Be o, Ma ia; Thomas, Ma hew W.
A icle
Asymme ic all-pay auc ions wi h spillo e s
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Be o, Ma ia; Thomas, Ma hew W. (2024) : Asymme ic all-pay auc ions wi h
spillo e s, Theo e ical Economics, ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 19,
Iss. 1, pp. 169-206,
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Theo e ical Economics 19 (2024), 169–206 1555-7561/20240169
Asymme ic all-pay auc ions wi h spillo e s
Ma ia Be o
Depa men o Economics, No hwes e n Uni e si y
Ma hew W. Thomas
Depa men o Economics, No hwes e n Uni e si y
When opposing pa ies compe e o a p ize, he sunk e o playe s exe du ing
hecon lic cana ec he alueo hewinne ’s ewa d.Thesespillo e scanha e
subs an ial in luence on he equilib ium beha io o pa icipan s in applica ions
such as lobbying, wa a e, labo ou namen s, ma ke ing, and R&D aces. To un-
de s and his in luence, we s udy a gene al class o asymme ic, wo-playe all-pay
auc ions whe e we allow o spillo e s in each playe ’s ewa d. The link be ween
pa icipan s’ e o s and ewa ds yields no el e ec s; in pa icula , playe s wi h
highe cos s and lowe alues han hei opponen s some imes ex ac la ge pay-
o s.
Keywo ds. All-pay, con es s, auc ions, spillo e s, wa o a i ion.
JEL classi ica ion. C65, C72, D44, D62, D74.
1. In oduc ion
All-pay auc ions, o con es s, model s a egic in e ac ions among playe s who mus ex-
pend some non- e undable e o in o de o win a p ize. They ha e been applied in
di e se se ings such as labo (Rosen (1986)), R&D aces (Che and Gale (1998), Dasgup a
(1986)), and li iga ion (Baye, Ko enock, and de V ies (2005)). Fo ac abili y, he e-
cen li e a u e mos ly assumes ha playe s’ ac ions a ec hei opponen ’s p obabili y
o winning, bu no he alue o he p ize. Ye in many se ings, such spillo e e ec s on
he p izes hemsel es a ise na u ally.
Fo example, conside he se ing in Che and Gale (1998), whe e wo lobbyis s com-
pe e in an all-pay auc ion o win an incumben poli ician’s a o h ough campaign con-
ibu ions. I he poli ician we e ins ead a candida e unning o o ice, hen she would
only be able o p o ide he ewa d i success ully elec ed. In his case, i is na u al o
assume ha o al campaign con ibu ions inc ease he candida e’s chances o p e ail-
ing. The e o e, each lobbyis ’s con ibu ions inc ease he opponen ’s alue o winning
he poli ician’s poli ical a o . This aises new ques ions: is i be e o cu b one’s own
con ibu ions o make hei opponen lose in e es o is i p e e able o amp up he
compe i ion? These ques ions ha e been la gely le unanswe ed.
Ma ia Be o: [email p o ec ed]
Ma hew W. Thomas: [email p o ec ed]
We hank Wojciech Olszewski, Alessand o Pa an, Ma ciano Siniscalchi, B uno S ulo ici, and Ashe Wolin-
sky o in aluable commen s h oughou he w i ing p ocess.
©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/TE5108
170 Be o and Thomas Theo e ical Economics 19 (2024)
In o he se ings, spillo e s may be designed. Conside an all-pay e sion o a s an-
da d labo ou namen in which di ision manage s apply e o owa d some p oduc-
ion echnology in o de o win a p omo ion awa ded o he mos p oduc i e di ision.
To maximize agg ega e e o , a p incipal migh choose o make he alue o his p omo-
ion depend on e e yone’s pe o mance in he con es . Fo example, i he p omo ion is
o a pa ne ship o in ol es s ock op ions, he p ize will be inc easing in he e o s o
all playe s. The e ec ha such compensa ion schemes ha e on he equilib ium has no
ye been s udied.
This pape ully iden i ies he equilib ium s a egies and payo s in gene al wo-
playe auc ions wi h spillo e s and es ablishes hei uniqueness.1We conside games
wi h (i) comple e in o ma ion, (ii) de e minis ic p izes, (iii) a leas pa ially sunk in es -
men cos s, and (i ) a gene al dependence o each pa icipan ’s alue o he p ize on
bo h playe s’ ac ions. The key con ibu ion o his pape lies in inco po a ing (i ). In-
deed, all-pay con es s wi hou spillo e s we e ex ensi ely s udied by Siegel (2009,2010).
These pape s ully cha ac e ize equilib ium s a egies and payo s in games whe e con-
es an s incu some (pa ially) un eco e able cos , such as e o , in o de o compe e
o p izes. We gene alize he wo-playe single-p ize e sion o hei model o allow o
gene al spillo e s o a ec he winne ’s payo . Ou pape also has some o e lap wi h
he symme ic linea con es s wi h spillo e s s udied in Baye, Ko enock, and de V ies
(2012). Unlike hei wo k, howe e , we es ic a en ion o he all-pay case, bu allow o
asymme ic equilib ia and nonlinea payo s. E en in he symme ic, linea all-pay auc-
ion wi h spillo e s, we no e ha no p e ious pape ha we a e awa e o has es ablished
equilib ium uniqueness.
The addi ion o spillo e s can ha e a signi ican impac on equilib ium beha io .
Fi s , playe s wi h s ic ly highe cos s can ha e highe payo s han hose wi h lowe
cos s, e en i hei alue unc ions o he p ize a e iden ical. In ac , in some se ings,
playe s could inc ease hei payo s i hey we e allowed o commi o a schedule o
cos ly handicaps (see Sec ion 4). Thus, ying o a o an “unde dog” pa icipan in a
con es by means o educing hei cos s may e y well ha e he opposi e e ec , and
in ac dec ease hei wel a e in equilib ium. This is also impo an in se ings in which
playe s can commi o inc easing hei cos s (e.g., by selec ing an ine icien echnology),
as hey may choose o do so.
Ano he con ibu ion o his pape is he p ocedu e o cons uc equilib ium s a egy
p o iles. The equilib ium s a egy dis ibu ions o asymme ic all-pay con es s ha e wo
dis inc pa s: he densi ies and a mass-poin a 0. In he li e a u e on all-pay con es s
wi hou spillo e s, s a ing wi h Baye, Ko enock, and de V ies (1996), expec ed payo s
a e ob ained independen ly o he equilib ium dis ibu ion. This independence is ex-
ploi ed o de i e he p obabili y mass a 0 o he weake playe om he payo s, which
is hen used o compu e he densi ies. In he p esence o spillo e s, howe e , a playe ’s
payo s canno be de i ed wi hou he equilib ium s a egy o hei opponen . Because
o his, he same p ocess canno be ollowed. To o e come his di icul y, we in oduce
1This pape also es ablishes he exis ence o equilib ium, hough his esul has al eady been p o en;
see Olszewski and Siegel (2023), o example. Ou me hod, howe e , di e s subs an ially om he p e ious
li e a u e.
Theo e ical Economics 19 (2024) Asymme ic all-pay auc ions 171
an algo i hm ha wo ks in exac ly he opposi e o de : i s , i sol es o he densi y in-
dependen ly o he mass-poin , and hen uses his densi y o ind he p obabili y mass
a 0.
Ou me hod capi alizes on he heo y o Vol e a in eg al equa ions (VIEs), which
a e in eg al equa ions wi h a unique ixed-poin ha can be ob ained ia i e a ion. To
he bes o ou knowledge, hese echniques ha e no p e iously been applied o he
de e mina ion o equilib ium mixed-s a egy p o iles.2
The game we s udy is gene al enough o encompass many di e en applica ions in
which spillo e s ma e . In pa icula , in es men wa s, con es s wi h winne ’s eg e ,
and mili a is ic con lic s all i ou amewo k, since spillo e s a e key in each o hese
se ings. Ou model also subsumes a na u al ex ension o he wa o a i ion. which,
unlike he classical model, yields a unique equilib ium on a bounded suppo . We a e
also able o use he same amewo k o desc ibe wa s o a i ion whe e a ional agen s
ace uncomp omising (ne e -yielding) ypes wi h posi i e p obabili y, as in Ab eu and
Gul (2000)andKambe (2019). Ou app oach iden i ies why hese games admi unique
equilib ia when he egula wa o a i ion does no : he addi ion o an uncomp omis-
ing ype in oduces an una oidable cos ha depends on a playe ’s own sco e, and we
show ha his single cha ac e is ic is su icien in ensu ing a unique equilib ium.
Finally, we ex end he analysis o mo e han wo playe s. The uniqueness esul does
no hold when he numbe o bidde s exceeds wo. We a e none heless able o cha ac e -
ize a class o asymme ic equilib ia when (app op ia ely no malized) cos s a e anked.
In his case, we show ha only wo playe s pa icipa e in equilib ium. In addi ion, we
a e able o ully cha ac e ize he unique symme ic equilib ium o all-pay auc ions wi h
(i) mo e han wo iden ical playe s, (ii) mul iple homogeneous p izes, and (iii) spillo e s
gene a ed by he i s unne -up. This se ing accommoda es a b oad class o games in-
cluding wa s o a i ion and auc ions wi h winne ’s eg e wi h any numbe o playe s
and p izes. This ex ends he use ulness o ou no el me hodology.
The pape is o ganized as ollows. We in oduce he model, he equilib ium con-
cep , and he assump ions in Sec ion 2. We cons uc he equilib ium and p o e i s
uniqueness in Sec ion 3.Sec ion4p esen s su icien condi ions unde which a playe
has a posi i e expec ed payo . This includes an example whe e a playe wi h highe
cos s and lowe alues ecei es a posi i e expec ed payo , while he opponen ecei es
0. Sec ion 5con ains use ul esul s on closed o ms ha allows o simpli ied equilib-
ium compu a ions in ce ain special cases. We illus a e hei usage in he ollowing
Sec ion 6, which is dedica ed o applica ions. Sec ions 6.2,6.4,and6.3 in pa icula
showcase closed- o m solu ions. Sec ion 6.1 in oduces a gene al pe u ba ion o he
classic wa o a i ion ha ensu es he equilib ium is unique. This pe u ba ion admi s
he wa o a i ion wi h he possibili y o an uncomp omising ype as a special case. In
Sec ion 7, we ex end he analysis o con es s wi h mo e han wo playe s. Uniqueness no
longe holds gene ally, hough we a e s ill able o ind he unique symme ic equilib ium
o a n-playe , m-p ize all-pay auc ion wi h spillo e s. Finally, in Sec ion 8,we e iew he
ela ed li e a u e and discuss he esul s.
2Few o he wo ks in economics use VIE me hods in gene al. We no e McA ee, McMillan, and Reny (1989)
and McA ee and Reny (1992) as some ea ly examples. Mo e ecen ly, Gomes and Sweeney (2014)alsoused
VIEs, o compu e he unique e icien equilib ium bidding unc ions in gene alized second-p ice auc ions.
172 Be o and Thomas Theo e ical Economics 19 (2024)
2. Model
We ocus, o now, on auc ions wi h wo pa icipan s. Ex ensions wi h mo e playe s a e
conside ed in Sec ion 7, whe e we show ha he symme ic equilib ium o an auc ion
wi h any numbe o iden ical playe s and p izes is jus a ans o ma ion o he equilib-
ium o he wo-playe case.
An asymme ic auc ion wi h spillo e s is a amily {I,{˜
Si}i∈I,{ui}i∈I},whe e he ol-
lowing s a emen s hold:
(a) The index se o playe s is I:={1, 2}.
(b) Fo each i∈I,˜
Si:=[0, ∞)is playe i’s ac ion space, i.e., he se o a ailable sco es
(o bids). We use a ilde because a la e assump ion will allow us o eplace he
ac ion se wi h a bounded in e al. We le s−iand ˜
S−ideno e he ac ion and ac ion
space, espec i ely, o playe j=i.
(c) Fo each i∈I,ui:˜
S→Ris playe i’s payo , whe e ˜
S:=i∈I˜
Si.Le s:=(si;s−i)
deno e an a bi a y elemen o ˜
S. Then, o each (si;s−i), we u he de ine
ui(si;s−i):=pi(si;s−i) i(si;s−i)−ci(si),
whe e (i) pi(si;s−i)deno es he p obabili y ha iwins he p ize gi en he sco e
p o ile (si;s−i),wi hpi(si;s−i)=1−p−i(s−i;si)and
pi(si;s−i)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
1i si>s
−i
αi∈[0, 1]i si=s−i
0i si<s
−i;
(ii) i:˜
S→R+maps each sco e p o ile (si;s−i) o playe i’s alue i(si;s−i) om
winning he p ize; (iii) ci:˜
Si→R+ou pu s playe i’s p i a e cos ci(si)gi en he
submi ed sco e si.
De ini ion 1 (Two-Playe All-Pay Auc ion Wi h Spillo e s). A wo-playe all-pay auc-
ion is said o ha e spillo e s i , o some i∈Iand si∈˜
Si, he eexis ss−i,ˆ
s−i∈˜
S−isuch
ha
i(si,s−i)= i(si,ˆ
s−i),
i.e., he p ize’s alue o a leas one playe and an ac ion o ha playe a e no cons an
in hei opponen ’s ac ion.
Accommoda ing spillo e s is he dis inguishing ea u e o ou analysis. As is s an-
da d, we a e in e es ed in cha ac e izing he Nash equilib ium o hese gene al con es s.
De ini ion 2 (Bes Responses). Conside a wo-playe all-pay auc ion {I,{˜
Si}i∈I,
{ui}i∈I}. Fo each i∈I,le ˜
Sideno e he se o p obabili y dis ibu ions on ˜
Siand le
˜
S:=i∈I˜
Si.Playe i’s bes esponse se bi(G−i) o G−i∈˜
S−iis gi en by
bi(G−i):=a gmax
s∈˜
Si˜
S−i
ui(s;s−i)dG−i(s−i).
Theo e ical Economics 19 (2024) Asymme ic all-pay auc ions 173
De ini ion 3 (Nash Equilib ium). Conside he wo-playe all-pay auc ion {I,{˜
Si}i∈I,
{ui}i∈I}. A Nash equilib ium o his game is a p o ile G:=(G
i)i∈I∈i∈I(˜
Si),whe e,
o each i∈I,G
i’s induced p obabili y measu e assigns measu e 1 o bi(G
−i).
2.1 Assump ions
The ollowing assump ions a e imposed h oughou whene e a wo-playe all-pay auc-
ion is in oked. Appendix III shows ha none o hese assump ions is supe luous o ou
esul s.
Assump ion 1 (A1, Smoo hness). The unc ion i(si;y)is con inuously di e en iable in
siand con inuous in y o all i∈I,si∈˜
Si,andy∈˜
S−iwi h si≥y. The unc ion ci(si)is
con inuously di e en iable in si o all i∈I,si∈˜
Si.
Assump ion 2 (A2, Mono onici y). Fo all i∈Iand si>0,c
i(si)>0and
i(si;y)<c
i(si)
o almos all y,whe e
i(s;y):=∂ i(si;y)/∂si.
Assump ion 3 (A3, In e io i y). Fo all i∈I,
i(0, 0)>c
i(0)=0and lim
si→∞ sup
y∈˜
S−i
i(si;y)<lim
si→∞ci(si).
Ve sions o Assump ions A1,A2,andA3a e adop ed by mos pape s in he all-pay
auc ion li e a u e. Assump ion A2 o malizes he sense in which hese con es s a e all-
pay, since bids a e cos ly o bo h he winne and he lose .3Assump ion A3ensu es ha
bids a e posi i e and bounded.
No e ha , o each i∈I, he eexis Ti∈˜
Sisuch ha playe iwill ne e choose a sco e
s≥Ti. Thus, we can es ic he ac ion space o Si:=[0, Ti].
Assump ion 4 (A4, Discon inui y a Ties). Fo all i∈Iand s∈Si∩S−i,
i(s;s)>0.
Assump ion A4is a no el, ye na u al assump ion. I s a es ha agen s would p e e
o win a ie han o lose one. I is sa is ied i he p ize is always aluable (i.e., winning
is be e han losing) o i he e a e no spillo e s. To see ha i is ne e iola ed in he
absence o spillo e s, no e ha Tiis less han o equal o any xsa is ying i(x;y)≤ci(x)
o all y≤x. I he e a e no spillo e s and i(s)≤0 o somes≤Ti, henci(s)≤0. The e-
o e, s=0, which iola es Assump ion A3. No e ha his assump ion is equi alen o
assuming a discon inui y in payo s a ies because Assump ions A1and A3gua an ee
ha i(s;s)=0 implies Assump ion A4.
3We no e ha Assump ion A2does exclude si ua ions whe e a highe sco e is no necessa ily mo e cos ly.
Siegel (2014) discusses con es s wi h nonmono onic cos s, allowing o compe i o s wi h head s a s and
he p o ision o pe o mance-based subsidies. These con ingencies a e excluded om ou analysis.
174 Be o and Thomas Theo e ical Economics 19 (2024)
3. Cha ac e iza ion o equilib ium
By s anda d a gumen s con ained in he Appendix, any pai o equilib ium s a egies
will be mixed wi h suppo on some in e al [0, ¯
s], and a mos one playe will ha e a
mass-poin a 0. Playe s mus , he e o e, be indi e en be ween all poin s o hei in e -
al suppo :
¯
ui(G−i):=s
0
i(s;y)dG−i(y)−ci(s) o all s∈[0, ¯
s].(1)
Any pai o dis ibu ions (G1,G2) ha sa is y(1) is an equilib ium. This pape ’s main
con ibu ion o he li e a u e is in cha ac e izing he solu ion o his sys em o equa ions
and in showing ha i is unique.
Theo em 1. E e y wo-playe all-pay auc ion has a unique Nash equilib ium (G
i)i∈I∈
i∈I(Si)in mixed s a egies. Fu he mo e,
G
i(s)=s
0˜
gi(y)dy +¯
si
¯
s˜
gi(y)dy,(2)
whe e ˜
gi(s)sol es
˜
gi(s)=c
−i(s)
−i(s;s)−s
0
−i(s;y)
−i(s;s)˜
gi(y)dy,(3)
¯
sisol es ¯
si
0˜
gi(y)dy =1,and¯
s=mini∈I¯
si. The solu ion admi s he ep esen a ion
˜
gi(s)=c
−i(s)
−i(s;s)+s
0
−i(s;y)c
−i(y)
−i(y;y)dy,
whe e
−i(s;y):=−k0
−i(s;y)+k1
−i(s,y)−k2
−i(s;y)+···
o k0
−i(s;y):=
−i(s;y)
−i(s;s)and kn
−i(s;y),n=1, 2, , de ined ecu si ely by
kn
−i(s;y):=s
y
−i(s;z)
−i(s;s)kn−1
−i(z;y)dz.
We ou line he p oo he e wi h an emphasis on he gene al me hodology. We show
in he Appendix ha in any equilib ium, playe s choose s ic ly inc easing, con in-
uous mixed s a egies wi h common suppo on some in e al [0, ¯
s],asin(1), and
ha a mos one pa icipan can ha e a mass-poin a 0. Mo eo e , di e en ia ing (1)
yields (3), which mus be sa is ied on [0, ¯
s]in equilib ium o some ¯
s(Lemma 0in he
Appendix).
The key s ep is ecognizing ha we can apply esul s abou Vol e a in eg al equa-
ions (VIEs) o show ha (3) has a unique solu ion. The ele an esul is summa ized in
he ollowing lemma. Fo a p oo , see, e.g., B unne (2017).
Theo e ical Economics 19 (2024) Asymme ic all-pay auc ions 175
Lemma 1(Vol e a (1896)). Le K(s;y)and (s)be con inuous unc ions. Then he in e-
g al equa ion
g(s)= (s)+s
0
K(s;y)g(y)dy o all s∈[0, ¯
s](4)
has a solu ion g ha is unique almos e e ywhe e. Mo eo e , (4) de ines a con ac ion
mapping, implying he solu ion can be ound by i e a ion. This i e a ion educes o
g(s)= (s)+s
0
R(s;y) (y)dy,
whe e R(s;y)is he unique esol en ke nel de ined by
R(s;y)=∞
m=0
Km(s;y),
whe e K0≡Kand Kmis de ined ecu si ely o m=1, 2, as
Km(s;y)=s
y
Km−1(s;z)K(z;y)dz.
No e ha (4)is hesameas(3) o (s):=c
−i(s)/ −i(s;s)and K(s;y):=−
−i(s;y)/
−i(s;s). So Lemma 1implies ha only one pai o unc ions (˜
g1,˜
g2)sol es (3). Nex
we show ha he unique solu ions a e densi ies, i.e., o each i he e is an in e al [0, ¯
si]
whe e ˜
giis nonnega i e and in eg a es o 1.
Lemma 2. Assume a wo-playe all-pay auc ion whe e (˜
gi)i∈Isa is ies he indi e ence
condi ion in (3). Then, o each i∈I, he eexis s¯
si∈Sisuch ha
¯
si
0˜
gi(y)dy =˜
Gi(¯
si)=1(5)
and ˜
gi(s)is posi i e o s≤¯
si.
Lemma 2is p o en in he Appendix. We mus now ensu e ha he wo densi ies
ha e he same suppo . The nex key insigh is ha he e is exac ly one way o do his.
Recall ha a mos one playe can ha e an mass-poin and ha his mass-poin mus
be a 0 (Lemma 0). I ¯
s1=¯
s2, hen he e is a unique equilib ium wi hou any mass-
poin . O he wise, o de he playe s such ha ¯
s1<¯
s2. Then gi e playe 2 a mass-poin o
size 1 −˜
G2(¯
s1). By cons uc ion, bo h playe s’ densi ies in eg a e o 1 on he common
suppo [0, ¯
s1].
The abo e can be pe o med ia he ollowing s eps:
S ep 1. Find each ˜
gi(s).4
S ep 2. In eg a e each ˜
gi(s) o ind ¯
sigi en by (5).
4Analy ically, i can be exp essed as a se ies o in closed o m when possible (see Sec ion 5)o nume i-
cally (see Appendix IV).
176 Be o and Thomas Theo e ical Economics 19 (2024)
S ep 3. Take ¯
s=mini¯
siand gi e each playe a mass-poin a 0 o size
1−˜
Gi(¯
s),
which is posi i e o a mos one playe .
The h ee s eps a e illus a ed by Figu e 1.
Since he cumula i e dis ibu ion unc ions a e use ul, we some imes use he al e -
na e exp ession p esen ed in Co olla y 1.
Co olla y 1. Conside a wo-playe all-pay auc ion whe e i(s;y)is con inuously di -
e en iable in bo h a gumen s o all i∈I(Assump ion A1gua an ees di e en iabili y in
he i s a gumen ). Then we can al e na i ely exp ess he unique equilib ium as
Gi(s)=˜
Gi(si)−˜
Gi(s)+˜
Gi(s),
whe e
˜
Gi(s)=c−i(s)
−i(s;s)+s
0
∂ −i(s;y)
∂y ˜
Gi(y)
−i(s;s)dy.(6)
The solu ion admi s he se ies ep esen a ion
˜
Gi(s)=c−i(s)
−i(s;s)+s
0
c−i(y)
−i(y;y)
R−i(s;y)
−i(s;s)dy,
whe e
R−i(s;y):=K0
−i(s;y)+K1
−i(s,y)+K2
−i(s;y)+···
o K0
−i(s;y):=∂ −i(s;y)/∂y and Kn
−i(s;y),n=1, 2, , de ined ecu si ely by
Kn
−i(s;y):=s
y
∂
−i(s;z)
∂z
Kn−1
−i(z;y)
−i(z;z)dz.
We end his sec ion wi h a no e on pa allels be ween ou me hodology and ha used
in he incomple e-in o ma ion all-pay auc ion li e a u e. The simila i ies a e o mal in
na u e, and a ise because bo h p oblems in ol e sol ing a pai o di e en ial (in he
incomple e-in o ma ion case) o in eg al (in ou case) equa ions.
In he incomple e-in o ma ion se ing o , e.g., Amann and Leininge (1996), he
unique equilib ium—in pu e s a egies—is ob ained as he solu ion o a pai o di e en-
ial equa ions, which a ise om aking i s -o de condi ions o each playe s’ expec ed
payo s. To back ou he mass o playe s ypes’ ha bid 0, he au ho s hen make use
o he bounda y condi ion whe e each playe s’ op ype mus , in equilib ium, choose
iden ical op bids.
In ou se ing wi h comple e in o ma ion, he unique equilib ium is ins ead in mixed
s a egies. I is ob ained as he solu ion o a pai o in eg al equa ions, which a ise om
indi e ence, a he han i s -o de condi ions: playe s’ payo s mus be in a ian o any
choice o bids wi hin hei mixed-s a egy suppo s (1). Wea e henable opindown
Theo e ical Economics 19 (2024) Asymme ic all-pay auc ions 183
P oposi ion 4 (Mul iplica i e Ma gin o Vic o y Spillo e s). Assume a wo-playe con-
es such (i) o some i,( i(s;s))−1(∂ i(s;y)/∂y)=:ψi(s−y)depends only on he sco e
di e en ial s−y, and (ii) ∂ i(s;y)/∂y and ψiand ci(s)/ i(s;s)a eo exponen ialo de .
Then, o all s∈(0, s],
˜
g−i(s)=L−1⎧
⎪
⎪
⎨
⎪
⎪
⎩
xLci(s)
i(s;s)
1−Lψi(s)⎫
⎪
⎪
⎬
⎪
⎪
⎭
and
˜
G−i(s)=L−1⎧
⎪
⎪
⎨
⎪
⎪
⎩
Lci(s)
i(s;s)
1−Lψi(s)⎫
⎪
⎪
⎬
⎪
⎪
⎭
,
whe e Land L−1deno e he Laplace and in e se Laplace ans o ms, espec i ely.
P oposi ion 4can be used whene e he p ize alue is o he o m i(s;y)=
1
i(s) 2
i(s−y). Fo an applica ion whe e we sol e an all-pay auc ion wi h winne ’s eg e ,
see Sec ion 6.4.
6. Applica ions
6.1 Wa o a i ion wi h cos ly p epa a ion
The canonical wa o a i ion (WoA) is a game be ween wo playe s i=1, 2. Each picks a
sco e, which ep esen s an exi ime, in [0, ∞)and he playe i o selec he la ges sco e
siwins an amoun ha is dec easing in he lose ’s choice s−iand cons an in he own.
Aplaye ’spayo unc ionis husgi enby
ui(si;s−i)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
i(s−i)i si>s
−i
i(si)i si<s
−i
αi i(s−i)+(1−αi)i(si)i si=s−i,
whe e iand ia e s ic ly dec easing, con inuously di e en iable unc ions such ha
i(s)>
i(s),lims→∞i(s)=−∞,i(0)=0, and αi=1−α−i∈(0, 1).
The ypical WoA admi s mul iple equilib ia and, he e o e, does no sa is y he as-
sump ions in Sec ion 2.1. In pa icula , i iola es mono onici y (Assump ion A2)and
in e io i y (Assump ion A3) because he payo o he winne is cons an (and, he e o e,
nondec easing) in he playe ’s own sco e.
We p opose a gene al pe u ba ion ha selec s a unique equilib ium o he WoA, and
show ha such a pe u ba ion is sol able unde ou amewo k.8
8The p oblem o equilib ium selec ion in WoAs has been widely s udied in he li e a u e (Geo giadis,
Kim, and Kwon (2022), Mya (2005)). One way o selec a unique equilib ium is o unca e he game, as in
Ghemawa and Nalebu (1985), so ha a some poin in ini e ime bo h playe s p e e o exi . A di e en
184 Be o and Thomas Theo e ical Economics 19 (2024)
Suppose he winne ’s ou come is dec easing in he own sco e, e en i his depen-
dence is minimal,
ui(si;s−i)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
i(s−i)−εi(si)i si>s
−i
i(si)−εi(si)i si<s
−i
αi i(s−i)+(1−αi)i(si)−εi(si)i si=s−i
o any s ic ly inc easing con inuously di e en iable unc ion εiwi h εi(0)=0and
lims→∞εi(s)>
i(0) o all i.
We denomina e his a ian a WoA wi h cos ly p epa a ion, as he e is some small
p epa a ion cos ε(s)incu ed o se sco e s, i.e., he maximum amoun o ime s o
which one wishes o pa icipa . Fo example, a company engaged in a p ice wa migh
ha e o build up in en o y in ad ance o secu e a cos ly line o c edi .
A WoA wi h cos ly p epa a ion i s he wo-playe all-pay auc ion wi h spillo e s
whe e
i(si;s−i):= i(s−i)−i(si)
ci(si):=εi(si)−i(si),
which sa is y Assump ions A1–A4. The e o e, his game has a unique equilib ium, and
he e exis s some ¯
ssuch ha no playe bids abo e ¯
s.Theo em1 u he allows us o
cha ac e ize he equilib ium and P oposi ion 2gi es a closed- o m exp ession o he
equilib ium s a egies.
As he p epa a ion cos s become small (wi h ε
i(s)→0 uni o mly o all s), he unique
equilib ium o a WoA wi h cos ly p epa a ion app oaches he mixed-s a egy equilib-
ium o he classic WoA. This is p o en in he Appendix.
The WoA wi h cos ly p epa a ion gene alizes o he pe u ba ions ha ha e a unique
equilib ium. Fo example, Ab eu and Gul (2000)andKambe (2019)ex end heWoA o
le a a ional playe ’s opponen be o an uncomp omising ype wi h posi i e p obabili y,
whe e “uncomp omising” desc ibes someone who bids (o exi s a ) in ini y. Le zide-
no e he (known) p obabili y ha playe iis o an uncomp omising ype. Agains such
an opponen , a a ional o comp omising playe loses wi h ce ain y. This is a special
case o he WoA wi h cos ly p epa a ion whe e εi(s):=−(z−i/(1−z−i))i(s).
This ela ionship sheds ligh on he uniqueness o equilib ium ound in he WoA
wi h an uncomp omising ype. Indeed, by adding he possibili y o a ne e -yielding op-
ponen , we e ec i ely in oduce an una oidable cos ha depends on he playe ’s own
sco e. As was shown in he WoA wi h cos ly p epa a ion, his cha ac e is ic is ac ually
su icien o a unique equilib ium.
way o selec o an equilib ium, which we discuss in mo e de ail, is o in oduce a small p obabili y ha
a playe ne e exi s. See, o example, Ab eu and Gul (2000), Kambe (2019), Ko nhause , Rubins ein, and
Wilson (1989).
Theo e ical Economics 19 (2024) Asymme ic all-pay auc ions 185
6.2 O ensi e/de ensi e balance
Mili a y s a egis s gene ally ag ee ha wa a e is na u ally asymme ic: he de ending
pa y can usually p e ail wi h less expendi u e o esou ces han he a acke (Clausewi z
(1982)). Mo e gene ally, schola s ha e ied o iden i y which ac o s in luence he so-
called o ensi e/de ensi e balance, ha is, he many elemen s o mili a y echnology
ha gene a e ei he o ensi e o de ensi e ad an ages, and hus a ec he p obabili y o
wa (Le y (1984)). Ou model is able o cap u e bo h he de ensi e ad an age and he
ole o he p ize-deple ing na u e o wa in he o ensi e/de ensi e balance deba e.
An a acke (a) in ades a de ende ’s (d) e i o y, which is wo h V. Bo h comba an s
pu chase cos ly sco es in [0, ∞), and he comba an wi h he highe sco e wins. A sco e
o sicos s cisi,whe eci>0 is a posi i e cons an , o playe i∈I:={a,d}. Fu he mo e,
a’s sco e in lic s δasadamage o he e i o y. Assuming he de ende also in lic s a cos
o δdsdon o he a acke does no change he analysis. I he a acke wins, i in e nalizes
all cos s aced by he de ende , as hese cos s e ec i ely deple ed he esou ces a ailable
om he e i o y. Conside he payo unc ions ua:[0, ∞)→R o he a acke ,
ua(sa,sd)=pa(sa,sd)(V−δasa)−casa,
and he payo unc ion ud:[0, ∞)→R o he de ende ,
ud(sa,sd)=1−pa(sa,sd)(V−δasa)−cdsd,
whe e pa(·):[0, ∞)2→[0, 1]deno es he p obabili y ha he a acke is ic o ious.
Acco dingly, we le pa(sa,sd)=1whene e sa>s
d,pa(sa,sd)=0whensa<s
d,and
pa(sa,sd)=λ∈[0, 1]whene e sa=sd.
When we ans o m his model in o ou amewo k, we ge ci(si):=cisiand
a(sa;sd)= d(sd;sa):=V−δasa.
Assume i cos s weakly mo e o a ack han o de end (i.e., ca≥cd). The a acke does
no ha e any spillo e s, while he de ende is ha med by he opponen .
We a e able o le e age he linea i y o payo s in his case o ob ain a closed- o m
solu ion o he p oblem using P oposi ion 2. The de ende ecei es posi i e payo s i
and only i
¯
sd=V
ca+δa
<V
δa1−exp−δa
cd=¯
sa,
which holds whene e δa>0andca≥cd.In hiscase,
Ga(s)=1+cd
δa
logcaV
(ca+δa)(V−δas)and Gd(s)=cas
V−δas.
The p obabili y P(sa>s
d|δa,ca,cd) ha he a acke succeeds, in equilib ium, is gi en
by
P(sa>s
d|δa,ca,cd)=cd
δ2
aδa+calogca
ca+δa<cd
2ca≤1
2,
186 Be o and Thomas Theo e ical Economics 19 (2024)
whe e he sup emum is eached as δa→0. I he wa damages he e i o y a leas as
much as i cos s he a acke o in lic such damage (δa≥ca), a igh e bound is ob ained:
P(sa>s
d|δa,ca,cd)<1−log(2)<1
3.
E en i ca=cd, he de ende is mo e han wice as likely o win han he a acke is.
In ou model, he s onge posi ion o he de ensi e pa y comes as a byp oduc o he
in e se ela ionship be ween he a acke ’s s eng h and he e osion o he p ize’s alue.
This p o ides an al e na e explana ion o why i is ypically easie o de end han o
a ack, some hing usually a ibu ed o he high cos s o main aining long supply lines
and o keeping seized e i o ies (Glase and Kau mann (1998)). The de ende ’s s onge
posi ion also sugges s ha any posi i e pa icipa ion cos in a wa con es imposed on
he agg esso would be e ec i e in discou aging agg ession.9
6.3 Wa o in es men
In es men has long been conside ed as a me hod o commi ing o en y de e ence
(Dixi (1980)), while he wa o a i ion is a popula model o exi (Fudenbe g and Ti ole
(1986)). Ou model can combine he wo a ibu es in o a single model o compe i ion
in con inuous ime, whe e playe s in es o s ay in he game, bu a e able o ecoup
pa o ha in es men i hei opponen in es s less. Wa s o in es men can also be
used o model Cold-Wa -s yle de ense spending and compe i ion be ween echnology
companies and R&D aces.
Assume wo compe i o s, 1 and 2, in es in capi al sia cos ci(si).Thecapi alisnec-
essa y o engage in compe i ion and dep ecia es a a cons an a e. Compe i ion esul s
in ze o p o i s. Howe e , he winne is able o ex ac monopoly p o i s and bene i s om
he emaining capi al acco ding o an inc easing unc ion i(si−s−i). Mo e conc e ely,
assume payo s a e
u(si;s−i)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
i(si−s−i)−ci(si)i si>s
−i
−ci(si)i si<s
−i
αi i(0)−ci(si)i si=s−i
o any αi∈[0, 1).
I Assump ions A1–A4a e me , he e is a unique equilib ium o capi al in es men s
in mixed s a egies on ini e suppo . Mo eo e , he equilib ium admi s a closed- o m
solu ion by P oposi ion 3.
Example 2. Le i(s;y):=eρi(s−y)ωiand ci(s)=eρis−1, whe e ωi,ρi∈(0, 1) o each
i∈I:={1, 2}.Then
˜
gi(s)=ρ−i
ω−i
,
9In he mo e gene al nonlinea model, whe e he alue o he e i o y a e in asion is gi en by δ(sa)
and he cos o choosing sco e si o playe iis gi en by a con inuously di e en iable unc ion ci:[0, ∞)→
R+sa is ying he equi ed Assump ions A1–A4,cd(s)≤ca(s)is su icien o ensu e ha ˜
Ga(s)<˜
Gd(s) o
all s>0. This gua an ees he de ende ’s payo emains posi i e, wi h Gd(s)=ca(s)/ δ(s).
Theo e ical Economics 19 (2024) Asymme ic all-pay auc ions 187
so he equilib ium s a egies, excluding he possible mass-poin a 0, will be uni o m
wi h ˜
Gi(s)=(ρ−i/ω−i)s.
The pai o a ios ωi/ρiis, he e o e, a su icien s a is ic o he equilib ium o his
game. Assume, wi hou loss o gene ali y ha his a io is weakly la ge o playe 1.
Then he maximum du a ion o he game is playe 2’s a io ¯
s=ω2/ρ2.
The equilib ium is ully cha ac e ized by he o e all s eng h o he playe s ¯
sand he
compe i i e balance δ:=(ω2/ρ2)/(ω1/ρ1)∈(0, 1].
Because he s a egies a e uni o m, playe 1’s a e age commi men du a ion is hal
o he s eng h. Playe 2 on he o he hand has a mass-poin o size
G2(0)=1−δ,
which dec eases as he compe i ion becomes mo e balanced.
O e all, he con lic is expec ed o las o
Emin(s1,s2)=¯
s
01−G1(y)1−G2(y)dy =δ¯
s
3
o al pe iods. The ela ionship be ween o e all powe and wa du a ion is one o one.
The du a ion is also inc easing in he compe i i e balance. So a la ge s eng h di e -
en ial implies he con lic will ypically be sho -li ed, whe eas close con es s can ha e
delayed esolu ions. ♦
6.4 All-pay auc ion wi h winne ’s eg e
Winne ’s eg e is he emo se ha he winne has om spending mo e han is necessa y
o win a con es o auc ion. This phenomenon has mos ly been s udied in he con ex
o winne -pay i s -p ice, auc ions (Engelb ech -Wiggans (1989), Filiz-Ozbay and Ozbay
(2007)). We ins ead apply ou amewo k o model winne ’s eg e in an all-pay auc ion.
Le each playe i∈I:={1, 2}choose a sco e in [0, ∞). Suppose i alues he p ize
a μi(si)[1−hi(si−s−i)],whe eμi(si)is he playe ’s objec i e alue o he p ize and
hi(si−s−i)is he sha e o he winnings ha is unapp ecia ed due o eg e . Each playe
pays he cos ci(si)whe he he/she wins o loses. So payo s a e
u(si;s−i)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
μi(si)1−hi(si−s−i)−ci(si)i si>s
−i
−ci(si)i si<s
−i
αiμi(si)−ci(si)i si=s−i
o any αi∈[0, 1). We assume all unc ions a e con inuously di e en iable wi h c(s)>0
and h
i(s)≥0. Mo eo e , μ(0)>h
(0)=c(0)=0andc(s)>μ
(s) o each s,so ha
lowe bids a e p e e able e en wi h no eg e . In ui i ely, he eg e unc ion hshould no
exceed 1, hough his is no a echnical equi emen . We can sol e o he equilib ium
wi h wo playe s using P oposi ion 4and we can ex end his equilib ium o he game
wi h any numbe o iden ical playe s o p izes using Theo em 4.
188 Be o and Thomas Theo e ical Economics 19 (2024)
Example 3. Le μi(s):=ωi∈(0, 1
2),hi(s)=s2/2, and ci(s):=s−s2/2 o s∈[0, 1].Then
˜
gi(s)=e−s
ω−i
.
Wi hou loss o gene ali y, le ω1≥ω2, implying playe 1 ecei es a nonnega i e pay-
o . Playe 1 will hus play a unca ed exponen ial dis ibu ion wi h pa ame e 1 and
suppo [0, −log(1−ω2)]. He expec ed sco e will be
E[s1|ω1,ω2]=1+1−ω2
ω2log(1−ω2),
which depends nega i ely on he opponen ’s payo scaling ac o ω2.Thisislowe han
in he same game wi hou eg e .
The playe wi h ze o expec ed payo s will place a mass-poin a 0 o size
G2(0)=1−ω2
ω1
,
which is exac ly he same size as i he e we e no eg e . Playe 2 will ha e an expec ed
sco e
E[s2|ω1,ω2]=ω2
ω11+1−ω2
ω2log(1−ω2),
which is also less han in he same game wi hou eg e . The expec ed sum o he wo
sco es sco e is
E[s1+s2|ω1,ω2]=1+ω2
ω11+1−ω2
ω2log(1−ω2),
which is dec easing in ω1and inc easing in ω2. In con es s such as labo ou namen s,
a la ge p oduc i i y di e en ial be ween pa icipan s in he o m o a high ω1and low
ω2dep esses agg ega e e o . This is ue in a con es wi h no spillo e s, bu he pa -
ial de i a i e o ω1is la ge in absolu e alue when he e is eg e . Tha is, he e ec
is exace ba ed by he ac ha he s onge playe is penalized o winning by a la ge
ma gin. ♦
7. Mo e playe s
In con es s wi h spillo e s and mo e han wo playe s, many o he esul s conside ed
he e a e iola ed. Exis ence s ill holds (see Olszewski and Siegel (2023)), bu uniqueness
does no . Mo eo e , expec ed payo s will now depend on which equilib ium is played.10
When he no malized cos s a e anked, Theo em 2 in Siegel (2010) and Theo em 2 in
Siegel (2009) show ha only wo playe s e e pa icipa e in he equilib ium o a con es
o a single p ize. This e ec i ely collapses he p oblem in o a wo-playe con es .
A e sion o his condi ion holds in ou se ing. We s ill equi e no malized cos s o
be anked in some sense, bu in a way ha akes he spillo e s in o accoun .
10This is also ue o con es s wi h no spillo e s i mono onici y does no hold (Siegel (2009, Example 2)).
Theo e ical Economics 19 (2024) Asymme ic all-pay auc ions 189
Theo em 3. Assume iand j,i= j,a e woo hen>2playe s in a con es sa is ying
Assump ions A1–A4. Suppose ha playe ihas a posi i e payo in he wo-playe con es
whe e iand ja e he pa icipan s, and ha he ollowing “ anked cos s” condi ion holds
o all k/∈{i,j},s∈˜
Sk,si∈˜
Si,andsj∈˜
Sj
ck(s)
k(s;s{i,j})≥cj(s)
j(s;s{i}), (11)
whe e sHis a ec o o opponen sco es ha is 0 o all playe s no in se H. Then he e
exis s an equilib ium whe e only playe s iand jpa icipa e.
To unde s and condi ion (11), conside he candida e equilib ium whe e playe s i
and jcompe e using hei wo-playe s a egies, and playe kdoes no pa icipa e. By
no pa icipa ing, playe kea ns a payo o 0— he same payo as playe j.Condi-
ion (11) says ha i she en e s, playe k’s no malized cos will be highe a e e y poin
han playe j’s al eady is. The e o e, he payo om pa icipa ing is a mos 0 (playe j’s
payo ). So he e is no p o i able de ia ion o any playe .
No e ha i is possible o kjand jkin he sense o (11) when spillo e s dec ease
he alue o he p ize. In his case, he e a e mul iple equilib ia whe e di e en pai s o
playe s pa icipa e.
In he absence o spillo e s, mul iple equilib ia also a ise wi h h ee o mo e play-
e s. Howe e , i payo s a e asymme ic, he e can be a mos one equilib ium whe e
he suppo o each playe ’s s a egy is a union o in e als. Addi ionally, he payo s o
each playe a e consis en ac oss all equilib ia. Nei he o hese p ope ies holds in con-
es s wi h spillo e s. Payo s gene ally a y ac oss equilib ia in which di e en playe s
pa icipa e.
7.1 Symme ic equilib ia
The same me hod used o ind he equilib ium o wo-playe auc ions wi h spillo e s
can be applied mo e gene ally o ind symme ic equilib ia o all-pay auc ions wi h n>2
iden ical playe s and m<np izes, whe e he alue o he p ize o any gi en pa icipan
depends on his/he own sco e and on he sco e o he i s unne -up ( he playe wi h
he m+1 h highes bid). Mo e speci ically, each p ize has alue (s;y),whe esis he
playe ’s own sco e and yis he sco e o he i s unne -up. When he e is only one p ize,
his amoun s o saying ha i s alue depends only on he wo highes bids. Spillo e s
depend only on he sco e o he unne -up in many games such as he all-pay auc ion
wi h winne ’s eg e and any game wi h a s uc u e ha esembles a wa o a i ion. Fo
example, ba gaining games and ee iding games equen ly ha e his s uc u e whe e
he las holdou o comply delays he p ize o he winne s. In he case whe e he e is
one p ize, spillo e s ha depend on he i s unne -up cap u e he ma gin o ic o y,
which is ele an in many applica ions including elec ions and R&D aces.
Fo mally, we de ine a symme ic auc ion wi h unne -up spillo e s as a amily
{I,P,{˜
Si}i∈I,{ui}i∈I}, whe e he ollowing condi ions hold:
(i) The index se o playe s is I:={1, 2, ,n}wi h n≥2.
190 Be o and Thomas Theo e ical Economics 19 (2024)
(ii) The index se o p izes is P:={1, 2, ,m}wi h m<n.
(iii) Fo each i∈I,˜
Si:=[0, ∞)is playe i’s ac ion space. We le s−ideno e an a bi a y
elemen o ˜
S−i:=j=i˜
Sj.We u he le s(j)deno e he j h highes sco e.
(i ) Fo each i∈I,ui:˜
S→R,whe e ˜
S=i∈I˜
Si. Fo each s:=(si;s−i)∈˜
S,we u he
de ine
ui(s):=pi(s) (si,s(m+1))−c(si),
whe e (a) pi(s)deno es he p obabili y ha iwins a p ize gi en he sco e p o ile
s,wi hi∈Ipi(s)=mand
pi(s)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
1i si≥s(m)>s
(m+1)
1
{k∈I:sk=s(m)}i si=s(m)=s(m+1)
0i si≤s(m+1)<s
(m);
(b) :[0, ∞)2→R+maps each pai o sco es (si,s(m+1)) o playe i’s alue
(si;s(m+1)) om winning he p ize; (c) c:[0, ∞)→R+ou pu s playe i’s p i a e
cos c(si)gi en he submi ed sco e si.
In con as wi h he wo-playe case in oduced in Sec ion 2, he e we assume all play-
e s a e symme ic in he sense ha hey ha e iden ical alue ( )andcos (c) unc ions.
Mo eo e , all p izes a e equally aluable o each playe icondi ional on (si,s(m+1)).
In his con ex , we a e able o use he wo-playe , one p ize equilib ium cha ac e -
ized in Theo em 1 o cons uc he symme ic equilib ia o a symme ic n-playe , m-
p ize all-pay auc ion wi h spillo e s
Theo em 4 (Equilib ium o a Symme ic n-Playe , m-P ize All-Pay Auc ion Wi h Run-
ne -up Spillo e s). Conside a symme ic n-playe , m-p ize all-pay auc ion wi h unne -
up spillo e s. Assume and csa is y Assump ions A1–A4.Le ˆ
Gbe de ined as in Co ol-
la y 1. Tha is, le ˆ
Gbe he equilib ium cumula i e dis ibu ion unc ion o a wo-playe
all-pay auc ion wi h spillo e s,
ˆ
G(s)=c(s)
(s,s)+s
0
c(y)
(y,y)
R(s,y)
(s,s)dy,
wi h
R(s,y)=K0(s,y)+K1(s,y)+K2(s,y)+···
o K0(s,y)=∂ (s,y)/∂y and K (s,y), =1, 2, , de ined ecu si ely by K (s,y):=
s
y(∂ (s,z)/∂z)(K (z,y)/ (z,z))dz.
Then he symme ic equilib ium o he n-playe , m-p ize all-pay auc ion wi h unne -
up spillo e s is gi en by he unique G ha sol es
ˆ
G(s)=
n−1
j=n−mn−1
jG(s)j1−G(s)n−j−1. (12)
Theo e ical Economics 19 (2024) Asymme ic all-pay auc ions 191
To see why Theo em 4holds, conside he expec ed payo o a playe iwho bids s,
s
0
(s;y)dˆ
G(y)−c(s),
whe e ˆ
Gis he p obabili y measu e o he m hla ges sco eou o hen−1playe sin
I {i}. In a symme ic equilib ium, ˆ
Gis he n−m h o de s a is ic o a sample o n−1
d aws om he equilib ium dis ibu ion G, gi ing us (12). The igh -hand side o (12)is
inc easing in G(s)and, hus, may be in e ed o ob ain Ggi en ˆ
G.
A hesame ime, ˆ
Gis he equilib ium o a symme ic wo-playe auc ion, since each
playe ’s indi e ence condi ion is iden ical o (3). This allows us o use Theo em 1 o
ind ˆ
G.
Theo em 4shows ha he equilib ium in he wo-playe case is also he symme ic
equilib ium o he game wi h any numbe o playe s and p izes, subjec o a pa icula
mono one ans o ma ion. In ui i ely, inc easing he numbe o playe s (o dec easing
he numbe o p izes) educes he sco es o each playe .
Asymme ic equilib ia a e mo e di icul o cha ac e ize when ei he spillo e s o
incomple e in o ma ion is p esen . In a wo ld wi hou spillo e s bu comple e in o -
ma ion, he equilib ium o an n-playe , m-p ize con es is unique unde mild condi-
ions, and Siegel (2010) was able o p o ide an algo i hm o i s cons uc ion. Unde
incomple e-in o ma ion, asymme ic equilib ia wi h mo e han wo playe s a e no o i-
ously di icul o analyze. A gene al cha ac e iza ion is s ill an open ques ion, as a as
we a e awa e.11
8. Rela ed li e a u e and conclusions
Th oughou his pape , we cha ac e ized and es ablished uniqueness o he equilib-
ium o wo-playe con es s using echniques om he heo y o in eg al equa ions. We
hen ex ended hese esul s o symme ic con es s wi h mo e playe s and p izes o cha -
ac e ize he symme ic equilib ium. This allowed us o de i e insigh s on equilib ium
payo s, winne s and lose s, and on he impo ance o spillo e s o applica ions. This
model does no equi e o imply ha he esul s o a con es a e known in ad ance. In
ac , playe s a e always unce ain o hei own ic o y. Howe e , his unce ain y s ems
om no knowing he esou ces ha you opponen dedica ed o he con es . The ac
ha anked no malized cos s a e no enough o es ablish dominance demons a es how
spillo e s can a o high-cos , low- alue playe s who ne e heless ha e a ma ginal cos
ad an age o e hei opponen when bids a e high. In pa icula , he esul s in his pape
sugges se e al po en ial consequences o legal s uc u es, con lic s and compe i ion.
This pape is mos closely ela ed o wo o he s. Baye, Ko enock, and de V ies (2012)
also conside s spillo e s in wo-playe con es s, bu ocuses on symme ic equilib ia and
linea symme ic cos s and alua ions. We show ha he e a e no asymme ic equilib ia
11See Ki kegaa d (2013) and Pa ei as and Rubinchik (2010) o analyses on pa icula equilib ia o N≥3
playe s all-pay incomple e-in o ma ion auc ions.
192 Be o and Thomas Theo e ical Economics 19 (2024)
in his wo-playe case and ex end he analysis o include asymme ic playe s and gen-
e al unc ional o ms o he p ize alues. This allows us o es ablish equilib ium unique-
ness, exp ess no el esul s abou payo s, and cha ac e ize he equilib ium in di e en
applica ions (Sec ion 6).
The second pape ha app oaches a simila ques ion o ou own is Xiao (2018). The
au ho , howe e , ocuses on cons an p ize alues and sepa able spillo e s in he cos
unc ions, which a e independen o winning o losing. This independence signi ican ly
es ic s he equilib ium e ec s o he spillo e s. Linea ly sepa able spillo e s on he
cos ha e no e ec on he equilib ium, while mul iplica i ely sepa able spillo e s scale
he cos o bids by an endogenous cons an . This is no ue when spillo e s a e in he
p ize alue.
Fu and Lu (2013) also analyze wo-playe all-pay auc ions wi h linea spillo e s in
he cos s. They assume ha each con es an is a i m wi h a mino i y s ake in hei op-
ponen ’s p o i s. As such, e en when a i m loses he auc ion, hey s ill ge o keep a sha e
o he p ize. On he o he hand, ega dless o winning o losing, hey mus also sha e in
he cos o e o incu ed by hei opponen ; hence, he exis ence o cos spillo e s. Be-
cause hese spillo e s a e linea and do no a ec he p ize o he winne , hey ha e no
e ec on he equilib ium dis ibu ions, as was also no ed in Xiao (2018).
Ou pape is also connec ed mo e b oadly o he li e a u e o spillo e s in o he auc-
ion and auc ion-like amewo ks. Hodle and Yek a¸s(2012), o example, use a linea
i s -p ice auc ion wi h spillo e s o model wa . The au ho s e e o his as an all-pay
con es , bu only he winne ac ually pays because o he way unds a e handled.
No ably, spillo e s ha e been gi en compa a i ely mo e a en ion in he Tullock con-
es amewo k. In hese con es s, each pa icipan s’ p obabili y o winning is gi en by
pi(si;s−i)=s
i/(s
i+s
−i)i (s1,s2)=(0, 0)and by pi(si;s−i)=1/2i (s1,s2)=(0, 0).He e
∈(0, ∞)is a pa ame e ha con ols how much one’s p obabili y o winning esponds
o an inc ease in sco es. The all-pay auc ion is a Tullock con es whe e =∞;when
=1, we ha e a Tullock lo e y ins ead.
Chowdhu y and She eme a (2011a) s udy a gene alized Tullock lo e y in which
payo s linea ly inco po a e one’s own e o and he e o o he i al. Thei pape
s udies symme ic payo and cos s uc u es, and, as is usual in Tullock- ype con-
es s, bo h playe s a e able o ex ac posi i e payo s. The au ho s ob ain asymme ic,
pu e-s a egy equilib ia, e en when playe s a e iden ical (Chowdhu y and She eme a
(2011b)). In con as , ou all-pay amewo k yields a unique symme ic mixed-s a egy
equilib ium when playe s a e iden ical, and expec ed payo s a e ze o.
Damiano , Sande s, and Yildizpa lak (2018) allows o playe s’ e o s o p oduce ei-
he posi i e (p oduc i e) o nega i e (des uc i e) ex e nali ies in a wo-playe Tullock
lo e y. The au ho inds ha spillo e s can ei he accen ua e o educe he compe i-
i e balance be ween pa icipan s when con as ed o a compa able ixed-p ize con es .
Howe e , unlike ou esul s, no e e sal can e e occu : he “ a o ed” playe is always
mo e likely o win and has a highe expec ed payo .12
12The e a e many o he examples o spillo e s in Tullock- ype con es s; see, e.g., Chung (1996) and Hi ai
and Szida o szky (2013).
Theo e ical Economics 19 (2024) Asymme ic all-pay auc ions 199
All ha emains is o show ha Gcan be eco e ed om ˆ
G(s). To see ha his is he case,
we ew i e (12)as ˆ
G(s)=E[G(s)],whe e
E[p]=
n−1
j=n−mn−1
jpj(1−p)n−j−1.
Tha is, E[p]is equal o he su i al unc ion o he binomial dis ibu ion e alua ed a
n−m. This unc ion is known o be s ic ly inc easing in p o p∈[0, 1]. The e o e, Eis
in e ible.
Appendix II: Op imal con es design
In his sec ion, we conside how a designe should bias a con es o inc ease he sco es.
Se e al pape s ha e analyzed his p oblem o assigning p izes o maximize o al sco es,
o he a e age sco e o he winne . Fo example, Mealem and Ni zan (2014)conside
p ize edis ibu ion in a wo-playe all-pay auc ion wi h ixed alues and symme ic
cos s. They show ha equalizing he p ize alues maximizes he o al sco es and ha he
con es yields weakly mo e o al sco e han any simila Tullock- ype lo e y con es . Che
and Gale (2003) in es iga e he op imal design o con es s o inno a ion p ocu emen
and ind ha he p ocu e migh wan o limi he maximum p ize a ailable o he mos
e icien i ms—e ec i ely elimina ing any posi i e en s—so as o inc ease hei own
expec ed maximum su plus. The p oblem o op imal con es design in all-pay auc ions
wi h spillo e s has no been p e iously analyzed.
This is ele an because p incipals a e cons ained in he p izes ha hey can o -
e . Many o he ools ha p incipals use o make p izes ha e spillo e s. Fo example,
i an employe chooses o cons uc a compensa ion package using a cash bonus and
s ock op ions, hen he inclusion o he s ock op ions will gene a e spillo e s. This sec-
ion analyzes he op imal p ize choice when p izes can be cons uc ed om mul iple
ins umen s.
Le i⊂R˜
Sideno e he se o p ize unc ions a ailable o he designe o playe i,
and le V:i∈I˜
Si×i∈Ii→Rdeno e he designe ’s payo unc ion; i.e., gi en he pai
o sco es s:=(s1,s2)and he pai o alue unc ions =( 1(·;·), 2(·;·)),V(s, )deno es
he designe ’s de i ed ne bene i om he con es .
We make he ollowing (mild) assump ions.
Assump ion 1 (Comple eness, D1). Fo each i∈I, he se o p izes iis con ex and i s
closu e con ains an elemen wi h i(·;·)≡0.
Assump ion 2 (P oduc i e Sco es, D2). Fo each i∈Iand ∈i∈Ii, hedesigne ’s
objec i e unc ion V(s, )is s ic ly inc easing in si.
Assump ion 3 (Cos ly P izes, D3). Fo each i∈I,s∈i∈I˜
Siand −i∈−i,V(s, )is
dec easing in i.14
14Tha is, i i,ˆ
i∈ia e such ha i(s;y)≤ˆ
i(s;y) o all (s,y)∈˜
Siט
S−i, henV(s,( i, −i)) ≥
V(s,(ˆ
i, −i)).
200 Be o and Thomas Theo e ical Economics 19 (2024)
The p ima y complica ion wi h he cons uc ion in his pape is ha he mass-poin
is di icul o compu e. Fo una ely, i he mechanism designe can disc imina e be-
ween he wo playe s, an op imal mechanism will ha e no a oms in many speci ica-
ions. This is o malized in he ollowing p oposi ion.
P oposi ion 5. Assume a wo-playe con es whe e a ully in o med p incipal wi h pay-
o unc ion Vchooses he p ize i∈i o each i∈I. Assume ha iand Vsa is y As-
sump ions D1–D3, and ha o all iand all i∈i, Assump ions A1–A4hold. Then no
con es an in equilib ium can ha e a posi i e payo . Equi alen ly, no playe will ha e a
poin -mass as pa o his/he s a egy.
P oposi ion 5implies ha he e will be no s ic ly dominan playe in any disc im-
ina ing con es design p oblem whe e he p incipal bene i s om he e o s o pa ic-
ipan s and pays o p izes. This p oposi ion comes om he ac ha he equilib ium
s a egy o he dominan playe is locally in a ian o changes in he p ize alue. In u-
i i ely, o any con es wi h a s ic ly dominan playe , he e exis s a mo e compe i i e
con es whe e his/he p ize is educed and sco es a e la ge .
P oo o P oposi ion 5. Take an op imal choice o :=( i)i∈I∈i∈Ii. Suppose, by
con adic ion, ha playe ihas a s ic ly posi i e payo . He s a egy is de ined by
˜
gi(s)=c
−i(s)
−i(s;s)−s
0
−i(s;y)
−i(s;s)˜
gi(y)dy,
which does no depend on i. Because playe −ihas an a om, we know ha ˜
Gi(¯
s)−
˜
G−i(¯
s)>0. The e o e, he e exis s a γ∈(0, 1)such ha ˜
Gi(¯
s)=˜
G−i(¯
s)/γ.
Then he p incipal could o e (γ i, −i)wi hou changing he equilib ium s a egy
o playe i. By cos ly p izes, Assump ion D3, his is weakly p e e able gi en a ixed dis-
ibu ion o s−i.
By cons uc ion, playe 2’s new equilib ium s a egy is ˜
G−i(¯
s)/γ. This i s -o de
s ochas ically domina es playe −i’s o iginal s a egy. In ac , i is he same dis ibu-
ion, bu wi h he mass-poin emo ed. The p oduc i e sco es assump ion implies ha
his mechanism is s ic ly p e e ed.
P oposi ion 5demons a es ha he expec ed wel a e o all agen s is 0 in a la ge
class o con es design p oblems.15 I also sugges s he op imali y, om a design pe -
spec i e, o handicapping he mos e icien playe s (as in he playe s wi h lowe cos s
and lowe ma ginal cos s) The idea is e y much analogous o he conclusion in Che
and Gale (2003); o example, handicapping he playe who has he echnological uppe
hand causes he less e icien playe o become mo e agg essi e and o choose highe
sco es han hey would o he wise.
15Which is no o say ha he e a e no se ings whe e i would no apply. Fo example, he designe could
wish o maximize he agen s’ expec ed wel a e. In his case, he p incipal’s objec i e unc ion would iola e
cos ly p izes. I would usually also iola e p oduc i e sco es.
Theo e ical Economics 19 (2024) Asymme ic all-pay auc ions 201
Appendix III: Remo ing assump ions
An auc ion ha sa is ies all bu one o Assump ions A1–A4may ha e equilib ia ha ail
o mee ou cha ac e iza ion. An auc ion ha ails o sa is y any o Assump ions A2–A4
can ha e mul iple equilib ia.
Elimina ing Assump ion 1(Smoo hness). We assume con inuous di e en iabili y
o iand ci. Con inui y is no su icien o ensu e ha he equilib ium has in e al sup-
po . Fo example, conside he case whe e he p ize is ixed a i=1 and he cos s a e
gi en by he densi y unc ion o some dis ibu ion ha uni o mly assigns p obabili y 1 o
a dense subse o [0, 1]wi h Lebesgue measu e 0.16 This cos unc ion is con inuous be-
cause he dis ibu ion assigns uni o m weigh o in ini ely many poin s. I is also s ic ly
inc easing because he suppo is dense. Howe e , i is no absolu ely con inuous. Then
he a o emen ioned dis ibu ion is an equilib ium, which has suppo only on a se o
measu e 0.
Elimina ing Assump ion 2(Mono onici y).Thecasewhe e
i(si;y)>c
i(si) o
some siis conside ed in Siegel (2014) wi hou spillo e s. In his case, he equilib ium
dis ibu ion has gaps and is hus no an in e al. In he p esence o spillo e s, non-
mono onici y may gene a e pu e-s a egy equilib ia o esul in non-uniqueness. Fo
example, conside he symme ic game whe e
1(s;y)= 2(s;y)= (s;y)=%1+s−yi s≤1
(3−s)s−yi s>1
and c1(s)=c2(s)=c(s)=s. No e ha his p ize alue sa is ies all assump ions excep
o mono onici y, which is iola ed on [0, 1]. The e a e wo asymme ic pu e-s a egy
equilib ia whe e one playe bids 0 and he o he playe bids 1.17
Elimina ing Assump ion 3(In e io i y). Conside he symme ic all-pay auc ion
wi h spillo e s, whe e 1(s;y)= 2(s;y)= (s;y)=2√yand c1(s)=c2(s)=c(s)=s.
Then he e is a pu e-s a egy equilib ium whe e bo h playe s play 0. Mo eo e , he e
is also a mixed-s a egy equilib ium a
G
1(x)=G
2(x)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
√xi x∈[0, 1]
0i x<0
1i x>1,
i.e., equilib ia a e no longe unique.
In he o he case whe e cos s a e no highe han he p ize alue in he limi ( ha
is, limsi→∞supy∈˜
S−i i(si;y)≥limsi→∞ci(si)), playe s migh ind i p o i able o submi
unbounded bids. This can esul in nonexis ence.
16Fo example, a uni o m dis ibu ion o e he coun able union o can o se s shi ed by each o he
a ionals modulo 1.
17This game also has a symme ic mixed-s a egy equilib ium.
202 Be o and Thomas Theo e ical Economics 19 (2024)
Elimina ing Assump ion 4(Discon inui y a Ties). Conside he symme ic all-
pay auc ion wi h spillo e s, whe e 1(s;y)= 2(s;y)= (s;y)=1y≤14&1−yand c(s)=s.
Then he e is a symme ic equilib ium whe e
G
1(x)=G
2(x)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
1−√1−x
2i x∈[0, 1)
0i x<0
1i x≥1
such ha bo h playe s ha e an a om a 1, he poin whe e he p ize is wo h 0 in he e en
o a ie. The usual a gumen ha he wo playe s canno ha e a oms a he same poin
ails he e because a small inc ease in ei he playe ’s bid does no inc ease he p obabili y
o winning a p ize o posi i e alue. The e a e also wo asymme ic equilib ia in his
game o he o m
G
i(x)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
1−√1−x
2i x∈[0, 1)
0i x<0
1i x≥1
G
−i(x)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
1−√1−x
2i x∈[0, 1]
0i x<0
1i x>1.
Tha is, one playe has an a om a 0 while he o he has an a om a 1. Any con ex com-
bina ion o he symme ic equilib ium and he abo e is also an equilib ium.
Appendix IV: Nume ical app oxima ion
I e a ion me hod I is possible o app oxima e he solu ion by i e a ing nume ically on
he sequence
˜
gn+1(s)=1
(s;s)c(s)−s
0
(s;y)˜
gn(y)dy
s a ing om ˜
g0=0 o ind he ue ˜
g. The e is a much simple and as e way.
Ma ix me hod 1 Conside ou o iginal equa ion
s
0
−i(s;y)˜
gi(y)dy =c(s)
and conside his 3 ×3 disc e e app oxima ion o his p oblem o s∈[0, 1]:
1
3
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
−i1
3,1
300
−i2
3,1
3 −i2
3,2
30
−i1, 1
3 −i1, 2
3 −i(1, 1)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
! "
V
·⎡
⎢
⎢
⎢
⎢
⎣
˜
gi1
3
˜
gi2
3
˜
gi(1)
⎤
⎥
⎥
⎥
⎥
⎦
! "
g
≈⎡
⎢
⎢
⎢
⎢
⎣
c−i1
3
c−i2
3
c−i(1)
⎤
⎥
⎥
⎥
⎥
⎦
! "
c
.
So we can app oxima e ˜
gi(s)wi h
g=3V−1c.
Theo e ical Economics 19 (2024) Asymme ic all-pay auc ions 203
Ma ix me hod 2 To ge a good es ima e, we do he same hing wi h an N×Ng id o
Nla geonsomein e al[0, T]:18
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
˜
gi1
N
˜
gi2
N
.
.
.
˜
gi(T)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
≈N
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
−i1
N,1
N0··· 0
−i2
N,1
N −i2
N,2
N··· 0
...
−iT,1
N −iT,2
N··· −i(T;T)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
−1
·
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
c−i1
N
c−i2
N
.
.
.
c−i(T)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
Ge ing he ac ual s a egies Once you ge (˜
g1,˜
g2), you jus ha e o pe o m he ollow-
ing s eps:19
S ep 1. Take he cumula i e sum and di ide by N o ge (˜
G1,˜
G2):
G1, G2 = cumsum(g1)/N, cumsum(g2)/N
S ep 2. T unca e bo h dis ibu ions a he las alue whe e bo h a e ≤1:
G1, G2 = G1[G1 <= 1 & G2 <= 1], G2[G1 <= 1 & G2 <= 1]
S ep 3. Add o each cumula i e dis ibu ion unc ion ec o so ha bo h end wi h 1
(add he a om):
G1, G2 = (G1 - G1[-1] + 1), (G2 - G2[-1] + 1)
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Co-edi o Ma ina Halac handled his manusc ip .
Manusc ip ecei ed 9 No embe , 2021; inal e sion accep ed 6 Janua y, 2023; a ailable on-
line 26 Janua y, 2023.
