Candidateentryandpoliticalpolarization:
Anexperimentalstudy*
JensGroßer(FloridaStateUniversity)a
ThomasR.Palfrey(CaliforniaInstituteofTechnology)b
December14,2016
Abstract
Wereporttheresultsofalaboratoryexperimentbasedonacitizen‐candidatemodelwithprivate
informationaboutidealpoints.Inefficientpoliticalpolarizationisobservedinalltreatments;that
is, citizens with extreme ideal points enter as candidates more often than moderate citizens.
Second, less entry occurs, with even greater polarization, when voters have directional
information about candidates’ ideal points, using ideological party labels. Nonetheless, this
directionalinformationiswelfareenhancingbecausetheinefficiencyfromgreaterpolarization
is outweighed by lower total entry costs and better voter information. Third, entry rates are
decreasingingroupsizeandtheentrycost.Thesefindingsareallimpliedbypropertiesofthe
uniquesymmetricBayesianequilibriumoftheentrygame.Quantitatively,weobservetoolittle
(too much) entry when the theoretical entry rates are high (low). This general pattern of
observedbiasesinentryratesisimpliedbylogitquantalresponseequilibrium.
*WewouldliketothankparticipantsatthePublicPolicyandSocial&EconomicBehaviorconference,UniversityofCologne,
the1stSouthwestExperimentalandBehavioralEconomics(SWEBE)conference,UCIrvine,andtheNorth‐AmericanESA
meetinginTucson,andatseminarsatCaltechandtheHertieSchoolofGovernance,Berlinfortheirhelpfulcomments.We
arealsogratefulforfinancialsupportfromtheCEC‐COFRSAward,FloridaStateUniversity.Palfreyacknowledgessupport
from NSF (SES‐1426560), the Gordon and Betty Moore Foundation (SES‐1158), and a Russell Sage Foundation Visiting
ScholarFellowship(2014‐15),
aDepartmentsofPoliticalScienceandEconomics,FloridaStateUniversity,Tallahassee,FL‐32306,[email protected]
b Division of the Humanities and Social Sciences, California Institute of Technology, Pasadena, CA‐91125,
[email protected]
1 1.Introduction
Whorunsforoffice?Howmanycandidatescanweexpecttocompeteinawinner‐take‐allelection?
Arethosewhorunforpoliticalofficerepresentativeoftheviewsofthegeneralpolity?Howmight
entrydependupontheroleofpoliticalorganizations,suchasparties,intheselectionofcandidates?
Here, we examine these and related questions in a laboratory experiment by testing predictions
derivedfromacitizen‐candidateentrygameandcomparingentrybehavioracrossseveraldifferent
environments. The citizen candidate model, which originates in Besley and Coate (1997) and
OsborneandSlivinski(1996),departsfromthecanonicalspatialmodelofelectoralcompetitionwith
exogenouspoliticians(Downs1957;Hotelling1929)intwoimportantways.1First,thecandidates
arecitizenswithpolicypreferences(asinWittman1983)whovoteintheelectionand,onceelected,
implementtheirowntasteasthecommonpolicy.Second,thevotingstageisprecededbyanentry
stage where each citizen decides whether to throw her hat in the ring. Thus, a citizen’s objective
functionnotjusttakesintoaccountthebenefitsofholdingoffice(“spoilsofoffice”)asinthecanonical
model,butisalsoincludesthecostofcandidacyandtheindirectbenefitofreducingthepossibility
of less desired policies that would be implemented by other potential candidates. Because the
citizens themselves decide on whether to run for office, both the number and the ideological
composition of entering candidates are modeled as equilibrium outcomes. Crucially, their entry
decisions are asymmetric since citizens with different policy preferences will anticipate different
benefits from policy implementation. Coordination problems are also present due to nontrivial
strategicuncertainties.
Inthestandardcitizencandidatemodel,allcitizensareendowedwithcompleteinformation
about the exact location of all others’ ideal points, and hence can infer the exact location of each
entering candidate. However, many empirical studies indicate that citizens tend to have limited
knowledgeaboutthecandidates’exactstancesonpolicyissues(e.g.,Campbelletal.1960;Palfrey
and Poole 1987). We can think of various reasons why this is the case. For example, time and
willpower is scarce so that many citizens simply cannot be as well informed about the policy
intentionsofcandidatesas,say,specialinterestgroups.Or,politiciansoftenremainquietabouttheir
trueintentionsduringcampaignsduetostrategicincentivesanditisalmostimpossibleforcitizens
todiscoverthesetastes,eveniftheyarewillingtoexerteffort.Morerealistically,citizenshaveonly
1Thecitizencandidateapproachhasitsrootsinworkonpolicy‐motivatedcandidates(e.g.,Wittman1983),strategicentry
(e.g.,Feddersen,Sened,andWright1990;Palfrey1984),andDuverger’slaw(e.g.,Feddersen1992;Palfrey1989).Fora
surveyofcitizencandidatemodels,seeBol,Dellis,andOak(2015).
2 incompleteinformationaboutthelocationoftheenteringcandidates,andthisleadsustoadopta
Bayesiangameformulationoftheentrygame.
Theexperimentisbasedonalaboratoryimplementationofthefollowingcitizen‐candidate
entrygamewithincompleteinformation(GroßerandPalfrey2014).Anelectorateconsistingof citizens is electing a leader to implement a policy outcome by plurality voting. Each citizen has a
privatelyknownidealpointinaonedimensionalpolicyspace.Theseidealpointsareiiddrawsfrom
a commonly known, uniform distribution over the policy space. A citizen’s utility from the policy
outcomedeclineslinearlyintheabsolutedistancefromheridealpoint.Thegamehastwodecision‐
makingstages.Inthefirststage(Entry),citizensdecideindependentlyandsimultaneouslywhether
and pay a cost
0tobecomeacandidateintheelection, or not enter and bear no cost. In the
secondstage(Voting),eachcitizencastsavoteforexactlyoneofthecandidates.Inthebaselinemodel
citizensmustvotewithoutanyadditionalinformationaboutthecandidates’idealpoints(ofcourse,
thecontendersknowtheirownidealpoint).Thecandidatewiththemostvotesbecomestheleader
andreceivesabonus
0(i.e.,thespoilsofoffice)andheridealpointisautomaticallyimplemented
asthecommonpolicy.Tiesarebrokenrandomly.Finally,ifnobodyenters,thenaleaderisrandomly
selectedfromallthecitizensandheridealpointisimplementedasthecommonpolicy.
Because symmetric BNE in cutpoint strategies is unique, potentially difficult issues of
equilibriumselectionareavoided.Bycontrast,citizencandidatemodelswithcompleteinformation
aboutcandidateidealpoints(e.g.,BesleyandCoate1997;OsborneandSlivinski1996)usuallyhave
multipleequilibria,andthereforearemorecomplicatedtoevaluateempiricallyeveninthelab.A
second advantage of the incomplete information approach is that distributional predictions are
qualitativepredictionsaboutpolarizationandthenumberofentrantsismorerobusttoawiderange
of environmental parameters one expects to encounter in the field. In fact, existing empirical
evidencestronglysuggeststhatpolicypreferencesofpoliticiansaremorepolarizedthanthecitizens
theyrepresent(e.g.,BafumiandHerron2010;DiMaggio,Evans,andBryson1996;Fiorina,Abrams,
andPope2006;McCarty,Poole,andRosenthal2006).Beyondthisempiricalsupport,ourapproach
offersatheoreticalfoundationforpossiblemechanismsthatcanleadtopoliticalpolarization.The
experimentprovidesadditionalevidencebygeneratingdataonentrybehaviorincarefullycontrolled
environments,inordertoassesstheplausibilityofthesetheoreticalmechanisms.
This citizen‐candidate entry game with incomplete information yields sharp predictions
about the distribution of the entrants’ ideal points and the rate of entry. The key property of
equilibrium is political polarization in the sense that candidate entry is from the extremes of the
policy space, contrary to usual centrist predictions of most models of political competition.
3 Specifically, the unique symmetric Bayesian Nash equilibrium (BNE) in the baseline entry game
consistsofaleftandrightcutpoint,
,
,whereeachcitizenwithanidealpointateithercutpoint
ortotheleftof ortotherightof entersthepoliticalcompetition,whileeverycitizenwithan
idealpointbetweenthetwocutpointsdoesn’tenter.2Basedonthisequilibrium,onecanalsocompute
expectedeconomicwelfareanditsvariouscomponents,andderivecomparativestaticspredictions
of interest such as the effects of electorate size, entry costs, benefits of holding office, and the
distributionofidealpointsonentrydecisionsandwelfare.
The intuition for why asymmetric information about citizen and candidate ideal points
createspoliticalpolarizationcanbeexplainedbyasimpleexample.Suppose,forexample,thepolicy
spaceis
1,1 andtherearethreeentrantswithidealpointsat 1,0,and1,respectively.Then,each
candidate has a one‐third chance of winning the election (i.e., they each vote for themselves and,
becauseidealpointsareprivateinformation,eachnon‐candidatevotesrandomlyforoneofthem).
Withidenticalentrycostsandoffice‐holdingbenefits,theyonlydifferintheirexpectedpolicylosses
if a rival candidate happens to win. In our example, for each of the two extreme candidates the
expectedlossequals1,whiletheexpectedlossisonly2/3forthemoderatecandidate.Thisillustrates
whatturnsouttobeageneralpropertyofthemodel:extremecitizenshaveastrongerincentiveto
enterthepoliticalcompetitionthanmoderatecitizens,andisthebasisfortheemergenceofpolitical
polarizationinthemodel.Thisresultholdsmoregenerallyforanysmoothprobabilitydistribution
ofidealpoints(GroßerandPalfrey2014)andweaklyconcavepreferencesofvoters.Importantly,
polarizationiswelfarereducingsinceexantetheexpectedtotalpolicylossisminimizedwhenthe
commonpolicyisacentristidealpoint.
Whileincompleteinformationissurelyanimportantconsiderationinelections,themodel
described above explores a polar case where citizens are completely uninformed about the
candidates’idealpoints,exceptfortheinferencetheycanmakefromequilibriumstrategies,towit,
thatentrycomesfromtheextremes.Itisinterestingtoexploreanintermediatecaseofincomplete
information that also corresponds to the widely observed phenomenon that ideologically‐based
partiesactasgatekeepersinthecandidateentrystage.Asaresultmostcitizensareawareofthe
partyaffiliationsofcandidates,whichareforexamplecommunicatedvianominatingconventions,
andsincepartiesareideologicaltheycanusethiscrucialpieceofinformationasacrediblecueabout
a candidate’s ideal point, for their voting decision (e.g., Ansolabehere, Rodden, and Snyder 2008;
SnyderandTing2002).
2IntheVotingstage,eachcandidateoptimallyvotesforherselfandeachnon‐candidateoptimallyvotesrandomlyfora
candidate.Theexpectedpolicyoutcomedoesnotchangeifabstentionisallowed.
4 Toaccountforrelevantpartycues,weextendthebasiccitizen‐candidateentrygamewhere
a left and a right party each nominates a candidate from the pool of entrants on their side of the
politicalspectrum,andcitizensareinformedabouteachnominee’spartyaffiliation.Thisprovides
some useful voting information, although the exact ideal points of the party nominees are not
revealed.This“directionalinformation”hastwoimportanteffects.First,itleadstofewerandeven
morepolarizedentrantsthanintheabsenceofparties.Thisisbecauseacitizenalwaysvotesforher
preferredpartynomineeso,availingtheownvote,herupdatedbeliefthatthisnomineeprevailsin
theelectionisgreaterthan50percentforoursymmetricdistributionofidealpoints(cf.Goereeand
Großer2007).Asaconsequence,ceterisparibus,theprobabilityofbecomingtheleaderisgreater
withthanwithoutparties(weassumethateachentrantisequallylikelyherparty’snominee)which,
inequilibrium,translatesintoalowerincentivetorunforoffice,moreextremecutpoints,andthus
fewerentrants.Second,politicalpartiesenableimplicitvotecoordination;thatis,citizensvoteforthe
nomineewhoseidealpointisfromthesamedirectionastheownone.Importantly,suchcoordination
iswelfareenhancinginexpectationsincethemajoritypartyismorelikelytowin.Noticethatwhile
themajoritycansometimesalsobedefeatedifnoneofitscitizensrunsforoffice,thismustnotbe
inefficient as they are all more moderate than the respective cutpoint. Overall, in expectation,
politicalpartiesraisewelfaresincethelowertotalentryexpenseandgreaterchancesofthemajority
partyoutweighthegreatertotalpolicylosscausedbymoreextremeleaders.
Byincludingtreatmentsbothwithandwithoutpoliticalparties,ourexperimentoffersaclean
testofthehypothesisthatparty‐organizedelectionsincreasepoliticalpolarizationonaveragebutat
the same time do not reduce welfare. The experiment also varies the environment in two other
dimensions: electorate size and entry cost. Both of these have intuitive qualitative effects, that is,
expectedentryrates(i.e.,entryasafractionoftheelectoratesize)decreaseinbothelectoratesize
and entry cost. The decrease in entry rates arises from the equilibrium cutpoints becoming more
extreme,whichimmediatelyimpliesthatpoliticalpolarizationisincreasinginboththeelectorate
sizeandtheentrycost.
Looking ahead at the results briefly, in all treatments conducted in the experiment, we
observethekeypolarizationeffect:theprobabilityofcandidateentryisincreasinginthedistance
between the median and a citizen’s ideal point. All of the model’s primary comparative static
propertiesofentrybehaviorfindsupportinthedata.And,allthemodel’sprimarycomparativestatic
propertiesaboutwelfarearealsosupported.Quantitatively,relativetothetheoreticalequilibrium,
we observe higher rates of entry for those treatments where entry is predicted to be below 50
percent and weakly lower entry rates for those treatments where predicted entry is above 50
5 percent.ThispatternofdeparturefromBNEisconsistentwithpastexperimentsonentryinmuch
differentsettings(seeGoereeandHolt2005)andisageneralpropertyofregularquantalresponse
equilibriuminthesegames(Goeree,HoltandPalfrey2005,2016).
2.Relatedliterature
Weareawareofjustthreeothercitizencandidateexperiments,whichallstudypluralityelections
with complete information about candidate ideal points and vary the entry cost (Cadigan 2005;
Elbittar and Gomberg 2009; Kamm 2016).3 Specifically, Cadigan (2005) uses a pen‐and‐paper
experimentwithelectoratesoffiveparticipantswhohavedistinctidealpointsandindependently
andsimultaneouslydecideonwhethertobecomeacandidate.Theelectoralcompositionandideal
points are constant throughout, but after each election ideal points are reallocated among the
participants.Further,participantsautomaticallyvotesincerelyforthecandidatenearesttotheown
locationtoselecttheleader,whoreceivesabonusandwhoseidealpointisdeclaredthecommon
policy.Ifnobodyenters,thenoneparticipantisrandomlyappointedtheleader.ElbittarandGomberg
(2009)andKamm(2016)employsetupssimilartotheonejustdescribed,butafewdifferencesare
worth mentioning. For example, their experiments are computerized and sincere votes are
exclusively cast by an infinite number of non‐candidate robots with uniformly distributed ideal
points over the continuum 0,100 . Also, Elbittar and Gomberg use electorates of three or five
participantslocatedatthreefeasiblepolicies,andthedefaultpolicyifnoneofthementersisthatall
must pay a large penalty.4 We can summarize the three main common results of these citizen
candidateexperimentsasfollows.First,therearemorecandidatesonaveragewhentheentrycostis
lower. Second, relative to Nash equilibrium (NE), there is over‐entry on average. Third, the
qualitative predictions of entry are mostly supported by the data and some learning towards
equilibriumplayisobserved.Lookingmorecloselyattheirresults,iftheentrycostishightheunique
NEisthatonlythemediancitizenenters(ElbittarandGomberghavetwoequilibria,eachwhereone
of two median citizens enters). The median participant does indeed enter often but, against the
3Ourstudyisalsorelatedtosimplerentryexperiments.Forexample,FischbacherandThöni(2008)examineawinner‐
take‐allmarketwhereamonetaryprizegoestoarandomlyselectedentrant,andtheexpectedamountfallsinthenumber
ofentrants.Theyfindover‐entryrelativetoNashequilibrium,andmoreseveresoinlargergroups.Or,CamererandLovallo
(1999) find under‐entry with asymmetric, randomly allocated rank‐based payoffs. Both studies contain features also
presentinourwork,namely,theprobabilityofgettingthebonusfallsinthenumberofentrantsandexpectedpayoffsare
asymmetric due to various different ideal points. Finally, Goeree and Holt (2005) use quantal response equilibrium
(McKelveyandPalfrey1995,1998)tomakesenseofbothover‐andunder‐entryinvariousentryexperiments.
4Inthecitizencandidatemodeladefaultpolicyisnecessarytoensurethatacommonpolicyisexecutedinequilibrium.But
out‐of‐equilibriumplayoccursinthelabandinElbittarandGomberg(2009)thepenaltyresultedinbankruptcyofsome
participants.SeeGroßerandPalfrey(2014)foradiscussionofdifferentdefaultpolicies.
6 prediction, so do her or his immediate neighbors (albeit to a much lesser extent in Cadigan). By
contrast,twoequilibriaariseforalowentrycost,onewithonlythemediancitizenenteringandone
withthemedian’simmediateneighborsentering(again,ElbittarandGomberghavemultiplesuch
equilibria).However,intheexperimentscoordinationononeofthesepurestrategyequilibriausually
doesn’t occur. Next, in addition to plurality elections Elbittar and Gomberg (2009) study run‐off
electionsandKamm(2016)examinesproportionalrepresentation.ElbittarandGombergobservea
predictedshiftinaverageentrytowardsthemedianinrun‐offelectionsrelativetopluralityvoting.
Kamm adopts proportional representation à la Hamlin and Hjortlund (2000), where the common
policyisthevote‐weightedaverageofthecandidateidealpointsandtheleaderbonusisgiventothe
contenderwithmostvotes(tiesarebrokenrandomly).Aspredicted,hefindsmorepolarizedentry
thanwithpluralityvoting.Althoughwetooexplorepluralityvotingandhowtheentrycostaffects
the decision to run for office, our study is very different from these other citizen candidate
experiments. In particular, we explore incomplete information about candidate ideal points, as
opposed to the complete information they study, which yields mostly unique distributional
predictionsofwhoenters.And,wealsopresentthefirstcitizencandidateexperimentexamininghow
partiesandelectoratesizeinfluencepoliticalpolarizationandwelfare.
3.Themodel
WeadapttheGroßerandPalfrey(2014)citizencandidatemodelwithacontinuouspolicyspacefor
thecaseofadiscretepolicyspace,whichisimplementedintheexperiment.Anelectorateof citizens
iselectingaleadertoimplementacommonpolicy fromthesetΓ
1,2, … ,100 .Eachcitizenihas
aprivatelyknownidealpoint thatisaniiddrawfromauniformdistributionalsooverΓ,wherei’s
payofffromthepolicyoutcome,
,
|
|,islinearlydecreasingintheabsolutedistance
betweenheridealpointandthepolicyoutcome, .
3.1.Equilibriumwithoutparties
Wefirstdescribeandanalyzethecasewheretherearenoparties.Inthefirststage(Entry),citizens
independentlyandsimultaneouslydecidewhethertoenterasacandidateandpayacost
0,or
notenterandbearnocost.Inthesecondstage(Voting),eachcitizen(includingeachoftheentrants)
votesforoneofthecandidates,possiblyherself.Thecandidatewiththemostvotesiselectedand
receives an office holding benefit of
0. Ties are broken randomly. If no citizen enters, then a
defaultpolicy,d,takeseffect,randomlyselectingonecitizenastheleaderwhoreceives butdoes
not pay . The leader’s ideal point is implemented as the policy outcome. Summarizing, the total
payoffofcitizen isgivenby
7 |
, , ,
whereKisaconstant,
1ifsheentered(
|
, 1 0otherwise)and
1ifsheistheleader(
0
otherwise).Weassumecitizen isriskneutralandmaximizestheexpectedvalueof .
The perfect Bayesian equilibrium (PBE) of our citizen candidate game has the following
properties.5 In the Voting stage, each candidate votes for herself and each non‐candidate votes
randomlywithequalprobabilityforoneofthecontenders.InthesymmetricBNEoftheEntrystage
eachcitizen followsthecutpointstrategy
̌
where
,
0
∈
1, … ,
1
∈ 1, … ,
is an ideal point pair with 1
1
, … ,100 ,
∪
50 and
2 101
.6 That is, the cutpoint
strategy ̌ dictatesthateachcitizenwithanidealpointatormore“extreme”than or runsfor
office, and each citizen with an ideal point more “moderate” than and does not run. The
equilibrium cutpoints are derived by comparing a citizen’s expected payoffs for entering and not
entering, given that other individuals are using such cutpoints (see appendix for details). For the
specificationassumedhere,ifallothercitizens
,
areusingcutpointstrategy
,theoptimal
entrystrategyofacitizentype istoenterifandonlyif
1
1
| ∈
,
1
1
1, … ,
1
1
1
3 ,
| ∉
1, … ,
1
,
wheretheleft‐handside(LHS)givesthedifferencebetweentheexpectedbenefitfromenteringand
expectedbenefitfromnotentering,excludingthecostofentry,whichappearsontheright‐handside
(RHS).Weusethenotation
≡∑
todenotethenumberofentrantsand todenotetheex‐ante
probabilitythatarandomlyselectedcitizen
enters.Ifnobodyenters,thenthedefaultpolicyd
takes effect, where the expected loss from the absolute distance in citizen ’s ideal point and the
commonpolicy,orpolicyloss,isgivenby
,
| ∈
1, … ,
1
1
1
|
|
100
, 4 5FordetailsseeAppendixAandGroßerandPalfrey(2014).
6Thesymmetryofthecutpointsaroundthemedianidealpointarisesbecausetheuniformdistributionofidealpointsis
symmetricaroundthemedian.Ingeneral,thecutpointscanbeasymmetricallylocatedifthedistributionisasymmetric.See
GroßerandPalfrey(2014)fordetails.
8 andifatleastonecitizen
,
enters,therespectiveexpectedpolicylossisgivenby
| ∉
1, … ,
1
1
|
|
100
|
|
100
. 5 TheLHSof 3 hasastraightforwardinterpretation.Thefirsttermcorrespondstotheeventthatno
citizen
enters,whichoccurswithprobability 1
.Theintuitionisthatifonlycitizen enters,thenshecanensureleadershipbyenteringandsoreceives andavoidsanexpectedloss 4 from someone else’s policy decision. Note that if does not enter, these two payoffs accrue with
probability1/ and
1 / ,respectively,duetothespecificationofthestochasticdefaultpolicy,
d.ThesecondtermontheLHSof 3 representstheeventwhereatleastoneothercitizen enters.
Foreachpossiblenumberofentrants
2,includingherself,citizen bothreceives andavoidsa
lossfrompolicywithprobability1/ .Theexpectedpolicylossisdifferentdependingonwhether
theleader’sidealpointisinthesamedirectionas ’sidealpoint,whichiscapturedbythetwoterms
inbracketsin 5 .Finally,ourexperimentalparametersyieldinteriorequilibriumentrycutpoints,
whichiscomputedbysetting
and
andthensolving(3)atequality.
3.2.Equilibriumwithparties
In elections where the entry stage is organized by ideological political parties, the two decision‐
making stages have the following differences. First, all citizens with an ideal point
∈ 1, … ,50 ( 51, … 100 ) automatically belong to the Left (Right) Party. If one or more citizens from a party
choose to enter, then one of them becomes the party nominee in the election. For simplicity we
assumeeachcandidatefromthepartyisselectedastheparty’snomineewithequalprobability.The
partyaffiliationofeachnominee,albeitnotexactidealpoint,isthenrevealedtoallcitizens.Further,
eachcitizenvotesforanominee,possiblyherself.Ifonlyonepartyhasanominee,everyonemust
voteforher.Ifnobodyenters,thenthedefaultpolicy isactivated.Asinthecasewithnoparties,the
chosenleader’sidealpointisimplementedasthepolicyoutcome.
ThePBEofthecitizencandidategamewithpartieshasthefollowingstructure.IntheVoting
stage,eachnomineevotesforherselfandeachnon‐nominee,entrantornot,votesforthenominee
whoyieldsherthehighestexpectedpayoff.Thiswillbethenomineefromtheirownparty(whose
idealpointisexpectedtobeclosertotheowntaste),ifthereisone.InthesymmetricBNEequilibrium
oftheEntrystageeachcitizenfollowsacutpointstrategyasin 2 .Analogoustoexpression(3),the
optimalentrystrategyofacitizenwithidealpoint intheRightPartyistoenterifandonlyif(and
similarforacitizenintheLeftParty):
9 1
1
| ∈
,
1
1
1
| ∈ 1, … ,
1
1
1
2
,
1
1
1
2
1, … ,
| ∈
1
1
2
6 1
2
.
enteringfromeitherdirectionisdenotedby ,and
and
thenumberofentrantsfromtheLeftandRightPartyisdenotedby
, … ,100
1
2
, … ,100
Theex‐anteprobabilityofarandomcitizen
| ∈
2
,
,
,respectively,with
≡
. Also note that the probability a random citizen enters as Left Party candidate (or Right
Partycandidate)is /2sincethedistributionofidealpointsisuniform.Thewinprobabilityofthe
RightPartyisdenotedby
0
1/2
1
with
0
0,andtheexpectedpolicy
0
lossesintherespectivetermsaregivenby
,
| ∈
1, … ,
,
,
1
| ∈ 1, … ,
| ∈
, … ,100
|
1
1
|
100
1
/2
|
1
/2
|
|
100
; 8 |
100
; 7 . 9 ThefirsttermontheLHSof 6 isthesameasin 3 andrepresentsthecasewherenocitizen
enters.Thesecondtermgivesthecaseswhereatleastone alsoentersfromtheRightParty,butno
oneentersfromtheLeftParty.Intheseevents,citizen anticipatesgains /
avoidance,1/
,
| ∈
, … ,100 because one of the
randomlyappointedthenomineeofthisparty.Thethirdtermon
least one citizen
Right Party entrants is
6 givesthecaseswhereat
enters from the Left Party, but only citizen enters from the Right Party,
10 and,frompolicyloss
so
1andshesecurestheRightPartynomination.Duetosymmetry,eachofthe
1non‐
entrantswithidealpointsstrictlywithintheequilibriumcutpointpairpreferstheLeftorRightParty
nomineewithprobabilityone‐halfforeach,andvotesaccordingly(asaccountedforbytheindex of
the summation). Since citizen is in the Right Party, her expected net gains from entry are
and
,
| ∈ 1, … ,
(i.e., the policy loss avoided if the opponent nominee runs
unopposed). Note that declines with each Left Party entrant, who is expected to vote for this
party’snominee.ThefourthtermoftheLHSof 6 representsthecaseswhereatleastonecitizen
enters from each direction, which yields a mix of the second and third terms. Finally, our
experimentalparametersyieldinteriorequilibriumentrycutpointscharacterizedbysolutionsto(6),
atequality.
4.Experimentaldesign
The experiment was conducted at the Experimental Social Science Laboratory at Florida State
University.7Atotalof148studentsparticipatedineightsessionsof16or20participantseach,with
eachsessionlastingabout1.5hours.Earningswereexpressedinpointsandexchangedforcashfor
$1per250pointsattheendofasession.Participantsearnedonaverage$22.91,including$7for
showingup.
In a 222 treatment design, we varied the “entry cost” (
subjects and “group size” (
4 and 10) and “party mode” (
10 and 20 points) within
and Party) between
subjects.Eachsessionhadtwopartsof30decisionperiodseach,wheretheentrycostchangedfrom
oneparttothenextandthecostorderchangedacrosssessions.Participantsknewtherearetwo
parts,butwereinstructedaboutthesecondpartonlyaftercompletingthefirstone.Inalltreatments,
at the start of each period the subject pool was randomly divided into separate 4‐ or 10‐person
groupsthatdidnotinteractwithoneanotherinthisperiod,andeachparticipantreceivedanewideal
pointand,entirelyindependently,anewletterIDlabel.Theywereinformedthatidealpointsareiid
randomdrawsfromauniformdistributionovertheintegers 1, 2, … , 100 andprivateinformation
(i.e.,notshowntoothers),andthatletterIDsareiiddrawsfromauniformdistributionoverthewhole
alphabetandrevealedtoeveryoneinthegroup(albeittheparticipantbehindaletterIDremained
anonymous).Inagivengroupandperioddifferentindividualscouldhavethesameidealpointbut
neverthesameletterID.
Eachperiodconsistedoftwoconsecutivestageswheretheparticipantsindependentlyand
simultaneouslymadetheirdecisionswithoutcommunication.IntheEntrystage,eachgroupmember
7
The software was programmed as server/client applications in Java, using the experimental open source package
Multistage(http://multistage.ssel.caltech.edu:8000/multistage/).Torecruitparticipants,weusedORSEE(Greiner2015).
11 decidedonwhethertoenterthepoliticalcompetitionandpaycpoints,ornotenterandbearnocost.
IntheVotingstage,inNoPartytheletterIDforeach(independent)candidatewasdisplayedonthe
computer screen with a button labeled with her or his letter ID (a candidate’s own label was
highlightedinred).InParty,oneoftheentrantswithanidealpoint
∈ 1, … , 50 ( 51, … , 100 )was
randomly selected as the Left (Right) Party nominee, with equal probability for each, and a lone
entrantwasthenomineeoutright.Ifnobodyenteredfromaparty,thenthepartyhadnonominee.
EachnomineewasdisplayedonthecomputerscreenwithabuttonlabeledwithherorhisletterID
(anominee’sownlabelwashighlightedinred).8Thus,everyonewasinformedthatanominee’sideal
pointisfromtheleftorrightsubsetofidealpoints,buttheexactlocationwasnotrevealed.Next,each
participant voted by clicking one of the candidate or nominee buttons and could not abstain.9
Candidatesandnomineeswerenotforcedtovoteforthemselves.Thecandidateornomineewiththe
most votes was appointed the leader and received a bonus of
5 points, with ties broken
randomly.Ifnobodyentered,thenoneparticipantwasrandomlyandequiprobablyappointedthe
leader (and received
5 points but did not pay c). Either way, the leader’s ideal point was
implementedasthepolicyoutcome.Aftertheelection,everyonewasinformedaboutthenumberof
votes for each candidate or nominee, the leader’s letter ID, the policy outcome, the own period
earnings,andremindedwhethersheorheenteredandwasaleader(andthuspaidcandreceived
b).10Inaddition,thebottomofthescreencontainedahistorypanelwhereatanytimeparticipants
could view this information from all previous periods. Participants were paid for all 2
30
60
periods.Oneunpaidpracticeroundwasconductedtofamiliarizethemwiththeuserinterface.11
Table 1 summarizes our experimental design (first six columns) and quantitative BNE
predictions of the relevant observable variables (last five columns; denoted by an asterisk and
subscript e for expected values). Each treatment
∗
,
∗
, ,
has a unique symmetric cutpoint pair
that determines the individual entry probability, ∗ (and thus the expected number of
8Intheexperiment,aparticipant’sidealpointwastermed“yourbestoutcome.”InParty,welabeledleftandrightas“low
number” and “high number” and citizens and nominees as “low/high number members“ and “low/high number
candidates,“ respectively. Also, with two nominees the button of the Left Party nominee was always to the left of the
opponent’sbutton.InNoParty,“candidate”buttonswerecenteredandorderedrandomlyfromlefttoright,independent
ofletterIDs.Infact,inagroupandperioddifferentparticipantscouldseedifferentIDorderings.
9Sinceintheorynon‐nomineesinPartystrictlyprefervotingtoabstainingwhilenon‐candidatesinNoPartyareindifferent
betweenthetwooptions,wechosemandatoryvotingtokeeptheNoPartyandPartydesignsassimilaraspossible.Future
experimentscanexplorevoluntaryvoting.
10Whiletheleader’sexactidealpointwasalwaysrevealedbythepolicyoutcome,wedidnotdiscloseanyoneelse’sideal
point.However,inPartywithtwonomineestheirvotetalliesindicated,albeitsomewhatimperfectlyduetounexpected
votinginthelab,howmanyidealpointswerefromtheleftandrightdirection,respectively.Wechosetogiveparticipants
relativelylittlefeedbackinordertokeeptheexperimentaldesignclosertothetheory.
11Theonlinesupportingmaterialincludesinstructionsandsamplescreenshotsofthecomputerdisplay.Duetoaminor
programmingerrorthatwelearnedaboutonlyafterallthedatawascollected,theidealpoint100neveroccurred.Our
analysisofthedataassumesthatparticipantswereunawareofthis.
12 ∗
entrants,
∗
), for which we also compute the ex‐ante expected individual payoff,
/ ,where
∗
∗
100and ∗ denotestheexpectedpolicyloss.
Table1:ExperimentaldesignandsymmetricBNEpredictionsintheentrygame
n
4
4
4
4
10
10
10
10
Design
#Subjects
#
(Sessions) Elections
36(2)
270
32(2)
240
36(2)
270
32(2)
240
40(2)
120
40(2)
120
40(2)
120
40(2)
120
c Party
10
No
10
Yes
20
No
20
Yes
10
No
10
Yes
20
No
20
Yes
BNEpredictions
#Obs.
1,080
960
1,080
960
1,200
1,200
1,200
1,200
Cutpoints
[ ∗, ∗]
[42,59]
[34,67]
[20,81]
[17,84]
[21,80]
[14,87]
[10,91]
[08,93]
∗
0.84
0.68
0.40
0.34
0.42
0.28
0.20
0.16
∗
66.98
69.09
64.59
66.69
59.71
61.24
57.59
59.90
∗
25.87
25.36
28.66
27.76
36.59
36.46
38.91
37.40
∗
8.40
6.80
8.00
6.80
4.20
2.80
4.00
3.20
Note:Allsessionshadtwoparts,eachwithadifferententrycostfor30periods.Eachtreatmentusedaleader
bonusof
5pointsandauniformdistributionofidealpointsovertheintegers 1,2, … ,100 .Notethatthe
twoPartytreatmentswith
10pointshaveanadditionalBNEwithtwocutpointpairs(seefootnote17).
5.Hypotheses
ThefirsthypothesiscapturesthemostimportantpropertyoftheBNEintheEntrygame:
H1:PoliticalPolarization.Ineverytreatment,theentryratesareaweaklyincreasingfunctionof
thedistancebetweenidealpointsandthemedianofthepolicyspace.
The next four hypotheses specify the primary comparative statics derived based on BNE,
from directly comparable pairs of treatments that differ only with respect to one variable (as
comparedtosecondaryqualitativepredictionswheretreatmentsdifferintwoorthreevariables).
H2:PartyEffect.Holdingtheentrycostandgroupsizeconstant,expectedequilibriumentryislower
withparty‐mediatedelectionsthanwithoutparties.Thisimpliesfourspecifichypothesesintermsof
pairwisecomparisonsfor
∗
∗
∗
∗
4,10,
4,20,
10,10,
10,20,
∗
∗
, ,
(theeffectisthesamefor
4,10,
;
4,20,
;
∗
10,10,
;
∗
10,20,
.
, ,
):
H3:SizeEffect.Holdingtheentrycostandpartymodeconstant,inequilibrium,theprobabilityof
entry isdecreasingin .Thisgivesfourspecifichypothesesintheformofpairwisecomparisons
for :
a) Entryprobability:
∗
, 10,
∗
, 10,
;
13 ∗
, 20,
∗
∗
∗
, 20,
;
, 10,
∗
, 10,
;
, 20,
∗
, 20,
.
H4: Cost Effect. Holding the group size and party mode constant, expected equilibrium entry is
decreasingin ,whichimpliesfourhypothesesintermsofpairwisecomparisonsfor (theeffectis
):
thesamefor
∗
∗
∗
∗
4,
,
10,
4,
∗
,
∗
,
10,
4,
∗
10,
4,
∗
,
,
,
,
10,
;
;
;
,
.
H5: Welfare Effect. The hypotheses for equilibrium expected welfare, measured by expected
individualpayoffs,
,havethesamesignsasfor inH3(sizeeffect)andH4(costeffect),butthe
oppositesignsinH2(partyeffect).
a) Party:
∗
b) Groupsize:
4,10,
∗
4,20,
∗
10,10,
∗
10,20,
∗
∗
4,10,
;
∗
, 10,
∗
, 10,
;
4,20,
;
∗
, 20,
∗
, 20,
;
∗
10,10,
;
∗
∗
10,20,
.
∗
, 10,
∗
, 10,
;
, 20,
∗
, 20,
.
c) Entrycost:
∗
4,
∗
,
10,
∗
4,
∗
10,
∗
,
∗
,
4,
∗
4,
∗
,
,
10,
;
,
,
10,
;
;
,
.
As an even more stringent test of the equilibrium model, the BNE of the entry game also
generates predictions about the complete order of qualitative predictions across all treatments,
varyingallthetreatmentvariablessimultaneously.
H6:Entryrateordering.Inequilibrium,theorderingof acrossalltreatmentsis:
∗
4,10,
∗
∗
∗
4,10,
∗
10,10,
10,10,
10,20,
∗
H7:Welfareordering.Inequilibrium,theorderingof
∗
4,10,
∗
4,10,
∗
10,20,
∗
∗
10,10,
14 4,20,
∗
10,20,
4,20,
;
acrossalltreatmentsis:
∗
4,20,
∗
4,20,
∗
∗
10,20,
.
10,10,
6.Experimentalresults:AnexaminationofH1‐H7
This section presents and analyzes the aggregate data as it relates specifically to the seven
hypotheseslistedabove.Inthenextsection,wetakeadeeperlookattheindividualleveldata.
6.1Thepolarizationhypothesis(H1)
Thepolarizationhypothesisspecifiesthatmoreextremecitizensare(weakly)morelikelytoenteras
candidates.ThisisimpliedbytheBNEoftheentrygameforalltreatmentsintheexperiment.An
exactcomparisonofthedatatothetheoryclearlyrejectsBNE,whichmakesthesharppredictionthat
entryratesshouldbeeitherzerooronedependingonwhetheracitizen’sidealpointissufficiently
extreme.Ofcoursethedataisnotdiscontinuouslikethis.Therefore,wefitalogitregressionmodel
oftheprobabilityofentryasafunctionofthedistanceofanidealpointfromthemedian.Sincethis
isastrategicgameratherthanasimpleindividualchoice,sothatiftheplayers’entryfunctionsare
logitfunctionsratherthanstrictcutpointpairs,thisinturnchangesalloftheplayers’responsesin
thegame.Thus,weanalyzethedatausinglogitquantalresponseequilibriumofthegame,orlogit
QRE(McKelveyandPalfrey1995,1998;Goeree,Holt,andPalfrey2016).
QRE is a statistical generalization of NE that allows for decision‐making errors that are
systematic in the sense that more lucrative decisions are made more often than less lucrative
decisions. In the logit specification of QRE, the parameter
0 represents the slope of the logit
responsefunction,withlowervaluesindicatingaflatterresponse(“highererror”)andhighervalues
indicateasteeperresponse.If
0,decisionsarepurelyrandomsoeachcitizentype ∈ 1, … ,100 enterswithprobabilityone‐half.Therationalitylevelstrictlyrisesin until
∞,whereeveryone
is virtually fully rational and follows the BNE cutpoint strategy. In particular, each citizen type enterswithprobability
∈ 0,1 strictlyinbetweenzeroandone,whichdependssmoothlyon
theidealpointandisnolongeracutpointstrategythatdictatesa“zeroorone”binarychoiceforall
. This leads to a set of equilibrium conditions that are somewhat different from 2 to 9 (see
Appendix A). The QRE entry probabilities are computed by simultaneously solving one‐hundred
differentconditions,oneforeachpossible .12GiventheseQREentryprobabilitiesasafunctionofλ,
weestimateλbymaximumlikelihood.Toavoidoverfitting,theestimatedparameterisconstrained
tobeequalacrossalltreatments.13
12Notethat
101
inequilibriumduetoouruniformdistributionofidealpoints,soweneedonlysolveforfifty
conditions.Also,wecomputeQREentryprobabilitiesassumingnoerrorsintheVotingstagesinceunexpectedvotesare
quiterareinthelab(asdocumentedinthenextsection).
13WealsoestimatedtheQREmodelusinganout‐of‐sampleapproach.Ratherthanconstrainingλtobeconstantacross
treatments,weestimateditseparatelyforeachtreatmentusingonlytheodd‐numberedperiods,andthenusedthoseodd‐
periodestimatestocomparepredictedandactualvaluesfortheeven‐numberedperiods.Theconclusionsaresimilar,and
thein‐samplefitsforthatalternativeapproacharesignificantlybetter.
15 Table2showsforeachtreatmenttheobservedentryrate,
,andtherespectiveBNEand
QREentryrates(columns4‐6),whereQREentryratesareevaluatedattheestimatedvalueof .The
theoreticalpredictionsareexact,inthesensethattheyarebasedontheactualdrawsofidealpoints
realizedintheexperiment(i.e.,empiricaldistribution),andhenceareindicatedbysubscriptemp,
whilestillassumingthatcitizensrespondtothetheoreticaluniformdistribution.14Importantly,the
complete order of qualitative predictions across all treatments is preserved when changing to
empirical BNE and QRE. The observed rates of entry are averaged over all periods and QRE
predictions use
0.083, the maximum likelihood estimate for all periods and treatments
combined.Furthermore,thescatterplotinFigure1depictsforeachtreatmenttheBNEentryrate
∗
on the horizontal axis against the average observed rate
(markers) and QRE rate
(markerslinkedbydottedline)ontheverticalaxis.
Table2:Entry‐Predictionsanddata
n
4
4
4
4
10
10
10
10
c Party
10
10
20
20
10
10
20
20
No
Yes
No
Yes
No
Yes
No
Yes
.687
.673
.560
.496
.519
.445
.426
.321
∗
.844
.671
.417
.364
.426
.256
.181
.152
.602
.570
.459
.436
.465
.423
.330
.302
Note:Standarderrorsof
areallintherange .013, .016 .BNEisindicatedbyanasteriskandQREby ,the
maximumlikelihoodestimateofthedegreeoferror.
AninterestingpatterninthedatathatisclearlyseeninFigure1isthat,relativetoBNEwe
find over‐entry for the treatments where
∗
where
∗
1/2 and (weak) under‐entry for the treatments
1/2. This is not just a coincidence, but is a general property of regular QRE in these
games. The independent random noise in QRE flattens out the treatment response in entry rates
comparedwithBNE,bypullingtheratesawayfromBNEinthedirectionof
1/2(seeGoereeand
Holt2005;Goeree,HoltandPalfrey2016).
14Forexample,inNoPartytheempiricalBNEentryrateinatreatmentiscomputedbydividingthenumberofobserved
idealpointsatormoreextremethanthetwotheoreticalBNEcutpointsbythetotalnumberofobservedidealpoints.And,
theempiricalQREentryrateinatreatmentistheaverageofalltheoreticalQREratesper idealpoint,weightedbythe
respectiverelativefrequenciesofobservedidealpoints.
16 Figure1:Entryrates–Predictionsanddata
Note:ThedatamarkersandempiricalQRE
0.083 entryratesuseallperiods.
Figure2displaysforeachtreatmenttheobservedandempiricalQREentryratesperblockof
tenidealpoints(solidlines),theBNEcutpointpair(crossmarkersatthetop),andthetheoretical
QRE entry rate function (dashed lines) at the estimated value of .15 With pure noise,
dashedlinewouldbeahorizontalthrough
0.5andinBNE,
0, the
∞,itwouldbeastepfunction
withentryratesequaltooneforallidealpointsatormoreextremethanthetwocutpoints(cross
markers)andequaltozeroforallidealpointsstrictlywithinbothcutpoints.16(seeappendix).Not
surprisingly, the sharp BNE cutpoint pairs are not found in the aggregate data.17 Instead, in all
treatmentstheentrycurvesareU‐shaped,whichisageneralpropertyofQREinthesegames.18
15FigureB1inAppendixBdisplaystheaverageentryratesfornormalizedidealpoints,thatis,thedifferencebetweenown
idealpointandthecloserofthetwoBNEcutpoints.
16FigureA1inAppendixAdisplaystheQREentryprobabilitiesforvariousdegreesoferrorfortheNoParty,n=4,c=20
treatment.Theseentryprobabilitiesforothertreatmentsvaryasexpected,butwithsimilaroverallshapes.
17Wereportananalysisofindividualcutpointsinthenextsection.Behaviorinvoterturnoutexperimentsissomewhat
moresimilartoBNEcutpointsthaninourstudy(e.g.,LevineandPalfrey2007).Morenoiseindecisionsheremayarise
frommorecomplexityinthecitizencandidategameandlittlefeedbackaboutotherparticipants’idealpoints(seefootnote
10),whichhamperslearningtowardsBNE.
18NoticethesmallhillsinQREaroundthemedianinParty(lowertwopanelsinFigure2)whereindividualshaveastronger
incentivetoenterthantheirsomewhatmoreextremeneighbors.Forsufficientlylowentrycosts,thisleadstoanadditional
BNEwithtwocutpointpairs.Thefirst“outer”pairdictatesthatallcitizenswithidealpointsatormoreextremethanthese
cutpointsenter,andthesecond,narrower“inner”pairaroundthemediandictatesthatallcitizenswithidealpointsator
withinthispairenter.ThetwoPartytreatmentswithalowentrycosthavesuchanadditionalBNE.Notethatweobserve
verysimilargeneralentrypatternsinallourtreatments.
17 Figure2:Entryratesperidealpoint(dataaveragedinblocksoften)–Predictionsanddata
18 ThestatisticalsignificanceoftheU‐shapeofentryratesissupportedbyalogitregression
(clusteredbyindividuals;seeTable3)ofentrydecisionsonextremenessofacitizen’sidealpoint,
measuredby
/49,where
,
∈ 50,51 dependingonwhichiscloserto
,
(we
chosenotthe“true”medianof50.5tonormalizethecoefficientbydividingbythemaximumdistance
of 49
50
1or100
51).The regressionalsocontrols forthethreetreatmentvariablesanda
measureofexperience(firstfifteenperiodsversuslastfifteenperiodsineachpart).Thecoefficient
of
/49ispositiveandhighlysignificantsothemoreextremetheownidealpoint,the
,
morelikleyaparticipantistorunforoffice.ThisprovidesstrongsupportforH1.Wenextturntothe
hypothesesaboutspecifictreatmenteffects.
Table3:Random‐effectslogitregression(alldata)
Dependentdummyvariable:Entrydecision(1if
Constant
Coeff.&Const. 0.637***
(Std.error)
(0.211)
,
1)
,
49
1.242***
(0.087)
Dummyindependentvariables
Entrycost Groupsize Party
(1if
20)
(1if
10)
(1ifParty)
‐0.714*** ‐1.054*** ‐0.425*
(0.051)
(0.231)
(0.230)
Blockof15
periods(1if2nd)
‐0.028
(0.050)
Note:*(**;***)indidcatesaone‐tailed5%(1%,0.1%)significancelevel.Thedataisclusteredattheindvidual
level.
6.2.Treatmenteffectsonentryratesandwelfare(H2‐H5)
Entryrates(p)
AsclearlyseeninTable2andFigure1,alltwelvepredictedprimarytreatmenteffectsonentryrates
findsupportinthedata.Forthepartyeffect(H2),holdingconstantthegroupsizeandentrycost,the
observed entry rates are always greater in No Party than Party. For the size effect (H3), holding
19 constanttheentrycostandpartymode,theyarealwaysgreaterwith
4than10.And,forthecost
effect(H4),holdingconstantthegroupsizeandpartymode,theyarealwaysgreaterwith
10than
20points.TheresultsofthelogitregressionreportedinTable3supportthesetreatmenteffects:the
coefficientsofParty,Groupsize,andEntrycostareallnegativeandstatisticallysignificant.Whilethe
regressionusesallthedata,thesameresults occurwhenonly therespectivesessionsofprimary
treatment comparisons are employed, except for the Party dummy with
4 and
10 (the
coefficientisinsignificant).Overall,ourexperimentprovidesstrongevidenceinfavorofH2toH4
withrespecttoentrydecisions.Finally,whetherentrydecisionsaremadeinthefirstorsecond15
periodsinatreatmentmakesnodifference(i.e.,thecoefficientofBlockof15periodsisinsignficant).19
Welfare(
)
We next turn to H5, which addresses the comparative statics predictions of aggregate welfare, as
measuredbyaveragepayoffs.Table4givespertreatmentobservedandpredicted(usingempirical
idealpointdistributions)averageindividualpayoffs.Notethatallprimaryqualitativepredictionsof
payoffsareidenticalforBNEandQRE
,independentofwhethertheyaretheoreticalorempirical
predictions(seeappendix).Forthewelfareeffect(H5),alltwelveprimaryqualitativepredictionsfind
support in the data. In terms of quantitative comparisons with the equilibrium predictions, in all
treatmentstheactualaveragepayoff
isgreaterthaninBNEandweaklysmallerthaninQRE
,
butthedataandQREpredictionstendtobemuchclosertooneanother.20
Table4:Averageindividualpayoffs–Predictionsanddata
n
4
4
4
4
10
10
10
10
c
10
10
20
20
10
10
20
20
Party
No
Yes
No
Yes
No
Yes
No
Yes
69.26
70.42
64.22
68.03
64.44
68.31
62.61
64.35
Note:Standarderrorsfor
∗
67.00
69.06
64.00
66.56
60.28
63.98
59.25
61.89
70.67
72.14
66.95
68.73
65.01
68.31
63.24
65.91
areintherange .71, .82 .
Next,Table5givestheresultsofanOLSregression,clusteredbyindividualsandpoolingall
data,withtheindividualpayoffinperiodtasthedependentvariableandthesamefiveindependent
19Wefindnosignificantlearningeffectsinthedatawhenusingotherspecificationsoftime.
20RecallthattheQREwasestimatedtofitthechoiceprobabilitiesoftheparticipants,notbyfittingtheexpectedpayoffsin
thegame.TableB3inAppendixBshowsaveragepayoffs,brokendownbypolicylosses,entryexpenses,andthespoilsof
office.TableB4showstheobservedvaluesforleadersandnon‐leaders,respectively.
20 variablesasinTable3.Ascanbeseen,allofthepredictedeffectsarehighlysignificantandlargein
magnitude.And,asinthelogitregressionforentryrates,thereisnoevidenceoflearning.
Table5:Random‐effectsOLSregressions(alldata)
Dependentvariable:Individualperiodpayoff
|
Constant
Coeff.&Const. 75.44***
(Std.error)
(0.76)
Dummyindependentvariables
| Entrycost
.
49
‐14.14***
(0.91)
(1if
20)
‐3.24***
(0.53)
Groupsize
Party
(1if
(1ifParty)
Blockof15
periods(1if2nd)
2.59***
(0.53)
‐0.03
(0.53)
10)
‐3.18***
(0.54)
Note:*(**;***)indicatesaone‐tailed5%(1%,0.1%)significancelevel.Thedataisclusteredattheindvidual
level.
Thereasonforthewelfaregainsfromparty‐organizedelectionsisthat,comparedtoNoParty,
majoritycandidatesinPartywinmoreoftenonaveragesinceiftherearetwonominees,eachcitizen
votesfortheonelocatedinthesamedirectionasherself(belowweshowthatparticipantsmostly
voteinthisway).Thatis,partylabelsprovidevaluableinformationtoallthecitizenssotheoutcome
morecloselyreflectsthetruedistributionofpreferences.Figure3indicatesthatforboth
4and
10(leftandrightpanel,respectively)themajoritywinsindeedsubstantiallymoreofteninPartythan
No Party in situations where it might also lose.21 The only exception are majorities of nine
participants,whichalwaysprovidedtheleaderinbothpartymodes.
Figure3:Actualwinproportionandvotecoordinationadvantageofmajorityparties
21FigureB2inAppendixBdisplaysthefrequencydistributionsofthenumberofparticipantsperdirection.
21 6.3Completeorderingofentryratesandwelfareacrosstreatments(H6andH7)
As noted earlier, because BNE produces quantitative predictions about entry and welfare for any
parameter configuration, it also generates hypotheses about comparisons across treatments that
differintwoorthree of thetreatmentvariables.Infact,asstatedinH6 andH7,BNEgeneratesa
completestrictorderovertheeighttreatmentswithrespecttobothentryratesandwelfare.
Forentryrates,thisismostclearlyseeninFigure2bytheleft‐rightorderingofthelabeled
datapointsforeachtreatment:withonlyoneexception,thedatamarkersareincreasingintheBNE
entryrate.Infact,outofall28possiblequalitativecomparisonsonlyonehasanunpredictedsign,
namely
4,20,
10,10,
,forwhichthepredictions
∗
0.417and
0.426areveryclosetooneanother.ThisprovidesstrongevidenceinfavorofH6.Itisalsoworth
mentioningthatwhiletheBNEandQRE
rates, except for
,
4,
modelsgeneratethesametreatmentorderingofentry
10 the latter predictions are always nearer to the data.
Interestingly, relative to BNE we find over‐entry if
∗
0.5 and (weak) under‐entry if
∗
0.5, a
patternthatwasalreadyreportedinvariousotherbinarychoice,entryexperimentsandexplained
usingQRE(GoereeandHolt2005).Asnotedbefore,thelogitQREmodelalsogeneratesthisentry
pattern.However,itisalsothecasethattheobservedentryratesareshifteduprelativetotheQRE
fittedestimates.Finally,thecompleteorderofexpectedwelfare,asmeasuredbyaverageindividual
payoffsgiveninTable4,isalsomostlyconsistentwiththeBNEandQRE
predictions(notethat
thetwomodelspredictsomewhatdifferentorders).Outof28possiblequalitativecomparisons,24
and25arecorrect,respectively.Thisincludesalltwelveoftheone‐variabletreatmentcomparisons
discussed in the last section, and twelve and thirteen out of sixteen of the comparisons between
treatmentsthatdifferedinmorethanonedimension.Thus,thedataprovidesomesupportforH7,
butweakerthanthesolidsupportthatwefindforH6.
7.Experimentalresults:Individualbehavior
7.1Votingbehavior
The predicted voting behavior in BNE is quite simple: 1 each candidate in No Party and each
nomineeinPartyvotesforherself; 2 withtwonomineeseachnon‐nomineevotesfortheonewhose
idealpointisfromherownsubsetofidealpoints,leftorright.Welabelvotingdecisionsthatare
inconsistentwiththesepredictionsasunexpected.22Table6showstheobservedaverageindividual
rateofunexpectedvotingforeachtreatment.Therateofeachparticipantisequallyweightedand
22Ofcourse,unexpectedvotesneveroccurfornon‐candidatesinNoParty(whoarepredictedtovoterandomly)norfor
electionswithzerooroneentrant,sothesesituationsareexcludedfromtheanalysis.
22 computedbydividinghernumberofunexpectedvotesbythenumberofcasessheorheisacandidate
respectivelynomineeornon‐nominee.Ascanbeseen,candidatesandnomineesdoindeedmostly
voteforthemselves.Specifically,inNoParty(Party)unexpectedvotesbycandidatesornomineesare
observedonly0.8to4.2(0to3.5)percentofthetime.Similarly,non‐nomineesinPartyrarelycast
unexpectedvotes(only2.0to6.5percentofthetime).Thus,overallvotingbehaviorisverycloseto
BNE.
Table6:Observedunexpectedvotes
n
4
4
10
10
4
4
10
10
c Party
10
No
20
No
10
No
20
No
10
Yes
20
Yes
10
Yes
20
Yes
Averageindividualratesof
unexpectedvotes(std.errors)
Candidates/
Nominees
Non‐nominees
.042(.017)
‐
.026(.015)
‐
.025(.014)
‐
.008(.004)
‐
.035(.025)
.052(.020)
.007(.007)
.020(.011)
.000(.000)
.057(.018)
.000(.000)
.065(.021)
Note:Electionswithzerooroneentrantareexcluded.Non‐nominee’sratesareforthePartytreatmentsonly.
Standarderrorsarecomputedusingthedifferencesineachindividual’srateandtheaverageindividualrate.
Figure5displaysthedistributionofthenumberofunexpectedvotesacrossallparticipants,
with the number of such votes on the horizontal axis (from zero to the maximum observed of
eighteen)andtherespectivefractionofindividualsontheverticalaxis.Thefigurealsoseparatesout
the observations for independent candidates, nominees, and non‐nominees as they face different
decision tasks.23 The diagonal axis shows combinations of party mode and group size, with data
pooledforbothentrycosts.Thefigureindicatesthatonlyveryfewparticipantsvotedunexpectedly.
Specifically,77.8and82.5percentoftheindependentcandidatesin4‐and10‐persongroupsalways
votedaspredicted,andthesenumbersare87.5and100percentfornomineesand71.9and57.5
percentfornon‐nominees,respectively.Andoftheparticipantswhocastatleastoneanomalousvote,
manydidsojustonceorlittlemorethanthis.Hence,thefewdeviationsfromequilibriumvotingare
duetothebehaviorofonlyahandfuloftheparticipantsintheexperiment.Forexample,thethree
largestindividualcountsofunexpectedvotesareseventeenbyacandidatein
,
4 and
23 Compared to nominees in Party, there can be more than two contenders to choose from by candidates in No Party,
includingthemselves.And,non‐nomineesinPartymustrealizethattheirexpectedpayoffisgreateriftheyvoteforthe
contenderwhoseidealpointisfromthesamedirectionastheownone,whilenomineessimplyvoteforthemselves.
23 thirteen and eighteen by two non‐nominees in
,
10 , where the latter of them never
entered.
Figure5:Numberofunexpectedvotesperindividual
Note:Thefractionsareshownperindependentcandidates,nominees,andnon‐nomineesandarepooledfor
bothentrycosttreatments,sothefigureconnectsonlythepartymodeandgroupsize.
Next, we analyze whether the observed rates of unexpected voting depend on the ideal
point.24Figure6showstheratesontheverticalaxisandtheabsolutedistanceintheownidealpoint
andthecloser“median”idealpoint,50or51,onthehorizontalaxis.Thus,atzeroonthehorizontal
axisbothidealpointscoincideandat49thedistancebetweenthemismaximal.Thefigureshowsthe
data (lines with spikes) and respective logarithmic trend lines for candidates and nominees
combined and for non‐nominees (thick black and gray lines, respectively), and also separate
logarithmictrendsforcandidatesandnominees(dashedblacklines)andnon‐nomineeswhodidand
did not enter (dashed gray lines). As can be seen, unexpected voting of candidates and nominees
doesn’t depend on the own ideal point (Spearman’s
0.037and 0.034 for each role, respectively;
differenceisinsignificant,
0.018 for both roles combined and
0.799) and is higher for candidates (but the
0.115,individuallevelone‐tailedWilcoxon‐Mann‐Whitneytest,fifteen
andfourparticipantsperrole).
24Thefollowinganalysispoolsalldataandutilizesonlyparticipantswhocastatleastoneunexpectedvoteintheentire
session.Usinginsteadallparticipantsinthenonparametrictestsyieldsmanytiesanddecreasesthep‐values,exceptfor
twoincreaseswheretheresultsarestatisticallysignificantwhetherornotallindividualsareconsidered.
24 Figure6:Unexpectedvotingperabsolutedistanceinownandmedianidealpoints
Note:Thefiguredepictsunexpectedvotingratesperabsolutedistanceinownandmedianidealpointsfornon‐
nomineesandforcandidatesandnomineescombined.Atzero,theownidealpointis50or51,andat49(50
1or100 51)thedistanceismaximal.Thedashedlinesshowtherespectivelogarithmictrends.
Bycontrast,weobserveanegativeassociationbetweenunexpectedvotingandtheabsolute
distanceintheownandmedianidealpointsfornon‐nominees(
0.352 for entrants and non‐entrants;
0.424overall,and 0.363and
0.012). In Party, note that anomalous voting of non‐
nomineestendstobesmallerwhentheyentered(
0.006,one‐tailedWilcoxonsigned‐rankstest,
25individuals)andespeciallyhighforidealpointswithinabouttenpointsofthemedian.Also,the
rates are always greater in the non‐nominee than nominee role (
0.003, same test, 26
individuals).Overall,ourresultsindicatethatamongthosewhovoteunexpectedly,candidatesand
nominees make “plain” errors while for non‐nominees models that incorporate the pecuniary
consequencesoferroneousvotingandbeliefsaboutnomineeidealpointsseemmoresuitable.25This
also makes sense, since the expected payoff‐maximizing vote is more obvious for candidates and
nominees than for non‐nominees,26 which is also supported by learning towards BNE voting of
25Forexample,QRE(McKelveyandPalfrey1995,1998)allowsfordecision‐makingerrorsandthusunexpectedvoting.It
alsopredictsthatsuchvotesgetmorefrequenttheneareranidealpointisto50or51,astheexpectedpolicylossifsomeone
elseiselecteddecreasestowardsthemedian.
26Table6suggeststwomorepatternsofaverageindividualratesofunexpectedvotingforprimarycomparisons.First,the
ratesarealwaysweaklygreaterwithalowerthanlargerentrycostforcandidatesandnominees(
0.035,one‐tailed
Wilcoxonsigned‐rankstestforbothrolescombined,eighteenindividuals),butnopatternisseenfornon‐nominees(albeit,
0.074infavorofgreaterrateswithalowercost,sametest,26individuals).Second,theratesarealwaysgreaterin
smallerthanlargergroupsforcandidatesandnominees(
0.029,one‐tailedWilcoxon‐Mann‐Whitneytest,twelveand
sevenindividualsin4‐and10‐persongroups,respectively),andthereverseisseenfornon‐nominees(butthedifference
25 participantsinthelatterrole:whileunexpectedvotingdoesn’tdependontheperiodforcandidates
andnomineescombinedusingall60periodsorthefirstandlast30periodsonly(Spearman’s
0.112, 0.247,and 0.029;
0.188),itdoessonegativelyfornon‐nomineesforall60andlast
30periods(
0.312and 0.378;
(
0.402).
0.159,
0.093and0.003,respectively)butnotthefirst30periods
7.2.Individualentrybehavior
Here we present individual level data of entry behavior. Figure 7 depicts cumulative
frequency distributions of actual average individual entry rates for
and
,
20,
10 ,whichhavewith
∗
,
10,
4 0.844and0.152themostextremeBNEentry
Figure7:Cumulativedistributionofindividualentryrates(forlowestandhighestBNEentry)
probabilities. The distributions of the six other treatments tend to fall within these two
distributions.27Clearly,thereismarkedheterogeneityinentryratesamongparticipants.Further,the
50percenthorizontallineintersectsthetwodistributionsintheexpectedorder,butmoretotheleft
andrightrelativeto
∗
0.844and0.152,respectively.ThisisconsistentwithQRE,whichpulls
theentryratesawayfromBNEtowards1/2.
ThescatterplotinFigure8shows,foreachparticipant,theaverageentryratewith
10
and 20 points on the horizontal and vertical axis, respectively. Each marker represents one
is not significant,
0.241,same test, nine and seventeen individuals). Finally, for the two possible within‐subject
comparisons,unexpectedvotingisneitherassociatedbetweenbothentrycostsforcandidatesandnomineescombinedand
fornon‐nomineesnorbetweenthenomineeandnon‐nomineerolesinParty(Spearman’s
0.026,0.105,and0.058;
0.611).Duetofewunexpectedvotesbyfewindividuals,allthesefindingsarehardtointerpretaswewouldneedto
controlfor,say,idealpointsandexpectedpayoffs.
27SeeFigureB3inAppendixB.Thecumulativedistributionsofentryratesof
,
10,
4 and
,
10,
4 intersectonce.
26 individual,wheredifferentsymbolsindicatedifferentcombinationsofthepartymodeandgroupsize
(a few markers are somewhat magnified in proportion to the number of individuals at that same
coordinate).Asexpected,independentofpartymodeandgroupsize,mostindividualsentermore
often when it costs less (i.e., have markers below the diagonal; one‐tailed Wilcoxon signed ranks
tests,
0.001foreachpartymodeandgroupsizecombination).Specifically,only28outofall148
participantsenteredmoreoftenwithalargercost,andmostofthemarefoundclosetothediagonal.
Also, fourteen participants have the same entry rates with both costs (i.e., with markers on the
diagonal),ofwhomoneneverenteredandsixalwaysentered.
Figure8:Entrycosteffect
Note:Eachparticipant’saverageentryratewith
10and20pointsisshownonthehorizontalandvertical
axis,respectively.Eachmarkerrepresentsoneparticipant,whereafewmarkersaresomewhatmagnifiedin
proportiontothenumberofindividualsatthatcoordinate.
Next,weexploretheextenttowhichobservedentrydecisionsareconsistentwithcutpoint
strategies,whicharegenerallyoptimalbestresponsesinthisgame.Specifically,foreachparticipant
and treatment we estimate a cutpoint pair as follows, assuming that individuals use such a
decisionrule.Foreachparticipantandtreatment,wehave
anidealpointandentrydecision,
101
,
,
,
.Fixingacutpointpair
,
,
,with1
50and
duetosymmetry,observation intreatment ismarkedconsistentwiththiscutpoint
pairif
27 30periodsorobservationalpairsof
,
,
,
,
,
⋁
and
,
,
,
0,or
and
,
1,
andmarkedaserrorotherwise.28And,asanestimatorofparticipant ’scutpointpairwechoosethe
onethatminimizesthetotalnumberoferrors,andiftherearemoresuchpairswetaketheaverage
ofthem.Usingthisprocedure,wecomputethedistributionofindividualclassificationerrorratesand
cumulativefrequencydistributionsofestimatedindividualcutpointpairs.
Figure9:Distributionofclassificationerrorrates
Note:Individualerrorratesarepooledforbothentrycosts.
Figure9depictstheoveralldistributionofindividualerrorrates,whicharepooledforboth
entrycosts.Forabout25(50;75)percentoftheparticipantstheerrorrateis
0.1(0.2;0.3),and
onlythreepercenthaveerrorratesof0.4orhigherbutnonereachesthe0.5.Figure10showsthe
cumulative distributions of estimated individual cutpoint pairs for
,
20,
,
10,
4 and
10 , which have the most moderate and most extreme BNE cutpoint pairs of
42, 59 and 8, 93 ,respectively.Duetosymmetryweonlyshowtheleftcutpoints,superimposing
the data from both directions.29 There is marked heterogeneity among the estimated individual
cutpointpairs.Finally,the50percenthorizontallineandtwodistributionsintersectintheexpected
order,andclosetotheeightandsomewhatclosertothemedianthanthe42predicted,respectively.
28Acitizen’sdiscreteidealpointmatchesacutpointwithstrictlypositiveprobabilityand,forourparameters,inequilibrium
shecanraiseherexpectedpayoffbyentering.Bycontrast,GroßerandPalfrey(2014)usecontinuoustypessoacitizen
locatedatacutpoint,whichalmostsurelyneveroccurs,isindifferentbetweenenteringandnotenteringandassumedto
enter.
29Thedistributionsoftheothersixtreatmentsfallwithinthesetwodistributions.SeeFigure4BinAppendixB.
28 Figure10:Cumulativedistributionsofindividualcutpointpairs(forlowestandhighestBNEentry)
Note:Duetosymmetry,weonlypresenttheleftcutpoints,superimposingthedatafrombothdirections.
Table 7 gives the fraction of average individual positive differences in the estimated “left
cutpointwith
10pointsminusleftcutpointwith
20points.”Thefractionrangesfrom0.58to
0.78,comparedto1inBNE,andaveragedifferencesrangefrom4.27to8.64(inbrackets).
Table7:Fraction(average)ofpositiveindividualcutpointdifferences
20
Leftcutpoint
10leftcutpoint
NoParty
Party
4
0.61(8.64)
0.60(6.66)
10
0.78(8.77)
0.58(4.27)
Note:Partywith
10hastwoindividualswithzerodifference,andeachoftheotherthreecombinationsof
thepartymodeandgroupsizehasoneindividualwithzerodifference.
Overall,participantsdonotfollowsharpcutpointstrategies.Whilethisisinconsistentwith
optimizingbehaviorandwithBNE,itisverymuchinlinewiththebehavioraltheorybehindregular
QRE. Participants in this experiment do not always optimize, but generally choose better entry
actionsmoreoftenthanworseones.Thisisalsoverymuchinlinewithresultsfromcutpointanalysis
inbinarychoiceturnoutgames(LevineandPalfrey2007),wherethevote/abstainchoiceissimilar
innaturetoenter/notenter.
8.Conclusions
Thispaperreportsalaboratorystudyofacitizen‐candidateentrygamewithincompleteinformation
about the ideal points of citizens and candidates. Ideal points are privately observed iid random
draws from a uniform distribution over the set of feasible common policies. Without ideological
politicalparties,citizenshavenoextrainformationabouttheidealpointsofindependentcandidates
at the time of voting. By contrast, with parties they learn whether a party nominee’s ideal point
belongstotheleftorrighthalfofthecommonpolicyset,butnotherexactidealpoint.Thestudy
29 comparesbothpartymodesin4‐or10‐persongroupsandwithtwodifferententrycosts.Intheentry
game, symmetric BNE makes sharp and mostly unique predictions of cutpoint pairs. That is, in
equilibriumeachcitizenwithanidealpointatormoreextremethanaleftorrightcutpointrunsfor
office,whileeveryonewithanidealpointstrictlyinbetweenthetwocutpointsdoesn’trun.Thus,the
modelpredictspoliticalpolarizationinthesensethattheidealpointsofpoliticiansaremoreextreme
thanthoseinthegeneralpolity.Finally,thecleardistributionalBNEentrypredictions,fromwhich
wealsoderiveimplicationsaboutwelfare,havetheadvantageofbeingstraightforwardtotestinthe
laboratory.
Themainexperimentalresultscanbesummarizedasfollows.First,allprimarycomparative
staticspredictionsofentryratesandeconomicwelfarearesupportedbythedata.Mostimportantly,
inefficient political polarization arises in all treatments. Second, participants appear to follow
cutpointstrategieswithsomeerror,andthedistributionofestimatedcutpointsindicatesignificant
heterogeneity.Consequently,insteadofstepfunctions,actualentryratesareU‐shapedfunctionsof
theidealpointsinalltreatments,withover‐entrywhentheBNEentryprobabilityissmallerthan50
percent and (weak) under‐entry when it is greater than 50 percent. Because participants with
moderateidealpointssometimesenterandwin,weobservelesspoliticalpolarizationandthuson
averageasmallertotalpolicylossandgreatereconomicwelfarethanpredicted(withover‐entry,the
greater total expense is exceeded by the smaller policy loss). The primary comparative static
predictionsoflogitQREareallsupportedinthedata,astheyarethesameasthoseofBNE,butin
addition QRE tracks the levels and patterns of entry and welfare much better. Third, ideological
partiesleadtomorepolarization,butatthesametimealleviatesomeoftheinefficienciescausedby
extreme policies because knowledge of a nominee’s party affiliation enables implicit vote
coordinationinfavorofthemajority,whichismorelikelytowinthanintheabsenceofparties.
Overall,thisstudyshowsempiricallythatincompleteinformationinelectionscanindeedlead
toinefficientpoliticalpolarizationthroughtheinformationaleffectsonentry.Inordertocheckthe
robustnessofourfindings,futureresearchcouldforexampleexaminevariousdifferentdistributions
ofidealpoints(e.g.,asymmetricdistributions)anddefaultpolicies.Anotherinterestingdirectionis
to compare different voting systems (e.g., Bol, Dellis, and Oak, forthcoming) and to study more
explicitlytheformationofpartiesandhowtheyselecttheirnominees,suchasviaprimaries(e.g.,
Hansen2014).Wehopethatthesefindingsmayinspirefurthertheoreticalandempiricalworkto
shed more light on the non‐trivial, realistic political processes such as those explored in citizen‐
candidateentrygames.
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33 AppendixA:TheoreticalDerivations
Bestresponseentrystrategies
Considerourcitizen‐candidateentrygamewithoutpartiesandwithadiscrete,uniformcumulative
probabilityfunctionofidealpoints
citizens
/100, ∈ 1,2, … ,100 anddensity
1/100.Ifall
areusingcutpointstrategy(cf. 2 inthemaintext)
̌
0
∈
1, … ,
1
∈ 1, … ,
1
, … ,100 ,
∪
thentheexpectedpayoffofacitizentype forenteringthepoliticalcompetition, ̌
| , ̌
1
1
1
1
andherexpectedpayofffromnotentering, ̌
| , ̌
0
1,isgivenby
1
1
1
1
1
1
,
| ∉
1, … ,
1 1
2
0,by
,
| ∈
1
1, … ,
| ∉
,
,
1, … ,
3
1 .
Theexpectedpolicylossterms
,
text,respectively.Relating
2 andrearrangingyieldsthebestresponseentrystrategy 3 1 and
and
1
arespecifiedin 4 and 5 inthemain
inthemaintext.
Withparties,ifallcitizens
areusingcutpointstrategy
1 ,thentheexpectedpayoffof
a citizen type in the Right Party for entering the political competition, ̌
1, is given by (and
analogousforacitizenintheLeftParty)
| , ̌
1
1
1
1
1
1
2
1
2
1
1
1
,
| ∈ 1, … ,
34 ,
1
2
| ∈
, … ,100
4 1
1
2
,
| ∈ 1, … ,
1
2
1
0
1
1
1
1
1
2
1
1
2
,
1
1
,
| ∈
,
| ∈
,
, … ,100
1
5 , … ,100 1
2
, … ,100
.
1
2
| ∈ 1, … ,
Thethreeexpectedpolicylossterms
| ∈
,
1, … ,
| ∈ 1, … ,
2
1
0,isgivenby
andherexpectedpayofffromnotentering, ̌
| , ̌
1
2
1
| ∈
,
, . withpartiesarespecifiedin 7 to 9 andthewin
probabilityoftheRightParty, ,isdescribedonp.10inthemaintext.Then,relatingbothconditions
andrearrangingyieldsthebestresponseentrystrategy 6 inthemaintext.
Averageexpectedwelfare
Without parties, the ex‐ante (i.e., before citizen ideal points are randomly drawn), the average or
∑
expectedindividualpayoff
,
|
isgivenby
1
1
1
,
|
|
1
1 ∙
1
2
|
1
|
1
|
|
|
andwithpartiesitisgivenby
,
1
∙
1
|
1
35 |
7 6 2
1
2
∙
1
2
1
|
50
|
|
|
|
where
2
|
50
|
50
1
|
1
1
2
1
|
1
|
|
|
1
|
|
|
1
|
|
|
|
50
|
,
istheLeftParty’sprobabilityofwinningand
0
1/2
1
0
0.
0
QREentryconditions
Here, we derive the logit QRE conditions of entry probabilities, allowing for erroneous binary
decisionsateachidealpoint ∈ 1,2, … 100 .If
0,theneachcitizentype makespurelyrandom
decisions(i.e.,enterswithprobabilityone‐half).If
∞,theneveryonefollowstheBNEcutpoint
strategy.Sincethereareone‐hundreddifferentcitizentypes ,wemustsimultaneouslysolveone‐
hundredconditions.Withoutparties,theQREconditionforacitizentype isgivenby
1
,
1
TheLHSgivesherentryprobability
andonRHS
.
8 denotesthevectorofentryprobabilitiesofall
feasibletypes ∈ 1,2, … ,100 thateveryothercitizen
maypossess.Sinceidealpointsareiid
random draws from a uniform distribution, each occurs with probability 1/100 so the average
entryprobabilityofeachothercitizen isgivenby
1
100
36 .
9 Further,RHScontainscitizen ’sexpectednetpayofffromentering(cf. 3 inthemaintext),
whichisgivenby
,
1
1
| ∈ 1, … ,100
,
1
1
1
1
| ∈ 1, … ,100
,
10 ,
with
,
1
| ∈ 1, … ,100
|
1
1
|
11 ,
12 100
and
,
wheretheexpectedpolicylosses
| ∈ 1, … ,100
11 and
|
1
|
100
12 accountforallfeasibleidealpointsofothers(i.e.,
notjustthoseofthemoreextremeentrantsasdictatedbytheBNEcutpointpair).
Then, for a given the one‐hundred equilibrium conditions of the form
simultaneouslysolvedfor
8 are
1, … ,100todeterminetheQREvectorofentryprobabilities,
.
Next, with parties we need to distinguish between entrants from the Left and Right Party,
respectively, so we replace the average entry probability of each other citizen 9 by the
probabilitiesthat entersfromtheleftorrightdirection,respectively:
1
50
13 , 14 .
15 and
1
50
wheretheaverageindividualentryprobabilityis
TheRHSof
8 containscitizen ’sexpectednetpayofffromentering(cf. 6 inthemain
text),whichforarighttype ∈ 51, … ,100 isgivenby(andsimilarforalefttype ∈ 1, … ,50 )
,
1
1
1
1
| ∈ 1, … ,100
,
1
∙
1
,
37 | ∈ 51, … ,100
16 1
| ∈ 1, … ,50
,
1
1
1
2
1
1
1
2
1
| ∈ 51, … ,100
,
,
with
,
1
| ∈ 1, … ,100
1
| ∈ 51, … ,100
,
|
1
1
|
,
17 ,
18 19 100
|
|
50
and
,
whereheexpectedpolicylosses
| ∈ 1, … ,50
17 to
1
50
18 accountforallfeasibleidealpointsofothers,and
0
1/2
1
givesthewinprobabilityoftheRightPartywith
for a given the one‐hundred equilibrium conditions of the form
for
1, … ,100todeterminetheQREvectorofentryprobabilities,
0
0.Then,
0
8 are simultaneously solved
.
FigureA1:QREdistributionsofentryprobabilities‐Example
38 AppendixB:SupplementaryTablesandFigures
TableB1:Entryrates‐Predictionsanddata
n
4
4
4
4
10
10
10
10
c Party
10
10
20
20
10
10
20
20
No
Yes
No
Yes
No
Yes
No
Yes
∗ (
.687
.673
.560
.496
.519
.445
.426
.321
∗
)
.840(.844)
.680(.671)
.400(.417)
.340(.364)
.420(.426)
.280(.256)
.200(.181)
.160(.152)
(
)
.603(.602)
.569(.570)
.457(.459)
.434(.436)
.465(.465)
.424(.423)
.331(.330)
.303(.302)
Note:Standarderrorsof
areallintherange .013, .016 .BNEisindicatedbyanasteriskandQREby ,the
maximumlikelihoodestimateofthedegreeoferror.
TableB2:Averagepayoffs–Predictionsanddata
n
4
4
4
4
10
10
10
10
c
10
10
20
20
10
10
20
20
Party
No
Yes
No
Yes
No
Yes
No
Yes
∗ ( ∗ )
66.98 (67.00)
69.09(69.06)
64.59 (64.00)
66.69(66.56)
59.71 (60.28)
61.24(63.98)
57.59 (59.25)
59.90 (61.89)
69.26
70.42
64.22
68.03
64.44
68.31
62.61
64.35
Note:Standarderrorsfor
(
)
69.91(70.67)
71.96(72.14)
66.65 (66.95)
68.41(68.73)
65.45(65.01)
68.36 (68.31)
63.20(63.24)
65.94 (65.91)
areintherange .71, .82 .
TableB3:Averagepayoffs,policylosses,andentryexpenses–Predictionsanddata
Treatment
n
4
4
4
4
10
10
10
10
c
10
10
20
20
10
10
20
20
Party
No
Yes
No
Yes
No
Yes
No
Yes
Payoffs
69.26
70.42
64.22
68.03
64.44
68.31
62.61
64.35
Note:Standarderrorsfor
and .13, .34 ,respectively.
∗
67.00
69.06
64.00
66.56
60.28
63.98
59.25
61.89
Policylosses
70.67
72.14
66.95
68.73
65.01
68.31
63.24
65.91
̅
25.12
24.10
25.83
23.30
30.87
27.74
29.38
29.74
̅
25.81
25.48
28.91
27.42
35.96
33.96
37.63
35.58
areintherange .71, .82 ,andfor ̅
39 ∗
Entryexpenses
̅
24.55
6.87
23.41
6.73
25.12 11.20
23.81
9.92
30.84
5.19
27.96
4.45
30.66
8.52
28.56
6.42
and
∗
8.44
6.71
8.33
7.27
4.26
2.56
3.62
3.03
Bonus
6.02
5.70
9.18
8.71
4.65
4.23
6.60
6.04
/
1.25
1.25
1.25
1.25
0.50
0.50
0.50
0.50
theyareintherange .69, .80 TableB4:Observedaveragepayoffs,policylosses,andentryexpensesofleadersandnon‐leaders
Leader
No
Yes
n
4
4
4
4
10
10
10
10
4
4
4
4
10
10
10
10
c
10
10
20
20
10
10
20
20
10
10
20
20
10
10
20
20
Party
No
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
Yes
60.65
62.23
57.19
61.88
61.04
65.34
60.12
62.00
95.11
95.00
85.30
86.50
95.00
95.08
85.00
85.50
Note: For non‐leaders, standard errors of
, ̅ , and
.15, .36 ,respectively.Forleaders,standarderrorsof
and
̅ 33.49
32.13
34.44
31.07
34.30
30.82
32.64
33.04
0
0
0
0
0
0
0
0
5.86
5.64
8.37
7.06
4.66
3.84
7.24
4.96
9.89
10.00
19.70
18.50
10.00
9.92
20.00
19.50
‐
‐
‐
‐
‐
‐
‐
‐
5
5
5
5
5
5
5
5
are in the range . 74, .98 , . 70, .89 , and
arebothintherange .00, .34 .
FigureB1:Entryratesperidealpointsfornormalizeddistancestomedianidealpoint–
Predictionsanddata
40 41 FigureB2:Frequencyofparticipantswithidealpoints 51, … ,100 FigureB3:Cumulativefrequencydistributionsofaverageindividualentryrates
42 FigureB4:Cumulativefrequencydistributionsofestimatedindividualcutpointpairs
43 Onlinesupportingmaterial
SampleInstructions(Party,
4,
10)
Instructions
Thank you foragreeing toparticipateinthisdecision‐makingexperiment.Youwillreceive$7for
participating,plusadditionalearningsduringtheexperimentthatdependonyourowndecisions,the
decisionsofothers,andchance.Yourearningsintheexperimentareexpressedinpoints.250points
are worth $1. At the end of the experiment, your total earnings in points will be exchanged into
dollarsandpaidtoyouincash.Nootherparticipantwillbeinformedaboutyourpayment!
Pleaseswitchoffyourcellphone,remainquiet,anddonotcommunicatewithotherparticipantsduring
theentireexperiment!Raiseyourhandifyouhaveanyquestions,andoneofuswillcometoyouto
answerthem.
PartsandDecisionRounds
Theexperimentconsistsoftwoparts,labeledPart1andPart2.Eachparthas30decisionrounds.We
willreadyoutheinstructionsforPart1now.AftercompletingPart1wewillreadinstructionsfor
Part2.
InstructionsPart1
YourGroup
Atthebeginningofeachround,allparticipantswillberandomlydividedintogroupsof4.Thus,in
additiontoyourself,therewillbethreeothermembersinyourgroup.Notethatyouwillnotknow
whotheseothermembersare.Also,pleasenotethatthegroupsarecompletelyindependentofeach
other.Inanyparticularroundyouwillhavenointeractionatallwithparticipantsintheothergroups.
GroupDecisionProblem
Ineachround,yourgroupwilldecideonagroupoutcome,whichcanbeanyintegerbetween1and
100.Thisisdonebyelectingagroupleader,whosebestoutcomewillbeimplementedasthegroup
outcome.Eachofyouwillbetoldwhatyourownbestoutcomeisinthatround,anddifferentgroup
memberswillgenerallyhavedifferentbestoutcomes.Youreceivethehighestbenefitsifthegroup
outcomeequalsyourownbestoutcome,andreceivelowerbenefitsthefurtherthegroupoutcomeis
fromyourownbestoutcome.Wewillexplaintheexactpayoffdetailsshortly.
RandomAssignmentofBestOutcomes
Howisyourownbestoutcomeassignedinaround?Thisisdonebythecomputer.Itwillrandomly
assigneachmemberinyourgroupabestoutcomebychoosingoneoftheintegersfrom1to100,with
eachintegerbeingequallylikely.Thecomputerdoesthiscompletelyindependentlyforeachgroup
member,sotypicallydifferentmemberswilleachhaveadifferentbestoutcome.Youareonlytold
yourownbestoutcome.Youarenottoldthebestoutcomeofanyothergroupmember.Therefore,
knowing your own best outcome gives you no information whatsoever about anybody else’s best
outcome.Allyouknowaboutanothergroupmember’sbestoutcomeisthatitissomeintegerfrom1
to 100, with an equal chance of being any of those integers. Importantly, best outcomes are
reassignedindependentlyineachround,soyourownbestoutcomewilltypicallyvaryfromroundto
round, and your past assigned best outcomes have nothing to do with your future assigned best
outcomes.
44 LowNumberandHighNumberMembers
If your best outcome is 50 or less, then we refer to you as a “Low number” member. If your best
outcomeis51orgreater,thenwerefertoyouasa“Highnumber”member.Thesameholdsforthe
othermembersinyourgroup.
Decision‐MakingStages
Eachroundconsistsoftwodecision‐makingstages,labeledStage1andStage2.
Stage1
Eachgroupmemberwilldecideonwhetherornottoenterasacandidateintheupcomingelection
forgroupleaderofthecurrentround.Whoeverwillbecomethegroupleaderreceivesabonusof5
points.However,ifyouchoosetoenterasacandidateforleadership,thenyoumustpayanentryfee
of10points.Ifyouchoosenottoenter,thenyoudonotpayanyfee(0points).Thewinnerofthe
electionwillbethegroupleader,andthegroupoutcomecoincideswithherorhisbestoutcome.
Stage2
In this stage, if more than one low number member entered in Stage 1, then the computer will
randomlyselectoneofthemfortheelectionwithanequalchanceforeach.Thisselectedmemberis
called“LowNumberCandidate.”Similarly,ifmorethanonehighnumbermemberenteredinStage
1,thenthecomputerwillrandomlyselectoneofthemfortheelectionwithanequalchanceforeach.
Thisselectedmemberiscalled“HighNumberCandidate.”Eachgroupmembercastsasinglevotefor
exactlyonecandidate.Thecandidatesareindicatedonthecomputerscreen,representedbydecision
buttons labeled with their member ID label letter and whether they are from Low or High. For
example,ifmemberXandmemberQaretherespectivelowandhighnumbercandidates,thenthere
willbetwodecisionbuttonswithlabels“LowNumberCandidateX”and“HighNumberCandidateQ”,
respectively.Ifnolow(high)numbermemberenteredinStage1,thenthereisnolow(high)number
candidate.
Eachgroupmember,whetheracandidateornot,thenvotesforoneofthecandidatesbyclickingon
therespectivedecisionbutton,possiblyforher‐orhimself.Ifyouareacandidateyourself,thenthe
labelonyourdecisionbuttonishighlightedinred.Thecandidatewiththemostvotesinyourgroupis
theelectedgroupleader.Ifthecandidateshavethesamenumberofvotes,thenoneofthemwillbe
randomlyselectedasthegroupleader,withanequalchanceforeach.
Specialcase:IfnogroupmemberenteredasacandidateinStage1,thenthereisnovoting.Instead,
oneofthefourgroupmemberswillberandomlyselectedasthegroupleader,withanequalchance
foreach.Pleasenotethatthisrandomlyselectedgroupleaderdoesnotpaytheentryfee(because
sheorheactuallydidnotenter)butnonethelessstillreceivestheleaderbonusof5pointsandthe
groupoutcomeequalsherorhisbestoutcome.
Itisimportanttorememberthatthegroupoutcomeisalwaysexactlyequaltothebestoutcomeof
the group leader, regardless of whether she or he entered and won the election or was selected
randomlyafternobodyentered.
YourRoundEarnings
45 Yourroundearningswilldependonthreefactors:thedistancebetweenyourownbestoutcomeand
thegroupoutcome,whetheryouchosetoentertheelectionasacandidate,andwhetheryouarethe
groupleader.Thereareonlyfourpossibilities:
(i)Youwereacandidatebutnotelectedtobegroupleader
Inthiscase,yourroundearningsequal“100pointsminustheabsolutedistancebetweenyourown
bestoutcomeandthegroupoutcome,minusthe10pointsentryfee”or:
,
|
100
|
10.
Example:Yourownbestoutcomeis60,andthegroupoutcomeis91,thenyourroundearningsare
100 |60 91| 10 59points.
(ii)Youwereacandidateandelectedtobegroupleader
Inthiscase,yourroundearningsareequaltoexactly95points,or:
|
0
100
100
5
|
10
5
10
95.
(iii)Youwerenotacandidateandwerenotthegroupleader
Inthiscase,yourroundearningsequal“100pointsminustheabsolutedistancebetweenyourown
bestoutcomeandthegroupoutcome”or:
|
100
|.
Example:Yourownbestoutcomeis60,andthegroupoutcomeis15,thenyourroundearningsare
100 |60 15| 55points.
(iv)Nobodyenteredasacandidateandyouwererandomlyselectedtobegroupleader
Inthiscase,yourroundearningsareequaltoexactly105points,or:
100
100
|
0
,
5
105.
|
5
Observethatifyouareacandidate,yourexpectedroundpayoffishighestifyouvoteforyourself.
Thisisbecauseincaseyouwillbeelectedgroupleader,youreceivetheleaderbonusof5pointsand
avoidanylossesinpointsfromtheabsolutedifferencebetweenyourbestoutcomeandthegroup
outcome,asyourbestoutcomewillbethegroupoutcome.
NotethatyourtotalearningsinPart1areequaltothesumofallyourroundearningsinthatpart.
Eachofthe30decisionroundsinPart1willfollowtherulesjustdescribed.Rememberthatyouare
randomlyre‐matchedintonew4‐persongroupsandrandomlyreassignedyourownbestoutcomes
between each round. At the bottom of your computer screen there will be a full summary of the
historyofyourexperienceandpayoffsinallpriorrounds.
46 Decisionscreens(Party,
4,
10points)
Entrydecision
Votingdecision
47 Electionresults
48 Decisionscreens(NoParty,
10,
10points)
Entrydecision
Votingdecision
49 Electionresults
50