Research Article Altruism, Noise, and the Paradox of Voter Turnout: An Experimental Study

23 Pages·2015·2.52 MB·English

Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2015, Article ID 972753, 22 pages
http://dx.doi.org/10.1155/2015/972753
Research Article
Altruism, Noise, and the Paradox of Voter Turnout:
An Experimental Study
SarahA.Tulman
UnitedStatesDepartmentofAgriculture,EconomicResearchService,1400IndependenceAvenueSW,MailStop1800,
Washington,DC20250-0002,USA
CorrespondenceshouldbeaddressedtoSarahA.Tulman;[email protected]
Received4December2014;Revised17March2015;Accepted20March2015
AcademicEditor:ArthurSchram
Copyright©2015SarahA.Tulman.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,
whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
Thispaperaddressestheparadoxofvoterturnout,whereinobservedvotingparticipationratesarefargreaterthanwhatrational
choicetheorywouldpredict.Votersfacemultiplevotingchoices,stochasticvotingcosts,andcandidatesofferingdifferenteconomic
platforms.Acombinationoftwoapproachesattemptstoresolvethisparadox:quantalresponseequilibrium(QRE)analysis,which
introducesnoiseintothedecision-makingprocess,andthepossibilityofethical(altruism-motivated)voting.Aseriesoflaboratory
experimentsempiricallyteststhepredictionsoftheresultingmodel.Participantsintheexperimentsarealsogivenopportunitiesfor
communicatingonlinewiththeirimmediateneighbors,inordertoenhancethechancesthatsubjectswouldrealizethepossibility
ofethicalvoting.Theresultsshowthatethicalvotingoccursbutgainsmomentumonlyinthepresenceofavocaladvocateand
eventhenitmostlydissipatedbythesecondhalfofthesession.TheQRE-basedmodelwasabletoexplainsomebutnotallofthe
overvotingthatwasobserved,relativetotheNashequilibriumprediction.Thereisevidencetosuggestthatcommunicationviathe
chatfeaturegeneratedsomeofthevotingandalsosomeoftheethicalvoting.
1.Introduction tobesomeothermotivationforvotingbeyondtheprobability
ofone’svotebeingpivotal.
Theparadoxofvoterturnout,whereinobservedturnoutrates Why does this paradox matter? As the fields of election
in elections are far greater than what is predicted under forecasting and polling analysis have expanded in order to
rational choice theory, has been a focus of researchers in takeadvantageoftheexplosioninavailabledata,interestin
economics and political science for over 50 years (see Geys these forecasts has rapidly increased. During the 2012 US
[1] for a survey of the theoretical literature) and remains presidentialcampaigntherewasanendlessflowofpolitical
unresolved.Underrationalchoicetheory,theprobabilityof and polling analysis covered in almost every media outlet,
casting a pivotal vote—making or breaking a tie—is a key and the presidential campaigns devoted vast resources to
component of the motivation to vote. If the electorate is “get out the vote” efforts while using the rapidly increasing
large then the probability of casting the pivotal vote falls quantitiesofavailabledatatomicrotargetvoters.Inspiteof
to almost zero, and rational choice theory predicts that no this increasing sophistication, one thing that can and often
onewillvote.Thisprobabilitydeclinesquickly,sothateven doescauseelectionoutcomestodivergefromforecastsisthe
electionswithrelativelysmallelectoratesfacethisissue.And matterofwhoactuallyvotesonElectionDay.
yet, observed turnout rates in elections are far greater than The model presented in this paper and the laboratory
zero.Forexample,inthe2012generalelectionintheUnited experimentswhichtestitspredictionsfocusontheeconomic
States the estimated turnout rate among eligible voters was motivations for voting. The payoffs that are offered by the
58.6%.(AllturnoutratesarefromtheUnitedStatesElection candidates, and the equitability (or lack thereof) of these
Project,http://www.electproject.org.)Eveninyearswithouta payoffs across groups, can be thought of as their economic
presidentialelection,whenturnoutisgenerallylower,itisstill platforms.Votersalsofaceacostassociatedwithvotingthat
considerablyhigherthanzero,suchasthe41.8%turnoutrate models the explicit and implicit costs of going to the polls,
inthe2010HouseandSenateelections.Therefore,therehas such as transportation costs, lost wages due to taking time
2 JournalofAppliedMathematics
offfromwork,ortheopportunitycostoftime.Idrawfrom Groupidentityandcommunicationwithinthosegroups
severalthreadsoftheliteraturewhichareattheintersection havealsobeenshowntoincreasevotingparticipation,asseen
of economics and political science, and my model and inpapersbyMorton[7],GroßerandSchram[8],Schramand
experimentsaugmentpreviousworkonvotingbehavior. vanWinden[9],SchramandSonnemans[10],andCharness
Thefirstthreadfromwhichmymodelandexperiments etal.[11].Kitteletal.[12],whichwasdevelopedconcurrently
aredrawnfocusesontheuseofquantalresponseequilibrium with this paper, incorporated multiple-candidate elections,
(QRE),ageneralizationoftheNashequilibriumthatallows costly voting as in Levine and Palfrey [4], and prevoting
fordecisionerrors,torelaxtheassumptionofperfectratio- communication. Although the focus and context of their
nality.BuildingupontheframeworkpresentedinMcKelvey paper differ from mine, they found that the distribution
andPalfrey[2],andalsodrawingfromGoereeandHolt[3], of earnings was more equitable when intergroup commu-
Levine and Palfrey [4] used laboratory experiments to test nication was allowed (analogous in some respects to the
voter turnout predictions in the presence of heterogeneous mixed-typechatgroupsinthispaper).Theirresultsalsoshow
and privately known participation costs. However, in their that voting participation increases when communication is
paper the subjects could only choose between voting and allowed,whichmatchesthepositiveeffectofcommunication
abstaining, whereas in most elections potential voters can onvotingturnoutthatIfound.
choose between at least two candidates, plus the option of The main findings of this paper are summarized as
abstaining.Wouldtheirresultsholdupifanotherchoicewere follows. High rates of voter turnout were observed in the
added so that potential voters could choose between three experimental data, especially when a subject voted for the
options:voteforonecandidate,votefortheothercandidate, candidate offering that subject’s voter type a higher payoff
andabstain?Themodelandexperimentaldesignpresented thanwhattheothercandidatewasoffering.Someethicalvot-
hereextendthoseofLevineandPalfrey[4]inthismanner. ingwasobserved,butnotenoughtoexplaintheovervoting.
Anotherthreadoftheliteraturefocusesontheroleofeth- Also, most of the overvoting occurred when voters of each
ical,oraltruism-motivated,votingingeneratingvoterturn- typevotedintheirowneconomicself-interestand,byitsvery
out. Morton and Tyran [5] separated the motivations for nature,ethicalvotingisnotapossiblemotiveforthat.
votingchoicesintoselfishvotingandethicalvoting,following One feature that could potentially explain overvoting,
the terminology used in the literature—voting for one’s unexpectedly, is the chat feature. (This feature was not
economicself-interestversusvotingaltruisticallyifoneofthe included in the experimental design for the purpose of
candidates offers a lower payoff but also a more equitable increasing voter participation but rather as a method to
distribution of payoffs across groups of voters; however, hopefully spread the word about the possibility of ethical
their equilibrium predictions were qualitative in nature. By voting and also to gain insight into the motivations behind
incorporatingasimilarpayoffstructureintotheQRE-based decisions.Thisresultwasunexpected.)Becausethechatwas
frameworkfromLevineandPalfrey[4],mymodelcangen- repeatedbeforeeveryroundandthegroupsremainedfixed
eratequantitativepredictionsofturnoutratesagainstwhich throughout the session it set up a dynamic in which there
the experimental results can be compared. Additionally, it may have been some accountability between chat partners.
can be argued that incorporating bounded rationality and Forexample,onemightasktheotherswhethertheyhadvoted
noise into the decision-making process may lead to a more in the previous round, and if so then for which candidate.
realistic representation of a world in which individuals do The transcripts also showed that subjects would often urge
notalways(indeed,almostnever)behaveperfectlyrationally. others to vote in a certain way or would agree to vote in
ThiscombinationwasalsousedinGoereeetal.[6]. a certain way and sometimes would ask about whether/for
AnotheraspectofMortonandTyran’s[5]approach,albeit whomothershadvotedinthepreviousround.Additionally,
one that was necessary given the nature of the experiment, inspiteofthedifferencesindesignandcontext,afewpatterns
wasthattheirexperimentwasaone-shotgame.Althoughthis were observed that had also appeared in the group identity
isamoreaccuraterepresentationofelectionsthanarepeated literature,suchastheemergenceofsubjectswhopersistently
game,inanactualelectionvotersusuallyhaveseveralweeks encouraged others to vote (analogous to Schram and van
or months leading up to the election during which voters Winden’s[9]producersofsocialpressure).
can learn about and discuss the issues and the candidates’ The rest of the paper proceeds as follows. Section2
platforms.Myexperimentsallowedfortheselearningeffects describes the theoretical model and results and the exper-
during the experiment, through the inclusion of 20 rounds imental design, which is influenced by the predictions
(elections). generated by the theoretical model. Section3 discusses the
Myexperimentsalsoincorporatedanonlinechatfeature empiricalresults.Finally,Section4concludes.
which allowed subjects to communicatebefore each round.
The initial purpose of this was an attempt to increase the 2.MaterialsandMethods
probability that the subjects would grasp the possibility of
ethical voting, in treatments where that was relevant—the 2.1. Model. This model, in which voter turnout is a partici-
hope was that the subjects could clarify this (and other pationgame(PalfreyandRosenthal[13]),isanextensionof
points) for each other. In the end, although this effect was thelogitquantalresponseequilibrium(QRE)modelusedin
indeed observed in certain cases, it also provided insight LevineandPalfrey[4]andGoereeandHolt[3],incorporating
intosubjects’motivations.Thechatfeaturewasalsofoundto the possibility of ethical voting along the lines of Morton
increasevotingparticipationrates. andTyran[5].WhereasLevineandPalfrey[4]onlyallowed
JournalofAppliedMathematics 3
Table1:Payoffs(symmetric). for the candidate that offers the higher payoff to his or her
type.
Type1 Type2
The aggregate voting probabilities for each type-candi-
Candidate𝑋 𝑞 𝑞−𝜃 datecombinationare
Candidate𝑌 𝑞−𝜃 𝑞
∞ 𝑐∗
𝑝∗ =∫ 𝜏 (𝑐)𝑓(𝑐)𝑑𝑐=∫𝑖𝑗 𝑓(𝑐)𝑑𝑐=𝐹(𝑐∗) (1)
𝑖𝑗 𝑖𝑗 𝑖𝑗
−∞ −∞
potentialvoters(alsoreferredtosimplyas“voters,”unlessa
foreachcombinationofvotertype𝑖andcandidate𝑗.
distinctionisneededbetweenactualandpotentialvoters)to
Thereisacutpointcostlevel𝑐∗belowwhich,intheNash
choosebetweenvotingandabstaining,heretheyhavethree 𝑖𝑗
choices: vote for Candidate 𝑋, vote for Candidate 𝑌, and equilibrium,theprobabilityofvotingequalsoneandabove
abstain. This better captures the reality of most elections, which the probability of voting is zero. At the cutpoint, the
in which voters choose between two or more candidates voterisindifferent.Thesecutpointsareexpressedas
in addition to the option of abstaining from voting. This
(𝑉 −𝑉 )
estxrtuecntsuiorens,oifnctlhuedimngodoenleawlshoicahllomwasyfionrduacreanetgheicoafl vpoatyionffg 𝑐𝑖∗𝑗 = 𝑖𝑗 2 𝑖𝑚 ×(Pr(make tie)𝑖𝑗+Pr(break tie)𝑖𝑗),
throughthepresenceofa“selfish”candidatewhooffersahigh (2)
payofftoonegroupofvotersandalowpayofftotheotherand
an“ethical”candidatewhooffersanequitabledistributionof where 𝑉𝑖𝑗 is the payoff received by a member of voter type
payoffsacrossgroupsofvoters. 𝑖 if candidate 𝑗 wins the election, and where 𝑚 indexes the
Inthismodel,thevotingruleisasimpleplurality.Tiesare other candidate. This is the difference between the payoff
brokenfairly,withthewinnerbeingpickedrandomly(e.g.,by receivedandthepayoffthatwouldhavebeenreceivedifthe
acointoss).Payoffs,thedistributionofwhichcanbethought othercandidatehadwon(thepayoffdifference).IntheNash
ofasthecandidates’economicplatforms,arepaidaccording equilibrium nobody would vote for the candidate offering
towhichcandidatewinsregardlessofwhetherorforwhom one’s type the lower payoff because the payoff difference
the participant voted. The payoff structure for each type is wouldbenegative,implyinganegativecostcutpointinwhich
commonknowledgetoeveryoneregardlessoftype. case no cost would be low enough to induce voting. These
Potential voters are separated into two “types,” which cutpointsaresettozerobecausecostscannotbenegative.
determine the payoff that each one will receive depending The term (1/2) × (Pr(make tie)𝑖𝑗 + Pr(break tie)𝑖𝑗) is
on the winner of the election. Each candidate offers differ- the probability that a vote cast by a member of type 𝑖 for
ent payoffs to the two different voter types (Table1). This candidate 𝑗 will be pivotal (henceforth referred to as the
table shows payoffs that are symmetric in both the payoff “pivotal probability”). The one-half is there because, in the
differences and the actual payoffs. This symmetry is not presenceoftwocandidates,ifsomeone’svotemakesatiethen
theoretically necessary—the main takeaway from this table thereisa50%chancethatthisvoter’schosencandidatewill
ishowthecandidatesofferdifferentpaymentstoeachtype— win(andthatthevotewillhavebeenpivotal)—thewinnerof
but matches the experimental design used in this research. a tie is determined randomly. If someone’s vote breaks a tie
With this payoff structure, voters of both types would face thenthereisa100%chancethathisorherchosencandidate
identical incentives to vote for the candidate offering their willwin.However,becausetherewasalreadya50%chance
typethehigherpayoff. thatthiscandidatewouldhavewonintheeventofatie,there
Voting is modeled as being costly. This mirrors the wouldonlybea50%chancethatthevotethatbrokethetie
costs associated with voting in real-world elections such as changed that candidate’s outcome from losing to winning.
transportation costs, childcare, lost wages due to time off Furtherdetailsaboutthecalculationofthepivotalprobability
from work, or the opportunity cost of whatever someone canbefoundinAppendixA.
would otherwise do. This cost is only incurred if someone Theexpectedpayoffdifferencesofvotingforcandidate𝑗
votes.Costsareindependentdrawsfromthesameuniform asamemberoftype𝑖,netofcosts,areexpressedas
distribution, varying across both voters and elections, and
both voter types face the same distribution of costs. In 𝜋𝑘,𝑖𝑗 =(𝑉𝑖𝑗−𝑉𝑖𝑚)×Pr(pivotal)𝑖𝑗−𝑐𝑘, (3)
each election, each voter knows his or her own cost before
deciding whether to vote but only knows the distribution where 𝑉𝑖𝑗 is the payoff received by a member of type 𝑖 if
from which others’ costs—and his or her own costs during candidate𝑗winstheelectionand𝑐𝑘 isthevotingcostfaced
futureelections—aredrawn. byagivenpersoninagivenelection.𝜋𝑘,𝑖𝑗 canbethoughtof
Following Levine and Palfrey [4], a quasi-symmetric asacomponentof𝜋̃𝑘,𝑖𝑗 =𝜋𝑘,𝑖𝑗+𝜀𝑘,𝑖𝑗,where𝜀𝑘,𝑖𝑗isindependent
equilibrium in this model is a set of four turnout strategies andidenticallydistributedextremevalue.Thisproducesthe
(𝜏1𝑋,𝜏1𝑌,𝜏2𝑋,𝜏2𝑌)specifyingtheprobabilitiesthatamember logitmodelthatisshownbelow.
of type 𝑖 will vote for candidate 𝑗, as a function of the Quantal response equilibrium (QRE) analysis, intro-
votingcost.Foraquasi-symmetricequilibrium,itisassumed duced in McKelvey and Palfrey [2], allows for “noise” in
thatwithineachtypeeveryonewillusethesamestrategy— thedecision-makingprocessforvoting.Thisprocessisoften
althoughitispossibleformembersofeachtypetovotefor influenced by factors such as emotions, perception biases,
eithercandidate,intheNashequilibriumanyvotingwillbe voter error, unobserved individual heterogeneity, and other
4 JournalofAppliedMathematics
sources of noise. The QRE introduces decision errors via a 1
logitprobabilisticchoiceruleaccordingtowhichparticipants 0.9
maketheirvotingchoices.
0.8
UnderQRE,thebestresponsefunctionsareprobabilistic
0.7
rather than deterministic. Although “better” responses are
more likely to be observed than “worse” responses, the e) 0.6
“worse” responses are still observed because choice has a ot 0.5
v
stochastic element. Thus, the assumption of perfect ratio- Pr ( 0.4
nality is relaxed in favor of bounded rationality. As the
0.3
amount of noise decreases—and the QRE approaches the
0.2
Nashequilibrium—votersbecomeincreasinglylikelytomake
choicesthatareconsistentwiththeNashequilibrium.Inits 0.1
most general form, the logit probability of choice “𝑥” takes 0
theform𝑃(𝑥)=𝑒𝜆𝜋𝑥/∑𝑖𝑒𝜆𝜋𝑖,where𝜋𝑖istheexpectedpayoff c∗
frommakingchoice𝑖and𝜆isthelogitprecisionparameter.
The denominator ∑𝑖𝑒𝜆𝜋𝑖 sums across all possible choices, NSoamshe enqouiisleibrium
summingto1. High noise
Astheprecisionparameter𝜆increases,thelevelofnoise
Figure1:Logitresponsefunctionrelativetocostcutpoint𝑐∗.
decreases and the level of precision in decision making
improves. As this happens, the ratio of “better” responses
to “worse” responses improves. When 𝜆 is very large (very Table2:Payoffs(asymmetric).
high precision/very low noise), the solution approaches the
Nash equilibrium, with all or almost all decisions match- Type1 Type2
ing the Nash equilibrium outcome. (Strictly speaking, this Candidate𝑋 𝑞+𝜃 𝑞−𝜃
convergence only holds for the principal branch of the Candidate𝑌 𝑞 𝑞
QRE when there are multiple equilibria. However, for the
model presented here the results were robust across a wide
range of starting points.) As 𝜆 approaches zero (very low In the presence of random costs, these response functions
precision/very high noise), voting participation decisions have to be averaged across all possible costs (for a given
approach randomness (50% if there are two choices, 33% type-candidatecombination)inordertoarriveattheaverage
if there are three choices, etc.) as the probabilities of “bet- response probability of type 𝑖 voting for candidate 𝑗. After
ter” responses and “worse” responses become increasingly averaging across costs, the probabilities are (𝑝𝑖𝑋,𝑝𝑖𝑌,1 −
similar. 𝑝𝑖𝑋 − 𝑝𝑖𝑌), where the third probability is the probability of
In the context of this model, a Nash equilibrium would abstaining.
correspond to a sharp drop in the response function at the The logit precision parameter, 𝜆, is estimated using the
costcutpoint,wheretheprobabilityofvotingwouldequalone likelihoodfunction:
iffacingacostbelow𝑐∗andwouldequalzeroiffacingacost
above𝑐∗.Foranintermediatelevelofnoise,thecostcutpoint log𝐿=∑(𝑁1𝑋,𝑙×ln(𝑝1𝑋,𝑙)+𝑁1𝑌,𝑙
would affect the decision of whether or not to vote but the 𝑙
responsefunctionwouldnotbeassharpbecausenoisewould
leadtosomeinstancesofvoting(orabstaining)evenwhenthe ×ln(𝑝1𝑌,𝑙)+(num obs1,𝑙−𝑁1𝑋,𝑙−𝑁1𝑌,𝑙)
ncoositseisleabveolv/vee(royrlboewlolwev)ethleofcuptrpecoiisniot.nA)vweoruylldowco𝜆rr(evseproynhdigtho ×ln(1−𝑝1𝑋,𝑙−𝑝1𝑌,𝑙)+𝑁2𝑋,𝑙 (6)
acutpointthatiseffectivelyirrelevant,wherethesizeofthe ×ln(𝑝2𝑋,𝑙)+𝑁2𝑌,𝑙×ln(𝑝2𝑌,𝑙)
costwouldhaveveryminimalornoeffectonthedecisionof
whetherornottovote.ThiscanbeseeninFigure1. +(num obs2,𝑙−𝑁2𝑋,𝑙−𝑁2𝑌,𝑙)
The logit quantal response function, which can be
thought of as the logit probability of a member of type 𝑖 ×ln(1−𝑝2𝑋,𝑙−𝑝2𝑌,𝑙)),
makingvotingchoice𝑗(wherethechoicesarevotefor𝑋,vote
for𝑌,andabstain)andfacingcost𝑘,isshownby where𝑁𝑖𝑗,𝑙 isthetotalnumberofvotesbymembersoftype
𝑖 for candidate 𝑗 that were observed during treatment 𝑙 (or
𝑝𝑘,vote𝑖𝑗 = 𝑒𝜆𝜋 +𝑒𝑒𝜆𝜆𝜋𝜋𝑘,𝑖𝑗 +𝑒𝜆𝜋 . (4) sfoerssisoonm)eaonndenoufmtyopbes𝑖𝑖,𝑙iins tthreeattomtaelnntu𝑙m. Thberesoef oobbsseerrvvaattiioonnss
𝑘,𝑖𝑋 𝑘,𝑖𝑌 𝑘,𝑖𝑎
include decisions for any of the three possible choices: vote
Thiscanbesimplified,foratype𝑖voterchoosing𝑗(wherethe forCandidate𝑋,voteforCandidate𝑌,andabstain.
choicesareonlyvotefor𝑋orvotefor𝑌)andfacingcost𝑘,as Oneofthecontributionsofthispaperistheincorporation
ofethical(altruism-motivated)votingintoaQREframework.
𝑒𝜆𝜋𝑘,𝑖𝑗
𝑝 = . (5) Table2showsthepayoffsforanethicalcandidateandaselfish
𝑘,𝑖𝑗 1+𝑒𝜆𝜋𝑘,𝑖𝑋 +𝑒𝜆𝜋𝑘,𝑖𝑌 candidate(followingtheterminologyusedintheliterature).
JournalofAppliedMathematics 5
The selfish candidate offers a high payoff to one voter type expected payoff difference equations, for the other type-
and a low payoff to the other type. The ethical candidate candidatecombinations,remainthesameasin(3).
offerspayoffsthathaveamoreequitabledistributionacross For ethical instrumental voting, utility is gained if the
types.Inthisrespect,thepayoffstructurefollowsMortonand ethical candidate wins. Therefore, the utility/payoff gain is
Tyran[5].InTable2,theethicalcandidateofferspayoffsthat tiedtotheprobabilityofone’svoteforthatcandidatebeing
areequalacrosstypes.Also,thedifferencebetweenthehigh pivotal.Thisiscapturedinthe𝛿termintheequationbelow:
andlowpayoffsisthesameforbothtypes.Neitherofthese
features—theethicalcandidateofferingexactlyequalpayoffs 𝜋1𝑌 =(𝑉1𝑌−𝑉1𝑋+𝛿)×Pr(pivotal)1𝑌−𝑐𝑘. (8)
orsettingpayoffssothatthepayoffdifferenceisthesamefor
both types—are necessary, but it matches the experimental In the presence of either ethical instrumental or ethical
designthatwasusedinthisresearch. expressivevoting,theestimationprocedurefor𝜆and𝛼orfor
InTable2,𝑞and𝜃standforthebaselineequitablepayoff 𝜆and𝛿isthesameasbeforeexceptthatnowtwoparameters
and the payoff difference, respectively. Candidate 𝑋 is the arebeingestimated.Ineachcaseeither𝛼or𝛿,respectively,is
selfish candidate and Candidate 𝑌 is the ethical candidate. identifiedthroughanexclusionrestrictionbecausetheethical
Candidate 𝑌’s payoffs maximize the minimum payoff and votingparameteronlyappearsinoneofthefournetexpected
alsominimizethedifferencebetweenthepayoffsreceivedby payoffdifferenceequations.Therefore,𝜆ispinneddownby
the two types. Members of Type 1 receive a higher payoff theotherthreenetexpectedpayoffdifferenceequationsand
if Candidate 𝑋 wins but may be motivated by altruism to then𝛼or𝛿ispinneddowngiven𝜆.
voteforCandidate𝑌eventhoughtheywouldreceivealower Inthismodel,itisnotpossibletosimultaneouslymodel
payoffifthiscandidatewontheelection.Thisisethicalvoting. bothethicalexpressiveandethicalinstrumentalpreferences.
Ontheotherhand,membersofType2receiveahigherpayoff In that case, the net expected payoff difference equation
ifCandidate𝑌winsandwouldhavenorationalmotivefor wouldbe
votingforCandidate𝑋becausethiscandidateoffersthema
lowerpayoffandoffersinequitablepayoffs. 𝜋1𝑌 =(𝑉1𝑌−𝑉1𝑋+𝛿)×Pr(pivotal)1𝑌−𝑐𝑘+𝛼. (9)
Selfish voting happens when economic self-interest is
acted upon by voting for the candidate who offers one’s However,because𝛼 = 𝛿×Pr(pivotal)1𝑌thereareaninfinite
typeahigherpayoff.ThiswouldhappenwhenType1votes number of possible combinations and 𝛼 and 𝛿 cannot be
for Candidate 𝑋 or when Type 2 votes for Candidate 𝑌. separatelyidentified.
As discussed above, ethical voting happens when altruism
outweighs economic self-interest and this would happen— 2.2.ExperimentalDesign. Thesetheoreticalpredictionswere
atleastpotentially—whenType1votesforCandidate𝑌.Itis empiricallytestedthroughaseriesoflaboratoryexperiments.
alsopossiblethatthiswouldoccurbecauseofnoise.Thisis All sessions were conducted in the Veconlab Experimental
oneexampleofhowthechatfeatureprovesusefulforgaining Economics Laboratory at the University of Virginia on
insightintothemotivesbehindvotingdecisions.Thereisno undergraduatestudents,usingVeconlabsoftware.Therewere
rationalmotiveforType2tovoteforCandidate𝑋;therefore, 180 subjects—109 male and 71 female—across a total of 12
thisvotingchoicewouldbeduetonoise. sessions.Eachsubjectcouldonlyparticipateinonesession;
One difference between this design and Morton and therefore, all subjects were inexperienced at the start of
Tyran’s[5]isthatinthisdesigntheaggregatepayoffsarethe their respective sessions. Additionally, each session utilized
sameacrosscandidates.Idesigneditthiswaysothat,inthe only one treatment, so each subject only participated in
absenceofethicalmotives,equilibriumpredictionswouldbe one treatment. The average payout per subject was $35,
thesameacrosssymmetricandasymmetricpayoffstructures including a $6 payment for showing up, and individual
foragivenelectoratesizeandagivennoiselevel. payouts depended upon individual decisions as well as the
Ethicalvotingcanbedividedintotwocategories:ethical collectivevotingoutcome.
expressive and ethical instrumental. For ethical expressive Each session consisted of 20 rounds, or elections. Each
voting, utility (or, in this model, implicit payoff) is gained subject was randomly assigned to one of two voting types,
fromtheactofvotingfortheethicalcandidateregardlessof Type 1 and Type 2, at the start of the session and remained
theoutcomeoftheelection.Thisisalongthelinesofawarm at that same type throughout the entire session in order to
glowparameter,asinAndreoni[14].Itiscapturedbythe𝛼 avoidreciprocityeffects.Withineachsession,equalnumbers
term in the net expected payoff difference equation below ofsubjectswereassignedtobothtypes.
for someone of Type 1 voting for Candidate 𝑌 and facing a Therewerethreepossiblechoices:voteforCandidate𝑋,
person-andelection-specificcost𝑘: voteforCandidate𝑌,andabstain.Anyonefromeithervoter
typecouldvoteforeithercandidate.Intheexperiment,elec-
𝜋1𝑌 =(𝑉1𝑌−𝑉1𝑋)×Pr(pivotal)1𝑌−𝑐𝑘+𝛼. (7) tionswerereferredtoasroundsandcandidateswerereferred
toasoptions,asin“Option𝑋”and“Option𝑌,”inanattempt
Because ethical expressive voting is an additive term, it is tokeepthelanguageasneutralaspossible.Throughoutthis
independent of the pivotal probability. Therefore, it would paper, “round” and “election” will be used interchangeably
increase the turnout rate even as the pivotal probability and the options will be referred to as “Candidate 𝑋” and
decreases,mitigatingorevenpossiblyoutweighingtheeffect “Candidate𝑌”exceptininstancesinwhichthewordchoice
of the declining pivotal probability. The other three net isrelevanttothediscussionathand.
6 JournalofAppliedMathematics
Asdiscussedintheprevioussection,whichevercandidate The backstories for asymmetric payoffs read as follows:
garnered the most votes within a round was declared the “Option 𝑋 will implement an investment that pays $2 to
winner, with ties decided with a virtual coin flip. After Type 1 voters and $1 to Type 2 voters” and “Option 𝑌 will
the conclusion of each round, all subjects were told which implementaninvestmentthatpays$1.50tobothType1and
candidate had won and the total number of votes for each Type2voters.”Forsymmetricpayoffs,thebackstorieswereas
candidate (but not the breakdown of how many of those follows: “Option 𝑋 will implement an investment that pays
werecastbymembersofeachtype).Eachcandidateoffered $1.50toType1votersand$1toType2voters”and“Option
differentpayoffstoeachvotertype,andsubjectswerepaidthe 𝑌willimplementaninvestmentthatpays$1toType1voters
amountthathadbeenpromisedtotheirtypebythecandidate and$1.50toType2voters.”
who won in that round, regardless of whether or for which Beforethevotingbeganineachround,subjectswereable
candidatetheyhadvoted.Attheendofthesession,subjects to chat online with their “neighbors” according to the ID
werepaidtheircumulativeearnings. number that had been assigned upon first logging into the
In each round, each subject drew a voting cost from program.Forexample,ID3couldchatwithID2andID4.
the same distribution. Costs ranged from $0 to 0.42, in Meanwhile,ID4couldchatwithID3andID5.Thesubjects
incrementsof$0.02.Thisincrementwaschosensothatthere with ID numbers at either end—ID 1 and ID 12 or ID 18,
were 22 possible costs and 20 rounds, meaning that, on dependingonthesessionsize—wereinthesamechatgroup,
average,therewouldbeagoodchanceofasubjectdrawing creatingacircularnetwork.Subjectscouldalsoseethevoter
almostallofthepossiblecostsoverthecourseofthesession. typeofeachoftheothertwosubjectsintheirchatgroups.The
The upper bound of the range of costs was chosen chatperiodlastedexactlyoneminuteineachround,andit
to generate at least some abstention. Along the lines of wasnotpossibleforanyonetosubmitavotingdecisionuntil
LevineandPalfrey[4],thepayofffrommakingorbreaking afterthechatended.
a tie would equal (𝑉𝑖𝑗 − 𝑉𝑖𝑚)/2 for a member of type 𝑖 This design models some features of real world discus-
voting for candidate 𝑗, where 𝑚 is the other candidate. For sions about politics. For example, most people only discuss
both the asymmetric and symmetric treatments, the payoff anupcomingelectionwithpeoplewhomtheyknowdirectly
from making or breaking a tie would equal $0.25 for both (friends, family members, possibly coworkers, etc.). Even
types,accordingtothepayoffsdetailedbelow.Therefore,for for those who broadcast their views on social media, the
subjects who drew a cost greater than this, abstention was number of people who are on the receiving end of that is
the strictly dominant strategy. If there were a cost equal to stillsmallrelativetothetotalsizeoftheelectorate.Giventhe
$0.25thenabstainingwouldbetheweaklydominantstrategy small electorate size in this experiment, limiting a subject’s
of a subject who drew this cost, but this was not a possible chat circle to only two other people roughly mirrors this.
drawbecauseofthe$0.02incrementsize.Thehighestpossible Additionally,theoverlappinggroupstructuremirrorsthefact
cost was set above the payoff from making or breaking a that social groups are not self-contained, so that ideas or
tie so that there would not exist an equilibrium in which information can potentially be spread to a large number of
the probability of voting was equal to one (i.e., in order to people.
generateatleastsomeabstentioninequilibrium)Notethat𝑉𝑖𝑗 Furthermore,mostofthechatgroupscontainedmembers
isassumedtobethehigherpayoff.Asdetailedearlier,there of both voting types. This was done to reflect the fact
would be no cross-voting in the Nash equilibrium because that most people have family members, friends, colleagues,
thepayoffdifferenceswouldbenegative,leadingtoanegative acquaintances,andsoforthwhoholddifferentpoliticalviews
costcutpoint. thantheydo.(Thisdiffersfromthefocusofmuchofthegroup
At the end of the session, subjects participated in an identity literature on the effects of communication within
exercisetoelicitthesubjects’perceivedprobabilitiesthattheir groups of voters.) Some subjects were part of a majority
votewas/wouldhavebeenpivotalinthefinalround,totest withintheirchatgroupandothersweretheminority,andin
therelationshipbetweenvotingturnoutandthe(perceived) afewcasesthesubjectwasinagroupconsistingentirelyof
probabilityofcastingapivotalvote.Ananalysisoftheelicited thatsubject’stype.
probabilities showed a significant and positive relationship Finally, the chat transcripts are extremely useful for
between the elicited probabilities and voting decisions— providinginsightintothesubjects’motivationsforvotinga
subjects who thought that they had a higher probability of certainwayorforabstainingaltogether.
beingpivotalweremorelikelytohavevotedinthefinalround Theexperimentwasdividedintofourtreatments,which
andwithlowerelicitedprobabilitieswerelesslikelytohave were designed to test the effects of electorate size—and
voted.Thissectionisavailablefromtheauthoruponrequest. therebytheroleoftheprobabilityofcastingapivotalvote—
The subjects’ computer screens showed short “backsto- and the possibility of ethical voting. First, the number of
ries” next to each candidate’s offered payoffs in the type- subjects per session—which can also be thought of as the
candidate payoff chart. The purpose of these was to help size of the electorate—was varied. Half (six) of the sessions
the subjects in the treatments with asymmetric payoffs had12subjectspersession,andtheremainingsessionshad
realize the possibility of ethical voting. However, in order 18subjects.Accordingtotheunderlyingtheory,theturnout
toavoidinadvertentlyencouragingorleadingsomesubjects rate should fall as the electorate size increases because the
to vote in a certain way—as had happened in some of the probabilityofone’svotebeingpivotaldecreases.Second,the
pilotexperiments—neutralwordingwasusedhere.Thenew payoffswerevariedsothatinhalfofthesessionsthesubjects
backstoriesmerelyrepeatedthepayoffsinwords. facedsymmetricpayoffsandintheotherhalfthepayoffswere
JournalofAppliedMathematics 7
Table3:Symmetricandasymmetricpayoffs. Table4:(a)Predictedturnoutprobabilitiesandcostcutpoints(12
subjects).(b)Predictedturnoutprobabilitiesandcostcutpoints(18
Asymmetric Symmetric subjects).
Type1 Type2 Type1 Type2
(a)
Candidate𝑋 $2.00 $1.00 $1.50 $1.00
Candidate𝑌 $1.50 $1.50 $1.00 $1.50 𝜆=100(Nash) 𝜆=7(LandP,2007) 𝜆=0.2(highnoise)
𝑝1𝑋 0.2996 0.3069 0.3333
𝑝1𝑌 0 0.0793 0.324
asymmetric.Thesepayoffs,whichwerecommonknowledge 𝑝2𝑋 0 0.0793 0.324
withinasession,canbeseeninTable3. 𝑝2𝑌 0.2996 0.3069 0.3333
Inthetreatmentswithasymmetricpayoffs,Candidate𝑋 𝑐1𝑋 0.122
was the selfish candidate, offering a high payoff ($2.00) to 𝑐1𝑌 0
Type1subjectsandalowpayoff($1.00)toType2subjects.The
ethicalcandidate,Candidate𝑌,offeredequalpayoffstoboth 𝑐2𝑋 0
votertypes.Thesepayoffsmaximizetheminimumpayoffand 𝑐2𝑌 0.122
minimizethedifferencebetweenthepayoffsreceivedbythe (b)
twogroups—indeed,inthisdesignmembersofbothgroups
𝜆=100(Nash) 𝜆=7(LandP,2007) 𝜆=0.2(highnoise)
would receive the same payoff. The payoff of $1.50 for each
group that was received by subjects of both voter types if 𝑝1𝑋 0.2608 0.3069 0.3333
Candidate𝑌wonwaslowerthanwhatType1subjectswould 𝑝1𝑌 0 0.0793 0.324
receiveandhigherthanwhatType2subjectswouldreceive 𝑝2𝑋 0 0.0793 0.324
ifCandidate𝑋won.ThisdesignwasinspiredbyMortonand 𝑝2𝑌 0.2608 0.3069 0.3333
Tyran[5]. 𝑐1𝑋 0.122
To account for the possibility that Candidate 𝑋 being 𝑐1𝑌 0
listedfirstonthetable(or𝑋comingbefore𝑌alphabetically) 𝑐2𝑋 0
might have affected voting decisions, the roles of ethical 𝑐2𝑌 0.122
andselfishcandidateswerereversedinroughlyeveryother
asymmetric session so that Candidate 𝑋 offered the equal
payoffs and Candidate 𝑌 offered the inequitable payoffs. Thesebothdecrease(increase)astheelectoratesizeincreases
Everythingelseremainedthesame.Nomeaningfuldifference (decreases).Therefore,thepredictedturnoutprobabilitiesfor
was observed in the data. In the final data, 𝑋 and 𝑌 were asessionwith18subjects—regardlessofwhetheritisasession
reversedbackforthesesessionstomakethedataconsistent withasymmetricorsymmetricpayoffs—arelowerthanthose
withtheothersessions. forsessionswith12subjects,allelseequal.
Under this payoff design, members of Type 1 may face Tables4(a)and4(b)showthepredictedturnoutproba-
conflictingincentivesinchoosingacandidate.Type1subjects bilitiesfortheNashequilibrium(extremelyhigh𝜆),for𝜆=7
wouldmaximizetheirearningsifCandidate𝑋wonbutmight (thevaluethatwasestimatedinLevineandPalfrey,2007),and
alsodecidetovoteforCandidate𝑌foraltruisticreasons. forextremelyhighnoise(verylow𝜆).Thepredictionsfora
In the symmetric payoff design, the two candidates sessionwith12subjectsareshownin(a),andthepredictions
offered payoffs that were mirror images of each other. This forasessionwith18subjectsareshownin(b).
meant that members of both voter types had the same In Tables 4(a) and 4(b), 𝑝𝑖𝑗 is the predicted probability
economic incentives to vote for the candidate who was that a member of type 𝑖 would vote for candidate 𝑗. The
offering one’s type the higher payoff. Each candidate was probabilityofamemberoftype𝑖abstainingis(1−𝑝𝑖𝑋−𝑝𝑖𝑌).
equally inequitable, so there was no possibility of ethical As shown in Tables 4(a) and 4(b), the probability of
votinghere.Thesetreatmentsservedasacontrol,tomeasure voting is increasing in the amount of noise (decreasing in
the effect of noise in the decision-making process in the the amount of precision). This is true for both own-voting
absenceofanypotentialforethicalvoting. andcross-voting,whereown-votingisdefinedasvotingfor
Eachofthefourtreatmentswasacombinationofsession thecandidateofferingone’stypethehigherpayoffandcross-
size(12or18)andpayoffstructure(asymmetricorsymmet- voting is defined as voting for the candidate offering one’s
ric). typethelowerpayoff.However,itismuchmorepronounced
The payoffs shown in Table3 were set such that the forcross-voting.IntheNashequilibrium,thereisnocross-
equilibriumpredictionsofvotingturnoutrateswerethesame voting because in the presence of extremely high levels of
acrossasymmetricandsymmetricpayoffstructuresbecause precision(lownoiselevels)inthedecision-makingprocess,
the difference between the two candidates’ offered payoffs and in the absence of altruism, nobody would choose to
is identical across the two payoff structures so that the net voteforthecandidateofferingthemalowerpayoffbecause
expectedpayoffdifferenceinfunctionsfrom(3)remainsthe it would imply a negative expected payoff difference and a
same. negativecostcutpoint.Asthelevelofnoiseincreases/thelevel
Meanwhile, the number of subjects per session would of precision falls, it becomes increasinglylikely that at least
affecttheturnoutpredictionsthroughtheeffectoftheprob- someofthesubjectswouldcross-voteinspiteofthenegative
abilityofone’svotebeingpivotalonthenetpayofffunction. expectedpayoffdifference.
8 JournalofAppliedMathematics
Table5:Predictedturnoutprobabilitiesfordifferentlevelsofethicalpreferences(𝛼).
For𝜆=7 𝛼=0 𝛼=0.1 𝛼=0.25 𝛼=0.5 𝛼=0.75 𝛼=1
Type1forCandidate𝑋 0.3071 0.2708 0.1606 0.0295 0.0052 0.0009
Type1forCandidate𝑌 0.0793 0.1575 0.3988 0.8232 0.9637 0.9934
Type2forCandidate𝑋 0.0793 0.0818 0.1059 0.1629 0.1702 0.1707
Type2forCandidate𝑌 0.3071 0.2921 0.2347 0.175 0.171 0.1708
Candidate𝑋(%ofallvoters) 0.1932 0.1763 0.13325 0.0962 0.0877 0.0858056
Candidate𝑌(%ofallvoters) 0.1932 0.2248 0.31675 0.4991 0.56735 0.5821
For a session with 12 subjects (Table4(a)) the cost hadnosignificanteffectandturnoutwassignificantlyhigher
cutpointis$0.122,whichmeansthattherearesevenpossible in sessions with symmetric payoffs relative to those with
costs—between $0.00 and $0.12—that are low enough that asymmetricpayoffs.(However,thelackofasizeeffectisnot
someonewouldvote(forthecandidateofferingone’stypethe entirelysurprisingconsideringthatthetheoreticaldifferences
higherpayoff)withprobabilityof1.Sevenoutof22possible in economically self-interested voting rates were not very
costs equal approximately 30%, which is in line with the large, as seen in Tables 4(a) and 4(b).) Third, some ethical
turnoutprobabilitiesintheNashequilibrium.Forasession voting (where motives were confirmed through the chat
of18subjects(Table4(b)),thecostcutpointis$0.1047,which transcripts) and potentially ethical voting were observed;
means that there are six possible costs—between $0.00 and however,itwasmostlyfoundonlyundercertainconditions
$0.10—that are low enough to induce someone to vote. Six andnotenoughwasobservedtoexplaintheovervoting.
outof22possiblecostsequalapproximately27%,whichisin
linewiththeturnoutprobabilitiesintheNashequilibrium. 3.1. Terminology. Before getting into the analysis, some ter-
Oneotherthingtonoteisthatasthenoiselevelincreases minology that is used throughout this section needs to be
tothepointwheresubjectsaredecidingrandomly,thehighest defined.“Turnout”referstoallvoting,regardlessoftypeor
possiblepredictedturnoutratewillbe33%(33%ofeachtype candidate. “Favored” refers to votes for the candidate who
wouldchooseCandidate𝑋,33%wouldchooseCandidate𝑌, offersthatsubject’stypeahigherpayoffthanwhattheother
and 33% would abstain). This is much lower than the 50% candidate offers: Type 1 for Candidate 𝑋 and Type 2 for
probabilitiesthatwouldbethecaseiftheycouldonlychoose Candidate 𝑌. Similarly, “unfavored” refers to votes against
betweenvotingandabstaining,suchasinLevineandPalfrey their economic self-interest—Type 1 for Candidate 𝑌 and
[4].ThiswillaffecttheabilityofQREtoexplainlargeamounts Type 2 for Candidate 𝑋. The unfavored category does not
ofovervoting,aswillbediscussedintheresultssection. distinguishbetweenpotentiallyethicalvoting—Type1voting
Additionally, the model predicts that increasing levels for𝑌inasessionwithasymmetricpayoffs—andcross-voting
of ethical preferences can increase the percentage of the thatisdoneforotherreasons.
electorate voting for the ethical candidate and decrease the
percentage voting for the selfish candidate (Table5). (As in 3.2.VotingTurnoutRates
Table4,thefirstfourrowsofTable5measurethepercentage
ofthatvoters’type whovoteforagiven candidate. Thelast 3.2.1.Overview. Theoverallturnoutrates,averagedacrossall
tworows,capturingthevotingdecisionsacrossvotertypes, rounds and broken down by type-candidate combinations,
measurethepercentageofallvoters.)Asthelevelofaltruism canbeseeninTable6(a).
increases, ethical voting would reduce the closeness of the The turnout rates for the candidates offering each type
race,whichinturnwoulddecreasetheprobabilityofcasting a higher payoff—Candidate 𝑋 if Type 1 and Candidate 𝑌 if
apivotalvote.Thislossofpivotalitywouldnotmatterforthe Type2—aremuchhigherthaneventhehighestturnoutrates
ethicalvoters,whoseincreasinglystrongaltruisticmotivation predicted under QRE for any type-candidate combination,
wouldoutweighthis,butitwouldleadthosewhowouldhave whichtoppedoutat33%eveninthepresenceofthehighest
votedfortheselfishcandidatetoabstain. possiblelevelsofnoise.Possiblereasonsforthisovervoting,
even relative to QRE predictions, will be addressed later.
3.ResultsandDiscussion The second observation is that potentially ethical voting—
Type1votingforCandidate𝑌intreatmentswithasymmetric
Thereareseveralthemesthatappearthroughouttheresults. payoffs—is higher than nonaltruistic cross-voting in these
Thefirstisthatvoterturnoutfarexceededeventhehighest same treatments, which is Type 2 for Candidate 𝑋. In
possibleturnoutpredictedbyQRE.Thiswaslargelyconcen- the treatments with symmetric payoffs, where there is no
trated in votes for the candidate offering the higher payoff possibilityofethicalvoting,thisisnotthecase.
tothatsubject’svotertypeandassuchisnotduetoethical Therearetwoothernotablerelationshipsinthedata.The
voting. This indicates a strong economic self-interest. In first is that the turnoutrate is higher, forall type-candidate
addition, contrary to the predictions of the model—which combinations, in the treatments with symmetric payoffs
werethat,intheabsenceofaltruism,thenumberofsubjects than in those with asymmetric payoffs. This contradicts
(sizeoftheelectorate)wouldaffectturnoutbutthesymmetry the model’s prediction, which was that there would be no
of the payoff structure would not—the number of subjects difference in turnout between the two payoff structures
JournalofAppliedMathematics 9
Table6:(a)Votingturnoutrates(allrounds).(b)Votingturnoutrates(rounds11–20).(c)Votingturnoutrates(rounds1–10).
(a)
Alltreatments Symmetric Asymmetric 12 18
Type1forCandidate𝑋 0.4911 0.5467 0.4356 0.4500 0.5185
Type1forCandidate𝑌 0.0639 0.0656 0.0622 0.0597 0.0667
Type2forCandidate𝑋 0.0544 0.0733 0.0356 0.0653 0.0472
Type2forCandidate𝑌 0.5250 0.5533 0.4967 0.5417 0.5139
(b)
Alltreatments Symmetric Asymmetric 12 18
Type1forCandidate𝑋 0.4622 0.5022 0.4222 0.3972 0.5056
Type1forCandidate𝑌 0.0344 0.0533 0.0156 0.0167 0.0463
Type2forCandidate𝑋 0.0211 0.0200 0.0222 0.0222 0.0204
Type2forCandidate𝑌 0.4911 0.5533 0.4289 0.5250 0.4685
(c)
Alltreatments Symmetric Asymmetric 12 18
Type1forCandidate𝑋 0.5200 0.5911 0.4489 0.5028 0.5315
Type1forCandidate𝑌 0.0933 0.0778 0.1089 0.1028 0.0870
Type2forCandidate𝑋 0.0878 0.1267 0.0489 0.1083 0.0741
Type2forCandidate𝑌 0.5589 0.5533 0.5644 0.5583 0.5593
except for possibly higher turnout for Type 1 voting for WithinFigure2, thetoprowshowstheturnoutratesfor12
Candidateintreatmentswithasymmetricpayoffs(potentially subjects(a)andfor18subjects(b),andthebottomrowshows
ethical voting). Also, there is no systematic difference in theturnoutratesforasymmetricpayoffs(c)andsymmetric
turnout between the two session sizes. This contradicts the payoffs(d).
model’s predictions, in which turnout would be higher in A few trends appear in these figures. First, in all treat-
sessionsof12thaninsessionsof18. mentstheturnoutratedecreasesastheroundsgoon.Second,
Theturnoutratesinthesecondhalfofthedata(rounds there is no systematic difference between the 12- and 18-
11–20)canbeseeninTable6(b). subject turnout rates. This is contrary to the theoretical
This focus on the second half of the sessions (largely) results, which showed higher turnout for smaller electorate
sidesteps the effect of the learning curve that the subjects sizes. And finally, there is a clear difference between the
experienced in the early rounds. Once again, the effects of treatments with asymmetric versus symmetric payoffs. As
size and symmetry run counter to the model’s predictions. before, turnout in treatments with symmetric payoffs is
Turnout is higher in treatments with symmetric payoffs almostalwayshigher.This,too,iscontrarytothetheoretical
relativetothosewithasymmetricpayoffs,andsessionsizehas results, which were identical across payoff structures for a
nosystematiceffect. givenelectoratesize.
In rounds 11–20 the turnout rate for potentially ethical
voting—Type 1 voting for Candidate 𝑌 in treatments with 3.2.2. Nonparametric Permutation Test. A nonparametric
asymmetric payoffs—is even lower than the nonaltruistic permutation test (also known as a randomization test) was
cross-voting done by Type 2 for Candidate 𝑋, again in the used to determine whether the turnout rates are different
asymmetric treatments. This points to altruism dissipating acrosstreatments.Thisnonparametrictestwasusefulwhen
and economic self-interest taking over fairly quickly. Given looking at session averages because there are only 12 data
that any possible altruism died out by the second half, it points. The results from the permutation tests showed the
would be useful to look at the first half of the data (rounds same trends that were observed in the stylized facts: the
1–10).ThisisshowninTable6(c). number of subjects per session does not significantly affect
Hereweseepotentialaltruism,withaturnoutprobability the probability of voting (except for symmetric sessions
of 0.1089 for Type 1 subjects voting for Candidate 𝑌 in acrossallrounds),andsubjectsintreatmentswithsymmetric
asymmetric treatments versus 0.0489 for Type 2 subjects payoffsaresignificantlymorelikelytovotethanaresubjects
votingfor Candidate𝑋 in those same treatments.Also, the intreatmentswithasymmetricpayoffs.
sizeandsymmetryrelationshipsarethesameasbefore,albeit Detailsaboutthetestandtheresultscanbefoundinthe
notasstrongly. Appendix.
Itisalsohelpfultoseethetrendsovertime,notjustthe
averages. Figure2 shows the turnout rate—overall turnout, 3.2.3.EconometricAnalysis. Inthelogitanalysis,thedepen-
rather than being broken down into the different type- dentvariablesarethedifferentcategoriesofvoterturnout—
candidatecombinations—overthecourseoftheexperiment. (“turnout”),favored,andunfavored.Ineachofthefollowing
10 JournalofAppliedMathematics
1 1
0.8 0.8
0.6 0.6
Pr (vote) 0.4 Pr (vote) 0.4
0.2 0.2
0 0
1 3 5 7 9 11 13 15 17 19 1 3 5 7 9 11 13 15 17 19
Round Round
12,asymmetric 18,asymmetric
12,symmetric 18,symmetric
(a) (b)
1 1
0.8 0.8
0.6 0.6
e) e)
ot ot
v v
Pr ( 0.4 Pr ( 0.4
0.2 0.2
0 0
1 3 5 7 9 11 13 15 17 19 1 3 5 7 9 11 13 15 17 19
Round Round
12,asymmetric 12,symmetric
18,asymmetric 18,symmetric
(c) (d)
Figure2:Votingturnoutrates(toprowsymmetricversusasymmetric,bottomrow12versus18).
regressions, a random effects term was used to account for favored𝑖𝑡 =𝛾0+𝛾1(average vote difference)𝑖𝑡
correlationwithineachsubject’s20decisionsthatcannotbe
attributed to any of the other independent variables—some +𝛾2(# of previous losses)𝑖𝑡+𝛾3cost𝑖𝑡
subjects were inherently more (or less) likely to vote, or to
voteinacertainway,thanothersubjects. +𝛾4type𝑖𝑡+𝛾5size𝑖𝑡+𝛾6asymmetric𝑖𝑡
Table7showstheresultsforthefollowingregressions:
turnout𝑖𝑡 =𝛽0+𝛽1(average vote difference)𝑖𝑡 +𝛾7round𝑖𝑡+𝛾8(type 1×asymmetric)𝑖𝑡
+𝛽2(# of previous losses)𝑖𝑡+𝛽3cost𝑖𝑡 +𝛾9(# same type)𝑖𝑡
+𝛽4type𝑖𝑡+𝛽5size𝑖𝑡+𝛽6asymmetric𝑖𝑡 +𝛾10(# group favored votes)𝑖𝑡
+𝛽7round𝑖𝑡+𝛽8(type 1×asymmetric)𝑖𝑡 +𝛾11(# group unfavored votes)𝑖𝑡
+𝑤 +𝑢 ,
+𝛽9(# same type)𝑖𝑡 favored,𝑖 favored,𝑖𝑡
+𝛽10(# group favored votes)𝑖𝑡 unfavored𝑖𝑡 =𝜃0+𝜃1(average vote difference)𝑖𝑡
+𝛽11(# group unfavored votes)𝑖𝑡 +𝜃2(# of previous losses)𝑖𝑡+𝜃3cost𝑖𝑡
+𝑤turnout,𝑖+𝑢turnout,𝑖𝑡, +𝜃4type𝑖𝑡+𝜃5size𝑖𝑡+𝜃6asymmetric𝑖𝑡