Richelle, Y es; Ga ella, Paolo
Wo king Pape
Exi , Sunk Cos s and he Selec ion o Fi ms
Quade ni - Wo king Pape DSE, No. 214
P o ided in Coope a ion wi h:
Uni e si y o Bologna, Depa men o Economics
Sugges ed Ci a ion: Richelle, Y es; Ga ella, Paolo (1995) : Exi , Sunk Cos s and he Selec ion o
Fi ms, Quade ni - Wo king Pape DSE, No. 214, Alma Ma e S udio um - Uni e si à di Bologna,
Dipa imen o di Scienze Economiche (DSE), Bologna,
h ps://doi.o g/10.6092/unibo/amsac a/5112
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/159057
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Exi , Sunk Cos s and he Selec ion o Fi ms
Y es Richel le
Uni e si yo La al (Queb ec)
and
Paolo G. Ga el la
Uni e si a di Bologna
ma ch 1995
JEL Classica ion : L13, L41
Abs ac
The pape analyzes he ques ion o whic h cos cha ac e is ics a e exhibi ed
by he  ms ha exi an oligop olis ic ma ke when cos s a e asymme ic and
 ms can c edibly b e o ced ou by he emaining comp e i o s. The main e-
sul s a e: (i) i een y is imp ossible (due o he p esence o la ge sunk cos s),
hen he  m wi h he highes ma ginal cos unc ion d ays in; i een y is
cos less hen he  m wi h he highes a e age cos exi s. Consequen y sunk
cos s no only aec he numb e o  ms in an indus y, bu hey also en e
he de e mina ion o he yp e o  ms ha esis p eda ion.
Keywo ds: endogenous coali ion o ma ion, exi , sunk cos s.
1
1 In o duc ion
In ma k e s whe e  ms die as o hei cos unc ions is i p ossible o p edic
wha a e he cos cha ac e is ics o he  ms ha s ay o ha exi ? In p e -
ec ly comp e i i e ma ke s one can p edic ha he  ms exi ing he ma ke
a e hose wi h highes a e age cos s. This p edic ion has b een ex ended by
Ghema wa and Nalebu (1985), (1990) and Fudenbe g and Ti ole (1986) o
he case o declining indus ies wi h ew comp e i o s. Thei analyses show
indeed ha , in a wa o a i ion, he less ecien  m will b e he  s o
exi
1
. Howe e many ecen wo ks (see chap e s 8 and 9 in Ti ole (1988)
and Wilson (1992)) show ha exi can o ccu in a wide a ie y o ci cum-
s ances. We a e he e o e led o ask i he abo
e p edic ion con inue o hold
in imp e ec ly comp e i i e ma k e s whe e  ms a e no engaged in a wa o
a i ion.
This pape a gues ha he exi ing  m may be he one wi h he
lowes
a e age cos unc ion. To iden i y he basic a gumen leading o his conclu-
sion, conside he ollowing example. Th ee  ms decide, a a  s s age, o
s ay in he ma k e o o exi and, a a second s age, hose ha s a
y decide
howmuch o p o duce. All  ms ha e iden ical xed cos s. They also ha e
cons an ma ginal cos s wi h  m
i
's ma ginal cos b eing s ic ly smalle
han  m
j
's ma ginal cos whic h, in u n, is s ic ly smalle han  m
k
's.
Fi ms can he e o e b e anked acco ding o hei a
e age cos unc ion wi h
 m
i
ha ing he lowes one.
Then suppose ha i all  ms s a
y in he ma k e , each o hem will ob ain
a s ic ly nega i e p o a he Cou no equilib ium while, i only wo  ms
s ay in, hei Cou no p o is p osi i e and he hi d  m ecei es a ze o
p o . I immedia ely ollows ha o each couple o  ms o s ay in he
ma ke and o p o duce hei Cou no qua i y is an equilib ium o he wo
s age game. They a e he e o e h ee equilib ia and a p edic ion on he cos
cha ac e is ics o he exi ing  m canno b e based only on his w
o s age
game. No e, inciden ally, ha hese equilib ia a e all Pa e o ecien so ha
one canno use coali ional p o o ness (see Be nheim, Peleg and Whins on
1
Fo ins ance, Ghema wa and Nalebu (1985) show ha in a wa o a i ion wi h
comple e in o ma ion whe e  ms die acco ding o hei p o duc ion capaci y, he bigges
 m is he  s o exi . Bu hese au ho s assume ha  ms incu only a ow main enance
cos which is p op o ional o hei capaci y. Acco dingly, he bigges  m is he one wi h
he highes a e age cos unc ion.
2
(1987)) o selec one o he o he .
A p ossible ou e o ollo w o ob aining a p edic ion is indica ed by
he li e a u e on endogenous coali ion o ma ion, as in Aumann and My-
e son(1988), Gul (1989), and esp ecially Blo c h (1990a) and (1990b). These
wo ks use a non-coope a i e sequen ial game o analyze he o ma ion o
coali ion s uc u es. In he same way, one can assume ha a coali ion o ma-
ion game p ecedes he play o a game o he kind illus a ed by he example
abo e. We he e adop a sp ecica ion o he coali ion o ma ion game whe e
each  m in u n makes a decla a ion consis ing o (i) a se o  ms ha s ay
in, (ii) a payo ec o o he h ee  ms ha can b e ob ained by he play
o a non-coop e a i e equilib ium o he wo s age game. One can in e p e
hese decla a ions as oe s", and we mo del he accep ance ( e usal) o an
oe as he making o an iden ical (die en ) decla a ion. Since a decla a ion
co esp onds o one equilib ium, i wo  ms make he same decla a ion hey
ag ee o play he same equilib ium. This de e mines whic h equilib ium is
played and he payos o all  ms i esp ec i e o he decla a ion made by
he hi d  m.
I is imp o an o ealize ha once wo  ms ha
e adop ed hei equilib-
ium s a egies he hi d  m has no b e e al e na i e han he play o i s
own b es eply o hose s a egies, whic h coincides wi h he s a egy sp ecied
in he equilib ium chosen by he o he wo  ms. The e o e he wa
ypayos
a e de e mined has no hing o do wi h he applica ion o a ma jo i
y ule in
collec i e decision making.
Fo u he e e ence we call his sequen ial game he ca el o ma ion
game". The equilib ium o his game gi es a p edic ion o he  m ha exi s.
Fo each o de in whic h  ms decla e, he e will b e a unique subgame p e ec
equilib ium ou come in he ca el o ma ion game. Bu , as one can exp ec ,
he equilib ium ou come will in gene al dep end on he o de o decla a ion.
We a e ne e heless able o show ha , as long as a  m exi s he ma ke
a he equilib ium, he cos cha ac e is ics o his  m can b e iden ied and
a e indep enden o he o de o decla a ion. The ca el o ma ion game will
he e o e p o ide a s ong p edic ion on he cha ac e is ics o he exi ing
 m.
In he example he unique equilib ium is wi h  ms
j
and
k
making he
same decla a ion o he o m (i)
j; k
g
, and (ii) payo ze o o  m
i
, and
Cou no payos o
j
and
k
. A p oo o his s a emen is i ial. Indeed he
Cou no p o o a  m is inc easing in he ma ginal cos o i s i al which
3
implies ha  m
j
mak es he highes equilib ium p o when
k
s ays on he
ma ke and  m
k
makes he highes equilib ium p o when
j
s ays on.
Hence he  m exi ing he ma ke is he one wi h he
lowes
a e age cos s,
and no wi h he highes , as i would b e p edic ed in a wa o a i ion o in
p e ec comp e i ion.
One is led o wonde i he die ence in p edic ion could disapp ea i
 ms play a supe game ins ead o a one-sho game. Indeed, in a sup e game,
 ms a e gene ally able o maximize join p o s and, since join p o max-
imiza ion equi es he minimiza ion o a iable cos , hey will b e induced o
in e nalize he gain made by ha ing an ecien pa ne .
In wha ollows we shall gene alize he example gi en ab o e by conside -
ing a gene al cos unc ion and a p o duc ion game consis ing o an inni e
ep e i ion o he wo-s age game o he example. In his game, we sa
y ha
a wo- m ca el is easible i he e exis s an equilib ium o he p oduc ion
game whe e hese  ms s ay in and he hi d s ays ou along he equilib-
ium pa h. Ob iously he analysis is in e es ing only i he p o duc ion game
displays a leas wo die en easible ca els.
The main esul s a e ha (i) i een y is imp ossible, hen he  m wi h
he highes ma ginal cos unc ion s ays on a all equilib ia o he ca el o -
ma ion game, o any o de o decla a ion, while xed cos s only de e mine
he se o easible ca els. (ii) I een y is p ossible hen he  m wi h he
highes a e age cos exi s. These esul s imply ha he die ence in p edic-
ion do es no dep end on he p ossibili y o no o collude, bu a he dep ends
up on he exis ence o no o sunk cos s o een y
.
Ano el implica ion o he p esence o sunk cos s app ea s he e: no only,
as i is al eady well known om he li e a u e on en y p eemp ion, hey can
de e mine he numbe o  ms, bu hey also en e he de e mina ion o he
yp e o  ms ha s ay in a ma ke .
The pape is o ganised as ollows: in he nex Sec ion we in o duce ou
assump ions ela i e o he cos and demand unc ions and we analyse he
equilib ium ou comes o he p oduc ion game. In Sec ion 3, he ca el o -
ma ion game is o mally p esen ed. Ou esul s a e s a ed in Sec ion 4 o
he unp o able een y case and in Sec ion 5 o he case o cos less een y.
In Sec ion 6 we es he obus ness o he esul s o he case o unp o able
een y o changes in he ca el o ma ion game. The esul s s a ed in sec ion
4 a e shown o go h ough. Sec ion 7 p esen s some concluding conside a ions
o ela e he esul s o he li e a u e on ansac ion cos economics.
4

2 The p o duc ion game
We conside a sup e game in ol ing h ee  ms. We shall deno e his game
by 0

and he se o  ms by
N
.0

consis s o he inni e ep e i ion o
he wo-s age game whe e (i) a he  s s age each  m decides o s ayin o
o s ay ou o he ma k e and (ii) a he second s age he  ms whic h ha e
decided o s ay in he ma ke , he ea e e e ed o as he ac i e  ms, play
an usual Cou no game whils an inac i e  m p o duces no hing. A each
s age decisions a e made simul aneously and ac ions aken a he  s s age
a e p e ec ly obse ed by all  ms b e o e hey choose hei p o duc ion a he
second s age. The scala

, belonging o he op en in e al (0
;
1), deno es he
discoun ac o common o all  ms.
The pu p ose o his p epa a o y sec ion is w
o old. On he one hand, we
gi e a p ecise con en o he concep o a easible ca el. On he o he hand,
o eac h easible ca el
s
, we c ha ac e ize he se o all pa
yo ec o s ha
 ms can ob ain a a subgame p e ec Nash equilib ium o 0

whe e along
he equilib ium pa h only he  ms in ca el
s
s ay in he ma ke a eac h
p e io d.
To simpli y he exp osi ion we shall p oceed in h ee s eps. Fi s , in sub-
sec ion 2.1., we shall in o duce he assump ions on he cos and demand
unc ions. In he second s ep, in subsec ion 2.2., we shall igno e he  s
s age o he cons i uen game and concen a e on he game 0

(
s
) consis ing
o he inni e ep e i ion o he Cou no game whe e he se o play
e s is gi en
by
s
(i.e.  ms in
s
decide o s ay in he ma k e a e e y p e io d and he  m
ou side
s
,i
s
6
=
N
, decides o s ay ou o he ma k e a e e y pe io d). We
can he eby use he esul s om he li e a u e on inni ely ep ea ed games
o b ing o h a c ha ac e iza ion o he se o equilib ium payo ec o s o
0

(
s
) . In he las s ep, subsec ion 2.3., we in o duce he p ossibili y o each
 m o exi he ma k e . This will allo w us o dene wha we mean by a
easible ca el and cha ac e ize, o each easible ca el, he se o a ainable
payo ec o s
V

(
s
).
2.1 Assump ions
An ac i e  m has o pay a ( ime-in a ian ) xed cos
F
i
as well as a a iable
cos gi en by he unc ion
c
(
q
i
;

i
;

q
i
) whe e

i
and 
q
i
a e ( ime-in a ian )  m-
specic pa ame e s and
q
i
s ands o quan i y ( he ime index ha should b e
5
assigned o he quan i y a iable is omi ed as long as his do es no c ea e
con usion). I a  m decides o s ay ou , i p oduces no hing and incu s no
cos . Fu he mo e i a  m, say
i
, has decided o s ay ou a p e io d
0
1, i
mus pay a een y cos ,
R
i
, i i decides o s ay in he ma k e a p e io d
.
We simpli y he analysis by conside ing in u n wo p ola cases namely:
Assump ion 1
Reen y is unp o able i.e.
R
i
is as la ge as we wan , o
i
=1
;
2
;
3
.
Assump ion 2
Reen y is cos less i.e.
R
i
=0
, o
i
=1
;
2
;
3
.
The a iable cos unc ion
c
dep ends up on wo  m-sp ecic pa ame e s,

i
and 
q
i
. 
q
i
s ands o he  m
i
's capaci y cons ain whic h means ha ,
o a gi en

i
,  m
i
canno p o duce mo e han 
q
i
and acco dingly
c
is only
dened o 0

q
i


q
i
. On he o he hand,

i
is a con enien way o ank
 ms acco ding o hei ma ginal cos unc ion. We shall indeed suppose ha
o any quan i y
q
such ha he ma ginal cos o p o duce his quan
i y is
well dened o  ms
i
and
j
,  m
j
's ma ginal cos is s ic ly g ea e han
he one o  m
i
i and only i

j
>
i
. Mo e p ecisely way, le
X
i
=[0
;

q
i
],
hen ou assump ions ega ding he a iable cos unc ion o any  m a e he
ollowing:
Assump ion 3
Le

q
i
>
0
. The a iable cos unc ion is wicecon inuously
die en iable wi h espec o
q
i
and

i
on
X
i
2
R
++
. In addi ion,
c
sa ises
he ol lowing p ope ies:
1.
c
(0;

i
;

q
i
)=0
,
8

i
2
R
++
,
2.
0

@c
(
q
i
;

i
;

q
i
)
=@ q
i


i
,

i
2
]0
;
1
[
, and
@c
(
q
i
;

i
;

q
i
)
=@ 
i
>
0
,
8
(

i
;q
i
)
2
R
++
2
X
i
,
3.
@
2
c
(
q
i
;
;

q
i
)
=@ q
i
@
i
>
0
,
8
(

i
;q
i
)
2
R
++
2
X
i
.
Now le
Q
s and o agg ega e ou pu . A each p e io d, he in e se demand
unc ion o he homogeneous goo d, deno ed
(
Q
), sa ises:
Assump ion 4
Fo al l
Q
2
[0
;
P
3
i
=1

q
i
]
,
is wicecon inuously die en-
iable wi h
(
Q
)

0
and
@ =@Q <
0
.
6
No e ha we supp ose ha  ms p o duce p e ec subs i u es in o de o
b e able o concen a e ou sel es only up on he inuence o he cos cha ac-
e is ics on ma ke s uc u e.
Fo an ac i e  m he p o unc ion (g oss o he een y cos ) in he
Cou no game will b e w i en as:

i
(
q
i
;Q
0
i
)=
(
q
i
+
Q
0
i
)
q
i
0
c
(
q
i
;

i
;

q
i
)
0
F
i
(1)
whe e
Q
0
i
=
Q
0
q
i
.We shall assume:
Assump ion 5
Fo al l
i
2
N
,

i
is s ic ly quasi-conca e on
X
i
2
[0
;
P
j
6
=
i

q
j
]
.
Fu he mo e he e exis s
(
q
1
;q
2
;q
3
)
2
X
1
2
X
2
2
X
3
such ha , o al l
i
,

i
(
q
i
;Q
0
i
)
>
0
.
Ob iously hese assump ions, oge he wi h he es ic ion ha an
y ac i e
 m
i
mus cho ose a quan i y in [0
;

q
i
], a e sucien o he exis ence o a
Cou no equilib ium. The second pa o Assump ion 5 will ensu e, as we
shall see la e on, ha he e exis s some ag eemen s" b e ween he h ee
 ms wi h all o hem emaining on he ma ke . This could b e assumed
away, in ac simpli ying he analysis wi hou changing he esul s, bu i is
kep o he sake o gene ali y.
The las es ic ion on he cos and demand unc ions is ha whene e
only wo  ms a e ac i e hen, o any quan i y i s opp onen can p o duce, a
 m can achie e a p osi i e p o . Fo mally, le us dene
w
i
(
s
) wi h
s
=
i; j
g
as he minimal pay o  m
i
can gua an ee o i sel when i aces  m
j
, i.e.:
w
i
(
i; j
) = min
q
j
2
X
j
max
q
i
2
X
i

i
(
q
i
;q
j
)
Unde assump ion 4

i
is a s ic ly dec easing unc ion o
q
j
. The e o e,
dening
q
R
i
(
q
j
) = a g max
q
i
2
X
i

i
(
q
i
;

q
j
), we ha e:
w
i
(
i; j
)=

i
(
q
R
i
(
q
j
)
;

q
j
)
We shall equi e:
Assump ion 6
Fo al l
i; j
2
N
,
w
i
(
i; j
)

0
and
q
R
i
(
q
j
)
<

q
i
.
This will gua an ee on he one hand ha he ma ke canno b e mo-
nopolized and on he o he hand ha he e exis couples (
q
i
;q
j
) such ha

i
(
q
i
;q
j
)
>w
i
(
i; j
). Wenow u n o he c ha ac e iza ion o he se o sub-
game p e ec equilib ium payo ec o s o he inni ely epea ed game 0

(
s
)
.
7
2.2 Equilib ium payos in
0

(
s
)
The ypical payo o  m
i
in he game 0

(
s
) is gi en by:
P
i
=(1
0

)
1
X
=0


i
(
q
i
;Q
0
i
)
Inni ely ep ea ed games wi h discoun ing ha e b een ex ensi ely analysed
in he li e a u e. I has b een es ablished (see, o ins ance, Theo em 3.2
in So in (1992)) ha he se o subgame p e ec Nash equilib ium payo
ec o s o 0

(
s
) con e ges (wi h esp ec o he Haussdo opology) o he
se o indi idually a ional and easible payo ec o s o he cons i uen game
as he discoun ac o ends o one
2
. This is one o he e sion o he so-called
Folk Theo em. Acco dingly, o ou pu p oses we only need o c ha ac e ize
he se o indi idually a ional and easible payo ec o s o he Cou no
one-sho game whe e he se o playe s is gi en by
s
. This se is deno ed
W
(
s
).
To b egin wi h le us deno e he se o easible pa
yo ec o s wi h h ee
ac i e  ms by
F
(
N
) and he one wi h wo ac i e  ms by
F
(
i; j
). Le
X
=
X
i
2
X
j
2
X
k
,
F
(
N
) and
F
(
i; j
) a e gi en by:
F
(
N
) = con ex hull
(
P
1
;P
2
;P
3
)
j 9
(
q
1
;q
2
;q
3
)
2
X
such ha
P
i
=

i
(
q
i
;Q
0
i
) o
i
=1
;
2
;
3
g
:
F
(
i; j
) = con ex hull
(
P
i
;P
j
)
j 9
(
q
i
;q
j
;
0)
2
X
such ha
P
i
=

i
(
q
i
;Q
0
i
)
and
P
j
=

j
(
q
j
;Q
0
j
)
g
:
When he h ee  ms a e ac i e, eac h ac i e  m can gua an ee o i sel a
p o gi en by:
w
i
(
N
)= min
q
j
2
X
j
;q
k
2
X
k
max
q
i
2
X
i

i
(
q
i
;q
j
+
q
k
)
Acco dingly he se o indi idually a ional and easible pa
yo ec o s when
 ms in
s
b eing ac i e and #
s

2 is simply:
W
(
s
)=
(
P
1
;P
2
;P
3
)
j
(
P
i
)
i
2
s
2F
(
s
)
;P
k
= 0 o
k
62
s
and
P
i

w
i
(
s
)
8
i
2
s
g
:
(2)
2
P o ided he se o indi idually a ional and easible p oin s has a non-emp y in e io .
This is clea ly he case unde Assump ion 6.
8
s ic ly nega i e o any
q
j
2
(
q
R
(
q
i
)
;

q
j
] and s ic ly p osi i e o any
q
j
2
[0
;q
R
(
q
i
)). Mo eo e , by assump ion 3, ma ginal cos is inc easing in

so
ha
@c
(
q
j
;

j
;

q
j
)
=@
j
0
@c
(
q
R
(
q
i
);

j
;

q
j
)
=@
j
is s ic ly p osi i e o any
q
j
2
(
q
R
(
q
i
)
;

q
j
] and s ic ly nega i e o any
q
j
2
[0
;q
R
(
q
i
)). I hen ollo ws
ha
dq
j
=d
j
is s ic ly nega i e o any gi en
q
i
2
[0
;

q
i
) and
q
j
sa is ying

j
(
q
j
;q
i
)=
w
j
(
i; j
). This es ablishes ha he isop o cu e

j
(
q
j
;q
i
)=
w
j
(
i; j
) shi s igh wa ds in he co o dina e (
q
j
;q
i
)as

j
dec eases. As shown
in Figu e 1 he esul hen ollo ws.
2
As shown by he p o o , he xed cos does no ma e since i aec s
b o h sides o he cons ain in he maximiza ion p og am in he same way.
Hence, o any quan i y p oduced by  m
i
, he quan i y equi ed o sa is y
he cons ain is indep enden o he xed cos . On he o he hand, he le el
o

, i.e. he le el o he ma ginal cos o a gi en quan
i y p o duced, aec s
he cons ain i n wo ways. Fi s , i he igh -hand side o he cons ain
we e indep enden o

hen, in he co o dina es o Figu e 1,  m
k
's isop o
cu e will b e en i ely belo w he  m
j
's one as long as

j
<
k
. This eec
eec s he ad an age o o m a ca el wi h a low ma ginal cos  m. Second,
howe e , he minimal payo equi ed by a  m o pa icipa e in a ca el
wi h  m
i
clea ly dec eases wi h

. This ansla es he in ui ion ha a lo w
ma ginal cos  m will be mo e g eedy han a  m wi h a highe ma ginal
cos . Wha he Lemma s a es is ha he second eec domina es he  s
one.
Acco ding o his esul , o

k
la ge han

j
,  m
k
can always gi e o
 m
i
a g ea e payo han he highes pay o  m
i
can ob ain wi h  m
j
.
On he o he hand i ca el
i; j
g
o ms hen  m
k
will ecei e a ze o payo
while i will ob ain a leas
w
k
(
i; k
)
>
0 i ca el
i; k
g
o ms. Hence, lo osely
speaking,  m
k
has always he oppo uni y and he willingness o p e en
he o ma ion o ca el
i; j
g
so ha his ca el canno o m.
Rema k ha we canno exclude he o ma ion o he g and ca el,
N
,
o all o de s o decla a ion. Indeed conside , o ins ance, he case whe e
S
=
i; j
g
;
i; k
g
;N
g
wi h

j
<
k
and  m
i
is he  s  m o decla e.
I
w
k
(
i; k
) is sucien ly la ge i could happ en ha
W
i
(
i; k
) is s ic ly
smalle han he g ea es payo  m
i
can ob ain in
V

(
N
). Consequen ly,
 m
i
will p opose he o ma ion o he g and ca el and he bes ei he  m
j
o  m
k
(o b o h) can do is o mak e a decla a ion compa ible wi h ha
15

o  m
i
8
. This shows ha he a ailabili y o a p eda o y s a egy is no
sucien o p eda ion o o ccu .
Finally, Pa 3 o P op osi ion 1 co esp onds o he example gi en in he
In o duc ion abo e, excep o he easibili y o he g and ca el. Howe e ,
o ha e he same p edic ion, i.e. ha he lo w ma ginal cos  m is excluded
o all o de s o decla a ions, wo addi ional equi emen s a e needed. The
 s is ha he maximal capaci y o  m
j
is smalle han ha o  m
i
;
he second is ha he maximal payo ha
j
can ob ain in he ca el
i; j
g
is la ge han he one i can ob ain in ca el (
N
). These condi ions seem
ai ly un es ic i e: as  m
i
has a lowe ma ginal cos unc ion han
j
i
is easonable o assume ha i has ins alled a highe capaci y; while i is
qui e plausible ha a  m can ob ain mo e in a wo- m han in a h ee-
 m ca el. ob iously, i ca el (
N
) was no easible, as in he in o duc o y
example, hen his second condi ion is i ially me .
As i can be seen om he P o o o P oposi ion 1 in he Appendix, hese
wo condi ions a e supe uous o all o de s o decla a ion excep when
k
is
he  s o decla e. In his case
k
in o de o induce
j
o en e he ca el
j; k
g
m us gi e o
j
apayo a leas as g ea as he maxim um pa
yo ha
j
could ob ain in he ca el
i; j
g
,
W
j
(
i; j
). The same is ue i
k
wan s o
induce
i
o en e he ca el
i; k
g
, ha is
k
mus gi e
W
i
(
i; j
). Thus,
k
will
p e e he ca el
j; k
g
i i ge s a highe payo in i a he han in
i; k
g
,
gi en he cons ain s imp osed by wha he m us oe o
j
and
i
. This is he
case i he condi ions in Pa 3 o P oposi ion 1 a e me , as i is sho
wn by
Lemma 4 in he App endix.
5 Equilib ia o he ca el o ma ion game
wi h cos less een y
I has b een shown in sec ion 2.3, ha he case wi h cos less een y die s
om he one wi h unp o able een y only by he ac ha in he o me
case he minimal payo equi ed by a  m, say
j
, o pa icipa e o a wo-
 ms ca el, say
i; j
g
, is equal o max
0
;w
j
(
N
)
g
while in he la e case
8
No e ha i he ca el
j; k
g
we e also easible hen he g and ca el could o m wi h
 m
i
b eing he  s  m o decla e and  m
j
( esp.  m
k
) he second one p o ided
ha
W
j
(
j; k
) ( esp.
W
k
(
j; k
)) is s ic ly smalle han he highes payo  m
j
( esp.
 m
k
) can ob ain in
V

(
N
).
16
i equals
w
j
(
i; j
). Acco dingly i ca els
i; j
g
and
i; k
g
a e easible, ha
is, i
w
j
(
N
) and
w
k
(
N
) a e nega i e, hen he minimal payo ob ained by
 ms
j
and
k
in hese ca els a e equal o ze o. The e o e, he highes payo
 m
i
can ob ain in a ca el,
W
i
(
s
), dep ends on b o h he ma ginal and xed
cos s o i s pa ne . Mo e p ecisely, le
q
0
h
b e he i al's ou pu which leads
o ze o p o o  m
h
when i plays i s b es eply,
q
R
h
(
q
o
h
), ha is,
q
o
h
is
such ha

h
(
q
R
h
(
q
0
h
)
;q
0
h
) = 0. Fu he mo e, o all
q
2
X
0
h
wi h
X
0
h
=[0
;q
0
h
),
le ^
q
h
(
q
) b e he smalles quan i y p o duced by
h
which gi es i a ze o p o
whene e i s i als p o duce
q
, ha is, ^
q
h
(
q
) is such ha :

h
(^
q
h
(
q
)
;q
)

0
and
@
h
(^
q
h
(
q
)
;q
)
=@ q
h
>
0. Weha e:
Lemma 2
Suppose al l ou assump ions excep assump ion 1 hold. The e
exis s

<
1
such ha , i ca els
i; j
g
and
i; k
g
a e easible and
(

j
;F
j
)
and
(

k
;F
k
)
a e such ha
^
q
k
(
q
)
>
^
q
j
(
q
)
o al l
q
2
X
0
j
X
0
k
, hen
W
i
(
i; j
)
>
W
i
(
i; k
)
>W
i
(
N
)
o al l

2
(
;
1)
.
Conside ing Figu e 2, he p oo o his esul is clea ly qui e ob
ious
and is hus omi ed. I mus b e no iced ha a necessa y and sucien
condi ion o
W
i
(
i; j
)
>W
i
(
i; k
) o hold would in ol e a compa ison o he
cos s uc u e o he h ee  ms. We hus c ho ose o s a e ou esul s in e ms
o a sucien condi ion which ac ually equi es only he compa ison o  ms
j
and
k
a e age cos unc ion.
Clea ly, Lemma 2 he e will play he ole o Lemma 1 in he case o no-
een y. I he e o e ollows:
P op osi ion 2
Suppose al l ou assump ions excep assump ion 1 hold. The e
exis s

<
1
such ha o al l

2
(
;
1)
and o any o de o decla a ion we
ha e:
1. A ca el o ms,
2. le
S
=
i; j
g
;
i; k
g
;N
g
and
^
q
j
(
q
)
<
^
q
k
(
q
)
o al l
q
2
X
0
j
X
0
k
, hen
ca els
i; k
g
and
N
do no o m,
3. le
S
=
i; j
g
;
i; k
g
;
j; k
g
;N
g
and
(
i
)^
q
i
(
q
)
<
^
q
j
(
q
)
o al l
q
2
X
0
i
X
0
j
,
(
ii
)^
q
i
(
q
)
<
^
q
k
(
q
)
o al l
q
2
X
0
i
X
0
k
,
(
iii
)^
q
j
(
q
)
<
^
q
k
(
q
)
o al l
q
2
X
0
j
X
0
k
hen ca els
i; k
g
;
j; k
g
and
N
do no o m.
17
This P oposi ion
9
con as s wi h ou p e ious esul s in wo ways: Fi s ,
he g and ca el,
N
, does no o m, so ha i seeing ha he e exis s a
p eda o y s a egy i will b e played i.e.
p eda ion occu s
. This comes om he
ac ha , as long as ca els
i; j
g
and
i; k
g
a e easible, he minimal payo
 ms
j
and
k
will ob ain in b o h a wo- m ca el and in he g and ca el is
equal o ze o. I hen ollows ha  m
i
can always ob ain a la ge payo
in a wo- m ca el han in he g and ca el (see Lemma 3). Consequen ly
i  m
i
is he  s  m o decla e i will ne e p op ose he o ma ion o he
g and ca el. On he o he hand i i is  m
j
( esp.  m
k
) whic h is he
 s o decla e hen i will ne e p opose he o ma ion o he g and ca el.
Indeed i i do es so hen b o h  m
i
and  m
k
( esp.  m
j
) can ob ain a
highe payo han he one p oposed in  m
j
's ( esp.  m
k
's) decla a ion
by making compa ible decla a ions which p opose he o ma ion o he ca el
i; k
g
( esp.
i; j
g
).
The second die ence b e ween he esul s wi h cos less een y and he
ones wi h unp o able een y can b e illus a ed i we suppose ha  ms
ha e iden ical xed cos s
10
. In his case ^
q
j
(
q
)
<
^
q
k
(
q
) o all
q
2
X
0
j
X
0
k
will hold i and only i

j
<
k
. Then P oposi ion 2 s a es simply ha he
 m wi h he highes ma ginal cos unc ion will b e p eda ed. The e o e
wi h cos less een y, con a y o wha happ ens in he unp op able een y
case, a low ma ginal cos cons i ues a s ong ad
an age o ace p eda ion.
On he o he hand, i we suppose ha

i
=

j
=

k
hen he condi ions
used in P op osi ion 2 will b e sa ised i and only i
F
i
<F
j
<F
k
. Hence we
nd back a esul s a ed  s by Ghema w
a and Nalebu (1985) o declining
indus ies acco ding o which he  m wi h he la ges capaci ies i.e. wi h he
highes xed cos le el is he  s  m o exi he ma k e . Suc h conclusion
has also b e d awn by Fudenb e g and Ti ole (1986) om he analysis o an
incomple e in o ma ion game.
To conclude wi h, i we a e able o ank he  ms wi h esp ec o hei
a e age cos unc ion hen P op osi ion 2 s a es ha he exi ing  m is he
one wi h he highes a e age cos unc ion.
9
The p o o o his P oposi ion ollo ws so closely ha o P op osi ion 1 ha i s omi ed.
10
Recall ha ou esul s in he case o unp o able een y do no dep end on he  ms
xed cos s.
18
6 Robus ness o he esul s wi h unp o able
een y
One sp ecic ea u e o he ca el o ma ion game p esen ed ab o e is ha
each  m in i s decla a ion p op oses
simul aneously
a pa icula ca el and
he payos ha each mem b e o he ca el will ecei e. As a consequence,
he ca el o ma ion game gi es o
al l
 ms a s ong inuence on he way
payos a e allo ca ed among ca el mem b e s.
This seems easonable when een y cos s a e negligible. In his case
indeed he p oduc ion game emains a h ee playe s game e en i a  m exi s
he ma ke . Howe e when een y cos s a e la ge, he p o duc ion game
b ecomes a wo playe s game once a  m decides o s ay ou o he ma k e .
In his case one can ask he ques ion i he ca el o ma ion game does no gi e
o he exi ing  m an un ealis ically excessi e inuence on he equilib ium o
he esul ing wo  ms p o duc ion game which shall b e play
ed. In o de o
p o ide an answe , we shall analyze he sensi i i y o he ineciencies s a ed
in P op osi ion 1 o he way  ms a e supposed o co o dina e.
Toin es iga e his issue we lo ok a a wo s ep co o dina ion p o cess whe e
he exi ing  m has no inuence on he wa
y he emaining  ms will sha e he
gains om co ope a ion in he p oduc ion game. This co o dina ion p o cess
cons i u es a game: i s  s s ep is a subs i u e o he ca el o ma ion game
p esen ed be o e. The only die ence is ha i is no
w supposed ha a  m
decla a ion only consis s o a easible ca el,
s
. I all decla a ions die he
game ends and each  m ecei es i s ese
a ion payo 
g
i
. O he wise one
mo es o he second s ep.
The second s ep consis s o a nego ia ion b e ween
he membe s o he
ca el gi en in he iden ical decla a ions o he  s s ep
,say
s
, o de e mine
apay o ec o ,
p
, belonging o
V

(
s
). I a  m does no b elong o
s
hen i s
ac ion se in his s ep is simply
do no hing
g
.
We shall no sp eci y explici ely he ba gaining game p o cedu e. W
e as-
sume ins ead ha , he gains om co ope a ion (i.e. he ac ual payo min us
he sum o app op ia ely discoun ed Cou no p o s o he one-sho quan i y
game) a e sha ed acco ding o a ba gaining solu ion. The ba gaining solu ion
we adop he e b elongs o he amily o egali a ian (also called p opo ional)
19
solu ions as axioma ized by Kalai (1977) and Kalai and Same (1985)
11 12
.
To b e p ecise, le us  s assume ha :
Assump ion 7
Fo any easible ca el,
s
, he Cou no equilib ium in he
quan i y game is unique.
Then, le

c
i
(
i; j
) deno e he  m
i
's Cou no equilib ium p o when only
 ms
i
and
j
a e ac i e on he ma ke . Fu he mo e deno e by (
q
e
i
(
i; j
)
;q
e
j
(
i; j
))
he quan i y ec o whic h maximizes
P
i
sub jec o
P
i
0

c
i
(
i; j
)=
P
j
0

c
j
(
i; j
)
and le
P
e
i
(
i; j
) ( esp.
P
e
j
(
i; j
)) b e gi en by (1
0

)
P
1
=0


i
(
q
e
i
(
i; j
)
;q
e
j
(
i; j
))
( esp. (1
0

)
P
1
=0


j
(
q
e
j
(
i; j
)
;q
e
i
(
i; j
)) ). Ob iously (
P
e
i
(
i; j
)
;P
e
j
(
i; j
)) is
he symme ic egali a ian solu ion
13
o he co op e a i e ba gaining game de-
ned by a se o ou comes gi en by
F
(
i; j
) and a s a u-quo p oin gi en by
(

c
i
(
i; j
)
;
c
j
(
i; j
)).
We can immedia ely s a e:
Lemma 3
Le al l ou assump ions excep 2 be sa ised. Fu he mo e, o
any easible wo- ms ca el, say
h; l
g
, suppose ha
(
q
e
h
(
h; l
)
;q
e
l
(
h; l
))
belongs
o
]0
;

q
h
[
2
]0
;

q
l
[
and ha he e exi s
(
q
h
;q
l
)

0
which maximizes
P
h
+
P
l
.
Then he e exis s

<
1
such ha , o al l
>


,
P
e
i
(
i; k
)
>P
e
i
(
i; j
)
i and
only i

k
>
j
.
11
This kind o s uc u e has al eady b een used in he li e a u e. F
o ins ance, in G oss-
man and Ha (1986), wo agen s  s c
ho ose non-co op e a i ely and sim ul aneously a
le el o in es men and hen, gi en hese in es men s, ake ac ions such ha he gains
om enego ia ion, which co espond o he gains om co op e a ion in ou amewo k,
is sha ed equally. In hei con ex , his co esponds also o he Nash ba gaining solu-
ion. The G o osman and Ha 's analysis has b een ex ended by Ha and Mo o e (1990)
o many agen s and he ba gaining solu ion adop ed he e o sha e he gain om ade is
he Shapley alue.
We adop he e an egali a ian solu ion one he one hand because i is m uch mo e ac able
han he o he ones (in pa icula he Nash ba gaining solu ion), and on he o he hand
b ecause he egali a ian solu ions a e he only ones which, in he p esence o o he s an-
da d equi emen s, sa is y he mono onici y p op e
y (see Kalai and Same (1985)). This
condi ion simply s a es ha i he easible se o one coali ion inc eases and he easible
se s o all o he coali ions emain he same, hen none o he mem b e s o his coali ion
should b ecome wo se o b ecause o his change.
12
No e ha simila esul s could b e ob ained by using he symme ic Nsah ba gaining
solu ion.
13
I will b e ob ious o e i y ha he esul s p esen ed b elow will hold i we ake an
asymme ic egali a ian solu ion p o ided he weigh o  m
i
in he solu ion dep ends
nega i ely on

i
and is indep enden on he xed cos s le el.
20

No ice ha he  s addi ional assump ion in his Lemma will simply
gua an ee ha he e exis s a easible payo ec o s ic ly g ea e han he
Cou no equilib ium p o s ec o .
Again an inc ease in

j
will ha e wo eec s on he co op e a i e ba gaining
game in ol ing  ms
i
and
j
: On he one hand, i leads o a mo dica ion in
he se o easible ou comes whic h aec s nega i ely he payo o  m
i
a he
egali a ian solu ion while, on he o he hand, i inc eases ( esp. dec eases)
 m
i
's ( esp.  m
j
's) s a u-quo payo whic h will ise he  m
i
's payo a
he egali a ian solu ion. The Lemma
14
s a es simply ha he posi i e eec
a ising om he mo e in he s a u-quo payo domina es he nega i e eec
coming om he educ ion in he se o easible ou comes.
This esul will play, o P oposi ion 3 b elow, he ole play
ed by Lemma 1
and 2 o P oposi ion 1 and 2 esp ec i ely
.To see his i suces o ealize
ha he se o subgame p e ec equilib ia o he game de i ing om he wo
s ep p o cedu e he e conside ed coincides wi h he one o he ca el o ma ion
game whe e a  m
i
's decla a ion consis s o a easible ca el,
s
i
, o whic h  m
i
b elongs and o a payo ec o whic h gi es o each  m in
s
i
he symme ic
egali a ian payo dened abo e
15
and a ze o payo o a  m (i any) which
does no b elong o
s
i
.Fo mally he se o  m
i
's decla a ions is now
D
i
=
(
s; p
)
j
i
2
s; s
2S
;
o all
h
2
sp
h
=
p
e
h
(
s
) and, o
l
62
s; p
l
=0
g
.
The e o e we ha e:
P op osi ion 3
Le al l assump ions in Lemma 3 hold. Then he e exis s

<
1
such ha o al l

2
(
;
1)
and wha e e he o de o decla a ion we
ha e:
1. A ca el o ms,
2. i
S
=
i; j
g
;
i; k
g
;N
g
and

j
<
k
, hen ca el
i; j
g
does no o m,
3. i
S
=
i; j
g
;
i; k
g
;
j; k
g
;N
g
and

i
<
j
<
k
, hen ca els
i; j
g
and
i; k
g
do no o m.
14
The p o o o his esul comes qui e s aigh o w a dly om he applica ion o he
en elop e heo em as well as he Folk heo em. Hence i will b e omi ed.
15
To sa e space we do no dene o mally he egali a ian pay o when he h ee  ms
a e ac i e. Howe e his can easily be done e en i one wan s o conside a coali ion o m
game ins ead o a co op e a i e ba gaining game. Anyway his do es no ma e o ou
analysis.
21
This shows he obus ness o ou conclusions wi h esp ec o he inuence
o he exi ing  m on he waypay os a e allo ca ed in he p oduc ion game.
7 Concluding ema ks
Weha e conside ed in his pape a dynamic p o duc ion game in ol ing h ee
 ms whic h a e die en ia ed acco ding o hei cos unc ion. Mo e p ecisely
we h a e assumed ha  ms can b e anked unam biguously acco ding o hei
ma ginal cos unc ion and ha hei xed cos ma y die . Fu he mo e we
suppose ha one  m can b e c edibly o ced o s ay ou o he ma ke by
he wo o he s and ha a leas w
o  ms can b e pu unde such a h ea .
We hen in es iga e he cos cha ac e is ics o he exi ing  m unde wo
al e na i e hyp o hesis conce ning he p ossibili y o een y namely he case
whe e een y is unp o able in any ci cums ances due o he p esence o
la ge sunk cos s, and he one whe e een y is cos less.
Weha e ob ained wo p edic ions (whic h appea s qui e obus o he
specica ion o he ca el o ma ion game). Fi s i een y is alwa
ys unp o -
i able hen he exi ing  m has he lowes ma ginal cos unc ion as compa ed
wi h he ma ginal cos unc ion o he  ms which can c edibly be p eda ed.
Fu he mo e his esul do es no dep end on he le el o xed cos s
16
. Ac-
co dingly, in his case, cos ineciencies will a ise since he exi ing  m is
he one which uses he mos ecien echnology
.
A second esul is ha when een y is cos less and when we can ank
 ms acco ding o hei a e age cos unc ion hen he exi ing  m has he
la ge a e age cos unc ion as compa ed o he a e age cos unc ion o he
 ms whic h can b e pu unde he h ea o p eda ion. The e o e in his case
cos ineciencies do no appea .
The esul ob ained in he no- een y case lo oks s ange since i goes
agains he common belie ha he mos ecien  m will emain on he
ma ke . Bu his b elie has b een de elopped in he con ex o neo-classical
economics". I ins ead we look a his esul om he p oin o iew o ans-
ac ion cos economics" (as de elopped in Williamson (1985) o ins ance)
hen hey app ea a he unsu p ising. Indeed in his con ex such kind o
ineciencies a e equen ly ob ained. I is wo hwhile emphasizing he deep
16
Howe e he se o  ms which can b e p eda ed depend ob iously on he le el o xed
cos s.
22
ela ionship b e ween ou analysis and he ansac ion cos app oach. In-
deed al hough he la e app oach o cuses mainly on he in e nal o ganisa-
ion o he  m he p esen s udy shows ha he basic p oin s whic h dis in-
guish ansac ion cos economics om o he economic app o c hes a e also well
sui ed o s udy he comp osi ion o an indus y and mo e gene ally o mak e
subs an ial p og esses in he unde s anding o he o ma ion and comp osi ion
o g oups o coali ions on a ma ke .
Roughly speaking ansac ion cos economics seeks o analyse si ua ions
in ol ing agen s cha ac e ized by
oppo unism
and
bounded a ionali y
whe e
(
i
) agen s will mee
equen ly
,(
ii
) agen s do no ely on cou s o se ling
dispu es among hem i.e.
p i a e o de ing
p e ails, (
iii
) agen s ha e he
oppo uni y o mak e
asse specic in es men s
and (
i
) agen s e ol ein an
unce ain
en i onmen .
In he p esen analysis we ha e uled ou b o h unce ain y and b ounded
a ionali y since hese cha ac e is ics appea unessen ial o ou esul s. No e
u he mo e ha equency will no b e ele an he e as he example gi en
in he In o duc ion p oin s ou . The die ence b e ween oppo unism and
sel -in e es ed b eha io does no ma e he e b ecause he se o subgame
p e ec Nash equilib ia and he se o Nash equilib ia o he p oduc ion game
coincides o a discoun ac o sucien ly close o ze o. We shall howe e
a gue ha i we mak e abs ac ion o he p esence o ei he p i
a e o de ing
o asse specic in es men s hen he cos ineciencies ob ained in he pap e
disappea .
Le us b egin wi h p i a e o de ing. Many exchange analysis suppose ha
ecacious ules o law a e in place so ha any disag eemen ega ding he
execu ion o a con ac is se led by cou s in a ully in o med and low-cos
way. This assump ion o cou o de ing is e y con enien since i allows o
dis ega d he ex-p os side o a con ac . In ou con ex ,  ms canno ely
on cou since he kind o con ac hey a e willing o do is simply illegal.
An immedia e consequence o p i a e o de ing is ha we canno dis ega d
he execu ion phase o he con ac since he la e m us b e sel en o cing.
This en ails ha  ms, as is supposed in he ca el o ma ion game, will only
conside payo ec o s which can b e associa ed wi h a subgame p e ec Nash
equilib ium o he p o duc ion game.
Bu suppose o he con a y ha  ms can ely cos lessly on cou o
en o ce an ag eemen . This implies ha he se o payo ec o s ha m us
now b e conside ed in he ca el o ma ion game coincides wi h he one co -
23
esp onding o he cos less een y case. Indeed, a  m whic h can b e o ced
o exis can commi o obey an ag eemen in whic h i ecei es a ze o payo.
Wi hou his p ossibili y o commi men , such an ag eemen is no c edible
in he no- een y case while i is in he case o cos less een y. Conse-
quen ly, he esul s a ed in Lemma 2 will hold e en i een y is unp o able
and he exi ing  m is he one wi h he highes a e age cos unc ion (see
P oposi ion 2 o a mo e p ecise s a emen ). The e o e he cos ineciencies
disappea once cou o de ing is allowed o . Rema k ha his clea ly shows
ha conside ing aci co op e a ion b e ween  ms as illegal is p ossibly cos ly.
Le us now u n o he asse sp ecic cha ac e o in es men s. In es -
men s a e said wholly asse sp ecic i hey a e un edeplo yable. Acco dingly
in es men cos s a e sunk o wholly asse sp ecic in
es men s while hey
a e xed when in es men lo oses i s asse sp ecic cha ac e . The main con-
sequence o he p esence o asse specic in
es men is he occu ence o he
undamen al ans o ma ion
. The la e concep e e s o he ans o ma ion
in he na u e o he comp e i ion p e ailing b e o e and a e he adop ion o
he con ac .
In ou con ex , he sunk een y cos we ha
e in oduced can simply
b e in e p e ed as he cos o un edeploy
able in es men s. Mo e p ecisely,
he unp o able een y case co esp onds o he si ua ion whe e la ge asse
specic in es men s m us b e achie ed b e o e b eing ac i e on he ma ke
while in he cos less een y case such in
es men s a e negligible. When
een y is unp o able he undamen al ans o ma ion o ccu s since, once
a  m exi s, he p oduc ion game b ecomes a w
o playe s game. I ins ead
een y is cos less his ans o ma ion does no o ccu . Indeed, in his case
e en i a  m exi s i can pa icipa e o he punishmen o a de ia ion om
he equilib ium pa h by one o he wo  ms which emain on he ma k e .
In o he wo ds he p o duc ion game s ill in ol es h ee playe s e en i a  m
exi s he ma k e . As we ha e shown, cos ineciencies app ea only in he
case o unp o able een y which means ha he p esence o la ge asse
specic in es men s is a necessa y condi ion o such cos ineciencies o
occu .
24
8.2.4 Supp ose ha  m
k
is he las  m o decla e:
1. Le  m
j
b e he second  m o decla e. Fo he ca el
i; j
g
o o m
i mus b e he case ha
d
i
=
d
j
,
d
i
2D
j
nD
k
. Bu i  m
i
makes a
decla a ion b elonging o
D
j
nD
k
,  m
j
will ob ain a mos
W
j
(
i; j
)
by decla ing
d
j
=
d
i
while i will ecei e
W
j
(
j; k
) i i decla es
d
k
j
=
(
j; k
g
;
(0
;W
j
(
j; k
)
;w
k
(
j; k
))), since o such decla a ions (
d
i
;d
k
j
)  m
k
will maximize i s payo by decla ing
d
k
=
d
k
j
. By Lemma 1 we know
ha , o

sucien ly close o one,
W
j
(
j; k
)
>W
j
(
i; j
) and he e o e
ca el
i; j
g
does no o m.
Fo ca el
i; k
g
o o m i mus be he case ha
d
i
62 D
i
D
j
. I  m
j
makes a decla a ion whic h induces  m
k
o decla e
d
k
=
d
i
hen i
will ecei e a ze o payo. Howe e o any
p
ik
2
[
w
k
(
i; k
)
;W
k
(
i; k
)]
we know by Lemma 1 ha , o

sucien ly close o one, he e exis s
d
j
2D
j
D
k
such ha
p
jk
>p
ik
and
p
jj
>w
j
(
j; k
)
>
0. The e o e
ca el
i; k
g
does no o m.
2. Le  m
i
b e he second  m o decla e. Fo he ca el
i; j
g
o
o m i m us b e he case ha
d
i
=
d
j
,
d
j
2D
i
nD
k
. Bu i  m
j
mak es such a decla a ion  m
i
will ecei e a mos
W
i
(
i; j
)by
decla ing
d
i
=
d
j
while i will ecei e
W
i
(
i; k
) i i decla es
d
k
i
=
(
i; k
g
;
(
W
i
(
i; k
)
;
0
;w
k
(
i; k
))) since o such (
d
j
;d
k
i
)  m
k
will max-
imize i s payo by decla ing
d
k
=
d
k
i
. By Lemma 1, o

sucien ly
close o one,
W
i
(
i; k
) is s icly g ea e han
W
i
(
i; j
) and he e o e
ca el
i; j
g
does no o m.
Fo ca el
i; k
g
o o m i mus b e he case ha
d
i
2D
k
nD
j
and
d
i
62 D
j
D
k
and
d
j
is such ha
p
ik

p
jk
( he equali y b e ween
p
ik
and
p
jk
is allo wed only i  m
k
, acing wo indie en al e na i es, cho oses
o decla e
d
k
=
d
j
). In his si ua ion  m
j
will ob ain a ze o payo
while  m
k
will ob ain a mos
W
k
(
i; k
). Howe e , by Lemma 1, we
know ha , o

sucien ly close o one, he e exis s
d
j
2D
j
D
k
such ha
p
jk
>W
k
(
i; k
) and
p
jj
>w
j
(
j; k
)
>
0. Consequen ly ca el
i; k
g
does no o m.
Summing up, i
k
is he las  m o decla e, Lemma 1 is sucien o
ensu e ha he ca el which o m is ei he
j; k
g
o
N
.
31

8.2.5 Supp ose ha  m
j
is he las  m o decla e:
1. Le  m
k
b e he second  m o decla e. Fo he ca el
i; j
g
o o m
i m us b e he case ha
d
i
2D
j
nD
k
and
d
k
is such ha
p
kj

p
ij
( he equali y b e ween
p
kj
and
p
ij
is allowed only i  m
j
, acing wo
indie en al e na i es, c ho oses o decla e
d
j
=
d
i
). In his case  m
j
will ob ain a mos
W
j
(
i; j
) while  m
k
will ob ain a ze o payo.
Howe e , by Lemma 1, i

is sucien ly close o one hen o any
d
i
which do es no b elong o
D
i
D
k
he e exis s
d
k
2D
j
D
k
such ha
p
kj
>p
ij
and
p
kk
>w
k
(
j; k
)
>
0. The e o e ca el
i; j
g
does no
o m.
Fo ca el
i; k
g
o o m i mus b e he case ha
d
i
2D
k
nD
j
and
d
i
=
d
k
. Bu since
d
i
62 D
j
 m
j
will maximize i s payo by decla ing
d
j
=
d
k
as long as
d
k
2D
k
and
p
kj
>
0. The e o e once
d
i
62 D
i
D
j
 m
k
can ob ain a payo o a leas
W
k
(
j; k
) while i ob ains a mos
W
k
(
i; k
) by decla ing
d
k
=
d
i
. By Lemma 1, o

sucien ly close
o one,
W
k
(
i; k
)
<W
k
(
j; k
) and consequen ly ca el
i; k
g
does no
o m.
2. Le  m
i
b e he second  m o decla e. Dene
D
R
i
(
d
k
)=
d
i
2
D
i
j
p
ii

p
ki
and (
d
i
;d
k
) is such ha  m
j
's b es - esp onse is
d
j
=
d
i
:
g
.Fo ca el
i; j
g
o o m a necessa y condi ion is ha  m
k
makes
a decla a ion such ha
D
R
i
(
d
k
)
nD
k
6
=
;
. Bu i  m
k
makes such a
decla a ion i will ob ain a ze o payo while we know by Lemma 1
ha , o

sucien ly close o one, he e exis s
d
k
2D
k
nD
i
such ha
i ob ains a payo s ic ly g ea e han
w
k
(
j; k
)
>
0 and o which
D
R
i
(
d
k
)
nD
k
=
;
. Consequen ly ca el
i; j
g
does no o m.
Now o ca el
i; k
g
o o m i is necessa y ha  m
k
makes a dec-
la a ion such ha
d
k
62 D
j
and
p
ki

max
W
i
(
i; j
)
;W
i
(
N
)
g
. Indeed
i
d
k
62 D
j
hen  m
j
will decla e
d
j
=
d
i
as long as
d
i
2D
j
and
p
ij
>
0. Acco dingly o
d
k
such ha
d
k
62 D
j
 m
i
can ob ain ei he
W
i
(
i; j
) by decla ing:
d
j
i
=(
i; j
g
;
(
W
i
(
i; j
)
;w
j
(
i; j
)
;
0)) o he max-
imal payo, deno ed by
P
N
i
(

)
;
ha  m
i
can ob ain in
V

(
N
) when
P
j
=
>
0 and
P
k
= 0, by decla ing
d
N
i
=(
N;
(
P
N
i
(

)
;;
0).
P
N
i
(

)
ends o
W
i
(
N
) when

ends o ze o. I ollows ha i
d
k
62 D
j
and
p
ki
<
max
W
i
(
i; j
)
;W
i
(
N
)
g
hen  m
i
will maximize i s payo
by decla ing ei he
d
j
i
o
d
N
i
and ca el
i; k
g
does no o m. Con-
32
sequen ly i
W
i
(
i; k
)
<W
i
(
N
) ca el
i; k
g
does no o m while i
W
i
(
i; k
)

W
i
(
N
) he maximal payo  m
k
can ob ain when ca el
i; k
g
o ms is equal o smalle han ~
w
k
(
i; k
).
On he o he hand o ca el
j; k
g
o o m i is sucien ha  m
k
's
decla a ion b e such ha
d
k
62 D
i
and
p
kj
>
max
W
j
(
i; j
)
;W
j
(
N
)
g
.
Indeed o such  m
k
's decla a ion he e does no exis
d
i
2D
i
such
ha
p
ij

p
kj
and hus  m
j
will decla e
d
j
=
d
k
. The assump ion ha
W
j
(
i; j
)

W
j
(
N
) oge he wi h Lemma 1 ensu e ha , o

sucien ly
close o one, we ha e
W
j
(
j; k
)
>
max
W
j
(
i; j
)
;W
j
(
N
)
g
=
W
j
(
i; j
).
Hence he e exis s
d
k
2D
k
nD
i
such ha
p
kj
>
max
W
j
(
i; j
)
;W
j
(
N
)
g
.
Fu he mo e, o

sucien ly close o one, we also ha e by Lemma 4
ha ~
w
k
(
i; k
)
<
~
w
k
(
j; k
). This implies he exis ence o
d
k
2D
k
nD
i
such ha
p
kj
>
max
W
j
(
i; j
)
;W
j
(
N
)
g
and
p
kk
>
~
w
k
(
i; k
). Conse-
quen ly ca el
i; k
g
does no o m.
8.2.6 Supp ose ha  m
i
is he las  m o decla e:
The a gumen s o p o e he esul s s a ed in he p oposi ion a e so close han
hose used in he p e ious case ha we omi hem he e.
9 Lemma 4
Le he maximal payo ha  m
k
can ob ain in
V

(
i; k
) sub jec o
P
i
=
W
i
(
i; j
) b e deno ed by ~
w
k
(
i; k
).Simila ly,~
w
k
(
j; k
) s ands o he maximal
payo ha  m
k
can ob ain in 2 sub jec o
P
j
=
W
j
(
i; j
).
Lemma 4:
Suppose all ou assump ions excep assump ion 2 hold.
Fu he mo e le
S
=
i; j
g
;
i; k
g
;
j; k
g
;N
g
,

i
<
j
<
k
,
q
j


q
i
and
W
j
(
i; j
)

W
j
(
N
). Then he e exis s 
<
1 such ha
o all

2
(
;
1) ~
w
k
(
i; k
)
<
~
w
k
(
j; k
).
The p oo is a ailable up on eques .
33