Auditing, disclosure, and verification in decentralized decision problems

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ContentslistsavailableatScienceDirect

Journal

of

Economic

Behavior

&

Organization

jo u r n al ho me p ag e :ww w . e l s e v i e r . c o m / l o c a t e / j e b o

Auditing,

disclosure,

and

veriﬁcation

in

decentralized

decision

problems

夽

Luca

Anderlini

a,∗

_,

_Dino

_Gerardi

b

_,

_Roger

_Lagunoff

a

a_Georgetown_University,_USA b_Collegio_Carlo_Alberto,_Italy

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received5February2016

Receivedinrevisedform30August2016 Accepted9September2016

Availableonline30September2016 JELclassiﬁcation: C73 D63 D72 D74 H11 Keywords: Auditing Disclosure Agencybias Ideologicalbias

a

b

s

t

r

a

c

t

Westudytherelativeperformanceofdisclosureandauditingindecentralizedinstitutions. Weconsidertheinformationtransmissionproblembetweentwodecisionmakerswhotake actionsinsequenceattwodecisiondates.Thefirstdecisionmakerhasprivateinformation aboutastateofnaturethatisrelevantforbothdecisions,andsendsamessagetothe second.Theseconddecisionmakercancommittoonlyobservethemessage(disclosure),or canretaintheoptiontoobservetheactionofthefirstdecisionmaker(auditing)or,atsome cost,toverifythestate.Inequilibrium,stateverificationwillneveroccurandthesecond decisionmakereffectivelychoosesbetweenauditinganddisclosure.

Whenthemisalignmentinpreferencesreﬂectsanagencybias–abiasinadecision maker’sownactionrelativetothatoftheother–thentheseconddecisionmakerchooses toauditinequilibrium.Whenthemisalignmentinpreferencesreﬂectsanideologicalbias –onedecisionmakerprefersallactionstobebiasedrelativetotheotherdecisionmaker –then,foralargeenoughmisalignment,theseconddecisionmakerchoosesdisclosurein equilibrium.Ourresultsindicatethattheabilitytocommittoforegotheaudithasvaluein thelattercase.

1. Introduction

Inanyongoingorganizationorsocietytherearetwoprimarywaysinwhichinformationisrevealed.Oneisdisclosure– transmissionofinformationbytheinformedpartiestouninformedreceivers.1_Disclosure_consists_of_all_forms_of

manufac-turedtransmissionincludingoralandwritten(ﬁles,archives,etc.)communication.Theactofdisclosinginformationmayor maynotbeobligatory.2_The_key_is_that_its_content_can_be_freely_chosen_by_the_informed_sender.

夽 Apreviousdraftwascirculatedunderthetitle:“DoActionsSpeakLouderthanWords?Auditing,Disclosure,andVeriﬁcationinOrganizations.”Wewish tothankTomGresik,anAssociateEditor,andtwoanonymousrefereesfortheirhelpfulcommentsandsuggestions.Wealsothankseminarparticipants atJohnsHopkins,NorthCarolina,PUC(Chile),HKUST,theIADB,UniversidadAlbertoHurtado,Vanderbilt,andtheMidwestTheoryMeetingsatIndiana University.AllthreeauthorsaregratefultotheNationalScienceFoundationforﬁnancialsupport(GrantSES-0617789).Someoftheworkwascarriedout whileLucaAnderliniwasvisitingEIEFinRome.Hegratefullyacknowledgestheirgeneroushospitality.

∗ Correspondingauthorat:GeorgetownUniversity,37thandOStreetsNW,Washington,DC20057,USA. E-mailaddress:luca@anderlini.net(L.Anderlini).

1_Notice_that_while_we_use_the_term_disclosure_as_synonymous_with_cheap_talk,_it_has_also_been_used_in_the_literature_as_revelation_of_information_that_is veriﬁablebythereceiver.

2_Record_keeping_requirements_in_firms_fit_this_definition.

http://dx.doi.org/10.1016/j.jebo.2016.09.002

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Theotherformofrevelationisdiscretionaryauditing–theattemptbytheuninformedpartytoinvestigateorverifythe informationinsomefashion.AuditingmaytakeplaceundertheguiseofaformalprocessorinstitutionsuchastheGeneral AccountabilityOfﬁceintheU.S.government,oritmaytaketheformofaninformalmonitoringarrangementinacollective actionproblem.

Thispaperstudiestherelativeperformanceofdisclosureandauditingindecentralizeddecisionsettings.Inthesesettings, itisoftendecisionmakersthemselveswhomakechoicesaboutwhichofthetwoinformationchannelsisbest.

Inﬁrms,forinstance,differentmanagersarepromotedtokeydecision-makingpostsovertime.Itisstandardpractice foracurrentmanagerialteamwithinside knowledgetoprovideadetailedprospectusofthecompany’soverallhealth periodically.Evenso,itspolicies,expenditures,andacquisitionsmayalsobeauditedtoprovidesomeexternalveriﬁcation toshareholdersandsubsequentgenerationsofmanagers.

Politicalsystemsoperateinmuchthesameway.Rivalpoliticalpartiesvieforpowerovertime.Arulingpoliticalparty holdssomeinsideinformation(forinstance,knowledgeoftheeffectsofaparticularregulation)andisrequiredtodisclose whatitknowsorobserves.Arivalpartyeventuallyreachespowerandmustthenchoosetoeitheraccepttheoutgoingparty’s wordsorinsteadinvestigatetheoutgoingparty’spoliciesanddecisions.

Otherexamplesincludecommonpoolproblemsinwhichuserswithcurrentaccesstotheresourcemustdecideonthe bestmeanstoextractinformationaboutthestockfromearlierusers,andregulatoryenvironmentsinwhichregulatorsmust decideonhowstructurereportingrequirements.

Withtheseexamplesinmind,wepositatwo-periodpolicysettingmodelthatattemptstocapturesomeofthekeytrade offsinthechoiceoftransmissionmechanism.Weassumethatthepolicyateachdateischosenbyadistinctdecisionmaker. Disclosureisassumedtotaketheextremeformofcheaptalk,whileauditingtakestheextremeformofbeingabletodirectly observetheactionstakenbyothers.Inperiod1,theinitialdecisionmaker–DM1(the“sender”)–choosesanaction(a “policy”)andacheap-talkmessageafterprivatelyobservingtherealizationofanunderlyingstatevariablethataffectsthe preferencesofbothDMs.Inperiod2,theseconddecisionmaker–DM2(the“receiver”)–choosesapolicy.

ThechoicesofpossibleinformationchannelsopentoDM2areasfollows.Shecanchoosetocommittoonlyobserve DM1’smessagebeforeDM1takesanyactionorsendshismessage.Werefertothisasacommitmenttorelyondisclosure. ThedecisionbyDM2ofwhetherornottorelyondisclosureisobservedbyDM1whothenchooseshispolicy.IfDM2chooses nottocommit,sheretainstheoptiontoobserveDM1’saction,whichwerefertoasauditing,ordirectlyverifythestate– referredtosimplyasveriﬁcation.Forsimplicity,auditingisassumedtobecostlessandstateveriﬁcationhasavanishing small(lexicographic)cost.3_Once_period₂_arrives,_DM2_is_in_charge_of_the_organization_and_her_action_is_then_taken.

Misalignedpreferencesbetweenthetwotemporallyseparatedagents(e.g.,twomanagerialteams,twopoliticalparties, ortworesourceusers)distortincentivesawayfromproﬁt-maximizing,orwelfaremaximizingpolicies.Theeffectofthis misalignmenthereisthatfulldisclosuredoesnotoccur.Inthissenseoursetupisrelatedtotheclassiccheaptalkenvironment inCrawfordandSobel(1982)(henceforth“CS”).TheirsenderisourDM1,whileDM2istheirreceiver.UnlikeintheCS environment,policydecisionsherearemadebybothplayersandthereceivermaychoosetoauditbeforetakingherown decision.

Weaskwhendisclosurewillbeused,andwheninsteadorganizationsrelyonauditing,orevenveriﬁcation,inequilibrium. Theanswerisnuancedanddependsontheparticularcharacteristicsoftheorganization.

Ourfirstresult,Proposition1,showsthatDM2willneverchoosethedirectverificationoptioninequilibrium.Intuitively, ifDM1expectsthestatetobeverifieddirectly,hewouldchoosehismostpreferredfirstperiodpolicytailoredtothestate. Butbecausehispolicywouldfullyrevealthestate,theuninformedDM2wouldthenswitchtothelesscostlypolicyaudit. Thisisshowninanypurestrategyequilibrium,butwelaterverifythatitholdsgenerically(inthelevelofbias/misalignment) formixedstrategyequilibriaaswell.4

AsaresultofProposition1,DM2’seffectivechoiceisbetween“words”(disclosure)andthe“actions”(policyaudit)ofher predecessor,DM1.Thechoicebetweenwordsandactionsultimatelydependsonthewedgeinpreferencesbetweenthetwo parties.Werefertothismisalignmentasthe“bias”(followingCS)andexaminethefollowingtwocanonicalcases.

Inthecaseofideologicalbias,theﬁrstdecisionmakerDM1alwayspreferssystematicallyhigherpoliciesthanDM2, regardlessofwhichpartychoosesorwhichperiodthechoiceismade.Theideologicalbiasesarisenaturallyinpoliticswhere competitionbetweenideologicallydifferentiatedpartiestakesplace.

Ideologicalbiascontrastswiththecaseofagencybiaswherebytheactivedecisionmakerprefersahigher(orlower)policy thanhiscounterpart,thepassivestakeholder.ThismeansthatDM1(DM2)prefers,say,ahigheractioninperiod1(period2) thanDM2(DM1).Arguably,manyprivatefirmsconformtothiscasesincethefuturedecisionmaker(DM2)isoftenselected internallyfromexistingmanagement,andthepreviousmanagerholdsstockoptionsthatincentivizeshisfuturedecisions. Thesetwoformsofbiasgiverisetoquitedifferenttransmissionprotocols.Withagencybias,DM2willgenerallychoosea policyaudit.However,underlargeenoughideologicalbias,DM2willcommitinadvancetouseonlythedisclosedinformation. Inotherwords,wefindthatunderideologicalbias,theuninformedpartywillrelysolelyontheinformedparty’scheaptalk inequilibriumandthusforegoallsubsequentauditandstateverificationopportunities.

3 _The_assumption_of_costless_policy_audits_is_shown_to_be_without_loss_of_generality._All_the_results_go_through_if_audits_costs_are_small._Our_main_result (Proposition3(i))goesthroughwitharbitraryauditingcosts.SeeSection5foradetaileddiscussion.

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Togainsomeintuitionfortheseresults, weobserveﬁrstthatequilibriainwhichDM2reliesonlyonmessageswill resemblethoseinstandardcheaptalkmodels(e.g.CS).Thatis,whileDM1can,inprinciple,reportfalseinformation,she choosesnottoinequilibrium.Instead,thesender’smessageispartially,butnotfully,informative.Thelackoffulldisclosure isofcoursebadforDM2.Incontrast,whenapolicyauditoccursinequilibrium,DM1chooseshisactiontakingintoaccount bothitsdirectpayoffeffect,drivenbythebias,anditssignalingvaluedrivenbyDM1’spreferencesoverDM2’sactioninthe secondperiod.

Whenthewedgebetweenpreferencestakestheformofanagencybias,thesetwoeffectsroughlyoffsetoneanother, therebymitigatingtheeffectofthesender’sbias.However,inthecaseofideologicalbias,thedirectpayoffeffectandthe signalingvaluereinforceoneanother;bothinduceactionsthataretoolargefortherecipient.Ifthereinforcedeffectofthe biasislargeenough,theauditbecomesundesirableforDM2whothenavoidsitinequilibrium.Infact,weshowthatforgoing theauditinganddirectveriﬁcationoptionsinfavorofdisclosureisnotjustpreferablefortherecipientunderideological bias,butitisParetoimprovingforallpartiesiftheideologicalbiasislargeenough.

AcriticalpartoftheargumentisthatDM2cancommitinadvancetoforegooratleastlimitthescopeofadiscretionary audit.Tosomeextent,thiscanbeviewedasasimplifyingassumptionsincethecommitmenttoforegoauditingisshownto bepreferredbybothdecisionmakersinmostcases.Thecommitmentbecomesinstitutionalizedwhenaconsensusexists amongallthepartiesexante.Standardcorporatepolicy,forinstance,placesauditingauthorityinthehandsoftheﬁrm’s boardofdirectors,ratherthantheCEO.5 _Governments_make_allowances_as_well_to_prevent_public_audits_for_reasons_of

nationalsecurity.

Naturally,insomecasescommitmentisnotalwayspossible.Theresultsofthepaperinformthesecasesaswell:without commitment,discretionaryauditingwilloccurmakingallpartiesworseoffunderalargeenoughideologicalbias.Atthis pointthelossofwelfarerelativetothedisclosureregimemaywellbecomeapparenttotheparticipants,butitsimplyistoo latetoavoidit.

Inourviewthisisatleastpartofthereasonwhy“audits”inthepoliticalsphereareoftenviewedas“partisanwitch hunts”where“allactorsinvolvedinaccountabilityprocessesuseavarietyofstrategiestoarguetheircaseandapportion blame”(Boinetal.,2008).A“witchhunt”ofthistypecanbeusedtouncoverembarrassingfactsaboutone’spoliticalrival. Inthelanguageofthemodel,themorepreciseistheembarrassinginformation,theeasieritisfortheinvestigatingpartyto implementitsidealpolicychoice.

Indeed, the frequency of partisan investigations is an indication that institutional commitments to avoid audit-ing/investigatingaremoredifﬁcultinpoliticalsystems.

“Partisaninvestigationshavehistoricallybeenapoxonbothhouses[intheU.S.Congress]–embarrassingthe inves-tigatorsasmuchastheinvestigatedandwoundingthemajorityparty.[...]Toooften,investigativehearingsturnout tobeﬂightsofvanityforacommittee’smemberstospeechifyandhumiliatewitnesses.”–Goldberg(2010)

Thedestructivenatureofsuchinvestigationssuggestsaparadox.Inﬁrms(whereagencybiasisprevalent),commitmentto forgoauditingmaybepossiblethoughunnecessarysinceauditswillalwaysbechoseninequilibriumregardlessofwhether commitmentispossible.Inpolitical systems,auditsoftenassumetheformofapartisaninvestigation.Inthis casethe commitmenttoforgoauditingwouldbechoseniffeasibleandwouldinfactbepreferableforallpartiesifthebiasislarge enough.Unfortunately,commitmentsofthistypearenotusuallycodiﬁedinpoliticalsystemsandmaythusbeinfeasible.

Theremainderofthepaperisorganizedasfollows.Section2discussesrelatedliterature.Section3setsoutthemodel.In Section4wecharacterizeequilibriaunderbothideologicalbiasandagencybias.Section5examinestherobustnessofthe mainresults,includingtheextensiontomixedstrategies.Section6concludeswithabriefdiscussionoftheaccountability systemsfoundinﬁrms,polities,andinformalcollectiveactionscenarios.AllproofsarerelegatedtoanAppendixtothe paper.Apreﬁxof“A”inthenumberingofequationsandsoonindicatesthattherelevantitemcanbefoundinAppendix.

2. Relatedliterature

Weexaminetheequilibriumchoiceofdisclosure,auditing,ordirectveriﬁcationbytheuninformedagent.Withsome exceptions(discussedbelow)mostoftheliteratureexaminesoneofthesetransmissionmechanisms,butdonotavailall theoptionstothemodeledagents.

Thereisawellestablishedliteratureon(costly)stateverificationbeginningwithTownsend(1979)(seeBoltonand Dewatripont,2005forfurtherreferences).Bywayofcomparison,therelevantfeatureinourmodelisthatstateverification doesnotariseinequilibriumwhenauditinganddisclosureareavailable,evenwhenverificationcostsaresmall.

DisclosureisthemainfocusofthecheaptalkcommunicationliteraturebeginningwithCS.6_The_present_paper_contains

elementsofdynasticcheaptalkreminiscentofSpector(2000)andAnderlinietal.(2012)(AGL)whichexaminetheeffectsof biasamongmultipledecisionmakersandtheincentivestodisclosethecontentsofone’sobjectivesignals(viacheaptalk) toothersinthetemporalchain.Inthesepapers,cheaptalk(disclosure)wastheonlyoptionfortransmittinginformation; neitherauditingordirectveriﬁcationwereavailabletotheuninformedagents.

5_See_[13]_.

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ConditionallyonDM2notcommittingtodisclosure,thepresentmodelisconnectedtothevastliteratureonsignaling startedbySpence(1973),sinceauditinginvolvestheveriﬁcationofpayoffrelevantdecisions.Amodelrelatedtooursin thesignalinggenrecomesfromCarrilloandMariotti(2000)who examineasingleagentproblemwithpresent-biased preferences.Giventhebias,theirmodelmaybere-interpretedasamultipleagentdecisionproblem–asinthepresent model.Theirfocusison“voluntaryignorance,”i.e.,when/whetheranagentchoosestostoplearning.Inasense,thepresent modelalsoconcernsvoluntaryignorance–inthiscasebythereceiverwhomightchoosetocommittodisclosureandhence notto“learn”fromthesender.DaughetyandReinganum(2010)focusonarelatedquestionfromasocialwelfareperspective. Theyexamineaplanner’schoiceofprotocolinapublicgoodsprovisionproblembetweenprivacyrightsofanindividualand thesocialbeneﬁtofexposingfreeriders.Our“wordsversusactions”tradeoffsomewhatresemblestheir“privacyversus publicinformation”trade-offeventhoughthereisno(cheaptalk)communicationinthelatter.

Anumberofpapershavecomparedoutcomeswhendifferenttypesofobservationtechnologiesarevaried.Prat(2005)

comparesdifferentmodelsinwhichaprincipalcanobserveeithertheactiondirectlyorinsteadanoisysignal.Otherpapers haveexamineddifferenttypesofcommunicationprotocolsthatmaybechosenbythesender.Austen-SmithandBanks (2000)andKartik(2007)allowthesendertochooseamonganarrayofbothcostlessandcostlymessages,thelatterreferred toas“moneyburning.”Theirresultsdemonstratehowtheprecisionofcheaptalkincreaseswiththeadditionofthemoney burningoption.KaramychevandVisser(2011)goastepfurtheringivingafullcharacterizationoftheoptimalexante equilibriumforthesenderinthemodelwithbothcheaptalkandmoneyburning.Kartik(2009)incorporateslyingcosts intothecheaptalkmodeltherebyturningcheaptalkintocostlysignaling.Heshowsthatequilibriaexistexhibitingfull separationincertainregionsofthetypespace,somethingthatisnotpossibleinthestandardCSmodelwithacontinuum oftypes.

Ourpaperdiffersfromtheseinthatweexaminethesignalingvalueofpoliciesratherthanthatofpuremoneyburning. Inthissense,thepresentpaperismorerelatedtoDaughetyandReinganum(2008)whostudytheendogenouschoiceof protocolbyﬁrmsthatattempttorevealqualityoftheirproducts.Intheirmodel,aﬁrmcandisclosequalitythroughdirect claims,oritcansignalqualitythroughitsproductchoices.

Themaindifferencebetweenthesepapersandoursisthatweplacethechoiceofcommunicationprotocolinthehandsof thereceiverratherthanthesenderoraplanner.Whileprotocoldecisionsbythesenderarequitenaturalinamarketsettings –ﬁrmsvisavisconsumers–studiedbyDaughetyandReinganum(2008)andAusten-SmithandBanks(2000)amongothers, wethinkitquitenaturalthatthereversewouldbelikelywithinﬁrmsandgovernmentswheredynasticconsiderationsplaya role.Inthesecases,theactorsfeartheirpoliciesmaybeundonebyfuturedecisionmakersiftheinformationwererevealed. Hencetheiractionswouldnotordinarilycometolightunlessanexplicitauditmakesitthecase.

3. Themodel

TherearetwoDecisionMakers–DM1andDM2.Theﬁrsthasdecisionauthorityattimet=1andthesecondattimet=2. TheactionschosenbythetwoDMsinthetwotimeperiodsaredenotedbya1 ∈Randa2 ∈Rrespectively.

3.1. Payoffs

Thesymbol ∈Rdenotesthevalueofastateofnaturethatisdrawnonceandforallfromastrictlypositivecontinuous densityfdeﬁnedontherealline_R.Thestateisdrawnbeforeanythingelsetakesplace.

Thetwodecisionmakersaredifferentiatedbyabiasparameterb>0.Thisparameteristhesourceofthepreference misalignmentbetweenthetwoDMs.Hence,itisthekeydriverofallequilibriumdecisionsinthemodel.Weconsidertwo canonicalcases.

TheﬁrstwecallIdeologicalBias,withpreferencesofDM1andDM2,respectively,givenby

V1=−(a1−−b)2−(a2−−b)2, and V2=−(a1−)2−(a2−)2 (1)

Here,theidealpolicyofDM1ineachperiodt=1,2isat=+bwhereastheidealpolicyforDM2isat=.WerefertoDM1as

being“biased”inbothperiodssincehisidealpolicyis“distorted”upwardbyb.7

WetermthesecondcanonicalcaseAgencyBias,withpreferencesofDM1andDM2,respectively,givenby

V1=−(a1−−b)2−(a2−)2, and V2=−(a1−)2−(a2−−b)2 (2)

Underagencybias,thepreferencemisalignmentisdrivenbytherolethateachDMhasineachperiod:activepolicymaker orpassivestakeholder.EachactiveDMisbiasedfortheactionthatcorrespondstotheperiodinwhichheistheactivepolicy maker,andunbiasedwhenheisa“passivestakeholder.”Hence,DMtideallypreferspolicyat=+bineachperiodt=1,2,

whereasDM( /=t)ideallyprefersat=.Ourterminologyinthiscaseismotivatedbythefactthatagencyproblemsarise

fromthemisalignmentofpreferencesbetweensomeonewhoactuallycontrolsapayoff-relevantvariable(theagent,orthe

7 _Of_course,_the_“bias”_label_is_relative._Moreover,_the_actual_direction_of_the_preference_misalignment_–_DM1’s_ideal_point_is_larger_than_DM2’s_–_is inessential.Wecouldallowforb<0.

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activeDMineachperiodinourmodel)andapassivestakeholderwhoisaffectedbythechoicebuthasnodirectcontrol overit(theprincipal,or,inourmodel,DM2inperiod1andDM1inperiod2).

Ourterminologyisintendedtocaptureinherentincentiveconflictsinvariouscollectivedecisionsettings.Asastylized representation,ideologicalbiascanbefoundinpoliticalinstitutionsinwhichanoutgoingpoliticalpartyhassystematically differentpolicypreferencesthanitsnewlyelectedrival.Agencybiascanbefoundinfirmsinwhichamanager’spreferences aremisalignedwithhisorhersuccessororpredecessor,andDM2isselectedinternallyfromexistingmanagement.This newmanager,moreover,willhavepreferencesovertheuseofmanagerialperks(sayprivatejets),whileapastmanager whoholdsstockoptionsawardedduringhistenureasamanageronlyhaspreferencesformanagerialactionsthatinduce stock-appreciation.Theagencybiascasecanalsoapplytopoliticiansprovidedtheyareprimarilyoffice-motivatedrather thanideologicallydriven.Bothtypesbiascanbefoundininformalgroupsthatcoalescetosolvecollectionactionproblems.8

ThebiasesofthethetwoDMsinthetwoperiodsinthetwocanonicalcaseswehaveidentiﬁedarerepresented schemat-icallybelow.

(3) Notethatineithertheideologicalortheagencybiascase,asb→0theconﬂictbetweentheplayersvanishes.

3.2. Information,auditing,anddisclosure

Inthissubsection,weintroducetheformalauditingmodel.Thefocusisonhowthedegreeofconﬂict–asrepresented herebyb–distortstheincentivesofdecisionmakersateachdecisiondate.

Crucially,weassumethatneitherthestatenorthefirstperiodpolicyisautomaticallyobservedbyDM2.Specifically, thestateandthefirstperiodpolicya1areknownonlytoDM1,unlessDM2makesitapointtoverifythem.DM2therefore

assumespowerint=2knowingonlythatisrealizedfromacontinuousstrictlypositivedensityfdeﬁnedonR.

Afterobservingandchoosinghispolicya1,theinformedagentDM1choosesamessagem∈Rthatcaninprinciplebe

usedtocommunicateor“disclose”somethingaboutthevalueofthestate.WerefertomessagemasDM1’sdisclosure statement.This,aswediscussedearlier,cantakeeitheroralorwrittenform.

Obviously,ifthedisclosurestatementfullyrevealedthestate,thenDM2wouldhavenoreasontoinvestigateor other-wiseexpendanyefforttoverifytheinformation.However,thebiascreatesamisalignmentinpreferencesoverthesecond periodpolicy.Hence,justasinthewellknowncheaptalkenvironments,themessagebyDM1willneverbefullyinformative. Knowingthisinadvance,theuninformedDM2hasachoicebeforeanypoliciesordisclosurestatementsareundertaken. ShecanchoosetorelyexclusivelyonDM1’smessageandtherebyforgoanyattempttoverifytheinformation.Alternatively, shecanholdopentheoptiontoseekadditionalinformationlateronafterthepoliciesanddisclosureoccurred.

Formally,welet∈{0,1}denotethe“commitment”choiceofDM2.Thechoice=1denotesthecommitmentbyDM2 toforgobothapolicyauditandstateveriﬁcationandhencetorelyonmalone,while=0denotes“keepingonesoptions open.”Choosing=0,DM2choosestoexercisediscretionaryauthorityatalaterdateoverwhatinformationshewillseek.

Iftheuninformedagentoptsfordiscretionaryauthority(=0),shesubsequentlyhasthreechoices,i.e.,threemethods ofobtaininginformationafterDM1’sdecisions.DM2can(i)stillchoosenottoaudit,andthereforerelyonlyonthemessage (wedenotethischoicebyNAfor“noaudit”),(ii)undertakeacostlessaudittoverifythepolicychoicea1ofDM1(wedenote

thischoicebyPA,for“policyaudit”),or(iii)undertakeacostlyaudittoverifythestatedirectly(wedenotethischoicebySV, for“stateveriﬁcation”).Tosimplify,weassumethatbothSVandPAare“perfect”inthesensethatSVfullyrevealsthestate ,whilePAfullyrevealstheﬁrstperiodpolicychoicea1.Weletthechoiceacrossthesethreepossibilitiesbedenotedbyd

sothat

d∈{NA,PA,SV} (4)

Otherthingsequal,DM2wouldsurely(weaklyatleast)preferStateVerificationoveraPolicyAuditsincethestate,not theprioraction,isrelevanttoherpolicychoiceinperiod2.Ontheotherhand,theactofverifyingthestatedirectlymay bedifficultorcostlyinmanycommonsituations.Moreover,thefirstperiodpolicy,whilehavingnodirecteffectonthe

8_One_example_suggested_to_us_by_a_referee_concerns_a_common_pool_problem_with_an_unknown_resource_stock._Initial_users_of_the_stock_(e.g.,_ﬁsherman) observeinformationaboutthestock(theﬁshingthatdaywasgood)whichcanbecommunicatedormonitored(audited)bysubsequentresourceusers. Agencybiasariseswhenthe“catch”isapurelyprivategood,whereasideologicalbiasariseswhentheusershavedifferentpreferenceintensitiesovera “sharedcatch.”

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secondperiodpayoff,maynonethelesshavesignalingvalueforthedecisionmakers.Inotherwords,sinceaffectsDM1’s preferencesoverhischoiceofa1,itisclearthataPolicyAuditcanrevealinformationabout.

Ofcourse,ifSVismuchmorecostlythanPA,thenitisnotaviableoption.TocapturetheideathatSVisonlymarginally morecostlythanSV,weassumethatPolicyAuditsarecostlesswhileStateVeriﬁcationcarriesalexicographiccost.9_The

assumptionofzerocostforPAiswithoutlossofgeneralityasitpertainstoourmainresult.Speciﬁcally,themainresulthere refersthesurprisingoutcomethatDM2maycommittodisclosureandthusforegoPAandSV.AddingacosttoPAtherefore onlyreinforcesthisresult.Moreover,ifthePAcostissufﬁcientlysmall,allourresultsgothrough.10

Thetimingisdepictedschematicallybelow.

3.3. Equilibrium

Weleta1=˛1(,)denotethechoiceofpolicybyDM1givenhisobservationofandDM2’schoiceof.Similarly,we

letm=(,)denoteDM1’schoiceofmessagem.Wealsodenotebyz(,d)theactualobservedoutcomecorrespondingto thetransmissionchoicesanddbyDM2.Accordingtothemodel,z(1,d)=mforalld(i.e.,DM2choosestocommittoview onlymessages),z(0,d)=mwhend=NA,z(0,d)=whend=SV,andz(0,d)=a1whend=PA.Finally,leta2=˛2(z(,d))denote

DM2’schoiceofpolicygivenherobservationz(,d).

WeevaluatekeypropertiesofthePerfectBayesianEquilibria(PBE)ofthegame.FornowwerestrictattentiontoPBEin purestrategies,deﬁnedformallybelow.Asidefromthesimpliﬁcationthisaffords,weshowinSection5thatmixedstrategy PBEarenon-genericinthebiasparameterb.Moreoverinthose(non-generic)caseswheremixedPBEmayexist,onecan constructapurestrategyPBEthatispayoffequivalentforDM2.

Inthedeﬁnitionbelow,wewillrefertoaPBEissimplyan“equilibrium.”

Deﬁnition1. Equilibrium:Anequilibriumisa5-tuple(˛1,,,d,˛2)suchthat

(1)DM1choosesthepair(˛1,)optimallygivenherobservationofandandDM2’s(correctlyanticipated)choiceofd

and˛2.

(2)DM2choosesthetriple(,d,˛2)optimally.Inparticular,anddarechosengivenherbeliefsaboutandhercorrect

anticipationofDM1’schoiceof˛1and.DM2’schoiceof˛2issimilarlyoptimal,reﬂectingherupdatedbeliefsafter

observingz(,d).

(3)DM2’supdatedbeliefssatisfyBayes’Rulewhereverpossible.

Weﬁrstestablishabaselinepartialcharacterizationofequilibriathatwillbehelpfulforsubsequentresults.

Proposition1. Fullrevelationwithoutcommitment:Consideranyequilibriumineithertheideologicalortheagencybiasmodel. SupposethatintheequilibriumDM2chooses=0.Thatis,DM2doesnotcommittoobservingonlymessagemandinsteadretains theoptiontoconductapolicyauditorverifythestateatalaterstage.Then

(i)DM1’schoiceofa1=˛1(0,)isfullyrevealingofthestate.

(ii)DM1’smessagestrategy(0,·)inthecontinuationgamefollowing=0coincideswithsomePBEstrategyintheCScheap talkgame.

(iii)DM2choosesapolicyauditsothatd=PAandz(0,PA)=˛1(0,).

(iv)DM2choosesheroptimalsecondperiodactiona2=˛2(z(0,PA))=˛2(˛1(0,))=.

OnesigniﬁcantimplicationofProposition1isthatinaworldwherethecommitmenttoforgopolicyauditingandstate veriﬁcationisnotpossible,theuninformedagentwouldalwayschoosetoundertakeapolicyaudit.Shewouldneverrelyon messagesalone,andshewouldneverchoosetoverifythestatedirectly.

Intheinterestofbrevity,weomitafullformalproofforProposition1.Theargumentisstandardandwhatfollowsisa sketchofthestepsinvolved.

9 _That_is,_PA_is_costless_to_implement_by_DM2,_and_DM2_chooses_SV_if_and_only_if_the_payoff_after_observing_the_state_is_strictly_larger_than_the_payoff_after observingDM1’saction.

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Assumethatinsomeequilibrium,=0.WhenDM2choosesditisnotpossiblethatshewillsetd=NAandonlyobserve m.Inthiscase,giventhata1issunk,themodelwouldboildowntoaversionoftheCSworld,andthemessagewouldnotbe

fullyrevealingofthestate.HenceDM2coulddeviatetosettingd=SV,observe,andhenceattainahigherpayoff.

CoulditthenbethatDM2setsd=SVinequilibrium?InthiscaseDM1wouldanticipatethathischoiceofa1playsno

(informational)roleinDM2’schoiceofa2.HenceDM1wouldsimplychooseanactionthatmaximizeshis=1payoff,and

hencehischoiceofa1wouldbefullyrevealingofthestate.ButsincebothSVandPAwouldinducefullyrevealingactionsby

DM1andarethusequallyinformative,thepolicyauditPAispreferabletoDM2asthecheaperoption.Hence,DM2would thendeviatetochoosingPA.

ThisinturnmeansthatonlypossibilityisthatDM2choosesPAwhengiventhechance.Coulditbethatthisisthecase andDM1’schoiceofa1isnotfullyrevealingof?Ifthiswerethecase,giventhatlexicographiccostofchoosingSV,DM2

wouldthengainbydeviatingtochoosingSV.Thiswouldgivehermoreinformationandhenceallowhertoimproveher payoff.Hence,weconcludethatinanyequilibriumif=0ischosenthend=PAisalsochosenbyDM2,andfurthermorethat thechoiceofa1onDM1’spartmustbefullyrevealingofthestate.

FromProposition1,theonlycandidatesforequilibriumareeitherforDM2toeitherrelyonthe“words”(thecaseof=1) orthe“actions”(thecaseof=0andd=PA)ofDM1.NoticethatwhileSVwillneverbechoseninequilibrium,itsavailability iscrucialtotheargument.Withoutit,d=NAcouldbechosenbyDM2and,consequently,thepolicyofDM1neednotbefully revealing.

4. Whendoactionsspeaklouderthanwords?

4.1. Ideologicalbias

Webeginwiththecaseofideologicalbias.InthiscaseDM1andDM2differintheirobjectivesinthesamewayacrossthe twoperiods.ForanygivenDM1’sidealpolicyislargerthanthatofDM2.Asweremarkedbefore,thistypeofmisalignment onpreferencesisastylizedrepresentationofpoliticalcompetitionbetweentwopartiesthatholdpowerinsequencebut haveapre-setdifferenceofopinionregardingthedesirablevaluesofagivenpolicyvariable.

FromProposition1weknowthatinthecontinuationofthegamefollowing=0(nocommitment),DM2choosesd=PA andDM1’schoiceofa1mustbefullyrevealingofthestate.Giventhatincheaptalkmodelswithacontinuumofstates(à

laCS)fullyinformativeequilibriaaregenerallyruledout,thepicturethatemergesfromProposition1seemstosaythatin equilibriumweshouldnotobserveDM2actuallycommittoonlyobservingthecheaptalkmessagem.

Yet,thisintuitionturnsouttobefalseinarobustsetofenvironments.ThemainresulttellsusthatwithIdeologicalBias andfor“largeenough”valuesofb,everyequilibriumwillhavethepropertythatDM2commitstoobservethemessages. ThegistoftheargumentreliesonﬁndingupperandlowerboundsforDM2’spayofffollowingachoiceof=0and=1 respectively.Thisisthereasonweﬁrststateapreliminarycharacterizationofequilibria,whichsharpenstheconclusionof

Proposition1inthecaseofIdeologicalBias.

Proposition2. Linearpareto-dominantequilibrium:ConsidertheIdeologicalBiascase.IfthereisanyequilibriuminwhichDM2 chooses=0,thenthereisanequilibriumasfollows.

(i)DM2chooses=0.

(ii)DM1’schoiceofpolicyisgivenby˛1(0,)=+2b.

(iii) ThepayoffsassociatedwiththisequilibriumPareto-dominatethoseofallotherequilibriainwhichDM2chooses=0. Forlackofabetterterm,wewillrefertotheequilibriumdescribedinthePropositionasthelinearequilibriumunder IdeologicalBias.Wearenowreadytostateourﬁrstmainresult.

Proposition3. Sometimeswordsdospeaklouderthanactions:ConsidertheIdeologicalBiascase.Thenthereexistsa ¯b>0such that

(i)Ifb> ¯b,theninanyequilibriumDM2selects=1.Inotherwords,inallequilibriaDM2commitstoobservingmonlyand forgoestheoptiontoconductapolicyauditortoverifythestateatalaterstage.

(ii)Ifb≤ ¯b,thenthereexistsanequilibriuminwhichDM2selects=0,subsequentlychoosestoauditDM1’spolicy(selectsd=PA) andDM1playsaccordingtothelinearequilibriumofProposition2,thussettinga1=˛1(0,)=+2b.

Obviously,thestrongerandmoresurprisingclaiminProposition3ispartiwhichappliestoeveryequilibriumofthe gamewhenb> ¯b.Thus,withalargeenoughideologicalbiasDM2commitstoobserveonlym.Wordsspeaklouderthan actions.

Weemphasizethatthecommitmentoptiontoforegoadiscretionaryauditappearsinmanyenvironments,including nationalsecurityissuesandstandardcorporatepractices.ThePropositionprovidesonerationaleforwhytheoptionwillbe exercised.WelatershowthattheresultholdsupevenifthecommitmentbyDM2requiredtheconsentofDM1.Forbias largeenough,bothpartieswouldprefertoavoidtheaudit(seeSection5).

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ThefullProposition3isinAppendix.Tounderstandtheresult,however,weﬁnditusefultoﬁrstdeveloptheintuition forparti.Later,wecommentonpartii.

Sinceweareinthecaseofideologicalbias,foranygivenlevelof,DM1’sidealactionsare+bbothinperiod=1and =2.TheseconddecisionmakerDM2favorsactionsequaltoinbothperiods.

Asthe“receiver”ofinformation,DM2choosestoset=1ifbislargeenough.Insodoing,DM2commitstoobserve messagemonly.NotethatDM2losesvaluableinformationsince,justasinCS,misnotfullyinformativeof.Infactthe degreeofinformativenessofmdecreasesasthesizeofbincreases.

AftertheDM2chooses=1,DM1effectivelychoosesa1 “insecret.”HenceDM1willsimplyseta1=+b.Therefore,

regardlessoftheﬁrstperiodpayoffofDM2is−b2_(see₍₁₎_)._DM2_then_chooses_a

2afterobservingm,andhence(using(1)

again)hersecondperiodexpectedpayoffcontingentonmis−Var(|m).11

IfDM2weretoset=0instead,weknowfromProposition1thatitwillsubsequentlychoosetosetd=PA,andhence observea1directly.Thekeytoseeingthat,forlargeenoughb,thiswillbeworsethansetting=1,istonotethatafterDM2

sets=0,DM1knowsthata1willbeauditedbyDM2inthesecondperiod(againweobservethefactthatSVisavailable

anddominatesNAiscriticaltothisargument,eventhoughSVisneverchoseninequilibrium).ByProposition1,DM2will havefullinformationaboutonceitobservesa1.ThereforeDM2setsa2=,andthisgivesDM2asecondperiodpayoffof0.

KnowingthatDM2willusetheinformationyieldedbythePolicyAudittoseta2=willinduceDM1to“over-compensate”

duetothemarginaleffectofitsﬁrstperiodactiononDM2’saction.DM1thereforesetsavalueofa1thatisabove+b.In

fact,weknowfromProposition2thatinthebestcaseforDM2(and,forthatmatter,forDM1aswell),a1willbesettoequal

+2b.Hencetheﬁrst-periodpayofftoDM2isnowworsethantheonewecalculatedfollowing=1sinceitis−4b2_.12_Since,

aswenotedabove,thesecondperiodpayofftoDM2iszero,thismeansthatitsoverallexpectedpayoffaftersetting=0is atbest−4b2_.

Tosumup,DM2’sexpectedpayoffaftersetting=1isboundedbelowby−b2₋_Var(),_while_after_setting₌₀_it_is

boundedaboveby−4b2_._Since_the_distribution_of_is_given,_it_is_then_clear_that_when_b_is_{sufﬁciently}_large_DM2_must_prefer

toset=1andthusforgoanyPolicyAuditorStateVerificationandrelyonlyonthecheaptalkmessagemsentbyofDM1. Wordsdospeaklouderthanactionsinthiscase.Infact,itisnothardtoextendtheargumentusedtoproveProposition3to showthatifbissufficientlylarge,thenDM2’scommitmenttoobserveonlymisbeneficialtobothplayers.

ThestatementinPart(ii)ofProposition3isnotassharpasonewouldideallylikebecauseoftheunderlyingmultiplicity ofcontinuationequilibriafollowingboth=0and=1inthecasewherebissmall.

TheequilibriumidentiﬁedinPart(ii)ofProposition3issupported(seeAppendixforthedetailsoftheargument)by choosingababblingequilibriumcontinuationfollowinga choiceof=1andthelinear(Pareto-dominant)equilibrium identiﬁedinProposition2followingachoiceof=0.

InthemodelweanalyzeitisDM2whochooseswhetherthegamecontinueswithanoptiontoperformPolicyAudit orStateVeriﬁcationortoshutdowntheseoptionsandcommittoobservingmonly.Itisthereforelegitimatetoaskwhat happensif,inthespiritofforwardinductionarguments,13_we_restrict_attention_to_{“continuation}_{equilibria”}_that_give_the

bestpossiblepayofftoDM2bothafterachoiceof=0andafterachoiceof=1(thelatterisnotababblingequilibrium).

Deﬁnition2. Forwardinductionproofequilibrium:Consideranequilibrium ofthegeneralgame.We deﬁneaforward inductionproofequilibrium(henceforthaFIPE)asaPBEsuchthatthereisnootherPBEwhichyieldsalargercontinuation payofftoDM2followingeitherofitscommitmentactions,either=0or=1.

Inotherwords,aPBEisaFIPEifitmaximizes,amongallPBE’s,DM2’scontinuationpayofffollowinghercommitment decision.Inthespiritofforwardinduction,DM2indicatesherintentionsaboutwhichactionshe’lltakefromherinitial commitmentdecision.Off-pathbeliefsofDM1acceptstheseintentionsandactsaccordingly.

Beforestatingtheresult,werequireonemoreassumption.

Assumption1. Constantdensity:Thestrictlypositivecontinuousdensityffromwhichisdrawnisconstantonsome non-degenerateinterval[, ¯].

Proposition4. ConsidertheIdeologicalBiascaseandletAssumption1hold.Thenthereexistsab>0suchthatforanyb≤b themodelhasauniqueFIPEinwhichDM2sets=0,andsubsequentlychoosesd=PA.

Clearly,b< ¯bwhere ¯bistheboundinProposition3.So,withtherefinementandasmallenoughideologicalbias,apolicy auditmustoccur.Combiningthisresultwithpart(i)ofthepreviousproposition,weobtainasharpcharacterizationofthe chosenvehicleforinformationtransmissionwhenideologicalbiasiseithersufficientlylargeorsufficientlysmall.

11 _The_actual_value_of_Var(|m)_depends_on_the_entire_message_function_(·|0)_chosen_by_DM1_in_equilibrium. 12 _Since_we_know_from_Proposition₁_that_a

1isfullyrevealingofthestate,wecanwritetheincentive-compatibilityconstraintforDM1asbalancingthe marginal“tangible”payofffroma1(asitentersdirectlyDM1’sﬁrstperiodpayoff)anditsmarginaleffectonthebeliefsofDM2aboutandhenceona2. Thisgivesrisetoasimpledifferentialequation(see(.5)inAppendix)whichunderpinsthelinearequilibriumofProposition2.

13 _For_reasons_of_space_and_ease_of_reading,_we_stay_clear_of_details_about_how_existing_forward_induction_reﬁnements_{speciﬁcally}_apply_to_our_model. Instead,werefertheinterestedreadertothecontributionsbyvanDamme(1989),andmorerecentlybyGovindanandWilson(2009),andMan(2012).

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4.2. Agencybias

Ourmodelbehavesquitedifferentlywhenthemis-alignmentinpreferencesbetweenDM1andDM2takestheformof AgencyBias.Recall(see(3))thatinthiscaseDM1favorsahigherlevelofa1thanDM2does,andsimilarly,DM2favorsa

higherlevelofa2thanDM1does.Themanagerwhoisactiveinanyperiod,foranygiven(theexternalmarketfactors

affectingtheﬁrm),favorsahigherlevelofmanagerialperksthanintheperiodinwhichsheisnottheactivemanagerbut holds,say,stockintheﬁrm.

JustasinthecaseofIdeologicalBias,fromProposition1weknowthatonlytwoalternativesareviableinequilibrium. EitherDM2choosestocommittoobservingonlymandthusforgoesanypossibilitytoauditDM1’spolicyortoverifythestate (DM2sets=1),orshedoesnotcommitinsuchaway(DM2sets=0)andsubsequentlychoosestoobservea1(d=PA).

Recallalsothat,justasinthecaseofIdeologicalBias,Proposition1furthercharacterizesthissecondpossibilitytellingus thatinthiscaseDM1’schoiceofa1mustbefullyrevealingofthestate.

WeﬁrststateourmainresultforthecaseofAgencyBiasandthenelaborateonitsmeaningandtheintuitionbehindit.

Proposition5. Underagencybiasactionsspeaklouderthanwords:ConsiderthecaseofAgencyBias.Then,foranylevelofb, thereisauniqueForwardInduction-Proofequilibrium(FIPE)inwhich

(i)DM2chooses=0therebynotcommittingtoobservingmaloneandretainstheoptiontocarryoutaPolicyAuditorState Veriﬁcation.

(ii)DM2choosesapolicyauditsothatd=PAandz(0,PA))=˛1(0,).

(iii) DM2enjoyshergloballyoptimalpayoffof0.

NotethatcombiningProposition5withProposition1weknowthatintheequilibriumsingledoutbyProposition5,DM2 proceedstosetd=PAandsotoauditDM1’spolicyandobservea1directly,thusgainingfullknowledgeoftheactualvalue

of.

JustasinthecaseofProposition3,theremaybeamultiplicityofPBE.However,FIPErulesoutanycontinuationother thantheoneinwhich(a)thefullyrevealing(linear)equilibriumisplayedfollowingthechoiceof=0,and(b)thebestCS equilibrium(fromDM2’spointofview)isplayedfollowingthechoiceof=1.Facedwiththesetwocontinuations,DM2 willchoose=0intheagencybiasmodel.

Theintuitionfortheresultcanbeunderstoodbyjuxtaposingtheagencybiasmodelwiththeideologicalbiasmodel.In bothmodels,theactionoftheﬁrstdecisionmaker,DM1,playstworoles.Firstly,ithas“tangible”valueforbothDM1and DM2sinceitentersdirectlytheirpayoffs.Secondly,ithasasignalingrolesinceitpotentiallyrevealstheactualvalueofto DM2–infact,byProposition1,itactuallydoesrevealitinanyequilibriuminwhich=0.Inbothmodels,themarginaleffect (viasignaling)ofDM1’sactiononDM2’sactionwillbeweightedbyDM1togetherwiththetangiblevalueofheractionasit entersherﬁrstperiodpayoff.

Unliketheideologicalbiasmodel,underagencybiasDM1wouldliketodistortDM2’sactiondownwardratherthan upward.Inordertodothisviasignaling,hisownactionshouldbelessdistortedthanwouldbeoptimalfromapurely tangiblepayoffpointofview.Thatistosay,thesignalingvalueofDM1’sactionworksinoppositiontoitstangiblevalue. Thisisquitedifferentthantheideologicalbiascasewheretangiblevalueandsignalingvaluearereinforcing–bothmotivate DM1todistorthisactionupward.

UnderagencybiasthetwoopposingforceseffectivelycanceloutmakingDM1’schoiceequaltotheonethatisthe overallfavoriteofDM2,namelya1=.14HenceDM2endsupenjoyingheroveralloptimalpayoffunderaPolicyAuditthatis

anticipatedbyDM1.Henceshechoosesnottocommittoobservingmonlywhengiventhechance.Actionsdoindeedspeak louderthanwordsintheAgencyBiascase.

5. Robustness

Anumberofextensionsor“robustnesschecks”areworthdiscussing.Weintroducewhatwebelievearethemostcritical onesandanalyzeeachinturn.

Mixedstrategies.Theanalysisthusfarfocussedexclusivelyonpurestrategies.Wegiveabriefargumentdemonstrating themainresultsextendtothemixedstrategycase,althoughwedonotgiveaprecisecharacterizationofallmixedequilibria ofthemodel.

ConsiderﬁrstProposition1,theprecursortosubsequentresults.SupposeweareatthestageinwhichDM2didnotalready committodisclosureandmustnowchoosebetweenPA(auditing),NA(messages),andSV(stateveriﬁcation)followingDM1’s choices.

IsthereanequilibriuminwhichDM2mixesatthisstage?NoticeattheoutsetthatDM2mustassignzeroprobabilityto NA.IfDM2assignspositiveprobabilitytoNA,thenDM1’smessageisnotfullyinformative(asshowninCS).Inthatcase, DM2strictlyprefersSVtoNAsincetheveriﬁcationcostissmallerthanitsbeneﬁts.NoticealsothatDM2willnotrandomize

14_The_differential_equation_embodying_DM1’s_Incentive_{Compatibility}_constraint_(see_(.17)_in_Appendix₎_now_yields_a

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betweenPAandSV.IfDM1’sactionisfullyrevealing,thenDM2strictlyprefersPA.Iftheactionisnotfullyrevealingthen DM2strictlyprefersSV.Hence,inequilibriumDM2choosesPA.

WillDM2actuallybewillingtorandomizeatthecommitmentstage?Infact,thiscanhappenonlyifDM2isindifferent betweenPAandNA.Intheideologicalbiascasewithblarge,babblingisstrictlybetterforhimthanthefullyrevealing continuation.15_Similarly,_with_either_agency_bias,_or_ideological_bias_with_b_small,_the_fully_revealing_continuation_is_strictly

betterthananyother.

Inshort,indifferencebyDM2isnon-generic.Moreover,iftheredoesexistanequilibriuminwhichDM2mixes,thereis apurestrategyequilibriumthatispayoffequivalent.

Auditingcosts.Themodelassumesthatauditingcostsarezero.Considerwhathappenswhenweallowforpositive auditingcostsinthemodel.Theresultsholdupifauditingcostsareintroduced,providedtheyaresmallenough.Thekey observationisthat,aslongasstateveriﬁcationismarginallycostlierthanthepolicyaudit,thelatterwillalwaysinduce fullyrevealingactionsbyDM1.Inthatcase,thepolicyauditwillalwaysbethebetteroption.Consequently,Proposition1, thegatewaytothesubsequentresults,continuestohold.Moreover,ifthePAcostissmallenough,thenitwillstillbethe preferredoptionoverdisclosureinthosecaseswherePAwaschoseninthemodelwithzeroauditingcosts.16

Noticethatifmixedstrategiesareintroducedinthemodelwithpositiveauditingcosts,theywillbenongenericforthe samereasonsoutlinedabove.Notealsothatif=1ischosenunderzeroPAcosts,thenitwillbechosenwhensuchcosts arepositive.Themoresubtlecaseiswhen=0andd=PA.SincethefullyrevealingcontinuationunderPAisstrictlybetter thananyotherwhenauditingcostsarezero,itwillbesowhencostsarelowenoughtopreservethestrictinequality.17

Discounting.Inadynamicmodel,itisnaturaltoaskwhathappensiftheplayersdiscountthefuture,sothatinourcase thesecondperiodpayoffsothatitismultipliedbyafactorı∈(0,1).Thegistofourresultssurvivesintactthediscounting ofsecondperiodpayoffsbyDM1andDM2.

Propositions2and3continuetoholdwiththeonlyamendmentthatthelinearequilibriumisnow˛1(0,)=+(1+ı)b.

Thethresholds ¯b andb< ¯b needtoberecalculatedforthediscountingcase,buttheirexistenceremainsexactlyasstatedin

Propositions3and4.Proposition5partialsocontinuestohold.

InshorttheoverallpictureframedbyPropositions3–5continuestoholdwhendiscountingisallowed.Wordsspeak louderthanactionsinthecaseofIdeologicalBiasandblarge,whileactionsspeaklouderthanwordswhenbissmalland weareintheIdeologicalBiascase,orforanyvalueofbwhenweareinthecaseofAgencyBias.

Theabilitytocommit.Weremarkedearlierthatcommitmentstoforegoauditsarecommonpracticeinﬁrms.Toa morelimitedextent,theyalsoexistinpolities–particularlyinareasofnationalsecurity.Onecouldtakeissuewiththefact that,unlikeourmodelwhereDM2makesthecommitment,thesecommitmentsareinstitutionalinnature.Theyareoften agreeduponeitherbynon-partisancommittees(e.g.,theﬁrm’sBoardofDirectors),orunanimousagreementbytheparties themselves.Ourresultscanbeextendedtoaccommodatethispossibility.

ConsideranextrastepinthemodelwherebyaproposedcommitmenttoforegoauditingbyDM2requirestheapproval ofDM1–aParetocriterionofsortsthatdetermineswhetherthecommitmenttousedisclosureistaken.Nowconsider twoinstitutions(formally,twodistinctgames).Inonegame,theonewe’veanalyzedsofar,thereexistsadecisionnodein whichDM2cancommittousedisclosure.Intheothergame,nosuchcommitmentoptionisavailable.Thatis,withoutthe commitmentoption,DM2mustsimultaneouslychoosebetweenallthreeoptions-disclosure,auditing,directveriﬁcation afterDM1’saction.

Intheideologybiasmodelwiththecommitmentoption,whenbiasbislarge,Proposition3showsthatDM2prefers tocommittodisclosure.ButthisisalsotrueforDM1.Inotherwords,forblargebothdecisionmakersprefertomakethe commitment.Toseewhy,observethatunderauditing,DM1’stotalpayoffis−2b2_._Under_disclosure,_the_worst_can_happen_to

DM1isbabblingandDM1’slowerboundinthiscaseis−Var()−b2_._Thus_for_b_large,_disclosure_–_the_commitment_to_avoid

auditing–Paretodominatestheoptiontoaudit.Inthissense,thetwoagentspreferaninstitutioninwhichacommitment optionexiststoavoiddiscretionaryaudits.

Bycontrast,whenbissmall,thenbothpartiespreferauditing.Inthiscasethepresenceorabsenceofthecommitment optioninthemodeldoesnotaffecttheoutcome.Thepartiesarethereforeindifferentaboutwhichofthetwogames,i.e., whichinstitutionalenvironment,theyarein.

Nowconsidertheagencybiasmodelwiththecommitmentoption.Here,whenthebiasbissmall,thenbothpartieswould preferauditing.AgainthereisaclearParetorankingbutthepresenceorabsenceofthecommitmentoptioninthemodel doesnotaffecttheoutcome.Thepartiesareonceagainindifferentaboutwhichofthetwogames,i.e.,whichinstitutional environment,theyarein.

Forthesakeofcompleteness,weobservethatwhenbislargeintheagencybiasmodel,DM2andDM1haveopposing preferencesonthecommitmentdecision.ByProposition4,thedecisioninthehandsofDM2resultsinauditing.18

15 _We_emphasize_that_DM1’s_choice_depends_on_the_outcome_of_DM2’s_mixed_strategy,_not_on_the_mixed_strategy_itself._For_this_reason,_DM1_does_not_mix, andsoneitherdoesDM2exceptonanongenericsetofb.

16 _Roughly,_we_can_bound_the_payoff_of_the_most_informative_cheap_talk_equilibrium_of_the_CS_cheap_talk_game._Since_the_action_by_DM1_in_the_PA continuationisfullyrevealing,itwillbestrictlypreferrediftheauditingcostisverysmall.

17 _There_may_of_course_be_an_auditing_cost_to_create_an_indifference_between_PA_and_NA,_but_this_case_would_be_non-generic.

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Hence,exceptforthislastcase,bothplayerswouldprefer,atleastweakly,tobeinaninstitution(agame)inwhichthe commitmentoption–theoptiontoforegodiscretionaryauditing–isavailable.

Quadraticlosspayoffs.Throughout,wehaveworkedwithpreferencesrepresentedbytime-separablequadratic-loss functions.SinceCS,quadraticpreferenceshavebeenthecanonicalbenchmarkcasethroughouttheliteratureon cheap-talk.Acrossthisﬁeld,quadraticpreferencesalsohavetheprivilegedpositionofguaranteeingtractability,yieldingexplicit solutionsinmanycases.Theyplayessentiallythesameroleinourmodel.

Itislegitimatetoaskhowcrucialtheirroleisinthequalitativeﬂavoroftheresultsweobtainhere.Wedonotdemonstrate ageneralresultinthepaper,however,thenatureoftheresultsindicatethatthesepreferencesarenot“knife-edged.”

AsimplewaytoseethisistoobservethatDM2’stransmissionchoicesarediscreteandgenericallyherbestresponsesin equilibriumarestrict.Thismeansthattheargumentsshouldgothroughunchangedinawelldeﬁnedcontinuousparametric neighborhoodofthequadraticpreferences.Totakeoneexample,supposeWi=Vi+(1−)UiwhereViisthequadraticpayoff

speciﬁedbeforeandUiisasmoothfunctionofactions,stateandbiasandstrictlyconcaveintheaction.Fortheresultto

extend,onewouldshowthattheresultsremainintactforWiifiscloseenoughtoone.19Westronglyconjecturethisis

thecase,butdonotproveitformally.

6. Auditsandpartisaninvestigations

Thispaperhasemphasizedtheimportanceofendogenouschoiceoftransmissionprotocolindecentralizeddecisions problems.Therearemanysuchproblemsinwhichtheinterestsofrelevantdecisionmakersdonotcoincide.Wefocuson twocleavages–ideologicalandagency–thatdifferentiateauthorityﬁguresatdifferentdecisiondates.

Theresultssuggestthatthecommitmenttoasystemofdisclosureismorevaluablewhenthecleavagebetweendecision makersisideological.Wearguedthat,broadlyspeaking,theIdeologicalBiascaseﬁtsbetterthepoliticaleconomyscenarioin whichtwopartiessuccessivelyholdpower,whiletheagencycaseisarguablybettersuitedtorepresentprivateorganizations inwhichtwodecisionmakersarechoseninternallytoheadtheoperationsinsuccession.20

This,inourview,talliessignificantlywiththefactthatAuditingdoesinfactprevailinthecaseoffirms.Audits(internal andexternal)arealmostalwaysthoughtofas“goodmanagementpractices”inthecaseofthefirms,21_whereas_audits_in_the

politicalsphereareoftenviewedas“partisanwitchhunts.”Whenanewpoliticalpartyoccupiespowerithaswidepowers torenegeonpreviouspromises,ifnecessarybychangingthelaw.Onenotableexceptionisintheareaofnationalsecurity. Institutionalcommitmentstoavoidpublicauditsofsensitivesecurityexistandareusuallyagreeduponbyallparties.

Itisworthemphasizingthatdecisionproblemsofthisnaturealsoariseinmoreamorphoussettings.22_Whether_the

institutionisformalorinformal,wepointoutthatiflegal/institutionalcommitmentsarefeasible,thenbothdecisionmakers atleastweaklyprefertheminallcases,agencyorideologicalbias,largeorsmallb,exceptunderagencybiaswhenbislarge. Finally,theresultsindicatethatwhensuchcommitmentisnotpossibletheoutcomewillinfactbeanAuditoftheactions ofthepreviouspartybythenewholderofpower.Atthispointthelossofwelfarerelativetothedisclosureregimemaywell becomeapparenttotheparticipants,butitsimplyistoolatetoavoidit.Afutureexplorationintowhen,how,andwhysuch commitmentsarepossibleinsomeenvironmentsandnotinotherswouldbewellworththeeffort.

Appendix.

ProofofofProposition2. Consider theIdeologicalBiascase,andassumethatthereisanequilibriumwith=0.By

Proposition1thenthereisanequilibriumwith=0,d=PAand˛1(0,)fullyrevealingof.Letthisequilibriumbedenoted

by(˛∗₁,∗,∗,d∗,˛∗₂).Foreveryvalueof,DM1’spolicychoicea1=˛∗1(0,)mustsolve

max

a1

−(a1−−b)2−[˛∗2(a1)−−b]2

Assuming˛∗₂isdifferentiable,23_we_can_write_the_ﬁrst_order_condition_as

˛∗₁(0,)−−b+[˛∗₂(˛∗₁(0,))−−b]˛∗₂(˛∗₁(0,))=0

Since˛∗₁(0,)isfullyrevealingandDM2’soptimala2isequaltoforanygiven,wemusthavethat˛∗2[˛∗1(0,)]=

˛∗₁ −1[˛∗₁(0,)]=.FromtheInverseFunctionTheorem,wethenknowthat˛∗₂[˛∗₁(0,)]=1/˛∗₁(0,).

19_The_exact_cutoffs_in_b_would_obviously_be_different.

20_These_{representations}_are,_of_course,_approximate._For_instance,_{ofﬁce-motivated}_politicians_–_as_opposed_to_the_{ideology-driven}_ones_we_have_in_mind here–mayﬁtbetterthecaseofAgencyBias.

21_See_Arter₍₂₀₀₀₎_. 22_See_footnote₈_.

23_For_this_step_of_the_argument_of_course_it_is_sufficient_to_verify_ex-post_that_function_we_find_is_indeed_{differentiable.}_In_the_proof_of_part_iii)_below_this isnotaviableoption.However,sinceweknowfromProposition1thatwearedealingwithfullyrevealingequilibriaitiseasytoseethatDM1’sdecision mustbecontinuousandstrictlymonotoneinandhencedifferentiablealmosteverywhere.Thisissufficientforourpurposesbelow,thoughthedetails areomittedforthesakeofbrevity.

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Hence,theﬁrstorderconditionimpliesthatthefollowingdifferentialequationmustbesatisﬁed

[˛∗₁(0,)−−b]˛∗₁(0,)−b=0 (.5)

Noticethat˛∗₁(0,)=+2bisonesolutiontothisdifferentialequation(and,pointbypoint,satisﬁestherequiredsecond ordercondition).

ThissufﬁcestoprovepartiiofProposition2.Forfuturereference,atthispointwenoticethat˛∗₁(0,)=+2banda2=

foreveryimplyanequilibriumpayoffforDM2of−4b2_.

ToshowpartsiandiiiofProposition2,usingpartiofProposition1,itissufﬁcienttoshowthatthepayoffsassociatedwith thelinearequilibriuminwhich=0and˛∗₁(0,)₌₊2bPareto-dominatethoseofanyotherfullyrevealingequilibrium inwhichDM2sets=0.

Consideragainthedifferentialequationin(.5),whichweknowanyfullyrevealingequilibriumtobeconsideredhere mustsatisfy(almosteverywhere).

Deﬁne

g

=˛1(0,)−−b (.6)

andnotice

g()=˛∗₁(0,)−1 (.7)

Hence,wecanrewrite(.5)as

g

1+g

=b (.8)

Theng*₍₎₌_b_is_one_solution_to_(.8)_,_giving_rise_to_the_linear_equilibrium_strategy_˛∗

1(0,)=+2bthatweidentiﬁedabove.

In general, however, there are inﬁnitely many solutions to (.8), corresponding to inﬁnitely many fully revealing equilibria.24

Oncewehaveagthatsolves(.8),foranygiventhepayoffstothedecisionmakersinthecorrespondingequilibrium, canberespectivelywrittenas

V1=−g()2−b2 and V2=−[b+g()]2

Moreover,observethat,wheneverg()>0,bothpayoffsaredecreasinging().

Therestoftheproofconsistsofshowingthatifgisasolutiontothedifferentialequationin(.8)andg /= g*_,_then_g()_≥_b

forevery.Thiswillsufﬁcetoshowthatinanyotherequilibriumtobeconsideredthepayoffstobothdecisionmakersare lowerthanthoseonthelinearequilibrium,asrequired.

Consideragainthedifferentialequationin(.8).Beginbynotingthatdirectlyfromthedifferentialequationitselfweknow thatitisimpossiblethatg()=0forany.Hence,sincegiscontinuousanddifferentiablealmosteverywhere,ifg(ˆ)is positive(negative)forsome ˆtheng()mustbepositive(negative)forall ∈R.

Now,bywayofcontradiction,considerasolution ˜g to(.8)suchthatforsome ˆ wehave ˜g(ˆ)<b.Weneedtodistinguish betweentwocases–ﬁrst ˜g(ˆ)>0,andthen ˜g(ˆ)<0.

Considerthecaseof ˜g(ˆ)>0ﬁrst.Sincebyourcontradictionhypothesis ˜g(ˆ)<b,directlyfrom(.8)wegetthat ˜g()< ˜g()<bforallpairs{,}satisfying<and< ˆ.

Restrictingattentiontoanappropriatesubsequenceifnecessary,wethenhavethat lim

→−∞˜g

₍₎₌_0._{Since ˜g()}_<_b_for_all

< ˆ,thisimpliesthatforsomesufﬁcientlysmall ¯ itmustbethat

˜g( ¯)[1+ ˜g_{( ¯)]}_<_b _(.9)

whichdirectlyimpliesthat ˜gcannotbeasolutionto(.8).

Wenowturntothesecondcase–so,assume ˜g(ˆ)<0.Similarlytowhatwearguedinthepreviouscase,thisimplies that ˜g()< ˜g()<0forallpairs_{,_}satisfying<and< ˆ.

Justasinthepreviouscase,andagainrestrictingattentiontoanappropriatesubsequenceifnecessary,wethenhavethat lim

→−∞˜g

₍₎₌_0._{Since ˜g()}_<₀_for_all_{< ˆ,}_this_implies_that_for_some_{sufﬁciently}_{small ¯}_it_must_be_that

˜g( ¯)[1+ ˜g_{( ¯)]}_≤₀ _(.10)

whichdirectlyimpliesthat ˜g cannotbeasolutionto(.8).

24 _Another_solution_is,_for_{example, ˆg()}₌_{b[W (e}1−/b₎₊_1]_where_W_is_the_so-called_Lambert_W_function_which_is_implicitly_deﬁned_by₌_W

_eW(_). Thefunction ˆgsatisfies ˆg (0)=2b(whichcanbeinterpretedastheinitialcondition).Thefunction ˆgthereforegivesrisetoanotherfullyrevealingequilibrium inwhichtheinformedDM’spolicyis˛ˆ1(,0)= ˆg()++b.Itisalsoeasytocheckthatthereareinfinitelymanyothersolutionsgtothedifferentialequation in(.8).Suppose ¯gisasolutionto(.8).Fixanddefine ˜gbyletting ˜g()= ¯g(+)forevery.Then ˜gisalsoasolution.

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LemmaA.1. ConsiderthecaseofIdeologicalBias.SupposethatDM2sets=1andDM2chooses˛1andoptimally.Thenby

choosinga2(m)=E()foreverym,DM2achievesanoverallexpectedpayoffof−b2−Var().25

Proof. GiventhatDM2chooses=1,DM1’schoiceofa1willnotbeobservedbyDM2.Hence,from(1)itisevidentthatit

isoptimalforDM1toseta1=+bforevery ∈R.HenceDM2’sﬁrstperiodexecutedpayoffis−b2.Againdirectlyfrom(1),

ignoringmandsettinga2=E(),itisclearthatDM2canachieveasecond-periodexpectedpayoffofVar().

LemmaA.2. ConsiderthecaseofIdeologicalBias.InanyequilibriuminwhichDM2sets=0heroverallexpectedpayoffis boundedaboveby−4b2_.

Proof. FromProposition1weknowthatsuchequilibriummustentailachoiceofd=PA,andmustbefullyrevealing.Hence DM2willbeabletoinferfromobservinga1.Therefore,from(1),DM2willchoosea2=forevery,andhenceachievea

secondperiodexpectedpayoffof0.This,ofcourse,isDM2’sgloballyoptimalexpectedpayoffinthesecondperiod. FromProposition2weknowthatDM2’soverallpayoffcannotbeabovetheonesheobtainsinthelinearequilibriumin whichDM1setsa1=+2bforevery ∈R.ThisgivesDM2aﬁrstperiodexpectedpayoffof−4b2.

ProofofofProposition3. Let ¯b=

Var()/3.Notethatthistriviallyimpliesthatforeveryb> ¯bwemusthavethat −4b2_<₋_b2₋_Var(),_and_for_every_b_{≤ ¯b we}_have_that_−4b2_≥₋_b2₋_Var().

Weprovepartiﬁrst.Suppose,bywayofcontradiction,thatb> ¯bandthatthereexistsanequilibriuminwhichDM2 chooses=0.ByLemmaA.2,insuchaputativeequilibriumDM2’spayoffisboundedaboveby−4b2_._By_Lemma_A.1_,_by

deviatingandsetting=1,DM2willachieveapayoffthatisboundedbelowby_−b2₋_Var()._This_is_evidently_a_proﬁtable

deviationfromtheputativeequilibriumandhencetheproofofpartiisnowcomplete.

Nowfortheproofofpartii.Letanyb≤ ¯b begiven,andconstructanequilibriumasfollows.DM2chooses=0,and followingthischoice,on-path,thelinearequilibriumofProposition2isplayed.IfDM2deviatesandsets=1,ababbling equilibriumisplayedinwhichDM1choosesaﬁxed ¯m regardlessofandDM2choosesa1=E().DM2’spayoffintheproposed

equilibriumis−4b2_._If_instead_he_deviates_and_sets₌₀_his_payoff_is_−b2₋_Var()._Since_b_{≤ ¯b this}_is_not_a_proﬁtable_deviation

andhencetheproofofpartiiisnowcomplete.

PreliminaryresultsfortheproofofProposition4.BeforeproceedingwiththeproofofProposition4,weletϕdenote theprobabilitythat ∈[, ¯]asrequiredbyDeﬁnition1.

DeﬁnitionA.1. Boundedpartitional:Ancontinuationequilibriumofthegamefollowing=1isboundedpartitionalifand onlyifDM1’sstrategypartitionsthesetofpossible(namelyR)intodisjointintervals(overwhichthemessagedoesnot change),andthelengthofsuchintervalsisboundedawayfromzero.

LemmaA.3. Foranyb>0,allcontinuationequilibriaofthegamefollowing=0areboundedpartitional.

Proof. ThisisastraightforwardadaptationofargumentsinCS(tothecaseof ∈Rratherthanaboundedinterval),and thedetailsareomitted.

LemmaA.4. SupposethatAssumption 1holds.Letadecreasingsequenceofstrictlypositivebiasparameters{bk}∞k=0 with

lim

k→∞bk=0begiven.

Consideranysequenceofassociatedcontinuationequilibriaofthegamefollowing=1,andletV_2k∗ bethesequenceofassociated payoffsforDM2.

AssumethatthatforeveryktheequilibriumstrategyofDM1partitions[, ¯]intoatmosttwosets(atmosttwodistinct messagesaresentinequilibriumasrangesover[, ¯]).

Then,thereexistsa ¯ksuchthatforeveryk≥ ¯k,wehavethatV_2k∗ <−4b2 k.

Proof. Letabbegiven.ConsiderastrategyforDM1thatentailssendingeitheroneortwomessagesasrangesover[, ¯]. Supposeonlyonemessageissent,say ˆm.Conditionalon ∈[, ¯],DM2’spayoffisthenboundedaboveby−Var(| ∈[, ¯]). Supposenextthatonlytwomessagearesent,say ˆm and ˆˆm.ThenthepayofftoDM2isboundedaboveby−Var(| ∈ [, ¯]andm= ˆm)−Var(|∈[, ¯]andm= ˆˆm).Notethattheseboundsdonotdependonb,butonlyonthedistributionof .Letthebestpossibleboundbedenotedby− (with >0).ThenifDM1onlysendsoneortwomessagesasrangesover [, ¯]thesecondperiodpayofftoDM2isboundedaboveby− ϕ.ThisisbecauseDM2’spayoffconditionalon∈/[, ¯]isat most0.Sinceinanycontinuationequilibriumofthegamefollowing=1,DM1obviouslychoosesa1=+b,DM2’spayoff

inanysuchequilibriumis−b2_._Hence,_we_can_conclude_that_if_DM1_only_sends_one_or_two_messages_as_ranges_over_{[, ¯]}

inanycontinuationequilibriumofthegamefollowing=1,theoverallpayofftoDM2isboundedaboveby−b2₋_ϕ.

Since− ϕ doesnotdependonb,forbsufﬁcientlysmallitisevidentthat −b2₋_ϕ_<_−4b2_._The_claim_then _follows

immediately.

ProofofofProposition4. Bywayofcontradictionassumethatthereexistsadecreasingsequenceofbiasparameters {bk}∞k=0with_klim_→∞bk=0,suchthatforeachbkinthesequencethemodelhasaFIPEinwhichDM2actuallychooses=1.

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NoticeﬁrstthatbyPropositions1and2weknowthatinanyFIPEtheplayers’strategiesconditionalon=0giveriseto thelinearequilibriumofProposition2andhenceyieldapayoffofV_2k∗ =−4b2

ktoDM2.

ByLemmaA.4,itmustbethat,contingenton=1,foreverykinthesequence(discardtheinitialelementsifnecessary) DM1’sstrategyinvolvessendingatleastthreedifferentmessagesasrangesover[, ¯].

OurnextendeavoristocomputeanupperboundtoDM2’spayoffcontingenton=0–orequivalentlyinthecontinuation equilibriumofthegamefollowing=1.

Foranygivenbk,considerthepartitionofthesetofpossible(namelyR)inducedbyDM1’sstrategy(seeLemmaA.3).

Considertheleft-mostcellofsuchpartitionsuchthatitsupperboundisnosmallerthan.Letsuchboundbedenotedbydk.

Considernexttheright-mostcellofsuchpartitionsuchthatitslowerboundisnolargerthan ¯.Letsuchboundbedenoted by ¯dk.Withoutlossofgenerality(bytakingsubsequencesifnecessary)weassumethatbothsequences{dk}∞k=0and{¯dk}

∞ k=0

convergeask→∞.

Noticenextthatitmustbethecasethat lim

k→∞( ¯dk−dk)=>0 (.11)

Thisissimplybecauseotherwisewecouldobtainacontradictionusingthesameargument(inthelimit)asintheproof ofLemmaA.4.Letkbetheprobabilitythat ∈[dk, ¯dk]and lim

k→∞k=.NotethatbyAssumption1itmustbethat=

ϕ/( ¯−)andhence>0.

Noticethatatthispointweknowthatcontingenton ∈[d_k, ¯dk_],_the_continuation_equilibrium_of_the_game_following

=1isexactlythemodelinCSinthespecialcaseofauniformdensity.26

LetU(d_k, ¯dk_,_b

k)beDM2’ssecondperiodpayoffunderthemostinformative(theonethatgivesDM2thehighestpayoff),

contingenton _∈[d_k, ¯dk_]._Overall,_DM2’s_equilibrium_payoff_in_the_continuation_equilibrium_of_the_game_following₌₁_is

thenboundedaboveby −b2

k+kU(dk, ¯dk;bk) (.12)

ThisissimplybecauseDM2’sﬁrstperiodpayoffinanyequilibriumofthecontinuationequilibriumofthegamefollowing =1is−b2

kandwhenever∈/[dk, ¯dk]DM2’ssecondperiodpayoffobviouslycannotexceed0.

Ourcontradictionhypothesisand(.12)implythatthereexistsn∈Nsuchthatforeveryk≥n, −b2 k+ 2U(dk, ¯dk,bk)≥−4b 2 k (.13) whichimplies U(d_k, ¯d_k;b_k)_≥₋6b 2 k (.14)

DirectlyfromCSintheconstantdensitycase27_we_get_that

U(d_k, ¯dk,bk)=−Akb2k−Bk (.15)

forsomepositiveAkandBk.AlsodirectlyfromCS,thereexistsm∈Nsuchthatforeveryk≥mwehavethatAk>6/.Hence,

fork≥max{n,m}itmustbethat U(d_k, ¯dk,bk)=−Akb2k−Bk<−

6b2 k

(.16)

whichcontradictsinequality(.14)andhenceconcludestheproof

WecontinuewithapreliminaryresultthatwillbeusedtoproveProposition5.

LemmaA.5. ConsidertheAgencyBiascase.IfthereisanyequilibriuminwhichDM2chooses=0,thenthereisanequilibrium asfollows.