Matematcal Socal Scences 45 (003) 85 03 www.elsever.com/locate/econbase a On coalton formaton: durable coalton structures a b, Salvador Barbera `, Anke Gerber * Departament d Economa d Hstora ` Economca ` and CODE, Unverstat Autonoma ` de Barcelona, 0893 Bellaterra, Barcelona, Span Insttute for Emprcal Researc n Economcs, Blumlsalpstrasse 0, CH-8006 Zurc, Swtzerland Receved June 00; accepted 30 January 003 b Abstract We defne a soluton to te problem of coalton formaton tat apples to purely edonc games. Coalton structures satsfyng our requrements are called durable, and we nterpret tem as muc more lkely to last tan tose coalton structures not satsfyng te requrements, wc we call transent. Durablty results from a combnaton of foresgt and extreme rsk averson on te part of agents, wen consderng to on oters to dsrupt an exstng structure n searc of ger gans. Agents calculatons are also constraned to satsfy a strong consstency requrement, wc s reflected n te recursve structure of our defnton. We prove tat durable coalton structures always exst, and we provde examples of edonc games were our solutons apply ncely. 003 Elsever Scence B.V. All rgts reserved. Keywords: Hedonc games; Foresgt; Rsk averson JEL classfcaton: C7. Introducton Gven te nterests of agents n a socety, and ter possbltes to cooperate, wll durable socal arrangements exst? If so, ow durable wll tey be? Wat s t tat keeps te members of groups togeter and apart from oter groups? Te answers to tese questons and te predctons we obtan from tem are relevant for many economc and socal problems, n partcular for te study of local publc goods suc as, for example, n Guesnere and Oddou (98), Greenberg and Weber (986, 993) and Demange (994). Despte ts relevance, te ssue of coalton formaton as been neglected untl very *Correspondng autor. Tel.: 4--634-3708; fax: 4--634-4907. E-mal address: agerber@ew.unz.c (A. Gerber). 065-4896/03/$ see front matter 003 Elsever Scence B.V. All rgts reserved. do:0.06/ S065-4896(03)0005-8
86 S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 recently. In te context of caracterstc functon games, were eac coalton can coose from a set of feasble utlty allocatons, te focus was on te queston of te coce of a utlty allocaton for some exogenously gven coalton structure. Among te notable exceptons tat also address te ssue of coalton formaton are Senoy (979), Hart and Kurz (983), Bennett and Zame (988), Zou (994) and Gerber (000). In ts paper we concentrate on te aspect of coalton formaton and defne a soluton concept tat apples to edonc coalton formaton games: tat s, to stuatons were agents derve utlty from belongng to a group, and do not care about te arrangements among people outsde ter own group. Unlke Dreze ` and Greenberg (980), wo ntroduced te edonc aspect of coalton formaton, we restrct ourselves to stuatons tat are purely edonc n te sense tat, wtn a coalton, tere s no dstrbutonal ssue to solve. Rater, an agent s utlty only depends on te composton of te group e belongs to. Suc edonc games ave recently been studed by Bogomolnaa and Jackson (00) and Baneree et al. (00). Bot papers provde suffcent condtons for te nonemptness of te core, wc s by no means guaranteed n ts context. Alcalde and Revlla (00) gve anoter condton ensurng tat te core s not empty, and also tat te drect mecansms tat assgn partcpants n coalton formaton games to coaltonal structures n te core wll be strategy-proof. Papa (000) provdes an even more strngent condton on famles of coalton formaton games, wc guarantees te exstence of a unque core allocaton n all games wtn any famly meetng er sngle-lappng requrement. Bogomolnaa and Jackson (00) also propose te weaker notons of ndvdual and Nas stablty and analyze te exstence of coalton structures satsfyng tese requrements. We propose a new soluton concept wc captures te noton of maxmn beavor on te part of te players and we prove tat our soluton selects a nonempty set of wat we call durable coalton structures for any purely edonc game. Wen te core of te coalton game s nonempty, all coalton structures n te core wll be declared to be durable. Wat makes a coalton structure durable s a combnaton of foresgt and rsk averson. Agents wll not dsrupt a coalton structure f tey can foresee tat tey may end up n a stuaton tat makes tem worse off. Our defnton qualfes ts statement and makes t precse. Suppose tat a coalton R s consderng to form, tus dsruptng a coalton structure 6 were te agents of R do not form a group. Wat coalton structure wll ts cange lead to? We consder tat t wll lead to any of te coalton structures 69 avng te propertes. R s one of te coaltons n 69,. coaltons n 6 not affected by te formaton of R reman n 69, and 3. agents wo were prevously assocated wt members of R, but wo are not part of R, are assocated n a coalton substructure tat would be durable f tey were alone n socety. Requrement 3 ntroduces a recursve element nto our defnton of durablty. We sall defne wat coalton structures are durable on te bass of a constructon It s often assumed tat a grand coalton s formed.
S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 87 wc already assumes wat coalton structures would be durable would a smaller socety form coaltons of ts own. Ts recursve element ncorporates a noton of consstency n te calculatons of agents: wen udgng weter certan coaltons mgt form, tey apply te same equlbrum concept to subsocetes as tat resultng for socety as a wole from ter own calculatons. Jont devatons from 6 by several coaltons wll drectly lead to anoter coalton structure 69 by smlar consderatons. We defne te transtve closure of ts relaton, and say tat 6 leads to 69 f te two stand n ts new relaton. 6 wll be durable f any pat away from t eventually leads to some coalton structure 69 were one of te agents dsruptng 6 would be worse off. We are aware tat our noton of durablty ncorporates some asymmetry n te degree of ratonalty tat eac agent attrbutes to mself, as compared to tat e attrbutes to oters. We elaborate on ts pont after te formal defnton of durablty n Secton, were we empasze te nterpretaton and te arguments tat lead us to propose and defend our concept. Its merts sould also be udged aganst te background of te alternatve proposals n te currently growng lterature on coalton formaton. Tere are a few papers tat ncorporate farsgtedness n a context were coaltons can nduce certan alternatves from oters. Cwe (994) proposes te noton of a consstent set, wc sares te dea of maxmn beavor wt our noton of durablty. Accordng to Cwe, a set of outcomes (coalton structures n our context) s consstent f any devaton from an outcome n ts set s deterred, because t may lead to anoter outcome n te set tat makes some devator worse off. In te same framework, Xue (998) utlzes Greenberg s (990) teory of socal stuatons and ntroduces te noton of a stable standard of beavor. Kons and Ray (999) model coalton formaton as a dynamc process and analyse equlbrum devaton scemes, tereby capturng perfect foresgt on te part of te players. Battacarya (00) ntroduces a furter requrement n Cwe s analyss, by mposng a credblty constrant over te type of outcomes tat can domnate oters. None of tese concepts, toug, s drectly applcable to a edonc game, or, more generally, to a caracterstc functon game. Te reason for ts s tat tese models take as gven an effectveness relaton wc descrbes te set of outcomes a devatng coalton can nduce from a gven alternatve. However, n a context were te utlty of a player depends on te group e belongs to, tere s no naturally gven effectveness relaton, and t s too nave to assume tat tose players wo face a coaltonal devaton by oters (te resduals ) eter stay togeter or dssolve nto sngletons. It s even more crtcal to assume tat a devatng coalton can mpose any coalton structure on te resduals. Hence we beleve tat one mportant aspect of our soluton concept s te development of a consstent teory of were coaltonal devatons lead to. Anoter dfference between our soluton and te concepts mentoned above s tat our noton of durablty can be referred drectly to eac possble coalton structure, wereas, for example, Cwe s consstency s not te property of any ndvdual coalton structure, but refers to sets of suc obects. Damantoud and Xue (000) extend several of te Ts dfference s analogous to tat between te core and te von Neumann Morgenstern stable set (von Neumann and Morgenstern, 944). Wle beng a core member s a property of eac mputaton, te von Neumann Morgenstern stable set refers to a set taken as a wole.
88 S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 pre-exstng soluton concepts by ncludng foresgt n te calculatons of potental colluders n edonc games, along te lnes proposed by Greenberg (990) and Cwe (994). Tey ave to address, as we do, te ssue of potental coalton formaton among agents wo are not part of devatng coaltons. Tey assume tat suc agents stay put and take no mmedate acton. Ts s n contrast to our consstent treatment of coalton formaton among resdual members wo ave lost some of ter partners. Te dea of recursve consstency tat s nerent n our noton of durablty can be found n several game teoretc soluton concepts. In te context of strategc form games t underles te defnton of a coalton-proof Nas equlbrum n Bernem et al. (987). For caracterstc functon games, Ray (989) proposes a modfcaton of te core, were te proposals of blockng coaltons are tested for stablty n te same way as s te orgnal utlty allocaton. It turns out tat te core and te modfed core are dentcal. Dutta et al. (989) ntroduce te consstent barganng set, wc also requres a test of counterobectons for valdty, and ts process s taken to te lmt. For games n wc te utlty of a player depends on te wole coalton structure, Ray and Vora 3 (997) ave proposed te concept of an equlbrum bndng agreement. Te recursve element of ter defnton s smlar to ours. If a coalton consders dsruptng some gven coalton structure t s neter assumed to be very pessmstc nor to be very optmstc concernng te partton tat t expects to arse after te devaton. Rater, te resultng partton s expected to be consstent wt te equlbrum noton one s gong to defne. Unlke Ray and Vora (997), wo only consder devatons tat lead to refnements of a gven coalton structure, we allow for arbtrary coaltonal devatons. A drawback of ter defnton wen appled to purely edonc games s tat te set of equlbrum bndng agreements concdes wt te core: ts leaves us wt te usual problem of emptness. To conclude te lterature revew let us remark tat our noton of durablty as an nterestng parallel n votng models, for wc Rubnsten (980) as ntroduced te 4 stablty set. Te stablty set captures prudent beavor on te part of te voters: a voter wll not vote for alternatve y over alternatve x f y s ten beaten by some alternatve z wc s worse for tan alternatve x. Hence, as n our defnton of durablty, voters are assumed to beave conservatvely. However, ter farsgtedness s lmted snce tey only look one step aead wle we assume te players to look arbtrarly far aead. Summarzng, we are aware of some of te sortcomngs of our proposal, but we feel tat t fares well relatve to oter solutons wc are well establsed n te lterature. In partcular, we value te fact tat t ncorporates a lmted but well-defned degree of ratonalty and foresgt for conservatve players, and tat t allows us to classfy every sngle coalton structure, per se, as beng durable or transent. Te paper proceeds as follows. After prelmnares, n Secton we provde a defnton of durablty followed by motvatonal remarks. We provde an exstence proof 3 In fact, te framework n Ray and Vora (997) s even more general, snce tey analyze games n strategc form, were bndng agreements can be wrtten between members of a coalton but not across coaltons. 4 See also Martn and Merln (00) for an analyss of te relatonsp of te stablty set wt oter socal coce correspondences and ts normatve propertes.
S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 89 and prove some nterestng propertes of durable structures. In Secton 3 we present examples to llustrate te potental of our proposed defnton for specal games or classes of games. Fnally, Secton 4 concludes.. Durable coalton structures We consder a edonc game, (N;(K ) [N), were N s te fnte set of players, and K s a complete and transtve preference relaton on S (N) 5 S, N u [ S for all [ N. Strct preference and ndfference wll be denoted by s and, respectvely. A set 5 ± S, N s called a coalton. If (N;(K ) [N) s a edonc game and T s a coalton, ten te restrcton of (N;(K ) [N) to te player set T s agan a edonc game T T and s gven by (T;(K ) [T), were K s te restrcton of K to S (T ). A coalton structure 6 on N s a partton of N nto dsont coaltons. We denote te set of all coalton structures on N by P. Let S(,6 ) be te coalton n 6 [ P tat contans player. We wll propose te noton of a durable coalton structure as motvated n te Introducton. In order to gve te basc dea, let us assume tat we ave explaned ow te players wll partton temselves after some coaltons ave devated from a gven coalton structure. Knowng tat ter devaton mgt be te startng pont of furter devatons by oters, tat s, knowng te grap on te set of coalton structures tat s defned by all potental coaltonal devatons, wll tese coaltons actually devate? Our assumpton s tat tey wll only devate f no matter wc coalton structure s reaced later, tey wll never be worse off tan n ter present coalton structure. Tus, we wll consder as potentally unstable tose coalton structures wc can be dsrupted by some coaltons wtout te rsk of a future loss for any of te devatng players. Formally, we recursvely defne te noton of a durable coalton structure as follows. Let (N;(K ) [N) be a edonc game and assume tat te noton of a durable coalton structure as already been defned for all edonc games wt strctly less tan unu 5 players. Ten, a coalton structure 6 on N drectly leads to a coalton structure 69 va m coaltons T,T,...,T (m $ ), and we wrte f m T,...,T 6 6 9. T [ 6 9 and T s S(,6 ) for all [ T and for all 5,...,m,. @ 5 T u T [ 6 and T > T 5 5 for all 5,...,m, 6 9, m 3. 6 9\( < 5T << S[@ 6 ) s eter empty or durable n te edonc game m obtaned by restrctng (N;(K ) ) to te player set N\( < T << 6 ). [N 5 6 [@ 5 By uau we denote te cardnalty of a set A.
90 S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 A coalton structure 6 drectly leads to a coalton structure 69 and we wrte 6 6 9 f m tere exst coaltons T,...,T suc tat T,...,Tm 6 6 9. Fnally, we defne to be te transtve closure of te relaton,.e. 6 6 9, f tere 0 t t exst coalton structures 6,...,6 suc tat 6,6 5 6 9 and 6 6 for 5 0,,...,t. We ten say tat 6 leads to 69. Defnton.. A coalton structure 6 on N s transent f tere exst coaltons m T,...,T, suc tat for all coalton structures 69 wt m T,...,T 6 6 9, and for all coalton structures 60 wt 6 9 6 0 t s true tat S(,6 0) K S(,6 ) for all [ T and for all 5,...,m. A coalton structure 6 s durable f t s not transent. Observe tat te core of te edonc game (N;(K ) [N) s always a subset of te set of 6 durable coalton structures. Let us reflect upon te defnton of durablty for a moment. Frst observe tat te noton of a durable coalton structure s well defned even f te set of durable coalton structures were empty for some of te restrcted edonc games appearng n te defnton of te relaton drectly leadng. However, n te followng we wll sow tat tere always exst durable coalton structures. It s not stragtforward to predct te way n wc players partton temselves nto coaltons after some coalton structure as been dsrupted. But f we were to predct wc coalton structure would arse, ten t sould be consstent wt our teory. Terefore, we assume tat te players n coaltons wo are not affected by a devaton stay togeter and tat te members of coaltons tat are broken and wo temselves are not part of any devatng coalton (f any) form a coalton structure tat s durable n ter subsocety. Ts defnes te bnary relaton drectly leadng. Ten, coaltons wll estate to devate from a durable coalton structure snce ts may trgger furter devatons and fnally lead to a partton n wc some of te orgnally devatng players are worse off. On te oter and, coalton structures tat can be safely dsrupted by some coaltons wll only be transent. Now tat we ave presented our defntons of durable coalton structures and before turnng to te proof of exstence, let us elaborate on ter nterpretaton and on some of te crtcsms tey are open to. Frst, we comment on te use of te maxmn crteron and ts proper nterpretaton. Maxmn beavor s an expresson of extreme rsk averson, wt agents attacng not only very low utltes to ter least preferred outcomes, but also attrbutng to tem some probablty of coce, owever small. Agents compute te 6 For any edonc game (N;(K ) ) te core s te set of all coalton structures 6 [ P suc tat tere exsts [N no coalton T wt T s S(,6 ) for all [ T.
S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 9 mn over outcomes wc are possble, ten coose actons tat maxmze ts mn. In order to be consstent wt ts nterpretaton, t s best to tnk of durablty as a matter of degree, not as an all-or-notng property. Durable coalton structures are lkely to stay, for all te reasons we provde. Non-durable coalton structures are lkely to be epemeral, at best, but stll possble. Ts relatvstc vew s more agreeable to us tan any determnstc, predctve nterpretaton. We are not clamng tat durable structures wll appen for sure, and last forever. Next, we address a crtcsm of our defnton wc as been advanced by Debra Ray. Suppose I am a player wo s consderng to cooperate n dsruptng a gven coalton structure. Suppose I am deterred by te treat tat ts devaton leads to anoter I dslke, after a few steps. Tese steps may be taken by agents wo, wen lookng aead, would also be deterred by a smlar treat, f tey used exactly te same reasonng. Yet, our defnton does not sever te lnk between te current coalton structure and my treat, even wen te ntermedate devants mgt n fact feel treatened wen takng te actons we attrbute to tem. Ts s not a trval obecton. Yet, te crtcsm would be especally botersome f we were tnkng of durablty as te bass for a zero one classfcaton between wat can appen and wat cannot. Under suc a radcal nterpretaton, non-durable structures sould not play any role n defnng durable ones, oter tan beng ter complement. We leave ts as an open puzzle, one tat s far from our motvaton. Under our more relatvstc vew we can actually defend our noton of durablty. If agents are maxmners tey wll attac postve probablty, owever small, to any utlty-enancng move tat leads away from a durable coalton structure. Tus, we delberately do not sever any lnks n te orgnal grap defned by te relaton drectly leadng. We now turn attenton to te queston of te exstence of durable coalton structures. We begn by a remark. Remark.. (a) If N 5, ten N s durable. Ts s because N s te unque coalton structure on N wc, by defnton, does not lead to any coalton structure on N. (b) If N 5,, ten we get te followng tree cases. (b) If N s for 5,, ten N s te only durable coalton structure. (b) If tere exsts [ N suc tat s N, ten, s te only durable coalton structure. (b3) If N K for 5, and N for at least one [ N, ten bot coalton structures N and, are durable. Te above remark proves tat te set of durable coalton structures s non-empty, meanngful and easy to compute wen tere are one or two agents. Before provng exstence for an arbtrary number of agents we provde a useful lemma. In words, t states te followng: consder a cycle of coalton structures n te grap defned by te relaton. Ten tere exsts at least one agent partcpatng n a devaton from some coalton structure 6 n te cycle wo loses utlty compared to 6 at some oter coalton structure n te cycle. More formally:
9 S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 J Lemma.. Let (N;(K ) [N) be a edonc game and let 6,...,6 be a fnte sequence of coalton structures on N suc tat te followng condton s satsfed. For all 5,...,J, tere exst coaltons T,...,T, wt. T,...,T m( ) 6 6, m( ). for all coalton structures 6 9 suc tat 6 6 9, t s true tat S(,6 9) K S(,6 ) for all [ T and for all 5,...,m( ). J Ten 6 ± 6. Proof. Let (N;(K ) [N) be a edonc game and assume by way of contradcton tat J tere exsts a fnte sequence of coalton structures 6,...,6 wc fulflls te J condtons n te statement of te lemma but for wc 6 5 6. Consder one of te T coaltons T for some [,...,m(). Clearly, T [ 6 5 6 and terefore tere exsts a maxmal [,...,T suc tat T [ 6. Snce T [ 6, by defnton tere exsts 9 [,...,m( ) and [ T > T 9. Hence, condton mples tat S(,6 ) K S(,6 ) 5 T. However, ts leads to a contradcton snce T s S(,6 ). Teorem.. Let (N;(K ) ) be a edonc game. If a coalton structure [N 6 s transent, ten tere exsts a durable coalton structure 6 suc tat 6 6. Proof. Let (N;(K ) ) be a edonc game and let [N 6 be a transent coalton structure. Defne @ 5 6 < 6 u 6 6. Observe tat @ \6 ± 5 snce a necessary condton for 6 to be transent s tat tere exsts a coalton structure tat 6 drectly leads to. Suppose te clam s false, so tat 6 s transent for all 6 [ @. Ten, for any 6 [ @ m tere exst coaltons T,...,T and some 6 9 [ @ suc tat m T,...,T 6 6 9, and S(,6 0) K S(,6 ) for all [ T, for all 5,...,m, and for all 6 0 suc tat 6 9 6 0. Snce ts s true for all 6 [ @ and snce @ s fnte, tere exsts a fnte T T sequence of coalton structures 6,...,6 wt 6 5 6, wc fulflls condtons and n Lemma.. Ts contradcts Lemma.. Corollary.. For any edonc game (N;(K ) structures s nonempty. [N ) te set of durable coalton Proof. Te clam mmedately follows from Teorem..
S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 93 We now provde a frst smple test tat our defntons are sensble, by sowng tat coalton structures tat volate ndvdual ratonalty can never be durable. Defnton.. A coalton structure 6 s sad to be ndvdually ratonal f S(,6 ) K for all [ N. Teorem.. Let (N;(K ) [N) be a edonc game. Ten, any ndvdually ratonal coalton structure leads to ndvdually ratonal coalton structures only. Moreover, f 6 s durable ten 6 s ndvdually ratonal. Proof. We wll sow nductvely over unu tat f 6 s an ndvdually ratonal coalton structure for te edonc game (N;(K ) [N), ten t leads to ndvdually ratonal coalton structures only, and f 6 s durable, ten 6 s ndvdually ratonal. If unu 5, ten notng as to be proved. Let unu. and assume tat, for all edonc games (T;(K ) [T ) wt utu, unu, te clam as been proved. Let 6 be an ndvdually ratonal coalton structure on N and assume by way of contradcton tat tere exsts [ N and a coalton structure 6 9 on N suc tat m T,...,T 6 6 9 and s S(,6 9). Ten [ T for all [,...,m. Terefore, eter S(,6 ) [ 6 9, wc gves an mmedate contradcton, or, for @ 5 T u T [ 6 and T > T 5 5 for all 5,...,m we fnd tat m 5 S(,6 9) [ 6 9\( < T << S), S[@ were te latter s a durable coalton structure n te restrcton of te edonc game m (N;(K ) [N) to te player set N\( < 5T << S[@ 6 ). By te nducton ypotess, ts mples tat S(,6 9) K, wc s a contradcton. Terefore, any ndvdually ratonal coalton structure 6 on N leads to ndvdually ratonal coalton structures only. Let 6 be a non-ndvdually ratonal coalton structure on N and let J 5 [ N u s S(,6 ). Ten 6 drectly leads to some ndvdually ratonal coalton structure 6 9 va te coaltons, [ J. We ave sown above tat 6 9 leads to ndvdually ratonal coalton structures 6 0 only,.e. S(,6 0) K s S(,6 ) for all [ J. Terefore, 6 s transent. A drect consequence of Teorem. s te followng corollary wc proves to be very useful for computng durable coalton structures. Corollary.. Let (N;(K ) [N) be a edonc game. Ten tere does not exst a durable coalton structure 6 and a coalton T suc tat [ S and T s for all [ T. Proof. Let (N;(K ) [N) be a edonc game. Assume by way of contradcton tat tere exsts a durable coalton structure 6 and a coalton T suc tat [ 6 and T s for all [ T. Ten, tere exsts some coalton structure 6 9 suc tat
94 S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 T 6 6 9. By Teorem. te coalton structures 6 and 6 9 are ndvdually ratonal and 6 9 leads to ndvdually ratonal coalton structures only. Hence, S(,6 0) K 5 S(,6 ) for all [ T and all coalton structures 6 0 suc tat 6 9 6 0. Hence, 6 s transent, wc s a contradcton. 3. Furter propertes and examples In ts secton we provde furter results on durable coalton structures and present dfferent examples of edonc games wc llustrate te versatlty of te soluton concept we ave proposed. In te followng we wll frequently use a smplfed notaton for coalton structures. For example, we wrte [ u 3 u 456] for te coalton structure,,3,4,5,6 on te player set N 5,,...,6. Also, for any edonc game (N;(K ) [N) we wll only lst te preferences of te players over ndvdually ratonal coaltons, were a coalton S s ndvdually ratonal f S K for all [ S. We begn by studyng te relatonsp between te set of durable coalton structures and te core, wc, as we ave already seen, s always a subset of te set of durable coalton structures. Example 3.. Ts s a smple example of a roommate problem taken from Gale and Sapley (96). Tere are N 5,,3,4 students wo can dvde nto pars of roommates. Everyone prefers to sare a room wt someone over beng alone. Coaltons of sze greater tan two are unfeasble. We can represent te students preferences over roommates by preferences over coaltons f we defne tese preference relatons suc tat unfeasble coaltons are not ndvdually ratonal. Te preferences we 7 are gong to study are as follows:, s,3 s,4 s,,3 s, s,4 s,,3 s,3 s 3,4 s 3, 3 3 3,4 s 3,4 s,4 s 4. 4 4 4 By Teorem. and Corollary. te only canddates for durable coalton structures are [ u 34], [3 u 4] and [4 u 3]. Accordng to Remark. te coalton structure s durable relatve to, for all, [,,3,4, ±. Hence we get te cycle llustrated n Fg.. Wle te core of ts game s obvously empty, we can mmedately see tat all coalton structures n ts cycle are durable snce tere s always one devator wo s deterred snce se may end up wt roommate 4 after a furter devaton. Example 3.. Ts example llustrates tat te core, even f nonempty, may be a strct 7 Actually, all we need n ts example s tat s most preferred roommate s, tat s most preferred roommate s 3, tat 3 s most preferred roommate s and tat,, 3 rank 4 last.
S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 95 Fg.. Example 3.. subset of te set of durable coalton structures. Let N 5,,3 and let te preferences K be gven as follows:,3 s,,3 s, s,, s,,3 s,3 s,,,3 s 3,3 s 3,3 s 3 3. By Teorem. and Corollary. we can concentrate on te coalton structures [ u 3], [3 u ], [ u 3] and [3]. Accordng to Remark. te coalton structure s durable relatve to. Terefore, we get te grap presented n Fg.. Te core s gven by te coalton structure [3] wle te set of durable coalton structures s gven by [ u 3],[3 u ],[3]. To see ts, note tat, for example, [ u 3] s durable snce t drectly leads to [3 u ] va,3 and ts drectly leads to [ u 3] wc s worse for player tan [ u 3]. In bot examples te set of durable coalton structures contans more tan one element and s a strct superset of te core. A stragtforward queston ten s weter Fg.. Example 3..
96 S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 one can construct an example wt a unque durable coalton structure and an empty core. Te next teorem sows tat ts s mpossble. Teorem 3.. Let (N;(K ) [N) be a edonc game. If tere exsts a unque durable coalton structure 6, ten te core s nonempty and contans as ts unque element te coalton structure 6. Proof. Let (N;(K ) [N) be a edonc game and let 6 be ts unque durable coalton structure. Assume tat te core s empty. Snce te core s always a subset of te set of durable coalton structures te teorem s proved once we ave establsed tat ts assumpton leads to a contradcton. If te core s empty any coalton structure 6 9 drectly leads to anoter coalton structure 6 0. Snce 6 9 s transent for all 6 9 ± 6 we can construct a pat 6,6,... as follows. Snce 6 s not n te core tere exsts a coalton T and a coalton structure 6 9 wt T 6 6 9. m( ) Let 6 5 6 and 6 5 6 9. For $ and 6 ± 6 let T,...,T, and 6 be suc tat T,...,T m( ) 6 6 ˆ ˆ ˆ and 6 K S(,6 ) for all [ T, for all 5,...,m( ), and for all 6 wt 6 6. If 6 5 6 for some $, let 6 5 6 5 6 9. Snce te core s empty and te set of coalton structures s fnte, te pat defned above necessarly ends up n a cycle. To smplfy te notaton let te coalton structures J n ts cycle agan be numbered 6,6,...,6, wt J 5. Ten T,...,T m( ) 6 6 T f 6 ± 6 and 6 6 f 6 5 6. By Lemma. tere exsts # # J suc tat 6 5 6. W.l.o.g. let 5. Snce T 6 6 t s true tat T s S(,6 ) for all [ T. Moreover, snce T [ 6 tere exsts a maxmal [,...,T suc tat T [ 6. Terefore, tere exsts [,...,m( ) and a player [ T > T. By constructon ts mples S(,6 ) K S(,6 ) 5 T, wc s a contradcton. Anoter way of statng ts result s to say tat tere exst at least two durable coalton structures wenever te core s empty. One case n wc tere exsts a unque durable coalton structure and ence te core and te set of durable coalton structures concde s te one were te edonc game satsfes te top-coalton property ntroduced by Baneree et al. (00). Defnton 3.. Let (N;(K ) [N ) be a edonc game and let V, N be a nonempty set of
S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 97 players. A nonempty subset S,V s a top-coalton of V, f for any [ S and any coalton T,V wt [ T, we ave S K T. Te edonc game satsfes te top-coalton property f for any nonempty set of players V, N, tere exsts a top-coalton of V. Observe tat f (N;(K ) [N) s a edonc game tat satsfes te top-coalton property, T ten, for any T, N te game (T;(K ) [T ) obtaned by restrctng te player set to T satsfes te top-coalton property as well. Baneree et al. (00) sow tat f a edonc game satsfes te top-coalton property and f preferences are strct, ten te core s nonempty and contans as ts unque element te coalton structure 6 * defned as 8 follows. For any V, N let TOP(V ) 5 S, N u S s a top-coalton of V. Defne V0 5 N m() and let TOP(V 0) 5 S,...,Sm(). Defne V 5 N\ < 5 S and let TOP(V ) 5 m(k) k S,...,Sm(). In te same way defne Vk5 N\ < 5 S and let TOP(V k) 5 k k S,...,Sm(k) for eac k. Snce N s fnte tere exsts K wt V K ± 5 and VK 5 5. Let 6 * 5 S,...,S,S,...,S,...,S,...,S. () K K m() m() m(k ) As te followng teorem sows, 6 * s te unque durable coalton structure. Hence, n a case were tere s a unque reasonable outcome of te coalton formaton game te core and te noton of durablty gve te same predcton. Teorem 3.. Suppose (N;(K ) [N) s a edonc game tat satsfes te top-coalton property, and tat preferences are strct. Ten tere exsts a unque durable coalton structure and t s te unque coalton structure n te core. Proof. We prove te teorem by sowng tat a durable coalton structure must consst of te set of coaltons contaned n 6 * as defned n expresson (). Te proof s by nducton over te number of players. Let (N;(K ) [N) be a edonc game tat satsfes te top-coalton property. If unu 5, ten obvously te sngle person coalton s te only element n te core and t s also te only durable coalton structure. Let unu 5 n and assume tat te clam as been proved for all edonc games wt strctly less tan n players. Let 6 be a durable coalton structure n te edonc game (N;(K ) [N). If S [ 6 for some [,...,m(), ten, snce preferences are strct, tere exsts 6 wt S 6 6. Snce S s a top-coalton of N, for no [ S tere exsts a strctly mprovng devaton. Moreover, by te nducton ypotess, for any TmN wt S, T t s true tat S [ 6 T*, were 6 T* s te unque durable coalton structure n te edonc game T (T;(K ) ). Hence, S [ 6 9 for all 6 9 wt 6 6 9. Ts mples tat 6 s transent, [T 8 Papa (000) ntroduces te sngle-lappng property wc also guarantees unqueness of te core. Snce ts property mples te top-coalton property we do not treat t separately ere.
98 S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 wc s a contradcton. Terefore, S [ 6 for all 5,...,m(), and by te same m() argument as above < 5 S, 6 9 for all 6 9 wt 6 6 9. If S [ 6 for some [,...,m(), ten by te prevous argument S(,6 ), N\ m() < 5 S for all [ S. Hence, snce preferences are strct and S s a top-coalton of m() N \ < S, tere exsts 6 wt 5 S 6 6. m() 5 m() 5 T* m() 5 T* T [T m() 5 k We ave sown above tat < S, 6 9 for all 6 9 wt 6 6 9. Moreover, for no [ S tere exsts a strctly mprovng devaton nvolvng only players n N\ < S. Fnally, from te nducton ypotess t follows tat S [ 6 for any T wt S, T, N\ < S, were 6 s te unque durable coalton structure n te edonc game (T;(K ) ). Terefore, S [ 6 9 for all 6 9 wt 6 6 9, from wc t follows tat 6 s transent, wc s a contradcton. Hence, S [ 6 for all 5,...,m(), and te same argument sows tat < S, 6 9 for all 6 9 wt 6 6 9. In te same way, one proves tat S [ 6 for all 5,...,m(k), and all k 5,...,K. Hence, 6 5 6 *, wc proves te teorem. Te followng tree examples consder matcng problems wc can be nterpreted as edonc games by defnng te preference relatons suc tat unfeasble coaltons are not ndvdually ratonal. Stable matcngs ten correspond to coalton structures n te core. As Gale and Sapley (96) ave sown, te core s nonempty for any one-to-one matcng problem. By contrast, tere exst many-to-one matcngs problems for wc no stable matcng exsts. Our examples wll llustrate tat te noton of durablty gves rse to reasonable predctons for bot types of matcng problems. Example 3.3. Ts s an example of a one-to-one matcng problem. Let N 5 W < M, were W 5 w,w,w 3,w4 s te set of women and M 5 m,m,m 3,m4 s te set of men. Eac w [ W as a strct preference orderng over M < w wc we represent by an ordered lst P(w): P(w ) 5 mmmw, 4 P(w ) 5 mmmw, 3 P(w ) 5 mw, 3 3 3 P(w ) 5 mmw. 4 4 4 Te nterpretaton of, for example, P(w 3) s tat w3prefers m3over beng alone, wle all oter men are strctly worse for w3 tan beng alone. Snce unacceptable mates do not play any role we omtted tem from te lst. Smlarly, te strct preference orderng of eac m [ M over W < m s represented by an ordered lst P(m): P(m ) 5 wwwm, 4
S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 99 P(m ) 5 wwm, P(m ) 5 wwm, 3 3 3 P(m ) 5 wwm. 4 4 4 As mentoned above t s stragtforward to formulate ts matcng problem as a edonc game f one defnes te preference relatons K over S (N) suc tat unfeasble coaltons,.e. coaltons were more tan one player belongs to eter W or M, are not ndvdually ratonal. A matcng m s a one to-one functon from W < M onto tself suc tat m () 5 for all [ W < M. Any matcng defnes a coalton structure on W < M n an obvous way. A matcng m s stable f tere s no [ W < M wo prefers beng unmatced over beng matced wt s mate at m and f tere s no w [ W and m [ M tat are not matced to one anoter at m but would eac prefer one anoter to ter mates at m. Its mmedate to see tat a matcng m s stable f and only f te correspondng coalton structure s n te core of te edonc game defned by te matcng market. In ts example tere s a unque stable matcng wc corresponds to te coalton structure [w m u wm u wm u wm]. However, ts s not te unque durable coalton 4 3 3 4 structure. As Fg. 3 sows, te coalton structure 6 5 [w m u wm u wm u wm] s 3 3 4 4 also durable snce te only possble devaton s by coalton w,m and t leads to Fg. 3. Example 3.3.
00 S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 [w m u wm u wm u w u m ] wc s strctly worse for m tan 6. Hence te core can 4 3 4 3 be a strct subset of te set of durable coalton structures (cf. Example 3.) also for te specal case of a matcng problem. Example 3.4. Te followng example of a many-to-one-matcng problem for wc tere exsts no stable matcng s taken from Dutta and Masso (997). Let tere be two frms F, F, and four workers w, w, w 3, w 4, so tat N 5 ^ < 0 were ^ 5 F,F and 0 5 w,w,w,w. Eac frm F as a preference orderng over F < W u 5 ± 3 4 W, W and eac worker w as a preference orderng over (^ 3 0 ) < w, were 05 S u S, 0,w [ S. In ts example we assume tat workers preferences are worker-lexcograpc,.e. w s rankng over colleagues determnes w s preference orderng over frm-co-worker pars n wc te set of co-workers s dstnct. Hence, for our purposes t s enoug to lst workers preferences over colleagues only. Te preferences of frms and workers are gven n Table, were elements are ranked n descendng order of preference and agan only acceptable partners are lsted. Agan one can represent ts matcng market as a edonc game so tat te stable matcngs correspond to coalton structures n te core. As Dutta and Masso (997) argue, te only canddates for stable matcngs are [F ww u Fww], 3 4 [F ww u Fww], 3 4 [F www u Fw]. 3 4 However, all tree matcngs are unstable. More precsely, we obtan te cycle n Fg. 4. It s stragtforward to see tat all coalton structures n ts cycle are durable snce all devatons are deterred. For example, [Fww 3 4 u Fww] s durable snce w s not wllng to on F, w 3, w4 and devate to [Fwww 3 4 u Fw] knowng tat ts may lead to [F ww u Fww] wc s worse for w tan [F ww u Fww]. 3 4 3 4 Table Example 3.4 F F w w w w 3 4 w,w,w w,w w,w w,w,w w,w,w w,w 3 4 3 3 4 w,w w,w w,w w,w,w w,w,w w,w 3 4 3 4 3 4 4 w,w w w,w w,w,w w,w,w w,w,w 3 4 4 3 4 3 4 3 4 w,w w w w,w w,w w,w,w 4 3 3 4 w,w w w,w w,w w,w 4 3 3 3 3 4 w w w,w w,w w 4 4 3 4 4 w w w 4 3 w 3 w
S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 0 Fg. 4. Example 3.4. Example 3.5. Te fnal example s agan a many-to-one matcng problem wt frms and workers. As before tere are two frms F, F and four workers w, w, w 3, w 4,of wom (w,w ) s a couple and w and w are sngles. Te frms are located n towns far 3 4 away from eac oter, so te couple strctly prefers beng unemployed over workng for dfferent frms. Frms and workers preferences are gven n Table, were we agan only lst ndvdually ratonal coaltons n descendng order of preference. Eac frm as a most preferred worker and tere are no returns to employ an addtonal worker once ts worker as been apponted. Tere s a unque stable matcng wc corresponds to te coalton structure [F w u F w u w u w ] n wc 3 4 te couple s unemployed. Accordng to Teorem. and Corollary. tere are only fve canddates for durable coalton structures. In Fg. 5 we present te grap on tese coalton structures tat s defned by te relaton drectly leadng. It s stragtforward to see tat, apart from te core structure, tere exsts one addtonal durable coalton structure n wc te couple s matced to F and te sngles are matced to F. Clearly, te couple wll prefer te suboptmal frm F and refuse to cange to F wen ts leads to a matcng were tey are bot unemployed. Te full employment n te durable coalton structure [F ww u Fww] seems to be a muc more appealng 3 3 4 outcome of te matcng problem tan te stable matcng. Table Example 3.5 F F w ( 5,) w w 3 4 F,w F,w F,w,w F,w,w F,w,w 3 4 3 4 3 4 F,w,w F,w,w F,w,w F,w F,w,w 3 3 4 F,w,w F,w,w w F,w,w F,w 3 4 3 4 3 4 4 F F w w 3 4
0 S. Barbera, ` A. Gerber / Matematcal Socal Scences 45 (003) 85 03 Fg. 5. Example 3.5. 4. Concluson We ave proposed a soluton concept for edonc coalton formaton games wc assumes some form of maxmn beavor on te part of players. Players are farsgted n tat tey take nto account tat ter ntal devaton may trgger furter devatons and tey are extremely pessmstc n tat tey attrbute to eac coalton structure a postve probablty tat t may last forever. Our defnton of durablty provdes an mplct teory about wc coalton structures may arse followng a coaltonal devaton. Our assumpton about te rearrangement of members of dsrupted coaltons s consstent wt our teory and gves rse to a recursve defnton of our soluton concept. We ave not concealed te sortcomngs of our soluton concept wc are manly due to te asymmetry n te degree of foresgt and ratonalty tat players attrbute to oters and to temselves. But te proposal as many attractve features. It always selects a nonempty set of coalton structures as beng durable, t satsfes a very strong consstency noton and t does well n smple applcatons. It always selects coalton structures n te core wen tey exst. It cooses core allocatons alone n smple cases were ts coce s very compellng. In oter smple examples t also selects as durable some addtonal coalton structures not n te core, and wen ts appens te retaned structures always turn out to be plausble. We ope to ave provded a sensble and well grounded tool for te analyss of possble outcomes of coalton formaton games. Te callenge of computng our soluton (or any oter) for large, complex games, stll les aead. Acknowledgements We are grateful to te partcpants of te 999 Manresa Conference on Networks, Groups and Coaltons for ter comments, and very specally to our dscussant tere,
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