Voting Models for the Council of Ministers of the European Union

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Voting Models for the Council of Ministers of the European Union

Marc Feix, Dominique Lepelley, Vincent Merlin, Jean-Louis Rouet

To cite this version:

Marc Feix, Dominique Lepelley, Vincent Merlin, Jean-Louis Rouet. Voting Models for the Council of Ministers of the European Union. 2004. �halshs-00003483�


European Union

M.R. Feix

, D. Lepelley y

, V. Merlin z

and J.L. Rouet x

August 23, 2004


An important question for the European Union is to know whether its institutions

will permit it to esape from politial deadloks eah time a question is at stake. Two

studies [1 , 7 ℄ suggest, by using the Impartial Culture assumption to model the voting


mehanisms. TheImpartialCulturemodelhasbeenritiizedfrom atheoretial pointof

view[5℄,apolitialone[12 ℄anddoesnott withexperimentaldata[10 ℄. Thegeneralized


thisrstmodel. Weherestudytheprobabilityofapprovalunderthisassumption,rstfor

theasymptotiase(reahedwhen thenumberofountries N goesto innity),and next

with omputer enumerations and Monte Carlo simulationsforthe European Union with

27 members. We onsiderboththeTreaty of Nie and some proposalsfor theEuropean


JEL lassiation: D7

1 Introdution

In the last ve years, a onsiderable body of researh on the hoie of the best voting rules

for federal union have been inspired by the debates on the Treaty of Nie and the projets for

an European Constitution. Without being exhaustive, we an mention the work by Baldwin,

Berglof, Giavazzi and Widgren [1℄, Barbera and Jakson [2℄, Bobay [4℄, Feix, Lepelley, Merlin

and Rouet [5℄, Felsenthal and Mahover [7, 9℄, Laruelleand Widgren [11℄. All these ontribu-

tions sharea ommonorganization: the authorspropose a voting model, andthen seekfor the

voting rule orthe onstitutionthat better ts aording tosome normativeriteria.

SUBATECH,Eole desMinesdeNantes,Lahanterie, 4rueA Kastler,BP20722,F-44307Nantes edex







Laboratoirede Mathematique,Appliationset PhysiqueMathematique -UMR6628, Universited'Orleans,

UFRdesSienes,F-45067OrleansCedex 2,Frane


and sientists (Dan Felsenthal and Moshe Mahover) has developed [12, 8℄. At the origin, the

sienti analysis of the Treaty of Nie [1,8℄ laims that the need of 255 (or 258) votes (on a

total of 345) to approve a proposition will result in a serious deadlok at the ounil of min-

isters with an a priori probability of approval of 2%. A. Moberg disagrees strongly, pointing

out that the result ignores the \strong onsensual ulture of the EU". Who is right ? In fat,

the sienti analysis given in [1, 8℄ is only a part of the full story : it is based on the use

of the Impartial Culture(IC hereafter) model, whih states that eah ountry hooses to vote

`yes' or`no'independentlywith equalprobability. Inother words, eahountryips afairoin

to take a deision! But another modelexists, whih is more subtle and less easy to ompute.

This model, alled Impartial Anonymous Culture (IAC), asserts that all the distributions of

the votes at the Union levelare equallylikely. 1

The aim of this study is to show that the use of a model related to the IAC one is able to

give answers whih are loser to the reality of the European Union with 27 members (EU27

hereafter)and,insomeway,takesintoaounttheonsensualharaterofthevote. Bydepart-

ing from the ommonIC assumption, we obtain a theoretial probability of passing a motion

that turns out to be higher. Our result onerns not only the Treaty of Nie with its famous

74.8% majority rule (one key vote), but also one of the deisionshemes that have been sug-

gested duringthe debates for the European Constitution (doublekey vote: amotionis passed


Thepaperisorganizedasfollows. Insetion2,wepresentageneralizedIACmodel(GIAC)and

webriey disussitsadequationtothevote atthe ounil. Insetion3,wegivethetheoretial

probability ofapprovalinthe asymptotilimit,i.e. whenthe number ofountries (denotedby

N in what follows) goes to innity. Setion 4 heks the relevane of this asymptoti solution

for EU27,by providingnumerialsimulations, and we present our onlusions insetion 5.

2 The dierent models

Weonsider binaryissue votes `yes' or`no' forthe N states(elsewhere voters,MPs,et...) ofa

federal union. Inthe simplest IC model,eah vote isindependentof the others and eah voter

says `yes' or`no' with equalprobabilityp=1=2. IChas seriousdrawbaks. It desribes avote

where either everybody isundeided (no exhange of points of viewallowing the emergene of

amajorityhas taken plae)orthe existeneoftwobloksofstritly equalimportane. In both

ases, the vote will be won by a margin going as N 1=2

. This explains the low probability of

approval with a quota of 258/345 i.e. 74:8% in the Treaty of Nie deision sheme. The idea

is onsequently to introdue a model where a probability p dierent from 1/2 has emerged.

Moreover, our knowledge of p is itself of a probabilistinature, it is mathematiallydesribed

by the funtion f(p) whihis the probabilitydistribution ofp. Theemergene of aprobability

p dierent from 1/2 seems rather natural in an assembly where ertainly long disussions,

explanations,ompromises,pakage deals,et...preede eahvote (the\onsensualulture"of

A.Moberg). Notiethat,allthesedisussionsareresumedinap6=1=2andthatthesubsequent


Notiethat thewidelyusedBanzhaf powerindexrelies upontheICprobabilityassumption. Foritspart,

theIACmodelanbeassoiatedtotheShapley{Shubikpowerindex. Thelinkbetweentheprobabilitymodels

andpowerindieswas rstemphasizedbyStraÆn[13℄ andBerg[3℄.


f(p) 0 and R



f(p)dp =1. The funtion f(p) =1 for all p gives the IAC model. With this

model, the average number of votes for whih n voters of equal importane, on a total of N,

have voted `yes' reads

C n



0 p


(1 p) N n

dp= 1


: (1)

All values of n (from 0 toN) have the same probability 1=(N+1),onsequently, for the IAC

model, the probability to have n `yes' on a total of N voters is a at urve. It is also easily

provedthat, ifbothn andN gotoinnitywhiletheration=N iskeptonstant,theprobability

distribution of n is torst orderin =N


N (n)=


N f(



): (2)

This resultisadiret onsequeneofthe possibilityof interpretingaprobabilityasafrequeny

when the number of random drawings goes to innity and would be trivial if we were not

dealing with adouble probability onept as mentioned above.

Now let us suppose that eah of the N voters has one vote and that Q votes are needed

for an approval. Letq = Q=N. Then for the IAC model, the probability of approval is 1 q,

independentofN. Forq=:75,forexample,theIACmodelgivesa25%haneforanapproval,

while the omputation using the IC modelgives0:3% for N =27 voters.

3 The probability of approval in the asymptoti limit



votes(or mandates);moreover,twokindsofmandates

have been proposed in the EU onstitution : eah voter has two mandates a

i and b


, and his

(her)vote (`yes' or`no') isused intwoqualiedmajority gamesA andB, therespetivequotas



and Q


. Notie thatforeahplayeri, itisthe samevote (`yes' or`no')whihisused

toompute thenumberofmandates obtainedrespetivelywith keys Aand B. Thetwoquotas

must bereahed for nal approval, eah one being related toa ertaintypeof legitimay. The

EU Constitution projet proposes for ountry i to take a


=1 and b


equal to the population

of state i.

Inordertobeabletoperformanalytialomputations,toseethe roleoff(p)andthe inuene

of the dierent quotas, we suppose that N is large enough to use asymptotialulations. At

the end, wewillomparethe obtained asymptotiresults tonumerialsimulationsand willsee

that, as already stated in [6℄, EU27 an be fairly approximated by this limit at least for the

one key vote (the details of the alulations are given inappendix).

For the GIAC model, haraterized by f(p), the distribution funtion for x mandates in favor

of approvalin the single key ase reads


N (x)=



f(x=A); (3)

while for the double key ase, we obtain




(x;y)= 1



A y


)f p= x


= y


; (4)

whereÆ isthe Diradistributionand xand yare thenumbersof mandatesinfavorof approval

for keys A and B respetively. Notethat x and y are stritly orrelated ( x


= y


). In formulae

(3) and (4), A and B are dened by

A= N


i=1 a


; B =



i=1 b


: (5)

For the IAC model (f(p) = 1) and a single key vote, equation (3) means a at density from

x = 0 to x = A. Still for the IAC modelbut for the double key vote, the points are loated

on the segment joining(x=0;y= 0)and (x=A;y =B) with a uniform distribution on this

segment. Inthe approximation usedtoobtain(3) and (4), votes haraterizedbyp givepoints

loatedat x=pA and y=pB.

NotethattheseresultsholdforN goingtoinnity. Itanbeshownthattherstorretion(N

large but not innite) provides adiusion aroundthese pointsinN 1=2

. Whilethis sattering

slightly modies the atness of F


(x) for the one key vote, it transforms the segment of the

twokey vote into a long ellipse with aratio long over small axes inN 1=2

. Comingbak to the

segment struture, we see that inthe doublekey vote ase, with two unequal quotas, it is the

one with the highestquota whih willset up the frequeny of `yes' votes. Also, we must point

out that equation (3) is a generalization of equation (2) obtained for a set of voters with one

mandate eah (inthat ase A =N). Notie also that integration of (4) respetively ony and

x gives


N (x)=


A f(



); and F

N (y)=


B f(



); (6)

in agreement with the results of the one key vote.

Conerning the doublekey vote, it is worth notiing that, for any N (not neessarilygoing to

innity)andforquotasequaltoA=2andB=2,thevotingpowerofastateX (i.e. itsprobability

of being pivotal) is given by

P(X)= P



B (X)



where P


(X) and P


(X) are the voting powers of X with (respetively) keys A and B. This

result is validfor IC, IACand GIAC models when f(p) =f(1 p).

To end this setion, a omparison with the results of the IC model is in order. The above

treatmentwithf(p)=Æ(p 1=2)onentratesallthe pointsatthe entralpoint(x=A=2;y=

B=2). This onrms the quikly dereasing probability of approval when the quotas are not

very losed to 1=2. For the double key vote, with the IC model, the authors have arried the

omputationsofthe next termtoobtainthe sattering aroundtheentralpoint. The approval

probability obtained analytially for the two relativequotas equalto1=2 reads

P = 1





1 r

) (8)


wherer = P


i=1 a

i b


= P


i=1 a


i P


i=1 b


i 1=2

istheorrelation fatorbetween vote A andvote B.

P varies from 1=4 (r=0)to 1=2 (r=1,obtained for b




, in fata single vote).

4 Numerial simulations

In this setion, the results of numerial simulations will be shown for the IAC ase. Beause

we want to reahthe asymptotilimit whih supposes both animportantnumberof eletions

and alargenumberofvoters,MonteCarlo methodshouldbeused. Atually,itisnot possible,

when the number of voters is large, to enumerate, stok and ompute the 2 N


beauseof lak of memoriesand omputationtime.

In addition,the MonteCarlotehnique willillustratelearly thedoubleprobabilistiharater

ofthe IACmodel. Thismethodonsiststomakearandomsamplingamongallthevoteong-

urations,butwithouttakingallofthem. Thentheontributionofallthesamplesaregathered.

The methodhas two steps. First a probabilityp is hosen at randominthe distributionfun-

tion f(p). Seond avote ongurationishosenaordingly this probabilityp: foreahof the

N voters, a randomnumberis taken ina uniform distribution, if this number islowerthan p,

the voter givesits mandates (it is a `yes' vote) while he doesn't if the number is higher. This

is in fat an aeptation-rejetion method and if the number of voters is large, the number of

`yes' voters divided by N will tend toward p. This proess is repeated for a large number of

eletions with,at eaheletion, ahoie ofa new p intof(p) and soon.

Notie that the results of the IC modelould also be obtained by this tehnique. The proba-

bility p of the N voters isthen equalto 1=2 whih orresponds tof(p)=Æ(p 1=2).

First we give the obtained results for a large number of voters (N = 100) and M = 50000

eletions, both for a single key and a double key vote using the Monte Carlo tehnique. For

the singlekey ase,gure 1shows thehistogram ofthe numberofongurations, asafuntion

of the relatednumberof mandates. The mandates ofthe N states havebeen taken atrandom

in a uniform distribution between 1 and 5, then the sum has been normalized to 100. This

normalization does not hange the ratio 5 between the highestvalue of the mandates and the

smallest one. The histogramis at inagreement with equation (3)and the probability of ap-

proval is very lose to (1 q).

For the double key ase, gure 2 gives the distribution of the M eletions performed in the

plane (x;y),one pointrepresenting oneeletion. Forbothkeys, themandates havebeen taken

at random in a uniform distribution between 1 and 5, then the sum has been normalized to

100. Beause allthe pointshavethe sameweight,their density givesthe value ofF



equation (4)). As expeted, the points are roughly distributed on the segment delimited by

the two points(0;0) and (A;B). In addition tothis global behavior, the distribution shows a

ertainsattering. Consequently the results obtainedin the asymptotilimitare reovered (at

least for N = 100 voters). We have heked that, for this ase, the probability of approval is

losely given by 1 sup(Q




)as shown table 1. Forexample, for Q




=70%, we get

29:3% of approval and for Q


= 50% and Q


= 80%, we get 20:4%.For gure 3 the number

of voters has been inreased up to N = 1000. We observed that the sattering of the points


14 17 19 22 25

50% 48.66 40.15 30.34 20.53 10.59

60% 40.08 39.11 30.34 20.53 10.59

Key B 70% 30.42 30.42 29.40 20.53 10.59

80% 20.49 20.49 20.49 19.58 10.59

90% 10.55 10.55 10.55 10.55 9.92

Table 1: Double key vote. Perentage of approval for the simulation presented gure 1 as a

funtion of the twoquotas Q


and Q

B .

dereases as expeted.

Figure 1: One key vote. Distribution of the

results of the votes for N = 100 voters and


nique. The mandates are hosen at random

in a ratio 1 to 5 and the sum is normalized

to 100.

Figure 2: Double key vote. Distribution of

the results of the votes for N = 100 voters

and 50;000 eletions using the Monte Carlo

tehnique. The mandates are hosen at ran-

dominaratio 1to5and the sumis normal-

ized to 100. Eah point represents an ele-


Now, the question istoknowwhether ornot the asymptotilimitisagoodapproximationfor

the EU27 2

. It is here possible to enumerate the 2 27

vote ongurations (taking are of their

dierentweights). Forthe singlekeyase,gure4showsthehistogramofthenumberofong-

urations asafuntion of the relatednumberof mandates, whih have been taken proportional

tothesquare rootofthestate populations. This hoieisinaordanewiththe prinipleused

in the EU15 and onstitutes a goodompromise between the state legitimay and the itizen

legitimay(see [4℄). Again, theurveisratherat, atleastforqbetween 0.2and0.8,indiating


Populationdataan befoundinMoberg[12℄.


50;000 eletions using the Monte Carlo tehnique. The mandates are hosen at random in a

ratio 1 to5 and the sum isnormalized to100. Eah point represents aneletion.

Key A

14 17 19 22 25

50% 45.44 38.27 31.92 21.42 10.71

60% 39.56 35.60 30.86 21.33 10.71

Key B 70% 31.56 30.20 27.73 20.68 10.71

80% 22.75 22.54 21.86 18.40 10.55

90% 13.64 13.64 13.61 12.80 9.03

Table 2: Double key vote. Perentage of approval for the EU27 as a funtion of the two

quotas Q


and Q


. The results have been obtained by omplete enumeration of all the vote


that the asymptotilimitould be used for this singlekey vote.

Forthe doublekeyase, weturn bak toMonteCarlosimulations(althoughomplete enumer-

ationispossible)beauseeahpointhas thesameweight. Then,itiseasiertointerpretgure5


one eletion). ForkeyA,allthemandates areequalto1(statelegitimay)whileforkeyB, the

numberof mandates of a state is proportional to its population. The sum of the mandates of

key B has been normalized to 100. Beause of the disretenature of the key A mandates, the

pointsare alignedon vertial lines distant of 1. The sattering of the points,not negligible,is

ompatible with the N 1=2

law as stated before. Neverthless the rule 1 sup(Q




) for the

approvalis fairly satisedas shown by table 2.