Department of EconomicsUniversity of CologneAlbertus-Magnus-PlatzD-50923 KölnGermany

http://www.wiso.uni-koeln.de

U N I V E R S I T Y O F C O L O G N E

W O R K I N G P A P E R S E R I E S I N E C O N O M I C S

No. 64

THE CURSE OF UNINFORMED VOTING: ANEXPERIMENTAL STUDY

JENS GROßERMICHAEL SEEBAUER

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Thecurseofuninformedvoting:Anexperimentalstudy

JensGroßer(FloridaStateUniversityandInstituteforAdvancedStudy,Princeton)

MichaelSeebauer(UniversityofErlangen‐Nuremberg)

ABSTRACT

We studymajority voting over two alternatives in small groups. Individuals have identical preferences but are

uncertainaboutwhichalternativecanbetterachievetheircommoninterest.Beforevoting,eachindividualcanget

informed, towit, buy a valuable but imperfect signal about the better alternative. Voting is either voluntary or

compulsory. In the compulsorymode, each individual can vote for either of the two alternatives, while in the

voluntarymode they can also abstain. An uninformed random vote generates negative externalities, as itmay

overrideinformativegroupdecisionsinpivotalevents.Inourexperiment,participantsingroupsofthreeorseven

get informedmoreoftenwithcompulsory thanvoluntaryvoting, and in thiswaypartly counteract thecurseof

uninformed voting when they cannot avoid it by abstaining. Surprisingly, uninformed voting is a common

phenomenoneveninthevoluntarymode!Aconsequenceofsubstantialuninformedvotingispoorgroupefficiency

inalltreatments,indicatingtheneedtoreconsidercurrentpracticesofjuryandcommitteevoting.

Thisversion:August2013

JensGroßerisAssistantProfessorattheDepartmentsofPoliticalScienceandEconomics,FloridaStateUniversityandExperimentalSocialScienceatFloridaState,531BellamyHall,Tallahassee,FL32306‐2230,andamemberofthe Institute for Advanced Study, School of Social Science, Princeton in the academic year 2012‐13.(jgrosser@fsu.edu).MichaelSeebauerisPhDCandidateattheDepartmentofEconomics,UniversityofErlangen‐Nuremberg,LangeGasse20,90403Nuremberg,Germany(michael.seebauer@fau.de).This isasubstantiallyrevisedversionofour2006workingpaper“Votingandcostly informationacquisition:Anexperimental study,” ofwhich an abstract appeared in an article in a researchmethods volume (Großer2012).Financial support from the GEW Foundation Cologne and the German Science Foundation (DFG) is gratefullyacknowledged.WealsothanktheEconomicsLaboratoryoftheUniversityofColognefortheirhelpandhospitality,and participants at seminars at Princeton University, the University of Hawai’i, and the Institute for AdvancedStudy inPrinceton,aswell asat theGEW2005Tagung inCologne, theESA2006Meeting inTucson,SEA2008Meeting in Washington D.C., and the Experiments in Political Economy Workshop, Caltech in 2013 for theirvaluablecomments.

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1Introduction

Groups of individuals make decisions in almost every realm of life. An important rationale for this

phenomenonistheirpotentialtoaggregateinformationthatwouldotherwiseliedispersedanddormant

among the individuals. Accordingly, it iswidely argued thatmore heads have, or can generate,more

knowledge than fewerheads,and therefore, largergroups tend tomakebetterdecisions thansmaller

groups. The importance of knowledge accumulation and aggregation in groups is evident in the

following examples: juries consider evidence to acquit or convict the accused; legislative committees

gather know‐how of experts and special interests groups in hearings; hiring committees evaluate

documents of applicants; and editors ask referees for their opinions about research papers. But do

larger groups reallyperformbetter than smaller groups in thepresenceof free rider incentives (e.g.,

whengathering information is time‐consumingandcostly)?Andhowdoes theanswerdependon the

specific information aggregation mechanism (e.g., the communication or voting procedure)? Do

individuals get informedmore oftenwith compulsory than voluntary voting?At first glance, banning

abstention, such as for jurors in criminal trials, seems to be a practical solution to improve group

decision‐making, but the impact on informational efforts and thus performance is virtually unknown.

We tackle these and similar questions using game theory and a laboratory experiment by studying

knowledge accumulation and aggregation in different sized groups where individuals have identical

preferences, information acquisition is costly, and consensus is obtained by voluntary or compulsory

majoritarianvoting.

Theaggregationofprivateinformationviavotingisakeyfunctionofdemocraticdecision‐making.1

When individuals have identical preferences but different opinions because they are uncertain about

which of several alternatives can best achieve the common interest, then pooling their private

information leadstoamutualagreementaboutthecorrectalternative(supposeonlyoneisbest).The

importantCondorcet JuryTheorem (1785)uses statistical inference to show thatmajority voting can

assume this pooling function, and therefore, the group’s decision is more likely correct than any

individual’s decision and almost surely correct in very large groups. In themost basic version of the

1Moregenerally,Piketty(1999)putstheimportanceofinformationaggregationinelectionsonalevelwiththeimportanceofinformationaggregationinmarkets.

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modelunderlyingthistheorem,agroupchoosesbetweentwoalternativesviacostlessmajorityvoting

anduncertaintyisgivenbyacommonpriorprobabilityofone‐half,indicatingthateachalternativehas

thesamechanceofbeingthebetterone.Moreover,eachindividualgetsacostlessprivatesignal,towit,

an independent Bernoulli trial showing the correct alternative with probability ∈ , 1 and the

incorrect alternativewith probability 1 . Thus, signals are valuable but imperfect and hencemay

induce conflicting opinions. Using this simplemodel, the intuition of Condorcet’s Jury Theorem is as

follows: the pool of private information growswith each sincere vote (i.e., itmirrors the signal) and,

since , the expected difference in votes grows in favor of the correct alternative so that the

probability of a correct group decisionweakly increases in the group size.2 And, by the law of large

numbers,inverylargegroupsthefractionofcorrectsignalsapproaches withprobabilityone,sothat

thevoteoutcomealmostsurelymatchesthecorrectalternative.

Theverypromisingresultsofthetheoremrelyonseveralassumptions,chiefamongwhicharethat

informationviaprivatesignalsiscostlessandthatvotingissincereandcostless.Butitignoresthefree

rider incentives that are generated if for example, more realistically, costs of signal acquisition are

involved (Downs 1957; Olson 1965).When the incentive to rely on informational efforts of others is

strongenough, theresultcanbe“rational ignorance”where theendogenous informationpoolshrinks

ratherthanincreasesinthegroupsize,say,ifthesizeisdoubledbutaverageeffortdropsbymorethan

half due to stronger free riding in the larger group (Martinelli 2006;Mukhopadhaya 2003). In other

words,Condorcet’sJuryTheoremneednotholdinthepresenceoffreeriderincentives.

In thepresentpaper,weexaminehowcostly informationacquisitiondependsonthegroupsize

(threeandseveninourexperiment)andifcompulsoryvotinginducesindividualstogetinformedmore

often than voluntary voting.3 In our game, they first decide independently and simultaneously on

whethertobuyaprivatesignalandthereaftervote,atnocost,independentlyandsimultaneouslyover

two alternatives usingmajority rulewith random tie‐breaking (in the voluntarymode, they can also

2 According to Austen‐Smith and Banks (1996), an “informative” vote mirrors the signal and a “sincere” vote maximizesexpectedpayoffsandmayopposethesignal.Forourparameters, thetwokindsofvotesalwayscoincidesowecanusebothtermssynonymously.And,anextrasincerevoteweaklyincreasestheprobabilityofacorrectgroupdecision,becauseityieldsanincreaseifanoddnumberofvotesisreachedbutdoesnotchangetheprobabilityifanevennumberisreached.3 For amore general discussion of voluntary versus compulsory voting and the incentive effects, see for exampleAbraham(1955),Birch(2009),andLijphart (1997).Moreover,Börgers(2004)andGroßerandGiertz (2009),amongothers,comparevoluntaryandcompulsorycostlyvotingwithprivatepreferences.

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abstain). Hence, we depart from Condorcet’s basic model in two important ways, to wit, by using

endogenous costly signals and by comparing situations where individualsinformed and

uninformedareeither freeorobligedtovote(incontrast, in thebasicmodeleveryone isassigneda

costlesssignalandthuspreferstovoteanyway).Individualsbasetheirdecisionsonlyongroupevents

that are both pivotal and informative, the latter meaning that the combined signals in the group

conveyedbysincerevotesfavoroneofthealternatives.Intheoneextreme,ifeveryoneisuninformed

andvotesrandomlyorabstains,thennosignalcostsaccrueandthegroupdecisioniscorrecthalfofthe

time.Intheotherextreme,ifeveryoneisinformedandvotessincerely,thenitiscorrectwithmaximum

possible probability, but the total signal costsmay outweigh the higher expected benefits.We utilize

quantal response equilibrium (QRE; McKelvey and Palfrey 1995), which generalizes Bayesian Nash

equilibrium (BNE) by allowing for decision‐making errors, in order to derive individual signal

acquisitionprobabilitiesinbetweenthesetwoextremes,includingforlargegroups.Wealsoderivethe

effectsoftheseprobabilitiesoncorrectgroupdecisionsandefficiency,usingcost‐benefitanalysis,and

testthe(comparativestatics)predictionsforourexperimentalparametersagainstthedata.

Whenvoting isvoluntary, in theory,uninformed individualsrationallyabstainas theirrandom

votemayoverrideaninformativegroupdecisioninpivotaleventsandlettheinformeddecide,whose

sincerevotesincreasetheprobabilityofacorrectgroupdecisionandthusexpectedbenefits.Thisform

ofdelegationisnamedthe“swingvoter’scurse”(FeddersenandPesendorfer1996).4Bycontrast,with

compulsoryvotingthecurseofuninformedvoting is tolerated. If itprevails, theexpectednetbenefits

(i.e., the increase in benefits) of voting informatively are too small compared to the costs and fewer

individualsgetinformedvis‐à‐visvoluntaryvoting.However,theanticipationofthecursecanalsoboost

signalacquisitionasthenetbenefitsareaugmentedbythevalueof“curseavoidance,”andthishappens

morelikelyinsmallergroupswherefreeriderincentivesareweaker.Suchendogeneityeffectsarevery

differentfromthesituationwithfixedinformationlevels,whereexpectedefficiencyisgreaterwiththe

opportunity toabstain(e.g.,Bhattacharya,Duffy,andKim2013;KrishnaandMorgan2011,2012)but

4 The intuition of the curse is as follows: suppose that individual is uninformed, and informed (uninformed) others votesincerely(abstain).Inpivotaleventswhereonealternativeleadsbyonevoteofothers,if abstains, thentheprobabilityofacorrectgroupdecisionis .However, if votesrandomly, thenwithprobabilityone‐halfhervoteturnsawinofthemorepromisingalternativeintoatie(whichisdecidedbyacoinflip)andwithprobabilityone‐halfhervoteonlyraisesitswinningmargin,sothegroupdecisioniscorrectwithprobability ,whichissmallerthanthe forabstaining.

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might be further improved, for example, by a turnout‐enhancing lottery prize for a random voter

(Gerardi et al. 2009). Another important differencewith endogenous costly signals is that observing

someone’svoluntaryorcompulsoryvotedoesnotrevealwhethertheindividualhasmadeinformational

efforts,butinthevoluntarymodeaspottedabstainermaybedeemedafree‐rider.Towit,uninformed

abstainingissociallydesirable,however,beingignorantinthefirstplacemaynotbeseenlikethat.By

contrast, with exogenous signals only the favorable view of curse avoidance is valid. Hence, with

endogenouscostlysignals,thereareincentivesfortheuninformedtovoteduetothepublicgoodnature

of information and the possibility that otherswill punish abstainers in oneway or another (see also

DellaVignaetal.2013;Funk2010;andBernstein,Chadha,andMontjoy2001foroverreportingvoting).

Sincethemid‐1990s,gametheoristshavebeguntorelaxtheassumptionsofCondorcet’sJuryTheorem

andexaminedavarietyofdifferentchangestothebasicmodel.5Austen‐SmithandBanks(1996)study

the theorem as an incomplete information game and show that BNE generally involves some

uninformativevoting,notingthatrationalindividualsonlycareaboutpivotalevents(Downs1957),infer

aggregateinformationfromtheirownsignalandothers’votesinpivotalevents,andbasetheirvoteon

this information and the pivot probabilities.6 However, for our parameters, everyone voting

informatively is actually a BNE (Austen‐Smith and Banks 1996, p. 38). Feddersen and Pesendorfer

(1996)allow forabstention fromvotinganddistinguishbetween “partisans”whoalwaysvote for the

samealternativeand“independents”whowanttoselectthecorrectalternativeandareeitherinformed

oruninformed.They show that uninformed independentshave incentives to abstain even if voting is

costless(i.e.,theswingvoter’scurse)andtovoteinordertocounterbalanceanypartisanbiassoasto

maximizethevoteimpactoftheirinformedallies(seealsoFeddersenandPesendorfer1997,1999).The

Feddersen‐Pesendorfermodelfindssupportinthelaboratory(Battaglini,Morton,andPalfrey2010;and

withqualification,MortonandTyran2011)andfromobservationalstudies(e.g.,CoupéandNoury2004;

Lassen 2005;Matsusaka 1995;McMurray2010; Palfrey andPoole 1987;Wattenberg,McAllister, and

5Forsurveys,seeGerlingetal.(2005)andPiketty(1999).Forexample,Ladha(1992)studiescorrelatedvotingandfindsthatCondorcet’s Jury Theorem still holds under fairly general conditions in large groups (small groups require substantialrestrictions),andtheprobabilityofacorrectgroupdecisionisinverselyrelatedwithvotecorrelation.6SeealsoMcLennan(1998)andWit(1998).FeddersenandPesendorfer(1998)showthatwithunanimityrulethe innocentmaybeconvictedmoreoften in larger thansmaller juries,and theprobabilityof falseverdicts increases compared toothervoting rules. In the laboratory,Guarnaschelli,McKelvey, andPalfrey (2000) find fewer incorrect convictionswithunanimitythanmajorityvoting.

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Salvanto 2000). All these studies use exogenous information. By contrast,we examine voluntary and

compulsorymajoritarianvotingwhen individualsdecideonwhetherornot toget informed.Also, the

effectsofpotential ignoranceonefficiencyarenot easilyquantifiable in the field (an exception is the

studyofBartels1996,whereuninformedvotingleadstobiasedresultsinU.S.PresidentialElections).By

contrast, our controlled laboratory study allows for amore detailed analysis of themutual effects of

endogenous costly signals and abstention, and their consequences for efficiency. Our findings with

regardtotheserelationshipshaveimportantimplicationsforthedesignofjuriesandcommittees.

Rational choice scholars (Downs 1957; Olson 1965) have pointed out that, generally, gathering

information about policy alternatives is costly. Therefore, individuals have incentives to free ride on

others’ informational efforts, which can result in rational ignorance. Mukhopadhaya (2003) revisits

Condorcet’sJuryTheorembystudyingendogenouscostlysignalsandfindsthatinBNEtheinformation

pool can shrink in the group size. In the samevein,Martinelli (2005) shows that if the signal quality

dependsontheamountinvested,individualeffortsgotozeroastheelectorategoestoinfinity.However,

if the marginal costs are near zero for nearly irrelevant information, then the vote outcome is

nonetheless representative of the majority. Feddersen and Sandroni (2006) study costly signal

acquisition in an ethical voter model and show that a significant fraction of a large electorate gets

informed, and successful information aggregation depends on the signal costs and quality. Oliveros

(2012) presents amodel inwhich costs are increasing in the signal quality and finds that BNE exist

where, forsomeindividuals,abstention ispositivelyrelatedwithquality.Moreover,GerardiandYariv

(2008), Gershkov and Szentes (2009), and Persico (2003), among others, examine costly signal

acquisitionasamechanismdesignproblem,wherethedesignerseekstoimplementasociallyoptimal

committee size and voting rule in order to overcome the free rider problem. The following recent,

independent studies are most closely related to ours. Shineman (2013) experimentally studies the

voluntary and compulsory modes as an individual decision‐making problem with costly voting and

signal acquisition, and Tyson (2013) analyzes this situation as an incomplete information game and

allows for partisans and independents. Sastro and Greiner (2010) investigate abstention and vote

invalidationwithexogenoussignalsinbothvotingmodesinthelaboratory.Finally,Elbittaretal.(2013)

experimentally study endogenous signal acquisitionwithmajority and unanimity voting for different

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groupsizes.Toourknowledge,wewere the first to study theeffectsofvoluntaryversus compulsory

votingonendogenoussignalacquisitionanduninformedvoting,andultimatelyongroupefficiency.

2Thegameandpredictions

Thegame

Anodd‐sizedgroupof2 1individuals,labeled 1, … ,2 1,ischoosingbetweentwoalternatives,

and ,throughmajorityvoting(acoinflipbreaksatie).Weassumethateachindividualisriskneutral

andmaximizesherexpectedownpayoffs.Moreover,theyallhaveacommoninteresttoalignthegroup

decisionwithanunobservable“truestate”oftheworld,whichisalsoeitherAorB,withequalcommon

priorprobabilityforeachstate.Forexample,ajurywantstoacquittheinnocentandconvicttheguilty.

Duetotheuncertaintyaboutthetruth,however, individualsmayhavedifferentopinionsaboutwhich

alternative is correct in this regard. Formally, their payoffs are identical ex post and given by

, , 1 and , , 0, where the first argument of denotes the group

decisionandthesecondargumentdenotesthetruestate.Thus,everyoneprefersA(B)ifthetruestateis

A (B).Each individual caneitherstayuninformed(i.e., continue toknownomore than thecommon

prior), 0, or get informed (i.e., learnmore than the prior), 1, by acquiring a private signal,

∈ , at a cost 0, ∀ . A signal is an independent Bernoulli trial from a state‐dependent

distribution with Pr | Pr | ∈ , 1 and Pr | Pr | 1 .

Thus,asignal isnoisybut informativebecausetheprobabilitythat itcorrectly indicatesthetruestate

exceedstheprior,or .(Intheexperiment,wefocuson .)

The game has two stages. In the first stage (Information), individuals independently and

simultaneouslydecideonwhetherornottogetinformed.Inthesecondstage(Voting),eithervoluntary

or compulsory independent and simultaneous costless voting takes place (henceforth also labeled

, ).Withvoluntaryvoting,eachindividualcanvoteforalternativeAoralternativeB,orabstain

from voting. By contrast, with compulsory voting the option to abstain is not available. Note that

everyoneinformed and uninformedmakes a decision in this stage, and thereafter learns the vote

outcomebutnotwhetherotherswere informed. In the following,weanalyzeequilibriumbehavior in

theVotingandInformationstagesinturn,usingbackwardinduction.

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Voting

In the Voting stage, we focus on the following symmetric BNE, which have been shown to predict

costlessvotingquitewellinmanyexperiments(e.g.,Battaglini,Morton,andPalfrey2010;McKelveyand

Ordeshook1990),thoughuninformedcompulsoryvotinghasnotbeenstudiedpreviously:

Proposition1 (Voting): In symmetricBNE, informed individuals vote sincerely in both votingmodes,

while theuninformedabstainwithvoluntaryvoting,andvote randomlywithequalprobability foreach

alternativewithcompulsoryvoting.

Proof:Seeappendix.

Signalacquisition

In the Information stage, we derive individual signal acquisition decisions using quantal response

equilibrium (QRE; McKelvey and Palfrey 1995), which has been shown to predict binary choices in

gamessimilartooursmuchbetterthanBNE(e.g.,GoereeandHolt2005;GroßerandSchram2010).QRE

generalizes BNE by including stochastic decision‐making errors that are systematic in the sense that

more lucrative decisions are made more often than less lucrative decisions (i.e., best responses are

smooth,notsharpasinBNE).Aparameter 0controlsthedegreeofnoise.Wefocusonsymmetric

QRE where everyone has the same probability of getting informed, , ∀ . Using the logit

specificationofGoereeandHolt(2005;henceforthlogitequilibrium),intheoneextremewithoutnoise,

0 andQRE turns intoBNE,while in the other extremewithpurenoise, ∞ anddecisions are

entirelyrandom,or .Wemainlyfocuson 0,implyingthat ∈ 0,1 .

The condition for a logit equilibrium ∗ to exist is derived in the appendix and given by

ln Π , , ∗ , , , , where superscript e denotes the expectation operator. The left‐

hand side (henceforth ) is identical inbothvotingmodesanddealswith theerrors. For 0, it

strictlyincreasesin ,approaches ∞( ∞)if approaches0(1),andalwaysequalszeroat .Given

thateveryonevotesàlaProposition1(denotedbythevector ∗ ),theright‐handside(henceforthRHS)

givesindividual ’sexpectednetpayoffs,orincreaseinpayoffs,ofgettinginformedifeveryoneelsegets

informedwithprobability ,whichdependsonthevotingmodeasspecifiednext.

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Withvoluntaryvoting,theinformedvotesincerelyandtheuninformedabstain(Proposition1).The

conditionforasymmetric ∗ toexistinthismode,assumingwithoutlossofgeneralitythatindividual

receivesana‐signalwhengettinginformed,isthengivenby:

1 22

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1 . 1

1 is described above, and 1 gives individual ’s expectednet payoffs of getting informed,

Π , , ∗ , , , ,whichconsidersthateach getsinformedwithprobability .Hence,each

votes sincerely for the correct [incorrect] alternative with probability [ 1 ], and stays

uninformed and abstainswithprobability1 . 1 includes ’s expectednetbenefits of getting

informed(i.e.,thesumterm)minusthesignalcost,c,wherethesumover 0, … , contains: the

probabilityofanevennumber2 ofothersbeinginformed, , , , ≡ 1 ;

given2 ,theprobabilityof beingpivotal, ,∗ , , ≡ 1 (i.e.,eachalternativehask

votes,sothatherA‐voteturnsatieintoawinforA);7and herexpectednetgainsinthesepivotal

events, ∗ , , 1 , , , , which considers that, if

she gets informed and uses Bayesian updating, her sincereA‐vote yields a correct [incorrect] group

decisionwithprobability [1 ],andsheonlyknowsthepriorifuninformed.

With compulsory voting, the informed vote sincerely and the uninformed vote randomly

(Proposition 1). The condition for a symmetric ∗ to exist in this mode, assuming without loss of

generalitythatindividual receivesana‐signalwhengettinginformed,isthengivenby:

1 21 1

11 1

. 2

2 is described above, and 2 gives individual ’s expectednet payoffs of getting informed,

Π , , ∗ , , , .Notethatwithcompulsoryvotingtheonlypivotaleventsarewith votesofothers

informed or uninformedfor each alternative (i.e., herA‐vote turns a tie into awin forA). IfB is

7Individual isalsopivotalifAisshortofonevoteofothers(i.e.,whenherA‐voteturnsalossintoatie).However,intheseeventsopposing signals (includinghera‐signal) eveneachotherout, so that sheknowsnomore than theprior.Hence,herinformedvotewouldonlyreplacethecoinflipandthesignalcostbewasted.

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correct,thenherpivotalA‐voteresultsinanincorrectgroupdecision,whichyields , 0sothat

thisevent is ignored (as in 1 ). IfA is correct, 2 considers that each gets informed

withprobability andreceivesana[b]‐signalwithprobability [1 ],andstaysuninformedwith

probability1 .Hence,each votessincerelyforA [B]withprobability [ 1 ]andvotes

randomlywithprobability 1 for each alternative. 2 contains ’s expectednetbenefits of

getting informed (i.e., the sum term)minus the signal cost, . In pivotal events, for any 0, … ,

informative votes for , , theremust be uninformed random ‐votes, and for a givenpair

, ) the pivot probability is ,∗ , , , , ≡ 2 1 . Since

pivotal events also involve uninformed random votes, individual ’s expected net gains from voting

informatively in these events vary and are equal to ∗ , , , ≡ ,

namely,herBayesianupdatedprobabilitythatAiscorrectgiven 1, ),includingherownA‐vote,

times , 1andminus , ifshevotesuninformed.8

Proposition 2 (Information acquisition): In symmetric logit equilibrium, for 0 individuals get

informed with probability ∗ ∗ , , , , 0 ∈ 0,1 , , . For large enough degrees of noise,

0, this probability is unique and decreasing in the odd group size, 2 1, decreasing in the

signalcost, ,anditmovesclosertoone‐halfwhen increases.

Proof:Seeappendix.

The uniqueness requirement that is large enough is due to convex and concave properties of

1 and 2 ,respectively.Moreover,thepropositiondoesnotgivecomparativestaticsresults

fortheeffectsofthevotingmode,whichdependon and .Italsoignoreschangesintheprobabilityof

a correct signal, sincewhile raising ∈ , 1 usually increases ∗ , there is a regionof theparameter

spacewherethereverseholds.Thereasonisthatraising hastwooppositeeffects,towit,itdecreases

thepivotprobability, , . ,andincreasestheexpectednetgainsfromvotinginformatively, . .

Finally,BNE( 0)isdiscussedinfootnote30intheappendix.

8Notethatineventswithmoreb‐thana‐signalsinthegroup,individual preferstovoteagainsthera‐signal.But,exante,sheexpectsotherstohaveonaveragemorea‐thanb‐signalssovotinginformativelyisheruniquebestresponse.

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Efficiencyandsocialoptimum

In order to compare ex ante group efficiencydefined by the group’s expected total benefits minus

expectedtotalsignalcostsacrossvotingmodesandgroupsizes,wemustderivetheprobabilityofa

correct group decision, , | , ∗ , , for , and , , where | denotes the event

that ischosenconditionalon being the truestate.Thisprobabilityuses and isdescribed in the

appendix.Given ∗ , note that increasing increases , . .Moreover, expected total signal costs

equal 2 1 in both voting modes, so ex ante expected efficiency is given by

2 1 , . , 2 1 , . , , . Finally, in the social optimum

with noise‐free decision‐making, maximization with respect to yields the implicit solution of :

, . / ,wheresuperscript denotestheoptimum.

Below,wederive ∗ , ,andtherespective , . forvarying forourexperimentaltreatmentsand

parameters(seealsoFiguresA2.1‐3intheappendix),andadditionalpredictionsfor 0and0.06and

variousgroupsizesaregiveninTablesA1andA2intheappendix.

Largegroups

Althoughwefocusonsmallgroups,ourgamehasalsointerestingimplicationsforlargeelectorates:

Proposition3(Largegroups):Asthegroupsize,2 1,goestoinfinity,if 0and ∈ , 1 ,thenthe

uniquelimitingsymmetriclogitequilibriumindividualsgetinformedwithprobability

∗ , , 01

1 ⁄ ∈ 0,1

for , andthegroupdecisionsisalmostsurelycorrect.If 0,theninthelimit ∗ , , 0

0for , andthegroupdecisioniscorrecthalfofthetime.

Proof:Seeappendix.

Theproposition states that in both votingmodes, Condorcet’s JuryTheoremalso holds for large

groupswith endogenous costly signals if there is somedecision‐making noise, 0, but not in BNE

without any noise. To see the intuition, first note that since 0 ∈ 0,1 , in large groups the

respective numbers of informed and uninformed individuals are large too. Then, by the law of large

numbers,withcompulsoryvotinganeven(odd)numberofuninformedindividualsalmostsurelycast

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equallymanyvotesforeachalternative(onemorevoteforeitheralternative),andwithvoluntaryvoting

the uninformed abstain. But since 1 ⟺ 2 1, in the limit, noting that and 0, the

informed individuals’ fraction of correct votes exceeds their fraction of incorrect votes by an infinite

margin,thatislim → 2 1 2 1 ∞ 1for ∈ 0,1 ,souninformedvotesareirrelevantand

thegroupdecision isalmostsurelycorrect.Finally,since thepivotprobability isnil in largegroups,a

noise‐freevoterchoosesnottobuyasignal, 0 0.

3Experimentaldesignandpredictions

ThecomputerexperimentwasrunattheUniversityofCologne’sLaboratoryforEconomicResearch.9In

total,220studentsattendedeightsessionsof27or28participantsthatlastedabout1.5hours.Earnings

wereexpressedinpointsandexchangedforcashfor€1per800points.Participantsearnedanaverage

of€11.79,including€2.50forshowingup.Instructionsareinouronlineappendix.10

We employed a 2 2 treatment design, varying the voting mode (voluntary and compulsory)

within‐subjects in one dimension, and group size (three and seven) between‐subjects in the other

dimension. Treatments are labeled Voluntary voting‐3, Voluntary voting‐7, Compulsory voting‐3, and

Compulsoryvoting‐7. Each sessionhad twopartsof30decisionperiodseach,whereeachpartuseda

different voting mode and the order was changed across sessions. At the beginning of a session,

unknowntotheparticipants,subjectpoolsof27(28)wererandomlydividedintothree(two)separate

matching groups of nine (fourteen). The composition ofmatching groupswas fixed during the entire

session, and therewas no interactionbetween them.Weused a randommatching protocol:within a

matchinggroup,participantswererandomlyput intogroupsofthree(seven)atthebeginningofeach

period.All our statistical tests are run at thematching group level, forwhichwehave twelve (eight)

independentobservationsforgroupsofthree(seven),halfofthempereachorderofthevotingmodes.11

9Weusedz‐Tree(Fischbacher2007)toprogramtheexperimentandORSEE(Greiner2004)torecruitparticipants.10Theonlineappendixisavailableathttps://sites.google.com/site/jwghome/research.11Observedaverageratesofacquiredsignalsarehigher inthefirstthansecondpart inVoluntaryvoting‐3,andviceversa inCompulsory voting‐3 ( 0.021,Wilcoxon‐Mann‐Whitney tests, one‐tailed), but no difference is found for the larger groups( 0.343 and 0.557).Note that the sixmatching groups that first faced the voluntarymode have statistically significantlyhigher rates averaged over both parts than the other sixmatching groups, but no difference is found for the larger groups( 0.008and0.557,sametests).Asaresult,inthesmallergroupsaverageratesarenotdifferentbetweenvotingmodesinthefirstpart,andhigherwithcompulsoryvotinginthesecondpart( 0.120and0.004,sametests).Inlargergroups,averageratesarehigherwithcompulsorythanvoluntaryvotinginbothparts( 0.014and0.014,sametests).Importantly,theleveleffectofratesforthesmallergroupsdoesnotaffectouranalysisandmainresults.

12

Eachperiod consisted of two stages. At the beginning of eachperiod, the true state,A orB,was

selectedwithprobabilityone‐half foreachbutnotrevealedtothegroup.Inall treatments, inthefirst

stage, participants independently and simultaneously decided onwhether to acquire a private signal

aboutthetruestateandpay 25points,orstayuninformedandbearnocost.Asignalindicatedthe

true (false) state with probability (1 ). In the second stage, voluntary or compulsory

majoritarian voting with random tie‐breaking took place, where all participantsinformed and

uninformedindependentlyandsimultaneouslyvoted,atnocost, foralternativeAorBor, if feasible,

abstained.12Aftervoting,thetruestate,groupdecision,andnumberofvotesforeachalternativewere

announced. Each participant’s period earnings were determined as follows: she or he received 200

points if the group decisionmatched the true state and 0 points otherwise, independent of the own

signalacquisitionandvotedecisions,minus25pointsifsheorheboughtasignal.Formoredetailsof

theprocedures,seetheinstructionsintheonlineappendix.

Figure1:Symmetriclogitequilibriumsignalacquisitionprobabilities

Note:Thefigureshows, forourexperimental treatmentsandparameters,symmetric logitequilibrium

signalacquisitionprobabilitiesfor ∈ 0,0.1 and ∈ 1,10,100 .Thoughnotshown(butseeTableA1),

Compulsoryvoting‐3hasanadditionalBNEwherenoonegetsinformed.

Figure 1 shows, for our experimental treatments and parameters, symmetric logit equilibrium

signal acquisition probabilities per treatment for ∈ 0,0.1 and the discrete cases ∈ 1,10,100 .

12Intheexperiment,theseoptionswerelabeledYellow,Blue,andNoColorinsteadofA,B,andabstain,respectively.Notethatduetooddgroupsizes,tieswerenotpossibleinthecompulsorymode.

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Though not shown in the figure (but see Table A1 in the appendix), Compulsory voting‐3 has an

additionalBNE( 0)withuniversalrationalignorance.Ascanbeseen,predictionsvarymarkedlyin

BNE, andmove closer to each other towards one‐halfwhen increases.With compulsory voting, for

bothgroupsizesthereisaBNEwherecursedvotingprevails(i.e.,nobodybuysasignal),andingroups

of three there is anotherBNEwhere it is counteracted by the highest predicted ∗ . In this voting

mode, 0has theeffectofpromotingsignal acquisition in the former twoequilibria (i.e.,universal

rationalignoranceisovercome)13andoppressingitinthelatterequilibrium.Moreover,giventhevoting

mode,individualsarealwaysmorelikelytogetinformedinsmallerthanlargergroups(seeProposition

2).And,giventhegroupsize,compulsoryvotingalwaystriggershigher ∗ thanvoluntaryvotingin

groups of three, but in larger groups this only holds for about 0.025while the reverse holds for

degrees of noise below this value. Using our ∗ , we can derive the groups‘ expected number of

acquiredsignals(or,informationpool),probabilityofacorrectdecision,andexpectedefficiency,which

are shown in Figures A2.1 to A2.3 in the appendix. Their comparative statics depend on and are

thereforediscussedfurtherafterestimatingthedegreesofnoiseforourdata.

Besides comparing group performances to universal ignorance, 0, it is also interesting to

confrontthemwiththesocialoptimum,whichis 1inalltreatments(i.e.,acornersolutionwhere

themarginaltotalcostsdonotexceedthemarginaltotalbenefitsevenifeveryonegetsinformed),and

withasingledelegate,D,whomakesanon‐noisygroupdecision(henceforthnoise‐freedelegate),also

always 1. These three benchmark performances are compared to our logit predictions in the

experimental results section (see also Figures A2.1 to A2.3). Moreover, since ∗ 1 in all

treatments,anysignalacquisitionprobabilitygreater than the logitprediction isefficiencyenhancing.

Thus, raising 0 strictly increases expected efficiency in all treatments but Compulsory voting‐3,

where it decreases efficiency. We can use this exception to test whether deviations from BNE are

consistenteitherwithlogitequilibriumorimplicitcooperationtowardshigherpayoffs.

13InCompulsoryvoting‐3,theBNEwithuniversalignoranceisnotrobusttoeventhesmallestnoise,soincreasing leadstoajumptotheequilibriumpathwithhighsignalacquisitionprobabilities.

14

4Experimentalresults

Inthissection,wepresentandanalyzeourexperimentalresults.Wediscussinturnsignalacquisition,

correctgroupdecisions,efficiency,uniformedvoting,andlearningaboutclosevoteoutcomes.

Figure2:SignalacquisitionAveragerates Averagenumbers

Signalacquisition

Figure 2 shows observed average rates (left panel) and average numbers (right panel) of acquired

signalsperfive‐periodblockpertreatment.TheoverallrateishigherinCompulsoryvoting‐3and‐7than

inVoluntary voting‐3 and ‐7 (0.47 and 0.38 versus 0.35 and 0.27). By contrast, the overal number is

higher inCompulsoryvoting‐7 andVoluntaryvoting‐7 than in the respective smallergroups (2.67and

1.90versus1.40and1.05).Withonlyoneexception,thesameordersofratesandnumbersofacquired

signals also hold per five‐period block. Finally, signal acquisiton tends to decline over time in all

treatments.14

Experimental result 1 (Signal acquisition): Larger groups acquire almost twice asmany signals as

smaller groups (81.3% and 90.7% more with voluntary and compulsory voting, respectively), but

individualsget informedat thesamerates.Forbothgroupsizes,compulsoryvoting inducesmoresignal

acquisitionsthanvoluntaryvoting(33.8%and40.7%more,respectively).

14Averageratesandnumbersofacquiredsignalsarehigherinperiods1‐15thanperiods16‐30inalltreatments( 0.059,Wilcoxonsignedrankstests,one‐tailed).Moreover,usingprobitregressionswithrandomeffectsatthematchinggrouplevelwithsignalacquisitionasdummydependentvariableandperiods1‐30orperiods1‐60(i.e.,overbothparts)asindependentvariable,wefindnegativeandstatisticallysignificantcoefficientsoftheperiodinalltreatments( 0.048,two‐tailed),exceptinVoluntaryvoting‐7forperiods1‐60( 0.206).

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15

Support: Thenullhypothesis ofnodifference in average acquired signalsbetweengroupsof sevenand three is rejected in

favor of greaternumbers in larger groupswith voluntary and compulsory voting ( 0.0003 and 0.0002,Wilcoxon‐Mann‐

Whitney tests, one‐tailed), but not for rates for both votingmodes ( 0.140 and 0.409). And, the null of no difference in

averageacquiredsignalsbetweenthetwovotingmodesisrejectedinfavorofgreaterratesandthusnumberswithcompulsory

votingforgroupsofsevenandthree( 0.001and0.004,Wilcoxonsignedrankstests,one‐tailed).Forcompleteness,thenull

isnotrejectedforratesforVoluntaryvoting‐3versusCompulsoryvoting‐7( 0.109,Wilcoxon‐Mann‐Whitneytest,one‐tailed),

but it is rejected in favor of greater rates in Compulsory voting‐3 versus Voluntary voting‐7 and greater numbers for both

comparisons( 0.032,sametests).

Figure 3 confronts our logit predictionswith the data.We compute best predictors of the degree of

noise, ̂ ∗, usingmaximum likelihoodestimations.15This isdone for thepooleddataof all treatments,

eachweightedbyone‐fourth, forall thirtyperiods(largesymbols)andforthefirstandsecondhalfof

periodsonly(smallsymbols),whichyields ̂ ∗ 0.06respectively ̂ ∗ 0.07and ̂ ∗ 0.05.The

dropinthebestpredictorfromthefirsttothesecondhalfofperiodshintstolearningtowardsBNE(see

alsoGoereeandHolt2005;McKelveyandPalfrey1995).16AsseeninFigure3,observedratesofsignal

acquisition are predicted poorly for about 0.025 (BNE predicts worst) and much better for

∈ 0.025,0.1 .17Forthelatterinterval,wenotonlypredictgreaterinformationpoolswithcompulsory

thanvoluntaryvotingpergroupsize,butalsoinlargerthansmallergroupspervotingmode(seeFigure

A2.1).Overall,ourestimatescorrectlypredictfiveoutofsix(sixoutofsix)possiblecomparativestatics

ofobservedaveragerates(numbers)ofacquiredsignals.18Bycontrast,BNEonlypredictsfouroutofsix

(threeoutofsix)comparativestatics.Notably, itoverratesfree‐ridingandthusfailstopredictgreater

informationpoolsinlargergroups(Experimentalresult1).Finally,observedaverageratesaregreater

15Denoteby thelogitequilibriafor .Let betheobservedsignalacquisitionratefor observations.TheLog‐likelihood

that yields is ln ln 1 ln 1 ,which ismaximized for if such

exists,otherwiseacornersolutionmaximizestheLog‐likelihood.16Wealsofindadropfromthefirsttothesecondpart( ̂

∗ 0.07and ̂ ∗ 0.06,respectively).Notethattheobserved

averagesignalacquisitionrateinCompulsoryvoting‐3 isinbetweenthetreatment’stwoBNE,solearningistowardstheBNEwithuniversalignorance.WhilethismayreflectthetensionbetweenthetwoBNE,noticethatalthoughtheratepermatchinggroupvariesmoreinCompulsoryvoting‐3 thantheothertreatments,onlyoneofitsmatchinggroupsisamongthesixwithalowestrateofabout0.2(Voluntaryvoting‐3andCompulsoryvoting‐7havethreeandtwo,respectively).17Thisalsoholdspertreatment,exceptforVoluntaryvoting‐3wheretheratesarebetterpredictedbylow .Tofinda ̂ ∗pertreatment,simplymovethedatapointsinFigure3horizontallyuntiltheylieontherespectivepredictionline.18Theonlyexceptionisanunpredicted,slightlyhigheraverageratefoundinCompulsoryvoting‐7 thaninVoluntaryvoting‐3.However,thesetreatmentsarenotdirectlycomparable,andtheoneexceptionintheorderofobservedratesreportedearlieroccursinthelastblockoffiveperiods(leftpanelofFigure1),inlinewithourbestpredictors.

16

thanBNEinalltreatmentsbutCompulsoryvoting‐3(i.e.,thehigher,robustBNE),whichismoreinline

withlogitequilibriumthanimplicitcooperationtowardsthesocialoptimum.

Figure3:Signalacquisitionandestimateddegreesofnoiseinlogitequilibrium

Note:Thefigureshowsmaximumlikelihoodestimationsofthedegreeofnoise, ̂ ∗,forthepooleddataof

alltreatments,eachweightedbyone‐fourth,forallthirtyperiods(largesymbols)andforthefirstand

secondhalfofperiodsonly(smallsymbols).

Correctgroupdecisions

Next, we examine how voting translates acquired signals into group decisions. On average, groups

choose thecorrectalternative62.0%and66.0%(68.8%and69.0%)of the time inVoluntaryvoting‐3

and‐7(Compulsoryvoting‐3and‐7),respectively.

Experimentalresult2(Correctgroupdecisions):Onaverage,62.0%to69.0%ofthegroupdecisionsare

correct,wheresmallergroupswithvoluntaryvotingdoworst.

Support:Thenullhypothesisofnodifferenceinaveragecorrectgroupdecisionsbetweenvoluntaryandcompulsoryvotingis

rejectedinfavorofgreateraveragesforgroupsofthreewithcompulsoryvoting,butnot forgroupsofseven( 0.006and

0.237,Wilcoxonsignedrankstests,one‐tailed).And,thenullofnodifferenceisnotrejectedforgroupsizecomparisonsinboth

the voluntary and compulsorymode ( 0.472 and 0.179,Wilcoxon‐Mann‐Whitney tests, one‐tailed). For comparison, it is

alsonotrejected forCompulsoryvoting‐3versusVoluntaryvoting‐7,butrejected in favorofgreateraverages forCompulsory

voting‐7thanVoluntaryvoting‐3( 0.372and0.042,Wilcoxon‐Mann‐Whitneytests,one‐tailed).

0

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Typically,inalltreatmentsaveragecorrectgroupdecisionspermatchinggroupareinbetweenthe

50% achievedwith universal ignorance and the 75% achieved by a single noise‐free delegate,19 and

henceliemarkedlybelowthesocialoptimumof84.4%ingroupsofthreeand92.9%ingroupsofseven

(seeFigureA2‐2intheappendix).Theserelativeperformancesarepredictedfor ∈ 0.025,0.1 ,albeita

delegate in Voluntary voting‐7 should do worse than the group for about 0.05. With regard to

comparative statics between treatments for ∈ 0.025,0.1 shown in Figure A2‐2, we should see on

averagefewercorrectgroupdecisionsinVoluntaryvoting‐3 than‐7(whichwedo,butnotstatistically

significantly) and about the same averages inCompulsory voting‐3 and ‐7 (whichwe do). Also, there

should bemore correct groupdecisionswith voluntary than compulsory voting for both group sizes,

whichwereject.Overall,whileinthecompulsoryvotingtreatmentsaveragecorrectgroupdecisionsare

similar to those predicted for ∈ 0.025,0.1 , in the voluntary voting treatments they are markedly

smaller than predicted. Against Proposition 1, this hints to cursed uninformed voluntary voting! For

anotherhint,groupdecisionsarenotmoreoftencorrectinVoluntaryvoting‐7thanCompulsoryvoting‐3,

even though in the former treatment the information pool was greater on average and participants

couldabstain.Wefurtherdiscussuninformedvoting,below.

Efficiency

Figure4depictspertreatment,averagedoverindividuals, theobservedsignalcosts(dashedline);net

benefits(darksolidline),whichequal200pointstimestheaveragefractionofcorrectgroupdecisions

minus100pointsfromarandomgroupdecisionwithuniversalignorance;andnetpayoffs(darkbars),

whichareequaltothenetbenefitsminusthecosts.(Thelightlineandbarsarediscussedbelow.)Ascan

be seen, individual net payoffs are smallest in Voluntary voting‐3 (15.3 points), and about equal in

Voluntaryvoting‐7 andCompulsoryvoting‐3 and ‐7 (25.3,25.9, and28.4points;note thatnetbenefits

andcostsaresomewhatsmallerinVoluntaryvoting‐7thaninthetwoothertreatments).

191(0)outof12matchinggroupshad 50%( 75%)correctgroupdecisionsinVoluntaryvoting‐3,andthesenumbersare0(3)outof12inCompulsoryvoting‐3,1(1)outof8inVoluntaryvoting‐7,and0(1)outof8inCompulsoryvoting‐7.

18

Figure4:Efficiency

Note: The figure shows per treatment, averaged over individuals, observed signal costs (dashed line), net

benefits (solid lines),andnetpayoffs(bars).Thenetvaluesaredepartingfromuniversal ignorance,where

eachindividualhasexpectedbenefitsandpayoffsof100points.Foradjustedvalues,allrealizeduninformed

votesarereplacedbyhypotheticalabstentions(ifthiscreatesatie,netvaluesaresettozeropoints).

Experimentalresult3(Efficiency):Inalltreatments,average individualnetpayoffsareinbetweenthe

benchmarksof0pointswithuniversalignoranceand41.7and46.4pointswithasinglenoise‐freedelegate

ingroupsofthreeandseven,respectively.Individualsinsmallergroupswithvoluntaryvotingearn10.0to

13.0pointslessthanthoseintheothertreatments.

Support:Thenullhypothesisofnodifference inaverageefficiencybetweenvoluntaryandcompulsoryvoting is rejected in

favorofgreateraveragesforcompulsoryvotingingroupsofthree,butnotforgroupsofseven( 0.026and0.320,Wilcoxon

signed ranks tests, one‐tailed). And, the null hypothesis of no difference for group size comparisons is rejected in favor of

greater averages for Voluntary voting‐7 than ‐3, but not in the compulsory mode ( 0.084 and 0.472, Wilcoxon‐Mann‐

Whitney tests, one‐tailed). For comparison, it is rejected in favor of larger averages forCompulsory voting‐7 thanVoluntary

voting‐3,butnotforCompulsoryvoting‐3versusVoluntaryvoting‐7( 0.062and0.472,sametests).

Note thatalmostalwaysdogroupsof threeandsevenachieve loweraveragenetpayoffs than in

socialoptimum(43.8and60.9points)andwithasinglenoise‐freedelegate(41.7and46.4points,i.e.50

minus25/groupsize),20andrecallthatnetpayoffsareequaltozerowithuniversalignorance(seealso

201(0;0)outof12matchinggroupsinVoluntaryvoting‐3earnedonaveragelessthanwithuniversal ignorance(morethanwithasinglenoise‐freedelegate;morethaninsocialoptimum),andthesenumbersare0(1;1)outof12inCompulsoryvoting‐3,1(1;0)outof8inVoluntaryvoting‐7,and0(1;0)outof8inCompulsoryvoting‐7.

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voting‐7Signal costs and net benefits and payoffs in points

Net payoffs

Net payoffs(adjusted)

Net benefits

Net benefits(adjusted)

Signal costs

.

19

FigureA2.3intheappendix).Finally,asaconsequenceofsmalleraveragecorrectgroupdecisionsthan

predictedinthevoluntaryvotingtreatments,observednetpayoffsarealsosmallerthanpredicted.By

contrast,predictedandrealizednetpayoffsaresimilarinthecompulsoryvotingtreatments.

Uninformedvoting

Here,weinvestigateuninformedvotinginmoredetail.Table1showspertreatmentaveragefractionsof

sincere voting, insincere voting, and abstention of the informed, and voting and abstention of the

uninformed(notethatwecannotdistinguishbetweensincereandinsincereuninformedvoting).

Table1:Fractionsofinformedanduninformedvoting

TreatmentInformed Uninformed

VotingAbstention Voting Abstention

Sincere InsincereVoluntaryvoting‐3 0.341 0.007 0.001 0.252 0.399Compulsoryvoting‐3 0.461 0.006 ‐ 0.533 ‐Voluntaryvoting‐7 0.268 0.003 0.000 0.293 0.436Compulsoryvoting‐7 0.377 0.004 ‐ 0.618 ‐

As seen in the table,observeddecisionsmainlysupportProposition1,albeitwithone important

exception.Aspredicted, the informedalmostnever abstain in thevoluntarymodeandalmost always

votesincerelyinbothvotingmodes.21However,assuspected,wefindsubstantialvoluntaryvotingofthe

uninformed(38.7%and40.2%oftheirdecisionsinVoluntaryvoting‐3and‐7,respectively).22Figure4

alsoshowsadjustednetbenefitsandnetpayoffsaveragedoverindividuals,forwhichwereplacedeach

realizeduninformedvotewithahypotheticalabstention(graylineandbars).23Theadjustmentallows

ustoevaluatetheactualnegative impactof theswingvoter’scurseongroupdecisionsandefficiency,

butnaturally itcannotcapturehowdecisionsinthevoluntarymodewouldhavebeenmadeabsentof

uninformedvoting.Thefigurerevealsasubstantialdamageinalltreatments:theadjustmentraisesthe

21Outof3,240(3,360)decisions,weonlyfind4(1)informedabstentionsbyoneparticipantinVoluntaryvoting‐3(‐7),and22and18(9and15)insincerevotesinVoluntaryvoting‐3andCompulsoryvoting‐3(‐7and‐7),respectively.22Uninformedvoteswereslightlybiased: ineachtreatment,54%forBlueversus46%forYellow ( 0.019,binomial tests,one‐tailed).Moreover,in40.4%and21.0%(29.7%and18.8%)oftheelectionsinVoluntaryvoting‐3and‐7(Compulsoryvoting‐3and‐7),uninformedvotesreplacedacoinflipforatieofinformedvotes,in4.4%and12.1%(7.4%and13.3%)theyturnedawinofthecorrectcolorintoaloss,andin1.5%and3.3%(1.9%and3.5%)theyturnedalossofthecorrectcolorintoawin.Finally,between54.1%and65.6%(56.3%and70.6%)of these incorrect (correct) changeswere forBlue, except for38.1%correctchangesinCompulsoryvoting‐3.23Whenanadjustmentcreatesatie,weusenetbenefitsofzeropointsinsteadofacoinflip.Thischoicehasvirtuallynoeffectonthereportedaverages.

20

fractionofcorrectgroupdecisionsby3.4and8.4(3.7and9.4)percentagepointsinVoluntaryvoting‐3

and ‐7 (Compulsory voting‐3 and ‐7),which raises average individual benefits and payoffs by 6.8 and

16.9(7.4and18.8)points.Finally,wecancompareadjustedvalues in thevoluntarymode,wherethe

uninformed are predicted to abstain, to the respective realized values in the compulsory mode. In

groupsofthree(seven),adjustedaverageindividualnetbenefitsandnetpayoffswithvoluntaryvoting

are30.8and22.1(49.0and42.2),andtherespectiverealizedvalueswithcompulsoryvotingare37.6

and 25.9 (37.9 and 28.4). These numbers accentuate the generally poor performance in Voluntary

voting‐3, and suggest high forgone gains inVoluntary voting‐7 where, without cursed voting, groups

couldhavedonemarkedlybetterthaninthecompulsorymode.

Figure5:Uninformedvoluntaryvotingperparticipant

Groupsofthree Groupsofseven

Note: The figure shows for eachparticipant (dot) the fraction of periods inwhich she or hewasinformedonthehorizontalaxis,andthefractionofperiodsinwhichsheorhevotedontheverticalaxis.Largercirclesizesindicatelargernumbersofparticipantsatthesamecoordinates.

Next,weexamineuninformedvoluntaryvotingattheindividuallevel.Figure5showsscatterplots

forVoluntaryvoting‐3(leftpanel)andVoluntaryvoting‐7(rightpanel),whereeachdotrepresentsone

participant (the size of their agglomeration at the same coordinates is indicatedby the size of larger

circles). The horizontal axis shows the fraction of periods inwhich she or hewas informed, and the

verticalaxisshowsthefractionofperiodsinwhichsheorhevoted.Hence,individualsonthediagonal

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(gray line) never voted uninformed,24 and those above the diagonal voted uninformed to various

extents. Clearly, uninformed voluntary voting is not restricted to just a few individuals: inVoluntary

voting‐3and‐7,morethanhalfofthemvoteduninformedtosomeextent(59outof108and60outof

112participants,respectively).

Experimentalresult4 (Uninformedvoting):There isa sizablenumberofuninformedvoluntaryvotes

(25.2%and29.3%outofalldecisions ingroupsof threeandseven),nearlyhalfasmanyasuninformed

compulsoryvotes(53.3%and61.8%).Uninformedvoluntaryvotingiscommonamongindividuals.

Thereareseveralpossibleexplanationswhysomanyuninformedparticipantsdonotabstainwhen

they can. For example,people areboundedly rational inmanyaspects, and thereforemay simplynot

realize that random voting can negatively affect the group decisionwhich is less intuitive than the

positiveeffectofinformedsincerevotingorignorepivotaleventsaltogether(e.g.,EspondaandVespa

2012;Martinelli2011).Onewould thenexpect toseesome learningof theswingvoter’scurse inour

experiment, towit, a decline inuninformedvoluntary voting.However, our results in this regard are

mixed(seeFigureA3),25whichhintstoafractionofparticipantswhowereconsciousofthecursewhen

casting their uniformed voluntary vote.26Thesepeoplemayvotebecause abstaining basically reveals

their ignorance and hence can lead others to reciprocate by getting informed less often in future

periods.27Finally, in linewithoverreportingofvoting inelectionsurveys(e.g.,Bernstein,Chadha,and

Montjoy2001),somepeoplederivepridefromsignalingtoothersthattheyvotedandfeelembarrassed

to admit their failure to do so (DellaVigna et al. 2013),which toomay explain uninformed voluntary

voting.Ourexperimentisnotdesignedtofindouttheexactcausesofanomalousuninformedvoting,so

weleavethistasktofutureresearch.

24Thetwoparticipantswhoabstainedinformed(seefootnote21)virtuallyalwaysvoteduninformedotherwise.25Usingprobitregressionswithrandomeffectsatthematchinggroup,inVoluntaryvoting‐3wefindnostatisticallysignificanteffectoftheperiodonuninformedvotingforbothpartstogether( 0.956,two‐tailed),andnegativeandpositiveeffectsforthe first and second part, respectively ( 0.090). In Voluntary voting‐7, we find no significant effect for the first part( 0.490),butnegativeeffects forthesecondpartandbothpartstogether,respectively( 0.070).Recall that inthetwocompulsoryvotingtreatmentssignalacquisitiondeclinesovertime(footnote14),souninformedvotingincreasesovertime.26Participantscouldnotdirectlyobserveifothersvotedinformed.Itislikelythattheyanticipatedsomeextentofuninformedvoluntaryvoting,especiallyiftheydidsotoo.Thoughlearningthisextentisverydifficult,theycouldupdatetheirbeliefsaboutuninformedvotingaftereachperiod,usingthenumbersofcorrectandincorrectvotesofothers.Thisiswhatwestudybelow.27Inourrandommatchingprotocol,participantscouldinferthattheyencounteraparticularotherparticipantonaverage2.31and 6.67 times per part in groups of three and seven (unknown to them, theymetmore often due to thematching groupproceduredescribedearlier),soabstainingcouldindeedhavedirectandindirecteffectsonrelevantfuturegroupdecisions.

22

Next,weanalyzeuninformedvoluntaryvotingusingmodelsofboundedrationality.First,applying

individualrationalchoicetooursituation,noise‐freeplayersdecideoptimallyasiftheywerealone.By

ignoringsocialinteractionentirely,theyarealwayspivotalintheirlimitedrationalminds.Consequently,

foralltreatments,theyshouldconsistentlygetinformedandvotesincerely(asournoise‐freedelegate),

and off the equilibrium path, the uninformed randomly choose A or B or, if feasible, abstention. By

contrast,ourparticipantsareoftenuninformed,andatratesthatdependonthetreatmentwhichalso

noisybehaviorcouldnotexplain(butuninformeddecisionsinthevotingstageappeartoberandom,as

predicted;seeFigure5and footnote22).Also,belowweshowthatparticipantsconsiderclosenessof

thenextvotewhenmakingtheirsignalacquisitiondecisions(seeTable2).Overall,individual“rational”

choiceisnotaplausibleexplanationofourdata.Second,turningtologitequilibrium,itcannotimprove

on theBNEpredictionof sincerevoting (Prediction1), as this iswhat informedparticipantsvirtually

always do (Table 1). However, it can capture the observed substantial uninformed voluntary voting,

whichBNEcannot.Toseethis,notethattheuninformedsurelyabstaininthenoise‐freeextremeoflogit

equilibrium,andchooseAorBorabstentionwithprobabilityone‐third foreach in theotherextreme

withpurenoise.Then,onthelogitequilibriumpath,raisingthedegreeofnoiseshiftsprobabilitydensity

fromuninformedabstentioninequalsharestouninformedA‐respectivelyB‐votes,whichisconsistent

withthe38.7%and40.2%ofuninformedvotes(abouthalfforeachAandB;seefootnote22)foundin

Voluntaryvoting‐3and‐7.Ifoneagreesthatsincerevotingismuchmoreintuitivethantheswingvoter’s

curse, so one can justify decision‐making noise only for the uninformed, then logit equilibrium is a

plausibleexplanationofourdata(butnotethatthenthreedifferentdegreesofnoisearetobeestimated,

includingforsignalacquisitiondecisions).Thenotionof intuitivesincerevotingissupportedbyother

prominentmodelsofboundedrationality,suchask–levelreasoning(e.g.,Camerer,Ho,andChong2004),

cursedequilibrium(EysterandRabin2005),andanalogy‐basedexpectationequilibrium(Jehiel2005).As

shown by Battaglini,Morton, and Palfrey (2010),whose analysis also applies to our voting stage, all

three models predict that any informed playereven a fully unsophisticated onevotes sincerely

(differentbehaviorispredictedforuninformedplayers:thefullyunsophisticatedchooserandomly,asin

theindividualchoicemodelofftheequilibriumpath,andeveryoneelseabstains).Asmentionedearlier,

ourdataarealsoconsistentwiththenotionthatsomeparticipantsconsciouslytaketheriskofcastinga

23

cursedvoluntaryvote,whichlikelyincludesthosewhorealizethatothersdosoandthenreciprocateby

doingjustthesame.Atthispoint,wemustleaveittofutureresearchtofindouttherespectivefractions

of noisy and conscious cursedvoting.Knowing these fractions canhelp to improvepredictions, since

noisy and conscious types have different expected benefits from voting and thus different signal

acquisition probabilities in equilibrium. Ideally, these predictions also take into account imperfect

informationaboutthedistributionofthevarioustypes.Thelikelyresultisthatlogitequilibriumsignal

acquisition probabilities are in between those derived in this paper, that is, where the uninformed

always abstain in the voluntary mode and always vote in the compulsory mode. However, realizing

cursedvoluntaryvotingofothersmighttriggerstrongnegativereciprocityinindividualsanddragdown

informationpoolsbelowpredictedlevels,somethingwhichmayhaveoccurredinVoluntaryvoting‐3.

Learningaboutclosevotes

Next,weanalyzewhetherparticipantsbasetheircurrentsignalacquisitiondecisionsonclosenessofthe

nextvote,whichtheymayprojectfromothers’previousvotingdecisions.Wedosobyrunningforeach

treatment separatelyprobit regressionswith randomeffects at the level of thematching group (mg).

The dependent dummy variable, , , equals 1 if individual acquired a signal in period , and 0

otherwise. As independent variables, we use one dummy variable, two quantitative variables, and

various interactions of these variables. First, the lagged dependent dummy variable, , , separates

individualswhowereinformedinthepreviousperiod, 1,fromthosewhowerenot.Thisallowsusto

inspectwhether thosewho tend to be informed and uninformed respond to information in previous

voting decisions of others differently. Second, the first quantitative variable, # /2 ∀ , , is

definedas thepreviousnumberofinformedanduninformedvotesbyall individualsother than ,

∀ ,labeled# ∀ , ,dividedbythenumberofotherindividuals,2 .Naturally,thisvariable

isonlyusedwithvoluntaryvoting.Third,thesecondquantitativevariable, ∆ /# ∀ , ,is

definedas thenumberofcorrectvotesminusthenumberof incorrectvotesby∀ in 1, labeled

∆ ∀ , , divided by # ∀ , . The two quantitative variables are normalized using 2

respectively # ∀ , as denominators in order to account for the differences in group size.

Fourth,weuseavariablethatinteractsthetwoquantitativevariables,andthreevariablesthatinteract

24

the lagged dependent variable with each quantitative variable and their interaction variable (again,

variables using # /2 ∀ , are not applicable in the compulsorymode). Finally,we include

twoerrorterms, , and ,wherethelatterisarandomeffectusedtocorrectforthepanelstructure

in thedata.The twoquantitativevariableswere chosen, since inour randommatchingprotocol they

revealbasicinformationthataparticipantcanavailforupdatingherorhisprojectionofaclosevotein

thecurrentperiod.Tobeprecise,foranygiven,unknownstrictlypositiveprobabilitywithwhichother

individualsbuyasignal,agreaternumberoftheirvotesleadstoagreaterexpectedmarginofvictoryfor

the correct alternative (recall that informedparticipants virtually always voted sincerely).And, given

thenumberofothers’votes,agreatermarginofvictoryforthecorrectalternativemeansthattheyhave

agreaterprobabilityofbuyingasignal,onaverage.Therefore,ifanindividualpreferstoavoidthesignal

costwhenbelievingthatthenextvoteoutcomemightnotbeclose,thenboth # /2 ∀ , and

∆ /# ∀ , shouldnegativelyaffecthercurrentsignalacquisitionprobability.Moreover,

the interaction of both quantitative variables should strengthen the negative effect, since a greater

numberofothers’votesjointlywithagreaterprobabilityofthemgettinginformedandcastingasincere

votemakesacorrectgroupdecisionevenmorelikely.

Table2:Probitestimationsofsignalacquisitionswithrandomeffectsatthematchinggrouplevel

Constantandindependentvariables

CoefficientsVoluntaryvoting‐3

Compulsoryvoting‐3

Voluntaryvoting‐7

Compulsoryvoting‐7

Constant ‐1.048***(0.099) ‐1.002***(0.094) ‐1.000***(0.119) ‐1.227***(0.045)∆# ∀ ,

0.131(0.151) ‐0.141**(0.063) 0.203(0.155) ‐0.278**(0.113)#

∀ , ‐0.258**(0.114) ‐0.379*(0.204)

∆#

#

∀ , ‐0.471**(0.230) ‐0.700**(0.311)

, 1.962***(0.119) 1.983***(0.067) 1.472***(0.186) 2.182***(0.067)

,∆# ∀ ,

0.228(0.226) 0.049(0.091) ‐0.300(0.261) 0.173(0.163)

,#

∀ , 0.089(0.179) 0.706**(0.323)

,∆#

#

∀ , ‐0.099(0.344) 0.257(0.511)

Note:Thedependentvariableistheindividuals’binarydecisiononwhethertobuyasignal( 1),ornotbuy(

0).Theindependentvariables(firstcolumn)aredefinedinthemaintext.*(**;***)indicatessignificanceatthe

10%(5%;1%)level,andabsolutestandarderrorsaregiveninparentheses.

25

Table 2 reveals some interesting learning effects.28 First, in all treatments, previously informed

individualshaveamuchhigherbaselineprobabilityofgettinginformedinthecurrentperiodthanthe

previously uninformed (all constants are negative and large, and statistically significant, and all

coefficients of , are positive and large, and significant). Second, focusing first on the previously

uninformed, in both compulsory voting treatments they get informed less likely when the value of

∆ /# ∀ , increases (both coefficients are negative and statistically significant). By

contrast, in both voluntary voting treatments, they avail the extra informationof varyingnumbers of

others’ votes by getting informed less likely when the value of # /2 ∀ , increases (both

coefficients are negative and statistically significant). Moreover, they only respond to ∆ /

# ∀ , bygettinginformedwithamuchlowerprobabilitywhenthevalueoftheinteractionof

the two quantitative variables increases (both coefficients of ∆ /# ∀ , are

insignificant, andboth coefficients of the interaction variable are large andnegative, and significant).

Third, compared to theiruninformedcounterparts, thepreviously informedgenerallydonot respond

differently to information in previous voting decisions of others, except that a greater value of

# /2 ∀ , markedlyandstatisticallysignificantlyraisestheirprobabilityofgettinginformed

inthecurrentperiod,which(allothercoefficientsareinsignificantinalltreatments).Overall,itseems

that apart from the higher baseline signal acquisition probabilities of individuals who tend to be

informed,bothpreviouslyinformedanduninformedparticipantsgenerallyrespondtovotingdecisions

of others in similar ways. In particular, our probit estimations suggest thatmany of them avoid the

signalcostinanticipationofaclosevote.

Experimentalresult5 (Learning and close votes): Inall treatments,previously informedparticipants

aremore likely to get informed in the current period than those who were not informed. Moreover,

independentoftheirprevioussignalacquisitiondecision,participantsaregenerallymore(less)likelytoget

informedwhen the information inothers’previousdecisions indicates that thenextvotemight (not)be

close.

28Thoughnotincludedinthepanelmodels,uninformedvoteswereslightlybiasedbythegambler’sfallacy.Overall,afterthetruestatewasYellow,55%votedforBlue( 0.023exceptfor 0.352inVoluntaryVoting‐7,binomialtests,one‐tailed)andafteritwasBlue,55%votedforYellow( 0.036exceptfor 0.159inCompulsoryVoting‐3,sametests).

26

5Conclusions

Westudymajorityvotingovertwoalternativesinsmallgroups.Individualshaveidenticalpreferences

but are uncertain aboutwhich alternative is correct (both are equally likely correct). Prior to voting,

eachindividualcanacquireacostlysignalthatisinformativebutimperfect.Towit,itshowsthecorrect

alternative with probability ∈ , 1 , and the incorrect one with probability 1 . We investigate

voluntaryandcompulsoryvoting.Anuninformedvotecannegativelyimpactthegroupdecisionasitcan

overridean informative groupdecision.Withvoluntary voting,uninformed individuals canavoid this

cursebyabstaining,whichisnotafeasibleoptioninthecompulsorymode.

Inourexperiment,weemploygroupsofthreeandseven.Wefindthat,onaverage, largergroups

havegreaterinformationpoolsthansmallergroupsinbothvotingmodes,andthatthepoolsaregreater

with compulsory than voluntary voting. The results are predicted by logit equilibrium for degrees of

decision‐making errors commonly observed in related collective action experiments, but not byBNE.

Surprisingly, in both voluntary voting treatments there is substantial uninformed voting by many

participants, with strong negative consequences for efficiency. Overall, the fraction of correct group

decisionsandefficiencyareinbetweenthosepredictedforuniversalignorance(i.e.,thegroupdecision

is correcthalfof the time,atnosignal costs)and fora singlenoise‐freedelegatewhodecides for the

groupandwhoseexpectedperformanceislowerthaninsocialoptimum.Moreover,smallergroupswith

voluntaryvotingdoworst,whilegroupsinallothertreatmentsfareaboutequallyunsatisfactory.While

lowperformancesarepredictedby logit equilibrium in the compulsorymodewhere someuniformed

votingistolerated,theyarelowerthanpredictedinthevoluntarymodeduetotheswingvoter’scurse.

Ourresultsaredisillusioningandsuggestthatthecursemustbetakenveryseriouslyinthecurrent

practiceof juryandcommitteevoting.Forexample, inorder topreventuninformedvoluntaryvoting,

onemightconsiderintroducingsomevotingcosts(seealsoTyson2013)orrandommonitoringofthe

knowledgeofdecisionmakers.And,animportantissuesistherobustnessofourfindings.Inthisregard,

Elbittaretal (2013)andShineman(2013)providecomplementaryevidenceofuninformedvoluntary

votingindifferentsetups.Overall,oursmallgroupstudyofmajoritarianvotingwithendogenouscostly

informationhints tosevereefficiency lossesassociatedwith thecurseofuninformedvoting.Buteven

withoutthecurse,comparedtothesocialoptimum,predictedandobservedefficienciesarelowdueto

27

thecollectiveactionproblemsinvolved.Therefore,ourstudysuggeststhatthereisroomforimproving

existing designs of jury and committee voting, and hopefully it can inspire further research in this

direction.

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Appendix

ProofofProposition1

IntheVotingstage,notethatindividual ispivotalifotherindividuals cast equallymanyvotes

foreachalternative; onemorevoteforA;or onemorevoteforB;andsheisnon‐pivotalinall

31

otherevents.Wesaythat “strictlyprefers”adecisiontoanotheroneif,inpivotalevents,itselectsthe

alternative indicatedby strictlymore signals in the group (includingher own signal), and that she is

“indifferent”innon‐pivotaleventsorif, inpivotalevents,thetwoalternativesareindicatedbyequally

manysignalsinthegroup.Wenowanalyzevoluntaryandcompulsoryvotinginturn.

With voluntary voting, we must show that there exists a symmetric BNE where each informed

individualvotessincerelyandeachuninformedindividualabstains.Supposethateach informedother

individual votessincerelyandeachuninformed abstains.If isinformed,assumingwithoutlossof

generality thatshehasana‐signal, thensincerevoting isheronlyweaklydominantstrategybecause:

first,itweaklydominatesabstainingasinpivotalevents itisstrictlypreferred,andinpivotalevents

and and in non‐pivotal events she is indifferent; second, sincere voting weakly dominates

votingforBasinpivotalevents and itisstrictlypreferred,andinpivotalevents )andinnon‐

pivotaleventssheisindifferent.If isuninformed,thenabstainingweaklydominatesvotingforA[B]as

inpivotal events [ ] abstaining is strictlypreferred, and inpivotal events and [ and

]andinnon‐pivotaleventssheisindifferent.Thus,giventhatothersvotesincerelyifinformedand

abstain ifuninformed, ’sonlyweaklydominantstrategy is tousethesamestrategyaseveryoneelse.

ThiscompletesourproofofexistenceofoursymmetricBNEwithvoluntaryvoting.

With compulsory voting,wemust show that there exists a symmetric BNEwhere each informed

individualvotessincerelyandeachuninformedindividualvotesrandomlywithprobabilityone‐halffor

eachalternative.Aseveryonemustvoteandthegroupsizeisodd,2 1,individual isonlypivotalif

otherindividuals cast votesforeachalternative(seepivotalevents ),andsheisnon‐pivotal

inallotherevents.Supposenowthateachinformed votessincerelyandeachuninformed votes

withprobabilityone‐half foreachalternative.If is informed,assumingwithout lossofgeneralitythat

shehasana‐signal, thenneitherAnorBdominates theotherdecisionbecause: innon‐pivotalevents

andinpivotaleventswithonemoreB‐thanA‐votesofinformedotherssheisindifferentbetweenvoting

forAandforB;inpivotaleventswithatleastasmanyA‐thanB‐votesofinformedotherssheprefersto

votesincerely;andinpivotaleventswithatleasttwomoreB‐thanA‐votesshepreferstovoteagainst

herownsignal.However,usingherowna‐signal,sheexpectsthatinformedothershavemorea‐thanb‐

signals, on average, and thus her unique best response is to vote according to her signal. If is

32

uninformed, then there isalsonodominationbecause: if she ispivotal, sheprefers tovote forA [B] if

therearemoreA‐thanB‐votes[B‐thanA‐votes]ofinformedothers,andsheisindifferentifsheisnon‐

pivotalandifeachalternativehasequallymanyvotesofinformedothers.As hasnosignal,sheexpects

that informedothershaveequallymanya‐andb‐signals,onaverage,and thusanyprobabilitymixof

voting for A and for B is a best response, including , . Thus, given that others vote sincerely if

informedandvotewithprobabilityone‐halfforeachalternativeifuninformed,it isabestresponseof

individual touse the same strategy as everyone else,which showsexistenceof our symmetricBNE

withcompulsoryvoting.∎

ProofofProposition2

We first derive the symmetric logit equilibrium conditions 1 and 2 for the signal acquisition

probability ∗ , , andthereafterproveourcomparativestaticsresults.

FigureA1:Logitequilibrium

Note: The left panel shows 1 for variousdegreesofnoise.The rightpanel shows 1 for 0.06,

1 , and the signal cost (i.e., the lower bound) for our experimental treatments and parameters. In all

treatments, both sides of 1 intersect only once so that each has a unique ∗ 0.06 : 0.409 and 0.277 in

Voluntaryvoting‐3and‐7,and0.517and0.355inCompulsoryvoting‐3and‐7,respectively.

33

(Logit equilibrium condition) Anticipating equilibrium voting, ∗ (Proposition 1), individual ’s

expected payoffs of getting informed and not getting informed, 1, , ∗ , , and

0, , ∗ , , ,areaugmentedbythestochasticterms and ,respectively,where and

are iid random variables (recall that the parameter 0 controls the degree of noise). To keep

thingssimple,inthefollowingweomittheindexmwheneverouranalysisappliestobothvotingmodes

orthemodeisobvious.Then,ifeveryoneelsebuysasignalwithprobability ,individual getsinformed

ifandonly if 1 0 ⇔Π , ∗, , , / ,29whichoccurs

withprobability Π . / ,whereΠ . ≡ 1 0 denotesherexpectednet

payoffsofgettinginformedandFisthedistributionfunctionofthedifference .Takingtheinverse

of . and multiplying both sides by yields Π . . Finally, using the logistic

distribution, 1/ 1 ,givesourlogitequilibriumcondition

1Π , ∗, , , . 1

If 0,then 1 | ∈ , 0and 1 turnsintotheBNEcondition.If 0,then 1 hasthe

following properties: regarding , 0 for ∈ 0,1 , lim → 1 ∞,

1 | / 0, and lim → 1 ∞; and regarding , 0 for ∈ 0, , 0

for , and 0 for ∈ , 1 . The left panel of Figure A1 illustrates these properties for

∈ 0.01, 0.1, 0.5, 10}.Itdepicts onthehorizontalaxis,and 1 ontheverticalaxis.For 1 ,

theexpectednetpayoffsof getting informedareboundedby Π . , that is, in the lower

bound ’spivotprobabilityapproacheszero,andintheupperbounditequalsoneand 1.

(Existence)Weuse IntermediateValueTheorem to show that ∗ ∈ 0,1 , , alwaysexists

for 0.Note that 1 and 1 are continuous for ∈ 0,1 . Then, as lim → 1

∞ 1 ∈ , lim → 1 ∞,bothsidesof 1 and thusbothsidesof logit

equilibrium conditions 1 and 2 intersect at least once,which proves that for 0 there always

existsa ∗ ∈ 0,1 inbothvotingmodes.30

29Weassumewithoutlossofgeneralitythatanindifferentindividual choosestogetinformed.30ForBNE( 0), itisreadilyverifiedthat ∗ 0[ 1]ifthesignalcostistoohigh[low]sothatbothsidesof 1 donotintersectorintersectat 1.Moreover, ∗ ∈ 0,1 forintermediatecostsandbothsidesintersectatleastonce.

34

(Uniqueness) The right panel of Figure A1 shows 1 for 0.06 and 1 for our

experimentaltreatmentsandparameters.Ascanbeseen, 1 hasconvexandconcaveproperties

with voluntary and compulsory voting, respectively, which for some parameter constellations yields

multipleequilibria(thoughnot forourexperimentalparameters). Inparticular,multiplicitycanarises

when 1 is flat (i.e., is small).31 To keep things simple, we focus on unique equilibria by

assumingforbothvotingmodesthatthedegreeofnoiseislargeenough,or 0,where gives

therespectivelowerboundthatensuresuniqueness.Notethatcomputationsshowthat fallsbelow

thetypicalestimateof ̂ inbinarychoiceexperimentssimilartoours.

Next,wederivecomparativestaticspredictionsforunique ∗.

(Groupsize)Wemustshowthatincreasingthegroupsizefrom2 1to2 3decreases ∗ in

theregionoftheparameterspaceofuniqueequilibrium.Forbothvotingmodes, 1 isconstantin

sothatitsufficestoprovethat 1 | 1 | for ∈ 0,1 .If 1 strictlydecreases

with for all ∈ 0,1 , then ∗ must indeeddecrease since 1 is constant in and strictly

increasingin ∈ 0,1 .

Withvoluntaryvoting, ,theterms and on 1 areconstantinboth and .Then,

toseethat 1 decreaseswith ,itsufficestoshowfor

, ∗ , ,22

12

1 2

that 1 0 for all ∈ 0,1 . First, for individual ’s pivot probability, if 2 others are

informed, we have ,∗ , , 1 ,

∗ , , 1 1 ⇔

1 , since 1 for ∈ , 1 and ⇔ 2 2 0 for

0, … , 1. Second, the cdf of , , , 1 first‐order stochastically

dominatesthecdfof , , 1, 1 for 0. . . 1and ∈ 0,1 .Thus,

combing both results, 2 and hence 1 decrease for ∈ 0,1 if the group increases by two

individuals,sothat ∗ ∗ 1 withvoluntaryvoting.

31 For example, for 0.001 and 0.75, we compute ∗ ∈ 0.541, 0.932, 0.99999 for 1 and 0.08, and ∗ ∈0.00002, 0.035, 0.225 for 23and 0.04.

35

Withcompulsoryvoting, , the term on 2 is constant in .Then, to see that 2

decreaseswith ,itsufficestoshowfor

, ∗ , ,2

1 1 3

11 1

that 1 0forall ∈ 0,1 .First,thefractionofpivotaleventsoutofallpossibleevents

decreaseswith ,thatis 2 2 21 for ∈ 0,1 ,whichworkstodecrease 3 .Second,

let’sdenote ∗ , , , , ≡ 1 , ≡ ,andrewrite ∗ , , , ≡

as ∗ , , 0 0if 0and ∗ , , 0

0if 0,with– 1 1(i.e.,theBayesianupdatedvaluedepends

on , but not on the particular levels of and that determine ). Then, for ∈ , 1 and all

, 0, … , and , … , wehave

, , 1, , , , ;

, , 1, 1, 1 1 , , ;

, , 1, 1, 1 , , 1 ;

, , 1, , 1 1 1 , , 1 ,

where the inequalities always hold because 1 , 1 1 , 2 1 , and

2 1 1 ,respectively.Third,thecdfof 1 first‐orderstochastically

dominatesthecdfof 1 for 0, … , ,with , and ∈ 0,1 .Thus,combing

allthreeresults, 3 andhence 2 decreasefor ∈ 0,1 ifthegroupincreasesbytwoindividuals,

sothat ∗ ∗ 1 withcompulsoryvoting.

(Signalcost)Anincreasein 0decreasestheunique ∗ bythefollowingargument: 1

is constant in ,while 1 0 for ∈ 0,1 . Since 1 strictly increaseswith ∈ 0,1 ,

bothsidesof 1 mustintersectfurthertotheleftifcincreases,sothat ∗ decreaseswith .

(Degreeofnoise)Anincreasein 0, , moves ∗ 0.5closertoone‐halfbythe

followingargument:increasing yieldsasteeper 1 ,whichalwaysgoesthrough 1 |

36

0 (where it is independent of ) and 1 is constant . Thismeans that both sides of 1 must

intersectatanewequilibrium ∗ ′ ∈ ∗ , if ∗ 0.5and ∗ ′ ∈ , ∗ if ∗ 0.5.

Thus, ∗ 0.5movesclosertoone‐halfif increases.∎

Probabilityofacorrectgroupdecision

Ifthetruestateis ,theneachinformedindividualhasmorelikelya ‐thana– ‐signal,where ,

and .Withvoluntaryvoting,theprobabilityofacorrectgroupdecisionisthengivenby:

, | , ∗ , ,2 1

1

1

12

2 12

1 2

1 . 4

Thefirsttermconsidersoutrightwinsofalternative (i.e.,therearemorevotesfor than outof

informed sincere votes, , and the remaining 2 1 individuals abstain), and the second

term considers ties (i.e., there are informed sincere votes for each alternative, and 2 1 2

abstentions),which are broken by a coin flip.Moreover,with compulsory voting, the probability of a

correctgroupdecisionsisgivenby:

, | , ∗ , ,2 1 2 1

5

1 1 ,

where denotesthetotalnumberofj‐votes,and [ ]thenumberofinformedsincerej[ ]‐votes.

Alternativejwinsoutrightiftherearemoreinformedsincereanduninformedrandomvotesforjthan

, or 1.Unlike in 4 ,there isno second term in 5 because tiesare impossible inodd‐

sizedgroupswithcompulsoryvoting.

ProofofProposition3

If goestoinfinity,thelogitequilibriumcondition 1 gets

lim→

1lim→

Π , , ∗ , , , , 6

37

, , where denotes the limiting equilibrium signal acquisition probability. First, note that

6 lim → and lim → 0 in Π , . on 6 . For

0,whichimplies 0,if goestoinfinity,thenindividual ’spivotprobabilityapproacheszero,

sothatlim → Π , , ∗ , , , for , (i.e.,aslim → 1 0for ∈ 0,1

and 0, … , and,forthecompulsorymode, . 1).Therefore, 6 gets

1⇔

1⁄ ⇔ ∗ , , 0

11 ⁄ 0 7

for , , asstated in theproposition.Next, for 0and ∈ , 1 , if increaseswithoutbound,

then the difference in expected numbers of informed votes for the correct minus the incorrect

alternative is lim→

2 1 2 1 1 lim→

2 1 2 1 ∞, the expected

numberofuninformedrandomvotesequalszerowithvoluntaryvoting(i.e.,theuninformedallabstain),

and lim→

2 1 1 ∞with compulsory voting. Thus, the uninformedby themselves create a

suretieinthevoluntarymodeand,bythelawoflargenumbers,theyalmostsurelycreateatieiftheir

numberisevenandaone‐voteleadforeitheralternativeiftheirnumberisodd.Butthesetiesandone‐

vote leadsdonotmatter, as thereare infinitelymorevotes for the correct than incorrect alternative.

Therefore,for 0and ∈ , 1 ,largegroupsmakeacorrectgroupdecisionalmostsurely.Finally,for

0(i.e.,BNE),wehave 6 0.If 0 0,thenlim → Π , , ∗ , , , ,which

is a contradiction for 0, since individual would choose not to buy a signal, 0 0!

Therefore,inequilibriumitmustbethat ∗ , , 0 0(“universalrationalignorance”),andthus

thegroupdecisioniscorrecthalfofthetime,whichcompletesourproofofProposition3.∎

38

FiguresA2:Groupperformances

FigureA2.1:Expectedinformationpool

FigureA2.2:Probabilityofacorrectgroupdecision

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 1 10 100

μ (degree of noise)

Lo

git

-QR

E e

xp

ec

ted

nu

mb

er

of

ac

qu

ire

d s

ign

als

pe

r g

rou

p

Voluntary voting-3 Voluntary voting-7 Compulsory voting-3 Compulsory voting-7

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 1 10 100

μ (degree of noise)

Pro

ba

bili

ty o

f a

co

rre

ct

gro

up

de

cis

ion

Voluntary voting-3 Voluntary voting-7 Compulsory voting-3 Compulsory voting-7

Social optimum-3 Social optimum-7 Delegate-3 and -7

39

FigureA2.3:Expectedefficiency

Note:Forour logitequilibriumsignalacquisitionprobabilities(Figure1), the linesdepictpertreatment

the expected number of acquired signals per group (Figure A2.1), the probability of a correct group

decision(A2.2),andexpectedefficiencyperindividual(i.e.,groupefficiencydividedbygroupsize;A2.3).

In social optimum, as 1, the expectednumberof acquired signalsper groupequals the respective

groupsize (notshown inA2.1),andwithasinglenoise‐freedelegate,as 1, itequalsone(alsonot

shown). Theprobabilities of a correct groupdecision andaverage expected efficiencyper individual in

socialoptimum(forasingledelegate)areshownascircles(triangles)inA2.2andA2.3,respectively.

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 1 10 100

μ (degree of noise)

Ex

pe

cte

d a

ve

rag

e in

div

idu

al

eff

icie

nc

y

Voluntary voting-3 Voluntary voting-7 Compulsory voting-3 Compulsory voting-7

Social optimum-3 Social optimum-7 Delegate-3 Delegate-7

40

TableA1:BayesianNashequilibrium

BNE predictions ( ) Voluntaryvoting Compulsoryvoting

Groupsize

∗ Efficiency ∗ Efficiency

1 1 .750 .625 1 .750 .6253 .321 .675 .635 .615

0.723.500

.647

.5005 .181 .662 .639 0 .500 .5007 .125 .657 .641 0 .500 .5009 .096 .654 .642 0 .500 .50011 .078 .652 .642 0 .500 .50013 .065 .651 .642 0 .500 .50015 .056 .650 .643 0 .500 .50017 .049 .649 .643 0 .500 .500

TableA2:Logitequilibrium

Logitpredictions ( . ) Voluntaryvoting Compulsoryvoting

Groupsize

∗ Efficiency ∗ Efficiency

1 .889 .722 .611 .889 .722 .6113 .409 .705 .654 .517 .690 .6255 .322 .732 .692 .408 .686 .6357 .277 .754 .719 .355 .688 .6449 .250 .771 .740 .323 .692 .65111 .230 .786 .757 .301 .696 .65813 .216 .799 .772 .284 .700 .66515 .205 .810 .784 .272 .705 .67151 .140 .914 .896 .193 .756 .732

41

FigureA3:Fractionsofinformedvoting,abstention,anduninformedvotingoverperiods

Bothparts Part1 Part2

Voluntaryvoting‐3

Voluntaryvoting‐7

Compulsoryvoting‐3

Compulsoryvoting‐7

Periods

0.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30