Universality in voting behavior: an empirical analysis

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by  Arnab Chatterjee

Universality in voting behavior: an empirical analysis
∗
Arnab Chatterjee, Marija Mitrovi´c, and Santo Fortunato
Department of Biomedical Engineering and Computational Science,
Aalto University School of Science, P.O. Box 12200, FI-00076, Finland
Electiondatarepresentaprecioussourceofinformationtostudyhumanbehavioratalargescale.
In proportional elections with open lists, the number of votes received by a candidate, rescaled
by the average performance of all competitors in the same party list, has the same distribution
regardlessofthecountryandtheyearoftheelection. Hereweprovidethefirstthoroughassessment
ofthisclaim. Weanalyzedelection datasetsof 15countrieswithproportional systems. Weconfirm
3 that a class of nations with similar election rules fulfill the universality claim. Discrepancies from
1 thistrendinothercountrieswithopen-listselectionsarealwaysassociated withpeculiardifferences
0 in the election rules, which matter more than differences between countries and historical periods.
2 Our analysis shows that the role of parties in the electoral performance of candidates is crucial:
n alternative scalings not taking intoaccount party affiliations lead to poor results.
a
Keywords: elections,universality
J

4
2 I. INTRODUCTION berofitsmembers[19],theclassificationofelectoralcam-

paigns [20], etc.
]
h The most studied feature is the distribution of the
Weknowfromstatisticalphysicsthatsystemsofmany
p numberofvotesofcandidates[21–35]. Inthefirstanaly-
particlesexhibit,intheaggregate,abehaviorwhichisen-
-
c forcedby a few basic features of the individual particles, sisby CostaFilho etal.[21], the distributionofthe frac-
o tion of votes received by candidates in Brazilian federal
but independent of all other characteristics. This result
s and state elections seems to decay as a power law with
. isparticularlystrikingincriticalphenomena,likecontin-
s exponent−1inthecentralregion. Followingthisfinding
uous phase transitions and is known as universality [1].
c
severalsimilar analyses have been performed on election
i Empirical evidence shows that a number of social phe-
s data of various countries, like India [25], Indonesia [26]
nomenaarealsocharacterizedbysimpleemergentbehav-
y
and Mexico [28].
h ioroutoftheinteractionsofmanyindividuals. Themost
p striking example is collective motion [2–4]. Therefore, However, Fortunato and Castellano observed that the
[ in the last years a growing community of scholars have analysis by Costa Filho et al. treats all candidates

been analyzing large-scale social dynamics and propos- equally, neglecting the role of the party in the electoral
2
v ing simple microscopic models to describe it, alike the performance [30]. This assumptions appears too strong
2 minimalistic models usedin statisticalphysics. Suchsci- andunjustified,asthefinalscoreofthecandidateislikely
4 entific endeavour, initially known by the name of socio- to depend on whether his/her party is popular or not.
1 physics[5–7],hasbeenmeanwhileaugmentedbyscholars For this reason Fortunato and Castellano argued that
2 and tools of other disciplines, like applied mathematics, characterizing and modelling the competition of candi-
.
2 social and computer science, and is currently referred to datesofthesamepartyismorepromising,asthe perfor-
1 as computational social science [8]. mance of the candidates would be mostly depending on
2 their own activity, rather than on external factors. Such
Elections are among the largest social phenomena. In
1
competition occurs in a peculiar type of voting system,
: India, USA and Brazil hundreds of million voters cast
v viz. proportional elections with open lists, where people
their preferences on election day. Fortunately, datasets
i may vote for a party and one or more candidates. In
X can be freely downloaded from institutional sources, like
this system, people may actually choose their represen-
theMinistryofInternalAffairsofmanycountries. There-
r
tatives by voting directly for them, whereas the number
a fore, it is not surprising that elections have been among
of candidates entering the Parliament for a given party
themoststudiedsocialphenomenaofthelastdecade[9].
typicallydepends onthe strengthofthe partyatthe na-
Bynow,severalaspectsofvotingbehaviorhavebeenex-
tional and/or regional level. In these elections, it was
amined, likestatistics ofturnoutrates[10,11], detection
found that the distributions of the number of votes of a
of election anomalies [12, 13], polarization and tactical
candidate, divided by the average number of votes of all
voting in mayoralelections [14, 15], the relationbetween
partycompetitorsinthesamelist,appeartobethesame
partysizeandtemporalcorrelations[16],therelationbe-
regardlessofthecountryandtheyearoftheelection[30].
tween number of candidates and number of voters [17],
This claim has been recently disputed by Araripe and
the emergenceofthirdpartiesinbipartisansystems[18],
Costa Filho, who found that the universal curve com-
thecorrelationbetweenthescoreofapartyandthenum-
puted in Ref. [30] does not follow well the profile of the
distributionofBrazilianelections,whicharealsopropor-
tional and with open lists.
∗Email: santo.fortunato@aalto.fi Here we carry out the first comprehensive analysis of
2
the distribution of candidates’ performance, using elec- their‘initial’preference,whichdependsoninternalparty
tion results of 15 countries. We focus on proportional rankings, etc. Candidates conquer parliamentary seats
elections, as they feature the open-list system that al- in the order they are ranked in the list, from the first
lowvoterstochoosetheirrepresentatives,enablingareal to the last. However, if a candidate gets a number of
competition between candidates. We conclude that the votes exceeding a threshold, then he/she climbs up the
relative performance, i.e. the ratio between the number ranking even if he/she was initially at the bottom of the
of votes of a candidate and the average score of his/her list. The final order of the candidates is decided based
partycompetitorsinagivenlisthasindeedthesamedis- on the ‘initial’ ordering and the actual votes received by
tribution for countries with similar voting systems, and the candidates. Sweden, Slovakia, Czech Republic, Bel-
thatdiscrepanciesfromtheuniversaldistributionemerge gium, Estonia (until 2002) and Netherlands fall in this
when the election has markedly different features (e.g. category.
largedistricts,compulsoryvotingandweakroleofparties Closed lists In the closed list system the party fixes
inBrazil). Wealsoshowthatpartyaffiliationscannotbe the order in which the candidates are listed and elected.
neglected: statistics of the absolute performance of can- The voter casts a vote to a party as a whole and cannot
didates of different parties, like that investigated in the express his/her preference for any candidate or group of
original analysis by Costa Filho et al., do not compare candidates. Therepresentativesarethenselectedasthey
well between countries. appear on the list, in the order defined before the elec-
tions. Countries voting with this system include Russia,
Italy (since 2006), Spain, Angola, South Africa, Israel,
II. RESULTS Sri Lanka, Hong Kong, Argentina, etc. We did not con-
sider this type of elections in our analysis, as there is no
A. Proportional elections real competition between the candidates.
The allocation of seats to the parties takes place
according to some pre-defined method, e.g. d’Hondt,
The electoral system we wish to study is proportional
Hagenbach-Bischoff, Sainte-Lagu¨e,orsomemodifiedver-
representation (PR) [36]. We analyze data from parlia-
sion of these [37].
mentary elections of 15 countries: Italy (before 1992),
Poland, Finland, Denmark, Estonia, Sweden, Belgium,
Switzerland,Slovenia,CzechRepublic, Greece,Slovakia,
B. Distribution of candidates’ performance: open
Netherlands, Uruguay and Brazil. The basic principle
lists
is that all voters deserve representation and all politi-
cal groups deserve to be represented in legislatures in
In every proportional election, the country is divided
proportion to their strength in the electorate. In order
into districts and each party presents a list with Q can-
to achieve this ‘fair’ representation, the country is usu-
didates. Voters typically choose one of the parties and
ally divided into multi-member districts, each district in
express their preference among the candidates of the se-
turn allocating a certain number of seats. Most coun-
lectedparty. The seatallocationdepends onthe country
tries having a PR system use a party list voting scheme
(see SectionA ofAppendix) andhasa largeinfluence on
to allocate the seats among the parties – each political
how voters choose who they will vote for. The data sets
party presents a list of candidates for each district. On
we considered contain information about the number of
the ballot the voters indicate their preference to a polit-
votes v that each candidate i received and the number
ical party by selecting one or more candidates from the i
of candidates Q of the party list l including candidate
list. Thenumberofseatsassignedtoeachpartyinadis- i i
i. Fromthisinformationonecanderivethetotalnumber
trict is proportional to the number of votes. The party
of votes N collected by the Q candidates of list l . By
list systems canbe categorizedinto open, semi-open and li i i
summing over all party lists in the district Di of candi-
closed.
Open lists Open lists enable voters to express their dateiweobtainthenumber ofvotesNDi inthe district.
The totalnumber ofvotescastduring the wholeelection
preferencenotonlyamongpartiesbutalsoamongcandi-
is indicated as N .
dates. A party presents an unordered, random or alpha- T
Ouranalysisconsistsincomputingtheprobabilitydis-
betically orderedlistofcandidates. Voterschooseoneor
tribution of the number of votes received by candidates,
morecandidates,andnottheparty. Thepositionofeach
suitably normalized. We use the following normaliza-
candidate depends entirely on the number of votes that
tions:
he/she receives. In this category, we have studied data
from Italy (before 1994, when a new system was intro-
• The scaling by Fortunato and Castellano [30],
duced),Poland,Finland,Denmark,Estonia(since2002), where the number of votes vi of a candidate is di-
Greece, Switzerland, Slovenia, Brazil, Uruguay. vided by the average number of votes v =N /Qi
0 li
Semi-open lists Semi-open lists impose some restric-
of all candidates in his/her party list. We shall in-
tions on votersdirectly orindirectly. The votervotes for
dicate it as FC scaling.
either a party or a candidate within a party list. The
parties usually put up a list of candidates according to • ThescalingbyCostaFilho,Almeida,Andradeand
3
100 A100 B100 C
10-1 10-1 10-1
)v/v0 10-2 10-2 10-2
P( 10-3 11997528 10-3 2001 10-3 1995
10-4 11997769 Italy 10-4 22000057 Poland 10-4 12909093 Finland
10-5 1987 10-5 2011 10-5 2007
100 10-2 10-1 100 101 D100 10-2 10-1 100 101 E100 10-2 10-1 100 101 F
10-1 10-1 10-1
)v/v0 10-2 11999940 10-2 10-2 Italy 1987
P( 10-3 12909081 10-3 10-3 FPionllaanndd 22001013
10-4 22000057 Denmark 10-4 22000037 Estonia I 10-4 DEensmtoanriak 22000075
10-5 2011 10-5 2011 10-5
10-2 10-1 100 101 10-2 10-1 100 101 10-2 10-1 100 101
v/v v/v v/v
0 0 0
FIG. 1: Distribution of electoral performance of candidates in proportional elections with open lists, according to FC scaling.
Italy (until 1992), Poland, Finland, Denmark and Estonia (after 2002) follow essentially the same rules, which is reflected by
the data collapse of panel F. The historical evolution of the countries does not seem to affect the shape of the distribution
(panels A to E).
101 A 101 B 101 C
100 100 100
)v0 10-1 10-1 10-1
P(v/ 1100--32 2004 1100--32 1100--32
10-4 2008 Slovenia 10-4 2007 Greece 10-4 2007 Switzerland
10-5 2011 10-5 2009 10-5 2011
101 10-4 10-3 10-2 10-1 100 D101 101 10-4 10-3 10-2 10-1 100 E101 101 10-4 10-3 10-2 10-1 100 F101
100 100 100
)v0 10-1 10-1 10-1
P(v/ 11110000----5432 222000010602 Brazil 11110000----5432 22000049 Uruguay 11110000----5432 SwSUitGzlroeurBvregelIraeutanancazileadyyil 122222900000800000779827
10-4 10-3 10-2 10-1 100 101 10-4 10-3 10-2 10-1 100 101 10-4 10-3 10-2 10-1 100 101
v/v v/v v/v
0 0 0
FIG. 2: Same analysis as in Fig. 1, for Slovenia, Greece, Switzerland, Brazil and Uruguay. Curves are fairly stable at the
national level,buttheydonotcomparewell across countriesandwith theuniversalcurvesofFig.1 (representedin panelFby
thedistributionfortheItalian elections in1987). Suchdiscrepanciesarelikelytobeduetothedifferentelection rulesofthese
countries as compared to each other and to theones examined in Fig. 1, although they all adopt open lists.
Moreira (CAAM) [21], where one considers the ported here. The results for CAAMn are shown in
fraction of votes received by a candidate. Since the Appendix (Figs. B.1, B.2, B.3).
itis unclear to us what one exactly means by that,
we consider two possible normalizations: a) the
The universality discovered in Ref. [30] referred to elec-
fraction of the total votes in the district, vi/NDi; tions held in Finland, Poland and Italy in various years.
b) the fraction of the total votes in the country
Here we confirm the result with a larger number of data
vi/N . We shall refer to them as to CAAMd and
T sets(Fig. 1). PanelsA,BandCdisplaythedistributions
CAAMn, respectively. We rule out the fraction of
for Italy, Poland and Finland, respectively, in different
votes in the party list because the authors made
years. Thestabilityofthecurvewithinthesamecountry
clear that they do not consider party affiliations.
is remarkable, especially on the tail. In panel F we com-
The most sensible definition appears the normal-
pare the distributions across the countries, yielding the
ization at the district level, which will be thus re-
collapse found in Ref. [30]. Elections data in Denmark
4
and Estonia (detailed in panels D and E), appear to fol- type of list is classified as free list.
low the universalcurve as well. We indicate this class of In Brazil, like in Greece, voting is compulsory, and we
countriesasGroupUinthefollowing. InRef.[30]itwas cannot exclude that this plays a role on the shape of
shown that this universal curve is very well represented the distribution. In addition, each state is just one dis-
by a log-normal function. trict, which then comprises a number of voters orders of
Italy (until 1992), Poland, Finland, Denmark and Es- magnitude larger than the typical districts in all other
tonia (after 2002)use open lists [36], inwhichvoterscan elections. This explains why the Brazilian curve spans a
expresstheirpreferencetowardcertaincandidateswithin much largerrange of values for the performance variable
the party list and have a direct influence on the list or- than all other curves. The huge number of voters in the
dering. These lists use the plurality method for the al- same district also explains why parties present very long
location of the seats within the party lists: candidates lists of candidates (often with over one hundred names).
withthelargestnumberofnominativevotesaredeclared Finally,theroleofpartiesisveryweak;thepoliticalcon-
elected. Therearejustsmalldifferencesinthenumberof stellationfrequentlychanges,withnewpartiesbeingcre-
candidates that a voter can indicate, the ordering of the ated and old ones being reshaped.
candidates on the ballot, but the systems are basically In Uruguay voters cannot choose candidates, but lists
the same, justifying the observed universality. ofcandidates presentedbythe parties,the so-calledsub-
Other countries using open lists are Slovenia, Greece, lemas. Thereforeouranalysisfocusesonthedistribution
Switzerland,Brazil,Uruguay. TheresultsoftheFCscal- of performance of sub-lemas, instead of that of single
ing are illustrated in Fig 2. While there is a histori- candidates.
cal persistence of the distribution at the national level, Figs. 3 and 4 show the analogues of Figs. 1 and 2 ob-
the curves do not really follow a common pattern, and tained by using CAAMd scaling. The historicalstability
do not match well the behavior of the universal distri- of the corresponding distributions at the national level
bution found for Italy, Poland, Finland, Denmark and holds, however the comparison across countries is poor:
Estonia. We distinguish here two classes of behaviors: curvesappeartocross,nottocollapse(panelF).Accord-
Slovenia, Greece and Switzerland are characterized by a ingtoCostaFilhoetal.[21]thecentralpartoftheBrazil-
pronouncedpeak atv/v =1,andtheir tails matcheach ian curve follows a power law, with exponent close to 1;
0
otherquitewell. BrazilandUruguayexhibitamonotonic power law fits of the centralregionof the other distribu-
pattern, quite different from the other three curves. The tions yield exponents sensibly different from each other,
Braziliancurvefollowsquitecloselytheprofileoftheuni- which confirms the crossing of the curves (see Table C.1
versal curve of Fig. 1 on the tail (v/v >1). of Appendix). In particular, we cound not identify any
0
We conclude that open list systems do not guaran- portion of the Polish curve resembling a power law. We
tee identical distributions, but can be grouped in classes conclude that the fraction of votes v/ND collected by a
of behaviors. A close inspection of the election sys- candidateinhis/herelectoraldistrictdoesnotfollowthe
tems, however, may explain why we observe discrepan- same probability distribution in different countries, not
cies. Slovenia divides its territory into eight districts evenwhentheyhaveessentiallyidenticalvotingschemes,
whichinturnarepartitionedinto11electoralunits,each as in Figs. 1 and 3.
giving one candidate in the district. The voters can cast
the votefor anyofthe candidatesinthe district, butthe
electionofthecandidatedependsonthenumberofvotes C. Distribution of candidates’ performance:
he/she won in his/her unit, i.e. the performance of the semi-open lists
candidateintheunitismoreimportantthanthenumber
of votes won in the district, which may affect both the The other countries we considereduse semi-openlists,
candidates’ campaigns and the voters’ choices. with different thresholds for the number of preferences
Greece uses a very complex seat allocation method that candidates are required to collect in order to se-
among party lists and individual candidates. Although cure a seat in the Parliament. The higher the electoral
the ranking of the candidates on the list and the seats quota is, the harder is for a candidate to reach the re-
reallocationdepends onthe numberofvotescollectedby quired number of votes. In this case the position of the
thecandidate,ifoneofthe candidateshappens tobethe candidate within the party, as it appears on the ballot,
headofa partyoracurrentorex PrimeMinister he/she has more influence on his/her final rank than the num-
issetautomaticallyatthetopofthepartylist,regardless ber of votes he/she collected. This can drastically effect
of his/her electoral performance. Additionally, voting is the motivation of the candidate to lead a personal cam-
compulsory, so many people cast a vote because they paign. Also, high quotas diminish the influence of the
have to, without an informed opinion and/or motivation voter on the final list ordering, which affects both the
to participate in the election. degree of a candidate’s involvement in his/her personal
InSwitzerland,votersmaycastasmanyvotesasthere campaignandthewaypeoplecasttheirpreferencevotes.
areseatsinthedistrict. Theymayvoteforallmembersof Therefore there is hardly an open competition between
the list,orforcandidatesofmorethanoneparty. Voters candidates,andthis may be reflectedinthe shape ofthe
are also allowed to cast two votes per candidate. This distribution of performance. Figure 5 shows the prob-
5
103 A103 B103 C
102 102 102
)ND 101 101 101
v/ 100 100 100
P( 10-1 10-1 10-1 1995
10-2 11997568 Italy 10-2 22000071 Poland 10-2 12909093 Finland
10-3 1979 10-3 2011 10-3 2007
103 10-5 10-4 10-3 10-2 10-1 D103 10-5 10-4 10-3 10-2 10-1 E103 10-5 10-4 10-3 10-2 10-1 F
102 102 102
)P(v/ND 111000-101 111999999048 111000-011 111000-011 FPionllIaatannlddy 212090170163
10-2 22000057 Denmark 10-2 22000037 Estonia I 10-2 DEensmtoanriak 21091918
10-3 2011 10-3 2011 10-3 Brazil 2010
10-5 10-4 10-3 10-2 10-1 10-5 10-4 10-3 10-2 10-1 10-5 10-4 10-3 10-2 10-1
v/N v/N v/N
D D D
FIG. 3: Same analysis as in Fig. 1, with CAAMd scaling. Curves are stable at the national level, but they do not compare
well across countries.
110034 A110034 B110034 C
)ND 110012 110012 110012
P(v/ 1100-10 1100-01 1100-01
10-2 22000048 Slovenia 10-2 2007 Greece 10-2 2007 Switzerland
10-3 2011 10-3 2009 10-3 2011
110034 10-5 10-4 10-3 10-2 10-1 D100110034 10-5 10-4 10-3 10-2 10-1 E100110034 10-5 10-4 10-3 10-2 10-1 F100
)ND 110012 110012 110012
P(v/ 1100-10 1100-01 1100-01 SGlorveeenciea 22000191
10-2 2006 Brazil 10-2 2004 Uruguay 10-2 SwitzeBrlraanzdil 22001101
10-3 2010 10-3 2009 10-3 Uruguay 2009
10-5 10-4 10-3 10-2 10-1 100 10-5 10-4 10-3 10-2 10-1 100 10-5 10-4 10-3 10-2 10-1 100
v/N v/N v/N
D D D
FIG. 4: Same analysis as in Fig. 2, with CAAMd scaling. Curves are stable at the national level, but they do not compare
well across countries.
ability density distributions for different countries with of performance of Dutch candidates follow an approxi-
semi-open lists, according to FC scaling. The elections mate power-law behavior over most of the range of the
in Czech Republic held in 2010 had the lowest electoral performance v/v (Fig. 5F).
0
quota and P(v/v ) (Fig. 5D) turns out to be very sim-
0 Besides the values of the electoral threshold, these
ilar to the curve obtained for Greek elections (Fig. 3B).
countries also differ in the number of nominative pref-
The country with the highest electoral quota are the
erences a voter can cast, in the size and number of
Netherlands, where each candidate has to win 10% of
multi-member districts, as well as in the electoral for-
votes cast on the national level in order to be directly
mulathatdeterminesthefinalrankings(seeSectionAof
elected. Voters in Netherlands have little or no influence
Appendix). Anychangeintheelectoralsystem,i.e. these
on the orderingof candidates, which is essentially frozen
severalfactors,mightinfluencetheshapeofP(v/v ). For
0
bytheparty,andthey oftenvoteforthe top-rankedcan-
instance, in 1994 Slovakia changed the number of multi-
didate andthe firstseveralnamesonthe list,astheyare
member districts, leading to appreciable changes in the
the mostpopularandappreciatedmembersof the party.
shape of the distribution (Fig. 5C). The change in the
This resembles the rich-gets-richer effect, which is char-
electoral quota and the number of nominative votes de-
acterized by power-law behavior of the distribution of
cided in Czech Republic in 2006,may be the responsible
the relevant quantities [38–42]. Indeed, the distribution
for the variation of the curve before and after that year
6
101 A 101 B 101 C
100 100 100
)v0 10-1 10-1 10-1
P(v/ 10-2 10-2 10-2 1994
10-3 10-3 10-3 12909082
10-4 22000160 Sweden 10-4 22000170 Belgium 10-4 22001102 Slovakia
101 10C-2 zech10-1 Repu100blic 101 D102 101 10-2 10-1 100 101 E102 101 10-2 10-1 100 101 F102
100 100 100
)v0 10-1 10-1 10-1
P(v/ 10-2 10-2 10-2
10-3 2002 10-3 1992 10-3
10-4 22000160 10-4 11999959 Estonia II 10-4 22001102 Netherlands
10-2 10-1 100 101 102 10-2 10-1 100 101 102 10-2 10-1 100 101 102
v/v v/v v/v
0 0 0
FIG. 5: Distribution of electoral performance of candidates in proportional elections with semi-open lists, according to FC
scaling. Voters may express preferences for the candidates, but this plays a role for the final seat assignments only if the
number of votes obtained by a candidate exceeds a given threshold, which varies from a country to another. At the national
level curves are mostly stable, significant discrepancies correspond to changes in the election rules, like in Slovakia (C), Czech
Republic (D) and Estonia (E). The apparent power law of the Dutch curve (F) might be generated by a rich-gets-richer
mechanism, since the threshold is very high (10% at the national level) and voters typically tend to support the candidates
based on their popularity. We stress that Estonia since 2002 has adopted open lists, which is why distributions of Estonian
elections after 2002 are illustrated in Figs. 1 and 2 (labeled as Estonia I).
104 A104 B104 C
103 103 103
)D 102 102 102
P(v/N 110001 110001 110001 11999948
10-1 10-1 10-1 2002
10-2 22000160 Sweden 10-2 22000170 Belgium 10-2 22001102 Slovakia
10-3 10-3 10-3
104 10-6 10-5 10-4 10-3 10-2 10D-1 104 10-6 10-5 10-4 10-3 10-2 10E-1 104 10-6 10-5 10-4 10-3 10-2 10F-1
103 103 103
)D 102 102 102
P(v/N 110001 Czech Republic 110001 110001
10-1 2002 10-1 1992 10-1
10-2 22000160 10-2 11999959 Estonia II 10-2 22001102 Netherlands
10-3 10-3 10-3
10-6 10-5 10-4 10-3 10-2 10-1 10-6 10-5 10-4 10-3 10-2 10-1 10-6 10-5 10-4 10-3 10-2 10-1
v/N v/N v/N
D D D
FIG. 6: Sameas Fig. 5, with CAAMd scaling.
(Fig. 5D). The transition from semi-open to open lists D. Estimating the similarity of the distributions
introduced in Estonia in 2002, might explain why the
curves before and after that year look different (Fig. 1E
Sofartheestimationoftheagreementordisagreement
versus Fig. 5E). Interestingly, after the introduction of
of different curves has been basically visual. In this sec-
open lists in Estonia, the distribution of performance
tion we would like to attempt a quantitative assessment
matches the universal distribution of the other countries
of this issue. We build two matrices, whose entries are
withsimilarelectionsystems(Fig.1F),whilebefore2002
thevaluesoftheaveragedistanceD andthemaximum
avg
we find clear discrepancies.
distance D between the distributions for any pair of
max
The corresponding distributions with CAAMd scaling countriesfor whichwe gatheredelectiondata (see Meth-
also show marked differences between different countries ods). The dissimilarity values for elections in the same
(Fig. 6). countryarereportedononediagonalofthematrix. Since
7
FC CAAMd CAAMn
A average distance C average distance E average distance
Nl 0.9 Nl 1 Nl 1
Cz 0.8 Cz 0.9 Cz 0.9
Sk Sk Sk
Be 0.7 Be 0.8 Be 0.8
SUey 0.6 SUey 0.7 SUey 0.7
Br Br 0.6 Br 0.6
CGhr 00..45 CGhr 0.5 CGhr 0.5
Si Si 0.4 Si 0.4
EEee I II 0.3 EEee III 0.3 EEee III 0.3
Dk 0.2 Dk 0.2 Dk 0.2
Pl Pl Pl
Fi 0.1 Fi 0.1 Fi 0.1
It 0 It 0 It 0
ItFiPlDkEe IEe IISiGrChBrUySeBeSkCzNl ItFiPlDkEe IEe IISiGrChBrUySeBeSkCzNl ItFiPlDkEe IEe IISiGrChBrUySeBeSkCzNl
B maximum distance D maximum distance F maximum distance
Nl 0.9 Nl 1 Nl 1
Cz 0.8 Cz 0.9 Cz 0.9
Sk Sk Sk
Be 0.7 Be 0.8 Be 0.8
SUey 0.6 SUey 0.7 SUey 0.7
Br Br 0.6 Br 0.6
CGhr 00..45 CGhr 0.5 CGhr 0.5
Si Si 0.4 Si 0.4
EEee I II 0.3 EEee III 0.3 EEee III 0.3
Dk 0.2 Dk 0.2 Dk 0.2
Pl Pl Pl
Fi 0.1 Fi 0.1 Fi 0.1
It 0 It 0 It 0
ItFiPlDkEe IEe IISiGrChBrUySeBeSkCzNl ItFiPlDkEe IEe IISiGrChBrUySeBeSkCzNl ItFiPlDkEe IEe IISiGrChBrUySeBeSkCzNl
FIG. 7: Quantitative assessment of the similarity between distributions at the national level and between countries. The
matrices in the top row indicate the values of the average K-S distance between datasets of different countries. On the
bottom row themaximum distances are reported. Each column corresponds to oneof thethreetypesof distributions we have
examined, by using FC, CAAMd and CAAMn scaling. A color code is adopted to better distinguish the low values of the
distance (indicated by the blue), indicating a big similarity between the curves, from the larger values, corresponding to poor
collapses. Thebluishsquareon thebottom left ofthematrices obtained viaFCscaling confirmthatthedistributionsof those
countriesareprettyclose toeach other,asillustrated in Fig. 1F.Conversely,thesimilarity between distributionsobtained via
CAAMd and CAAMn scaling appears rather modest for most pairs of countries.
we have adopted three different types of scaling for the fer in the initial part, especially the Brazilian distribu-
electoral performance of candidates, FC, CAAMd and tions. The strong similarity between the results of elec-
CAAMn, we end up with six matrices, which are illus- tions in the nations of Group U persists even if we con-
trated in Fig. 7. In each column we display the pair of siderthemaximumdistance(panelB),asthedarkblock
matrices corresponding to one type of scaling, the first isstillthere,thoughblurred. SloveniaandGreeceappear
row contains the average distances, the second row the verysimilarto eachotherbut sensibly differentfromthe
maximum distances. We built 16×16 matrices, even if other countries. The diagonal from bottom left to top
we studied 15 countries, because we considered two sets right shows the values of the distance for datasets in the
of elections for Estonia, because of their transition from same country. In general, the distances are pretty low,
semi-open lists (Ee II) to open lists (Ee II), which took but we also find fairly large values. These correspondto
place in 2002. countries which introduced changes in the election rules,
reflected in the shape of the distributions, as described
Potential data collapses are indicated by low values of
above.
D andD ,whichareeasiertospotbyusingacolor
avg max
code, aswe did inthe figure. Numericalvalues arelisted If we move to CAAMd scaling (panels C and D) the
in the Appendix in the Tables C.2, C.3, C.4, C.5, C.6, scenario is considerably worse, in that the curves are
C.7. Dark squares (black-blue) correspond to the lowest much more dissimilar to each other than the ones ob-
valuesofD andD ,sotoverysimilardistributions. tainedwith FC scaling. In panelC, the averagedistance
avg max
The data collapsefor the countries ofGroupU (Fig.1F) between the countries of Group U is still low, though
is illustrated by the bottom left block of A and B. In- higherthanforFCscaling(panelA),butwhenonemoves
terestingly, we see that only the Estonian elections held tothemaximumdistancetheblockdisappears(panelD).
after 2002 (Ee I) are very similar to the other curves of For CAAMn (panels E and F) the curves are even more
GroupU;before2002Estoniansusedsemi-openlists,the dissimilar to each other.
correspondingcurvesdo not matchwell with the univer-
We are not giving here any indication on the signifi-
sal distribution.
cance of the measured values of the K-S distance. Large
We see thatalso the Brazilianand the Uruguayandis- valuesindicatewithcertaintythatthecorrespondingdis-
tributions are fairly similar, on average, to the universal tributionsarereallydifferentcurves,butlowvaluescould
curve,mostlyonthetail,althoughtheyconsiderablydif- still have high significance. As a matter of fact, all val-
8
ues that we found, for all types of scaling, indicate a larger data collections and holds for Denmark and Esto-
significantdiscrepancybetweenthe correspondingdistri- nia as well.
butions. However,we stressthathere weareconsidering Different patterns are found for countries adopting
the whole profile of the distribution, from the lowest to semi-open lists, in which in principle voters can choose
the highest value of the performance variable. The most the candidates, but the main ranking criterion is still
interestingpartofthedistributions,andtheonewhichis imposed by their party, regardless of the final electoral
likelytoreflectcollectivesocialdynamics,iscertainlythe scoreofthecandidate,unlessitexceedsagiventhreshold.
tail, because it is where one has the largest cascades of In this system the competition among the candidates is
votesforthesameindividual. Onthecontrary,theinitial therefore not really open, and it is no wonder that the
partofthecurvecorrespondstopoorlyvotedcandidates, distribution of electoral performance does not follow the
andtherearemanywaystogettosuchmodestoutcomes profile of the curves of Group U.
(like being voted solely by closest family members and Ingeneralwefoundthattheshapeofthedistributionis
friends),hardlysusceptibleofamathematicalmodelling. muchmoresensitivetothespecificelectionrulesadopted
But at this stage we did not want to identify the most in the countries than to the historical and cultural con-
“interesting”partofthe distributionbyconstrainingthe text where the election took place. This is evident when
range of the variable, which is always tricky. Therefore oneconsiderstheevolutionintimeofdistributionsofany
we decided to compare the full distributions. givencountry,whichremainessentiallyidenticalevenaf-
We finally remark that in social dynamics one can ter many years, if the voting system does not change,
hardly get the same striking data collapses obtained in but display visible variations following the introduction
physical systems and models. Even if the social atom and/or modification of election rules as it happened in
hypothesis implies that just a few features of the social Estonia in 2002, Slovakia in 1994, Czech Republic in
actors and their interactions determine the large-scale 2006. The case of Estonia is spectacular: before 2002
behavior, the complexity of human nature and context- it used semi-open lists, and the distributions of relative
dependent factors may still have some influence, albeit performanceofacandidatewithrespecttohis/herparty
small. For instance, in the Polish distributions of Fig. competitors did not compare well with the curves of the
1B there is a hump for v/v ≈ 5, which occurs system- other countries of Group U. After the introduction of
0
atically at the national level, but which is absent in the openlists, instead,the distributions became verysimilar
other distributions of the same class. Therefore, obtain- to the universal curve. Such sensitivity of the distribu-
ing the agreementof the distributions shown in Fig. 1F, tions might allow to detect anomalies, e.g. large-scale
despite all differences between countries and historical fraud, in future elections [12, 13].
ages, is truly remarkable. Our analysis proves that the success of a candidate,
measured by the number of votes, strongly depends on
the party he/she belongs to, and that only when one
III. DISCUSSION considersthe competition among candidates of the same
party universal signatures may emerge. Indeed, neglect-
ing the party affiliation does not seem to take us very
We have performed an empirical analysis of elections
far: the two party-independent normalizations we have
held in 15 countries in various years. We focused on the
considered, following the procedure by Costa Filho et
competition between candidates, which is a truly open
al.[21,23,32],donotseemtorevealstrongcommonfea-
competition when the voters can indicate their favourite
turesamongdistributionsofdifferentcountries,noteven
representatives in the ballot and candidates with the
when the latter follow nearly identical election schemes
largest number of votes are ranked the highest. This
(e.g. the nations of Group U).
occurs in proportional elections with open lists. Of the
countries for which we found data, 10 adopt open lists.
Five of them (Group U), Italy, Finland, Poland, Den-
IV. METHODS
markandEstonia(since 2002)haveverysimilar election
rules, the other five are characterized by important dif-
ferences (e.g. compulsory vote, huge districts and weak A. Election data
role of parties in Brazil), which are likely to affect the
behavior of voters and candidates, leading to measur- Here we consider the data sets for parliamentary
able differences in the statistical properties of the elec- elections from 15 countries with open and semi-open
toral outcomes. Indeed, the distribution of the number lists: Italy (1958, 1972, 1976, 1979 and 1987) [43],
of votes received by a candidate, normalized by the av- Poland (2001, 2005, 2007 and 2011) [44], Finland
erage number of votes gained by his/her competitors in (1995, 1999, 2003 and 2007) [45], Denmark (1990,
the same partylist, seems to be the same for the nations 1994, 1998, 2001, 2005, 2007 and 2011) [46], Estonia
of Group U, while there are marked differences from the (1992, 1995, 1999, 2003, 2007 and 2011) [47], Slovenia
curves obtained from the other countries. This result, (2004, 2008 and 2011) [48], Greece (2007 and 2009)
originally found by Fortunato and Castellano for Italy, [49], Switzerland (2007 and 2011) [50], Brazil (elec-
Finland and Poland [30], is confirmed here on a much tions for state deputies in 2002, 2006 and 2010) [51],
9
Uruguay (2004 and 2009) [52], Sweden (2006 and 2010) Since we have multiple datasets for each country, in
[53], Belgium (2007 and 2010) [54], Slovakia (1994, ordertocompute the dissimilarityofthe distributionsat
1998, 2002, 2010 and 2012) [55, 56], Czech Republic the nationallevelandacrosscountriesweproceedas fol-
(2002, 2006 and 2010) [57] and the Netherlands (2010 lows. For a given country X we compute the distance
and 2012) [58]. Further details and sources for each between any two distributions for elections of X. For a
file are given in Table C.8 in Appendix, while the pair of countries X and Y we compute the distance be-
compiled and cleaned data maybe be downloaded at tween any pair of distributions P and P , correspond-
X Y
http://becs.aalto.fi/en/research/complex_systems/elinegcttiooonnse/.dataset of X and one of Y, respectively. In
both cases we take the average D and the maximum
avg
D of the resulting values. In this way we estimate
max
B. Comparing distributions the average and the maximum distance between distri-
butionsofthesamecountryandbetweendistributionsof
WeusetheKolmogorov-Smirnov(K-S)distance[59]to two different countries.
measure the dissimilarity of two empirical distributions.
The K-S distance D is defined as the maximum value of
theabsolutedifferencebetweenthecorrespondingcumu-
Acknowledgments
lative distribution functions, i.e.
D =max|SN1(x)−SN2(x)| (1) We thank Lauri Loiskekoski for helping us to collect
x
the electiondata. We alsothankClaudioCastellanoand
where S (x) and S (x) are the cumulative distribu- Raimundo N. Costa Filho for useful comments on the
N1 N2
tions for two data sets of size N and N . manuscript.
1 2
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