listeningtoleaders,andthenbasetheiractionsonwhattheyhear.Wecharacteriz - PDF document

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listeningtoleaders,andthenbasetheiractionsonwhattheyhear.Wecharacterizethein-uenceofleaderswithdifferentskillsbothwhentheiraudiencesareexogenousandalsowhentheirfollowerscanchoosetowhomtolisten.Acentralmessageemergingfromouranalysisisthataleader'sabilitytocommunicateclearlytothemassisrelativelymoreimportantthanherabilitytodiscoverthebestcourseofactionforthem.Inaextensiontoourmodelwealsoalloweachleadertochooseendogenouslytheclarityofhercommuni-cation.Beforefullydescribingourformalmodelandourresults,intheremainderofthisintroductorysectionofthepaperweexplainthemotivationforthiswork.Leadershipcanbeimportantwhenpoliticalactorswishtocoordinate.AssuggestedbyCalvert(1995),Myerson(2004),andDewanandMyatt(2007),aleadercanbefocal:whenaleadercommunicatesshehelpstounifyexpectationsabouthowamasswillact.3Lead-erscanalsohelppeopletolearn.AsLevi(2006)arguedrecently,“leadership...providesthelearningenvironmentthatenablesindividualstotransformorrevisebeliefs.”Asanillustrativeexample,whichwewillusethroughoutthepaperasavehicleforde-scribingourgeneraltheoryofleadership,considerthefollowingstylizedrepresentationofapoliticalpartypopulatedbyamassofactivists.Anactivistadvocatesthepolicyhebelievestobedesirable.Hemay,however,notknowwhichpolicyisbest.Heisalsoconcernedwiththecohesionofhisparty.Apartyismoresuccessfulwhenitsmembersadvocatesimilarpolicies,andlesssowhenthereiswidespreaddiscordance.Becauseofthisapartyactivistwouldliketoadvocateapolicythatisinlinewithothers;intheab-senceofcommonexpectationsthe“partyline”maybehardtodiscern.Inthissituationapartyleaderhasinuenceviahercommunication.Shemightconveyinformationtoac-tivists,perhapsviaadirectspeechtothepartymembershiporviaothermediachannels,andsoaidthemintheiradvocacy.Thisalsohasfocalproperties:herwordscouldcreateacommonviewpointaroundwhichsupportcancoalesce.Thisisimportant,sinceanac-tivistfacesuncertaintynotonlyaboutwhichisthebestpolicy,butalsoaboutwhatothersthinkisthebestpolicy.Successfulcoordinationdependsuponaccurateassessmentsofothers'beliefs;leadershiphelpstoprovidesuchassessments.Withinthisframework,agoodleaderhelpsamassofactorstoachievetheirgoals:hercommunicationfosterstheunderstandingthatisneededforthemtochoosetherightac-tions,andtochoosethemtogether.Ontheotherhand,asuccessfulleaderisonewhohasinuence:herwordsimpactupontheactionstakenbyherfollowers.Theperformanceofaleaderonbothdimensionsdependsonherqualities.AsLevi(2006)suggested,“[the] 3Thusaleadermightbeinterpretedasacoordinating“rationalritual”inthesenseproposedbyChwe(2001).HereweagreewithChwe(2001)thatcollective-actionproblemsextendbeyondthefreeridingemphasizedbyOlson(1965)andincludemanysituationsinwhichcoordinationisrequired. 4 varianceintheclarityofcommunication.Forexample,intheUnitedKingdomtheshifttoaudio-broadcastingofparliamentarydebatesin1978,theintroductionoftelevisedde-batesin1989,andsimilarchangestothecoverageofparty-conferencespeeches,allowedawideraudiencetolistendirectlytothespeechesmadebyleaders.Toassesstheeffectofthesetwokeyleadershipqualitiesweanalyzeagameinwhichfollowerswishtochoosethebestactionwhileconformingascloselyaspossibletotheactionschosenbyothers.Theylistenandrespondtoleaders.Inequilibrium,therelativeinuenceofaleaderandherfollowers'aggregateperformanceincreasewithhersenseofdirectionandclarityofcommunication;goodandsuccessfulleadershipcoincide.Anemphasisuponoratoricalabilitymayseemquaint,belongingmoretotheworldofCicerothantothemodernworldofpoliticalcommunication.However,ourresultsrevealthataleader'sclarityofcommunicationisrelativelymoreimportantthanhersenseofdirection:heuristically,aleaderwhocanperfectlycommunicateanimperfectopinionhasmoreinuencethanaleaderwhoimperfectlycommunicatesaperfectone.Drivingthisisthedesireforunity:whenaleaderspeaksclearly,followersrallyaroundacommonlyunderstoodso-called“partyline”eventhoughitmaydifferfromtheideal.Theimportanceofclaritysuggestsafurtherquestion:howmightsuchclarityendoge-nouslyarise?Theclarityofaleader'smessageisaffectedbywhetherfollowerslistentoher:iftheypaycarefulattentionthentheyunderstandwhatshehastosay.However,payingattentiontooneleaderentailsbeinglessattentivetoanotherandsoleadstoagameinwhichfollowersendogenouslydecidetowhomtolisten.Indeed,thedesiretocoordinatemeansthatafollowerlistenstothoseleaderswhoalreadyattracttheatten-tionofothers.Thissuggeststhatanelitesubsetofclearoratorsmayattractattentionandobtaininuencebyactingasfocalpointswhileothersareignored.Ofcourse,anambitiousleaderdesirespowerandinuence.Shemayadaptherrhetor-icalstrategytoattractattentiontoherviews.Thisdrivesawedgebetweengoodandsuccessfulleadership:agoodleaderhelpsfollowerstotaketherightactions,whereasasuccessfulleaderenjoysdecisiveinuenceorattractsthebiggestaudience.Thepromi-nenceofclearcommunicatorsintheelitewhoattractattentionsuggeststhatleaderswillspeakasclearlyaspossible.However,anear-perfectcommunicatordeliverstheessenceofhermessageinashortperiodoftime.Afollowerneednotlingerinheraudience;havingheardwhatheneedstohear,hemovesontolistentoothers.Thislogicsuggestsaroleforobfuscation:aleadermightdeliberatelychooseanopaqueformofwords,avoidspeakingviatransparentmedia,orcommunicateviainterlocutors. 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Keynes(1936)believedthatthislogicreectedelementsofprofessionalinvestmentinthestockmarket.Inrecentyearseconomictheoristshavebuiltformalmodelsofthebeauty-contestparable(MorrisandShin,2002;AngeletosandPavan,2007)andusedthemtoanalyzepublicannouncementsinmonetaryeconomies(Amato,Morris,andShin,2002;Hellwig,2005),complementaryinvestments(AngeletosandPavan,2004),andtheroleofhigherorderbeliefsinassetpricing(Allen,Morris,andShin,2006).Aswewillshow,thebeauty-contestparablecanbeusedtodevelopimportantinsightsintothefocalroleofleadership.Wenditusefultodescribeourownversionoftheparableasavehicletoconveyourideas.Ratherthanthinkofcontestantschoosingtheprettiestfaces,wecanthinkofmembersofapoliticalpartyadvocatingandcampaigningforthebestpolicies.Thepressureforunityandconformitywithinpoliticalpartiesthenprovidesanincentivetobackapolicythatislikelytobepopular,whichsubsequentlyleadstotheanticipationoftheaverageopinion.Totellthisstorymoreformallywebuildasimultaneous-movegameplayedbyaunitmassofactivistpartymembersindexedbyt2[0;1].4Anactivistadvocatesapolicyat2R.Thismightbeinterpretedasthepositionhesupportsatapartyconference,orthepolicyhepromotesduringanelectioncampaign.Drawingtogethertheactionsofallpartymembers,the“partyline”istheaveragepolicyadvocated:aR10atdt.Apartyactivistpursuestwoobjectives.Firstly,hewouldliketoadvocatethepolicythatbestmeetstheparty'sneeds.Secondly,hewishestocoordinatewithothersinhisparty.Thatis,aconcernforpartyunitydriveshimtoconformtothepartyline.Werepresentthesetwinconcernsviaapairofquadraticlossfunctions:ut=u�(at�)2| {z }(i)concernforpolicy�(1�)(at�a)2| {z }(ii)desireforconformity:indexesanactivist'srelativeconcernforchoosingtheidealpolicycomparedtomain-tainingpartyunity.5Toensurethatbothconcernsarepresentweassumethat01.Whenactivistssharecommonknowledgeofthenitisoptimalforthemalltoadvocatethesameidealpolicyat=.Infact,thisistheuniqueNashequilibriumofthegame.6Whenthisissothereisnotensionbetweentheactivists'twinobjectives.Itistemptingtoconcludethatthereisnodisagreementbetweenpartymembers. 4Thefundamentalresultsofthispapercontinuetoholdwithanitenumberofpartymembers.5Thelossfunction(at�a)2capturesthepenaltyofnon-conformity.Adifferentmeasureofmis-coordinationistheaggregatedistanceR10(at�at0)2dt0betweenanactivist'sactionandthepoliciesadvocatedbyothers.Asweexplainintheappendix,theuseofthismeasureofpartydisunitydoesnotchangeactivists'behavior.6Forthiswerequire�0.When=0activistscareonlyaboutcoordinatingandsothereareinnitelymanyNashequilibria:itisanequilibriumforallactiviststobackanyarbitrarypolicya6=. 8 sothat~st=~st0,thentheysharethesamebeliefsabouttheidentityofthebestpolicyandaboutthelikelysignalsofotheractivists;putsuccinctly,activistsareexantesymmetric.Whatevertheinformativesignalsrepresent,anactivist'sadvocacystrategyisamappingfromsignalrealizationstopolicychoices;formally,at=At(~st):Rn!R.Astrategyprolemightinvolvetheuseofdifferentstrategiesbydifferentplayers.However,onceweseek(BayesianNash)equilbriaitiswithoutlossofgeneralitytorestrictattentiontosymmetricstrategyproles,sothateveryplayerusesthesameadvocacystrategyA(
).Thisisbecauseeachindividualisnegliblysmallandso,conditionalonobservingthesamesignalrealization,twodifferentactivistsseetheirworldinthesameway.Sinceanybestreplyisunique(andhencestrict)thisimpliesthattheybehavesimilarly.AnadvocacystrategyyieldsaBayesianNashequilibriumwhenitspeciesanoptimalchoiceforanactivist,givenhisbeliefs,andwhenthosebeliefsareconsistentwiththeparty-wideuseofthestrategy.GivenactivistsuseastrategyA(
),anactivist'sexpectationofthepartylineisE[aj~st]=E[A(~st0)j~st]fort06=t.Similarly,hisexpectationoftheidealpolicyisE[j~st].HencethestrategyA(
)formsanequilibriumifandonlyifA(~st)=E[j~st]+(1�)E[A(~st0)j~st]:(?)Thusanactivist'sstrategyisaweightedaverageofhisexpectationoftheidealpolicyandhisunderstandingoftheaveragepolicyadvocatedbypartymembers.7ToobtainasolutiontoEquation(?)weneedtospecifyfullyhowsignalshelpanactivisttoformbeliefsabouttheidealpolicyandbeliefsaboutthesignalsseenbyotheractivists.Todothis,weturnourattentiontothedetailsofthemechanismviawhichactivistslearn.LEARNINGFROMLEADERSLeadershelpafollowertodevelophisbeliefsabouthisworldandaboutthelikelymovesofothers.Onceagain,ratherthanexplainhowfollowerslearnfromleadersinanabstractsetting,wedescribeourmodelwithinthecontextoftheparty-activiststory.Activistsbeginwithnosubstantiveknowledgeoftheidealpolicy:theyshareadiffusepriorover.8Theylearnbylisteningtonpartyleadersindexedbyi2f1;:::;ng.Theterm“leader”canbeviewedasalabelforaninformativesignal;indeed,thesourcesofinformation 7Ifactivistsaresubjecttoprivatepolicybiases,asdiscussedearlier,thentheequilibriumadvocacystrategychangesinastraightforwardway.Specically,supposethattheidealpolicyofactivisttis+btwheretheprivatepolicybiasbthaszeromeanandisindependentofactivists'informationsources.Inthissetting,anequilibriumadvocacystrategytakestheformat=A(~st)+btwhereA(
)isthesolutiontoEquation(?).8ItisstraightforwardtoextendouranalysistoaworldinwhichactivistsshareacommonpriorN(;2). 10 aboutwhatissaidbutratheraboutwhatismeant.Thus,2iwillbelargewhenaleaderuseswordstowhichtheactivistswhofollowherattachvaryingdenitions.Despitethisinterpretation,itmightseemextremetoimposelimitsontheclarityofthecommunicationprocess;presumablyaleadercanchooseunambiguouswordsandsaypreciselywhatshemeans,sothat2i=0.Weoffersomejusticationsforsetting2i�0.Firstly,aleader'sclarityisrestrictedwheneversheisunabletospeakdirectlyandsimul-taneouslytoactivists.Indeed,theabilitytospeakdirectlytoanentiremassisarelativelyrecentphenomenon.Priortoradioandtelevisionaleaderwouldneedtotravelfarandwidetoreachanaudience,herabilitytocommunicatehinderedbythephysicaldemandsofsuchagruelingschedule.Asasubstitutefordirectcommunicationaleadermightrelyoninterlocutorstoprovideaccountsofherwords.Forexample,intheUnitedKingdomtheLobbyservesasthegatewaytotheHouseofCommons.Ministerialbriengsaredeliveredtolobbyjournalists.Althoughlobbymembersobservethesameevents,theiraccountofproceedingswilldiffer;imprecisereportingcompromisesclarity.Secondly,evenwhenaleadercancommunicatedirectly,comprehensionmaybelimited.Giventime,anarticulatespeakermaygetherpointacrossbyremoving,clausebyclause,anyvagueness.Butsuchalegalisticapproachcanprovetheantithesisofclarity—thelanguagebecomeslongandtortuousandthetaskofabsorbingthemessageinitsentiretymovesbeyondmostlisteners.Thelimitstotheattentionspanofalistenermaythenimposeaboundontheoverallclarityofthemessagereceived.Suchlimitsmightbeendogenouslychosenbythelistener:evenifthereisenoughtimeforhimtoabsorballofaleader'swords,doingsohasanopportunitycostsinceitpreventshimfrompayingcarefulattentiontootherleaders.Thirdly,theremaybeendogenouslimitstoaleader'sclarity.Evenifaleaderisabletospeaksuccinctlyusingunambiguouswords,shemaychoosenottodoso.Indeed,inlatersectionswearguethatattention-seekingleaderswillintentionallyobfuscate.Rhetoricalstrategiesthateschewclarityhavebeenstudiedelsewhere,notablybyZeckhauser(1969)andbyShepsle(1972)whobothconsideredthestrategicadvantagerealizedbyacandi-datewhoisvagueoverthepolicyshewouldimplementifelected.Shepsle(1972),inparticular,highlightedthe“politician'sadvantageinspeakinghalf-truthsandinvaryinghisappealswithvariationsinaudienceandpoliticalclimate.”9 9TheconsiderationofequivocationrelatesourworktothatofAragon´esandNeeman(2000)whostudiedequilibriumlevelsofcandidates'ambiguity.OurworkcontrastswiththatofMierowitz(2005).Hecon-sideredpoliticianswholearnaboutvoters'preferencesduringprimaryelections.Candidateswhorefrainfromcommittingtospecicpolicieshavegreaterexibilityuponreceivinginformationfromtheprimary. 12 FOLLOWINGTHELEADERSWenowaskhowactivistsreacttothespeechestheyhear.Whenthereisonlyoneleader,anactivistcandonobetterthantofollowtheadvicegiveninherspeech;thisadviceyieldsanunbiasedestimateoftheidealpolicyandthepartyline,solongasothersbehaveinthesameway.However,whenthereisnoclearleaderapparent,differentleadersmustbeassessedaccordingtotheircompetencies.ThisassessmentiscapturedbyanequilibriumpolicyadvocacystrategyA(
)satisfyingEquation(?).Inprinciple,anequilibriumstrategycouldtakeacomplicatedfunctionalform.Fortu-nately,however,weareabletofocusourattentiononasimple,robust,andeasilyinter-pretedclassofstrategies.ActivistsemployalinearstrategyifA(~st)=Xni=1wi~sitwherewiisacoefcientattachedtothespeechoftheithleader;thisprovidesaconvenientmeasureofthisleader'sinuenceontheactionsofthemass.Ourfocusonlinearadvocacystrategiesstemsfromtheuseofthenormaldistributioninthespecicationofourmodel.NormalityensuresthattheconditionalexpectationsoftheidealpolicyE[j~st],ofotheractivists'signalsE[~st0j~st],andoftheleaders'underlyingsignalsE[sj~st]arealllinearin~st.Ifanotheractivistusesalinearstrategy,thenthecondi-tionalexpectationE[A(~st0)j~st]ofhisactionisalsolinearin~st.Thisimpliesthatifallotheractivistsusealinearstrategythenabestreplyistousealinearstrategy.(Thatis,theclassofsymmetricandlinearadvocacystrategiesisclosedunderbestreply.)Pushingfurther,weobtainthefollowinglemma(formalproofsarecontainedinthetechnicalappendix.) Lemma1. ThereisauniqueBayesianNashequilibriuminvolvingtheuseoflinearstrategies,sothatA(~st)=Pni=1wi~sit.Thisequilibriumsatiseswi�0foralliandPni=1wi=1.Whentheuniquelinearequilibriumisplayed,anactivistadvocatesaweightedaverageofthepolicyrecommendationsthathehears.Theweightwiplacedonaspeechactsasanindexoftheorator'seffectiveness;itcanmeasurethesuccessofaleader.Ofcourse,thepossibilityofnon-linearequilibriaremains.Nevertheless,inthetechnicalappendixweexplainhowamildfurtherrestrictiononadvocacystrategiesrulesoutnon-linearequilibria.Furthermore,evenifnon-linearequilibriaexist(weconjecturethattheydonot)thenfurthercriteriasuggesttheselectionoftheuniquelinearequilibrium.11 11WehavenotedthatweleanheavilyupontheworkofMorrisandShin(2002)andindeedtheirmodelisobtainedbysettingn=2,21=0and22=0.Theyclaimedthatnon-linearequilibriadonotexist.However,thereisasmallchinkintheprooftheyused.Weexplainfurtherinourtechnicalappendix;theproblemwehighlightwasalsonotedinarecentpaperbyAngeletosandPavan(2007,p.1112). 14 Proposition1. TheuniquelinearBayesianNashequilibriumadvocacystrategysatiseswi=^ i Pnj=1^ jwhere^ i=1 2i+2i:Partywelfareisu�1=[Pni=1^ i].Aleader'sinuence,indexedby^ i,increaseswithbothhersenseofdirectionandherclarityofcommunication.Therelativeinuenceofbettercommunicatorsincreasesasactivists'concernforpartyunitygrows:if2i2jthen i= jisdecreasingin.Theindex^ imeasuresinuenceandthereforesuccessfulleadership;however,sincewel-fareincreaseswithPni=1^ iitalsomeasuresgoodleadership.Aninuentialleader(withahighvalueof^ i)clearlycommunicateshersharpsenseofdirection.Unsurprisingly,aleaderwhoexcelsonbothdimensionsenjoysmoreinuence.Nevertheless,andsincewi&#x-277;0foralli,evenunskilledleadersenjoysomeinuence.Sowhichskillismoreimportant?Thepresenceofintheindex^ isuggeststhataleader'sabilitytogiveclearexpressiontoherviewsismoreimportantthanherabilitytounder-standthepoliticalenvironment.Ofcourse,thisclaimreliesonanimplicitassumptionthatitisappropriatetocomparedirectlythevariances2iand2i.Evenifthiscompari-sonisinappropriate,thenalclaimofProposition1revealsthattherelativeinuenceofclearcommunicatorsgrowsaspartycohesionloomslargerinthemindsofactivists.SincePni=1wi=1thisnecessarilymeansthattheabsolutefollowingofpoorercommunicatorsmustfallasthatofskilledoratorsgrows.Furthermore,aninspectionof^ irevealsthatinuenceisentirelydeterminedbyclarityasvanishestozero. Corollary. Anincreaseinactivists'concernforpartyunityshiftsinuenceawayfrompoorcom-municatorsandtowardclearcommunicators:ifleaders'labelsareinstrictorderofdecreasingclarity,sothat2122


2n,thenthereissomeksuchthatwiislocallydecreasinginforikandlocallyincreasinginfori&#x-277;k.Ifissufcientlysmallthenw1&#x-277;w2&#x-277;


&#x-277;wn&#x-277;0.(Appropriatelymodiedclaimsmaybemadewhenever2i=2jforsomei6=j.)Toobtainfurtherinsightrecallonceagainthataleaderhelpsactiviststolearnaboutpolicyandtocoordinate.Hermessageaboutpolicyismuddledbytwosourcesofnoise:anyerrorsofjudgementonherpart(thevariance2i)plusanymisunderstandingofwhatshesays(thevariance2i).Combiningthesesourcesofnoise,~sitjN(;2i+2i)sothat i1 2i+2i=QualityofInformation:Ifactivistscaredonlyaboutdiscoveringthebestpolicy(sothat=1)thenthetwocomponentsofaleader'sskillsetwouldbeequallyimportant.Eachactivistwouldchoose 16 Sinceistheweightplacedonanydeviationfromtheidealpolicy,theremainder1�isthedesireforpartyunity.Proposition2revealsthedeterminantsofgoodleadership:Leadership=QualityofInformation 1�[CorrelationofMessagesDesireforUnity]:Fixingthequalityofinformationprovided,coherentcommunicationdeterminestheef-fectivenessofleadership,andmoresowhenthereisagreaterdesireforpartyunity.Infact,itisusefultocompareaperfectcommunicator(i1,sothat^ i i=)withaper-fecttechnocrat(j0,sothat^ j j).Asvanishes,sothatonlypartyunitymatters,theperfectcommunicatorbecomesfarmoreinuentialthantheperfecttechnocrat.LISTENINGTOLEADERSTointerpretourmodelwemightthinkofpartymembersattendingalargepartyconfer-encewhereeachlistenstospeechesmadefromtheconferenceplatform.Animplicitas-sumptionisthatactivistsformacaptiveaudience.Underthisassumption,weconcludedthattheclearestcommunicatorsenjoyrelativelymoreinuence.Ofcourse,speechesconveyinformationonlyiftheyareheard.Activistsmayabstainfromlisteningtoaparticularspeech,ormaynotdevotetheirfullattentiontoit.Theclarityofaleader'smessagedependsonthewillingnessofheraudiencetolisten;butthedecisiontolistenisendogenous.Thisisimportantwhenbeinginformediscostlyas,forexample,whenactivistshavelimitedattentionspansandcannotlistentoaleaderindenitely.Furthermore,therewillbestrategicinteractionamongstactivistswhentheydecidetowhomtolisten.IncommonwithHellwigandVeldkamp(2008)werecognizethattheremaybecoordinationmotivesininformationacquisitionsothat,usingtheirlanguage,ifanactivistwantstodowhatothersdo(coordinationofpolicyadvocacy)thenhewantstoknowwhatothersknow(coordinationofattention).Toanalyzetheseeffectsweextendourmodel.Activistsaregivenasingleunitoftime(perhapsthedurationofapartyconference)toallocatetodifferentleaders:activisttspendsaproportionxitofhistimelisteningtowhatleaderihastosay.Wethinkofhimasobservingasampleof(noisy)observationsoftheleader'sviews.Inthissense,thetimespentlisteningrepresentsthesamplesize.Intheusualway,thesamplevariancedeclineswiththesamplesize;equivalently,theprecisionoftheaggregatesignalislinearlyincreasinginxit.Thisleadsustothespecication~sitjsiNsi;2i xitandsoxit 2i=ClarityofMessage; 18 Whenallleaderssharethesamecommunicationskills(i=jforalli6=j)thenallleadersenjoyanaudience,andtheattentionpaidtoeachisproportionaltohersenseofdirection.However,whenleadersdifferintheircoherencericherresultsemerge:activistsgravitatetowardtheclearestcommunicators.Correspondingly,oncealeader'sclarityfallsbelowathreshold(thatis,wheni�Km)activistswillignoreher;suchaleadercanhavenoinuence.Whilstintuitivelyonemightthinkthatagoodsenseofdirectionwoulddemandattention,ourresulthighlightstheimportanceofgettingthemessageacross.13Despitethisnding,communicatingtooclearlycandeectattentiontowardothers:wheniKm=2forim(thenoiseinaleader'sspeechisrelativelylow)anincreaseinherclarityreducestheattentionpaidtoher.Wereturntothisissueinduecourse,whenweevalutealeader'sincentiveeithertoclarifyortoobfuscatewhenspeakingtoherparty.Nevertheless,sufcientclarityisapre-requisiteforsuccessfulleadership.OnepossibilityemergingfromProposition3isthatm=1,sothatactivistslistenonlytothebestcommu-nicator.Suchaleader,shouldsheexist,enjoysundividedattentionandthusundilutedinuence;shebecomesadefactodictator.Butwhenwillsuchaleaderemerge? Proposition4. Recallthatwehave(withoutlossofgenerality)orderedtheleadersbydecreasingclarity,sothat21


2n.Theclearestcommunicatorisadefactodictatorifandonlyif22211+21 212:Thisfailswhen21issufcientlysmall.Hence,fortheclearestcommunicatortoenjoyexclusiveattentionsheneedstocommunicateimperfectly.Theclarity21=21whichbestsupportsherdic-tatorship(byminimizingtheright-handsideoftheinequality)increaseswithhersenseofdirectionandthedesireforunity.If2122,thensheenjoysexclusiveattentionifissmallenough.Adefactodictatormustbetheclearestcommunicator(Proposition3).Herpowerandinuencearemaximizedonlywhentheattentionofherfollowersisnotdivertedtoothers.Forhertoenjoythisexclusiveattention,however,theclarityofherclearestcompetitormustbesufcientlylow;equivalently,22(and2iforotherleadersi&#x-277;2)mustbelarge.Beingtheclearestcommunicatorisnotenough;2122issufcientfordictatorshipinonlytwocases.Therstcaseiswhen!0,sothatactivistscareonlyaboutpartyunity,andtheclearestcommunicatorisbestabletodescribeafocalpolicyaroundwhichthepartycanrally.Thesecondcaseiswhen21!0,sothatthebestcommunicatoralsoenjoysanexcellentsenseofdirection;sheisaChurchillianleaderwhotrumpsallothers. 13Proposition3differsfromthecentralresultofHellwigandVeldkamp(2008),sincetheyndmultipleequilibria.Thereasonisthatouractivistscanvarycontinuouslytheattentionpaidtoaleader,whereastheirmodelisequivalenttooneinwhichaleadercanbelistenedtoforaxedlengthoftime,ornotatall. 20 himtodothis,butgiventimecontraintshewillnotlistentoaleaderlongerthanheneedsto.Whenaleaderisagoodcommunicator,anactivistcandiscernherpositioninashortperiodoftime;withtimetospare,hemovesontogathermoreinformation.Itisalsoworthwhilenotingthatthefocusofactivists'attentiondependsupontheirrel-ativepreferenceforpolicyversuspartyunity.FromProposition3,mdeclineswiththepolicy-concernparameter.Hence,asthedesireforunitygrows(sothatfalls)thesizeoftheeliteshrinks;Proposition4revealsthatadefactodictatoremergeswhenissmallenough.Theintuitionisnatural:whenactivistscareonlyaboutunitythentheinforma-tionregardingpolicyprovidedbyleadersisirrelevant,andallthatmattersisndingaclearfocalpointaroundwhichthemembershipcancoalesce.OBFUSCATIONSofarthecharacteristicsofleadershavebeenexogenous.Thisseemsreasonableforaleader'ssenseofdirectionwhich,atthetimeofspeakingtoherfollowers,islikelytobebeyondhercontrol.However,aleader'sclarityofcommunicationmaybemorema-nipulable:theoverallclarityofamessagedependsendogenouslyonbothspeakerandaudience.Forinstance,aleadermightbeabletoreduceherclarity;shemayobfuscate.Hereweextendourmodelbyallowingforendogenousobfuscation.Ofcourse,ifaleaderchoosesherclaritythenwemustconsiderherobjectives.Forinstance,abenevolentpartyleadermaywishtomaximizethewelfareofherparty'smembership.Sinceactivistsmakeefcientdecisionswhenchoosingtowhomtolistenandwhomtofollow,theleaderhasnoreasontodistorttheirbehavior.Indeed,shecanbestenhancetheirwelfarebymaximizingtheinformationavailabletothem,andshedoessobyspeakingasclearlyaspossible.Ofcourse,theobjectivesofanambitiousleadermayextendbeyondbenevolence,andthereforetheremaybeavarietyofreasonsforhertodistortactivists'decisions;obfusca-tionisonewaytodothis.Forinstance,apower-hungryleadermaywishtomaximizeherinuence.Wehavealreadynotedthatsuchinuenceismaximizedwhenaleaderenjoysapositionasadefactodictator,andwehaveseen(Proposition4)thatthisrequireslessthanperfectclarity.Thesourceofthisrequirementisthefactthatareductioninclaritycanincreasetheattentiondevotedtoaleader.Indeed,aleadermustattractattentionasitisanecessarycomponentofexercisinginuence.Toexplorefullythedesireforatten-tionwefocusonthisdimensionofaleader'sambitionandask:whatrhetoricalstrategymaximizestheattentionpaidtoher?(Laterinourpaper,wewillreturntoconsiderotherpossibleobjectivesandshowthattheytooresultinobfuscatoryrhetoric.) 22 isequivalenttoapricehike:itdirectlyincreasesherrevenue(intheformoftheatten-tionpaidtoher)foragivenquantity(clarityofmessage);ontheotherhand,obfuscationpromptsanactivisttolowerhisdemandforherproduct(speech)bysubstitutingtooth-ers.Balancingthetwoeffectsofachangeinclaritygeneratesanintermediatesolution.Lemma3revealsthatleaderswillsometimesobfuscate.Tomovefurther,however,wendtheuniqueequilibriumofourattention-seekinggame. Proposition5. Thereisauniquepure-strategyNashequilibriumoftheattention-seekinggame.Orderingleaderssothat 21


 2n,theirclaritiessatisfy21


2n.Thereisaunique~m2f0;1;:::;ngsuchleadersi�~mspeakwithmaximumclarity(2i= 2i)whereasleadersi~mobfuscate(2i� 2i).Thosewhoobfuscatechoosetheclarity:ifij~mthen2i=2j.If 2i~2foralli(sothatallleadershavegoodcommunicationskills)then~m=n(theyallobfuscate)andtheuniqueNashequilibriumissymmetric,satisfying2i=~2foralli.Hence,ClarityofCommunication=1 ~2=1 nXi=11 2i=AggregateSenseofDirection 1�[DesireforUnity]:Anincreaseinanyleader'ssenseofdirectionandactivists'desireforunity:(i)(weakly)increaseseveryleader'sclarityofcommunication;(ii)(weakly)reducesthenumber~mofleaderswhoobfus-cate;and(iii)(weakly)reducesthesizemoftheelitewhoattractanaudience.LinkingPropositions3and5,thereare(potentially)threegroupsofleaders.Firstly,thoseleadersi~mmhaveexcellentcommunicationskillsandyetdonotexploitthem;withoratoricalairinabundance,theyneverthelessattractmaximumattentionbyobfuscat-ing.(Infact,theirchoicesallsatisfyi=Km=2.)Sincethese~mleaderschoosethesameclarity,therelativeinuenceofeachindividualisdeterminedbyhersenseofdirection.Bycontrast,leadersinthesecondgroup~mimhavelessdevelopedoratoricalskillsandmuststraintobecleartothebestoftheirabilitiesinordertoattractanaudience.Theexogenouslimitstotheiroratoricalskillsmeanthattheydonotobfuscate,buttheyareheardsolongas iKm.Aleaderi&#x-306;mtalksonlytoherself.Toenjoyanaudienceshemustimproveheroratoricalskills:onlywhenthenoiseinherspeechisnogreaterthantwicethatoftheobfuscating(andhenceclearest)communicatorswillherviewsbeheard.Ofcourse,thethreedifferentclassesofleadermaycollapsetoasinglegroup.Forinstance,when 2n~2everyleaderwillobfuscate.Thiswillbesowhentherearefewexogenouslimitstoclarity;perhapsaworldinwhichaleaderisabletodeliveracommonlyheardspeechdirectlytotheentireactivistmass.Perhapssurprisingly,theuniqueequilibriumidentiedbyProposition5(illustratedinFigure2)revealsthatallleadersspeakwiththesameclarity,eventhoughtheydonotshareacommonsenseofdirection. 24 Forexample,whenleadersareconstrainedtouseprimitivemedia(forinstance,aseriesofprivateaudiences)onlythemostoratoricallyskilledleaders,whocanspeakclearlyandconcisely,attractattention.Astechnologyimproves,sothatthesameleadershipspeechcansimultaneouslybebroadcasttoallactivists,thenexceptionaloratoricalairisnolongerapre-requisiteforsuccess.Thus,whilstaleaderinthemouldofTheodoreRoosevelt—combiningphysicalstamina,oratoricalair,andtheabilitytodeliveranef-fectivestumpspeechtomanydifferentaudiences—enjoysinuencedespitetechnologyconstraints,others,notsowellendowed,canourishonlyastechnologyadvances.Forexample,itiswelldocumentedthatCalvinCoolidgelackedthecapacitytosucceedonthestump.Heowedmuchtobroadcasting;inhisownwords(CornwellJr,1957): “IamveryfortunatethatIcameinwiththeradio.Ican'tmakeanengaging,rousing,ororatoricalspeechtoacrowd...butIhaveagoodradiovoice,andnowIcangetmymessagesacrosstothemwithoutacquaintingthemwithmylackoforatoricalability.”Ofcourse,iftechnologyallowsallleaderstocommunicateclearly,thenallwillobfuscate.Thisrhetoricalbehaviorhas(perhapssupercial)similaritieswiththe“garbling”ofmes-sagesinsender-receivergamesanalyzedbyCrawfordandSobel(1982)andextendedtopoliticalsettingsbyGilliganandKrehbiel(1987)andLi,Rosen,andSuen(2001),amongstothers.Inthese“cheaptalk”scenariosaninformedpoliticianisrestrictedtosendinggarbledmessagesduetohercommonlyunderstoodpolicybias.Ourleadershavenoinherentpolicybias,butthestrategicincentivetoobfuscatearisesnevertheless.Althoughaleadermayobfuscate,herpayoffdoesnotdependonthepoliciesadvocatedbyactivists;shehasnodirectincentivetomisrepresentthetruthassheseesit.Heronlystrategicmoveistochangetheprecisionoftheinformationshetransmits.Thusourlead-ers'speechesaresignalsinthestatistical,ratherthanthegame-theoretic,sense;anactivistneednotanticipateanybias.17Ourmodelthuslacksthestrategictensionsthatarisewhenleadershavepolicypreferencesthatdifferfromthoseoftheirfollowers.Nevertheless,suchbiasescouldariseendogenouslyfromattention-seekingbehavior.Forexample,supposethatwhenfallswithinaspecicrangeofvaluesinformationismoreeasilyconveyedthanwhenfallsoutsidethatrange.Thismightbesowhenthereisaparty-wideconsensusaboutthecorrectpolicy.Forexample,apolicythatformsthestatusquomighteasilybedescribedtoanaudience,whereaselucidatinganalternativepathrequiresalongerexposition.Alternatively,aparticularpolicy(sayleftorright)mayrequiresmore(orless)clarication.Insuchscenariosthereisanincentiveforaleader 17Incontrast,Hermalin(1998)describedatheoryof“leadingbyexample”inwhichthecostlyactionofaleader(sender)actsasatextbooksignalofherprivateinformationtoherfollowers(receivers). 26 Weobservethefullimportanceofthispointwhenweconsiderthewelfareandpolicy-performanceimplicationsofattention-seekingleaders.Recall(Proposition1)thatpartywelfareincreaseswithPni=1^ i,andsotheindex^ imeasuresbothgoodleadershipandsuccessfulleadership;itreactspositivelytoaleader'sclarityaswellashersenseofdi-rection.Allowingtheattentionpaidtoleaders(andhencetheoverallclarityoftheirmessages)tobedeterminedendogenously,thesituationbecomesmorecomplex.Clar-ityofcommunicationremainscriticalinensuringthataleaderreceivessomeattention.However,whereasincreasedclaritybenetsactivistsandisthusacomponentofgoodleadership,anattempttoseekattentionortomonopolizetheagendamayinducealeadertoreduceherclarity;asuccessfulleader(asopposedtoagoodleader)mayobfuscate.Thevanityofattentionseekersseparatesgoodandsuccessfulleadership.Allowingourleaderstoplayagameinwhichtheysimultaneouslychoosetheirrhetoricalstrategiesmightbeexpectedtocomplicatethingsfurther.Inpracticeitsimpliesmatters.Since(when 2n~2)allleaderschoosethesameclarity,theattentionpaidtoeachleaderisproportionaltohersenseofdirection.Theleadershipindexreducesto^ i=1 2i+[2i=xi]=1 22iandsopartywelfareincreaseswiththecombinedjudgementoftheleadership,butfallswithactivists'relativeconcernforpolicy.Theeffectofisunsurprising,sincethepolicycomponentofanactivist'slossfunctionreactstotwosourcesofnoiseratherthanone;evenxingthebehaviorofallactors,ourwelfaremeasurewillfallwithanincreasein.Amoresurprisinginsightisobtainedbyconsideringanobjectiveperformanceindex.Apartyexhibitsgoodpolicyperformanceifthepoliciesadvocatedbyitsmembersareclosetotheidealpolicy.AnappropriatemeasurehereisthelossfunctionR10(at�)2dt=E[(at�)2].Theinverseofthisprovidesourmeasureofpolicyperformance. Proposition6. If 2i~2foralli,sothatallleaderschoosethesameclarity,thentheequilibriuminuenceofaleaderisproportionaltohersenseofdirection.Furthermore,PolicyPerformance1 E[(at�)2j]=1 1+nXi=11 2i:whichincreaseswiththeleaders'combinedsenseofdirectionbutdecreaseswiththepolicy-concernparameter;agreaterdesireforpartyunityimprovesthepolicyperformanceoftheparty.Paradoxically,activistsbecomebetteratadvocatingthebestpolicyastheycarelessaboutdoingso.Recognizingtheendogenousqualityofleadershipprovidesthecorrectintu-ition.Whenactivistsdesireunitytheyseekoutacommonpartyline,andsoleaders 28 Wealsoaskedwhetherleaderscanbebothgoodandsuccessful.Agoodleaderenhancespartyperformance,aidingactivistsinpursuitoftheirtwingoalstothebestofherabilities.Asuccessfulleadercommandsattention.Onlywhenskillsareexogenousdogoodandsuccessfulleadershipnecessarilycoincide.Otherwise,aleadermayincreasehersuccessbyobfuscatinghermessage;activistsreceivelessinformationandconsequentiallyarelessinformedabouttheirenvironment.Perhapssurprisingly,thereislessobfuscationwhenactivistsplacemoreemphasisonfollowingthepartylinethanpursuingthebestpolicy.Whenapartyemphasizesunity,itprovidesleaderswiththenecessaryincentivessothatgoodandsuccessfulleadershipcoincideandpolicyperformanceimproves.Fewwoulddenythatleadersplayanimportantroleininuencingpeople'sactions,andindeedthereisexperimentalevidencethattheydo(Humphreys,Masters,andSandbu,2006;G¨uth,Levati,Sutter,andvanderHeijden,2004).Yettherehasbeennorecentformalworkwhichevaluatestheinuenceofdifferentleaders.Thispaperhasattemptedtollthisgap.BuildingonsuggestionsmadebyLevi(2006,p.10)that“leadership—bothofgovernmentandwithincivilsociety—providestheagencythatcoordinatestheeffortsofothers”wehaveexploredaworldwithmultiplepotentialleadersdifferentiatedbytheirskillsetsandextendedouranalysistoaworldwhereleadershipskillsemergeendoge-nously.Ourworkprovides,wehope,asmallstepinresponsetoLevi's(2006,p.11)claimthat“stilllackingisamodeloftheoriginsandmeansofensuringgoodleadership.”TECHNICALAPPENDIX UnityandConformity. InFootnote5wenotedthatR10(at�at0)2dt0mightbeabettermea-sureofpartydisunity.ItisstraightforwardtoconrmthatR10(at�at0)2dt0=(at�a)2+R10(at0�a)2dt0,andsoanactivist'spayoffwouldbecome~ut=ut�(1�)R10(at0�a)2dt0.Thesecondtermisindependentofatatthemarginandsoisirrelevanttothedecision-makingofactivistt.Hencethebehaviorstemmingfromthisrevisedspecicationwouldbeequivalenttothatarisingfromthespecicationusedinthetext.Oneelementofouranalysisis,however,affectedbytheuseoftherevised“partydis-unity”lossfunction.When~utisanactivist'spayoff,partywelfarebecomesE[~ut]=E[ut]�(1�)var[at0j]=E[ut]�(1�)nXi=1w2i2i:TheequilibriumadvocacystrategymaximizesE[ut]ratherthanE[~ut].Whatthismeansisthat,fromawelfareperspective,activistsplacetoolittleweightonrelativelyclearleaders.(Ofcourse,ifwespeciedpolicyperformanceasourwelfaremeasure,capturedbytheexpectedlossE[(at�)2],thenwewouldreachtheoppositeconclusion.) 30 andhenceinvertible.Thusthesolutionforw,andhencethelinearequilibrium,isunique.ThecoefcientsmustsatisfyPni=1wi=1toensurethattheacommonshiftinallsignalsresultsinthesameshiftinactivists'actions.(Solvingforwexplicitlyconrmsthis.) ProofofLemma2. Theeffectofchangingais@[R10ut0dt0]=@a=�2R10(at0�a)dt0=0. ProofofProposition1. FromtheproofofLemma1,theuniquelinearequilibriumsatisesw=[I�(1�)B0]�1b.Ratherthancalculatethisdirectly(thisapproachisusedinoursupplementaryappendix)weuseLemma2:ndingthe(unique)equilibriumboilsdowntominimizingPni=1w2i[2i+2i]subjecttoPmi=1wi=1.IntroducingtheLagrangemulti-pliertherst-orderconditionstaketheform2wi[2i+2i]=foreachi,orequivalentlywi=^ i=2;thejointsolutionyieldstheproposition'smainclaim.Thewelfaremeasurefollowsbysubstitution,andthecomparative-staticclaimsfollowbyinspection. ProofofProposition2. ^ ifollowsfromsimplealgebra,andthecomparative-staticclaimsregarding iandifollowbyinspection.Takinglogarithmsanddifferentiating,@log(^ i=^ j) @=�i i ((1�i)+i)2(1�i)+i i��j j ((1�j)+j)2(1�j)+j j=j (1�j)+j�i (1�i)+i0,i&#x]TJ/;༕ ;.9;U T; 12;&#x.426;&#x 0 T; [00;j;whichyieldsthenalclaimoftheproposition. Non-LinearEquilibria. InthetextwecharacterizedtheuniquelinearBayesianNashequi-libriumofthebeauty-contestgame,butnotedthatthepossibilityofnon-linearequilibriaremainedopen.Wealsosuggestedthatamildrestrictiononadvocacystrategieselimi-natesanynon-linearequilibria.Hereweexpanduponourclaims.Weusethefollowingnotation.(i)Thesubscripttindicatesanexpectationconditionalontheinformationofactivistt,sothatEt[
]E[
j~st].(ii)E[
]R10Et[
]dtistheaverageexpectationacrossthemassofactivists.(iii)ForanypositiveintegerkwedeneEk[
]inductively:E1[
]E[
]andEk+1[
]E[Ek[
]].(iv)BR[A(
)]:Rn7!RisthebestreplyofactivisttgiventhatallothersplayA(
).(v)Finally,foranypositiveintegerkwedeneBRk[A(
)]inductively:BR1[A(
)]BR[A(
)]andBRk+1[A(
)]BR[BRk[A(
)]].Theextraconditionwhichweimposeprevents“exploding”higher-orderexpectations. Denition. FixapolicyadvocacystrategyA(
)playedbypartymembers.Higherorderexpecta-tionsofthepartylinearenon-explosiveiflimk!1kEt[Ek[A(
)]=0forany2(0;1). 32 strategysatisesBRk[A(
)]=A(
)forallk,wecanbeassuredthatanynon-linearequilib-riamustgenerateexplosivehigher-orderexpectationsofthepartyline.ThelogicusedherewasemployedbyMorrisandShin(2002).Theyclaimedtondauniqueequilibrium,butdidnotprovethat(1�)k+1Et[Ek[A(
)]]!0;asAngeletosandPavan(2007)noted,theiranalysiswasnotquitewatertight.Indeed,itispossibletondstrategieswhichgenerateexplosivehigher-orderexpectations.Consider,forinstance,aworldinwhichn=1,21=0,and21=2�0.Sincethereisonlyoneleader,wedropthe“i”subscript.IfA(~st)=e~stforsome�0thenEt[Ek[A(
)]]=ek22+~st,andso(1�)k+1Et[Ek[A(
)]]divergesif(1�)e22�1.BeforepresentingtheproofofProposition3itisusefultoderivethepropertiesofKm. Lemma4. Withoutlossofgeneralityorderleaderssothat21


2nanddeneKi+Pii0=1[2i0=2i0] Pii0=1[i0=2i0]:(i)Forallji:Kiis(strictly)increasingin2jifandonlyifj()KiandKiis(strictly)increasingin2jifandonlyifj(&#x]TJ/;༕ ;.9;U T; 9.;ą ;� Td;&#x[000;)Ki=2.(ii)Foralli&#x]TJ/;༕ ;.9;U T; 9.;ą ;� Td;&#x[000;1:i()Kiifandonlyifi()Ki�1.(iii)Thereexistsauniquem2f1;:::ngsuchthatiKiforallimandiKiforalli&#x-380;m.(iv)Ki&#x-380;Ki+1forallimandKiKi+1forallim,andhenceKmKiforalli2f1;:::;ng.(v)muniquelysatisesiKii+1. Proof. (i)Kiis(strictly)increasingin2jifandonlyifitis(strictly)decreasingin[1=2j]:@Ki @[1=2j]=2j Pii0=1[i0=2i0]�j+Pii0=1[2i0=2i0] [Pii0=1[i0=2i0]]2()0,j()Ki:Next,wediffereniatewithrespecttojtoobtain@Ki @j=2j 2jPii0=1[i0=2i0]�+Pii0=1[2i0=2i0] 2j[Pii0=1[i0=2i0]]2=2j�Ki 2jPii0=1[i0=2i0](�)0,j(�)Ki 2:(ii)AstraightforwardbuttediousalgebraicexerciseconrmsthatKi=Ki�1ifandonlyifi=Ki�1.ThisimpliesthatKi�i=0whenevaluatedati=Ki�1.Now,@Ki @ii=Ki=i=2i Pii0=1[i0=2i0]1:ThisimpliesthatKi�iisstrictlydecreasinginiwhenevaluatedatKi�i=0.(iii)Denemtobelargestmemberoff1;:::;ngsatisfyingiKiforallim.(Suchanmcanbefound,since1K1byinspection.)Ifm=ntheclaimholds.Ifmnthenm+1Km+1.Sinceleadersareorderedbyclarity,m+2m+1Km+1andhence,byusingclaim(ii),m+2Km+2.Continuinginductively,jKjforalljm+1. 34 Weturntothecomparative-staticclaims,andfornowassumethatKmm+1,sothatalocalchangeinKmdoesnotchangethesizeoftheelite.Ifm=1thenx1=1isinvarianttolocalparameterchanges.Form2andimdifferentiatexiwithrespectto[1=2i]:@xi @[1=2i]=i(Km�i) +i 2i@Km @[1=2i]=i(Km�i) +i 2i"2i Pmj=1[j=2j]�i+Pmj=1[2j=2j] [Pmj=1[j=2j]]2#=i(Km�i) "1�i=2i Pmj=1[j=2j]#�0:Next,differentiatexiwithrespecttoi:@xi @i=1 2iKm�2i+i@Km @i=Km�2i 2i"1�i=2i Pmj=1[j=2j]#:Thisis(strictly)positiveifandonlyifiis(strictly)lessthanKm=2.Thecomparative-staticclaimsalsoholdwhenKm=m+1:foranylocalparameterchangewhichincreasesKm,andsointroducesleaderm+1intotheelite,wereplaceKmwithKm+1intheformulaederivedabove,notingthatKm=Km+1whenKm=m+1.Finally,thesizemoftheeliteisdeterminedbytheinequalitiesmKmm+1.Changesin2ifori&#x-277;mhavenoeffect.Afallinm+1(oriniforanyi&#x-277;m,followinganappropriaterenamingofplayers)canexpandtheelite.Anincreasein(byinspection)orin2iforim(fromLemma4)increasesKmandsomayexpandtheelite. ProofofProposition4. Forattentiontobefocusedontheclearestcommunicator,weneed2K1=+[21=21] 1=21=11+21 21:Squaringyieldsthelowerboundon22givenintheproposition.Now,@K21 @21=1+21 212�2211+21 2121 41=1�21 212:Thisisincreasingin21andhenceK21isconvexin21.Settingthederivativetozeroyields21=21,asclaimed.Theremainingclaimsfollowbyinspection. ProofofLemma3. Aleader'soptimallychosenclaritysatisesi=maxf i;Km=2g.Toprovethisclaim,notethatifi(&#x]TJ/;༕ ;.9;U T; 12;&#x.811;&#x 0 T; [00;)Km=2thenshewouldwishtoreduce(increase)herclarity.Thisisbecauseeither(i)xi2(0;1)islocallyincreasing(decreasing)ini(fromProposition3,or(ii)xi2f0;1gandtheadditionalassumptionsmadeforthesecasesapply.Henceif iKm=2theni=Km=2andif i�Km=2theni= i.Weconcludethatsheobfuscatesif iKm=2.ToobtainalowerboundtoKm=2letuschooseasetofclaritiesforallleaderstominimizeKm.FollowingProposition3thisisachievedwhen 36 wherethesecondequalityfollowsfromsubstitutionandsimplication.Thisincreaseswithanddecreaseswiththeaggregatesenseofdirectionoftheleaders. ConstructionofFigure1. UsestheformulafromProposition4. ConstructionofFigure2. Wecomputedthereactionfunctionforaleaderwithunconstrainedclarity.Notethatsuchaleaderichoosesherclarityofoptimallywheni=Km=2.Hence2i=Km=+Pmj=1[2j=2j] Pmj=1[j=2j]=2i+A+2i B+i,2i+Bi�(A+2i)=0;whereA2iPj6=i[2j=2j]andB2iPj6=i[j=2j].Solvingforthepositiveroot,i=�B+q A+B2+2i=vuut 2i+2iXj6=i2j 2j+4i"Xj6=ij 2j#2�2iXj6=ij 2j:Forthecaseoftwoplayersthisreducestoi=s 2i+2j2i 2j+4i 4j�j2i 2j;whichistheformulausedintheconstructionofFigure2.REFERENCES ALLEN,F.,S.MORRIS,ANDH.S.SHIN(2006):“BeautyContestsandIteratedExpectationsinAssetMarkets,”ReviewofFinancialStudies,19(3),719–52. ALVAREZ,R.M.,ANDC.H.FRANKLIN(1994):“UncertaintyandPoliticalPerceptions,”JournalofPolitics,56(3),671–88. AMATO,J.D.,S.MORRIS,ANDH.S.SHIN(2002):“CommunicationandMonetaryPolicy,”OxfordReviewofEconomicPolicy,18(4),495–503. ANGELETOS,G.-M.,ANDA.PAVAN(2004):“TransparencyofInformationandCoordinationinEconomieswithInvestmentComplementarities,”AmericanEconomicReview:AEAPapersandProceedings,94(2),91–8. 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