Nakao, Keisuke
A icle
Simul aneous ba les and sequen ial ba les in ba gaining
models o wa
Peace Economics, Peace Science and Public Policy (PEPS)
P o ided in Coope a ion wi h:
De G uy e B ill
Sugges ed Ci a ion: Nakao, Keisuke (2025) : Simul aneous ba les and sequen ial ba les in
ba gaining models o wa , Peace Economics, Peace Science and Public Policy (PEPS), ISSN
1554-8597, De G uy e , Be lin, Vol. 31, Iss. 1, pp. 1-20,
h ps://doi.o g/10.1515/peps-2024-0033
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h ps://hdl.handle.ne /10419/333336
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Keisuke Nakao*
Simul aneous Ba les and Sequen ial Ba les
in Ba gaining Models o Wa
h ps://doi.o g/10.1515/peps-2024-0033
Recei ed July 30, 2024; accep ed Oc obe 18, 2024; published online No embe 18, 2024
Abs ac : A wa consis s o mul iple ba les, ye he heo e ical li e a u e o
In e na ional Rela ions has gi en li le a en ion o how ba les ela e wi hin a wa .
By in eg a ing wo ul ima um games wi h p i a e in o ma ion played by an
Agg esso and a De ende , we de elop ba gaining models o wa wi h wo s uc u es:
(i) pa allel wa , whe e ba les occu simul aneously in wo domains, as he Agg esso
can access bo h di ec ly; and (ii) se ies wa , whe e a ba le in one domain (e.g., sea)
p ecedes a ba le in ano he (e.g., land), as he Agg esso mus fi s con ol he
o me domain o ins iga e conflic in he la e . In a heo e ical compa ison be ween
pa allel and se ies wa s, we demons a e ha al hough se ies wa imposes s uc u al
disad an ages on he Agg esso , se ies wa is mo e likely o b eak ou han pa allel
wa unde b oad ci cums ances. I p ewa ba gaining o se ies wa ails, he
Agg esso may in e ha u u e ba gaining is also likely o ail, leading him o ake a
g ea e isk o wa when issuing an ul ima um. Such dynamics a e absen in pa allel
wa . We also discuss u he de elopmen s in heo ies o wa (182 wo ds).
Keywo ds: ba gaining model; likelihood o wa ; ul ima um game
JEL Classifica ion: D74; F51; F52
1 In oduc ion
Fo mal heo is s in he mains eam o In e na ional Rela ions ha e po ayed wa as
a dynamic o wo bellige en s figh ing leng hy ba les. This app oach o modeling wa
is ound in ba gaining models (Fea on 2004, 2007; Le en oğlu and Slan che 2007;
Powell 2004a, 2004b, 2012; Slan che 2003a; Wagne 2000; Wol o d, Rei e , and
Ca ubba 2011) as well as a i ion models (Langlois and Langlois 2009, 2012; Nakao
2022) and andom-walk models (Fey and Ramsay 2011; Slan che 2003b; Smi h 1998;
I hank Hi oshi Uno, Yasu omo Mu asawa, and wo anonymous e iewe s o aluable commen s.
*Co esponding au ho : Keisuke Nakao, College o Business and Economics, Uni e si y o Hawaii a
Hilo, 200 W. Kawili S ., Hilo, HI 96720, USA, E-mail: [email p o ec ed]. h ps://o cid.o g/0000-0001-
9109-2542
Peace Econ. Peace Sci. Pub. Pol. 2025; 31(1): 1–20
Open Access. © 2024 he au ho (s), published by De G uy e . This wo k is licensed unde he
C ea i e Commons A ibu ion 4.0 In e na ional License.
Smi h and S am 2003, 2004). These models ypically p esume ha as a wa p oceeds, a
se ies o ba les e ol e along he dimension o ime.
In con as , heo is s in ano he s eam –pe haps less influen ial on In e na-
ional Rela ions bu mo e aligned wi h Economics –ha e ea ed wa as a clash o
mili a y o ces on mul iple ba lefields. This app oach has been adop ed mainly
by Blo o models o commande s deploying oops ac oss ba lefields (Bo el 1953;
Golman and Page 2009; Robe son 2006) and hei siblings in he li e a u e on coun e -
e o ism (Bie , Oli e os, and Samuelson 2007; Powell 2007a, 2007b, 2009). While he
heo ies in he mains eam (hence o h, dynamic models) ha e ocused on he ime
dimension, hose in he la e s eam (spa ial models) ha e placed mo e emphasis on
he geog aphic dimension o wa .
1
In eali y, wa s encompass bo h he ime and geog aphic dimensions –a wa can
las o mon hs o yea s and be waged ac oss mul iple ba lefields, sugges ing
ha exis ing models p esen only incomple e pic u es o wa . Howe e , i wa is
modeled along bo h he dimensions, he me i s o pa simony would be se iously
unde mined. In his a icle, we a emp a heo e ical compa ison be ween he
dynamic and spa ial models in he con ex o ba gaining. Mo e conc e ely, by
combining wo ul ima um games wi h p i a e in o ma ion in wo s uc u ally
con as ing ways, we de elop and compa e wo models o wa ha co espond o
he wo modeling app oaches men ioned abo e.
One o ou models depic s wha we label as “pa allel wa ,”whe e upon a
p ewa ba gaining ailu e, ba les a e ough simul aneously in wo domains (land
and sea).
2
This model p esumes ha because an Agg esso and a De ende a e
geog aphically con iguous (e.g., F ance s. Ge many), he Agg esso can di ec ly
access bo h he domains.
3
The o he model is o “se ies wa ,”whe e a ba le in one
domain p ecedes a ba le in he o he domain.
4
The la e model pos ula es ha
because he Agg esso and he De ende a e geog aphically dis an (e.g., he U.S. s.
Japan), he Agg esso mus win one domain (sea) in o de o p o oke a conflic in
he o he (land). To make a compa ison o pa allel and se ies wa s possible, we
ensu e he models sha e he same pa ame e alues and diffe only in he sequence
o ba les.
1Alongside ecen de elopmen s in Blo o models, he e has also been some dynamic models whe e
playe s alloca e esou ces ac oss a sequence o figh s (Rino , Sca sini, and Yu 2012; Sela and E ez
2013).
2The land and sea a e me e me apho s o he wo domains (o ba lefields). They can be eplaced
wi h he cybe and physical spaces in some ins ances, o con en ional and nuclea hea e s in o he s.
3Th oughou he a icle, we assign he masculine p onouns o he Agg esso and he eminine
p onouns o he De ende .
4Pa allel wa and se ies wa a e named a e he co esponding ci cui s in an elec ical ne wo k.
2K. Nakao
The compa ison p oduced an unexpec ed esul . A fi s glance, pa allel wa
looks mo e likely han se ies wa , because he Agg esso ’s en y in o he land ba le
o se ies wa is condi ional on his ic o y a sea. This condi ionali y can impede he
Agg esso ’s in asion o he land. Howe e , ou heo e ical analysis sugges s ha
se ies wa is mo e likely o b eak ou han pa allel wa unde a wide ange o
ci cums ances. In he p ewa ba gaining o se ies wa , he Agg esso ’s ul ima um
ma e s no only o he ou b eak o wa bu also o he condi ion o es o e peace. I
a gene ous ul ima um is p esen ed by he Agg esso bu is ejec ed by he De ende ,
he Agg esso would in e ha he De ende is so esol ed ha his u u e offe o end
he wa is also likely o be ejec ed. This in e ence induces he Agg esso o place a
oughe ul ima um a a g ea e isk o wa . In con as , pa allel wa lacks such
dynamic incen i es.
The es o he a icle p oceeds as ollows. A e configu ing he common se up o
he wo models, we p esen and analyze he model o pa allel wa fi s and he model
o se ies wa la e . Subsequen ly, he wo models a e compa ed. The las sec ion
discusses u he de elopmen s in heo ies o wa . The Appendix ou lines he
equilib ium condi ions in bo h he models.
2 Common Se up
To add ess how he ou come o ba gaining and he likelihood o wa a e influenced
by he s uc u al ela ions ac oss ba les in a wa , we de elop wo game- heo e ic
models o wa –one is o pa allel wa , and he o he o se ies wa . To make a
compa ison be ween he wo o ms o wa possible, hese models mus be iden ical
excep o he s uc u al ela ions. Hence, hey a e assumed o sha e he ollowing
common se up.
The e a e an Agg esso (A) and a De ende (D) in conflic (i∈A,D
{}
). They ha e
in e es s bL> 0 and bS> 0 a s ake in wo domains/ba lefields –land and sea
(d∈L,S
{}
), espec i ely. Upon a wa ’s b eakou , he land ba le is won by iwi h
p obabili y pL
i>0, and he sea ba le won by iwi h p obabili y pS
i>0 such ha
pd
A+pd
D=1 o each d∈L,S
{}
. The ba le ou comes ac oss domains a e independen
om each o he .
These ba les inflic cos s on he bellige en s as hey figh . In figh ing a ba le in
domain d,Aincu s cos cd
A.A’s expec ed payoff om figh ing in dcan be defined as:
πd
A≡pd
Abd−cd
A>0 o d∈L,S
{}
.
In con as , D’s cos s o figh ing ba les depend on he ype and a e unknown o
A. Pu mo e p ecisely, when p ewa ba gaining begins, Ais unce ain abou D’s
esol e (“ ype”)λ, bu Aonly knows ha λ ollows he uni o m dis ibu ion on 0,Λ
[]
Simul aneous Ba les and Sequen ial Ba les 3
(λ∼U0,Λ
[]
). D’s cos s o figh ing a e de e mined by λ:cd
D|λ≡λkd, whe e kd> 0 can be
in e p e ed as D’s ma e ial cos o figh ing in d∈S,L
{}
.D’s payoffs om figh ing
in d∈L,S
{}
will be shown as: πd
D|λ≡pd
Dbd−λkd. Thus, Dwi h a lowe λis mo e
esol ed o figh ba les. To ule ou equilib ia whe e no wa b eaks ou , he
ollowing es ic ions a e imposed on he dis ibu ion o λ, so ha he Agg esso is
willing o un he isk o wa :
5
Assump ion 1: The Agg esso is so unce ain o e he De ende ’s ype λ ha :
6
(i)Λ>cL
A+cS
A
kL+kS(1)
(ii)Λ>
2kS+pS
A
2kL
[]
cL
A
kL+cS
A
kS+pS
A
2kL.(2)
Bo h he models assume he simples possible ba gaining p o ocol a e e y s age –
he Agg esso makes an ul ima um, o which he De ende esponds ei he by
accep ing i in peace o by ejec ing i h ough figh ing. Whene e peace and figh ing
a e payoff-equi alen , he De ende always chooses peace.
3 Pa allel Wa
3.1 Ba gaining Model
The game o pa allel wa begins wi h Na u e choosing D’s ype λ. Wi hou knowing
he ue alue o λ,Aplaces an ul ima um θLS ∈0,bLS
[]
o D, whe e bLS ≡bL+bS.D’s
esponse o θLS is deno ed as σLS
λ.I Daccep s θLS, he game ends wi h payoffs
bLS −θLS,θLS
()
.I D ejec s θLS, he sea and land ba les a e ough simul aneously
be ween Aand D. The expec ed payoffs om figh ing he wa can be shown as
πS
A+πL
A,πS
D|λ+πL
D|λ
()
. The ex ensi e o m o pa allel wa appea s in Figu e 1.
5While he nex wo sec ions ocus on equilib ia whe e he p obabili y o wa is posi i e, o he
equilib ia and hei condi ions a e discussed in he Appendix.
6Ano he way o in e p e Assump ion 1 is ha he cos s cL
Aand cS
Aa e so small ha he Agg esso is
willing o figh ba les wi h posi i e p obabili ies.
4K. Nakao
3.2 Equilib ium
The equilib ium can be de i ed backwa d by finding he sequen ially- a ional
s a egy a e e y in o ma ion se . Any ype λo Daccep s he ul ima um θLS i and
only i i is la ge han o equal o he expec ed payoff om figh ing he wo ba les
(θLS ≥πL
D|λ+πS
D|λ). Le λ
LS θLS
()
be he h eshold o λ, wi h which Dis indiffe en
be ween accep ing θLS and figh ing. Fo λ=λ
LS θLS
()
,
θLS =πL
D|λ+πS
D|λ
=pL
DbL+pS
DbS−kL+kS
[]
λ
LS θLS
()
.
An icipa ing D’s esponse abo e, Aseeks he balance be ween a comp omise in peace
and he isk o wa so as o maximize his con inua ion payoffby choosing θLS:
ΠLS
AθLS
()
≡1−P Ba LS
()[]
bLS −θLS
[]
+P Ba LS
()
πL
A+πS
A
[]
,
whe e P Ba LS
()
is he p obabili y ha pa allel wa b eaks ou :
P Ba LS
()
≡P (λ<λ
LS θLS
())
=
λ
LS θLS
()
Λ.
The payoff-maximizing θLS can be de i ed om he fi s -o de condi ion o ΠLS
AθLS
()
.
The second-o de condi ion is gua an eed by he non-dec easing haza d a e o he
uni o m dis ibu ion o λ(c . Fudenbe g and Ti ole 1991: 267).
The equilib ium can be summa ized as ollows:
P oposi ion 1: In he ba gaining model o pa allel wa , he e is a unique pe ec
Bayesian equilib ium θLS*,σLS*
λ
()
such ha
θLS*=pL
DbL+pS
DbS−kL+kS
[]
Λ−cL
A+cS
A
[]
2
σLS*
λ=
accep o λ≥λ
LS θLS
()
igh o λ<λ
LS θLS
()
,
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
Figu e 1: Ex ensi e o m o pa allel wa .
Simul aneous Ba les and Sequen ial Ba les 5
whe e λ
LS θLS
()
is he h eshold o λ ha de e mines whe he D accep s θLS o figh s:
λ
LS θLS
()
=pL
DbL+pS
DbS−θLS
kL+kS.
Mo eo e , because he h eshold o λin he equilib ium is:
λ
LS θLS*
()
=pL
DbL+pS
DbS−θLS*
kL+kS
[]
Λ
=
pL
DbL+pS
DbS−pL
DbL+pS
DbS−kL+kS
[]
Λ−cL
A+cS
A
[]
2
()
kL+kS
[]
Λ
=1
2Λ−cL
A+cS
A
kL+kS
[]
,(3)
The p obabili y ha pa allel wa b eaks ou can be shown as:
P Ba LS*
()
=
λ
LS θLS*
()
Λ
=1
21−cL
A+cS
A
kL+kS
[]
Λ
⎡
⎢
⎣⎤⎥⎦,(4)
which is posi i e by Inequali y (1).
4 Se ies Wa
4.1 Ba gaining Model
In con as o he simul aneous land and sea ba les in pa allel wa , he sea ba le
p ecedes he land ba le in se ies wa . Tha means, he Agg esso mus fi s con ol
he sea o in ade and occupy he land a s ake.
The game o se ies wa also begins wi h Na u e choosing D’s ype λ.A hen issues
an ul ima um θS∈0,bLS
[]
o D.D’s esponse o θSis deno ed as σS
λ.I Daccep s θS, he
game ends wi h payoffsbLS −θS,θS
()
.I D ejec s θS, he sea ba le is ough . Based on
D’s decision, Aupda es i s belie abou λ.I Dwins he sea ba le, Dsecu es i s
in e es s bo h in he sea and in he land, whe eas Ano only ails in he sea bu also
abandons i s in asion o he land, so ha he game ends wi h payoffs−cS
A,bLS −cS
D|λ
()
.
I Awins he sea ba le, Agains bSand u he demands θL∈0,bL
[]
, o which
6K. Nakao
D esponds wi h σL
λ.I Daccep s θL, he game ends wi h payoffs
bLS −cS
A−θL,−cS
D|λ+θL
()
.I D ejec s θL, he land ba le is ough , esul ing in payoffs
bLS −cS
A+πL
A,−cS
D|λ+πL
D|λ
()
. The ex ensi e o m o se ies wa is shown in Figu e 2.
4.2 Equilib ium
The equilib ium o he game o se ies wa can also be de i ed backwa d – om
ba gaining θL,σL
λ
()
o ba gaining θS,σS
λ
()
.
Ba gaining o e he Land: The second s age (i.e., ba gaining igh be o e he
land ba le) esembles he game o pa allel wa bu diffe s om i in wo old: (a) only
he land ba le is ough ; and (b) a ac ion o λ∈0,Λ
[]
is sc eened ou in he fi s s age
(i.e., ba gaining be o e he sea ba le). Suppose ha hose ypes o Dwi h λ≥λ
SθS
()
accep θSin he fi s s age, and only hose wi h λ<λ
SθS
()
en e he second s age. Any
ype o Daccep s θLin he second s age i and only i θLis no less han he expec ed
payoff om figh ing he land ba le (θL≥πL
D|λ). The h eshold λ
LθL
()
o λ, which
de e mines whe he Daccep s θLo figh s, hen sa isfies ha :
θL=pL
DbL−kLλ
LθL
()
.
In esponse o σL
λwi h λ
LθL
()
,Achooses θL o maximize his con inua ion payoff om
figh ing he land ba le:
ΠL
AθL
()
≡1−P Ba L
()[]
bL−θL
[]
+P Ba L
()
πL
A
[]
,
whe e P Ba L
()
is he p obabili y o he land ba le condi ional on λ<λ
SθS
()
:
P Ba L
()
≡P (λ<λ
LθL
()
|λ<λ
SθS
())
=
λ
LθL
()
λ
SθS
()
.
Figu e 2: Ex ensi e o m o se ies wa .
Simul aneous Ba les and Sequen ial Ba les 7
A’s op imal θLcan be ob ained om he fi s -o de condi ion o ΠL
AθL
()
. No e ha
because he p obabili y o he land ba le depends on λ
SθS
()
, he op imal θLis a
unc ion o θS.
Lemma 1: In he second s age o se ies wa ollowing θS,λ
SθS
()()
, he e is a unique
pe ec Bayesian equilib ium o he θL*θS
()
,σL*
λ
()
,which sa isfies ha :
θL*θS
()
=pL
DbL−kLλ
SθS
()
−cL
A
2(5)
σL*
λ=accep o λ≥λ
LθL
()
igh o λ<λ
LθL
()
,
⎧
⎪
⎨
⎪
⎩(6)
whe e λ
LθL
()
is he h eshold o λ o D o accep θLo o figh :
λ
LθL
()
=pL
DbL−θL
kL.(7)
Mo eo e , he playe s’con inua ion payoffs om he equilib ium o he second s age
a e:
ΠL*
AθS
()
=πL
A+
kLλ
SθS
()
+cL
A
[]
2
4kLλ
SθS
() (8)
ΠL*
D|λθS
()
=
θL*θS
() o λ≥λ
LθL*θS
()()
πL
D|λ o λ<λ
LθL*θS
()()
,
⎧
⎪
⎨
⎪
⎩(9)
o which he h eshold o λin he equilib ium λ
LθL*θS
()()
is:
λ
LθL*θS
()()
=pL
DbL−θL*θS
()
kL
=
pL
DbL−(pL
DbL−kLλ
SθS
()
−cL
A
2)
kL
=1
2[λ
SθS
()
−cL
A
kL],(10)
8K. Nakao
To summa ize, he s uc u al ela ions o ba les in a wa could be complex,
unp edic able, and endogenous. These elemen s o wa a e po en ials o –as well as
obs acles o –inno a ing a new heo y o a med conflic .
Appendix
By d opping Assump ion 1, he Appendix explo es equilib ia and hei condi ions
wi h pa ame e s aking a b oade ange o alues.
A Pa allel Wa
In he model o pa allel wa , he condi ions o equilib ia a e a he i ial. I
Λ>cL
A+cS
A
kL+kS(Assump ion 1-(i)), he equilib ium akes an in e io solu ion o A’smaxi-
miza ion o ΠLS
AθLS
()
–a ac iono ypeso Dfigh , while o he s do no . In his in e io
equilib ium, θLS*=pL
DbL+pS
DbS−kL+kS
[]
Λ−cL
A+cS
A
[]
2,λ
LS θLS*
()
=1
2Λ−cL
A+cS
A
kL+kS
[]
>0, and
P Ba LS*
()
=1
21−cL
A+cS
A
kL+kS
[]
Λ
[]
>0 (P oposi ion 1). I Λ≤cL
A+cS
A
kL+kS, he equilib ium is placed
a he co ne –all he ypes o Daccep θLS*, and no wa akes place. In he co ne
equilib ium, θLS*=pL
DbL+pS
DbS,λ
LS θLS*
()
=0, and P Ba LS*
()
=0.
B Se ies Wa
In he model o se ies wa , he o m o equilib ium depends on whe he he solu ion
o A’s payoffmaximiza ion is in e io o co ne a each o he fi s and second s ages.
B.1 Second S age
A he second s age, a ional s a egies depend on λ
SθS
()
, o he ac ion o ypes o D
en e ing i . I λ
SθS
()
>cL
A
kL, he equilib ium is in e io : θL*θS
()
=pL
DbL−kLλ
SθS
()
−cL
A
2,
λ
LθL*θS
()()
=1
2λ
SθS
()
−cL
A
kL
[]
>0, and P Ba L*
()
=1
2⎡
⎣1−cL
A
kLλ
SθS
()
⎤⎦>0. I λ
SθS
()
≤cL
A
kL,
he equilib ium appea s a he co ne , so ha no land ba le occu s: θL*θS
()
=pL
DbL,
λ
LθL*θS
()()
=0, and P Ba L*
()
=0.
Simul aneous Ba les and Sequen ial Ba les 15
B.2 Fi s S age
A he fi s s age, A’s objec i e unc ion depends on whe he he second-s age equi-
lib ium is in e io o co ne :
ΠS
AθS
()
≡1−
λ
SθS
()
Λ
⎡
⎢
⎣⎤⎥⎦bLS −θS
[]
+
λ
SθS
()
ΛπS
A+pS
AΠL
AθL*θS
()()[]
,
o which
λ
SθS
()
=
pS
DbS+pS
ApL
D+pS
D
[]
bL+pS
A
cL
A
2−θS
kS+pS
A
kL
2
i λ
SθS
()
>cL
A
kL
pS
DbS+pS
D+pS
ApL
D
[]
bL−θS
kSi λ
SθS
()
≤cL
A
kL
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
ΠL
AθL*θS
()()
=
πL
A+[cL
A+kLλ
SθS
()]
2
4kLλ
SθS
() i λ
SθS
()
≤cL
A
kL
pL
AbLi λ
SθS
()
≤cL
A
kL.
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Acco dingly, he condi ions o fi s -s age equilib ia a y wi h he ela i e size
be ween λ
SθS
()
and cL
A
kL.
B.2.1 Fo λ
SθS
()
>cL
A
kL
I he second-s age equilib ium is in e io (λ
LθL*θS
()()
>0byλ
SθS
()
>cL
A
kL), he fi s -
s age equilib ium mus also be in e io (λ
SθS*
()
>0). By P oposi ion 2, i is:
θS*=pS
DbS+pS
ApL
D+pS
D
[]
bL+pS
A
cL
A
2−kS+pS
A
kL
2
[]
ΛkS+pS
A
kL
2
[]
−cS
A
2kS+pS
AkL
2
⎡
⎢
⎣⎤⎥⎦
λ
SθS*
()
=1
2Λ+
ΛpS
A
kL
4
[]
−cS
A
kS+pS
AkL
4
⎡
⎢
⎣⎤⎥⎦.
16 K. Nakao
The condi ion ha λ
SθS*
()
>cL
A
kLis ansla ed as Λ>
2kS+pS
A
2kL
[]
cL
A
kL+cS
A
kS+pS
A
2kL(Assump ion
1-(ii)).
B.2.2 Fo λ
SθS
()
≤cL
A
kL
I he second-s age equilib ium appea s a he co ne (λ
LθL*θS
()()
=0by
(λ
SθS
()
≤cL
A
kL), he fi s -s age equilib ium can be ei he in e io (λ
SθS*
()
∈0,cL
A
kL
())
o
a one o he wo co ne s (λ
SθS*
()
=0,cL
A
kL). The condi ion o he second-s age co ne
equilib ium ha λ
SθS*
()
≤cL
A
kLis equi alen o Λ≤
2kS+pS
A
2kL
[]
cL
A
kL+cS
A
kS+pS
A
2kL. In addi ion, i he fi s -
s age equilib ium is in e io , θS*=pS
DbS+pS
ApL
D+pS
D
[]
bL−1
2kSΛ−cS
A
[]
, and
λ
SθS*
()
=1
2Λ−cS
A
kS
[]
∈0,cL
A
kL
()
, which holds i cS
A
kS<Λ<2cL
A
kL+cS
A
kS. I i is a he lowe
co ne , θS*=pS
DbS+pS
ApL
D+pS
D
[]
bL, and λ
SθS*
()
=0, which holds i Λ≤cS
A
kS. I i is a he
uppe co ne , θS*=pS
DbS+pS
ApL
D+pS
D
[]
bL−kS
kLcL
A, and λ
SθS*
()
=cL
A
kL, which holds i
Λ≥2cL
A
kL+cS
A
kS.
B.3 Summa y o Equilib ium Condi ions
To ecap, he second-s age equilib ium depends on he ela i e size be ween λ
SθS*
()
and cL
A
kL.I λ
S
θS*
()
>cL
A
kL, he second-s age equilib ium is in e io (λ
LθL*θS*
()()
>0),
and he fi s -s age equilib ium mus be in e io (λ
SθS*
()
>0). I λ
SθS*
()
≤cL
A
kL,
he second-s age equilib ium is co ne (λ
LθL*θS*
()()
=0), and he fi s -s age
equilib ium depends on whe he λ
SθS*
()
is mo e han 0 and is less han cL
A
kL.
Fo g aphical illus a ion, Figu e A shows A’s objec i e unc ion a he fi s s age
ΠS
AθS
()
when: (a) Λis so la ge (Λ= 100) ha he second-s age equilib ium is in e io
(λ
LθL*θS*
()()
>0, λ
SθS*
()
>cL
A
kL); and (b) Λis small enough (Λ= 20) ha he second-s age
equilib ium is co ne (λ
LθL*θS*
()()
=0, λ
SθS*
()
≤cL
A
kL), wi h he ollowing pa ame e
alues: bL= 100, bS= 120, cL
A=10, cS
A=12, kL=1,kS=1,pL
A=0.5, pS
A=0.4. I can be
confi med ha o (a), λ
SθS*
()
=1
2Λ+ΛpS
A
kL
4
[]
−cS
A
kS+pS
A
kL
4
[]
=49.0909 …>cL
A
kL=10, and o (b),
λ
SθS*
()
=1
2Λ−cS
A
kS
[]
=4<cL
A
kL=10.
Simul aneous Ba les and Sequen ial Ba les 17
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Figu e A: A’s objec i e unc ion ΠS
AθS
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. (a) The equilib ium is in e io a bo h he fi s and second
s ages. (b) The equilib ium is in e io a he fi s s age and co ne a he second s age.
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