Symmetric reduced-form voting

Lang, Xu; Mish a, Debasis
A icle
Symme ic educed- o m o ing
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Lang, Xu; Mish a, Debasis (2024) : Symme ic educed- o m o ing, Theo e ical
Economics, ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 19, Iss. 2, pp. 605-634,
h ps://doi.o g/10.3982/TE5400
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Theo e ical Economics 19 (2024), 605–634 1555-7561/20240605
Symme ic educed- o m o ing
XuLang
Cen e o Economic Resea ch, Shandong Uni e si y
Debasis Mish a
Economics and Planning Uni , Indian S a is ical Ins i u e, Delhi
We s udy a model o o ing wi h wo al e na i es in a symme ic en i onmen .
We cha ac e ize he in e im alloca ion p obabili ies ha can be implemen ed by a
symme ic o ing ule. We show ha e e y such in e im alloca ion p obabili y can
be implemen ed as a con ex combina ion o wo amilies o de e minis ic o ing
ules: quali ied majo i y and quali ied an i-majo i y. We also p o ide analogous
esul s by equi ing implemen a ion by a symme ic mono one (s a egy-p oo )
o ing ule and by a symme ic unanimous o ing ule. We apply ou esul s o
show ha an ex an e Rawlsian ule is a con ex combina ion o a pai o quali ied
majo i y ules.
Keywo ds. Reduced- o m o ing, unanimous o ing, o dinal Bayesian incen i e
compa ibili y, mono one educed o m.
JEL classi ica ion.D82.
1. In oduc ion
In many mechanism design p oblems, he incen i e cons ain s and he objec i e unc-
ion o he designe can be w i en in he in e im alloca ion space. While a mechanism
desc ibes he ex pos alloca ion o he agen s, he solu ion o an incen i e cons ained
op imiza ion may desc ibe only in e im alloca ions. This aises a na u al ques ion,
“Which in e im alloca ions can be gene a ed by a (ex pos ) mechanism?” I he e is a
cha ac e iza ion o in e im alloca ions ha can be gene a ed by a mechanism, hen i
can be used as a cons ain in any incen i e cons ained op imiza ion. This app oach
o mechanism design is known as he educed- o m app oach. I was pionee ed in he
single objec auc ion li e a u e by Ma hews (1984)andMaskin and Riley (1984), leading
o he seminal cha ac e iza ion in Bo de ’s heo em (Bo de (1991)).
We analyze educed- o m o ing mechanisms in a simple model o o ing wi h wo
al e na i es: aand b. In ou model, each agen has wo possible ypes: (i) he a- ype
agen p e e s a ollowed by band (ii) he b- ype agen p e e s b ollowed by a.Wecon-
side a symme ic o ing en i onmen : he p obabili y o wo ype p o iles wi h he same
Xu Lang: [email p o ec ed]
Debasis Mish a: [email p o ec ed]
We a e g a e ul o A una a Sen, Zai u Yang, and wo anonymous e e ees o hei commen s. Xu
Lang hanks he Na ional Na u al Science Founda ion o China (NSFC72033004). Debasis Mish a
acknowledges inancial suppo om he Science and Enginee ing Resea ch Boa d (SERB G an No.
SERB/CRG/2021/003099) o India.
©2024 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/TE5400
606 Lang and Mish a Theo e ical Economics 19 (2024)
numbe o a ypes is iden ical. Hence, we ocus on symme ic o ing ules, which choose
a p obabili y dis ibu ion o e aand b o e e y numbe o a ypes. The in e im alloca-
ion p obabili y o choosing a(and b) o a- ype and b- ype agen s can be compu ed
om he symme ic o ing ule. The educed- o m o ing ques ion is, “Gi en he in-
e im alloca ion p obabili ies o choosing aand b o a- ype and b- ype agen s, is he e
a symme ic o ing ule ha can gene a e hese in e im alloca ion p obabili ies?”
We comple ely cha ac e ize hese in e im alloca ion p obabili ies, which we call
educed- o m implemen able symme ic o ing ules. The educed- o m implemen able
symme ic o ing ules a e cha ac e ized by a amily o 2(n+1)linea inequali ies, whe e
nis he numbe o agen s. The ex eme poin s o hese symme ic o ing ules a e (i) a
amily o (n+1)quali ied majo i y o ing ules and (ii) a amily o (n+1)quali ied an i-
majo i y o ing ules. A quali ied majo i y (an i-majo i y) o ing ule is cha ac e ized by
aquo aKand chooses al e na i e a( espec i ely, b) whene e a leas Kagen s o e o
a. As a co olla y, we show ha e e y symme ic o ing ule is educed- o m equi alen
(i.e., gene a ing he same in e im alloca ion p obabili ies) o a con ex combina ion o
quali ied majo i y and quali ied an i-majo i y o ing ules. Bo h hese amilies con ain
only de e minis ic o ing ules.
We ex end ou cha ac e iza ion o mono one o ing ules, i.e., o ing ules ha se-
lec awi h highe p obabili y as he numbe o a- ypes inc eases. Mono one o ing ules
a e s a egy-p oo (dominan s a egy incen i e compa ible). The educed- o m imple-
men able symme ic mono one o ing ules a e cha ac e ized by a amily o (n+2)
linea inequali ies. The ex eme poin s o hese ules a e he amily o (n+1)quali-
ied majo i y ules and a cons an ule ha selec s al e na i e ba all ype p o iles. We
use his esul o show ha an ex an e Rawlsian ule ( ha maximizes he minimum
o expec ed u ili y o a- ype agen s and b- ype agen s) is a con ex combina ion o a
pai o quali ied majo i y ules. We also in es iga e he educed- o m ques ion unde
a weake no ion o incen i e cons ain s, i.e., o dinal Bayesian incen i e compa ibili y
(OBIC) (d’Asp emon and Peleg (1988), Majumda and Sen (2004), Mish a (2016)). We
show i s connec ion o educed- o m implemen a ion by mono one o ing ules.
We ex end ou cha ac e iza ions o unanimous symme ic o ing ules: a o ing
ule is unanimous i i chooses a(b) whene e all he agen s ha e ype a( espec i ely, b).
Using his, we cha ac e ize he symme ic p io s o which OBIC is implied by symme y
and unanimi y. Fo independen p io s, his is he case when he p obabili y o an a
ype is su icien ly small o su icien ly high. I we allow o co ela ion (s ill main aining
symme y), he se o p io s whe e symme y and unanimi y imply OBIC con ains p io s
whe e ex eme ype p o iles wi h low and high numbe s o a ypes a e chosen wi h high
p obabili y.
We belie e ou esul s will be use ul in designing op imal mechanisms in a ious
models o o ing o e a pai o al e na i es. Indeed, Bo de ’s heo em is ex ensi ely used
in auc ion heo y and mechanism design o designing op imal auc ions wi h budge
cons ained bidde s (Pai and Voh a (2014)), o designing op imal e i ica ion mecha-
nisms (Ben-Po a h, Dekel, and Lipman (2014), Mylo ano and Zapechelnyuk (2017), Li
(2020,2021)), o designing symme ic auc ions (Deb and Pai (2017)), and so on. The
ad an age o using a educed o m app oach in mechanism design p oblems is ha i
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Theo e ical Economics 19 (2024) Symme ic educed- o m o ing 607
educes he dimensionali y o such p oblems. Fo ins ance, in he p oblem we s udy,
he educed o m is wo-dimensional, bu he (ex pos ) o ing ules a e n-dimensional,
whe e nis he numbe o agen s. Ou easy de i a ion o he ex an e Rawlsian ule illus-
a es his ad an age.
We gi e a de ailed e iew o he li e a u e in Sec ion 7, bu ela e ou esul s o Bo -
de ’s heo em he e. Conside Bo de ’s single objec alloca ion p oblem, bu whe e each
agen has wo ypes (possible alues o he objec ), {0, 1}. This is analogous o ou p ob-
lem whe e he e a e wo ypes: aand b. Howe e , he o ing p oblem in he cu en
pape is a public good p oblem: he p obabili y o choosing aand bis hesameac oss
all he agen s. The single objec alloca ion p oblem is a p i a e good p oblem whe e
he p obabili y o choosing aand bmay di e ac oss agen s. This makes he easibili y
cons ain s o alloca ion ules di e en in bo h p oblems.
Goe ee and Kushni (2023) use a geome ic app oach (using suppo unc ions o
con ex se s) o s udy implemen a ion in social choice p oblems. Thei abs ac o -
mula ion also cap u es ou p oblem and hei esul s can be used o desc ibe he sup-
po unc ions o ou educed- o m o ing ules. Bu his nei he desc ibes he ex eme
poin s no he necessa y and su icien condi ions ha cha ac e ize he educed- o m
o ing ules.1Indeed, i is no clea ha an analogue o Bo de ’s heo em can exis in he
o ing p oblem. In an impo an pape , Gopalan, Nisan, and Roughga den (2018)show
ha in a simple public good model wi h wo al e na i es, no compu a ionally ac able
cha ac e iza ion o educed- o m alloca ion ules is possible. Though his nega i e e-
sul applies o ou model, hey allow educed- o m implemen a ion ia asymme ic
mechanisms. By looking only a symme ic mechanisms, we o e come his impossi-
bili y: ou cha ac e iza ion admi s a compu a ionally ac able desc ip ion o educed-
o m p obabili ies by a sys em o (linea in numbe o o e s) linea inequali ies.
The es o he pape is o ganized as ollows. Sec ion 2in oduces he model. Sec-
ion 3p o ides he main esul o he pape : a cha ac e iza ion o he educed- o m
implemen able o ing ules. Sec ion 4ex ends he main esul by equi ing mono one
implemen a ion and p o ides an applica ion o inding a Rawlsian o ing ule. The
main cha ac e iza ion is ex ended wi h unanimi y in Sec ion 5and ex ended o la ge
economies in Sec ion 6.Sec ion7gi es a de ailed li e a u e e iew. The missing p oo s
a e p o ided in he Appendix.
2. The model
Le N={1, ,n}be a ini e se o agen s ( o e s), whe e n≥2. Le A={a,b}be he se
o wo social al e na i es ( o ins ance, a s a us quo and a new al e na i e). Each agen
has a s ic anking o A. Hence, he p e e ence o an agen can be exp essed by he op
anked al e na i e. We call his he ype o he agen . The ype o agen iis deno ed as
i∈{a,b}, which means ha iis he op anked al e na i e o agen i.Hence, hese o
all ypes ( ype space) is Aand he se o all ype p o iles is An. A ype p o ile in Anis
deno ed by ≡( 1,, n).
1They u he assume independen p io s, which we do no assume. They use hei suppo unc ion
cha ac e iza ion o ede i e Bo de ’s esul .
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608 Lang and Mish a Theo e ical Economics 19 (2024)
Exchangeable p io Le Gbe a p obabili y dis ibu ion o e ype p o iles. We assume G
o be exchangeable, i.e., o e e y ype p o ile and e e y pe mu a ion σ,G( )=G( σ),
whe e σis he pe mu ed ype p o ile. In his sense, he p obabili y o a ype p o ile
is only a unc ion o he numbe o agen s ha ing ype a.So, o e e yk∈{0, ,n},
o any se o kagen s, he p obabili y ha exac ly hese agen s ha e ype a(and o he
agen s ha e ype b)isgi enbyλ(k). By exchangeabili y, he p obabili y ha a ype p o-
ile has exac ly kagen s o ype ais C(n,k)λ(k),whe eC(n,k)deno es he numbe o k
combina ions om a se o nelemen s.
We deno e he ma ginal p obabili y o any agen ha ing ype aas πand ha ing ype
bas (1−π).
Vo ing ule A o ing ule is a map q:An→[0, 1],whe eq( )deno es he p obabili y
wi h which al e na i e ais chosen (and, hence, 1 −q( )is he p obabili y wi h which
al e na i e bis chosen) a ype p o ile . We conside only symme ic o anonymous
o ing ules, i.e., o any pe mu a ion σ,wewill equi eq( )=q( σ) o all ∈An,whe e
σis ype p o ile ob ained by pe mu ing using he pe mu a ion σ. Wi h a sligh abuse
o no a ion, we will w i e qas a map q:{0, 1, ,n}→[0, 1], i.e., q(k)∈[0, 1]deno es he
p obabili y wi h which al e na i e ais chosen a any ype p o ile wi h k o es o a.2We
discuss only symme ic o ing ules, and whene e we e e o a o ing ule om now
on, we mean a symme ic o ing ule.
Gi en a o ing ule q, we can compu e he in e im p obabili y o each al e na i e
being chosen. I an agen has ype a, he p obabili y ha al e na i e ais chosen by o ing
ule qis deno ed by Q(a). To ela e Qand q, deno e he p obabili y ha he e a e k
agen s o ype aas
B(k):=λ(k)C(n,k)∀k∈{0, ,n}.
No e ha
n

k=0
B(k)=1and
n

k=0
kB(k)=nπ.
The second equali y ollows because bo h nπ and kkB(k)deno e he expec ed num-
be o agen s who ha e ype a.
Using his, Qcan be compu ed om qas
nπQ(a)=
n

k=0
kq(k)B(k),
2We es ic ou sel es o o dinal o ing ules. Any ca dinal o ing ule in a wo al e na i e model mus
be o dinal i i is incen i e compa ible (Majumda and Sen (2004)). Since educed o ms a e usually used
along wi h incen i e cons ain s, es ic ing a en ion o o dinal o ing ules is wi hou loss o gene ali y in
his sense. E en wi hou incen i e cons ain s, Schmi z and T öge (2012) and Az ieli and Kim (2014)show
ha es ic ing a en ion o o dinal o ing ules is wi hou loss o gene ali y i he planne is op imizing o e
in e im u ili ies o agen s.
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Theo e ical Economics 19 (2024) Symme ic educed- o m o ing 609
whe e bo h he le -hand side and he igh -hand side compu e he expec ed numbe o
a ypes who ge a.Hence,
Q(a)=1
nπ
n

k=0
kq(k)B(k).
Simila ly, i an agen has ype b, he p obabili y ha al e na i e ais chosen by o ing ule
qis
Q(b)=1
n(1−π)
n

k=0
(n−k)q(k)B(k).
O cou se, 1 −Q(a)and 1 −Q(b)deno e he in e im p obabili ies wi h which al e na i e
bis chosen o ypes aand b, espec i ely.
3. Reduced- o m implemen a ion
The in e im alloca ion p obabili ies a e wo-dimensional. Hence, hey a e easy o wo k
wi h. Some in e im alloca ion p obabili ies a e clea ly no possible: o ins ance, Q(a)=
1andQ(b)=0 is impossible o n≥2 because any o ing ule o which Q(a)=1mus
choose aa some p o iles whe e o he agen s ha e ype b. By symme y, Q(b)=0. Then
he educed- o m ques ion is, “Wha in e im alloca ion p obabili ies a e possible?”
De ini ion 1. In e im alloca ion p obabili ies Q≡(Q(a),Q(b)) ∈[0, 1]2a e educed-
o m implemen able i he e exis s a o ing ule qsuch ha
1
nπ
n

k=0
kq(k)B(k)=Q(a)
1
n(1−π)
n

k=0
(n−k)q(k)B(k)=Q(b)
0≤q(k)≤1∀k∈{0, ,n}.
To see wha kind o condi ions a e necessa y o educed- o m implemen a ion,
conside he ollowing se ing. Suppose he e is a cos j∈{0, 1, ,n}o choosing al-
e na i e abu al e na i e bcos s ze o. Fo any a- ype agen , suppose he alue o al e -
na i e ais 1 and ha o al e na i e bis 0. The expec ed alue o a- ypes minus he cos
o choosing an al e na i e om a o ing ule qis
n

k=0
(k−j)q(k)B(k)=1
n(n−j)
n

k=0
kq(k)B(k)−j
n

k=0
(n−k)q(k)B(k)
=(n−j)πQ(a)−j(1−π)Q(b).(1)
The le -hand side o (1) is maximized by se ing q(k)=0i k<jand q(k)=1i
k≥j. Hence, an uppe bound o he le -hand side o (1)isn
k=j(k−j)B(k). Simila ly,
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610 Lang and Mish a Theo e ical Economics 19 (2024)
he le -hand side o (1) is minimized by se ing q(k)=1i k<jand q(k)=0i k≥j.
Hence, a lowe bound o he le -hand side o (1)isj
k=0(k−j)B(k).Thus, o any
j∈{0, 1, ,n},
n

k=j
(k−j)B(k)≥(n−j)πQ(a)−j(1−π)Q(b)≥
j

k=0
(k−j)B(k).(2)
So he inequali ies (2) a e necessa y o educed- o m implemen a ion. Ou main esul
says hey a e su icien .
Theo em 1. In e im alloca ion p obabili ies Qa e educed- o m implemen able i and
only i
j(1−π)Q(b)−(n−j)πQ(a)+
n

k=j
(k−j)B(k)≥0∀j∈{0, ,n}(3)
(n−j)πQ(a)−j(1−π)Q(b)+
j

k=0
(j−k)B(k)≥0∀j∈{0, ,n}.(4)
The su iciency pa o he p oo o Theo em 1and o he esul s a e p o ided in Ap-
pendix. I is p o ed by i s desc ibing he ex eme poin s o all educed- o m imple-
men able o ing ules (Theo em 2) and hen showing ha he ex eme poin s o he
sys em (3)and(4) co espond o exac ly he same o ing ules.
The educed- o m implemen able o ing ules a e desc ibed by 2(n+1)inequali ies,
ou o which ou co espond o nonnega i i y o Q(a)and Q(b), and uppe bounding
o Q(a)and Q(b)by 1. The es o he 2(n−1)inequali ies es ic he space o in-
e im alloca ion p obabili ies in he uni squa e. To see his, conside he uni o m p io
(independen p io ) wi h π=1
2and n=3. In his case, (Q(a),Q(b)) is educed- o m
implemen able i and only i
2Q(a)−Q(b)≤5
4,Q(a)−2Q(b)≤1
4,Q(b)−2Q(a)≤1
4,
2Q(b)−Q(a)≤5
4,Q(a),Q(b)∈[0, 1].
The poly ope enclosed by hese inequali ies is shown in Figu e 1. The e a e eigh ex-
eme poin s o his poly ope, wo o which co espond o he cons an alloca ion ules
((0, 0)co espond o balways chosen and (1, 1)co espond o aalways chosen). The
es o hem belong o a amily o o ing ules ha we call quali ied majo i y and quali-
ied an i-majo i y. We es ablish his esul nex . This allows us o show ha any educed-
o m implemen able o ing ule is “equi alen ” o a con ex combina ion o o ing ules
om his se .
De ini ion 2. Two o ing ules qand ˆ
qa e educed- o m equi alen i hey gene a e
he same in e im alloca ion p obabili ies: Q(a)=
Q(a)and Q(b)=
Q(b).
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Theo e ical Economics 19 (2024) Symme ic educed- o m o ing 611
Figu e 1. Poly ope o educed- o m implemen able o ing ules.
We now in oduce wo classes o o ing ules ha a e use ul o desc ibe he ex eme
poin s o educed- o m implemen able o ing ules.
De ini ion 3. A o ing ule q+is a quali ied majo i y i he e exis s j∈{0, ,n}such
ha o all k∈{0, ,n},
q+(k)=1i k≥j
0o he wise.
We call such a o ing ule a quali ied majo i y wi h quo a j.
A o ing ule q−is quali ied an i-majo i y i he e exis s j∈{0, ,n}such ha o all
k∈{0, ,n},
q−(k)=1i k<j
0o he wise.
We call such a o ing ule a quali ied an i-majo i y wi h quo a j.
The de ini ion o quali ied majo i y is simila o Az ieli and Kim (2014). The only
di e ence is ha i he quo a is j, hey allow q+(j) o ake any alue in [0, 1],bu we
b eak he ie de e minis ically.
I qjis a quali ied majo i y wi h quo a j, hen i s educed- o m p obabili ies a e
Qj(a)=1
nπ
n

k=0
kqj(k)B(k)=1
nπ
n

k=j
kB(k)
Qj(b)=1
n(1−π)
n

k=0
(n−k)qj(k)B(k)=1
n(1−π)
n

k=j
(n−k)B(k).
No ice ha when j=0, we ha e Q0(a)=Q0(b)=1. This co esponds o he cons an
o ing ule whe e ais chosen a e e y ype p o ile.
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612 Lang and Mish a Theo e ical Economics 19 (2024)
I ¯
qjis a quali ied an i-majo i y wi h quo a j, hen i s educed- o m p obabili ies a e
Qj(a)=1
nπ
n

k=0
k¯
qj(k)B(k)=1
nπ
j−1

k=0
kB(k)
Qj(b)=1
n(1−π)
n

k=0
(n−k)¯
qj(k)B(k)=1
n(1−π)
j−1

k=0
(n−k)B(k).
Deno e he se o all quali ied majo i y o ing ules by Q+and deno e he se o all
quali ied an i-majo i y o ing ules by Q−. No ice ha when j=0, we ha e Q0(a)=
Q0(b)=0. This co esponds o he cons an o ing ule whe e bis chosen a e e y ype
p o ile. Hence, Q+∪Q−con ains he wo cons an o ing ules.
Theo em 2. E e y symme ic o ing ule is educed- o m equi alen o a con ex combi-
na ion o o ing ules in Q+∪Q−.
We compa e ou esul s o some o he esul s in Az ieli and Kim (2014). They
conside a ca dinal o ing model wi h wo al e na i es, whe e he ype o an agen (a
one-dimensional numbe wi h ini e suppo ) gi es ca dinal u ili ies o wo al e na i es.
They conside ca dinal o ing ules and Bayesian incen i e compa ibili y (BIC). They
ha e wo main esul s wi h symme ic ca dinal o ing ules: (a) a u ili a ian maximize
in he class o symme ic BIC ules is a quali ied majo i y; (b) an in e im e icien and
symme ic BIC ule is a quali ied majo i y.3
While ela ed, hei esul s and ou esul s a e no compa able. Fi s , we conside
only o dinal o ing ules, while hey allow o ca dinal ules. Second, he ypes o agen s
in hei model a e independen , while we allow o co ela ed ypes; exchangeable dis-
ibu ions allow o co ela ion.
Thi d, Theo em 2says ha he ex eme poin s o he se o educed- o m imple-
men able o ing ules consis o quali ied majo i y and quali ied an i-majo i y ules. We
do no equi e incen i e compa ibili y o any addi ional axiom (like in e im e iciency)
o his esul . In he nex sec ion, we impose mono onici y (equi alen o dominan
s a egy incen i e compa ibili y) o o ing ules, and show ha he he ex eme poin s
o he se o mono one educed- o m implemen able o ing ules consis o quali ied
majo i y ules and a cons an ule. As we discuss in Sec ion 4.1, ou esul s a e use ul in
se ings whe e he objec i e unc ion o he planne is no linea .
Finally, we explo e he consequences o imposing unanimi y on he educed- o m
implemen a ion in Sec ion 5. Unanimi y is a much weake axiom han in e im e iciency
used in Az ieli and Kim (2014). Theo em 5desc ibes he ex eme poin s o educed- o m
implemen able ules sa is ying unanimi y and his con ains ules ha a e no quali ied
majo i y.
3They ha e analogues o hese esul s wi hou symme y oo. A weigh ed majo i y ule is in e im e icien
and BIC. Simila ly, a weigh ed majo i y ule is a u ili a ian maximize in he class o BIC ules.
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Theo e ical Economics 19 (2024) Symme ic educed- o m o ing 619
P oposi ion 2. E e y unanimous and symme ic o ing ule is OBIC i and only i
λ(j)≤minλ(1)+λ(n)
C(n−1, j−1),λ(0)+λ(n−1)
C(n−1, j)∀j∈{1, ,n−1}. (12)
Fu he , i he p io is independen , e e y unanimous and symme ic o ing ule is OBIC
i and only i
C(n−1, j−1)≤ π
1−πn−j
+π
1−π1−j∀j∈{1, ,n−1}. (13)
Using Co olla y 1, we can a gue ha when (13) holds and he p io is independen ,
e e y unanimous o ing ule is educed- o m equi alen o a s a egy-p oo o ing ule.
An immedia e co olla y o he abo e esul is ha when he e is a small numbe o
agen s, e e y unanimous o ing ule is OBIC i he p io is independen .
Co olla y 2. I he p io is independen and n=3, e e y unanimous and symme ic
o ing ule is OBIC.
P oo . Since π∈(0, 1),j∗=3π≤2. I j∗=1, we ge
B(1)=3π(1−π)2≤3ππ2+(1−π)2.
I j∗=2, we ge
B(2)=3π2(1−π)=3π
22π(1−π)≤3π
2π2+(1−π)2.
Hence, by P oposi ion 2, e e y unanimous o ing ule is OBIC.
To illus a e P oposi ion 2, suppose n=4. The condi ion (12)isgi enby
3λ(2)≤λ(1)+λ(4)
3λ(3)≤λ(1)+λ(4)
3λ(1)≤λ(0)+λ(3)
3λ(2)≤λ(0)+λ(3).
No ice ha o independen uni o m p io s, λ(k)=(1
2)4, he belie condi ions ail. Fo
su icien ly posi i ely co ela ed belie s whe e λ(0)and λ(4)a e la ge, he belie con-
di ions hold. This is in gene al ue. I λ(0)and λ(n)a e su icien ly la ge, (12)holds.
Simila ly, i λ(0)and λ(1)(o , λ(n−1)and λ(n)) a e su icien ly la ge, (12)holds.
6. La ge economies
In his sec ion, we apply ou esul s o la ge economies. Fo his, we assume indepen-
den and iden ically dis ibu ed ypes. So πdeno es he p obabili y ha an agen is a
ype. Le μ:=nπ deno e he mean o he binomial dis ibu ion.
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620 Lang and Mish a Theo e ical Economics 19 (2024)
The e a e wo ways in which we inc ease he alue o n. Fi s , we ix he alue o
πand inc ease n. This implies ha he expec ed numbe o a ypes (μ) also inc eases.
Second, we ix he expec ed numbe o a ypes a μand inc ease n. This implies ha he
alue o πdec eases wi h inc easing n. We show he implica ion o la ge non he se o
educed- o m implemen able o ing ules in bo h he cases.
Since nis a iable in his sec ion, o an a bi a y o ing ule, we deno e he in e im
alloca ion p obabili ies as (Q(a;n),Q(b;n)).Fo a ixedπand n, he in e im alloca-
ion p obabili ies co esponding o quali ied majo i y and an i-quali ied majo i y o -
ing ules will be use ul o ou analysis. In pa icula , pick a quali ied majo i y o ing
ule wi h quo a j>0.5Fo such a quali ied majo i y, he in e im alloca ion p obabili ies
sa is y
Qj(a;n)−Qj(b;n)=1
nπ
n

k=j
kB(k)−1
n(1−π)
n

k=j
(n−k)B(k)
=1
nπ
n

k=j
kC(n,k)πk(1−π)(n−k)
−1
n(1−π)
n

k=j
(n−k)C(n,k)πk(1−π)(n−k)
=
n

k=j
C(n−1, k−1)πk−1(1−π)(n−k)
−
n

k=j
C(n−1, k)πk(1−π)(n−k−1)
=C(n−1, j−1)πj−1(1−π)(n−j). (14)
Simila ly, o a quali ied an i-majo i y wi h quo a j>0, he in e im alloca ion p obabil-
i ies sa is y
Qj(b;n)−Qj(a;n)=C(n−1, j−1)πj−1(1−π)(n−j). (15)
This can also be seen om he ac ha o a ixed quo a j, he quali ied majo i y and
he quali ied an i-majo i y in e im alloca ion p obabili ies a e ela ed as Qj(a;n)=1−
Qj(a;n)and Qj(b;n)=1−Qj(b;n).
Depending on whe he we inc ease n o a ixed πo ixed μ, he igh -hand side
o (14)(and(15)) beha es di e en ly. In he o me case, i is app oxima ely equal o a
no mal dis ibu ion wi h anishing alues o densi y. In he la e case, i is ela ed o he
Poisson dis ibu ion. This leads o di e en con e gence esul s in hese cases.
5Quali ied majo i y wi h quo a j=0 co esponds o he cons an o ing ule whe e ais chosen a e e y
ype p o ile.
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Theo e ical Economics 19 (2024) Symme ic educed- o m o ing 621
P oposi ion 3. Suppose πis ixed and π∈(0, 1).Then, o e e y>0, he eexis sn0
such ha o e e y n-agen economy wi h n>n
0, i in e im alloca ion (Q(a,n),Q(b,n))
is educed- o m implemen able, hen
Q(a;n)−Q(b;n)<.
P oposi ion 3says ha in la ge economies, he only educed- o m implemen able
p obabili ies a e hose whe e Q(a;n)=Q(b;n).6I he numbe o agen s is la ge, he
in e im alloca ion p obabili ies ( o any o ing ule) a e less sensi i e o he ype o he
agen . Hence, bo h a ypes and b ypes ge he same in e im alloca ion p obabili ies wi h
la ge n.
Howe e , his is no he case i he economies become la ge wi h a ixed μ.I μ
is ixed, inc easing ndec eases π, so he p obabili y o a ypes dec eases, i.e., b ypes
domina e he economy. As a esul , depending on how sensi i e a o ing ule is o he
numbe o b ypes (o a ypes), we may ge qui e di e en in e im alloca ion p obabili-
ies Q(a;n)and Q(b;n). Fo ins ance, conside he simple ule ha chooses bwhen all
agen s ha e b ype and chooses ao he wise. Then, i an agen has a ype, he ule mus
choose Q(a;n)=1, bu i an agen has b ype, he ule chooses bi all o he (n−1)agen s
ha e b ype. Fo a ixed μ, he p obabili y ha a gi en agen has b ype is 1 −(μ/n),so
he p obabili y ha (n−1)agen s ha e b ype is (1−(μ/n))n−1, which con e ges o e−μ
o la ge n.So, o la gen,weha eQ(b;n)=1−e−μand Q(a;n)−Q(b;n)=e−μ>0.
The p oposi ion below uses a sligh ly mo e sophis ica ed o ing ule o come up wi h an
imp o ed bound on Q(a;n)−Q(b;n).
P oposi ion 4. Suppose μis ixed. Then he e is a posi i e cons an M(μ)such ha o
e e y >0, he eexis sn0such ha o e e y n-agen economy wi h n>n
0, he ollowing
s a emen s hold:
(i) In e im alloca ion p obabili ies (Q(a,n),Q(b,n)) exis ha a e educed- o m im-
plemen able and
Q(a;n)−Q(b;n)>M(μ)−.
(ii) In e im alloca ion p obabili ies (
Q(a;n),
Q(b;n)) exis ha a e educed- o m im-
plemen able and

Q(b;n)−
Q(a;n)>M(μ)−.
Combining P oposi ions 3and 4, and Co olla y 1, we conclude ha e e y educed-
o m implemen able ule is s a egy-p oo in a la ge economy o he ixed π, bu his is
no he case i μis ixed.
6Fo co ela ed p io s, i is well known ha he cen al limi heo em does no hold in gene al. Howe e ,
we conjec u e ha P oposi ion 3con inues o hold o he case o in ini e exchangeable p io s, whe e we
say an in ini e sequence X1,X2,X3, o andom a iables is exchangeable i o any ini e n, hejoin
p obabili y dis ibu ion o (X1,X2,,Xn)is hesameas ha o (Xσ(1),Xσ(2),,Xσ(n)) o any pe mu-
a ion σ.
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622 Lang and Mish a Theo e ical Economics 19 (2024)
7. Rela ion o he li e a u e
Bo de ’s heo em o single objec alloca ion p oblem was o mula ed in Ma hews
(1984)andMaskin and Riley (1984). The educed- o m cha ac e iza ion o his p ob-
lem was de eloped in Bo de (1991). The symme ic e sion o Bo de ’s heo em wi h an
elegan p oo using he Fa kas lemma wass de eloped in Bo de (2007). The e a e o he
app oaches o p o ing Bo de ’s heo em (which also makes i applicable in some con-
s ained en i onmen ): he ne wo k low app oach in Che, Kim, and Mie endo (2013)
and he geome ic app oach in Goe ee and Kushni (2023). Ha and Reny (2015)p o-
ide an equi alence cha ac e iza ion o Bo de ’s heo em using second-o de s ochas ic
dominance. Kleine , Moldo anu, and S ack (2021) u he de elop he majo iza ion
app oach and apply i o a a ie y o p oblems in economics. Bo de ’s heo em ap-
plies o p i a e alues single objec auc ions, bu Goe ee and Kushni (2016)ex endBo -
de ’s heo em o allow o alue in e dependencies. Zheng (2024) gene alizes educed-
o m cha ac e iza ions o alloca ion o mul iple objec s wi h pa amodula cons ain s.
Lang and Yang (2023) s udy a uni e sal implemen a ion o alloca ion o mul iple ob-
jec s. Yang (2021) conside s he consequences o inco po a ing ai ness cons ain s in
he educed- o m p oblem. Lang (2022) conside s a public good alloca ion p oblem bu
wi h only wo agen s (bu mul iple al e na i es). He p o ides an ex ension o Bo de ’s
heo em o his wo-agen p oblem. Ou o dinal o ing model o e wo al e na i es is a
public good model wi h a speci ic ype space, which is no co e ed in hese pape s.
Voh a (2011) s udies he combina o ial s uc u e o educed- o m auc ions by he
polyma oid heo y; see also Che, Kim, and Mie endo (2013), Alaei, Fu, Haghpanah,
Ha line, and Malekian (2019), and Zheng (2024). Ou cha ac e iza ion condi ion sha es
some simila i y wi h a polyma oid as i equi es only in ege - alued coe icien s in lin-
ea inequali ies. A he same ime, i di e s om a polyma oid in ha he inequali ies
con ain no only 0, 1 coe icien s bu mo e gene al in ege coe icien s.
The wo al e na i es o ing model has ecei ed a en ion in he li e a u e in social
choice heo y— om May’s heo em (May (1952)) o i s ex ensions, including a ecen
ex ension by Ba holdi, Hann-Ca u he s, Josyula, Tamuz, and Ya i (2021). Schmi z and
T öge (2012)iden i y quali ied majo i y ules as ex an e wel a e maximizing in he class
o dominan s a egy o ing ules. The esul s in Az ieli and Kim (2014) (which we dis-
cussed ea lie ) show ha ocusing a en ion o o dinal ules in his model is wi hou loss
o gene ali y in a ce ain sense; see Neh ing (2004)also.
Appendix:Missing p oo s
We i s p o e Theo em 2and hen Theo em 1.
A.1 P oo o Theo em 2
Reduced- o m p obabili ies (Q(a),Q(b)) a e implemen able i
1
nπ
n

k=0
kq(k)B(k)=Q(a)(16)
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Theo e ical Economics 19 (2024) Symme ic educed- o m o ing 623
1
n(1−π)
n

k=0
(n−k)q(k)B(k)=Q(b)(17)
0≤q(k)≤1∀k∈{0, 1, ,n}. (18)
Le Pbe he p ojec ion o his poly ope on o he (Q(a),Q(b)) space. Clea ly, Pis a
poly ope. Conside he linea p og am
max
QμaQ(a)+μbQ(b)
subjec o Q(a),Q(b)∈P.(LP-Q)
As we a y μaand μb, he solu ions o he linea p og am p og am (LP-Q)cha ac e ize
he bounda y poin s o P. Since each poin in Pis equi alen o inding a o ing ule q
ha sa is ies (16), (17), and (18), we can ew i e he linea p og am (LP-Q) in he space
o qas
max
qμa
nπ
n

k=0
kq(k)B(k)+μb
n(1−π)
n

k=0
(n−k)q(k)B(k)
subjec o 0 ≤q(k)≤1∀k∈{0, 1, ,n}. (LP-q)
Hence, he se o bounda y poin s o Pcan be desc ibed by he in e im alloca ion p ob-
abili ies o he o ing ules ob ained as a solu ion o he linea p og am (LP-q)aswe a y
μaand μb.
We now do he p oo in wo s eps.
S ep 1. We i s show ha e e y ex eme poin o Pis implemen ed by ei he a qual-
i ied majo i y o ing ule o a quali ied an i-majo i y o ing ule, i.e., e e y elemen o P
can be w i en as a con ex combina ion o quali ied (an i-) majo i y o ing ules.
I is su icien o show ha o e e y μaand μb, he e is a solu ion o (LP-Q) ha is
implemen ed by ei he a quali ied majo i y o a quali ied an i-majo i y o ing ule. To
show his, we show ha o e e y μaand μb, some quali ied (an i-) majo i y o ing ule
is a solu ion o (LP-q).
By deno ing ˆμa:=μa/(nπ)and ˆμb:=μb/(n(1−π)), we see ha he objec i e unc-
ion o (LP-q)is
n

k=0nˆμb+k(ˆμa−ˆμb)q(k)B(k).
We show ha nˆμb+k(ˆμa−ˆμb)is ei he weakly inc easing, in which case some quali ied
majo i y o ing ule is op imal, o weakly dec easing, in which case some quali ied an i-
majo i y o ing ule is op imal.
I nˆμb+k(ˆμa−ˆμb)>0 o allk, hen a solu ion o (LP-q)is ose q(k)=1 o all
k. This is he quali ied majo i y wi h quo a 0. I nˆμb+k(ˆμa−ˆμb)<0 o allk, hena
solu ion o (LP-q)is ose q(k)=0 o allk. This is he quali ied an i-majo i y wi h quo a
0. I nˆμb+k(ˆμa−ˆμb)=0 o allk, hen e e y o ing ule qis a solu ion.
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624 Lang and Mish a Theo e ical Economics 19 (2024)
I he sign o nˆμb+k(ˆμa−ˆμb)changes wi h k, hen we conside wo cases. I ˆμa>ˆμb,
hen he e is a cu o k∗such ha nˆμb+k(ˆμa−ˆμb)>0 o allk≥k∗and nˆμb+k(ˆμa−
ˆμb)<0 o allk<k
∗. Then he quali ied majo i y wi h quo a k∗is a solu ion o (LP-q).
On he o he hand, i ˆμa<ˆμb, hen he e is a cu o k∗such ha nˆμb+k(ˆμa−ˆμb)>0 o
all k≤k∗and nˆμb+k(ˆμa−ˆμb)<0 o allk>k
∗. Then he quali ied an i-majo i y wi h
quo a k∗is a solu ion o (LP-q).7No e ha in bo h cases abo e, i nˆμb+k(ˆμa−ˆμb)=0
o k=k∗, he (an i-) quali ied majo i y wi h quo a k∗is a solu ion o (LP-q).
S ep 2. We now show ha e e y quali ied (an i-) majo i y o ing ule implemen s a
dis inc ex eme poin o P. E e y ex eme poin in Pis ob ained by conside ing alues
o μaand μb ha gene a e a unique op imal solu ion o he linea p og am (LP-Q). I
is su icien o show ha e e y quali ied (an i-) majo i y o ing ule is unique op imal
solu ion o (LP-q) o someμaand μb. This is easily seen om ou analysis abo e ha
o almos all μaand μb,incaseanop imalsolu ion o(LP-q) exis s, i is unique and
co esponds o a quali ied majo i y o a quali ied an i-majo i y o ing ule.
Combining S eps 1 and 2, we see ha he se o ex eme poin s o Pis he se o
quali ied majo i y o ing ules and he se o quali ied an i-majo i y o ing ules.
A.2 P oo o Theo em 1
We know ha he necessa y condi ions o educed- o m implemen a ion a e (3)and
(4). Le P∗deno e he poly ope desc ibed by (3)and(4). We show ha he ex eme
poin s o P∗co espond o he quali ied majo i y and he quali ied an i-majo i y o ing
ules. F om Theo em 2, we know ha he ex eme poin s o Palso co espond o he
quali ied majo i y and he quali ied an i-majo i y o ing ules. Hence, P=P∗.
To show ha he ex eme poin s o P∗co espond o he quali ied majo i y and he
quali ied an i-majo i y o ing ules, we ollow wo s eps.
S ep 1: E e y q∈Q+∪Q−is an ex eme poin . Conside any quali ied majo i y o ing
ule wi h quo a j∈{1, ,n}. Using
nπQj(a)=
n

k=j
kB(k)and n(1−π)Qj(b)=
n

k=j
(n−k)B(k),
i is easy o e i y ha Qjsa is ies all inequali ies in (3)and(4), and inequali y (3)is
binding o jand (j−1)a Qj. Since Qj∈P∗and Qjis he in e sec ion o wo linea ly
independen hype planes, i gi es an ex eme poin o P∗. Since he quali ied majo i y
o ing ule wi h quo a 0 co esponds o a cons an o ing ule, i is also an ex eme poin .
An analogous a gumen shows ha he in e im alloca ion p obabili y o e e y qual-
i ied an i-majo i y o ing ule wi h a quo a j∈{0, ,n}is an ex eme poin .
S ep 2: No ex eme poin ou side Q+∪Q−. Conside an ex eme poin o P∗ ha is no a
quali ied (an i-) majo i y ule. Then wo non-adjacen cons ain s mus be binding, i.e.,
7When ˆμa=ˆμb, hesigno nˆμb+k(ˆμa−ˆμb)does no change wi h k.
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Theo e ical Economics 19 (2024) Symme ic educed- o m o ing 625
ei he (3)binds o somejand j+wi h >1, o (4)binds o somejand j+wi h
>1, o (3) binds o some jand (4)binds o some.
Assume i s ha (3)binds o jand j+,whe e>1. The equali y co esponding
o (j+)is
0=(j+)(1−π)Q(b)−(n−j−)πQ(a)+
n

k=j++1
(k−j−)B(k)
=πQ(a)+(1−π)Q(b)+j(1−π)Q(b)−(n−j)πQ(a)
+
n

k=j++1
(k−j)B(k)−
n

k=j++1
B(k).
Since inequali y (3)binds o j, subs i u e he equali y in o (3) o j+1,
πQ(a)+(1−π)Q(b)≥
n

k=j+1
B(k).
We ge
0≥
n

k=j+1
B(k)−
n

k=j++1
B(k)+
n

k=j++1
(k−j)B(k)−
n

k=j+1
(k−j)B(k)
=
j+

k=j+1
B(k)−
j+

k=j+1
(k−j)B(k)=
j+

k=j+1
(j+−k)B(k)>0,
which is a con adic ion. Hence, (3) canno bind o jand (j+) o >1. An analogous
p oo shows ha (4) canno bind o jand (j+) o >1.
Now assume (3) binds o jand (4) binds o . Hence, adding hose wo equali ies,
we ge
0=(j−)(1−π)Q(b)+(j−)πQ(a)+
−1

k=0
(−k)B(k)+
n

k=j+1
(k−j)B(k).
I j≥and (j,)= (n,0
), he igh -hand side is posi i e, gi ing us a con adic ion. I
j<and (j,)=(0, n), using πQ(a)+(1−π)Q(b)≤1, we ge
0=(j−)(1−π)Q(b)+πQ(a)+
−1

k=0
(−k)B(k)+
n

k=j+1
(k−j)B(k)
≥j−+
−1

k=0
(−k)B(k)+
n

k=j+1
(k−j)B(k)
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626 Lang and Mish a Theo e ical Economics 19 (2024)
=j1−
n

k=j+1
B(k)−1−
−1

k=0
B(k)+n

k=
kB(k)−nπ+nπ −
j

k=0
kB(k)
=
j

k=0
(j−k)B(k)+
n

k=
(k−)B(k)>0,
which also gi es us a con adic ion.
I (j,)=(n,0
)o (0, n), he wo equali ies de e mine (Q(a),Q(b)) =(0, 0)o (1, 1),
which co espond o he wo cons an o ing ules, which a e in Q+∩Q−.
A.3 P oo o Theo em 3
(i) ⇒(ii). Since Qis educed- o m mono one implemen able, i is educed- o m imple-
men able by a mono one o ing ule q. Hence, we can w i e
nπ(1−π)Q(a)−Q(b)=
n

k=0k(1−π)−(n−k)πq(k)B(k)=
n

k=0
(k−nπ)q(k)B(k)
≥qnπn

k=0
(k−nπ)B(k)=0,
whe e we use mono onici y o q o he inequali y. This shows Q(a)≥Q(b).
(ii) ⇒(iii). I Qis educed- o m implemen able, by Theo em 2, i can be exp essed
as a con ex combina ion o in e im alloca ion p obabili ies o quali ied majo i y and
quali ied an i-majo i y o ing ules.
Conside any quali ied an i-majo i y wi h quo a j∈{0, ,n}(quali ied an i-majo-
i y wi h quo a 0 co esponds o a cons an o ing ule). Fo each j∈{0, ,n}, de ine
δ(j):=Qj(a)−Qj(b)=1
nπ
j−1

k=0
kB(k)−1
n(1−π)
j−1

k=0
(n−k)B(k)
=1
nπ(1−π)
j−1

k=0
(k−nπ)B(k).
No e ha δ(0)=0andδ(n)=−n(1−π)B(n)<0.
Fo all j∈{0, ,n−1},wege
δ(j+1)−δ(j)=1
nπ(1−π)(j−nπ)B(j),
which is nonnega i e i j≥nπ and nega i e i j<nπ. Hence, he alue o δ(j)dec eases
wi h j o all j<nπand inc eases a e ha un il j=n. Since δ(0)=0andδ(n)<0, we
conclude ha δ(j)=Qj(a)−Qj(b)<0 o allj∈{1, ,n}and δ(0)=0.
On he o he hand, o any quali ied majo i y wi h quo a j,weha eQj(a)≥Qj(b).
The quali ied an i-majo i y wi h quo a ze o co esponds o a cons an o ing ule ha
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Theo e ical Economics 19 (2024) Symme ic educed- o m o ing 627
gene a es in e im alloca ion p obabili ies Q(a)=Q(b)=0. Hence, i Q(a)≥Q(b), hen
Qis educed- o m implemen able by con ex combina ion o quali ied majo i y o ing
ules and a cons an o ing ule ha selec s ba all ype p o iles.
(iii) ⇒(i ). E e y quali ied majo i y and quali ied an i-majo i y wi h quo a ze o gen-
e a es in e im alloca ion p obabili ies Q ha sa is y Q(a)≥Q(b). Hence, hei con ex
combina ion also sa is ies Q(a)≥Q(b).ByTheo em1,i Qis educed- o m imple-
men able, hen i sa is ies (5).
(i ) ⇒(i). The p oo o Theo em 1shows ha he se o ex eme poin s o (5)is
he se o quali ied majo i y o ing ules. The line Q(a)=Q(b)connec s wo cons an
o ing ules and all he quali ied majo i y o ing ules sa is y Q(a)≥Q(b).Asa esul ,
any Qsa is ying (5)and(6) mus be educed- o m equi alen o a con ex combina ion
o quali ied majo i y o ing ules and he wo cons an o ing ules. Hence, i is educed-
o m mono one implemen able.
A.4 P oo o P oposi ion 1
By Theo em 3, he ex an e Rawlsian ule sol es he op imiza ion p oblem
max
Qmin(πQ(a),(1−π)1−Q(b)
subjec o Q(a)≥Q(b)(19)
j(1−π)Q(b)−(n−j)πQ(a)+
n

k=j
(k−j)B(k)≥0∀j∈{0, ,n}. (20)
Conside he elaxed p oblem whe e we d op he inequali ies in (19). Fu he ,
change he a iables as ollows: x:=πQ(a)and y:=(1−π)(1−Q(b)). So he elaxed
p oblem (wi h inequali ies (19)in e mso x,y)is
max
x,ymin(x,y)
subjec o jy +(n−j)x≤j(1−π)+
n

k=j
(k−j)B(k)∀j∈{0, ,n}. (21)
No ice ha o any easible solu ion (x,y) o he abo e p oblem, he solu ion ˆ
x=ˆ
y=
min(x,y)is also a easible solu ion wi h he same objec i e unc ion alue. Hence, i is
wi hou loss o gene ali y o assume x=y. Hence, subs i u ing x=yon he le -hand
side o (21), we ge nx, and he p oblem simpli ies o
max
xx
subjec o nx ≤j(1−π)+
n

k=j
(k−j)B(k)∀j∈{0, ,n}. (22)
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628 Lang and Mish a Theo e ical Economics 19 (2024)
Fo e e y j∈{0, ,n},le H(j):=j(1−π)+n
k=j(k−j)B(k). Hence, he op imal so-
lu ion is gi en by
x=y=1
nmin
j∈{0,,n}H(j).
Fo j∈{1, ,n},wesee
H(j)−H(j−1)=1−π−
n

k=j
B(k).
Le j∗:=max{j∈{0, ,n}:n
k=jB(k)≥1−π}.ThenHis dec easing un il j∗and in-
c easing a e ha . So x=y=(1/n)H(j∗)is an op imal solu ion o he elaxed p oblem.
This op imal solu ion co esponds o
Q(a)=1
nπ j∗(1−π)+
n

k=j∗k−j∗B(k)
Q(b)=1
n(1−π)n−j∗(1−π)−
n

k=j∗k−j∗B(k).
This co esponds o sa is ying inequali y (22) o j∗.
Now de ine
α:=1
Bj∗1−π−
n

k=j∗+1
B(k).
By de ini ion o j∗,α∈[0, 1]. Using he exp essions o Qj∗(a)and Qj∗+1(a),i canbe
easily e i ied ha
Q(a)=αQj∗(a)+(1−α)Qj∗+1(a)
Q(b)=αQj∗(b)+(1−α)Qj∗+1(b).
This shows ha he op imal Qis a con ex combina ion o wo quali ied majo i y o ing
ules wi h quo as j∗and j∗+1.
Since each quali ied majo i y is mono one, Qis also mono one. Hence, he op imum
o he elaxed p oblem is a mono one o ing ule.
A.5 P oo o P oposi ion 2
By Theo em 5, e e y unanimous o ing ule is educed- o m equi alen o a con ex
combina ion o u-quali ied majo i y and u-quali ied an i-majo i y ules. Since a con-
ex combina ion p ese es OBIC, e e y unanimous o ing ule is OBIC i and only i e -
e y u-quali ied majo i y and u-quali ied an i-majo i y ule is OBIC. We know ha e e y
u-quali ied majo i y is OBIC (since hey a e s a egy-p oo ). Hence, e e y unanimous
o ing ule is OBIC i and only i e e y u-quali ied an i-majo i y ule is OBIC.
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