Ba die , Pie e; Xuan, Bach Dong; Nguyen, Van-Quy
Wo king Pape
Hoping o he bes while p epa ing o he wo s in
he ace o unce ain y: A new ypemo incomple e
p e e ences
Cen e o Ma hema ical Economics Wo king Pape s, No. 701
P o ided in Coope a ion wi h:
Cen e o Ma hema ical Economics (IMW), Biele eld Uni e si y
Sugges ed Ci a ion: Ba die , Pie e; Xuan, Bach Dong; Nguyen, Van-Quy (2025) : Hoping o he bes
while p epa ing o he wo s in he ace o unce ain y: A new ypemo incomple e p e e ences,
Cen e o Ma hema ical Economics Wo king Pape s, No. 701, Biele eld Uni e si y, Cen e o
Ma hema ical Economics (IMW), Biele eld,
h ps://nbn- esol ing.de/u n:nbn:de:0070-pub-30009222
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/312917
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h ps://c ea i ecommons.o g/licenses/by/4.0/
701
Janua y 2025
Hoping o he bes while p epa ing o he
wo s in he ace o unce ain y: a new ype
o incomple e p e e ences
Pie e Ba die , Bach Dong-Xuan and Van-Quy Nguyen
Cen e o Ma hema ical Economics (IMW)
Biele eld Uni e si y
Uni e si ¨a ss aße 25
D-33615 Biele eld ·Ge many
e-mail: [email p o ec ed]
uni-biele eld.de/zwe/imw/ esea ch/wo king-pape s
ISSN: 0931-6558
Unless o he wise no ed, his wo k is licensed unde a C ea i e Commons
A ibu ion 4.0 In e na ional (CC BY) license. Fu he in o ma ion:
h ps://c ea i ecommons.o g/licenses/by/4.0/deed.en
h ps://c ea i ecommons.o g/licenses/by/4.0/legalcode.en
Hoping o he bes while p epa ing o he wo s in he
ace o unce ain y: a new ype o incomple e p e e ences∗
Pie e Ba die †Bach Dong-Xuan‡Van-Quy Nguyen§
Janua y 2025
Abs ac
We p opose and axioma ize a new model o incomple e p e e ences unde unce -
ain y, which we call hope-and-p epa e p e e ences. An ac is conside ed mo e desi able
han ano he when, and only when, bo h an op imis ic e alua ion, compu ed as he
wel a e le el a ained in a bes -case scena io, and a pessimis ic one, compu ed as he
wel a e le el a ained in a wo s -case scena io, ank he o me abo e he la e . Ou
compa ison c i e ion in ol es mul iple p io s, as bes and wo s cases a e de e mined
among se s o p obabili y dis ibu ions. We make he case ha , compa ed o ex-
is ing incomple e c i e ia unde ambigui y, hope-and-p epa e p e e ences add ess he
ade-o be ween con ic ion and decisi eness in a new way, which is mo e a o able
o decisi eness.
Keywo ds: Decision heo y; Incomple e p e e ence; Mul iple-sel es; Non-ob ious ma-
nipulabili y.
JEL classi ica ion: D01; D81; D90
∗We deeply hank Ma c Fleu baey, An onin Macé, William Thomson, Xiangyu Qu, and F ank Riedel o
hei suppo and hei ad ice. We also hank he pa icipan s o he heo y semina o he Uni e si y o
Roches e , economic heo y lunch semina o he Biele eld Uni e si y, he TOM semina o he Pa is School
o economics, he heo y semina o he Ka ls uhe Technical Uni e si y, and he pa icipan s o he 17 h
Mee ing o he Socie y o Social Choice and Wel a e (Pa is), and he Time, Unce ain ies & S a egies X
con e ence (Pa is).
†Pa is School o Economics, École No male Supé ieu e de Pa is (Pa is Sciences e Le es). E-mail:
[email p o ec ed]
‡Cen e o Ma hema ical Economics, Biele eld Uni e si y. Financial suppo by he Ge man Re-
sea ch Founda ion (DFG) [RTG 2865/1 – 492988838] is g a e ully acknowledged. E-mail: bach.dong@uni-
biele eld.de
§Uni e si é Pa is-Saclay, Uni E y, EPEE, F ance. Facul y o Ma hema ical Economics, Na ional Eco-
nomics Uni e si y, Vie nam. E-mail: nguyen [email p o ec ed]
1
1 In oduc ion
“Hoping o he bes , p epa ed o he wo s , and unsu p ised by any hing in
be ween.”
- Maya Angelou, I Know Why he Caged Bi d Sings.
The complexi y o economic decisions is likely o esul in agen s’ inabili y o unwilling-
ness o decide o e he unce ain op ions hey a e supposed o compa e. In his ega d, he
es ic i eness o he assump ion ha indi idual p e e ences be comple e was ea ly acknowl-
edged,1and was ecen ly highligh ed by empi ical s udies.2We p opose and cha ac e ize a
new incomple e decision c i e ion acco ding o which, in he ace o Knigh ian unce ain y
(Knigh (1921)), agen s bo h hope o he bes and p epa e o he wo s .
We s udy p e e ences o e ac s :S→X, which a e mappings om s a es o he
wo ld o ou comes, and we in oduce and axioma ize p e e ences admi ing he ollowing
ep esen a ion:
g⇐⇒
minp∈C∫u( )dp > minp∈C∫u(g)dp
maxp∈D∫u( )dp > maxp∈D∫u(g)dp
,(1)
whe e uis a nume ical ep esen a ion o p e e ences o e ou comes, and Cand Da e se s o
p obabili y dis ibu ions o e he s a es, in e p e ed as se s o di e en scena ios.3Thus, a
decision make (DM) ollowing such a c i e ion anks an ac abo e an ac gi and only i
p o ides a highe expec ed u ili y han gin he wo s -case scena io in Cas well as in he
bes -case scena io in D.
Ou c i e ion is based on he conjunc ion o an op imis ic (o ambigui y-seeking) assess-
men and o a pessimis ic (o ambigui y-a e se) assessmen .4We hen in e p e a DM wi h
such a p e e ence as hoping o he bes scena io o ealize, while also p epa ing o he wo s
one o happen, when e alua ing each op ion: we hus e e o a p e e ence ela ion admi -
ing such a ep esen a ion as a hope-and-p epa e p e e ence. As a b ie illus a ion, hink
1Fo ins ance, Aumann (1962) w o e: “O all he axioms o u ili y heo y, he comple eness axiom is
pe haps he mos ques ionable. Like o he s o he axioms, i is inaccu a e as a desc ip ion o eal li e; bu
unlike hem, we ind i ha d o accep e en om a no ma i e iewpoin .”Schmeidle (1989), commen ing on
his cha ac e iza ion o he maxmin c i e ion, depic ed he comple eness axiom as “ he mos es ic i e and
demanding assump ion.”
2See Ce olin and Riedl (2019), Nielsen and Rigo i (2022).
3The unc ion u:X→Ris non-cons an , affine and unique up o posi i e affine ans o ma ion. The
se s Cand Da e unique, non-disjoin , compac and con ex.
4Cand Dbeing non-disjoin , he expec ed u ili y in he bes -case scena io is highe .
2
o a company conside ing launching a new p oduc . Typically, such a dual policy o deci-
sion making would a o in es men o p oduc ion s a egies ha p esen p omising p o i
oppo uni ies, in case he p oduc cap u es an impo an ma ke sha e, and a subs an ial
sa egua d, in case he p oduc does no .
We shall gi e special a en ion in ou analysis o he conco dan case in which C=D
— o which we also p o ide an axioma iza ion. Ac s a e hen e alua ed acco ding o he
in e al o all expec ed u ili y le els ha hey induce ac oss all possible scena ios. Mo e
p ecisely, an ac is p e e ed o an ac gi and only i any expec ed u ili y le el ha is
a ainable om gbu no om is below any expec ed u ili y le el ha is a ainable om
, and he e exis s a leas one le el ha is indeed a ainable om gbu no om .This
in ui i e c i e ion o compa ing anges o expec ed u ili y le els wo ks as a s ic e sion o
he s ong se o de , which is, a guably, he mos common way o compa e in e als.
Impo an ly, hope-and-p epa e p e e ences ea he op imis ic and he pessimis ic as-
sessmen s symme ically: he e o e, hey do no sys ema ically display a pa icula a i ude
owa d ambigui y, which is consis en wi h ex ensi e empi ical e idence (see T au mann and
an de Kuilen (2015) o a su ey).
The conjunc ion o a bes -case e alua ion and o a wo s -case e alua ion a play in ou
c i e ion is akin o he one a play in he no ion o ob ious manipula ion (T oyan and Mo ill
(2020)), de ined o e ela ion games in which he unce ain y aced by an agen conce ns
o he s’ messages. Acco dingly, he signi ican p ac ical ele ance o he no ion o ob ious
manipula ion p o ides suppo o ou c i e ion wi hin unce ain s a egic en i onmen s.
This no ion gi es an explana ion, o ins ance, o un u h ul epo ing s a egies ha ha e
been consis en ly obse ed in he Immedia e Accep ance mechanism, used o ma ch s uden s
wi h schools.5
The scope o applica ions o ou c i e ion goes beyond s a egic in e ac ions. The idea
ha bo h wo s -case and bes -case scena ios se e as e e ence poin s is ecognized o a ious
social and economic domains whe e ambigui y is p esen . In his ega d, le us simply
men ion he e alua ion o inancial asse s (Bossae s e al. (2010), Sch öde (2011), Ahn
e al. (2014)), o he e alua ion o di e en medical ea men s by physicians and pa ien s
(Back e al. (2003), Taylo e al. (2017)); we discuss a hi d example in mo e de ail.
I is no unusual o p ac i ione s, epo e s o ans o e alua e “young p ospec s” pa ic-
ipa ing in he annual D a in No h-Ame ican spo s leagues —we ake he example o he
5See Pa hak and Sönmez (2008) and Du e al. (2018).
3
Na ional Baske ball Associa ion league (NBA)— acco ding o “ceiling and loo scena ios.”6
This can be o mula ed in ou amewo k. The e is po en ially a my iad o pa ame e s ha
he agen conside s ele an o he e alua ion o p ospec s: a s a e is a pa icula con ig-
u a ion o pa ame e s.7In his complex en i onmen , he agen aces ambigui y and mus
compa e p ospec s on he basis o a se Co p obabili y dis ibu ions o e con igu a ions
o pa ame e s. Loosely speaking, each playe is iden i ied wi h an ac , indica ing hei
o e all pe o mance in each s a e, which is hen e alua ed acco ding o a u ili y unc ion
u, and, o e e y scena io p∈C, he agen can compu e he expec a ion o u( )acco ding
o p. Then, he incomple eness o a c i e ion such as ou s e lec s he necessi y o ha e
sufficien con ic ion when decla ing ha a playe is mo e p omising han an o he one. On
he o he hand, a c i e ion should no be oo incomple e; le us illus a e his poin by com-
pa ing ou c i e ion o wo al e na i e ones. Gi en uand C, he agen could equi e, o
a “playe ” o be decla ed mo e p omising han a “playe g”, ha , o each scena io in
C, he expec ed u ili y associa ed wi h gbe lowe han he expec ed u ili y le el associa ed
wi h (Bewley (2002)). One could e en equi e ha any expec ed u ili y le el a ainable
om gbe lowe han any expec ed u ili y le el a ainable om (Echenique e al. (2022))
— ha he “ceiling” o gbe lowe han he “ loo ” o . Bo h o hese condi ions a e s onge
han condi ion (1), exp essing a mo e demanding no ion o sufficien con ic ion. Howe e ,
i may e y well be he case ha only “gene a ional alen s” such as Vic o Wembanyama,8
(who was p esen in he 2023 D a ) be dis inguished om o he playe s on he basis o hese
mo e conse a i e c i e ia, and ha o a he homogeneous coho s such as he 2024 coho ,
he agen ail o ank any playe abo e an o he one.9In p ac ical e ms, acco ding o ou
c i e ion, a “playe ” is decla ed mo e p omising han a “playe g” i and only i any hing
ha gcould achie e and ha could no is conside ed wo se han any hing could achie e.
Wi h hope-and-p epa e p e e ences, which, in his case, compa e playe s on he basis o he
associa ed anges o expec ed u ili y, in a way ha is eminiscen o he s ong se o de , he
ade-o be ween decisi eness and con ic ion is add essed in a way ha is mo e a o able o
6See, o example, James Hansen, “Wha makes an NBA D a p ospec high ceiling o high loo ?”, SLC
Dunk, June 2023, and Kyle Boone, “NBA D a 2024 ceiling and loo scena ios: The bes o wo s case
p ojec ions o i e op p ospec s”, CBS Spo s, June 2024. We no e ha he use o he exp essions “ceiling”
and “ loo ” sugges s ha any case lying “in be ween” is conside ed possible.
7A s a e may hus encompass he oas e s o coaches and playe s, a he beginning o he season and a e
he win e “ ade” pe iod, o each anchise, hei inancial capaci ies, he pe o mance o playe s al eady
in he league, he p og ession o each o hese p ospec s, he app oach o officia ing a o ed by he league’s
execu i es, e c.
8See, o example, Sam Ha is, “Why ‘alien’ Wembanyama is F ance’s nex big hing - li e ally”, BBC
Spo s, July 2024.
9See, o example, Adam Finkels ein, “No s a s ha e e ealed hemsel es in he 2024 NBA D a , bu
his o y ells us hey’ e hiding in plain sigh ”, CBS Spo s, June 23 2024.
4
decisi eness.
The wo o iginal axioms in ol ed in ou cha ac e iza ion a e in e p e ed along his line:
we p opose in Axiom 6a ela i ely s ong sufficien condi ion o incompa abili y —so ha
Axiom 6is sa is ied by he as majo i y o incomple e c i e ia de ined on single ac s p oposed
in he li e a u e— and a ela i ely weak sufficien condi ion o compa abili y in Axiom 7.
Ou axioma iza ion main ains he assump ion ha p e e ences a e comple e o e cons an
ac s, deemed as he simples ones. Axiom 6unde sco es he ole o cons an ac s as bench-
ma ks o decision making: i he DM is unable o compa e he ac g o he cons an ac x
whene e she is unable o compa e x o , hen she is no able o compa e and g. Acco d-
ing o Axiom 7, i i) he DM canno compa e o he cons an x, while she decla es xmo e
desi able han gand, on he o he hand, ii)she canno compa e g o he cons an ac y, while
she decla es mo e desi able han y, hen she decla es mo e desi able han g. Thus, wo
speci ic aligned pieces o e idence a e enough o conclude ha an ac is be e han an o he
one, and Axiom 7may be seen as o mula ing a minimal depa u e om he comple eness
o a s anda d expec ed u ili y p e e ence ela ion —we e e he eade o Sec ion 3.1.1 o
a mo e p ecise discussion.
Fu he mo e, in o de o accoun o ypical si ua ions in which agen s ha e o choose
be ween wo op ions, e en i hey lack con ic ion o exp ess a clea p e e ence be ween hem
in he i s place, we s udy he comple ion o hope-and-p epa e p e e ences.10 We demon-
s a e ha he in a ian bisepa able comple e ex ension o a hope-and-p epa e p e e ence
admi s an asymme ic11 α-maxmin expec ed u ili y (α-MEU) ep esen a ion —and a s an-
da d α-maxmin ep esen a ion i he hope-and-p epa e p e e ence is conco dan . No ably,
he asymme ic α-MEU e ains much o he ac abili y o he s anda d α-MEU —which is
bene icial o applica ions— while emaining lexible enough o accommoda e mixed ambigu-
i y a i udes (Chand asekhe e al. (2022)).12 Impo an ly, in he ep esen a ion we ob ain,
he weigh αdoes no depend on he conside ed ac s, and is unique whene e he ex ended
hope-and-p epa e p e e ence is incomple e.
Finally, answe ing wo na u al ques ions o compa a i e s a ics ha eme ge om he
p oposi ion o a new ype o incomple e p e e ence unde ambigui y, we compa e he deg ee
o incomple eness o ou c i e ion o ha o Bewley p e e ences (Bewley (2002)) and o
wo old p e e ences (Echenique e al. (2022)), and we p o ide a way o compa e he ambigui y
10F om a heo e ical poin o iew, s udying a comple ion o an incomple e p e e ence ela ion enables o
use s anda d ma hema ical ools, o example o u ili y maximiza ion and wel a e analysis.
11“Asymme ic” e e s o he ac ha bes and wo s cases may be aken on di e en se s o scena ios.
12Speci ically, i cap u es ambigui y-a e se beha io o la ge/mode a e-likelihood e en s, ambigui y-
seeking beha io o small-likelihood e en s, and sou ce-dependen ambigui y a i udes (Chand asekhe e al.
(2022)).
5
a i udes o wo hope-and-p epa e p e e ences.
Ou pape is o ganized as ollows: we de ine he o mal amewo k and in oduce ou
c i e ion in Sec ion 2. In Sec ion 3, we gi e he main ep esen a ion esul and explo e he
case in which he se s o scena ios used in he wo assessmen s a e equal. In Sec ion 4, we
in es iga e he comple ion o ou c i e ion. Sec ion 5is dedica ed o he compa a i e s a ics
ques ions men ioned abo e. Sec ion 6p o ides an illus a i e compa ison o conco dan
hope-and-p epa e p e e ences o Bewley p e e ences, in he con ex o he agg ega ion o
opinions o expe s. The conclusions a e p esen ed in Sec ion 7. All p oo s can be ound in
he appendix.
1.1 Rela ed li e a u e
A DM hoping o he bes while also p epa ing o he wo s esponds o unce ain y by
combining opposi e ambigui y a i udes. In his pe spec i e, one may in e p e a DM wi h
a hope-and-p epa e p e e ence as equi ing ha he op imis ic (ambigui y lo ing) sel and
he pessimis ic (ambigui y a e se) sel be unanimous o he o ank some ac abo e an
o he one. The idea ha he DM consis s o mul iple (s a egic) sel es appea s equen ly
in beha io al economics, in pa icula in models o dynamic choice o choice wi hin isky
en i onmen s.13 In ecen wo ks, Chand asekhe e al. (2022) and Xia (2020) p o ided
axioma iza ions o p e e ences in ol ing wo sel es, called by he o me dual-sel expec ed
u ili y. Thei ep esen a ion di e s om ou s in ha he agen ’s inal decision is o be
in e p e ed as he esul o a speci ic leade - ollowe game be ween an op imis ic sel and a
pessimis ic sel , whe eas, in ou ep esen a ion, i is induced by a equi emen o unanimi y
imposed by he agen he sel on he assessmen s o he wo sel es.14
Ou ep esen a ion is also mo i a ed by he concep o ob ious manipula ion p oposed
in he con ex o mechanism design by T oyan and Mo ill (2020). A e ela ion mechanism
is said o be non-ob iously manipulable i , o any agen and any po en ial un u h ul epo
om he , e ealing he own ype leads o a mo e desi able ou come in bo h o he ollowing
cases: when he o he s’ epo s a e he mos a ou able o he , and when hey a e he leas
a ou able. In ou model, in he same spi i , an op ion —such as an un u h ul epo in he
p e ious example— is only abandoned o an al e na i e i his al e na i e leads o p e e ed
13Thale and She in (1981), Bénabou and Pycia (2002), Fudenbe g and Le ine (2006), B ocas and Ca illo
(2008).
14A a high -le el, he di e ence o ou app oach wi h ha o “P epa ing o he Wo s bu Hoping o
he Bes : Robus (Bayesian) Pe suasion” (Dwo czak and Pa an (2022)) is simila o he di e ence wi h
Chand asekhe e al. (2022): in he c i e ia s udied in bo h o hese pape s, one o he (pessimis ic o
op imis ic) e alua ions cons ains he o he . I is no he case wi h hope-and-p epa e p e e ences in which
bo h e alua ions a e ea ed symme ically.
6
ou comes in bo h he bes and he wo s scena ios among gi en se s o p obabili y measu es.
The ela ion o ou con ibu ion o he concep o non-ob ious manipula ion mi o s ha o
Echenique e al. (2022) o he concep o ob ious dominance, due o Li (2017): in o mally,
when he se o scena ios acco ding o which all ac s a e e alua ed is he simplex, ac is
p e e ed o ac gby a wo old mul i-p io p e e ence i and only i ob iously domina es g,
and, on he o he hand, is p e e ed o gby a hope-and-p epa e p e e ence i and only i
domina es gin he sense o T oyan and Mo ill (2020).15
Hope-and-p epa e p e e ences de ine a pa ial o de on ac s. Pionee ing wo k by Au-
mann (1962), Bewley (2002) and Dub a e al. (2004) s udied he ep esen a ion o incom-
ple e p e e ences unde isk and unce ain y. Incomple e p e e ences in non-de e minis ic
en i onmen s ha e been he objec o a g owing li e a u e: see, o example, Nascimen o
and Riella (2011)Galaabaa a and Ka ni (2012), E e e al. (2012), Fa o (2015), Mina di and
Sa ochkin (2015), Hill (2016), Ka ni (2020), Cusumano and Miyashi a (2021) and Echenique
e al. (2022). The closes model o incomple e p e e ence o ou s, apa om hose s udied
in Bewley (2002) and Echenique e al. (2022), bo h compa ed o ou s in he in oduc ion,
is in oduced in Nascimen o and Riella (2011). As a special case o hei main esul , hey
s udy a c i e ion in which he DM conside s se e al se s o scena ios, in each o which he
pe o mance o an ac is e alua ed acco ding o he wo s -case expec ed u ili y le el. Then,
an ac is p e e ed o an o he one i and only i i pe o ms be e in each se o scena -
ios. Hope-and-p epa e p e e ences enable o cap u e a di e en ype o ambigui y a i ude,
h ough he conside a ion o he op imis ic assessmen . We discuss in mo e de ails how ou
wo k ela es o Bewley (2002), Nascimen o and Riella (2011) and Echenique e al. (2022) in
he nex sec ions.
In line wi h Hu wicz’s app oach o decision making unde comple e igno ance (Hu wicz
(1951)), he α-MEU model was p oposed o cap u e he idea ha , unde ambigui y, wo s
and bes expec ed u ili y le els, o e one se o p obabili y measu es, can se e as sufficien
s a is ics o he DM: she hen compu es an α-weigh ed a e age o hese le els (Ma inacci
(2002), Kopylo (2002), Ghi a da o e al. (2004)).16 Among he ecen explo a ions o ( a i-
an s o ) he α-maxmin model,17 he one o F ick e al. (2022) is pa icula ly impo an o
he way we cha ac e ize he asymme ic α-maxmin model as ep esen ing he comple ion o
15Wi h his same se o scena ios, one can also eco e he concep o s a egy-p oo ness om Bewley
p e e ences (Bewley (2002)).
16Le us men ion wo al e na i es o he s anda d α-MEU model. The geome ic α-MEU model (Binmo e
(2009)) uses a geome ic weigh ed a e age. Mo e ecen ly, G an e al. (2020) in oduced and cha ac e ized
a gene al agg ega ion o bes -case and wo s -case expec ed u ili y ep esen a ions, e e ed o as o dinal
Hu wicz expec ed u ili y.
17Cha eauneu e al. (2007), Eichbe ge e al. (2011), Gul and Pesendo e (2015), F ick e al. (2022),
Klibano e al. (2022), Ha mann (2023), Hill (2023), Cha eauneu e al. (2024).
7
o unce ain y: when he DM has sufficien con ic ion o decla e wo unce ain ac s less
desi able han he cons an ac x, hen he DM also conside s wi h sufficien con ic ion ha
an ac ob ained h ough hedging be ween he wo is less desi able han x.
Axiom 5. Fo all , g ∈ F, i (s)g(s) o all s∈S, hen g.
Acco ding o Axiom 5, i he ou come o an ac is conside ed mo e desi able han he
ou come o an o he ac in each s a e o he wo ld, hen he i s ac is p e e ed o he second
one. In o he wo ds, acco ding o a p e e ence ela ion sa is ying Axiom 5, he s a e-wise
dominance o an ac o e an ac gp o ides sufficien con ic ion o ank abo e g. In he
pe spec i e o he ade-o be ween decisi eness and con ic ion, we see his p ope y as an
in ui i e limi a ion o incompa abili y. While i is imposed in mos app oaches close o ou s,
he s ong deg ee o conse a ism, o indecisi eness, o wo old p e e ences is oo ed in he
ac ha hey iola e i .32
We p opose in Axiom 6a ela i ely s ong sufficien condi ion o incompa abili y —
equi alen ly, a ela i ely weak necessa y condi ion o compa abili y— so ha Axiom 6is
sa is ied by almos all ( he asymme ic pa o ) he incomple e c i e ia compa ing single ac s
men ioned in Sec ion 1.1, ha is, almos all he incomple e c i e ia de ined in a classical
Anscombe-Aumann amewo k men ioned in Sec ion 1.1. Mo e p ecisely, he (asymme ic
pa o ) he c i e ia p oposed in Bewley (2002), Nascimen o and Riella (2011), E e e al.
(2012), Fa o (2015), Cusumano and Miyashi a (2021) and Echenique e al. (2022) all sa is y
Axiom 6(see Appendix A).33 On he o he hand, we impose a ela i ely weak sufficien
condi ion o compa abili y in Axiom 7.
We join ly discuss hese axioms a e we b ie ly p esen hem.
Axiom 6. Fo all , g ∈ F, i o all x∈X, ’ximplies g’x, hen ’g.
Axiom 6unde sco es he ole o cons an ac s as benchma k ac s based on which com-
pa isons o mo e complex ac s a e made: o he DM o exp ess a p e e ence be ween he
ac s and g,i is necessa y ha he e exis s a cons an ac x ha he DM p e e s o ei he
o g, while she canno compa e xwi h he o he ac .34
Axiom 7. Fo all , g ∈ F, and o all x, y ∈X, i ’x,xg,g’yand y hen
g.
32See Cusumano and Miyashi a (2021) and Echenique e al. (2022).
33The e is one incomple e c i e ion men ioned in Sec ion 1.1 ha is de ined on single ac s and ha may
no sa is y Axiom 6, he one p oposed in Hill (2016).
34No e ha we do no impose ha whene e he e exis s x∈Xsuch ha ’xand g’x, hen ’g
(which is Axiom 5 in Echenique e al. (2022)). Ac ually, ou c i e ion does no sa is y his p ope y in
gene al.
14
While he DM canno compa e o he cons an x, she decla es xmo e desi able han g.
On he o he hand, while she canno compa e g o he cons an ac y, she decla es mo e
desi able han y. Axiom 7implies ha in he p esence o such consonan conclusions as o
he compa ison o and g, he DM conside s , wi h sufficien con ic ion, mo e desi able
han g.
As we al eady highligh ed, gi en he complexi y in ol ed in he e alua ion o unce ain
ac s, cons an ac s, which a e he simples ac s, a e likely o be used as compa ison de ices.
A s aigh o wa d way o use hem in compa ing wo ac s, when p e e ences may be incom-
ple e, hen consis s in looking o a cons an ac ha is incompa able o one o hem and
compa able o he o he one. Each such cons an ac hen p o ides a piece o e idence as
o he compa ison be ween he wo unce ain ac s — he ques ion is hen o de e mine wha
a e sufficien pieces o e idence.
Consequen ly, gi en wo ac s and g, he DM we model compa es o gall cons an ac s
ha a e incompa able o , and ice e sa. This p ocess gi es ise o h ee possible cases:
(i) o all x∈Xsuch ha ’x,g’x;
(ii) o all x∈Xsuch ha g’x, ’x; and
(iii) he e a e x, y ∈Xsuch ha [ ’xand gand xa e compa able] and [g’yand and
ya e compa able].
In he i s wo cases, he e is no piece o e idence on which he DM may base he
compa ison: Axiom 6implies ha and ga e incompa able.
In he las case, he e a e ou possible si ua ions; i suffices o conside he ollowing
wo, o which he o he ones a e symme ic:
(a) [ ’xand xg]combined wi h [y and g’y]; and
(b) [ ’xand xg]combined wi h [ yand g’y].
In case (a), he i s piece o e idence a o s while he second a o s g. In con as , in
case (b), he wo pieces o e idence go in he same di ec ion, a o ing : acco ding o Axiom
7, his is sufficien o conclude ha is mo e desi able han g.
The e is a sense in which Axiom 7exp esses, o an asymme ic and incomple e p e e ence
ela ion, a minimal depa u e om he comple eness o weak o de s o which all ac s admi
ace ain y equi alen .35 Using he p e ious o mula ion, o hese weak o de s, one piece o
35Tha is, bina y ela ions Áwhich a e e lexi e, ansi i e and comple e, such ha , o all ∈ F, he e
is x∈Xsuch ha ∼x.
15
e idence is sufficien : i ∈ F has a ce ain y equi alen x∈Xand xis s ic ly p e e ed
o g∈ F, hen is s ic ly p e e ed o g. Fo an asymme ic and incomple e p e e ence
ela ion, Axiom 7in ol es no mo e han one piece o e idence based on a cons an ac
incompa able o and one piece o e idence based on a cons an ac incompa able o g.
Axiom 7is iola ed by wo old p e e ences, Bewley p e e ences and N&R p e e ences in
gene al. Th ough he sa is ac ion o bo h Axioms 6and 7, in pa icula , hope-and-p epa e
p e e ences add ess he ade-o be ween decisi eness and con ic ion in a new way.
We some imes e e o he classical Axioms 2,3and 5as con inui y, ce ain y indepen-
dence and mono onici y.
3.1.2 Fi s cha ac e iza ion heo em
Theo em 1. A bina y ela ion sa is ies Axioms 1-7i and only i he e exis
•a non-cons an affine unc ion u:X→R, unique up o posi i e affine ans o ma ion,
•a unique pai (C, D)o non-disjoin con ex compac subse s o ∆,
such ha , o all , g ∈ F,
g⇔
minp∈C∫u( )dp > minp∈C∫u(g)dp
maxp∈D∫u( )dp > maxp∈D∫u(g)dp
,
ha is, admi s he hope-and-p epa e ep esen a ion (u, C, D), whe e Cand Da e unique,
and uis unique up o posi i e affine ans o ma ion.
We now gi e a b ie ske ch o he p oo and highligh some in e es ing p ope ies o
ha we de i e.36 Fi s o all, Axioms 1-3gua an ee ha he e exis s a non-cons an affine
unc ion u:X→R, unique up o affine ans o ma ion, ep esen ing on X.
The p oo consis s in de ining wo bina y ela ions on F, deno ed pand o, such ha
o any , g ∈ F, gi and only i pgand og—we p o ide he p ecise de ini ions
o hese ela ions below. In ha pe spec i e, he ollowing wo lemmas a e c ucial.
Lemma. Fo all ∈ F, he se {x∈X:x’ }is non-emp y.
Lemma. Fo all ∈ F, and x, y, z ∈X, i x’ , y, and z , hen zxy.
This second esul has an in e es ing in e p e a ion. While he DM canno asse wi h
sufficien con ic ion ha is mo e desi able han he cons an ac x, she conside s wi h
36The ollowing lemmas a e no p esen ed he e in he o de in which hey a e p o ed.
16
sufficien con ic ion ha is mo e desi able han he cons an ac yand wo se han he
cons an ac z. We show ha in such a case, he DM conside s, wi h sufficien con ic ion,
ha zis mo e desi able han x, and ha xis mo e desi able han y.
F om he o iginal ela ion , we de ine wo p e e ence ela ions on Fas ollows:
gp ⇐⇒ gxand x’ o some x∈X,
go ⇐⇒ x’gand x o some x∈X.
The subsc ip s pand oa e used o deno e espec i ely a pessimis ic and an op imis ic
assessmen , based on , whe e hese wo e ms a e jus i ied gi en he way he incompa a-
bili y o a cons an ac is ea ed. Le us desc ibe he in e p e a ion o p: his ela ion
is pessimis ic in he sense ha o he de aul ac , whene e he e is a cons an ac x
such ha canno be compa ed wi h sufficien con ic ion o x, while gis conside ed mo e
desi able han xwi h sufficien con ic ion, hen pdecla es o be wo se han g.
We hen p oceed by showing ha pand oa e asymme ic and nega i ely ansi i e.
This enables us o de ine ∼pby ∼pgi and only i čpgand gčp , o all , g ∈ F, and
o de ine Ápby Ápgi and only i ei he pgo ∼pg, o all , g ∈ F. We de ine in
he same way ∼oand Áo. Then i is clea ha Ápand Áoa e weak o de s,37 and we show
ha hey a e con inuous and mono one, ha hey sa is y he classical p ope ies o ce ain y
independence, and, espec i ely, a e sion o ambigui y and p e e ence o ambigui y.38
As a consequence, Ápcan be ep esen ed by he unc ion 7→ minp∈C∫up( )dp, and
Áocan ep esen ed by he unc ion 7→ maxp∈D∫uo( )dp, whe e Cand Da e non-emp y
con ex compac subse s o ∆, and upand uoa e wo affine unc ions on X. We conclude
ha he e is no loss o gene ali y in assuming up=uo=u, and ha C∩D6=∅, using he
sepa a ing hype plane heo em on hese subse s o ∆endowed wi h he weak* opology.
No e ha in his ske ch o p oo , he ela ion be ween and he wo weak o de s pand
ois es ablished be o e he minmax and maxmax ep esen a ions o pand o: Axioms 1-3
and Axioms 5-7a e necessa y and sufficien o a gene al ep esen a ion ha we desc ibe in
Appendix B.
37They a e non- i ial asymme ic and nega i ely ansi i e bina y ela ions.
38De ini ions o hese p ope ies o weak o de s a e p o ided in he appendix.
17
3.2 Cha ac e iza ion o conco dan hope-and-p epa e p e e ences
3.2.1 Axioms
The necessa y and sufficien condi ions iden i ied in Echenique e al. (2022) o he iden i y
C=D o hold in hei wo old mul ip io p e e ence ep esen a ion a e also necessa y and
sufficien in ou ep esen a ion.39 Be o e in oducing hem, le us speci y ha , as sugges ed
in he ske ch o he p oo o Theo em 1, when sa is ies Axioms 1-3, we de ine on X he
ela ion Áby xÁyi and only i yčx o all x, y ∈X. Clea ly, Áon Xis asymme ic and
nega i ely ansi i e; and ’is equi alen o ∼, he symme ic pa o Á, on X.
We use he no ion o complemen a y ac s (Siniscalchi (2009)) o iden i y compa isons ha
a e, unde Axioms 1-7, cha ac e is ic o he unce ain y a e sion o he agen ’s pessimis ic
e alua ion, and o he p e e ence o unce ain y o he op imis ic e alua ion, espec i ely.
Two ac s and ga e complemen a y i hey pe ec ly hedge agains each o he in he sense
ha hei equal-weigh -mix u e is equi alen o a cons an ac :
1
2 (s) + 1
2g(s)∼1
2 (s′) + 1
2g(s′) o all s, s′∈S.
Axiom 8. I and gin Fa e complemen a y, hen 1
2 +1
2gimplies 1
2 +1
2gg.
Conside wo complemen a y , g ∈ F, and a p e e ence wi h ep esen a ion (u, C, D)
on F. Assume 1
2 +1
2g, as in Axiom 8and le x∈Xdeno e a cons an ac such ha
x∼1
2 +1
2g. I canno be he case ha g1
2 +1
2g, because his would imply xand
gx, and hus 1
2 +1
2gx—a con adic ion.
In o he wo ds, i 1
2 +1
2g, hen ei he 1
2 +1
2g’go 1
2 +1
2gg. Axiom 8
equi es ha he second case hold, and his equi emen is in e p e ed as a consequence
o he simplici y o cons an ac s. Indeed, by ansi i i y, in his second case, one has, by
ansi i i y, g, so ha Axiom 8s a es ha whene e 1
2 +1
2g, one has g,
ha is, i should always be easie o he DM o assess whe he is mo e desi able han he
essen ially cons an ac 1
2 +1
2g han o assess whe he is mo e desi able han g.
The in e p e a ion o Axiom 9is simila : i s a es ha o complemen a y ac s , g ∈ F,
i should always be easie o he DM o assess whe he he essen ially cons an ac 1
2 +1
2g
is mo e desi able han g han o assess whe he is mo e desi able han g.
Axiom 9. I and gin Fa e complemen a y, hen 1
2 +1
2ggimplies 1
2 +1
2g.
39The p oo o he ollowing esul is a di ec adap a ion o he p oo o P oposi ion 1 in hei pape .
18
3.2.2 Second cha ac e iza ion heo em
Theo em 2. The ollowing s a emen s hold:
(i) A hope-and-p epa e p e e ence , wi h unique ep esen a ion (u, C, D), sa is ies Axiom
8i and only i D⊆C.
(ii) A hope-and-p epa e p e e ence , wi h unique ep esen a ion (u, C, D), sa is ies Axiom
9i and only i C⊆D.
In pa icula , a bina y ela ion sa is ies Axioms 1-9i and only i he e exis
•a non-cons an affine unc ion u:X→R, unique up o posi i e affine ans o ma ion,
•a unique con ex compac subse o ∆, deno ed C, such ha , o all , g ∈ F,
g⇐⇒
minp∈C∫u( )dp > minp∈C∫u(g)dp
maxp∈C∫u( )dp > maxp∈C∫u(g)dp
.
When admi s a conco dan ep esen a ion, ac s a e e alua ed acco ding o he mini-
mum and he maximum expec ed u ili y le el a ained on a common se o scena ios. On
he o he hand, when sa is ies bo h Axiom 8and 9, o any simple complemen a y ac s
and g, 1
2 +1
2gi and only i 1
2 +1
2gg. In o he wo ds, o complemen a y ac s, i is
always as easy o de e mine whe he hei equal-weigh -mix u e is mo e desi able han one
o hem as i is o de e mine whe he one o hem is mo e desi able han he mix u e.
As a ecall, acco ding o a conco dan hope-and-p epa e p e e ence ela ion, ac s a e
e alua ed acco ding o he in e al o all expec ed u ili y le els ha hey induce ac oss all
scena ios in a gi en se . Mo e p ecisely, an ac is p e e ed o an ac gi and only i any
expec ed u ili y le el ha is a ainable om gbu no om is below any expec ed u ili y
le el ha is a ainable om , and he e exis s a leas one le el ha is indeed a ainable
om gbu no om .
4 Comple e ex ension o hope-and-p epa e p e e ences
In his sec ion, we will explo e he ex ension o hope-and-p epa e p e e ences o comple e
p e e ences. We will ocus on he in a ian bisepa able comple e ex ension; hese de ine a
b oad class o comple e p e e ences ha nes s he majo i y o p e e ences s udied in he
li e a u e.
19
We e e o an asymme ic comple e and nega i ely ansi i e bina y ela ion on Fsa -
is ying Axioms 2,3and 5as in a ian bisepa able.40 When i is, in addi ion, a weak o de ,
i sa is ies he axioms cha ac e izing expec ed u ili y, apa om he independence axiom,
which is weakened o he ce ain y independence p ope y in oduced in Gilboa and Schmei-
dle (1989).
De ini ion 5. A p e e ence ela ion on Fadmi s an asymme ic α-MEU ep esen a ion
i he e exis α∈[0,1], wo non-disjoin compac con ex subse s Cand Do ∆, and a
non-cons an affine unc ion u:X→Rsuch ha o all , g ∈ F,
g⇐⇒ αmin
p∈C∫u( )dp + (1 −α) max
p∈D∫u( )dp
> α min
p∈C∫u(g)dp + (1 −α) max
p∈D∫u(g)dp.
We will e e o such ep esen a ion as a (u, C, D, α) ep esen a ion.
Rema kably, Chand asekhe e al. (2022) show ha he asymme ic α-MEU, while e-
aining he ac abili y p ope y o he s anda d α-MEU, is lexible enough o accommo-
da e ambigui y-a e se o la ge/mode a e-likelihood e en s bu ambigui y-seeking o small-
likelihood e en s and sou ce-dependen ambigui y a i udes.
S anda d α-MEU c i e ia a e ob ained i C=Din De ini ion 5, and he ollowing esul ,
as a pa icula case, cha ac e izes hem as in a ian bi-sepa able ex ensions o conco dan
hope-and-p epa e p e e ences.
Theo em 3. The ollowing condi ions a e equi alen when is a hope-and-p epa e p e e ence
wi h unique ep esen a ion (u, C, D):
(i) ∗is an in a ian bisepa able p e e ence and an ex ension o .
(ii) ∗admi s an α-maxmin expec ed u ili y ep esen a ion (u, C, D, α)in which αis unique
whene e is no comple e.
40Ghi a da o e al. (2004) o iginally used he exp ession “in a ian bisepa able p e e ences” when s udying
weak-o de s. Fo an asymme ic comple e and nega i ely ansi i e bina y ela ion on F, as o pand
oin Sec ion 3, we de ine ∼by ∼gi and only i čgand gč , o all , g ∈ F, and Áby Ági
and only i ei he go ∼g, o all , g ∈ F. Then, in he p oo o Theo em 3, we show ha Á∗is an
“in a ian bisepa able p e e ence” in he sense o Ghi a da o e al. (2004).
20
5 Compa ison o incomple e c i e ia
5.1 Deg ee o incomple eness
We ha e s a ed ha wi h hope-and-p epa e p e e ences, in compa ison o Bewley p e e ences
and wo old p e e ences, he ade-o be ween decisi eness and con ic ion is add essed in
a way ha is mo e a o able o decisi eness. The c i e ion we use o de e mine whe he a
bina y ela ion is mo e conse a i e han an o he one pe ains o hei espec i e deg ee o
incomple eness.
De ini ion 6. Gi en wo p e e ence ela ions 1and 2on F, we say ha 1is mo e
conse a i e han 2i 2is an ex ension o 1, ha is, o all , g ∈ F,
1gimplies 2g.
The nex p oposi ion iden i ies necessa y and sufficien condi ions unde which a hope-
and-p epa e p e e ence ela ion is an ex ension o a Bewley o o a wo old p e e ence ela ion.
P oposi ion 1. Le Hbe a hope-and-p epa e p e e ence wi h unique ep esen a ion (u, CH, DH).
Le Tbe a wo old mul ip io p e e ence wi h unique ep esen a ion (u, CT, DT). Le B
be a Bewley p e e ence wi h unique ep esen a ion (u, CB). Then,
(i) he p e e ence ela ion Bis mo e conse a i e han Hi and only i CH∪DH⊆CB;
(ii) he p e e ence ela ion Tis mo e conse a i e han Hi and only i CH⊆CTand
DH⊆DT.
Rema k 1. A di ec consequence o his p oposi ion and P oposi ion 4 in Echenique e al.
(2022) is ha i CH∪DH⊆CB⊆CT∩DT, in pa icula i CH=DH=CB=CT=DT,
hen Tis mo e conse a i e han B, which is mo e conse a i e han H.
5.2 Ambigui y a i udes
We a e able o compa e ambigui y a i udes displayed by di e en hope-and-p epa e p e e -
ences using he classical compa a i e s a ics no ions o Ghi a da o and Ma inacci (2002).
De ini ion 7. Gi en wo p e e ence ela ions 1and 2on F,
(i) 1is mo e ambigui y a e se han 2i , o all ∈ F and x∈X, 1ximplies
2x.
21
(ii) 1is mo e ambigui y lo ing han 2i , o all ∈ F and x∈X,x1 implies
x2 .
An agen is mo e ambigui y a e se han an o he one i she is less inclined o choose an
unce ain ac o e a cons an ac x. On he o he hand, an agen is mo e unce ain y
lo ing han an o he one i she is mo e inclined o s ick o an unce ain ac han o swi ch
o a cons an ac x. The nex esul cha ac e izes ambigui y a i udes o hope-and-p epa e
p e e ences.
P oposi ion 2. Le 1and 2be wo hope-and-p epa e p e e ence ela ions wi h unique
ep esen a ion (u, C1, D1)and (u, C2, D2), espec i ely. Then,
(i) 1is mo e ambigui y a e se han 2i and only i C2⊆C1.
(ii) 1is mo e ambigui y lo ing han 2i and only i D2⊆D1.
Fo a hope-and-p epa e ep esen a ion (u, C, D), he wo se s o p io s Cand D ep esen
he le el o pessimism and op imism ela ed o he DM’s ambigui y a i udes. Mo e p ecisely,
he ela ionship C2⊆C1means ha , in he wo s scena io, he le el o wel a e a ained by
he agen i she has p e e ence ela ion 1is lowe han he one a ained i she has p e e ence
ela ion 2. Simila ly, D2⊆D1means ha , in he bes scena io, he le el o wel a e a ained
by he agen i she has p e e ence ela ion 1is highe han he one a ained i she has
p e e ence ela ion 2.
Based on P oposi ion 2(i), by compa ing he conco dan p e e ence wi h ep esen-
a ion (u, C, C) o he non-conco dan p e e ence 1wi h ep esen a ion (u, C1, C), wi h
C1⊂C, we can say ha 1is mo e ambigui y a e se han i is ambigui y lo ing. Simila ly,
he non-conco dan ep esen a ion 2wi h ep esen a ion (u, C, D2), wi h D2⊂C,can be
said o be mo e ambigui y lo ing han i is ambigui y a e se. Then, a DM wi h conco -
dan p e e ences is as ambigui y lo ing as she is ambigui y a e se, o , in o he wo ds, he
pessimis ic e alua ion is as pessimis ic as he op imis ic e alua ion is op imis ic.
We end his subsec ion by b ie ly discussing he ela ion be ween he deg ee o conse -
a ism o a hope-and-p epa e p e e ence ela ion and he a i ude owa ds ambigui y ha
i displays. I is easy o see ha i 1and 2a e hope-and-p epa e p e e ences, and i 1
is mo e conse a i e han 2, hen 1is bo h mo e ambigui y a e se and mo e ambigui y
lo ing han 2. Does he con e se s a emen hold ? This ques ion is all he mo e na u al
ha i 1and 2a e wo old p e e ences, hen 1is mo e conse a i e ha 2i , and
only i ,1is mo e ambigui y a e se and mo e ambigui y lo ing han 2.41 An example in
Appendix Cshows ha he answe is nega i e o hope-and-p epa e p e e ences.
41See Co olla y 1 in Echenique e al. (2022).
22
6 Agg ega ing he opinion o expe s wi h hope-and-
p epa e p e e ences
Nume ous economic decisions unde unce ain y, such as hose ela ed o iscal policy and
hose add essing clima e change, o en hinge on he guidance p o ided by g oups o expe s,
who equen ly hold con lic ing “opinions.” We p opose a simple illus a ion, in he con ex
o he agg ega ion o con lic ing opinions among expe s, in which he ac ha he planne ’s
decisions a e aken acco ding o a hope-and-p epa e p e e ence ela ion a he han acco ding
o a Bewley one e lec s he p e e ence o decisi eness.
Due o he complexi y o he issue a hand, he opinions o expe s may encompass
se e al p obabili y dis ibu ions (scena ios) o e payo -con ingen s a es. Following Danan
e al. (2016), we assume ha expe s ha e Bewley p e e ences, which exp esses, gi en a se
o plausible scena ios, he need o expe s o ha e a s ong con ic ion in o de o epo o
he planne ( he DM) ha an op ion is be e han an o he one:42
“[...] a gi en indi idual may also conside mo e han one model o be plausible—o
ha e an imp ecise belie . Fo such an indi idual, which o wo policies yields he highes
expec ed u ili y may depend on he model conside ed. When a policy yields a highe
expec ed u ili y han ano he one o all plausible models, we say ha he indi idual
unambiguously p e e s he o me policy o he la e . Unambiguous p e e ences a e
hus obus o belie imp ecision.”
Le N={1,2, . . . , n}be a ini e se o expe s. Expe j∈Nhas a p e e ence j
on F. We use 0 o deno e he DM’s p e e ence on F. We suppose ha , o all i∈N,
expe i’s p e e ence is a Bewley p e e ence wi h unique ep esen a ion (u, Ci). We hus
assume in pa icula ha he e is no di e si y o p e e ences o e ou comes, which is a
dis inc i e elemen o he heo y o he agg ega ion o opinions, compa ed o he heo y o
he agg ega ion o p e e ences. We s udy how 0should depend on (j)j∈Nand impose he
wo ollowing condi ions:
Axiom 10 (Pa e o).Fo all , g ∈F, i ig o all i∈N, hen 0g.
Axiom 11 (Cau ion o incompa abili y).Fo all ∈Fand x∈X, i he e exis s
i∈Nsuch ha ’ix, hen ’0x.
The Pa e o condi ion is he s anda d one. I asse s ha he DM should ollow he com-
pa isons exp essed by expe s when hey a e unanimous: i all expe s p e e ac o ac g,
42In pa icula , gi en his se o scena ios, he condi ion unde which hey ha e sufficien con ic ion ha
an op ion is be e han an o he one is s onge han i hey had a conco dan hope-and-p epa e p e e ence
ela ion.
23
holds: xg. Le y∈Xsuch ha g’y. F om Lemma 1,xy, implying y. Bu
hen, pg, which is a con adic ion. The e o e, pis nega i ely ansi i e.
S ep 2. gi and only i pgand og.
Le us i s p o e ha o , g ∈ F such ha g, one has pgand og,
gi ing he explici a gumen exclusi ely o pg, as ogis p o ed symme ically. By
con adic ion, assume ha čpg, hen o all x’g, one has čx, ha is, ei he ’x
o x . Bu i x , hen xgby ansi i i y, which con adic s x’g. Thus, o all
x∈X, i x’g, hen x’ . By Axiom 6,g’ , a con adic ion. We ha e hus p o ed
pg.
Suppose now pgand og, and le us show g. By de ini ion o pand o,
he e exis x∈Xsuch ha xand x’g, and y∈Xsuch ha y’ and yg. Axiom
7 hen implies g.
De ine ∼pby ∼pgi and only i čpgand gčp , o all , g ∈ F, and de ine Ápby
Ápgi and only i ei he pgo ∼pg, o all , g ∈ F. The ela ions ∼oand Áoa e
simila ly de ined. I is clea ha Ápand Áoa e comple e and ansi i e. We say ha Áp
( esp. Áo) is con inuous i p( esp. o) is con inuous, which is equi alen o he closedness
o {α∈[0,1] : α + (1 −α)gÁph}and {α∈[0,1] : hÁpα + (1 −α)g}.
S ep 3. Ápand Áoa e con inuous and sa is y mono onici y and ce ain y independence.45
We only p o ide he p oo ha Ápis con inuous and sa is ies mono onici y and ce ain y
independence, whe e mono onici y, when allowing o indi e ence, means ha o all , g ∈ F
such ha (s)Ápg(s) o all s∈S, Ápg, and ce ain y independence means ha o all
, g ∈ F, all x∈X, and all α∈(0,1), Ápgi and only i α + (1−α)xÁpαg +(1 −α)x.
We i s show ha Ápis con inuous. Le , g, h ∈ F and x∈X; deno e Ax he se o
α∈[0,1] such ha α + (1 −α)gxand x’h. Ei he Axis emp y o i coincides wi h
{λ∈[0,1] : λ +(1−λ)gx}. Then Axis open by Axiom 2. The e o e, {α∈[0,1] : α +(1−
α)gph}=∪x∈XAxis open. Simila ly, one can show ha {α∈[0,1] : hpα + (1 −α)g}
is open; hus, Ápis con inuous.
Nex , we p o e ha Ápsa is ies mono onici y. Le , g ∈ F such ha (s)Ápg(s) o
all s∈S, which clea ly implies (s)Ág(s) o all s∈S. Suppose gp , which means ha
he e exis s x∈Xsuch ha gxand x’ . This is a di ec con adic ion as, by Lemma
4, o any x′∈Xsuch ha x′’ ,gčx′. Thus, Ápg.
45The de ini ion o hese p ope ies o a weak o de a e eminded in he ollowing lines.
30
Las ly, we es ablish ha Ápsa is ies ce ain y independence. Le , g ∈ F,x∈X, and
α∈(0,1). We i s show ha Ápgimplies α + (1 −α)xÁpαg + (1 −α)x. Since Ápis
a weak o de , Ápgis equi alen o gčp , which holds i , and only i , o all y∈Xsuch
ha gy, and ya e compa able. Unde Axiom 3, i is sufficien o p o e ha , o all
y∈Xsuch ha αg + (1 −α)xy,α + (1 −α)xand ya e compa able. Le y∈Xsuch
ha αg + (1 −α)xyand suppose by con adic ion α + (1 −α)x’y.
Claim: Fo such y∈X, he e is z∈ {z:z’ }wi h u(z) = in {u(z) : z’ }such ha
yÁαz + (1 −α)x.
We ha e shown in S ep 2 ha he e exis s z∈ {z∈X:z’ }such ha u(z) = in {u(z) :
z’ }. Since ’sa is ies ce ain y independence (Lemma 2), α +(1−α)x’αz +(1−α)x.
We claim yÁαz + (1 −α)x. Indeed, i he e exis s z∈Xsuch ha zz, hen i
ollows om Axiom 3and Lemma 1 ha zβ:= βz + (1 −β)z o all β∈(0,1). Using
Axiom 3again yields α + (1 −α)xαzβ+ (1 −α)x. I hen ollows om Lemma 1 ha
yαzβ+(1−α)x o all β∈(0,1). Le ing β end o 1, one concludes, since uis affine, ha
yÁαz +(1−α)x. I zÁz o all z∈X, hen α (s)+(1−α)xÁαz +(1−α)x o all s∈S
(o he wise, Axiom 3implies z (s), which is a con adic ion). Since α + (1 −α)x’y,
Lemma 4implies αz + (1 −α)xčy, which is equi alen o yÁαz + (1 −α)x.
Since αg + (1 −α)xyand yÁαz + (1 −α)x, Lemma 3implies
αg + (1 −α)xαz + (1 −α)x,
which is equi alen o gzby Axiom 3. Hence, by de ini ion o z,gp , a con adic ion.
The e o e, α +(1−α)xand ya e compa able, which yields α +(1−α)xÁpαg +(1−α)x.
We now show he con e se implica ion. Fo α∈(0,1), and any wo , g ∈ F,α + (1 −
α)xÁpαg +(1−α)xi , and only i , o all y∈Xsuch ha αg +(1−α)xy,α +(1−α)x
and ya e compa able. Le y∈Xsuch ha gy, hen one has αg+(1−α)xαy+(1−α)x
by Axiom 3. Thus, α + (1 −α)xand αy + (1 −α)xa e compa able, implying ha and y
a e compa able by Axiom 3. Hence, gčp , which is equi alen o Ápg. We ha e p o ed
ha Ápsa is ies ce ain y independence.
S ep 4. An agen wi h p e e ences Ápon Fis a e se o ambigui y, i.e., o all , g ∈ F,
∼pgimplies α + (1 −α)gÁp . An agen wi h p e e ences Áoon Flo es ambigui y,
i.e., o all , g ∈ F, ∼pgimplies Áoα + (1 −α)g.
We only p o e ha Ápdisplays ambigui y a e sion. Le , g ∈ F such ha ∼pg,
i.e., čpgand gčp . In o he wo ds, o all x∈Xwi h x,xis compa able wi h
g, and o all x∈Xwi h gx,xis compa able wi h . Le x∈Xsuch ha x.
31
I xg, hen g, and hen, by S ep 2, pg, which is a con adic ion; hus, one
mus ha e gx. This implies {x∈X: x}⊆{x∈X:gx}. Analogously,
{x∈X:gx} ⊆ {x∈X: x}; he e o e, {x∈X: x}={x∈X:gx}.
Le α∈(0,1), we claim ha α +(1−α)gÁp . Since Ápis a weak o de , i is sufficien
o p o e čpα +(1−α)g, which holds i , o all x∈Xsuch ha x,α +(1−α)gx.
Ye , we ha e jus p o ed ha xi and only i gx. Axiom 4 hen di ec ly en ails
α + (1 −α)gx; which concludes.
Conclusion. I is well-known since Gilboa and Schmeidle (1989) ha a weak o -
de de ined on Fsa is ying he p ope ies s a ed in S ep 3can be ep esen ed by 7→
minp∈C∫up( )dp i i displays ambigui y a e sion, such as Áp, and by 7→ maxp∈D∫uo( )dp
i i displays lo e o ambigui y, such as Áo, whe e Cand Da e unique non-emp y con ex
compac subse s o ∆,upand uoa e wo affine unc ions on X, unique up o posi i e affine
ans o ma ion. Clea ly, o all x, y ∈X,xÁpyi and only i xÁy, and xÁoyi and only
i xÁy. Thus, upand uoa e posi i e affine ans o ma ions o u, and one may assume ha
up=uo=u. Finally, i emains o p o e ha C∩D6=∅.
Claim: Cand Da e non-disjoin i , and only i , o all ∈ F,minp∈C∫u( )dp ≤
maxp∈D∫u( )dp.
We only p o e he i pa , he o he di ec ion being i ial. We p oceed by con aposi ion.
Suppose ha C∩D=∅. By he sepa a ing hype plane heo em, he e exis s a bounded
measu able unc ion φ:S→Rsuch ha minp∈C∫φdp > maxp∈D∫φdp. Ye , he e exis s
a sequence o simple unc ions {φn} ha con e ges (in supno m opology) o φ. Since
bo h ˜φ7→ minp∈C∫˜φdp and ˜φ7→ maxp∈D∫˜φdp a e con inuous, he e is n∈Nsuch ha
minp∈C∫φndp > maxp∈D∫φndp. As aφn+balso sa is ies his las inequali y o all a > 0
and b∈R, one can choose a > 0and b∈Rsuch ha aφn(s) + b∈u(X) o all s∈S, which
implies φn=u( ) o some ∈ F:
min
p∈C∫u( )dp > max
p∈D∫u( )dp.
As a consequence, he ac ha , o all ∈ F,minp∈C∫u( )dp ≤maxp∈D∫u( )dp, implies
C∩D6=∅.
Based on his claim, i emains o p o e ha minp∈C∫u( )dp ≤maxp∈D∫u( )dp o all
∈ F in o de o conclude ha Cand Da e no disjoin .
Le us show ha he inequali y minp∈C∫u( )dp ≤maxp∈D∫u( )dp holds o all ∈ F
i and only i , o all x∈X, o all ∈ F, pximplies ox.
Suppose ha o all x∈X, o all ∈ F, pximplies ox. Suppose, by
32
con adic ion, ha he e is ∈ F such ha minp∈C∫u( )dp > maxp∈D∫u( )dp. Clea ly,
one has u(x∗)≤minp∈C∫u( )dp ≤u(x∗), whe e x∗and x∗a e de ined as in he p oo o
Lemma 5. Then, since u(X)is con ex, minp∈C∫u( )dp belongs o u(X). Simila ly, one can
deduce ha maxp∈D∫u( )dp lies in u(X). Then, he con exi y o u(X)implies ha he e
exis s x∈Xsuch ha
min
p∈C∫u( )dp > u(x)>max
p∈D∫u( )dp,
which is a con adic ion as i implies, as minp∈C∫u(x)dp = maxp∈D∫u(x)dp =u(x), px
and xo . The o he di ec ion o he equi alence is i ial.
I emains o show ha , indeed, pximplies ox, o all x∈X, and all ∈ F. Ye ,
pximplies x. Indeed, pxi and only i he e exis s y∈Xsuch ha yand
y’x; hen Lemma 3implies x. By S ep 2, we conclude ha ox.
We ha e hus p o ed ha minp∈C∫u( )dp ≤maxp∈D∫u( )dp o all ∈ F, and, hus,
ha Cand Da e non-disjoin .
I pa . Assume ha admi s a hope-and-p epa e ep esen a ion. One can eadily check
ha Axioms 1 o 5a e sa is ied.
Fo all ∈ F, deno e p ∈a g maxp∈D∫u( )dp and p ∈a g minp∈C∫u( )dp. De ine
also he cons an ac s =∫ dp and =∫ dp . Clea ly, ’ and ’ ; mo eo e ,
x⇐⇒ u( )> u(x),
x ⇐⇒ u(x)> u( ),
’x⇐⇒ u( )≥u(x)≥u( )
.(2)
We p o e ha Axiom 6is e i ied by con adic ion. Conside , g ∈ F such ha o all
x∈X, ’ximplies g’x. I g, hen u( )> u(g). Howe e , by assump ion, g’ ,
which implies u(g)≥u( )≥u(g), a con adic ion. The same a gumen applies o p o e
ha g canno hold. The e o e, ’g.
Axiom 7easily ob ains om he compa isons in (2). Indeed, le , g ∈ F and x, y ∈X
such ha ’x,g’y,xg, and y. Using (2), one ge s
u( )≥u(x)≥u( ),
u(g)≥u(y)≥u(g),
u(x)> u(g), u( )> u(y)
.(3)
33
Then u( )≥u(x)> u(g)and u( )> u(y)≥u(g), ha is g, by de ini ion o a
hope-and-p epa e p e e ence.
D.2 P oo o Theo em 2
By assump ion,
g⇐⇒
minp∈C∫u( )dp > minp∈C∫u(g)dp
maxp∈D∫u( )dp > maxp∈D∫u(g)dp
,
whe e uis an affine unc ion de ined on X, unique up o affine ans o ma ion, Cand Da e
wo unique compac and con ex subse s o ∆wi h C∩D6=∅. I emains o p o e ha
admi ing such a ep esen a ion sa is ies Axiom 8and 9i and only i C=D. We show in
a e y simila way o Echenique e al. (2022) ha i sa is ies Axiom 8i and only i D⊆C
— he o he inclusion being equi alen o Axiom 9is shown in a symme ic way.
Only-i pa . Suppose by con aposi ion ha DĘC: he e is some p∗∈Dsuch ha p∗/∈
C. Then, by he sepa a ing hype plane heo em and he a gumen gi en in he Conclusion
s ep o he p oo o Theo em 1, he e is an ac ψand k∈Rsuch ha
min
p∈C∫u(ψ)dp > k > ∫u(ψ)dp∗.(4)
By scaling ψand kapp op ia ely, as uis affine, one can ind , h ∈ F and x∈Xsuch ha
u( ) = 1
2u(ψ), u(h) = −u(ψ)and u(x) = 2k.46 Le g=1
2h+1
2x:
u´1
2 +1
2g¯=1
4u(ψ) + 1
2ˆ−1
2u(ψ) + k˙=k
2,
ha is, and ga e complemen a y, and 1
2 (s) + 1
2g(s)∼y o some y∈Xsuch ha
u(y) = k
2, o all s∈S. Since u( ) = 1
2u(ψ), Equa ion (4) implies
min
p∈C∫u( )dp=1
2min
p∈C∫u(ψ)dp > k
2=u(y).
In addi ion, as D∩C6=∅,maxp∈D∫u( )dp≥minp∈C∫u( )dp>u(y). As a consequence,
y.
46We abuse no a ion in a s anda d way when w i ing u( ) = , o ∈R, o ac ually deno e u( (s)) =
o all s∈S.
34
Fu he mo e, u(g) = u`1
2h+1
2x˘=−1
2u(ψ) + k. Since p∗∈D, Equa ion (4) implies
max
p∈D∫u(g)dp≥∫u(g)dp∗=−1
2∫u(ψ)dp∗+k > −k
2+k=k
2=u(y),
om which yčg. One hus has 1
2 (s) + 1
2g(s)∼y o all s∈S, y, and yčg, which is
a iola ion o Axiom 8.
I pa . Suppose ha D⊆C. Conside wo complemen a y ac s , g ∈ F such ha
1
2 (s) + 1
2g(s)∼x o some x∈X, o all s∈S, o , 1
2u( ) + 1
2u(g) = k, wi h u(x) = k.
Assume x, which is is equi alen o
minp∈C∫u( )dp > k
maxp∈D∫u( )dp > k
⇐⇒
1
2minp∈C∫u( )−u(g)dp > 0
1
2maxp∈D∫u( )−u(g)dp > 0
⇐⇒
1
2maxp∈C∫u(g)−u( )dp < 0
1
2minp∈D∫u(g)−u( )dp < 0
.
Since D⊆C, he las inequali ies yield
1
2maxp∈D∫u(g)−u( )dp < 0
1
2minp∈C∫u(g)−u( )dp < 0
.
Plugging u( ) = 2k−u(g), one ob ains
2 maxp∈D∫u(g)−kdp < 0
2 minp∈C∫u(g)−kdp < 0
⇐⇒
maxp∈D∫u(g)dp < k
minp∈C∫u(g)dp < k
.
As k=u(x), his means xg. The e o e, sa is ies Axiom 8.
D.3 P oo o P oposi ion 1
i)Le Hbe a hope-and-p epa e p e e ence wi h unique ep esen a ion (u, CH, DH), and le
Bbe Bewley p e e ence wi h unique ep esen a ion (u, CB).
Fi s , suppose ha CH∪DH⊆CB. I Bg, hen o all p∈CH∪DH,
∫u( )dp > ∫u(g)dp,
35
which implies, as CHand DHa e no disjoin ,
minp∈CH∫u( )dp > minp∈CH∫u(g)dp,
maxp∈DH∫u( )dp > maxp∈DH∫u(g)dp.
The e o e, Hg. Thus, Bis mo e conse a i e han H.
Con e sely, suppose Bis mo e conse a i e han Hand suppose, by con adic ion, ha
he e exis s p∗∈CH CB. By he sepa a ion a gumen we al eady used in he Conclusion
s ep o he p oo o Theo em 1, he e a e ∈ F and x∈Xsuch ha
∫u( )dp∗> u(x)>max
p∈CB∫u( )dp.
I ollows ha xB bu xčH , a con adic ion. Simila ly, suppose he e exis s
p∗∈DH CB. Then he e a e ∈ F and x∈Xsuch ha
min
p∈CB∫u( )dp > u(x)>∫u( )dp∗.
In his case, we ha e Bxbu čHx, an o he con adic ion. The e o e, CH∪DH⊆CB.
ii)Le Hbe a hope-and-p epa e p e e ence wi h unique ep esen a ion (u, CH, DH),
and Tbe a wo old mul ip io p e e ence wi h unique ep esen a ion (u, CT, DT).
Fi s , suppose ha CH⊆CTand DH⊆DT. Since DT∩CT6=∅and DH∩CH6=∅,
CH∩DT6=∅and DH∩CT6=∅. I Tg, hen
min
p∈CT∫u( )dp > max
p∈DT∫u(g)dp,
which implies
min
p∈CH∫u( )dp ≥min
p∈CT∫u( )dp > max
p∈DT∫u(g)dp ≥max
p∈DH∫u(g)dp,
max
p∈DH∫u( )dp ≥min
p∈CT∫u( )dp > max
p∈DT∫u(g)dp ≥max
p∈DH∫u(g)dp.
Since DH∩CH6=∅, one ge s
max
p∈DH∫u( )dp ≥min
p∈CH∫u( )dp > max
p∈DH∫u(g)dp ≥min
p∈CH∫u(g)dp.
The e o e, Hg. Thus, Tis mo e conse a i e han H.
36
Con e sely, suppose Tis mo e conse a i e han Hand suppose, by con adic ion,
ha he e exis s p∗∈CH CT. The e a e ∈ F and x∈Xsuch ha
min
p∈CT∫u( )dp > u(x)>∫u( )dp∗,
om which i ollows ha Txbu čHx, a con adic ion. To p o e ha DH⊆DT,
suppose he e exis s p∗∈DH DT. The e a e ∈ F and x∈Xsuch ha
∫u( )dp∗> u(x)>max
p∈DT∫u( )dp.
In his case, xT bu xčH , an o he con adic ion.
D.4 P oo o P oposi ion 2
Clea ly, o each i∈ {1,2}, and all x∈X, ixi and only i ip x, whe e ip is he
pessimis ic ela ion de ined, as in he p oo o Theo em 1, by ip gi and only i iy
and y’ig o some y∈X. Thus, 1is mo e ambigui y a e se han 2i and only i 1p
is mo e ambigui y a e se han 2p. When p o ing Theo em 1, we ha e shown ha ip is
ep esen ed by a maxmin expec ed u ili y unc ional; he e o e, 1pis mo e ambigui y a e se
han 2pi and only i C2⊆C1.
Simila ly, o each i∈ {1,2}, and all x∈X,xi i and only i xio , whe e io
is he op imis ic ela ion de ined, as in he p oo o Theo em 1, by io gi and only i
’iyand yig o some y∈X. As we ha e p o ed ha io admi s a maxmax expec ed
u ili y ep esen a ion, one ob ains ha 1is mo e ambigui y lo ing han 2i and only i
D2⊆D1.
D.5 P oo o Theo em 3
We will only p o e ha (i)implies (ii), he in e se implica ion being ou ine.
Lemma 6. A weak o de ela ion on Fsa is ies Axioms 2,3and 5i and only i he e
exis s a mono onic, cons an -linea unc ional I:B0(Σ) →Rand a non-cons an affine
unc ion u:X→Rsuch ha , o all , g ∈ F,
g⇐⇒ I(u( )) > I(u(g)).
Mo eo e , Iis unique and uis unique up o posi i e affine ans o ma ion.
37
P oo . As be o e, de ine Áby Ági and only i gč o all , g ∈ F. Clea ly, Á
is comple e and ansi i e, and ’is an equi alence ela ion (see Theo em 2.1 in Fishbu n
(1970)). The weak o de Áis con inuous i , o all , g, h ∈ F,{α∈[0,1] : α +(1−α)gÁh}
and {α∈[0,1] : hÁα + (1 −α)g}a e closed. Clea ly, Áis con inuous and non- i ial.
I is mono one i and only i , o all , g ∈ F, i (s)Ág(s) o all s∈S, hen Ág.
Since Lemma 4holds, in pa icula , o a weak o de o which he asymme ic pa sa is ies
Axioms 2,3and 5, and since ’is an equi alence ela ion, Áis mono one.
Now, we check ha ha Ása is ies ce ain y independence: o all , g ∈ F and x∈X,
Ág⇐⇒ gč
⇐⇒ αg + (1 −α)xčα + (1 −α)x
⇐⇒ α + (1 −α)xÁαg + (1 −α)x.
As a consequence, by Lemma 1 in Ghi a da o e al. (2004),47 he e exis s a mono onic,
cons an -linea unc ional I:B0(Σ) →Rand a non-cons an affine unc ion u:X→Rsuch
ha , o all , g ∈ F,
Ág⇐⇒ I(u( )) ≥I(u(g)).
Mo eo e , Iis unique and uis unique up o posi i e affine ans o ma ion.
Lemma 7. Suppose ha I, I′, I′′ :B0(Σ) →Ra e mono onic and cons an -linea wi h
I′≤I′′. Then he ollowing s a emen s a e equi alen :
(i) Fo all ϕ, φ ∈B0(Σ), i I′(ϕ)> I′(φ)and I′′(ϕ)> I′′(φ), hen I(ϕ)> I(φ).
(ii) The e exis s α∈[0,1] such ha , o all φ∈B0(Σ),I(φ) = αI′(φ) + (1 −α)I′′(φ).
P oo . The ollowing p oo closely ollows he p oo o Lemma A.3 o F ick e al. (2022). We
only p o e ha (i)implies (ii); he o he implica ion is easily checked. By (i), he e is an
inc easing unc ion W:{(I′(φ), I′′(φ)) : φ∈B0(Σ)} → Rsuch ha W(I′(φ), I′′(φ)) = I(φ).
Le φ∈B0(Σ) be such ha I′(φ) = I′′(φ) = k. We will show ha I(φ) = k. Since I′
and I′′ a e mono onic and cons an -linea , k+ε=I′(k+ε)> I′(φ)> I′(k−ε) = k−εand
k+ε=I′′(k+ε)> I′′(φ)> I′′(k−ε) = k−ε. Thus, by (i),k+ε=I(k+ε)> I(φ)>
I(k−ε) = k−ε. Le εcon e ge o 0, hen I(φ) = k. Thus, I(φ) = k, which implies ha
I(φ) = αI′(φ) + (1 −α)I′′(φ) o all α∈R.
47Axiom 2implies he “A chimedean axiom” in Ghi a da o e al. (2004).
38
Now, conside φ∈B0(Σ) such ha I′(φ)< I′′(φ). The e exis s α(φ)∈Rsuch ha
I(φ) = α(φ)I′(φ) + (1 −α(φ))I′′(φ). By a simple compu a ion, one ob ains
α(φ) = I(φ)−I′′(φ)
I′(φ)−I′′(φ)=−I(ϕ) = −W(I′(ϕ), I′′(ϕ)),
whe e ϕ=φ−I′′ (φ)
I′′ (φ)−I′(φ). Clea ly, I′(ϕ) = −1and I′′(φ) = 0. Thus, α(φ) = −W(−1,0), which
is independen o φ. Le α=−W(−1,0). Then, I(φ) = αI′(φ) + (1 −α)I′′(φ) o all
φ∈B0(Σ).
We now p o e ha α∈[0,1]. By con adic ion, assume ha α < 0. Fo any φ∈B0(Σ)
such ha I′(φ)< I′′(φ), we ha e I(φ)> I′′(φ). The e exis s ε > 0such ha I(φ)> I′′(φ)+ε.
Mo eo e , I′′(φ) + ε=I′(I′′(φ) + ε)> I′(φ)and I′′(φ) + ε=I′′(I′′(φ) + ε)> I′′(φ). By (i),
I′′(φ) + ε=I(I′′(φ) + ε)> I(φ), which is a con adic ion. Thus, α≥0. One can simila ly
show ha α≤1.
Assume ha is a hope-and-p epa e p e e ence and ∗is an in a ian bisepa able
ex ension o . Le u:X→Rbe a non-cons an affine unc ion, and le Cand Dbe wo
compac con ex subse s o ∆wi h C∩D6=∅such ha
g⇐⇒
minp∈C∫u( )dp > minp∈C∫u(g)dp
maxp∈D∫u( )dp > maxp∈D∫u(g)dp
.
F om he uniqueness esul o Theo em 1,uis unique up o posi i e affine ans o ma ion,
and Cand Da e unique.
I ollows om Lemma 6 ha he e exis a mono onic, cons an -linea unc ional I:
B0(Σ) →Rand a non-cons an affine unc ion u′:X→Rsuch ha , o all , g ∈ F,
∗g⇐⇒ I(u′( )) > I(u′(g)).
Mo eo e , Iis unique and u′is unique up o posi i e affine ans o ma ion.
I i ially ollows om he ex ension p ope y ha , o all x, y ∈X,u(x) = u(y)i and
only i u′(x) = u′(y), which implies ha uis a posi i e affine ans o ma ion o u′. Thus,
one can assume wi hou loss o gene ali y u=u′.
De ine I′:B0(Σ) →Rand I′′ :B0(Σ) →Rby I′(φ) = minp∈C∫φdp and I′′(φ) =
maxp∈D∫φdp o all φ∈B0(Σ). Clea ly, I′and I′′ a e mono onic, cons an -linea unc ion-
als; and since C∩D6=∅,I′′ ≥I′.
Now, le ϕ, φ ∈B0(Σ) such ha I′(ϕ)> I′(φ)and I′′(ϕ)> I′′(φ). We deno e by
B0(Σ, u(X)) he se o all unc ions in B0(Σ) ha ake alues in u(X). Since u(X)is an
39
