Simultaneous Nash bargaining with consistent beliefs

Simul aneous Nash Ba gaining wi h Consis en Belie s Robe o Bu gue yand Ramon Caminalz This d a : No embe 2010 Abs ac We p opose and analyze a new solu ion concep , he Rsolu ion, o h ee-pe son, ans e able u ili y, coope a i e games. In he spi i o he Nash Ba gaining Solu ion, ou concep is ounded on he p edic ed ou comes o simul aneous, wo-pa y nego ia- ions ha would be he al e na i e o he g and coali ion. These possibly p obabilis ic p edic ions a e based on consis en belie s. We analyze he p ope ies o he Rsolu ion and compa e i wi h he Shapley alue and o he concep s. The Rsolu ion ex- is s and is unique. I belongs o he ba gaining se and o he co e whene e he la e is no emp y. In ac , when he g and coali ion can simply execu e one o he h ee possible bila e al ades, he Rsolu ion is he mos egali a ian selec ion o he ba gaining se . Finally, we discuss how he Rsolu ion changes impo an conclusions o se e al well known Indus ial O gani- za ion models. Keywo ds: coope a i e games, ba gaining, endogenous all- back op ions, consis en belie s, Rsolu ion. JEL classi…ca ion numbe s: C71, C78, L14. We hank Albe Banal, Oli ie Comp e, Ma hew Ellman, Jo di Massó, Cla a Pon- sa í, Deb aj Ray and Józse Sáko ics o use ul commen s. Also, we acknowledge he suppo o he Ba celona GSE, Gene ali a de Ca alunya, and Spanish Minis y o Sci- ence and Inno a ion (p ojec ECO2008-01850). yIns i u d’Anàlisi Econòmica CSIC, and Ba celona GSE, e-mail: obe o.bu gue @iae.csic.es zIns i u d’Anàlisi Econòmica CSIC, and Ba celona GSE, e-mail: a- [email protected] 1 1 In oduc ion When bila e al ba gaining is one o he componen s o an economic model, mos au ho s use he Nash Ba gaining Solu ion (NBS) as a educed o m ha maps he undamen als o he model in o nego ia ed ou comes. Since we o en know e y li le abou how agen s ac ually ba gain in he eal wo ld, a black-box app oach seems jus i…ed. A e all, he p inciples and in ui ions implici in he NBS a e e y con incing. Howe e , such b oad consensus does no exis when ba gaining in ol es h ee playe s and di¤e en pai s o playe s can achie e by hemsel es di¤e en ag eemen s.1This is he case when one (o mo e) playe (s) may ade o each an ag eemen wi h wo al e na i e, po en ial pa ne s. When analyzing such p oblems, some au ho s ake a non-coope a i e app oach and assume a pa icula ba gaining p o ocol. An al e na i e is o in oke solu ion concep s bo owed om coope a i e game heo y. The Shapley alue is he mos popula choice, as a simple alue cha ac e ized by seemingly na u al axioms. Ye , he Shapley alue p edic s ou comes ha in some cases a e con o e sial, o say he leas .2 This pape p esen s a new solu ion concep o h ee-playe coope a i e games ha can be eadily applied o p edic ing he ou come o h ee-pa y nego ia ions. Ins ead o a emp ing o iden i y sensible axioms ha single ou one ou come o conside ing a pa icula p o ocol ha would do he job, ou app oach is based on a ew mains eam ideas in economics. The … s is ha he NBS is a sa is ac o y p edic ion o wo-playe ba gaining o in gene al o wha a e called pu e ba gaining games, whe e he only coali ion ha adds some su plus is he g and coali ion.3The second is ha when playe s ba gain hey also o m belie s abou wha would happen i ag eemen is no eached in ha pa icula nego ia ion. The hi d one is 1Examples o economic models ha include h ee-playe ba gaining abound. In Sec ion 4 we discuss in de ail some pa icula examples. 2See, o ins ance, De Meza and Sel aggi (2007), page 89. 3See K ishna and Se ano (1996) o a non-coope a i e mo i a ion o his solu ion. 2 ha hese belie s should sa is y some no ion o consis ency wi h payo¤s. 4 Conside one o he simples o hese h ee-pe son ba gaining si ua ions, ha o a buye ha has o choose among wo po en ial selle s. A p edic ion o any such model should include a (possibly p obabilis ic) p edic ion o which o he wo ades will ake place and how playe s would spli he su plus in each o he wo po en ial ades.5Also, i he la e p edic ion is o be made acco ding o he NBS, hen disag eemen poin s o each o he wo nego ia ions should be speci…ed. Fo he buye , he disag eemen payo¤s should be endogenous. Indeed, he allback op ion in each nego ia ion is he possibili y o ade wi h he al e na i e selle . As we allow o mo e complex in e ac ions, we will need o conside he case whe e all wo-playe nego ia ions esul in some posi i e su plus. This is known as he h ee-playe / h ee-cake p oblem (see Binmo e, 1985). In his case, disag eemen poin s and payo¤s will need o be simul aneously and endogenously de e mined o all h ee playe s in all h ee al e na i e wo- playe nego ia ions. Mo eo e , now he (possibly p obabilis ic) p edic ion o wha nego ia ion will end in an ag eemen will be necessa y in o de o consis en ly calcula e (expec ed) allback op ions. Finally, wha is p edic ed o he h ee-playe / h ee-cake p oblem may lea e gains ha he h ee playe s may ealize by coo dina ing. In o he wo ds, he o al su plus ha he g and coali ion can ealize may exceed he su plus expec ed om bila e al nego ia ions. Tha may be so because o syne gies ha can be ealized only wi h he pa icipa ion o all h ee playe s o jus because, absen coo dina ion, playe s an icipa e ha ine¢ cien bi- 4In ou p e ious esea ch on labo con ac ing (Bu gue e al., 2002) we also had o decide how o p edic he ou come o nego ia ions among h ee playe s. In ac , in he Appendix o ha pape we imidly s a ed o ou line some o he ideas ha we ully de elop he e. 5The Shapley alue p edic s ha he buye will buy om he mos e¢ cien selle , ye he non ading selle will s ill ecei e a posi i e paymen a he expense o he ading pa ne s. Such a posi i e payo¤ is some imes in e p e ed as he b ibe ha he non- ading selle ecei es in o de o allow he implemen a ion o he e¢ cien ade. We will show ha such a jus i…ca ion makes sense only in some games bu no in his pa icula example. 3 la e al ag eemen s may occu wi h posi i e p obabili y. In his case, playe s may be able o a oid ine¢ cien ou comes h ough h ee-pa y nego ia ions and hen we expec hem o sha e he ex a su plus acco ding o he (gen- e alized) NBS.6In pa icula , he disag eemen poin o his h ee-playe nego ia ion should be he playe s’expec ed payo¤s in he al e na i e o he g and coali ion ag eemen : he p edic ed ou come o bila e al nego ia ions. As we ha e men ioned, ou solu ion concep equi es ha agen s o m (and sha e) belie s on he p obabili ies o success o each al e na i e nego- ia ion. This is an impo an ea u e o ou concep . In addi ion, we will impose a consis ency equi emen on his sys em o belie s: pa ies should no expec a wo-playe nego ia ion o succeed when bo h pa ies o ha nego ia ion p e e hei al e na i e one. In Sec ion 2 we p esen ou solu ion concep , he Rsolu ion, as a o maliza ion o hese ideas. We show ha he Rsolu ion exis s and is unique. Tha is, i u ns ou ha hese simple ideas a e su¢ cien o p edic he di ision o su plus in hese games. Mo e- o e , compu ing he Rsolu ion is a s aigh o wa d exe cise. We p o ide hese compu a ions o all pa ame e alues. The idea ha disag eemen poin s in h ee-pa y nego ia ions should emana e om he al e na i e o hese nego ia ions, ha is, he p edic ed ou comes o simul aneous, bila e al nego ia ions, is p obably non con o- e sial. The same applies o assuming ha disag eemen poin s in simul- aneous, bila e al nego ia ions should be endogenous. Mo eo e , he ideas a e no no el. Benne ’s (1997) app oach o he analysis o such nego ia- ions is he closes o ou s in spi i (also, see Binmo e, 1985, and e e ences in Benne , 1997). Indeed, Benne also a gues ha disag eemen payo¤s should be ob ained endogenously, bu in he solu ion playe s do no o m and sha e belie s abou he p obabili y o success o each bila e al nego ia- 6Th ee-pa y nego ia ions may no be easible due o ou side cons ain s. In Sec ion 4, we conside one case when his is so. 4 ion.7Indeed, in Benne ’s app oach, when wo pa ies nego ia e bo h use as a allback op ion hei own ag eemen wi h he hi d playe . Tha is equi alen o assuming ha di¤e en playe s assign p obabili y one o wo di¤e en , mu ually exclusi e ou comes. On he con a y, a cen al piece o ou concep is he endogenously de e mined, cohe en sys em o belie s ha playe s use o compu e hei endogenous allback op ions.8 We analyze he p ope ies o he Rsolu ion in Sec ion 3. We show ha he Rsolu ion sa is…es symme y, e¢ ciency, and he dummy playe axioms. Thus, i has o iola e he addi i i y axiom since he Shapley alue is he only solu ion concep ha sa is…es all ou . Indeed, he Rsolu ion is no addi i e. We a gue ha , a he han a weakness, his non addi i i y is a desi able p ope y o he concep o p oblems like he one discussed abo e. The seemingly innocuous addi i i y axiom implici ly imposes oo much s uc u e on wha "p o ocols" a e easible o he playe s. Fo ins ance, in ou one-buye , wo-selle s example, i implici ly imposes ha he buye canno a emp bundling o make join o¤e s o wo goods when dealing wi h he same wo po en ial selle s o hese wo goods. The Rsolu ion le s he p imi i es o he p oblem speak abou such possibili ies. Con a y o he Shapley alue, he Rsolu ion is a selec ion o he co e when he la e is no emp y. When he co e is emp y, he Aumann-Maschle ba gaining se (BS) is he mos popula gene aliza ion. The BS con ains he co e and is ne e emp y. We show ha , again con a y o he Shapley alue, he Rsolu ion is a selec ion o he BS. In ac , o supe addi i e, 7In Benne (1997), a solu ion should speci y he di ision o su plus in each al e na i e bila e al nego ia ion. The disag eemen poin in each nego ia ion is he payo¤ ha each playe would ob ain in he al e na i e nego ia ion. Thus, he disag eemen poin in some nego ia ions may be ou side he easible se o ha nego ia ion, which Benne in e p e s as ailu e o he nego ia ion. A p edic ed ou come speci…es wha nego ia ion will succeed and hen sha ing o he su plus acco ding o he NBS (o any o he concep ) gi en he co esponding disag eemen poin . 8In Sec ion 3 we also discuss al e na i e app oaches o endogenizing allback op ions, which a e implici in he no ion o consis ency p oposed by Ha and Mas-Colell (1989) and Se ano and Shimomu a (1998). 5 h ee-playe TU-games, he BS ( o he g and coali ion) coincides wi h he co e when he la e is no emp y, and is a single on when he co e is emp y. Thus, he Rsolu ion coincides wi h he BS in he la e case. Mo eo e , i bila e al ba gaining is all he e is in he game, ha is, i he g and coali ion does no add any addi ional su plus, he Rsolu ion is he mos egali a - ian selec ion in he BS. Thus, i is mo e egali a ian han o he , di¤e en selec ions o he co e o he BS, like he nucleolus.9 We pos ula e he Rsolu ion as a sa is ac o y, uni ying concep ha can be used o analyze models ha include h ee-pa y nego ia ions. In Sec ion 4 we illus a e he use o ou concep in some leading models in he Indus ial O ganiza ion li e a u e. Exclusi e con ac s (Segal and Whins on, 2000), endogenous me ge s (Ho n and Pe sson, 2001), and he p ope y- igh s he- o y o he … m (Ha and Moo e, 1990) ha e been analyzed in models wi h a enego ia ion s age, bu using some o he , di e se solu ion concep s. In Sec ion 4 we also discuss he use and implica ions o he Rsolu ion in hese cases. Sec ion 5 o¤e s some closing discussions. Finally, mos o he p oo s a e elega ed o an Appendix. 2 The Rsolu ion o a h ee-pe son game Le N= 1;2;3gbe he se o playe s, and le 2N ep esen he se o subse s o N. An elemen Z22N ep esen s a coali ion. A TU game in cha ac e is ic o m is he pai (N; ), whe e : 2N!Rsa is…es (?) = 0. We assume o be supe addi i e. Assump ion 1 (supe addi i i y): I Z; Z02Nand Z Z0=?, hen (Z) + (Z0) (Z[Z0). To sa e some space, we will use an abb e ia ed no a ion o he unc- ion. Thus, we will le ij = ( i; jg), i= ( ig)and V= ( 1;2;3g). 9The nucleolus is also a selec ion o he BS. Thus, when he co e is emp y, he nucleolus and he RSolu ion coincide. Howe e , when he co e is no emp y and se - alued, he wo concep s di¤e . Mo e on his in Sec ion 3. 6 Also, e e y ime we w i e " o all i; j" o " o all i; j; k" we mean o all i; j = 1;2;3; i 6=j, and o all i; j; k = 1;2;3,i6=j6=k; i 6=k, espec- i ely. Tha is, di¤e en sub/supe indices in he same exp ession will al- ways deno e di¤e en playe s. Wi hou loss o gene ali y, we will assume ha 12  1 2 13  1 3 23  2 3. In o he wo ds, coali ion 1;2gis he (weakly) mos "e¢ cien " among he wo-playe coali ions and coali ion 2;3gis he (weakly) leas e¢ cien . The hea o ou solu ion concep is a p edic ion o he ou comes o he h ee possible bila e al nego ia ions, including a p edic ion o which o hese nego ia ions would succeed (wi h wha p obabili y), should h ee- playe nego ia ions ail.10 In many cases his is in ac all ha will be needed o p edic ing he ou come o he whole game. We begin by de…ning his p edic ion o he ou come o simul aneous, bila e al nego ia ions. Fo each playe iin each bila e al nego ia ion ij, we deno e i’s p edic ed payo¤ by uij i. Also, we ep esen by pij he p edic ed p obabili y ha playe s iand ja e he ones whose nego ia ion succeeds and hen " ade". Finally, since ou concep is based on he wo-playe NBS, o each playe iin each bila e al nego ia ion ij, we will de…ne i’s disag eemen payo¤ o allback op ion, which we will ep esen by ij i. Be o e de…ning ou solu ion, we explain he consis ency equi emen s on hese alues ha will de…ne ou solu ion concep o simul aneous, bila e al nego ia ions. i) Gi en he allback op ions, ij i, playe s iand jsha e any ex a su plus equally, p o ided his su plus is posi i e. Tha is, uij i= ij i+1 2 ij  ij i ij j= 1 2 ij + ij i ij j, i ij  ij i+ ij j. Howe e , i hei disag eemen payo¤s sum up o an amoun in excess o he wo h o he coali ion, ij < ij i+ ij j, hen playe s will no be willing o each an ag eemen . In his case, uij i= ik i. In he nex pa ag aph we discuss he easons and in e p e a ion o his spec- 10 As in he one-buye / wo-selle s example o in he h ee-playe / h ee-cake game, we assume ha only one o he wo-playe coali ions could o m, i he g and coali ion canno o m. See Sec ions 4 and 5 o mo e on his. 7 i…ca ion. ii) The disag eemen payo¤s a e compu ed acco ding o he payo¤s p e- dic ed in, and he p obabili y dis ibu ion o e al e na i e, wo-pa y nego i- a ions. In pa icula , assume ha he nego ia ion be ween iand j‡ounde s, and playe s con empla e hei op ions in he la ge pic u e o all wo-playe nego ia ions. As playe s calcula e wha hey expec o ge in his scena io, ij i, hey p edic ha , (a) wi h p obabili y pij wha hey ace is p ecisely his de aul , ij i; (b) wi h p obabili y pik coali ion (i; k)will each an ag ee- men , and playe i’s payo¤ will be uij i; and (c) wi h p obabili y pjk i will be coali ion (j; k)who will ag ee, and hence i’s payo¤ will be i. Thus, ij i=pij ij i+pikuik i+pjk i. I pij <1we can ew i e his exp ession as: ij i=pikuik i+pjk i 1pij : Thus, playe i0s allback op ion in he nego ia ion wi h jis he expec ed payo¤ in al e na i e nego ia ions, whe e he expec a ion is "condi ional" on he nego ia ion wi h jha ing come o a hal .11 I he sum o he disag eemen poin s in he nego ia ion be ween playe s iand kexceeds he wo h o ha coali ion, ik < ik i+ ik k, hen uij i= ik i, and hen he de…ni ion abo e implies ha ij i= i. In o he wo ds, i an ag eemen be ween iand kis no iable, hen when playe s iand jnego ia e hey an icipa e ha i hey do no each an ag eemen hen playe s jand kwill do so and sha e jk wi h p obabili y one, so ha playe i’s payo¤ will be i.12 Thus, playe i’s payo¤ when (hipo he ically) dealing wi h ki he nego ia ion wi h ja e suspended coincides wi h he payo¤ when dealing wi h junde he same assump ion, i.e., ij i. iii) pij is ( i ually) ze o i uik iuij iand ujk juij j, wi h one s ic in- 11 In con as o ou app oach, Bene (1997) assume ha playe s iand jbelie e ha each one o hem will be able o each an ag eemen wi h playe kwi h p obabili y one, in case nego ia ions be ween iand j ail. 12 No e ha i coali ion (i; k)is no iable hen uik iwill no en e in o he compu a ions o expec ed payo¤s, and will only ma e in he de e mina ion o ij i: 8 equali y. Tha is, an ag eemen be ween playe s iand jcanno be eached (wi h non-negligible p obabili y) i bo h playe s p e e hei al e na i e ag ee- men , one o hem s ic ly. Thus, we will build on he NBS by de…ning endogenous allback op ions o each nego ia ion. O en, ou solu ion will p edic ha some coali ion would o m wi h p obabili y one, should he h ee-playe coali ion ail o o m. Howe e , p obabili y one e en s lea e oo many deg ees o eedom wi h espec o wha a e consis en ou comes in he es o e en s. In o de o a oid his inde e minacy, we will p oceed in he s anda d way o … s conside ing only p obabili y dis ibu ions ha assign o each wo-playe ne- go ia ion a p obabili y o success bounded away om 1. De…ni ion 1 Fo  > 0, an P edic ion o simul aneous, bila e al ne- go ia ions o he h ee-playe game (N; ),PSBN o sho , is a iple nuij i(); ij i(); pij ()oi;j=1;2;3 ha sa is…es: 1) uij i() = (1 2 ij + ij i() ij j()i ij  ij i() + ij j(); ik i()o he wise; 2) ij i() = pij () ij i() + pik ()uik i() + pjk () i, o all i; j; k; 3) p12 () + p13 () + p23 ()=1;pij ()1 o all i; j; and o all i; j; k,pij ()<  i uij i()uik i()and uij j()ujk j(), wi h one s ic inequali y. Ou p edic ion o simul aneous, bila e al nego ia ions is he limi ing alue o p edic ions as he uppe bound on pij ends o 1. De…ni ion 2 A P edic ion o simul aneous, bila e al nego ia ions o he h ee-playe game (N; ), PSBN o sho , is a iple nuij i; ij i; pijoi;j=1;2;3 ha sa is…es lim!0nuij i(); ij i(); pij()oi;j=1;2;3=nuij i; ij i; pijoi;j=1;2;3. 9 he g and coali ion coincides wi h he co e i he la e is no emp y. I he co e is emp y, hen he ba gaining se o he g and coali ion is a single on. The p oo o his popula lemma is gi en in he Appendix. This lemma allows us o conside only he ela ionship be ween he Rsolu ion and he BS. P oposi ion 2 The Rsolu ion belongs o he ba gaining se ( o he g and coali ion) and so o he co e i he la e is no emp y. P oo . Fi s , we s udy he co e. An elemen o he co e is a posi i e ec o (x1; x2; x3)such ha : (i) x1+x2+x3=Vand (ii) xi+xj ij o all i; j. Adding up hese las h ee condi ions, we ob ain x1+x2+x3 12+ 13+ 23 2, which combined wi h condi ion (i) gi es: V 12 + 13 + 23 2:(2) When 12  13 + 23, i.e., in Regions 1 and 2, his is sa is…ed i ially. I is hen immedia e o check ha he Rsolu ion sa is…es (i) and (ii) in Regions 1 and 2. Thus, in Regions 1 and 2 he Rsolu ion belongs o he co e and hen o he BS. In Region 3 he co e may be emp y, ha is, (2) may no hold. Thus, we will show ha he Rsolu ion belongs o he BS. Remembe ha in Region 3 Ui=V+ ij + ik2 jk 3. Since Ui i o all i, and since he g and coali ion canno be pa o an objec ion, we need only conside objec ions ha use wo-playe coali ions. Thus, conside an objec ion o iagains j, o i= 1;2;3, and j6=i. Tha is, conside a di ision o ik,x= (xi; xk) whe e k6=i; j:xi+xk= ik, such ha xi> Ui, and xk> Uk. We show ha he e is a coun e -objec ion o j, ha is, a di ision y= (yj; yk)o jk whe e yj+yk= jk, such ha yjUjand ykxk. Conside in pa icula yj=Uj, so ha yk= jk Uj. I xi> Ui, hen xk= ik xi< ik Ui. Bu hen ykxk> jk Uj( ik Ui) = 0: 16 Thus, i xis an objec ion hen yis a coun e -objec ion. QED. In Region 3, when V < 12+ 13+ 23 2, since he BS is a single on and he Rsolu ion belongs o he BS, we conclude ha he Rsolu ion coincides wi h he BS, and so wi h any selec ion o subse o he BS, in pa icula he nucleolus and he ke nel (Nash se ). We nex discuss he Rsolu ion wi h ega d o hese concep s and he "consis ency" mo i a ions behind hem. Fo he es o his sec ion, le us es ic a en ion o he case V= 12. i.e., suppose ha he g and coali ion does no add su plus. In his domain, he Rsolu ion can be cha ac e ized om a pe haps su p ising pe spec i e. Indeed, le us label an alloca ion as he mos egali a ian in a se i i Lo en z- domina es he es o alloca ions in he se . P oposi ion 3 I V= 12, he Rsolu ion coincides wi h he selec ion o he mos egali a ian alloca ion in he ba gaining se . Thus, i also coincides wi h he selec ion o he mos egali a ian alloca ion in he co e, when he co e is no emp y. P oo . No e ha U1U2U3. Thus, a mo e egali a ian alloca ion would equi e o inc ease he payo¤ o playe 3o , a leas , o inc ease he payo¤ o playe 2by educing he payo¤ o playe 1. We show … s ha in Region 1and Region 2any alloca ion xin he BS o , equi alen ly in hese egions, in he co e assigns a payo¤ x3= 0. Assume o he wise x3>0. Then x1+x2= 12x3< 12, so ha he alloca ion would no be in he co e. This immedia ely p o es ha he Rsolu ion is he mos egali a ian alloca ion in he BS o Region 1. Now suppose ha we a e in Region 2and ha he e is an alloca ion x ha is mo e egali a ian han he Rsolu ion. Since x3= 0, his implies ha x2> 12 13, so ha x1+x3= 12 x2< 13 iola ing he condi ions o x o be in he co e. Thus, he Rsolu ion is he mos egali a ian alloca ion in he BS in Region 2. Finally, in Region 3 he co e is emp y, so ha he BS is a single on. Thus, he Rsolu ion is he only alloca ion in he BS. QED 17 Thus, he Rsolu ion is he mos egali a ian among he s able (in he sense o Aumman-Maschle ) alloca ions. Tha is, he mos egali a ian among he alloca ions ha canno be blocked in he sense o he (g and coali ion) BS. An al e na i e selec ion in he BS is he nucleolus. Fo hese games, he nucleolus is also a selec ion o he ke nel, i sel a subse o he BS. Thus, as we men ioned abo e, in Region 3 he ou concep s, BS, nucleolus, ke nel, and Rsolu ion, coincide. In egions 1 and 2 and when 12 =V( he co e is no emp y), he nucleolus is (x1; x2; x3) = 1 2( 12 + 13  23);1 2( 12 + 23  13);0.18 Bo h he ke nel and he Shapley alue coincide wi h he NBS o wo- playe , TU games. Mo eo e , each o he wo concep s has been shown o be he unique gene aliza ion o he NBS, in he sense ha each sa is…es a di - e en concep o in e nal consis ency (Se ano and Shimomu a, 1998; Ha and Mas-Colell, 1989).19 Fo he p esen discussion, he concep o in e nal consis ency means ha i x= (x1; x2; x3)is he co esponding solu ion i sa is…es he ollowing p ope y: o any pai o playe s i; j,(xi; xj)is he NBS o a educed game (N0; 0), whe e N0= i; jgand 0(N0) = xi+xj. The di - e ence be ween he wo consis ency c i e ia lies in wha 0( ig)and 0( jg) a e. Tha is, he disag eemen poin in he educed nego ia ion be ween i and j. Keeping he no maliza ion i= 0, o he ke nel (and nucleolus, since o hese games bo h concep s coincide), 0( ig) = max ik xk;0g(Se ano and Shimomu a, 1998), whe eas o he Shapley alue 0( ig) = 1 2 ik (Ha and Mas-Colell, 1989). Tha is, in bo h cases i wo playe s i; j ba gain o e how o sha e he o al ha he solu ion alloca es o hem, xi+xj, hey s ill ag ee on he di ision (xi; xj), p o ided he disag eemen poin is as speci- 18 See Leng and Pa la , 2010. 19 Comp e and Jehiel (2010) de…ne ano he ex ension o he NBS, he Coali ional Nash Ba gaining Solu ion, as he alloca ion ha maximizes he p oduc o payo¤s in he co e. No e ha , wi h h ee playe s and 12 =V, all co e alloca ions gi e a p oduc o payo¤s equal o 0. When he co e con ains an in e io (which equi es V > 12 ), he Coali ional Nash Ba gaining Solu ion and he mos egali a ian selec ion o he co e coincide. Ye he Rsolu ion is no he mos egali a ian selec ion in his case. 18 …ed. This la e poin is he c ucial di¤e ence be ween he ke nel and he Shapley alue on one hand, and he Rsolu ion on he o he . Jus like in he concep s p oposed by Benne (1997), nei he o he disag eemen pay- o¤s o he nego ia ion be ween iand jde…ned abo e come om a easible, al e na i e ag eemen . Fo ins ance, o i; j = 1;2, he disag eemen poin ha sus ains he ke nel is ( 13; 23)and he one ha sus ains he Shapley alue is ( 13 2; 23 2). Al hough hey imply a di¤e en di ision o he su plus wi h playe 3, in bo h cases he disag eemen payo¤s esul om playe 1 and also playe 2" ading" wi h playe 3. Bu hose wo ades a e mu- ually exclusi e, and in ha sense playe s’expec a ions a e no consis en . Ins ead, in he Rsolu ion, i wo playe s i; j ba gain o e how o sha e hei o al payo¤, Ui+Uj, hey s ill ag ee on he di ision (Ui; Uj)p o ided ha he disag eemen poin is a lo e y o e he payo¤s ( ik Uk;0) and (0; jk Uk), whe e he lo e y is pa o he solu ion. Tha is, he NBS and he Rsolu ion a e also consis en , bu in a way ha is i sel based on consis en ly compu ed disag eemen poin s.20 4 Applica ions In his sec ion we s udy in some de ail how ou solu ion concep changes he p edic ions o well-known Indus ial O ganiza ion models in which ba gain- ing among h ee playe s plays a c ucial ole. We s a wi h a model (Segal and Whins on, 2000) ha … s pe ec ly wi hin he se o games conside ed in p e ious sec ions. Nex , we discuss an example (Ho n and Pe sson, 2001) whe e bila e al ag eemen s gene a e ex e nali ies ( he wo h o an indi idual coali ion depends on whe he o no he o he wo playe s each an ag ee- men ). We a gue ha he Rsolu ion can also be applied o his ype o games (pa i ion unc ion o m) by simply aking in o accoun he alue o 20 In Region 3, all concep s coincide. The eason is ha in ha case he " easible" disag eemen poin and he "in easible" one lie on he same 45 deg ee line in he payo¤ space o any pai i; j. 19 indi idual coali ions condi ional on he ag eemen be ween he o he wo playe s. Mo eo e , in his example he g and coali ion canno o m and hence in his case he na u al solu ion concep is no he Rsolu ion bu he PSBN. Finally, we a gue ha he ideas con ained in he Rsolu ion can easily be ex ended o ma ch he h ee-playe example discussed by Ha and Moo e (1990). The main issue in his example is ha bila e al ades a e no mu ually exclusi e. 4.1 Exclusi e con ac s Segal and Whins on (2000), SW, s udy he impac o exclusi e con ac s. Thei main insigh is ha an exclusi e con ac enhances he abili y o he incumben selle o cap u e en s in he ex-pos ba gaining game, bu i is i ele an in p o ec ing his ela ion-speci…c in es men , unless such in es - men gene a es an ex e nali y on he en an . This is a somewha coun e in ui i e esul ha con adic s he con en ional wisdom (see, o ins ance, Klein, 1988, Ma el, 1982, o Mas en and Sneyde , 1993). He e we discuss a e sion o he model p esen ed in hei Sec ion 2. The e a e h ee playe s B; S; and E: Playe B(buye ) de i es a po en ial u ili y o 1 om one uni o he good ha can be p o ided by ei he S( he incumben selle ) o by E( he en an ). The e a e h ee pe iods: 0;1;and 2. In pe iod 0,Sand Bmay o may no sign an exclusi e con ac . In pe iod 1, playe S akes a cos ly in es men decision, x2[0;1], which a¤ec s he incumben selle ’s cos s. Also in pe iod 1, once xis …xed, playe s lea n he ealiza ion o a andom a iable y2[0;1], which in‡uences he en an ’s cos and is dis ibu ed acco ding o he cumula i e unc ion H(y)and has expec a ion by. In pe iod 2p oduc ion and ade ake place, and playe s ecei e hei payo¤s. Playe s Sand Ecan p oduce one uni o he good a a cos cs(x)and ce(y), espec i ely. Fo simplici y, we assume cs(x)=1x and ce(y)=1y. I in pe iod 0playe s Sand Bhad signed an exclusi e 20 con ac , hen in pe iod 2playe Bcanno pu chase om Ewi hou S’s pe mission. In bo h cases, wi h and wi hou an exclusi e con ac , B; S; and Eba gain in pe iod 2abou who p oduces he good and how he su plus is dis ibu ed. In hei gene al model SW use a gene aliza ion o he Shapley alue as he solu ion concep o he enego ia ion in pe iod 2:As an illus a ion o hei ideas le us apply he Shapley alue o he abo e simple e sion o hei model.21 In he absence o any con ac , he wo h o a ious coali ions is as ollows: V= max x; yg; SB =x; BE =y: (3) The es o coali ions ha e a wo h o 0. Unde he exclusi e con ac , he only di¤e ence is ha BE = 0: Acco ding o he Shapley alue, wi hou exclusi i y S0s payo¤ equals Une S=1 3max xy; 0g+x 6:Thus, S’s ma ginal e u n on in es men is 1 3H(x)+ 1 6:No e ha he ma ginal e u n on in es men o he pai (B; S) is 2 3H(x) + 1 3. Hence, om he poin o iew o he pai (B; S) he e is unde in es men ( he classic hold up p oblem). Su p isingly, unde he Shapley alue an exclusi e con ac does no help educing he unde in es men p oblem. Mo e speci…cally, unde exclusi i y playe S’s payo¤ is equal o Ue S=1 3max x; yg+x 6and he ma ginal e u n on in es men is also 1 3H(x) + 1 6:22 Conclusion 1 Unde he Shapley alue, an exclusi e con ac does no a - ec in es men incen i es. 21 In hei Sec ion 2, SW conside he case o a compe i i e en an who is willing o supply he good a a p ice pe= 1 y; and gi en such an ou side op ion playe s Band Sengage in ba gaining and he ou come is de e mined by he NBS. I u ns ou ha exclusi i y is also neu al wi h espec o in es men incen i es. 22 The ma ginal social e u n on in es men is H(x):Hence, he equilib ium le el o in es men may be below o abo e he … s bes le el. 21 I is impo an o emphasize ha he neu ali y esul hinges on he speci…c way he Shapley alue is compu ed. An exclusi e con ac only changes playe S’s payo¤ by changing his ma ginal con ibu ion o he g and coali ion. In he absence o exclusi i y S0s ma ginal con ibu ion o he g and coali ion is max xy; 0gand unde exclusi i y i is max x; yg:The di¤e ence be ween hese wo alues is y: Hence, unde exclusi i y S0spayo¤ inc eases by 1 3y(whe e 1 3is he weigh o he g and coali ion in payo¤s), bu S’s ma ginal e u n on in es men emains unchanged. Le us now analyze he same p oblem when we use he Rsolu ion o p edic payo¤s in pe iod 2. In his case, playe S’s payo¤ is Une s=8 < : x 2;i yx 2 xy; i xyx 2 0;i x < y Thus, S’s ma ginal e u n on in es men is H(x)1 2Hx 2. Once again, he e is unde in es men om he poin o iew o he pai (B; S): he ma - ginal e u n on in es men o he pai (B; S)is equal o min H(2x);1g: Unde exclusi i y, playe S’s payo¤ is he same ha we ound when we used he Shapley alue: Ue S=x 2;i yx; x 6+y 3, i yx: Thus, S’s ma ginal e u n on in es men is 1 3H(x) + 1 6. The e o e, unde exclusi i y in es men incen i es may be enhanced o dep essed wi h espec o he no con ac case. No e ha unde exclusi i y he Rsolu ion and he Shapley alue coin- cide. Hence, we need o unde s and why hese wo solu ion concep s deli e di¤e en payo¤s in he absence o a con ac . In he la e case, i y > x he Shapley alue g an s playe Sa payo¤ o x 6. I we hink in e ms o he sequen ial a i al in e p e a ion o he Shapley alue, such payo¤ esul s om he ac ha Smakes a posi i e con ibu ion in case he a i es second a e playe B. Howe e , acco ding o he Rsolu ion playe Sis edundan 22 and should ge a ze o payo¤ (he is playe 3and he only hing ha he migh do is o in‡uence he way yis spli be ween Band E). Thus, unde he Rsolu ion incen i es o in es will be enhanced i y > x is a likely sce- na io; i.e., i in es men cos s a e ela i ely high so ha xis low. Howe e , i x 2< y < x hen playe S’s ma ginal con ibu ion o he coali ion wi h B is x(weigh 1 6) and he ma ginal con ibu ion o he g and coali ion is xy (weigh 1 3). Hence, acco ding o he Shapley alue Sis able o app op ia e one hal o his in es men e¤o s. In con as , he Rsolu ion g an s playe Sa payo¤ o xy(in his case playe Sis playe 2), and hence he is able o app op ia e he en i e e u n on in es men . Thus, unde he Rsolu ion incen i es o in es a e dep essed i x 2< y < x is a likely scena io; i.e., i in- es men cos s a e ela i ely low and xis high. In his case, he pa adoxical esul ob ained by SW is magni…ed.23 Conclusion 2 Unde he Rsolu ion, an exclusi e con ac enhances in- es men incen i es i he cos o in es men is ela i ely high, bu he oppo- si e holds i he cos is ela i ely low. In o he wo ds, unde he Rsolu ion exclusi i y helps p o ec ing ela ion- speci…c in es men s only when he selle ’s compe i i e posi ion is su¢ cien ly weak. Exclusi i y is use ul only when he e is a lo o p o ec .24,25 4.2 Endogenous me ge s Ho n and Pe sson (2001), HP, p esen a model o endogenous me ge o - ma ion. He e we ocus on he example discussed in hei Sec ion 2.1, which 23 I y < x 2S’s ma ginal e u n on in es men is equal o 1 2;unde bo h he Shapley alue and he Rsolu ion. 24 I in es men cos s a e su¢ cien ly high (xlow); hen he le el o in es men unde exclusi i y is ine¢ cien ly high. In o he wo ds, om a social poin o iew an exclusi e con ac may ac ually o e p o ec ela ion-speci…c in es men s. 25 De Meza and Sel aggi (2007) also show ha SW’s conclusions a e no obus o changes in he solu ion concep o he ba gaining game. They se up a non-coope a i e ba gaining game ha deli e s di¤e en p edic ions han he Rsolu ion and show ha exclusi i y always enhances in es men incen i es. 23 conside s a ma ke ini ially popula ed by h ee oligopolis ic … ms, 1, 2, and 3. They a e allowed o me ge, bu no o o m a monopoly. In o he wo ds, he e a e ou possible ma ke s uc u es: no me ge , 1 and 2 me ge, 1 and 3 me ge, and 2 and 3 me ge. Al hough … ms a e symme ic be o e any me ge , he syne gies gene a ed by al e na i e me ge s a e asymme ic. Fi ms’p o… s in he no me ge case a e no malized o 0. P o… s o he … m esul ing om he me ge be ween … ms iand j, and he non-me ged … m ka e deno ed by ij and k espec i ely and a e: 12 = 70; 3= 50; 13 = 100; 2= 0; 23 = 90; 1= 5: In p e ious sec ions we de…ned he Rsolu ion o games in cha ac e - is ic o m, whe e he alue o a coali ion is independen o he ag eemen s eached by playe s no included in he coali ion. Howe e , in HP he alue o s and-alone coali ions do depend on whe he o no he o he wo play- e s ha e eached an ag eemen . Thus, his model can be desc ibed as a game in pa i ion unc ion o m (Lucas and Th all, 1963). In a h ee-playe game, we need o speci y wha playe ican ob ain i no coali ion is o med, wi ig; jg; kgg;and wha playe iob ains i he o he wo playe s do o m a coali ion, wi ig; j;kgg. Mye son (1977) ex ended he Shapley alue o pa - i ion unc ion o m games. In his ex ension, playe i’s payo¤ depends on bo h, wi ig; jg; kgg and wi ig; j;kgg. On he con a y, he de…ni ion o he Rsolu ion al eady akes in o accoun possible ex e nali ies. The s and- alone wo h plays a ole only in he de…ni ion o he alues ij iand ik i. These alues a e ob ained as a p obabili y dis ibu ion o e he e en s ha can be expec ed as an al e na i e o i o ming coali ion wi h jo k, e- spec i ely. The only such e en ha has is anding alone is he o ma ion 24 o coali ion j; kg. Thus, only wi ig; j;kgg ma e s, and hen ishould be in e p e ed as his alue. Summa izing: Rema k 2 The Rsolu ion de…ned o games in cha ac e is ic o m can also be applied o games in pa i ion unc ion o m, simply by eplacing he wo h o indi idual coali ions, i, wi h he wo h o indi idual coali ions condi ional on he o he wo playe s o ming a coali ion, wi ig; j;kgg: The ne su plus c ea ed by each me ge is gi en by: 12 12= 65; 13 13= 45; 23 23= 35: Thus, he mos e¢ cien me ge ( om he poin o iew o … ms’p o… s) is he one be ween … ms 1and 2. HP use as a solu ion concep he se o ma ke s uc u es ha a e no domina ed om he poin o iew o decisi e playe s. In o he wo ds, in an Equilib ium Owne ship S uc u e (EOS) he sum o p o… s achie ed by all decisi e playe s mus be a leas as high as in any o he ma ke s uc u e. Since in his example all playe s a e decisi e when we compa e al e na i e duopolies (all … ms ha e a di¤e en posi ion in each possible ma ke s uc u e esul ing om a me ge ) hen he only ma ke s uc u e which is undomina ed is he esul ing om he me ge be ween … ms 1and 2:In o he wo ds, HP p edic ha he mos e¢ cien ma ke s uc u e will occu wi h ce ain y. Conclusion 3 Unde he no ion o Equilib ium Owne ship S uc u e he e¢ cien me ge occu s wi h p obabili y one. HP do no allow any ans e be ween he me ged and non-me ged … ms. Hence, since he g and coali ion canno be o med, we canno di ec ly ap- ply he Rsolu ion o his pa icula model. Howe e , we can s ill p edic 25 Moo e (1990) pape . In he spi i o ou solu ion concep , he se o possible e en s when he g and coali ion ails o o m should in‡uence he way pa ies sha e he su plus i he g and coali ion does o m. In a companion pape , we ex end he Rsolu ion o games wi h any such se o possible e en s. In gene al, he in o ma ion con ained in he cha ac e is ic unc ion o a game is no su¢ cien o de e mine ha se . The e o e concep s ha a e de…ned on only he in o ma ion con ained in he cha ac e is ic unc ion, like he Shapley alue, will be insensi i e o a ia ions in he se o possible e en s. The s udy o games in ol ing mo e han h ee playe s poses new ques- ions ha a e no p esen in he cu en analysis. One se o such ques ions has o do wi h he hie a chy o coali ions and is ela ed o he discussion in he p e ious pa ag aph. As we ha e jus men ioned, in his pape we ha e assumed ha i he g and coali ion b eaks down hen only one ade be ween wo playe s can be ealized. In ac , he e a e h ee al e na i e wo-playe coali ions and each one o hem is expec ed o s ike a deal wi h ce ain p obabili y. The e o e, compu ing he allback op ion o each playe in each coali ion is ela i ely s aigh o wa d. Howe e , in a ou -playe game, i he g and coali ion ails hen he ele an al e na i es a e no so easy o ob ain e en i we impose ha only disjoin coali ions can o m. The al e na i e o he g and coali ion may be a one h ee-playe coali ion, excluding he ou h playe bu i may also be wo disjoin wo-playe coali ions. Speci ying he allback op ion o a pa icula playe in an a bi a y coali ion can s ill be done along he lines discussed in Subsec ion 4.3, bu i in ol es a highe deg ee o complexi y. We lea e he analysis o games wi h mo e han h ee playe s o u u e esea ch. 6 Re e ences Benne , E. (1997), "Mul ila e al Ba gaining P oblems", Games and Eco- nomic Beha io 19, 151-179. 32 Binmo e, K. (1985) "Ba gaining and Coali ions." Chap e 13 in Game- heo e ic Models o Ba gaining, ed. Al in Ro h. Camb idge: Camb idge Uni e si y P ess, 269-304. Binmo e, K., A. Shaked, and J. Su on (1989) "An Ou side Op ion Ex- pe imen ", Qua e ly Jou nal o Economics 104 (4), 753-770. Binmo e, K., M. Osbo ne, and A. 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Whins on (1995), On he E¢ ciency o P i a ely S ipu- la ed Damages o B each o Con ac : En y Ba ie s, Reliance, and Rene- go ia ion, The RAND Jou nal o Economics 26, (2) (Summe ), 180-202. Win e , E (2002) "The Shapley alue", chap e 53 in R.J. Aumann & S. Ha (ed.) Handbook o Game Theo y wi h Economic Applica ions, 2025- 34 2054. Webe , R. (1988) "P obabilis ic Values o Games", chap e 7 in A. E. Ro h (ed.) The Shapley alue: essays in hono o Lloyd S. Shapley, 101-120. 7 Appendix 7.1 P oo o P oposi ion 1: Fi s we p opose an PSBN o he game (N; ) o small enough. This will show exis ence. To sa e in no a ion, we will dispose o he ()index o he solu ion, and speci y i we e e o he limi ins ead. 1) Le 1 2 12  13. 1.a) I 1 2 12 > 13 (so ha 12  13 + 23 is also sa is…ed), conside u12 1=u12 2=1 2 12, and uij i= 0 o all o he alues o i; j. Also, le p12 = 1, p13 =p23 = 2. Finally, le 12 1= 12 2= i3 3= 0 and i3 i=1 2 12 o i= 1;2. No e ha lim!01 2 12 =1 2 12 > 13  13. Thus, o su¢ cien ly small, his sa is…es he de…ni ion o an PSBN. 1.b) I 1 2 12 = 13 > 23, conside u12 1=u12 1=1 2 12 (= 13), and u23 2=u23 3= 0. Also, le p12 = 1 and p13 = 0,p23 =. Then, 12 1= 23 3= 12 2= 13 3= 0 and 23 2=1 2 12 > 23. To comple e he de…ni ion o an PSBN we need only 13 1= (1 )1 2 12,u13 1=1 2( 13 + 13 1 13 3) = (1  2) 13 and u13 3= 2 13. 1.c) I 13 = 23 =1 2 12, conside u12 1=u12 2=1 2 12 (= i3,i= 1;2), p12 = 1 and p13 =p23 = 2. Then i3 i=(1) 12 2< i3,i= 1;2. Also, conside u13 3=u23 3=A > 0. Thus, i3 3,i= 1;2, will ha e o sa is y: i3 3=A 2;and A=1 2 i3(1 ) 12 2+A 2; and sol ing o A aking in o accoun ha 1 2 12 = i3, we ob ain A= i3 43; which is smalle han i3 o small . No e ha o small i3 3+ i3 i< i3, i= 1;2. Also, no e ha gi en hese alues o ui3 i, we should de…ne 12 1= 12 2=  2( i3A) =( 122A) 4, and 12 1+ 12 2< 12. This sa is…es he de…ni ion o an PSBN. 2) I 12  13 + 23 bu 13 >1 2 12, hen conside u12 1=u13 1=u1, o be ob ained la e , wi h 0< u1< 13, and u23 2=u23 3= 0. Thus, u12 2= 12 u1> u23 2and u13 3= 13 u1> u23 3. Consequen ly, le p23 =. Then 35 p12 = 1 p13. Finally, u23 2=u23 3= 0 implies ha 12 2= 13 3= 0, and we can hen check ha 12 2+ 12 2< 12, whe eas 13 1+ 13 3=p12u12 2 1+p13u13 3 1 = ( 12 u1)p13( 12  13) 1: We will p opose u1su¢ cien ly close o 13 so ha 13 1+ 13 3 13. In ha case, ushould sa is y u1=1 2( 13 +(1 p13)u1 1p13 ) = 1 2( 12 +p13u1 +p13 ): This is a sys em o wo equa ions wi h wo unknowns. No e ha i we ha e a ( alid) solu ion o his sys em, hen as app oaches 0 he … s equa ion app oaches u1=1 2( 13 +u1)whose only solu ion is 13 =u1. (Fo posi i e , indeed u1< 13.) Thus, o small enough, 13 1+ 13 3< 12u1= 122 13 + 13 + ( 13 u1)and he igh hand side con e ges o 12 2 13 + 13 < 13. Also, sol ing o u1, we can w i e he sys em as 13 1 +  p13 += 12 1 +  1p13 : This is a quad a ic equa ion in p13 wi h one posi i e oo ha con e ges o 0as con e ges o ze o. Thus, we ha e an PSBN o small enough. And o small, p12 is close o 1. 3) I 12 < 13 + 23, hen p opose uij i=uik i=ui>0, o all i; j; k. Then he de…ni ion o uij i equi es ha ui+uj= ij o all i; j. This is a sys em o h ee linea (independen ) equa ions wi h solu ion ui= ij + ik jk 2. Also, ij i=pikui 1pij . Finally, pshould sa is y ui=1 2( ij +pikui pik +pjk pjkuj pik +pjk ) o all i; j; k. Taking in o accoun ui+uj= ij, hese equa ions can be w i en as p13u2+p23u1= 0; p12u3+p13u2= 0; p12u3+p23u1= 0: No e ha he hi d equa ion is simply he sum o he p e ious wo. Tha is, he e a e only wo linea ly independen equa ions. Thus, wo o hese equa ions plus p13 +p23 +p23 = 1 o m a linea sys em wi h a unique solu ion. The solu ion is a p obabili y dis ibu ion, since all h ee a iables ake posi i e alues. Indeed, he … s wo equa ions can be w i en as p13 u1= 36 p23 u2and p12 u2=p13 u3, so ha all solu ion ec o s o hese wo equa ions ha e ei he all posi i e componen s o all nega i e. And no solu ion wi h all nega i e componen s sa is…es he equa ion p13 +p23 +p23 = 1. Finally, no e ha ij j+ ij i=pjkuj pjk+pik pikui pjk+pik , so ha since bo h uj; ui< ij, indeed ij j+ ij i< ij. This concludes he p oo o exis ence. Nex , we can simply check ha i we selec he PSBN ha we ha e jus cha ac e ized o each possi- ble alues o ij o all ij, hen he lim!0 u(); (); p()gis as s a ed in he P oposi ion. Thus, we only need showing ha he e is no o he iple u; ; pg ha is he limi o a sequence o PSBN as app oaches 0. Fi s we p o e a handy esul . Lemma 2 In a PSBN, cycles canno occu . Tha is, i canno be ha uij iuik i;ujk juij j;uik kujk k o some alues o i; j; k. Mo eo e , uij i= uik i;ujk j=uij j;uik k=ujk kcan only occu i 12  13 + 23. P oo o Lemma: Fi s , assume ha we ha e such cycle wi h a leas one s ic inequali y, and such ha ij i+ ij j ij o all ij. In any such cycle, uij i=1 2 ij + ij i ij j o all i; j; k. Subs i u ing o ij =uij i+uij j, and also subs i u ing o ij i=pik 1pij uik i(6) we can w i e his exp ession as (uij iuij j)(1 pij) = pikuik ipjkujk j(7) Adding hese h ee equa ions, o all h ee pai s, his implies ha (uij iuij j)+(uik kuik i)+(ujk jujk k) = 0; ha is, uij i+uik k+ujk j=uij j+uik i+ujk k, which iola es he inequali ies de…ning he cycle i he e is one ha is s ic . Second, assume ha ij i+ ij j> ij o some ij, bu ik i+ ik k ik, and jk j+ jk k jk. Gi en he cycle, his implies ha uij i=uij j=uik i= 0, so ha also uik k= ik. Thus, equa ions (7) o he pai jk become (ujk jujk k)(1 pjk) = pik ik. Since pjk <1, ha implies ujk kujk j. No e, howe e , ha jk j= 0, since uij j= 0, so ha ujk jujk k. These wo inequali ies hen imply bo h ujk j= ujk k= jk 2, and pik = 0. Since he cycle inequali ies include uik kujk k, hen we mus ha e ik  jk 2. Bu subs i u ing o ujk k= jk 2and pik = 0 in 37 (6) co esponding o ik k, we also ha e ha ik k=pjk jk 2< jk 2 ik. This con adic s ha uik k= ik. Thi d, assume ha ij i+ ij j> ij and ik i+ ik k> ik o some ij and ik bu jk j+ jk k jk. Tha implies ha uij i=uij j=uik i=uik k= 0, which implies ha jk j= jk k= 0, so ha ujk k= jk 2> uik k, which con adic s he inequali ies in he cycle. Thus, he only cycle ha may exis is uij i=uik i;ujk j=uij j;uik k=ujk k, wi h ij i+ ij j ij o all ij. Bu he sys em ui+uj= ij, o all ij has a alid solu ion only in Region 3, and coincides wi h he one ound abo e. QED Thus, an PSBN mus sa is y: uij iuik i;ujk juij j;uik kujk k;(8) and excep o he one we used in 3) abo e, a leas wo inequali ies mus be s ic . Also, gi en pa h ee o he de…ni ion o PSBN, pik <  unless uij i=uik iand uik k=ujk k. Thus, in any bu he PSBN cons uc ed in 3) abo e, pik < . Thus, in a sequence ha con e ges as !0, we mus ha e lim!0pik = 0. Conside such a sequence o PSBN so ha lim!0pij >0and lim!0pjk > 0. F om (8) and pa h ee o he de…ni ion o PSBN, ha implies ha o small uij j=ujk j. Thus, since a leas wo inequali ies need o be s ic , uij i> uik iand uik k< ujk k. These las inequali ies imply ha ujk k+ujk j= jk and uij i+uij j= ij. Also, as app oaches 0,pjk pjk+pik app oaches 1, as does pij pij +pik , so ha applying pa one o he de…ni ion o a PSBN, uij j=ujk j!uj=1 2( ij +uj) = 1 2( jk +uj): This canno occu unless ij = jk. In he la e case, uj= ij = jk, which implies ha bo h uij iand ujk kcon e ge o 0, and so ik i+ ik kcon e ges o 0, in which case uik icon e ges o ik 2> uij i o small and when ik >0. This is a con adic ion unless ik = 0. Bu i ij = jk, ik = 0, he limi o such a sequence coincides wi h he PSBN cons uc ed in 3) abo e. Thus, we mus ha e ha bo h lim!0pik = 0, and ei he lim!0pjk = 0 o lim!0pij = 0. Bu i lim!0pij = 0 hen lim!0pjk >0, and his con- adic s pa 3 o he de…ni ion o an PSBN since uij iuik iand uij jujk j wi h a leas one inequali y. Thus, assume ha lim!0pik = lim!0pjk = 0. We conside wo possible cases: 1) Assume ha jk j+ jk k> jk in all he e ms o he sequence29 as  con e ges o 0, so ha ujk j=ujk k= 0 = ij j, o each small enough in he 29 No e, in gene al, ha excep in i ial cases, ei he his is sa is…ed o close o 0o else he sequence canno con e ge. 38 sequence conside ed. Thus, om (8), we mus also ha e ha uik k= 0. Since only one inequali y in (8) may be non s ic , and ujk k=uik kwe mus ha e uij i> uik i, and since ujk j= 0, we mus also ha e ha uij j> ujk j. These wo inequali ies imply ha uij i+uij j= ij. Since ij j= 0, we mus hen ha e ha uij i ij 2. Thus: 1.a) I ij 2> ik, since ik icon e ges o uij i ij 2, hen o small we mus also ha e ha ik i+ ik k> ik, so ha uik i= 0, and hen ij j= ij i= 0, and hen uij j=uij i= ij 2. No e ha jk juij jand jk k= 0. Thus, o jk j+ jk k> jk, i mus be ha ij 2> jk. This equi es ha ij = 1;2and also ha we a e in Region 1. Thus, he limi o such sequence is he one s a ed in he P oposi ion. 1.b) I ij 2 ik, as be o e, i ik i+ ik k> ik, hen uij j= ij 2, and since ik k= 0, his would imply ha ij 2 ik i> ik which is a con adic ion. Thus, we mus ha e ik i+ ik k ik. Thus, since uik i= 0, we mus ha e uik k= ik. Since ujk k= 0, his con adic s he inequali y uik kujk kin (8) unless ik = 0. In he la e case, since ij 2 ik, ij = 0 and we ha e a con adic ion wi h jk j+ jk k= 0 > jk. 2) Assume ha jk j+ jk k jk in all he e ms o he sequence as  con e ges o 0. 2.a) I ij i+ ij j ij, hen ujk j jk j=pij pij +pik uij j; whe e he igh hand side con e ges o uij j. F om, (8),uij jujk j. Thus, he limi o any such sequence should sa is y lim!0uij j= lim!0ujk j= lim!0 jk j. Tha implies ha lim!0ujk k= 0, and equi es ha ij  jk, and lim!0uij i= ij  jk. Since ujk kuik k, hen we also ha e lim!0uik k= 0. Bu i lim!0ujk k= 0, hen lim!0 ik k= 0, whe eas ik iuik i. Thus, i ik > ij  jk, hen lim!0 ik i+ ik k< ik and hen lim!0uik k> ik( ij  jk) 2>0, which is a con adic ion. The e o e, ik  ij  jk. Since lim!0 ik i= lim!0uij i= ij  jk, hen lim!0uik k= 0 = lim!0ujk k. Thus, lim!0uik i> 0, only i lim!0uik i= ij  jk. In his case, we would ha e ui= lim!0uij i= lim!0uik i,uk= lim!0uik k= lim!0ujk k, and uj= lim!0uij j= lim!0ujk j. This equa ion, oge he wi h ui+uk= ik,ui+uj= ij uj+uk= jk has a solu ion only in Region 3( ik = ij  jk). Thus, i ik < ij  jk,uik i= 0 o small, so ha ik i= 0, so ha uij j ij 2, and hen jk  ij 2. This is Region 2, and he limi coincides wi h he one s a ed in he P oposi ion. 2.b) I ij i+ ij j> ij, hen uij i(=uij j)= 0, so ha ik i= 0, and since om (8) uij iuik i, hen uik i= 0. On he o he hand, ik kapp oaches 0as !0, 39 and hen ik i+ ik kapp oaches 0, which con adic s uik i= 0 unless ik = 0. Mo eo e in his la e case jk j= jk k= 0, so ha ujk j=ujk k= jk 2, so ha ujk j> uij jwhich con adic s (8). 7.2 P oo o Lemma 1 Wi hou loss o gene ali y, assume ha i= 0, o all i= 1;2;3. Assume he co e is no emp y, ha is, condi ion (2) holds, and ha xdoes no belong o he co e. We will show ha xdoes no belong o he BS o he g and coali ion. We do no need o conside alloca ions whe e xi<0 o some i, o whe e x1+x2+x3< V , since hey canno be in he BS. Thus, assume ha xi+xj< ij o some i; j, so ha xk> V  ij , o k6=i; j: Conside an objec ion yo iagains kwhe e yi+yj= ij, wi h yi> xiand yj> xj. A coun e -objec ion zo kagains iwould ha e o sa is y ha zjyj, and zj+zk= jk, so ha zk jk yj= jk ( ij yi). Also, zkxk> V  ij. The e o e, i jk ( ij yi)< V  ij; o yi< V  jk; hen he objec ion ywould ha e no coun e -objec ion and xwould no belong o he BS. I xi< V  jk we can always cons uc such y, and hen a necessa y condi ion o x o belong o he BS is ha xiV jk. Swi ching he subsc ip s iand j, we could conside an objec ion y0o jagains k, and epea he a gumen o show ha a necessa y condi ion o x o belong o he BS is ha xjV ik. Thus, a necessa y condi ion is ha xi+xj2V jk  ik  ij; whe e he las inequali y ollows om condi ion (2). This con adic s ha xi+xj< ij and p o es ha he BS coincides wi h he co e when he la e is no emp y. Now assume ha condi ion (2) is no sa is…ed. In pa icula , his implies ha we a e in Region 3. We ha e shown abo e ha he Rsolu ion belongs o he BS. So we only need o show ha any o he alloca ion does no belong o he BS. No e ha (2) implies ha o any easible alloca ion (including he e¢ cien ones), i xi=Ui+(in Region 3), hen xj+xk jk , o any  > 0. So, conside an e¢ cien alloca ion such ha his is he case o some , and an objec ion yo jagains i, wi h yj=xj+ 2and yk= jk yj= jk xj 2. A coun e -objec ion zo i agains jshould sa is y ha zkykbu also zixi, so ha zk ik xi. Thus, o i o indeed ha e a coun e -objec ion agains ji is equi ed ha ik xi= ik Uiyk= jk xj 2; ha is, xj jk  ik +Ui+ 2=Uj+ 2, whe e he las equali y ollows om he de…ni ion o Ui. Thus, his is a necessa y condi ion o x o be in he BS. Swi ching he subsc ip s jand k, we would also conclude ha ano he necessa y condi ion is ha xkUk+ 2. Thus, a necessa y condi ion is ha xi=VxjxkVUjUk=Ui. And his con adic ion p o es he esul . QED 40 7.3 P oo o Rema k 4 Exis ence: Fo small, le u1i 1=u1i i= 1i 2 o i= 2;3,p= 12, and p1i= o i= 1;2. Thus, 1i i= 0, and u1i i=u1i 1= 1i 2. Also, 23 i= (1 ) 1i 2, so ha 23 2+ 23 3= (1 ) 12+ 13 2. I 12+ 13 2> 23, hen o small u23 i= 0. Hence, 12 2= 13 3= 0. I 12+ 13 2= 23, hen i mus be ha 12 = 13 = 23. Then u23 i= 23 2. S ill, 12 2= 13 3= 0. In bo h cases, we ha e an PSBN wi h uindependen o . In he limi , p= 1. Thus, applying he de…ni ion o he Rsolu ion, U1= 12+ 13 2, and Ui= 1i 2. Uniqueness: Conside a limi o PSBN whe e u12 1=u12 2= 0. This means ha o small 12 1+ 12 2= 12 2=p23u23 2 1(p+p12)> 12. This is a con a- dic ion, unless p23 =p13 = 0. since u23 2 23  12 and p23 1(p+p12). Also p23 =p13 = 0 implies p+p13 = 1, which iola es he hi d condi ion in he de…ni ion o an PSBN, so indeed u12 1=u12 2= 0 canno be he limi o a sequence o PSBN. Fo he same a gumen , we canno ha e u13 1=u13 3= 0. Thus, since 1i 1= 0, we ha e ha u12 2 12 2, and u13 3 13 2. Also, u1i 1>0, since p23 1, and so 1i i< 23 o all  > 0. Thus, p1i , o i= 2;3. Thus, 23 2+ 23 3=(p+p12)u12 2 1p23 +(p+p13)u13 3 1p23 con e ges o u12 2+u13 3 as con e ges o 0, so ha o small, u12 2+u13 3> 12 2+ 13 2 23. Thus, o small, u23 i= 0, and hen 1i i= 0, o i= 2;3, and p23 . Thus u1i 1; u1i ishould con e ge o 1i 2, and pshould con e ge o 1. This comple es he p oo . QED 41