Buying voters with uncertain instrumental preferences

Louis-Sidois, Cha les; Musol , Leon And eas
A icle
Buying o e s wi h unce ain ins umen al p e e ences
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Louis-Sidois, Cha les; Musol , Leon And eas (2024) : Buying o e s wi h
unce ain ins umen al p e e ences, Theo e ical Economics, ISSN 1555-7561, The Econome ic
Socie y, New Ha en, CT, Vol. 19, Iss. 3, pp. 1305-1349,
h ps://doi.o g/10.3982/TE4658
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Theo e ical Economics 19 (2024), 1305–1349 1555-7561/20241305
Buying o e s wi h unce ain ins umen al p e e ences
Cha les Louis-Sidois
Depa men o Economics, Vienna Uni e si y o Economics and Business
Leon Musol
The Wha on School, Uni e si y o Pennsyl ania
We analyze a o e-buying model whe e he membe s o a commi ee o e on a
p oposal impo an o a o e buye . Each membe incu s a p i a ely-d awn disu-
ili y i he p oposal passes. We cha ac e ize he cheapes combina ion o b ibes
ha gua an ees he p oposal passes in all equilib ia. When membe s o e simul-
aneously, he numbe o b ibes is a leas 50% la ge han he numbe o o es
equi ed o pass he p oposal ( o e h eshold). The numbe o b ibes inc eases
wi h he dispe sion o he disu ili y dis ibu ion and all membe s a e b ibed wi h
su icien dispe sion. A p opo ional inc ease in he numbe o membe s and he
o e h eshold leads o a less- han-p opo ional inc ease in cap u e cos , and he
cos may inc ease wi h he o e h eshold. Wi h sequen ial o ing and disu ili y
dis ibu ion U[0, 1], all membe s a e b ibed and b ibes a e equal. Finally, sequen-
ial o ing inc eases cap u e cos in small commi ees and dec eases i in la ge
commi ees.
Keywo ds. Vo e buying, legisla u es, poli ical economy.
JEL classi ica ion. D71, D72.
1. In oduc ion
Go e nmen s o en in oduce bills ha go agains he in e es s o pa liamen membe s,
such as a law limi ing dual manda es.1To o e come membe s’ opposi ion, he go e n-
men can o e ewa ds o hose who suppo he bill, e.g., in es men s in legisla i e
dis ic s.
Cha les Louis-Sidois: [email p o ec ed]
Leon Musol : [email p o ec ed]
We hank h ee anonymous e e ees o e y de ailed commen s ha g ea ly imp o ed he pape . We also
hank Alessand o Riboni, Ana Luiza Du a, And ea Ma ozzi, B endan Lucie , Ch is oph Ro he, Elia Sa o i,
Eme ic Hen y, E nes o Dal Bo, E genii Sa ono , Fa uk Gul, F ançoise Fo ges, F anz Os izek, Gio anni An-
d eo ola, Hans Pe e G üne , Ian Ball, Jeanne Hagenbach, Joao The eze, Jonas Mülle -Gas ell, Leaa Ya i ,
Ma ga e Meye , Ma kus Mobius, Ma ias Ia yczowe , Nicole Immo lica, Nicole Kliewe , Pie o O ole a,
Rachel K an on, Roland Benabou, Se gei Gu ie , S ephen Mo is, Thomas Rome , Thomas T oege , Tyle
Maxey, Wol gang Pesendo e , as well as discussan s a semina s a P ince on, Science Po, Mannheim, and
Yale o many insigh ul commen s.
1A dual manda e, o double jobbing, is he p ac ice in which elec ed o icials se e in mo e han one pub-
lic posi ion simul aneously. Fo example, mo e han 80% o pa liamen membe s in F ance held ano he
o ice be o e a law p ohibi ing dual manda es was passed in 2013.
©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/TE4658
1306 Louis-Sidois and Musol Theo e ical Economics 19 (2024)
We de elop a o e-buying model o analyze hese si ua ions. A commi ee o es on a
p oposal ha a o s he in e es s o a o e buye . Howe e , commi ee membe s p e e
he p oposal no o pass. To gain suppo , he o e buye o e s b ibes o membe s in ex-
change o hei o es. Ou key inno a ion is in oducing unce ain y o membe s’ p e -
e ences. Fo ins ance, in he case o a dual manda e p ohibi ion, his unce ain y e lec s
each membe ’s unce ain u u e ( e)elec ion p ospec s. Ou i s example illus a es how
a o e buye exploi s he implica ions o his unce ain y o pi o al p obabili ies.
Example 1. A h ee-membe commi ee o es on a p oposal. The p oposal passes i a
leas wo membe s o e o i . In his example, membe s o e simul aneously. Mem-
be s dislike he p oposal. C ucially, each membe d aws his disu ili y ip i a ely a he
beginning o he game: i
i.i.d.
∼U[0, 1].
A o e buye ( eminine p onoun) is in e es ed in he p oposal passing. Be o e he
o e, she publicly commi s o paying a b ibe o some membe s i hey indi idually o e
o he p oposal. We assume he alue o he b ibe is b≥0 and ha i is he same o
all b ibed membe s. The o e buye knows he dis ibu ion o membe s’ disu ili ies bu
does no obse e hei ealiza ions. The p oposal is impo an o he so she wan s o
gua an ee ha i passes wi h ce ain y in all equilib ia o he o ing subgame. Subjec o
his condi ion, she minimizes he cos o b ibes.
We compa e wo s a egies o he o e buye . Fi s , suppose she b ibes wo mem-
be s. We assume he unb ibed membe plays his weakly dominan s a egy and o es
agains he p oposal. The p oposal passes wi h ce ain y i he wo b ibed membe s
o e o ega dless o hei disu ili y. This s a egy p o ile is clea ly he unique equilib-
ium i b>1 because o ing o is a dominan s a egy o all disu ili ies. Howe e , i
b<1, b ibed membe s wi h disu ili y i>bwould de ia e and he s a egy p o ile is no
an equilib ium. Mo eo e , as will be es ablished, he e exis s an equilib ium whe e he
p oposal is ejec ed wi h a posi i e p obabili y. Thus, he cheapes b ibe such ha he
p oposal passes wi h ce ain y in any equilib ium is b=1 o he womembe s,which
yields a cos o 2.
Ins ead, suppose he o e buye b ibes all h ee membe s. We will show ha i b> 8
27 ,
he e is no equilib ium whe e he p oposal is ejec ed wi h posi i e p obabili y; ha is,
buying a hi d membe is cheape o he o e buye . Fo each membe , o ing o he
p oposal gua an ees b ibe paymen . Howe e , i he membe is pi o al (i.e., i exac ly
one o he membe o es o he p oposal), i also leads o he passing o he p oposal.
Deno ing he pi o al p obabili y by π,membe i o es o i b>
i×π.
The equilib ium o he o ing subgame akes a cu o o m: a membe o es o he
p oposal i his disu ili y is below a h eshold. Fo now, ocus on symme ic s a egies
and call he common cu o .Thenπ( )=2 (1− )and an equilib ium cu o ∈(0, 1)
sa is ies
b= π( ).
In Figu e 1, we plo he igh -hand side o his equa ion. Fo small b ibes like b1, wo
equilib ia exis wi h cu o s 1and 2. Mo eo e , he e is a hi d equilib ium whe e all
membe s accep he b ibe: commi ee membe s a e no pi o al and ha e no incen i e
Theo e ical Economics 19 (2024) Buying o e s 1307
Figu e 1. The s uc u e o equilib ium in he o ing subgame. No es:As hecu o usedby
o he membe s changes, so does he alue o π( )(solid blue). The alue o he maximum is
8
27 ( eached o =2
3). Fo a gi en b ibe b1below 8
27 , he e a e h ee equilib ia o he o ing
subgame: one wi h cu o 1, one wi h cu o 2,andoneinwhichallmembe s o e o he
p oposal. Fo b ibes abo e 8
27 , only he la e exis s.
o de ia e. Th oughou he pape , we assume commi ee membe s play he equilib ium
in which he p oposal is ejec ed wi h he highes p obabili y. Fo ins ance, aced wi h
b1, hey would play 1as his lowe cu o implies he lowes p obabili y o passing.
When bis la ge han he maximum o π( ), he hi d equilib ium, whe e he p o-
posal passes wi h ce ain y, is he only equilib ium o he o ing subgame. He e, he
maximum is 8
27 . Thus, i is su icien o pay sligh ly mo e han 8
9,whichis hecos o
b ibing all h ee membe s, o gua an ee he e is no equilib ium whe e he p oposal is
ejec ed wi h posi i e p obabili y. In ui i ely, b ibing mo e membe s educes membe s’
pi o al p obabili ies, o cing hem o accep smalle b ibes. ♦
We cha ac e ize he cheapes combina ion o b ibes equi ed o passing he p o-
posal wi h ce ain y in all equilib ia. We conside a ious ac o s such as he disu ili y
dis ibu ion, he numbe o commi ee membe s, and he o e h eshold. Speci ically,
we examine simul aneous o ing in Sec ion 2.Fi s ,inSec ion2.1, we assume sym-
me ic s a egies and equal b ibes o all membe s b ibed. Ou main inding is ha he
cheapes cap u e always in ol es a numbe o b ibes a leas 50% highe han he o e
h eshold. Fu he mo e, he numbe o b ibes inc eases wi h dispe sion, and all mem-
be s a e b ibed when he e is enough dispe sion. As o he cap u e cos , inc easing
he o e h eshold and he numbe o membe s p opo ionally esul s in a less- han-
p opo ional inc ease in cos (because membe s a e less likely o be pi o al in a la ge
commi ee), while inc easing only he o e h eshold inc eases he cap u e cos i mo e
han hal o he membe s mus o e o o pass he p oposal.
In Sec ion 2.2, membe s may play asymme ic s a egies. Depending on he dis i-
bu ion, he e may exis asymme ic equilib ia whe e he p oposal can be ejec ed when
membe s ecei e he b ibes o Sec ion 2.1. This is he case when he disu ili y dispe -
sion is small, bu no when i is la ge. Sec ion 2.3 conside s unequal b ibes. Wi h la ge
dispe sion, we show wi h an example ha unequal b ibes can yield a lowe cap u e cos .
Howe e , we es ablish ha i i
i.i.d.
∼U[0, 1]as in Example 1, he cap u e cos is minimized
by equal b ibes.
1308 Louis-Sidois and Musol Theo e ical Economics 19 (2024)
We s udy sequen ial o ing wi h i
i.i.d.
∼U[0, 1]in Sec ion 3. The o e buye also ex-
ploi s pi o al conside a ions and he cheapes cap u e equi es o e ing he same b ibe
o all membe s. Finally, Sec ion 4shows ha compa ed o sequen ial o ing, simul a-
neous o ing yields a highe cap u e cos i he commi ee is la ge, while he opposi e is
ue o small commi ees o e y high o e y low o e h esholds.
The model has a a ie y o applica ions. Ou se up p ima ily applies o decision-
making in o ganiza ions. Fo example, a CEO may wan o pe suade boa d membe s
o make a decision a o ing his in e es s. I boa d membe s expec he decision o be
app o ed ega dless o hei o e, he CEO can ob ain hei suppo in exchange o
small a o s. Al e na i ely, conside he applica ion o Genico and Ray (2006)inwhich
a aide akes o e a company. In such a case, he pos - akeo e alue o non ende ed
sha es could be dilu ed, ha ming all sha eholde s.2Ne e heless, i sha eholde s expec
he akeo e o happen ega dless o hei selling decision, sha es could be bough a li -
le cos . Finally, ou model has implica ions o lobbying and o e-buying in commi ees
o expe s (like FDA commi ees) o ju ies.
We con ibu e o he o e-buying li e a u e by combining a single o e buye wi h
commi ee membe s who ca e abou he o e’s ou come bu a e unce ain abou each
o he ’s p e e ences. The combina ion is no el, hough li e a u e on each ing edien ex-
is s.
Se e al pape s s udy o e-buying when membe s ha e publicly known p e e ences
o e ou comes. Dal Bo (2007) shows ha a o e buye b ibes a commi ee a no cos by
condi ioning he b ibes on he comple e o ing p o ile. She o e s o pay an in ini esimal
amoun i membe s a e no pi o al and a la ge b ibe i o es a e decisi e. By con as ,
we exclude any con ac s based on he join ealiza ion o o es. Mo eo e , he models
o Rasmusen and Ramseye (1994)andDahm and Glaze (2015) ea u e some equilib ia
un a o able o commi ee membe s in which a supe majo i y accep s small b ibes be-
cause no membe is pi o al. Ins ead, we allow membe s o coo dina e on hei p e e ed
equilib ium. Cheap cap u e also occu s in Genico and Ray (2006)andChen and Zápal
(2020) whe e he o e buye app oaches membe s sequen ially and exploi s he iming
o o e s. On he con a y, he o e buye makes all o e s a he same ime in ou model,
bo h in simul aneous and sequen ial o ing.
We ocus on he p obabili y o a o e being decisi e and do no conside in o ma ion
agg ega ion (Fedde sen and Pesendo e 1996,1997,1998). Fedde sen and Pesendo e
(1998) highligh ha unanimi y, which in ou se up maximizes cap u e cos , and makes
in o ma ion ha de o agg ega e. Hen y (2008)andFelgenhaue and G üne (2008)com-
bine o e-buying and in o ma ion agg ega ion. In Hen y (2008), each commi ee mem-
be ecei es a signal abou he quali y o a common alue p oposal. B ibes de e mine
he numbe o membe s who o e in o ma i ely, shaping membe s’ in e ences condi-
ional on being pi o al. Simila ly, in Ekmekci and Laue mann (2019)anelec iono ga-
nize chooses u nou o manipula e he in o ma ion agg ega ed. These pape s conside
a common alue p oposal while we ocus on p i a e alues.
2Fo ins ance, his happens in G ossman and Ha (1980) whe e he aide uses he dilu ion o o ce
a omis ic sha eholde s o sell, bu Bagnoli and Lipman (1988) show ha dilu ion does no necessa ily hap-
pen wi h a ini e numbe o sha eholde s.

Theo e ical Economics 19 (2024) Buying o e s 1309
The mechanism exploi ed by he o e buye in ou model elies on pi o ali y and is
no p esen in he li e a u e on o e-buying wi h exp essi e p e e ences, e.g., in Zápal
(2017), membe s’ esponses o a b ibe a e unce ain, bu pi o al conside a ions a e ab-
sen because membe s do no ake in o accoun he e ec o hei o e on he ou come.
G oseclose and Snyde (1996), Banks (2000), Dekel, Jackson, and Wolinsky (2008), Mo -
gan and Vá dy (2011), and Ia yczowe and Oli e os (2017) also assume exp essi e p e -
e ences and in oduce a second o e buye . They ind ha he i s mo e b ibes a la ge
coali ion o inc ease he cos o he ollowe .
Ou pape p oposes a new explana ion o he high empi ical equency o supe -
majo i ies. While ea ly heo ies o coali ion o ma ion p edic ed minimal winning coali-
ions (Axel od (1970)), some la e pape s p edic supe majo i ies (Koehle (1975), Wein-
gas (1979), Shepsle and Weingas (1981), Ba on and Die meie (2001)). The closes o
us is Ca ubba and Volden (2000), in which a la ge - han-necessa y coali ion ensu es no
membe can p e en he cos ly passing o o he membe s’ bills. Supe majo i ies a e also
ound in he li e a u e on legisla i e ba gaining (Volden and Wiseman (2007), Tsai and
Yang (2010), Dahm, Du , and Glaze (2014)); o an o e iew, see E aslan and E doki-
mo (2019). Fo ins ance, No man (2002) cha ac e izes he nonsymme ic equilib ia o
he classical model o Ba on and Fe ejohn (1989) and shows ha some p oposals can be
unanimously app o ed.
Chen and E aslan (2013,2014) look a he o he side o he p oblem and s udy a
o e-selling model whe e membe s wi h unce ain p e e ences send messages o he
o e buye o in luence he p oposal.
Finally, we a e also ela ed o he la ge li e a u e on unique implemen a ion wi h
mo al haza d. In Win e (2004)andWin e (2006), agen s sepa a ely pe o m indi idual
asks o a p ojec ha succeeds i all agen s succeed. In case o success, he p inci-
pal ewa ds agen s who suppo he p ojec . Con ibu ions a e simul aneous in Win e
(2004) and sequen ial in Win e (2006). Ou pape di e s as membe s’ p e e ences a e
unce ain. Win e ’s p incipal aims o p e en asymme ic equilib ia whe e he p ojec
ails and Win e es ablishes ha disc imina o y ewa ds can be op imal. Wi h su icien
unce ain y, we ind ha asymme ic equilib ia canno be sus ained, and equal b ibes
may be p e e ed.
2. Simul aneous o ing
We conside a commi ee o nmembe s o ing simul aneously on a p oposal. The o e
h eshold mis he minimum numbe o o es o equi ed o pass he p oposal. We ex-
clude m=n(unanimi y equi ed o pass he p oposal) and m=1 (unanimi y equi ed
o ejec i ).3A he beginning o he game, commi ee membe s d aw hei disu ili ies
om he passing o he p oposal. These disu ili ies a e d awn p i a ely and indepen-
den ly om a common dis ibu ion: i
i.i.d.
∼F(·),whe e iis he disu ili y o membe i.
F(·)has suppo [ min, max]wi h min ≥0and max ini e. We assume ha he disu il-
i y dis ibu ion F(·)is con inuously di e en iable on ( min, max), and has an inc easing
3We discuss unanimous o e h esholds a he end o Sec ion 2.1; hey equi e echnical modi ica ions o
he esul s bu do no a ec ou conclusions.
1310 Louis-Sidois and Musol Theo e ical Economics 19 (2024)
gene alized haza d a e:
∂
∂  F( )
1−F( )≥0.
O he models (e.g., La i ie e (2006)) use his assump ion, which is sa is ied by all Uni-
o m and Be a dis ibu ions.
Be o e he o ing subgame, a o e buye who a o s he p oposal publicly o e s
b ibes (b1,,bn),whe ebi≥0ismembe i’s paymen i he o es o he p oposal. We
assume he p oposal is impo an o he o e buye , so she minimizes he cap u e cos ,
i.e., he amoun spen on b ibes, subjec o he p oposal passing wi h ce ain y.
We ocus on Bayesian Nash equilib ia o he o ing subgame. When mul iple equi-
lib ia exis , we assume commi ee membe s play one o he equilib ia whe e he p o-
posal passes wi h he smalles p obabili y. This assump ion is in he spi i o Win e
(2004)andGenico and Ray (2006). Fi s , i ules ou equilib ia whe e he p oposal
passes wi h a bi a ily small b ibes because all b ibed membe s accep and a e no pi -
o al. Second, i ollows na u ally i a o e buye o whom he p oposal is impo an is
unce ain abou which equilib ium will be played. Thi d, i selec s an equilib ium p e-
e ed by commi ee membe s.
The game’s iming is as ollows. Fi s , commi ee membe s p i a ely obse e hei
disu ili y. Then he o e buye o e s b ibes (b1,,bn). Membe s obse e he b ibes
and simul aneously choose whe he o o e o o agains he p oposal. Finally, he p o-
posal passes i a leas mmembe s o e o i .
2.1 Symme ic o ing s a egies and equal b ibes
This subsec ion ocuses on equal b ibes: he o e buye b ibes kmembe s who all e-
cei e he same b ibe b. Thus, he combina ion o b ibes is cha ac e ized by (b,k).Fo
commi ee membe i,as a egyσi: i→[0, 1]is a mapping om disu ili y iin o a
p obabili y o o ing o he p oposal. We only conside membe s o whom he o e
buye o e s a b ibe; unb ibed membe s a e assumed o use hei weakly dominan s a -
egy and o e agains he p oposal. We ocus on symme ic equilib ia, i.e., equilib ia in
which b ibed membe s play he same s a egy.
We i s sol e he o ing subgame. Gi en a combina ion o b ibes (b,k), i a membe
is no pi o al, he payo di e ence be ween o ing o and agains is he b ibe’s alue. I
he is pi o al, a o e o he p oposal makes i pass and he incu s his disu ili y. We deno e
he pi o al p obabili y o commi ee membe iby πi. He accep s he b ibe and o es
o he p oposal i b>
iπi,whe e iπiis he expec ed cos o o ing o he p oposal.
Mo eo e , he o es agains i b<
iπiand can o e o wi h any p obabili y i b= iπi.
Thus, equilib ium s a egies ake a cu o o m. Since we ocus on symme ic equilib-
ia, all membe s o e o he p oposal i hei disu ili y is smalle han some cu o
de e mined in equilib ium.
Ou i s lemma cha ac e izes he equilib ium o he o ing subgame whe e he p o-
posal passes wi h he smalles p obabili y. When ewe membe s a e b ibed han he
o e h eshold (k<m), he e exis s an equilib ium o he o ing subgame whe e he
Theo e ical Economics 19 (2024) Buying o e s 1311
p oposal is always ejec ed, and hence no b ibe can gua an ee ha he p oposal passes
wi h ce ain y in any equilib ium. As a esul , we ocus on k∈{m,,n}.
Lemma 1. Suppose he o e buye o e s a b ibe b>0 o kmembe s wi h m≤k≤n.In
he symme ic equilib ium o he o ing subgame in which he p oposal passes wi h he
smalles p obabili y:
(a) I b≤max ∈[ min, max] πk( ), b ibed membe s o e o he p oposal i hei disu ili y
is smalle han a cu o ha sa is ies =min{ ∈[ min, max]:b= πk( )}whe e
πk( )=k−1
m−1F( )m−11−F( )k−m.
Mo eo e , hey o e agains i hei disu ili y is la ge han and a membe wi h
i= can o e o wi h any p obabili y.
(b) I b>max ∈[ min, max] πk( ), all b ibed membe s o e o he p oposal ega dless o
hei disu ili y: >
max.
(P oo in Appendix A.2.) Fi s , conside he case whe e mmembe s a e b ibed.
πm( )is inc easing in and πm( )→1as → max.Fo b≤ max, Lemma 1(a) cha ac-
e izes he unique equilib ium cu o and he p oposal is ejec ed wi h posi i e p obabil-
i y. Fo b>
max, he s a egy p o ile desc ibed in Lemma 1(b) is he unique equilib ium,
and he p oposal passes wi h ce ain y.
Now conside k>m. As es ablished in Lemma A.2.1 in Appendix A.2, inc easing
gene alized haza d a es imply ha πk( )is single-peaked in o ∈[ min, max].
By he in e media e alue heo em, he equa ion πk( )=badmi s wo solu ions i
b<max ∈[ min, max] πk( ), one i b=max ∈[ min, max] πk( ), and none o he wise. Thus,
equilib ium cu o s a e illus a ed by Figu e 1. The equilib ium whe e he p oposal
passes wi h he smalles p obabili y is associa ed wi h he smalles cu o , and his cu -
o is cha ac e ized by Lemma 1(a). I we le ∗
k:=a gmax ∈[ min, max] πk( ), he small-
es b ibe such ha he p oposal passes wi h ce ain y in any symme ic equilib ium is
b∗
k= ∗
kπk( ∗
k).4Fo Example 1, and hence in Figu e 1,b∗
3=8
27 .
We now u n o he o e buye ’s p oblem. As jus es ablished, i he o e buye o e s
kb ibes, she needs o o e b∗
k o make he p oposal pass wi h ce ain y. Hence, he cos
c(k)is de e mined by he equilib ium whe e he cu o is ∗
k,
c(k)=k×max
∈[ min, max] πk( )=k× ∗
kπk ∗
k=k×b∗
k.
We wan o de e mine a cos -minimizing numbe o b ibes a gmink∈{m,,n}c(k).5In-
ui i ely, while b ibing addi ional membe s equi es paying mo e b ibes, i also makes i
4Mo e p ecisely, b∗
kis he “smalles numbe abo e ∗
kπk( ∗
k),” which is no de ined because b ibes a e on
a con inuum, bu makes sense as he limi o a g id.
5As khas o be an in ege , he e a e dis ibu ions o which a g mink∈{m,,n}c(k)is no unique. In pa ic-
ula , c(k)can be minimized o wo consecu i e in ege s.
1312 Louis-Sidois and Musol Theo e ical Economics 19 (2024)
ha de o commi ee membe s o be pi o al wi h a high p obabili y. Hence, i dec eases
b∗
k. Which e ec domina es depends on he numbe o b ibes, on he o e h eshold,
and on he disu ili y dis ibu ion. Ou main esul cha ac e izes a g mink∈{m,,n}c(k).
P oposi ion 1.
(a) Fo any disu ili y dis ibu ion, any cos -minimizing numbe o b ibes is a leas
min{3
2m−1, n};
(b) Fo any numbe o b ibes k∈Nsuch ha min{3
2m+1, n}≤k≤n, he eexis sa
disu ili y dis ibu ion such ha kis a cos -minimizing numbe o b ibes.
(P oo in Appendix A.2.) P oposi ion 1implies ha he o e buye always wan s o
o e a numbe o b ibes subs an ially la ge han he o e h eshold. I she could o e
any numbe o b ibes, she would choose a leas k=3
2m−1, which o mla ge ep esen s
a numbe o b ibes 50% la ge han he o e h eshold. Howe e , he numbe o b ibes
canno exceed he numbe o membe s. As a esul , when he e a e ewe membe s han
3
2m−1, his cons ain binds and he o e buye b ibes all membe s. Wi h mo e han
3
2mmembe s, i can s ill be he case ha all membe s a e b ibed, bu i depends on he
disu ili y dis ibu ion. This is ue e en when he numbe o membe s is a bi a ily la ge:
o some dis ibu ions, he o e buye ’s cos is always dec easing in he numbe o b ibes
and she o e s as many b ibes as possible.
We now show P oposi ion 1in h ee s eps. Fi s , we es ablish ha he o e buye
b ibes mo e membe s when he disu ili y dis ibu ion is mo e dispe sed. Second, we
show ha e en when dispe sion is small, any cos -minimizing numbe o b ibes is a
leas 3
2m−1. Finally, we demons a e ha wi h a su icien ly dispe sed dis ibu ion, all
membe s a e b ibed ega dless o hei numbe . The de ini ion o dispe sion used o
his analysis is om Shaked and Shan hikuma (2007, p. 213).
De ini ion 1. ˜
F(·)is mo e dispe sed han F(·)i he a io o he in e se CDFs,
˜
F−1(q)/F−1(q), is nondec easing in q o all q∈(0, 1).Insuchacase,wew i eF≤∗˜
F.6
An example o dis ibu ions anked in his o de a e U[1
2−α,1
2+α]wi h α∈(0, 1
2],
which become mo e dispe sed as αinc eases.7We use hese uni o m dis ibu ions o
simula e he cos -minimizing numbe o b ibes a g mink∈{m,,n}c(k)in Figu e 2. In line
wi h P oposi ion 1, he smalles cos -minimizing numbe o b ibes is app oxima ely 3
2m
and is ob ained o small dispe sion (α→0). Mo eo e , he cos -minimizing numbe o
b ibes inc eases wi h dispe sion, which is no speci ic o uni o m dis ibu ions: o all
dis ibu ions ha can be anked in ou dispe sion o de ,
6The inc easing gene alized haza d a e assump ion implies ha F( )>0, so he CDF is s ic ly mono-
one. Hence, F−1(q)is well-de ined o q∈(0, 1).
7These dis ibu ions a e cen e ed a ound 1
2, bu some dis ibu ions wi h di e en means can also be dis-
pe sion anked. In pa icula , mo ing he uni o m suppo o he igh on he eal line dec eases he a i-
ance ela i e o he mean, which esul s in less dispe sion. Howe e , no e ha he ≤∗o de is no comple e
and some dis ibu ions canno be anked.
Theo e ical Economics 19 (2024) Buying o e s 1319
membe s o e o wi h he same p obabili y p( he equilib ium is o mally de i ed in he
p oo o Lemma A.2.3). An equilib ium p obabili y ¯
p∈[0, 1]sol es
b=1
2π(¯
p)=¯
p(1−¯
p).
This exp ession is single-peaked and i s maximum is 1
4.Aslongasb≤1
4, he eisan
equilib ium wi h ¯
p<1 whe e he p oposal is ejec ed wi h a posi i e p obabili y. Thus,
i he o e buye o e s (sligh ly mo e han) b=1
4,shepays3
4and he p oposal passes
wi h ce ain y in all equilib ia whe e membe s play symme ic s a egies.
Now we allow o asymme ic s a egies when he h ee membe s ecei e b=1
4.
The e is an equilib ium whe e one membe accep s his b ibe wi h a p obabili y o one
and he o he wo decline wi h a p obabili y o one. Thus, he ocus on symme ic s a e-
gies is no wi hou loss. Indeed, wi h no dispe sion, he cheapes b ibes such ha he
p oposal passes wi h ce ain y in all equilib ia a e b=1
2o e ed o wo membe s. ♦
Fo su icien ly dispe sed dis ibu ions, howe e , he e is no equilib ium o he o -
ing subgame whe e he p oposal is ejec ed wi h a posi i e p obabili y i he o e buye
o e s he cos -minimizing b ibes de i ed in Sec ion 2.1.
P oposi ion 3. Suppose he dis ibu ion is a leas as dispe sed as U[0, 1]. O e ing b∗
n
o nmembe s, which minimizes he cap u e cos i membe s use symme ic s a egies,
ensu es he p oposal passes wi h ce ain y in any equilib ium o he o ing subgame.
(P oo in Appendix A.3.) In ui i ely, dispe sion makes he beha io o o he mem-
be s ha de o p edic , which p e en s he exis ence o asymme ic equilib ia. Fo mally,
he p oposi ion’s p oo elies on an i e a ed dele ion o s ic ly domina ed s a egies.
Membe i’s pi o al p obabili y is maximized i o he s spli sui ably be ween wo ex eme
cu o s. In pa icula , i m−1 o he membe s always accep (cu o a max)andn−m
always decline (cu o a min), membe iis pi o al wi h ce ain y. E en hen, membe i
s ill o es o i i<b
∗
n.Hence,cu o sbelowb∗
na e no a ionalizable. Once hose s a e-
gies ha e been elimina ed, membe icanno an icipa e being pi o al wi h ce ain y, and
ano he se o cu o s is no a ionalizable. Fo dis ibu ions a leas as dispe sed as
U[0, 1],e en ually,nocu o below max is a ionalizable, and he p oposal passes wi h
ce ain y in all equilib ia.
We use U[0, 1]as a benchma k o p o ide a lowe bound on dispe sion o he p opo-
si ion o be ue. Howe e , no all dis ibu ions a e anked in ou dispe sion o de . Thus,
his lowe bound is su icien bu no necessa y and he e exis o he dis ibu ions o
which o e ing b∗
n o nmembe s would also ensu e ha he p oposal passes wi h ce -
ain y in any equilib ium; e.g., see Example 3.
2.3 Unequal b ibes
Can unequal b ibes educe he cap u e cos ? Example 3illus a es ha he o e buye
can “di ide and conque ” o some dis ibu ions, and hence ha unequal b ibes can
yield a lowe cap u e cos .

1320 Louis-Sidois and Musol Theo e ical Economics 19 (2024)
Example 3. Le (m,n)=(2, 3)and i
i.i.d.
∼Be noulli(1
2).10 Fo b ibed membe i,as a egy
consis s o a p obabili y o accep ing he b ibe i i=0, and a p obabili y o accep -
ing i i=1. We i s cha ac e ize he cos -minimizing b ibes o k=3andk=2i he
o e buye o e s he same b ibe o all b ibed membe s. Nex , we show ha he e exis
unequal b ibes ha yield a lowe cap u e cos .
Suppose he o e buye o e s he same b ibe b>0 o all h ee membe s. Each ac-
cep s i i=0. Hence, he pi o al p obabili y o membe iwould be maximized i he
o he wo membe s o e agains i hei disu ili y is 1. Then πi=1
2.Thus,i b<1
2, he e
is an equilib ium whe e membe s o e o i hei disu ili y is 0 and o e agains i hei
disu ili y is 1. As a esul , he b ibe needs o be a leas b=1
2. I i is (sligh ly mo e han)
1
2, an i e a ed dele ion o domina ed s a egies p o es ha no equilib ium o he o ing
subgame exis s whe e he p oposal is ejec ed wi h posi i e p obabili y. A membe wi h
disu ili y i=0 o es “ o .” Thus, no membe can be pi o al wi h p obabili y la ge han
1
2, and all o hem accep . Hence, cap u e cos s 3
2.
Now, suppose wo membe s a e b ibed. The unb ibed membe o es “agains .”
Thus, a b ibed membe would always be pi o al i he o he accep s ega dless o his
disu ili y. As a esul , he o e buye needs o o e 1 o bo h membe s o gua an ee ha
he p oposal passes wi h ce ain y in all equilib ia; hence, cap u e cos s 2.
Howe e , he unequal b ibes (b1,b2,b3)=(0.51, 0.51, 0.01)yield a lowe cap u e
cos . All membe s o e “ o ” i hei disu ili y is 0, and no pi o al p obabili y can ex-
ceed 1
2. Thus, membe s 1 and 2 always accep . In u n, membe 3 is no pi o al and also
accep s. As a esul , he p oposal is accep ed wi h ce ain y in all equilib ia. ♦
Ne e heless, equal b ibes can be p e e ed o o he dis ibu ions. In pa icula ,
equal b ibes do minimize cap u e cos in Example 1.
Example 4. Conside (m,n)=(2, 3)wi h i
i.i.d.
∼U[0, 1]and allow he o e buye o o -
e unequal b ibes (b1,b2,b3). Deno ing by i he equilib ium cu o o membe i,an
equilib ium o he o ing subgame whe e all cu o s a e in (0, 1)sa is ies:11
1π1( 2, 3)=b1,
2π2( 1, 3)=b2,(1)
3π3( 1, 2)=b3.
When b ibes a e la ge enough, an equilib ium sa is ying (1) does no exis and all com-
mi ee membe s o ing o ega dless o hei disu ili y is he unique equilib ium. Thus,
he o e buye o e s he cheapes (b1,b2,b3)such ha (1) has no solu ion. To iden i y
10In his example, we elax he assump ion ha F(·)is con inuously di e en iable and has an inc easing
gene alized haza d a e o p o ide he clea es illus a ion.
11Appendix A.4 conside s cases whe e some cu o s a e 0 o 1. They do no a ec ou conclusion: i he e
exis s a local pe u ba ion o he b ibes ha gua an ees he e is no equilib ium whe e he p oposal can be
ejec ed, hen he b ibes a e mo e expensi e han bi=8
27 o all membe s.
Theo e ical Economics 19 (2024) Buying o e s 1321
hese b ibes, i is use ul o look a he Jacobian o (1):
J=⎛
⎜
⎝
2(1− 3)+(1− 2) 3 1(1−2 3) 1(1−2 2)
2(1−2 3) 1(1− 3)+(1− 1) 3(1−2 1) 2
(1−2 2) 3(1−2 1) 3 1(1− 2)+(1− 1) 2
⎞
⎟
⎠
Fo gi en b ibes (b1,b2,b3), suppose he e exis ( 1, 2, 3)sol ing (1). As long as Jis
nonsingula , ollowing any -pe u ba ion o (b1,b2,b3) he e also exis s a solu ion o
(1). By con as , when he de e minan o Jis 0, we can ind an -pe u ba ion o he
b ibes such ha (1) has no solu ion wi hin a neighbo hood o ( 1, 2, 3)and, po en-
ially, no solu ion a all.
The cos canno be minimized i some b ibes a e la ge han needed o pass he p o-
posal. The e o e, he cheapes b ibes such ha he p oposal passes wi h ce ain y in
equilib ium mus be a bi a ily close o a (b1,b2,b3) o which he e is an -pe u ba ion
o he b ibes such ha (1) has no solu ion ( he su icien condi ion). This can only be he
case when he de e minan o Jis ze o ( he necessa y condi ion). In he ollowing, we
i s es ablish ha he b ibes o Example 1(bi=8
27 o all membe s) sa is y he necessa y
condi ion, and hen show hey also sa is y he su icien condi ion. Finally, we a gue ha
hey a e he cheapes b ibes sa is ying he necessa y condi ion.
To begin wi h, compu ing he de e minan o Jgi es 2 1 2 3(2− 1− 2− 3), i.e., he
ma ix is singula i 1+ 2+ 3=2.12 Thus, i a combina ion o b ibes is associa ed wi h
an equilib ium sa is ying 3
i=1 i=2, hen he e exis s an -pe u ba ion o he b ibes
ha ensu es ha (1) has no solu ion wi hin a neighbo hood o ( 1, 2, 3).Inpa icula ,
3
i=1 i=2 is sa is ied o 1= 2= 3=2
3. Plugging hese alues in o (1)shows ha
hese cu o s a e an equilib ium i bi=8
27 o all membe s; hence, hese b ibes sa is y
he necessa y condi ion.
Mo eo e , he i e a ed dele ion o domina ed s a egies used o P oposi ion 3gua -
an ees ha i all b ibes a e (sligh ly mo e han) 8
27 ,nocu o in[0, 1)is a ionalizable.
Hence, he -pe u ba ion consis ing in ma ginally inc easing all b ibes gua an ees ha
(1) has no solu ion, and bi=8
27 o all membe s sa is ies he su icien condi ion.
Finally, we es ablish ha bi=8
27 o all membe s a e he cheapes b ibes sa is ying
he necessa y condi ion. We look o ( 1, 2, 3)∈[0, 1]3,3
i=1 i=2 ha minimizes
3
i=1bi. Wi hou loss o gene ali y, suppose 1< 2< 3. We now show ha dec easing
he di e ence be ween cu o s dec eases 3
i=1bi. In pa icula , le us inc ease 1and
dec ease 3by he same amoun : d 1=1, d 3=−1andd 2=0, which keeps 3
i=1 i
cons an . Using he Jacobian, he change in he cap u e cos is
3

i=1
dbi=( 1− 3)(6 2−2),
which is nega i e because 1< 3and 2>1/3.13 As a esul , unde he cons ain
3
i=1 i=2, 1= 2= 3=2
3, minimize 3
i=1bi. These cu o s a e an equilib ium i
12The de e minan is also 0 i some cu o s a e 0. As i≥bi, i=0 can only be pa o an equilib ium i
bi=0. Bu hen he p oposal passes wi h p obabili y 1 i he wo o he membe s ecei e a b ibe o 1, which
does no minimize he cap u e cos .
13As 1+ 2+ 3=2 and 3<1, we ob ain 1+ 2≥1. Combining wi h 2> 1,wemus ha e 2>1/3.
1322 Louis-Sidois and Musol Theo e ical Economics 19 (2024)
bi=8
27 o all membe s. Thus, he equal b ibes iden i ied in Example 1do minimize
he cap u e cos . ♦
Is i gene ally ue ha he o e buye o e s he same b ibes when i
i.i.d.
∼U[0, 1]?Ex-
ample 4es ablished his o (m,n)=(2, 3)by demons a ing ha he Jacobian o (1)is
no in e ible i and only i n
i=1 i=m.We ind hesamecondi ion o (m,n)=(1, 2),
(1, 3),(3, 4),and(1, 4). Hence, in all hese cases, equal b ibes minimize he cap u e
cos . Howe e , we could no ind a gene al o mula o he de e minan and p o e he
esul o any (m,n).
E en i Example 3showed ha he es ic ion o equal b ibes is no always wi hou
loss o gene ali y, Example 4indica es ha he main model gene a es economic insigh s
going beyond he equal b ibes assump ion. To see his, no ice ha equal b ibes did no
p e en he o e buye om se ing k<n(i.e., o e ing some b ibes o ze o). Res ic ed
o o e ing equal b ibes, he o e buye did no wan o exploi his ex eme o m o in-
equali y wi h i
i.i.d.
∼U[0, 1]and ins ead b ibed all membe s. Example 4 u he es ablishes
ha o (m,n)=(2, 3), she ne e bene i s om any o m o inequali y in he b ibes.
3. Sequen ial o ing
We now conside a commi ee o ing sequen ially. The p oposal passes i a leas mo
nmembe s o e o i . Membe s d aw hei disu ili ies om he passing o he p oposal
a he beginning o he game: i
i.i.d.
∼U[0, 1]. The o de o o es is known in ad ance and
membe s obse e p e ious o es, like in he US Sena e whe e membe s o e in alpha-
be ical o de . The o e buye minimizes he cap u e cos subjec o he p oposal passing
wi h ce ain y. B ibes a e simul aneously and publicly o e ed o all membe s be o e he
o e begins, and a b ibe is paid i a membe o es o he p oposal. B ibes can be un-
equal, bu hey canno depend on he numbe o o es s ill needed o pass he p oposal
when he membe o es.14
This sec ion shows ha he o e buye also o e s a numbe o b ibes la ge han he
o e h eshold o exploi pi o al conside a ions wi h sequen ial o ing.
P oposi ion 4. When o ing is sequen ial and i
i.i.d.
∼U[0, 1], he o e buye b ibes all
membe s equally, o e ing b=1/(n−(m−1)) o all nmembe s.
(P oo in ex below.) Hence, wi h i
i.i.d.
∼U[0, 1], b ibing all membe s equally ensu es
ha he p oposal passes wi h ce ain y in all equilib ia a he lowes possible cos o
bo h simul aneous and sequen ial o ing.
14As in Genico and Ray (2006), his assump ion ules ou b ibes, which depend on he numbe o o he
membe s accep ing. Howe e , i has e y di e en implica ions because we conside b ibes o e ed be o e
he o e, while membe s can be app oached sequen ially in Genico and Ray (2006). The e o e, in hei
se up, b ibes may depend on he numbe o o es s ill needed o pass he p oposal. In ou model, his
would imply ha he o e buye o e s 1 o all emaining membe s i all o hei o es a e equi ed o pass
he p oposal, and small b ibes when he e a e mo e membe s. I n>m, all membe s would o e o and
ecei e small b ibes on he equilib ium pa h.
Theo e ical Economics 19 (2024) Buying o e s 1323
Table 1. Equilib ium o he o ing subgame, an example.
(x,y)p(x,y)
x=0x=1x=2x=0x=1x=2
y=11 b111 b10
y=21 b2
1−b1
b2
b11b1+b2b2
No e: Equilib ium cu o s (le panel) and passing p obabili ies ( igh panel) o b2≤b1and b1+b2≤1.xis he numbe o
o es equi ed o pass he p oposal and y he numbe o membe s s ill o o e.
To es ablish his esul , we i s cha ac e ize he equilib ium o he o ing subgame,
which i sel has o be decomposed in o mul iple subgames. Wi hou loss o gene ali y,
we ocus on membe s who ecei e s ic ly posi i e b ibes. De ine S(x,y)as he subgame
whe e x o es a e needed o pass he p oposal and ymembe s s ill ha e o o e. The
o ing subgame begins in S(m,n).I inS(x,y) he membe o es o he p oposal, S(x−
1, y−1)is eached while a o e agains leads o S(x,y−1). Membe s and b ibes bya e
now indexed by y∈{1, ,n}, wi h he numbe o membe s s ill o o e.
Membe s use backwa d induc ion o in e hei pi o al p obabili y as in Spenkuch,
Mon agnes, and Magleby (2018, July). Le (x,y)be he cu o played in S(x,y)and
p(x,y)be he p obabili y ha he p oposal passes gi en ha S(x,y)is eached. We
join ly cha ac e ize (x,y)and p(x,y) o ind he equilib ium o he o ing subgame,
beginning wi h wo-membe commi ees. Table 1gi es an example o he exp essions
o (x,y)in he le and p(x,y)in he igh panel. When a membe o es o he p o-
posal, he subgame loca ed No h-Wes is eached; i he o es agains , we mo e No h.
Example 5. Le (m,n)=(1, 2)and assume membe s ecei e posi i e b ibes wi h b1+
b2≤1. We sol e he game backwa d and s a wi h he las membe , y=1. I membe
y=2 o ed “ o ,” y=1 o esinS(0, 1). The p oposal passes ega dless o he o e o
membe y=1, who accep s wi h ce ain y. Thus, (0, 1)=1andp(0, 1)=1. I membe
y=2 o ed agains , membe y=1 o esinS(1, 1), whe e he is pi o al. He o es o i
b1>
1and we ha e (1, 1)=p(1, 1)=b1.
Mo ing backwa ds, membe y=2s a sinS(1, 2). A o e “ o ” passes he p oposal.
Al e na i ely, i he o es agains , S(1, 1)is eached, whe e he p oposal passes wi h p ob-
abili y b1. Thus, membe y=2 o es “ o ” i
b2− 2>− 2b1⇐⇒ 2<b2
1−b1
,
so ha (1, 2)=b2/(1−b1)and he p oposal passes wi h p obabili y
p(1, 2)=b2
1−b1+1−b2
1−b1b1=b1+b2.
Thus, b ibes a e subs i u es om he pe spec i e o he o e buye : he p oposal passes
wi hce ain y o anyb ibessuch ha b1+b2=1a acos o 1. ♦
Using a ecu si e cha ac e iza ion o (1, y)and p(1, y), his subs i u abili y o
b ibes gene alizes when one o e is needed o pass he p oposal.
1324 Louis-Sidois and Musol Theo e ical Economics 19 (2024)
Lemma 5. In equilib ium, p(1, y)=min{y
s=1bs,1
}.
(P oo in Appendix B.) Wi h i
i.i.d.
∼U[0, 1], he o e buye is exac ly indi e en be-
ween b ibing he i s membe o o e o a membe o ing la e . Fo gene al dis ibu-
ions, he p oblem is no ac able, bu he e is s ill a adeo . On he one hand, he i s
membe can always de e mine he passing o he p oposal: as x=1, his membe is pi -
o al. On he o he hand, a membe who o es la e is less likely o be pi o al ( he p oposal
may be al eady accep ed), bu his cu o a ec s membe s who o e ea lie : hey o ecas
ha o ing agains is less likely o make he p oposal ejec ed when la e membe s e-
cei e highe b ibes. Hence, hei cu o s also inc ease, as we can see in Example 5whe e
he cu o o he i s membe (1, 2)=b2/(1−b1)inc eases wi h he b ibe o he second
membe b1.
The nex example shows ha when mo e han one o e is needed o pass he p o-
posal, b ibes a e no pe ec subs i u es.
Example 6. Le (m,n)=(2, 2)and assume b1+b2≤1. We s a wi h b2<b
1.Fi s ,
conside membe y=1. I membe y=2 o ed “ o ,” S(1, 1)is eached, o which we
ha e es ablished (1, 1)=p(1, 1)=b1.I membe y=2 o ed agains , he p oposal will
be ejec ed and membe y=1 o es “ o .” Tu ning o membe y=2, he s a s in S(2, 2)
and o es “ o ” i
b2− 2b1>0⇐⇒ 2<b2
b1
so ha (2, 2)=b2/b1and he p oposal passes wi h p obabili y
p(2, 2)= (2, 2)p(1, 1)=b2.
B ibes a e no subs i u es anymo e: only b2, he smalle o he wo b ibes, a ec s he
p obabili y o passing.
Ins ead, suppose b2≥b1. The s a egy o membe y=1isasbe o eandmembe
y=2 o es “ o ” i 2<b
2/b1.Asb2/b1>1, he always o es “ o ” and he p oposal passes
wi h p obabili y p(1, 1)=b1. Again, only he smalle o he b ibes a ec s he p obabili y
o passing.
As a esul , he o e buye o e s equal b ibes. I no , he la ges b ibe does no a ec
he p obabili y o passing and should be dec eased. Finally, gi en ha he p obabili y
o passing is equal o he smalle b ibe, his b ibe mus be 1 o make he p oposal pass
wi h ce ain y. Hence, cap u e cos is minimized when b2=b1=1. ♦
The example gene alizes as ollows.15
Lemma 6. Le b(s)
ybe he s h o de s a is ic (i.e., he s h lowes alue) among {b1,,by}.
Then, o x≥1, in equilib ium p(x,y)=min{y−(x−1)
s=1b(s)
y,1
}.
15This lemma uses he con en ion ha emp y sums e alua e o ze o.

Theo e ical Economics 19 (2024) Buying o e s 1325
(P oo in Appendix B.) Hence, he p obabili y o passing is he sum o he n−(m−1)
smalles b ibes. In ui i ely, o gi en b ibes, a membe is mo e likely o accep i he
o es ea ly. Fo ins ance, in Example 6, when he membe who ecei es he la ges b ibe
o es i s (b2≥b1), he accep s ega dless o his disu ili y and ee- ides on he second
membe , elying on him o ejec he p oposal. This inding gene alizes: in S(x,y)wi h
x>1, i byis one o he x−1 la ges b ibes among membe s s ill o o e, membe y
accep s ega dless o his disu ili y. Now, suppose he i s m−1membe s ecei e he
la ges b ibes. They accep ega dless o hei disu ili y and S(1, n−(m+1)) is eached
wi h a p obabili y o one. We ha e he same pa e n when all b ibes a e equal: he m−1
i s membe s accep and ee- ide on he n−(m−1)las membe s, who can po en ially
ejec he p oposal. Then we ha e x=1 and, by Lemma 5, he p obabili y o passing is
he sum o he emaining b ibes, which a e he n−(m−1)lowes b ibes.
Ins ead, i he membe who ecei es he la ges b ibe o es las in Example 6(b2<
b1), bo h membe s decline i hei disu ili y is la ge enough. In such cases, inc easing
he la ges b ibe b1has wo coun e ailing e ec s on he p obabili y o passing. On he
one hand, inc easing b1di ec ly inc eases he p obabili y o passing because i aises he
cu o o he second membe (1, 1)=b1. On he o he hand, b1dec eases he p oba-
bili y o passing h ough he i s membe : he o ecas s ha o ing “ o ” is mo e likely o
make he p oposal pass, and hence becomes mo e likely o decline and make he p o-
posal ejec ed. When i
i.i.d.
∼U[0, 1], he wo e ec s cancel ou and an inc ease in b1does
no a ec he p obabili y o passing. This mechanism gene alizes and he p obabili y o
passing is also he sum o he n−(m−1)lowes b ibes i he membe s who ecei e he
la ges b ibesdono o e i s .
We can now conside he p oblem o he o e buye . Gi en ha o ing s a s in
subgame S(m,n), he o e buye chooses he cheapes combina ion {by}n
y=1such ha
p(m,n)=1. Using Lemma 6, we can w i e his p oblem as
min
{by}n
y=1
n

y=1
bysuch ha
n−(m−1)

s=1
b(s)
y≥1.
The m−1 la ges b ibes do no a ec he p obabili y o passing. Thus, he cos is
minimized when he m−1 la ges b ibes a e equal o b(n−(m−1))
y, he maximum o he
n−(m−1)smalles b ibes. As he sum o hese b ibes mus be 1 o make he p oposal
pass wi h ce ain y, he smalles b(n−(m−1))
yis achie ed when hey a e all equal o 1/(n−
(m−1)). The e o e, he lowes b ibes a e 1/(n−(m−1)), which implies ha all membe s
a e b ibed. Fu he mo e, he m−1 la ges b ibes a e also equal o he n−(m−1)smalles
b ibes. As a esul , all b ibes a e equal and we ob ain P oposi ion 4.
We conclude his sec ion wi h compa a i e s a ics o he cap u e cos . Gi en ha
he o ebuye paysb∗
n=1/(n−(m−1)) o nmembe s, he esul ing cos is
Cseq(m,n)=n
n−(m−1).
Compa a i e s a ics a e simila o P oposi ion 2.I wemul iplymand nby he same
scala λsa is ying (λm,λn)∈N2
+, he cos is mul iplied by less han λ.Mo eo e , he
e ec o mis now clea ly posi i e. Finally, he cap u e cos dec eases wi h n.
1326 Louis-Sidois and Musol Theo e ical Economics 19 (2024)
Figu e 4. Cos compa ison. No es: Simula ion o he lowes cos as a unc ion o mand n.
4. Cos compa ison
We compa e he cap u e cos s unde simul aneous and sequen ial o ing o i
i.i.d.
∼
U[0, 1]. Wi h simul aneous o ing, all membe s a e b ibed and he cos is
Csim(m,n)=nn−1
m−1m
nm1−m
nn−m
.
Which o ing iming minimizes cap u e cos depends on he numbe o membe s nand
on he o e h eshold m. The esul o he compa ison, illus a ed in Figu e 4,isas ol-
lows.
P oposi ion 5. Suppose i
i.i.d.
∼U[0, 1].
(a) I i akes one o all bu one o es o pass he p oposal, he cap u e cos is lowe wi h
simul aneous o ing: Csim(m,n)<Cseq(m,n) o m=1and m=n−1.
(b) I unanimi y is no equi ed o pass he p oposal (i.e., m<n), he e is a λ∗such ha
Cseq(λm,λn)<Csim(λm,λn)wi h λ>λ
∗and (λm,λn)∈N2
+.
(P oo in Appendix C.) The models wi h sequen ial and simul aneous o ing di e
in mul iple aspec s. Howe e , he equilib ium s uc u e o he o ing subgame wi h se-
quen ial o ing p o ides an in ui ion o he cos compa ison. When all b ibes a e equal
o b∗
n, all membe s accep on he equilib ium pa h, which shu s down some in e ac-
ions and p e en s a clea exposi ion o he unde lying mechanisms. Ins ead, suppose
membe s ecei e equal b ibes sligh ly lowe han b∗
n. Thecos o heseb ibesisclose
o he cap u e cos , bu some membe s can o e agains on he equilib ium pa h. As
explained a e Lemma 6, he g oup o he m−1 i s membe s accep hei b ibes and
ely on he g oup o he n−(m−1)las membe s o po en ially ejec he p oposal. In u-
i i ely, he i s g oup ee- ides on he second g oup and his ee- iding dec eases he
Theo e ical Economics 19 (2024) Buying o e s 1327
cap u e cos . Mo eo e , ee- iding is pa icula ly p onounced i bo h g oups a e la ge:
he e should be bo h membe s who ee- ide and membe s o ee- ide on o make he
cap u e cos lowe unde sequen ial o ing.
I one o he wo g oups is small, he e is a limi ed e ec o ee- iding and we ind
ha he cap u e cos is smalle wi h simul aneous o ing. In pa icula , i m=1, one
o e is su icien o pass he p oposal and he g oup o ee- ide s is emp y. Indeed, we
ha e Cseq(1, n)=1. Meanwhile, Csim(1, n)=(1−1/n)n−1is 1
2 o n=2 and dec eases in
n.16 Thus, Cseq(1, n)>Csim(1, n)as s a ed in P oposi ion 5(a).
Now conside a o e h eshold close o unanimi y m=n−1 ( o m=n, pi o al con-
side a ions canno be exploi ed o bo h sequen ial and simul aneous o ing and he
cap u e cos is mei he way). The g oup o he n−(m−1)las membe s is emp y and
he e is no one o ee- ide on. Hence, we also ind ha he cap u e cos is lowe wi h
simul aneous o ing in P oposi ion 5(a). Fo mally, Cseq(n−1, n)=n/2andCsim(n−
1, n)=n×(1−1/n)n.As(1−1/n)nis inc easing in n,and1
2>e
−1=limn→∞(1−1/n)n,
we ha e Cseq(n−1, n)>Csim(n−1, n).
Tu ning o P oposi ion 5(b), he esul simply s a es ha he cap u e cos is smalle
unde sequen ial o ing i mand na e su icien ly la ge and he o e h eshold is no
one o he ex eme cases al eady discussed. This esul can also be explained wi h ee-
iding: he size o he wo g oups is limi ed when mand na e small, and Figu e 4con-
i ms ha he cap u e cos is smalle wi h simul aneous o ing. As we mul iply bo h m
and nby a gi en λsuch ha (λm,λn)∈N2
+, he size o he wo g oups inc eases and he
cap u e cos becomes e en ually smalle wi h sequen ial o ing because o ee- iding.
In he limi , we ha e
lim
λ→∞Cseq(λm,λn)=1
1−m
n
<∞= lim
λ→∞Csim(λm,λn).
Hence, ee- iding e en implies ha he cos g ows bounded wi h sequen ial o ing,
which is no he case wi h simul aneous o ing. Wi h sequen ial o ing, he cos de-
pends on he sha e o membe s in he wo g oups. The p oposal is accep ed wi h ce -
ain y i he sum o he b ibes in he g oup o he n−(m−1)las membe s is one. These
b ibes ep esen a sha e (n−(m−1))/n o he cap u e cos because all membe s ecei e
he same b ibe, and he o al cos is he in e se o his sha e. As λbecomes la ge, his
sha e con e ges o (n−m)/n and ee- iding implies ha he cos is bounded.
5. Concluding ema ks
When membe s ha e unce ain p e e ences, he o e buye b ibes supe majo i ies o ex-
ploi pi o al conside a ions. As we conside ed b ibes condi ioned on indi idual o ing
decisions, we conclude wi h a discussion o o he con ac ual en i onmen s.
I b ibes a e condi ioned on he passing o he p oposal, a pi o al o e also decides
he paymen o he b ibes. Thus, i is a weakly dominan s a egy o o e agains i he
16As min =0, he assump ion m>1inSec ion2.1 plays no ole and Csim(m,n)accu a ely de ines he
cap u e cos o m=1.
1328 Louis-Sidois and Musol Theo e ical Economics 19 (2024)
disu ili y exceeds he b ibe. The o e buye canno exploi pi o al conside a ions wi h
such con ac s: o make he p oposal pass wi h ce ain y, she has o o e max o mmem-
be s. Hence, condi ioning on passing is bad o he o e buye . Ins ead, suppose b ibes
depend on he numbe o o es “ o .” A o e ma e s o he b ibe e en when i is no
pi o al o he passing o he p oposal. In a p e ious e sion o he pape (Louis-Sidois
and Musol 2023), we p oposed an example whe e he o e buye exploi ed pi o al con-
side a ions: she b ibed a numbe o membe s la ge han he o e h eshold and paid
less han when she only condi ioned on passing.
In an un es ic ed con ac ual en i onmen , b ibes can be con ingen on he en i e
ec o o o es. In such a case, Dal Bo (2007) has es ablished ha cap u e occu s a no
cos : he o e buye p omises a b ibe max i a membe is pi o al and an a bi a ily small
b ibe o he wise. Vo ing “ o ” is hen a dominan s a egy, and when mo e han mmem-
be s ecei e such o e s, he p oposal always passes. I such con ac s a e allowed, ou
solu ion is s ill ele an o a budge -cons ained o e buye : e en i membe s a e ne e
pi o al in equilib ium, he o e buye mus be able o pay he la ge pi o al b ibes o
Dal Bo’s s a egy o be c edible. The e o e, while ou solu ion is mo e expensi e (as he
o e buye ac ually pays he b ibes), i would ne e heless be easible o lowe budge
cons ain s.
We ha e conside ed o e s isible o all. I o e s a e p i a ely communica ed o each
membe , he o e buye canno c edibly claim o ha e b ibed mo e membe s han nec-
essa y and he numbe o b ibes is equal o he o e h eshold in equilib ium. Each b ibe
is equal o max, and cap u e is mo e expensi e wi h p i a e o e s. To see why o e ing
mo e b ibes han he o e h eshold is no c edible, conside a o ing p o ile whe e mo e
han mmembe s o e o wi h ce ain y. The o e buye would de ia e and p opose ex-
ac ly mb ibes. This de ia ion canno be de ec ed by membe s who con inue ecei ing
he b ibe, bu in equilib ium, a b ibed membe canno belie e he e a e mo e han m−1
o he b ibes. The b ibe mus be equal o max o him o always o e “ o .”
Finally, ou model can be ein e p e ed wi h punishmen s o membe s who o e
agains ins ead o b ibes. Fo he o e buye , en o cing punishmen is likely o be cos ly,
which implies she e ec i ely pays o membe s who o e agains he p oposal. I she has
enough esou ces o punishmen , she uses he s a egy o his pape . Cap u e is cos -
less because all app oached membe s o e “ o .” The cap u e cos we compu ed co e-
sponds o he minimum esou ces needed o secu e ce ain passing o he p oposal.
Appendix A: P oo s (simul aneous o ing)
A.1 Help ul ac s
Be o e p oceeding o he p oo s, we es ablish some necessa y p e equisi es. Recall ha
(·)is a con inuous ex ension o he ac o ial unc ion. In pa icula , (x)=(x−1)! o
x∈N.Thus,
k
m=explog(k+1)−log(m+1)−log (k−m+1).
Theo e ical Economics 19 (2024) Buying o e s 1335
Using he p ope ies o he digamma unc ion gi en in Appendix A.1 and no ing ha o
any dec easing unc ion b
s=a+1g(s)<b
ag(s)ds,
ψ(k)−ψ(k−m+1)+1
k=
m

s=1
1
k−m+s
<m
0
1
k−m+sds =−logk−m
k.
Thus, he cos s ic ly dec eases in kso ha a g mink∈{m,,n}c(k)=n.
Lemma A.2.5. Fo i∼U[1
2−α,1
2+α], he cos unc ion c(k)has a unique global eal
minimize .
P oo . Recall ha
dlogc(k)
dk =1
k+ψ(k)−ψ(k−m+1)+log1−F ∗
k.
We es ablish ha his exp ession c osses he ho izon al axis a mos once, and neces-
sa ily om below. To do so, we will show ha when he FOC is sa is ied (i.e., when
dlogc(k)
dk =0), d2logc(k)
dk2>0.17
1. Conside :
d2logc(k)
dk2=−1
k2+ψ(k)−ψ(k−m+1)−F ∗
k
1−F ∗
k
d ∗
k
dk .
The exp ession has he same sign as
k−1
k2+ψ(k)−ψ(k−m+1)−F ∗
k
1−F ∗
k
d ∗
k
dk .(7)
We now a gue ha o i∼U[1
2−α,1
2+α],(7) is inc easing in αso ha i we wan
o show ha i is posi i e, we only need o do so o α→0.
2. Fo i∼U[1
2−α,1
2+α], he FOC o he choice o ∗
k educes o
2(k−m)
2α−2 ∗
k+1=2(m−1)
2α+2 ∗
k−1+1
∗
k
(8)
Hence,
∗
k=(4αk −8αm +4α−2k−2)2−161−4α2k−4αk +8αm −4α+2k+2
8k.(9)
17One may wo y ha he e could be mul iple local minima. Howe e , his canno be he case as he
de i a i e is con inuous: ∗
kis con inuous, and hence so is dlog c(k)
dk . This means c(k)has (a mos ) one local
minimum (o he wise i would ha e o ha e a leas one local maximum, which is uled ou by he p oo ).

1336 Louis-Sidois and Musol Theo e ical Economics 19 (2024)
Fu he mo e, we can implici ly di e en ia e (8) o ge an exp ession o d ∗
k
dk and
plug he alue o ∗
k om (9) in o F( ∗
k)
1−F( ∗
k) o ind
−F ∗
k
1−F ∗
k
d ∗
k
dk
=m44α2−1k+−2α(k−2m+1)+k+12−(2α−1)k(m−2)+αm(4m−2)+m
2k(k−m)44α2−1k+−2α(k−2m+1)+k+12.(10)
The de i a i e o his las exp ession wi h espec o αis
d
dα−F ∗
k
1−F ∗
k
d ∗
k
dk =8(m−1)(1−2α)
44α2−1k+−2α(k−2m+1)+k+123/2>0.
Hence, i is inc easing in α, and so is (7).
3. Fo α→0, (10)implies
−F ∗
k
1−F ∗
k
d ∗
k
dk →m−1
(k−1)(k−m)
and (7) becomes
k−1
k2+ψ(k)−ψ(k−m+1)+m−1
(k−1)(k−m).
We only need o show his exp ession is posi i e o ksuch ha he FOC (dlog c(k)
dk =
0) is sa is ied. We es ablish a s ic ly s onge claim: we p o e ha a di e en ex-
p ession (Zbelow) ha educes o his exp ession when he FOC is sa is ied is pos-
i i e o all k:
Z=ψ(k)−ψ(k−m+1)+log1−m−1
k−1
+kψ(k)−ψ(k−m+1)+m−1
(k−1)(k−m)
Fo α→0, F( ∗
k)→m−1
k−1and he FOC is sa is ied i 1
k+ψ(k)−ψ(k−m+1)+log(1−
m−1
k−1)=0. In his case, Z=(7).
4. To show Z>0, we i s u ilize he bounds o Qi e al. (2005) o p o ide a lowe
bound o he i s line (and hen o Z). To his end, no e ha hei Co olla y 8
implies
1
2x+log(x)−1
12x2≤ψ(x+1)≤1
2x+log(x),
and hence
ψ(k)−ψ(k−m+1)+log(k−m)−log(k−1)
Theo e ical Economics 19 (2024) Buying o e s 1337
≥− 1
2(k−m)−1
12(k−1)2+1
2(k−1)
≥1
k−1−1
k−m.
Plugging his back in o Z, i now su ices o show
1
k−1−1
k−m+kψ(k)−ψ(k−m+1)+m−1
(k−1)(k−m)≥0.
This simpli ies o
m−1
k(k−m)+ψ(k)−ψ(k−m+1)≥0.
We again u ilize bounds om Co olla y 8 o Qi e al. (2005), his ime
1
x−1
2x2+1
6x3−1
30x5≤ψ(x+1)≤1
x−1
2x2+1
6x3.
Thus, we need o show
1
m−k−1
k+1
3015
(k−m)2+5
(m−k)3−15
(k−1)2+5
(k−1)3−1
(k−1)5+30
k−1≥0.
This holds o k≥m+1. As we ha e shown in he p oo o Lemma A.2.4 ha he
cos dec eases be ween k=mand k=m+1, he FOC canno be sa is ied o k∈
[m,m+1].Hence,whene e dlogc(k)
dk =0, d2logc(k)
dk2>0.
P oposi ion 1.
(a) Fo any disu ili y dis ibu ion, any cos -minimizing numbe o b ibes is a leas
min{3
2m−1, n};
(b) Fo any numbe o b ibes k∈Nsuch ha min{3
2m+1, n}≤k≤n, he eexis sa
disu ili y dis ibu ion such ha kis a cos -minimizing numbe o b ibes.
P oo .No ice ha his p oo does no ollow he o de o he ex : i builds on Lem-
ma a 2,3,and4.
(a) See Lemma 3.
(b) We es ablish he claim using uni o m dis ibu ions: i
i.i.d.
∼U[1
2−α,1
2+α].Fo his
p oo only, le k∗:=a g mink∈[m,n]c(k)be he eal (as opposed o in ege ) num-
be o b ibes ha minimizes he cos ; his numbe is unique by Lemma A.2.5.
We i s show ha k∗is con inuous in α. Then we es ablish ha limα→0k∗≤
3
2m+1
2and limα→1
2k∗=n. Hence, by he in e media e alue heo em, o all
k≥3
2m+1
2, he e exis s an α∈(0, 1
2)such ha k∗=k. Finally, o any in e-
ge k∈{m,,n},k∗=kimplies k∈a g minκ∈{m,,n}c(κ).Hence,i mis odd,
3
2m+1
2∈Nand o all k∈Nsuch ha min{3
2m+1
2,n}≤k≤n, he e exis s an α
1338 Louis-Sidois and Musol Theo e ical Economics 19 (2024)
such ha k∈a g minκ∈{m,,n}c(κ).I mis e en, 3
2m+1∈Nand o all k∈Nsuch
ha min{3
2m+1, n}≤k≤n, he e exis s an αsuch ha k∈a g minκ∈{m,,n}c(κ).
(i) k∗is con inuous in α.To begin wi h, ∗
kis con inuous in αas he oo s o
a polynomial a e con inuous unc ions o i s coe icien s and (2), he FOC
de ining ∗
k, simpli ies o a polynomial when i
i.i.d.
∼U[1
2−α,1
2+α]:
4α2−1=(−2+4α−2k+4αk −8αm) ∗
k+4k ∗
k2.
This, in u n, implies ha c(k)is con inuous in α, which implies ia Be ge’s
maximum heo em ha i s minimize k∗is con inuous in α.
(ii) lim
α→0k∗≤(3/2)m+1/2. As i∼U[1
2−α,1
2+α]implies F( )= −(1/2−α)
2α o
∈[1
2−α,1
2+α],weha eF−1(p)=1
2−α+2αp o p∈[0, 1]and we can
exp ess he maximiza ion de ining b∗
kin e ms o p=F( ):
b∗
k=max
p∈[0,1]1
2−α+2αp×k−1
m−1pm−1(1−p)k−m.
As α→0, we ha e [1
2−α+2αp]→1
2 o all p∈[0, 1]: as he dis ibu ion
con e ges o a mass poin , all i s quan iles con e ge o his poin . By Be ge’s
maximum heo em, his con e gence implies he con e gence o he maxi-
mum:
b∗
k→max
p∈[0,1]
1
2×k−1
m−1pm−1(1−p)k−m.
Hence, all b∗
kcon e ge o he alues hey ha e in Lemma A.2.3 (wi h δ=1
2)
whe e he disu ili y dis ibu ion has no dispe sion. By Be ge’s maximum he-
o em, k∗ hus con e ges o he minimize o he cos unde no dispe sion as
α→0. Finally, Lemma A.2.4 es ablishes ha he cos unde no dispe sion
inc eases o k≥3
2m+1
2. Hence, we also ha e k∗≤3
2m+1
2when α→0.
(iii) lim
α→1/2k∗=n.This ollows om he p oo o Lemma 4, which es ablishes ha
c(k)dec eases wi h kwhen α=1
2.
A.3 O he simul aneous o ing p oo s
P oposi ion 2.
(a) P opo ional inc eases in o e h eshold mand numbe o commi ee membe s n
aise cap u e cos subp opo ionally: Csim(λm,λn)<λCsim(m,n)wi h λ∈N+.
(b) Suppose only a majo i y can pass he p oposal, i.e., n≤2m−1. Then, o any num-
be o b ibes k,b∗
k,andCsim(m,n)inc ease in he o e h eshold m.
Theo e ical Economics 19 (2024) Buying o e s 1339
P oo .
(a) Suppose he o e buye b ibes kmembe s in a commi ee (m,n).Asλ∈N+,she
can b ibe λk membe s in a commi ee (λm,λn). While λk b ibes need no min-
imize cos in he la ge commi ee, hey gi e an uppe bound on i s minimized
alue. Thus, ecalling p=F( ), i su ices o show ha i
b(λ,p)=F−1(p)
p×λk −1
λm −1pm(1−p)k−mλ,
hen b(λ,p∗)=maxp∈[0,1]b(λ,p)dec eases in λ. The log de i a i e o b(λ,p∗)is
dlogbλ,p∗
dλ =∂logbλ,p∗
∂λ +∂logb(λ,p)
∂p p=p∗
dp∗
dλ
=(A)
∂logbλ,p∗
∂λ
=logp∗m1−p∗k−m−(k−m)ψ(λk −λm +1)
+kψ(λk)−mψ(λm)
=(B)logp∗m1−p∗k−m−(k−m)ψ(λk −λm +1)
+kψ(λk +1)−mψ(λm +1)
≤(C)logm
km1−m
kk−m−(k−m)ψ(λk −λm +1)
+kψ(λk +1)−mψ(λm +1)
=(D)mg(m)−kg(k)
<(E)0,
whe e
(A) le ing p∗=a g maxp∈[0,1]b(λ,p),∂log b(λ,p)
∂p |p=p∗=0 by he en elope heo em,
(B) uses xψ(λx)=x[ψ(λx +1)−1
λx ]=xψ(λx +1)−1/λ,
(C) uses he ac ha pm(1−p)k−m≤maxp∈[0,1]pm(1−p)k−m=(m
k)m(1−m
k)k−m,
(D) de ines g(x):=[log(x)−log(k−m)] −[ψ(λx +1)−ψ(λk −λm +1)],and
(E) ollows because g(·)is inc easing. To see his, no e
g(x)=1
x−λψ(λx +1)
=(i)
1
x−λψ(λx)+λ
λ2x2
>(ii)
1
x−λ1
λx +1
λ2x2+λ
λ2x2=0,
1340 Louis-Sidois and Musol Theo e ical Economics 19 (2024)
whe e (i) uses ψ(u+1)=ψ(u)−1
u2and (ii) uses ψ(u)<1
u+1
u2 om Guo and
Qi (2010, Lemma 3, p. 107).
(b) I n≤2m−1, we show ha b∗
kis inc easing in m o all k∈{m,,n}.Hence,
mink∈{m,,n}c(k)mus also be inc easing in mas c(k)=kb∗
k. By he en elope he-
o em, ∂logb∗
k
∂ | = ∗
k=0and
dlogb∗
k
dm =∂logb∗
k
∂m +∂logb∗
k
∂  = ∗
k
d
dm =∂logb∗
k
∂m
=ψ(k−m+1)−ψ(m)+logF ∗
k−log1−F ∗
k.
Lemma A.2.3 implies F( ∗
k)≥m−1
k−1.Thus,
dlogb∗
k
dm ≥ψ(k−m+1)−ψ(m)+logm−1
k−1−logk−m
k−1
=log(m−1)−log(k−m)−ψ(m)−ψ(k−m+1).
n≤2m−1impliesm−1≥k−m.Then
log(m−1)−log(k−m)=m−1
s=k−m
1
sds
>
m−1

s=k−m+1
1
s=ψ(m)−ψ(k−m+1),
whe e he las equali y esul s om he p ope y o he digamma unc ion a he
beginning o he p oo sec ion. As a esul , dlogb∗
k
dm >0.
P oposi ion 3. Suppose he dis ibu ion is a leas as dispe sed as U[0, 1]. O e ing b∗
n
o nmembe s, which minimizes he cap u e cos i membe s use symme ic s a egies,
ensu es he p oposal passes wi h ce ain y in any equilib ium o he o ing subgame.
P oo . Recall ha oo no e 4de ines b∗
nas he smalles numbe abo e ∗
nπn( ∗
n).Fo
his p oo , we assume he e exis s a ixed minimum cu ency >0, so ha b∗
n=
∗
nπn( ∗
n)+. We use a simul aneous i e a ed dele ion o s ic ly domina ed s a egies
o a gue ha when nmembe s a e b ibed wi h b∗
n, he p oposal passes wi h ce ain y in
any equilib ium.
We elimina e cu o s in inc easing o de . Le 
ibe he smalles a ionalizable cu o
o membe ia e i e a ion .Then +1
iis he smalles a ionalizable cu o o membe
iwhen no o he membe jplays a cu o below 
j. A each i e a ion, we simul aneously
elimina e cu o s o all membe s. We ha e ∀i:0
i= min and, as he disu ili y dis ibu-
ion is he same o all membe s, he same se o cu o s is elimina ed o all membe s a
each s ep. Thus, 
i= ∀i.
Le x( y)be he p obabili y ha among all membe s bu y he e a e exac ly x o es
o (w i ing y=( 1,, y−1, y+1,, n) o he ec o o cu o s o all membe s o he

Theo e ical Economics 19 (2024) Buying o e s 1341
han y). Wi hou loss o gene ali y, conside membe 1. Le πmax()deno e he maximal
pi o al p obabili y he can expec i e e y o he membe has a cu o o a leas :
πmax():=max
{( 2,, n):∀i∈{2,,n}≤ i≤ max} m−1 1.
Then he smalles a ionalizable cu o  +1a i e a ion +1sol es
 +1×πmax =b∗
n.
The emainde o he p oo shows ha i  <
max, hen
 +1− πmax =b∗
n− πmax ≥, (11)
whence  +1≥ +(as πmax ≤1) and all cu o s smalle han max a e e en ually elim-
ina ed. We p oceed by bounding  πmax( ).
1. Fo any i=1, we can ew i e m−1( 1)as
m−1 1=F( i) m−2 1,i+1−F( i) m−1 1,i,
whe e 1,i=( 2,, i−1, i+1,, n)is he ec o o cu o s o all membe s o he
han 1 and i.Hence, hesigno
∂ m−1( 1)
∂ iis independen o i.Thus, he eisa
solu ion o he maximiza ion p oblem πmax( )wi h i∈{ , max} o all i.Inligh
o his, le πn,h( )be he alue o he pi o al p obabili y i exac ly ho he n−1
o he b ibed membe s choose a cu o o max and n−1−hchoose a cu o o  ;
hen
πmax =max
h∈{0,,n−1}πn,h .
2. To bound  maxhπn,h( ) om abo e, no e
 max
h∈{0,,n−1}πn,h ≤max
h∈{0,,n−1}, ∈[ min, max] πn,h =max
h∈{0,,n−1}ˆ
bn,h,
whe e ˆ
bn,his such ha when o e ing any amoun s ic ly abo e ˆ
bn,h o n−hmem-
be s wi h o e h eshold m−h, he e is no equilib ium whe e membe s o e agains
he p oposal wi h posi i e p obabili y. Thus, ˆ
bn,h=0 o h≥mand else
ˆ
bn,h=max
∈[ min, max] n−h−1
m−h−1F( )m−1−h1−F( )n−m
 
πn,h( )
.
3. By de ini ion, ˆ
bn,0 =b∗
n−. When he dis ibu ion is mo e dispe sed han U[0, 1],
we now show ha ˆ
bn,h<b
∗
n− o all h∈{1, ,m}.Weha e
∂log ˆ
bn,h
∂h =−ψ(n−h)−ψ(m−h)+logF ∗
n,h, (12)
whe e ψis he digamma unc ion and we de ine ∗
n,h:=a gmax ∈[ min, max] πn,h( ).
1342 Louis-Sidois and Musol Theo e ical Economics 19 (2024)
•I i
i.i.d.
∼U[0, 1],F( ∗
n,h)=m−h
n−hand (12)is
−ψ(n−h)−ψ(m−h)−log(n−h)−log(m−h).
Using he p ope ies o he digamma unc ion gi en in Appendix A.1 and no -
ing ha o any dec easing unc ion b−1
s=ag(s)>b
ag(s)ds,
ψ(n−h)−ψ(m−h)=
n−1

s=m
1
s−h
>n
m
1
s−hds =log(n−h)−log(m−h).
The e o e, (12) is nega i e o U[0, 1].
•By Lemma A.2.2,F( ∗
n,h)is la ge o mo e dispe sed dis ibu ions. Thus, (12)
mus also be nega i e o dis ibu ions mo e dispe sed han U[0, 1].
Pu ing hese s eps oge he ,18
 ×πmax =(1) max
h∈{0,,n−1}πn,h 
≤(2)max
h∈{0,,n−1}ˆ
bn,h
=(3)ˆ
bn,0
=(3)b∗
n−.
Plugging his in o (11) indeed yields  +1− ≥.
A.4 Bounda y solu ions in Example 4
When some membe s ha e cu o s a he bounda y, i.e., when o a leas one i∈{1, 2, 3},
i∈{0, 1}, he nonsingula i y o he Jacobian is no in o ma i e abou he exis ence o
nea by equilib ia o any local pe u ba ion o he b ibes: while local pe u ba ions such
ha equa ion (1) con inues o hold mus exis , hese local pe u ba ions may ake some
iou side o he easible egion [0, 1]. To add ess his conce n, we conside b ibes asso-
cia ed wi h an equilib ium wi h cu o s a he bounda y and such ha a local pe u ba-
ion o he b ibes gua an ees he e is no equilib ium whe e he p oposal can be ejec ed.
We show ha such b ibes a e mo e expensi e han bi=8
27 o all membe s:
•Cu o s a 1.
–( 1, 2, 3)=(1, 1, 1)is an equilib ium o any (b1,b2,b3), bu i is i ele an o
he exis ence o o he equilib ia.
–( 2, 3)=(1, 1). 1<1onlyi b1=0. Wlog, suppose b2≤b3. Then, i b2<1,
( 1, 2, 3)=(0, b2,1
)is an equilib ium. Hence, he cos is a leas 2 o make he
p oposal pass wi h ce ain y.
18The index on equali ies/inequali ies e e s o he ele an s ep in he p oo .
Theo e ical Economics 19 (2024) Buying o e s 1343
– 3=1, 1≤ 2<1. Then ( 1, 2)sa is y
1(1− 2)=b1; 2(1− 1)=b2. (13)
The o e buye minimizes b1+b2+b3.No ice 3=1 equi esb3≥π3= 1(1−
2)+ 2(1− 1)=b1+b2.Thus,b1+b2+b3is a leas
2(b1+b2)=2( 1+ 2−2 1 2). (14)
The Jacobian o (13)is
J=1− 2− 1
− 21− 1.
The de e minan is 1 − 1− 2,whichis0i 1+ 2=1. Unde his condi ion,
he cos in (14) is minimized o 1= 2=1
2 o which i is equal o 1. Thus, he
cheapes b ibes such ha (13) has no solu ion a e necessa ily mo e expensi e
han 8
9( he cap u e cos wi h equal b ibes).
•Cu o s a 0.
– i=0onlyi bi=0. Wi h wo o h ee cu o s (and hence b ibes) a 0, he p oposal
ne e passes.
– 1=0and0< 2≤ 3 equi es b1=0, b2>0, and b3>0. An equilib ium would
ha e o sa is y 2 3=b2and 3 2=b3.I b2=b3, his sys em does no ha e a so-
lu ion. Suppose b2=b3. Then, i b2<1, ( 1, 2, 3)=(0, b2,1
)is an equilib ium.
Hence, wi h one cu o a 0, he cos is a leas 2 o make he p oposal pass wi h
ce ain y.
Appendix B: P oo s (sequen ial o ing)
Lemma 5. In equilib ium, p(1, y)=min{y
s=1bs,1
}.
P oo . In gene al, i membe y o es o , his expec ed u ili y is by− yp(x−1, y−1)
while a o e agains gi es − yp(x,y−1). The e o e, in S(x,y), he membe o es o o
he p oposal i his disu ili y is la ge han a cu o (x,y)de ined by
(x,y)=minby
p(x−1, y−1)−p(x,y−1),1
.
Fo all y,p(0, y−1)=1. Thus, he cu o o membe yin S(1, y)is
(1, y)=minby
1−p(1, y−1),1
.
I by
1−p(1,y−1)<1, he p obabili y o passing is
p(1, y)= (1, y)p(0, y−1)+1− (1, y)p(1, y−1)
1344 Louis-Sidois and Musol Theo e ical Economics 19 (2024)
=by
1−p(1, y−1)×1+1−by
1−p(1, y−1)×p(1, y−1)
=by+p(1, y−1)
We use an induc ion o comple e he p oo . No ice ha he lemma holds o y=1
and assume ha i holds o y−1. Then p(1, y−1)=min{y−1
s=1bs,1
}and we do ha e
p(1, y)=min{y
s=1bs,1
}, which p o es he claim.
Lemma 6. Le b(s)
ybe he s h o de s a is ic (i.e., he s h lowes alue) among {b1,,by}.
Then, o x≥1, in equilib ium p(x,y)=min{y−(x−1)
s=1b(s)
y,1
}.
P oo . We p oceed by induc ion on x. Lemma 5p o es he base case (x=1). Suppose
he esul holds o x−1. We p o e ha i also holds o x. To do so, we use an induc ion
on y.
•Base case: y=1. I x≥2, p(x,1
)=0; i x=1, hen p(x,1
)=b1as equi ed ( ecall
we use he con en ion ha emp y sums e alua e o ze o).
•Induc i e s ep. Rea anging he equa ion o a membe ’s cu o and hen employing
he induc i e hypo hesis, we ha e
(x,y)=minby
p(x−1, y−1)−p(x,y−1),1

=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
minby
b(y−(x−1))
y−1
,1
i p(x−1, y−1)<1,
min$by
1−
y−x

s=1
b(s)
y−1
,1
%i p(x−1, y−1)=1
and p(x,y−1)<1,
1i p(x,y−1)=1.
We now e i y he exp ession o p(x,y) ollowing hese cases:
– Assume p(x−1, y−1)<1.
∗Assume by>b
(y−(x−1))
y−1.Then (x,y)=1so ha
p(x,y)=p(x−1, y−1)
=min$y−(x−1)

s=1
b(s)
y−1,1
%
=min$y−(x−1)

s=1
b(s)
y,1
%.
∗Assume by≤b(y−(x−1))
y−1.Then
p(x,y)=p(x,y−1)+ (x,y)p(x−1, y−1)−p(x,y−1)