Appointed learning for the common good: Optimal committee size and monetary transfers

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Games and Economic Behavior

journal homepage:www.elsevier.com/locate/geb

Appointed learning for the common good: Optimal committee size and monetary transfers ^✩

Hans Gersbach

^a

, Akaki Mamageishvili

^b,∗

, Oriol Tejada

^c

aCER-ETH–CenterofEconomic,ResearchatETHZurichandCEPR,Zürichbergstrasse18,8092Zurich,Switzerland bCER-ETH–CenterofEconomic,ResearchatETHZurich,Zürichbergstrasse18,8092Zurich,Switzerland cFacultyofEconomicsandBusiness,UniversitatdeBarcelona,Diagonal690-696,08034Barcelona,Spain

a rt i c l e i nf o a b s t ra c t

Articlehistory:

Received15December2020 Availableonline27June2022

JELclassification:

C72 D71 D8

Keywords:

Voting Committee

Informationacquisition Monetarytransfers Majorityrule

Incompletesocialcontracts

Apopulationofidenticalindividualsmustchooseoneoftwoalternativesunderuncertainty aboutthestateoftheworld.Individualscanacquiredifferentlevelsofcostlyinformation and complete contracts are not feasible. For such a setup, we investigate how vote delegation to acommitteeand suitable monetary transfersforits members canensure thathighoroptimallevelsofinformationare(jointly)acquired.Weshowthatfora(stable) committeethatusesthemajorityruletomaximizethe probabilityofchoosingtheright alternativeandthentominimizeaggregateinformationacquisitioncosts,itssizemustbe smallinabsoluteterms(iffulllearningispossible)andsmallrelativetopopulationsize(if onlypartiallearningispossible).Yetcommitteesmustneverbemadeupofonemember, sothetyrannyofasingledecision-makercanbeavoided.Ouranalysisidentifiesboththe potentialandsomeofthelimitationsofmonetarytransfersincommitteedesign.

©^{2022TheAuthor(s).PublishedbyElsevierInc.Thisisanopenaccessarticleunderthe} CCBYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

From atheoretical andempiricalviewpointalike,ithasbeenwell understoodfora longtime thatagentshavestrong incentivestofree-rideon others’effortstofindout whichalternativeisbestforthecommongood (Downs, 1957).Incen- tives tobecome informed beforevotingare low because individualsbear the costof acquiringtheinformation buttheir probability ofbeingpivotalissmall,particularlyinlargepopulations. Theproblemoftherebeingtoolittleinformationat theindividuallevelandtheaggregatelevelalikeispervasivebothinpoliticsandincorporategovernance.Itisalsosalient fornumerouspanelsofappointedexpertssuchasscientificreferees,juries,publicprocurementcommittees,monetarypol- icycommittees,andhiringcommittees.Howshouldcommitteesof(expert) decision-makersbe designedto ensurethata sufficientlyhighlevelofinformationisacquiredandlaterexpressedthroughvoting?

Anintuitivewayofremedyingtheunderinvestmentininformationinalloftheabovescenariosistoincreasethechances foragentstobepivotaland/ortoreducetheprivatecostsassociatedwithinformationacquisition.Inthispaper,weexplore thisdoubleavenuebyinvestigatingmechanismsthatfirstappointacommitteeofacertain sizeandsecond setasuitable

✩ WearegratefultoDavidBasin,BardHarstad,CésarMartinelli,PaulMaunoir,LaraSchmid,MaikSchneider,JanZápal,andallparticipantsofthe3^rd ETHWorkshoponPoliticalEconomy,the2020ConferenceonMechanismandInstitutionDesign,andseminarsatCityUniversityofLondonandUniversitat RoviraiVirgiliforinsightfulcomments.Wealsothanktwoanonymousreviewersfortheircommentsthatimprovedourpaper.Allerrorsareourown.

*

Correspondingauthor.

E-mailaddresses:[email protected](H. Gersbach),[email protected](A. Mamageishvili),[email protected](O. Tejada).

https://doi.org/10.1016/j.geb.2022.04.005

0899-8256/©^{2022TheAuthor(s).PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).

rewardschemeforitsmembers.Acommittee isa(randomly)chosensubsetofagents,allofwhomaregiventheexclusive righttovote.Arewardscheme isapopulation-wide,budget-balancedvectorofmonetarytransfers.

Fortheanalysis,weusethemodelof Martinelli(2006).Therearetwoexante equallylikelyalternativesandeachagent (of thecommittee)receives an informativesignal aboutthestate oftheworld. Thesignal’squality canbe increasedata cost.Conditionalonknowingthestateoftheworld,allagentsoftheentirepopulationwouldliketoimplementthesame alternative.Committeemembersareanonymousandthereforewedonotconsiderthepossibilityofcommunication.Thisis areasonableassumptionforanumberofsetups,including,butnotlimitedto,groupsofcitizenswhohavebeenrandomly chosenfromthecitizenry,nodesinablockchain,andscientificreviewers.¹

Wefollowtheincompletesocialcontractperspective (seee.g.AghionandBolton,2003)andassumethatitisimpossible to conditiontransfers bothonthecorrectstate oftheworldandontheindividual votescast. Rewardscannot dependon the correctstate of theworld when the latteris revealedmuch later than paymentsneed to be executed orwhen it is neverrevealed independentlyofthecommitteedecision,since itisprecisely thetaskofanexpertcommitteetofindout thestateoftheworld.Transfersthattreatmembersofthemajoritydifferentlyfrommembersoftheminority(withinthe committee)dependingonthevotecastmayhavesomedrawbacks.First,theymayentailabreachofprivacysinceindividual votesmust beknown tothe mechanism.This canbe especiallyundesirable ifcommitteemembers worrytheir decisions mayimpacttheircareers(seee.g.GersbachandHahn,2008,2012;Levy,2007).Votesecrecyappliesinmanyproceduresin law.²Second,contractingonindividualvotingbehaviormaypunishthecitizenwhoexertseffortbutisunluckyandreceives thewrongsignal.Atthesametime,however,itmayrewardanindividualwhoexertsthesameeffortlevelbutreceivesthe rightsignal. Adler andSanchirico(2006) and Fleurbaey(2010), amongothers,haveput forwardseveralreasonswhyrisk shouldbeevaluatedexpost.

We furtherproceedontheassumption thatcontractscannot bewrittenthat arecontingent ontheinformationacqui- sitioncosts incurredby theagents.Forexample,thisoccurswhensuch costsareprivate information,inwhichcaseevery individualhasincentivestoclaimcoststhatarehigherthanthoseactuallyincurred.

Tochanneltheagents’incentivestoacquireinformation,weconsiderthefollowingclassofrewardschemes:Eachmem- ber ofthecommitteereceivesa(positiveornegative)transferthat dependsonthevotetallydifferencebetweenthetwo alternativesobtainedafterallcommitteemembershavevoted.³Thismeansthatwefocusonpivotalevents,ase.g.in Persico (2004) and DalBo(2007),withthedifferencethatourmechanismscannotuseindividualvotesandrelyonlyonaggregate informationaboutthevotetally.Theparticularstructureoftherewardschemetobechosendependsonwhetherfulllearn- ingisfeasibleornot.Ineithercase,werequiretherewardstobeuniformlyfinancedbyall membersofthepopulationto ensurethat,likewisecommitteemembers,allcitizenswhoarenotcommitteemembersaretreatedequally.

We show that an adequate reward scheme guaranteesthat theright alternativeis implemented withprobability one (if full learning isfeasible) or that such probability convergesto one withpopulation size(if full learning is unfeasible).

Toachieve theseoutcomesatthesmallestaggregateinformationacquisitioncosts, committeesmustbesmall, whetherin absoluteterms(iffulllearningispossible) orinrelativeterms(iffull learningisimpossible).Roughlyspeaking,monetary subsidiesimprovedecisionsregardlessofcommitteesizeandenable smallcommitteestoyieldinformeddecisions.Impor- tantly, committee sizecan never be smallerthan three,so the possibilitythat a single citizendictates policy isavoided, whichcanbeundesirableforavarietyofreasons.Ourinsightscarryovertomoregeneralsetupsthanourbaselinemodel includingasymmetricpriors,asymmetricpreferences,andprivatevalues.⁴

Thepropertiesofourfamilyofmechanismsaremaintainedundertherestrictionthatallcitizens—andnotjustcommittee members—mustkeeptheir righttovote. Givingtherighttovotetoallcitizens ofthepopulationisanaturalrequirement fordecisionsinademocracy.Toshowthisresult,weaddasecondvotingstagetothemechanismsconsidered,whichthen correspondsto(amodifiedversionof)AssessmentVoting (Gersbachetal.,2021).Thisisatwo-roundmechanismthatsplits thepopulationintotwogroupsthatvotesequentially,withalltheindividualsinthesamegroupvotingsimultaneously;the firstvotinggroupcanbethoughtofasthecommittee(orAssessmentGroup,AG).Allindividualscastonevote,membersof thesecondvotingroundknowtheoutcomeofthefirstvotingroundbeforetheyvote,andthealternativethatreceivesmore votes inthetwo votingrounds combinedisimplemented. Tobe consistentwithour framework,we allow allcitizens to acquirecostlyinformationaboutthestateoftheworld,nomatterwhichroundtheyvotein.UnderAV,weshowthatonly themembersoftheAGacquireinformation,sothey endogenouslybecometheonlyexpertsofthesociety.Theremaining

1 Wealsoassumeinformativevoting,i.e.,individualsdonotstrategizeintheuseoftheinformationtheyget(regardlessofquality),butsimplytranslate itintoavote.Formostofouranalysis,thiscanbedonewithoutlossofgenerality.FromAusten-SmithandBanks(1996),weknowthat,intheabsenceof transfers,informativevotingisrationalwhenthepriorissymmetricandthevotingruleissimplemajority.

2 Forexample,constitutionalcourtsofcountriessuchasAustria,Belgium,France,orItalybanthepossibilityforindividualvotestobemadepublic;only theverdictofthecourtasawholeispublic.Seehttps://www.venice.coe.int/webforms/documents/default.aspx?pdffile=CDL-AD(2018)030-e,retrievedon12 February2022.

3 Financingisguaranteedwhentaxationcanbeenforced,inwhichcasewedonotneedtobeconcernedwithparticipationincentiveconstraints(i.e., with individualrationality).Evenifsuchconcernsmatter, weshowthatprovidedthereareenoughcitizens,at theinterim stageindividuals whoare notcommitteememberswillprefertotakepartinthemechanismthantoexit.Committeemembers,fortheirpart,arecontentwithbeingpartofthe committee,sothatcommitteesarestable.

4 Wealsodiscusshowtochooseoneparticularmechanismfromourfamilyofmechanisms,dependingonotherfactorssuchasrobustnessagainsterrors andthepossibilityforcommitteemembersnottoreceivetransfersatallortoevenhavetopayfineswithsomeprobability.

citizens donot acquire anyinformationby themselvesandsimplyvalidate the committee’schoice withtheir votein the secondvotinground.

Thepaperisorganizedasfollows:InSection2wediscussthepapersthataremostrelevantforourmodel.InSection3 we introduce our model andset up notation. In Section 4 we analyze our family of reward schemes. In Section 5 we considerthecasewherecitizenscannot bedeprivedoftheir righttovote. InSection 6wediscussreal-worldapplications ofourfamilyofmechanisms.InSection 7weinvestigatesome extensionsofourbaseline setup.Section 8concludes.The proofsofthemainbodyofthepaperareintheappendix.

2. Relationtotheliterature

Our paper is related to several strands of the literature. First, the formal literature on the Condorcet jury theorem (see e.g. Austen-SmithandBanks,1996;Castanheira, 2003; FeddersenandPesendorfer,1996; Gratton,2014;Krishna and Morgan,2012;Ladha,1992;Persico,2004;Razin,2003;Triossi,2013;Young,1995)investigateswhetherlarge elections(or committees) canaggregatedispersedinformation.Whentheanswerispositive,astrongrationaleisprovidedforworking together(e.g.asindemocracy).Wecontributetothisliteraturebyshowinghowmonetarytransfersandvotedelegationto asmall committeecanbeusedjointlytoovercomelowlevelsofinformationacquisition inthepopulationatlargearising whentheaccuracyofinformationischosenbyeachindividual.

Thepossibilitythatwithendogenousinformationvotedelegationtoasmallcommitteecouldbebetterthanone-round universal suffrage was already pointedout by Martinelli(2006). However, our insights are broaderbecause besides vote delegation toa committeewe consider monetary transfers to thecommitteemembers madefromthe restof thepopu- lation.Doingsosufficesto yieldinformeddecisionsthat areexpressedthrough votingforgeneralinformationacquisition costfunctionsandpopulationsizes(notjustlargeelectorates).Inparticular,ourasymptoticresultsguaranteeingaggregate learning butindividualignorance forlarge populationsapply morebroadlythanthosein Martinelli(2006).Moreover, our approachyields stablecommitteesandcanbe extendedtogivevotingrighttoallcitizens(notjust committeemembers).

Theprincipleofone-citizen,one-voteisparamountfordemocracy.

If all citizens have voting power butcommittee members votefirst as in AssessmentVoting (Gersbach etal., 2021), committeemembersbecomeendogenouslytheonlyexpertsinsociety.Ourmechanismsarethusfirstcharacterizedbythe featurethatvotingpowerisfully delegatedtoacommittee,whetherdejure ordefacto.Atleastsince GilliganandKrehbiel (1987),itiswellknownthattheparentbody(inthiscase,therestofthepopulation)shouldhavelimitedrightstoamend thedecisiontakenbyacommittee.Thisisinlinewithourinsights.

The secondmainfeatureofourfamilyofmechanismsisthat theyusemonetary transfers.⁵ Winter(2004) arguesthat evenifallagentsare exante equal,monetarytransfers, intheformofasymmetricrewardschemes,canprompt agentsto exertsociallyefficienteffortlevels (seealsoBernsteinandWinter,2012).Ourpapersharesthisinsightinthecaseofvoting:

Givingrewardstomembersofthecommitteeonlymaybedesirableforthecommongood.⁶

Acquiringcostly informationthat benefitseverybodyandcanbe usedlaterforvotingisa publicgood.Hencethepos- itive externalitiesassociated withacquiringinformationmaylead toan underprovisionof information.Forgeneralpublic goods,itisknownthatintroducingmonetarytransfersfromsomeagentstootherscanintroducenegativeexternalitiesthat compensate the positive externalities set out above (Morgan, 2000). For committeedesign, Persico (2004) analyzes how (sufficientlylarge)side-paymentscanbeusedtoensurefullinformationacquisitionincommitteesbyconditioningrewards onpivotalevents(see alsoDalBo,2007,forasimilar mechanismina setupwithprivate values).Whileourmechanisms also build onpivotalevents,our approachdiffers insofarasthe rewardschemes we considertreat every memberofthe votingbodyequallyexpost,sincetheydonotuseindividualvotes.⁷

Some papers have examined optimal committee size by analyzing how voting rules should be designed not only to aggregate dispersed information efficientlybutalso toinduce agents toacquire sufficientinformation (see Gerling etal., 2005,forasurveyoncommitteedesign).Bylettingthevotethresholdtoupsetastatusquovary, Persico(2004) alsoshows that boththethresholdandthecommitteesize relativetothe totalpopulationshould bedetermined bytheinformation levelacquiredinequilibrium,andthusbegenericallyaboveonehalf,buttypicallybelowone.While Persico(2004) considers thevotingthreshold,westicktothemajorityruleandintroducetransferschemes,whichproducesquitedifferentresults.

Finally, ourpaper isalso relatedto a strand of themechanism designliterature that studies optimalinformation ac- quisitionmechanismsforcommittees.InGerardiandYariv(2008) communicationispossible,andthemechanismdesigner who participatesinthe decisionchooses boththeoptimalcommitteesize andhowindividual decisionstranslateintothe final outcome. GershkovandSzentes(2009) characterizetheoptimalstoppingruleina sequentialmechanisminwhicha

5 Ifsmallcommitteesareexposed toexternalinfluence,outside paymentstoswayvoters maybedirected atcommitteemembers.Thishas been thoroughlystudiedboththeoreticallyandempiricallyforlobbyinggroupstryingtobuythevoteoflegislators (seee.g.Austen-SmithandWright,1994;

FelgenhauerandGrüner,2008;Wright,1990)orofcentralbankcommitteemembers (GersbachandHahn,2008).AccordingtoName-CorreaandYildirim (2018),largecommitteesarebetterthansmallonesatavoidingcapturefromanoutsider.

6 Morerecently, Azrieli(2021) hasexaminedasetupsimilartoours,buthefocusesonindividualcontractsinsteadofvotingbythemajorityrule.

7 Theserewardschemesalsofeatureminimalmonetarytransfersensuringhighlevelsofinformationacquisitionforstablecommittees.Thisisimportant forpracticalimplementation,ashighmonetarytransferscancrowdoutthedirectpriceeffectprovidedbymonetarytransfers(seee.g.Gneezyetal.,2011;

Meier,2007).

randomlyselectedvoterisaskedtoacquiresomeinformationineachround.⁸ KoriyamaandSzentes(2009) studythemin- imumsizeofcommitteesguaranteeingexante efficiency.Wedifferfromthesepapersinsofarthatweallowsidepayments andconsiderdifferent(continuous) levels ofinformationacquisition.Thisleads to resultsaboutthe optimalsizethat are notinaccordwiththeabovepapers,aswesuggestalowercommitteesize.

3. Model

Our model builds on Martinelli (2006). There is a population of 2n+1 individuals who must choose between two alternatives, say A and B.Onealternative isthe“right” one,i.e., ityields higherutility forallindividualsthan theother.

Forsimplicity,wenormalizetheutility obtainedfromtherightalternativetooneandtheutility obtainedfromthewrong alternative tozero.However, therightalternative isunknown, asitdependsonthe (unknown)state ofthe world.Ifthe state of the world w is A (B), then A (B)is the rightalternative. We assume that exante, the probabilities that either alternativeisrightareequalto1/2.Individualscanobtainasignalaboutthelikelihoodoftherightalternative,i.e.,asignal aboutthestateoftheworld,yetatsomecost.Specifically,thereisan(informationacquisition)costfunction

c

:



0

,

¹

2



→ R

₊

∪ {∞},

wherec(x) isthecost inutilityunitsforanyindividual i ofreceiving asignal si∈ {^A,B}^{aboutthestate oftheworld of} quality1/2+^{x (x}∈ [⁰,¹₂]^{).Thismeansthatfor y}∈ {^A,B}^,

P rob



s_i

=

^y

|

^w

=

^y



=

¹ 2

+

^x

.

Signalsare stochasticallyindependentacross individualsandc(x) isstrictly increasing (andconvex, seebelow).Notethat c(1/2)canbeinfinite,inwhichcaseanindividualcannotinformhimself/herselfwithcertaintyaboutthestateoftheworld.

Thereisalsoacommittee,whichisasubsetof 2m+1 individualschosen(randomly)fromthegeneralpopulation.This meansthat m≤^n.Eachmember i ofthecommitteecandecideabouthis/herownlevelofinformation,whichwe denote by xi.Oncecommitteemembershavereceivedtheirsignals,allofthemvotesimultaneouslyforoneofthealternatives.The alternative that receives atleastm+^{1 votes is}implemented. Wealso assume that noabstention occurs. While thisisa verystrongassumptiontoimposeontheentirepopulationif n islarge,itislessdemandingforlowvaluesofm.⁹Without abstention,thealternativewiththelargestnumberofvotesisimplemented.

Finally,thereisamonetarytransferscheme(tobespecifiedlater)accordingtowhichtotalpayoffsarerealized.Atransfer scheme isavector(v_i)²ⁿ_i=⁺₁¹∈ R^{2n+1specifyingapayoffforallmembersofthe}population,withthepropertythatitisbudget balanced, i.e.,2n+¹

i=1 v_i=^0.Individualshavelinearutility inmoney.This meansthat whenalternative y is implemented andtherightalternativeis z,thetotalutilitythatindividual i derivesis

u_i

(

z

,

y

,

x_i

,

v_i

) := 1

z

(

y

) −

^c

(

x_i

) +

^vi

,

where1z(y)=^{1 if z}=^{y and}1z(y)=0 otherwise.

Tosumup,weanalyzeamechanismthatspecifiesthefollowingsequenceofevents:

1. Thecommitteeisformed.

2. Committeemembersdecidehowmuchinformationtheywanttoacquire.

3. Committeemembersreceiveasignalandcastavote.

4. Thestateoftheworldisrealized.

5. Atransferschemeisappliedandtotalpayoffsarerealized.

We assume thatindividuals alwaysvotein favorofthealternative that isinterim most likely,giventheir signal. Ifan agent exerts some effort toacquire information,the signal is informative and votingin favorof the other alternative is weaklydominated.Ifbothalternativesareequallylikely(i.e.,theonlyinformationtheagentshaveistheprior),individuals vote according to their signal. This rules out implausible equilibria, with or without transfer schemes, such as the one wherenoagentprocuresinformationandallagentsvoteforagivenalternative.Inoursetupwithmonetary transfersand anonymousvoting,thelatterassumptioncapturesthefactthatcollusionisdifficult.

Following all the above, the (static) game we consider is one inwhich the player set iscomposed ofall committee membersandeachplayer’s strategysetistheinterval[⁰,1/2]^{.Wedenotethisgameby}G^{m,asthecommitteeismadeup} of 2m+^{1 members. Fortheanalysis, we focus on}symmetric Nash equilibria, i.e., we assume that all playerschoosethe sameinformationlevel x,withx∈ [⁰,1/2]^{.Thisisreasonableinoursetupinwhichallagentsareex}ante equal.Westress

8 Ageneralmultipartycomputationsettingisanalyzedin SmorodinskyandTennenholtz(2006).

9 Abstentioninacostlyinformationacquisitionsetupisanalyzedin McMurray(2013) and Oliveros(2013).

that ifwe takethedescriptionofourgameatfacevalue, acitizenchoosesthelevelofinformations/hewantstoacquire butdoesnot choosethealternatives/hevotesfor.Thisfollowsfromtheassumptionmadeabovethataftereachagent i has receiveda(possiblyuninformative)signal,s/hesimplyfollowstherecommendationgivenbythesignal.¹⁰

Finally,we make some assumptions onthe cost functionc(x) beyondthe fact that itis strictlyincreasing in [⁰,1/2). First, c(x) is twice continuously differentiable inthe interval [⁰,1/2). Thisassumption is oftechnical nature andsimply facilitatestheanalysis.Second,aslongasc(x)isfinite,c(x)isstrictlyconvex,i.e.,c^(x)isstrictlyincreasing.Thisrulesouta multiplicityofsymmetricequilibriawithoutrewardschemes.Third,c(0)=^{0,sothatacquiringzero}informationiscostless.

Fourth,c^(0)=^{0.Thislastassumptionguaranteestheexistenceofequilibriawithpositive}informationacquisition.¹¹ 4. Thresholdrewardschemes

AsmentionedintheIntroduction,weassumethroughoutthatthecoststoacquireinformationareprivate.Alternatively, accrued informationacquisition costs may beobservable but notcontractible, as extensivelydiscussed inthe incomplete socialcontractsliterature.Ifvotingtakesplaceprivately,asiscustomaryindemocraticsocieties,thentheonlyinformation thatispubliclyavailableisthevotingtally,i.e.,thenumberofvotesinfavorofalternatives A and B.

Weshallshowthat adequatecontractingonthevotetallysufficestopromptthecommitteememberstoeitherinform themselves fully or atleast to ensure that the probability of selecting the correct alternative converges to one. In such cases,the transferschemecanact asadevice coordinatingcommitteemembers toacquire ahigherlevel ofinformation.

Contracting on thevotetally guarantees that,in equilibrium, theincentives linked tovoting pivotalityare either zeroor closeto zero.The onlyincentivesthat matterthen arethose linkedto theside-paymentscontingent onvotetally, which leadtodesirableoutcomes.

Toelaborate,let k denotethevotetallydifferencebetweenalternatives A and B.Thatis,ifk>0,alternative A received k more votes from the committeemembers than alternative B in the votinground. Since there is no abstention andthe committeehas 2m+^{1 members,itmustbethecasethatk}∈Dm,whereDm:=

2l+¹|^l∈ Z, −^m−¹≤^l≤^m

.Foreachm, withm∈ N^{,wethenconsiderfunctions}

t^m

:

Dm

→ R

k

→

^tm

(

k

),

with

t^m

(

k

) =

^tm

( −

^k

)

for k

∈

Dm

.

(1)

Any such functiondetermines a (positiveornegative) rewardto anyvotingoutcome basedonthe absolutedifference in termsofvotesbetweenalternatives,andhencenomatterwhichalternativereceivesmostvotes.

AsarguedalsointheIntroduction,wefocusontransferschemesthat,fromanexpost perspective,treatmembersofthe committeeequallyandindividualsoutsidethecommitteealsoequally.Thatis,givenavotetallydifferencek∈Dm,foreach i∈ {¹,. . . ,2n+^1},welet

v_i

=

t^m

(

k

)

if i is committee member

,

−

₂^2m_(n−⁺_m¹₎

·

^tm

(

k

)

if i is not committee member

.

⁽²⁾

Ift^m(k)>0,rewardstothecommitteemembersarefinancedhomogeneouslybytherestofthepopulation.Ift^m(k)<0,the committeememberssubsidize therestofthepopulation.Atransferschemedefinedasin (2) iscalledathreshold(reward) scheme, orTS forshort.TheonlyinformationaTSusesbeyondthevotetallyiswhetheragivenindividual belongstothe committeeornot.

With aTS, by accountingforall possible events,the expectedutility ofany member i ofthe committee(for a given size 2m+^{1)whens/hechooses}informationacquisitionlevelx_i∈[0,1/2] andalltheother2m membersofthecommittee choose x∈ [⁰,1/2]^{,asdemandedbyournotionofsymmetric}equilibrium,is

U_i

(

x_i

|

^x

) =

2m

m



·

1

2

+

^x



m

1 2

−

^x



m

1 2

+

^xi



−

^c

(

x_i

) + χ +



m k=¹

2m

m

+

^k



1

2

+

^x



m+^k

1 2

−

^x



m−^k



1 2

+

^xi



·

^tm

(

2k

+

¹

) +

1

2

−

^xi



·

^tm

(

2k

−

¹

)



10 AnexceptionisSection7.1.

11 Ifc^(0)>0,themechanismsweproposecouldbeusedtoincreasethepivotalityofthecommitteememberstogeneratepositiveinformationacquisition inequilibrium.

+ 

m k=¹

2m

m

+

^k



1

2

+

^x



m−^k

1 2

−

^x



m+^k



1 2

+

^xi



·

^tm

(

2k

−

¹

) +

1

2

−

^xi



·

^tm

(

2k

+

¹

)



+

2m

m



1

2

+

^x



m

1 2

−

^x



m

·

^tm

(

1

),

(3)

where

χ

is a componentof utility that doesnot depend on xi. Tounderstand Equation (3), note that thefirst term on theright-hand sidecapturesindividual i’sexpectedutility forcastingavotefollowinghis/hersignal ifs/heincurscost x_i, conditionalonbeingpivotal.Theprobabilityofthelattereventwhenallcommitteemembersotherthanindividual i choose informationlevel x isequalto

Px

[

tie

] =

2m

m



·

1

2

+

x



m

·

1

2

−

x



m

.

Thesecond termontheright-handsideofEquation (3) isthecosttoacquireinformationlevel xi.Then,thetwosumson theright-hand side ofEquation (3) capture theexpectedtransferindividual i willreceive whencastinghis/hervoteafter havingacquiredasignalofquality1/2+^xi,conditionalonthevotesoftheremainingindividualsofthecommitteeyielding agivenmargininthevotetally.Thesemarginsareindexedby k.Avotecanincreaseordecreasethevotemargin.Thefirst sumcapturesthecasewhenthereareatleastasmanyvotesfortherightalternativeasthereareforthewrongalternative, withthesecond sumcapturingthecasewhenthere areatleastasmanyvotesforthewrong alternativeasthere arefor therightalternative.Finallynotethat ift^m(k)=^{0 forallk}∈Dm,Equation (3) reducestothecaseofstandardvotingwith turnoutequalto2m+^1.12

Nextrecallthatweareassuming (1).Therefore,ifwedenote

φ

^m

(

l

) =

^tm

(

2l

+

¹

) −

^tm

(

2l

−

¹

)

for all l

∈ {

⁰

, . . . ,

m

}

itfollowsthatforallxi∈ (⁰,1/2), U_i^

(

x_i

|

^x

) =

2m

m



·

1

4

−

^x2



m

−

^c

(

x_i

) +



m l=1

2m

m

+

^l



· φ

^m

(

l

) ·

1

4

−

^x2



m−^l

·

1

2

+

^x



2l

−

1

2

−

^x



2l



and

U_i^

(

x_i

|

^x

) = −

^c

(

x_i

) <

0

.

Accordingly, anecessaryandsufficient conditionforastrategyprofile (x,. . . ,x),withx∈ (⁰,1/2),tobe anequilibriumof gameG^mis

c^

(

x

) =

:=^A(x)

  

2m

m



·

1

4

−

^x2



m

+ 

m l=¹

2m

m

+

^l



· φ

^m

(

l

) ·

1

4

−

^x2



m−^l

·

1

2

+

^x



2l

−

1

2

−

^x



2l



  

:=^B(x)

.

(4)

Ifm>0,twoimportantobservationsfollowreadilyfromtheaboveexpression.First,sincec^(0)=^0,ifφ^m(l)=^{0 foralll}∈ {¹,. . . ,m}^,thenEquation (4) hasexactlyonesolution,say x.Thereasonisthatc^(·)^isstrictlyincreasing, A(·)^isdecreasing, A(1/2)=^0,andc(0)=^0.Hence,(x,. . . ,x)istheonlysymmetricequilibriumofG^{m.Thiscapturesone-round,}simultaneous votingwithin acommittee.Ifφ^m(l) =^{0 forsomel}∈ {¹,. . . ,m}^{,bycontrast,neitheruniqueness norexistenceof(interior)} symmetricequilibriaisguaranteed.Clearly,becausec^(0)=^{0 and A}(0)+^B(0)>0,thestrategyprofilewherenocommittee memberacquiresinformation,namely(0,. . . ,0),cannotbeanequilibrium.If,moreover,

A

(

x

) +

^B

(

x

) >

c^

(

x

)

for all x

∈ (

⁰

,

1

/

2

),

(5)

theonlysymmetricequilibriumofG^{misthestrategyprofileinwhichallcommitteeagentsacquirefull}information,namely (1/2,. . . ,1/2).Bycontrast,if (5) doesnothold,thenthereisatleastone(interior) symmetricequilibrium(x,. . . ,x),with x∈ (⁰,1/2).Buttheremaybeotherequilibria,including(1/2,. . . ,1/2).

12 Thisisthecaseanalyzedby Martinelli(2006).

The second important observationfrom (4) is that φ^m(0)=^t(1)−^t(−¹)=^{0, so theexact value of t}(1) andt(−¹) is immaterial for equilibrium. This is becausein the event that a citizen breaks a tie, s/he will create a difference of one vote, no matterwhich alternatives/he votesfor. Whilethisis relevantforthe utility citizensderive fromthe alternative implemented,itdoesnotaffecttheincentivestobeinformed,conditionalonbeingpivotal.Thisisreminiscentoftheswing voters’curse (FeddersenandPesendorfer,1996;Herreraetal.,2019a,b).Aswediscussbelow,theexactvalueoft(1)=^t(−¹) does matterfortheextentoftransfersandforparticipationincentives,i.e.,forindividualrationality.

Foreachintegerm≥^{0,weconsiderinthefollowingfunctionssuchas}

φ

^m

: {−

¹

,

0

,

1

, . . . ,

m

} → R

l

→ φ

^m

(

l

),

withφ^m(0)=^0.Since φ^m(·) ^determinestm(·)^{withone degreeoffreedom, we alsoadopttheconvention that} φ^m(−¹)= t^m(1).Thisguaranteesthatthereisaone-to-onecorrespondencebetweenφ^m(·)^andtm(·)^{,anditallowsustowritetm}(2l+ 1)=l

l^=−¹φ^m(l^)foralll∈ {⁰,1,. . . ,m}^{.Westressthat}φ^m(l),withl>0,rewards(orpunishes)amarginalvoteevenifit doesnotbreakatieintheelection.Suchrewardsmayariseinnormalelectionsifvoterscareaboutthevictorymargin,and theymightbesignificant (Herreraetal.,2019b).Inoursetup,theserewardsarechosenbydesign.

Forthesubsequentanalysis,weproceedwithageneralinformationacquisitioncostfunction c(x)fulfillingourassump- tionsanddistinguishtwocases,dependingonthevalueof c^(1/2).

4.1. Fullinformation

First,weconsiderthecasewherec^(1/2)<∞^{.Thismeansthatitis}theoreticallypossibletofindoutwithfullprecision whattherightalternativeis(atafinitecost).Forthiscase,westartbyconsideringthefollowingrewardscheme:

φ

^m

(

l

) =

c^

(

1

/

2

)

if l

=

^m

,

0 otherwise

.

⁽⁶⁾

TheTSdefinedin (6) onlyrewardsdecisionsthat arereachedunanimouslywithinthecommittee.Ifunanimityisreached, eachofthecommitteemembersisgivenatransferthatamountsto c^(1/2).Aunanimousdecisioncanopt forthecorrect alternative ortheincorrectone. Unlessall themembers ofthecommitteeinformthemselves perfectly,both optionswill occurinequilibrium,albeitpossiblywithdifferentprobabilities.

Westartbyshowingthefollowingresult:

Proposition1.SupposetheTSisdefinedby (6) foragiveninteger m≥^{1.Thenthestrategyprofileinwhichallmembersofthe} committeeacquirecompleteinformation,(1/2,. . . ,1/2),isanequilibriumofG^m.

Proof. Seeappendix. 

From theproofofProposition1,oneimmediatelyseesthat providedc^(1/2)<∞^{,anecessaryandsufficientcondition} for(1/2,. . . ,1/2)tobeanequilibriumofG^mis

φ

^m

(

m

) ≥

^c

1

2



.

(7)

The intuition for thisresult is clear. The TS must reward the marginal effort of acquiringanother piece of information beyond x sufficiently.Thiseffortishighestwhen x is(infinitely)closeto1/2 becausec(x)isconvex.Assumingthatφ^m(l)≥⁰ forl∈ {−¹,0,. . . ,m−¹}^{impliesthatthe transfersthat eachmemberwho isnot partofthecommitteemustpaytothe} membersofthecommitteetoensurefullinformationacquisitioncanneverbelowerthan

2m

+

1 2

(

n

−

^m

) ·

^c

1

2



.

(8)

InProposition 1,wehaveassumed m≥^{1,sothatthe committeeconsistsofatleastthreemembers.If m}=^0,andhence the committeeconsistsofonlyone member,the rewardschemedefinedin (6) hasnobearingon theunique committee member’scalculusformaximizinghis/herutility(seeEquation (3)),whichisaccordingly

max

x∈[0,1/2]

[

^x

−

^c

(

x

)].

The solution to the above problem is generically different from 1/2. Why is the case with more committee members different? Ifthereareatleastthreemembers inthecommittee,all theincentivesforbecoming informeddueto(voting) pivotality disappear if the strategy profile of Proposition 1 is played. This is because all individuals informthemselves

Fig. 1. Function(A(x)+B(x))−c^(x)—seeEquation(4)—whenφ²(0)= φ²(1)=0 andtherewardforunanimity,φ^m(2),takesdifferentvalues.Theblue (uppermost)curvecorrespondstoφ²(2)=^2c(¹₂),thegreen(secondfromthetop)curvecorrespondstoφ²(2)=^c(¹₂)(seeExample1),theyellow(third fromthetop)curvecorrespondstoφ²(2)=¹2c^(¹₂),andthered(lowest)curvecorrespondstonorewards,i.e.,φ²(2)=^0.Aninteriorequilibriumexists whenthecurveintersectsthex-axisbetween0 and1/2.Afullinformationequilibriumexistswhenthecurveisnotnegativeatx=¹/2.

perfectlyandvotefortherightalternative.Asaconsequence,nosingle individualcan changethevotingoutcome (which isdetermined by themajorityrule). Theonlyincentivesleft forinformingoneself are thoselinked totheTS.If φ^m(m)≥ c^(1/2),convexity of c(x) guarantees that the strategy profile of Proposition 1 is an equilibrium. In this equilibrium, all members ofthecommitteeinformthemselvesfully, sothat theequilibriumis inefficientinthat duplicateinformationis acquired. Conditionalon implementing the rightalternative with probability one, such (welfare) inefficiency is therefore minimizedform=^1.

WhileProposition1ensuresthatfullinformationacquisitionisanequilibrium,theremaybeotherequilibriawithpartial informationacquisition.Thisisshownnextandwarnsusthatill-designedtransfersmayhaveconsequencesforoutcomes.

Example1.Consider c(x)=^3x3+^{2x2 and m}=^{2, and assume that the TS defined by (6) is used. Then game} G^{m has} two equilibria:one equilibriuminwhichall citizenschoose x=⁰.5 and anotherequilibriuminwhichall citizenschoose x≈0.24.Fig.1illustratesthismultiplicityofequilibriabyplottingfunction(A(x)+B(x))−c^(x)whentheTSdefinedby (6) isused,aswellaswhenotherrewardsforunanimityareconsidered—seeEquation (4).

Todealwithuniquenessof(symmetric)equilibria,weconsidernowthefollowingTS:

φ

^m

(

l

) =

4^m−1

2m

·

^rm if l

=

^m

,

0 otherwise

.

⁽⁹⁾

In (9), (rm)m≥¹ isa certain increasing sequencethat dependson costfunction c(x),is boundedfromabove,andsatisfies thatforallm≥^1,13

4^m−1

2m

·

^rm

≥

^c

1

2



.

(10)

ItisimportanttonotethatInequality(10) neednotbetight.

Weobtainthefollowingresult:

Theorem1.SupposetheTSisdefinedby(9) foragiveninteger m≥^{1.Thenthestrategyprofileinwhichallmembersofthecommittee} acquirecompleteinformation,(1/2,. . . ,1/2),istheonlysymmetricequilibriumofG^m.

Proof. Seeappendix. 

Theabovetheoremshowsthatregardlessofthesizeofthecommittee,onecanalwaysfindfiniterewardsforitsmem- bersthat induceeach of themto acquirefull information.Such rewardsexclude thepossibilitythat other equilibriaexist

13 Thevalueofrmisindependentofn,providedthatn≥^{m.Hence,sequence}(rm)^∞_m=0implicitlyassumesthatasm grows,somustn.Werefertothe proofofTheorem1fordetailsaboutsuchasequence.

withsuboptimal informationacquisition levels. Theydoso by rewarding unanimityenough so that, givenanysuch sub- optimal information level, all individuals prefer to marginally acquire more information and increase the probability of unanimity beingreached, and hence ofreceiving the reward. While this result is intuitive forarbitrarily large rewards, Theorem1achievesmore,asitsetsanupperboundtotherewardsensuringaunanimousdecisionwithprobabilityone.

Using thefact that(rm)m≥¹ is boundedfromabove,we letr denotethe infimumof anysuch bound.Then, individual rewardsforcommitteemembersarebounded(fromabove)by

4^m−1 2m

·

r

.

Thismeansthat theamounteachindividual whoisnot amemberofthecommitteeneedstocontribute canbe bounded (fromabove)by

4^m−1

·

^r

(

n

−

^m

) .

Hence,aslongasm canbechosensuch thatm= (^{ln n}),individualcontributionstotherewardsforcommitteemembers donotgrowunbounded.Thismeansthat iffullinformationacquisitionispossible,asufficient conditionforcommitteesto implementthecorrectalternativewithprobabilityonethroughafeasiblerewardschemeisthatthesizeofthecommittee grows logarithmically withthe total population. Thisis importantfor real-world implementationof the mechanismthat usestheTSdefinedby(9),andmoregenerallyfortheuseofmonetarytransfersincommitteedesign.Ontheotherhand,it isstraightforwardtoseethatasfarasthemechanismusingtheTSdefinedby (9) isconcerned,itisalwayspreferablefrom awelfareperspectivetochoosem=^{1 overanym}>1.Thereasonisthatbothoptionssetcommitteesthatimplementthe correctalternativewithprobabilityone,butm=^{1 involvesthelowestaggregate}informationcosts.

In the following, we show that it is possible to compensate the committee members for the costs associated with acquiringfull informationand,atthesametime, approximatetheboundgivenin (7). Tothisend,considerthe following rewardscheme:

φ

^m

(

l

) =

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

c^



₁

2



if l

=

^m

,

t_m

>

0 if l

=

^l

,

−

^tm

−

^z

<

0 if l

= −

¹

,

0 otherwise

,

(11)

wherel^∈ {¹,. . . ,m−¹}^{,and t}m>0 and z∈ R^{areparametersthatneedtobe}determined.Implementingthemechanism thatusestheTSdefinedby (11) requiresm>2,i.e.,thecommitteemustconsistofatleastsevenmembers.

Weobtainthefollowingresult:

Proposition2.SupposetheTSisdefinedby(11) foragiveninteger m>2.Thenthestrategyprofileinwhichallmembersofthe committeeacquirecompleteinformation,(1/2,. . . ,1/2),istheonlysymmetricequilibriumofG^{m,providedthatparametert}mis sufficientlylarge.

Proof. Seeappendix. 

The above result guarantees that, under uniqueness ofequilibria, the bound definedby (7) is attainable. Thatis, the marginalrewardforbeingpivotaltoattainunanimityis c^(1/2).TheTSdefinedby (11) worksbysettinglargepunishments ifslimmajorities arereached. While thesemajoritieswill notoccur onthe equilibriumpath(ofthe uniqueequilibrium), such punishmentsarecrucialofftheequilibriumpathtoensurethat nootherequilibriaexistbeyondthefullinformation acquisition equilibrium. Ascapturedby Theorem1andProposition2,there isthereforeatrade-off betweenloweringthe rewardforunanimityandincreasingthepunishmentiftheoutcomedeviatesfromunanimity.

ItremainsanopenquestionwhethertheresultofProposition2,togetherwiththepropertythattheboundgivenin (7) isbinding,canbeobtainedwithoutany(off-equilibrium)fines(i.e.,withoutthepossibilitythatcommitteemembersmake monetarytransferstotherestofthepopulation).Nevertheless,belowwediscusstherelevanceoftheoff-equilibriumfines associated with the TS definedby (11) from the perspective of a social planner with preferences equal to those of the citizens.

Supposenow, forinstance,thatl^=^{1 fortheaboveTS.Thenamajorityofonevoteforeither}alternativeinthevoting tallyispunishedby−^tm−^{z,amajorityofatleastthreevotesthatisshortofunanimityispunishedby}−^{z,andunanimity} isrewardedbyc^(1/2)−^z.Since z onlyentersinφ^m(−¹),theexactvalueof z doesnotmatterforequilibrium.Thisimplies that thetransfers committeemembersexpecttoreceive,namelyc^(1/2)−^{z,canbe settoanyvalue,inparticulartozero} if z=^c(1/2).Insuch acase, therewillbe notransfersinequilibrium. Aswe discussbelow,the(expected)extent ofthe