Critical dimensions in the empirical measurement of common shareholding

Rosa i, Nicole a; Bomp ezzi, Pie o; Ma inez Cille o, Ma ia
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C i ical dimensions in he empi ical measu emen o
common sha eholding
Resea ch in In e na ional Business and Finance
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Sugges ed Ci a ion: Rosa i, Nicole a; Bomp ezzi, Pie o; Ma inez Cille o, Ma ia (2024) : C i ical
dimensions in he empi ical measu emen o common sha eholding, Resea ch in In e na ional
Business and Finance, ISSN 1878-3384, Else ie , Ams e dam, Vol. 70, Iss. Pa A,
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C i ical dimensions in he empi ical measu emen o common
sha eholding✩
Nicole a Rosa i a,b,∗, Pie o Bomp ezzi c, Ma ia Ma inez Cille oa
aEu opean Commission, Join Resea ch Cen e (JRC), Via En ico Fe mi 2749, 21027 Isp a (VA), I aly
bCEMAPRE, Rua do Quelhas 6, 1200 – 781, Lisbon, Po ugal
cUni e si y o Milan-Bicocca, Piazza dell’A eneo Nuo o 1, 20126 Milano, I aly
ARTICLE INFO
JEL classi ica ion:
C18
D21
D22
G11
G32
L40
Keywo ds:
Common owne ship
Co po a e go e nance
Ne wo ks
Mobile ne wo k ope a o s
An i-compe i i e p ac ices
ABSTRACT
The deba e on common sha eholding and i s po en ial an i us e ec s is cu en ly on he agenda
o majo ins i u ions wo ldwide. Discussions poin o he need o imp o ed empi ical quan i i-
ca ion o his phenomena. This wo k p esen s a lexible, mul i ace ed s a is ical amewo k o a
se o new common sha eholding indica o s, co e ing bo h i m and in es o pe spec i es, which
can be adop ed unde di e en economic models. Many indices cu en ly used in he li e a u e
all wi hin his amewo k as special cases. Agg ega ion a ma ke le el yields sui able indus y-
le el indica o s, p o iding policymake s wi h ools o e alua e he ex en o common owne ship
in s a egic ma ke s. The indices a e es ed using i m-le el da a o Eu opean Mobile Ne wo k
Ope a o s in 2007–2021, showing a sec o wi h concen a ed owne ship unde la ge co po a e
g oups, bu also he p esence o ins i u ional in es o s wi h ex ensi e owne ship ac oss he
majo i ms.
1. In oduc ion
In Decembe 2017, he OECD o ganised in Pa is a Compe i ion Policy Round able o discuss he ise o he ela i ely new
phenomenon o common owne ship.1Acco ding o his epo , he p e ious en yea s had been cha ac e ised by a ‘‘ apid g ow h in
passi ely-managed in es men unds, [which] has had a signi ican impac on he owne ship s uc u e o la ge i ms in se e al indus ies’’.
Common owne ship, o he simul aneous owne ship o sha es in many i ms ac i e in he same ma ke , was also on he ada o o he
indus y wa chdogs and s akeholde s.2Subsequen ly in May 2018, he Eu opean Co po a e Go e nance Ins i u e (ECGI) dedica ed
a ocus panel o i s Annual Membe s’ Mee ing o ‘‘Common Owne ship: An i us Mee s Co po a e Go e nance’’. The ECGI E en
Repo 3no ed a endan s aised conce ns ega ding ‘‘po en ial collusion be ween compe ing i ms ha ing he same sha eholde s’’. The
✩Disclaime : The iews exp essed a e pu ely hose o he au ho s and may no in any ci cums ances be ega ded as s a ing an o icial posi ion o he
Eu opean Commission, o o any o he a ilia ion.
∗Co espondence o: LEAR, Via di Monse a o 48, 00186 Roma, I aly.
E-mail add esses: [email p o ec ed] (N. Rosa i), [email p o ec ed] (P. Bomp ezzi), [email p o ec ed]
(M. Ma inez Cille o).
1The epo is i led ‘‘Common owne ship by ins i u ional in es o s and i s impac on compe i ion’’ and is accessible a h p://www.oecd.o g/da /compe i ion/
common-owne ship-and-i s-impac -on-compe i ion.h m. This Round able buil on a p e ious one held in 2008 en i led ‘‘Mino i y Sha eholdings and In e locking
Di ec o a es’’.
2In he li e a u e, common sha eholde s a e mos ly known as ‘‘common owne s’’. The e m ‘‘common owne s’’ can be somehow misleading, as hese in es o s
do no ac ually own companies, hey a he own (usually small) pa icipa ions in many companies. The wo e ms will be used in e changeably in his pape .
3h ps://ecgi.global/si es/de aul / iles/e en s/2018_annual_membe s_mee ing.pd
h ps://doi.o g/10.1016/j. iba .2024.102315
Recei ed 12 Decembe 2022; Recei ed in e ised o m 28 No embe 2023; Accep ed 4 Ma ch 2024
Resea ch in In e na ional Business and Finance 70 (2024) 102315
2
N. Rosa i e al.
same yea , he ‘‘Viewpoin ’’ o he In e na ional Co po a e Go e nance Ne wo k issued in Oc obe (see ICGN,2018), and a public
hea ing o ganised by he Fede al T ade Commission (FTC) in Decembe in he US4concluded ha he impac o common owne ship
on compe i ion equi ed u he unde s anding and analysis.
These kind o deba es a e now becoming mo e common among policymake s. Al hough adi ionally common owne ship has no
been seen as an an i us issue, in ecen yea s esea che s and policy make s ha e s a ed o conside i s po en ial an icompe i i e
e ec s. Following seminal wo k on an icompe i i e e ec s o common owne ship among U.S ai lines (Aza e al.,2018), nume ous
empi ical s udies ha e analysed he impac in speci ic sec o s, among which ag i ood (Clapp,2019;Backus e al.,2021a;To shizi
and Clapp,2021), ai lines (Kennedy e al.,2017;Aza e al.,2018;Schmalz,2018;Dennis e al.,2022), banking (Schmalz,2018;
Aza e al.,2021), ene gy (A gen esi e al.,2021) and pha maceu icals (Newham e al.,2018); Banal-Es añol e al. 2021; (Xie,
2021)). These a ious s udies unde line he ele ance o common owne ship, bu also highligh empi ical challenges and di e ing
e idence ega ding he e ec s o common sha eholding on compe i ion. In 2021, he An i us Bulle in dedica ed i s Ma ch special
issue o he opic o common owne ship and i s an icompe i i e e ec s. In his issue, Schmalz (2021) p esen s a comp ehensi e
su ey o ecen s udies on his opic, concluding ha empi ical e idence has con i med an icompe i i e e ec s and ha he s udy
o he economic channels implemen ing an icompe i i e incen i es has ad anced. Along his line, Tzanaki (2022) looks om a
co po a e go e nance pe spec i e a a ie ies and mechanisms o common sha eholding, and a he plausibili y o common owne s’
an icompe i i e s a egies, discussing possible policy implica ions.
Deba e on he ele an me hodological opics is ongoing. A he 2020 Associa ion o Compe i ion Economics special panel session
on common owne ship, op schola s deba ed a leng h he ques ion o empi ical measu emen . They highligh ed ha esea ch
dealing wi h he empi ical quan i ica ion o common sha eholding is a an ad anced s age, bu a amewo k o quan i ying common
owne ship emains a key objec i e. Few ecen schola ly a icles p opose en a i e solu ions o he measu emen challenge, such
as p o i weigh s (Backus e al.,2019,2021a,b,c); (An ón e al.,2023), model-based measu es (Gilje e al.,2020), o da a-de i ed
measu es (He and Huang,2017).5Schmalz (2021) highligh s ha any p oposed measu e has an in insic syn he ic na u e, which
will cap u e in u n di e en aspec s o an indus y o ma ke , and ha he e o e one single bes measu e o common owne ship
does no exis , encou aging esea che s o conside he ele an economic con ex when adop ing a ce ain measu e.
This pape con ibu es o he quan i ica ion o common owne ship wi h a lexible, mul idimensional amewo k ha can be
adop ed wi hin di e en economics models o s udying he impac on ma ke ou comes. Many indices cu en ly used in he li e a u e
all wi hin his amewo k as special cases. This new amewo k exploi s solely he owne ship links be ween ma ke ac o s h ough
indices based on spa se ma ix heo y and ne wo k analysis, a oiding he sho comings o o he common owne ship measu es,
such as subjec i e assump ions abou con ol weigh s o he compu a ion o ma ke sha es.6The indices explo e bo h he i m’s
and he in es o ’s pe spec i e, conside ing in e ac ions be ween he wo bu also wi hin pee g oups. The e a e a numbe o use ul
applica ions o his p oposed amewo k, which a e illus a ed he e h ough a eal-da a example o he Mobile Telecoms ma ke
in he EU.7
In he i s pa o his pape (Sec ion 2), we e iew he cu en knowledge, oge he wi h i s g owing c i iques, and iden i y he
main measu emen issues o be ackled. In Sec ions 3and 4a se ies o new indices o common sha eholding a e p oposed, based on
balance shee and owne ship i m-le el da a, unde a uni ying s a is ical amewo k wi h de ailed ma hema ical p ope ies. Many
o he indices cu en ly used in he li e a u e can be iden i ied as special cases alling wi hin his amewo k. The new indices co e
bo h he i m and he in es o ’s pe spec i es, and a e hen agg ega ed o ob ain sui able indus y-le el measu emen s o common
owne ship. Sec ion 5goes o e he applica ion o hese new indices in he con ex o common owne ship. Finally, Sec ion 6p esen s
an empi ical applica ion o he p oposed measu es using i m-le el inancial and owne ship da a o Mobile Ne wo k Ope a o s
(MNOs) ac i e in Eu ope o e he pe iod 2007–2021. The las Sec ion concludes.
2. Measu ing common owne ship
All deba es men ioned ea lie poin o he inc eased need o de eloping sound measu es o common owne ship and o i s
po en ial impac s. This Sec ion e iews some o he measu emen app oaches used in pas li e a u e, oge he wi h hei main
d awbacks, and ecen de elopmen s in his a ea.
Table 1 summa ises he main measu es ou lined below, oge he wi h hei main limi a ions.
The mos popula ool used o assess he e ec s o common sha eholding was, un il ecen ly, he so-called Modi ied He indahl–
Hi schman Index (MHHI), a ma ke -le el indica o ha cap u es he dis o ion in oduced in ma ke compe i ion by he p esence o
common sha eholde s. I does so by co ec ing he He indahl–Hi schman Index (HHI) o compe i ion acco ding o he owne ship
and con ol sha es o common sha eholde s in compe ing companies. The MHHI howe e p esen s se e al d awbacks. Some ela e o
he compu a ion o he MHHI i sel — i equi es compu ing he ma ke sha es o he i ms, and also con ol weigh s o sha eholde s
in i ms which a e di icul o de e mine in p ac ice. In addi ion, he equa ions gene ally used o compu e he index in he empi ical
4US FTC public hea ing on ‘‘Common owne ship’’, 6 Decembe 2018, h ps://www. c.go /news-e en s/e en s-calenda / c-hea ing-8-compe i ion-consume -
p o ec ion-21s -cen u y.
5Sec ion 2below discusses mo e in de ail hese measu es.
6The s a is ical amewo k was ini ially de eloped in he Eu opean Commission epo ‘‘Common Sha eholding in Eu ope’’ (Rosa i e al.,2020), whe e i
was illus a ed h ough a simpli ied ma ke example. The p esen wo k la gely d aws om Rosa i e al. (2020).
7Rosa i e al. (2022a) p esen an applica ion o he new indices p oposed he e o es he possible e ec s o common owne ship on compe i i eness in he
EU be e ages indus y.
Resea ch in In e na ional Business and Finance 70 (2024) 102315
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N. Rosa i e al.
Table 1
Summa y o common owne ship (CO) measu es.
Index Type o measu e Desc ip ion Limi a ions
MHHI ma ke -le el
concen a ion
measu e
compe i ion measu e; di ec policy
in e p e a ion
ma ke and no i m-le el; needs
he calcula ion o ma ke sha es
and con ol weigh s; no a
measu e o CO; misspeci ica ion
and endogenei y issues in
empi ical applica ions
P o i weigh s (Backus e
al.)
model-based;
i m-le el measu e
de ine weigh a i m pu s on
compe i o s’ p o i s
need assump ions on con ol
weigh s
GGL (Gilje e al.) model-based;
i m-le el measu e
measu es manage s’ incen i es shi s
due o CO
ules ou s a egic in e ac ions o
i ms; no sui able o some
indus ies
Desc ip i e measu es (He
and Huang)
da a-d i en; ma ke -
and i m-le el
i m-le el explana o y a iables used
o model e ec s o CO
no model-based
Ou app oach da a-d i en;
ma ke -, i m- and
in es o -le el
mul idimensional amewo k;
includes mos abo e measu es as
special cases; measu es dis o sions
and links c ea ed by CO a i m– i m,
in es o –in es o , i m–in es o
le els; measu es ma ke s uc u es
due o CO; measu es ne wo k e ec s
no model-based
li e a u e su e om a misspeci ica ion p oblem, which may gene a e a (posi i e) co ela ion be ween p ice and he measu e o
common sha eholding, e en in he absence o a causal e ec o common sha eholding on p ice (O’B ien and Waeh e ,2017). Finally,
some o he ac o s ha d i e p ices may also a ec ins i u ional in es o s’ s ock pu chasing decisions, and consequen ly he inancial
sha es o in es o s, which hen become endogenous (Kennedy e al.,2017;O’B ien and Waeh e ,2017). Mo e impo an ly, i ails
o measu e di ec ly he ex en o common sha eholding i sel .
Mo e ecen ly, Backus, Conlon and Sinkinson in oduced he ‘‘p o i weigh s’’ (see Backus e al.,2019,2021a,b,c), a measu e
ep esen ing he weigh a i m pu s on i s compe i o s’ p o i s. These weigh s a ise wi hin a i m’s objec i e unc ion unde common
owne ship, whe e he i m maximises a combina ion o i s own p o i s oge he wi h i s compe i o s’ p o i s, duly weigh ed by he
p oposed weigh s. In he au ho s’ own wo ds he ‘‘p o i weigh s [...] a e he channels h ough which common owne ship [...] a ec s
i m beha iou ’’. Simila ly o he MHHI, he p o i weigh s need o hei calcula ion he choice o con ol weigh s (Pa e o weigh s)
o sha eholde s on i ms, ep esen ing he in luence o he in es o s on i m decisions. The ypical choice h oughou he li e a u e
has been one o p opo ional con ol (whe e he con ol weigh s equal he owne ship sha es), bu o he al e na i es a e possible.
The p o i weigh s ha e been used in some ecen empi ical s udies such as An ón e al. (2023) and Bolle and Sco Mo on (2020).
Gilje e al. (2020) de i e a bi-di ec ional, pai -le el measu e o common owne ship aiming a cap u ing he ex en o which
common owne ship shi s manage s’ incen i es o in e nalise ex e nali ies. The index accoun s o he sha es held by common
in es o s in wo compe ing companies, as well as o he ela i e weigh o each i m in he in es o s’ po olios. The measu e
is based on a model whe e assump ions a e made on how manage s deal wi h ex e nali ies imposed on one ano he by commonly-
owned i ms, speci ying a unc ion cap u ing how a en ion is alloca ed ac oss po olio companies. Howe e , as poin ed ou by
Schmalz (2021), he measu e is no sui able o cap u ing he compe i i e e ec s o common owne ship in ce ain indus ies, gi en
ha he model unde lying his measu e ules ou s a egic in e ac ions o i ms.
As al e na i es, o he s udies ha e gene ally limi ed he measu emen o common sha eholding o a small se o desc ip i e
measu es. Examples a e he p opo ion o common sha eholde s among all he in es o s p esen in a ma ke ; he p opo ion o
i ms ha a e c oss-held by a common holde , a a ce ain le el o owne ship; he numbe o compe i o s linked h ough a common
sha eholde ; he p opo ion o a i m’s sha es held by common sha eholde s, o s ill he sha es held by common sha eholde s in
a i m’s compe i o s, and so on (see o ins ance, He and Huang,2017). Such desc ip i e measu es ha e been used as i m-le el
explana o y a iables in models ying o cap u e he e ec o common sha eholding on ma ke s, oge he wi h o he measu es
cap u ing he co po a e owne ship s uc u e, such as he p opo ion o a omis ic sha eholde s o a i m.
Howe e , se e al o he aspec s o in es o s’ beha iou and o po olios’ composi ion can help d aw a mo e p ecise pic u e o
he phenomenon. The same applies o he analysis o he i ms’ sha eholding s uc u es, which can e eal in e es ing pa e ns o
o e lap in a gi en ma ke . Focusing on he beha iou o indi idual in es o s, we conside se e al dimensions o in e es . A gene al
o e iew o he deg ee o connec edness o a ma ke due o he p esence o common sha eholde s is a s a ing poin , bu he
in es men decisions a e d i en by a a ie y o objec i es, which de e mine no only how many and which i ms o include in
an in es o ’s po olio, bu also he amoun o be held in each o he chosen companies. The dis ibu ion o in es men s wi hin
a po olio can also a y, being mo e o less concen a ed a ound ew playe s a he han equally sp ead ac oss all chosen i ms,
e ealing di e en sha eholde s’ s a egies. Ano he aspec o in e es is he compa ison o po olios o concu en in es o s. This
allows o he analysis o possible ma ke -le el s uc u es, in pa icula conside ing whe he a ma ke is spli in o segmen s allo ed
o di e en s akeholde s o – on he opposi e side – o al access o any company is a ailable o all po en ial in es o s. Finally, he
Resea ch in In e na ional Business and Finance 70 (2024) 102315
4
N. Rosa i e al.
conside a ion o he sha eholde s’ ype (such as indus ial company, inancial company, public au ho i y, indi idual, e c.) is also o
in e es o in es iga e possible di e en ia ion o in es men s ac oss ce ain g oups.
A complemen a y pe spec i e o he one p oposed abo e ega ds he p esence o common sha eholde s ac oss indi idual i ms’
owne ship s uc u e. He e, he objec i e becomes he s udy o he sha eholde s uc u e o a gi en i m, and he assessmen o
he deg ee o o e lap wi h o he compe i o s’ owne ship in o ma ion. The s onge he simila i y o he sha eholding s uc u es o
compe i o s, he s onge he po en ial dis o ion in compe i ion due o common in es o s. I is c ucial hen o assess he s eng h
o he i m– i m links induced by common owne s. Fu he mo e, he deg ee o o e lap will be con ingen on he sha eholde s
conside ed o a gi en i m, and will a y based on he h eshold o equi y conside ed in he analysis.
We conside in he ollowing some me hodological s a egies o cons uc indices o he ex en common owne ship, which cap u e
he a ious aspec s men ioned abo e.
2.1. Spa si y and ne wo ks me hods in he con ex o common owne ship
A s a ing poin o he analysis o common owne ship is he ma ix ep esen a ion o a gi en ma ke . The u ili y o his
ep esen a ion is e iden when he analysis shi s in o a mo e o mal amewo k whe e he p ope ies o hese ma ices can be
le e aged o compu e he ele an indices.
A simpli ied ep esen a ion o he owne ship s uc u e o a ma ke can be ob ained h ough a able, whe e – o ins ance – each
ow co esponds o a sha eholde and each column o a i m:
F1 F2 ⋯
SH1 𝑒11 𝑒12 ⋯
SH2 𝑒21 𝑒22 ⋯
⋯ ⋯ ⋯ ⋯
The elemen s 𝑒𝑖𝑗 o he able can ei he epo he co esponding owne ship sha e, in which case we can name i owne ship ma ix
(OM), o simply epo a alue o one i a link exis s be ween a i m and an owne , ze o o he wise ( ela ion ma ix - RM). The
empi ical s uc u e o his ma ix in he owne ship con ex plays a ele an ole in he choice o app op ia e s a is ical echniques
o i s analysis. In ac , he numbe o in es o s is ypically way la ge han he numbe o i ms, wi h an a e age sha eholding
s uc u e easily p esen ing dozens o owne s; on he o he hand, he la ge majo i y o sha eholde s only in es in one i m, hence
displaying no link o he emaining compe i o s.8This gi es ise o a e y la ge ma ix whe e he majo i y o he elemen s a e ze o.
Se e al di e en s a is ical echniques ha ex ac pa e ns om gi en ma ices a e a ailable, bo h o he case o nume ical and
o bina y ( ela ional) ma ices. Such echniques allow o he iden i ica ion o ma ices’ cha ac e is ics, as well as o he calcula ion
o indices quan i ying speci ic aspec s o he ela ionships ep esen ed in he ma ix, and a e he e o e a aluable s a ing poin o
he analysis o CO. Gi en he mul iplici y o possible ma ix aspec s o be conside ed, we shall analyse in his wo k he measu es
ela ed o he concep o spa si y o a ma ix, which is mo e di ec ly linked o he CO p oblem. In ac , he concep o spa si y has
o do wi h he ep esen a ion o a phenomenon whe e only a small numbe o coe icien s con ain a la ge p opo ion o he o al
in o ma ion, he emaining elemen s o he ep esen a ion being negligible, in mos cases conside ed jus noise. In ma ix language,
a spa se ma ix o ec o is such ha mos o i s elemen s a e ze o, jus like he ypical empi ical s uc u e o a ma ke ep esen ed
hough he owne ship and ela ion ma ices — as jus no ed abo e.
The dis ance o simila i y be ween ma ices will also be analysed. The simila i y be ween ma ices is de ined acco ding o
a speci ic me ic used o de e mine he dis ance be ween wo gi en ma ices. I one o he wo ma ices is a benchma k – o
example he mos spa se ma ix in a speci ic con ex – he dis ance o simila i y measu e can be used o iden i y he deg ee o a
ce ain phenomenon wi h espec o he gi en benchma k. In he s udy o CO, a speci ic benchma k ma ix can be easily de ined,
ep esen ing o ins ance absence o CO, a he han o al in e connec ion be ween owne s and i ms, o any o he ma ke s uc u e
o in e es .
Finally, ne wo k me hods will also be conside ed, applied sepa a ely o he ne wo k o in es o s and o he ne wo k o i ms.
The assessmen o he s eng h o he links exis ing in hese wo ne wo ks will be pe o med applying he s anda d ne wo k indices,
bu also using some o he ma ix me hods men ioned abo e, applied o he ma ix ep esen a ion o he ne wo k links.
In he ollowing sec ions, he a o emen ioned concep s and s a is ical me hods will be e iewed, and hei ele ance in he con ex
o CO measu emen analysed.
3. Spa si y
The concep o spa si y is o en linked o de ini ions o inequali y o di e si y o he dis ibu ion o a phenomenon in a popula ion
o size 𝑁, say (𝑐1,…, 𝑐𝑁). Al hough he li e a u e p esen s di e en in e p e a ions and measu es o spa si y,9a common ag eemen
is ha a dis ibu ion wi h all i s in o ma ion concen a ed in one coe icien , and all o he ze o, is he mos spa se. On he o he
8Fo example, in he EU Mobile Telecoms indus y s udied in Sec ion 6, mo e han wo- hi ds o he in es o s a e ‘‘single owne s’’ i.e. hold s akes only o
one o he i ms ac i e in his indus y, consis en ly h oughou he pe iod o obse a ion. This is in line wi h wha obse ed in o he EU indus ies, such as
Oil&Gas and Elec ici y (Rosa i e al.,2022b), o Be e ages (Rosa i e al.,2022a), bu also wi h empi ical indings o he US lis ed i ms (He and Huang,2017).
9A comp ehensi e e iew can be ound in Hu ley and Ricka d (2009).

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Table 2
Some common spa si y measu es and hei p ope ies.
No. Measu e De ini ion RH Sc RT Cl BG Ba
1. 𝓁0∕𝑁#{𝑘∶𝑐𝑘= 0}∕𝑁✓(✓)✓
2. 𝓁0
𝜖∕𝑁#{𝑘∶𝑐𝑘≤𝜖}∕𝑁(✓)✓
3. −𝓁1∕𝑁−1
𝑁∑𝑘𝑐𝑘✓(✓) (✓)
4. −𝓁𝑝∕𝑁−1
𝑁(∑𝑘𝑐𝑝
𝑘)1∕𝑝,0< 𝑝 < 1✓ ✓ (✓)
5. 𝓁2∕𝓁1√∑𝑘𝑐2
𝑘∕(∑𝑘𝑐𝑘)✓ ✓ ✓
6. − log∕𝑁−1
𝑁∑𝑘log(1 + 𝑐2
𝑘)✓(✓) (✓)
7. 𝑁𝜅4𝑁∑𝑘𝑐4
𝑘∕(∑𝑘𝑐2
𝑘)2✓ ✓ (✓)✓(✓)
8. Hoye 1
√𝑁−1 (√𝑁−∑𝑘𝑐𝑘
√∑𝑘𝑐2
𝑘)✓ ✓ ✓ ✓ ✓
9. 𝑝𝑞-mean −(1
𝑁∑𝑘𝑐𝑝
𝑘)1∕𝑝
∕(1
𝑁∑𝑘𝑐𝑞
𝑘)1∕𝑞
𝑝≤1, 𝑞 > 1✓ ✓ ✓ ✓ ✓
𝑝 < 𝑞
10. Gini 1 − 2
𝑁∑𝑐𝑘∑𝑁
𝑘=1(𝑁−𝑘+1
2)𝑐(𝑘)✓ ✓ ✓ ✓ ✓ ✓
No es: P ope ies as p esen ed ea lie : RH = Robin Hood; Sc = Scaling; RT = Rising Tide; Cl = Cloning; BG = Bill Ga es; Ba = Babies. P ope ies in b acke s a e
only alid o he no malised e sion o he measu es.
hand, he e is ag eemen ha he leas spa se dis ibu ion is ound when he in o ma ion is e enly sp ead o e all coe icien s. In
he ollowing, his will be he e e ence de ini ion o spa si y. In he case o co po a e owne ship, a spa se in es men beha iou
would co espond gene ically o a po olio wi h high concen a ion o owne ship in ew i ms, o , om a i m’s pe spec i e,
a sha eholding s uc u e wi h ew sha eholde s owning la ge s akes. Low spa si y would be obse ed, ins ead, in case o mo e
widesp ead in es men s ac oss he ma ke , ypically wi h mino i y pa icipa ion.10
3.1. Some common spa si y indices
The e is a la ge se o spa si y measu es in he li e a u e, coming mainly om he ields o signal p ocessing and in o ma ion
heo y. The mos common a e he Ku osis, he Gini Index, he Hoye measu e, he 𝑝𝑞-means, he 𝓁𝑝no ms and hei combina ions.
These and o he measu es a e discussed, o ins ance, in Ka anen and Cichocki (2003), Ricka d and Fallon (2004), Hu ley and
Ricka d (2009), Zonoobi e al. (2011), Pas o e al. (2013,2015).
The choice be ween al e na i e measu es can be mo i a ed by hei ma hema ical p ope ies, which ep esen minimum c i e ia
a ‘‘good’’ measu e o spa si y should sa is y. Six main p ope ies a e ecognised o be desi able o a spa si y measu e11: he Dal on’s
Laws (‘‘Robin Hood’’, scaling, ‘‘ ising ide’’ and ‘‘cloning’’), he ‘‘Bill Ga es’’ and he ‘‘Babies’’ p ope ies. Thei aim is o gua an ee
ha a spa si y index goes in he igh di ec ion when a change occu s in he unde lying dis ibu ion. The p ope ies a e desc ibed
in de ail in Table A.1 in Appendix A, whe e hei meaning and ele ance in he con ex o common owne ship is also discussed.
Table 2 p esen s some among he mos common spa si y measu es wi h hei p ope ies. Again, he coe icien s o he dis ibu ion
unde analysis a e deno ed by 𝑐𝑘,𝑘= 1 … , 𝑁, while hei o de ed se is indica ed by 𝑐(𝑘)(in inc easing o de ). Measu es 1–4 and
6–7 a e p esen ed in hei no malised e sion, which accoun s o he leng h 𝑁o he ec o ep esen ing he dis ibu ion.
The measu es in Table 2 do no p esen a uni ied no a ion in he li e a u e; we ollow he e he sugges ions o Hu ley and Ricka d
(2009), whe e some measu es ha e been modi ied – ei he wi h a minus sign o by changing he di ec ion o some inequali y – in
o de o ob ain an homogeneous in e p e a ion in he sense ha an inc ease in spa si y yields a posi i e inc ease in he co esponding
measu e. In pa icula , no ice ha measu e 1 is usually de ined in he li e a u e as he coun o non-ze o alues, bu his would go
in he opposi e di ec ion, inc easing when spa si y dec eases. Fo his eason he e he de ini ion is e e sed, coun ing he numbe o
ze os ins ead. Fo all hese measu es, he less spa se he dis ibu ion, he smalle he alue o he index, he alue inc easing wi h
spa si y.
As we can see om Table 2, mos measu es ul il many o he p esen ed p ope ies, i no all. Ka anen and Cichocki (2003)
compa e measu es 1–4 and 6–7. Qué é and F élico (2012) compa e 7 and 8 h ough a se o simula ions on bina y (0–1)
dis ibu ions, es ing hei pe o mance in he con ex o uzzy pa i ions. Measu e 10 is conside ed, among o he s, in Ricka d and
Fallon (2004) and Zonoobi e al. (2011). Hu ley and Ricka d (2009) and Pas o e al. (2015) p opose mo e comp ehensi e accoun s
o o iginal and no malised measu es, o he addi ional measu es and p ope ies no discussed he e, as well as p oo s o he ul ilmen
o he espec i e p ope ies.
10 A mo e de ailed analysis o he applica ion o spa si y concep s o co po a e owne ship is discussed in Appendix A.
11 See o example Hu ley and Ricka d (2009).
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Measu es 1 and 2 simply compu e he p opo ion o ze o o negligible elemen s o he dis ibu ion; he highe he p opo ion,
he mo e concen a ed he dis ibu ion, i.e. he highe he spa si y. In measu e 6, any ze o coe icien gi es a ze o log alue in
he summa ion, con ibu ing owa ds a smalle o al; he minus sign e e s he di ec ion o his e ec , so ha he mo e he null
coe icien s in he sum, he highe he spa si y measu e. Measu e 7 is based on he Ku osis coe icien 𝜅4– named by analogy o
he measu e o peakedness o a p obabili y dis ibu ion – while measu e 10 is he well-known Gini Coe icien o inequali y.
Measu es 3–5 and 8–9 a e based on 𝓁𝑝- ype no ms, which a e sums o he coe icien s each aised o a ce ain powe 𝑝, sum ha
in u n is aised o he powe 1∕𝑝in o de o go back o he o iginal scale o he coe icien s. The 𝓁𝑝- ype no ms do no ha e in
gene al an in ui i e in e p e a ion, excep o he 𝓁1no m which is simply he a e age. Howe e , a special no e should be de o ed
o wo measu es, namely 𝓁0and 𝓁2, om which se e al o he measu es in he able a e de i ed.
The 𝓁0index is he base o he popula densi y measu e, gi ing he p opo ion o non-ze o elemen s o a ec o . The densi y
concep is complemen a y o spa seness, and has se e al applica ions bo h o ma ices and o ne wo ks, which will be discussed
la e . Measu e 2 es ic s he a en ion o ele an coe icien s only, hus igno ing all hose o a negligible size. A co ec ed densi y
index can be p oposed acco ding o his mo e es ic i e exclusion c i e ion, only conside ing alues abo e a ce ain h eshold.
The 𝓁2no m is also known as Euclidean no m, and o ec o 𝐶= (𝑐1,…, 𝑐𝑁)o coe icien s is gi en by:
𝓁2(𝐶) = √
√
√
√𝑁
∑
𝑘=1
𝑐2
𝑘
This is one o he mos popula no ms in se e al ields o applica ion; i is commonly used o compu e he ‘‘leng h’’ o ec o s o
size 𝑁, since i co esponds o he dis ance om he o igin o he poin 𝐶in an 𝑁-dimensional space. The ec o s p esen ing la ge
coe icien s will ha e a la ge 𝓁2no m, showing ha hey a e u he away om he o igin ( he null ec o ). Based on his no m,
se e al dis ance measu es ha e been de eloped, especially in e o minimisa ion con ex s, such as he leas squa es c i e ion (which
minimises he squa e o he 𝓁2no m), o he mean squa e e o measu emen (di ides by 𝑁 he squa e o he 𝓁2no m).
No ice ha he 𝓁2no m gi es highe weigh o la ge coe icien s, by squa ing hei alues, con a y o he simple a e age 𝓁1.
Fo his eason, he de i ed measu e numbe 5, gi en by he a io o 𝓁2 o 𝓁1, can be seen as assessing he ela i e weigh o la ge
coe icien s o e he coe icien s’ o al, hence showing la ge alues o mo e concen a ed dis ibu ions, i.e. o highe spa si y. Fo
example, conside ing h ee dis ibu ions wi h same o al alue and inc easing concen a ion – say (2, 2, 2), (4, 1, 1) and (5, 1, 0)
– measu e 𝓁2∕𝓁1 akes alues 0.58, 0.71 and 0.85 espec i ely. In he spa ses case, i.e. (6, 0, 0), we ge 𝓁2∕𝓁1= 1.
3.1.1. Ex ension o spa si y measu es o owne ship and ela ion ma ices
Spa si y measu es can be o pa icula in e es in he amewo k o CO when conside ing a gi en in es o and he ela ionship
wi h all he i ms in i s po olio. Re e ing o he ma ix ep esen a ion o a ma ke in oduced in Sec ion 2.1, he dimension o
in e es is he so called ow-wise spa si y,12 gi en ha each ow in he ma ix co esponds o an in es o ’s po olio, epo ing he
ull sha eholde - i m ela ionships. I we look a a ow, he spa ses case occu s when an owne only holds sha es o one i m in he
ma ke (wha e e he pe cen age), he minimum spa si y being achie ed when he owne owns sha es in all i ms in he ma ke ,
a a cons an pe cen age, i.e. does no show p e e ence o any i m in pa icula . This also ep esen s he mos ex eme o m o
common owne ship, an in es o eaching all compe i o s ac i e in a ma ke .
Conside now he applica ion o he spa si y measu es o ei he he OM o RM ma ix, say 𝑋, whose elemen s shall be deno ed
by 𝑥𝑖𝑗 , whe e again index 𝑖= 1 … , 𝐼 spans he ows i.e. he owne s, and index 𝑗= 1 … , 𝐽 he i ms in he columns. Recall ha he
elemen s ep esen , espec i ely, ei he he owne ship sha e o he p esence/absence o a owne – i m link.
Any o he spa si y measu es o Table 2 can be applied o each ow 𝑖o 𝑋, and hen agg ega ed ac oss ows acco ding o some
c i e ion. Deno ing gene ically by 𝑆(𝑥𝑖)a spa si y measu e applied o ow 𝑖(i.e. o elemen s 𝑥𝑖𝑗 , wi h ixed 𝑖), an o e all measu e
o ow-wise spa si y o ma ix 𝑋is gi en by 𝑆(𝑋)de ined as ollows:
𝑆(𝑋) =
𝐼
∑
𝑖=1
𝑆(𝑥𝑖)
This ype o agg ega ion is men ioned in Ricka d and Fallon (2004) as a common measu e o ma ix spa si y, howe e o he ways
o summa ising ows in o ma ion can be conside ed.
Table 3 p oposes some al e na i e agg ega ion me hods o ow spa si y indices 𝑆(𝑥𝑖), discussing he in e p e a ion o he esul ing
ma ix measu es. Besides he sum o ow indices, he a e age ac oss ows is conside ed, as well as he median and o he ele an
pe cen iles; as al e na i es, he maximum o minimum ow spa si y a e also o in e es , ep esen ing, espec i ely, he alue o
spa si y co esponding o he owne wi h mos concen a ed in es men s (in he limi no a common owne ), and o he mos
‘‘democ a ic’’ owne , in es ing mo e equally ac oss i ms in he ma ke (in p inciple a common owne ). In gene al, low alues
o 𝑆(𝑋) aise conce ns in he CO con ex , showing mo e e enly sp ead in es men s o owne s ac oss i ms in he ma ke , going in
he di ec ion o CO.
Among he possible ow-spa si y measu es ha can be used o cons uc he ma ix indices, measu e 5 om Table 2 will be used
o discuss he applica ion o he OM and RM ma ices, he emaining measu es p esen ing in gene al analogous in e p e a ions. The
special case o measu e 1, gi en i s links wi h o he ields in he li e a u e, will be discussed sepa a ely below.
12 See mo e de ails in Appendix A.
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Table 3
Ma ix spa si y measu es 𝑆(𝑋)cons uc ed applying di e en ow agg ega ion c i e ia.
No. C i e ion 𝑆(𝑋)In e p e a ion
1. Sum
𝐼
∑
𝑖=1
𝑆(𝑥𝑖)To al owne s’ spa si y. A high alue deno es high
concen a ion in he in es men beha iou , i.e.
owne s end o hold sha es o ew i ms. A low
alue deno es endency o owne s o dis ibu e
in es men s ac oss i ms.
2. A e age 1
𝐼
𝐼
∑
𝑖=1
𝑆(𝑥𝑖)A e age owne s’ spa si y. Same as abo e, bu
no malised by 𝐼, he numbe o owne s in he
ma ke .
3. Median Med 𝑆(𝑥𝑖)Median spa si y. 50% o owne s ha e an
in es men beha iou wi h spa si y lowe han
𝑆(𝑋). I 𝑆(𝑋)is high, hen owne s do no
di e si y much in es men s; i low, he e is
s onge endency o CO.
4. 𝑝 h pe cen ile 𝑄𝑝𝑆(𝑥𝑖)Same as abo e, wi h now 𝑝% o owne s ha ing
in es men s wi h spa si y lowe han 𝑆(𝑋).
Th eshold ha de e mines he deg ee o spa si y o
he 𝑝% mos ‘‘democ a ic’’ owne s.
5. Minimum min𝑖𝑆(𝑥𝑖)Spa si y o mos ‘‘democ a ic’’ ma ke owne ,
holding a e y simila p opo ion o sha es in all
ma ke i ms.
6. Maximum max𝑖𝑆(𝑥𝑖)Spa si y o mos ‘‘unequal’’ ma ke owne , holding
e y di e en p opo ion o sha es ac oss ma ke
i ms, in he limi ha ing in es ed only in one
i m.
Measu e 5 is he 𝓁2∕𝓁1spa si y index; he exp ession applied o one ow o he OM o RM is gi en by:
𝑆(𝑥𝑖) =
𝓁2(𝑥𝑖)
𝓁1(𝑥𝑖)=√∑𝐽
𝑗=1 𝑥2
𝑖𝑗
∑𝐽
𝑗=1 𝑥𝑖𝑗
The gene al meaning o his measu e was discussed in Sec ion 3.1; in he CO applica ion, he 𝓁2measu e conside s one owne 𝑖a
a ime, sums he squa es o he sha es he owne holds in each subsidia y 𝑗in he ma ke – gi ing mo e weigh o la ge sha es –
and inally akes he squa e oo o he o al. This measu es he (Euclidean) ‘‘dis ance’’ o he owne ’s in es men beha iou om
he ‘‘null’’ owne , i.e. an owne ha holds ze o (o negligible) sha es in all i ms in he ma ke . The a io o 𝓁2 o he 𝓁1measu e
(simple sum o he sha es held by he owne in all i s subsidia ies), escales such dis ance acco ding o he owne ’s o al in es men ,
gi ing a ela i e measu e o how concen a ed is he owne ’s in es men beha iou ac oss he ma ke . Such indi idual beha iou
can hen agg ega ed acco ding o any me hod p oposed in Table 3.
A li le mo e a en ion should be de o ed o measu e 1, whose link o he ec o densi y measu e was discussed in Sec ion 3.1.
I we choose 𝑆(𝑥𝑖) = 𝓁0∕𝐽i.e. he p opo ion o null elemen s o ow 𝑖, hen he a e age agg ega ion c i e ion would gi e:
𝑆(𝑋) = 1
𝐼
𝐼
∑
𝑖=1
𝑆(𝑥𝑖) = 1
𝐼
𝐼
∑
𝑖=1
#{𝑗∶𝑥𝑖𝑗 = 0}
𝐽=#{(𝑖, 𝑗) ∶ 𝑥𝑖𝑗 = 0}
𝐼𝐽
This is he o e all p opo ion o null elemen s o he 𝑋ma ix, he complemen o he well-known ma ix densi y index – an index
e y widely used in ma ix analysis and also in he ne wo ks li e a u e – i.e. he p opo ion o non-ze o elemen s o he ma ix:
densi y(𝐶) = #{(𝑖, 𝑗) ∶ 𝑥𝑖𝑗 ≠0}
𝐼𝐽
A ho ough discussion o his index in he analysis o ne wo ks will ollow in Sec ion 4.2.
In he owne ship applica ion, howe e , all lines o he OM o RM mus ha e a leas one non-ze o elemen , i.e. each owne is
included in he ma ix i and only i i owns a leas one i m in he ma ke . The e o e, he spa si y index o each ow should compu e
he p opo ion o null ow elemen s excluding hose ha a e s uc u ally non-ze o. i.e. should ake he exp ession 𝑆(𝑥𝑖) = 𝓁0∕(𝐽−1),
o he wise he ow index would ne e each he maximum o one in he case o maximum spa si y (absence o CO). This yields he
ollowing CO-co ec ed ma ix spa si y measu e:
𝑆CO(𝑋) = #{(𝑖, 𝑗) ∶ 𝑥𝑖𝑗 = 0}
𝐼(𝐽− 1)
Fo he same eason, a CO a ia ion o he ma ix densi y index should also be conside ed as ollows:
densi yCO(𝑋) = #{(𝑖, 𝑗) ∶ 𝑥𝑖𝑗 ≠0} − 𝐼
𝐼(𝐽− 1)
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Table 4
Some common ma ix no ms.
No. No m De ini ion — Single ma ix De ini ion — 𝑑(𝐴, 𝐵)
1. 𝓁𝑝- ype (∑𝑖∑𝑗|𝑐𝑖𝑗 |𝑝)1∕𝑝(∑𝑖∑𝑗|𝑎𝑖𝑗 −𝑏𝑖𝑗 |𝑝)1∕𝑝
2. 𝐿2,1∑𝑖√∑𝑗|𝑐𝑖𝑗 |2∑𝑖√∑𝑗|𝑎𝑖𝑗 −𝑏𝑖𝑗 |2
3. F obenius √∑𝑖∑𝑗|𝑐𝑖𝑗 |2√∑𝑖∑𝑗|𝑎𝑖𝑗 −𝑏𝑖𝑗 |2
4. 𝐿𝑝,𝑞 (∑𝑖(∑𝑗|𝑐𝑖𝑗 |𝑝)𝑞∕𝑝)1∕𝑞(∑𝑖(∑𝑗|𝑎𝑖𝑗 −𝑏𝑖𝑗 |𝑝)𝑞∕𝑝)1∕𝑞
No es: The elemen s o he e e ence ma ix a e deno ed by 𝑝𝑖𝑗 ,𝑖= 1 … , 𝐼,𝑗= 1 … , 𝐽 , we e 𝑝=𝑎, 𝑏 o 𝑐,
acco ding o he case. In he hi d column no ms applied o a pai o ma ices 𝐴and 𝐵in o de o de e mine
hei dis ance 𝑑(𝐴, 𝐵).
which compu es he p opo ion o non-ze o elemen s only among he en ies in he ma ix which can ac ually be ze o, hus excluding
he cases ha a e bound o be non-ze o. This modi ied densi y index will each he minimum alue o ze o in case o absence o
common owne s, hence allowing o ha e a ze o-densi y benchma k o a ma ke wi h only single owne s.
3.2. Simila i y measu es o compa ison o benchma k scena ios
A possible al e na i e app oach o he measu emen o he ex en o CO, as men ioned ea lie , is he compa ison o he obse ed
ma ke s uc u e wi h an hypo he ical benchma k scena io o in e es . As long as i is possible o exp ess he desi ed benchma k
h ough a speci ic ma ix, hen he compa ison wi h he benchma k can be pe o med h ough he compu a ion o a simila i y
measu e be ween he benchma k ma ix and he ma ix ep esen ing he ac ual ma ke .
Ma ix simila i y measu es a e in ended o assess he ‘‘dis ance’’ 𝑑(𝐴, 𝐵)be ween wo ma ices 𝐴and 𝐵, wi h iden ical numbe
o ows and columns, no necessa ily squa e. Simila i y measu es a e usually calcula ed en y-wise, ha is he elemen s o he wo
ma ices in he same posi ion a e compa ed (usually calcula ing di e ences), and hese alues a e hen agg ega ed h ough a ma ix
no m, i.e. a ma ix measu e simila o he ec o measu es conside ed ea lie .
In o de o compu e a ma ix no m, he ma ix is ea ed as a long ec o , whe e he columns (o ows) ha e been s acked
oge he ; he no ms ea a 𝐼×𝐽ma ix as a ec o o size 𝐼𝐽, and apply any ec o measu e o i . Among he mos popula , we
ha e he 𝓁𝑝- ype no ms, he F obenius no m and he mo e gene al amily o 𝐿𝑝,𝑞 -no ms, whose o mulas a e gi en in Table 4. The
s uc u e o hese no ms is e y simila o he spa si y measu es p esen ed ea lie , he main di e ences being ha summa ion is
now pe o med o e he wo ow and column indices, 𝑖and 𝑗 espec i ely, o span all elemen s o he ma ix.
Bo h he 𝐿2,1and he F obenius no m a e pa icula cases o he amily o 𝐿𝑝,𝑞 no ms, since he F obenius is ac ually an 𝐿2,2
no m. The 𝐿2,1no m is a popula e o unc ion used in obus da a analysis and in spa se coding, gi en ha he e o o each da a
poin ( he ma ix ow in his case) is no aised o any powe , bu simply summed o e all poin s ( ows).
The F obenius no m is also a he popula , as being he Euclidean no m on he 𝐼×𝐽space o he ma ix elemen s. Bo h no ms
a e in a ian unde o a ions o he ow and/o o he columns, i.e. he o de in which he i ms and he owne s a e a anged
in o he ma ix is i ele an . Fo a ulle accoun o ma ix heo y and applica ions, see o ins ance Zhang (2017), o Boyd and
Vandenbe ghe (2018).
Fo he calcula ion o he dis ance be ween ma ices, he chosen no m is no applied o he o iginal ma ices, bu o he
ans o med ma ix (o di e ences). In he igh column o Table 4, he no ms p esen ed abo e a e used o he measu emen
o ma ices simila i y, based on he ma ix o di e ences 𝐴−𝐵.
I we wish o use he densi y index (o i s a ia ion p esen ed ea lie ) in he case o compa ison o ma ices, we can ei he compu e
he densi ies o he wo ma ices and hen compa e hem, o compu e di ec ly he densi y o he ma ix o di e ences. Since he
densi y is no a linea ope a o , in gene al Densi y(𝐴) − Densi y(𝐵)≠Densi y(𝐴−𝐵)(unless 𝐴=𝐵, in which case hey a e bo h null).
In he i s compu a ion on he le -hand side, we assess he di e ence in spa si y be ween 𝐴and 𝐵; i he di e ence is posi i e, hen
ma ix 𝐴is mo e dense i.e. has mo e non-ze o elemen s han 𝐵, he e e se applying in case o a nega i e alue. On he o he hand,
he densi y o 𝐴−𝐵will always be a posi i e alue, compu ing he p opo ion o non-ze o alues o he di e ence ma ix, i.e. he
p opo ion o cases o which he elemen s o 𝐴and 𝐵a e no equal. A small alue indica es ha he wo ma ices coincide in mos
en ies, while a alue close o one means ha mos o he elemen s o he wo ma ices a e di e en , hence p oducing non-ze o
en ies in he di e ence ma ix.
4. Ne wo k me hods
Social ne wo k analysis s udies he empi ical s uc u e o social ela ions and associa ions ha may be exp essed in ne wo k
o m. I can he e o e be applied o he analysis o he co po a e owne ship s uc u e o a ma ke , which can be easily ep esen ed
h ough a ne wo k o ela ionships be ween owne s and i ms. A ligh in oduc ion o social ne wo ks can be ound in Bo ga i e al.
(2009), while König and Ba is on (2009) p esen he main ea u es o economic ne wo ks. Fo a comp ehensi e accoun o social
ne wo ks analysis see, o ins ance, Sco (2017)o Bo ga i e al. (2018).
A social ne wo k is cons i u ed by a se o nodes (o ac o s), and a se o ies (o links) ha connec pai s o nodes. The nodes can
be all o he same ype, o example when s udying ela ionships be ween indi iduals, gi ing ise o a so-called one-mode ne wo k.
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Table 5
Summa y s a is ics o i ms ac i e in he EU28 in he MNOs sec o be ween 2007 and 2021.
Yea Numbe Numbe (%) i ms Numbe Numbe (%) Numbe (%)
Fi ms C oss-held by BH o SH Common SH Single SH
2007 84 61 (72.62) 640 188 (29.38) 452 (70.63)
2008 87 64 (73.56) 652 193 (29.60) 459 (70.40)
2009 90 63 (70.00) 662 187 (28.25) 475 (71.75)
2010 91 67 (73.63) 717 194 (27.06) 523 (72.94)
2011 94 65 (69.15) 639 191 (29.89) 448 (70.11)
2012 95 63 (66.32) 657 169 (25.72) 488 (74.28)
2013 94 67 (71.28) 645 185 (28.68) 460 (71.32)
2014 94 67 (71.28) 617 177 (28.69) 440 (71.31)
2015 96 72 (75.00) 642 205 (31.93) 437 (68.07)
2016 100 76 (76.00) 616 209 (33.93) 407 (66.07)
2017 100 78 (78.00) 593 188 (31.70) 405 (68.30)
2018 100 79 (79.00) 595 179 (30.08) 416 (69.92)
2019 96 75 (78.13) 712 185 (25.98) 527 (74.02)
2020 95 71 (74.74) 692 164 (23.70) 528 (76.30)
2021 95 73 (76.84) 500 139 (27.80) 361 (72.20)
No es: Pe cen age o i ms c oss-held by block-holde s (BH) a minimum 5%. Pe cen age o common sha eholde s = Pe cen age
o sha eholde s wi h mo e han one i m in hei po olio. Single owne s = Those holding sha es only o one i m.
2007 and 2021.18 All o hese ope a o s a e based in Eu ope, 97% being egis e ed in he EU28, while he emaining 3% (i.e. 3
i ms) a e egis e ed in No way.
Table 5 p esen s he summa y igu es o he sha eholde s uc u e o he indus y o e he pe iod o obse a ion. The numbe
o i ms in his indus y has been inc easing slowly, s abilising jus unde 100 in 2021. The sample includes an a e age o 642
sha eholde s pe yea . The p opo ion o common owne s has been a he s eady a ound a mean o 29% h oughou he pe iod.
The pe cen age o MNOs ha sha e block holde s wi h compe i o s has inc eased be ween 2007 and 2021 o almos 77% o i ms.
By de ini ion, wo i ms a e c oss-held by a block-holde i he common in es o holds a leas 5% in bo h i ms. The empi ical
e idence o ou sample implies ha he as majo i y o i ms a e linked o a leas one o he company in he ma ke h ough a
subs an ial amoun o sha es in he hand o a common owne . This is mainly due o he peculia s uc u e o his ma ke , whe e
mos companies belong o la ge co po a e g oups such as Voda one o O ange, whe e all subsidia ies sha e a common pa en . In
ac , hese igu es a e ma kedly highe han EU and US indica o s calcula ed on a a ie y o ma ke s o he same pe iod in Rosa i
e al. (2020) and He and Huang (2017), espec i ely, epo ing p opo ions o block c oss-held i ms well below 70%. No ice ha
ins i u ional in es o s hold e y la ge po olios in his ma ke , as will be de ailed la e , and hence a e esponsible o linking a
subs an ial numbe o compe i o s; howe e , he sha es held a e usually below he block h eshold o 5% conside ed he e, so hei
holdings do no in luence his indica o .
6.1. Main playe s
Since mos o he MNOs included in ou sample a e egis e ed in he EU28, he e a e only ew la ge playe s om ou side
(No way). The e o e, in he analysis o he main i ms ac i e in his indus y en EU28-based companies and only wo non-EU28
co po a ions will be conside ed. Table 6 displays summa y in o ma ion o he op 10 la ges en e p ises in EU28 and op 2 ou side
EU28, by coun y o egis a ion. The i ms a e anked acco ding o he alue o hei o al asse s (TOAS, in Bln e) in 2021. No e
ha some o he op i ms a e con olled subsidia ies; in hese cases, he owne ship in o ma ion e e s o he mo he companies (see
‘‘Con olled by’’ column).
As a single in es o , BlackRock domina es he MNOs indus y, wi h a he la ge sha es in some o he op playe s. This
beha iou is in line wi h o he empi ical s udies o ins i u ional in es o s in s a egic ma ke s, which has in u n been associa ed o
an icompe i i e beha iou .19 Vangua d and No way lag behind, wi h ela i ely small pa icipa ions, while S a e S ee plays only a
mino ole. F ance is also p esen wi h small sha es in only a hand ul o op compe i o s. To he bes o ou knowledge, his pape
also ep esen s he i s ins ance whe e he ole o so e eign in es men s unds in common owne ship is me hodically quan i ied.
The addi ional associa ion o an icompe i i e beha iou o so e eign unds could be an in e es ing opic o u he esea ch.
6.2. Densi y and uni o mi y
Table 7 now p esen s summa y s a is ics o he densi y-based and uni o mi y indica o s calcula ed on 2021 da a o MNOs.20
These s a is ics ep esen indus y-le el empi ical es ima es o he p esence o common owne ship. The densi y index indica es ha
18 Mos o he companies (82%) classi y hei co e ac i i y wi hin he NACE Di ision ‘‘61: Telecommunica ions’’. The o he 18% o i ms a e sca e ed ac oss
di e en ac i i ies (e.g. Re ail, Pos al and Cou ie Ac i i ies and Head O ices).
19 See o example Aza e al. (2018), Banal-Es añol e al. (2021), and Aza e al. (2021) which iden i y Black ock as one o he key common sha eholde s
in he ai lines, pha maceu icals, and banking sec o s, espec i ely.
20 Tables displaying he ull se o ma ix indices calcula ed o all yea s be ween 2007 and 2021 a e a ailable upon eques om he co esponding au ho .

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Table 6
La ges i ms o MNOs sec o and selec ed owne ship da a on hei main common owne s.
Coun y TOAS, Black F ance No way S a e Vangua d
2021 ock s ee
(Bln e)
Regis e ed in EU28 Con olled by Sha es held in 2021 (%)
VODAFONE GROUP UK 153.95 9.11 0.13 3.09 2.40 3.77
ORANGE FR 99.46 4.87 29.41 1.65 0.27 2.00
TELECOM ITALIA IT 57.48 1.55 0.75 0.21 1.34
TELIA COMPANY SE 20.25 3.00 1.12 0.18 1.90
WIND TRE IT 16.41 CK HUTCHISON HOLD. (KY) 4.90 1.07 0.91 2.04
EE LIMITED UK 13.60 BT GROUP PLC (UK) 4.55 2.99 2.16 3.31
KONINKLIJKE NL 12.74 4.84 2.80 0.38 2.20
O2 HOLDINGS LTD UK 12.58 (50%) LIBERTY GLOBAL (UK) 3.49 0.92 1.50 2.02
(50%) TELEFONICA (ES) 2.90 1.89 2.49
TELENET GROUP BE 9.76 TELENET GROUP HOLD. (BE) 0.64 0.99 0.14 1.13
PROXIMUS SA BE 9.23 4.88 1.14 0.15 1.17
TELEFONICA DE DE 7.59 1.78 0.90 0.10 0.64
Regis e ed ou side EU28 Con olled by Sha es held in 2021 (%)
TELENOR ASA NO 20.03 1.60 53.97 0.80 1.25
TELIA NORGE AS NO 3.76 TELIA COMPANY AB (SE) 3.00 1.12 0.18 1.90
No es: Di ec o indi ec pa icipa ion sha es (%) held in 2021 a e displayed.
Table 7
Summa y s a is ics o MNOs sec o common sha eholding indices (yea 2021).
Index Mean 75 h pe cen ile 99 h pe cen ile Maximum
Densi y 2.05 2.11 12.11 17.89
Uni o mi y 11.45 19.19 74.78 76.54
TOAS densi y 12.67 26.40 67.25 68.48
TOAS weigh ed densi y 0.25 0.15 3.68 5.73
MKT CAP densi y 16.35 28.04 96.61 98.98
MKT CAP weigh ed densi y 0.20 0.08 4.22 8.77
No es: Pe cen age poin s displayed, index alues be ween 0 and 100.
he MNOs in es o s end o ha e small po olios, including a mos 18% o he i ms in he ma ke . The op 1% o la ges po olios
a e ac ually a he smalle , holding sha es in jus abo e 12% o he ma ke i ms. As no ed ea lie , his is due o he p esence,
in his indus y, o la ge g oups mo he ed by Telecom gian s such as Voda one, O ange o Deu sche Telekom, which ha e li le
i no o e lap in hei po olios, and ha e basically pa i ioned he se o i ms ac i e in his indus y in o independen domains.
F om his pe spec i e o he sha eholde s hen, common owne ship in 2021 emained a ela i ely con ained phenomenon. This is
also con i med by o e all lowe le els o uni o mi y indices,21 whe e he maximum did no each 77% in 2021, showing a highe
concen a ion o in es men s a egy. This sugges s ha in es o s a e less ‘‘democ a ic’’ and end o a ge highe pa icipa ions in
a smalle se o i ms, a he han widesp ead low in es men ac oss all ma ke .
In e ms o alue-based indices, he e a e e iden di e ences in le el be ween o al asse s and ma ke capi alisa ion esul s,
gi en ha he se o lis ed i ms in his indus y is a he educed, and he e o e he second se o indices is compu ed on a e y
es ic ed pa o he ma ke . We see ha he op 1% o in es o s hold s akes in i ms ha ep esen almos 68% o he To al Asse s
o he indus y, showing ha such op in es o s end o p i ilege he la ges en e p ises. Gi en he high pa icipa ion sha es in
his ma ke , he weigh ed TOAS index eaches high alues, wi h he op in es o holding 5.73% o he ma ke o al asse s h ough
sha es. A simila pic u e is ound o lis ed i ms, whe e he la ges po olios hold s akes in i ms ep esen ing basically he en i e
ma ke capi alisa ion (almos 99%), and h ough hei pa icipa ions con ol abou 9% o he ma ke alue. Unlike he unweigh ed
indices, hese indus y measu es imply a deg ee o common owne ship ha could p esen an icompe i i e conce ns, as a ew op
in es o s exe in luence o e a la ge po ion o he ma ke asse s.
6.3. Common owne ship indices o op in es o s
Table 8 p esen s he densi y indices o he op in es o s engaged in common owne ship in he MNO sec o in 2021. These
a e mainly la ge Telecom g oups, wi h se e al subsidia ies in di e en coun ies. Fo example, O ange holds 2.38% o he ma ke
To al Asse s h ough i ually ull owne ship o i s subsidia ies in Belgium, Poland, Romania, Slo akia o Spain. Simila ly, Deu sche
Telekom holds 1.89% o he To al Asse o he indus y h ough con ol o i ms in Aus ia, Check Republic, C oa ia, Hunga y,
Poland and Slo akia.
21 Compa ed o o he ma ke s. See o example Rosa i e al. (2020).
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Table 8
Top in es o s in MNOs sec o (yea 2021).
SH name SH No. Densi y TOAS W TOAS La ges subsidia ies and sha es held (%)
coun y Subs. densi y den.
FRANCE FR 9 9.47 46.39 5.73 ORANGE (29.41); VODAFONE GROUP (0.13); ORANGE
ESPAGNE (25.01); ORANGE POLSKA (25.01); ORANGE
BELGIUM (25.01); ORANGE ROMANIA (25.01); ORANGE
SLOVENSKO (25.01)
DEUTSCHE BANK DE 12 12.63 67.46 3.82 VODAFONE GROUP (13.68); ORANGE (0.48); TELECOM
ITALIA (0.17); TELIA COMP (0.16); TELENOR (1.62);
KONINKLIJKE (0.64); PROXIMUS (0.63)
CK-HUTCHISON HOLDING KY 6 6.32 3.89 3.82 WIND TRE (98); HUTCHISON DREI AUSTRIA (100); THREE
IRELAND (98); CK HUTCHISON UK (98); HI3G ACCESS AB
(98); HI3G DENMARK APS (98)
BLACKROCK US 12 12.63 67.46 3.79 VODAFONE GROUP (9.11); ORANGE (4.87); TELECOM
ITALIA (1.55); TELIA COMP (3.00); TELENOR (1.6);
KONINKLIJKE (4.84); PROXIMUS (4.88)
TELEFONICA ES 3 3.16 4.14 3.74 O2 HOLDINGS LTD (100); TELEFONICA DEUTSCHLAND
(69.93); TELEFONICA MOVILES (100)
NORWAY NO 17 17.89 68.48 3.35 TELENOR (53.97); VODAFONE GROUP (3.09); KONINKLIJKE
(2.80); ORANGE (1.65); TELECOM ITALIA (0.75); TELIA
COMP (1.12); PROXIMUS (1.14)
ORANGE FR 7 7.37 2.94 2.38 ORANGE ESPAGNE (100); ORANGE POLSKA (50.67);
ORANGE BELGIUM (78.32); ORANGE ROMANIA (99.20);
ORANGE SLOVENSKO (100); ORANGE ROMANIA COMM.
(53.58); ORANGE COMM. LUX. (78.32)
VODAFONE GROUP UK 9 9.47 2.41 2.22 VODAFONE ITALIA (100); VODAFONE PORTUGAL (38.62);
VODAFONE MAGYARORSZAG (100); VODAFONE CZECH
REPUBLIC (100); VODAFONE ROMANIA (100); VODAFONE
LIBERTEL (50); VODAFONE ENTERPRISE GERMANY (100)
SWEDEN SE 15 15.79 58.73 1.97 TELIA COMP AB (39.50); VODAFONE GROUP (0.18);
ORANGE (0.27); TELENOR (0.13); KONINKLIJKE (0.23);
PROXIMUS (0.19); TELEFONICA DEUTSCHLAND HOLDING
(0.12)
TELIA COMP SE 6 6.32 1.94 1.90 TELIA SVERIGE (100); TELIA FINLAND (100); TELIA NORGE
(100); TELIA LIETUVA (88.2); TELIA EESTI (100); LATVIJAS
MOBILAIS TELEFONS (50.01)
DEUTSCHE TELEKOM DE 8 8.42 2.40 1.89 MAGYAR TELEKOM (40.77); T-MOBILE POLSKA (100);
T-MOBILE AUSTRIA (100); HT-HRVATSKI
TELEKOMUNIKACIJE (52.17); SLOVAK TELEKOM (100);
T-MOBILE CZECH REP. (100); TELEKOM DEUTSCHLAND
(100)
VANGUARD US 12 12.63 67.46 1.68 VODAFONE GROUP (3.77); ORANGE (2.00); TELECOM
ITALIA (1.34); TELIA COMP (1.90); TELENOR (1.25);
KONINKLIJKE (2.20); PROXIMUS (1.17)
No es: SH coun y: Coun y o sha eholde . Las column displays he i ms wi h la ges o al asse s in po olio and espec i e quan i y o sha es held.
Ra he impo an a e also he S a es, wi h F ance and No way anking amongs he op in es o s, holding he la ges pe cen ages
o he ma ke To al Asse s, wi h 5.73% and 3.35% espec i ely. As o he Funds, only BlackRock has la ge enough pa icipa ion
(i.e. 3.79% o he ma ke To al Asse s) o appea in he op posi ions, while Vangua d is le behind wi h a lowe , ye s ill sizeable,
sha e o To al Asse s held (abou 1.68%).
While ou indices do documen a deg ee o common owne ship among s a e in es o s (such as F ance, No way and Sweden), he e
is s ill no e idence in he li e a u e o go e nance ela ed channels which ie he p esence o common owne ship o an icompe i i e
beha iou .
6.4. Ne wo k indices
The esul s o sha eholde s’ and i ms’ ne wo ks in he MNOs sec o a e epo ed in Tables 9 and 10, espec i ely. The e a e
s ong in e connec ions be ween ac o s o he same ype. The p opo ion o pai s o po olios ha a e linked h ough commonly
held i ms (i.e. p opo ion o non-ze o co ela ions in Table 9) has been oscilla ing o e ime be ween 4 and 5%, wi h a sligh
decline in he las ew yea s. On he o he hand, among he la ge22 connec ed po olios he links a e a he s ong, ha ing obse ed
22 Po olios in his sec o a e conside ed la ge i hey hold s akes in a leas 10% o he ma ke ’s i ms.
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Table 9
Sha eholde s’ ne wo k indices o MNOs sec o .
Yea To al No. P op. Non-ze o No. High Co . P op. High Co .
Common SH Co ela ions High Dens. High Dens.
2007 188 4.57 21 6.46
2008 193 5.02 35 6.34
2009 187 4.37 38 7.25
2010 194 4.24 14 4.07
2011 191 5.27 19 6.33
2012 169 4.23 12 4.08
2013 185 4.51 9 4.46
2014 177 4.72 14 5.74
2015 205 5.05 22 7.97
2016 209 5.28 20 6.67
2017 188 4.73 14 6.67
2018 179 4.30 12 7.84
2019 185 3.26 12 7.02
2020 164 2.72 19 13.97
2021 139 3.06 21 23.08
No es: High-densi y po olios a e conside ed as such i he densi ies a e highe han 10%, i.e. i hey
hold s akes in a leas 10% o he ma ke ’s i ms.
Table 10
Fi ms’ ne wo k indices o MNOs sec o .
Yea To al No. i ms P op. Non-ze o No. o high P op. o high
c oss-held by BH co ela ions co ela ions co ela ions
2007 61 3.96 73 52.90
2008 64 4.49 99 58.93
2009 63 4.39 99 56.25
2010 67 4.37 98 54.75
2011 65 3.84 88 52.38
2012 63 4.32 105 54.40
2013 67 4.12 97 53.89
2014 67 4.37 87 45.55
2015 72 3.84 82 46.86
2016 76 3.31 88 53.66
2017 78 3.25 95 59.01
2018 79 3.41 101 59.76
2019 75 3.55 86 53.09
2020 71 3.85 85 49.42
2021 73 3.54 83 52.53
No es: Block-holde s (BH) hold a leas 5% o sha es.
an inc easing p opo ion o highly co ela ed in es o s, on a e age abou 7% o owne s showing almos coinciden in es men
s a egies, wi h po olio o e laps o 80% o mo e.
The i ms’ ne wo k is also highly in e connec ed h ough block holde s, wi h be ween 3.5% and 4% o pai s o i ms connec ed
h ough some common sha eholde . Mo e impo an ly, s eadily o e hal o hese linked po olios p esen a co ela ion o mo e
han 80%. Rega dless, his e y high ac ion o highly co ela ed owne ship s uc u es is p obably he esul o he p esence o
clus e s o i ms belonging o he la ge Telecom g oups, which a e almos all wholly owned o a leas con olled by he pa en ,
showing he e o e e y simila sha eholde s’ s uc u es. Again, compa ed o o he sec o s like Oli&Gas, Elec ici y o Be e ages (see
Rosa i e al.,2020), he MNOs show s onge ne wo k in e connec ions bo h be ween i ms and be ween sha eholde s.
As men ioned ea lie , he sou ce o he quan i ied links be ween i ms is ele an o compe i ion conce ns. I he la ge sha e o
highly co ela ed owne ship among i ms de i es om co po a e g oups, i would no necessa ily indica e an icompe i i e beha iou
s emming om common owne ship as he empi ical li e a u e ypically inds. The indices p oposed p o ide insigh in o hese
di e en dimensions o indus y links.
7. Conclusions
The exis ence o common sha eholde s among compe ing companies in a gi en indus y has aised he conce n o academics and
policy make s wo ldwide, due o i s possible e ec s on ma ke e iciency and compe i ion. The main empi ical wo ks in es iga ing
his issue co e only a limi ed numbe o indus ies, and a mo e gene al e ec on he economy has ye o be analysed. This is in pa
because he measu es used so a o assess he e ec s o common owne ship in a gi en ma ke ha e been subjec o c i icism om
se e al schola s, aising he need o he de elopmen o a sound measu emen amewo k. Besides speci ic echnical sho comings
o he known me ics, c i ics unde line ha each measu e will cap u e di e en aspec s o an indus y o ma ke , and encou age
esea che s o conside he ele an economic con ex when adop ing a ce ain measu e. The absence o a single one-size- i -all
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‘‘bes ’’ measu e o common owne ship poin s owa ds a plu al app oach, whe e a se o indices a e able, oge he , o cap u e he
complexi y o he phenomenon.
This wo k conside ed some no el mul i ace ed me hodological s a egy o be applied o he measu emen o common owne ship.
The me hods a e aken in pa om he heo y o spa se ma ices and in pa om social ne wo k analysis, and a e adap ed o e lec
he need o p ope measu emen o he phenomenon o common sha eholding. F om spa si y heo y, a se ies o concen a ion
indica o s a e de i ed o he analysis o a sha eholde s’ po olio, desc ibing i s in es men beha iou bo h a indus y le el and
a in es o s’ le el. Value-based indices a e also p oposed, whe e he size o he i ms held in each po olio is aken in o accoun .
The s eng h o he ela ionships exis ing wi hin he g oups o i ms and o in es o s is also s udied, based on pu pose-buil ne wo k
indices. The ne wo k analysis looks, in u n, a simila i ies be ween po olios o pai s o sha eholde s, o a o e laps o sha eholde s
s uc u es o pai s o i ms. The p oposed me hodologies can be applied bo h o he se o owne – i m ela ions induced by co po a e
owne ship, as well as o he ull in o ma ion abou he amoun o pa icipa ion sha es. Compa ison wi h gi en benchma ks such
as he case o absence o common owne ships a e also conside ed. Because he amewo k p oposed he e abs ac s om p e ious,
heo y-based app oaches, he e a e limi a ions o wha he measu emen s can say abou he ansmission mechanisms o common
owne ship ends o an icompe i i e beha iou . We a gue howe e ha hei model- ee na u e makes hem mo e objec i e and
compa ible wi h a ange o di e en economic models, as hey do no equi e a numbe o he assump ions made by p e ious
me ics. As a ma e o ac , many o he indica o s cu en ly used in he li e a u e a e iden i ied as special cases alling wi hin his
amewo k.
The measu es a e es ed using i m-le el da a o Eu opean Mobile Ne wo k Ope a o s be ween 2007–2019, wi h special
a en ion dedica ed o he ole o he la ges in es o s. Resul s show ha in es o s’ po olios on a e age in his indus y a e a he
specialised, due o he p esence o la ge co po a e g oups con olling coun y-speci ic subsidia ies, such as Voda one. Howe e ,
ce ain ins i u ional in es o s ypically associa ed wi h common-owne ship- ela ed an icompe i i e conce ns, such as Black ock,
p esen a sha e o i ms held and a o al asse sha e compa able o hese co po a e g oups. Mo eo e , he subsample o la ge in es o s
wi h ex ended in e es s (again gene ally co esponding o ins i u ional in es o s), shows a a he s ong ne wo k connec ion wi h
highly-co ela ed po olios. This is no he case, ins ead, o he co po a e g oups, which p esen non-o e lapping po olios, c ea ing
clus e s o owne ship segmen ing he ma ke . This highligh s he impo ance o a holis ic app oach o common sha eholding as
p oposed he e.
The amewo k p oposed in his pape , based on di ec ly obse ed linkages be ween i ms and in es o s p o ides a da a-d i en
measu e o common owne ship. I s mul idimensional app oach helped disen angle he di e en ial beha iou o common owne s
o di e en na u e, such as he pa en s o co po a e g oups as opposed o ins i u ional in es o s. The policy implica ions in e ms
o possible an i-compe i i e beha iou o hese in es o s a e dis inc , so he app oach p o ides policy make s wi h a mo e comple e
pic u e o he ma ke dis o ions in oduced by di e en a ie ies o common owne s. The empi ical applica ion p o ed use ul also
in highligh ing new ele an ends o esea ch. In pa icula , ou indings unde sco ed he g owing ole o s a es as common
sha eholde s, some hing which me i s close a en ion in u u e esea ch.
CRediT au ho ship con ibu ion s a emen
Nicole a Rosa i: Concep ualiza ion, Da a cu a ion, Fo mal analysis, In es iga ion, Me hodology, P ojec adminis a ion, So -
wa e, Supe ision, Valida ion, W i ing – o iginal d a , W i ing – e iew & edi ing. Pie o Bomp ezzi: Da a cu a ion, Fo mal
analysis, In es iga ion, So wa e, Valida ion, W i ing – o iginal d a , W i ing – e iew & edi ing. Ma ia Ma inez Cille o: Da a
cu a ion, Fo mal analysis, In es iga ion, So wa e, Valida ion, W i ing – o iginal d a , W i ing – e iew & edi ing.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing inancial in e es s o pe sonal ela ionships ha could ha e appea ed
o in luence he wo k epo ed in his pape .
Da a a ailabili y
The au ho s do no ha e pe mission o sha e da a.
Acknowledgemen s
This pape bene i ed om aluable con ibu ions om Jean Be ge in, Ma ia Elena Despo , José En ique Elías Cab e a, Sean
G eenaway, Issam Hallak, Cy il Ha i on, Igo Jelinski, Michela Na do, Co nelius Schmid , Joanna Siko a-Wi nebel and An hea
Wingende . The au ho s also hank aluable commen s om semina pa icipan s a he Essex Finance Cen e and om con e ence
pa icipan s a Eu opean Economics and Finance Socie y Annual Con e ence and a he In e na ional Con e ence on Eu opean
S udies. Au ho Nicole a Rosa i was pa ially suppo ed by he P ojec CEMAPRE/REM - UIDB/05069/2020 inanced by FCT/MCTES
h ough na ional unds. The o he au ho s did no ecei e any speci ic g an om unding agencies in he public, comme cial, o
no - o -p o i sec o s.
Appendix A. Applica ion o he spa si y amewo k o common owne ship
In o de o apply he spa si y concep o he CO p oblem, and in pa icula o he wo ma ices de ined in Sec ion 2.1 ( he
owne ship ma ix OM and he ela ion ma ix RM), i is necessa y o iden i y he mos and leas spa se scena ios and hei meaning.
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The main issue o be conside ed is he applicabili y o he spa si y concep o a ma ix, since he concep i sel , as well as he
measu es, we e ini ially de eloped in ela ion o a ec o o coe icien s — ep esen ing he dis ibu ion o weal h.
The ex ension o a ec o measu emen o a ma ix can be done in h ee di e en ways, which in u n gi e ise o di e en
benchma k scena ios, as discussed below. Le index 𝑖= 1,…, 𝐼 deno e he owne s, and 𝑗= 1,…, 𝐽 he i ms. The spa si y o a gi en
owne ship s uc u e can be s udied looking a he ollowing dimensions in he OM and RM:
(1) column dimension:spa si y o he sha eholde s’ dis ibu ion o a gi en i m.
In he analysis o a column, he maximum spa si y is achie ed when a gi en i m 𝑗p esen s only one owne who holds 100%
o he sha es, he leas spa se dis ibu ion co esponding ins ead o ha ing 𝑛𝑗sha eholde s, each wi h a p opo ion o 1∕𝑛𝑗o
he sha es. Fo he RM, hese cases co espond, espec i ely, o a ec o wi h one uni elemen and all he es ze o, o o a
ec o o all ones.
(2) ow dimension:spa si y o he in es men dis ibu ion o a gi en owne in o he i ms cons i u ing he ma ke .
I we look a a ow, he spa ses case occu s when an owne only holds sha es o one i m in he ma ke (wha e e he
pe cen age), he minimum spa si y being achie ed when he owne owns sha es in all i ms in he ma ke , a a cons an
pe cen age (does no show p e e ence o any i m in pa icula ).
(3) o e all dimension:spa si y o he i m–owne links p esen in he ma ke .
In he o e all analysis o he ma ices we can de ine maximum spa si y ei he o all he ows (one i m only pe each owne ,
o all owne s; one i m can ha e mo e owne s, 𝐼≥𝐽), o o all he columns (one owne only pe each i m, o all i ms; he
owne can be common o o he i ms, 𝐼≤𝐽), o o bo h a he same ime (squa e diagonal ma ix, one owne only pe each
i m and one i m only pe each owne , 𝐼=𝐽).
De ining maximum spa si y o a ma ix looking a he column dimension would conside only he sha eholde s’ s uc u e o a
gi en i m, and no he in e -linking be ween i ms caused by common owne s, so does no co espond o he p ima y objec i e
o ou s udy. Howe e , his app oach can be used o assess o he ma ke cha ac e is ics in a la e s age. The las app oach
(join maximum spa si y o bo h ows and columns) again imposes he column maximum spa si y, which is no o cen al
in e es a his s age, so i will be le o la e analyses.
I ollows ha he ow-wise app oach o he de ini ion o o e all maximum spa si y o a ma ix seems mo e app op ia e o
ou case, as being di ec ly connec ed o he s udy o CO. In ac , in his app oach he mos spa se ma ix co esponds o he
absence o common owne s, i.e. each ow only p esen s one non-ze o elemen , while he opposi e happens when each ow
is comple ely illed wi h posi i e and equal alues, ha is each owne being linked o all i ms in he economy wi h equal
sha es (leas spa se scena io). No ice ha his las case, being epea ed o all owne s, implies ha he sha eholde s uc u e
o all i ms is iden ical. The ela i e weigh o each sha eholde in a i m is no cons ained o a speci ic alue, howe e i
will depend on i s inancial capaci y.
P ope ies o spa si y measu es
Table A.1 p esen s he de ini ion and in e p e a ion o he mos common p ope ies o spa si y indices.
Desi able c i e ia ha a spa si y measu e should ul il a e, among o he s, he so-called ‘‘Robin Hood’’ p ope y, he scaling o
homogenei y, he ‘‘ ising ide’’ p ope y, and he ‘‘cloning’’ p ope y. These ou c i e ia we e ini ially p oposed by Dal on (1920)
in he inancial se ing o inequali y measu emen o weal h dis ibu ion, bu a e nowadays gene ally ecognised o be minimum
c i e ia a ‘‘good’’ measu e o spa si y should sa is y, being e e ed o as ‘‘Dal on’s Laws’’.
Mo e ecen ly, Ricka d and Fallon (2004) add wo ex a p ope ies, named ‘‘Bill Ga es’’ and ‘‘Babies’’, espec i ely. Addi ional
axioms and a ibu es, whose analysis goes beyond he scope o his no e, can be ound in Pas o e al. (2013,2015).
Mo e de ails abou he abo e p ope ies can be ound in Hu ley and Ricka d (2009); he pape also p esen s six een spa si y
measu es and p o e which o hem sa is y which p ope ies. The main esul is ha only wo measu es sa is y all six p ope ies,
namely he Gini Index and he 𝑝𝑞-means (see de ini ions in Table 2), while all emaining measu es sa is y some, bu no all o hem.
P ope ies e isi ed in he con ex o common owne ship
Gi en ha he p ope ies we e ini ially conside ed in a wel a e inequali y con ex , hey need o be c i ically eassessed in he CO
amewo k, speci ically in he sense discussed abo e, i.e. looking a ow-wise spa si y (inc ease in spa si y = dec ease o he ex en
o CO; dec ease in spa si y = s onge p esence o CO). Each p ope y will be analysed ini ially looking a a change in beha iou
o one single owne , subsequen ly ex ending he same beha iou o all owne s in he ma ke (i meaning ul) in o de o assess he
o e all e ec on he OM and RM. This exe cise will also help exclude spa si y measu es ha a e inapp op ia e o ou pu pose.
The e ised p ope ies a e hose p esen ed in Table A.1. Summa ising, all six p ope ies o spa seness measu es a e o e all s ill
meaning ul in he con ex o CO, al hough he scaling and cloning p ope ies show some limi a ions when applied o he OM, being
less p oblema ic in he RM case.
Robin Hood: in case an owne holding a la ge sha e in a i m decides o di es , edi ec ing he in es men owa ds o he i ms
whe e i holds smalle (o ze o) sha es, he measu e cap u es he change as a dec ease in spa si y, i.e. a mo emen om absence
o low le el o CO owa ds a highe le el o CO. The same applies i all owne s change hei beha iou in an analogous way: he
measu e would de ec a educ ion in spa si y. This is a desi able p ope y as i goes in he di ec ion o he e ec we would like o
measu e. The e o e spa si y measu es ha ul il he Robin Hood p ope y a e app op ia e in ou con ex .

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Table A.1
P ope ies o spa si y measu es.
No. P ope y De ini ion Desc ip ion
1. Robin Hood A ‘‘ ai ’’ eadjus men o
coe icien s dec eases
spa si y
Taking some amoun om he la ge coe icien s and
‘‘ edis ibu ing’’ owa ds smalle coe icien s yields a less
concen a ed i.e. mo e equal dis ibu ion.
2. Scaling A change in scale o he
coe icien s does no
change spa si y
Mul iplying weal h o all uni s by an equal ac o does no
change he le el o (in)equali y o he dis ibu ion.
3. Rising Tide An iden ical inc ease in all
coe icien s dec eases
spa si y
Adding a ixed amoun o each coe icien educes he
ela i e di e ence be ween la ge and small alues, i.e. yields
a mo e equal dis ibu ion.
4. Cloning Doubling he numbe o
coe icien s by cloning he
same alues does no
change spa si y
I a second popula ion is cloned, ep oducing he same
alues as he i s one, hen he le el o (in)equali y o he
me ged popula ions equals ha o he ini ial popula ion.
5. Bill Ga es A signi ican inc ease in
one coe icien inc eases
spa si y
When one coe icien becomes e y la ge, he le el o
inequali y inc eases.
6. Babies The addi ion o ex a ze o
coe icien s inc eases
spa si y
Adding indi iduals wi h ze o weal h o a popula ion
inc eases inequali y, by concen a ing he (posi i e) o al
weal h in he hands o a smalle sha e o indi iduals.
Scaling: I an owne , say, doubles he owne ship sha es held in all i s subsidia ies, he measu e does no de ec a change in spa si y,
i.e. he ex en o CO is conside ed unchanged. Fo a single ow, his change in in es men does no al e he p esence o CO, since
he numbe o exis ing links is unchanged. Howe e , gi en ha he s eng h o he links inc eased, and ha he column sum o he
sha es is cons ained o be ≤100%, his implies a dec ease o he s eng h o he links o he owne s ha e wi h he same subsidia ies.
Such dec ease in he limi could e en lead o a ze o sha e, he e o e elimina ing an exis ing link, and al e ing he spa si y. I ollows
ha his p ope y is somewha con o e sial, especially in he case o he OM, which migh su e an (unde ec ed) eadjus men
o all ows when one owne changes i s in es men s. The p ope y seems mo e accep able o he RM, which in gene al will be
unchanged unde his scena io, al hough in an ex eme case he ac ual numbe o links migh be a ec ed, as men ioned ea lie .
The e o e, a measu e ul illing his p ope y aises some conce ns, and should be es ed u he . The si ua ion whe e all owne s
would double hei sha es is an impossible e en gi en he abo e cons ain on he column o als, he e o e is no analysed.
Rising Tide: I an owne inc eases i s owne ship sha es by 𝑘 > 0poin s in all i ms in he ma ke (e en in hose whe e i had
p e iously no sha es), hen he spa si y dec eases. In he limi , an owne ha only owned sha es in one i m, becomes common
owne o all companies in he economy, so he e is a change owa ds an inc eased le el o CO. This p ope y is in line wi h he
dynamics o CO, he e o e is accep able o ou s udy. I all owne s made a simila change in in es men s, he RM would immedia ely
become he spa ses one p oposed ea lie , he e o e going again in he di ec ion o his p ope y; howe e , in he case o he OM
we canno add inde ini ely ex a sha es o all elemen s, due o he column o al cons ain , so his case will no be con empla ed.
Cloning: I he numbe o i ms in he ma ke doubles, and an owne in es s in he new i ms exac ly he same amoun s o
sha es held in he o iginal se o i ms, spa si y does no change. Duplica ing (‘‘cloning’’) he ini ial ec o o in es men s does no
change hei ela i e concen a ion, since he p opo ion o i ms held by he owne is unchanged. Howe e , he absolu e numbe
o i ms linked by he owne doubled, in oducing mo e in e -connec ions be ween i ms. In he mos ex eme case, an owne who
had in es ed only in one i m (and he e o e who was no a common owne ) will in oduce a connec ion be ween wo i ms, and
become a common owne . The e o e, measu es wi h his p ope y a e accep able i we seek o measu e he ela i e ex en o CO,
bu a e less sui able o absolu e measu emen s.
Bill Ga es: I one owne inc eases la gely i s in es men in one i m, spa si y inc eases. The owne will ha e o di es wi h espec
o o he i ms held, in o de o mo e unds massi ely o ha speci ic i m. The e o e all emaining exis ing links o ha owne will
dec ease in s eng h, some e en eaching a ze o alue, i.e. some links may disappea . The same will happen a column le el, as
a la ge sha e held by he owne unde conside a ion makes all o he sha es o ha speci ic i m dec ease la gely, again possibly
causing some links o ha i m wi h o he owne s o anish. In bo h cases his implies an inc ease in spa si y, he e o e he p ope y
is in line wi h he ma ke dynamics and is accep able o ou analysis.
Babies: Adding an ex a i m in he ma ke whose owne s a e no common o any o he i m inc eases spa si y. This amoun s
o adding a new column wi h he sha eholde s’ s uc u e o he added i m, and some new lines co esponding o he new added
owne s, which we e no p esen be o e in he ma ke , since hey a e no common o any o he i ms. The e o e, he column will
be illed wi h ze os, excep o he las ew elemen s con aining he sha es held by he new owne s. This implies ha each line
co esponding o an exis ing owne will ha e an ex a ze o, hence inc easing he spa si y o ha line. On he o he hand, since he
new owne s a e no common o o he i ms, he deg ee o CO dec eases. I ollows ha he ma ke dynamics in his scena io goes
in he di ec ion p edic ed by he p ope y, which he e o e is admissible in he CO amewo k.
Appendix B. New measu es o CO summa y ables
See Tables B.1 and B.2.
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Table B.1
Common sha eholding indica o s o sha eholde s’ po olios, and ma ke le el coun e pa s.
Indica o De ini ion In e p e a ion Ma ke le el
Densi y Nsubs/N i ms Numbe o i ms in a sha eholde ’s po olio o e o al numbe
o i ms in he ma ke . Rep esen s he sha e o he ma ke o
which an in es o has access h ough sha eholding.
A e age, median,
maximum, pe cen iles
TOAS densi y To al TOAS
subs/To al ma ke
TOAS
Sum o he TOAS o all i ms in a sha eholde ’s po olio o e
sum o TOAS o all i ms in he ma ke . Rep esen s he
ela i e weigh o he i ms chosen by a speci ic in es o o e
he whole ma ke .
A e age, median,
maximum, pe cen iles
TOAS weigh ed densi y Sum weigh ed TOAS
subs/To al ma ke
TOAS
Sum o he TOAS o i ms in po olio, each weigh ed by he
ac ual owne ship sha e o in es o , o e sum o TOAS o all
i ms in he ma ke . Rep esen s he ac ual sha e o TOAS o
he ma ke in he hand o an in es o h ough he speci ic
owne ship sha es held.
A e age, median,
maximum, pe cen iles,
sum o ‘Big 3’
MKT CAP densi y To al MKT CAP
subs/To al ma ke
MKT CAP
Sum o he MKT CAP o all lis ed i ms in a sha eholde ’s
po olio o e sum o MKT CAP o all lis ed i ms in he
ma ke . In e p e a ion is same as o TOAS densi y, bu e e s
o ma ke capi alisa ion. Only lis ed i ms in po olio.
A e age, median,
maximum, pe cen iles
MKT CAP weigh ed densi y Sum weigh ed MKT
CAP subs/To al
ma ke MKT CAP
Sum o he MKT CAP o lis ed i ms in po olio, each
weigh ed by he ac ual owne ship sha e o in es o , o e sum
o MKT CAP o all lis ed i ms in he ma ke . In e p e a ion is
same as o TOAS weigh ed densi y, bu e e s o ma ke
capi alisa ion. Only lis ed i ms in po olio.
A e age, median,
maximum, pe cen iles,
sum o ‘Big 3’
Uni o mi y 1 - √∑𝐬𝐡𝐚𝐫𝐞𝐬𝟐
∑𝐬𝐡𝐚𝐫𝐞𝐬 One minus he ollowing a io: (Squa e oo o he) Sum o he
squa es o he sha es in po olio, o e sum o all sha es in
po olio. Index assesses he ela i e weigh o la ge
pa icipa ion sha es o e he sha es o al, showing smalle
alues o mo e concen a ed dis ibu ions.
A e age, median,
maximum, pe cen iles
Numbe o Block-holdings Numbe o holdings
> 𝑝%
Numbe o pa icipa ions in po olios wi h sha e alue abo e
a ce ain pe cen age 𝑝. Rep esen s he numbe o mo e
in ensi e in es men s o po olio. Compu ed o 𝑝= 3,5,10.
Sum o ‘Big 3’, sum
o all SH
Table B.2
Common sha eholding indica o s o sha eholde s’ and i ms’ ne wo ks.
Sha eholde s’ ne wo k
Indica o De ini ion In e p e a ion
Co ela ion Pea son’s 𝜌Pea son’s co ela ion coe icien 𝜌be ween pai s o po olios. Re eals he le el o o e lap
be ween wo sha eholde s’ in es men s.
P opo ion o non-ze o
co ela ions
𝐍𝐨.𝐧𝐨𝐧−𝐳𝐞𝐫𝐨 𝜌
𝐧(𝐧−𝟏)∕𝟐Numbe o non-ze o co ela ions o e o al numbe o possible connec ions be ween pai s o
po olios. 𝑛is he o al numbe o po olios. Rep esen s he p opo ion o exis ing links in
he sha eholde s’ ne wo k, i.e. he ne wo k’s densi y.
Numbe o highly
co ela ed high-densi y
po olios
No. o 𝜌 > 80% Numbe o co ela ions highe han 80% be ween pai s o la ge (high densi y) po olios
(holding mo e han 10% o he ma ke ’s i ms). Coun s he numbe o e y s ong links
among la ge sha eholde s’ po olios.
P opo ion o highly
co ela ed high-densi y
po olios
𝐍𝐨. 𝜌>𝟖𝟎%
𝐤(𝐤−𝟏)∕𝟐Numbe o co ela ions highe han 80%, o e o al numbe o possible connec ions be ween
pai s o la ge (high densi y) po olios (holding mo e han 10% o he ma ke ’s i ms). 𝑘is
he o al numbe o high densi y po olios. Rep esen s he p opo ion o e y s ong links
among la ge sha eholde s’ po olios, i.e. he deg ee o simila i y o hei in es men s.
Fi ms’ ne wo k
Indica o De ini ion In e p e a ion
Co ela ion Pea son’s 𝜌Pea son’s co ela ion coe icien 𝜌be ween pai s o SH s uc u es o i ms. Re eals he le el
o o e lap be ween wo i ms’ SH s uc u es.
P opo ion o non-ze o
co ela ions
𝐍𝐨.𝐧𝐨𝐧−𝐳𝐞𝐫𝐨 𝜌
𝐧(𝐧−𝟏)∕𝟐Numbe o non-ze o co ela ions o e o al numbe o possible connec ions be ween pai s o
i ms. 𝑛is he o al numbe o i ms. Rep esen s he p opo ion o i ms ha a e connec ed
h ough some common sha eholde , i.e. he i ms’ ne wo k densi y.
Numbe o highly
co ela ed SH s uc u es
No. o 𝜌 > 80% Numbe o co ela ions highe han 80% be ween pai s o i ms. Coun s he numbe o e y
s ong links among i ms’ SH s uc u es.
P opo ion o highly
co ela ed SH s uc u es
𝐍𝐨. 𝜌>𝟖𝟎%
𝐤(𝐤−𝟏)∕𝟐Numbe o co ela ions highe han 80%, o e o al numbe o non-ze o connec ions be ween
pai s o i ms. 𝑘is he o al numbe o connec ed i ms. Rep esen s he p opo ion o e y
s ong links among connec ed i ms, i.e. he deg ee o simila i y o SH s uc u e.
Resea ch in In e na ional Business and Finance 70 (2024) 102315
23
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