The Achilles Heel of Plurality Systems: Geography and Representation in Multiparty Democracies

17 Pages·2015·0.81 MB·English

The Achilles Heel of Plurality Systems: Geography
and Representation in Multiparty Democracies
Ernesto Calvo
University of Maryland
Jonathan Rodden
Stanford University
BuildingontheunfinishedresearchprogramofGudginandTaylor(1979),weanalyticallyderivethelinkagebetweena
party’sterritorialdistributionofsupportandthebasicfeaturesofitsvote-seatcurve.Wethendemonstratetheusefulness
ofthecorrespondingempiricalmodelwithananalysisofelectionsinpostwarGreatBritain,focusinginparticularonthe
transformationoftheLiberalsfromaterritoriallyconcentratedtoadispersedpartyinthe1970s.Weshowthatmajoritarian
biasesincreasewiththenumberofparties,andmajoritariansystemsharmsmallpartieswhentheirvoteismoredispersed
thanaverage,andlargepartieswhentheirvoteismoreconcentratedthanaverage.Moreover,theevolvingexperiencesof
LabourandConservativesdemonstratehowaparty’sterritorialsupport,andhenceitsexpectedseatpremiumorpenalty,
changeswithitselectoralfortunes.Thismodelhasawidevarietyofapplicationsinmultipartymajoritariandemocracies
aroundtheworld.
Three party elections, it seems, can enter the political geography as the “Achilles heel” of plurality
realms of fantasy, making the bias of the cube systems of representation. They saw this not only as a
(orsimilar)lawseemaverytameaffair. normative problem of democratic representation, but
alsoasapuzzleforpositivistsocialscience.
—GudginandTaylor(1979,93)
It remains both. Much of the literature on majori-
S
cholars and pundits are relatively good at pre- tarian democracies since then views third parties as
dicting the transformation of votes to seats in annoyances that must be assumed away in order to un-
periodsofstabletwo-partycompetitioninmajori- derstandthetransformationofvotestoseatsamongthe
tariandemocracies.Election-nightdramaunfolds,how- twomajorpartiesthatstandachanceofformingagov-
ever, along with striking asymmetries in the translation ernment.Butthisapproachmissesthemarksubstantially
of votes to seats, when third parties experience surges inthevastmajorityofmajoritariandemocraciesoutside
insupport.Undertheseconditions,the“winning”party the United States, where winner-take-all districts yield
mighthopetoformanoutrightlegislativemajoritywith differentconstellationsofpartycompetitionindifferent
less than 40% of the vote. The Canadian Conservatives regionsanddistricts,andwherethemajorpartiessurge
achievedthisin2011,andtheBritishConservativescame and decline in ways that are not mirror images of one
closein2010.IncountrieslikeIndia,wheremultiplepar- another.Moreover,“third”partiessometimestransform
ties surge and decline and geographic support bases are themselves from small, geographically concentrated
constantlyinflux,thetransformationofvotestoseatscan regional parties to bona fide national parties that
bequitedifficulttopredict. compete in all districts. Prime examples are Labor and
After observing some sudden and surprising asym- SocialDemocraticpartiesinearly20th-centuryEurope.
metries in the transformation of votes to seats in Great Likewise,apartythatseemstobefullynationalizedaftera
Britain in the 1970s, Gudgin and Taylor (1979) referred periodofelectoralsuccesscan,inaperiodofmisfortune,
to multiparty competition in their classic book on quicklyretreattoitsoriginalgeographicbailiwick.
ErnestoCalvoisAssociateProfessor,GovernmentandPolitics,UniversityofMaryland,3140-BTydingsHall,CollegePark,MD20742
([email protected]). JonathanRoddenisProfessor,DepartmentofPoliticalScience,SeniorFellow,HooverInstitution,StanfordUniversity,
EncinaHall,616SerraStreet,Stanford,CA94305([email protected]).
Replication data and code are available at http://www.gvpt.umd.edu/calvo/data.htm. The authors wish to thank Jim Adams, Bernie
Grofman,andAlbertoPenade´sforhelpfulcomments,andtheUniversityofMaryland,theInstituteforResearchintheSocialSciencesat
Stanford,andtheHooverInstitutionforfinancialsupport.
AmericanJournalofPoliticalScience,Vol.59,No.4,October2015,Pp.789–805
(cid:2)C2015,MidwestPoliticalScienceAssociation DOI:10.1111/ajps.12167
789
790 ERNESTOCALVOANDJONATHANRODDEN
Such developments have large and predictable More importantly, we provide an empirical model
implications for the transformation of votes to seats, that election scholars can implement in a wide variety
and they can sometimes create outcomes that seem ofsettings.Researcherscanuseourmodeltoassessand
stunningly unfair. We know remarkably little about predict the transformation of votes to seats associated
the implications of these patterns of geographic surge with a wide range of scenarios as parties enter or leave
and decline for legislative representation in multiparty theelectoralarenaandastheirgeographicsupportbase
plurality systems. The literature got off to a good start expands and contracts. Our model can help shed light
whenGudginandTaylor(1979)laidthegroundworkfor on the transformation of votes to seats associated with
acoherenttheorylinkingthegeographyofpartysupport historical situations like the rise of workers’ parties and
to patterns of legislative representation. While shying thesqueezingofLiberalpartiesintheearly20thcentury,
away from a general model and drawing on historical or the entry and geographic expansion of third parties
examples,theymadesomeimportantobservations. like the Cooperative Commonwealth Federation (even-
“Gudgin and Taylor showed us the way in the late tuallytheNewDemocraticParty)andtheReformParty
1970s,”asJohnston(2002,28)putit,“butveryfewhave in Canada. It can also help with pre-election prognosti-
followedtheirsignposts.”Theyprovidedsomeinductive cationsaboutseatsharesincomplexmultipartysystems
observationsthathavestoodthetestoftime,butsurpris- likeIndia’s.
ingly,theirintuitionshaveyettobetransformedintoan Above all, our model also helps shed light on the
explicittheoreticalandempiricalmodel.Thus,theliter- dynamicsofvotesandseatsinmultipartysystemswhere
atureismissingsomethingratherbasic:aclearpredictive thebasicunderlyinggeographicsupportbaseisdifferent
model linking parties’ evolving territorial support with foreachoftheparties,oftenbecauseofdeeplyingrained
theexpectednumberoflegislativeseats. aspectsofpoliticalgeography,andwherethepartiescan
This article picks up where Gudgin and Taylor left expectverydifferentgeographicsupportdistributionsas
offandprovidessuchamodel.Weestablishgeneralthe- theyexpandandcontract(seeRodden2013).
oreticalandempiricalrulesthatcharacterizethelinkage We apply the model to British elections from
between a party’sterritorialdistributionof supportand World War II to the present in order to demonstrate its
the basic features of its seat-vote curve. We analytically usefulness in two distinct ways. First, we show how the
derive an expected majoritarian rate from the mean and modelcapturessomethingthathasalreadybeenintuited
varianceofchoiceprobabilitiesofthemultinomialdistri- by observers of British politics: The Liberal Party has
bution.Wedefinetheterm(cid:2)˙ asthenaturalmajoritarian transformed itself from a geographically concentrated
biasanddescribeitsapplicationinempiricalmodels.We minor party to a dispersed minor party, and after the
provideestimatesofitseffectandasubstantiveexample criticaljunctureofthe1974election,theLiberalsmoved
ofitsuse. from being somewhat advantaged by their relative
Astheterritorialdistributionofpartyvoteschanges concentration of votes to being severely penalized after
overtime,sodotheexpectedmajoritarianandpartisan theirexpansiontonewelectoraldistricts.
biases of electoral rules. We show that majoritarian Second,ouranalysisofUKelectionsalsoilluminates
biases can be approximated by the mean concentration some interesting subtleties about the larger parties that
of votes for all parties, whereas partisan bias can be have not been widely discussed in the literature. The
approximated by party-specific deviations from the dominant approach in the literature has been to ignore
meanvoteconcentration. the“minor”partiesandrelyonanunrealisticassumption
Wepresentthreemainanalyticfindings: ofuniformswingbetweenLabourandtheConservatives
in order to infer the parties’ seat shares in hypotheti-
1. Majoritarianbiaseswillincreasewiththenum- cal tied elections. This approach has been increasingly
berofparties. untenablesincetheresurgenceoftheLiberals.Weshow
2. Majoritarian systems will benefit small parties that swings in votes between parties are far from uni-
whentheirvoteismoreconcentratedthanaver- formacrossdistricts,andthegeographicdistributionof
age and penalize them when their vote is more support for Labour and the Conservatives, and hence
dispersedthanaverage. theirexpectationsregardingthetransformationofvotes
3. Majoritariansystemswillpenalizelargerparties toseats,arequitedifferentastheygrowandcontract.
when their votes are more concentrated than Geography-basedpartisanbiasesbenefitedtheCon-
averageandrewardthemwhentheirvoteismore servatives until recently, since they have been a large
dispersed. partywithavotethatwasconsistentlymoreterritorially
GEOGRAPHYANDREPRESENTATION 791
dispersedthantheotherparties.YetweshowthatLabour’s in equivalent scenarios, the seat differential is classified
concentratedpocketsofsupportintraditionalworking- aspartisanbias,whichwithsomealgebracanbedecom-
classstrongholdsallowittosufferfewerseatlossesdur- posedintocomponentsthatarecausedbycross-district
ingitsperiodsinthewilderness(e.g.,theThatcheryears) asymmetriesindistrictsize,turnout,andpartysupport.
than do the Conservatives during their lean times (e.g., This approach has yielded important insights.
theBlairyears).Moreover,Labour’splatformmoderation For example, low turnout in Labour strongholds and
inthe1990sallowedittoradicallytransformitsgeogra- overrepresentation of Scotland generate consistent
phy, gaining support in moderate Southern districts to bias in favor of Labour, whereas the asymmetries in
achieveheretoforeimpossibleseatpremiums. the geography of support favored the Conservatives
Thisexampledemonstrateshowourmodelcanhelp (relativetoLabour)intheinitialpostwarperiodbecause
researchers understand the dynamics of votes and seats Labour’ssupportwasmoregeographicallyconcentrated
aspartiesexpandandcontractinotherpluralitysystems. (in urban and mining constituencies) than that of the
By focusing on hypothetical tied elections, existing Conservatives(Johnston2002).
researchmissesoutonanimportantsubtletyilluminated Theentireframeworkwasbasedontheassumption
by our model: Depending on its geographic support oftwoparties,whichwasnolongertenableaftertheresur-
distribution, a plurality system that creates asymmetric gence of the Liberals in the mid-1970s. Borisyuk et al.
seat bonuses for a party during good times can create (2010)thusextendtheoriginalBrookesapproachbycon-
asymmetric penalties for the same party during bad sideringadifferenttypeofnotionalcomparisonelection
times(andviceversa). builtonhypotheticalscenariosinwhicheachofthethree
majorpartiesswapsitsnationalvoteshares,withthevote
swings applied uniformly across all districts. Borisyuk
etal.(2010)showthatthisapproachyieldssubstantially
Geography,Votes,andSeats:An
different estimates of electoral bias than the two-party
UnfinishedResearchAgenda
approach: In particular, the new estimates suggest that
Labour received a substantial seat bonus in the 1980s
Gudgin and Taylor were part of a larger ongoing effort thatwasnotcapturedinthesimpletwo-partyanalysis.
to understand the properties of vote-seat curves and Anaggingweaknessinthisliteratureisthatitdoesnot
explain the origins of observed disjunctures between provideamodeloftheoverallvote-seatcurve.Rather,a
votes and seats. An important goal in this literature is greatdealofworkisdonebythebluntandoftenunrealis-
to distinguish between majoritarian bias, whereby the ticassumptionthatvotesswinguniformlyacrossdistricts
largestpartyalwaysreceivesaseatpremium,andpartisan from one election to the next. By design, this approach
bias, whereby some parties are expected to receive a cannotcapturetheeffectsofdifferencesintheterritorial
larger seat share than other parties with a similar share distribution of support—and hence partisan bias—that
of the vote. Partisan biases can result from asymmetries areexperiencedbythesamepartywhenitissurgingversus
inthesizeofdistricts(malapportionment),asymmetries whenitisdeclining.Asweshowbelow,thesedifferences
inturnoutacrossconstituencies,andasymmetriesinthe can be substantial. Our approach is to explicitly model
partisan distribution of support across constituencies. therelationshipbetweenterritorialsupportandpartisan
Thisarticlefocusesonthelatter,whichcancomeabout bias,whichaffordsus(1)analternativewayofmeasuring
either because of “natural” geographic patterns of party-specific biases, owing to electoral geography that
partysupportorintentionalgerrymandering.Johnston, doesnotrelyonhighlyunrealisticassumptions,and(2)
Rossiter, and Pattie (1999) and Borisyuk et al. (2010) aframeworkforexplainingthesebiases.
have provided evidence that the territorial distribution OurapproachisalsoadeparturefromtheAmerican
of support across districts is by far the most extensive political science literature, which models the relation-
sourceofpartisanbiasintheUnitedKingdom. shipbetweenvotesandseatsbutlargelyignoreselectoral
TheearlyworkofBrookes(1959,1960)setthestage geography and multiparty systems. Much of this litera-
foraBritishandCommonwealthliteraturethatattempts ture builds from Kendall and Stuart’s (1950) statement
tomeasureanddecomposeelectoralbias(Johnston1979, of the cube law and searches for more general ways of
2002;Johnston,Rossiter,andPattie1999).Theirapproach characterizing the relationship between votes and seats
wastoapplya“uniformswing”todistrict-levelelection (e.g.,Grofman1983;King1990;KingandBrowning1987;
results to examine hypothetical scenarios such as equal Taagepera1973;Tufte1973).Thisliteraturehasgivensur-
orreversedoverallnationalvotesharesforthetwomajor prisingly little attention to the geography of party sup-
parties. If one party receives more seats than the other port. Until recently, the empirical literature on electoral
792 ERNESTOCALVOANDJONATHANRODDEN
biashasreliedtoasurprisingextentonaggregatenational seats, which result from differences in the mean party
dataratherthandistrict-leveldata(e.g.,Ansolabehereand concentrationofdistrict-levelvotes,frompartisanbiases
Snyder2008;King1990;Tufte1973). rewarding or penalizing individual parties, which result
Morerecently,inthespiritofBrookes,Gelmanand fromcross-partydeviationsfromthatmean.Thismodel
King (1990) have used transformations of district-level allows us to describe the evolution of majoritarian and
votes to estimate bias and responsiveness, but with the partisanbiasesovertimeasparties’supportdistributions
exceptionofGrofman,Koetzle,andBrunell(1997),this evolve.
literature has not attempted to disentangle the various
sources of partisan bias. Moreover, while U.S. scholars
pointoutthatobservedasymmetriesinthetransforma-
TheGeographyofSeatsandVotesin
tionofvotestoseatsareexplainedbythemoreefficient
MultipartySystems
geographic distribution of Republicans across districts
(e.g.,ChenandRodden2013;Erikson2002),theydonot
explicitlymodelelectoralgeography,andforobviousrea- Letusbeginwithanexample.Imagineanelectoralsystem
sons,U.S.empiricalstudieshavelittletosayaboutminor withtwoparties,aConservativeandaLiberalparty,com-
parties. In fact, because of the influence of “Duverger’s peting in 100 single-member districts that are perfectly
Law,”effortstoextendclassicvote-seatmodelstomulti- apportioned. Every registered citizen casts a vote, with
partysystemstypicallyfocusontheimpactofincreasing the Conservatives collecting 55% and the Liberals 45%.
thedistrictmagnitudeandmovingtoproportionalrep- Let us imagine two different scenarios for this national
resentation(e.g.,Taagepera1986;TaageperaandShugart election:
1993).
1. In the first scenario, parties are perfectly dis-
Another closely related literature examines the
persed over the space. In any given district, the
geographic concentration of support for parties across
Conservatives win the same vote share, 55%,
districts as one of the factors driving the vote share a
while the Liberals collect the remainder, 45%.
small insurgent party needs in order to win its first seat
Given that the Conservatives win a majority of
(Taagepera1989,2002).
thevoteineverydistrict,withjust55%ofthevote
WebuildmostdirectlyontherecentworkofCalvo
they collect 100% of the seats. In this scenario,
(2009)andLinzer(2012).Inattemptingtomodelthedis-
a perfectly dispersed Conservative party wastes
tortions introduced by the entry of new parties in early
no votes and wins every single contest. That is,
20th-century Europe during the era of franchise expan-
single-memberdistrictsproduceawinner-takes-
sion,CalvointroducedageneralizedversionofKingand
allallocationofseats,andtheelectoralrulesdis-
Browning(1987)toexaminetheimpactofpartyentryin
playdramaticmajoritarianbiases.
thecontextofsingle-memberdistrictswithastabletwo-
2. Inthesecondscenario,partiesareperfectlycon-
party system.However, Calvo (2009) did not model the
centrated by district. Therefore, the Conserva-
centralconceptofthisarticle:cross-partyandtime-series
tives collect 100% of the vote in 55 districts
variationinthegeographicconcentrationofvotes.Then,
while the Liberals collect 100% of the vote in
more recently, Linzer (2012) proposed a model of seats
theremaining45districts.Becausevotesareper-
and votes that considered the empirical distribution of
fectlyconcentrated(inperfectlyapportioneddis-
vote shares at the district level, effectively incorporating
tricts),theConservativeswin55%ofthevoteand
district-levelsupportinamodelofseatsandvotes.How-
55% of the seats, whereas the Liberals receive a
ever,Linzer(2012)doesnotprovideageneraltheorythat
seatsharethatisequaltotheirvoteshare.Inthis
explainstheallocationofseatsastheterritorialdistribu-
scenario,aperfectconcentrationofvotesresults
tion of support changes. Instead, he takes the territorial
in a strictly proportional allocation of seats with
distribution of votes as given and conditions the alloca-
nomajoritarianbiases.
tionofseatstothedifferentempiricalexamplesobserved
in the data. Our objective is to provide a more general Thedescriptionofthesetwoscenariosillustrateshow,
rationaletoexplaintheexpectedallocationofseatsasthe inatwo-partysystem,theterritorialconcentrationordis-
territorialdistributionofpartyvoteschanges. persion of the vote has a dramatic effect on the majori-
Inthenextsection,weintroducereaderstoamodel tarianpropertiesofsingle-memberdistrictFTPrules.
thatexplainsmajoritarianandpartisanbiasesasafunc- Letusdescribethefirstscenarioasafullynationalized
tion of the geographic distribution of votes. We dis- party system, where a party wins roughly the same vote
tinguish majoritarian biases in the overall allocation of sharesineverydistrict,andthesecondscenarioasafully
GEOGRAPHYANDREPRESENTATION 793
denationalizedorsegmentedpartysystem,whereparties In Equation (1), S describes the total number of
cL
controldifferentregions.Aspoliticalsystemsevolve,the seats that Party L expects to collect in the national elec-
territorialdistributionofallparties’voteschangesandso tion c as a function of the probability of winning K
dothepropertiesofelectoralrules. single-member districts, the majoritarian parameter (cid:2),
Similargeographiceffectswillbeshownwhenthere andtheoverallvotesharev .Propertiesoftheelectoral
cL
are more than two parties competing for votes. Table 1 rulesdeterminethevalueofthemajoritarianparameter
previews the expected majoritarian and partisan biases (cid:2), which takes the value of 1 if the system allocates seat
withmorethantwoparties.Aswewillshow,whenthedis- shares in proportion to vote shares, v =s = S /K.
cL cL L
tributionofpartyvotesisroughlysimilaracrossdistricts Meanwhile,valueslargerthan1,(cid:2) >1,provideseatpre-
(dispersedvote),majoritarianbiaseswillbelarger.When miumstothewinningparty,v > 1,andseatpenalties
cL 2
partiesconcentratetheirvoteinafewdistricts,majoritar- to the losing party, v < 1. Substituting (cid:2) =3 in our
cL 2
ianbiasesareattenuated.Whenthevoteofasmallparty two-partysystemallocatesseatsaspredictedbytheven-
j ismoreconcentratedthantheaverageparty,itwillwin erablecubelaw,withPartyLcollecting23%ofseatswith
a seat premium (positive partisan biases). By contrast, 40% of the vote and Party R collecting 77% of the seats
whenthevoteofasmallparty j ismoredispersedthan with60%ofthevotes.
the average party, it will be penalized (negative party DepartingfromKingandBrowning,wewillderivea
biases). naturalmajoritarianparameter(cid:2)˙ insingle-memberdis-
Next,weprovideamoretechnicaldescriptionofour tricts solely from the properties of the binomial distri-
approach,modelingmajoritarianbiasesasafunctionof bution. We will regard (cid:2)˙ as the Natural Majoritarian
the mean dispersion of votes and partisan biases as a Rate in single-member FTP systems, where the param-
functionofparty-specificdeviationsfromthismeanter- eter (cid:2)˙ is explained by the expected mean and variance
ritorialdistribution.Webeginwithageographiccharac- of the binomial distribution in K districts. Given that
terizationofmajoritarianbiasesintwo-partysystemsand weknowthemeanandvarianceofthevoteforPartyL,
thenmoveontomultipartysystems. we can write the square of the coefficient of var(cid:8)iation,1
CV2 = (cid:4)2 = vcL(1−vcL) = 1−vcL so that CV = 1−vcL,
(cid:3) v v v
cL2 cL2 cL cL
andanalyticallysolvethefollowing:
Two-PartySystems
(cid:2) (cid:3) (cid:4) (cid:5)(cid:6)(cid:7)
v −1
Ourbasicapproachistomodifytheinverselogisticequa- SL = K 1+exp −(cid:2)˙ln 1−cLv (2)
cL
tion (King and Browning 1987) that allocates seats to
(cid:9) (cid:10) (cid:12)
parties as a function of vote shares, allowing majoritar- (cid:11)
1 1−v 1 v
ian biases to be expressed as a function of the expected log((cid:2)˙)=(cid:5) cL + cL (3)
geographicdistributionofvoteprobabilities. 2 vcL 2 1−vcL
Let us begin by defining two parties, J ≡{L,R},
Equation(3)rewritesthechangeinthemajoritarian
competing for a majority of votes in k ∈ K single- parameter (cid:2)˙ as a function of the mean coefficient of
member districts in an electoral contest c ∈C. For
variation in district-level vote shares, using as input the
simplicity, we will assume that each district k selects
expected mean and expected variance of the binomial
Party L with probability (cid:3) and Party R with proba-
L distribution for Party L and for Party R. We also
bility (1−(cid:3)L), where vote share vL ∼ B(1,(cid:3)L) is a add a parameter (cid:5) that will allow us to rescale the
random variable with a binomial distribution selecting natural majoritarian parameter (cid:2)˙ for overdispersed or
one party candidate per district, with expected mean
underdispersed distributions of votes. For the moment,
E [vL]= K(cid:3)L, and expected variance VAR[vL]= wewillconstrain(cid:5)=1toderiveamajoritarianbiasthat
K(cid:3) (1−(cid:3) ).
L L is purely a function of the expected mean and variance
Wewillalsoassumethatthewinnerofeachdistrict intheterritorialdistributionofvoteshares,(cid:2)˙.
is elected by a simple majority of votes, with an overall
allocation of seats for Party L in contestc derived from
the inverse logistic distribution described in King and 1Inprobabilitytheory,thecoefficientofvariation(CV)isameasure
Browning(1987): ofdispersionofaprobabilitydistribution.Whiletherearemany
differentmeasuresofdispersionthatcanbeusedtoassessthegeo-
(cid:2) (cid:3) (cid:4) (cid:5)(cid:6)(cid:7) graphicdistributionofvotes,thecoefficientofvariationprovides
v −1
S = K 1+exp −(cid:2)ln cL (1) asimplestrategytoincorporatetheexpecteddistributionofparty
cL 1−v votes in the estimation of majoritarian biases. We discuss some
cL advantages to using the Gini coefficient in the empirical section
below.
794 ERNESTOCALVOANDJONATHANRODDEN
TABLE1 TheGeneralIntuition:MajoritarianBiases,PartisanBiases,andtheTerritorial
ConcentrationorDispersionofPartyVotes
MeanTerritorialDistributionofAllParties’Votes
PartySpecificDeviationsfromthe
MeanConcentrationofVotes TerritoriallyConcentrated TerritoriallyDispersed
(cid:13) (cid:13)
(cid:13)C −C¯(cid:13)>0Partyjvoteismore (cid:129)Smallmajoritarianbias (cid:129)Largemajoritarianbias
j
concentratedthantheaverage (cid:129)Positivepartisanbiasifjisasmallparty (cid:129)Positivepartisanbiasifjisasmallparty
party (cid:129)Negativepartisanbiasifjisalargeparty (cid:129)Negativepartisanbiasifjisalargeparty
(cid:13) (cid:13)
(cid:13)C −C¯(cid:13)<0Partyjvoteisless (cid:129)Smallmajoritarianbias (cid:129)Largemajoritarianbias
j
concentratedthantheaverage (cid:129)Negativepartisanbiasifjisasmallparty (cid:129)Negativepartisanbiasifjisasmallparty
party (cid:129)Positivepartisanbiasifjisalargeparty (cid:129)Positivepartisanbiasifjisalargeparty
FIGURE1 Seat-VotesandtheTerritorialDistributionoftheVoteina
Two-PartySystem
Note:Majoritarianbiasisderivedfrom(cid:14)th(cid:8)emeancoef(cid:8)ficiento(cid:15)fvariationofthebinomialdistri-
butionsforPartyLandR,log((cid:2)˙)=(cid:5) 1 1−vcL + 1 vcL .
2 vcL 2 1−vcL
Figure 1 plots the expected allocation of seats as a regionsoftheseat-votecurve.Thatis,giventhatweexpect
functionofvotesharesfromEquations(2)and(3).The thevarianceinvotesharestobedifferentwhentwopar-
proposedspecificationmakesanumberofimprovements ties approach 50% of the vote or when they are in each
overpreviousalternatives.First,itallowsforsmallcorrec- extreme (90%–10%), our model induces small correc-
tionsinmajoritarianbiasesthatresultfromthedifferent tionsthatimproveonmodelfit.Forexample,setting(cid:5)=
level of concentration of district-level votes in different 1, we may compute the natural majoritarian bias when
GEOGRAPHYANDREPRESENTATION 795
FIGURE2 Seat-VotesandtheTerritorialDistributionoftheVotein
aFour-PartySystem
Note: Majoritarian bias is derived from the mean coe(cid:8)fficient of variation of the binomial
(cid:16)
distributionsforPartyLandR,log((cid:2)˙)=(cid:5)(1 J 1−vcj).Thesolidlinedescribesthe
J j=1 vcj
natural majoritarian rate, (cid:2)˙. Profile for the allocation of votes to all parties is given by
v ={v =v ,v =(1−v )∗.6,v =(1−v )∗.3,v =(1−v )∗.1}.
A 1 1 2 1 2 1 2 1
PartyLcollects30%ofthevote(andPartyRtheremain- ofpartyvotesapproximatesthemeanandvarianceofthe
ing70%)as(cid:2)˙ =2.97,veryclosetothecubelaw,(cid:2) =3. binomialdistribution.
Jc
Ontheotherhand,whenPartyLcollects40%ofthe
vote,thereisasmalldeclineinmajoritarianbias,where
(cid:2)˙ =2.77. These small differences in seat premiums MultipartySystems
resultfromadjustmentsintheexpectedvariationinvote
Wenowextendthemodeltoamultipartysetting,allow-
shares in different areas of the seat-vote curve: that is,
ing for different territorial distributions of party votes,
from small differences in the probability that a losing
which,aswewillshow,inducedistinctmajoritarianand
party with mean vote v < 1 will be able to win more
cL 2 partisanbiases.Asinthetwo-partymodel,majoritarian
votesthanv > 1 inatleastafewdistricts.
cL 2 biasesresultfromdifferencesinthemeanterritorialcon-
More importantly, Equations (2) and (3) estimate
centration of party votes. Different from the two-party
a theoretical seat-vote majoritarian bias in FTP single-
model,interpartydifferencesintheterritorialconcentra-
member districts that can be analytically compared
tion of vote shares will induce partisan biases. We will
to empirical distributions and to natural majoritarian
provide structure to these partisan biases, showing that
biasesderivedfromotherelectoralrules.Ourmodelwill
largepartiesthatareconcentratedpayaseatpenaltywhile
alsoallowustosystematicallyexploremajoritarianbiases
smallpartiesthatareconcentratedreceiveaseatpremium.
whenvotesaremoreorlessdispersedthan(cid:3) (1−(cid:3) ),
L L This finding is analytically derived, and its comparative
byrescaling(cid:2)˙ asafunctionof(cid:5).
staticsprovideadeeperunderstandingofthemechanical
Itisimportanttohighlightthatwecanapproximate
propertiesofelectoralsystems.
theseat-votepropertiesofsingle-memberdistrictswith-
PerEquation(3),readerscanverifythatinthetwo-
outanyempiricalinformationabouttheactualallocation
party system the value of the majoritarian parameter (cid:2)˙
ofseats.Indeed,wewillshowbelowthatourmodelcan
will be identical when a party collects v votes or col-
approximate the distribution of seats of electoral rules cL
lectsitscomplement1−v (i.e.,40%or60%).Indeed,
simplybyknowingthedegreetowhichthedistribution cL
with only two parties competing for votes, the expected
796 ERNESTOCALVOANDJONATHANRODDEN
varianceswillbesymmetricforeachofthetwopartiesand For any election, we can map a vote profile,
the natural majoritarian bias (cid:2)˙ will capture all possible v =v ...v , to derive the natural majoritarian bias
c1 cJ
allocations of seats. In two-party systems, the distribu- (cid:2)˙. Let us consider a four-party system as depicted
tion of party votes will be symmetric even in the pres- in Figure 2, where we map the vote share of Party 1
enceofoverdispersionorunderdispersion,toberescaled while adjusting the other parties’ vote at a fixed rate.
by (cid:5). If the vote of one party is territorially dispersed, We begin with a vote profile for election c where
the vote for the other party will be dispersed as well. votesareallocatedaccordingtothefollowingrule:v =
A
Therefore,symmetryintwo-partysystemswillallowthe {v =v , v =(1−v ) ∗ .6, v = (1−v ) ∗ .3, v =
1 1 2 1 2 1 2
majoritarianparameter(cid:2)˙ tocaptureallbiasesintheallo- (1−v )∗.1}.
1
cationofseats.2 However,thisisnolongerthecasewith For example, if Party 1 collects 50% of the votes,
morethantwopartiesunlesstheexpectedvoteprobabil- then party v collects v =(1−.5)∗.6=.3=30% of
2 2
itiesinthemultinomialdistributionarestrictlyindepen- the votes, party v =(1−.5)∗.3=.15=15%, and
2
dentfromeachother. v =(1−.5)∗.6=.1=5%. This profile allows us to
2
Let us begin by generalizing the binomial distribu- map the expected allocation of seats to votes as Party 1
tion used in our two-party election to a multinomial increases its vote share from 0% to 100%. It will also
designformorethantwoparties.WenowconsiderJ >2 allow us to show how the majoritarian properties of
parties with posi(cid:16)tive vote probabilities, v =(cid:3)c1...(cid:3)cJ, single-member FTP rules change as a function of the
thataddupto1, J (cid:3) =1.Theexpectednumberof territorialdistributionofvotes.
j=1 cj
districtsthatwillelectacandidatefromPartyL—outof Figure 2 describes the natural majoritarian rate as
allpossibleKdistricts—is E [v ]= K(cid:3) ,andexpected a solid line constraining the parameter to 1, (cid:5)=1.
L L
variance VAR[v ]= K(cid:3) (1−(cid:3) ). However, the Consistent with Calvo (2009), as we increase the num-
L L L
off-diagonalentries in the covariancematrix will reflect ber of parties from 2 to 4, comparative statics show a
the constraint that an increase in the probability of sharpincreaseinmajoritarianbiases,withthecutpoint
electing a candidate of one party will decrease the thatdivideswinnersandlosersmovingtotheleft.3
probabilityofelectingacandidatefromanotherparty,so Given the vote profile, we can see that the cut
that COV[v ,v ]=−K(cid:3) (cid:3) . This will be important pointbetweenwinnersandlosersisapproximately37%
1 2 1 2
forthestatisticalimplementationofthemodelbutdoes of votes, with parties above this threshold winning a
notaffectthemoregeneralresultsthatfollow. premium in seats and parties below this threshold suf-
As in the previous section, we can derive a natural feringapenalty.Becausethevarianceofthemultivariate
majoritarianbias(cid:2)˙ foreverycombinationofvoteshares, normal is narrower at the tails, majoritarian biases
given that we know the expected mean and variance of rapidly increase as the vote becomes more fragmented.
each probability in the multinomial distribution. As in Indeed, a rapid increase in the coefficient of variation
thetwo-partyexample,wecanevaluatethemultinomial can be equated with more homogeneity across districts
seat-voteequation: andmoredramaticseatpremiums.
e(cid:2)˙ln(vc1) Plugging some values in Equations (4) and (5) will
S = K(cid:16) (4) be illustrative. Consider, for example, that we have
1 Jj=1e(cid:2)˙ln(vcj) four parties with vote shares, v ≡{A=.4,B=.3,
⎛ (cid:10) ⎞ C=.2,D=.1}. Given this profile, we may compute
log((cid:2)˙)=(cid:5)⎝1 (cid:19)J 1−vcj⎠ (5) an e1ffective=n3u.3m3)bearnodf, esluebctsotirtaultipnagrtpiersopoefr3v.3a3lue(se.gin.,
J v .42+.32.22.12
j=1 cj (4) and (5), note that Party A wins 51.3% of the vote
with40%ofseats.Inthenextelection,imaginethathalf
InEquations(3)and(4),theexpectedseatsofParty1
of Party B voters defect to Party D, so that vote shares
resultfromamultinomialdistributionwithpartyvotes,
v , the natural majoritarian parameter (cid:2)˙, and the dis- areallocatedv ≡ {A=.4,B=.2,C=.2,D=.2}.The
c1
fact that Party A has a larger lead has several different
trict magnitude, K. As in the two-party system, the
majoritarianparameter(cid:2)˙ isafunctionofthem(cid:8)eancoef- implications: First, the effective number of electoral
ficientofvariationofallparties’votes, 1 (cid:16)J 1−vcj. partiesincreasesto3.57(e.g.,.42+.212.22.22 =3.57).Second,
J j=1 vcj the increase in the effective number of electoral parties
increases the majoritarian bias in the system by moving
2Inotherwords,ourtheoreticalmodelintentionallyturnsablind
eyetopartisanbiasarisingfromskeweddistributionsofpartyvotes 3Calvo(2009)showsthatthec(cid:16)utpointbetweenwinnersandlosers
intwo-partysystems.Wereturntothisissuebelow. movesexactlytocutpoint = v2.
i
GEOGRAPHYANDREPRESENTATION 797
theseat-votecurvetotheleft.Third,substitutingthenew majoritarianbiasandofpartisanbiases.Becausethevari-
values into Equations (4) and (5), we can see that Party ance will no longer be constant across districts, we will
A has increased its seat advantage to 53.3%: 2% more now consider a mean district vote share v for Party j
ckj
seats than in the previous election. Indeed, although thatisdifferentfromthenationalvotesharev ,suchthat
cj
Party A’s vote total has remained unchanged, increased v (cid:7)=v . Furthermore,to simplify the notation, let us
ckj cj
fragmentation of the vote among the three opposition define a coefficient of variation of party j that may dif-
. .
partiesincreasestheseatpremiumofPartyA. ferfromitsnaturalrate,CVC(cid:22)V = CV,bysomemulti-
j (cid:6)
Let us explain in further detail why an increase in plicativefactor(cid:6) ,capturingvariancesthatarelargerthan
j
the number of parties yields larger majoritarian biases its natural rate, (cid:6) <1, or smaller than its natural rate,
j
in single-member FTP systems. Readers will have al- (cid:6) >1.
j
ready noticed that th(cid:11)e natural majoritarian parameter Todescribehowasymmetriesinthegeographiccon-
(cid:16)
log((cid:2)˙)=(cid:5)(1 J (1−vcj)) will increase as the vari- centration of the vote affect expected party seats, let us
J j=1 vcj augmentourmodelsothat
ance(1−v )becomeslargerrelativetothemeanparty
cj
vote,vcj.Inatwo-partysystem,thecoefficientofvaria- S = K(cid:16)e(cid:2)˙ln(vjc) and (6)
tionislargeforthelosingpartyandsmallforthewinning jc Jj=1e(cid:2)˙ln(vjc)
party.However,asthenumberofpartiesincreases,there
⎛ ⎞
aremorepartieswhosemeanvotesharefallsbelow50%.
(cid:23) (cid:24) (cid:19)C (cid:19)J
Tcohmateiss,smthaellmereaanndvothteesvhaarrieanincethlaergdeern,olemadininagtotrovlcajrbgee-r log (cid:2)cj =(cid:5)1⎝J ∗1C c=1 j=1C(cid:2)Vjc⎠
valuesofCV.Thisfeatureofthemodelcapturesthees-
(cid:2)
sentialcharacteristicofmultipartycompetitioninpolities eCVjc
−(cid:5) (cid:16) . (7)
with single-member districts: A party that collects 60% 2 Jj=1eC(cid:2)Vjc
of the vote and faces four parties with 10% of the vote
each will do considerably better than a party with 60% Ourfinalmodelincludesachangeinmajoritarianbi-
ofthevotethatfacesasingleoppositionpartycollecting asesthatisafunctionofthemeanincre(cid:16)aseint(cid:16)heterritorial
40%ofthevote.Indeed,votefragmentationincreasesthe concentrationofpartyvotes,(cid:5) ( 1 C J C(cid:2)V ).
1 J∗C c=1 j=1 jc
rewardsforsuccessfulparties,asthecutpointforvictory However, Equation (7) also includes a term measuring
moves to the left of 0.5, making large margins of vic- the relative concentration of Party j with respect to all
tsomraylilnpcarretaiesisn.glyvaluabletolargepartiesandaburdento otherparties,−(cid:5)2(cid:16)Jje=(cid:2)C1Vej(cid:2)CcVjc,where−(cid:5)2 isnegativegiven
In real applications, as we will show, votes tend to thatamoreconcentratedpartyvote,C(cid:2)Vjc >C(cid:22)V¬jc,will
vary more widely across districts, yielding majoritarian result in a lower partisan bias for party j in election c,
biasesthatdeviatefromthenaturalmajoritarianrateof (cid:2)cj <(cid:2)c¬j.
the multinomial distribution. As in the two-party ex- Figure 3 describes two parties with different levels
ample, we can adjust the natural majoritarian bias by of concentration of party votes: a more concentrated
allowing (cid:5) to take a number of different values that party, CVcj >CV¬jc, and a more dispersed party,
rescale (cid:2)˙. When (cid:5)=0, then (cid:2)˙ =1 and the allocation CVjc <CV¬jc. The lines intersect at the cut point that
ofseatsisproportionaltovotes.When(cid:5)=1,themodel dividesthewinnersfromthelosersoftheelectoralcontest,
reduces to the expected allocation of seats analytically withpartiesabovethecutpointreceivingpremiumseats
derivedfromthemeanandvarianceofthemultinomial when dispersed and losses when concentrated. Mean-
distribution,(cid:2)˙. while,smallpartiesreceivepremiumseatswhenconcen-
Partiesoftenhaveadistributionofdistrict-levelvotes tratedandseatlosseswhendispersed.
thatismoreconcentratedordispersedthanthatoftheir Equations (6) and (7) provide a formal explana-
competitors,theresultofcontextualandbehavioralfac- tionofwhywinningpartieswithaconcentratedvoteare
torsaffectingtheprobabilityrateindifferentgeographic penalized and winning parties with a dispersed vote are
regions.Heterogeneityandhomogeneityindistrict-level rewardedinmultipartyelections.Inthenextsection,we
votesresultsincoefficientsofvariationthatarelargeror estimatetheproposedmodelusingdatafromtheUnited
smaller than expected—that is, coefficients of(cid:8)variation Kingdomsince1950.Weshowparameterestimatesthat
·
thatdifferfromthenaturalrate,C(cid:22)V (cid:7)=CV = 1−vcj. adjust to expectations and provide a systematic expla-
v
cj nation for (1) the larger seat penalties suffered by the
Equations(6)and(7)describethefullmodel,which
Liberalssince1974and(2)thecontrastingexperiencesof
incorporates both the geographic determinants of the
798 ERNESTOCALVOANDJONATHANRODDEN
FIGURE3 Seat-VotesandtheTerritorialDistributionoftheVoteina
Four-PartySystem,withPartisanBiasforaPartyjandaParty
¬j fromDifferencesintheCoefficientofVariationamong
Parties
Note: Partisan bias is derived from differences in the mean coefficient of variation among parties,
log((cid:2))=−(cid:5)2(cid:16)Jje=(cid:2)C1Vej(cid:2)CcVjc.
LabourandConservativesduringtheirrespectiveperiods withLiberalschippingawayitsvotesinBirminghamand
ofdominanceanddistress. ConservativeschippingawayvotesinSoutherndistricts.
More recently, New Labour has been able to dramat-
ically (but temporarily) reduce its concentration, even
achievingaslightlymoredispersedsupportbasethanits
AnExample:TheGeographic
competitors during the Blair years for the first time in
DistributionofVotesintheUK
history.Figure4alsorevealsthattheGinicoefficientsof
LabourandLiberalsaremuchmorevariablethanthatof
To demonstrate the relationship between the territorial theConservatives.
distributionofvotesandtheallocationofseats,weanalyze More importantly, we see that since 1974, all three
theevolutionofpartyvotesintheUnitedKingdomsince major parties are roughly within the same band, as the
1950.Figure4describestheconcentrationofpartyvotes Liberalshaveachievedafarmoregeographicallydispersed
withGinicoefficients.Eachofthethreemainpartieshas support base. The Liberals have undergone a dramatic
followeddifferenttrajectories. transformationfromalosingbutgeographicallyconcen-
Beginning in 1950, the Conservatives have consis- tratedpartytoalosingbutgeographicallydispersedparty,
tently maintained a nationalized party vote, with rela- withimportantimplicationsfortheentirepartysystem.4
tively low but slowly increasing levels of concentration.
Labour,ontheotherhand,startedwithanidenticallevel
ofnationalization,butithassteadilybecomemorecon-
4FormoreinformationontheevolutionofGinicoefficients,see
centrated than the Conservatives, especially after 1970,
theLorenzcurvesinonlineAppendixA.