Contests with sequential entry and incomplete information

Deng, Shanglyu; Fu, Qiang; Wu, Zenan; Zhu, Yuxuan
A icle
Con es s wi h sequen ial en y and incomple e
in o ma ion
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Deng, Shanglyu; Fu, Qiang; Wu, Zenan; Zhu, Yuxuan (2024) : Con es s wi h
sequen ial en y and incomple e in o ma ion, Theo e ical Economics, ISSN 1555-7561, The
Econome ic Socie y, New Ha en, CT, Vol. 19, Iss. 2, pp. 705-742,
h ps://doi.o g/10.3982/TE5367
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Theo e ical Economics 19 (2024), 705–742 1555-7561/20240705
Con es s wi h sequen ial en y and incomple e in o ma ion
Shanglyu Deng
Depa men o Economics, Uni e si y o Macau
Qiang Fu
Depa men o S a egy and Policy, Na ional Uni e si y o Singapo e
Zenan Wu
School o Economics, Peking Uni e si y
Yuxuan Zhu
School o Economics, Peking Uni e si y
This pape p o ides a gene al s udy o a con es modeled as a mul iplaye
incomple e-in o ma ion, all-pay auc ion wi h sequen ial en y. The con es con-
sis s o mul iple pe iods. Playe s a i e and exe e o s sequen ially o compe e
o a p ize. They obse e he e o s made by hei ea lie opponen s, bu no
hose o hei con empo aneous o u u e i als. We es ablish he exis ence and
uniqueness o a symme ic pe ec Bayesian equilib ium (PBE) and ully cha -
ac e ize he equilib ium. Based on he equilib ium esul , we show ha a la e
mo e always secu es a la ge ex an e expec ed payo . Fu he , we endogenize
he iming o mo es and show ha all playe s choose o mo e in he las pe iod
in he unique equilib ium ha su i es i e a ed elimina ion o s ic ly domina ed
s a egies (IESDS).
Keywo ds. Con es wi h sequen ial en y, all-pay auc ion, la e -mo e ad an-
age, endogenous iming.
JEL classi ica ion. C72, D43, D44, D82, L13.
Shanglyu Deng: [email p o ec ed]
Qiang Fu: [email p o ec ed]
Zenan Wu: [email p o ec ed]
Yuxuan Zhu: [email p o ec ed]
Zenan Wu is he co esponding au ho . Deng, Fu, Wu, and Zhu a e co- i s au ho s. We hank Law ence
M. Ausubel, Aislinn Boh en, Jidong Chen, Zhuoqiong Chen, Ma hias Dahm, Hong Feng, Bå d Ha s ad, Kai
Kon ad, Ming Li, Bin Liu, John Mo gan, Emel Filiz-Ozbay, Roland S ausz, And ew Swee ing, Qian eng Tang,
Ta -How Teh, Felix Vá dy, Daniel R. Vincen , Jun Yu, Huaxia Zeng, Jun Zhang, and semina /con e ence
pa icipan s a he Uni e si y o Ma yland, Conco dia Uni e si y, he Chinese Uni e si y o Hong Kong,
Shenzhen (CUHK-Shenzhen), Ha bin Ins i u e o Technology (Shenzhen), Renmin Uni e si y o China, he
Shanghai Uni e si y o Finance and Economics (SUFE), Zhejiang Uni e si y, Wo kshop on Indus ial O -
ganiza ion and Compe i ion Policy (UIBE), Se en h Annual Con e ence on Con es s: Theo y and E idence,
and 2022 Beijing In e na ional Wo kshop on Mic oeconomics: Empi ics, Expe imen s, and Theo y o help-
ul discussions, sugges ions, and commen s. Fu hanks he Singapo e Minis y o Educa ion Tie -1 Aca-
demic Resea ch Fund (R-313-000-139-115) o inancial suppo . Wu hanks he Na ional Na u al Science
Founda ion o China (72222002, 72173002, and 71803003), he Wu Jiapei Founda ion o he China In o -
ma ion Economics Socie y (E21100383), and he Resea ch Seed Fund o he School o Economics, Peking
Uni e si y, o inancial suppo . Any e o s a e ou own.
©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/TE5367
706 Deng, Fu, Wu, and Zhu Theo e ical Economics 19 (2024)
1. In oduc ion
Many compe i i e ac i i ies esemble a con es , in which con ende s s i e o leap og
and hei e o s a e non e undable ega dless o win o loss. Such phenomena a e
widesp ead in socioeconomic con ex s, anging om elec o al campaigns (Snyde
(1989)), lobbying (Che and Gale (1998), Baye, Ko enock, and De V ies (1993)), in e nal
labo ma ke s inside i ms (Lazea and Rosen (1981), Rosen (1986), G een and S okey
(1983)), and spo ing e en s (B own (2011)) o R&D aces (Lou y (1979), Lee and Wilde
(1980), Taylo (1995), Fulle on and McA ee (1999), Che and Gale (2003)).
Con es -like compe i ions in p ac ice a e o en inhe en ly sequen ial, in ha con-
ende s en e and ac in succession. Fi ms may en e a ace successi ely o an inno a-
i e echnology. Conside , o ins ance, he ecen ace o de elop Co ona i us accines.
Mode na/NIH, China’s CanSino Biologics, and he Uni e si y o Ox o d/As aZeneca
PLC ook he lead in en y.1P omising esul s in ea ly ials spa ked s ong en husiasm
and encou aged a massi e global e o , wi h mo e han 200 candida es jumping on he
bandwagon. In an R&D p ojec , a i m o en has o decide on he in ensi y o i s e o s
(e.g., he numbe o ials) be o e esea ch p og ess ma e ializes due o budge equi e-
men s and esou ce planning, which canno la e be lexibly adjus ed. Fu he , i ms’
ac ions a e o en subjec o disclosu e equi emen s o leaked o compe i o s. Fo in-
s ance, EU coun ies ypically equi e manda o y disclosu e o i ms’ R&D ac i i ies (La
Rosa and Libe a o e (2014)). In he Uni ed S a es, he Hones Leade ship and Open Go -
e nmen Ac o 2007 amended he Lobbying Disclosu e Ac o 1995, which s eng hened
public disclosu e equi emen s ega ding lobbying ac i i ies and unding. On Taskcn,
a leading c owdsou cing pla o m, a pa icipan is gi en access o ea lie submissions
(Liu, Yang, Adamic, and Chen (2014)), and an ea lie en an is ully awa e o he in o -
ma ion spillo e o u u e con ende s.
Dynamic in e ac ions a ise in such scena ios. La e mo e s condi ion hei ac ions
on p io mo es, and an ea lie mo e shapes hei s a egies in an icipa ion o u u e
opponen s’ eac ions. These complica e s a egical analysis o he con es game. The
complexi y can be u he compounded when he con es allows o iche iming a -
chi ec u es: Fo ins ance, mul iple playe s can en e and ac in a single pe iod simul a-
neously; hey obse e p io ac ions bu no con empo aneous ac ions, which embeds
simul aneous compe i ions in a dynamic s uc u e. Conside a biopha maceu ical i m
ha en e ed he ace o Co ona i us accines in mid-2020; i s esea ch can p esumably
le e age he e o s o pionee s, bu no hose o he many en an s ha looded in o
he a ena wi hin he sho ime window. A ull- ledged analysis in ol es subs an ial an-
aly ical sub le y, which con ines he majo i y o p e ious s udies o limi ed se ings, o
example, a wo-playe , wo-pe iod s uc u e.2,3
Sequen ial mo es ha e spawned wo ela ed classical ques ions in oligopoly heo y
(Ami and S epano a (2006)). The i s conce ns playe s’ compa a i e payo s wi h e-
spec o hei iming posi ions; ha is, he ea lie - e sus la e -mo e ad an age (see,
1See h ps://www.ny imes.com/2020/05/20/heal h/co ona i us- accines.h ml.
2See, o example, Dixi (1987), Baik and Shog en (1992), and Ho mann and Ro a-G aziosi (2012).
3Hinnosaa (2024) p o ides a ema kable excep ion.
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Theo e ical Economics 19 (2024) Con es s wi h sequen ial en y 707
e.g., Gal-O (1987); Dow ick (1986); Dixi (1987)). The second iews he iming a chi-
ec u e o an oligopoly as he endogenously de e mined ou come o playe s’ s a egic
choices (see, e.g., Hamil on and Slu sky (1990); Ami (1995); Mo gan (2003)), which ad-
d esses he classical Cou no /S ackelbe g deba e. The con en ional wisdom ob ained
in he usual duopolis ic se ings, howe e , does no eadily ex end unde mo e gene al
sequen ial s uc u es and dese es o be eexamined. Shinkai (2000), o ins ance, con-
side s a h ee- i m, h ee-pe iod model. He shows ha playe s’ payo s can be non-
mono one along he sequence, which p ecludes a con enien answe in gene al o he
ques ion ega ding ea ly- o la e -mo e ad an age in oligopoly.
We conside a gene al con es game wi h sequen ial en y ha imposes no es ic-
ions on he numbe o playe s and accommoda es a ull spec um o iming a chi ec-
u es. Analogous o s anda d s a ic all-pay auc ion models (e.g., Moldo anu and Sela
(2001)andMoldo anu, Sela, and Shi (2007)), ex an e symme ic playe s s i e o a com-
monly alued p ize and he highes bidde wins; playe s’ p i a e ypes (abili ies) a e
independen ly and iden ically dis ibu ed, wi h highe abili y yielding lowe ma ginal
e o cos . The con es p oceeds in mul iple pe iods, and mul iple playe s can be clus-
e ed in a single pe iod; all playe s wi hin each pe iod ac simul aneously and hey ob-
se e ea lie mo es. A ully sequen ial con es and he s anda d simul aneous bench-
ma k boil down o special cases o ou model. The un es ic ed iming a chi ec u e
in oduces subs an ial game- heo e ical sub le ies ha would be absen in he usual
duopolis ic se ings.4,5The li e a u e has ye o p o ide an equilib ium analysis o his
game, and ou pape ills he gap. The equilib ium esul u he enables us o ackle he
wo a o emen ioned classical ques ions.
Findings and implica ions: Summa y Ou pape i s conduc s a comp ehensi e equi-
lib ium analysis o he con es game wi h sequen ial en y desc ibed abo e. To mee
he analy ical challenges posed by he dynamic in e ac ions, we ake ad an age o he
ecu si e p ope y o he payo s uc u e and con e he game in o one ha esem-
bles a simul aneous-mo e, all-pay auc ion wi h an endogenously de e mined p ize. The
pseudo-p ize is shaped by playe s’ abili y dis ibu ion unc ion and can be exp essed
as a unc ion o a playe ’s bid. Ou model does no impose speci ic equi emen s on
he cu a u e o playe s’ abili y dis ibu ion. This may cause i egula i y in hei payo
unc ions and, he e o e, discon inui y in hei bidding s a egies. Despi e he nuance,
we es ablish ha he e exis s a unique symme ic pe ec Bayesian equilib ium (PBE)
in he game and p o ide a comple e equilib ium cha ac e iza ion (Theo em 1). The
4In a simple wo-playe , sequen ial-mo e con es , he second mo e , upon obse ing he i s mo e ’s
e o , ei he simply ma ches he ea lie e o o s ays inac i e. This p ope y g ea ly simpli ies he equilib-
ium analysis. This, howe e , no longe holds when a hi d playe is in oduced o he con es . Imagine a
simple case wi h h ee playe s and ully sequen ial mo es. Now he second mo e canno simply ma ch he
ea lie e o , which allows him o de ea he i s mo e bu may no be op imal gi en he h ea om he
hi d. The op imal esponse depends on his expec a ion o he u u e compe i ion. The li e a u e has ye o
p o ide an equilib ium analysis o his game, and ou pape ills he gap.
5Sege and Sela (2014) and Jian, Li, and Liu (2017) allow o mul iple playe s bu assume a ully sequen ial
s uc u e. As p e iously no ed, Hinnosaa (2024) p o ides a ema kable excep ion o he li e a u e ha
allows o an un es ic ed iming a chi ec u e bu assumes a lo e y con es , which di e s om ou se ing.
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708 Deng, Fu, Wu, and Zhu Theo e ical Economics 19 (2024)
equilib ium esul enables h ee applica ions ha shed ligh on he undamen als o he
con es game wi h sequen ial en y.
We i s in es iga e whe he a playe who mo es la e would ecei e a highe (lowe )
payo han his ea lie opponen s. We es ablish ha a payo mono onici y a ises: Re-
ga dless o he p e ailing con es a chi ec u e, a playe ends up wi h a highe ex an e
expec ed payo in a la e iming posi ion is-à- is an ea lie one (Theo em 2). Ou e-
sul hus p o ides a o mal a gumen o an unambiguous la e -mo e ad an age in he
con ex o mul iplaye con es s.
We hen allow playe s o simul aneously commi o he iming o hei mo es p io
o he con es , which endogenizes he iming a chi ec u e o he con es . I is wo h no -
ing ha despi e he inhe en o e lap, he abo e-men ioned analysis—which es ablishes
a la e -mo e ad an age—does no add ess a playe ’s iming choice. The la e -mo e
ad an age is ob ained by compa ing playe s’ ex an e expec ed payo ac oss di e en
pe iods unde a gi en iming a chi ec u e. A playe ’s iming choice, howe e , a ec s
he iming a chi ec u e o he con es ; as a esul , he analysis equi es ha we compa e
a playe ’s equilib ium expec ed payo s ac oss di e en iming a chi ec u es. We o -
mally e i y ha all playe s choose he las pe iod o hei mo es, which cons i u es he
unique equilib ium ha su i es i e a ed elimina ion o s ic ly domina ed s a egies
(Theo em 3). A ully simul aneous con es a ises when each playe makes au onomous
iming choices.
Finally, we gene alize he model o allow o a hyb id paymen ule ha in ol es bo h
winne -pay and all-pay elemen s. Speci ically, he winne o he con es is obliged o pay
he ull cos o his own e o , while a lose may only pay a ac ion o ha . Ou analysis
can eadily be adap ed o accommoda e his ex ension o cha ac e ize he equilib ium,
as in Theo em 1. The main implica ions o he equilib ium a e summa ized in Theo-
em 4and a e consis en wi h he insigh s ob ained in Theo ems 2and 3. This indica es
ha ou main p edic ions do no ely on he all-pay ea u e o he baseline model.
Link o he li e a u e This pape belongs o he small bu bu geoning li e a u e on se-
quen ial con es s. Dixi (1987), Baik and Shog en (1992), Mo gan (2003), and Ho mann
and Ro a-G aziosi (2012) all conside comple e-in o ma ion Tullock con es s in which
wo playe s mo e sequen ially. Mo gan and Vá dy (2007) adop a simila amewo k bu
assume ha he ollowe has o bea a small cos o obse e he leade ’s e o . Glaze
and Hassin (2000) allow o h ee-pe iod sequen ial plays. Analysis o mul i-playe se-
quen ial con es s in ol es subs an ial echnical di icul ies, because s anda d backwa d
induc ion is o no a ail. Kahana and Kluno e (2018) apply an “in e ed bes esponse”
app oach o a ully sequen ial lo e y con es wi h mul iple symme ic playe s. Hin-
nosaa (2024) allows o a gene al se up ha imposes no es ic ions on he p e ailing
iming a chi ec u e. Rema kably, he gene alizes and o malizes Dixi ’s hesis ha ea lie
playe s exe s ic ly highe e o s and a e ewa ded wi h s ic ly highe payo s, which
esul s om he s a egic subs i u abili y o e o s in a symme ic sequen ial lo e y con-
es .
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Theo e ical Economics 19 (2024) Con es s wi h sequen ial en y 709
Ou pape examines a adically con as ing game heo e ical con ex (i.e., all-pay
auc ions) and p o ides a gene al and comp ehensi e analysis ha imposes no es ic-
ions on iming a chi ec u es and allows o a b oade class o abili y dis ibu ion unc-
ions. All-pay auc ions do no gene a e a con inuous and well-beha ed bes - esponse
co espondence, unlike a lo e y o a Tullock con es . Ou esul s di e ge om ha o
Hinnosaa (2024): We es ablish a la e -mo e ad an age. Sege and Sela (2014)andJian,
Li, and Liu (2017) bo h conside ully sequen ial incomple e-in o ma ion all-pay auc-
ions. Sege and Sela (2014), assuming conca e dis ibu ion unc ions, in es iga e how
ex an e he e ogeneous playe s’ expec ed highes e o depends on he numbe o play-
e s and abili y dis ibu ions. Jian, Li, and Liu (2017), assuming ha playe s’ ype dis-
ibu ion unc ion akes a powe unc ional o m, compa e ex an e symme ic playe s’
winning p obabili ies wi h espec o he o de o mo es. Kon ad and Leininge (2007)
conside wo-s age mul iplaye comple e-in o ma ion all-pay auc ions. They show ha ,
as in simul aneous-mo e con es s, only he playe wi h he lowes cos ends up wi h a
posi i e expec ed payo , while he payo depends on his own iming posi ion is-à- is
hose o he o he s.
This pape con ibu es o he ex ensi e li e a u e on playe s’ compa a i e payo s
wi h espec o hei iming posi ions in sequen ial-mo e games—such as Gal-O (1985,
1987), Dixi (1987), Dow ick (1986), Daughe y (1990), Denecke e and Ko enock (1992),
Ami and G ilo (1999), Van Damme and Hu kens (1999,2004), Ami and S epano a
(2006), and on S engel (2010), among many o he s—in a ious con ex s, anging om
quan i y/capaci y o p ice-se ing compe i ions.6As s a ed abo e, his s and o he li -
e a u e ypically ocuses on duopolis ic i al y. Shinkai (2000) ex ends he amewo k o
a h ee- i m, h ee-pe iod se ing and illumina es he nuance caused by he mo e ex en-
si e sequence. To he bes o ou knowledge, ou pape and Hinnosaa ’s (2024)a e he
ew excep ions in he li e a u e ha examine ea lie -/la e -mo e ad an age unde an
un es ic ed iming a chi ec u e.
Ou analysis adds o he li e a u e on endogenous iming in oligopoly, such as
Hamil on and Slu sky (1990), Maila h (1993), Ami (1995), and Ami and S epano a
(2006). A hand ul o s udies explo e his issue in con es se ings, including Baik and
Shog en (1992), Leininge (1993), and Mo gan (2003). All o hese s udies conside wo-
playe models. Kon ad and Leininge (2007) allow o mul iple con es an s, bu impose
a wo-pe iod s uc u e.
The es o he a icle is o ganized as ollows. Sec ion 2se s up he model. Sec ion 3
cha ac e izes he equilib ium. Sec ion 4 u he del es in o he undamen als o his
con es game and applies ou equilib ium esul s o he ex ended se ings. Sec ion 5
concludes. P oo s a e elega ed o he Appendix.
2. The model
A con es in ol es N≥2 ex an e iden ical isk-neu al playe s, indexed by i∈N≡
{1, ,N}. The playe s a i e sequen ially and each exe s e o upon a i al o com-
pe e o a p ize wi h uni alue. The con es p oceeds in T≥1 pe iod(s), and he playe s
6Kemp and Ro a-G aziosi (2010) conside a se ing in which wo ju isdic ions se ax a es and endoge-
nize leade ship in ax compe i ions.
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710 Deng, Fu, Wu, and Zhu Theo e ical Economics 19 (2024)
a e acco dingly pa i ioned in o Tg oups. Deno e by N he se o playe s in pe iod ,
and le n :=|N |≥1 indica e he numbe o playe s in N . A playe obse es he e -
o s sunk by his ea lie opponen s bu no hose in con empo aneous o u u e pe i-
ods. The a chi ec u e o he con es is ully desc ibed by a ec o n:=(n1,,nT),wi h
N=T
=1n .7The con es is ully sequen ial wi h n=(1, ,1
), while i degene a es o
a ully simul aneous one wi h n=(N).
Aplaye i, when exe ing an e o (o , in e changeably, a bid) bi≥0, incu s a cos
c(bi)=bi/ai,whe eai>0 measu es one’s abili y and is p i a ely known.8Abili ies a e
d awn independen ly om an in e al (0, 1]acco ding o a common dis ibu ion unc-
ion F(·). We assume ha F(·)admi s a posi i e and con inuous densi y (·)≡F(·)and
is piecewise analy ic on [δ,1
] o all δ∈(0, 1).
Winne selec ion mechanism and payo s The compe i ion is modeled as an all-pay
auc ion. The playe wi h he highes e o wins. Speci ically, a playe i∈N ,whenexe -
ing an e o bi≥0, is he sole winne i and only i (i) his e o is g ea e han o equal
o hose in ea lie pe iods (i.e., bi≥bj o j∈ −1
k=1Nk) and (ii) his e o is s ic ly la ge
han hose in con empo aneous and u u e pe iods (i.e., bi>b
j o j∈T
k= Nk {i}).
In he e en ha (i) mul iple playe s in pe iod place he same highes bid and (ii) no
u u e playe s ma ch ha , he p ize is andomly dis ibu ed among hem. To pu i mo e
o mally, ixing a se o e o en ies b≡(b1,,bN),con es an i’s winning p obabili y
is9
pi(b):=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
1, i bi≥max
j∈ −1
k=1Nkbjand bi>max
j∈T
k= Nk {i}bj,
1/m,i bi≥max
j∈ −1
k=1Nkbj,bi>max
j∈T
k= +1Nkbj,
and bjis among he mhighes o bjj∈N wi h a ie,
0, i bi<max
j∈
k=1Nk {i}bjo bi≤max
j∈T
k= +1Nkbj,
(1)
and his ex pos payo , o a gi en abili y le el ai,is
pi(b)−bi/ai, o alli∈N.(2)
Equilib ium concep We conside he solu ion concep o pe ec Bayesian equilib ium
(PBE) o he con es game wi h sequen ial en y h oughou he pape . We ocus on he
symme ic equilib ium in which all playe s in he same pe iod adop he same bidding
s a egy.
7We igno e pe iods in which no playe s en e .
8We ollow he adi ion in he con es li e a u e and accommoda e playe he e ogenei y in hei cos
unc ions (e.g., Moldo anu and Sela (2001,2006); Moldo anu, Sela, and Shi (2007); B own and Mino
(2014)). I is no ewo hy ha he model is isomo phic o an al e na i e se ing in which e o is in e p e ed
as bid in he auc ion li e a u e: Playe s alue he p ize di e en ly bu bea he same e o cos s. All o ou
esul s emain quali a i ely unchanged unde his model speci ica ion.
9The ie-b eaking ule in (1) is asymme ic, which is commonly assumed in he li e a u e (see, e.g., An-
d eoni, Che, and Kim (2007), Simon and Zame (1990), Maskin and Riley (2000)). The asymme y ensu es
well-de ined bes esponses and he exis ence o an equilib ium, as in an asymme ic Be and duopoly
game.
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Theo e ical Economics 19 (2024) Con es s wi h sequen ial en y 711
Mo e o mally, we ix an e o p o ile (bj)j∈ −1
k=1Nk. De ine β :=maxj∈ −1
k=1Nk{bj} o
∈{2, ,T},andle β1≡0. In wo ds, β is he maximum e o in he con es p io o
pe iod . A symme ic PBE is deno ed by {b∗
(a;β )}T
=1,whe eais a playe ’s abili y and
b∗
(a;β )is he equilib ium bidding s a egy o a playe in pe iod . I is no ewo hy ha
he in o ma ion a ailable o a pe iod- playe iis summa ized by his own ype aiand
he highes e o β ≡maxj∈ −1
k=1Nk{bj}ins ead o he bidding his o y p io o pe iod
, ha is,(bj)j∈ −1
k=1Nk: Only he maximum p e ious bid ma e s o a playe in an all-
pay auc ion wi h sequen ial en y (see Equa ion (1)), so β can be iewed as a su icien
s a is ic o (bj)j∈ −1
k=1Nk.
Besides he usual es ic ions o PBE, we equi e ha each playe no upda e his
belie abou con empo aneous and u u e i als upon obse ing pas e o le els and
his own ype, ei he on o o he equilib ium pa h. This condi ion is sensible because
playe s’ abili ies a e independen ly dis ibu ed.10 Fu he , playe s’ payo s in ou se ing
do no depend on hei belie s abou ea lie mo e s’ abili ies. As a esul , we do no
speci y a belie sys em explici ly o de ine he PBE.
3. Equilib ium analysis
In his sec ion, we i s lay ou he undamen als o he analysis and hen o mally cha -
ac e ize he equilib ium.
3.1 P elimina ies o equilib ium analysis
This sec ion se s up impo an p imi i es ha lay he ounda ion o ou equilib ium
analysis. We i s in oduce se e al pieces o no a ion ha pa e he way o ou analysis
and discussion. We hen p esen ou p elimina y esul s (Lemmas 1 o 4) ha unde -
pin he main equilib ium esul s. Lemma 1depic s he undamen als o he bidding
s a egies in a hypo he ical symme ic PBE and enables subsequen analysis ha elies
on he ecu si e na u e o his con es game wi h sequen ial en y. Lemma 2na ows
he se o equilib ium e o s. Lemmas 3and 4iden i y a po en ial discon inui y in he
equilib ium bidding s a egies.
Fixing a con es a chi ec u e n≡(n1,,nT), we de ine a sequence o unc ions
{Q (b),a∗
(β),˜π (b,a)}T
=1, ecu si ely, as ollows:
QT(b)≡1, Q −1(b):=Q (b)Fn a∗
(b),∀b∈[0, 1],(3)
a∗
(β):=max0<a≤1: ˜π (b,a)≤0, ∀b∈[β,1
],(4)
˜π (b,a):=Q (b)Fn −1(a)−b/a.(5)
The sequence o unc ions {Q (b),a∗
(β),˜π (b,a)}T
=1is key o equilib ium cha ac-
e iza ion, and hei implica ions will be e ealed as he analysis un olds. As a head
10No e ha o all o ally mixed small pe u ba ions o belie s, a playe ’s belie s abou con empo ane-
ous and u u e i als mus be equal o he p io , which implies ha his belie sa is ies he consis ency
equi emen unde he solu ion concep o sequen ial equilib ium.
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712 Deng, Fu, Wu, and Zhu Theo e ical Economics 19 (2024)
s a , he unc ion a∗
(β)allows us o de i e he h eshold abili y abo e (below) which a
pe iod- playe s ays ac i e (inac i e) in equilib ium. Le b≥0 be he ealized highes e -
o by he end o pe iod ∈T.ThenQ (b)gi es he p obabili y o e o b’s exceeding all
subsequen e o s in a symme ic equilib ium. The unc ion ˜π (b,a)lays a ounda ion
o equilib ium payo cha ac e iza ion; i plays a c i ical ole in iden i ying he ele an
ange o equilib ium e o s and po en ial discon inui y in playe s’ bidding s a egies.
I can be e i ied om he abo e de ini ion ha a∗
(β)and Q (b)a e well-de ined
and sa is y he ollowing p ope ies:
(a) a∗
(β)is con inuous, piecewise di e en iable, and weakly inc easing on [0, 1],sa -
is ying a∗
(β)≥β o all ∈T;and
(b) Q (b)is con inuous, piecewise di e en iable, and s ic ly inc easing on [0, 1],wi h
Q (0)=0andQ (1)=1 o all ∈T {T}.11
We he eby p esen ou lemmas ha pa e he way o ou equilib ium esul .
Lemma 1delinea es use ul p ope ies o playe s’ bidding s a egy in a hypo he ical sym-
me ic PBE.
Lemma 1 (P ope ies o Equilib ium Bidding S a egy). Conside a con es wi h sequen-
ial en y n≡(n1,,nT), and suppose ha a symme ic PBE exis s. A pe iod- playe ’s
equilib ium bidding s a egy b∗
(a;β )sa is ies he ollowing p ope ies:
(i) b∗
(a;β )is inc easing in aon (0, 1);
(ii) b∗
(a;β )=0 o a≤a∗
(β )and b∗
(a;β )≥β o a>a
∗
(β );
(iii) b∗
(a;β )s ic ly inc eases wi h aon (a∗
(β ),1
)i n ≥2.
Lemma 1(i) is in ui i e: A s onge playe ends o bid mo e agg essi ely. Lemma 1(ii)
e eals he na u e o a∗
(·):Ape iod- playe would s ay ac i e (inac i e) in equilib ium
i his abili y a exceeds ( alls sho o ) he h eshold a∗
(β ). Recall ha a∗
(·)inc eases
wi h i s a gumen s and β is he maximum e o p io o pe iod- , which implies ha
highe ea lie e o ele a es he h eshold o ac i e bidding, he eby discou aging u-
u e compe i ion. By Lemma 1(iii), when a pe iod in ol es wo o mo e playe s, one’s
e o s ic ly inc eases wi h his abili y p o ided ha he is willing o place a posi i e bid,
ha is, a>a
∗
(β ). No e ha he s ic mono onici y does no necessa ily hold in he
case wi h n =1. To see his, conside a wo-playe , sequen ial-mo e con es n=(1, 1).
The la e mo e simply ma ches he i s mo e ’s e o β2i espec i e o his own abili y,
p o ided ha i exceeds a∗
2(β2)=β2, ha is,b∗
2(a;β2)=β2 o a>a
∗
2(β2).
Recall ha b≥0 deno es he ealized highes e o by he end o pe iod ∈T,which
leads o an e en ual win i and only i i exceeds all subsequen e o s om pe iod +1.
Lemma 1(ii) allows us o de i e he equilib ium p obabili y o his e en . By Lemma 1,b
ends up as he e en ual winning e o i and only i all subsequen playe s s ay inac i e
(i.e., e e y pe iod-playe ’s abili y alls below he h eshold a∗
(b),∀∈{ +1, ,T}),
11See he p oo o Lemma 8in he Appendix o mo e de ails.
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Theo e ical Economics 19 (2024) Con es s wi h sequen ial en y 719
Figu e 3. Fi s mo e ’s bidding s a egy in wo-pe iod, sequen ial-mo e con es s unde di e -
en abili y dis ibu ions.
unde F3(·). Recall ha he i s -mo e ’s winning p obabili y is gi en by F3(b)and his
expec ed payo is F3(b)−b/a. The cu a u e o he CDF o abili y cap u es he mag-
ni ude o he ma ginal e u n on his e o . In he con ex egion o he abili y dis ibu-
ion, he i s mo e enjoys inc easing ma ginal e u ns on his e o . As a esul , his bid
would no all in he egion o [1
4,1
2]. Mo eo e , as shown by Figu e 3(c), he discon inu-
i y would pe sis when an addi ional playe is added in he i s pe iod, which yields a
simul aneous compe i ion (n1=2).15
4. Discussions and ex ensions
In his sec ion, we apply ou equilib ium esul s o u he del e in o he undamen als
o his con es game wi h sequen ial en y and demons a e he e sa ili y o ou ap-
p oach. Fi s , we o mally es ablish he mono onici y o playe s’ ex an e expec ed pay-
o s wi h espec o hei iming posi ions in he gene al se ing. Second, we endogenize
playe s’ mo ing o de in he con es . Finally, we allow o a gene al paymen ule, such
ha each playe may no bea he ull cos o his e o .
15No e ha he jump in he i s mo e ’s bidding s a egy unde F3(·)is no d i en by he kinks in he
CDF’s de i a i es; a he i is caused by he change in he conca i y/con exi y o he CDF. Mo e o mally,
we can cons uc an example o a dis ibu ion unc ion such ha all de i a i es a e di e en iable on (0, 1)
and a jump eme ges in he equilib ium bidding unc ion.
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720 Deng, Fu, Wu, and Zhu Theo e ical Economics 19 (2024)
4.1 Mono one payo anking
We now o mally add ess he ollowing esea ch ques ion: Holding ixed he con es
a chi ec u e, does a playe bene i om being an ea lie /la e mo e ? Answe ing his
ques ion equi es ha we compa e playe s’ expec ed payo s wi h espec o hei im-
ing posi ions. We es ablish ha playe s’ equilib ium expec ed payo s can be anked
mono onically.
Le ∗
deno e a pe iod- playe ’s equilib ium expec ed payo , wi h ∈T,inacon es
wi h sequen ial en y n≡(n1,,nT). Recall ha T0≡{ ∈T:Q (b)≤b,∀b∈[0, 1]}
indica es he se o he pe iods in which playe s always s ay inac i e in equilib ium, wi h
0≡maxT0. The ollowing esul can be ob ained.
Theo em 2 (La e -Mo e Ad an age in Con es s Wi h Sequen ial En y). Conside a
con es wi h sequen ial en y n≡(n1,,nT). A playe ’s expec ed payo is highe han
hose o all ea lie mo e s in he unique symme ic PBE. To pu his o mally, 0=∗
1=
···=∗
0<
∗
0+1<···<
∗
T.
Recall ha 0=0 unde a conca e abili y dis ibu ion and 0=T−1unde acon ex
dis ibu ion. The ollowing esul can immedia ely be ob ained.
Co olla y 2 (La e -Mo e Ad an age Wi h Conca e/Con ex Abili y Dis ibu ions).
Conside a con es wi h sequen ial en y n≡(n1,,nT). The ollowing s a emen s hold
in he unique symme ic PBE:
(i) Suppose ha F(·)is con inuous, wice di e en iable, s ic ly conca e, and sa is ies
lima0[ (a)a]=0.Then0<
∗
1<···<
∗
T.
(ii) Suppose ha F(·)is con inuous, wice di e en iable, and weakly con ex. Then
∗
1=···=∗
T−1=0<
∗
T.
Theo em 2and Co olla y 2 o mally es ablish la e -mo e ad an age in a mul i-
playe all-pay auc ion wi h sequen ial en y. We ske ch he p oo as ollows. Fo ease
o exposi ion, le us conside a ully sequen ial con es , wi h N=T(one en an pe
pe iod). Recall ha he p o ile o equilib ium bidding s a egies is deno ed by b∗:=
{b∗
1(a;β1),,b∗
T(a;βT)}. Fix an a bi a y pe iod τ∈{ 0+1, ,T−1}.Weconduc he
ollowing hough expe imen . Le us modi y he pe iod-(τ+1)playe ’s bidding s a egy
om b∗
τ+1(a;βτ+1) o
b†
τ+1(a;βτ+1):=b∗
τ(a;βτ).
In o he wo ds, he hypo he ically igno es he pe iod-τplaye ’s e o and eplica es
he la e ’s equilib ium s a egy (no his e o ). Deno e playe s’ expec ed payo s un-
de he cons uc ed s a egy p o ile b†:={b∗
1(a;β1),,b∗
τ(a;βτ),b†
τ+1(a;βτ+1),b∗
τ+2(a;
βτ+2),,b∗
T(a;βT)}by (†
1,,†
T).
The key is o show ha he pe iod-τplaye would be s ic ly be e o wi h he
pe iod-(τ+1)playe ’s hypo he ical de ia ion, ha is, ∗
τ<
†
τ. The in ui ion is as ol-
lows. A la e mo e , ce e is pa ibus, ends o be mo e agg essi e in compe i ion han an
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Theo e ical Economics 19 (2024) Con es s wi h sequen ial en y 721
ea lie mo e : The o me needs o bea a smalle numbe o u u e opponen s o a win
han he la e , which encou ages he la e mo e . Thus, when he pe iod-(τ+1)playe
de ia es and eplica es his immedia e p edecesso ’s s a egy, he would be less likely o
ou pe o m he la e . This ob iously bene i s he pe iod-τplaye .
To ix ideas, conside a pe iod-τplaye , wi h abili y aτ>a
∗∗
τ(βτ), o agi enβτ.By
Theo em 1, he would exe an e o s ic ly abo e βτ. When he pe iod-(τ+1)playe
mimics he pe iod-τplaye , he o me can de ea he la e i and only i he pe iod-
(τ+1)playe is o a highe ype, which occu s wi h p obabili y 1−F(aτ). Unde he equi-
lib ium s a egy p o ile, in con as , he pe iod-(τ+1)playe ou pe o ms he pe iod-τ
playe as long as he o me chooses o s ay ac i e: He does so whene e his abili y ex-
ceeds he h eshold a∗
τ+1(βτ+1)=a∗
τ+1(b∗
τ(aτ;βτ)), which occu s wi h a p obabili y o
1−F(a∗
τ+1(b∗
τ(aτ;βτ))). We o mally show in he Appendix ha aτ>a
∗
τ+1(b∗
τ(aτ;βτ)):16
In o he wo ds, he pe iod-(τ+1)playe beha es less agg essi ely when he mimics his
immedia e p edecesso .
To comple e he p oo , i s no e ha †
τ+1≤∗
τ+1by he de ini ion o PBE. We u -
he ha e †
τ≤†
τ+1by he cons uc ion o b†
τ+1=b∗
τ.17 Combining hese inequali ies
yield ∗
τ<
†
τ≤†
τ+1≤∗
τ+1, which concludes ha a pe iod-(τ+1)playe ecei es a
highe equilib ium payo han a pe iod-τplaye .
Ou p edic ion s ands in sha p con as o ha o Hinnosaa (2024). He es ablishes
ha an ea lie mo e exe s a highe e o and secu es a la ge expec ed payo . We
ne e heless obse e he opposi e mono one payo anking in ou se ing. Hinnosaa
(2024) conside s a lo e y con es , in which ea lie and la e e o s can be s a egic sub-
s i u es nea he equilib ium. In a lo e y con es , one is emp ed o p eemp u u e
opponen s. Howe e , his does no occu in an all-pay auc ion: The la e mo e is
awa ded an in o ma ion ad an age since he can obse e p e ious e o s; he winne -
selec ion mechanism o an all-pay auc ion allows him o ou bid ea lie opponen s by
simply ma ching hei e o s. As a esul , s a egic complemen a i y could a ise in ou
bidding game, which as we es ablish, discou ages ea ly bidde s. Ou s udy hus comple-
men s ha by Hinnosaa (2024).
4.2 Endogenous iming
Ou equilib ium esul s enable us o explo e how he a chi ec u e o he con es game
could a ise endogenously. Le he con es be p eceded by a iming-choice s age, in
which playe s simul aneously commi o he iming o hei mo es. Each playe picks
one om L≥2 a ailable pe iods, deno ed by L:={1, ,L}, be o e he lea ns his e-
alized ype and ac s acco dingly. Be o e he con es begins, he a chi ec u e ˜
nis an-
nounced publicly, and each playe lea ns his own abili y p i a ely. The con es wi h
16We show in he p oo o Theo em 2 ha aτ>a
∗
τ+1(b∗
τ(aτ;βτ)) holds o an a bi a y con es a chi ec-
u e.
17No e ha he s ic inequali y may hold (i.e., †
τ<
†
τ+1) due o he ie-b eaking ule, despi e he ac
ha pe iod-τand pe iod-(τ+1)playe s employ he same s a egy. To see his, conside a ully sequen ial
con es (1, 1, 1)wi h a conca e abili y dis ibu ion and le he hi d mo e eplica e he second mo e ’s
equilib ium s a egy. In he e en ha playe s 2 and 3 choose o ma ch playe 1’s e o , playe 3 wins,
which occu s wi h a posi i e p obabili y and esul s in he s ic inequali y.
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722 Deng, Fu, Wu, and Zhu Theo e ical Economics 19 (2024)
sequen ial en y akes place as desc ibed in Sec ion 2 he ea e , and Theo em 1 ully
cha ac e izes he unique PBE o he con es subgame.18
I is no ewo hy ha he la e -mo e ad an age es ablished in Theo em 2does no
imply ha choosing a la e pe iod is a dominan s a egy o each playe . To be mo e spe-
ci ic, he la e -mo e ad an age is ob ained by compa ing di e en playe s’ expec ed
payo s wi h espec o hei iming posi ions unde a p ede e mined con es a chi ec-
u e. Wi h endogenous iming o mo es, howe e , a playe ’s au onomous iming choice
would eshape he esul an con es a chi ec u e and a ec all playe s’ equilib ium pay-
o s. Unde s anding a playe ’s iming choice equi es ha we compa e a gi en playe ’s
equilib ium payo s ac oss di e en con es a chi ec u es. The subsequen analysis
akes up his challenge.
The analysis begins wi h playe s’ equilib ium winning p obabili ies. Fix an a bi a y
con es a chi ec u e n≡(n1,,nT),wi hn ≥1 o ∈{1, ,T}. Conside a pe iod-
playe o abili y a∈(0, 1)and deno e by WP∗
(a;n)his expec ed equilib ium winning
p obabili y in he unique symme ic PBE. The ollowing lemma can be ob ained.
Lemma 5. Conside wo a bi a y con es a chi ec u es n≡(n
1,,n
T)—wi h n
≥1,
∈{1, ,T},T≥2,andT
=1n
=N—and n ≡(n
1,,n
T )—wi h n
≥1, ∈
{1, ,T},T ≥2,andT
=1n
=N. Fo almos e e y a∈(0, 1), we ha e
maxWP∗
1a;n,WP∗
1a;n<FN−1(a)<minWP∗
Ta;n,WP∗
Ta;n.
Tha is, o almos e e y a, a playe is mo e likely o win when ac ing in he las pe iod
o he con es han being one o he i s mo e s, ega dless o he p e ailing con es a -
chi ec u e. The compa ison is b idged h ough FN−1(a), which is a playe ’s equilib ium
winning p obabili y in a simul aneous con es .
This inequali y pa es he way o a compa ison o equilib ium payo s. We in oke
he s anda d payo -equi alence a gumen o di ec mechanisms. A pe iod- playe ’s
equilib ium payo in a con es wi h sequen ial en y n≡(n1,,nT)—which we de-
no e by ∗
(n)—can be pinned down by his equilib ium expec ed winning p obabili y as
ollows:
∗
(n)=E1
aa
0
WP∗
(x;n)dx=1
0a
0
WP∗
(x;n)
adxdF(a).
We u he de ine SIM :=1
0a
01
aFN−1(a)dxdF(a), which is one’s expec ed payo in a
simul aneous con es . Lemma 5can hen be ansla ed in o a compa ison o equilib-
ium payo s:
max∗
1n,∗
1n<
SIM <min∗
Tn,∗
Tn.(9)
By his inequali y, we a e eady o explo e playe s’ incen i es in hei iming choices. The
ollowing esul ensues.
18Theo em 1is es ablished unde he assump ion ha each pe iod possesses a leas one playe . Wi h
endogenous iming, his assump ion may no be sa is ied due o he possibili y ha no playe s choose o
mo e in a ce ain pe iod. In such a scena io, we can simply emo e hese pe iods and elabel he es o
in oke Theo em 1.
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Theo e ical Economics 19 (2024) Con es s wi h sequen ial en y 723
Lemma 6 (S ic ly Domina ed S a egy Wi h Endogenous Mo ing O de ). Fo e e y
playe , choosing o mo e in pe iod 1is s ic ly domina ed by choosing o mo e in pe iod L.
Lemma 6implies ha he equilib ium in he iming-choice s age is sol able by i e -
a ed elimina ion o s ic ly domina ed s a egies (IESDS).
Theo em 3 (Unique Equilib ium Wi h Endogenous Mo ing O de ). All playe s’ choos-
ing o mo e in he las pe iod cons i u es a Nash equilib ium o he i s -s age game ha
uniquely su i es IESDS.
When playe s a e allowed o pick he iming o hei mo es, all playe s will choose
he las pe iod and a simul aneous con es endogenously eme ges. Zhang (2024)ap-
plies he mechanism design app oach o op imal con es design wi h con ex (o linea )
e o cos and iden i ies a su icien and necessa y condi ion o he s a ic single-p ize
con es o be e o maximizing. Wi h linea e o cos , he condi ion degene a es o
Mye son’s (1981) classical egula i y condi ion o nondec easing i ual alue, ha is,
wi h a−[1−F(a)]/ (a)being nondec easing in ain ou con ex . Ou Theo em 3, o-
ge he wi h Zhang (2024), indica es ha a p ocess o decen alized decision on imings
o mo es leads o a simul aneous con es and gene a es he maximum amoun o ex-
pec ed o al e o unde he egula i y condi ion.
4.3 Hyb id paymen ule o lose s
Ou main esul s do no ely on he all-pay ea u e. Speci ically, we allow each lose o
bea only a po ion o his e o cos . The associa ed paymen ule is speci ied as ollows:
The winne in he con es is obliged o pay he ull cos o his own e o , while a lose
pays θ∈[0, 1]o ha .19,20 To pu his o mally, ixing a con es an i’s abili y aiand he
e o p o ile b≡(b1,,bN), his ex pos payo is
pi(b)1−bi/ai−1−pi(b)θbi/ai, o alli∈N.
The abo e exp ession degene a es o (2) in he baseline se ing as θ=1, and he con es
game u ns in o a i s -p ice auc ion wi h sequen ial en y as θ=0.21 Aθ∈(0, 1)depic s
a hyb id paymen ule ha in ol es bo h winne -pay and all-pay elemen s.
19See Amann and Leininge (1996) and Baye, Ko enock, and De V ies (2005,2012) o simila pa ame e -
iza ion.
20No e ha a bid b∈(0, β )always leads o a loss and is subop imal o a pe iod- playe o θ>0. In
con as , when θ=0, bidding b∈(0, β )is s a egically equi alen o bidding ze o because a bid does no
incu a cos o a lose . In his case, we impose he es ic ion ha pe iod- playe s bid 0 o weakly abo e β
wi hou any loss o gene ali y when cha ac e izing he symme ic PBE.
21To he bes o ou knowledge, he p e ious s udies o i s -p ice auc ions ha e ye o accommoda e a
se ing o un es ic ed iming s uc u e like ou s.
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724 Deng, Fu, Wu, and Zhu Theo e ical Economics 19 (2024)
Fixing a con es a chi ec u e n≡(n1,,nT)and θ∈[0, 1], a sequence o unc ions
{Q (b;θ),a∗
(β;θ),˜π (b,a;θ)}T
=1in pa allel wi h (3), (4), and (5) can be de ined ecu -
si ely as ollows:22
QT(b;θ)≡1, Q −1(b;θ):=Q (b;θ)Fn a∗
(b;θ),∀b∈[0, 1], (10)
a∗
(β;θ):=max0<a≤1: ˜π (b,a;θ)≤0, ∀b≥β, (11)
˜π (b,a;θ):=Q (b;θ)Fn −1(a)1−(1−θ)b/a−θb/a. (12)
Wi h sligh abuse o no a ion, le T0(θ):={ ∈T:Q (b;θ)≤θb
1−b+θb ,∀b∈[0, 1]}and
de ine 0(θ):=maxT0(θ). Again, we can ob ain 0 ≤ 0(θ)≤T−1.
Theo em 4 (Con es s Wi h a Gene alized Paymen Rule o Lose s). Fix θ∈[0, 1]and
conside a gene alized con es wi h sequen ial en y n≡(n1,,nT)unde a ie-b eaking
ule as speci ied in (1). The eexis sauniquesymme icPBEo hecon es game. In he
equilib ium, all playe s in pe iods 1 h ough 0(θ)choose o s ay inac i e ega dless o
hei abili y and he p e ious maximum bid; mo eo e , a playe ’s expec ed payo is highe
han hose o all ea lie mo e s. I playe s can choose he iming o hei mo e, hen all
playe s’ choosing o mo e in he las pe iod cons i u es a Nash equilib ium o he i s -
s age game ha uniquely su i es IESDS.
Theo em 4 eins a es he main esul s o ou baseline model unde he hyb id pay-
men ule. Ou analysis and p edic ions ex end o all he al e na i e se ings wi h
θ∈[0, 1], such as s anda d i s -p ice auc ions. The main esul s a e no an a i ac o
he all-pay ea u e. Ins ead, he s a egic complemen a i y in hese bidding games is he
key d i e o he esul s.
5. Concluding ema ks
In his pape , we conduc a gene al analysis o an incomple e-in o ma ion con es wi h
sequen ial en y in he o m o ( i s -p ice) all-pay auc ions. Ou model allows o a
lexible a chi ec u e, such ha mul iple playe s can be clus e ed in a single pe iod: They
mo e simul aneously wi hin he pe iod, while obse ing ea lie e o s and an icipa ing
u u e compe i ions. Ou analysis ully cha ac e izes he unique symme ic equilib ium
unde a gene al abili y dis ibu ion, which adds o he con es li e a u e since a gene al
analysis o con es s wi h sequen ial en y emains sca ce.
Based on ou equilib ium analysis, we o mally es ablish a la e -mo e ad an age, in
ha one secu es a highe ex an e expec ed payo when he is assigned o a la e iming
posi ion is-à- is an ea lie one. We u he allow playe s o choose he iming o hei
mo es in a p e-con es s age. The unique equilib ium ha su i es i e a ed elimina ion
o s ic ly domina ed s a egies equi es ha all playe s choose he las pe iod. Finally,
we demons a e ha he all-pay ea u e is no c ucial o ou analysis and ha all o he
esul s ex end o con es s wi h a hyb id paymen ule o lose s.
22We add θ o {Q (b),a∗
(β),˜π (b,a)}T
=1 o highligh he ac ha he de ined sequence o unc ions de-
pends on θ.
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Theo e ical Economics 19 (2024) Con es s wi h sequen ial en y 725
La ge oom o ex ensions emains. Fo ins ance, a model o mul iple p izes wi h se-
quen ial mo es dese es se ious schola ly e o . A second-p ice all-pay auc ion (e.g.,
K ishna and Mo gan (1997); Bulow and Klempe e (1999); Ha e (2006); Be gemann,
B ooks, and Mo is (2019)) also dese es se ious esea ch e o unde a sequen ial im-
ing a chi ec u e, and will be a emp ed in he u u e. Fu he , ou analysis assumes ex
an e symme ic playe s. This allows us o iden i y he e ec o iming posi ions on play-
e s’ expec ed payo . An equilib ium analysis o con es s wi h sequen ial en y and ex
an e he e ogeneous playe s unde a gene al iming a chi ec u e is echnically challeng-
ing, bu wa an s se ious esea ch e o . Also, ou pape assumes ha each playe com-
mi s o his e o upon en y. One na u al a ia ion is o allow hem o add o hei bids
in u u epe iodsasinYildi im (2005).23 Such an analysis en ails eno mous compli-
ca ions in ou se ing: Wi h incomple e in o ma ion, playe s’ bidding s a egies igge
complica ed in o ma ion upda ing and gi e ise o a challenging and sub le signaling
game. Finally, we assume ha playe s’ imings o mo es a e well known be o e hey sink
hei e o . I is in iguing o assume ins ead ha playe s’ iming posi ions a e andomly
assigned, so hey do no know p ecisely he imings o u u e opponen s’ en ies while
obse ing he his o y o p e ious bids. This se ing also causes echnical di icul y: The
gene al and andom iming a chi ec u e can lead o nume ous possibili ies o u u e
compe i ions—which is his o y-dependen —and, in u n, complexly and e lexi ely e-
shape ea lie bidding.
Appendix A: P oo s
A.1 P oo o Lemma 1
P oo . We p o e Lemma 1along wi h he ollowing lemma.
Lemma 7 (Equilib ium Winning P obabili y o a P o isional Winne ). Conside a con es
wi h sequen ial en y n≡(n1,,nT)and suppose ha a symme ic PBE exis s. Le b≥0
be he ealized highes e o by he end o pe iod ∈T.ThenQ (b)gi es he p obabili y
o e o b’s exceeding all subsequen e o s in equilib ium.
I is use ul o p o e se e al in e media e esul s.
Lemma 8. The ollowing s a emen s hold:
(i) a∗
(β)is con inuous, piecewise di e en iable, and weakly inc easing on [0, 1], sa -
is ying a∗
(β)≥β o all ∈T;and
(ii) Q (b)is con inuous, piecewise di e en iable, and s ic ly inc easing on [0, 1],wi h
Q (0)=0and Q (1)=1 o all ∈T {T}.
23Rela edly, Quin and Hend icks (2018) le a selle use indica i e bids (i.e., nonbinding p elimina y bids)
be o e a s anda d English auc ion o selec a subse o bidde s o conduc ing due diligence and elici ing
binding o e s. In hei model, bidde s simul aneously send cheap- alk messages o he selle , who subse-
quen ly uses hese messages o selec pa icipan s o he auc ion. The chosen bidde s hen pa ake in he
auc ion.
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726 Deng, Fu, Wu, and Zhu Theo e ical Economics 19 (2024)
P oo . We p o e he lemma by induc ion. No e ha piecewise analy ici y implies
piecewise di e en iabili y. The e o e, o show ha a∗
(β)and Q (b)a e piecewise di e -
en iable, i su ices o show ha hey a e piecewise analy ic. Deno e by Gn(·) he in e se
unc ion o aFn−1(a) o an a bi a y posi i e in ege n∈N+. I can be e i ied ha Gn(·)
is s ic ly inc easing, piecewise analy ic, and di e en iable on [0, 1],wi hGn(0)=0and
Gn(1)=1.
Base case: By de ini ion, QT(b)=1. The e o e, ˜πT(b,a)≡QT(b)FnT−1(a)−b/a =
FnT−1(a)−b/a. These ac s, oge he wi h (4), imply ha
a∗
T(β):=max0<a≤1: ˜πT(b,a)≤0, ∀b∈[β,1
]=GnT(β)
and QT−1(b)=FnT(GnT(b)). I is s aigh o wa d o e i y ha a∗
T(β)sa is ies pa (i) o
he lemma and QT−1(b)sa is ies pa (ii).
Induc i e s ep: Suppose ha Q (b)sa is ies pa (ii) o he lemma o some ≤T−1.
I su ices o show ha a∗
(β)sa is ies pa (i) o he lemma and Q −1(b)sa is ies pa (ii).
Fixing b∈(0, 1],˜π (b,a)s ic ly inc eases wi h a∈(0, 1). De ine ˘
a (b)as ollows:
˘
a (b):=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
Gn min1
Q
(0),1
,i b=0,
1, i b∈(0, 1]and ˜π (b,1
)<0,
he unique solu ion o ˜π (b,a)=0, o he wise.
I can be e i ied ha a∗
(β)=minb≥β˘
a (b)and ˘
a (b)is con inuous on [0, 1].Thisin u n
implies ha a∗
(β)is con inuous, piecewise analy ic, and weakly inc easing on [0, 1].
Fu he , o b≥β,weha e
˜π (b,β)=Fn −1(β)Q (b)−b/β ≤0,
which indica es ha β∈{0<a≤1: ˜π (b,a)≤0, ∀b≥β},and husa∗
(β)≥β.Tosum-
ma ize, a∗
(β)sa is ies pa (i) o he lemma.
Because Q (b)sa is ies pa (ii) o he lemma by assump ion and a∗
(β)sa is ies
pa (i), we can conclude ha Q −1(b)=Q (b)Fn (a∗
(b)) sa is ies pa (ii). This com-
ple es he induc i e s ep.
Conclusion: By he p inciple o induc ion, a∗
(β)sa is ies pa (i) o Lemma 8 o all
∈Tand Q (b)sa is ies pa (ii) o all ∈T {T}. This concludes he p oo .
Lemma 9. The ollowing s a emen s hold o all ∈T:
(i) I a∗
(β)<a<1, hen he e exis s b∈[β,1
]such ha ˜π (b,a)>0.
(ii) I 0<a<a
∗
(β), hen ˜π (b,a)<0 o all b∈[β,1
] {0}.
P oo . Pa (i) o he lemma is ob ious and i emains o p o e pa (ii). Fix 0 <a<
a∗
(β). Suppose, o he con a y, ha ˜π (b0,a)≥0 o someb0∈[β,1
] {0}. I ollows im-
media ely ha a≥b0. Fu he , we ha e ha ˜π (b0,a∗
(β)) >˜π (b0,a)≥0, which con a-
dic s wi h he ac ha ˜π (b0,a∗
(β)) ≤0 o allb∈[β,1
]. This comple es he p oo .
Now we can p o e Lemmas 1and 7by induc ion.
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Theo e ical Economics 19 (2024) Con es s wi h sequen ial en y 727
Base case: Conside he las pe iod, ha is, =T. I is e iden ha he ealized high-
es e o by he end o pe iod Twins he con es wi h ce ain y. By de ini ion, QT(b)=1.
The e o e, Lemma 7holds o =Tand i emains o show ha Lemma 1holds o he
las pe iod. We conside he ollowing wo cases:
(a) Suppose nT=1. Then he op imal bidding s a egy o he unique pe iod-Tplaye
is o bid βTi a>β
Tand bid 0 o he wise. The e o e, Lemma 1(i) and (ii) hold.
(b) Suppose nT≥2. We i s show ha b∗
T(a;βT)is inc easing in a. Suppose, o he
con a y, ha he e exis s an abili y pai (a,a),wi h0<a
<a
 <1, such ha
b :=b∗
T(a;βT)<b
:=b∗
T(a;βT). Deno e he equilib ium winning p obabili y
o bidding bby WP∗
T(b).I isob ious ha
WP∗
T(b)<WP∗
T(b);o he wise,a
ype-aplaye has a s ic incen i e o bid b. Mo eo e , om playe s’ incen i e
compa ibili y cons ain s, we ha e ha
WP∗
Tba−b≥WP∗
Tba−b,andWP∗
Tba −b ≥WP∗
Tba −b,
which is equi alen o
aWP∗
Tb−WP∗
Tb≥b−b,andaWP∗
Tb−WP∗
Tb≤b−b.
Combining he abo e inequali ies yield
a−a×WP∗
Tb−WP∗
Tb≥0,
which is a con adic ion gi en ha WP∗
T(b)<WP∗
T(b)and he pos ula ed
a<a
.
Le ¯
aT:=in {a:b∗
T(a;βT)>0}. We i s show ha b∗
T(a;βT)s ic ly inc eases
wi h a o a>¯
aT. Suppose, o he con a y, ha ¯
aT<a
<a
 <1andb:=
b∗
T(a;βT)=b :=b∗
T(a;βT). I ollows immedia ely ha b∗
T(a;βT)=b o a∈
[a,a].Thena ype-aplaye has an incen i e o de ia e om bidding b.Speci -
ically, he can aise his e o by an in ini esimal amoun o subs an ially inc ease
his winning p obabili y, which leads o an inc ease in his in e im expec ed payo .
A con adic ion.
Fu he , no e ha b∗
T(a;βT)≥βT o a>¯
aTand b∗
T(a;βT)=0 o a≤¯
aT,and
i hus emains o p o e ha ¯
aT=a∗
T(βT)≡max{0<a≤1: ˜πT(b,a)≤0, ∀b∈
[βT,1
]},whe e ˜πT(b,a)≡QT(b)FnT−1(a)−b/a. We conside he ollowing wo
cases:
(i) Suppose ha ¯
aT<a
∗
T(βT). Conside a ype-aplaye , wi h ¯
aT<a
<
a∗
T(βT). Recall ha b∗
T(a;βT)s ic ly inc eases wi h a o a>¯
aT. The e o e,
we ha e b∗
T(a;βT)>0. His equilib ium expec ed payo is
FnT−1a−b∗
Ta;βT
a=˜πTb∗
Ta;βT,a<0,
whe e he s ic inequali y ollows om b∗
T(a;βT)=0, b∗
T(a;βT)≥βT,and
Lemma 9(ii). Howe e , he can secu e a nonnega i e expec ed payo by bid-
ding ze o. A con adic ion.
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728 Deng, Fu, Wu, and Zhu Theo e ical Economics 19 (2024)
(ii) Suppose ha ¯
aT>a
∗
T(βT).Fixa∈(a∗
T(βT),¯
aT). I ollows immedia ely
om a<¯
aT ha b∗
T(a;βT)=0. No e ha bidding ze o mus gene a e ze o
expec ed payo o a ype-aplaye . O he wise, we mus ha e βT=0; o-
ge he wi h ¯
aT>0, we can conclude ha a playe whose ype alls below ¯
aT
can s ic ly inc ease his expec ed payo by exe ing an in ini esimal amoun
o e o . A con adic ion.
By Lemma 9(i), he e exis s some b∈[βT,1
]such ha ˜πT(b,a)>0.
Then ype-aplaye ’s expec ed payo o bidding bis bounded om below
by
FnT−1a−b
a=˜πTb,a>0.
The e o e, a ype-aplaye has a s ic incen i e o de ia e om exe ing
ze o e o , which is a con adic ion.
Induc i e s ep: Suppose ha he equilib ium bidding s a egy b∗
(a;β )sa is ies he
p ope ies s a ed in Lemma 1and Q (b)gi es he p obabili y o he e o b’s exceeding
all subsequen e o s in equilib ium, as p edic ed in Lemma 7, o some ≤T.Weshow
ha he same holds o pe iod −1.
Suppose ha he ealized highes e o by he end o pe iod −1isb. Then he p ob-
abili y o he e o b’s exceeding all subsequen e o s in equilib ium is Q (b)Fn (a∗
(b)),
which is exac ly Q −1(b) om (3).
Fo he case o n −1=1, no e ha he p oblem o he only pe iod-( −1)playe wi h
abili y ais maxb∈{0}∪[β −1,1][Q −1(b)−b/a]. I is hen s aigh o wa d o e i y ha (i)
b∗
−1(a;β −1)is inc easing in aon (0, 1), and (ii) b∗
−1(a;β −1)=0 o a≤a∗
−1(β −1)
and b∗
−1(a;β −1)≥β −1 o a>a
∗
−1(β −1). Fo hecaseo n −1≥2,by hesamea -
gumen as in he base case, we can show ha b∗
−1(a,β −1)sa is ies all p ope ies s a ed
in Lemma 1. This comple es he induc i e s ep.
Conclusion: By he p inciple o induc ion, b∗
(a;β )sa is ies all p ope ies s a ed in
Lemma 1 o all ∈T.Mo eo e ,Q (b)gi es he p obabili y o he e o b’s exceeding all
subsequen e o s in equilib ium o all ∈T, as p edic ed in Lemma 7.Thisconcludes
he p oo .
A.2 P oo o Lemma 2
P oo . See he main ex .
A.3 P oo o Lemma 3
P oo . I is use ul o p o e he ollowing in e media e esul .
Lemma 10. Suppose ha a∗
(β )<1.Then ˜π (s1
(a∗
(β );β ),a∗
(β )) =0.
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Theo e ical Economics 19 (2024) Con es s wi h sequen ial en y 735
Fo ease o exposi ion, deno e he cons uc ed p o ile o bidding s a egies by b†.Fu -
he , deno e playe i’s expec ed payo and a pe iod-τplaye ’s unde b†by i†
τ+1and †
τ,
espec i ely.
We i s show ha ∗
τ<
†
τ. Conside an indica i e pe iod-τplaye j,j∈Nτ,whose
abili y we deno e by aj. Deno e his in e im expec ed payo unde b∗and ha unde b†
by π∗
τ(aj;βτ)and π†
τ(aj;βτ), espec i ely.
I bj:=b∗
τ(aj;βτ)=0, hen he pe iod-τplaye loses unde bo h b∗and b†, indica ing
π∗
τ(aj;βτ)=π†
τ(aj;βτ)=0. I bj>0, hen we ha e bj≥βτand
π∗
τaj;βτ=Fnτ−1ajQτbj−bj/aj
=Fnτ−1ajQτ+1bjFnτ+1a∗
τ+1bj−bj/aj,
whe e he second equali y ollows om (3). Simila ly, we ha e
π†
τaj;βτ=F´
ajFnτ−1ajQτ+1bjFnτ+1−1a∗
τ+1bj−bj/aj,
whe e ´
ajis de ined as
´
aj:=a∗
τ(βτ),i nτ=1anda∗
τ(βτ)<a
j≤a∗∗
τ(βτ),
aj,o he wise,
and sa is ies b∗
τ(´
aj;βτ)=b∗
τ(aj;βτ)>0. I is s aigh o wa d o e i y ha
π∗
τaj;βτ<π
†
τaj;βτ⇐⇒ Fa∗
τ+1bj<F´
aj⇐⇒ a∗
τ+1bj<´
aj.(20)
I ollows immedia ely om b∗
τ(´
aj;βτ)=b∗
τ(aj;βτ)>0 ha π∗
τ(´
aj;βτ)>0, om which
we can conclude
Qτ+1bjFnτ+1−1a∗
τ+1bj−bj/´
aj>0. (21)
Fu he , i ollows om he de ini ion o a∗
τ+1(·)[see Equa ion (4)] ha
˜πτ+1bj,a∗
τ+1bj=Qτ+1bjFnτ+1−1a∗
τ+1bj−bj/a∗
τ+1bj≤0. (22)
Combining (21)and(22)yields
a∗
τ+1bj≤bj
Qτ+1bjFnτ+1−1a∗
τ+1bj<´
aj.
The abo e condi ion, oge he wi h (20), implies ha π∗
τ(aj;βτ)<π
†
τ(aj;βτ)and
∗
τ=Eπ∗
τaj;βτ<Eπ†
τaj;βτ=†
τ, (23)
whe e he expec a ion is aken wi h espec o bo h ajand βτ.
To comple e he p oo , i s no e ha ∗
τ+1≥i†
τ+1by he de ini ion o PBE. Mo e-
o e , i ollows immedia ely om he cons uc ion bi†
τ+1(a;βτ+1):=b∗
τ(a;βτ) ha i†
τ+1≥
†
τ. These inequali ies, oge he wi h (23), imply ha ∗
τ+1≥i†
τ+1≥†
τ>
∗
τ.
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736 Deng, Fu, Wu, and Zhu Theo e ical Economics 19 (2024)
A.9 P oo o Co olla y 2
P oo . The co olla y ollows immedia ely om Theo em 2and Co olla y 1.
A.10 P oo o Lemma 5
P oo . Fixing an a bi a y a chi ec u e n≡(n1,,nT),wi hn ≥1 o all ∈{1, ,T}
and T≥2, i su ices o show ha
WP∗
1(a;n)<FN−1(a)<WP∗
T(a;n), o almos e e y a∈(0, 1).
We i s p o e ha WP∗
1(a;n)<FN−1(a) o all a∈(0, 1). Conside a ep esen a i e
pe iod-1 playe i∈N1. Recall β1≡0. The inequali y ob iously holds i bi:=b∗
1(ai;β1)=
0, and i emains o conside he case whe e bi>0. Playe i’s expec ed equilib ium pay-
o is
π∗
1ai;n:=WP∗
1ai;n−bi
ai>0. (24)
Fixing ∈{2, ,T},weha e ha
WP∗
1ai;n=Fn1−1ai

=2
Fn a∗
biQbi≤Fn−1a∗
biQbi, (25)
whe e he equali y ollows om Lemma 1and Lemma 7. Combining (24)and(25)yields
Fn−1a∗
biQbi−bi
ai>0. (26)
F om (3), (4), and (5), we ha e ha ˜π(bi,a∗
(bi)) ≤0, which is equi alen o
Fn−1a∗
biQbi−bi
a∗
bi≤0. (27)
Compa ing (26)wi h(27) yields ha ai>a
∗
(bi) o all ∈{2, ,T}, which in u n im-
plies ha
WP∗
1ai;n=Fn1−1aiT

=2
Fna∗
bi<Fn1−1aiT

=2
Fnai=FN−1ai.
Nex , we p o e ha FN−1(a)<WP∗
T(a;n) o almos e e y a∈(0, 1).Fixa∈(0, 1),
β∈[0, 1],and( ,),wi h1≤ <≤T. Following a simila a gumen as in he p e ious
analysis, we can show ha i b∗
(a;β)>0, hen
a>a
∗
b∗
(a;β). (28)
Conside a ep esen a i e pe iod-Tplaye , j∈NT,wi habili yaj∈(0, 1).By(28), we
ha e aj>a
∗
T(b∗
1(aj;0
)).No e ha a∗
T(b∗
1(a;0
)) weakly inc eases wi h a, and hus is con-
inuous almos e e ywhe e. We can ocus on he case whe e a∗
T(b∗
1(a;0
)) is con inuous
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Theo e ical Economics 19 (2024) Con es s wi h sequen ial en y 737
a a=aj. The e o e, he e exis s >0such ha aj>a
∗
T(b∗
1(aj+;0
)).Le a:=aj+.I
ollows immedia ely ha
a>a
j>a
∗
Tb∗
1(a;0
). (29)
I is use ul o p o e he ollowing in e media e esul .
Lemma 12. Fix an a bi a y a chi ec u e n≡(n1,,nT)—wi h n ≥1 o all ∈
{1, ,T}and T≥2—and conside an indica i e pe iod-Tplaye j∈NT.Hewins he
con es in he unique PBE i a>a
j o all j∈N1and aj>a
j o all j∈N ({j}∪N1).
P oo . Fix an abili y p o ile a:=(a1,,aN)such ha a>a
j o all j∈N1and aj>a
j
o all j∈N ({j}∪N1).Le ıdeno e he index o he p o isional winne by he end
o pe iod T−1 gi en ha all playe s use he equilib ium s a egy and he pe iod he
mo es. Then βT=b∗
(aı;β ). E iden ly, he lemma holds i b∗
(aı;β )=0 and i emains
o conside he si ua ion whe e b∗
(aı;β )>0. We conside he ollowing wo cases:
(a) Suppose ≥2. Then we ha e
aj>a
ı>a
∗
Tb∗
aı;β =a∗
T(βT),
whe e he i s inequali y ollows om he pos ula ed ı/∈N1and he second in-
equali y om (28).
(b) Suppose =1. Then we ha e
aj>a
∗
Tb∗
1(a;0
)≥a∗
Tb∗
1aı;0
=a∗
T(βT),
whe e he i s inequali y ollows om (29).
To summa ize, i b∗
(aı;β )>0, hen aj>a
∗
T(βT), indica ing ha playe jplaces a
posi i e amoun o bid in equilib ium. The e o e, he ou bids all playe s up o pe iod
T−1. Nex , no e ha aj>a
j o all j∈NTby assump ion; oge he wi h Lemma 1(iii),
playe jou bids all o his con empo aneous i als in pe iod Tand wins he con es .
By Lemma 12,playe j’s expec ed winning p obabili y, WP∗
T(aj;n), can be bounded
om below by
WP∗
Taj;n≥Fn1(a)F(T
=2n )−1aj>FN−1aj.
This concludes he p oo .
A.11 P oo o Lemma 6
P oo . Fix an indica i e playe i∈Nand conside he ollowing wo cases:
(a) All o he playe s choose o mo e in he las pe iod. I playe ichooses o mo e in
pe iod 1, he esul an con es a chi ec u e is ˆ
n=(1, N−1)and his equilib ium
payo in his subgame is ∗
1(ˆ
n).I playe ichooses o mo e in he las pe iod,
all playe s mo e simul aneously in he second-s age game and his equilib ium
payo amoun s o SIM.By(9), we ha e ∗
1(ˆ
n)<
SIM.
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738 Deng, Fu, Wu, and Zhu Theo e ical Economics 19 (2024)
(b) A leas one o playe i’ opponen s chooses no o mo e in he las pe iod. Deno e
he esul an con es a chi ec u e when playe ichooses o mo e in pe iod 1 and
ha when he chooses o mo e in pe iod Lby ˆ
nand ˆ
n, espec i ely. No e ha ˆ
n
degene a es o a simul aneous-mo e con es i all o he playe s choose o mo e
in pe iod 1 and a sequen ial-mo e one o he wise. By (9), we ha e ∗
1(ˆ
n)≤SIM.
Nex , no e ha ˆ
n is a sequen ial-mo e con es . Deno e he numbe o pe iods
wi h a leas one playe by ˆ
T. Clea ly, playe i’s equilib ium payo unde ˆ
n
is ∗
ˆ
T(ˆ
n). Again, we can ob ain SIM <
∗
ˆ
T(ˆ
n) om (9). The e o e, we ha e
∗
1(ˆ
n)≤SIM <
∗
ˆ
T(ˆ
n).
To summa ize, mo ing in pe iod Lyields a s ic ly highe payo o playe i han
mo ing in pe iod 1. This concludes he p oo .
A.12 P oo o Theo em 3
P oo . The heo em ollows immedia ely om Lemma 6.
A.13 P oo o Theo em 4
P oo . Fixing a con es a chi ec u e n≡(n1,,nT)and θ∈[0, 1], deno e a symme ic
PBE o he con es game, i i exis s, by {b∗
(a;β )}T
=1, wi h sligh abuse o no a ion. Recall
ha he sequence o unc ions {Q (b;θ),a∗
(β;θ),˜π (b,a;θ)}T
=1is de ined by (10), (11),
and (12). By a gumen s simila o he case o θ=1, we can show ha Lemmas 1,3,4,
and 7ex end o θ∈[0, 1]. The p oo o he exis ence and uniqueness o symme ic PBE
esembles ha o Theo em 1,excep ha ˇπ (ˇ
a,a;β)is now de ined as
ˇπ (ˇ
a,a;β ):=Q b∗
(ˇ
a;β );θFn −1(ˇ
a)1−(1−θ)b∗
(ˇ
a;β )/a−θb∗
(ˇ
a;β )/a,
and he di e en ial equa ion ha go e ns a pe iod- playe ’s bidding s a egy b∗
(a;β )—
gi en ha n ≥2andb∗
(a;β )is con inuous in some in e al U˜
a=(˜
a,˜
a+)—is
(n −1)Q b∗
(a;β );θFn −2(a) (a)a−(1−θ)b∗
(a;β )
+b∗
(a;β )a∂˜π (b,a;θ)
∂b b=b∗
(a;β )=0.
The p oo s o he la e -mo e ad an age and he endogenous iming esul a e simila
o hose in Theo em 2and Theo em 3and omi ed o b e i y.
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