Kolo ilin, An on; Woli zky, Alexande
A icle
Dis ibu ions o pos e io quan iles ia ma ching
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Kolo ilin, An on; Woli zky, Alexande (2024) : Dis ibu ions o pos e io quan iles
ia ma ching, Theo e ical Economics, ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol.
19, Iss. 4, pp. 1399-1413,
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Theo e ical Economics 19 (2024), 1399–1413 1555-7561/20241399
Dis ibu ions o pos e io quan iles ia ma ching
An on Kolo ilin
School o Economics, UNSW Business School
Alexande Woli zky
Depa men o Economics, MIT
We o e a simple analysis o he p oblem o choosing a s a is ical expe imen
o op imize he induced dis ibu ion o pos e io medians o , mo e gene ally, q-
quan iles o any q∈(0, 1). We show ha a single expe imen — he q-quan ile
ma ching expe imen —implemen s all implemen able dis ibu ions o pos e io
q-quan iles, wi h di e en dis ibu ions spanned by di e en selec ions om he
se s o pos e io q-quan iles. A dense subse o implemen able dis ibu ions o
pos e io q-quan iles can be uniquely implemen ed by pe u bing he q-quan ile
ma ching expe imen . A linea unc ional is op imized o e dis ibu ions o pos-
e io q-quan iles by aking he op imal selec ion om each se o pos e io q-
quan iles. The q-quan ile ma ching expe imen is he only expe imen ha simul-
aneously implemen s all implemen able dis ibu ions o pos e io q-quan iles.
Keywo ds. Quan iles, s a is ical expe imen s, median ma ching, o e con i-
dence, ge ymande ing, pe suasion.
JEL classi ica ion. C61, D72, D82.
1. In oduc ion
Se e al p oblems o ecen economic in e es amoun o cha ac e izing he se o dis i-
bu ions o pos e io quan iles ha can be induced by some s a is ical expe imen o o
inding a dis ibu ion in his se ha maximizes some objec i e. These p oblems include
appa en o e con idence (Benoî and Dub a (2011)) (e.g., wha dis ibu ions o medians
o indi iduals’ belie s abou hei own abili ies a e consis en wi h Bayesian upda ing?),
pa isan ge ymande ing (F iedman and Holden (2008), Kolo ilin and Woli zky (2020b))
(e.g., wha is he highes dis ibu ion o dis ic median o e s a ained by any dis ic ing
plan?), and quan ile pe suasion (Kolo ilin and Woli zky (2020a)) (e.g., wha expe imen
maximizes he expec ed ac ion o a ecei e who minimizes he expec ed absolu e de i-
a ion o he ac ion om he unknown s a e o he wo ld?).1
An on Kolo ilin: [email p o ec ed]
Alexande Woli zky: [email p o ec ed]
1Yang and Zen e is (2024) explo e hese and o he applica ions. Kolo ilin and Woli zky (2020b)conside
a mo e gene al ge ymande ing model, which educes o op imizing he dis ibu ion o pos e io quan iles
in a special case. Kolo ilin and Woli zky (2020a, P oposi ion 2’) in oduce quan ile pe suasion as a special
case o a mo e gene al pe suasion model, which is u he de eloped in Kolo ilin, Co ao, and Woli zky
( o hcoming).
©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/TE6057
1400 Kolo ilin and Woli zky Theo e ical Economics 19 (2024)
Ou p oblem is as ollows. The e is a eal- alued s a e θ. A s a is ical expe imen
induces a dis ibu ion o e pos e io s μ.Fo anyq∈(0, 1), each pos e io μhas a leas
one q-quan ile. In gene al, a pos e io can ha e mul iple q-quan iles due o gaps in he
suppo o μ: o example, i μpu s equal weigh on wo s a es θ<θ
, hen he se o
medians o μis he en i e in e al [θ,θ]. An expe imen , oge he wi h a selec ion ule
o b eak ies o pos e io s wi h mul iple q-quan iles, induces a dis ibu ion o pos e io
q-quan iles. A dis ibu ion o pos e io q-quan iles is implemen able i i is induced by
some expe imen and selec ion ule; i is uniquely implemen able i i is induced by an
expe imen ha almos always induces pos e io s wi h unique q-quan iles. We ask wha
dis ibu ions o pos e io q-quan iles a e implemen able o uniquely implemen able,
how o implemen hem, and how o op imize a linea unc ional o e dis ibu ions o
pos e io q-quan iles.
We p o ide a simple solu ion o his p oblem. Fo any q∈(0, 1), he e is a single
expe imen — he q-quan ile ma ching expe imen — ha simul aneously implemen s
all implemen able dis ibu ions o pos e io q-quan iles, wi h di e en dis ibu ions
spanned by di e en selec ion ules. Fo example, i he s a e is uni o mly dis ibu ed on
[0, 1]and he ele an quan ile is he median, he q-quan ile ma ching expe imen is he
median ma ching expe imen ha , whene e he ue s a e is θ∈[0, 1/2], e eals only
ha he s a e is ei he θo 1/2+θ(and, hence, whene e he ue s a e is θ∈(1/2, 1],
e eals only ha he s a e is ei he θo θ−1/2).2In gene al, he q-quan ile ma ching ex-
pe imen pools pai s o s a es ac oss a q-quan ile o he p io in a posi i ely asso a i e
manne , wi h weigh qon he lowe s a e in each pai .
To see why he q-quan ile ma ching expe imen implemen s all implemen able dis-
ibu ions o pos e io q-quan iles, conside again he median ma ching expe imen
wi h a uni o m s a e. When he expe imen e eals ha he s a e is θo 1/2+θwi h equal
p obabili y, e e y alue x∈[θ,1/2+θ]is a pos e io median. The median ma ching ex-
pe imen hus simul aneously implemen s (i) he dis ibu ion H(x)=max{0, 2x−1}, (ii)
he dis ibu ion H(x)=min{2x,1
}, and (iii) e e y dis ibu ion Hsa is ying H≤H≤H.
Con e sely, simple Ma ko - ype inequali ies imply ha e e y implemen able dis ibu-
ion is bounded by Hand H. Mo eo e , he se o uniquely implemen able dis ibu ions
o pos e io quan iles is essen ially he same: any desi ed selec ion om each se o q-
quan iles induced by he q-quan ile ma ching expe imen can be uniquely selec ed by
mixing each pos e io unde he q-quan ile ma ching wi h he degene a e dis ibu ion
on he desi ed selec ion wi h p obabili ies 1 −εand ε, espec i ely. Finally, op imiz-
ing a linea unc ional o e dis ibu ions o pos e io quan iles simply equi es aking
he op imal selec ion om each se o q-quan iles induced by he q-quan ile ma ching
expe imen . See Figu e 1 o an illus a ion o ou esul s.3
2To ou knowledge, he median ma ching expe imen i s appea s in Kolo ilin and Woli zky (2020a,p.
29). I is closely ela ed o he median one- o-one ma ching in oduced by K eme and Maskin (1996) and
u he s udied by Leg os and Newman (2002); he i le o he p esen pape acknowledges his connec ion.
3Simila igu es in he li e a u e include Figu e 1 o Owen and G o man (1988), Figu e 2 o Kamenica and
Gen zkow (2011), and Figu e 3 o Yang and Zen e is (2024).
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Theo e ical Economics 19 (2024) Dis ibu ions o pos e io quan iles 1401
Figu e 1. Implemen able dis ibu ions o pos e io medians. When he p io Fis uni o m on
[0, 1],Hand Ha e he lowes and highes implemen able dis ibu ions o pos e io medians.
A dis ibu ion His implemen able i and only i H≤H≤H. Op imizing a linea unc ional
o e dis ibu ions o pos e io medians equi es aking he op imal selec ion om each ho i-
zon al do ed line. Fo example, he blue ( ed) do s a e he op imal selec ions o an inc easing
(dec easing) objec i e unc ion.
We also show ha he q-quan ile ma ching expe imen is he unique expe i-
men ha simul aneously implemen s all implemen able dis ibu ions o pos e io q-
quan iles. To see why, conside again a uni o m s a e, and compa e he median ma ch-
ing expe imen wi h he nega i e asso a i e ma ching expe imen ha , whene e he
ue s a e is θ∈[0, 1], e eals only ha he s a e is ei he θo 1 −θ.Thenega i eas-
so a i e ma ching expe imen simul aneously implemen s he lowes and highes dis-
ibu ions o pos e io medians, Hand H, bu i does no implemen all in e medi-
a e dis ibu ions, such as he dis ibu ion H1/2gi en by H1/2(x)=H(x) o x<1/4,
H1/2(x)=1/2 o x∈[1/4, 3/4),andH1/2(x)=H(x) o x≥3/4. Indeed, he nega i e
asso a i e ma ching expe imen induces pos e io s wi h medians be ween 1/4and3/4
when he ue s a e lies be ween 1/4and3/4, while H1/2assigns p obabili y 0 o hese
medians.
The cu en pape is closely ela ed o Benoî and Dub a (2011)andYang and Zen-
e is (2024). Bo h o hese pape s es ablish esul s ha a e e y simila o ou Theo em
1(albei Benoî and Dub a do so o disc e e expe imen s wi h ini ely many induced
pos e io s). Ou main con ibu ion is in oducing he q-quan ile ma ching expe imen ,
which yields a much simple p oo o Theo em 1,aswellasnew esul s(Theo ems2and
3).
2. Implemen able dis ibu ions o pos e io quan iles
This sec ion shows ha he q-quan ile ma ching expe imen implemen s all imple-
men able dis ibu ions o pos e io q-quan iles.
Le =[θ,θ]⊂R,wi hθ < θ, be a compac s a e space, le C()be he se o con-
inuous unc ions on ,le ()be he se o cumula i e dis ibu ion unc ions on ,
endowed wi h he weak opology, and le (()) be he se o p obabili y measu es
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1402 Kolo ilin and Woli zky Theo e ical Economics 19 (2024)
on (). Recall ha G∈()is a nondec easing, igh -con inuous unc ion sa is ying
G(θ)≥0andG(θ)=1. Le δx,wi hx∈, deno e he degene a e dis ibu ion a x,so
ha δx(θ)=1{θ≥x}.
Fix a p io dis ibu ion F∈()and a quan ile o in e es q∈(0, 1). Following Ka-
menica and Gen zkow (2011), de ine an expe imen as a dis ibu ion τ∈(())o pos-
e io dis ibu ions G∈()such ha Gdτ(G)=F. Fo each pos e io G, de ine he
se o q-quan iles o Gas
X(G)=x∈:Gx−≤q≤G(x),
whe e G(x−)deno es he le limi limθ↑xG(θ), wi h he con en ion G(θ−)=0. In addi-
ion, o each G, de ine i s gene alized in e se G−1as
G−1(p)=in θ∈:G(θ)≥p o all p∈[0, 1].
Tha is, G−1(p)is he smalles p-quan ile o G.
To de ine he q-quan ile ma ching expe imen , le ωbe uni o mly dis ibu ed on
[0, 1], and o each ω∈[0, q],le G=Gωbe he dis ibu ion ha assigns p obabili y q o
F−1(ω)and assigns p obabili y 1 −q o F−1(q+(1−q)ω/q).Theq-quan ile ma ching
expe imen is de ined as an expe imen τsuch ha o τ-almos all G, he eexis s
ω∈[0, q]such ha G=Gω.4Fo mally, τis de ined by
τ(M)=q
0
1qδF−1(ω)+(1−q)δF−1(q+(1−q)ω/q)∈Mdω/q o all M⊂().
While all o ou esul s hold o gene al Fand q, o simplici y we will p o ide in u-
i ion only o he uni o m-median case whe e Fis uni o m on [0, 1]and q=1/2.
A dis ibu ion Ho q-quan iles is implemen ed by an expe imen τi he e exis s a
(measu able) selec ion χ om he co espondence Xsuch ha he dis ibu ion o χ(G)
induced by τis H. A dis ibu ion Ho q-quan iles is uniquely implemen ed by an ex-
pe imen τi His implemen ed by τand X(G)is a single on o τ-almos all G.Le
Hand Hbe he se s o implemen able and uniquely implemen able dis ibu ions o
q-quan iles.
The ollowing heo em cha ac e izes Hand H.
Theo em 1. The ollowing s a emen s hold:
(i) We ha e H={H∈():H≤H≤H},whe eH(x)=max{0, (F(x)−q)/(1−q)}
and H(x)=min{F(x)/q,1
} o all x∈.
(ii) E e y H∈His implemen ed by τ.
4Fo example, when Fis a omless, we can le ω=F(θ), so ha he q-quan ile ma ching expe imen
induces pos e io s ha assign p obabili y q o θand assign p obabili y 1 −q o F−1(q+(1−q)F(θ)/q) o
θ∈[0, F−1(q)].
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Theo e ical Economics 19 (2024) Dis ibu ions o pos e io quan iles 1403
(iii) I Fhas a posi i e densi y on , henHis he closu e o H. In pa icula , o any
objec i e unc ion V∈C(), we ha e
sup
H∈H
V(x)dH(x)=max
H∈H
V(x)dH(x).(1)
Figu e 1illus a es he se H. The in ui ion o Theo em 1is s aigh o wa d. Fi s ,
by simple Ma ko - ype inequali ies, any implemen able Hmus sa is y H≤H≤H.Fo
example, i he pos e io median is less han xwi h p obabili y p, henθmus be less
han xwi h p obabili y a leas p/2. When F(x)=x, his implies ha p≤2x,so he
p obabili y ha he pos e io median is less han xis a mos min{2x,1
}=H(x).5
Con e sely, o see ha any Hsa is ying H≤H≤His implemen able, conside he
median ma ching expe imen τ ha induces only pos e io s Gθ ha assign equal p ob-
abili y o some θ∈[0, 1/2]and o 1/2+θ. The se o medians o such a pos e io is
X(Gθ)=[θ,1/2+θ]. A he same ime, H≤Himplies ha H−1(2θ)≥θ,andH≥Him-
plies ha H−1(2θ)≤1/2+θ,soweha eH−1(2θ)∈[θ,1/2+θ].Thus,χ(Gθ)=H−1(2θ)
is a selec ion om X(Gθ). Finally, he dis ibu ion o χ(Gθ)induced by τis H, because
he s a es ha induce medians below xunde τwi h selec ion χ(Gθ)a e p ecisely hose
in [0, H(x)/2]and [1/2, 1/2+H(x)/2], and he measu e o hese s a es is H(x).
As o unique implemen a ion, o any e∈(0, 1]and any implemen able and ab-
solu ely con inuous dis ibu ion Hwi h densi y h, we explici ly cons uc a modi ica-
ion o he median ma ching expe imen τ
e ha uniquely implemen s he dis ibu ion
(1−e)H+eF o medians, by making e e y pos e io Gθa con ex combina ion o he
median ma ching dis ibu ion (δθ+δ1/2+θ)/2 and he degene a e dis ibu ion δH−1(2θ)
a he unique median H−1(2θ)∈[θ,1/2+θ]. In ui i ely, o each θ∈[0, 1/2],τ
einduces
pos e io s Gθand GH(θ)/2wi h p obabili ies 1 −eand e; simila ly, o each θ∈(1/2, 1],
τ
einduces pos e io s Gθ−1/2and GH(θ)/2wi h p obabili ies 1 −eand e. Then pos e io
medians in [x,x+dx]a e induced a θ∈[H(x)/2, H(x+dx)/2]wi h p obabili y 1−e,a
θ∈[1/2+H(x)/2, 1/2+H(x+dx)/2]wi h p obabili y 1 −e,anda θ∈[x,x+dx]wi h
p obabili y e. Since H(x+dx)=H(x)+h(x)dx, he densi y o he pos e io median x
mul iplied by he pos e io a xis equal o (1−e)h(x)(δH(x)/2+δ1/2+H(x)/2)/2+eδx,as
equi ed. To comple e he p oo o Theo em 1, we p o ide a simple a gumen showing
ha any dis ibu ion in Hcan be app oxima ed by uniquely implemen able dis ibu-
ions (1−e)H+eF.6
The li e a u e con ains se e al close an eceden s o Theo em 1.F iedman and
Holden (2008) s udy pa isan ge ymande ing wi h a ini e numbe o legisla i e dis-
ic s. Benoî and Dub a (2011) s udy es ing o o e con idence in a sel - anking ex-
pe imen wi h a ini e numbe o bins. In ou no a ion, F iedman and Holden and
5This a gumen is closely ela ed o Kamenica and Gen zkow’s 2011 “p osecu o –judge” example. As in
hei example, he key obse a ion is ha i he p io p obabili y o an e en (e.g., he e en ha θ≤x)isx,
hen he maximum p obabili y ha he pos e io p obabili y o his e en is a leas 1/2ismin{2x,1
}.
6A comple e cha ac e iza ion o he se H emains an open p oblem. Two obse a ions a e ha His
ap ope subse o H(as Hand Hdo no belong o H) and ha no all uniquely implemen able dis ibu-
ions can be implemen ed by ou modi ica ion o q-quan ile ma ching. Fo example, H=δ1/2is uniquely
implemen ed by comple e pooling bu no by ou modi ica ion o median ma ching.
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1404 Kolo ilin and Woli zky Theo e ical Economics 19 (2024)
Benoî and Dub a conside disc e e expe imen s wi h ini ely many induced pos e i-
o s. F iedman and Holden show ha a disc e e e sion o His he highes imple-
men able dis ibu ion o pos e io medians. Benoî and Dub a show ha he se o
uniquely implemen able dis ibu ions o pos e io medians is a disc e e e sion o he
se {H∈():H<H<H}. In a gene al se ing wi h possibly in ini ely many induced
pos e io s in he con ex s o quan ile pe suasion and pa isan ge ymande ing, espec-
i ely, Kolo ilin and Woli zky (2020a)andKolo ilin and Woli zky (2020b) show ha H
is he highes implemen able dis ibu ion o pos e io medians. Finally, in a gene al
se ing, Yang and Zen e is (2024) show ha he se o implemen able dis ibu ions o
pos e io medians is {H∈():H≤H≤H}, and also cons uc a dense subse o dis-
ibu ions ha a e uniquely implemen able.7Rela i e o Benoî and Dub a and Yang
and Zen e is,Theo em1shows ha he q-quan ile ma ching expe imen implemen s
e e y H∈Hand also yields a much simple p oo .
Fa he a ield, Blackwell (1953), S assen (1965), and Kolo ilin (2018)cha ac e -
ize implemen able dis ibu ions o pos e io means. An in e es ing open ques ion is
whe he a use ul analogue o Theo em 1( o medians) and S assen’s heo em ( o
means) exis s o in e media e s a is ics ha in e pola e be ween he median and he
mean.8
3. Op imal dis ibu ions o pos e io quan iles
This sec ion uses he q-quan ile ma ching expe imen o cha ac e ize he dis ibu ions
o pos e io q-quan iles ha maximize a con inuous linea unc ional.
Theo em 2. Le V∈C().ThenH(uniquely) maximizes V(x)dH(x)on Hi and
only i H−1(p)(uniquely) maximizes Von [H−1(p),H−1(p)] o (almos ) all p∈[0, 1].
Consequen ly, he alue o he maximiza ion p oblem is
max
H∈H
V(x)dH(x)=1
0
maxV(x):x∈H−1(p),H−1(p)dp.(2)
Concep ually, Theo em 2 ollows easily om Theo em 1. Since he median ma ch-
ing expe imen τimplemen s all implemen able dis ibu ions o medians, op imiza-
ion jus equi es selec ing an op imal median χ(Gθ)∈a g maxx∈[θ,1/2+θ]V(x) o each
pos e io Gθinduced by τ, as illus a ed in Figu e 1. The alue o he maximiza ion
p oblem is, hus, 2 1/2
0maxx∈[θ,1/2+θ]V(x)dθ, and a dis ibu ion Ho mediansisop i-
mal i and only i H−1(2θ)∈a g maxx∈[θ,1/2+θ]V(x) o all θ∈[0, 1/2]. Tha is, op imal
solu ions can be ob ained by poin wise maximiza ion wi hou any i oning p ocedu e.
7To es ablish esul s simila o ou Theo em 1,Yang and Zen e is cha ac e ize he ex eme poin s o he
se {H∈():H≤H≤H}. As ecen ly emphasized by Kleine , Moldo anu, and S ack (2021), cha ac e iz-
ing a con ex se by i s ex eme poin s can be use ul o es ablishing some p ope ies o he se . In con as ,
we show ha di ec ly cha ac e izing he se o implemen able dis ibu ions o pos e io quan iles is much
easie han cha ac e izing he ex eme poin s o his se .
8Kolo ilin, Co ao, and Woli zky ( o hcoming) s udy he ques ion o cha ac e izing op imal dis ibu ions
o such in e media e s a is ics— he analogous p oblem o ha o Theo em 2in he cu en pape .
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Theo e ical Economics 19 (2024) Dis ibu ions o pos e io quan iles 1405
In gene al, by Theo em 1, o each H∈Hand p∈[0, 1],weha eH−1(p)≤H−1(p)≤
H−1(p). I we conside he elaxed p oblem o inding a measu able unc ion J:[0, 1]→
o
maximize 1
0
VJ(p)dp
subjec o H−1(p)≤J(p)≤H−1(p) o all p∈[0, 1],
one solu ion is
J(p)=mina gmaxV(x):x∈H−1(p),H−1(p) o all p∈[0, 1].
This unc ion Jis mono one; mo eo e , he p oo o Theo em 2shows ha he e exis s
H∈()such ha J=H−1,soHsol es he op imiza ion p oblem (2).
The closes an eceden o Theo em 2is Co olla y 4 o Yang and Zen e is (2024),
which sol es he maximiza ion p oblem (2) in he special cases whe e Vis quasi-
conca e o quasi-con ex. The solu ion ollows immedia ely om Theo em 2.Tosee
how, suppose ha Vis quasi-conca e wi h a maximum a x∈[0, 1]. Fo each in e -
al [θ,1/2+θ], i is op imal o selec xi x∈[θ,1/2+θ],θi x<θ,and1/2+θi
x>1/2+θ. This induces he dis ibu ion o pos e io medians
H(x)=H(x),x<x
,
H(x),x≥x.
Nex , suppose ha Vis quasi-con ex wi h V(x)=V(1/2+x) o some x∈[0, 1/2].
Then, o each in e al [θ,1/2+θ], i is op imal o selec θi x>θand 1/2+θi x<θ.
This induces he dis ibu ion o pos e io medians
H(x)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
H(x),x<x
,
2x,x∈[x,1/2+x),
H(x),x≥1/2+x.
F om he pe spec i e o op imiza ion, i is na u al o ask whe he each ex eme poin
o His exposed, meaning ha i is he unique maximize in Ho V(x)dH(x) o some
V∈C(). I u ns ou ha some ex eme poin s a e no exposed. To see his, no e ha
in he uni o m-median case, he dis ibu ion H=(δ1/4+δ1/2)/2 is an ex eme poin
o H, as he e a e no dis inc H1,H2∈Hsuch ha H=(H1+H2)/2. By Theo em 2,
i Hmaximizes V(x)dH(x)on H o some V∈C(), henV(1/4)≥V(x) o all x∈
[θ,1/2+θ]and all θ∈[0, 1/4], and, simila ly, V(1/2)≥V(x) o all x∈[θ,1/2+θ]and
all θ∈[0, 1/2].Thus,V(1/4)=V(1/2)≥V(x) o all x∈[0, 1]. Bu hen he dis ibu ion
δ1/2∈Halso maximizes V(x)dH(x), which shows ha His no an exposed poin o
H.9
9The dis ibu ion Hdoes uniquely maximize V(x)dH(x) o V=2·1{x=1/4}+1{x=1/2},which
is uppe semi-con inuous, bu no con inuous. An open ques ion is whe he each ex eme poin o H,
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1406 Kolo ilin and Woli zky Theo e ical Economics 19 (2024)
4. Unique p ope ies o he quan ile ma ching expe imen
Theo em 1shows ha he q-quan ile ma ching expe imen simul aneously implemen s
all implemen able dis ibu ions o pos e io q-quan iles. We now show ha i is he
unique expe imen o do so. Fo simplici y, in his sec ion we assume ha Fhas a posi-
i e densi y on .
We ac ually es ablish he s onge esul ha he q-quan ile ma ching expe imen
is he unique expe imen ha simul aneously implemen s all op imal dis ibu ions o
s ic ly quasi-con ex objec i e unc ions.
Theo em 3. The q-quan ile ma ching expe imen τis he unique expe imen τ ha , o
each p∈[0, 1], implemen s he dis ibu ion Hp∈Hgi en by
Hp(x)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
H(x),x<x
p,
p,x∈[xp,xp),
H(x),x≥xp,
whe e xp=F−1(qp )and xp=F−1(q+(1−q)p).
In o he wo ds, o any expe imen τ= τ, he eissomep∈[0, 1]such ha τdoes
no implemen Hp. Fo example, in he uni o m-median case, he nega i e asso a i e
ma ching expe imen does no implemen H1/2, as no ed in he In oduc ion.
An immedia e co olla y o Theo em 3is ha he q-quan ile ma ching expe imen is
he unique expe imen ha minimizes he maximum eg e o a designe who chooses
an expe imen τbe o e lea ning he objec i e V, bu chooses a selec ion χa e lea ning
V. Fo mally, o each expe imen τ∈(()) and each objec i e V∈C(), de ine he
designe ’s eg e as
(τ,V)=max
H∈H
V(x)dH(x)−sup
H∈H
V(x)dH(x):His implemen ed by τ.
No e ha (τ,V)≥0 o allτand V. Say ha a se o possible objec i e unc ions V⊂
C()is ich i , o all x0,x1∈, he e exis s a s ic ly quasi-con ex V∈Vwi h V(x0)=
V(x1). We hen ha e he ollowing esul .
Co olla y 1. I Vis ich, hen he q-quan ile ma ching expe imen τis he unique
expe imen τsuch ha (τ,V)=0 o all V∈V.
Appendix:P oo s
P oo o Theo em 1. Conside any expe imen τ∈(()) and any measu able se-
lec ion χ(G) om X(G).Le Hbe he dis ibu ion o χ(G)induced by τ. Then, o each
cha ac e ized in Theo em 1 o Yang and Zen e is (2024), is he unique maximize o V(x)dH(x) o some
uppe -semicon inuous V. This is a weake p ope y han exposedness, as he usual heo y o exposed
poin s elies on con inui y.
15557561, 2024, 4, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE6057 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
Theo e ical Economics 19 (2024) Dis ibu ions o pos e io quan iles 1413
Yang, Kai Hao and Alexande Zen e is (2024), “Mono one unc ion in e als: Theo y and
applica ions.” Ame ican Economic Re iew, 114, 2239–2270. [1399,1400,1401,1404,1405,
1406]
Co-edi o Rakesh Voh a handled his manusc ip .
Manusc ip ecei ed 26 Feb ua y, 2024; inal e sion accep ed 17 June, 2024; a ailable online 20
June, 2024.
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