Mononen, Lasse
A icle — Published Ve sion
Obse able in e pe sonal u ili y compa isons
Social Choice and Wel a e
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Mononen, Lasse (2025) : Obse able in e pe sonal u ili y compa isons, Social
Choice and Wel a e, ISSN 1432-217X, Sp inge , Be lin, Heidelbe g, Vol. 65, Iss. 3, pp. 629-644,
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Social Choice and Wel a e (2025) 65:629–644
h ps://doi.o g/10.1007/s00355-025-01584-z
ORIGINAL PAPER
Obse able in e pe sonal u ili y compa isons
Lasse Mononen1
Recei ed: 23 Feb ua y 2024 / Accep ed: 31 Janua y 2025 / Published online: 15 Ma ch 2025
© The Au ho (s) 2025
Abs ac
Ha sanyi’s seminal agg ega ion heo em axioma ized weigh ed u ili a ianism based
on expec ed u ili y heo y. Howe e , he weigh s assigned o each indi idual canno
be sepa a ed om he indi idual’s u ili y. We show ha once we depa om he
expec ed u ili y amewo k, i is possible o uniquely iden i y he u ili ies and he
weigh s. Speci ically, we show ha in he min-o -means social wel a e unc ion i
each indi idual has a ca dinal u ili y, unique up o a posi i e a ine ans o ma ion,
and any edis ibu ion o u ili ies changes he social wel a e o some ini ial alloca ion,
hen we can uniquely iden i y he u ili ies o he indi iduals and he weigh s o he
social wel a e unc ion.
1 In oduc ion
Ha sanyi’s seminal agg ega ion heo em (Ha sanyi 1955) axioma ized weigh ed u il-
i a ianism based on expec ed u ili y heo y. Howe e , he esul has been c i icised
since he weigh s a ibu ed o indi iduals a e no meaning ul because hey canno be
sepa a ed om he indi iduals’ u ili ies (Sen 1976; B oome 1987;Weyma k1991).
To o e come his iden i ica ion issue, Ha sanyi (1977) used di ec in e pe sonal u il-
i y compa isons. Howe e , in e pe sonal u ili y compa isons a e di icul o make and
ha e emained con o e sial in he li e a u e (Els e and Roeme 1991; G ea es and
Lede man 2018). Addi ionally, he assump ion o expec ed u ili y heo y in Ha sanyi’s
agg ega ion heo em has been c i icised (e.g. Diamond (1967); Sen (1970); B oome
(1987)).
We show ha once we depa om Ha sanyi’s expec ed u ili y amewo k, he
weigh s and u ili ies a e can be meaning ul e en wi hou di ec in e pe sonal u ili y
compa isons. Speci ically, we show ha in he min-o -means social wel a e unc ion
i each indi idual has a ca dinal u ili y, unique up o a posi i e a ine ans o ma-
The au ho hanks Niels Boissonne , A hu Dolgopolo , La y Eps ein, Ma c Fleu baey, Dominik Ka os,
F ank Riedel and he anonymous e iewe s o use ul commen s. This wo k was unded by he Deu sche
Fo schungsgemeinscha (DFG, Ge man Resea ch Founda ion)-P ojec -ID 317210226-SFB 1283.
BLasse Mononen
[email p o ec ed]
1Cen e o Ma hema ical Economics, Biele eld Uni e si y, PO Box 10 01 31, 33 501 Biele eld,
Ge many
123
630 L. Mononen
ion, and o e e y u ili y edis ibu ion, he e is some ini ial alloca ion such ha he
edis ibu ion changes he social wel a e, hen we can uniquely iden i y he u ili ies
o he indi iduals and he weigh s o he social wel a e unc ion. This shows ha he
social obse e is beha ing as i making in e pe sonal u ili y compa isons ha we
can obse e indi ec ly oge he wi h he ai ness o he socie y. Ou esul o malizes
Kaneko’s (1984) sugges ion o obse ing in e pe sonal u ili y compa isons om he
social wel a e unc ion.
He e, we iden i y he in e pe sonal u ili y compa isons om he non-linea i ies
o he social wel a e unc ion. Fo example, in he case o he Rawlsian social wel-
a e unc ion (Rawls 1971), he non-linea i ies cap u e he change o he wo s -o
indi idual ha allows us o iden i y u ili ies ac oss indi iduals. We gene alize his
iden i ica ion s a egy beyond he Rawlsian social wel a e unc ion.
We s udy he iden i ica ion o indi iduals’ u ili ies and weigh s in he min-o -means
social wel a e unc ion. This has been conside ed as cap u ing he igno an obse e in
Gajdos and Kandil (2008). Addi ionally, i has been conside ed in he con ex o income
inequali y in Ben-Po a h e al. (1997), Gajdos and Mau in (2004), C ès e al. (2011),
and ecen ly in Mongin and Pi a o (2021). This ep esen a ion includes u ili a ianism
and he Rawlsian social wel a e unc ion (Rawls 1971) as special cases.
The min-o -means ep esen a ion consis s o a ( on Neumann–Mo gens e n) u ili y
unc ion ui o each membe i∈I={1,...,n}and a se o weigh s o each membe
⊆(I)such ha he socie al alue o an al e na i e xis
min
λ∈
i∈I
λiui(x).
We show ha he se o weigh s and he u ili y unc ions a e iden i ied i and only
i any u ili y edis ibu ion om one membe o ano he changes he wel a e in some
si ua ion. Tha is i ( i)i∈I∈Rnis a u ili y edis ibu ion such ha he e exis membe s
iand jwi h i>0>
j, hen he e exis s an ini ial u ili y alloca ion (wi)i∈I∈Rn
such ha edis ibu ing he u ili y by ( i)i∈Ichanges he social wel a e,
min
λ∈
i∈I
λiwi= min
λ∈
i∈I
λi(wi+ i). (1)
This condi ion cap u es in e ms o he social wel a e unc ion ha he socie al
alue o e e y u ili y edis ibu ion depends on he con ex : Fo mally, o each u ili y
edis ibu ion ( i)i∈I∈Rnsuch ha he e exis membe s iand jwi h i>0>
j,
he e exis weigh s λ,λ∈such ha
i∈I
λi i=
i∈I
λ
i i.
Technically, his condi ion is equi alen o he se o weigh s ha ing a non-emp y
in e io .
This esul shows ha once we mo e away om he expec ed u ili y amewo k
and he socie al alue o e e y u ili y edis ibu ion depends on he cu en u ili y
dis ibu ion, in e pe sonal u ili y compa isons and weigh s assigned o indi iduals can
be obse able. Especially we show ha in he min-o -means ep esen a ion wi h a ine
u ili ies, ou iden i ying condi ion Eq. (1) is equi alen o he iden i ica ion o he se
o Pa e o weigh s and equi alen o he iden i ica ion o in e pe sonal u ili y
123
Obse able in e pe sonal u ili y… 631
Ma hema ically, he sepa a ion o weigh s and u ili y unc ions is symme ical o he
sepa a ion o p obabili ies and s a e dependen u ili ies in choice unde unce ain y.
He e, he min-o -means social wel a e unc ion co esponds o he s a e dependen
maxmin expec ed u ili y. Ou iden i ica ion esul ollows as a co olla y om he
iden i ica ion o p obabili ies and s a e dependen u ili ies in a s a e dependen maxmin
expec ed u ili y in Mononen (2024).
Ou second con ibu ion is ha we cha ac e ize he exis ence o he min-o -means
social wel a e unc ion by elaxing Ha sanyi’s assump ion ha he socie al p e e ences
sa is y he expec ed u ili y heo y. Ins ead, we allow o iola ions o expec ed u ili y
heo y when he al e na i es in ol e ade-o s ac oss he membe s and only assume
ha he socie al p e e ences sa is y expec ed u ili y heo y when he e a e no ade-o s
ac oss he membe s.1
Ou esul s a e closely ela ed o Gajdos and Kandil (2008). They s udy when an
impa ial obse e ’s ex ended p e e ences ha e a min-o -means ep esen a ion. In his
ex ended se ing, he obse e is especially able o make di ec in e pe sonal u ili y
compa isons and Ha sanyi’s u ili a ianism is ully iden i ied. We ins ead s udy obse -
able social p e e ences ha do no include di ec in e pe sonal u ili y compa isons. This
allows us o subs an ially simpli y he axioma iza ion o Gajdos and Kandil (2008)
and cla i y u he he di e ence be ween he min-o -means social wel a e unc ion
and u ili a ianism.
We ollow he single-p o ile o malism pionee ed by Ha sanyi (1955) ha s udied
p e e ence agg ega ion o e ixed p e e ences. This app oach has been used o exam-
ple in Mongin (1995), Gilboa e al. (2004), Chambe s and Hayashi (2006) and Gajdos
e al. (2008). This app oach is in con as o he mul i-p o ile o malism s udying p e -
e ence agg ega ion o e a ying p e e ences as pionee ed by A ow (1951) and Sen
(1970) and has been summa ized in d’Asp emon and Louis (2002).
In choice unde unce ain y, Mongin (1995) and Mongin (1998) s udy he agg e-
ga ion o indi iduals unde subjec i e expec ed u ili y heo y. Mongin shows ha i
he e is su icien p e e ence di e si y, hen unde s a e independen u ili y and Pa e o
mono onici y, his leads o a dic a o ial choice ule. Howe e , unde s a e dependen
u ili y, non-dic a o ial agg ega ion is possible. Ama an e and Ghossoub (2021)shows
he possibili y o non-dic a o ial agg ega ion when he agg ega ed p e e ences do no
ollow subjec i e expec ed u ili y heo y.
The solu ion in he li e a u e o he lack o iden i ica ion in Ha sanyi (1955) has
been o conside non-obse able ex ended lo e ies ha allow o di ec in e pe sonal
u ili y compa isons. This app oach was pionee ed in Ha sanyi (1977) and used in Ka ni
and Weyma k (1998), Gajdos and Kandil (2008), G an e al. (2010), and discussed in
Adle (2014) and G ea es and Lede man (2018).
Ano he solu ion o he lack o iden i ica ion has been o conside ela i e u ili-
a ianism ha was in oduced by Dhillon and Me ens (1999) and Segal (2000) and
p e e ence sa is ac ion. Bö ge s and Choo (2017) and Ka ni and Weyma k (2024)
s udy he elici a ion o Pa e o weigh s in ela i e u ili a ianism. These app oaches a e
discussed la e in Sec .2.3.
1Howe e , since we do no obse e in e pe sonal u ili y compa isons, we make a mo e gene al assump ion
and assume he e a e wo di e en lo e ies ha sa is y he expec ed u ili y heo y.
123
632 L. Mononen
Technically, ou esul s a e ela ed o he li e a u e on income inequali y measu e-
men Weyma k (1981), Yaa i (1988), Ben-Po a h e al. (1997). Howe e , he e we ocus
on he mo e gene al wel a e inequali y measu emen wi h subjec i e u ili y o each
membe .
The emainde o he pape p oceeds as ollows: Sec .1.1 o e s a simple example
highligh ing he in ui ion o ou iden i ica ion esul . Sec ion2s udies he iden i ica-
ions o he min-o -means social wel a e unc ion, Sec .2.3 compa es he iden i ica ion
o ela i e u ili a ianism and u ili a ianism. Sec ion3axioma ically cha ac e izes he
exis ence o he ep esen a ions and Sec .4concludes. The Appendix p o es all he
esul s.
1.1 An example o iden i ica ion
We begin wi h a simple example illus a ing ha wi h he weigh ed u ili a ian social
wel a e unc ion indi iduals’ u ili ies and weigh s canno be sepa a ed. Howe e , his
is only an uniden i ied special case. In he second pa o he example, we show ha
wi h he min-o -means social wel a e unc ion hese can be sepa a ed and iden i ied
om he iola ions o he independence axiom.
Fi s , we illus a e he lack o iden i ica ion in weigh ed u ili a ianism. Conside
a socie y consis ing o wo indi iduals 1 and 2 ha ha e p e e ences o e some se
o lo e ies (X)o e social al e na i es X. Each o he indi iduals has an a ine
on Neumann–Mo gens e n u ili ies u1,u2:(X)→R. These a e agg ega ed in o
weigh ed u ili a ian social wel a e wi h equal weigh o bo h o he indi iduals: The
social alue o al e na i e p∈(X)is
0.5u1(p)+0.5u2(p).
Now hese p e e ences ha e an al e na i e weigh ed u ili a ian ep esen a ion wi h
any weigh λ∈(0,1) o indi idual 1 since
0.5u1(p)+0.5u2(p)=λ0.5
λu1(p)+(1−λ)0.5
1−λu2(p)=λu1(p)+(1−λ)u2(p),
whe e he e ms inside he pa en heses de ine new u ili y unc ions u1,u2.
In his al e na i e ep esen a ion, we ha e eplaced he weigh o he indi idual
o he in ensi y o p e e ences. This highligh s he impossibili y o iden i ica ion in
weigh ed u ili a ianism since he in ensi ies o p e e ences a e insepa able om he
weigh s. This iola es ou iden i ica ion condi ion (1) since o u ili y edis ibu ion
( 1,
2)=(1
λ,−1
1−λ), we ha e o all (w1,w
2)∈R2,
λw1+(1−λ)w2=λ(w1+ 1)+(1−λ)(w2+ 2).
Nex , we mo e on o min-o -means social wel a e o e lo e ies pde ined by wo
weigh s λ∗<λ
∗ o indi idual 1 and a ine on Neumann–Mo gens e n u ili ies u1,u2
min
λ∈[λ∗,λ∗]λu1(p)+(1−λ)u2(p).
We show ha u ili ies a e iden i iable ac oss indi iduals om changes in he Pa e o
weigh s, ha is iola ions o he independence axiom.2Fo his, le pand qbe wo
2The independence axiom om on Neumann and Mo gens e n (1947) and Ha sanyi (1955) cha ac e izes
123
Obse able in e pe sonal u ili y… 633
Fig. 1 An example o iden i ying when he u ili ies ac oss indi iduals a e equal om he non-linea i ies.
Lo e ies pand qa e such ha u1(p)+u1(q)=u2(p)+u2(q)and u1(q)<u2(q). The x-axis changes p
o qwi h con ex combina ions. The y-axis is he min-o -means social wel a e o αp+(1−α)q.A α=0.5
u ili ies ac oss he indi iduals a e equal ha is obse able as a iola ion o he independence axiom
lo e ies such ha u1(p)+u1(q)=u2(p)+u2(q)and u1(q)<u2(q).Nex ,weshow
ha he e is a iola ion o he independence axiom a 0.5p+0.5q.
We ocus on he min-o -means social wel a e o αp+(1−α)qwhen αchanges
om 0 o 1 as in Fig.1. Fi s , be ween 0 and 0.5, by he a ine u ili ies, u1(α p+
(1−α)q)<u2(α p+(1−α)q). So he min-o -means wel a e uses he weigh λ∗
and i changes linea ly a he a e λ∗u1(p)−u1(q)+(1−λ∗)u2(p)−u2(q).
Second, be ween 0.5 and 1, u1(α p+(1−α)q)>u2(α p+(1−α)q). So he min-
o -means wel a e uses he weigh λ∗and his ime i changes linea ly a he a e
λ∗u1(p)−u1(q)+(1−λ∗)u2(p)−u2(q). Since u1(p)−u1(q)>u2(p)−u2(q),
he a e o change swi ches a α=0.5 and he e is a non-linea i y a ha poin . This
ep esen s a iola ion o he independence axiom a he lo e y 0.5p+0.5qwhe e
bo h o he indi iduals ha e he same u ili y.
Finally, he e can be iola ions o he independence axiom only i he u ili ies o
bo h indi iduals a e he same. The only si ua ions whe e he e can be non-linea i ies
as in Fig.1a e when he used Pa e o weigh changes. Howe e , he min-o -means
social wel a e unc ion wi h wo indi iduals always maximizes he weigh o he
indi idual wi h a lowe u ili y. Thus, he change in he used Pa e o weigh means ha
he u ili y o de o he indi iduals changed. Especially, in he e, he u ili ies o bo h
o he indi iduals a e exac ly he same. In summa y, he lo e ies whe e he u ili ies
o bo h indi iduals a e equal a e cha ac e ized by he iola ions o he independence
axiom and especially hey a e obse able.
The min-o -means social wel a e unc ion ules ou he p e ious iola ions o ou
iden i ica ion condi ion (1). Fi s , o many edis ibu ions, he axiom holds i ially. I
( 1,
2)is a u ili y edis ibu ion such ha 1<0< 2and minλ∈[λ∗,λ∗]λ 1+(1−λ) 2=
he linea i y o weigh ed u ili a ianism. I s a es ha o all lo e ies p,q, and α∈(0,1),
pq⇐⇒ αp+(1−α) αq+(1−α) .
123
634 L. Mononen
0, hen he condi ion holds o (˜w1,˜w2)=(0,0). Second, we ocus on edis ibu ions
ha p ese e he wel a e when edis ibu ing om (0,0)and show ha in ano he
con ex when edis ibu ing om (w1,w
2)=(−0.5 1,−0.5 2), he edis ibu ion
a ec s he wel a e. Now, we ha e since 1<0<
2and he edis ibu ion does no
a ec wel a e when edis ibu ing om (0,0),
min
λ∈[λ∗,λ∗]λ(w1+ 1)+(1−λ)(w2+ 2)=λ∗(0.5 1)+(1−λ∗)(0.5 2)=0.
Nex , since λ∗<λ
∗and 1<0<
2,weha e
0<λ
∗(0.5 1)+(1−λ∗)(0.5 2).
Hence,
0>λ
∗(−0.5 1)+(1−λ∗)(−0.5 2)=min
λ∈[λ∗,λ∗]λw1+(1−λ)w2.
This shows ha he iden i ica ion condi ion (1) holds also in his case. This illus a es
ha o mos o he u ili y edis ibu ions, he iden i ying condi ion holds i ially.
Howe e , he iden i ying condi ion assumes ha u ili y edis ibu ions ha a e wel a e
p ese ing in one con ex a ec wel a e in some o he con ex .
This iden i ica ion example is gene alized in ou main esul , Theo em 1, o ini ely
many indi iduals. The e we show ha i he social alue o e e y edis ibu ion depends
on he con ex , hen he indi iduals’ u ili ies a e obse able.
2 Iden i ica ion
2.1 P elimina ies and no a ion
We ollow he se ing om Ha sanyi (1955,1977). Socie y consis s o membe s I=
{1,...,n}.Xis a se o social-al e na i es. Each membe i∈Ihas p e e ences i
o e (simple) social-al e na i e lo e ies (X)and addi ionally, we obse e socie al
p e e ences 0o e (simple) social-al e na i e lo e ies (X). (No malized) weigh s
o he membe s a e p obabili y dis ibu ions on he membe s (I).(I)is equipped
wi h he Euclidean opology.
We conside he min-o -means social wel a e unc ion ollowing (Ben-Po a h e al.
1997; Gajdos and Kandil 2008) o e expec ed u ili y membe s as in Ha sanyi (1955,
1977).
De ini ion A ine u ili ies ui:(X)→R o each i∈Iand a con ex and closed
se o Pa e o weigh s ⊆(I)is a min-o -means ep esen a ion o ((i)i∈I,0)
i he ollowing wo condi ions hold:
1. o each i∈Iand p,q∈(X),weha e
piq⇐⇒ ui(p)≥ui(q).
2. o all p,q∈(X),weha e
p0q⇐⇒ min
λ∈
i∈I
λiui(p)≥min
λ∈
i∈I
λiui(q).
We o e an axioma ic cha ac e iza ion o he min-o -means social wel a e unc ion
la e on in Sec .3. We ocus especially on min-o -means ep esen a ions wi h he
smalles possible se o Pa e o weigh s as de ined nex .
123
Obse able in e pe sonal u ili y… 635
De ini ion A ine u ili ies ui:(X)→R o each i∈Iand a con ex and closed se o
Pa e o weigh s ⊆(I)is a minimal min-o -means ep esen a ion o ((i)i∈I,0)
i o any o he min-o -means ep esen a ion wi h he same u ili ies (ui)i∈Iand a se
o Pa e o weigh s ˜
,weha e⊆˜
.
The nex example shows he signi icance o minimal ep esen a ions since he e
can be weigh s ha he social wel a e unc ion ne e uses. We connec gene al and
minimal ep esen a ions in he nex sec ion.
Rema k (Non-minimal example) I n=2 and o all p∈(X),u1(p)<u2(p), hen
he se o Pa e o weigh s ={(λ, 1−λ)|λ∈[0,1
2]} is no minimal since o all
p∈(X)
min
λ∈[0,1
2]
λu1(p)+(1−λ)u2(p)=1
2u1(p)+1
2u2(p).
Nex , we de ine when he minimal se o weigh s and he u ili y unc ions a e
iden i ied.
De ini ion The se o weigh s in he minimal min-o -means ep esen a ion is iden i ied
i o all minimal min-o -means ep esen a ions ((ui)i∈I,) and (( ˜ui)i∈I,˜
),we
ha e
=˜
.
In con as o he se o weigh s ha a e only iden i ied o minimal ep esen a ions,
we iden i y he u ili y unc ions o all he min-o -means ep esen a ions.
De ini ion The u ili ies in he min-o -means ep esen a ion a e iden i ied up o a com-
mon posi i e a ine ans o ma ion i o all min-o -means ep esen a ions ((ui)i∈I,)
and (( ˜ui)i∈I,˜
), he e exis α>0 and β∈Rsuch ha o each i∈Iand p∈(X)
ui(p)=α˜ui(p)+β.
2.2 Uniqueness
Ou main esul cha ac e izes when he minimal min-o -means ep esen a ion is ully
iden i ied. This iden i ica ion is cha ac e ized by he ollowing condi ion.
Axiom 1 Fo any lo e ies pand qsuch ha he e exis i,j∈Iwi h piqand
qjp, he e exis s a lo e y and α∈[0,1]such ha
αp+(1−α) 0αq+(1−α)
He e, we conside edis ibu ing he u ili ies by (uk(q)−uk(p))k∈I ha bene i s
he membe jand makes he membe iwo se o . Howe e , we do no make any
es ic ions on how he edis ibu ion a ec s o he membe s. Then he axiom assumes
ha he e exis s an alloca ion αp+(1−α) wi h u ili ies
uk(α p+(1−α) )k∈I
such ha pe o ming he u ili y edis ibu ion o change he u ili ies o
uk(α p+(1−α) )+α(uk(q)−uk(p))k∈I=uk(αq+(1−α) )k∈I
123
636 L. Mononen
changes he wel a e. Tha is, o any u ili y edis ibu ion he e is some si ua ion such
ha he edis ibu ion changes he wel a e. Fo his in e p e a ion, i is c ucial ha he
membe s ha e a ine u ili ies.
The nex esul shows ha Axiom 1cha ac e izes he iden i ica ion o he minimal
min-o -means ep esen a ion.
Theo em 1 Assume ha ((i)i∈I,0)has a minimal min-o -means ep esen a ion
((ui)i∈I,)such ha
in ui(p)i∈Ip∈(X)= ∅.
Then he ollowing i e condi ions a e equi alen .
(1) ((i)i∈I,0)sa is y Axiom 1.
(2) in = ∅.
(3) Fo all ∈RIsuch ha he e exis i,j∈Iwi h i>0> j, he e exis λ,λ∈
such ha
i∈I
λi i=
i∈I
λ
i i.
(4) The se o weigh s in he minimal min-o -means ep esen a ion is iden i ied.
(5) The u ili ies in he min-o -means ep esen a ion a e iden i ied up o a common
posi i e a ine ans o ma ion.
Fi s , he equi alency be ween (1), (4), and (5) shows he iden i ica ion o weigh s
assigned o membe s and in e pe sonal u ili y compa isons: When any edis ibu-
ion changes wel a e in some si ua ion, hen he minimal se o weigh s assigned o
membe s and in e pe sonal u ili y compa isons can be iden i ied in he min-o -means
ep esen a ion om he socie al p e e ences. He e, he social obse e is beha ing
as i making in e pe sonal u ili y compa isons ha we can obse e indi ec ly. This
iden i ica ion was illus a ed in Sec .1.1. As in he example, Axiom 1gua an ees ha
any edis ibu ion iola es he independence axiom and he e is a change in he Pa e o
weigh o any edis ibu ion ha is used o he iden i ica ion.
Second, he equi alency be ween condi ions (1), (2), and (3) cha ac e izes when he
minimal min-o -means ep esen a ion sa is ies Axiom 1. This shows ha ou iden i y-
ing condi ion is equi alen o he socie al alue o e e y u ili y edis ibu ion depending
on he con ex o o he se o weigh s ha he social wel a e unc ion uses ha ing a
non-emp y in e io . Fo example, his esul shows ha i all he Pa e o weigh s ag ee
on he weigh o membe io i all he Pa e o weigh s ag ee ha he weigh o membe
iis wice as la ge as membe j, hen he min-o -means ep esen a ion does no sa is y
Axiom 1.
Theassump ion ha
in ui(p)i∈Ip∈(X)= ∅
is a s anda d iden i ica ion condi ion in he li e a u e. I has been used e.g. in Ha sanyi
(1955), Weyma k (1991), and Fleu baey and Mongin (2016). I is cha ac e ized by he
independen p ospec s axiom assuming ha o each i∈I, he e exis s lo e ies pand
qsuch ha piqand o each j= i,p∼jq.
The nex example shows he iden i ica ion o a con ex combina ion o u ili a i-
anism and he Rawlsian social wel a e unc ion as p oposed by Gajdos and Kandil
123
Obse able in e pe sonal u ili y… 643
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