Exit, Sunk Costs and the Selection of Firms

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 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by-nc/3.0/ 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