A general model of Bertrand-Edgeworth duopoly

Allison, Blake A.; Lepo e, Jason J.
A icle
A gene al model o Be and-Edgewo h duopoly
Games
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Sugges ed Ci a ion: Allison, Blake A.; Lepo e, Jason J. (2025) : A gene al model o Be and-Edgewo h
duopoly, Games, ISSN 2073-4336, MDPI, Basel, Vol. 16, Iss. 3, pp. 1-37,
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Ci a ion: Allison, B. A., & Lepo e, J. J.
(2025). A Gene al Model o Be and–
Edgewo h Duopoly. Games,16(3), 26.
h ps://doi.o g/10.3390/g16030026
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A icle
A Gene al Model o Be and–Edgewo h Duopoly
Blake A. Allison 1and Jason J. Lepo e 2,*
1Depa men o Economics, Emo y Uni e si y, A lan a, GA 30322, USA; baallison@emo y.edu
2Depa men o Economics, Cali o nia Poly echnic S a e Uni e si y, San Luis Obispo, CA 93407, USA
*Co espondence: jlepo [email protected]
Abs ac : This pape s udies a class o wo-playe all-pay con es s wi h ex e nali ies ha
encompass a gene al e sion o duopoly p ice compe i ion. This all-pay con es o mula ion
pu s li le es ic ion on p oduc ion echnologies, demand, and demand a ioning. The e
a e wo ypes o possible equilib ia: In he i s ype o equilib ium, he lowe bound o
p icing is he same o each i m, and he p obabili y o any p icing ie abo e his p ice is
ze o. Each i m’s equilib ium expec ed p o i is hei monopoly p o i a he lowe bound
p ice. In he second ype o equilib ium, one i m p ices a he lowe bound o he o he
i m’s a e age cos and o he i m p ices acco ding o a non-degene a e mixed s a egy.
This ype o equilib ium can only occu i p oduc ion echnologies a e su icien ly di e en
ac oss i ms. We de i e necessa y and su icien condi ions o he exis ence o pu e s a egy
equilib ium and use hese condi ions o demons a e he agili y o de e minis ic ou comes
in p icing games.
Keywo ds: p ice compe i ion; con es ; demand a ioning; capaci y cons ain s
1. In oduc ion
The de e mina ion o p ices in ma ke s wi h e y ew selle s has been a cen al subjec
o inqui y since he incep ion o ma hema ical economics . Edgewo h (1925) mo ed he
unde s anding o his subjec o wa d by app ecia ing he impac o consume a ioning
and he p ominence o p ice inde e minacy, o p icing cycles .
1
While he concep ual o igins
o he Be andâ
e
“Edgewo h (he ea e BE) model can be aced back o Edgewo h, his
basic insigh s we e i s o malized in o a game heo e ic model by Shubik (1959).
2
Shubik
ocused on unde s anding he ange o p icing in mixed s a egy equilib ium and he
cha ac e o pu e s a egy equilib ia when hey exis .
These p icing games ha e been widely s udied since Shubik’s o maliza ion. The
s anda d BE model in he li e a u e has he ollowing ea u es: i ms possess cons an
ma ginal cos s up o capaci y (an absolu e limi on p oduc ion), and consume s a e a ioned
acco ding o ei he he e icien o p opo ional a ioning ule.
3,4
This BE model has been
used o unde s and undamen al issues in p ice de e mina ion, including duopoly p icing
and capaci y in es men (Allen & Hellwig,1993;Da idson & Denecke e,1986;Denecke e &
Ko enock,1996;K eps & Scheinkman,1983;Lepo e,2009;Le i an & Shubik,1972;Osbo ne
& Pi chik,1986), sequen ial p icing (Allen,1993;Allen e al.,2000;Denecke e & Ko enock,
1992), la ge ma ke s (Allen & Hellwig,1986;Dixon,1987,1992;Vi es,1986), oligopoly
(De F ancesco & Sal ado i,2010;Hi a a,2009), and unce ain y (de F u os & Fab a,2011;
Lepo e,2008,2012;Reynolds & Wilson,2000). Only a ew pape s ha e in es iga ed BE
models wi h cos s uc u es ou side he cons an ma ginal cos case.
5
Dixon (1987) conside s
a model o BE oligopoly wi h s ic ly con ex cos s, showing he non-exis ence o pu e
s a egy equilib ium o he case o e icien and p opo ional a ioning. Yoshida (2006)
Games 2025,16, 26 h ps://doi.o g/10.3390/g16030026
Games 2025,16, 26 2 o 37
cha ac e izes equilib ium p icing in symme ic duopoly wi h con ex cos and e icien
a ioning.
6
While he li e a u e has p oduced in e es ing esul s, he models a e es ic i e
in e ms o p oduc ion echnologies and demand a ioning o consume s and asymme ies
ac oss i ms.
The pu pose o his pape is o analyze he p ope ies o equilib ia in a BE model
wi h a b oad ange o p oduc ion echnology, minimal es ic ion demand a ioning, and
asymme ies ac oss i ms. Ou app oach o he analysis is based on modeling p ice compe-
i ion as a pa icula ex ension o an all-pay con es (Siegel,2009,2010,2014). In o de o
con ex ualize ou analysis, we iden i y some o he abs ac p ope ies o he BE model wi h
hose o a s anda d all-pay auc ion.
7
In he BE model, i ms place bids in he o m o a p ice
in an a emp o win he la ge sha e o demand, which goes o he i m wi h he highes
bid (lowes p ice). The e a e wo undamen al dis inc ions be ween he BE model and he
all-pay con es o adi ional all-pay auc ion. Fi s , he payo o he losing playe ( he
i m wi h he highes p ice) may depend on he p ice o he winne h ough he a ioning
o esidual demand, while, adi ionally, he losing playe ’s payo depends only on he
commi ed bid. Second, he payo o bo h he winne and lose can be non-mono onic in
he bid, as a educ ion in p ice inc eases he quan i y demanded, possibly aising p o i s,
while an inc eased bid in adi ional con es s me ely commi s he winne o lose o a
lowe payo .8
In Sec ion 2, we p esen he gene al model and in oduce key no a ion. The model is
de ined based on abs ac p ope ies o he on -side p o i o he lowe -p iced winne i m
and he esidual p o i o he highe -p iced lose i m. The abs ac con es o mula ion
allows us o analyze a model wi h a b oad ange o unde lying speci ica ions: gene al
p oduc ion echnology including he case o
U
-shaped a e age cos o p oduc ion, minimal
es ic ion on demand allowing o a wide swa h o demand a ioning (including a ioning
ou comes ha a e equilib ia om consume sea ch), and asymme ies ac oss i ms.
9
In
o de o es ablish he bounds on equilib ium p ices and payo s, we de ine he ollowing
p elimina y objec s. Fi s , de ine he c i ical judo p ice as he highes p ice ei he i m can se
o gua an ee ha he o he i m would a he maximize i s esidual p o i han unde cu .
This e minology is based on he sequen ial p icing model o judo economics by Gelman
and Salop (1983).
10
The second impo an p ice we de ine is he c i ical sa e p ice, which is
he in imum o all p ices a which i ea ns a leas i s minâ
e
“max p o i i i s i al unde cu s.
The gene al esul s on he p ope ies o all equilib ia a e p esen ed in Sec ion 3. A e
es ablishing gene ic equilib ium exis ence,
11
we show ha he e a e only wo ypes o
possible equilib ia. The i s ype o equilib ium is such ha bo h i ms’ p icing dis ibu-
ions ha e he same lowe bound, and ies a p ices g ea e han he lowe bound occu
wi h p obabili y ze o. This ype o equilib ium includes he possibili y o pu e s a egy
equilib ium in which i ms play he lowe bound p ice wi h ce ain y. The second ype o
equilib ium can only occu i he i ms ha e su icien ly di e en p oduc ion echnology,
and a e such ha one i m plays a pu e s a egy p ice while he o he plays a nondegene -
a e mixed s a egy. Since a model speci ica ion can ha e mul iple non-payo equi alen
equilib ia, we show ha , in all equilib ia, he expec ed p o i s o each i m a e bounded
be ween i s monopoly p o i a he c i ical sa e p ice and he c i ical judo p ice. In he
p ocess o es ablishing he payo bounds, we p o ide abs ac bounds o he ange o
equilib ium p icing.
The esul s pa icula o pu e s a egy equilib ium a e p esen ed in Sec ion 4. We
p o ide necessa y and su icien condi ions o he exis ence o pu e s a egy equilib ium.
All pu e s a egy equilib ia mus be symme ic and only exis unde wo ci cums ances. The
i s case is a symme ic p icing p o ile a which each i m’s esidual p o i is maximized
and equal o he monopoly p o i a he same p ice. The second case is a symme ic p icing
Games 2025,16, 26 3 o 37
p o ile a which exac ly one i m’s monopoly p o i exceeds i s esidual p o i (weakly
exceeding i s maximum esidual p o i ), and he sha ing ule is such ha his i m ecei es
i s monopoly p o i wi h ce ain y. In his second case, he p ice mus maximize he o he
i m’s esidual p o i . The exis ence o a pu e s a egy equilib ium does no gua an ee
uniqueness, as he e may be addi ional mixed s a egy equilib ia ha exis concu en ly. We
show ha a pu e s a egy equilib ium p ice
x∗
is unique as he pu e s a egy equilib ium
p ice
x∗
is unique i , o each i m, his p ice is he unique maximize o esidual p o i
when he o he i m p ices a
x∗
, and each i m’s esidual p o i is noninc easing in he
o he i m’s p ice.
We p esen a special case model using a s anda d BE cons uc ion wi h all assump ion
made di ec ly on each i m’s p oduc ion echnology and demand in Sec ion 5. Fi s , we
explo e addi ional p ope ies o pu e s a egy equilib ium o his special case model, iden-
i ying he necessa y and su icien condi ions on he unde lying p imi i es o p oduc ion
echnology and demand. Second, we examine he impac s o demand and supply shi s on
he bounds o equilib ium p ices and p o i s. These shi s can accommoda e changes in
a ioning, p oduc ion cos , o capaci y. We demons a e ha an inc ease in esidual demand
will weakly inc ease he bounds on he lowes equilib ium p ice along wi h he bounds on
p o i s; howe e , by example, we show ha he uppe bound on p icing may be educed.
An inc ease in a i m’s supply weakly dec eases he bounds on he lowes equilib ium p ice
along wi h he bounds on he o he i m’s p o i s. A gene al p edic ion canno be made o
he bounds on he p o i o he i m wi h he supply inc ease, as he e a e coun e ailing
e ec s: a di ec e ec h ough which lowe cos s o highe capaci ies enhance p o i abil-
i y, and an indi ec compe i i e e ec h ough which hose changes inc ease he le el o
compe i ion, d i ing down p ices and p o i s.12
Finally, all p oo s o lemmas and p oposi ions a e loca ed in Appendix A.
2. The Model
In his sec ion, we lay ou he gene al model and hen p o ide a subsec ion o examples
o illus a e he scope o he model. All assump ions s a ed in his sec ion a e main ained
h oughou he emainde o he pape . Conside a homogeneous p oduc indus y wi h
wo i ms
i=
1, 2. We will use
j=
1, 2, o e e o he i m o he han
i
. The i ms
simul aneously and independen ly announce p ices. We deno e by
pi
he p ice o i m
i
and
by
p
he ec o o bo h i ms’ p ices. Since
p
is he ec o o p ices
(p1
,
p2)
, we will use
x
o
unambiguously deno e a single p ice when i is no associa ed wi h a pa icula i m. The
p o i ha each i m ecei es depends on whe he i has a lowe p ice han he o he i m.
The on -side p o i o he i m
i
wi h a lowe p ice han i m
j
is
φi(pj)
, while he esidual
p o i o he i m
i
wi h a highe p ice han i m
j
is
ψi(p)
. The domain o esidual p o i
ψi
is
{(pi
,
pj)∈R2
+
:
pi≥pj}
, as i need no be de ined o p ices such ha
pi<pj
since he
esidual p o i canno be ob ained a such p ices. We make he ollowing assump ions on
he p o i unc ions φiand ψi.
Assump ion 1. φi(x)≥ψi(x,x′)≥0 o all x ≥x′.
The assump ion ha he on -side p o i is a leas as la ge as he esidual p o i is
consis en wi h he no ion ha cus ome s p e e lowe p ices, and hus, he i m wi h he
lowes p ice has weakly g ea e po en ial o sell. This assump ion also implies ha he
lowe -p ice i m is no equi ed o sell uni s ha dec ease p o i .
Assump ion 2. Fo each i m
i
, he e exis s he la ges
ai
such ha
φi(x) = ψi(x
,
pj) =
0 o all
pj≤x≤ai. Fu he , ψi(pi,x) = φi(pi) o all pi≥x such ha x <aj.
Games 2025,16, 26 4 o 37
The p ice
ai
is he minimal p ice o which a i m is willing o p oduce, ypically he
in imum o he a e age cos o p oduc ion when explici ly modeled. As such, when a
i m
i
p ices below
ai
, i m
j
’s esidual p o i is equal o i s on -side p o i because i m
i
p oduces no hing.
Assump ion 3.
φi
has a unique maximize
b
pi>ai
wi h
φi(b
pi)>
0. On he in e al
(ai
,
b
pi)
,
φi
is posi i e alued, and s ic ly inc easing. Fu he ,
ψi(pi
,
pj) =
0 o all
pi≥pj≥b
pj
, and
pi>b
pj.
This assump ion on he on -side p o i is weake han assuming he s ic quasicon-
ca i y o
φi
as i does no es ic beha io a p ices
pi>b
pi
. The hi d pa o he assump ion
is ha he e is no esidual p o i o i m
i
i i m
i
p ices abo e
b
pj
. This assump ion i s he
wo cases ha demand is con inuous a b
pj, o ha he e is ze o quan i y demanded abo e
he p ice b
pj.
The ollowing assump ion is impo an o ou cha ac e iza ion.
Assump ion 4. Fo each i m i, he e exis s a p ice ρi∈[ai, min{b
pi,b
pj}]such ha
φi(x)>ψi(x,x) o all x ∈ρi,b
pii,
φi(x) = ψix,x′ o all x′≤x<ρi.
Fu he , i ai>aj, hen ρi≤ρj.
The maximum o he wo i ms’ p ices
ρi
is a p ima y objec used in he analysis ha
ollows; as such, we deno e ρ=max{ρ1,ρ2}.
Rema k 1. The exis ence o he p ices
ρi
is a na u al consequence o adi ional cons uc ions
o he BE model. I is common ha
ρi
co esponds o he p ice a which o al indus y supply is
equal o ma ke demand, as below such a p ice, he esidual demand would exceed he supply o he
high-p iced i m. Al e na i ely, in ha case ha he ma ginal cos o
i
is less han he ma ginal cos
o
j
,
ρi
can co espond o he (cons an ) ma ginal cos o i m
j
, as below ha p ice, i m
j
does no
p oduce, lea ing he ma ke demand o he esidual, while a o abo e ha p ice, i m
j
p oduces up
o i s capaci y, po en ially limi ing esidual p o i below he on side. Assump ion 4accommoda es
ei he o hese scena ios and gene ally allows mo e a ie y o ma ke s uc u es. Fo example, i
allows o si ua ions in which unsa ia ed demand is no a ioned o o he i ms, as may be he case
wi h di ec ed sea ch models.
We make he ollowing assump ion o ule ou he possibili y ha one i m has a
su icien ly compe i i e ad an age o ac as a monopoly.
Assump ion 5. Fo each i and j, b
pi>aj.13
Each i m i’s p o i is speci ied as ollows:
ui(p) = 




φi(pi)pi<pj
αi(p)φi(pi) + (1−αi(p))ψi(p)pi=pj
ψi(p)pi>pj
, (1)
whe e
αi(p)∈[
0, 1
]
and
α1(p) + α2(p)∈(
0, 2
)
. I we ins ead assume ha
α1+α2=
1, hen
his es ic s a en ion o sha ing ules ha assign one i m i s on -side p o i and he o he
i s esidual p o i , wi h some andomiza ion o e he assignmen . By pe mi ing he sum o

Games 2025,16, 26 5 o 37
he sha es o be g ea e (o less) han one, he model cap u es any sha e o demand a ies,
which can na u ally esul in each i m ecei ing a (non-s ochas ic) p o i s ic ly be ween
i s on -side and esidual p o i s.
We deno e he se o maximize s o
ψi
a any
pj
by
e
Pi(pj)
. Deno e he maximized
esidual p o i by e
ψi(pj), ha is,
e
ψi(pj) = max
pi≥pj
ψipi,pj.
Assump ion 6. The e exis s he lowes p ice
b
x
such ha
φi(b
x)>φi(x)
and
ψi(b
x
,
pj)≥ψi(x
,
pj)
o all p ices x and pjwi h x >b
x≥pj.
No e ha
b
x≥b
pi
o each i m
i
. Gi en Assump ion 6, he p ice
b
x
weakly domina es
all p ices x>b
x.
Assump ion 7. Bo h
φi
and
ψi
a e con inuous in
pi
on
[
0,
b
x)
and le con inuous a
b
x
.
ψi
is igh
uppe semicon inuous in
pj
, ha is,
lim supkψi(pi
,
xk)≤ψi(pi
,
x)
o any sequence
{xk}
such
ha pi≥xk>x and xk→x.
These con inui y assump ions a e sa is ied in mos BE models p e iously s udied. The
igh uppe semicon inui y cap u es he no ion ha a i m does no d as ically dec ease i s
p oduc ion when he o he i m’s p ice inc eases. The po en ial o discon inui ies a p ices
abo e b
xallows he model o accommoda e se ings wi h box demand.
De ine
i
o be i m
i
’s judo p ice, which is he lowe bound such ha he on -side
p o i o i m
i
is g ea e han he maximal esidual p o i o i m
i
when i m
j
uses any
p ice weakly g ea e . Fo mally,
i=in {x|φi(x)>supz≥xe
ψi(z)}.
De ine
i
o be i m
i
’s sa e p ice, which is he lowe bound o p ice such ha he on -
side p o i o i m
i
is g ea e han he highes p o i ha i m
i
can gua an ee i sel . Fo mally,
i=in {x|φi(x)>ui},
whe e ui=in pjsuppiui(pi,pj).
De ine he la ge o he wo i ms’ judo p ices o be c i ical judo p ice, deno ed by
=max i
. Simila ly, de ine he la ge o he wo i ms’ sa e p ices o be he c i ical sa e
p ice, deno ed by
=max i
. Based on he ac ha
ui≤supz≥xe
ψi(z)
and ha
φi
is s ic ly
inc easing when posi i e, he judo p ice is always weakly g ea e han he sa e p ice, ha
is, ≥ . No e u he ha ≤b
x.
De ine i m
i
’s judo p o i o be he on -side p o i o i m
i
a he c i ical judo p ice,
deno ed by
φi≡φi( )
. Simila ly, de ine i m
i
’s sa e p o i o be he on -side p o i o i m
i
a he c i ical sa e p ice, φi≡φi( ).
Fo equilib ium s a egies
µ=(µ1,µ2)
, we use
xi
and
xi
o deno e he in imum and
sup emum o he suppo o i m
i
’s s a egy, espec i ely. We will use
x
o deno e he
minimum o
x1
and
x2
, and
x
o deno e he maximum o
x1
and
x2
.
14
Fu he , we de ine
Fi
o be he dis ibu ion unc ion (CDF) o i m
i
’s mixed s a egies on
[x,x]
, wi h
F= (F1
,
F2)
.
Addi ionally, le u∗
ideno e i m i’s equilib ium expec ed p o i .
Be o e p oceeding wi h he analysis o he model, we discuss some unde lying speci i-
ca ions ha ou model con ains in he ollowing subsec ion.
Games 2025,16, 26 6 o 37
Examples Wi hin Ou F amewo k
To p o ide con ex o he abs ac model, we p esen h ee examples nes ed wi hin
ou amewo k. The amewo k o ou model pu s e y li le es ic ion on he unde lying
ma ke demand excep con inui y on
[
0,
b
x)
, e en allowing o inc easing demand. As such,
he demand s uc u e is easily unde s ood. To help con ey he gene ali y o ou model,
we p esen he ollowing examples o demons a e he ange o p oduc ion echnologies
and demand a ioning ules ha can be accommoda ed in ou model. Wi h his pu pose in
mind, we use ec angula uni demand in hese nume ical examples. Fo mally, he ma ke
demand is
D(x) = (0 i x>1
1 i x∈[0, 1].
The i s wo examples exhibi p oduc ion echnologies included in ou speci ica ion. The
hi d example exhibi s a esidual demand a ioning scheme based on di ec ed sea ch.
In each example, we speci y each i m’s cos o p oduc ion as a unc ion o quan i y
p oduced and use ha o de i e he supply co espondence o quan i ies ha maximize he
di ec p o i unc ion
πi(x
,
q) = xq −ci(q)
.
15
We hen use hese o de i e he co esponding
on -side and esidual p o i unc ions as well as he p ices b
x,b
pi,ai, and ρi.
Example 1 (Discon inuous supply).Fi m 1’s cos o p oduc ion is
c1(q) = (1
2q−1
8i q ≥1/2
1
4q i q ∈[0, 1/2],
while i m 2’s cos o p oduc ion is
c2(q) = 1
3q
. Nei he i m aces a capaci y cons ain . The supply
co espondence o each i m is hus
ϑ1(p1) =















{∞}i p1>1/2
[1
2,∞]i p1=1/2
n1
2oi p1∈(1/4, 1/2)
[0, 1
2]i p1=1/4
{0}i p1∈[0, 1/4)
,
and
ϑ2(p2) = 




{∞}i p2≥1/3
[0, ∞]i p2=1/3
{0}i p2∈[0, 1/3)
.
The on -side p o i s a e
φ1(p1) = 










0i p1>1
p1−1
41
2+p1−1
21
2i p1∈[1/2, 1]
p1−1
41
2i p1∈[1/4, 1/2)
0i p1∈[0, 1/4)
,
and
φ2(p2) = 




0i p2>1
p2−1
3i p2∈[1/3, 1]
0i p2∈[0, 1/3)
.
As he supply co espondence is no single- alued, he esidual p o i s o he i ms may depend on
he pa icula quan i y chosen. Ra he han p esen all possibili ies, we use he con en ion ha he
i ms p oduce he la ges quan i y possible in hei supply co espondence. Wi h his con en ion,
Games 2025,16, 26 7 o 37
he esidual p o i s o he i ms a e de ined wi h demand a ioned acco ding o he e icien (o
equi alen ly p opo ional) ule
ψ1(p1,p2) = (0i p2≥1/3
φ1(p1)i p2<1/3 ,
ψ2(p2,p1) = 










0i p2>1, o p1≥1/2,
o p2<1/3, o p1∈[1/4, 1/2)
p2−1
31
2i p2≥1/3, p1∈[1/4, 1/2)
φ2(p2)i p1<1/4
.
The key model pa ame e s o his example a e
b
x=b
pi=
1,
a1=
1
/
4,
a2=
1
/
3, and
ρi=
ρ=1/3.
The nex example includes i ms wi h a U-shaped a e age and ma ginal cos s.
Example 2 (U-shaped a e age and ma ginal cos s).Each i m i’s cos o p oduc ion is
ci(q) = (2
3q3−1
4q2+1
16 q+1
20 i q >0
0i q =0.
Nei he i m aces a capaci y cons ain . The supply co espondence o each i m
i
can be exp essed
as he unc ion
ϑ(pi) = (1+√32pi−1
8i pi≥0.194
0i pi∈[0, 0.194].
Thus, he on -side p o i o i m i is
φi(pi) = 




0i pi>1
piϑ(pi) + ci(ϑ(pi)) i pi∈[0.194, 1]
0i pi∈[0, 0.194)
,
and he esidual p o i o i m i (pi≥pj) is (again using e icien a ioning)16
ψi(pi,pj) = 




0i pj≥0.397
pimin{ϑ(pi), 1 −ϑ(pj)}+ci(min{ϑ(pi), 1 −ϑ(pj)})i pj∈(0.194, 0.397)
φi(pi)i pj<0.194
.
The key model pa ame e s o his example a e b
x=b
pi=1, ai=0.194, and ρi=ρ=0.3125.
The inal example has demand a ioning de e mined by a di ec ed consume
sea ch game.
Example 3 (Sea ch).Each i m has ze o cos o p oduc ion. Bo h i ms ha e he same capaci y
k∈(
1
/
2, 1
)
, wi h each i m limi ed o p oducing a quan i y o a mos
k
. Demand a ioning is
de e mined by he equilib ium o a di ec ed consume sea ch game. The e is a uni mass o consume s,
each o which demands a single uni o he good, which hey alue a 1. The consume s obse e he
p ices o he i ms, and hen simul aneously choose a i m o isi . I a mass
Wi≤k
o consume s
isi i m
i
, each o hose consume s ecei es a good a p ice
pi
. I a mass
Wi>k
o consume s isi s
i m
i
, hen each o hose consume s ecei es a good a p ice
pi
wi h p obabili y
k/Wi
. Consume s
ha do no ecei e goods ob ain a payo o ze o. In any pu e s a egy equilib ium o he consume
Games 2025,16, 26 8 o 37
game, i
pi<pj
, hen ei he all consume s shop a i m
i
, o he mass
Wi
o consume s ha shop a
i m i sa is ies
k
Wi
(1−pi)=1−pj.
The e o e, we can w i e
Wi=(0i pi>1
minn1, k1−pi
1−pjoi pi∈[0, 1],
and he mass o consume s who go o i m
j
a e
Wj=
1
−Wi
. No ice ha
Wi≥k
o all p ices
pi<pj. Thus, we can w i e he on -side and esidual p o i o i m i as ollows:
φi(pi) = (0i pi>1
pik i pi∈[0, 1],
and
ψi(pi,pj) = 


0i pi≥1
pimax0, 1 −k(1−pj)
1−pii pi∈[pj, 1).
The key model pa ame e s o his example a e b
x=b
pi=1, ai=ρi=ρ=0.
3. Mixed S a egy Equilib ia
In his sec ion, we es ablish some abs ac p ope ies o all equilib ia. We begin he
analysis by dealing wi h he p oblem o exis ence o equilib ium. As long as each i m’s
esidual p o i is lowe semicon inuous in he o he i m’s p ice, we a e able o show ha
an equilib ium exis s i
ρ1=ρ2
. When he e is i m
i
such ha
ρi<ρ
, ou p oo equi es
an addi ional condi ion ha his i m
i
ecei es i s on -side p o i wi h ce ain y a ies
below ρ.
P oposi ion 1. Assume ha (1)
αi(x
,
x) =
1 o any
x
such ha
φi(x)>ψi(x
,
x)
and
φj(x) = ψj(x
,
x)
and (2) ha
ψi
is lowe semicon inuous in
pj
. Then a mixed s a egy equi-
lib ium o he BE game exis s.
I he esidual p o i s
ψi
a e con inuous, hen he exis ence o equilib ium o he BE
duopoly ollows di ec ly om P oposi ion 2 in Allen and Lepo e (2014). A gene aliza ion
o his p oposi ion is p esen ed in Appendix A.1, which applies o he case in which he
esidual p o i s a e no con inuous. The equi emen ha
ψi
is lowe semicon inuous is
added no ou o necessi y o he exis ence o equilib ium, bu ou o necessi y o he
abs ac e i ica ion o he exis ence o equilib ium wi hou explici calcula ion.17.
We now u n o he analysis o he se o equilib ia o he BE game. The ollowing
p oposi ion pa i ions he se o equilib ia in o wo possible ypes and p o ides a pa ial
cha ac e iza ion o each ype.
P oposi ion 2. The e a e wo ypes o equilib ia: symme ic lowe bound and asymme ic
lowe bound.
Symme ic lowe bound equilib ia a e such ha
•x1=x2=x≥ρ;
•The p obabili y o an a om a any p ice x >ρis ze o;
•The p obabili y o a ie a x is posi i e only i φi(x) = ψi(x,x) o each i m i; and
•A mos one i m can p ice highe han b
x.
Games 2025,16, 26 15 o 37
I will be use ul o deno e he lowe bound o all p ices such ha i m
i
’s supply is a leas
as big as he demand by τi. Fo mally, τi=in {x:si(x)≥D(x)}.
Condi ion 7. piD(pi)−ci(D(pi)) is s ic ly quasiconca e on [τi,pc].
The es ic ion o he s ic quasiconca i y o only he in e al
[τi
,
pc]
allows some
addi ional eedom o he demand unc ion a p ices below
τi
. The p ope ies o
piD(pi)−
ci(D(pi))
a p ices
pi<τi
a e i ele an , as he he p o i o he i m does no co espond o
his exp ession a such p ices.
An immedia e consequence o Condi ion 7is ha he e is a unique maximize
b
pi
o
he on -side p o i o each i m
i
. The nex condi ion is a es a emen o Assump ion 5,
which we s ill need o gua an ee ha no single i m will monopolize he ma ke .
Condi ion 8. b
pi>aj o each i m i.
The ollowing lemma shows ha Condi ions 1–8 imply Assump ions 1–7 o he
gene al model.
Lemma 6. I Condi ions 1–8 a e sa is ied, hen Assump ions 1–7 a e sa is ied.
In his special case model, we can de ine he judo and sa e p ice o each i m
i
in a
sligh ly simpli ied way. The judo p ice o i m iis
i=in {x|φi(x)>e
ψi(x)}.
No e ha , based on he assump ions o his sec ion, ei he
φi( i) = e
ψi( i)
, o
i=aj
. The
sa e p ice o i m iis
i=min{x|φi(x)≥ui}.
5.1. Special Resul s o Pu e S a egy Equilib ium
As we desc ibed in he gene al model sec ion on pu e s a egy equilib ium, he e is an
in ui i e way o classi y he pu e s a egy equilib ium in o wo ypes. Pa icula ly, Type
B wi h
x∗=max{a1
,
a2}
and Type C wi h
x∗>max{a1
,
a2}
. The s uc u e added in his
sec ion allows us o say mo e abou hese ypes o equilib ium. Pa icula ly, in he ollowing
p oposi ion, we p esen he necessa y and su icien condi ions on supply, demand, and
esidual demand o he exis ence o pu e s a egy equilib ium.
P oposi ion 6. The p ice
x∗=ρ
is a pu e s a egy Nash equilib ium i and only i one o he
ollowing h ee condi ions holds:
B.1 ρ=a1=a2and di(x′,ρ) = 0 o each i m i, and all x′>ρ.
B.2
ρ=ai>aj
,
di(x
,
ρ) =
0 o all
x≥ρ
,
uj(ρ
,
ρ) = φj(ρ)
, and
ψj(x
,
ρ)≤φj(ρ)
o
all x ≥ρ.
C
ρ∈(max{a1
,
a2}
,
min{b
p1
,
b
p2}]
,
ρ∈e
Pi(ρ)
o each i m
i
, and
si(ρ) + sj(ρ)≤
D(ρ) o any i m i wi h αi(ρ,ρ)<1, whe e si(x) = min ϑi(x).
The speci ici y o he model in his sec ion allows us o b eak Type B equilib ium in o
wo ca ego ies. Type B.1 equi es ha each i m’s in imum o a e age cos is he same,
a1=a2
. In all Type B.1 pu e s a egy equilib ium, bo h i ms make ze o p o i . The e a e
wo di e en possibili ies o B.1 pu e s a egy p icing. The i s case, akin o Classical
Be and ma ginal cos p icing, is he case in which
min{s1(ρ)
,
s2(ρ)} ≥ D(ρ)
,, and hus
by Condi ion 6,
di(ρ
,
ρ) =
0 o bo h
i
. The second case allows o a i m’s supply o no
co e all o demand
si(ρ)<D(ρ)
as long as he e is no esidual demand o he o he i m

Games 2025,16, 26 16 o 37
dj(x′
,
ρ) =
0 o all
x′>ρ
. This case elies on no consume s going o he highe -p iced i m
jwhen i m ip ices a ρ.
Type B.2 is pu e s a egy p icing ha can only occu in he case ha he wo i ms ha e
di e en in ima o hei a e age cos s. In such an equilib ium, bo h i ms p ice a he highe
o he wo in imum a e age cos s, and i mus be ha he i m wi h he lowe in imum
a e age cos ob ains i s ull on -side p o i . This means ha he lowe -cos i m mus
ob ain a la ge-enough sha e o demand a his p ice o achie e i s on -side p o i , while
he highe -cos i m makes ze o p o i . This ype o pu e s a egy equilib ium was shown
in Theo em 2 o Denecke e and Ko enock (1996) o a model o i ms wi h asymme ic
cons an ma ginal cos s.
In all Type C equilib ia, bo h i ms make posi i e p o i s. Like B.1, Type C p icing has
wo di e en possibili ies. The i s case is simila o Classical Cou no ma ke clea ing
p ices (
s1(ρ) + s2(ρ) = D(ρ)
) abo e he in imum o a e age cos . The second case is such
ha o al supply is less han demand
s1(ρ) + s2(ρ)<D(ρ)
and esidual demand a ioning
does no pe mi a p o i inc ease by p icing highe han
ρ
. This case o Type C p icing
equi es ha each i m’s esidual demand is su icien ly low a p ices g ea e han ρ.
Nex , we es ablish a condi ion such ha he only possible pu e s a egy equilib ium is
Type C.
P oposi ion 7. Index he i ms such ha
a1≥a2
. I
s1(a1) =
0, hen only Type C pu e s a egy
p icing equilib ia a e possible.
F om his p oposi ion, in o de o ha e a Type B equilib ium, we see ha he minimum
a e age cos o he highe -cos i m mus be achie ed by a leas one posi i e p oduc ion
quan i y (i could be equal o he in imum o he a e age cos a ze o a well). This necessi-
a es ei he a la sec ion o his i m’s ma ginal cos o a U-shaped ma ginal cos cu e.
Now we make a ema k ega ding he cha ac e o Type C p icing in a di e en iable model.
Rema k 4. By adding s anda d di e en iabili y assump ions, i becomes clea ha he exis ence o a
pu e s a egy equilib ium is ema kably agile. I all key componen s o he model a e di e en iable
(demand, esidual demand, cos , and supply), hen any Type C equilib ium mus be such ha
s1(ρ) + s2(ρ) = D(ρ)
. This basic insigh on he non-exis ence o pu e s a egy equilib ium is i s
ound by Shubik (1959).
5.2. Some Compa a i e S a ics on Equilib ium Bounds
We examine he e ec s o changes in esidual demand a ioning and indi idual supply
on he bounds o equilib ium p ices and payo s. I should be clea ha , o small changes
o hese componen s, he ac ual equilib ium payo s need no ollow he bounds ( hough
pe haps hey a e likely o). Howe e , o su icien ly la ge changes, when he new ange o
p ices o p o i s does no in e sec he old, we a e able o p ecisely conclude how he ac ual
equilib ium p o i s a e a ec ed.
We begin by examining he ole o changes in he demand side o he ma ke . Speci i-
cally, we conside an inc ease in esidual demand a ioning o consume s, whe eby a leas
one i m has an inc ease in hei esidual demand. We use esidual demands
di
and
d′
i
; le
φ= (φ1
,
φ2)
and
φ′
deno e he co esponding on -side p o i s and
ψ= (ψ1
,
ψ2)
and
ψ′
deno e he co esponding esidual p o i s. Ou i s esul es ablishes ha an inc ease in
he esidual demand o ei he i m weakly inc eases he bounds on equilib ium payo s.
P oposi ion 8. I d′
i≥di, hen ′≥ , ′≥ , φ′≥φand φ′≥φ.
Games 2025,16, 26 17 o 37
Simply pu , his p oposi ion es ablishes ha a shi o a mo e gene ous a ioning
scheme inc eases he bounds on he p o i s o each i m.
Now we u n ou a en ion o unde s anding he impac o changes in p oduc ion
echnology. The ollowing p oposi ion shows condi ions such ha an inc ease in a i m’s
supply, which could esul om an inc ease in capaci y o educ ion in cos s, will weakly
educe i s judo and sa e p ice and weakly educe he p o i o he o he i m. Gi en supply
unc ions
s= (s1
,
s2)
and
s′
, le
φ= (φ1
,
φ2)
and
φ′
deno e he co esponding on -side
p o i s and ψ= (ψ1,ψ2)and ψ′deno e he co esponding esidual p o i s.
P oposi ion 9. Conside a weak cos educ ion o i m
i
ha esul s in a weak supply inc ease so
ha
s′
i≥si
. Suppose u he ha his esul s in a weak educ ion in esidual demand o i m
j
,
dj(x
,
y)≥d′j(x
,
y)
o all
x≥y
. Las ly, suppose ha
ci(q)−c′
i(q)
is nondec easing in
q
. Then
′≤ , ′≤ , φ′j≥φj, and φ′j≥φj.
No e ha his p oposi ion does no make any s a emen s ega ding he p o i s o he
i m whose supply shi s. The eason is ha he e ec is ambiguous. Tha is, a echnological
inc ease o a i m does no necessa ily imply an inc ease in equilib ium p o i s o ha
i m. The di ec ion o he change in p o i s is ins ead de e mined by he na u e o he shi
and ma ke condi ions. Two ex eme examples illus a e his poin .
Conside a duopoly in which iden ical i ms ha e cons an ma ginal cos s and capac-
i ies equal o hal he monopoly quan i y. In such a se ing, pu e s a egy p icing can be
sus ained wi h each i m ea ning hal he monopoly p o i . Now conside a echnology
shock ha inc eases he capaci y o bo h i ms so ha hei capaci y is nonbinding a any
p ice. This echnological inc ease ac ually lowe s each i m’s p o i om some hing s ic ly
posi i e o ze o.
The p e ious example in ol ed an indus y-wide capaci y shock; howe e , he same
esul may occu as a esul o a cos educ ion o a single i m. Conside a duopoly in
which i m 1 has cons an ma ginal cos
c=
0 while i m 2 has a s ic ly con ex cos o
p oduc ion wi h supply
s2(x)>
0 o all
x>
0 and
s2(
0
) =
0. Suppose ha he i ms a e
no capaci y cons ained. I ollows ha
ρ=
0. No e ha by choosing a p ice
x
a bi a ily
close o ze o, he igh con inui y o
s2
gua an ees ha
ψ1(x
, 0
)>
0. Thus,
p1=p2=
0
canno be an equilib ium, and any equilib ium mus be in mixed s a egies. The e o e, i
mus be ha i m 2 ecei es i s on -side p o i in equilib ium wi h posi i e p obabili y, in
which case, i ea ns posi i e p o i s. Conside a echnology inc ease o i m 2 ha educes
i s cos o ze o. Then he game becomes he classic Be and duopoly wi h ze o p o i s.
Thus, a educ ion in one i m’s cos may ac ually educe i s p o i s.
These examples highligh ha he e a e coun e ailing e ec s associa ed wi h a change
in echnology. The e is a p ima y cos e ec o capaci y e ec ha allows a i m o ea n a
highe p o i ma gin o p oduce mo e a any gi en p ice, bo h o which inc ease he p o i s
o ha i m. Al e na i ely, he e is a seconda y compe i ion e ec , whe eby he change in
cos o capaci y al e s he s a egic en i onmen and incen i izes he o he i m o p ice
mo e compe i i ely, d i ing he p ices o bo h i ms down and he eby educing p o i s.
Whe he he ne change in p o i s is posi i e o nega i e depends on he ela i e s eng h o
hese wo e ec s.
6. Concluding Rema ks
Re o mula ing p ice compe i ion as an all-pay con es wi h ex e nali ies has allowed us
o de i e esul s on he na u e o equilib ium o a class o BE p icing games a mo e gene al
han using con en ional me hods. The b oad ange o unde lying speci ica ions includes
many new speci ica ions (as U-shaped a e age cos o p oduc ion, minimal es ic ion
on demand, demand a ioning based on consume sea ch, and echnology asymme ies
Games 2025,16, 26 18 o 37
ac oss i ms) ha expand he possible policy applica ions o he BE model. Fu he , we
ha e p esen ed a me hodology ha can be ex ended o analyze BE oligopoly including he
possibili y o incomple e in o ma ion.
Au ho Con ibu ions: Concep ualiza ion, B.A.A. and J.J.L.; Fo mal analysis, B.A.A. and J.J.L.;
W i ing—o iginal d a , B.A.A. and J.J.L. Bo h au ho s ha e con ibu ed o all phases o he manusc ip .
All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding: The e was no unding ecie ed o his p ojec .
Da a A ailabili y S a emen : No new da a we e c ea ed o analyzed in his s udy.
Con lic s o In e es : The au ho s decla e no con lic s o in e es .
Appendix A
Appendix A.1. Exis ence o Equilib ium (P oo o P oposi ion 1)
We p o e he exis ence o equilib ium using he ollowing esul based on he wo ks
o Reny (1999) and Bagh and Jo e (2006).
Fac A1. I he mixed ex ension o a compac game is payo secu e and sa is ies weak ecip ocal
uppe semicon inui y (WRUSC), hen he game has a mixed s a egy Nash equilib ium.
Ve i ying payo secu i y and WRUSC in he mixed ex ension o a game is bu densome,
and so we ely on he ecen esul s o Allen and Lepo e (2014) and Allison e al. (2018),
which p o ide easily e i iable condi ions o games ha imply ha hese p ope ies a e
sa is ied in he mixed ex ension.
The ollowing de ini ion is om Allen and Lepo e (2014). Le
Xi
and
ui
deno e
playe
i
’s s a egy se and u ili y unc ion, espec i ely. De ine he discon inui y mapping
Di:Xi→X−isuch ha
Di(xi) = {x−i∈X−i:ui(xi,x−i)is discon inuous in x−ia (xi,x−i)}.
De ini ion A1. A game sa is ies disjoin payo ma ching (DPM) i , o each playe
i
and all
xi∈Xi, he e exis s a sequence {xk
i} ⊂ Xisuch ha
(1) lim in kui(xk
i,x−i)≥ui(xi,x−i) o all x−i∈X−i; and
(2) lim supkDi(xk
i) = ∅.22
Fac A2 (Allen and Lepo e (2014)).I a compac game sa is ies DPM, hen he mixed ex ension
o he game is payo secu e.
The p oblem in using his de ini ion o DPM o e i y he exis ence o equilib ium
in ou model is ha he payo unc ion
ui
can be discon inuous in
pj
h ough
ψj
a some
p ices
p
ega dless o he choice o
pi
.
23
As such, i may be impossible o sa is y pa 2 o
he de ini ion. A i ial modi ica ion is su icien o gene alize he exis ence esul . De ine
he discon inui y map D′
i:Xi→X−isuch ha
D′
i(xi) = {x−i∈Di(xi):ui(xi,x−i)is no lowe semicon inuous in x−ia (xi,x−i)}.
By eplacing
Di
wi h
D′
i
in he de ini ion o DPM, he p oo o he main esul o Allen
and Lepo e (2014) is una ec ed. Since
ψi
is assumed o be lowe semicon inuous in he
s a emen o P oposi ion 1, i ollows ha he discon inui y se s
Di(xi)
and
D′
i(xi)
coincide,
and so we will be able o use his modi ied de ini ion o DPM in ou model.
Games 2025,16, 26 19 o 37
Ve i ying ha he BE game sa is ies DPM is qui e simple: o any p ice
pi>
0, he
sequence o de ia ions
pk
i=pi−
1
/k
sa is ies he de ini ion, as hese de ia ions esul in
ei he he same p o i in he limi o a highe p o i by gua an eeing he on -side p o i i
he e would be a ie a
pi
. Fu he , he only poin s o discon inui y a which he payo s a e
no lowe semicon inuous a e ies, and
lim supkDi(xk
i) = ∅
since
Di(pk
i) = {pk
i}
, and hus,
Di(pk
i)∩Di(pk′) = ∅
o all
k=k′
. I
pi=
0, hen
ui(pi
,
pj) =
0 o all
pj
and
Di(pi) = ∅
,
so
pk
i=
0 o all
k
i ially sa is ies he de ini ion. Thus, he mixed ex ension o he BE
game is payo secu e.
We now e i y ha he mixed ex ension o he BE game sa is ies WRUSC. I will
be use ul o de ine he objec
ui(x) = lim supx′→xui(x′)
and
u
o be he ec o alued
unc ions whose indi idual componen s a e each ui.
Fac A3 (Allison e al. (2018)).Le
G= (N
,
X
,
u)
be a compac game. Suppose ha , (1) o
each playe
i
, he e exis s a sequence o Bo el measu able unc ions
Tk
i
:
Xi→Xi
such ha , o all
x∈X
,
lim in kui(Tk
i(xi)
,
x−i)≥ui(x)
, and (2) o any s a egy p o ile
x∈X
, i he e is some
sequence
{xk}
wi h
limku(xk) = u(x)
, hen
u(x) = u(x)
. Then he mixed ex ension o he game
sa is ies WRUSC.
These wo condi ions in ui i ely s a e ha (1) each playe can de ia e om any s a egy
so ha , gi en any s a egy p o ile o he o he playe s, he de ia ing playe ob ains he
highes easible payo nea ha s a egy p o ile, and (2) i i is easible o all playe s o
simul aneously ob ain hei highes easible payo nea a s a egy p o ile, hen he payo s
speci y ha hey all ecei e such a payo a ha s a egy. In he con ex o ou BE model,
(1) is sa is ied by he same de ia ions as wi h DPM:
Tk
i(pi) = max{pi−
1
/k
, 0
}
. This
sequence o de ia ions maximizes he i m’s chances o ob aining he on -side payo ,
which co esponds o
ui
. Fo (2), obse e ha since
ψi
is lowe semicon inuous in
pj
, hen
ψi=ψi
, whe e
ψi
is de i ed om
ψi
as
ui
is de i ed om
ui
. Thus,
ui(pi
,
pj) = ψi(pi
,
pj) =
ui(pi
,
pj)
o any
pi>pj
. I
pi<pj
, hen
ui(pi
,
pj) = φi(pi) = ui(pi
,
pj)
. Any iola ion o
condi ion (2) can hus only be a ies. No e ha i
φi(x)>ψi(x
,
x)
o bo h i ms
i
, i is no
easible ha bo h i ms
i
simul aneously ob ain
ui
. I
φi(x) = ψi(x
,
x)
o bo h i ms
i
, hen
ui(x
,
x) = ui(x
,
x)
. Las ly, i
φi(x)>ψi(x
,
x)
and
φj(x) = ψj(x
,
x)
, hen by he assump ion
in he s a emen o he p oposi ion,
αi(x
,
x) =
1, and so
ui(x
,
x) = φi(x) = ui(x
,
x)
and
uj(x
,
x) = φj(x) = uj(x
,
x)
. Thus, condi ion (2) is sa is ied. We conclude ha he mixed
ex ension o he BE game sa is ies WRUSC.
Appendix A.2. P oo o Lemmas and P oposi ions
Lemma A1. suppj∈[x0,x]ψi(x,pj)is igh uppe semicon inuous in x a x =x0.
P oo o Lemma A1.
Le
x0≥
0 and obse e ha
suppj∈[x0,x]ψi(x
,
pj) = ψi(x0
,
x0)
a
x=x0
. Le
ε>
0 and
xn→x0
be such ha
xn>x
o each
n
. Fo each
n
, le
yk
n
be
a sequence in
k
such ha
limkψi(xn
,
yk
n) = suppj∈[x0,xn]ψi(xn
,
pj)
. Fo each
n
, le
K(n)
be
such ha
ψi(xn,yk
n)−suppj∈[x0,xn]ψi(xn,pj)<ε/
3 o all
k>K(n)
and choose
yn=yk
n
o some k>K(n). Then no e ha
sup
pj∈[x0,xn]
ψi(xn,pj)−ψi(x0,x0)≤sup
pj∈[x0,xn]
ψi(xn,pj)−ψi(xn,yn) + ψi(xn,yn)−ψi(x0,x0)
<ε
3+ψi(xn,yn)−ψi(xn,x0) + ψi(xn,x0)−ψi(x0,x0).
Since
yn∈[x0
,
xn]
, i ollows ha
yn→x0
. Thus, by he igh uppe semicon inui y o
ψi
in
pj
, he e exis s an
N1
such ha
ψi(xn
,
yn)−ψi(xn
,
x0)<ε/
3 o all
n>N1
. Simila ly, by
Games 2025,16, 26 20 o 37
he con inui y o
ψi
in
pi
, he e exis s an
N2
such ha
|ψi(xn,x0)−ψi(x0,x0)|<ε/
3 o all
n>N2. Thus, o all n>max{N1,N2}, i mus be ha
sup
pj∈[x0,xn]
ψi(xn,pj)−ψi(x0,x0)<ε
3+ψi(xn,yn)−ψi(xn,x0) + ψi(xn,x0)−ψi(x0,x0)<ε.
The e o e,
sup
pj∈[x0,xn]
ψi(xn,pj)<ψi(x0,x0) + ε,
so by de ini ion, suppj∈[x0,x]ψi(x,pj)is igh uppe semicon inuous in xa x=x0.
P oo o Lemma 1.
Suppose o he con a y ha
µi([
0,
b
x]) <
1 o each i m
i
. Then o any
i m
i
and any
x>b
x
in suppo o
µi
, Assump ion 6gua an ees ha
φi(b
x)>φi(x)
and
ψi(b
x,pj)≥ψi(x,pj) o all pj≤b
x. Obse e ha
Zui(b
x,pj)dµj≥(1−Fj(b
x))φi(b
x) + Z[0,b
x]ψi(b
x,pj)dFj
≥(1−Fj(b
x))φi(b
x) + Z[0,b
x]ψi(x,pj)dFj
and (1−Fj(b
x)) >0 since µj([0, b
x]) <1. Thus, we ha e
Zui(b
x,pj)dµj≥(1−Fj(b
x))φi(b
x) + Z[0,b
x]ψi(x,pj)dFj
>(1−Fj(b
x))φi(x) + Z[0,b
x]ψi(x,pj)dFj
≥(1−Fj(x) + µj({x}))φi(x) + Z[0,x)ψi(x,pj)dFj
=Zui(x,pj)dµj.
The las inequali y ollows om he ac ha
Fj
is a CDF and hus nondec easing and om
Assump ion 1gua an eeing ha
φi(x)≥ψi(x
,
x′)
o
x′≤x
. This con adic s all
x>b
x
as
equilib ium s a egies.
P oo o Lemma 2.
Le
µ
be an equilib ium and
x
be such ha
φi(x)>ψi(x
,
x)
and
µi({x})>
0. Suppose ha
µj({x})>
0 and
αi(x
,
x)<
1. Conside a sequence o de-
ia ions by i m i o e
µn
ide ined by
e
µn
i(E) = (µi(E∪{x})i x−δn∈E
µi(E∖{x})o he wise ,
whe e each
δn
is chosen so ha 0
<δn<
1
/n
and
µj({x−δn}) =
0. Tha is,
e
µn
i
is he
measu e c ea ed om
µi
by shi ing all mass om he p ice
x
o he p ice
x−δn
. Then
no e ha
Zui(p)de
µn
i×µj=Zui(p)dµ+µi({x})Zui(x−δn,pj)−ui(x,pj)dµj.
We will show ha
limnRui(x−δn,pj)−ui(x,pj)dµj>
0 o su icien ly la ge
n
, which
will gua an ee a p o i able de ia ion o i m
i
, iola ing
µi
as an equilib ium s a egy.
No e ha

Games 2025,16, 26 21 o 37
ui(x−δn,pj)−ui(x,pj) = 












ψi(x−δn,pj)−ψi(x,pj)i pj<x−δn
φi(x−δn)−ψi(x,pj)i x−δn≤pj<x
φi(x−δn)−αi(x,x)φi(x)
−(1−αi(x,x))ψi(x,x)i pj=x
φi(x−δn)−φi(x)i pj>x
.
I ollows ha he poin wise limi as n→∞is
lim
nui(x−δn,pj)−ui(x,pj)=




0 i pj<x
(1−αi(x,x))(φi(x)−ψi(x,x))i pj=x
0 i pj>x
.
Thus , since |ui|≤φi(b
pi), hen by he Lebesgue domina ed con e gence heo em,
lim
nZui(x−δn,pj)−ui(x,pj)dµj=Zlim
nui(x−δn,pj)−ui(x,pj)dµj
=µj({x})(1−αi(x,x))(φi(x)−ψi(x,x)).
Since
φi(x)>ψi(x
,
x)
and
αi(x
,
x)<
1,
µn
i
is a p o i able de ia ion o i m
i
o su icien ly
la ge
n
, iola ing
µ
as an equilib ium. We conclude ha ei he
αi(x
,
x) =
1 o
µj({x}) =
0.
F om Lemma 1, we know ha
µi([
0,
b
x]) =
1. Obse e ha i
x∈(ρ
,
b
x]
, i mus be ha
µ({(x
,
x)}) =
0 since
φi(x)>ψi(x
,
x)
o each i m
i
a any p ice
x∈(ρ
,
b
x]
and
αi(x
,
x)<
1
o some i m ia any p ice x.
Nex , we show ha he equilib ium is in a ian o he choice o
α
a p ices
x∈(ρ
,
b
x]
.
Le
µ
be an equilib ium gi en he sha ing ule
α
wi h expec ed p o i s
= ( 1
,
2)
and
conside ano he sha ing ule
α′
such ha
α(x
,
x) = α′(x
,
x)
o all
x≤ρ
. Le
ui(x
,
µj)
deno e i m
i
’s expec ed payo when choosing a p ice
x
gi en
α
and
u′
i(x
,
µj)
he co e-
sponding payo gi en
α′
. To show ha
µ
is an equilib ium o he game wi h sha ing ule
α′
, i will su ice o show ha , o each playe
i
, (i)
u′
i(x
,
µj) = iµi
-almos e e ywhe e and
(ii) u′
i(x,µj)≤ i o all p ices x.
(i) No e ha he sha ing ule does no in luence he payo s a any p ice
x
such ha
µj({x}) =
0, and so
ui(x
,
µj) = u′
i(x
,
µj)
a all such p ices. Fu he , a all p ices
x≤ρ
,
ui(x
,
µj) = u′
i(x
,
µj)
since
α(x
,
x) = α′(x
,
x)
. The i s pa o his lemma demons a es ha
µi({x}) =
0 o all
x∈(ρ
,
b
x]
such ha
µj({x})>
0. Since
µj
has a mos coun ably many
a oms, hen
µi({x
:
µj({x})>
0
}) =
0. I ollows ha
u′
i(x
,
µj) = iµi
-almos e e ywhe e.
(ii) As we ha e shown in pa (i),
ui(x
,
µj) = u′
i(x
,
µj)
excep possibly a p ices
x∈(ρ
,
b
x]
such ha
µj({x})>
0. Since p ice abo e
b
x
is weakly domina ed by
b
x
, i is
su icien o examine p ices in
[
0,
b
x]
. Conside any such p ice
x
and le
{xk}
be a sequence
such ha
xk→x
,
xk<x
o all
k
, and
µj({xk}) =
0 o all
k
. Then no e ha he con inu-
i y o
φi
and
ψi
in
pi
on
[
0,
b
x]
om Assump ion 7implies ha
limku′
i(xk
,
µj)≥u′
i(x
,
µj)
.
Since
µj({xk}) =
0 o all
k
, hen
ui(xk
,
µj) = u′
i(xk
,
µj)
o all
k
. I
u′
i(x
,
µj)> i
, hen
ui(xk
,
µj)> i
o su icien ly la ge
k
, iola ing
µi
as an equilib ium s a egy wi h he
sha ing ule α. The e o e, u′
i(x,µj)≤ i o all x.
We conclude ha µis an equilib ium gi en he sha ing ule α′.
P oo o Lemma 3.
We i s a gue ha , in any equilib ium,
xi≥ai
o a leas one i m
i
. Suppose o he con a y ha
xi<ai
o each i m
i
. Then Assump ion 2implies ha
Rui(x
,
pj)dµj=φi(x) =
0 and
ψj(pj
,
x) = φj(pj)
o all
pj≥x
and all
x∈[xi
,
ai)
. I
ollows ha ei he playe
i
could choose a p ice o
b
pi
and ecei e a payo o
φi(b
pi)>
0 wi h
posi i e p obabili y since
µj([xj
,
aj)) >
0. This con adic s p ices in
[xi
,
ai)
as equilib ium
s a egies. We conclude ha xi≥ai o a leas one i m i.
Games 2025,16, 26 22 o 37
Le
µ
be an equilib ium wi h
xi<xj
. No e ha
Rui(x
,
pj)dµj=φi(x)
o
all
x∈[xi
,
xj)
. I
xj>ai
, hen Assump ions 2and 3imply ha
φi
, and hus
Rui(x
,
pj)dµj
is s ic ly inc easing on
[ai
,
xj)
, iola ing p ices in
[xi
,
xj)
as equilib ium s a egies o
i m
i
. Thus,
xj≤ai
and so
xi<ai
. Assump ion 2 hus implies ha
φi(x) =
0 o all
x∈[xi
,
xj)
, so i m
i
’s equilib ium p o i mus be ze o. F om he esul p o ed immedia ely
abo e, i mus be ha
xj≥aj
. Suppose ha
xj<ai
. Then Assump ion 2implies ha
Ruj(x
,
pi)dµi=φj(x)
o all
x∈[xj
,
ai)
. The e o e, since
xj≥aj
, Assump ion 3gua an ees
ha
φj
, and hus,
uj
is s ic ly inc easing on
[xj
,
ai)
, iola ing hese p ices as equilib ium
s a egies o i m
j
. Thus, i mus be ha
xj=ai
. I
µj
is nondegene a e, hen he e is
some
x>ai
such ha i m
j
p ices s ic ly highe han
x
wi h posi i e p obabili y. I i m
i
se s a p ice o his
x
, hen wi h posi i e p obabili y, i m
i
will ecei e
φi(x)
, which is
s ic ly posi i e by Assump ion 3, con adic ing ze o as i s equilib ium p o i . The e o e,
µj
is degene a e wi h
µj({ai}) =
1. Since he e is a posi i e p obabili y ha i m
i
chooses a
p ice
x<ai
, Assump ion 2implies ha he e is a posi i e p obabili y ha i m
j
will ob ain
a p o i
φj(b
pj)
i i se s i s p ice a
b
pj
. Consequen ly, i m
j
’s equilib ium p o i mus be
posi i e, and hus aj<ai.
Suppose ha
ai<ρ=ρj
, since
aj<ai
by Assump ion 4. Assump ions 3and 4
gua an ee ha
Ruj(x
,
pj)dµj=φj(x)
o all
x∈(ai
,
ρ)
and ha
φj
is s ic ly inc easing on
his in e al. Thus, i mus be ha ai=ρ.
Finally, suppose ha
µi([ρ
,
ρ+ε)) =
0 o some
ε>
0. Then i m
j
could se any p ice
x∈[ρ
,
ρ+ε)
and s ill ecei e
φj(x)
wi h ce ain y. Since
φj
is s ic ly inc easing, his would
iola e
ρ
as an equilib ium s a egy o i m
j
. We conclude ha
µi([ρ
,
ρ+ε)) >
0 o any
ε>0.
P oo o Lemma 4.
Lemma 3implies ha any nondegene a e mixed s a egy equilib ium
equi es ha
x1=x2=x
. Le
µ
be an equilib ium wi h
x1=x2=x
. Recall om he p oo
o Lemma 3 ha a leas one i m imus ha e xi≥ai.
Fi s , we show ha
x≥max ai
. Suppose o he con a y ha
x<ai
o some i m
i
.
Then i mus be ha
x≥aj
. By Assump ion 2, i m
j
can se any p ice
x∈[x
,
ai)
and ob ain
a p o i o
φj(x)
wi h ce ain y. F om Assump ion 3,
φj
is s ic ly inc easing on his in e al,
con adic ing hese as equilib ium s a egies. The e o e, i mus be ha x≥max ai.
Second, we show ha
x≥ρ
. Suppose o he con a y ha
x<ρ
. Then by Assump-
ion 4,
ψi(ρ
,
pj) = φi(ρ)
o all
pj<ρ
. Consequen ly, ei he i m
i
could choose any p ice
x∈(x
,
ρ)
and ea n
Rui(x
,
pj)dµj=φi(x)
. Since
x≥ai
, Assump ion 3gua an ees ha
φi(x)
is s ic ly inc easing on his in e al, iola ing hese p ices as equilib ium s a egies.
The e o e, i mus be ha x≥ρ.
Thi d, we show ha nei he i m
i
can ha e an a om a an
x
i
φi(x)>ψi(x
,
x)
.
Suppose o he con a y ha i m
j
has an a om a
x
,
µj({x})>
0, no ing ha
x≤b
x
since i
is in suppo o each i m’s s a egy. F om Lemma 2, we know ha
µi({x}) =
0; howe e ,
as we ha e jus shown,
xi=x
. Thus,
µi((x
,
x+δ)) >
0 o all
δ>
0. We will show ha
he e is some
x′<x
and neighbo hood
(x
,
x+δ)
such ha
Rui(x′
,
pj)dµj>Rui(x
,
pj)dµj
o all x∈(x,x+δ).
De ine
β=µj({x})>
0 and le
ε>
0 be such ha
ε<β(φi(x)−ψi(x,x))
. I i m
i
se s
a p ice
x<x
, hen i s p o i will be
φi(x)
wi h ce ain y. Since
x≤b
x
, Assump ion 7gua -
an ees ha
φi
is le con inuous, so he e exis s a
δ1>
0 such ha
|φi(x)−φi(x)|<ε/
2
o all
x∈(x−δ1
,
x)
. No e ha
Rui(x
,
pj)dµj≤(
1
−β)φi(x) + βsuppj∈[x,x]ψi(x
,
pj)
a any p ice
x>x
and obse e ha
suppj∈[x,x]ψi(x
,
pj) = ψi(x
,
x)
a
x=x
. No e
ha
suppj∈[x,x]ψi(x
,
pj)
is igh uppe semicon inuous by Lemma A1. I
x=b
x
, hen
Assump ion 6
gua an ees ha
φi(x)<φi(x)
o all
x>x
, and hus,
φi
is igh up-
pe semicon inuous a
x
. Al e na i ely, i
x<b
x
, hen Assump ion 7gua an ees ha
Games 2025,16, 26 23 o 37
φi
is con inuous and hus igh uppe semicon inuous. In ei he case, he unc ion
(
1
−β)φi(x) + βsuppj∈[x,x]ψi(x
,
pj)
is igh uppe semicon inuous in
x
a
x
, so he e exis s
δ2>0 such ha
(1−β)φi(x) + βsup
pj∈[x,x]
ψi(x,pj)<(1−β)φi(x) + βψi(x,x) + ε
2
o all x∈(x,x+δ2). Le x′∈(x−δ1,x)and no e ha φi(x′)>φi(x)−ε/2. Thus,
φi(x′)>(1−β)φi(x) + βφi(x)−ε
2
= (1−β)φi(x) + βψi(x,x) + β[φi(x)−ψi(x,x)] −ε
2.
Fo all x∈(x,x+δ2), we ha e
(1−β)φi(x) + βψi(x,x) + β[φi(x)−ψi(x,x)] −ε
2
>(1−β)φi(x) + βsup
pj∈[x,x]
ψi(x,pj) + β[φi(x)−ψi(x,x)] −ε
and he e o e,
φi(x′)>(1−β)φi(x) + βsup
pj∈[x,x]
ψi(x,pj) + β[φi(x)−ψi(x,x)] −ε.
Since
ε<β(φi(x)−ψi(x
,
x))
, his implies ha
Rui(x′
,
pj)dµj>Rui(x
,
pj)dµj
o all
x∈(x
,
x+δ2)
. This iola es such p ices as equilib ium s a egies o i m
i
. We conclude
ha µj({x}) = 0, so nei he i m ican ha e an a om a xi φi(x)>ψi(x,x).
P oo o P oposi ion 3.
We will p o e he p oposi ion o he wo ypes o equilib ia om
P oposi ion 2sepa a ely.
Case 1: Le µbe a symme ic lowe bound equilib ium.
Fo his case, we i s show ha
x∈[
,
]
in wo pa s. Fi s , we a gue ha
x≤
.
Suppose o he con a y ha
x>
. F om Lemma 2, a mos one i m can ha e an a om a
x
.
Wi hou loss o gene ali y, le i m
i
be such ha ha
x
is in suppo o
µi
and
µj({x}) =
0.
Choose {xk
i}in suppo o µisuch ha xk
i→x; hen no e ha
lim
k→∞Zx
xui(xk
i,pj)dµj=Zx
xψi(x,pj)dµj
since
ψi
is con inuous in
pi
om Assump ion 7. Nex , since each
xk
i
is in suppo o
µi
, i
mus be ha each Rx
xui(xk
i,pj)dµj=u∗
i. No e ha
Zx
xψi(x,pj)dµj≤Zx
xe
ψi(pj)dµj
by de ini ion o
e
ψi
. By de ini ion o
i
and he ac ha
x> i
,
φi( i)>e
ψi(x)
o all
x≥ i
o each i m
i
. Fu he , Assump ions 2and 3imply ha
φi
is nondec easing, so
φi(x)>e
ψi(x)
o all
x≥x
. This implies ha
Rx
xe
ψi(pj)dµj<φi(x)
, and hus
u∗
i<φi(x)
.
This iola es
µi
as an equilib ium s a egy since i m
i
has a p o i able de ia ion o
x−ε
o
su icien ly small ε ha would gua an ee a payo o φi(x). We conclude ha x≤ .
Second, we a gue ha
x≥
. Suppose o he con a y ha
x<
. Le i m
i
be such
ha
i=
. By de ini ion o
i
and he con inui y o
φi
om Assump ion 7, i mus be
ha
ui=φi( )
. F om Lemma 4,
x≥ρ
, and since
ρ≥ai
, Assump ion 2implies ha
φi(x)>φi(x)
o all
x>x
. Thus,
φi(x)<φi( ) = ui
. The con inui y o
φi
and
ψi
in
pi
Games 2025,16, 26 24 o 37
hus gua an ee ha
φi(x)<ui
and
ψi(x
,
x′)<ui
o all p ices
x∈[x
,
)
wi h
x′≤x
, so
Rui(x
,
pj)dµj<ui
o all such
x
. This iola es all
x∈[x
,
)
as equilib ium s a egies. We
conclude ha x≥ .
I ollows om Lemma 4 ha each i m
i
’s equilib ium expec ed p o i is
u∗
i=φi(x)
.
The s a emen o he p oposi ion hus ollows om
x∈[
,
]
and he ac s ha
φi
is s ic ly
inc easing on [ρ,b
pi]and ha ρ≤ .
Case 2: Le µbe an asymme ic lowe bound equilib ium.
Fi s , conside playe
i
. The lowe bound payo
ui=
0, which implies ha
φi( ) =
0.
Fu he , we know om Lemma 3 ha
u∗
i=
0. By cons uc ion,
φi( )≥
0; he e o e,
u∗
i∈[φi
,
φi]
. Nex , conside playe
j
. F om Lemma 3,
u∗
j=φj(ρ) = φj(ai)
. This payo is
possible o playe
j
o any p icing by playe
i
. Thus,
u∗
j=uj=φj( )
. By cons uc ion,
φj( )≥0; he e o e, u∗
j∈[φj,φj].
P oo o Lemma 5.
We use
pi(Fj)
and
pi(Fj)
o deno e he smalles and la ges condi ional
esidual maximize , espec i ely. Tha is,
pi(Fj) = min e
Pi(Fj)
and
pi(Fj) = sup e
Pi(Fj)
,
whe e he igh con inui y o
Fj
ensu es ha
e
Pi(Fj)
con ains a minimal elemen , while i
need no con ain a maximal elemen .
Le
µ
be an equilib ium wi h he co esponding CDF’s
F
. I
x=ρ
, hen om
Lemmas 3
and 4, ei he he equilib ium is degene a e o
xi<xj
o some i m
i
. I he equilib ium
is degene a e, hen he s a emen o P oposi ion 4(in Sec ion 4) gua an ees ha
ρ∈e
Pi(ρ)
o some i m
i
. I i ially ollows ha
min{p1(F2)
,
p2(F1)} ≤ x≤max{p1(F2)
,
p2(F1)}
.
Al e na i ely,
xi<xj
o some i m
i
; hen Lemma 3gua an ees ha
µj({ρ}) =
1 and
u∗
i=
0. In o de o
µ
o be an equilib ium, i m
i
canno ha e any p o i able de ia ions,
so i mus be ha
ψi(x
,
ρ) =
0 o all
x≥ρ
. Thus, by de ini ion,
e
Pi(ρ) = [ρ
,
∞)
, so
min{p1(F2),p2(F1)} ≤ x≤max{p1(F2),p2(F1)}.
Finally, le
x>ρ
and suppose ha ei he
x<min{p1(F2)
,
p2(F1)}
o
x>max{p1(F2)
,
p2(F1)}
. F om Lemma 2, a mos one i m may ha e an a om a
x
.
Le i m
i
be such ha
xi=x
and
µj({x}) =
0. Since
x
is in suppo o i m
i
’s s a egy, we
may choose {xk
i}in suppo o µisuch ha xk
i→x; hen no e ha
lim
k→∞Zx
xui(xk
i,pj)dµj=Zx
xψi(x,pj)dµj
=EFj[ψi(x,pj)|pj≤x].
Since each
xk
i
is a bes esponse o i m
i
, his implies ha
x
is also a bes esponse o i m
i. I ollows om ou supposi ion ha x/∈e
Pi(Fj). No e ha , o any p ice x,
ui(x,Fj)≥(1−Fj(x))φi(x) + Z[x,x]ψi(x,pj)dFj
= (1−Fj(x))φi(x) + Fj(x)EFj[ψi(x,pj)|pj≤x].
By de ini ion o
e
Pi(Fj)
,
EFj[ψi(x
,
pj)|pj≤x]>EFj[ψi(x
,
pj)|pj≤x]
o all
x∈e
Pi(Fj)
. Thus,
ui(x
,
Fj)>ui(x
,
Fj)
o all
x∈e
Pi(Fj)
since
φi(x)≥ψi(x
,
pj)
o all
pj
. This con adic s
x
as
a bes esponse. We conclude ha min{p1(F2),p2(F1)} ≤ x≤max{p1(F2),p2(F1)}.
P oo o P oposi ion 4.
Fi s , obse e ha , in any equilib ium
(p∗
1
,
p∗
2)
wi h co espond-
ing p o i s
( ∗
1
,
∗
2)
, i mus be ha
∗
i≥φi(min{p∗
j
,
b
x})
. To see why, no e ha
Assump ion 7gua an ees ha
φi
is le con inuous a
min{p∗
j
,
b
x}
. Thus, i m
i
can gua -
an ee i sel a payo o
φi(min{p∗
j
,
b
x}−ε)
by de ia ing o
pi=min{p∗
j
,
b
x}−ε
, wi h he
gua an ee ha φi(min{p∗
j,b
x}−ε)→φi(min{p∗
j,b
x}).
Games 2025,16, 26 31 o 37
ollows ha
ψi(x′
,
ρ) = πi(x′
, 0
) =
0 o all
x′>ρ
. Thus, nei he i m possesses a p o i able
de ia ion, so each i m p icing a x∗=ρis an equilib ium.
B.2 Conside i m
i
wi h
ai>aj
. As no ed in he (B.1) case, i m
i
has no p o -
i able de ia ions. F om he assump ions o B.2, obse e ha
uj(ρ
,
ρ) = φj(ρ)
. F om
Assump ions 2and 3,
φj
is nondec easing on
[
0,
b
pj]
, and so
φj(ρ)≥φj(x)
o all
x<ρ
.
Las ly, he assump ion in B.2 ha
ψj(x
,
ρ)≤φj(ρ)
o all
x≥ρ
gua an ees ha he e a e
no p o i able de ia ions o i m
j
o p ices highe han
ρ
. Thus, nei he i m possesses a
p o i able de ia ion, so each i m p icing a x∗=ρis an equilib ium.
C. F om Condi ion 6,
di(ρ
,
ρ) = D(ρ)−Qj(ρ)
. Since
Qj(ρ)≤sj(ρ)
, i ollows ha
di(ρ
,
ρ)≥D(ρ)−sj(ρ)
. Obse e ha i
αi(ρ
,
ρ) =
1, hen by de ini ion
ui(ρ
,
ρ) = φi(ρ
,
ρ)
.
Suppose ha
αi(ρ
,
ρ)<
1. Then om he assump ion o C,
si(ρ) + sj(ρ)≤D(ρ)
, so
D(ρ)−sj(ρ)≥si(ρ)
. I ollows ha
πi(ρ
,
Qi(ρ)) = πi(ρ
,
min{si(ρ)
,
di(ρ
,
ρ))
o each i m
i
, and so
ui(ρ
,
ρ) = φi(ρ)
. As no ed abo e,
φi
is nondec easing on
[
0,
b
pi]
, so he e a e
no p o i able de ia ions o p ices
x<ρ
. Fu he , since
ρ∈e
Pi(ρ)
, he e a e no p o i able
de ia ions o p ices
x>ρ
. Thus, nei he i m possesses a p o i able de ia ion, so each i m
p icing a x∗=ρis an equilib ium.
Nex , we p o e ha any pu e s a egy equilib ium mus sa is y ei he B.1, B.2, o C.
P oposi ion 4implies ha any pu e s a egy equilib ium mus be symme ic wi h
x∗=ρ. Fu he , by de ini ion, i mus be ha ρ≥aiand ρ<b
pi o each i m i.
B.1 Suppose ha
x∗=ρ=a1=a2
and ha
di(x
,
ρ)>
0 o some
x>ρ
o some i m
i
. Then since
x>ai
, om Condi ion 3, he e is some quan i y
z∈(
0,
ρ)
such ha
x>ci(z)
z
,
so
πi(x
,
z)>
0. Since
ψi(x
,
ρ)≥πi(x
,
z)
, i ollows ha
ψi(x′
,
ρ)>
0. This con adic s
x∗
as
an equilib ium since φi(x∗) = 0.
B.2 Suppose ha
x∗=ρ=ai>aj
. I
di(x
,
ρ)>
0 o some
x>ρ
, hen he p eceding
a gumen o he (B.1) case applies and ules ou
x∗
as an equilib ium. I
uj(ρ
,
ρ)<φj(ρ)
,
hen by con inui y o
φj
by Assump ion 7, he e exis s a p ice
x<ρ
such ha
uj(x
,
ρ) =
φj(x)>uj(ρ
,
ρ)
. This con adic s
x∗
as an equilib ium. Las ly, suppose ha
ψj(x
,
ρ)>
φj(ρ)
o some
x≥ρ
. Then since
ui(x
,
ρ)≥ψi(x
,
ρ)
, hen
x
is a p o i able de ia ion om
x∗, iola ing x∗as an equilib ium.
C. Suppose ha
ρ∈(max{a1
,
a2}
,
min{b
p1
,
b
p2}]
. I
ρ/∈e
Pi(ρ)
o some i m
i
, hen
by de ini ion, any p ice
x∈e
Pi(ρ)
is a p o i able de ia ion o i m
i
, iola ing
x∗
as an
equilib ium. Las ly, suppose ha
si(ρ) + sj(ρ)>D(ρ)
o some i m
i
wi h
αi(ρ
,
ρ)<
1.
Then since
di(ρ
,
ρ) = D(ρ)−sj(ρ)
by Condi ion 6, i mus be ha
di(ρ
,
ρ)<si(ρ)
, so
qi(ρ
,
ρ)/∈a g maxzπi(ρ
,
z)
. The e o e, since
ψi(ρ
,
ρ) = πi(ρ
,
qi(ρ
,
ρ))
and
αi(ρ
,
ρ)<
1,
i mus be ha
ui(ρ
,
ρ) = αi(ρ
,
ρ)φi(ρ)+ (
1
−αi(ρ
,
ρ))ψi(ρ
,
ρ)<φi(ρ)
. Then, since
φi
is
con inuous by Assump ion 7, he e exis s a p ice
x<ρ
such ha
ui(x
,
ρ) = φi(x)>ui(ρ
,
ρ)
.
This iola es x∗as an equilib ium.
P oo o P oposi ion 7.
Wi hou loss o gene ali y, we assume
a1≥a2
. Suppose o he
con a y ha he e is an equilib ium p ice
x∗=a1
wi h
s1(x∗) =
0. Then
Q1(x∗) =
0, so
Condi ion 6gua an ees ha
d2(x
,
a1) = D(x)
o all
x>a1
. I ollows immedia ely ha
ψ2(x
,
a1) = φ2(x)
o all
x≥a1
. F om Condi ion 8,
b
p2>a1
. The e o e, by de ini ion o
b
p2
,
we ha e ψ2(b
p2,a1) = φ2(b
p2)>φ2(a1), con adic ing x∗=a1as an equilib ium p ice.
P oo o P oposi ion 8.
Since
D
and
ci
a e unchanged, hen
φi(x) = φ′
i(x)
o all
x
.
No e ha
ψi(pi,pj) = max
z∈[0,min{ki,di(pi,pj)}]xz −ci(z)and
ψ′
i(pi,pj) = max
z∈[0,min{ki,d′
i(pi,pj)}]xz −ci(z).

Games 2025,16, 26 32 o 37
I ollows immedia ely ha
ψ′
i≥ψi
. The ac ha
′
i≤ i
and
′
i≤ i
ollows immedia ely
om hei de ini ions.
P oo o P oposi ion 9.
The ac ha
′j≤ j
and
′j≤ j
ollows di ec ly om
P oposi ion 8since any change in i m
i
’s supply has no impac on he on -side p o i o
i m
j
, so
φj(x) = φ′j(x)
o all
x
. I emains o show ha
′
i≤ i
and
′
i≤ i
. Obse e ha
di(x,y) = d′
i(x,y) o all p ices x≥y.
De ine
si(x) = sup ϑi(x)
, wi h
s′
i
de ined analogously o
π′
i
. Le
Qi(x) =
min{si(x)
,
D(x)}
and
qi(x
,
y) = min{si(x)
,
di(x
,
y)}
, wi h
Q′
i
and
q′
i
de ined analogously.
Then no e ha φi(x) = πi(x,Qi(x)). No e ha siis nondec easing.
Pa 1: ′
i≤ i
The p oo ha
′
i≤ i
is conduc ed in ou s eps. In S ep 1, we a gue ha
πi(b
pi
,
z)<
πi(b
pi
,
D(b
pi))
o all
z<D(b
pi)
and hen use ha ac o a gue ha we show ha
i<b
pi
. In
S ep 2, we show ha
qi(x
,
y)≤Qi(y)
o all
x≥y> i
. In S ep 3, we a gue ha
qi(x
,
y)≥
q′
i(x,y) o all x≥y> i, implying ha q′
i(x,y)≤Qi( i) o all x≥y> i. Finally, in S ep
4, we show ha i
′
i> i
, we can ind p ices
x≥y> i
such ha
ci(Qi(x)) −c′
i(Qi(x)) <
ci(q′
i(e
x
,
x)) −c′
i(q′
i(e
x
,
x))
, con adic ing he assump ion ha
ci(q)−c′
i(q)
is nondec easing
in q.
S ep 1: We i s show ha
πi(b
pi
,
z)<πi(b
pi
,
D(b
pi))
o all
z<D(b
pi)
. Suppose o
he con a y ha
πi(b
pi
,
z)≥πi(b
pi
,
D(b
pi))
o some
z<D(b
pi)
. As no ed in he p oo
o P oposi ion 6,
Qi(b
pi) = D(b
pi)
. Thus,
πi(b
pi
,
z) = φi(b
pi)
. Since
φi
is con inuous by
Assump ion 7, he e exis s a p ice
x>b
pi
such ha
z<D(x)
. Obse e ha
φi(x)≥
πi(x
,
z) = xz −ci(z)>b
piz−ci(z) = φi(b
pi)
. This con adic s
b
pi
as he maximize o
φi
. We
conclude ha πi(b
pi,z)<πi(b
pi,D(b
pi)).
We nex a gue ha
i<b
pi
. Suppose o he con a y ha
i≥b
pi
. Since
aj<b
pi
by Assump ion 5, we may choose a s ic ly inc easing sequence
{yk}
such ha
y0>aj
and
yk→b
pi
. By de ini ion o
i
,
φi(x)≤e
ψi(x)
o all
x< i
. Thus, since
e
ψi(x)
is
noninc easing as no ed ea lie , his implies ha
e
ψi(y0)≥φi(yk)
o all
k
. By con inui y o
φi
by Assump ion 7,
limkφi(yk) = φi(b
pi)
, and so
e
ψi(y0)≥φi(b
pi)
. Le
e
x∈e
Pi(y0)
and no e
ha
ψi(e
x
,
y0)≤φi(e
x)
by Assump ion 1. Obse e ha
φi(b
pi)≥φi(e
x)≥e
ψi(y0)≥φi(b
pi)
.
Since
b
pi
is he unique maximize o
φi
, i ollows ha
e
x=b
pi
. As demons a ed in he
p oo o P oposi ion 6,
Qi(b
pi) = D(b
pi)
and
sj(b
pi)>
0 since
b
pi>aj
. By Condi ion 6, his
implies ha
di(b
pi
,
y0)<D(b
pi)
, so
qi(b
pi
,
y0)<D(b
pi)
. F om he i s pa ag aph o his
s ep, his implies ha
πi(b
pi
,
qi(b
pi
,
y0)) <πi(b
pi
,
D(b
pi))
and hus ha
ψi(b
pi
,
y0)<φi(b
pi)
. A
con adic s o e
ψi(y0)≥φi(b
pi). We conclude ha i<b
pi.
S ep 2: We show ha
qi(x
,
y)≤Qi(y)
o all
x≥y> i
. Suppose o he con a y ha
qi(x,y)>Qi(y) o some x≥y> i. No e ha
ψi(x,y) = max
z∈[0,min{ki,di(x,y)}]πi(x,z).
Since Qi( i)<qi(x,y)≤di(x,y), i ollows ha
ψi(x,y)≥xQi(y)−ci(Qi(y))
≥yQi(y)−ci(Qi(y))
=φi(y).
This con adic s he de ini ion o
i
as
i=sup{x|φi(x)≤e
ψi(x)}
. We conclude ha
qi(x,y)≤Qi( i) o all x≥y> i.
S ep 3: We a gue ha
qi(x
,
y)≥q′
i(x
,
y)
o all
x≥y> i
. Le
x≥y> i
and suppose
o he con a y ha
qi(x
,
y)<q′
i(x
,
y)
. Then since
q′
i(x
,
y)≤di(x
,
y)
, i mus be ha
qi(x
,
y)<di(x
,
y)
. I ollows ha
qi(x
,
y) = si(x)
, and so
ψi(x
,
y) = πi(x
,
si(x)) = φi(x)
.
Games 2025,16, 26 33 o 37
Since
si
is nondec easing and
di
is noninc easing by Condi ion 5, i ollows ha
si(pi)<
di(pi
,
pj)
o all
pi<x
such ha
pi≥pj> i
. Thus,
ψi(pi
,
pj) = πi(pi
,
si(pi)) = φi(pi)
o all
pi
and
pj
such ha
x>pi≥pj> i
. Thus,
e
ψi(pj)≥ψi(pj
,
pj) = φi(pj)
o all
pj∈( i
,
x)
, con adic ing he de ini ion o
i
. We conclude ha
qi(x
,
y)≥q′
i(x
,
y)
o
all x≥y> i.
In summa y we ha e now es ablished ha , since
si
is nondec easing, hen o any
x≥y> i, i ollows ha q′
i(x,y)≤qi(x,y)≤Qi(y).
S ep 4: We a gue ha
′
i≤ i
. Suppose o he con a y ha
′
i> i
. Then o any p ice
x∈( i
,
′
i)
, i mus be ha
φ′
i(x)≤e
ψ′
i(x)
. Le
x∈( i
,
′
i)
and
e
x∈e
P′
i(x)
and no e ha om
abo e, qi(e
x,x)≤Qi(x). No e ha e
ψ′
i(x)is noninc easing in xsince
e
ψ′
i(x) = max
pi
max
z∈[0,min{ki,di(pi,x)}]πi(pi,z)
and
di(pi
,
x)
is noninc easing in
x
by Condi ion 5. F om abo e, we may choose he p ice
x∈( i
,
′
i)
such ha
x<b
pi
. Since
φi
is s ic ly inc easing on
(ai
,
b
pi)
by Assump ion 3
and om con inui y o
D
om Condi ion 4, we may choose
x
so ha
φ′
i(x)≤e
ψ′
i(x)
and
φi(x)>e
ψi(x). Thus, we ha e
φ′
i(x)−φi(x)<e
ψ′
i(x)−e
ψi(x). (A1)
No e ha
φi(x) = xQi(x)−ci(Qi(x))
and
φ′
i(x)≥xQi(x)−c′
i(Qi(x))
. Pu ing hese
oge he , we ha e
φ′
i(x)−φi(x)≥ci(Qi(x)) −c′
i(Qi(x)). (A2)
Nex , no e ha
e
ψ′
i(x) = e
xq′
i(e
x
,
x)−c′
i(q′
i(e
x
,
x))
and
e
ψi(x)≥e
xq′
i(e
x
,
x)−ci(q′
i(e
x
,
x))
. Pu ing
hese oge he yields
e
ψ′
i(x)−e
ψi(x)≤ci(q′
i(e
x,x)) −c′
i(q′
i(e
x,x)). (A3)
The inequali ies (A1), (A2), and (A3) oge he imply ha
ci(Qi(x)) −c′
i(Qi(x)) <ci(q′
i(e
x,x)) −c′
i(q′
i(e
x,x)),
which con adic s he assump ion ha
ci(z)−c′
i(z)
is nondec easing in
z≥
0. We conclude
ha ′
i≤ i.
Pa 2: ′
i≤ i
The p oo ha
′
i≤ i
is also done by con adic ion. Suppose o he con a y ha
′
i> i
. Recall ha
ui=suppiin pjui(pi
,
pj)
. As no ed abo e,
ψi
is noninc easing in
pj
. Fu he , by Assump ion 1, we can conclude ha
in pjui(pi
,
pj) = ψi(pi
,
pi)
, and hus
ui=suppiψi(pi
,
pi)
. Addi ionally, since
φi
is con inuous by Assump ion 7, i ollows ha
φi( i) = ui.
Le
{xk}
be a sequence such ha
ψ′
i(xk
,
xk)→u′
i
. We may wi hou loss o gene ali y
choose his sequence such ha
xk→x∗
o some p ice
x∗
. I ollows ha
u′
i=π′
i(x∗
,
q∗)
,
whe e
q∗=limkq′(xk
,
xk)
. By de ini ion o
ui
, i mus be ha
ui≥ψi(xk
,
xk)
o all
k
. Fu -
he ψi(xk,xk) = πi(xk,qi(xk,xk)) ≥πi(xk,q′
i(xk,xk)), and so ui≥πi(x∗,q∗). The e o e,
u′
i−ui≤π′
i(x∗,q∗)−πi(x∗,q∗) = ci(q∗)−c′
i(q∗). (A4)
We b ie ly a gue ha
x∗≥ ′
i
. To see his, suppose o he con a y ha
x∗< ′
i
. Then
no e ha
φ′
i( ′
i) = u′
i=π′
i(x∗
,
q∗)≤φ′
i(x∗)
. By de ini ion o
ai
, i mus be ha
a′
i≤ai
, and
since
ai≤ i< ′
i
, i ollows ha
′
i>a′
i
. Thus, Assump ion 3implies ha
φ′
i
is s ic ly
Games 2025,16, 26 34 o 37
inc easing on
(max{a′
i
,
x∗}
,
′
i)
. Since
φ′
i(x) =
0 o
x<a′
i
by Assump ion 2, i ollows ha
φ′
i(x∗)<φ′
i( ′
i), a con adic ion. We conclude ha x∗≥ ′
i.
We will now a gue ha q∗≤Qi(x) o all xin some neighbo hood ( i, i+δ).
We begin by a guing ha
Qi(y)≥di(x
,
x)
o all p ices
x>y> i
. Suppose o he
con a y ha
Qi(y)<di(x
,
x)
o some
x>y> i
. Recall ha
φi( i) = πi( i
,
Qi( i))
. Nex ,
since
πi(x
,
z)
is quasiconca e in
z
by Condi ion 2, i ollows ha
ψi(x
,
x) = xqi(x
,
x)−
ci(qi(x,x)), whe e qi(x,x) = min{si(x),di(x,x)}. No e ha by de ini ion o uiand si,
ui≥ψi(x,x)
=xqi(x,x)−ci(qi(x,x))
≥xQi(y)−ci(Qi(y)).
Since
y> i
and
i≥ai
, i ollows ha
φi(y)>
0, and so
Qi(y)>
0. Thus,
xQi(y)>yQi(y)
.
The e o e,
ui≥xQi(y)−ci(Qi(y))
>yQi(y)−ci(Qi(y))
=φi(y)
≥φi( i)
=ui.
This is a con adic ion. We conclude Qi(y)≥di(x,x) o all p ices x>y> i.
Now, suppose o he con a y ha he e exis s a sequence
{yn}
wi h
yn→ i
and
yn> i
such ha
q∗>Qi(yn)
o all
n
. Since
x∗≥ ′
i> i
, we may wi hou loss o gene ali y
assume ha
yn<min{xk
,
x∗}
o all
k
and
n
. Thus, om abo e,
Qi(yn)≥di(xk
,
xk)
o
all
k
and
n
. Since
q′
i(xk
,
xk)≤di(xk
,
xk)
, his implies ha
Qi(yn)≥q∗
, a con adic ion. We
conclude ha q∗≤Qi(x) o all xin some neighbo hood ( i, i+δ). Choose such a δ.
Now obse e ha by de ini ion o
′
i
,
φ′
i( i)≤u′
i
o any p ice
x∈( i
,
′
i)
. Thus,
o any p ice
x∈( i
,
b
pi)
, i mus be ha
φi(x)>ui
since
φi
is s ic ly inc easing by
Assump ion 3. Recall ha
i≤ i
, and as shown abo e,
i<b
pi
, so
( i
,
b
pi)
is nonemp y.
Le
x∈( i
,
min{ ′
i
,
b
pi
,
i+δ})
wi h
δ>
0 picked such ha
q∗≤Qi(x)
o all
x
in some
neighbo hood ( i, i+δ). No e ha
φ′
i(x)−φi(x)<u′
i−ui.
Obse e ha
φi(x) = xQi(x)−ci(Qi(x)),
and
φ′
i(x) = xQ′
i(x)−c′
i(Q′
i(x))
≥xQi(x)−c′
i(Qi(x)).
Pu ing hese oge he , we ha e
φ′
i(x)−φi(x)≥ci(Qi(x)) −c′
i(Qi(x)),
and hus
ci(Qi(x)) −c′
i(Qi(x)) <u′
i−ui.
Games 2025,16, 26 35 o 37
Recall ha om (A4)
u′
i−ui≤ci(q∗)−c′
i(q∗).
I ollows ha
ci(Qi(x)) −c′
i(Qi(x)) <ci(q∗)−c′
i(q∗),
which con adic ions he assump ion ha
ci(z)−c′
i(z)
is nondec easing in
z≥
0 since
Qi(x)≥q∗. We conclude ha ′
i≤ i.
No es
1Vi es (1986,1993) bo h p o ide excellen con ex o Edgewo h’s con ibu ion o oligopoly.
2
Be o e Shubik (1959), Shapley (1957) published an abs ac wi h a desc ip ion o esul s de i ed om a game heo e ic model o
p icing. O he ea ly con ibu ions o BE compe i ion we e made by Beckmann and Hochs ad e (1965),
Shapley and Shubik (1969)
,
and Le i an and Shubik (1972).
3
To con ex ualize he di e en a ioning schemes, imagine ha demand is composed o a con inuum o consume s wi h di e en
le els o willingness o pay o a single uni o he good. The e icien a ioning ule speci ies ha he low p ice i m se es he
consume s wi h he highes willingness o pay. Tha is, all a ioned consume s ha e a weakly lowe willingness o pay han all
consume s ha pu chase om he low p ice i m. The p opo ional ule speci ies ha all consume s willing o pay he low p ice
a e equally likely o be se ed by he low p ice i m, esul ing in a p opo ion o high willingness o pay being a ioned and hus
a la ge esidual demand han he e icien ule.
4
Almos all o he BE li e a u e also assumes ha he i ms ha e a symme ic, cons an ma ginal cos up o capaci y.
Denecke e and Ko enock (1996)
and
Allen e al. (2000)
a e he no able excep ions. These pape s ocus on he in e es ing case in
which i ms ha e cons an ma ginal cos s ha a e asymme ic. Addi ionally, he bulk o his li e a u e u he es ic s demand o
be such ha a i m’s monopoly p o i is conca e in i s p ice. Ou analysis is based on he conside ably weake assump ion ha a
i m’s monopoly p o i is s ic ly inc easing in i s own p ice up o i s unique p o i -maximizing monopoly p ice.
5
Hoe nig (2007) p o ides a ea men o classical Be and p ice compe i ion wi h gene al cos s uc u e and sha ing ules. In he
classical Be and speci ica ion, any i m ha does no ha e he lowes p ice ecei es no esidual demand.
6Yoshida (2002) p o ides a simila ea men o Yoshida (2006) o a model wi h linea demand and quad a ic cos .
7
The ela ionship be ween BE games and all-pay auc ions is discussed in Baye e al. (1996), he i s comp ehensi e ea men o all-
pay auc ions. Mo e ecen ly,
Chowdhu y (2017)
ely on echniques om he analysis o BE duopoly by
Osbo ne and Pi chik (1986)
and Denecke e and Ko enock (1996) o examine all-pay auc ions wi h non-mono onic payo s.
8
The wo dis inc ions be ween a adi ional all-pay auc ion and ou BE game ha e been each ea ed indi idually in he all-pay
auc ion li e a u e. In Baye e al. (2012), he issue o ex e nali ies o bids (con ingen on being a winne o a lose ) has been
add essed in he con ex o all-pay auc ions. Chowdhu y (2017) p o ides a ea men o all-pay auc ions in which he winning
payo is nonmono onic in a playe ’s own bid.
9
In e ms o ma ke demand es ic ions, we equi e only ha he monopoly p o i be s ic ly inc easing a p ices abo e minimum
a e age o al cos up o i s unique maximize .
10
Gelman and Salop (1983) show ha , in a wo-pe iod sequen ial game, a single po en ial en an can use judo capaci y es ic ion
and p icing o induce an uncons ained monopolis o allow en y. The ma hema ical objec ha we ha e deno ed as he c i ical
judo p ice has played a c i ical ole in he analysis o BE p ice compe i ion since i was i s used o cha ac e ize he Edgewo h
ange o p ice luc ua ion in (Shubik,1959, p. 96).
11
P og ess wi h he analysis o models wi h mo e gene al cos s and demand a ioning has been hinde ed by heo e ical p oblems
wi h he exis ence o equilib ium (pu e o mixed) in his se ing. Howe e , we u ilize ad ances in he li e a u e on he exis ence
o equilib ium in discon inuous games by Bagh (2010) and Allen and Lepo e (2014) ha allow o he s aigh o wa d e i ica ion
o he exis ence o equilib ium in as gene aliza ions o BE oligopoly.
12
This coun e ailing e ec o a supply inc ease is immedia e in he exis ing BE li e a u e wi h ega d o an inc ease in a i m’s
capaci y. This is also ela ed o he impac o an impo ade quo a in a duopoly wi h an in e na ional and domes ic i m, o
example K ishna (1989).
13
When his assump ion ails o hold, he equilib ium is i ial: one i m cha ges i s monopoly p ice, and he o he i m cha ges any
p ice and does no p oduce. While i would be easy o conduc he analysis in his pape wi hou his assump ion, i would ake
away om he cla i y o he esul s and would no meaning ully con ibu e o he s udy o duopoly.
14
He e, he bounds
xi
and
xi
a e inhe en ly dependen on he equilib ium s a egies, hough we supp ess no a ion indica ing his
o cla i y as he e is no ambigui y as o which s a egies hey co espond o.
15 The supply co espondence is aken o be a subse o he ex ended eal line.
Games 2025,16, 26 36 o 37
16
Obse e ha he esidual p o i may be ze o despi e he p esence o esidual demand as he cos o engaging in low le els o
p oduc ion may exceed he associa ed e enues, hus inducing he i ms o no p oduce.
17
Fo eade s amilia wi h Simon and Zame (1990), hei esul can be used o gua an ee ha an equilib ium exis s o some sha ing
ule o he game. The complica ion in his se ing is ha he sha ing ule in he Simon and Zame amewo k does no co espond
only o he di ision
αi
be ween
φi
and
ψi
a p icing ies. The sha ing ule in his se ing also e lec s speci ica ions o payo s a
poin s o discon inui y o
ψi
in
pj
. The esul s o Simon and Zame (1990) do no gi e any way o iden i ying he sha ing ule o
which an equilib ium exis s. I he sha ing ule wi h an equilib ium i s he speci ica ions o P oposi ion 1 and he co esponding
equilib ium happens o place mass a p ice ies as in P oposi ion 1 o a a poin o discon inui y o
ψi
, hen his s a egy p o ile
would no be an equilib ium o a sha ing ule ha iola ed he condi ions o P oposi ion 1 a hose p ices. Howe e , i he sha ing
ule wi h an equilib ium iola es he condi ions o P oposi ion 1, hen he co esponding equilib ium will also be an equilib ium
o a sha ing ule ha sa is ies he condi ions o P oposi ion 1. O he applicable esul s ha gua an ee he exis ence o equilib ium
ypically necessi a e he same condi ions. We ha e ound mo e gene al esul s o be in easible o applica ion o ou model.
18
Ties a a p ice
x<ρ
a e i ele an since each i m’s on -side p o i is iden ical o i s esidual p o i . Ties a p ices
x=ρ
may be
ele an by his no ion and a e co e ed by he lemma.
19
The e icien and p opo ional a ioning ules ha ha e ypically been used in s udying p ice compe i ion bo h lead o esidual
p o i unc ions being noninc easing in he i al i m’s p ice.
20
In his o mula ion o he model, demand is ini e a all p ices, so a i m ha is no capaci y-cons ained can be accommoda ed ia
an a bi a ily la ge capaci y.
21
Wi h any con inuous a ioning ule, such as e icien o p opo ional a ioning, i m
i
’s esidual demand will be lowe semicon in-
uous in pjso long as sjis uppe semicon inuous.
22 He e, he limi supe io o he sequence o se s Ak e e s o he se T∞
n=1S∞
k=nAk.
23
This occu s a p ices
pj
such ha i m
j
’s cos o p oduc ion is cons an and equal o
pj
o some le els o p oduc ion. As such, i
is possible ha i m j’s quan i y jumps up a such a p ice, causing a disc e e d op in he esidual p o i o i m i.
24 A co espondence ϑis nondec easing i , o any x≤x′and any y∈ϑ(x), he e exis s a y′∈ϑ(x′)such ha y′≥y.
25
No e ha
τ′
i
is inhe en ly a unc ion o
pj
. We choose no o in oduce no a ion o exp ess his as
pj
is ixed o he du a ion o he
p oo ha u ilizes τ′
i, and hus, he e is no possibili y o ambigui y.
Re e ences
Allen, B. (1993). Capaci y p ecommi men as an en y ba ie o p ice-se ing i ms. In e na ional Jou nal o Indus ial O ganiza ion,
11(1), 63–72. [C ossRe ]
Allen, B., Denecke e, R., Fai h, T., & Ko enock, D. (2000). Capaci y p ecommi men as a ba ie o en y: A be and-edgewo h
app oach. Economic Theo y,15(3), 501–530. [C ossRe ]
Allen, B., & Hellwig, M. (1986) Be and-edgewo h oligopoly in la ge ma ke s. Re iew o Economic S udies, 54(2), 175–204. [C ossRe ]
Allen, B., & Hellwig, M. (1993). Be and-edgewo h duopoly wi h p opo ional esidual demand. In e na ional Economic Re iew,
34(1), 39–60. [C ossRe ]
Allison, B. A., Bagh, A., & Lepo e, J. J. (2018). Su icien condi ions o weak ecip ocal uppe semi-con inui y in mixed ex ensions o
games. Jou nal o Ma hema ical Economics,74, 99–107. [C ossRe ]
Allison, B. A., & Lepo e, J. J. (2014). Ve i ying payo secu i y in he mixed ex ension o discon inuous games. Jou nal o Economic Theo y,
152, 291–303. [C ossRe ]
Bagh, A. (2010). Va ia ional con e gence o games, exis ence and app oxima ion o equilib ia in discon inuous games. Jou nal o
Economic Theo y,145(3), 1244–1268. [C ossRe ]
Bagh A., & Jo e, A. (2006). Recip ocal uppe semicon inui y and be e eply secu e games: A commen . Econome ica,76, 1715–1721.
[C ossRe ]
Baye, M. R., Ko enock, D., & de-V ies, C. G. (1996). The all-pay auc ion wi h comple e in o ma ion. Economic Theo y,8, 1–25. [C ossRe ]
Baye, M. R., Ko enock, D., & de-V ies, C. G. (2012). Con es s wi h ank o de spillo e s. Economic Theo y,51, 315–350. [C ossRe ]
Beckmann, J. M., & Hochs ad e , D. (1965). Edgewo h-be and duopoly e isi ed. In R. Henn. (Ed.), Ope a ions esea ch e ah en
(Vol. III, pp. 55–68). Hain.
Chowdhu y, S. M. (2017). The all-pay auc ion wi h nonmono onic payo . Sou he n Economic Jou nal,24(2), 375–390. [C ossRe ]
Da idson, C., & Denecke e, R. (1986). Long- un compe i ion in capaci y, sho - un compe i ion in p ice, & he cou no model. RAND
Jou nal o Economics,17(3), 404–415.
De F ancesco, M., & Sal ado i, N. (2010). Be and-Edgewo h compe i ion in an almos symme ic oligopoly. MPRA Pape No. 24228.
Munich Pe sonal RePEc A chi e.
de F u os, M. A., & Fab a, N. (2011). Endogenous capaci ies and p ice compe i ion: The ole o demand unce ain y. In e na ional
Jou nal o Indus ial O ganiza ion,29, 399–411. [C ossRe ]

Games 2025,16, 26 37 o 37
Denecke e, R., & Ko enock, D. (1992). P ice leade ship. Re iew o Economic S udies,59(1), 143–162. [C ossRe ]
Denecke e, R., & Ko enock, D. (1996). Be and-Edgewo h duopoly wi h uni cos asymme y. Economic Theo y,8(1), 1–25. [C ossRe ]
Dixon, H. D. (1987). App oxima e be and equilib ia in a eplica ed indus y. Re iew o Economic S udies,54(1), 47–62. [C ossRe ]
Dixon, H. D. (1992). The compe i i e ou come as he equilib ium in an Edgewo hian p ice-quan i y model. The Economic Jou nal,
102(411), 301–309. [C ossRe ]
Edgewo h, F. (1925). The pu e heo y o monopoly. Pape s ela ing o poli ical economy,1, 111–142.
Gelman, J., & Salop, S. (1983). Judo economics: Capaci y limi a ion and coupon compe i ion. Bell Jou nal o Economics,14(2), 315–325.
Hi a a, D. (2009). Asymme ic Be and-Edgewo h oligopoly and me ge s. The B.E. Jou nal o Theo e ical Economics,9(1), 25. [C ossRe ]
Hoe nig, S. H. (2007). Be and games and sha ing ules. Economic Theo y,31(3), 573–585. [C ossRe ]
K eps, D., & Scheinkman, J. (1983). Quan i y p ecommi men and be and compe i ion yield cou no ou comes. Bell Jou nal o
Economics,14(2), 326–337. [C ossRe ]
K ishna, K. (1989). T ade es ic ions as acili a ing p ac ices. Jou nal o In e na ional Economics,26(3–4), 251–270. [C ossRe ]
Lepo e, J. J. (2008). Cou no and Be and-Edgewo h compe i ion when Ri als’ cos s a e unknown. Economic Le e s,101(3), 237–240.
[C ossRe ]
Lepo e, J. J. (2009). Consume Ra ioning and he Cou no Ou come. The B.E. Jou nal o Theo e ical Economics,9(1). [C ossRe ]
Lepo e, J. J. (2012). Cou no ou comes unde Be and-Edgewo h compe i ion wi h demand unce ain y. Jou nal o Ma hema ical
Economics,48(3), 177–186. [C ossRe ]
Le i an, R., & Shubik, M. (1972). P ice duopoly and capaci y cons ain s. In e na ional Economic Re iew,13, 111–122. [C ossRe ]
Osbo ne, M., & Pi chik, C. (1986). P ice compe i ion in a capaci y-cons ained duopoly. Jou nal o Economic Theo y,38(2), 238–260.
Reny, P. J. (1999). On he exis ence o pu e and mixed s a egy Nash equilib ia in discon inuous games. Econome ica,67, 1029–1056.
[C ossRe ]
Reynolds, S., & Wilson, B. (2000). Be and-Edgewo h compe i ion, demand unce ain y, & asymme ic ou comes. Jou nal o Economic
Theo y,92(1), 122–141.
Shapley, L. S. (1957). A duopoly model wi h p ice compe i ion, (abs ac ). Econome ica,25, 354–355.
Shapley, L. S., & Shubik, M. (1969). P ice s a egy oligopoly wi h p oduc a ia ion. Kyklos,1, 30–43. [C ossRe ]
Shubik, M. (1959). S a egy and ma ke s uc u e; compe i ion, oligopoly, & he heo y o games. Wiley.
Siegel, R. (2009). All-pay con es s. Econome ica,77, 71–92.
Siegel, R. (2010). Asymme ic con es s wi h condi ional in es men s. Ame ican Economic Re iew,100, 2230–2260. [C ossRe ]
Siegel, R. (2014). Asymme ic all-pay auc ions wi h in e dependen alua ions. Jou nal o Economic Theo y,153, 684–702. [C ossRe ]
Simon, L. K., & Zame, W. R. (1990). Discon inuous games and endogenous sha ing ules. Econome ica,58, 861–872. [C ossRe ]
Vi es, X. (1986 ). Ra ioning ules and Be and-Edgewo h equilib ia in la ge ma ke s. Economics Le e s,21, 113–116. [C ossRe ]
Vi es, X. (1993). Edgewo h and mode n oligopoly heo y. Eu opean Economic Re iew,37, 463–476. [C ossRe ]
Yoshida, Y. (2002). Be and-Edgewo h duopoly wi h quad a ic cos unc ion. Mimeo Seikei Uni e si y.
Yoshida , Y. (2006). Be and-Edgewo h p ice compe i ion wi h s ic ly con ex cos unc ions. Mimeo Seikei Uni e si y.
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