Jou nal o Indus ial Enginee ing and Managemen
JIEM, 2015 – 8(1): 122-136 – Online ISSN: 2013-0953 – P in ISSN: 2013-8423
h p://dx.doi.o g/10.3926/jiem.1189
Compu ing Vo e T ansi ions:
The Elec ions o he Ca alan Pa liamen , om 2010 o 2012
Albe Co ominas, Amaia Lusa, M. Dolo s Cal e
Uni e si a Poli ècnica de Ca alunya (Spain)
albe .co [email p o ec ed], [email p o ec ed], [email p o ec ed]
Recei ed: June 2014
Accep ed: Feb ua y 2015
Abs ac :
Pu pose:
To es ima e he ansi ion a es co esponding o he 2010 and 2012 elec ions o he
Ca alan Pa liamen o he ou cons i uencies in which Ca alonia is di ided o his pu pose.
The main ea u es o he esul s, which a e ob ained by means o ma hema ical p og amming,
a e commen ed.
Design/me hodology/app oach:
Ma hema ical p og amming op imiza ion models a e
o mula ed in o de o ind he ansi ion a es ha yield a be e adjus be ween he ac ual
esul s in 2012 and hose compu ed applying he ansi ion a es o he 2010 esul s. The
ansi ion a e ma ices a e es ima ed o each one o he ou cons i uencies, since he se o
op ions is no he same o all hem. No o he assump ions ha hose o nume ical consis ency
a e adop ed.
Findings and O iginali y/ alue:
The ansi ion a e models p o ide sa is ac o y goodness o
i . Ma hema ical p og amming u ns ou o be an easy- o-use ool o es ima ing he ansi ion
a es and, a he same ime, e y lexible, since, i necessa y, i allows inco po a ing he
cons ain s co esponding o addi ional assump ions.
O iginali y/ alue:
The ansi ion a es om 2010 o 2012 in Ca alonia a e pa icula ly
in e es ing, since 2012 esul s implied a signi ican change in he composi ion o he Ca alan
Pa liamen . To he bes o ou knowledge, no o he scien i ic jou nal pape has deal wi h his
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ques ion. Ou esul s a e a ailable o he esea che s in o de o in e p e he change and y o
o esee u u e lows o o e s.
Keywo ds:
o e ansi ions, elec o al change, ma hema ical p og amming
1. In oduc ion
When conside ing he esul s o wo consecu i e elec ions in he same elec o al a ea, a usual
way o ying o in e p e he esul s o he la e polls is seeing hem as he consequence o
o e ansi ions om he op ions hey p e e ed in he o me . Poli icians, he media, poli ical
scien is s and mos ci izens a e in e es ed in he changes o he p e e ences o people ha ing
he igh o o e.
Acco ding o Hawkes (1969) he his o ian T e o Lloyd was he i s o s a ing he ques ion.
Be o e 1970, i was deal wi h in some sca ce wo ks (Benewick, Bi ch, Blumle & Ewbank,
1969; Be ing on, 1965; Bu le , 1952, 1953; Bu le & King, 1966;).
Fo mally, gi en he esul s o wo consecu i e (o e en simul aneous) elec ions o each one o
h e di isions (cons i uencies, municipali ies, polling s a ions o any o he pa i ion) o an
elec o al a ea, he p oblem is o ind he ma ix o ansi ion a es om he op ions a ailable in
he i s elec ion ( ows) o he op ions in he second one (columns). O cou se, i one conside s
only agg ega e esul s o he whole e i o y o o any se o cons i uencies, in gene al he e
a e in ini ely many solu ions o he ma ix. On he o he hand, applying a unique ansi ion
a e ma ix o di e se cons i uencies o g oups o cons i uencies, he compu ed esul s will no
always coincide wi h he eal ones. Clea ly, he elemen s o he ma ix mus be nonnega i e
and hose belonging o any gi en ile mus sum up o 1 ( hese wo condi ions imply ha he
elemen s mus be less han o equal o 1).
A su ey can be used o es ima e he elemen s o he ansi ion a e ma ix. Howe e , he
esul s a e highly un eliable, because o many easons ha a e discussed, o ins ance, in
B own and Payne (1986) and in Van de Ploeg, Van de Pol and Kampen (2006). Mo eo e ,
unless he numbe o elemen s in he sample is e y high, many elemen s o he ma ix ( hose
co esponding o small alues o he ansi ion a e) will be equal o ze o. The e o e, a he
expense o g ea e modelling and compu ing e o s, he use o he esul s o bo h elec ions is
a mo e eliable way o ob ain he ma ix.
Hawkes (1969) ied “ o es ima e he numbe o people o ing o a pa icula pa y a one
elec ion who subsequen ly o e o ano he speci ied pa y a he nex elec ion”. In o de o do
his, p oposed h ee me hods. Howe e , hese did no gua an ee ha he esul s ul il he
s a ed abo e condi ions. The e o e, he au ho concluded ha al hough “ he a emp has no
been as success ul as one would wish, some use ul esul s a e ob ainable”. Mille (1972) and
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Up on (1977) adop an app oach ha may be insc ibed in he same s eam ha Hawkes
(1969). O he ela ed wo ks a e Hayes (1976) and Moo es (1987).
Ins ead, I win and Mee e (1969) and McCa hy and Ryan (1977) use quad a ic p og amming
o es ima e he ansi ion a es, hus gua an eeing om he ou side he ul illing o he abo e
men ioned condi ions, which a e imposed by means o cons ain s in he ma hema ical
p og amming. Tzia e as (1986) uses an app oach simila o ha o McCa hy and Ryan (1977)
and a a ian o i , which consis s in using he absolu e alue o he de ia ions ins ead o he
squa es o hem.
Up on (1978), conce ning McCa hy and Ryan’s app oach, obse es ha i gi es a high
p opo ion o ze oes in he ma ix and concludes ha o e es ima es he p opo ion o s aye s
( o e s ha do no change hei p e e ences be ween he o elec ions –ac ually, elec o s ha
o e o op ions whose name is he same in bo h elec ions−, con as ing wi h mo e s). This
c i icism, which we do no deem ully jus i ied, has been assumed by o he au ho s, as, o
ins ance Johns on and Hay (1983).
The es ima ion o o e s’ ansi ion a es is o en seen as a pa icula case o he ecological
in e ence p oblem, i. e., o deduc indi idual beha iou s om agg ega e da a, a p oblem which
was deemed impossible, wi h he me hods a ailable a ha ime, in Robinson (1950). This
no wi hs anding, many me hods ha e been p oposed o dealing wi h i . Some o hem
assume, pe haps implici ly, ha he beha iou pa e n is he same o e y simila in all he
a eas (ecological eg essions; see: Goodman, 1953; Goodman, 1959; conce ning ansi ion
a es: Fülle, 1994; an de Ploeg e al., 2006). O he s, conside ha he beha iou pa e n
may depend on he a eas and usually adop a p obabilis ic app oach (ecological in e ence; see:
Glynn & Wake ield, 2010; G eine & Quinn, 2009; G o man & Me ill, 2004; King, 1997;
conce ning ansi ion a es –p obabili ies−: And eadis & Chadjipadelis, 2009; An weile , 2007;
B own & Payne, 1986; Johns on & Hay, 1983).
The pu pose o he p esen pape is o de e mine, o he 2010 and 2012 elec ions o he
Pa liamen o Ca alonia, a ansi ion ma ix o each cons i uency ha (i) ul ils he
nonnega i e and sum- o-one cons ain s; (ii) applied o he agg ega e esul s o he i s
elec ion gi e exac ly he esul s o he second one o each one o he a ailable op ions in his
la e and (iii) minimises a unc ion o he disc epancies be ween he esul s ob ained wi h he
ma ix and hose gi en by he coun o o es in e e y di ision. No e ha we nei he o mula e
any assump ion abou he di e ences o coincidences be ween he beha iou pa e ns o he
elec o s co esponding o di e se polling s a ions no adop a p obabilis ic poin o iew.
The e o e, he p oblem can be s a ed as a ma hema ical p og amming model, which o some
kinds o disc epancy unc ions is easy o sol e. E en hough ou pu pose is o ind ansi ion
ma ices o any se o di isions (and o di e en kinds o di isions), wi hou in oducing any a
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p io i assump ion abou he alues o he ansi ion a es, we will commen some esul s gi en
by he models in o de o acili a e hei in e p e a ion.
The layou o he es o he a icle is as ollows. Sec ion 2 p esen s he p oblem and i s
ma hema ical p og amming o mula ions. The da a and he ob ained esul s a e p esen ed and
commen ed in Sec ion 3. Sec ion 4 ends he pape wi h some sho conclusions.
2. S a emen o he P oblem and i s Ma hema ical P og amming Fo mula ion
I is assumed ha we ha e he esul s co esponding o wo elec ions in he same elec o al
a ea, such ha he i s one happened a and he second one a ’(≥ ).
When ’ is e y close o (o e en equal o ) i may be ha he censuses co esponding o
bo h elec ions a e iden ical. I ac ually hey a e no , howe e , i is usual o ci cum en his
di icul y assuming ha he beha iou o he elec o s no belonging o he in e sec ion o bo h
censuses is no di e en om hose ha belong o i . In ac , his is equi alen o assume ha
bo h censuses a e iden ical and his assump ion is easonable when ’ is no a om (B own
& Payne, 1986; Hawkes, 1969; McCa hy & Ryan, 1977), as happen wi h he wo elec ions
conside ed in his pape , sepa a ed by only wo yea s (2010, 2012).
The elec o al a ea is pa i ioned in o cons i uencies and hese, a he end, in o polling s a ions.
The e o e he esul s a e a ailable o all he polling s a ions o he elec o al a ea. I may
happen ha he polling s a ions belonging o a gi en cons i uency do no coincide om one
elec ion o ano he (because some a e c ea ed, supp essed o di ided) and in his case one
can only compa e he esul s co esponding o he polling s a ions common o bo h elec ions.
On ano he hand, he a ailable op ions (including blank o e, null o e and abs en ion) may be
di e en om one cons i uency o ano he (in he case o he elec ions o he Ca alan
Pa liamen , hey a e, since he candida es o win he sea s a e di e en , e en o he op ions
wi h he same name). The e o e, he cons i uencies ha e o be conside ed sepa a ely.
Hence, he da a mus e e always o a gi en cons i uency o a subse o polling s a ions
belonging o a gi en cons i uency. The conside ed polling uni s can be g ouped o o m
di isions ( he e o e, a di ision is a se o one o e mo e polling s a ions; e e y conside ed
polling s a ion mus belong o one and only one di ision). The di isions may be, o ins ance,
municipali ies, dis ic s o any se s o polling uni s ha be con enien o he analysis.
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The no a ion ha we use o he da a a e as ollows:
mNumbe o di isions (common o bo h elec ions).
n, n’ Numbe o op ions a and ’, espec i ely.
pik P opo ion o o es ob ained a by he op ion k in he di ision i (i = 1, …, m; k = 1, …, n).
pij P opo ion o o es ob ained a ’ by he op ion j in he di ision i (i = 1, …, m; j = 1, …, n’).
ciCensus (i.e., numbe o people ha ing he igh o o e) o di ision i a ’, assumed o be
equal o ha o (i = 1, …, m).
And o he decision a iables:
kj T ansi ion a e om op ion k a o op ion j a ’ (k = 1, …, n; j = 1, …, n’).
The ma ix R made up o he ansi ion a es kj mus belong o he se F de ined by he
ollowing cons ain s:
(1)
(2)
(3)
Equa ion 1 impose ha he ansi ion a es om an op ion a o e e y o he a ’ mus sum
up o one; Equa ion 2, ha he o al numbe o o es ob ained in he cons i uency by an op ion
a ’ equals he numbe ha esul s when applying he ansi ion a es o he numbe o o es
co esponding o (unde he assump ion ha he census a and a ’ a e he same);
Equa ion 3 en o ce he ob ious non-nega i i y condi ion.
The ma hema ical p og amming models p oposed in McCa hy and Ryan (1977) and Tzia e as
(1986) include cons ain s (1) and (3). Cons ain s (2) a e simila o hose p oposed in
Johns on and Hay (1983).
These cons ain s de ine a se o ma ices ha ing gene ally in ini ely many elemen s. One way
o selec one o hese elemen s is o minimise he disc epancies be ween he ac ual esul s o
elec ions a ’ and hose esul ing om he applica ion o he ansi ion a es. O cou se, he
selec ed ma ix depends on he used measu e o he disc epancies.
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This way we de ine he ollowing ou models:
The objec i e unc ions o he models a e (M1) he sum o he squa ed disc epancies be ween
ac ual and modelled numbe o o es, (M2) he sum o he squa ed disc epancies be ween
ac ual and modelled p opo ions, (M3) he alue o he maximum disc epancy be ween ac ual
and modelled numbe o o es and (M4) he alue o he maximum disc epancy be ween
ac ual and modelled p opo ions.
Since he cons ain s ha de ine he se F a e linea , M1 and M2 a e quad a ic p og ams. Fo
hei pa , M3 and M4 can be e o mula ed as linea p og ams as ollows:
As i is known, howe e , minmax p oblems ha e usually mul iple op imums, since he objec i e
unc ion does no ake in o accoun he alues o he disc epancies ha a e s ic ly less han
he op imum alue. The e o e, a e sol ing M3 and M4, mo e sa is ac o y solu ions may be
ob ained using a second c i e ion ( he sum o he absolu e alues o all he disc epancies); his
leads o he ollowing wo quad a ic p og amming models:
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Whe e * and

* a e, espec i ely, he op imum alues o he objec i e unc ion co esponding
o M3 and M4.
3. Da a and esul s
The da a se s used in he compu a ional expe imen co espond o he elec ions o he
Pa liamen o Ca alonia held on 28 No embe 2010 and 25 No embe 2012.
The cons i uencies coincide wi h he ou p o inces o Ca alonia: Ba celona, Gi ona, Lleida and
Ta agona.
Each p o ince is di ided in o coma ques, and, o i s pa , each coma ca is di ided in o
municipali ies. Ca alonia has 41 coma ques and 947 municipali ies.
The municipali ies, o elec o al pu poses, a e di ided in o sec ions, so ha in each sec ion he
numbe o egis e ed elec o s belongs o he in e al 500-2,000 (wi h he possible excep ion o
small ela i ely isola ed illages, in which he numbe o egis e ed elec o s may be less han
500). Acco ding o his, because o he changes in popula ion, om one elec ion o he nex a
sec ion can be di ided in o wo o mo e o can be inco po a ed o ano he . O cou se, he da a
used in he compu a ional expe imen a e only hose co esponding o sec ions ha emain
unchanged.
The alues o m, n and n’ co esponding o he ou cons i uencies in he used da a se a e
included in Table 1.
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Cons i uency Numbe o di isions (m) Numbe o op ions in 2010 (n) Numbe o op ions in 2010 (n’)
Ba celona 3573 32 19
Gi ona 523 29 19
Lleida 390 32 19
Ta agona 523 33 19
Table 1. Sizes o he da a se s used in he compu a ional expe imen . In all cases, he numbe o op ions
includes null and blank o es and abs en ion
To sol e he six ma hema ical p og amming models desc ibed in he p eceding sec ion o each
one o he cons i uencies he so wa e CPLEX 12.2 was used. The compu ing imes we e no
signi ican in any case. The ansi ion ma ices ob ained wi h models M3 and M4 can be
dis ega ded, since hose co esponding o models M3’ and M4’, espec i ely, a e always
p e e able; only he op imum alues o hei espec i e objec i e unc ions a e used (as inpu s
o models M3’ and M4’, espec i ely).
In o de o e alua e he goodness o i o he solu ions p o ided o he di e en models we
will use he ollowing c i e ia, espec i ely ela ed o he objec i e unc ions o models M1, M2,
M3’ and M4’:
•, coe icien o de e mina ion co esponding o he numbe s o o es, de ined as
ollows:
whe e
(i.e., he p opo ion o o es ob ained globally by he op ion j in he se o di isions).
•, coe icien o de e mina ion co esponding o he p opo ions:
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•max, he maximum disc epancy be ween ac ual and modelled numbe s o o es.
•

max, he maximum disc epancy be ween ac ual and modelled p opo ions o o es.
Coe icien s o de e mina ion, which a e use ul as complemen a y in o ma ion o assess he
goodness o i o he models, may be de ined o each op ion as well:
O cou se, models M1 and M2 will yield always he bes alues o and , espec i ely
(since he denomina o s in he de ini ion o hese c i e ia a e cons an s and he espec i e
nume a o s a e he objec i e unc ions o M1 and M2). In a simila way, models M3-M3’ and
M4-M4’ will be he bes o max and

max, espec i ely. The beha iou o he men ioned models
wi h ega d o he o he c i e ia is di icul o o ecas , excep ing ha M1 and M2 a e mo e
obus han M3’ and M4’ in ela ion o ou lie s.
The ansi ion a es ma ices and he alues o all he c i e ia desc ibed abo e o Ba celona,
Gi ona, Lleida and Ta agona, can be ound in:
h ps://dl.d opboxuse con en .com/u/6741065/RESULTS%20BARCELONA.xlsx,
h ps://dl.d opboxuse con en .com/u/6741065/RESULTS%20GIRONA.xlsx,
h ps://dl.d opboxuse con en .com/u/6741065/RESULTS%20LLEIDA.xlsx and
h ps://dl.d opboxuse con en .com/u/6741065/RESULTS%20TARRAGONA.xlsx, espec i ely.
Table 2 shows he alues o he ou c i e ia co esponding o he ou models M1, M2, M3’ and
M4’ and o he ou cons i uencies.
One can see ha he alues o and a e good (high). The la e , wi h M2, eaches 0.97 o
Ta agona, while he bes alue o is 0.86 (Ba celona and Gi ona). Conce ning hese wo
c i e ia, he esul s a e e y simila o bo h M1 and M2, al hough, o cou se, M1 ge s he bes
alues o and M2 o . The wo s alue, om hose gi en by M1, o is 0.77
(Ta agona) and he wo s o , om hose gi en by M2, is 0.75 (Lleida).
Fo hei side, he alues o max and

max a e high, e en o M3’ and M4’ and, al hough hey a e
no e y much be e han hose ob ained wi h M1 and M2, when he objec i e o minimising
hese c i e ia is imposed (M3’ and M4’) he alues o and de e io a e.
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