoerornot,shewillaccept6 R1foranym,(y(m);(m))=argmaxy;0(y;)(jm)dwhere(mj)f(m)(jm)=R1(mjt)f(t)dt0(SmakestheoerwhichmaximizeshisexpectedprotsasaresponsetothemessagesentbyB.)2.2SomePropertiesandDenitionsThissectionintroducessomepropertiesthatwillbeusedthroughoutthepaper.Denition1.Amonotonepartitionequilibriumisoneinwhichthebuyertypespace[0;1]isdividedintonintervalswithboundarypoints0=a10forallnii+12(a;a)then(mj0)=0forall02(a;a)foranyj6=i.ii+1jj+1Inamonotonepartitionequilibrium,eachmessageissentbytypesthatareconnectedinintervalsandnoothertypeoutsidetheseintervalsendsthesamemessage.Thisiscalleduniformsignaling(CrawfordandSobel,1982).Inthistypeofequilibrium,fewernumberofintervalsisequivalenttocoarsercommunication.Fromnowon,Iwilldenotethemessagesentbytypesinanintervaliasmiandsaythatintervalisendsmessagemi.Hence,uponreceivingamessagemi,thesellerknowsthatisfoundinintervali.Beforerestrictingattentiontomonotonepartitionequilibria,Iwillruleoutthepossibilityofnon-monotonepartitions.Anon-monotonepartitionhastypesinseparateintervalsthatpoolintheirmessages6.Aseachbuyertypewillhaveastrictlypreferredmessage,Irestrictattentiontopurestrategiesofthebuyer.Lemma1.If(mj1)=(mj2)=1,then(m;3)=1forall32(1;2):equilibriumstrategiesaremonotoneinanygameinwhichtradehappenswithcer-tainty.Hence,therecannotexistanynon-monotonepartitionequilibriainwhichallbuyertypesacceptanoer.7Asthepaperfocusesonecientequilibriainwhichtradehappensforalltypes,attentionwillberestrictedtomonotonepartitions.Throughoutthepaper,thetermanintervalisservedorcoveredwillbeusedtosaythattheoermadebythesellerasaresponsetothemessagesentbythatintervalisacceptedbyallthetypes.6thisdenitionisborrowedfromGolosovetal.20147Incasesometypesarealreadynotgettinganoer,thentheymightaswellbeindierenttosendingothermessageswhichwillalsogivethem0surplus.7 Inaddition,thefollowingconditionfortheboundarytypesineachintervaltobeindierentbetweentheoersinducedbythetwomessagesshouldbesatised:U(ai;y(mi1))(mi1)=U(ai;y(mi))(mi)foralli.3OnePeriodGameFirst,Iconsiderthegamewhichendsafteronlyoneroundofcommunicationandoer.Duetotheassumptionthatkissucientlysmall,iftheintervalwhichsendsamessageisnotneenough,thentheseller'soerwillexcludesometypes,inotherwordstradewillnottakeplaceforthesetypes.Giventhatthereisasingleoer,thebuyeriswillingtoacceptitaslongasU(;y)0ashisoutsideoptionis0.AsU(;y)=kf(y)isdecreasingasmovesawayfromy,andisconstant,thenthetypesforwhomU(;y)0arelocatedonaconnectedline.Denition2.Thethresholdtypesiandiarethetypesinintervaliwiththelowestutilityamongthosewhoaccepttheoer:(i;i)=argminU(;yi)suchthat(yi;i)=1.Thethresholdtypesarethehighestandlowesttypesinanintervalwhoaccepttheoer.Thetypeslocatedin(i;i)willaccept,whilethoselocatedoutsidetheintervalrejecttheoermadeasaresponsetothemessagesentbythetypesinintervali,astheirutilitiesfromtheoerarenegative.Theboundarytypescoincidewiththethresholdtypeswhenthewholeintervalisbeingserved.Asthethresholdtypesget0surplus,i=kf(yi)=kf(thetayi)andbythesymmetryaround0off,y=y,whichleadstoy=i+i.Incasetheiiiii2wholeintervalisserved,y=ai+ai+1.Figure1displaystheresponsetoagiveni2oer(yi;i)afteramessagesentbyanintervaliandtheutilitiesofthetypesinthatinterval.Proposition1.Anymonotonicpartitioncanbeobservedasanequilibrium.Proof.Iwillprovethisbyshowingthatnotypehasanincentivetodeviateinamonotonepartitionequilibrium.Takeamonotonepartition,andanintervalisendingthemessagemi.Uponreceivingthismessage,itisoptimalforthesellertoalwaysleavezerosurplustotheboundarytypesaiandai+1:eitherthesetypesareexcludedortheyarethethresholdtypeswhoaccepttheoerwith0surplusastheyarethefarthestfromyi.Ifthethresholdtypesarelocatedstrictlyinsidetheinterval,ai<iandai+1>i,thenU(;yi)i<0fortheboundarytypes.In8 Figure1:responseofintervalitooer(y(i);(i))U(;y(i))(i)y(i)aiai+1ii(y(i);(i))=1eithercase,forthetypeslocatedintheconsecutiveintervals,ai+1,U(;yi)i<0asfincreaseswiththedistancefromthegoodyi,sothesetypesdonothaveanincentivetodeviateandpoolwiththeintervali.Asthesameholdsforanyintervali,thispartitiondoesformanequilibriumpartition.Now,Iwillndthelargestintervalthattheselleriswillingtoserve.Thisistheintervalthattheseller'soerwillcoverinthebabblingequilibrium.Ababblingequilibriumisoneinwhichallbuyertypes2(0;1)sendthesamemessage.Denition3.x<1isthelargestintervalthattheseller'soerwillcoverinanyequilibrium.Uponreceivinganuninformativemessage,thesellerrespondsbyanoerthatwillbeacceptedbytypes2(;)where=x,andx<1duetotheR1assumptionthatkissmallenough.x=(y;)dwhereyandaregiven0bytheseller'sproblem:Z1(y;)=argmax(y;)d8y2;2<+0Hence,anyintervalsuchthatxxwillbecovered.Lemma2.Inanyinterval(i;i)where(yi;i)=1forall,theexpectedbuyerRisurplusisstrictlypositive:[U(;yi)i]d>0.iProof.Thethresholdtypesinanyintervaliget0surplus:kf(iyi)=kf(y)=wherey=i+i.AsU>0,U(;y)>0forall2(;)iiii212iiRiwhichleadsto(U(;yi)i)d>0.i8Foragiven(y;),thereisaninterval(;)forwhichU(;y)=U(;y)=0andU(;y)0.Then,outsidethisinterval,for<and>,U(;y)<0.ycanbeanyythatsatises=xwherey=+.Hence,xisthelargestpoolinginterval2suchthatthereisnoexclusion.9 Lemma2willbeusedformakingwelfarecomparisonsamongequilibria.Inanyinterval,thebuyersurplusisdecreasingasmovesawayfromyi,withthetype=yigettingthehighestsurplus.Hence,thetypesclosertothemiddleoftheintervalgetinformationrentsthatareincreasingintheirdistancefromthethresholdtypes.Nextpropositionestablishestheexistenceofafullyrevealingequilibrium.Proposition2.Thereexistsanequilibriumwhichisfullyrevealinginwhicheachsendsm()inducingtheoer(;k)andleaving0surplustothebuyer.Theexpectedbuyersurplusisstrictlypositiveinanyotherlessinformativeequilibria.Proof.First,considerthefullyrevealingequilibrium.Eachtypeofthebuyerhasnoincentivetodeviate,asincaseofsendinganothermessagem(0),theseller'soerwillbe(0;k)whichgivesutilityU(;0)f(x)and2d>0,equation3isalwayspositiveforanyxfor0xconvexf,(indeedholdsalsoformostconcavefunctionsf),sotheexpectedbuyersurplusisincreasingintheintervallengthaslongasnotypeisbeingexcluded.Whenthenumberofintervalsdecreasessubjecttotheconstraintthatthereisnoexclusion,thetotalbuyersurplusalwaysincreases.Then,theexpectedbuyersurplusincreasesinthelengthofthepartitionintervalsforxx.Thesellersurplusontheotherhandisdecreasinginthelengthofthepartitionintervalsforx,U(;y(m0))(m0)<0.Sothemessagesandsurplusesofthetypesintheinitialintervalxdonotchange.Forthenewtypesbeingserved,byRlemma2,(U(;y(m0))(m0))d>0.Hencethisnewequilibriumissuperiortotheinitialoneintermsofbuyersurplus.Repeatingthesameprocedure,anyequilibriumwheresometypesareexcludedisdominatedbyamoreinformativeequilibriuminwhichmoretypesareserved,untilreachinganequilibriuminwhichtradehappensforalltypes.Therstpartoftheproofisdone.Secondpartcon-sistsofshowingthatmovingtomoreinformativeequilibrialowersbuyersurplus.Lemma4showedthatanyequilibriumpartitionthathasmorethan1interval9Thispartitionequilibriumlookslike:1intervalsofsizex(where1isthebiggestintegersmallerthan1)xxxif(11x)>0,anotherintervalofsize11xxx12 oflengthsmallerthanxisinferiorintermsofbuyersurplus.ThiscompletestheproofthatthepartitionrulexdominatesanyothermoreorlessinformativeequilibriaintermsofbuyersurplusProposition3saysthattheequilibriumwhichisbuyer-optimalhasthelargestintervalssubjecttotheconstraintthatthereisnoexclusion.Inotherwords,aslargerintervalsimplylessinformativeequilibria,messagesarejustinformativeenoughfortradetotakeplacewithprobabilityone.Thebuyerdoesnotbenetfrombeingmoreinformative,asnotypeisbeingexcludedandtheexpectedbuyersurplusisthehighestpossibleunderthisrule.Inaddition,shedoesnotbenetfrombeinglessinformativeeither,becauseinanyequilibriumwhichislessinfor-mative,sometypeswillbeexcludedfromtrade.Hence,thebuyerbenetsfrombeingimpreciseaslongassheisstillallocatedagood.AppendixBsolvesfortheequilibriumpartitionruleusingthequadraticutilityqexample:U(;y)=k(y)2,andndsx=2k,Astheintrinsicvaluek3increases,thepoolingintervalsbecomewider,inotherwordslessinformationistransmittedinequilibrium.Whenpreferencesarelessimportantcomparedtotheintrinsicvalueofthegood,thebuyercanbelesspreciseandstillgetagood.Inaddition,asf0increases,meaningasfbecomesmoreconvex,xdecreases.Whenthecostofmismatchincreasesfaster,moreinformativemessagesarenecessaryinordertoensuretrade.Whilemovingfromfullyinformativeequilibriumtolessinformativeones,thesellersurplusisdecreasingandthebuyersurplusisincreasing,whiletotalwelfareisdecreasing.5Dynamicsofthegame:twoperiodcaseInthissection,thedynamicsofthegameareexploredbyintroducingasecondroundofcommunicationandoer.Thiscasealsoprovidestheintuitionforexten-siontotheinnitehorizongame.Theonlymodicationtotheoneperiodgameisthatnowwhenperiod1oerisrejected,thereisasecondroundofcommunicationandoer,andsurplusisdiscountedby.Thetypeofthebuyerdoesnotchangewhenthegamemovestothenextround,butshesendsasecondmessage.Thediscountfactoriswhatmakesdelaycostly.Thebuyer'sstrategyisnowdenotedasq(m1j),q(m2j)andtheseller'sstrategyh(m1);h(m2;m1).Theseller'speriod2oernowdependsonboththeperiod1andperiod2messages.Tounderstandthedynamicsofthegame,lookataperiod1intervalioflengthxisendingthemessagemi.Theseller'sbestresponseisanoer(yi;i)suchthatatleastsometypesintheintervalaccept,whichwillbedenotedasanintervalz(xi)withthethresholdtypes(i;i).Dependingontheseller'soer,theremaybesometypes,2biwherebi=xinz(xi)whorejecttheoer.Incasethe13 Figure2:Dynamicsofthegamet=1ayiai+1ayiai+1iiiibi1z(ai)bi2z(ai)birefuseandgoacceptt=1refuseandgoacceptatt=1refuseandgotoperiod2toperiod2toperiod2##yt=2yt=2yi1i2ibi1bi2bigamemovesontoperiod2,thesellerknowsthatthebuyerislocatedinintervalioutside(i;i),asincasewerefoundinthisregionthentheoerwouldbeacceptedatperiod1.Ifi=aiori=ai+1thesellerputsprobabilityonetothebuyerbeingfoundinthesingleintervalbi.Otherwise,thereare2separatedintervals,bi1=(ai;)andbi2=(;ai+1)insidewhichthebuyercouldbefound,sotheseller'sbeliefis2bi1[bi2atthebeginningofperiod2.Theperiod2oerstrategyofthesellerissimilartotheoneperiodgamegiventhatitisthelastperiod.Figure2givesademonstrationofthesedynamics,restrictedthecaseinwhichthereisnoexclusionoverthe2periodsandnodivisionofintervalsthatrefusetheperiod1oer.Itwillbeshownthroughoutthissectionthatthebuyeroptimalequilibriumhasthisstructure.Thepresenceofasecondperiodgivesthesellertheopportunitytobetteridentifythebuyer'stasteanddierentiateamongthetypesoverthetwoperiods.Givenidenticalpartitionintervals,theselleralwayschargesahigherpriceinperiod1ofthe2periodgamecomparedtowhathechargesinthesingleoergame,ashecanmakeanotheroeratperiod2incasetheperiod1oerisrejected.Therejectionalsogivesinformationaboutthebuyer'stype,asitleadstheintervalofpossibletypestoshrink.Hence,eveniftheperiod2messageisuninformative,thesellerhasbetterinformationaboutthebuyer'stypeduetotheperiod1messageandthefactthattheperiod1oerhasbeenrejected.Tosumup,thepresenceofperiod2givestwoinformationalbenetstotheseller:theshrinkingoftheintervalinwhichthebuyercanbefound,andanotherpossiblyinformativemessageinperiod2.NowIdenetheequilibriuminwhichthegamenevermovestoperiod2,whichiscalledtheno-delayequilibrium.14 Denition4.Ano-delayequilibriumisoneinwhichforeachmessagemisentatperiod1,theseller'soer(yi;i)issuchthat(yi;i)=1forall2(ai;ai+1)andforalli:anybuyertypeinanyintervalacceptstheperiod1oerandtheprobabilityofreachingperiod2iszero.Icalltheno-delaythresholdintervallengthxndsuchthatwheneverxxndforallintervals,tradehappenswithnodelay.Intheno-delayequilibrium,theperiod1partitionintervalsareneenoughthatthesellerdoesnothaveanincentivetoexcludeanytypeandmoveontoperiod2withpositiveprobability.Itistrivialthatxnd0,whichmeans2222as<1andfisconvex,bz,andz(x)x.Thelargestperiod1intervalsuch3thatalltypesareservedwithnosubdivisionisx=3x,whichgivesz(x)=b=x.5.1.2Whenperiod2isbabblingNowIconsidertheequilibriuminwhichonlytherstperiodmessagesareinfor-mative.Denition7.Ababblingstrategyinperiod2isoneinwhichallthebuyertypesinanintervalthatrejecttheperiod1oersendthesameperiod2messages.Lemma7.Ifthebuyer'sstrategyistobabbleatperiod2,thengivenamessagemi,thesellerweaklyprefers(yi;i)suchthateitheri=aiori=ai+1sothatthetypeswhichrejecttheperiod1oerarefoundinasingleinterval.Iftheexcludedtypesarefoundinasingleinterval,atperiod2thesellerknows2b=xnzandhewillserveatmostxmeasureoftypes.Incasethetypesiiithatrefusehisoerarefoundin2separatedintervalsbi1andbi2,thesellerknowsthat2bi1[bi2,sohemaynditoptimaltoserveonlyoneoftheintervals,meaningtrademaynotoccurforapositivemeasureoftypes.Incasehewantstoselltosometypesinbothintervals,hehastochargealowerpricethanhewouldiftheywerefoundinasingleinterval.11Atperiod1,thesellerchooseszandbtakingintoaccountthattheexcludedtypeswillpoolintheirperiod2messages(again,focusingonthecasewherebwillnotsubdivideinperiod2):zb(z;b)=argmaxz(kf())+b(kf())z;b22subjectto:z+b=xTheFOCgives:kf(z)zf0(z)=(kf(b)zf0(b))whenz;b>0,which2222givesbzandz(x)xforfconvex.Themaximumvaluexcantakesothat2thereisnoexclusionisx=2xwhichleadstoz(x)=b=x.Then,foralla2x,z(a)b.11Thereasonheisweaklybetteroisthattherecanbesomecasesinwhichthesellerwillbeindierentbetweenhavingtheexcludedtypesinasingleintervalortwoseparatedintervals.Forexampleifthemeasureofbuyertypesexcludedisbx,thenheisstrictlybetterowhenthesetypesarefoundinasingleinterval,ashewillchoosetoserveallofthem.Incasetheyareseparated,theneitherhewillserveonlyoneintervalorchargealowerpricetoserveallthetypes.However,incaseb>x,asthesellerwillsellonlytoxmeasureoftypesatperiod2,heisindierenttohavingasingleor2separateintervalsaslongasoneoftheintervalsmeasuresatleastx.18 5.1.3No-delayequilibriumTakeanequilibriumwhichisbabblingatperiod2.Incase@[(kf(z))z+(k@z2f(xz))(xz)]>0forz(x)=x,thennotypeisexcludedatperiod1andthereis2noexpecteddelay(theconditionisidenticalwhenperiod2isinformative).Themaximumxsuchthatthereisnodelayisfoundbyreplacingz(x)=xinthederivativeandsettingittozero,whichgives:xx0xk(1)=f()+f()222therighthandsideisincreasinginxwhereasthelefthandsideisconstant,hencethereisasinglexndsuchthatthisequationholdswithequality.Forallxxnd,thegamehasnodelay,andforallx>xnd,thereisexpecteddelay.Thelasttypeofequilibriumistheonewhichisbabblingatperiod1andinformativeatperiod2,whichisshowntobedominatedintheAppendix.Proposition4.Inanydynamicgame,buyersurplusisalwaysinferiortothesurplusinthebuyer-optimalequilibriumoftheoneperiodgame.Proof.Proposition3showedthatxistheintervallengthintheequilibriumwhichmaximizesthebuyersurplusintheoneperiod(singleoer)game.Inthe2periodsgame,givenziforallitheintervalswhichgettheoeratt=1,andbiforallitheintervalsthatgettheoeratt=2ineachintervali,theexpectedsurplusofthebuyerusingequation2is:"#XnzZf(zi)XnbZf(bi)2zi2bi2(f()f())d+2(f()f())d(4)0202i=1i=1PnzPnbPnzwherei=1zi+i=1bi1.Hence,i=1zi1.Asthesellerwillneverserveanyintervallargerthanx,zxforalli.ThesurpluswouldbemaximizedifPiz=xforalliandnzz=1:onlyifnotypewereexcludedandthegameendedii=1iatperiod1withcertainty,thesurpluswouldbeequaltothesurplusoftheoneperiodgame.However,whenxistheintervallengthatperiod1,thenz(x)0.PnzThen,i=1zi<1,andduetothediscountfactor,surplusisalwayslowerthaninthestaticgame.Finally,amongtheequilibriainwhichnotypeisexcludedatperiod1,x=xndisthelargestintervallength,whichalsogiveslowersurplusbecausexndx,thenthetypesinbdivideintonintervalsoflengthxandanaliiintervalofbnxbysendingseparatemessages.iAsintheoneperiodgame,thesurplusmaximizingequilibriumforthebuyerisoneinwhichthepartitionintervalsareneenoughtoensurethatnotypeisexcludedattheendofthegame.Ifperiod2isinformative,thenthetypeswhorejecttheperiod1oerwillsendaperiod2messagewhichisjustpreciseenoughthattradehappenswithcertainty.Ontheotherhand,ifperiod2messagesarebabbling,thentheperiod1partitionhastobeneenoughsothattradestillhappenswithcertaintyoverthe2periods.Proposition5.Inthebuyer-optimalequilibriumofthetwoperiodgame,tradehappensforanytypeeitheratperiod1oratperiod2.Proof.Thispropositionisadirectresultoflemma8.Eithertradehappensatt=1orotherwiseatperiod2,duetolemma8,bxforalli,hencethereisnoiexclusioninperiod2incaseitisreached.NowthatIconcludedthereisnoexclusion,intheexpressionofthebuyerPnzPnbsurplusinproposition4,Icanreplacei=1zi+i=1bi=1.Lemma9.Anyequilibriuminwhichmessagesareinformativeatperiod1andasubdivisionhappensatperiod2isParetodominatedbyanotherequilibriumwithnosubdivisionandamoreinformativeperiod1partition.20 Thisisprovenbyshowingthattheintervalswhichsubdivideatperiod2wouldhavebeenbetteroinanotherequilibriuminwhichtheyhadseparatedthemselvesatperiod1.Inthebuyer-optimalequilibrium,ifinformationisprovidedinperiod1,itshouldbesucienttomakesuretradeisensuredwithoutasubdivisionoverthe2periods.Lemma9tellsusthatequilibriawithsubdivisioncanbedisregardedinthesearchforthebuyer-optimalequilibrium.Togetherwithlemma8,itleadstotheconditionbxforalliinorderfortradetobeensuredatperiod2iwithoutasubdivision.Aftershowingthatthereisnosubdivision,inthesurplusfunctionofthebuyerInowreplaceb=xizi(xi)incaseperiod2isinformative,i2andbi=xiz(xi)incaseperiod2isbabbling.Givenapartitionruleofxatperiod1,theexpectedbuyersurpluscanbewrittenas:Zz(x)Zxz(x)12z(x)4xz(x)2(f()f())d+22(f()f())d+x0204000Zz(x)0Zxz(x)002z(x)4xz(x)2(f()f())d+22(f()f())d(5)0204fortheequilibriumwitht=2informative,and:Zz(x)Zxz(x)12z(x)2xz(x)2(f()f())d+2(f()f())d+x0202000Zz(x)0Zxz(x)002z(x)2xz(x)2(f()f())d+2(f()f())d(6)0202fortheequilibriumwitht=2babbling,wherex0=1x1isthenalasymmetricxinterval.Now,byanargumentsimilartothesingleoergame,Iwillconcludethatthepresenceofthislastasymmetricintervaldoesnotaecttheoveralloptimalityofthepartitionrule.Lemma10.Theoptimalpartitionruleaisgivenby:"Zz(x)12z(x)a=argmax2(f()f())dxx02Zxz(x)2xz(x)+22(f()f())d(7)0221 whent=2isinformative,and:"Zz(x)12z(x)a=argmax2(f()f())dxx02Zxz(x)2xz(x)+2f()f())d(8)02whent=2isbabbling.Nextpropositioncharacterizesthebuyer-optimalequilibriumofthetwoperiodgame.Proposition6.Inthebuyer-optimalequilibriumofthe2periodgame,period1messagesareinformative,for^,partitionintervalsatperiod1areoflength2x:{halfofthetypesineachintervalareservedatperiod1andtherestatperiod2{period2messagesarebabblingfor<^,no-delayequilibriumwithintervalsofxnd:{thegameendsatperiod1withcertainty.Ashortsketchoftheproofisasfollows:bylemma9andlemma8,Iknowthatinequilibriumtradewillhappenforanytypeofthebuyerwithnosubdivision.Thecandidateequilibriaeitherhaveinformativemessagesinbothperiods,orinformativemessagesinperiod1andbabblingmessagesatperiod2.Theequilibriawhichhavebabblingmessagesatperiod1canbeshowntobedominated.Thesurplusfunctionsinlemma10have2maximalpoints.Therstoneiswhenz(x)takesitsmaximumvaluesubjecttodelay.Inthiskindofequilibrium,z(x)=xwhenx=2xinthebabblingequilibriumandwhenx=3xintheinformativeequilibrium.Thisshowsthatmoretypesaregettingtheperiod1oerinthebabblingequilibriumatthesamepricethanintheinformativeequilibrium.Thesecondmaximumisachievedwhenperiod1intervaltakesitslargestvaluesubjecttono-delay,hencex=xnd.When<^,theno-delayequilibriumdominatestheequilibriawithdelay,andfor^thebabblingequilibriumdominatestheinformativeone.(TheextendedproofcanbefoundinAppendixB.)Intheno-delayregion^,buyersurplusisatrstdecreasingin,asthepartitionintervalsgetnerinordertosatisfythiscondition.Aswaitingistoocostlyinthisregion,thebuyerisbetterointheequilibriumhavingnerpooling22 intervals.When>^,thebuyeroptimalstrategyislessinformative,andtheintervalsarethelargestpossiblesubjecttotheconstraintthatalltypesgetservedoverthe2periods.Theintervalsareconstantinandthesurplusincreasesinonlybecausethecostduetodelayislower.Intheequilibriumwithdelay,thebuyerandsellersurplusesarebothlowerthanintheoneperiodgame.Thesellersurplushasajumpdownat=^whenshiftingtotheequilibriumwithdelay.When>^,thesellersurplusisalwayslowerthaninthestaticgame,asthebuyer'speriod1messageislessinformativeandthereisexpecteddelay.Ontheotherhand,when<^,hissurplusisevenhigherthaninthestaticgame,duetothenerpartitionintervalsandnodelayintrade.Inthisregion,thematchbetweenthegoodandthebuyerimprovescomparedtothesingleoergame.Theresultsaysthatthebuyerisbetteroprovidinginformationonlyatperiod1whenthereare2periodstoplay.Babblingstrategyservesisawaytodisciplinethesellerandincreasesthepossibilityoftradeatperiod1.Thesellersellstomoretypesatperiod1.Inreturn,theperiod1partitionhastobeinformativeenoughthattradestillhappenswithcertaintyasaresultofthe2periods.Whenthethesellerknowstheperiod1messageistheonlyinformationhewillget,hehaslessincentivetodelaybyaskingahigherpricethanwhenheexpectsaninformativemessageatperiod2.Having2roundsrstappearstobeanadvantageforthesellerwhocanbetteridentifythetypeofthebuyerduetotheopportunitytomakeasecondoerincasetheperiod1oerisrejected.Thisadvantageislessintheequilibriumwhichisbabblinginperiod2.Knowingthathewillnotreceiveanotherinformativemessage,thesellerismoreeagertotradeatperiod1comparedtothegameinwhichperiod2isinformative.Toconclude,revealinginformationgraduallyisnotprotableforthebuyerwhoistheownerofinformation.Theresultwhen>^suggeststhattheintroductionofasecondperioddoesnotimproveinformationtransmissioncomparedtothesingleperiodgameandinadditionresultsinex-pecteddelay.However,when<^,informationtransmissionandoverallwelfareincrease.Hence,thesecondroundofplayisbenecialforthegainsfromtradeonlyincasedelayisverycostly.Thegraphdisplaystheratioofbuyerandsellersurplusescalculatedbyusingthefollowingquadraticutility:B2U=k(y)(9)^=131k(1+)k(1+)for>,buyersurplusandsellersurplus.396for<1,buyersurplus2k(1)andsellersurplus2k+39323 sellerbuyer332^=136TheinnitehorizongameThissectiongeneralizesthegametooneinwhichtheroundsofcommunicationandoercontinueuntiltradetakesplace.IftradetakesplaceatperiodT,thenthebuyer'sutilityisT1(U(;y))andtheseller'sutilityT1fromaperiod0pointofview.Theseller'sinformationatperiodtandhencetheoerdependstonthehistorywhichofthepastmessagesplusthecurrentmessageht=fmsgs=0.Manypropertiesofthetwoperiodsgamecarryontotheinnitehorizon.Iwillfocusonasingleinterval,henceanintervalatperiodt,xitwillbeshort-enedasxt,andzitwhichistheintervaloftypesthataccepttheoeratperiodt,willbeshortenedasztandbtwilldenotethesetoftypesexcludedinperiodtinsideanintervali,whichagainmaybeasingleortwoseparateintervals.Asalreadystatedinthetwoperiodgame,inanyperiodinwhichallthepar-titionintervalssatisfyxxnd,tradewillhappenwithcertainty.Thisgeneralizestoinnitehorizongame:ifgivenanintervalthesellerhasnoincentivestoexcludeanytypeintherstperiodwhenthereisasecondperiod,thenhewouldnothaveanyincentivetoexcludewhenthereareaninnitenumberofperiodslefteither.Asxndisthelargestintervalsuchthatthereisno-delay,thenforanyintervalx>xnd,hewillalsoserveatleastxndmeasureoftypesbuttherewillbeexpecteddelay.Hence,inanyperiod,foranyintervalx>xnd,sometypesarenotserved:zxndandb>0.ttThissectionwewillmakeuseofthemorespecicutilityfunction:B2U=k(y)(10)Althoughtheresultscontinuetoholdunderthemoregeneralutilityfunction,thisspecicationmakescomputationeasierinthissection.Remark1.Theno-subdivisionconditionofthetwoperiodsgamecarriesontotheinnitehorizongame:inthebuyer-optimalequilibriumtherewillbenosubdivisioninanyfutureperiodt>1.24 Thiswasproveninthe2periodsgamebylemma9.Now,Igeneralizeittotheinnitegame.Bymakinguseofthelemma9,anyequilibriuminwhichsubdivisionhappensatt1Paretodominatesequilibriainwhichsubdivisionhappensatperiodt.Repeatingthesameargumentuntilperiod1,Iconcludethatthereisnosubdivisioninanyperiodt2.Twokindsofequilibriawhichsatisfythenosubdivisionconditionareofin-terest.Therstoneistheequilibriuminwhichthemessagesareinformativeineachperiod.Usinglemma6,iffutureperiodsareinformative,theseller'sweaklyoptimaloeristheoneinwhichthetypesthatrejecttheperiod1oerwillbefoundin2separatedandequalintervalsinperiod2,whichmeansthesellersetsy=ai+ai+1.Intheinnitehorizongame,thisisstillrelevant:thetypesinani2intervalthatrejecttheperiod1oerarefoundin2separateintervals.Continuinginthisfashion,fromaperiod1pointofview,thereare2t1intervalsofzthataretservedateachperiodtintheinformativecase.Thesecondtypeofequilibriumisthebabblingoneinwhichtheonlyinformativemessagesaresentatperiod1andthetypesthatrejecttheoerpoolineachcomingperiod.Inthebabblingequilibrium,bylemma7,asingleintervalztisservedateacht.Hence,fromaperiod1pointofview,thereisonly1intervalztservedineachperiodinthebabblingequilibrium.Givenaperiod1messagesentbyapartitionintervalx1,ifthegamelastsatmostTperiods,ateacht,xtisthelengthofthepoolingintervalwhileztisthemeasureoftypesthataccepttheoer.Theintervalsfollowxt+1=xtztinthebabblingequilibriumandx=xtztintheinformativeequilibrium.Givent+12therstperiodpartitionandthefuturemessagestrategyofthebuyer,thesellerdeterminesthefutureoers(y;)T.ttt=0Lemma11.IftradehappensatlatestatperiodT,z1z2::::zTandsop1p2p3::::pT.Proof.Thisisfoundbytakingtheseller'sproblemwhichndsthatthegamewillendatlatestatperiodT.Uponreceivingamessagesentbyanintervalxatperiod1andgivenwhetherornotthefutureperiodsareinformative,theseller'sproblemistomaximizehisprotbydeterminingztforeacht,andhenceTwhichisthelastperiodinwhichtradecantakeplaceforsometypes.Thediscountedexpectedsurplusofthesellerintheinformativecaseis:X1zi(z)=2t1t1z(kf(t))tt2t=1subjectto:X1t12zt=xt=125 TheFOCwrtztgives:ztzt0ztkf()f()222t1As<1,itfollowsthatz1z2z3:::zT,whereTdenotesthelatestperiodthatcanbereached.ThereisaperiodTsuchthatx=zxndsuchthatinTTcasethisperiodisreached,tradewilltakeplaceinthatperiodwithcertainty.Second,incaseallperiodst>1arebabbling,thesurplusgiventherstperiodintervalxis:X1zb(z)=t1z(kf(t))tt2t=1subjectto:X1zt=xt=1TheFOCwrtzt:ztzt0zt(kf()f()=222t1whichleadstoandz1>z2>:::>zTwhereagainTisthelatestperiodwhichcanpossiblybereached.RealizethattheFOCareidenticalintheinformativeandxbabblingcases.Incasez=x,thenz=z=:::=z=xas(kf(t)112T12x0xtf(t)=0,andforallothervalues,z>z>:::>z.2212Tqz24k(1)Ifweusethequadraticutility,therelationz=t3issatisedforqt+14k(1T1)T12allt1isdominatedforanybuyertypeandforthesellerbyanotheronewhichisinformativeonlyatperiod1andbabblinginallfutureperiodst>1.26 (ProofinAppendixA)TheproofproceedsbytakinganyinformativeequilibriumandshowingthatthereexistsacorrespondingbabblingequilibriumwhichalsolastsforTperiodsandhasidenticalzforallt2(1;T)andanerperiod1partitionrule,meaningxba.Lookatathresholdtype(k;ai)inintervaliwhosesurplusis0whentruthfullyrevealinghertypebypoolingwiththektypesinintervali.Howeverbypoolingwiththe(k;ai)type,shecanensureautilityofatleastkk(astheworstcasewouldbethatktypeisalsoathresholdtype).Hence,inaseparatingequilibrium,thethresholdtypeswithkshouldgetpositivesurplusuponrevealingthatthey27 arehightype.However,thisisnotcompatiblewiththeseller'soptimality:hewillleave0surplustothethresholdtypeswithkoncehereceivesthismessage,ashehasnocommitmentpower.So,thereisnoequilibriuminwhichthehighandlowtypesfollowdierentpartitionruleswitha>a.Toconclude,unlessktypesareexcludedbytheseller,thektypesalwayshaveanincentivetopoolwiththem.Thebuyer'sexpectedutilityismaximizedinthepartitionequilibriumwithxaslongasktypesarenotexcluded.Inthisequilibrium,kandktypespooltogetherinintervalsoflengthxwhichisafunctionofk.However,ifthedierencekkishighenough,thesellerwillthenexcludethektypes.Belowarethethreepossibleequilibriathatmayariseunderthiscondition.Thispartagainmakesuseofthefollowingutilityforthebuyer:2U(;y)=k(y)(11)qThersttwocasesarisewhenthesellerexcludesktypesincasex=2k3partitionisplayed.Thishappensaslongas:7k>k512Then,thereare2dierentpartitionsthatcanariseunderthiscondition:If7k.332ThenwehaveU(2;y(m))>U(1;y(m)),asUisdecreasingindistance.ThenU(2;y(m2))(m2)U(2;y(m))(m)>U(1;y(m))(m).First,assumey(m2)>y(m).Then,U(3;y(m2))>U(3;y(m)).AsU(2;y(m2))(m2)U(2;y(m))(m)andU(2;y(m))>U(2;y(m2)),weconclude(m2)<(m).ThenitshouldbethecasethatU(3;y(m2))(m2)>U(3;y(m))(m),whichmeans3wouldinthebeginningprefertosendthemessagem2.Second,ify(m2)U(1;y(m))andU(2;y(m2))(m2)>U(2;y(m))(m)whichresultsinU(2;y(m2))U(2;y(m))>(m2)(m),andduetoU11>0,U(1;y(m2))U(1;y(m))>U(2;y(m2))U(2;y(m))>(m2)(m),whichnallyresultsinU(1;y(m2))(m2)>U(1;y(m))(m).ProofofLemma6:Proof.Ialreadyknowbylemma3thatthebuyer'sexpectedutilityisincreasinginthelengthsoftheintervalsaslongasthereisnoexclusion.Leastnumberofintervalsisequivalenttolargestintervallengths,andxisthelargestpossibleintervallengthsubjecttonoexclusion.Incase1isnotaninteger,thentherexaren2intervalsofsizexand2consecutiveintervalsiandi+1respectivelyoflengthsxandxwithx;xxandx+x=x>x.Thesurplusofbuyer122112typesintheseintervals,usingthefactthatthethresholdtypesareatdistancexi,2is:Zx1Zx22x12x22(f()f())d+2(f()f())d(13)0202subjectto:x1+x2=x(14)thepartialderivativesof13are:f(x1)f0(x1)andf(x1)f0(x1).Asfisconvex,then2222equation13isalwaysincreasingwhenoneintervalwidensattheexpenseoftheother.However,asnoexclusionrequiresthatx;xx,thenoptimalpartition12willhavethelasttwointervalshavinglengthxandx+xx.1231 ProofofLemma6:Proof.Letuslookatacasewhenthesellerexcludesatperiod1atotalmeasurecoftypesinanintervali,whichcanbeasingleor2separatedintervals.Letusassumethatcconsistsof2separatedintervalsoflengthsbi1andbi2,withbi1+bi2=c.Restrictattentiontob;bxsoallthetypescanbeservedwithnosubdivision.i1i2Asweknowthataslongasb;bx,thebuyeroptimalequilibriumsuggestsi1i2nosubdivision.Theseller'sexpectedprotwhent=2isinformative,sothatthetypesinbi1andbi2sendseparatemessagesatt=2andbothintervalsarecovered,willbe:bi1cbi1bi1(kf())+(cbi1)(kf())22FOCwrtbi1:cbi1bi1cbi10cbi1bi10bi1f()f()+f()f()=0222222c=2bsatisestheequality,sob=b=c.Ifthesellerisgoingtoexcludeai1i1i22totalofcmeasureoftypes,itisweaklyoptimalthatthesetypesarefoundintwoseparatedintervalsofequallengthwhennextperiodwillbeinformative.ProofofLemma7:Proof.Irestrictattentiontothecasewhenthereisnoinformativemessagesatperiod2.Ifi>aiandixandthereisasingleinterval,thesellerwillcharge(kf(2))andxxxmeasureoftypeswillaccept,givinghimprotsof(kf())whereaswhenxizi2thereare2separatedintervals,heeitherchoosestoselltooneoftheintervalsbi1orbwhichmeanshemakeslowerprotsincasemax(b;b)x,iallthetypespoolintheirmessages.Astheseller'soercoversatmostxmeasureoftypesinagiveninterval,thereisbx>0oftypesforwhomtradewillnotihappen.AssumeWLOGthatthesetypesarelocatedontheleftendoftheinitialinterval.Then,byseparatingandsendingadierentmessageatperiod2,theywouldn'taecttheallocationofthexintervaloftypes,plustheexpectedsurplusofthesetypeswouldbepositive.ProofofLemma9:Proof.Figure5demonstrates2intervalsiwithasubdivisionhappeningatperiod2.Lookattheintervalsofmeasurex.Whenthesetwointervalsarepoolingwithintervalai,theyareexcludedatperiod1andgettheperiod2oer,hencethe33 Figure5:intervaliwithoutsubdivisioniyib#bxxz(jaij2x)expectedsurplusovertheseintervalsis:Zx2x2(f()f())d(15)02Now,consideranotherpartitionwheretheseintervalswerefoundconsecutivelyontherightsideofintervaliandseparatefromtheintervaliandadoptastrategyofpoolingtogetherinperiod1andbabblingatperiod2.Bytheseller'sproblemandtheequation??,xwouldbeservedatperiod1andtheotherxatperiod2.Then,thesurplusovertheinterval2xwillbe:Zx2x(1+)2(f()f())d(16)02then,(16)>(15).Fromtheseller'sproblem,lemma5andlemma??,weknowthatz=z(x2x)whichisnotin uencedbythechangeinthestrategyofiithesetwosubintervals.Inaddition,allthetypesinsidetheintervalsxareweaklybetteroaswell:halfofthetypesgetthesameoerbutnowatperiod1insteadofperiod2.Hence,thisnewpartitionwhichhas2intervalsoflengthsx2x,i2xanddominatesthepreviouspartition.Toconclude,anyequilibriumpartitionwithsubdivisionisdominatedbyanotheronewhichismoreinformativeatperiod1andhasnosubdivisionatperiod2.ProofofLemma10Proof.ThisproofmakesuseofthequadraticutilityUB=k(y)2wheregiven2Rz2anintervalz,=kzandbuyersurplus22(zf())d=1z3.Lookatthe4046surplusunderpartitionrulea:11az(a)a0z(a0)(z(a)3+2()3)+z(a0)3+2()36a22aistheintervallengthwhichmaximizesthesurplusovertheintervala(1).aLetustaketheasymmetricintervaloflengtha0=1a1andtheneigboura34 intervaloflengtha,andlookforanimprovement.Calla+a0=c,andsupposethatthiswholeintervalisdividedintotwointervalsofakandak+1withzk=z(ak)andzk+1=z(cak),wherewlogak>ak+1.Theexpectedsurplusofthetypesinthesetwointervalsis:13akz(ak)33cakz(cak)3z(ak)+2()+(z(cak))+2()622maximizingwrtakandrearranginggives:(az(a))2(caz(ca))2z0(a)z(a)2kkz0(ca)z(ca)2kk+kkkk2222[(akz(ak))(cakz(cak))=02First,asa>ca,andz(a)isanincreasingandconvexfunction,z0(a)>kkk02(akz(ak))22(cakz(cak))2z(cak).Inaddition,z(ak)2>z(cak)2.So,thersttwoexpressionsgiveapositivevalue.Finally,asakz(ak)>cakz(cak)),thethirdexpressionisalsopositive.Hence,thewholeexpressionispositiveaslongasak>cak,whichmeansthatsurplusisincreasingwhenoneintervalisincreasedattheexpenseoftheother,subjecttotheconstraintthataa.Itconcludeskthatthereexistsanoptimalpartitionrulea.Proofofproposition6Proof.Thisisprovenbycharacterizingallthecandidateequilibria.Byusinglemma8and9,Irestrictattentiontoequilibriawheretradehappensforalltypeswithoutasubdivision.Thisleavesthefollowingcandidateequilibria:1.period1isinformativeand:period2isinformativeperiod2isbabbling2.period1isbabblingandperiod2isinformative.ThisproofmakesuseofthequadraticutilityUB=k(y)2.Period1informativesubcase1:period2babblingRestrictedtothecasewherea2x,theconditionforthemarkettobecoveredoverthetwoperiodsgivennosubdivisionatperiod2.Asxisthehighestintervalthatthesellercovers,then2xistheupperboundontheperiod1partition.By35 usinglemma10,thepartitionrulethatgivesthehighestsurplustothebuyerwhenplayinganinformativestrategyatperiod1andbabblingatperiod2is:Xni33a=argmaxz(ai)+(aiz(ai))(17)aii=1subjectto:Xniai=1(18)i=1theFOCwrtaigives:2020z(ai)(3z(ai)aiz(ai))+(aiz(ai))(2ai3aiz(ai)+z(ai))=(19)First,a'sshouldbeidenticalamongiasz(a)2(3z0(a)az(a))+(az(a))2(2aiiiiiiii3az0(a)+z(a))=foralli.z(a)=aaslongasaand,sountiltheno-delayiiiqpartition,z0(a)=1.Ata=2k(1),z0(a)=0,whichmeansandgivesalocal3maximum.Aslongasz(a)=a,thesurplusisincreasingina.Firstcandidateisthenthemaximumvaluez(a)cantakesuchthata=z(a),whichisand.Second,qwhena>2k(1),10,meaningz(a)isincreasingina,32andtogetherwiththeresultthatthebuyersurplusisincreasingintheintervals,thesecondlocalmaximumiswhenz(a)takesitshighestvalue,meaninga=2x.Theoverallexpectedsurplusofthebuyercanbewrittenas:12a6whichgives2k(1)intheno-delayequilibriumandk(1+)whenperiod2is99reachedwithpositiveprobability.k2(1+)>k(1)99if>1,andfor<1theno-delayequilibriumgivesthehighestsurplus.33Toconclude,thebestpartitionequilibriumwhenonlyperiod1isinformativeis:1.when>1:3thepartitionrulea=2xhalfofthetypesineachintervalacceptingtheoeratperiod1:z=xtheotherhalfrefusingandmovingontoperiod2:b=x36 2.when<13q(1)kno-delaypartitioninterval:23Theseller'ssurplusis:(kx2)(1+)=k(1+)when>1and(2)kwhen42333>1.3subcase2:period2informativeInorderfortradetobeensuredwithoutanysubdivision,thepartitionintervalsshouldsatisfya3x(inperiod1atmostxtypescanbeserved,andatperiod2,therewillbe2separateintervalsofx.)Ifa>3x,thenaszx,b=az>x,2sosometypeswouldbeexcludedincaseofnosubdivision.Theequilibriumsurplusofthebuyerfromapartitionruleawhenperiod2isinformativeis:3(az)31z(a)+(4)(20)a6derivatingwrtz:3(az)202z(a)(z(a))6a4asz0(a)>0,thisexpressionisalwayspositiveaslongasz>a.Byreplacingz(a)3fromtheseller'sproblem,theconditionsimpliesto:96K>0hence,z(a)>aissatised.Comparedtotheprevioussection,wherethesurplus3ismaximizedwhenz(a)takesitsmaximumvalue,theonlydierencehereisthat(az(a))ismultipliedby1.Also,z0(a)>0and@2z=8k[1][63]>0.Then2@2a2optimalzisgivenbythehighestvalueofaundertheconstraintaz(a)2x.Giventhatsubdivisionissuboptimal,thehighestvalueacantakeis3x,inwhichcasez(a)=xandb=x.Then,thepartitionrulethatgivesthehighestbuyersurpluswhenbothperiodsareinformativeis:a=3xz(a)=xb=b=x12comparingcase1andcase2:37 Letuscomparethebuyersurplus,wheregivenaninitialpartitionintervalai,ziaretheintervalsservedatt=1andbitheintervalsservedatt=2.Inthebabblingcase:nbX33(zi+bi)(21)i=1Pnwherei=1(zi+bi)=1Intheinformativecase:niX0303zi+2bi(22)i=1Pn00wherei=1(zi+2bi)=1ThenIhavez=z0=xandb=b0=x,andai=3xandai0=2xwhichiiiileadstonb=3ni.Replacingniandnbinthesurplusfunctions,itistrivialto2concludethatthesurplusinthebabblingequilibriumishigher.Babblingatperiod1Alltypessendthesameperiod1message,hencethereisasingleintervaloflength1.Atperiod1,bylemma6theselleroersy;suchthatz(1)=xintervaloftypesaccept.Betweentheequilibriawheret=1isbabbling,byusinglemma8whichsaysthattradeshouldtakeplacewithcertainty,thebuyeroptimaloneisinformativeatt=2andhasthepartitionrulex.Letusshowthatthisequilibriumisdominated.Assumethaty=1andthatthetypes2(0;2x)areexcluded2atperiod1andinperiod2separateintointervalsofx.Theexpectedsurplusoftypesinthesetwointervalsis1(2x3).Consideranotherequilibriuminwhich6thesetypesseparatethemselvesatperiod1,andbabbleatperiod2.Thisdoesn'taectthesurplusofothertypes,duetolemma5andlemma??.Fromtheseller'sproblem,thismeansoneoftheintervals(0;x)or(x;2x)willbeservedatt=1andtheotheroneatt=2.Theoverallsurplusofthesetypeswouldbehigherthanbefore:1x3(1+)>12x3.Infact,thesurplusofeachbuyertypeis66weaklyhigher,andforsomeitisstrictlyhigher.Finally,playingbabblingatperiod1cannotbethebuyer-optimalstrategy.Nowweconcludethatthesurplusmaximizingequilibriumstrategyforthebuyeroverallcandidateequilibriaistheonewhereperiod1isinformativeandperiod2isbabblingincaseitisreached.Proofofproposition7Proof.TheFOCofthesellergivestheidenticalztundertheinformativeandzt+1babblingcasesforzt;zt+1>0.However,intheinformativestrategythenumberofz'sineachinitialintervalevolveby2t1whereasthereissinglezateachttt38 inthebabblingcase.Inaddition,asztisdecreasingint,forthesameintervalamoretypesacceptthegoodinearlyperiodsinthebabblingcasecomparedtotheinformativeone.Takeaninformativeequilibriumwithzt>0untilperiodT.Then,considerababblingequilibriumwiththesameztforalltwhichalsolastsatmostTperiods.Thismeanszb=ziforalltwherebdenotesbabbling,andittdenotesinformativeequilibrium.Usingthequadraticutility,thesurplusfromtheinformativeequilibrium:PTt12t1z3t=1tPT62t1ztt=1subjectto:XTt1i2zt=at=1whereaiistherstperiodintervallengthintheinformativecase.TheFOCwithP3t122t2zt2ztt122t2zt3respecttoztgives(6PT2t1zt)2whichispositiveanddecreasingint.t=1Thesurplusfromthebabblingequilibrium:PTt13t=1ztPT6t=1ztsubjectto:XTbzt=at=1P3t1zt2ztt1zt3TheFOCwithrespecttoztgives(6PTz2whichisalsopositiveandde-t)t=1creasingint.Hencethesurpluswouldbehigherwhenearlierztwouldbehigher.NextIwillshowthattheperiod1intervalsaarenerinthebabblingequilibrium:PTz1=ni:therearemoret=1tt=1tabaiintervalsatperiod1inbabblingequilibriumthanintheinformativeone.Nowthesurplusintheinformativecase:ni[PTt12t1z3]t=1t6subjectto:XTt1ni2zt=1t=1Thesurplusforthebabblingcase:bPTt13n[t=1zt]639 subjectto:XTnbzt=1t=1wherenini2t1t=1tt=1tandfort>t,nb0,then=xk223otherwise,=0,meaningthatthewholeintervalisserved.Then,thehighestxwhichguaranteestrade,meaning=0sothatnotypeisexcluded,is:rkx=23Nextpartshowsthatxisindeedthebuyer'soptimalpartitionpartitionrule.buyer'sproblemThesurplusoftypesinapartitionintervalofmeasurex,takingintoaccountdeterminedbytheseller'soer,is:Zx[(x)2(x)2]d22xqqThereare2casestoconsider:xk0andxk<0.2323qqCase1:Ifxk0,thenbytheseller'sproblem,=xk.Replacing2323inthebuyer'ssurplusgives:Zxpk+223kx2pz+zxdzxk3423xderivatingthisexpressionwrtxgives:p53Kkp<09341 qwhichsaysthatxshouldbeassmallaspossible.However,aswehave=xk23qwhen0,itshouldbethatxk0.Thesmallestpossiblevaluexinthis23regionisthen:rkx=23qNextstepistochecktheintervalxk0whichiswhen=0.23Case2:Nowwecanreplace=0inthebuyer'ssurplus:Rxx22x2[z+xz]dz044xderivativewrtxgivesxwhichisalwayspositive.Hence,thesurplusincreasesinq3xupto2k.Thisconrmsthestatementthatthebuyergainsfrompoolingas3longasnotypeisexcludedbytheseller,meaningtradeisensured.Finallythebuyer-optimalpartitionruleis:rkx=2(23)3Asp1maynotbeaninteger,followingtheproposition3,thepartitionrule2kq3x=2kwillhave:3qp1intervalsoflength2k2k33qqif(1p1(2k))>0,alastintervalof1p12k.2k32k333Thesolutionofseller'sprobleminSection??Inapartitionintervalofa,z(a)istheintervaloftypesthataccepttheperiod1oer,andbtheintervaloftypesthatreject,meaningb=anz(a).Whenperiod2isinformativeWithoutsubdivisionFirstwelookatthecasewherethemarketiscoveredwithoutasubdivisionatt=2,whichrequiresa2x.Figure6demonstratestheintervalaforthiscase.42 Figure6:intervalawhent=2isinformativeay=20a1bzbrefuseandgoacceptatt=1refuseandgotoperiod2toperiod2Theseller'sproblemuponreceivingamessagefromanintervalaatperiod1andwhenperiod2willbeinformative:z2(az)az2z(a)=argmaxz[k()]+[k()](24)z224wherezistheintervalthatisservedatperiod1andaziseachofthetwo2separatedintervalsservedatperiod2.Thewholeintervalaisservedoverthetwoperiodsinthebuyer-optimalequilibrium,whichmeansthatthebuyer'sstrategyisinformativeenoughthattradeisguaranteed.z(a)isthenfoundas:q3a+28k[1][63]+9a22z(a)=(25)123qqfora2k(1).Fora2k(1),z(a)=awhichmeansthegameendsat33period1andtherewillnotbeanydelay.Thederivative@z:@a18a3+q8k[1][63]+9a22whichsimpliesto:23(6a8k[1])(26)q(1)kwhena2,whichistheregionwhereperiod2isreachedwithpositive3probability,@z(a)>0.Inaddition,@2z=36a>0whichmakeszanincreasing@a@aaz(a)andconvexfunctionofa.Finally,eachofthe2excludedintervals,b=is2foundas:q6a8k[1][63]+9a2212343 Thepartialderivative@b:@a9a6q>08k[1][63]+9a22asexpected,az(a)isalsoincreasingina.Inaddition,theconditionzb,meaningzaissatisedas:3r3228k[1][6]+9a>a(2)2whichsimpliesintheendto:322228k[1][6]>4a13a+a2qandasa3x,replacingawithx=6kgives:396k>0anz(a)b=,whichiseachexcludedinterval,isalsoincreasingina.z(a)b(a)2andsoz(a)a3Fora3xtradetakesplaceforalltypesofthebuyerevenwithoutasubdi-vision.Thisthreshold3xisthehighestvaluethattherstperiodintervalacantakesuchthatbx.Abovethisthreshold,unlessthereisasubdivisionamongthetypesthatrejecttheperiod1oer,tradewillnothappenwithsometypesofthebuyer.WithsubdivisionIncase2xx.Then,thesetypeswouldsubdivideat2period2.Theintervalalooksasingure7.Theprotofthesellerfromanintervalaisnowgivenas:z(a2x)a2xz(a2x)22z(a2x)k()+2(a2xz(a2x))k()22x2+2x[k()](27)244 Figure7:subdivisionawithsubdivisiony=a2aiai+1xbxzbxxtypesthatacceptatt=1Figure8:sellerprotsunder2cases0.01250.01200.01150.01100.01050.01000.00950.500.520.540.560.580.60qwherex=2k.3Indeed,thereisathresholdasuchthat,fora>a,thesellerisbetterowhensubdivisionhappens,andforaa,hewouldchoosetheoersuchthattherewillbenosubdivisiongivenbyequation25intheprevioussection,wherethethresholdais2x0:538,heisbetteroundersubdivision:servingz(a2x)=z(a0:4)atperiod1andexcludingaz(a0:4)types.Thevaluesarea=0:538,z=0:18andb=0:178581,whichconrmourndings.Whenperiod2willbebabblingWithoutsubdivisionWhenperiod2willbebabbling,bylemma7thetypesexcludedatperiod1shouldbefoundinasingleinterval.Theconditionfortradetohappenforalltypesover45 Figure9:awhent=2isbabblingawhent=2babblingy=z2zbtypesthatbuyatt=1typesthatrefuseandmovetoperiod2the2periodswithoutasubdivisionisa2x.Figure9demonstratesthiscase.Theproblemoftheselleristhen:z2az2z(a)=argmaxz[k()]+(az)[k()](28)z22derivatingwrtzgives:qa+a2+4(1)2k3z(a)=(29)1qqfora2kmeaningabovetheno-delaypartitionrule.Fora2k,z(a)=33aanditdoesn'tmatterwhetherperiod2isinformativeornot.Theno-delaypartitionruleisindependantofperiod2informationstructurebecausewhenthesellerdecideswhethertoexcludethemarginaltype,itdoesnotmatterwhethernextperiodwillbeinformativeorbabbling.az(a)b=isthen:2qaa2+4(1)2k31qLetusshowthatzforalla2x=4k:3q2a2+4(1)2ka(1+)3>01simplifyingthiscondition:16222k(1)>a[1+2]3qAs[1+22]<1anda4k(1)=2x,theconditionissatised.346 qz0(a)=+pa>0andz00(a)=a2+4(1)2kpa2>0a2+4(1)2k3a2+4(1)2k33@z>0and@2z>0,meaningzisanincreasingandconvexfunctionofa.Also,@a@az(a)b(a)andsoz(a)a2WithsubdivisionIncasea>2x,thenthesellerhasachoiceofincludingonlyz(ax)att=1ifasubdivisionwillbehappeningatperiod2.However,rememberthatitisinthebabblingequilibrium.Herebabblingequilibriummeansthattheintervaloflengthbxontheleftwouldpooltogetherwiththebxontheright,andxontheleftwouldpoolwiththeintervalxontheright.However,weknowthatwhennextperiodisbabbling,theexcludedtypesarefoundinasingleseparatedinterval.Theintervalagivenperiod1oerlooksasinthegure.Thenthesurplusofthesellerwillbe:z[ax]az(ax)x222z(ax)k()+(az(ax))k()+xk()222(30)Asweshowthatsubdivisionisunderoptimal,thiscaseisnotgoingtooccurinequilibrium.47 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