Do al, Lau a; Sk e a, Vasiliki
A icle
Op imal mechanism o he sale o a du able good
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Do al, Lau a; Sk e a, Vasiliki (2024) : Op imal mechanism o he sale o a du able
good, Theo e ical Economics, ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 19, Iss.
2, pp. 865-915,
h ps://doi.o g/10.3982/TE4485
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Theo e ical Economics 19 (2024), 865–915 1555-7561/20240865
Op imal mechanism o he sale o a du able good
Lau a Do al
Economics Di ision, Columbia Business School, Columbia Uni e si y and CEPR
Vasiliki Sk e a
Depa men o Economics, Uni e si y o Texas a Aus in, Depa men o Economics, Uni e si y College
London, and CEPR
A buye wishes o pu chase a du able good om a selle who in each pe iod
chooses a mechanism unde limi ed commi men . The buye ’s alue is bina y and
ully pe sis en . We show ha pos ed p ices implemen all equilib ium ou comes
o an in ini e-ho izon, mechanism-selec ion game. Despi e being able o choose
mechanisms, he selle can do no be e and no wo se han i he chose p ices in
each pe iod, so ha he is subjec o Coase’s conjec u e. Ou analysis ma ies in-
sigh s om in o ma ion and mechanism design wi h hose om he li e a u e on
du able goods. We do so by elying on he e ela ion p inciple in Do al and Sk e a
(2022).
Keywo ds. Mechanism design, limi ed commi men , in o ma ion design, public
PBE, pos ed p ices, Coase conjec u e.
JEL classi ica ion. D84, D86.
1. In oduc ion
We cha ac e ize he equilib ium ou comes o an in ini e-ho izon, mechanism-selec ion
game be ween a du able-good selle and a p i a ely in o med buye unde limi ed com-
mi men , so ha he selle can commi o oday’s mechanism, bu no o he mecha-
nism he will o e i no sale occu s. Theo em 1shows ha all equilib ium ou comes can
be implemen ed ia pos ed p ices. We cons uc a pe ec Bayesian equilib ium o he
mechanism-selec ion game, which achie es he selle ’s unique equilib ium payo , and
we show ha i implemen s he essen ially unique equilib ium ou come.1In his equi-
lib ium, as long as a sale has no occu ed, he selle will choose a mechanism ha can
be implemen ed as a pos ed p ice. Despi e being able o choose om a ich se o mech-
Lau a Do al: [email p o ec ed]
Vasiliki Sk e a: [email p o ec ed]
We hank h ee anonymous e e ees o excellen commen s, which subs an ially imp o ed he pape . We
would like o hank Rahul Deb, F ede ic Koessle , Dan Quigley, Pablo Schenone, and especially Max S inch-
combe, as well as audiences a Cowles, SITE, and S ony B ook, o hough -p o oking ques ions and illumi-
na ing discussions. Vasiliki Sk e a is g a e ul o gene ous inancial suppo h ough he ERC consolida o
g an 682417 “F on ie s in design.” This esea ch was suppo ed by g an s om he Na ional Science Foun-
da ion (Do al: SES-2131706; Sk e a: SES-1851729).
1Whene e he equilib ium ou come is no unique, all equilib ium ou comes a e achie ed also ia a
sequence o pos ed p ices.
©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/TE4485
866 Do al and Sk e a Theo e ical Economics 19 (2024)
anisms, he selle can do no be e and no wo se han i he could only choose p ices in
each pe iod.
In ou game, an unin o med selle aces a p i a ely in o med buye , whose alua-
ion is bina y, ully pe sis en , and s ic ly abo e he selle ’s ma ginal cos . In each pe-
iod, as long as he good has no been sold, he selle o e s he buye a mechanism, he
ules o which de e mine he alloca ion o ha pe iod. A mechanism consis s o (i) a
se o inpu messages o he buye , and (ii) o each inpu message, a dis ibu ion o e
ou pu messages and alloca ions. Whe eas he selle obse es he ou pu message and
he alloca ion, he does no obse e he inpu message he buye submi s o he mech-
anism. Thus, when designing he mechanism, he selle ge s o design how much he
obse es abou he buye ’s choices, and hence design his belie s abou he buye ’s alue.
The combina ion o mechanism design and in o ma ion design elemen s is key o ou
cha ac e iza ion.
Ou analysis b idges he li e a u es on mechanism design and on he du able-good
monopolis , especially he wo k o Gul, Sonnenschein, and Wilson (1986). To see his, i
is use ul o e iew he main s eps in ol ed in he p oo o Theo em 1. Fi s , we cons uc
an assessmen ha is iden ical along he pa h o ha in Ha and Ti ole (1988), which we
dub he pos ed-p ices assessmen . In his assessmen , along he pa h o play, he selle
sells he good using a dec easing sequence o p ices, which e lec ha condi ional on
he good no being sold he selle assigns less p obabili y o he buye ’s alue being high.
Second, we a gue ha he selle ’s payo unde he pos ed-p ices assessmen is an
uppe bound on he selle ’s equilib ium payo in he mechanism-selec ion game. To do
so, we ely on an auxilia y p og am, ha only in ol es he selle (see (OPT)inSec ion4).
In his p og am, he selle maximizes he dynamic analogue o he i ual su plus,by
choosing a Bayes’ plausible dis ibu ion o e pos e io s and o each pos e io (i) a p ob-
abili y o ade and (ii) a ec o o equilib ium con inua ion payo s. We a i e a he
p og am de ined in (OPT) by elying on he ools in ou p e ious wo k, Do al and Sk e a
(2022). The main heo em in Do al and Sk e a (2022) allows us o simpli y he class o
mechanisms he selle o e s in any equilib ium o he game and he buye ’s equilib ium
beha io . This s ep educes he sea ch o he op imal sequence o mechanisms o hose
ha sa is y, loosely speaking, a sequence o pa icipa ion and u h- elling cons ain s,
allowing us o he mos pa o igno e he buye as a playe . Like in s anda d mechanism
design, he low- alua ion buye ’s u ili y and he high- alua ion buye ’s u h- elling con-
s ain de e mine an uppe bound on he e enue he selle can ex ac wi hin a pe iod
(Lemma 2). Replacing his uppe bound in he selle ’s payo p o ides us wi h a dynamic
analogue o he i ual su plus (Equa ion (4)), whe e he selle ’s payo is w i en as a
unc ion o he alloca ion, bu also he con inua ion payo s.
We show ha he alue o (OPT) coincides wi h he selle ’s payo in he pos ed-
p ices assessmen , and hence ha he selle canno do be e han in he pos ed-p ices
equilib ium. Because he auxilia y p og am (OPT) igno es he u h- elling cons ain
o he low- alua ion buye (i.e., i co esponds o he elaxed p og am in mechanism
design), ou esul implies ha he solu ion o (OPT) sa is ies he emaining cons ain s
and can hus be implemen ed as an equilib ium ou come. As we discuss in he con-
clusions, we expec ha in se ings wi h ans e able u ili y, he s udy o he analogous
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Theo e ical Economics 19 (2024) Op imal mechanism o he sale o a du able good 867
p oblem o (OPT) p o ides a na u al benchma k o unde s and he p ope ies o he
p incipal’s op imal mechanism, e en i in some se ings he solu ion o he analogue o
(OPT) may no deli e an implemen able ou come.
Finally, ollowing he logic in Gul, Sonnenschein, and Wilson (1986), we show ha
he selle ’s payo in he pos ed-p ices assessmen is a lowe bound on he selle ’s equi-
lib ium payo . Unde lying he a gumen in Gul, Sonnenschein, and Wilson (1986) ha
a unique equilib ium payo exis s in he gap case is he p ope y ha he minimum
p ice he selle chooses in equilib ium imposes an uppe bound on he maximum pay-
o he buye can ob ain. Relying once again on (OPT), we es ablish ha gua an ees on
he selle ’s equilib ium payo ansla e in o uppe bounds on he high- alua ion buye ’s
payo . A med wi h his esul , we show ha he selle can always unde cu he p ice in
he pos ed-p ices assessmen and ea n close o his payo in ha assessmen .
The signi icance o ou esul s is wo- old. Fi s , o he bes o ou knowledge, his is
he i s pape o cha ac e ize op imal mechanisms unde limi ed commi men and pe -
sis en p i a e in o ma ion in an in ini e-ho izon se ing. Because he se o ools a ail-
able o ackle he di icul ies wi h he e ela ion p inciple unde limi ed commi men
do no eadily apply o in ini e-ho izon se ings (see, e.g., he seminal wo k o Bes e
and S ausz (2001,2007), and he discussion in he ela ed li e a u e), such cha ac e iza-
ion has p o ed elusi e. In Do al and Sk e a (2022), we p o ide a e ela ion p inciple o
mechanism-selec ion games unde limi ed commi men ha applies o a b oad class o
games, including in ini e-ho izon ones. I is he applica ion o his ool ha allows us
o a gue ha he mechanism we cha ac e ize is he op imal one among all mechanisms
he selle could ha e o e ed he buye unde limi ed commi men .
Second, he op imali y o pos ed p ices should no be aken o g an ed, e en i i is
e oca i e o Sk e a (2006). Fi s , ou model is no an in ini e-ho izon e sion o ha in
Sk e a (2006), since we conside a la ge class o mechanisms han Sk e a (2006). Indeed,
he mechanisms in Sk e a (2006) p esume ha he selle mus obse e he buye ’s inpu
message (c . La on and Ti ole (1988), Bes e and S ausz (2001)), whe eas we conside
mechanisms in which he selle ge s o design how much he obse es abou he buye ’s
inpu message. In Do al and Sk e a (2022), we s udy a wo-pe iod e sion o he model
in Sk e a (2006), bu we allow he selle o o e mechanisms like hose in his pape . We
show ha when he selle is su icien ly pa ien i is no an equilib ium o he selle o
pos a p ice in each pe iod (Rema k 3explains why pos ed p ices may ail o be op imal
in Do al and Sk e a (2022)). I ollows ha we canno ake limi s using he equilib ium
ou come in Sk e a (2006) o analyze he equilib ium ou comes o he game we s udy,
e en a e showing ha he game ends in ini e ime. Second, B eig (2022)showsin
a bina y- alue model wi h a pe ishable good ha pos ed p ices may no be op imal:
Indeed, he selle may bene i om using andom deli e y con ac s.
Rela ed li e a u e: The pape con ibu es mainly o h ee s ands o li e a u e. The i s
s and, simila o his pape , de i es op imal mechanisms when he designe has limi ed
commi men . Mos pape s in ha li e a u e examine ei he ini e-ho izon se ings (see
La on and Ti ole (1988), Sk e a (2006,2015), Deb and Said (2015), Fiocco and S ausz
(2015), Beccu i and Mölle (2018)), o in ini e-ho izon se ings, imposing es ic ions on
he class o con ac s ha can be o e ed (e.g., Ge a di and Maes i (2020)), o on he
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868 Do al and Sk e a Theo e ical Economics 19 (2024)
solu ion concep (e.g., Acha ya and O ne (2017)).2Unde lying hese es ic ions a e
ha he esul s in bo h Bes e and S ausz (2001)andSk e a (2006) do no eadily ex end
o in ini e-ho izon se ings. Fo ins ance, he esul in Bes e and S ausz (2001) applies
only i he p incipal is ea ning his highes payo consis en wi h he agen ’s payo (see
Lemma 1 in Bes e and S ausz (2001)). Thus, implici in hei mul is age ex ension is a
es ic ion o equilib ia o he mechanism-selec ion game ha possess a Ma ko s uc-
u e, which as shown by Ausubel and Denecke e (1989), may no be enough o cha ac-
e ize he p incipal’s bes equilib ium payo . The app oach in Sk e a (2006)has head-
an age ha he se o incen i e easible ou comes has a well-unde s ood s uc u e. I is
no clea , howe e , how o inco po a e he p incipal’s sequen ial a ionali y cons ain s
in in ini e-ho izon se ings in a ac able way.
The second s and is he li e a u e ha ollows he obse a ion in Coase (1972) ha
he du able-good monopolis aces a ime-inconsis ency p oblem, which in u n lim-
i s his monopoly powe . The pape s in he du able-good monopolis li e a u e (S okey
(1981), Bulow (1982), Gul, Sonnenschein, and Wilson (1986), Sobel (1991), O ne (2017))
s udy p ice dynamics and es ablish (unde some condi ions) Coase’s conjec u e.3Re-
la ed o his li e a u e is he p oblem o dynamic ba gaining wi h one-sided incomple e
in o ma ion4(e.g., Sobel and Takahashi (1983), Fudenbe g, Le ine, and Ti ole (1985),
Ausubel and Denecke e (1989)). In all hese pape s, he unin o med pa y’s inabili y o
commi limi s his ba gaining powe .
Finally, as will become clea om he analysis, he pape con ibu es o he li e a-
u e on in o ma ion design (Aumann, Maschle , and S ea ns (1995)andKamenica and
Gen zkow (2011)), highligh ing i s po en ial o p o ide ac able cha ac e iza ions o
equilib ium ou comes in games. Con a y o mos o hese pape s, howe e , he selle
aims o pe suade his u u e sel , as opposed o ano he playe , highligh ing he ole o in-
o ma ion as a commi men de ice (see, e.g., Ca illo and Ma io i (2000), and ecen ly,
Habibi (2020)).
O ganiza ion The es o he pape is o ganized as ollows. Sec ion 2desc ibes he
model; Sec ion 2.1 summa izes he esul s in Do al and Sk e a (2022) used o simpli y
he analysis ha ollows. Sec ion 3p esen s he main esul o he pape , Theo em 1.
Sec ion 4in oduces he auxilia y p og am (OPT) and s udies i s p ope ies. Sec ion 5
e iews he main s eps o he p oo o Theo em 1.Sec ion6concludes. All p oo s no in
he main ex a e in he Appendix.
2. Model
P imi i es: Two playe s, a selle and a buye , in e ac o e in ini ely many pe iods. The
selle owns one uni o a du able good o which he a aches alue 0. The buye has
2Beccu i and Mölle (2018) lies somewha in be ween hese wo s ands because hey ake limi s o hei
ini e-ho izon esul s o d aw conclusions abou he in ini e-ho izon game. They do no show ha his limi
co esponds o he selle ’s e enue-maximizing equilib ium in he in ini e-ho izon game.
3O he pape s, like Wolinsky (1991), McA ee and Wiseman (2008), and Boa d and Pycia (2014), s udy
a ia ions on Coase’s o iginal p oblem and hei implica ions o Coase’s conjec u e. Rela edly, B zus owski,
Geo giadis-Ha is, and Szen es (2023) show ha sma con ac s help he selle a oid he Coase conjec u e.
4Recen ly, Peski (2022) s udies al e na ing ba gaining games whe e playe s can o e menus.
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Theo e ical Economics 19 (2024) Op imal mechanism o he sale o a du able good 869
p i a e in o ma ion: be o e he in e ac ion wi h he selle s a s, she obse es he alue
∈{ L, H}≡V,wi h0<
L<
H.Le ≡ H− Ldeno e he di e ence in alues. Le
μ0deno e he p obabili y ha he buye ’s alue is Ha he beginning o he game. In
wha ollows, we deno e by (V) he se o dis ibu ions on V.
An alloca ion in pe iod is a pai (q,x)∈{0, 1}×R,whe eqindica es whe he he
good is aded (q=1) o no (q=0), and xis a paymen om he buye o he selle . Le
Adeno e he se o alloca ions.5The game ends he i s ime he good is sold.
Payo s a e as ollows. I in pe iod , he alloca ion is (q,x), he low payo s a e
uB(q,x, )= q −xand uS(q,x)=x o he buye and he selle , espec i ely. The selle
and he buye maximize he expec ed discoun ed sum o low payo s. They sha e a
common discoun ac o δ∈(0, 1).
Mechanisms: To in oduce he iming o he game, we i s de ine he selle ’s ac ion
space. In each pe iod, he selle o e s he buye a mechanism. Following Do al and
Sk e a (2022), we de ine a mechanism as ollows. A mechanism, M=(MM,SM,ϕM),
consis s o a se o inpu messages MM,ase o ou pu messages SM, and a ansi ion
p obabili y ϕM om MM o SM×A.6Fo ins ance, MMcan be he se o buye alues, V,
and SMbe he se o selle belie s abou he buye ’s alue, (V). In his case, he mech-
anism associa es o each epo a dis ibu ion o e belie s and alloca ions. We endow
he selle wi h a collec ion (Mi,Si)i∈Io inpu and ou pu messages in which each Mi
con ains a leas wo elemen s, and each Sicon ains (V).7Deno e by MI he se o all
mechanisms wi h message se s (Mi,Si)i∈I.8
Mechanism-selec ion game: The selle ’s p io μ0and he collec ion (Mi,Si)i∈Ide-
ine a mechanism-selec ion game, deno ed GI(μ0), as ollows. In each pe iod ,aslong
as he good has no been sold, he game p oceeds as ollows. Fi s , he selle and he
buye obse e he ealiza ion o a public andomiza ion de ice, ω∼U[0, 1].Second, he
selle o e s he buye a mechanism, M. Obse ing he mechanism, he buye decides
whe he o pa icipa e o no . I she does no pa icipa e in he mechanism, he good is
no sold and no paymen s a e made. I she ins ead chooses o pa icipa e, she sends a
message m∈MM, which is unobse ed by he selle . An ou pu message and an alloca-
ion, (s,q,x),a ed awn omϕM(·|m)and a e obse ed by bo h he selle and he buye .
I he good is no sold, he game p oceeds o pe iod +1.
His o ies: The game GI(μ0)has wo ypes o his o ies: public and p i a e. Public
his o ies cap u e wha he selle knows h ough pe iod : he pas ealiza ions o he
5E en i he se o alloca ions is {0, 1}×R, we allow he selle o o e andomiza ions on A, and hence
induce ac ional assignmen s o he good.
6Th oughou , we assume ha MMand SMa e Polish spaces, ha is, hey a e sepa able, comple ely
me izable opological spaces. No e ha he se o alloca ions Ais also a Polish space and, he e o e, SM×A
is a Polish space. Fo a Polish space X,le (X)deno e he se o Bo el measu es on X.Weendow(X)
wi h he weak∗ opology. Thus, (X)is also a Polish space (Alip an is and Bo de (2013)). Fo any wo
measu able spaces Xand Y, a mapping ζ:X→ (Y)is a ansi ion p obabili y om X o Yi o any
measu able C⊆Y,ζ(C|x)is a measu able eal alued unc ion o x∈X.
7Because Vis ini e, aking MM o be ini e is wi hou loss o gene ali y (Do al and Sk e a,2022).
8We es ic he selle o choose mechanisms wi h inpu and ou pu messages in (Mi,Si)i∈I o ha e a
well-de ined ac ion space o he selle . This allows us o ha e a well-de ined se o de ia ions, a oiding
se - heo e ic issues ela ed o sel - e e en ial se s. The analysis in Do al and Sk e a (2022) shows ha he
choice o he collec ion plays no u he ole in he analysis.
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870 Do al and Sk e a Theo e ical Economics 19 (2024)
public andomiza ion de ice, his pas choices o mechanisms, he buye ’s pa icipa ion
decisions, and he ealized ou pu messages and alloca ions. We le h deno eapublic
his o y h ough pe iod and le H deno e he se o all such his o ies. Ins ead, p i a e
his o ies cap u e wha he buye knows h ough pe iod . Fi s , he buye knows he
public his o y o he game and he inpu messages in o he mechanism (hence o h, a
buye his o y). Second, he buye also knows he p i a e in o ma ion. We le h
Bdeno e a
buye his o y h ough pe iod and le H
B(h )deno e he se o buye his o ies consis en
wi h public his o y h .Thus,V×H
B(h )deno es he se o p i a e his o ies consis en
wi h public his o y h .
S a egies and belie s: A beha io al s a egy o he selle is a collec ion o measu -
able mappings ≡( )∞
=0, whe e o each pe iod and each public his o y h , (h )de-
sc ibes he selle ’s (possibly andom) choice o mechanism a h .9Simila ly, a beha io al
s a egy o he buye is a collec ion o measu able mappings (π ( ,·), ( ,·))∞
=0,whe e
o each pe iod , each p i a e his o y ( ,h
B), and each mechanism, M ,π ( ,h
B,M )
desc ibes he buye ’s pa icipa ion decision, whe eas ( ,h
B,M )desc ibes he buye ’s
choice o inpu messages in he mechanism, condi ional on pa icipa ion. We deno e
he uple (π ( ,·), ( ,·)) by (π , ), and he collec ion (π , )∞
=0by (π , ).
A belie o he selle a he beginning o ime ,his o yh , is a dis ibu ion μ (h )∈
(V×H
B(h )). The belie sys em, (μ )∞
=0, is deno ed by μ.
Solu ion concep : We a e in e es ed in s udying he pe ec Bayesian equilib ium
(hence o h, PBE) payo s o his game, whe e PBE is de ined in o mally as ollows. An
assessmen , ,(π , ) ∈V,μ, is a PBE i he ollowing hold:
1. ,(π , ) ∈V,μsa is ies sequen ial a ionali y, and
2. μsa is ies Bayes’ ule whe e possible.
Appendix Econ ains he o mal s a emen . Fo now, we no e ha i he selle ’s s a egy
space was ini e and he mechanisms used by he selle had ini e suppo , hen his
coincides wi h he de ini ion in Fudenbe g and Ti ole (1991b).10
The p io μ0 oge he wi h he s a egy p o ile (,(π , ) ∈V)induce a dis ibu ion
o e he e minal nodes V×H∞
B. We a e in e es ed ins ead on he dis ibu ion i induces
o e he payo - ele an ou comes, V×A∞. We say ha a dis ibu ion η∈(V×A∞)is
a PBE ou come i a PBE assessmen ,(π , ) ∈V,μexis s ha induces η.Wedeno e
by O∗
I(μ0) he se o PBE ou comes and by E∗
I(μ0)⊆R3 he se o PBE payo s o GI(μ0).
We deno e a gene ic elemen o E∗
I(μ0)by u≡(uL,uH,uS),whe euSis he selle ’s payo
and uL,uHdeno e he buye ’s payo when he alue is L, H, espec i ely.
Theo em 1cha ac e izes O∗
I(μ0)and E∗
I(μ0). In pa icula , we show ha he essen-
ially unique equilib ium ou come can be achie ed ia a sequence o pos ed p ices so
ha he selle o a du able good can do no be e and no wo se han by using pos ed
p ices.
9As we explain in Appendix E,MIis a Polish space—which we endow wi h i s Bo el σ-algeb a—and we
can ollow Aumann (1964) when de ining he selle ’s s a egy.
10The only di e ence be ween Bayes’ ule whe e possible and consis ency in sequen ial equilib ium is
he ollowing. Unde PBE, he selle can assign ze o p obabili y o one o he buye ’s alues and hen, a e
he buye de ia es, can assign posi i e p obabili y o ha same alue.
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Theo e ical Economics 19 (2024) Op imal mechanism o he sale o a du able good 871
2.1 Re ela ion p inciple
The game GI(μ0)is no simple o analyze o a leas wo easons. Fi s , he selle ’s ac-
ion space is la ge, and a p io i, i is no clea which mechanisms could be uled ou om
conside a ion. Second, ixing a selle ’s s a egy, and hence a sequence o mechanisms
aced by he buye , we s ill need o unde s and he buye ’s bes esponse in he game
induced by he sequence o mechanisms.
Le G(μ0)deno e he same game in he p e ious sec ion, excep ha in each pe iod
he selle ’s ac ion space is he se o canonical mechanisms, deno ed by MC, and de ined
as ollows. MCis he se o all mechanisms whe e he se o inpu and ou pu messages
a e he se o buye ’s alues and o selle ’s belie s abou he buye ’s alue, espec i ely.
Tha is, (M,S)=(V,(V)).
Le O∗(μ0)deno e he se o PBE ou comes and E∗(μ0)deno e he se o PBE pay-
o s o G(μ0). In wha ollows, a subse o he se o canonical mechanisms has special
signi icance: he se o di ec Blackwell mechanisms. A di ec Blackwell mechanism is a
canonical mechanism ϕ:V→ ((V)×A) ha can be decomposed in o a Blackwell
expe imen , β:V→ ((V)),andanalloca ion ule,α:(V)→ (A).11
Lemma 1summa izes he key implica ions o Do al and Sk e a (2022) o ou analy-
sis, which we explain below.
Lemma 1(Do al and Sk e a (2022)). Fo any PBE ou come o any mechanism-selec ion
game GI(μ0), an ou come-equi alen PBE o game G(μ0)exis s. Tha is, IO∗
I(μ0)=
O∗(μ0).
Mo eo e , le η∈O∗(μ0). Then a PBE assessmen ,(π , ) ∈V,μo G(μ0)exis s
ha induces ηand sa is ies he ollowing p ope ies:
(a) Fo all his o ies h , he buye pa icipa es in he mechanism o e ed by he selle a
ha his o y and u h ully epo s he ype, wi h p obabili y 1;
(b) Fo all his o ies h , i he mechanism o e ed by he selle a h ou pu s pos e io μ,
he selle ’s upda ed equilib ium belie s abou he buye ’s alue coincide wi h μ;
(c) Fo all his o ies h , he mechanism o e ed by he selle a h is a di ec Blackwell
mechanism;
(d) The buye ’s s a egy depends only on he p i a e alue and he public his o y.
Lemma 1has se e al implica ions. Pa (a) o Lemma 1implies he mechanisms cho-
sen by he selle in equilib ium mus sa is y a pa icipa ion cons ain and an incen i e
compa ibili y cons ain o each buye alue and each public his o y. As in he case o
commi men o long- e m mechanisms, pa (a) simpli ies he analysis o he buye ’s
11Tha is, o all measu able subse s U⊂(V)and A⊂A, we ha e ha o all ∈V,
ϕU×A| =U
αA|μβ(dμ| ).
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872 Do al and Sk e a Theo e ical Economics 19 (2024)
beha io , by educing i o a se ies o cons ain s he selle ’s equilib ium o e o a mech-
anism mus sa is y (see Equa ions (PCh , )and(ICh , , )inSec ion4.1).
Pa (b) implies ha he mechanism’s ou pu message encodes all o he in o ma ion
ha he selle has in equilib ium abou he buye ’s alue. In pa icula , condi ional on
obse ing he ou pu message, he alloca ion ca ies no mo e in o ma ion abou he
buye ’s alue. As a consequence, condi ional on he ou pu message, he alloca ion can
be d awn independen ly o he buye ’s epo . This, in u n, deli e s he decomposi ion
o ϕM as a di ec Blackwell mechanism desc ibed in pa (c).
Pa (c) implies ha he choice o mechanism a his o y h can be equi alen ly
hough o as he choice o a Blackwell expe imen , βM , and an alloca ion ule, αM .
Di ec Blackwell mechanisms allow us o sepa a ely op imize on he alloca ion gi en a
pa icula expe imen , and hen op imize on he expe imen . As in he li e a u e on in-
o ma ion design, i is con enien o wo k wi h he dis ibu ion o e pos e io s induced
by he expe imen βM ,whichwedeno ebyτM and is de ined as ollows. Fo all Bo el
subse s U⊆(V),
U
τM (dμ +1)=
∈V
μ h ( )βM U| ,(BC
μ (h ))
whe e μ (h )∈(V)is heselle ’sbelie abou hebuye ’s aluea h . Fu he mo e, as
in he li e a u e on mechanism design wi h quasilinea u ili ies, we can w i e αM (·|μ)as
an expec ed paymen , xM (μ), and a p obabili y o ade, qM (μ).
Pa (d) implies he se o PBE payo s o G(μ0)coincides wi h he se o Public PBE
payo s o G(μ0)(A hey and Bagwell (2008)). Relying on Ab eu, Pea ce, and S acche i
(1990), A hey and Bagwell (2008) show ha Public PBE ha e a ecu si e s uc u e and
we use his p ope y o a gue he assessmen we de ine in Sec ion 3is indeed a PBE
assessmen .
The es o he pape s udies he equilib ium ou comes and payo s o G(μ0)and
when we e e o a PBE assessmen , we mean one ha sa is ies he condi ions o
Lemma 1.
Rema k 1. Below, we abuse no a ion in he ollowing wo ways. Fi s , because alues a e
bina y, we can hink o an elemen in (V)(a dis ibu ion o e Land H)asanelemen
o he in e al [0, 1]( he p obabili y assigned o H). We use he la e o mula ion in
wha ollows. Tha is, whe eas he mechanism ou pu s a dis ibu ion o e Land H,we
index his dis ibu ion by he p obabili y o H. Second, e en hough β(·| )is a measu e
o e (V)(in his case a c.d. .), we some imes w i e β(μ| )when βhas an a om a μ.
3. Main esul
Sec ion 3con ains he main esul o he pape : Theo em 1cha ac e izes he equilib-
ium ou comes and payo s o G(μ0). To s a e Theo em 1, we p oceed as ollows. Fi s ,
we in o mally desc ibe he pos ed-p ices assessmen , ∗,(π∗
, ∗
) ∈V,μ∗, which is sin-
gled ou by he p oo o Theo em 1. Second, we explain why he ou come induced by
∗,(π∗
, ∗
) ∈V,μ∗can be implemen ed ia a sequence o pos ed p ices. Finally, we
s a e Theo em 1.
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Theo e ical Economics 19 (2024) Op imal mechanism o he sale o a du able good 879
P og am (OPT) allows us o ci cum en his di icul y: Because in (OPT) he selle can
choose bo h he mechanism and he con inua ion payo s, he selle does no need o
conside how his choice o mechanism may ad e sely a ec his con inua ion payo s.
Sec ion 4.1 con ains he de ails o he cons uc ion o he i ual su plus de ined in
Equa ion (4) and e i ies ha i is an uppe bound on he selle ’s equilib ium payo
(Lemma 2). The eade in e es ed in he p ope ies o he solu ion o (OPT)andhowi is
used in he p oo o Theo em 1can p oceed o Sec ion 4.2 wi h li le loss o con inui y.
4.1 De i a ion o he i ual su plus
We now de i e he i ual su plus de ined in Equa ion (4) by elying on Lemma 1and
show ha i is an uppe bound on he selle ’s equilib ium payo . To do so, conside a
PBE assessmen , ,(π , ) ∈V,μ. Fix a his o y h and le M deno e he mechanism
o e ed by he selle a h unde he assessmen . Le (τM ,qM )deno e he dis ibu ion
o e pos e io s and he p obabili y o ade associa ed wi h M . Fu he mo e, he PBE
assessmen speci ies con inua ion payo s uM when he selle o e s mechanism M a
his o y h . In wha ollows, we show he ollowing.
Lemma 2. The i ual su plus o mechanism M is an uppe bound on he sum o he
selle and he low- alua ion buye ’s payo s. Tha is,
USh +ULh ≤VS
τM ,qM ,uM ,μ h .
Lemma 2is he analogue o he esul in mechanism design ha he mechanism’s
alloca ion oge he wi h he lowes ype’s u ili y in he mechanism pin down he selle ’s
maximum e enue. Because he low- alua ion buye ’s payo is nonnega i e, Lemma 2
also implies ha he i ual su plus o M is an uppe bound on he selle ’s payo . The
inequali y in Lemma 2shows ha hemechanism’s i ualsu plusis hemaximumpay-
o he selle and he low- alua ion buye can sha e. As we explain in Sec ion 5,once
we show he selle cap u es he en i e y o he i ual su plus, he inequali y in Lemma 2
implies low- alua ion buye ’s payo is 0 in any equilib ium.
To see why Lemma 2holds, no e ha Lemma 1implies ha he selle ’s equilib ium
payo a h ,US(h ), can be w i en as
USh =(V)xM (μ +1)+1−qM (μ +1)δuM
S(μ +1)τM (dμ +1),(7)
whe e uM
S(μ +1)is sho hand no a ion o he selle ’s con inua ion payo when a his-
o y h , he o e s M and he ou pu message is μ +1.15 Equa ion (7) uses Lemma 1as
ollows. Fi s , he selle ’s payo om o e ing M is w i en unde he assump ion ha
he buye pa icipa es in he mechanism and u h ully epo s he alue. Second, i
uses Lemma 1 o w i e he mechanism in e ms o he dis ibu ion o e pos e io s τM
and he alloca ion (qM ,xM ).
15This con inua ion payo can also depend on h , bu we omi his dependence o simpli y no a ion.
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880 Do al and Sk e a Theo e ical Economics 19 (2024)
In pa icula , he mechanism M oge he wi h he con inua ion payo s (uM
L,uM
H)
sa is y he ollowing cons ain s. Fi s , he buye p e e s o pa icipa e in he mechanism
o bo h he alues, ha is, o ∈{ L, H} he ollowing holds:
(V) qM (μ +1)−xM (μ +1)+1−qM (μ +1)δuM
(μ +1)βM (dμ +1| )
≥uM
(∅),(PC
h , )
whe e he le -hand side o Equa ion (PCh , )is hebuye ’spayo a h ,U (h ),and
uM
(∅)is sho hand no a ion o he buye ’s con inua ion payo when a his o y h , he
selle o e s M and he buye ejec s. Also, he buye p e e s o u h ully epo he
alue o he mechanism, ha is, o ∈{ L, H}and = , he ollowing holds:
(V) qM (μ +1)−xM (μ +1)
+1−qM (μ +1)δuM
(μ +1)βM (dμ +1| )−βM dμ +1|
≥0. (ICh , , )
The abo e exp essions implici ly use Lemma 1in one mo e way. By Lemma 1, he
assessmen ,(π , ) ∈V,μis a public PBE, so ha he con inua ion payo ec-
o uM (μ +1)≡(uM
L(μ +1),uM
H(μ +1),uM
S(μ +1)) is an equilib ium payo ec o o
G(μ +1). Fo mally, uM (μ +1)∈E∗(μ +1).
Equa ions (PCh , )and(ICh , , ) a e analogous o he pa icipa ion and incen i e
compa ibili y cons ain s one would ob ain in mechanism design excep o he ollow-
ing: The pa icipa ion cons ain is po en ially ype-dependen ( he igh -hand side is
uM
(∅)). To sides ep his challenge, we igno e he igh -hand side o he low- alua ion
buye ’s pa icipa ion cons ain . Ins ead, we use he ollowing iden i y o sol e o he
ans e s gi en he low- alua ion buye ’s u ili y in he mechanism, UL(h ):
(V) LqM (μ +1)−xM (μ +1)
+1−qM (μ +1)δuM
L(μ +1)βM (dμ +1| L)=ULh .(8)
Equa ion (8) can be used o ew i e Equa ion (ICh , , ) o = Has ollows:
(V) HqM (μ +1)−xM (μ +1)+1−qM (μ +1)δuM
H(μ +1))βM (dμ +1| H)
≥(V) qM (μ +1)+1−qM (μ +1)δuM
H(μ +1)−uM
L(μ +1)βM (dμ +1| L)
+ULh .(9)
Equa ions (8)and(9) al eady impose cons ain s on he maximum e enue he selle
can make in pe iod when he low- alua ion buye ’s payo is UL(h ). Indeed, like in
s anda d mechanism design, he u ili y o Land he u h- elling cons ain o Hde-
e mine he maximum expec ed ans e he selle can ex ac om he buye in pe iod
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Theo e ical Economics 19 (2024) Op imal mechanism o he sale o a du able good 881
. Replacing he uppe bound on he expec ed ans e s ob ained om hese equa ions
in he selle ’s payo (Equa ion (7)), we ob ain an uppe bound on he selle ’s e enue
a h when he o e s mechanism M . I is immedia e o check ha his uppe bound
co esponds o VS
((τM ,qM ),uM ,μ (h ))−UL(h ). Lemma 2 hen ollows.
Ha ing es ablished ha (OPT) is an uppe bound on he selle ’s equilib ium pay-
o , Sec ion 4.2 s udies p ope ies o i s solu ion, which a e impo an o he p oo o
Theo em 1.
4.2 P ope ies o he solu ion o (OPT)
The main esul in his sec ion, P oposi ion 1, shows wo p ope ies o he solu ion o
(OPT) ha a e impo an o p o e ha he selle ’s unique equilib ium payo is u∗
S(μ0).
As we explain nex , a key di icul y ha (OPT) allows us o ci cum en is he analysis o
he belie dynamics in he game.
An impo an esul in he ba gaining li e a u e is he skimming lemma,which
s a es ha incen i e compa ibili y o he buye ’s beha io implies ha he expec ed dis-
coun ed p obabili y o ade o His highe han ha o L(see, e.g., Fudenbe g, Le ine,
and Ti ole (1985)). This p ope y immedia ely implies ha along he pa h o play he
selle ’s belie s all condi ional on no ade. As a consequence, p ices mus all along he
pa h o play.
Con as his o he game we analyze in which he selle o e s mechanisms ha en-
able him o design how much he obse es abou he buye ’s choices. In pa icula , he
selle can choose how as he lea ns abou he buye ’s alue condi ional on no ade; he
could e en choose o become mo e op imis ic abou he buye ’s alue condi ional on
no ade.16 On he one hand, his would allow he selle o a oid he belie dynamics
associa ed wi h pos ed p ices, and hence a oid he emp a ion o ade mo e o en wi h
Lin u u e ounds. On he o he hand, his comes a a cos . Lemma 1implies ha he
mechanism’s alloca ion is measu able wi h espec o he in o ma ion gene a ed by he
mechanism and his in o ma ion is, in u n, subjec o he Bayes’ plausibili y cons ain .
This implies ha o he selle ’s belie s condi ional on no ade o all slowly (o no all
a all), i mus be ha he selle is selling he good o Hwi h small p obabili y.
I u ns ou ha (OPT) is use ul o discipline belie dynamics. Whe eas i may no be
ob ious how o ule ou ha an equilib ium in which he selle ’s belie s may some imes
go up condi ional on no ade exis s, i u ns ou ha his is ne e he case in a solu ion
o (OPT). Indeed, as we es ablish in P oposi ion 1below, i is ne e op imal o no sell he
good and induce a belie abo e he p io . Fu he mo e, whene e μ0> μ1, condi ional
on selling he good wi h posi i e p obabili y, he selle sells he good only o he high-
alua ion buye .
16Whe eas he u h- elling equa ions (ICh , , ) can be used o de i e a “mono onici y” condi ion anal-
ogous o ha in he skimming lemma, his condi ion only implies ha on a e age he expec ed p obabili y
o ade o Hmus be highe han ha o L:
(V) qM (μ +1)+1−qM (μ +1)δuM
H(μ +1)−uM
L(μ +1)βM (dμ +1| H)−βM (dμ +1| L)≥0.
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882 Do al and Sk e a Theo e ical Economics 19 (2024)
P oposi ion 1. Suppose ha μ0≥μ1and le (τ0,q0),udeno e a solu ion o (OPT).
Then he ollowing hold:
(a) I is ne e op imal o induce a belie μ1≥μ0and no sell he good. Tha is,
[μ0,1]1−q0(μ1)τ0(dμ1)=0.
(b) Fu he mo e, i μ0> μ1and he selle induces μ1and sells he good (i.e., q0(μ1)>
0), hen μ1=1.
The p oo is in Appendix B. In wha ollows, we p o ide in ui ion o P oposi ion 1,
s a ing om pa (a). To see why no selling he good and a he same ime induce a
belie μ1≥μ0is no op imal, no e he ollowing. Fi s , associa ed o any con inua ion
payo , u(μ1), he e is a mechanism chosen by he selle when his belie is μ1, and con-
inua ion payo s o he selle and he buye in he e en ha he good is no sold. This
implies ha , condi ional on inducing a belie μ1, he selle wi h belie μ0could always
choose oday he mechanism and he con inua ion payo s associa ed wi h u(μ1)in a
solu ion o (OPT). Second, he selle wi h belie μ0below μ1pays en s o Hwi h lowe
p obabili y han he selle wi h belie μ1(a e g ouping e ms, he e m p e-mul iplying
uH(μ1)−uL(μ1)in Equa ion (6) is posi i e). I ollows ha he selle wi h belie μ0
p e e s o acc ue oday he payo om he mechanism (and con inua ion payo s) ha
induce u(μ1), con adic ing ha i is op imal o induce μ1and no sell he good.
Pa (b) ollows om he obse a ion ha a selle wi h p io abo e μ1p e e s o ade
wi h Lin a leas wo pe iods ( ecall ha ˆ
L(μ0)<0whenμ0> μ1). Thus, i is ne e
op imal o sell he good o Lwi h posi i e p obabili y oday. I ollows ha i q0(μ1)>0,
hen he selle mus assign he good o H, and hence μ1=1.
P oposi ion 1implies ha a solu ion o (OPT) ne e induces pos e io s in [μ0,1
).
This, in u n, deli e s he ollowing exp ession o he alue o (OPT), which in a sligh
abuse o no a ion, we deno e by VS
(μ0):
VS
(μ0)≡max
τ0,uτ0{1} H+[0,μ0)
δuS(μ1)+uL(μ1)
+μ1−μ0
1−μ0uH(μ1)−uL(μ1)τ0(dμ1). (10)
Equa ion (10) simply s a es ha he solu ion o (OPT) can be desc ibed by he p obabil-
i y o selling o H oday ( he p obabili y o inducing a belie μ1=1) and he p obabili y
wi h which he good is no sold and a belie below he p io is induced. One dis ibu ion
o e pos e io s is o pa icula in e es in wha ollows: he one ha spli s μ0be ween
1(wi hq0(1)=1) and μ1<μ
0(wi h q0(μ1)=0). Bayes’ plausibili y implies ha he
weigh s on 1 and μ1a e
μ0−μ1
1−μ1
and 1−μ0
1−μ1
,
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Theo e ical Economics 19 (2024) Op imal mechanism o he sale o a du able good 883
espec i ely. No e ha his is p ecisely he kind o mechanism ha he selle uses in he
pos ed-p ices assessmen . Co olla y 2shows ha hese a e essen ially he dis ibu ions
o e pos e io s ha sol e he p oblem in Equa ion (10).
Co olla y 2. The alue o (OPT), VS
(μ0), equals he alue o
max
G∈([0,μ0))max
u[0,μ0)μ0−μ1
1−μ1
H+1−μ0
1−μ1
δuS(μ1)+uL(μ1)
+μ1−μ0
1−μ0uH(μ1)−uL(μ1)G(dμ1). (11)
The p oo is in Appendix Band is a consequence o he cons ain ha τ0is Bayes’
plausible o μ0. Gi en he p eceding discussion, he e m in he squa e b acke s in-
side he in eg al in Equa ion (11) is he payo om spli ing μ0be ween 1 and μ1<μ
0.
Co olla y 2implies ha , o a ixed choice o con inua ion payo s, he solu ion o he
p oblem in Equa ion (10) is as i he selle we e andomizing o e pos e io dis ibu-
ions ha spli he p io be ween 1 (and selling he good) and μ1<μ
0(and no selling
he good). In o he wo ds, i pos e io s μ1and μ
1a e on he suppo o τ0, hen he selle
is indi e en be ween spli ing μ0be ween μ1and 1 and spli ing μ0be ween μ
1and 1.
As a consequence, o de e mine he op imal τ0i is enough o compa e he payo s o
he spli ings o μ0be ween μ1and 1 o di e en μ1.
Because condi ional on no selling he good he selle induces belie s below he p io ,
Equa ion (11) shows ha o a ixed dis ibu ion G, he selle ades o he sum uS+uL
agains he high- alua ion buye ’s en s, uH−uL, when choosing con inua ion payo s.
Indeed, i con inua ion payo s u,u∈E∗(μ1)exis such ha (uH−uL,uS+uL)(u
H−
u
L,u
S+u
L), he solu ion o he p og am in Equa ion (11) would choose uo e ui he
alue o he in eg and is highe o u. Tha is, he selle may o go maximizing uS+uL—
and hence, by Lemma 2po en ially o go he maximum con inua ion i ual su plus—
i his could lead o lowe en s o he high- alua ion buye . As we explain in Sec ion 5
below, we ci cum en his ade-o in he p oo o Theo em 1by showing ha o each
μ1<μ
0 he sum uS+uLis unique, which in u n elies on excluding he exis ence o
payo s like uand u.
5. P oo o Theo em 1:Key s eps
Sec ion 5o e iews he main s eps o he p oo o Theo em 1. Taking as gi en ha
∗,(π∗
, ∗
) ∈V,μ∗is a PBE assessmen , Sec ion 5.1 e iews he main s eps o show ha
i achie es he unique equilib ium payo o he selle and he low- alua ion buye , and
ha excep o he h eshold belie s, {μn}n≥1, a unique equilib ium ou come exis s, and
hence a unique equilib ium payo o he high- alua ion buye as well. Sec ion 5.2 e-
iews he main s eps o show ha ∗,(π∗
, ∗
) ∈V,μ∗is a PBE assessmen .
5.1 Cha ac e iza ion o he equilib ium payo s o G(μ0)
We show u∗
S(μ0) is bo h a lowe bound and an uppe bound on he selle ’s equilib ium
payo . Tha is, we show he selle can ne e do be e no wo se han i he we e limi ed
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884 Do al and Sk e a Theo e ical Economics 19 (2024)
o choose p ices in each pe iod so ha ha ing access o a iche ac ion space does no in-
c ease no dec ease he selle ’s payo . P og am (OPT)iskey oshowu∗
S(μ0) is he selle ’s
unique equilib ium payo . The p oo ha u∗
S(μ0) is an uppe bound on he selle ’s equi-
lib ium payo ollows om showing ha u∗
S(μ0) is he alue o (OPT). Ins ead, he p oo
ha u∗
S(μ0) is a lowe bound on he selle ’s payo uses a cons ained e sion o (OPT)
o hen apply he logic in Gul, Sonnenschein, and Wilson (1986): I an equilib ium in
which he selle ea ns less han u∗
S(μ0) exis s, he selle can always unde cu he p ice in
he pos ed-p ices assessmen and ea n close o u∗
S(μ0).
Using he p ope y in he pos ed-p ices assessmen ha along he pa h o play be-
lie s all, he p oo o Theo em 1p oceeds by induc ion on he in e al he selle ’s p io
belongs o. Fo each n≥0, we es ablish wo esul s. Fi s , o all μ0∈[μn,μn+1), hese
o equilib ium payo s co esponds o ha in Theo em 1(c . Equa ion (3)). Second, o
all μ0≥μn, he selle can gua an ee a ce ain payo , deno ed u∗
S(μ0,n), which coincides
wi h u∗
S(μ0) o μ0∈[μn,μn+1). This payo is ob ained by he selle emula ing he s a -
egy ha sells he good o Lin npe iods in he pos ed-p ices assessmen (Figu e 1(a)):
The selle uses a mechanism ha spli s his belie s be ween 1 and μn−1; when he be-
lie is 1, he good is sold a a p ice o L+(1−δn) , and when he belie is μn−1, he
good is no sold and play p oceeds acco ding o he pos ed-p ices assessmen . As we
explain below, his second esul is key o es ablish in he (n+1) h s ep o he induc-
ion ha he selle can gua an ee he payo om he pos ed-p ices assessmen when
μ0∈[μn+1,μn+2).
To illus a e he main s eps o he p oo ha u∗
S(μ0) is he selle ’s unique equilib ium
payo , ix n≥1.17 Suppose ha o all m<nwe ha e al eady shown ha (i) Theo em 1
holds o all μ0∈[μm,μm+1)and (ii) he selle can gua an ee u∗
S(μ0,m) o all μ0≥μm.
In wha ollows, we a gue ha (i) Theo em 1holds o μ0∈[μn,μn+1)and (ii) he selle ’s
payo is a leas u∗
S(μ0,n) o μ0≥μn.
Pos ed p ices maximize he i ual su plus: To show ha he alue o (OPT)co e-
sponds o he selle ’s payo in he pos ed-p ices assessmen , we a gue ha spli ing he
p io be ween 1 and μn−1domina es all o he spli ings. Co olla y 2implies ha his is
enough o show ha u∗
S(μ0)=VS
(μ0). In wha ollows, we i s a gue ha , condi ional
on inducing a belie μ1< μn, he selle places weigh on a mos μn−1(and on μn−2only
i μ0=μn). We hen a gue i is ne e op imal o induce a belie μ1∈[μn,μ0).
The induc i e hypo hesis—which iden i ies he con inua ion payo s below μn—and
he p ope ies o he pos ed-p ices assessmen imply ha condi ional on inducing a be-
lie in [0, μn), he solu ion o he p oblem in Equa ion (11)placesweigh ona mos
{μn−2,μn−1}. Fi s ,excep a he h esholdbelie s{μm}m≤n−1, he induc i e hypo he-
sis pins down he alue o he in eg and in Equa ion (11) o μ1< μn. Second, because
a he h eshold belie s he selle ’s and he low- alua ion buye ’s payo a e unique, he
selle hen chooses he con inua ion payo ha minimizes he high- alua ion buye ’s
en s, u∗
H(·), as is he case in he pos ed-p ices assessmen . Thi d, elying on he p ope -
ies o he pos ed-p ices assessmen , Lemma C.2 shows ha inducing belie s in [0, μn)
o he han {μn−2,μn−1}is no op imal. Fu he mo e, μn−2can be in he suppo o τ0
17The p oo also e i ies ha Theo em 1holds o μ0∈[0, μ1)(see Appendix C.2.1).
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Theo e ical Economics 19 (2024) Op imal mechanism o he sale o a du able good 885
only i μ0=μn. This is in ui i e. On he one hand, i can only be op imal o induce
he h eshold belie s {μm}m≤n−1:Fo m≤n−1, bo h μ1∈(μm,μm+1)and μmimply ha
ade happens wi h Lin mpe iods; howe e , inducing μmallows he selle o ade wi h
Hwi h a highe p obabili y. On he o he hand, he indi e ence condi ion ha de ines
he h eshold belie s implies ha because μ0≥μn, inducing belie s below μn−2canno
be op imal (see Lemma C.1).
To conclude he p oo ha u∗
S(μ0)is an uppe bound on VS(μ0), we show in
Lemma C.3 ha inducing pos e io s in [μn,μ0)is no op imal. Whe eas P oposi ion 1
implies ha a he solu ion o (OPT) he selle ’s belie s go down condi ional on no ade,
i does no say how as hey go down. Indeed, i could be op imal o induce belie s in
[μn,μ0)i a he induced belie s con inua ion equilib ia exis whe e he selle somehow
manages o slowly ade wi h Hso as o maximally delay ade wi h L.Asweshowin
Appendix C.2, his canno be op imal o μ0close o μn:Theclose oμn, he smalle
he p obabili y ha he selle wi h belie μ0can ade wi h Hi condi ional on no ade,
his belie s mus emain abo e μn. I ollows ha μ∗
0small enough exis s such ha i he
selle ’s p io μ0is in [μn,μ∗
0], i is be e o ade wi h Lin npe iods in exchange o
inc easing he p obabili y o ading wi h H oday.
Mo e o mally, no e ha Lemma C.2 implies ha o μ0∈[μn,μn+1), he alue o
(OPT), VS(μ0), is bounded abo e by
VS
(μ0)≤maxu∗
S(μ0),μ0−μn
1−μn
H+1−μ0
1−μn
δVS
[μn,μ0), (12)
whe e VS
[μn,μ0)is he sup emum o he alue unc ion VS
(μ)on [μn,μ0].Toseewhy
Equa ion (12) holds, no e he ollowing. Fi s , i he solu ion o (OPT) places posi i e
mass below μn, Lemma C.2 implies ha VS
(μ0)=u∗
S(μ0), since he selle places weigh
on μn−1(o μn−2i μ0=μn). The equali y ollows om Co olla y 2and ha u∗
S(μ0)co -
esponds o spli ing μ0be ween 1 and μn−1. Second, i he solu ion o (OPT)places
weigh on [μn,μ0), he second e m on he igh -hand side o Equa ion (12) is an uppe
bound o VS(μ0). A e all, (i) (μ0−μn)/(1−μn)is he la ges weigh ha can be as-
signed o Hwhile s ill emaining on [μn,μ0)and (ii) he emaining weigh co esponds
o some μ1∈[μn,μ0)wi h payo
uS(μ1)+uL(μ1)+(1−μ1)μ1
1−μ1
−μ0
1−μ0uH(μ1)−uL(μ1)
≤uS(μ1)+uL(μ1)≤VS
[μn,μ0),
whe e he i s inequali y ollows om μ1<μ
0and he second om Lemma 2and he
de ini ion o VS
[μn,μ0) oge he wi h μ1∈[μn,μ0).
Fo μ∗
0close o μn, he p obabili y o ading wi h H oday and a he same ime
emaining abo e μnis small and u∗
S(μ0)a ains he maximum on he igh -hand side
o Equa ion (12) o μ0∈[μn,μ∗
0](see Lemma C.3 o de ails). In o he wo ds, μ∗
0small
enough exis s such ha o all μ0∈[μn,μ∗
0], he alue o (OPT), VS
(μ0), coincides wi h
he selle ’s payo in he pos ed-p ices assessmen , u∗
S(μ0). Replacing μnwi h μ∗
0in
Equa ion (12), one can a gue ha o belie s μ0close enough o μ∗
0,u∗
S(μ0)is also an
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886 Do al and Sk e a Theo e ical Economics 19 (2024)
uppe bound on he selle ’s payo . P oceeding his way, one es ablishes ha u∗
S(μ0)is
an uppe bound on he alue o (OPT) o all belie s μ0∈[μn,μn+1).
Selle can gua an ee he payo om selling o Lin npe iods: We show in P oposi-
ion C.2 ha he selle can gua an ee he payo u∗
S(μ0,n) o μ0≥μn, and hence he
payo o he pos ed-p ices assessmen o μ0∈[μn,μn+1). The logic is simila o ha in
Gul, Sonnenschein, and Wilson (1986): We show ha he selle can always unde cu he
p ice in he pos ed-p ices assessmen —say, by o e ing a mechanism ha sells he good
o L+(1−δn) −δF o some small F>0—and ea n close o u∗
S(μ0,n).
P o ided he high- alua ion buye pa icipa es in he mechanism wi h posi i e
p obabili y (no e ha he low- alua ion buye always ejec s), we show ha Fsmall
enough can be chosen so ha he selle ’s belie s condi ional on ejec ion a e μn−1.Tha
belie s condi ional on ejec ion a e μn−1, in u n, implies he selle ea ns a payo close
o u∗
S(μ0,n): Fi s , i implies ha he high- alua ion buye ’s accep ance p obabili y coin-
cides wi h he p obabili y o selling he good in he pos ed-p ices assessmen . Second,
because μn−1< μn, he induc i e hypo hesis implies he selle ’s con inua ion payo s
coincide wi h hose in he pos ed-p ices assessmen . Thus, i su ices o show ha he
selle can gua an ee ha he high- alua ion buye pa icipa es in he mechanism.
P oposi ion 2is key o showing ha he high- alua ion buye does no ejec he
mechanism wi h p obabili y 1.
P oposi ion 2 (Buye ’s maximal en s o μ0≥μn). Fo all μ0≥μn,uH≤δn−1 .
The p oo is in Appendix C.3. P oposi ion 2implies he high- alua ion buye canno
ejec he p ice o L+(1−δn )−δF wi h p obabili y 1, as doing so can yield a payo o
a mos δδn−1 . Toge he wi h he a gumen in he p eceding pa ag aph, we conclude
ha as Fbecomes small he selle can secu e u∗
S(μ0,n) o μ0≥μn.
To p o e P oposi ion 2, we show ha i he high- alua ion buye makes a leas
δn−1 , he selle makes a mos u∗
S(μ0,n−1)(Lemma C.5). Because by he induc i e
hypo hesis, he selle can gua an ee u∗
S(μ0,n−1), and we conclude he high- alua ion
buye can make a mos δn−1 . To show Lemma C.5, we i s a gue ha whene e he
high- alua ion buye makes a leas δn−1 , he selle ’s payo is bounded abo e by he
alue o a cons ained e sion o (OPT) s a ed in Lemma C.4. In his p og am, he selle
maximizes he i ual su plus subjec o he cons ain ha he high- alua ion buye ob-
ains a leas δn−1 . Relying on P oposi ion 1and ha he alue o (OPT)a μnis he
selle ’s payo in he pos ed-p ices assessmen , Lemma C.5 shows ha he alue o his
cons ained p og am is exac ly u∗
S(μ0,n−1).
The uppe and lowe bound esul s o μ0∈[μn,μn+1)imply ha u∗
S(μ0) is he selle ’s
unique equilib ium payo . This is key o show ha he buye ’s payo is as in Theo em 1.
Buye ’s payo o μ0∈[μn,μn+1):We i s a gue ha he low- alua ion buye ’s payo
is 0 in any equilib ium.
P oposi ion 3(uL=0 o μ0∈[μn,μn+1)). Le μ0∈[μn,μn+1)and suppose he ollow-
ing hold. Fi s , uS≥u∗
S(μ0) o all u∈E∗(μ0).Second,u∗
S(μ0)=VS
(μ0). Then, o all
u∈E∗(μ0),uL=0.
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Theo e ical Economics 19 (2024) Op imal mechanism o he sale o a du able good 887
In o he wo ds, when, as we ha e a gued abo e, he selle can cap u e he en i e y o
he maximum i ual su plus, he e is no hing le o he low- alua ion buye .
P oo . Weha e ha o allu∈E∗(μ0), he ollowing holds:
u∗
S(μ0)+uL≤uS+uL≤u∗
S(μ0),
whe e he i s inequali y ollows om uS≥u∗
S(μ0), and he second inequali y ollows
om Lemma 2and ha u∗
S(μ0)is he alue o (OPT). I ollows ha uL=0 o allu∈
E∗(μ0).
As we show in Appendix C.2, ha he selle can cap u e he maximum i ual su plus
also implies he uniqueness o he high- alua ion buye ’s payo excep a μn,as he
solu ion o (OPT) is unique excep a μn.
5.2 ∗,(π∗
, ∗
) ∈V,μ∗is an equilib ium assessmen
The analysis so a has elied on he obse a ion ha (0, u∗
H(μ0),u∗
S(μ0)) is an equilib-
ium payo . The es o he p oo o Theo em 1shows ha ∗,(π∗
, ∗
) ∈V,μ∗is an
equilib ium assessmen (see Appendix D).Todo his,we i s comple e heequilib ium
assessmen by speci ying he selle ’s and he buye ’s s a egy a e e e y his o y (see Ap-
pendices D.1–D.2). We hen show ha gi en belie s and con inua ion payo s, nei he
he buye no he selle ha e a one sho de ia ion (Appendix D.3). The esul s in A hey
and Bagwell (2008) imply ha his is enough o conclude ha ∗,(π∗
, ∗
) ∈V,μ∗is an
equilib ium assessmen .
Selle ’s s a egy: Excep o he cu o belie s {μn}n≥1, we speci y ha he selle plays
he mechanism desc ibed in he pos ed-p ices assessmen on and o he pa h o play.
Ins ead, he selle ’s s a egy o he pa h o play when his belie s a e in {μn}n≥1needs o
be de e mined join ly wi h he buye ’s s a egy, o which we u n nex .
Buye ’s s a egy (Appendix D.1)To comple e he buye ’s s a egy, we i s classi y mech-
anisms acco ding o whe he hey sa is y he pa icipa ion and u h- elling cons ain s
o he buye gi en he con inua ion payo s unde ∗,(π∗
, ∗
) ∈V,μ∗.Fo mecha-
nisms ha sa is y hese cons ain s, we speci y ha he buye indeed pa icipa es and
u h ully epo s he alue o he mechanism.
To speci y he buye ’s s a egy o mechanisms ha ail o sa is y ei he cons ain ,
one needs o de e mine simul aneously he buye ’s bes esponse and he selle ’s belie s
condi ional on obse ing ei he he buye ejec he mechanism, o he buye accep he
mechanism and he ou pu message ha esul s om he buye ’s epo . On he one
hand, he buye ’s con inua ion payo depend on he selle ’s belie s, which a e de e -
mined by he buye ’s s a egy. On he o he hand, whe he he buye ’s s a egy is a bes
esponse depends on he con inua ion payo . We use he esul s in Simon and Zame
(1990) o sol e o his ixed poin . I is a his poin whe e he possibili y ha he selle
andomizes when indi e en be ween ading wi h Lin no n−1 pe iods a ises o
ensu e ha he buye ’s bes esponse is well-de ined.
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888 Do al and Sk e a Theo e ical Economics 19 (2024)
By speci ying he buye ’s s a egy in he way desc ibed abo e, we ensu e ha he
buye is bes esponding o he selle ’s s a egy gi en he con inua ion payo . I emains
o show ha he selle has no one-sho de ia ions.
The selle has no one-sho de ia ions (Appendix D.3): To show ha he selle does no
ha e one-sho de ia ions, we ely once again on (OPT). Fo conc e eness, suppose we
a e a a his o y h and le μ deno e he selle ’s belie a h .
We show in Appendix D.3 ha he payo om any de ia ion a his o y h is bounded
abo e by he alue o (OPT) e alua ed a μ0=μ subjec o he cons ain ha o pos-
e io belie s μ +1below μ , he con inua ion payo s o he buye and he selle a e
gi en by (0, u∗
H(μ +1),u∗
S(μ +1)). The esul s in Sec ion 5.1 imply ha he alue o his
p og am coincides wi h u∗
S(μ ). Since his is he selle ’s payo unde he equilib ium
s a egy a h , i ollows ha he selle has no one-sho de ia ions a h .
Two obse a ions a e key o show ha his uppe bound holds. The i s is one we
ha e al eady used in he solu ion o (OPT): Fo h eshold belie s μnbelow μ , he selle
wi h p io belie μ p e e s he con inua ion payo ec o in which he high- alua ion
buye ecei es he lowes equilib ium payo , u∗
H(μn). Thus, by selec ing con inua ion
payo s in his way, we exagge a e he payo ha he selle can gua an ee om a de ia-
ion. The second is ha any mechanism M oge he wi h he buye ’s bes esponse o
M de ine a new mechanism, M
, ha sa is ies he buye ’s pa icipa ion and u h- elling
cons ain gi en he con inua ion payo s associa ed o M . Thus, we can use mecha-
nism M
oge he wi h he con inua ion payo s speci ied by he assessmen when M is
o e ed o bound he e enue om M by i s i ual su plus. This comple es he desc ip-
ion o he main s eps in he p oo o Theo em 1.
6. Conclusions
This is he i s pape o cha ac e ize all equilib ium ou comes in an in ini e-ho izon,
mechanism-selec ion game be ween an unin o med designe and a p i a ely in o med
agen wi h pe sis en p i a e in o ma ion unde limi ed commi men . We do so by ma -
ying insigh s om he li e a u es on ba gaining and mechanism design. Following he
esul s in ou p e ious wo k, Do al and Sk e a (2022), we endow he selle wi h a class
o mechanisms ha enables he selle o design his pos e io belie s abou he buye ’s
alue. The combina ion o mechanism design and in o ma ion design elemen s was key
in ob aining a ac able cha ac e iza ion. The e ela ion p inciple in Do al and Sk e a
(2022) also allows us o simpli y he buye ’s equilib ium beha io , so ha o he mos
pa we we e able o ocus on he s a egic conside a ions ha pe ain o he selle .
P ope ies o he solu ion o (OPT) can p o ide a use ul benchma k o he analysis
o he designe ’s bes equilib ium payo in o he se ings wi h quasilinea u ili y. In such
se ings, he i ual su plus is an uppe bound on he designe ’s equilib ium payo . Fu -
he mo e, when he e is a con inuum o ypes, he applica ion o he en elope heo em
implies ha he designe ’s payo can be ep esen ed as he i ual su plus. P og am
(OPT) is exac ly like he elaxed p og am in s anda d mechanism design: I a solu ion
o (OPT) sa is ies he igno ed cons ain s, hen a PBE o he mechanism-selec ion game
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Theo e ical Economics 19 (2024) Op imal mechanism o he sale o a du able good 895
P oo o Lemma C.1. Recall ha μ0=0 and as a gued below Equa ion (C.2), μ1=
L/ H. Suppose we ha e shown ha μn−1> μn−2. We show ha μn> μn−1.
To do so, ix μ≥μn−1. We claim ha he di e ence
n(μ;μn−1,μn−2)
=μ−μn−1
1−μn−1
H
+1−μ
1−μn−1
δu∗
S(μn−1)+(1−μn−1)μn−1
1−μn−1
−μ
1−μu∗
H(μn−1)
−μ−μn−2
1−μn−2
H+1−μ
1−μn−2
δu∗
S(μn−2)
+(1−μn−2)μn−2
1−μn−2
−μ
1−μu∗
H(μn−2),(C.4)
is inc easing in μ.No e ha nis di e en iable in μ,and
∂
∂μn(μ;μn−1,μn−2)= H(μn−1−μn−2)
(1−μn−1)(1−μn−2)
−δ H
μn−1−μn−2
(1−μn−1)(1−μn−2)+δδn−2−δn−1 H>0.
The cu o μnis de ined by n(μn;μn−1,μn−2)=0. No e ha μn= μn−1i n≥1. I μn=
μn−1, hen
0=n(μn;μn−1,μn−2)=δu∗
S(μn−1)−u∗
S(μn−1)<0,
since δ<1. Because n(·;μn−1,μn−2)is inc easing, we conclude ha μn> μn−1.
Ha ing es ablished his, we can now de ine he selle ’s payo a any his o y h unde
∗,(π∗
, ∗
) ∈V,μ∗.I μ∗
(h )∈[μn,μn+1), hen
U∗
Sh
=μ∗
h −μn−1
1−μn−1
H+1−μ∗
h
1−μn−1
δu∗
S(μn−1)
+(1−μn−1)μn−1
1−μn−1
−μ∗
h
1−μ∗
h u∗
H(μn−1)
≡u∗
Sμ∗
h .(C.5)
In wha ollows, we simpli y no a ion by deno ing o any μ0,μ1∈(V):
R(τ∗,q∗)(μ1,μ0)=u∗
S(μ1)+(1−μ1)μ1
1−μ1
−μ0
1−μ0u∗
H(μ1),
and no e ha R(τ∗,q∗)(μ0,μ0)=u∗
S(μ0).
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896 Do al and Sk e a Theo e ical Economics 19 (2024)
Equa ion (C.3) implies ha unde he speci ica ion o equilib ium play, when he
selle ’s belie is μn, he is indi e en be ween a pos ed p ice o L+(1−δn) and a
p ice o L+(1−δn−1) . This, in u n, implies ha he high- alua ion buye may
ob ain any payo in [δn ,δn−1 ], i he selle we e o andomize be ween hese wo
pos ed p ices. This andomiza ion is impo an o he speci ica ion o he buye ’s and
he selle ’s s a egies o he pa h o play. Fo u u e e e ence, le U∗
Hdeno e he ollow-
ing co espondence:
U∗
H(μ0)=u∗
H(μ0)i μ0= μn,n≥1,
δn ,δn−1 i μ0=μn o some n≥1.
No e U∗
His uppe hemicon inuous, con ex- alued, and compac - alued.
C.2 Omi ed p oo s om Sec ion 5.1
Appendix C.2 comple es he s eps o show ha he se o equilib ium payo s o G(μ0)
is as desc ibed in Equa ion (3). In wha ollows, we i s s a e he induc i e hypo he-
sis. We hen p o e he base case, which es ablishes Equa ion (3) o μ0∈[0, μ1)(Ap-
pendix C.2.1). We hen p o e he induc i e s ep in Appendix C.2.2.Along heway,we
p o ide he omi ed p oo s o he s a emen s in Sec ion 5.1 ega ding he p ope ies o
he solu ion o (OPT) (Lemmas C.2 and C.3). In wha ollows, uS(μ0),uS(μ0)deno e he
selle ’s maximum and minimum equilib ium payo s when his belie is μ0.Mo eo e ,
o n≥0andμ0≥μn,wele u∗
S(μ0,n)deno e he selle ’s payo om “emula ing” he
s a egy in he pos ed-p ices assessmen ha sells he good o he low- alua ion buye
in npe iods om now. Fo mally,
u∗
S(μ0,n)=μ0−μn−1
1−μn−1
H+1−μ0
1−μn−1
δR(τ∗,q∗)(μn−1,μ0).(C.6)
Induc i e hypo hesis: Fix n∈N0. The induc i e hypo hesis P(n)is gi en by
P(n).The ollowing hold:
(n.1) Fo all μ0≥μn,uS(μ0)≥u∗
S(μ0,n).
(n.2) Fo all μ0∈[μn,μn+1),Equa ion(3)inTheo em1holds.
C.2.1 Base case We i s show ha P(0)=1.
P oposi ion C.1 (Selle payoffgua an ee o μ0≥μ0). Fo all μ0≥μ0,uS(μ0)≥
u∗
S(μ0,0
).
P oo o P oposi ion C.1. De ine MS={μ0∈(V):uS(μ0)<
L}.Towa dacon a-
dic ion, suppose ha MSis nonemp y and le μS=in MS. We conside wo cases.
Case 1: μS∈MS.Le uL(μS)=sup{uL:u∈E∗(μS)}. Suppose he selle wi h p io
μSo e s mechanism MF ha sells he good a p ice L−δF o F>0. We a gue ha
i MFis ejec ed wi h posi i e p obabili y in a PBE assessmen , hen he selle ’s belie s
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Theo e ical Economics 19 (2024) Op imal mechanism o he sale o a du able good 897
condi ional on ejec ion, μR, coincide wi h μS. Suppose ha MFis ejec ed wi h pos-
i i e p obabili y. Then Lemma A.4 implies ha μR≤μS. Fu he mo e, i canno be
ha μR=0 (Lemma A.2, (a) implies ha μS>0) because δuL(μR)=0<δF.Thus, he
high- alua ion buye mus ejec he mechanism wi h posi i e p obabili y. We inally
ule ou ha μR∈(0, μS). In ha case, we would need ha δF + ≤δuH(μR)and
δF ≤δuL(μR), con adic ing ha uS(μR)≥ L o μR<μ
S(Lemma A.3).
Thus, μR=μS,so ha o F=uL(μS)+/δ, he low- alua ion buye accep s MF
wi h p obabili y 1 and so does he high- alua ion buye (c . Lemma A.2, (b)). Then
uS(μS)≥ L−δuL(μS).(C.7)
I uL(μS)=0, we a i e a a con adic ion. Suppose hen ha uL(μS)>0. We now a gue
ha i he selle wi h p io μSo e s MF o F=δuL(μS)+/δ, hen he buye accep s
his mechanism wi h p obabili y 1. By he same logic as abo e, i MFis ejec ed wi h
posi i e p obabili y, hen μR=μSand
δF + ≤δuH(μR),δF ≤δuL(μR).(C.8)
Lemma A.3 implies ha uS(μS)<
L−δuL(μS), a con adic ion o Equa ion (C.7). P o-
ceeding i e a i ely, we conclude ha o all n,uS(μS)≥ L−δnuL(μS),so ha asn→∞
we ha e ha uS(μS)≥ L, con adic ing he de ini ion o μS.
Case 2: μS/∈MS.Fix η>0andle uL(η)=sup{uL:u∈E∗(μ0),μ0∈(μS,μS+η)}.Fix
μ0∈(μS,μS+η). Suppose he selle wi h p io μ0o e s MF ha sells he good a p ice
L−δF,F>0. Simila logic o Case 1 implies ha i MFis ejec ed wi h posi i e p oba-
bili y in a PBE assessmen , hen μR∈(μS,μ0]. In pa icula , Lemma A.2, (a) implies ha
[0, μS]is nonemp y, and hence we can ule ou ha only he low- alua ion buye ejec s
MF.
We conclude ha i F=uL(η)+/δ, he low- alua ion buye accep s wi h p obabil-
i y 1 and so does he high- alua ion buye . We conclude ha o all μ0∈(μS,μS+η),
uS(μ0)≥ L−δuL(η).(C.9)
I uL(η)=0, we a i e a a con adic ion since Equa ion (C.9)holds o allμ0∈(μS,μS+
η)con adic ing he de ini ion o μS.
Suppose hen ha uL(η)>0. We now a gue ha o μ0∈(μS,μS+η) he mecha-
nism MFmus be accep ed wi h p obabili y 1 o F=δuL(η)+/δ.Tosee his,no e
ha i MFis ejec ed wi h posi i e p obabili y, by he same logic as abo e μR∈(μS,μ0],
so ha Equa ion (C.8) holds. Lemma A.3 implies ha μR∈(μS,μ0]exis s such ha
uS<
L−δuL(η), a con adic ion o Equa ion (C.9). P oceeding i e a i ely, we conclude
ha o all nand all μ0∈(μS,μS+η),
uS(μ0)≥ L−δnuL(η), (C.10)
so ha as n→∞we ha e ha uS(μ0)≥ L o all μ0∈(μS,μS+η), con adic ing he
de ini ion o μS. I ollows ha o all μ0∈(V),uS(μ0)≥ L=u∗
S(μ0,0
).
We conclude ha pa (n.1) holds.
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898 Do al and Sk e a Theo e ical Economics 19 (2024)
We now show ha pa (n.2) holds o μ0∈[μ0,μ1), s a ing om he selle ’s payo :
uS≤u∗
S(μ0) o μ0∈[μ0,μ1):No e ha uS(μ0)≤ Las Lis he selle ’s payo in he
commi men solu ion o μ0∈[0, μ1). Since Lis he selle ’s payo in he pos ed-p ices
assessmen , hen uS(μ0)=u∗
S(μ0)= L. Toge he wi h P oposi ion C.1, his implies ha
he selle ’s equilib ium payo is unique and coincides wi h L.
uL=0in [μ0,μ1): P oposi ion C.1 and he abo e a gumen imply ha he assump-
ions in P oposi ion 3hold o μ0∈[μ0,μ1).Hence,uL=0 o allu∈E∗(μ0)
uH= in [μ0,μ1): I emains o show ha he high- alua ion buye ’s payo is
unique in [μ0,μ1). Tha he selle ’s payo is a leas L o μ0∈[μ0,μ1)and Lemma A.3
imply ha uH≤ . Mo eo e , ha he selle ’s payo is a mos Limplies ha uH= .
C.2.2 Induc i e s ep Fix n≥1 and suppose ha P(k)=1 o allk∈{0, ,n−1}.
Pa (n.1): Wenowshow ha P
(n)=1, s a ing om he lowe bound on he selle ’s
payo .
P oposi ion C.2 (Selle payoffgua an ee o μ0≥μn). Fo all μ0≥μn,uS(μ0)≥
u∗
S(μ0,n).
The p oo o P oposi ion C.2 elies on P oposi ion 2, he p oo o which is in Ap-
pendix C.3.
P oo o P oposi ion C.2.Fixμ0≥μnand assume u∈E∗(μ0)exis s such ha uS<
u∗
S(μ0,n).Le ,(π , ) ∈V,μdeno e a PBE assessmen wi h payo uand conside
he ollowing de ia ion o he selle . The selle o e s a mechanism ha sells he good a
p ice L+(1−δn) −δF, ha is, o ∈{ L, H},β(μ0| )=1, (q(μ0),x(μ0)) =(1, L+
(1−δn) −δF ),whe eF>0 sa is ies ha o n≥2, δn−1 < δn−1 +F<δ
n−2 .
Because − (1−δn)+δF < − (1−δn−1)<0, he low- alua ion buye ejec s he
mechanism wi h p obabili y 1, so ha he selle ’s belie s upon ejec ion μRa e de e -
mined by Bayes’ ule and sa is y ha μR∈[0, μ0].
Fo n=1, he high- alua ion buye mus accep he mechanism wi h p obabili y
1: P oposi ion 2implies δ is he la ges payo om ejec ion and he high- alua ion
buye ge s s ic ly mo e by accep ing he mechanism. The selle ’s payo is hen μ0( H−
δ −δF )+(1−μ0)δ L=u∗
S(μ0,1
)−δF, whe e we use Lemma A.2, (a) o de e mine he
selle ’s payo condi ional on ejec ion. Le ing F→0, p o es P oposi ion C.2 o n=1.
Fo n≥2, we a gue he high- alua ion buye mus andomize be ween accep ing
and ejec ing he mechanism. I canno be he case ha he high- alua ion buye ejec s
he mechanism wi h p obabili y 1: P oposi ion 2implies ha δuH(μR)≤δδn−1 <
δ(δn−1 +F),whe eμR=μ0. Simila ly, i canno be he case ha he high- alua ion
buye accep s he mechanism wi h p obabili y 1: In ha case, ejec ion e eals he low-
alua ion buye (i.e., μR=0) and he high- alua ion buye ’s con inua ion payo is δ ,
which is s ic ly la ge han δn +δF (c . Lemma A.2, (a)).
I ollows ha he high- alua ion buye mus be indi e en be ween accep ing and
ejec ing so ha he con inua ion payo s a e ejec ion sa is y ha δn−1 +F=
uH(μR), which can only be he case i μR=μn−1. Indeed, μR∈(μn−1,μ0)would yield
15557561, 2024, 2, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE4485 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) Op imal mechanism o he sale o a du able good 899
apayo o a mos δn−1 (c . P oposi ion 2)andμR< μn−1wouldyieldapayo o a
leas δn−2 . This pins down he high- alua ion buye ’s accep ance p obabili y o be
(μn−1/μ0)(μ0−μn−1/(1−μn−1)). Simple algeb a shows ha he selle ’s payo om
o e ing he abo e mechanism is
μ0−μn−1
1−μn−1
H+δ1−μ0
1−μn−1u∗
S(μn−1)+μn−1−μ0
1−μ0u∗
H(μn−1)+F
=u∗
S(μ0,n)−δμ0−μn−1
1−μn−1
F.
Le ing F→0 comple es he p oo o P oposi ion C.2 o n≥2.
Pa (n.2): We now p o e ha pa (n.2) holds o μ0∈[μn,μn+1). Because we ha e
al eady es ablished ha he selle can gua an ee he payo om he pos ed-p ices as-
sessmen o μ0∈[μn,μn+1), i emains o show ha u∗
S(μ0)is an uppe bound on he
selle ’s payo .
De ine Mn={μ0∈[μn,μn+1):Equa ion(3)doesno hold
}. Towa d a con adic ion,
suppose Mn= ∅ and le μ=in Mn. Then, o all >0, μ
0∈[μ,μ+)exis s such ha
ei he uS(μ
0)>u
∗
S(μ
0)o he buye ’s payo is no as in Equa ion (3).
Lemmas C.2 and C.3 below deli e ha he selle ’s payo in he pos ed-p ices as-
sessmen is he selle ’s maximal equilib ium payo in [μn,μn+1).Fixμ0∈[μ,μn+1).By
de ini ion, o μ1∈[0, μ), he selle ’s and he low- alua ion buye ’s payo s a e gi en
by u∗
S(μ1)and 0, espec i ely. Fu he mo e, since μ0≥μn, he selle p e e s o minimize
he high- alua ion buye ’s con inua ion payo a {μm}m≤n−1.Hence, o μ0∈[μ,μn+1),
he objec i e unc ion in Equa ion (11)equals
[0,μ)μ0−μ1
1−μ1
H
+1−μ0
1−μ1
δu∗
S(μ1)+(1−μ1)μ1
1−μ1
−μ0
1−μ0u∗
H(μ1)
R(τ∗,q∗)(μ1,μ0)
G(dμ1)
+[μ,μ0)μ0−μ1
1−μ1
H+1−μ0
1−μ1
δuS(μ1)+uL(μ1)
+(1−μ1)μ1
1−μ1
−μ0
1−μ0uH(μ1)−uL(μ1)G(dμ1). (C.11)
Lemma C.2. Fix μ0∈[μ,μn+1)and suppose Gmaximizes he exp ession in Equa ion
(C.11) and is such ha G([0, μ)) >0.ThenGplaces posi i e p obabili y on a mos
{μn−2,μn−1}.
P oo . I is immedia e o see ha no μ1∈n−1
m=0(μm,μm+1)can be on he suppo o
G:i μ1∈(μm,μm+1) o m≤n−1, his is domina ed by choosing μm:
μ0−μm
1−μm
H+1−μ0
1−μm
δR(τ∗,q∗)(μm,μ0)−μ0−μ1
1−μ1
H+1−μ0
1−μ1
δR(τ∗,q∗)(μ1,μ0)
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900 Do al and Sk e a Theo e ical Economics 19 (2024)
= Hμ0−μm
1−μm
+δ1−μ0
1−μm
μm−μm−1
1−μm−1
−μ0−μ1
1−μ1
−δ1−μ0
1−μ1
μ1−μm−1
1−μm−1>0.
A simila a gumen implies μ1∈[μn,μ)is domina ed by choosing μn−1. Thus, i i is
op imal o se G([0, μ)) >0, we can educe he p oblem o inding he op imal such G
o
max
G∈({μ0,,μn−1})
n−1
m=0μ0−μm
1−μm
H+1−μ0
1−μm
δR(τ∗,q∗)(μm,μ0)G{μm}. (C.12)
Howe e , o m≤n−2, Lemma C.1 implies ha
m+1(μ0;μm,μm−1)
=μ0−μm
1−μm
H+1−μ0
1−μm
δR(τ∗,q∗)(μm,μ0)
−μm−1−μ0
1−μm−1
H+1−μ0
1−μm−1
δR(τ∗,q∗)(μm−1,μ0)>0,
since μ0> μn−1. Thus, any solu ion o he p oblem in Equa ion (C.12) sa is ies ha α∈
[0, 1]exis s such ha he alue o his p oblem is gi en by
αμ0−μn−1
1−μn−1
H+1−μ0
1−μn−1
δR(τ∗,q∗)(μn−1,μ0)
+(1−α)μ0−μn−2
1−μn−2
H+1−μ0
1−μn−2
δR(τ∗,q∗)(μn−2,μ0)
=αn(μ0;μn−1,μn−2)+μ0−μn−2
1−μn−2
H+1−μ0
1−μn−2
δR(τ∗,q∗)(μn−2,μ0),
so ha unless μ0=μni is no op imal o se α<1. Ins ead, o μ0=μnany α∈[0, 1]is
a maximize .
Lemma C.3. A eal numbe >0exis s such ha o all μ0∈[μ,μ+),VS
(μ0)=
u∗
S(μ0).
P oo .Fo ∈(0, μn+1−μ),le VS
deno e he sup emum o VS
(·)o e [μ,μ+).
The same a gumen s as he ones a e Equa ion (12) imply ha o all μ0∈[μ,μ+),
he ollowing holds:
VS
(μ0)≤maxu∗
S(μ0),μ0−μ
1−μ
H+1−μ0
1−μ
δVS
. (C.13)
Taking he sup emum o e μ0∈[μ,μ+)on bo h sides o Equa ion (C.13), we ob ain
VS
≤maxuS,
1−μ
( H−δVS
)+δVS
, (C.14)
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Theo e ical Economics 19 (2024) Op imal mechanism o he sale o a du able good 901
whe e uS is he sup emum o u∗
S(μ0)o e μ0∈[μ,μ+), and he exp ession in he
second e m ollows om no ing ha μ0−μ<and H>δVS
.
We claim >0exis ssuch ha o all∈(0, ), he igh -hand side o Equa ion (C.14)
equals uS. Towa d a con adic ion, suppose no . Then, o all ∈(0, μn+1−μ), ()∈
(0, )exis s such ha
uS ()< ()
1−μ
( H−δVS
())+δVS
(). (C.15)
Since ()<, hen ()→0as→0. Equa ion (C.14) oge he wi h Equa ion (C.15)
imply
VS
()≤ ()
1−μ
( H−δVS
())+δVS
()⇒lim
→0VS
()≤δlim
→0VS
(),
a con adic ion, since o all ,VS
≥ L>0.22 I ollows ha >0exis ssuch ha
∀∈(0, ),VS
=uS.
We now claim ha VS
(μ0)=u∗
S(μ0) o all μ0∈[μ,μ+). Towa d a con adic-
ion, suppose μ0∈[μ,μ+)exis s such ha VS
(μ0)>u
∗
S(μ0). By con inui y o u∗
Son
[μn,μn+1)(see Equa ion (C.5)), η>0 exis s such ha le ing =μ0+η−μ,weha e
u∗
S(μ0+η)=uS <VS
(μ0)≤VS
,
whe e he i s equali y ollows om u∗
Sbeing inc easing on [μn,μn+1)(c . Equa ion
(C.5)). This is a con adic ion. Thus, o all μ0∈[μ,μ+)we ha e ha u∗
S(μ0)isan
uppe bound on he selle ’s payo .23
Lemmas C.2 and C.3 imply ha uS(μ0)=u∗
S(μ0) o all μ0∈[μ,μ+). By de ini ion
o μ,μ0∈[μ,μ+)exis s such ha E∗(μ0) ails o sa is y Equa ion (3) because o he
buye ’s payo . In wha ollows, we ule his ou by showing ha o all μ0∈[μ,μ+),
he buye ’s payo is as in Theo em 1 o μ0∈[μ,μ+).
Low- alua ion buye ’s payo is 0in [μ,μ+): P oposi ion C.2 and VS
(μ0)=
u∗
S(μ0) o μ0∈[μ,μ+)imply ha he assump ions in P oposi ion 3hold o μ0∈
[μ,μ+).Hence,uL=0 o allu∈E∗(μ0)and all μ0∈[μ,μ+).
High- alua ion buye ’s payo in [μ,μ+):Fo anysuchμ0, suppose u∈E∗(μ0)
exis s such ha u
S=u∗
S(μ0),bu u
H= u∗
H(μ0). This equilib ium payo , u, is associa ed
o a mechanism in pe iod 0, (τ
0,q
0), and con inua ion payo s, u. We a gue ha
u∗
S(μ0)=VS
τ
0,q
0,u,μ0. (C.16)
To see his, no e ha we mus ha e VS
((τ
0,q
0),u,μ0)≥u∗
S(μ0);o he wiseu
S≤
VS
((τ
0,q
0),u,μ0)<u
∗
S(μ0), whe e he i s inequali y ollows om Lemma 2and he
22No e ha he limi lim→0VS
()exis s up o a con e gen subne because VS
()is bounded.
23Thea gumen abo eis eminiscen o ha inFudenbe g and Ti ole (1991a)’s ea men o he equilib-
ium in he pos ed-p ices game wi h bina y alues.
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902 Do al and Sk e a Theo e ical Economics 19 (2024)
second inequali y is by assump ion. Since u
S=u∗
S(μ0), we ha e a con adic ion. Fu -
he mo e, i canno be he case ha VS
((τ
0,q
0),u,μ0)>u
∗
S(μ0), because (τ
0,q
0) o-
ge he wi h he con inua ion payo s ua e easible choices in (OPT). Equa ion (C.16)
hen ollows. Thus, ((τ
0,q
0),u)is also a solu ion o (OPT).
Howe e , he p oo ha u∗
S(μ0) is he alue o (OPT) implies ha he solu ion o
(OPT) is unique, unless μ0=μn. I hen ollows ha μ0=μn, so ha he e is a con-
inuum o solu ions o (OPT), wi h H’s payo anging om u∗
H(μ0) ou∗
H(μ0)/δ.This
comple es he p oo ha uHis as in Equa ion (3) o allμ0∈[μ,μ+).
We conclude ha Equa ion (3)holds o allμ0∈[μ,μ+), con adic ing he de i-
ni ion o μ. I ollows ha Mnis emp y, and hence pa (n.2) o he induc i e s a emen
holds.
C.3 P oo o P oposi ion 2
We now p o e P oposi ion 2, which p o ides he bound on he high- alua ion buye ’s
payo . Fo n=1, he esul ollows om P oposi ion C.1 and Lemma A.3,asuS(μ0)≥ L
implies ha uH≤ . I emains o es ablish P oposi ion 2 o n≥2. Thus, ix n≥2and
assume ha o k≤n−1, P(k)=1. The p oo o P oposi ion 2 elies on Lemmas C.4 and
C.5:
Lemma C.4 (Maximal selle ’s payoffgi en high- alua ion buye ’s en s). Fix n≥2and
μ0≥μn.Le >0be such ha an equilib ium payo u∈E∗(μ0)exis s such ha uH≥
δn−1 +δ. Then he selle ’s payo is bounded abo e by VS
(μ0,n−1, ),whe e
VS
(μ0,n−1, )≡max
(q,τ,u)VS
(τ,q),u,μ0(OPT(μ0,n−1, ))
such ha
(V) q(μ)+1−q(μ)δuH(μ)−uL(μ)β(dμ| L)
≥δn−1 +δ,(R
(n−1))
Eτ[μ]=μ0,(BP)
∀μ∈(V)u(μ)∈E∗(μ0).(Eqbm)
No e ha (OPT(μ0,n−1, )) does no educe o (OPT)when=0andμ0∈
(μn,μn+1). Indeed, o μ0∈(μn,μn+1), he solu ion o (OPT) deli e s en s δn o
he high- alua ion buye , which iola es (R(n−1)). Ins ead, when μ0=μn, a solu ion o
(OPT) exis s ha deli e s en s δn−1 o he high- alua ion buye .
P oo o Lemma C.4.Le μ0≥μn.Le u∈E∗(μ0)be such ha uH≥δn−1 +δ.Le -
ing Mdeno e he mechanism o e ed by he selle in he i s pe iod, we ha e ha M
sa is ies a leas he ollowing cons ain s:
(V) HqM(μ)+1−qM(μ)δuM
H(μ)βM(dμ| H)−xH
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Theo e ical Economics 19 (2024) Op imal mechanism o he sale o a du able good 903
≥δn−1 +δ,(PCH)
(V) HqM(μ)+1−qM(μ)δuM
H(μ)βM(dμ| H)−βM(dμ| L)
≥xH−xL,(ICH)
(V) LqM(μ)+1−qM(μ)δuM
L(μ)βM(dμ| L)
≥xL,(PCL)
whe e x =(V)xM(μ)βM(dμ| ). Equa ions (PCH), (ICH), and (PCL)a eonlyasub-
se o he cons ain s he mechanism Mmus sa is y. Indeed, we a e igno ing he low-
alua ion buye ’s u h- elling cons ain and (PCL) is only a necessa y condi ion o he
low- alua ion buye o pa icipa e in he mechanism.
P og am ().
The selle ’s payo , uS, is bounded abo e by he solu ion o he ollowing p og am:
Maximize he selle ’s payo by choosing (i) ans e s xH,xL, (ii) ade p obabili ies
q:(V)→ [0, 1], (iii) a Bayes’ plausible pos e io dis ibu ion τ, and (i ) con inua ion
payo s u(·)∈E∗(·), subjec o he cons ain s (PCH), (ICH), and (PCL). Because we
allow he selle o choose xLand xHins ead o x(μ), we gi e he selle mo e deg ees
o eedom han in he game. In wha ollows, we a gue ha he alue o ()equals
VS
(μ0,n−1, ).
I is immedia e o see ha xLis chosen so ha (PCL) binds. We can hen w i e (ICH)
as ollows:
(V) Hq(μ)1−q(μ)δuH(μ)β(dμ| H)−xH
≥(V) q(μ)+1−q(μ)δuH(μ)−uL(μ)β(dμ| L).(ICH
)
Wenowshow ha (ICH) mus bind a he solu ion o (),whichin u nimplies ha
(R(n−1)) holds. Towa d a con adic ion, suppose ha (ICH) does no bind. Then xH
mus be chosen so ha (PCH) binds. The binding cons ain s (PCH)and(PCL)imply
ha he alue o () ob ains om maximizing
(V)
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
q(μ)μ H+(1−μ) L−μ0
1−μ0δn−1 +δ
+1−q(μ)δuS(μ)+μuH(μ)
+(1−μ)uL(μ)−μ0
1−μ0δn−2 +
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
τ(dμ), (C.17)
subjec o(BP)and(Eqbm).
We a gue ha he solu ion o he abo e p oblem is no easible o (). Indeed, he
op imal alue o he objec i e in Equa ion (C.17)is
μ0 H+(1−μ0) L−μ0
1−μ0 δn−1+δ.
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904 Do al and Sk e a Theo e ical Economics 19 (2024)
To see his, no e ha Lemma A.1 and δ<1 imply ha , o all μ, he ollowing holds:
δuS(μ)+μuH(μ)+(1−μ)uL(μ)−μ0
1−μ0 δn−2+
≤δμ H+(1−μ) L−μ0
1−μ0 δn−2+
<μ
H+(1−μ) L−μ0
1−μ0δn−1 +δ.
Thus, o all μ, se ing q(μ)=1 is p e e ed o se ing q(μ)=0. The linea i y in μo he
e m associa ed o qimplies he esul . Howe e , his is no easible since q(μ)=1 o all
μand he binding (PCH) iola es(ICH). I ollows ha (ICH) mus hold wi h equali y
a a solu ion o ().
Replacing he binding (PCL)and(ICH) in he selle ’s payo yields ha he objec-
i e in ()isVS
((τ,q),u,μ0). Mo eo e , eplacing he binding (ICH)in(PCH)yields
Equa ion (R(n−1)). We conclude ha () coincides wi h (OPT(μ0,n−1, )).
Lemma C.5 (Value o (OPT(μ0,n−1, ))a =0). Fix n≥2and μ0≥μn.Then
VS
(μ0,n−1, 0)=u∗
S(μ0,n−1).
P oo o Lemma C.5.Le λdeno e he Lag ange mul iplie on he cons ain (R(n−1))
in he p og am (OPT(μ0,n−1, )) o =0. In a sligh abuse o no a ion, de ine μ0(λ)=
μ0−λ
1−λ. The Lag angian is gi en by
L(τ,q),u;λ=VS
(τ,q),u,μ0(λ)−λδn−1 .(L)
Tha is, up o a cons an , he Lag angian co esponds o he i ual su plus o a selle
wi h belie μ0(λ).Le λ∗be such ha μ0(λ∗)=μn. By he uppe bound p oo , we know
ha one o he solu ions o maximizing he i ual su plus o a selle wi h belie μnde-
li e s en s δn−1 . Howe e , his solu ion sa is ies he Bayes’ plausibili y cons ain o a
selle wi h belie μn, whe eas (OPT(μ0,n−1, )) equi es ha he dis ibu ion o e pos-
e io sa e ages oμ0. No e ha he policy ha deli e s a payo o u∗
S(μ0,n−1) o he
selle gi es en s δn−1 o he high- alua ion buye and sa is ies he Bayes’ plausibili y
cons ain a μ0. In wha ollows, we desc ibe he policy and a gue ha i is a solu ion o
(OPT(μ0,n−1, )). Impo an o his a gumen is ha he selle wi h belie μn inds i
op imal o gi e he high- alua ion buye en s equal o δn−1 .
Le (ˆτ,ˆ
q)deno e he ollowing mechanism:
ˆτ(1)=μ0−μn−2
1−μn−2
=1−ˆτ(μn−2),ˆ
q(1)=1=1−ˆ
q(μn−2),
and con inua ion payo s ˆ
u(μn−2)=(0, δn−2 ,u∗
S(μn−2)). The mechanism (ˆτ,ˆ
q) o-
ge he wi h he equilib ium con inua ion payo s ˆ
u(μn−2)sa is ies he cons ain s.
Fu he mo e, we now a gue ha ha o all ((τ,q),u) ha sa is y he cons ain s in
(OPT(μ0,n−1, )),
L(τ,q),u;λ∗≤L(ˆτ,ˆ
q),ˆ
u;λ∗. (C.18)
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Theo e ical Economics 19 (2024) Op imal mechanism o he sale o a du able good 911
is, u∗
H(μ +1). The las inequali y ollows by de ini ion. P oposi ion 1and he esul s
in Appendix C.2 imply he las equali y. We conclude ha he selle has no one-sho
de ia ions.
Appendix E: Pe ec Bayesian equilib ium:Fo mal s a emen
We in oduce in his sec ion he necessa y o malisms o de ine PBE. To simpli y no a-
ion, we assume in wha ollows ha MIis such ha Miis ini e o all i∈I.Thisis
wi hou loss o gene ali y when Vis ini e (c . Do al and Sk e a (2022)). I a o ds wo
impo an simpli ica ions. Fi s , MIis i sel a Polish space, which means ha we can
condi ion he buye ’s s a egy di ec ly on he mechanism, M, chosen by he selle (c .
Aumann (1964)). Second, gi en a public his o y h , he se o buye his o ies consis-
en wi h h ,H
B(h ), is ini e and, he e o e, he suppo o μ (h )∈(V×H
B(h )) is
ini e.
Gi en he buye ’s pa icipa ion and epo ing s a egy and a mechanism M de ine
a dis ibu ion o e (SM ×{0, 1}×R)such ha , o all measu able subse s S×A⊆
SM ×{0, 1}×R,
ρ(π, )S×A| ,h
B,M =π h
B,M
m∈MM
ϕM S×A|m h
B,M (m).
Fix an assessmen ,(π , ) ∈V,μ, a public his o y h , and a mechanism, M .The
selle ’s payo is gi en by
US,(π , ) ∈V,μ|h ,M
=
( ,h
B)
μ h ,h
B1−π h
B,M δEUS,(π , ) ∈V,μ|h ,z∅(M ),·
+
( ,h
B)
μ h ,h
BSM ×{0,1}×Rx+(1−q)δEUS,(π , ) ∈V,μ|h ,
z(s ,(0,x))(M ),·ρ(π, )d(s ,q,x)| ,h
B,M .
Simila ly, he buye ’s payo when he alue is , he his o y is h
B, and he selle o e s
mechanism M is gi en by
U ,(π , ),μ|h
B,M
=1−π h
B,M δEU ,(π , ),μ|h
B,z∅(M ),·+π h
B,M
×
m∈MM
h
B,M (m)SM ×A q −x+(1−q)δEU ,(π , ) ∈V,μ|h
B,m,
z(s ,(q,x))(M ),·ϕM d(s ,q,x)|m.
De ini ion 1. The assessmen ,(π , ) ∈V,μsa is ies sequen ial a ionali y i o all
pe iods , and all public his o ies h , we ha e:
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912 Do al and Sk e a Theo e ical Economics 19 (2024)
1. Fo all mechanisms M in he suppo o (h ),US(,(π , ) ∈V,μ|h ,M )≥
US(,(π , ) ∈V,μ|h ,M
) o all M
= M ,
2. Fo all ∈V,allbuye his o iesh
B∈H
B(h ), and all mechanisms M ,U (,(π , ),
μ|h
B,M )≥U (,(π
,
),μ|h
B,M ) o all al e na i e s a egies (π
,
).
De ini ion 2. The belie sys em μsa is ies Bayes’ ule whe e possible i o all public
his o ies h , and mechanisms M , he ollowing hold:
μ +1h ,z∅(M ) H,h
B,z∅(M )
∈V,h
B∈H
B(h )
μ h ,h
B1−π h
B,M
=μ h H,h
B1−π Hh
B,M ,
and, o all measu able subse s S×Ao SM ×A,
( ,h
B)
μ h ,h
BS×A
μ +1h ,· ,h
B,z(s ,(q ,x ))(M ),mρ(π, )d(s ,q,x)| ,h
B
=μ h ,h
Bπ h
B,M h
B,M (m)ϕM S×A|m.
De ini ion 3. An assessmen ,(π , ) ∈V,μis a pe ec Bayesian equilib ium i i is
sequen ially a ional and sa is ies Bayes’ ule whe e possible.
Re e ences
Ab eu, Dilip, Da id Pea ce, and Ennio S acche i (1990), “Towa d a heo y o discoun ed
epea ed games wi h impe ec moni o ing.” Econome ica, 58, 1041–1063. [872]
Acha ya, A idi and Juan O ne (2017), “P og essi e lea ning.” Econome ica, 85, 1965–
1990. [868]
Alip an is, Cha alambos and Kim Bo de (2013), In ini e Dimensional Analysis: A Hi ch-
hike ’s Guide. Sp inge -Ve lag, Be lin. [869]
A hey, Susan and Kyle Bagwell (2008), “Collusion wi h pe sis en cos shocks.” Econo-
me ica, 76, 493–540. [872,887,905]
Aumann, Robe J. (1964), “Mixed and beha io s a egies in in ini e ex ensi e games.”
Tech. ep., P ince on Uni e si y. [870,911]
Aumann, Robe J., Michael Maschle , and Richa d E. S ea ns (1995), “Repea ed games
wi h incomple e in o ma ion.” MIT p ess. [868]
Ausubel, Law ence M. and Raymond J. Denecke e (1989), “Repu a ion in ba gaining and
du able goods monopoly.” Econome ica, 511–531. [868]
Beccu i, Juan and Ma c Mölle (2018), “Dynamic ad e se selec ion wi h a pa ien selle .”
Jou nal o Economic Theo y, 173, 95–117. [867,868]
15557561, 2024, 2, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE4485 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) Op imal mechanism o he sale o a du able good 913
Bes e , Helmu and Roland S ausz (2001), “Con ac ing wi h impe ec commi men
and he e ela ion p inciple: The single agen case.” Econome ica, 69, 1077–1098. [867,
868]
Bes e , Helmu and Roland S ausz (2007), “Con ac ing wi h impe ec commi men
and noisy communica ion.” Jou nal o Economic Theo y, 136, 236–259. [867]
Boa d, Simon and Ma ek Pycia (2014), “Ou side op ions and he ailu e o he Coase
conjec u e.” Ame ican Economic Re iew, 104, 656–671. [868]
B eig, Zacha y (2022), “Repea ed con ac ing wi hou commi men .” Jou nal o Eco-
nomic Theo y, 204, 105514. [867]
B zus owski, Thomas, Alkis Geo giadis-Ha is, and Balázs Szen es (2023), “Sma con-
ac s and he Coase conjec u e.” Ame ican Economic Re iew, 113, 1334–1359. [868]
Bulow, Je emy I. (1982), “Du able-goods monopolis s.” Jou nal o Poli ical Economy, 90,
314–332. [868]
Ca illo, Juan D. and Thomas Ma io i (2000), “S a egic igno ance as a sel -disciplining
de ice.” The Re iew o Economic S udies, 67, 529–544. [868]
Coase, Ronald H. (1972), “Du abili y and monopoly.” The Jou nal o Law and Economics,
15, 143–149. [868]
Deb, Rahul and Mahe Said (2015), “Dynamic sc eening wi h limi ed commi men .”
Jou nal o Economic Theo y, 159, 891–928. [867]
Denicolo, Vincenzo and Paolo G. Ga ella (1999), “Ra ioning in a du able goods
monopoly.” The RAND Jou nal o Economics, 44–55. [875,876]
Do al, Lau a and Vasiliki Sk e a (2022), “Mechanism design wi h limi ed commi men .”
Econome ica, 90, 1463–1500. [865,866,867,868,869,871,876,888,891,911]
Dwo czak, Pio and Gio gio Ma ini (2019), “The simple economics o op imal pe sua-
sion.” Jou nal o Poli ical Economy, 127, 1993–2048. [889]
Fiocco, Ra aele and Roland S ausz (2015), “Consume s anda ds as a s a egic de ice
o mi iga e a che e ec s in dynamic egula ion.” Jou nal o Economics & Managemen
S a egy, 24, 550–569. [867]
Fudenbe g, D ew, Da id Le ine, and Jean Ti ole (1985), “In ini e ho izon models o ba -
gaining wi h one-sided unce ain y.” In Game Theo e ic Models o Ba gaining, olume
73, 79. Camb idge Uni e si y P ess. [868,873,881]
Fudenbe g, D ew and Jean Ti ole (1991a), Game Theo y. MIT P ess. [901]
Fudenbe g, D ew and Jean Ti ole (1991b), “Pe ec Bayesian equilib ium and sequen ial
equilib ium.” Jou nal o Economic Theo y, 53, 236–260. [870]
Gen zkow, Ma hew and Emi Kamenica (2016), “A Ro hschild–S igli z app oach o
Bayesian pe suasion.” Ame ican Economic Re iew, 106, 597–601. [889]
15557561, 2024, 2, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE4485 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
914 Do al and Sk e a Theo e ical Economics 19 (2024)
Ge a di, Dino and Lucas Maes i (2020), “Dynamic con ac ing wi h limi ed commi -
men and he a che e ec .” Theo e ical Economics, 15, 583–623. [867,906]
Gul, Fa uk, Hugo Sonnenschein, and Robe Wilson (1986), “Founda ions o dynamic
monopoly and he Coase conjec u e.” Jou nal o Economic Theo y, 39, 155–190. [866,
867,868,884,886]
Habibi, Ami (2020), “Mo i a ion and in o ma ion design.” Jou nal o Economic Beha -
io & O ganiza ion, 169, 1–18. [868]
Ha , Oli e D. and Jean Ti ole (1988), “Con ac enego ia ion and Coasian dynamics.”
The Re iew o Economic S udies, 55, 509–540. [866,873]
Kamenica, Emi and Ma hew Gen zkow (2011), “Bayesian pe suasion.” Ame ican Eco-
nomic Re iew, 101, 2590–2615. [868]
Kolo ilin, An on (2018), “Op imal in o ma ion disclosu e: A linea p og amming ap-
p oach.” Theo e ical Economics, 13, 607–635. [889]
La on , Jean-Jacques and Jean Ti ole (1988), “The dynamics o incen i e con ac s.”
Econome ica, 1153–1175. [867]
La on , Jean-Jacques and Jean Ti ole (1993), A Theo y o Incen i es in P ocu emen and
Regula ion. MIT p ess. [889]
McA ee, R. P es on and Thomas Wiseman (2008), “Capaci y choice coun e s he Coase
conjec u e.” The Re iew o Economic S udies, 75, 317–331. [868]
O ne , Juan (2017), “Du able goods monopoly wi h s ochas ic cos s.” Theo e ical Eco-
nomics, 12, 817–861. [868]
Peski, Ma cin (2022), “Ba gaining wi h mechanisms.” Ame ican Economic Re iew, 112,
2044–2082. [868]
Rocka ella , Ralph Ty ell (1970), Con ex Analysis. P ince on Uni e si y P ess. [889]
Simon, Leo K. and William R. Zame (1990), “Discon inuous games and endogenous
sha ing ules.” Econome ica, 861–872. [875,887,907]
Sk e a, Vasiliki (2006), “Sequen ially op imal mechanisms.” The Re iew o Economic
S udies, 73, 1085–1111. [867,868]
Sk e a, Vasiliki (2015), “Op imal auc ion design unde non-commi men .” Jou nal o
Economic Theo y, 159, 854–890. [867]
Sobel, Joel (1991), “Du able goods monopoly wi h en y o new consume s.” Econome -
ica, 59, 1455–1485. [868]
Sobel, Joel and Ichi o Takahashi (1983), “A mul is age model o ba gaining.” The Re iew
o Economic S udies, 50, 411–426. [868]
S okey, Nancy L. (1981), “Ra ional expec a ions and du able goods p icing.” The Bell
Jou nal o Economics, 112–128. [868]
15557561, 2024, 2, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE4485 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) Op imal mechanism o he sale o a du able good 915
Wolinsky, Ashe (1991), “Du able-good monopoly wi h in en o ies.” Economics Le e s,
37, 339–343. [868]
Co-edi o Simon Boa d handled his manusc ip .
Manusc ip ecei ed 25 Augus , 2020; inal e sion accep ed 26 July, 2023; a ailable online 1 Au-
gus , 2023.
15557561, 2024, 2, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE4485 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
