Expected balanced uncertain utility

G an , Simon; Roo da, B.; Yang, Jingni
A icle
Expec ed balanced unce ain u ili y
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: G an , Simon; Roo da, B.; Yang, Jingni (2025) : Expec ed balanced unce ain
u ili y, Theo e ical Economics, ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 20, Iss.
1, pp. 1-25,
h ps://doi.o g/10.3982/TE5404
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Theo e ical Economics 20 (2025), 1–25 1555-7561/20250001
Expec ed balanced unce ain u ili y
Simon G an
Resea ch School o Economics, Aus alian Na ional Uni e si y
Be end Roo da
Facul y o Beha iou al, Managemen and Social Sciences, Uni e si y o Twen e
Jingni Yang
School o Economics, Uni e si y o Sydney
We in oduce and analyze expec ed balanced unce ain u ili y (EBUU) heo y.
Ap io andabalanced ou come-se u ili y cha ac e ize an EBUU decision make .
Condi ional on a e e ence o “balancing alue,” he la e assigns a u ili y o each
ou come-se . The decision make associa es wi h each ac , i s en elope, hemin-
imal measu able mapping om s a es o ou come-se s ha con ains he ac . She
hen (implici ly) anks an ac acco ding o he balancing alue a which he ex-
pec ed balanced u ili y o i s associa ed en elope is ze o. As a consequence, he
isk p e e ences need only exhibi be weenness allowing o beha io ha can ac-
commoda e Allais- ype pa adoxes.
Keywo ds. Unce ain y, ambigui y, be weenness.
JEL classi ica ion. D80, D81.
1. In oduc ion
In he adi ion o he oluminous li e a u e ini ia ed by Ellsbe g (1961), conside a de-
cision make (he ea e , DM) who possesses only pa ial in o ma ion abou he unde -
lying s ochas ic p ocess ha de e mines he esolu ion o he unce ain y she aces. In
pa icula , his means she is no com o able quan i ying wi h a p ecise p obabili y he
unce ain y she associa es wi h each and e e y e en . The e does exis , howe e , a ich
collec ion o e en s she deems measu able o e which is de ined he p io , a unique
p obabili y ep esen ing he belie s o e hose e en s.
An objec o choice o ou DM is an unce ain p ospec o ac ha maps each s a e
o an ou come. Using he p io , any ac measu able wi h espec o he p io can be
mapped o a co esponding p obabili y dis ibu ion o lo e y o e ou comes. Thus he
es ic ion o he p e e ences o measu able ac s may be iewed as inducing a p e e ence
ela ion o e lo e ies, which we e e o as he DM’s isk p e e ences.
Simon G an : [email p o ec ed]
Be end Roo da: [email p o ec ed]
Jingni Yang: [email p o ec ed]
We hank audiences a Royal Holloway U. o London, ANU, E asmus U., SAET2021, and U. Alabama as well
as Jü gen Eichbe ge , Geo ge Maila h, Ron S aube , and Pe e Wakke o commen s and sugges ions. We
a e also e y g a e ul o he e e ees who pushed us o imp o e he exposi ion and igh en he analysis.
©2025 The Au ho s. Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0.
A ailable a h ps://econ heo y.o g.h ps://doi.o g/10.3982/TE5404
2G an , Roo da, and Yang Theo e ical Economics 20 (2025)
In Gul and Pesendo e ’s (2014) expec ed unce ain u ili y model, he DM deems
measu able p ecisely hose e en s in which he e en and i s complemen join ly sa is y
a e sion o Sa age’s (1954) s a e-sepa abili y pos ula e P2.1As a consequence, he isk
p e e ences con o m o expec ed u ili y. The Allais pa adoxes, howe e , clea ly illus a e
ha P2 is no only challenged when p obabili ies a e unknown. This is ce ainly he case
when i comes o desc ip i e modeling (see, e.g., T e sky and Sha i (1992) o empi ical
indings). Fu he mo e, e en on no ma i e g ounds P2 has no gone unchallenged (see,
e.g., Heukelom (2015) o an ex ensi e his o ical accoun ). Hence ou goal is o cha ac-
e ize a class o DMs who may pe cei e ambigui y bu whose isk p e e ences need no
con o m o expec ed u ili y heo y.
One app oach aken by G an , Rich, and S eche (2022) ha allows o isk-p e e en-
ces ha can accommoda e Allais s yle iola ions o expec ed u ili y is o assume he se
o measu able e en s a e exogeneously speci ied and hen axioma ize a amily o p e e -
ence ela ions in which he e alua ion o an a bi a y ac is cha ac e ized by a mapping
o an equi alen measu able ac along wi h a gene alized no ion o he ce ain y equi -
alen o a lo e y.2
The app oach aken he e is o conside an al e na i e p ope y an e en mus sa is y
in o de o i o be deemed measu able by he DM and hen explo e i s implica ions o
he co esponding isk-p e e ences. The one we p opose is simple as i co esponds o
G an , Kajii, and Polak’s (2000) (weak) decomposabili y p ope y (in i s s ic o m). Le -
ing deno e he DM’s p e e ences o e ac s and w i ing Eg o he ac ha ag ees wi h
he ac on he e en Eand wi h he ac gon i s complemen , an e en Ris decompos-
able, i o any pai o ac s and g:
[ Rggand gR g]=⇒ g.
G an , Kajii, and Polak (2000) con end ha by in e p e ing a s a emen like “ would
be p e e ed o gi he e en Rwe e known o ob ain” as only en ailing Rgg,decom-
posabili y may be in e p e ed as encapsula ing he ollowing easoning:
I he DM would p e e o gknowing Rob ains, and she would p e e o gknowing Rdoes
no ob ain, hen she should p e e o ge en hough she cu en ly does no know whe he R
will o will no ob ain.
Tha is, simila o he p ope y employed by Gul and Pesendo e o iden i y hose e en s
deemed measu able by he DM, decomposabili y p o ides a way o ope a ionalize Sa -
age’s (1954)ex alogical Su e-Thing P inciple (STP). Wi h measu able e en s classi ied
as hose ha sa is y decomposabili y, as G an , Kajii, and Polak (2000) es ablish, he co -
esponding isk-p e e ences need only exhibi he be weenness p ope y o Chew (1983)
and Dekel (1986), and hus can accommoda e Allais s yle iola ions o expec ed u ili y.
1Gul and Pesendo e e e o any such e en as ideal.
2G an , Rich, and S eche (2022) also conside endogenizing he se o e en s he DM iews as measu -
able by u ilizing Eps ein and Zhang’s (2001) p e e ence-based de ini ion o classi ying measu able e en s.
E en hen, howe e , in hei ep esen a ion heo em (Theo em 5, p. 14) he s uc u e o he se o measu -
able e en s needed o he e alua ion o a bi a y ac s is assumed as pa o he hypo hesis o he heo em
and no de i ed om he pos ula es hey p opose.
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Theo e ical Economics 20 (2025) Expec ed balanced unce ain u ili y 3
Ou second poin o depa u e conce ns he handling o nonmeasu able ac s. To
model he alua ion o such ac s, we adap Gul and Pesendo e ’s cons uc ion ha as-
signs o each ac a measu able en elope. We e ain hei no ion o a measu able spli
o he s a e space induced by he p eimage o he ac . Each elemen o his spli co e-
sponds o an ou come-se (wi h ini e ca dinali y), o which, gi en he limi ed in o ma-
ion abou he unde lying s ochas ic p ocess, ende s he incapable o a ibu ing any
ac ion o he p obabili y he p io assigns o ha elemen o he spli o any s ic sub-
se o he co esponding ou come-se . The essen ial di e ence is ha whe eas Gul and
Pesendo e de ine an en elope as a mapping om s a es o in e als o ou comes, de-
ined in e ms o leas - and mos -p e e ed ou comes o he co esponding ou come-se ,
we e ain he en i e ou come-se . Tha is, he en elope o he ac maps o an ou come-
se p ecisely hose s a es in he elemen o he measu able spli induced by he ac ’s
p eimage ha co esponds o ha subse o ou comes. This in u n means he en elope
o an ac can be cha ac e ized as he minimal (wi h espec o se -inclusion) measu able
mapping om s a es o ou come-se s ha con ains he ac . Indeed om a pe cep ual
pe spec i e, we con end i makes sense o iew he DM as incapable o dis inguishing
among ac s ha ha e a common en elope.3Theaxiomsweadop gua an ee heexis-
ence and uniqueness o en elopes.
The in e p e a ion o en elopes in e ms o belie and plausibili y unc ions, as de-
sc ibed in Gul and Pesendo e ’s, becomes e en mo e s aigh o wa d: he belie in a
pa icula ou come-se ob aining is he o al p obabili y assigned o ha ou come-se
and all i s subse s in he en elope, while i s plausibili y is he o al p obabili y o all sub-
se s con aining a leas one elemen o ha ou come-se . Mo eo e , he p io and he
en elope o an ac induce he ou come-se lo e y in which o each ou come-se , he
p obabili y assigned o ha ou come-se is gi en by he p obabili y he p io assigns o
he se o s a es ha he en elope maps o ha pa icula ou come-se .
Imposing ha he DM is indi e en among ac s inducing he same ou come-se lo -
e y, we in oduce Expec ed Balanced Unce ain U ili y (EBUU) p e e ences co espond-
ing o he amily o p e e ences ha admi an implici p obabili y equi alen (u ili y)
ep esen a ion cha ac e ized by a pai μ,U,whe eμis he DM’s p io de ined o e
hose e en s she deems measu able and a balanced ou come-se u ili y,U(Y,p), ha
speci ies he u ili y o an ou come-se Yin a lo e y ha has a p obabili y equi alen
o p, by which we mean any lo e y he DM iews as equally aluable as a bina y gamble
ha yields he bes ou come wi h p obabili y pand he wo s ou come wi h he comple-
men a y p obabili y 1−p. I exhibi s a na u al (ou come-se ) mono onici y wi h espec
o i s i s a gumen .
Thus we ob ain a clean sepa a ion o he ambigui y she pe cei es o be p esen gi en
he knowledge abou he andom p ocess go e ning he esolu ion o he unce ain y
she aces om he a i ude owa d isk (i.e., measu able unce ain y). The o me is
cha ac e ized by hose e en s ha lie ou side he domain o he p io while he es ic-
ion o he balanced ou come-se u ili y o single on ou come-se s encodes he la e .
3In his ega d, he measu able spli induced by an ac ’s in e se image is eminiscen o Ghi a da o’s
(2001, p. 249) second scena io o an unde speci ied s a e space as one possible way o in e p e his model
in which p e e ences a e de ined o e ou come-se ac s.
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4G an , Roo da, and Yang Theo e ical Economics 20 (2025)
Finally, he a i ude owa d (gene al) unce ain y in ol ing bo h isk and ambigui y is
embodied in he (un es ic ed) balanced ou come-se u ili y.
This in e p e a ion cla i ies why we do no impose decomposabili y o non-
measu able e en s. The p inciple elies on a sha p sepa a ion be ween ou comes o
an ac on an e en and i s complemen , bu his ge s blu ed when e en s a e nonmea-
su able, since ou comes on an e en may con ibu e o he ou come-se ou side ha
e en . We e e o Gul and Pesendo e (2014)(Sec ion5) o anillus a ionbywayo
he Ellsbe g pa adox o his e ec . Hence, hei mo i a ion no o impose P2 o non-
measu able e en s in essence also applies o es ic ing decomposabili y o measu able
e en s only.
We de elop he o mal de ini ion o EBUU p e e ences in Sec ion 2wi h i s axioma ic
cha ac e iza ion appea ing in Sec ion 3. We p o ide h ee examples in Sec ion 4.We
conclude in Sec ion 5. P oo s appea in he Appendix.
2. The model
Ou se ing is one in which he pu ely subjec i e unce ain y he DM aces is desc ibed
by a s a e space . The objec s o choice a e ac s ha o each s a e o na u e ω∈,
deli e an ou come x om a se X.Eachac is simple, ha is, i s image ()is a ini e
subse o X.
We deno e he se o all ac s by F. We iden i y any ou come x∈Xwi h he (con-
s an ) ac in which (ω)=x o all ω. And wi h u he (albei ai ly s anda d) abuse o
no a ion, Xwill also e e o he se o cons an ac s.
Fo any pai o e en s E,B⊆,B Eshall deno e he se o elemen s ha a e in B
bu no in E. Fo any pai o ac s and gin Fand any e en E⊂,wew i e Eg o he
ac ha ag ees wi h on Eand wi h gon  E.
The DM is cha ac e ized by he p e e ences o e ac s, a bina y ela ion on F,wi h
asymme ic and symme ic pa s deno ed by and ∼, espec i ely.
We begin ou desc ip ion o expec ed balanced unce ain u ili y p e e ences by i s
no ing he DM possesses ich cohe en belie s. This en ails he exis ence o a su icien ly
ich collec ion o ( isky) e en s, cons i u ing a σ-algeb a o subse s o , o e which can
be de ined a coun ably-addi i e and con ex- anged p obabili y measu e μ(he “p io ”)
wi h which he DM p ecisely quan i ies he unce ain y she associa es wi h each isky
e en . We deno e he domain o μby R.
Coun able-addi i i y equi es he p obabili y o he union o a coun able collec ion
o disjoin measu able e en s om Requals he in ini e sum o he p obabili ies o hese
e en s. Fo μ o be con ex- anged equi es o any e en Rin Rand any in (0, 1) he e
exis s a subse B⊂R ha is in Rand o which μ(B)= μ(R).Le Fμ⊂Fdeno e he se
o ac s ha a e measu able wi h espec o μ.
F om his poin on, he e m ou come-se will e e o any nonemp y ini e subse
o Xwi h gene ic elemen s deno ed by Y,Z,Y, e c. As we alluded o in he In o-
duc ion, he e is a na u al way o use his p io o iden i y wi h each ac i s ou come-se
en elope. Le Fμbe he se o measu able (wi h espec o μ) unc ions :→{Y⊂
X:Y= ∅,|Y|<∞}. We e e o elemen s o Fμas ou come-se ac s.
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Theo e ical Economics 20 (2025) Expec ed balanced unce ain u ili y 5
De ini ion 1 (En elope o an Ac ). The ou come-se ac ∈Fμis he en elope o i :
(i) (ω)∈ (ω) o all ω∈,and
(ii) o any ou come-se ac g∈Fμ:
(ω)∈g(ω) o all ω∈=⇒ μω∈: (ω)⊆g(ω)=1.
To cons uc he en elope o an ac , i is use ul i s o de ine he inne measu e o μ,
deno ed by μ∗, ha is de i ed om he p io by assigning o each e en E⊂ he weigh
μ∗(E)∈[0, 1] ha is he solu ion o
sup
R∈R,R⊆E
μ(R).
Since μis coun ably addi i e, he sup emum is a ained. We shall e e o he measu -
able e en [E]∗∈Ras he inne -slee e o E,i [E]∗⊆Eand μ([E]∗)=μ∗(E).4
Following Gul and Pesendo e , we associa e wi h each ac a measu able pa i ion o
he s a e space gene a ed by he ac ’s p eimage as ollows.
De ini ion 2 (Measu able Spli ). The measu able spli (o he s a e space) associa ed
wi h he ac :→Xand deno ed by {RY
∈R:Y⊆ (),Y= ∅}is induc i ely de ined
as ollows:
1. Fo each elemen x∈ (),se R{x}
:=[ −1(x)]∗.
2. Fo each Y⊆ ()such ha |Y|>1, se
RY
:= −1(Y)∗ 
Z⊂Y,Z=∅
RZ
.
We e e o RY
as he -ma ginal inne -slee e o he ou come-se Y.
To see how he en elope o an ac can be cons uc ed using he measu able spli
gene a ed by i s in e se image, i s conside a bina y ac xAy. The measu able spli is
he h ee elemen pa i ion o he s a e-space
R{x}
xAy
=
[A]∗
,R{y}
xAy
=
[ A]∗
,R{x,y}
xAy
=
 ([A]∗∪[ A]∗).
The i s ( esp., second) elemen co esponds o he la ges measu able subse in which
he bina y ac xAyyields he ou come x( esp., y). Fo he hi d elemen , all he DM can
disce n is ha he ou come will be ei he xo y. Howe e , she is unable o a ibu e any
ac ion o he p obabili y he p io assigns o his elemen o he spli o ei he xo y
4No ice ha he inne slee e is unique up o a se o μ-measu e 0.
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6G an , Roo da, and Yang Theo e ical Economics 20 (2025)
ob aining alone. Thus, i eadily ollows om De ini ion 1 ha he en elope o xAyis
he ou come-se ac ∈Fμ o which
(ω)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
{x}i ω∈[A]∗
{y}i ω∈[ A]∗
{x,y}o he wise
This me hod eadily ex ends o an a bi a y ac in F. Using he measu able spli
{RY
∈R:Y⊆ (),Y= ∅}, i s (essen ially unique) en elope is he ou come-se ac ∈
Fμcons uc ed by se ing (ω):=Ywhene e ω∈RY
.
An ou come-se lo e y is a ini e anged unc ion L:{Y⊂X:Y= ∅,|Y|<∞}→
[0, 1]sa is ying Y⊂X,|Y|<∞L(Y)=1. We associa e wi h he ac he ou come-se lo -
e y μ◦ −1.
Recalling he app oach o Demps e (1967)andSha e (1976), we shall in e p e he
ou come-se lo e y μ◦ −1as encoding how he DM weigh s ha pa o he e idence
suppo ing he belie ha he ac leads o an ou come in a gi en se o ou comes ob-
aining ha is no well speci ied enough o allow he o dis ibu e any o i ac oss any
o he elemen s o ha se o any o he o he s ic subse s o ha se o ou comes. In
o he wo ds, o each ou come-se Y⊆ ()we shall in e p e μ◦ −1(Y)(=μ(RY
)) as
he weigh assigned by he DM o e idence ha di ec ly suppo s he ac leading o
an ou come in Yob aining ha canno be u he e ined in e ms o any o he s ic
subse s o Y.
Analogous o G an ’s (1995, p. 163) endi ion o Machina and Schmeidle ’s (1992)
concep o p obabilis ic sophis ica ion, we equi e ha no ele an p e e ence in o ma-
ion is los by his associa ion.
De ini ion 3 (Cohe en Belie s). The p io μis a cohe en belie o he p e e ence ela-
ion , i o each pai o ac s and 
, wi h espec i e en elopes and , ∼
whene e
μ◦ −1=μ◦ −1.
One mo e elemen is needed, a balanced ou come-se u ili y ha speci ies he u ili y
o each ou come-se in a lo e y o a gi en alue. Res ic ed o single ons, we impose he
s anda d p ope ies o u ili y in be weenness models, bu o de e mine a use ul concep
o mono onici y o se s u ns ou o be a mo e delica e issue. I in ol es he choice
o a dominance ela ion be ween se s o ou comes. To impose a comple e dominance
ela ion, as o single ons, would o e ly es ic he model class. Howe e , i is na u al
o equi e he DM s ic ly ( esp., weakly) p e e one ou come-se o e ano he i he DM
s ic ly ( esp., weakly) p e e s all he ou comes in he o me o all he ou comes in he
la e .
To s eamline he exposi ion, we assume ha he e exis s a bes ou come ¯
xand a
wo s ou come x(i.e., ¯
xxx o all xin X), and impose a no maliza ion ha o any
bina y ac ¯
xRxwi h μ(R)=p, i s expec ed balanced u ili y is ze o.
De ini ion 4 (Balanced Ou come-Se U ili y). A balanced ou come-se u ili y is a unc-
ion U:{Y⊂X:Y= {∅},|Y|<∞}×[0, 1]→R ha :
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Theo e ical Economics 20 (2025) Expec ed balanced unce ain u ili y 7
1. Exhibi s ou come-se mono onici y, in he sense ha o any pin (0, 1]and any
pai o ou come-se s Yand Z,U(Y,p)>(≥)U(Z,p),whene e o all(y,z)∈Y×
Z:U({y},q)=U({z},p)=0=⇒ q>(≥)p;and,
2. Is no malized wi h espec o some maximal and minimal ou comes in X, deno ed
¯
xand x, espec i ely, in he sense ha :
pU{¯
x},p+(1−p)U{x},p≡0. (1)
I is deemed o be canonical i , in addi ion, U({¯
x},p)−U({x},p)≡1.
Fo all he examples p esen ed in Sec ion 4, he balanced ou come-se u ili y we
speci y will be canonical. This amoun s o se ing U({¯
x},p):=1−pand U({x},p):=
−p. Tha we always can choose Ucanonical, elies on he ac ha escaling ( o a
ixed p)U(·,p)by a s ic ly posi i e scala λphas no e ec in an EBUU ep esen a ion,
as i becomes e iden om he nex de ini ion.
De ini ion 5 (EBUU p e e ences). A p e e ence is EBUU i he e exis s a p io μand
a balanced ou come-se u ili y Usuch ha admi s a (p obabili y equi alen ) ep esen-
a ion V:F→[0, 1]de ined as he unique solu ion o

Y⊆ ()
UY,V( )μ −1(Y)=0, whe e is he en elope o .(2)
The le -hand side o equa ion (2) unc ions as a balance scale, in he sense ha , o
each p obabili y pand any bina y ac ¯
xRxwi h R∈Eand μ(R)=p:

Y⊆ ()
U(Y,p)μ −1(Y)≥0⇐⇒ ¯
xRx.
3. Cha ac e iza ion
We begin ou cha ac e iza ion o EBUU p e e ences by i s speci ying wha p ope y an
e en mus sa is y ha allows us o in e he DM deems i o be “ isky,” and hus “measu -
able.” As we no ed abo e in he In oduc ion,inGul and Pesendo e ’s (2014)expec ed
unce ain u ili y heo y, his equi es bo h i and i s complemen sa is y a e sion o Sa -
age’s (1954) pos ula e P2.
De ini ion 6 (Ideal E en s). An e en E⊆is ideal i o any ou ac s ,g,hand hin
F,[ EhgEhand hE hEg]=⇒ [ EhgEhand h
E h
Eg].
Ins ead, we p opose he ollowing p ope y o decomposabili y.
De ini ion 7 (Decomposable E en s). An e en R⊆is decomposable i o e e y pai
o ac s and gin F, Rggand gR g=⇒ g.
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8G an , Roo da, and Yang Theo e ical Economics 20 (2025)
Ou axioms will ensu e ha decomposabili y is equi alen o he c i e ion wi h 
eplaced by ≺, and en ails he nons ic a ian s wi h and as well.5We shall e e
o Ras he se o decomposable e en s. No ice ha by cons uc ion he se R(like he se
o ideal e en s) is closed unde complemen s. Also, i eadily ollows ha i is comple e
and ansi i e, hen any ideal e en is also decomposable.6The con e se, howe e , need
no hold.
We e e o an ac as decomposable i i is measu able wi h espec o R, ha is,an
ac gis decomposable i g−1({x})∈R o all x∈X. Hence, loosely speaking, a decom-
posable ac coincides wi h i s own en elope. Le G⊂Fdeno e he se o decomposable
ac s.
A subclass o decomposable e en s a e hose o which modi ying any ac on ha
e en lea es i in he same indi e ence se . These a e known as (Sa age-)null e en s.
De ini ion 8 (Null E en s). An e en N⊆is null i ∼gN o all ,g∈F.Le Nde-
no e he se o null e en s.
Se R+:=R N, he class o nonnull decomposable e en s. And o each non-null
decomposable e en R∈R+and each ac ∈F,se (R)+:={y∈ (R): −1(y)∩R/∈N}.
Tha is, (R)+con ains each elemen in he image o (R) o which i s p eimage has a
nonnull in e sec ion wi h R.
Analogous o he ole played by ideal e en s in Gul and Pesendo e (2014), we sup-
pose an EBUU DM uses elemen s o R+ o quan i y he unce ain y o any e en . So, i
seems na u al o iew an e en as maximally ambiguous i i and i s complemen con-
ain no elemen o R+. Adop ing he e minology o Gul and Pesendo e , we will e e
o such an e en (as well as i s complemen ) as di use.
De ini ion 9(Diffuse E en s). An e en D⊆is di use i , o e e y nonnull decom-
posable e en R∈R+,R∩D/∈N,andR∩( D)/∈N.Le Ddeno e he se o di use
e en s.
We say an ac his di use i i s in e se image gene a es a di use pa i ion o ,by
which we mean h−1(x)∈D o all x∈h()+.Le H⊂Fdeno e he se o di use ac s.7
Wi h hese p elimina ies in hand, we can now s a e he axioms. Ou i s is he s an-
da d o de ing axiom.
Axiom 1 (O de ing). The bina y ela ion is comple e and ansi i e.
We nex equi e he collec ion o e en s he DM deems unambiguous o be closed
unde conjunc ions. Tha is, we equi e o any pai o decomposable e en s Rand 
R
ha hei in e sec ion is also decomposable.
5Whe e is he ela ion de i ed om by se ing  i  , and ≺is he asymme ic pa o .
6I he e en Eis ideal, hen i ollows om Gul and Pesendo e ’s (2014) Lemma B0 (p. 25) ha i is also
le ideal, ha is, gE  implies g Ego equi alen ly, ¬(g Eg)implies ¬(gE  ). Thus, i ollows
om he comple eness o  ha Eggimplies gE .Hencei ,inaddi ion,weha egE g, hen g
ollows om he ansi i i y o , which in u n ollows om he comple eness and ansi i i y o .
7I will u n ou ha he en elope o any di use ac h∈Hwill be he cons an unc ion h(ω)=h()+ o
all ω∈.
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Theo e ical Economics 20 (2025) Expec ed balanced unce ain u ili y 15
he s anda d (i.e., unadjus ed) quasi-a i hme ic mean. Mo eo e , i e e y Be noulli
u ili y in he se (Y)is g ea e han ( esp., less han o equal o) p, hen he bal-
anced u ili y is equal o he di e ence be ween he quasi-a i hme ic mean o he
Be noulli u ili ies in he se (Y)and p.
Howe e , i γ>1andp esides in he open in e al ( Y,¯
Y), hen
Mp (Y)<φ
−11
|W|
∈ (Y)
φ( ).
Tha is, he quasi-a i hme ic mean is adjus ed downwa d, e lec ing he decision-
make ’s a e sion o ambigui y abou whe he he ou come she will ecei e om he
se Ywill p o e disappoin ing because i s Be noulli u ili y is less han p,o willbe
a cause o ela ion because i s Be noulli u ili y is g ea e han p.
The case o each example in which he balanced ou come-se u ili y is quasi-linea
wi h espec o p( hus allowing o a s aigh o wa d ea angemen o he implici ep-
esen a ion o ob ain an explici ep esen a ion) co esponds o β=0, α(p)≡0, and
γ=0, espec i ely. And, o each o he h ee, s anda d de ini ions o a e sion o ambi-
gui y om he li e a u e espec i ely co espond o u( ,¯
)≤( +¯
)/2; α(p)≥1/2 o all
p;φis conca e and γ≥0, espec i ely.
5. Concluding commen s
Like Gul and Pesendo e ’s (2014) expec ed unce ain u ili y, EBUU a o ds he ou side
obse e he abili y o in e hose e en s he DM deems measu able, solely om he
p e e ences. And jus as i was he case o ideal e en s in expec ed unce ain u ili y, de-
composable e en s may be iewed as ones o which Sa age’s (1954)su e- hing p inciple
applies.
Unlike Gul and Pesendo e , howe e , we ha e no “ope a ionalized” his p in-
ciple by imposing Sa age’s pos ula e P2. Ins ead, ollowing G an , Kajii, and Polak
(2000), we ha e in e p e ed s a emen s like “ would be p e e ed o gi he e en R
we e known o ob ain” as only en ailing Rgg. As a consequence and ollowing
as an immedia e co olla y o Theo em 1, o a DM who exhibi s ich cohe en be-
lie s, he isk p e e ences (o e ou come lo e ies) need only sa is y he be weenness
p ope y o Chew (1983)andDekel (1986), a he han ( ull) independence. Mo e-
o e , he hi d example om Sec ion 4p o ides us wi h a pa simoniously pa ame e -
ized model ha no only can accommoda e Ellsbe g-s yle choice pa e ns bu allows
he DM o exhibi he no el phenomenon o a e sion o ambigui y abou disappoin -
men .
Al hough a guably hey ha e ecei ed a easonable amoun o a en ion in he isk
li e a u e, p e e ences ha exhibi be weenness p ope ies a e almos comple ely absen
in he ambigui y li e a u e. Howe e , as decomposabili y p o ides us wi h a na u al way
o ope a ionalize he su e- hing p inciple, we con end EBUU heo y p o ides us wi h a
no ma i ely a ac i e app oach o modeling choice unde unce ain y.
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16 G an , Roo da, and Yang Theo e ical Economics 20 (2025)
Appendix
P oo o Theo em 1
To se he s age o he p oo , we i s desc ibe how e e y ac can be exp essed in e ms
o cons an s ac s o (and) di use ac s on i s measu able spli , as explained in Gul and
Pesendo e (2014). Wi hou loss o gene ali y, we ake ()= ()+.
Lemma 1. Fix a ac ∈Fwi h measu able spli {RY
∈R:Y⊆ (),Y= ∅}.Fo each
nonnull elemen RY
in hemeasu ablespli o ,i |Y|>1, he e exis s a di use ac
hY∈Hsuch ha hY
RY
= ,andi Y={y}, =yRY
.
P oo . Gi en a nondecomposable ∈F, choose a nonnull RY
wi h Y={y1,,yn},
n>1( hecasen=1 is ob ious). The e exis s a sequence o disjoin e en s {Bn}such
ha Bi= −1(yi)∩RY
o all i. Lemma A2 o GP show ha he e exis s a di use pa i ion
{D1,,Dn}∈Do . Now de ine
D∗
1=D1∩ RY
∪B1,
······
D∗
n=Dn∩ RY
∪Bn.
We nex show ha {D∗
1,,D∗
n}is a di use pa i ion o . Assume by way o con a-
dic ion ha D∗
iis no a di use e en o some i. Then he e is R∈R+such ha R∈D∗
i.
Since Ris a σ-algeb a, ( (RY
)) R∈R.Mo eo e ,( (RY
)) R∈Bi, which con adic s
Bicon aining no decomposable e en . Thus, D∗
iis a di use e en . I is easy o check ha
D∗
is a e all disjoin and hei union is . The e o e, {D∗
1,,D∗
n}is also a di use pa i ion
o .Se hY=(D∗
1:y1,,D∗
n:yn).ThenhY
RY
= .
Su iciency o he axioms
S ep 1: De i ing he p io Fi s , we show ha he se o decomposable e en s cons i-
u es a σ-algeb a: Tha is, (1) i con ains bo h he uni e sal se and he emp y se , ∅;
(2) i is closed unde complemen s; (3) i is closed unde in e sec ion; and (4) i is closed
unde coun able unions.
Lemma 2. The se o decomposable e en s Ris a σ-algeb a.
P oo . (1) F om he de ini ion o a decomposable e en , i is immedia e ha ∅∈Rand
∈R.(2)I R∈R, hen also by de ini ion we ha e  R∈R.(3)Axiom2ensu es Ris
closed unde in e sec ion.
(4) Finally, le {Rn}be a se o (inc easing) decomposable e en s wi h Rn⊂Rn+1.
Assume by way o con adic ion ha R∞:=Rnis no a decomposable e en . Tha is,
he e exis ,g∈Fsuch ha R∞ggand gR∞ gbu g . Bu since each Rnis
decomposable, his means o e e y nei he g Rngo ggRn (o bo h). Hence, we
can ind an in ini e subsequence {
Rn}o {Rn}wi h 
Rn=R∞, and o which:
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Theo e ical Economics 20 (2025) Expec ed balanced unce ain u ili y 17
(i) ei he g 
Rng(x) o alln,
(ii) o gg
Rn (x) o alln.
I (i) ( esp., (ii)) holds, axiom 5implies g R∞g( esp., ggR∞ ) con adic ing R∞gg
( esp., gR∞ g). Thus we ha e es ablished he e mus exis a leas one n o which
Rnggand gRn g, which since Rnisdecomposable,in u nimplies g—a con-
adic ion. Thus, Rnis a decomposable e en . The e o e, Ris a σ-algeb a.
The ollowing auxilia y esul is undamen al and used bo h he e and in subsequen
s eps.
Lemma 3. Fo all x∗x,all , ∈F,andR∈R+,i x∗R  xR , hen he e is a
R∈R o which (x∗Rx)R ∼ .
P oo .I ei he x∗R ∼ o ∼xR ,se R:=o se R:=∅. We now only need o
ind a decomposable e en Rwhen x∗R  xR .
Since x∗R  ,Axiom6implies he e is a decomposable e en R1⊂Rsuch ha
(xR1x∗)R  . Applying Axiom 6again implies he e is a decomposable e en R2⊂R
R1such ha (xR1∪R2x∗)R  . By epea ing he a gumen , his yields a se ies {Rn}⊂R+
o disjoin subse s o R ha sa is ies
xk
n=1Rnx∗R   o all k.(3)
Le Adeno e he collec ion o all such se ies, and de ine B:={∞
n=1Rn:{Rn}∈A}.
No ice ha B⊂R,and ha (x
Rx∗)R   o each 
R∈B,byAxiom5. We show ha we
can ake R=R Mwi h Ma maximal elemen o Bi i exis s, and in oke Zo n’s lemma
o es ablish ha o he wise we can ake Mas he uppe bound ou side Bo a chain in B.
Lemma (Zo n). Le Pbe a pa ially o de ed se in which each chain Chas an uppe
bound. Then Phas a leas one maximal elemen .
This applies o Bas a pa ially o de ed se , by se -inclusion. Fi s , assume Bhas a
maximal elemen M. We can exclude ha (xMx∗)R  , since o he wise he p ocedu e
abo e would de e mine ˜
R∈R+ o which s ill ((xM∪˜
Rx∗)E  , implying ha also M∪
˜
R∈B, as limi o he se ies (M∪˜
R,∅,)∈A, con adic ing ha Mis maximal o Bwi h
M⊂M∪˜
R.So,(x∗R Mx) ∼ .
Nex , suppose Bhas no maximal elemen . By Zo n’s lemma, Bmus con ain a
chain Cwi h uppe bound ˜
R∈C˜
R/∈B.Nowwecan akeMas his uppe bound, we
can again exclude ha (xM∪˜
Rx∗)R  .
To conclude, in bo h cases, we can ake Ras he decomposable e en R Msuch
ha (xMx∗)R ∼ .
Nex , we es ablish Machina and Schmeidle ’s (1992)AxiomP4∗holds on he es ic-
ion o  o G( he se o decomposable ac s).
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18 G an , Roo da, and Yang Theo e ical Economics 20 (2025)
Lemma 4. Fo any decomposable e en s R,
R,T∈R,R∪
R⊆T, any ou ou comes x∗x
and y∗yin X, and any pai o ac s , ∈F:
x∗
RxT x∗

RxT =⇒ y∗
RyT y∗

RyT .
P oo .F om(x∗
Rx)T (x∗

Rx)T xT , by applying Lemma 3we can ind a de-
composable e en R⊆Rsuch ha (x∗
Rx)T ∼(x∗

Rx)T .ThusbyAxiom4i ollows
(y∗
Ry)T ∼(y∗

Ry)T . And since R⊆Ri ollows om Axiom 3 ha (y∗
Ry)T (y∗
Ry)T .
The desi ed implica ion hen ollows om he ansi i i y o .
Now we conside he es ic ion o  o bina y be s on decomposable e en s in ol -
ing he bes ¯
xand wo s xou comes, deno ed as F{x,¯
x}, and es ablish i admi s an SEU
ep esen a ion. Ou s uc u al assump ion on X ha ¯
xximplies (Sa age’s) P5.Ax-
iom 1is P1.Axiom3di ec ly implies P3.P4 is edundan , and Axiom 6is P6. Finally,
in a se ing wi h exac ly wo ou comes ¯
xx,P4∗(as de i ed abo e om Axiom 4)is
equi alen o P2;c .Machina and Schmeidle (1992, p. 764).
Thus, he e is a ini ely addi i e, con ex- anged μon and a unc ion :{x,¯
x}→R
such ha V:F{x,¯
x}→R, de ined by V(¯
xRx)=μ(R) (¯
x)+(1−μ(R)) (x), ep esen s
on F{x,¯
x}. Wi hou loss o gene ali y, we can se (x):=0and (¯
x):=1. Hence
V(¯
xRx)=μ(R).(4)
Axiom 5implies ha V(¯
xR1∪···∪Rnx)con e ges o V(¯
x∞
n=1Rnx) o any disjoin se-
quence o decomposable e en s {Rn},whichyields
lim
n→∞
n

n=1
μ(Ri)=μ∞

n=1
Rn,
ha is, μis coun ably addi i e.
S ep 2: Va ian s o decomposabili y To p epa e o he cons uc ion and alida ion o
he EBUU ep esen a ion, we add ess some a ian s o decomposabili y condi ions.
Lemma 5. Le be a ela ion ha sa is ies Axiom 1and Axiom 3. Fo any decomposable
e en E∈Rand any pai o ac s ,g∈F:
1.  Egand gE implies g;
and u he mo e, i also sa is ies Axioms 2and 6, hen
2.  Egand gE implies g;and,
3. Eggand gE gimplies g.
P oo .FixE∈R. We show ha s a emen 1 holds by he same echniques in G an ,
Kajii, and Polak (2000). Assume by way o con adic ion ha he e exis wo ac s ,g∈F,
such ha  Eg, gE ,andg . We conside wo cases:
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Theo e ical Economics 20 (2025) Expec ed balanced unce ain u ili y 19
(a) Suppose EggE .Thenweha e
g  EggE .
Se 
:= Egand 
g:=gE .No ice
E
g= and 
gE
=g.Thus

gE

E
g

g.
Since Eis decomposable, 
gE

E
g
implies ha 
g
, which con adic s


g.
(b) Now suppose, gE  Eg.Thenweha e
g gE  Eg.
So again, se 
:= Egand 
g:=gE , and again no ice ha 
E
g= and 
gE
=g.
Thus

gE

E
g
g
.
Since Eis decomposable, 
gE

E
g
gimplies ha 

g, which con adic s

g
.
The e o e, we ha e es ablished ha s a emen 1 holds.
Fo s a emen 2, we i s show ha o all (a bi a y) ac s ∈Fand decomposable
ac s g∈G,
∼ Eg∼gE implies ∼g(5)
Assume by way o con adic ion ha ∼ Eg∼gE and [ go g ]:
(a) ∼ Eg∼gE and g.
(i) In case, E⊂g−1(¯
x)o  E⊂g−1(¯
x).I E⊂g−1(¯
x), hen
∼ Eg∼¯
xE g=¯
xEg, ha is, Eg¯
xEg
and Axiom 3is iola ed. Simila ly, i  E⊂g−1(¯
x), hen
∼ E¯
x∼gE g=gE¯
x, ha is,gE gE¯
x.
Again, Axiom 3is iola ed. The same a gumen applies when g(E)o g( E)
only has ou comes indi e en o ¯
x.
(ii) O he wise, Axiom 6implies he e is a pa i ion {Ri}such ha ¯
xRig o
all Ri. Since Eis nonnull, he e exis Rjsuch ha Ri∩Eis nonnull. Apply-
ing Axiom 6again, he e is a pa i ion {R
i}such ha ¯
xR
i(¯
xRjg) o all R
i.
Then he e is R
jsuch ha R
j∩Ecis non-null. Le R=Rj∪R
jand so ¯
xRg
wi h bo h R∩Eand R∩Ecnonnull. Toge he wi h Axiom 3, E(¯
xRg) and
(¯
xRg)E  ,andso ¯
xRg and we each a con ac ion.
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20 G an , Roo da, and Yang Theo e ical Economics 20 (2025)
(b) ∼ Eg∼gE and g . This case can be p o ed in he same way as wha we
ha e done in pa (a) by applying Axiom 6on x.
Nex , we show s a emen 2 holds. Assume o con adic ion ha he e a e ,gsuch ha
any o he ollowing holds:
(i) ∼ Eg∼gE and g .
(ii)  Eg, ∼gE ,andg .
(iii) ∼ Eg, gE and g .
We only need o show case (i) since he o he wo a e in a o o he di ec ion o ge
con adic ion. Applying Lemma 3, he e exis decomposable e en s R1
gand R2
gsuch ha
∼ E(xR1
g¯
x)∼(xR2
g¯
x)E
and so ∼xR1
g∪R2
g¯
xby Eq. (5). Simila ly, he e he e exis decomposable e en s R1
and
R2
such ha
∼gE(xR1
¯
x)∼(xR2
¯
x)Eg(6)
Now we le ∗=xR1
g∪R2
g¯
xand g∗=xR1
∪R2
¯
x, he abo e p e e ences a e educed o
∼ ∗∼ E ∗∼ ∗
E ∼gEg∗∼g∗
Eg∼ Eg∼gE
No ice ha ∗
E =( ∗
Eg∗)E(gE )and gEg∗=(gE )E( ∗
Eg∗)and ∗
Eg∗∈Gand so ∗
Eg∗∼ ∗
by exp ession (5). Simila ly, g∗
E ∗∼ ∗,so ∗∼g∗.F om(4), i ollows ha μ(R1
)=
μ(R1
g)and μ(R2
)=μ(R2
g). Exp ession (6) becomes
∼gE(xR1
g¯
x)∼(xR2
g¯
x)Eg∼xR1
g∪R2
g¯
x
and so g∼xR1
g∪R2
g¯
x∼ , which gi es us he con adic ion.
S a emen 3 can be p o ed in he same way as s a emen 2.
Lemma 5implies he condi ional independence o ac s on an indi e ence class.
P oposi ion 2. Fo any e en R∈Rand any ou ac s , , , ∗∈F:
R ∼ R  ∼ ∗
R =⇒ ∗
R  ∼ ∗
R ∼ R ∼ R .
P oo . By he nons ic c i e ion in Lemma 5, R  R  and R  ∗
R implies
R  ∗
R . Simila ly, R  R  and R  ∗
R implies R  ∗
R .Thus, ∗
R ∼
R implies ∗
R  ∼ ∗
R ∼ R ∼ R .
S ep 3: Cons uc ing he balanced ou come-se u ili y The p elimina y esul s abo e
(pa icula ly, Lemmas 3and 5) enable us o de ine he balanced ou come-se u ili y
U(·,·), as ollows. Se U({¯
x},p):=1−pand U({x},p):=−p, o allp∈[0, 1].
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Theo e ical Economics 20 (2025) Expec ed balanced unce ain u ili y 21
Fix an ou come se Yand p∈[0, 1]. We employ a sho hand no a ion ¯
xpx o an
ac o he o m ¯
xRxwi h μ(R)=p.WhenYis no a single on, choose a di use ac
hY∈H, o whichi sen elopehYis he cons an unc ion hY(ω)=Y o all ω∈.
When Y={y},se hY:=y.
(a) Fo he case hY¯
xpx, de e mine a decomposable e en Rsuch ha hY
Rx∼¯
xpx.
Such an Rexis s, since he balance p obabili y o hY
Rwi h R∈Ronly depends on
μ(R), again by Axiom 4, and we ake any R∈Rwi h μ(R)=q, o q he maximum
p obabili y o Rsuch ha hY
R¯
xpx.No ice ha qdoes no depend on he choice
o hY,byAxiom3(i). In o de o  o admi an EBUU ep esen a ion equi es
qU(Y,p)+(1−q)U{x},p
=pU{¯
x},p+(1−p)U{x},p=0.
Sol ing o U(Y,p)yields
U(Y,p):=1−q
q×p.
(b) O he wise, de e mine a decomposable e en Rsuch ha hY
R¯
x∼¯
xpx,andse q:=
μ(R). Again, qdoes no depend on he choice o Rand hY.Ino de o  o admi
a EBUU ep esen a ion equi es
qU(Y,p)+(1−q)U{¯
x},p=0,
which yields
U(Y,p):=−1−q
q×(1−p).
By cons uc ion, his unc ion sa is ies he wo p ope ies equi ed o a balanced
ou come-se u ili y.
S ep 4: Es ablishing he EBUU ep esen a ion I emains o e i y ha μand U, as spec-
i ied abo e, cons i u e a p ope EBUU ep esen a ion ha ep esen s he gi en o de -
ing on F.
Fix an a bi a y ac in F ha has associa ed wi h i he measu able spli {RY
:Y⊆
()+}, and he di use ac s hY
o which (hY
)RY
= .
The e exis s a p∈[0, 1],such ha ∼¯
xpx(again by Lemma 3, aking R=). Fo
each Y⊆ ()+, we can ind a decomposable sube en ¯
RY
⊆RY
o which [¯
x¯
RY
x]RY
∼
.Also, he eisA⊂ RY
in Rsuch ha ∼ RY
[¯
xAx], and P oposi ion 2gua an ees
ha also [¯
x¯
RY
x]RY
[¯
xAx]∼ .
To analyze hese exp essions, no ice ha he alue o an ac o he o m (hY
Rx)E¯
x o
measu able e en s R⊂E, only depends on μ(R)and μ(E), again by Axiom 4. Since we
can spli Rand Ein ksubse s Ri,Eiwi h equal p obabili ies, espec i ely μ(R)/k and
μ(E)/k,weha e(hY
Rix)Ei¯
x∼(¯
xAix)Ei¯
x o e en s Ai⊂Ei,wi halsoμ(Ai)independen
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22 G an , Roo da, and Yang Theo e ical Economics 20 (2025)
o i. F om P oposi ion 2, i ollows now ha (hY
Rx)E¯
x∼(¯
xAx)E¯
x. This means ha his
indi e ence in ac only p esc ibes he a io μ(R)/μ(A) o any E∈R.
Mo e gene ally, we know om Axiom 4 ha he a io μ(¯
RY
)/μ(RY
)mus be he same
in all ac s o he o m hY
RY
p  ha a e on he same indi e ence cu e as , being all indi -
e en o hY
RY
p[¯
xAx] o some A∈R. Since we de ined U(Y,p) om he ule hY
RY
px∼¯
xRpx
(when hY )andhY
RY
p
¯
x∼¯
xRpx(when hY), we can de e mine his a io as
μ¯
RY

μRY
=⎧
⎪
⎨
⎪
⎩
p
q=U(Y,p)+pi hY
q−1+p
q=U(Y,p)+po he wise.
So, he con ibu ion o each ou come se Y o he EBUU equa ion is
μRY
μ¯
RY

μRY
(1−p)+1−μ¯
RY

μRY
(−p)=μRY
U(Y,p),
as desi ed.
Necessi y o he axioms Axiom 1 ollows om he ac ha o any ∈F, he eexis sa
unique p∈[0, 1]such ha (2) holds ue o V( )=p.
Le Rdeno e he domain o μ,whichisaσ-algeb a. To show ha he e en s in Ra e
decomposable, conside an a bi a y R∈R, and ix a pai o ac s ,g∈F,wi h ∼xR¯
x∈
Xand μ(R)=p.I gR  , hen

Y∈X()
U(Y,p)μg−1(Y)∩R>
Y∈X ()
U(Y,p)μ −1(Y)∩R.(7)
And Rg implies ha

Y∈Xg()
U(Y,p)μg−1(Y)∩ R>
Y∈X ()
U(Y,p)μ −1(Y)∩ R.(8)
Adding inequali ies (7)and(8), we ge

Y∈Xg()
U(Y,p)μg−1(Y)>0.
Tha is, g .So,Ronly con ains decomposable e en s.
Nex , we show ha any e en ou side Ris no decomposable. Conside an e en
E⊂bu E/∈Rμ.Le
[E]∗( esp., [ E]∗) deno e he inne slee e o E( esp.,  E),
and de ine ˜
E:= (E∗∪[ E]∗). Since E/∈R, :=μ(˜
E)>0. By Lemma A2 (p. 22) and
he p oo o B11 (p. 31) in GP, we can pa i ion E [E]∗( esp., ( E) [ E]∗) in o wo
nonnull e en s B11 and B12 ( esp., B21 and B22). No ice by cons uc ion none o he ou
e en s B11,B12,B21,andB22 con ain any nonnull measu able e en .
To show Eis no decomposable, obse e ha U({¯
x},0
)>U
({x},0
)=0andU({¯
x},
0)≥U({x,¯
x},0
)≥U({x},0
)=0. Hence a leas one o ollowing wo inequali ies:
(i) U({¯
x,x},0
)>0 and (ii) U({¯
x},0
)>U({¯
x,x},0
)mus hold.
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Theo e ical Economics 20 (2025) Expec ed balanced unce ain u ili y 23
Conside i s , he case U({¯
x,x},0
)>U
({x},0
)and he ac =¯
xB11∪B21 x. Since
xE =¯
xB21 xand Ex=¯
xB11 x, i ollows ha ={x}E = E{x}(i.e., he en elope o each
o hose h ee ac s a e all he same). Since by cons uc ion, he measu e μis a cohe en
belie o he p e e ences gene a ed by he (implici ly de ined) EBUU unc ional, his
means ∼xE ∼ Ex. Howe e , since
1−μ[E]∗−μ[ E]∗u{¯
x,x},0
+μ[E]∗+μ[ E]∗u{x},0

>1−μ[E]∗−μ[ E]∗u{x},0
+μ[E]∗+μ[ E]∗u{x},0
(=0),
i ollows ha x,say ∼¯
xRpx o some Rpwi h μ(Rp)=p>0. To a i e a
a iola ion o he decomposabili y c i e ion, choose a measu able e en R⊂˜
Ewi h
0<μ
(R)<p,so ha  :=xR x. Since he en elopes o 
Exand xE a e he
same as hose o espec i ely Exand xE ,weha e 
Ex and xE  ,ye x.
So, Eis no decomposable.
So, now conside he case U({¯
x},0
)>U({¯
x,x},0
)and he pai o ac s =¯
xB11∪B21 x
and =x[E]∗∪[ E]∗¯
x. Since E =xB11∪B21∪B22 ¯
xand 
E =xB11∪B12∪B22 ¯
x, = E = 
E ,
ha is, all h ee ac s come om he same indi e ence se , and hence ∼ E and ∼

E . Howe e , since
1−μ[E]∗−μ[ E]∗u{¯
x},0
+μ[E]∗+μ[ E]∗u{x},0

>1−μ[E]∗−μ[ E]∗u{¯
x,x},0
+μ[E]∗+μ[ E]∗u{x},0
(=0),
i ollows  . A iola ion o he decomposabili y c i e ion o Ecan be es ablished
as abo e. So, also in his case, Eis no decomposable, and hence Ris he se o all
decomposable e en s.
Axiom 2now ollows di ec ly om he assump ion ha Ris a σ-algeb a. The ne-
cessi y o he es o he axioms ollows s aigh o wa dly om he EBUU ep esen a ion
combined wi h he ac ha o any pai o ac s and gwi h espec i e en elopes and g,
and any decomposable e en Rin R, he en elope o gR is gR . In pa icula , he en e-
lope o hR has ou come-se h()on R,andequals ou side R. The necessi y o Axiom 4
is now ob ious. Axioms 3and 5 ollow di ec ly om he co esponding p ope ies o U.
Axiom 6 ollows om he ac ha μis con ex- anged.
P oo o P oposi ion 1
Le be cha ac e ized by μ,U, and sa is y Axiom 7.Gi enh,h∈Hwi h h()+,
h()+⊂(,m),h≥h.Le

hn=h+εn∈Hwi h εn>0andlimn→∞ εn=0. Axiom 7
implies 
hnh o all n. Since 
hncon e ges o huni o mly wi h |
hn()=h()| o all n,
Axiom 5.1 implies 
hncon e ges o hin p e e ence, ha is, hh. Choose h=xDy∈H
wi h D∈Dand x,y∈(,m)wi h x>y. Lemma A2 o Gul and Pesendo e (2014)implies
he e a e disjoin D1,D2∈Dwi h D1∪D2=D. Simila ly, he e a e disjoin D
1,D
2∈D
wi h D
1∪D
2=Dc.Fo anyzwi h x>z>y, de ine h=xD1zD2yand h =xDzD
1y.We
ha e h ≥h≥hand so h hh. Since h()=h(),h ∼h, ha is,h ∼h∼h.
The same a gumen gi es ha o all h,h∈H,h∼hwhen maxh()=maxh(),
minh()=minh()and h(),h()⊂(,m).
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
24 G an , Roo da, and Yang Theo e ical Economics 20 (2025)
Le h,h∈Hwi h h()={m,}and h()={m,x1,,xn,}wi h m>x
1>,,>
xn>. Gi en a posi i e dec easing sequence {εn}such ha m−x1>ε
n,xn−>ε
n,and
o all nand εngoes o 0 as n→∞. De ine hn,h
nas ollows:
hn(ω)=h(ω)−εni ω∈h−1(m)h
n(ω)=h(ω)−εni ω∈h−1(m)
hn(ω)=h(ω)+εni ω∈h−1(l)h
n(ω)=h(ω)+εni ω∈h−1(l)
hn(ω)=h(ω)o he wise h
n(ω)=h(ω)o he wise
and so hn∼h
n o all n. Since hn,h
ncon e ges uni o mly o h,h espec i ely wi h
|hn()|=|h()|and |h
n()|=|h()| o all n.Axiom5implies h∼h. The e o e, o
all h,h∈H,h∼hi maxh()=max h()and min h()=min h().Andso o all
ou come-se s Yand all p∈(0, 1),U(Y,p)=U({minx∈Yx,maxy∈Yy},p).
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15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License