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Alvin E. Roth** iVIarilda Sotomayor PDF

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Preview Alvin E. Roth** iVIarilda Sotomayor

STABLE OUTCOMES IN DISCRETE AND CON­ TINUOUS MODELS OF TWO-SIDED MATCHING: A UNIFIED TREATMENT* AlvEi.nR oth** iVIaSroitlodmaa yor*** Resumo Aprescl1l1t1a1I1l ltolrSla ltlaidlfelHi lIcclcanWldt aoods esl el ioddeem lam,;t ching ded ailsa dqonshe l cmloldleidl iossc r(ettaCoiO:;sU llOl loddoe icaa sadmeGe anlteo 0 eS haplee ym)o decloanst i(ntlaCliOosI}; l mloOd edleda e sigdneaS Ghaaop el ey 0::; 0 Slllel mboidke dleJa e siggncauGearoa dleDi CL;laldHoeGl allleEg)uCc.! o ntrcaosmt c 0 ast raltlacluI:tUolst earsic oOrlelscp,ia nrsaolpcea!lriaaa sds o iC:-;O lljdleml oldletloasH sadoe rivaaqdldalosmi e slmloldcoo paa rdtaiisrU CShlilplo(;1t!Ie)vs Ieoss.tg rlatlel lOS asr esuletmaq dlolsc ssetganopc rlaltli,c C1dmalc 1uitped6,etq elslnee( lccloeiou cide 0 COllllili cdleefoi puoiJrdo om iufar<;:aaNoco al .l lOddoce alsoa lIhlicpuotdtoee fic 0 a qlalsep referseaneocs itar[sia ct�oa mqs H ees tdeasiC sO lljClolilllltecolisld qallllaln to nomso decloonst luJlolcioOsSI ljcnolilutcpoiosdr eqmna eg eltletpmer se fel'encias 01) 01) cOllteio uspl rleavpsoo sd eSmCa Ij'l lsctOaldlotsi ullalllellte. * Partiaslulpyp ortLeyd. 1.CSl.I ggeuhe:'.iJleUl lloFroiuanid atiCo?\nP,q -COllsel',i'haoc iollal deD esenvolvimento Cei eTuetclnfoilc6og iBcroa,z ialwj, by grantfsr omt heA lfrePd. S loan Fouuuatiaonnd t he?\ atiolSlcaile nFcOel illualt."iSoAl.l , ** Departllo1feE ucto nomicUsn,i versoiftP yi ttsburPgiht,t .suuPrA g1h5,2 00U,S A *** Il1stidteuE tcoo llomIinad ustrUilalli,v ersiFueaddeerd aolR iod eJ aneirAov,.P astel2l5l0' , -UrcaR,i od e. JaneiRr.Jo,B, r asil. Q R.d eE conometridaeJ anReiiovr .1o 6 n, 2, 1-24N ovem1b9r9o6 pp. Two-smiadtecdh ing Abstract Vvep reseal lttm ifiterde atmeolfla t c lasosft wo-sidleldl atclmioidlelltgsh ati u­ clllcleh' umLo<;ucerl(e:.;st uec ash them arrialglel odoefG la lea ndS hapleayn)d c Olltillll­ OilmSo del(ss llacsht hem isigullllueoluleoLlfS hapleayu dS hllbaimkI t heg enerali�eu assignmelnltl odoefDl e mangea udG ale)I.I cIo ntrawsitt hp reviotwr,e atllltehuet s, parallel cOllfcolrtn hsei otlwlos o sfle ltlso daerleds e rivheedr iel Llh es amew ay from thes amea ssllluptVvieo lslhSo.wt hatt he resilllqlll ttse stailofllo ll low clforsoelllyl thea ssllluptthiaottl hle c orec oillciwdietsht hec ored efinbeyd weakd ominatioll. Int he marrimaogdee lt,h ea sslllllopfts itorliplcr t;e ferenccaenss etsh estew os ettso coinciwdhei,l ien t hec OlltiulmloOdlelltSsh et wos etcso iucibdeec allasgee nthsa ve cOlltillplrteoflelrSe nacued:,; pricesb ea cda.ni nsctoeudL ill1lOnsly. Pala1l'l"G.s-Chiav:lvaet:c isItianbgll,le l atchcionrgel,,a ttiocpet,i lllllaliatchillg C6digJoE L: C78 1.I ntroduction. Oneo ft hpee rsipsutzezinlnte stt hheeoo rfty w o-smiadtecdh ing itsh ger esaitm iloaftr hirete ys uclotnsci enltg'1h c1eo roeft hgea me thahta vbee eonb taifnoterhd de i scmroedtee(l es.t gh.me a rriage modeolfG alaen Sdh ap(l1e9y6 a2n)fd)o c ro ntinmuoodue(sle s. g. thaes signmmoedneotlfS hapalneSdyh ub(i1k9 7a2nt)dh gee ner­ aliazsesdi gnmmoednoetflD emanagneGd a le (1985)), even though thtew koi nodfms o dealpsp etaohr a ve importanFtu rd­ifferences. thermtohrpeer ,o boyfw sh ich rtehseuhslaetvs seof abre eonb tained remaqiuni dtieff eraennsdte ,et mor eloynd ifferpernitn ciApsl es. BalliannsGdka il( e1 9p9.02 ,7 4n)o te: "Theirseb, y n owa, s ubstalnittiearloa ntt uhrees e problaenmdos n,ie ss trubcytk h fea ctth aatl moaslrtle ­ sulptrso vfeotdrh oer dicnaashlea vaen aloigntu hecesa r­ dincaalsa el,t hotuhtgeeh c hnoifqp ureoisont f h tew coa ses arieng eneqruaidlti eff eraenntdth ,ei rases y ento " unified theowrhyi"cc ohv ebrost h." 2 RevidseEt cao nomet1r6i( a2N )o vem1b9r9o6 AlvEi.nR ot&h M ariSlodtao mayor Them ospte rpleaxsipneogcft t h peu zzilste h atth sei milar conclufsotirho tenw sco l asosfge asm reesq uire atshsuaamtrp et ions nomte rely dbiuffater rieent n htfe,o llowing siemnvpseoerr yt ant nearolpyp osTiodt eer.im voesot ft hree suwlewt isbl elc oncerned withhe rient, h mea rrimaogdewe eln eetdoa ssutmheap tl ayhearvse strpircetf erie.netc.he asnt,op layiesir n diffebreetnwteb eeni ng matchteoad n pya iorfp ossmiabtleeIs nt. h ceo ntinmuooduesl s, howevtehares, s umpittsih oaentv eprlya yebrem acdaienn different betweepna iorfp ossmiabtlebesy a ppropardijautset ments of any prices. Perhtahpmeso sdtr amactoincc lcuosmimootnno t htew moo d­ elisst haitnt, h mea rrimaogdeew lh enno a genatrsie n different betwaeletne rnmaatticvhaeen sid,n t hceo ntinmuoodue(sli psna r­ ticutlhaaers signmmoednetl ) alwalhg eennm tasyb em adien dif­ ferbeenttw aenetynw moa tchtehsee,rx ei sftoesra, c shi doeft he markaep to,i ntth ceio nrw eh iicsoh p tifmoaarll a lg enotnts h at sidoeft he mairntk hesete ,n tshean toa geonntt hasti doeft he l markdeoteb se tattea rn oyt hceorro eu tco.Im tie as lrekandoyw n thatth emraeyn oetx iosntoe f t hoep tipmoailni tnts h ceo re if indiffeirsae lnlcoeiwsnte hdme a rrimaagrek (esteR,eo tahn d So­ tomay(o1r9 90A)f)tp.er re setnhtfeio nrgmm aold ewlew, i slhlo w bye xamptlheat th iasl shoa ppeinfis n diffeirspe rneccel iunad ed modeilnw himcohn eiysc ontinduiovuisslTiyhb iilsset .y piocfa l howt hvee rsyi micloanrc luasbiootunhtste w moo dedlois n f act depend odni svsieamrsiysl uamrp tions. Thipsa pperre saeu nntisfi terde atomfae c nlta osfts w o-sided matchmiondget lhsai tn cltuhdmeea sr riaangde continuous models, sucthh atthp ea raclolnecll ufsotirho tenw sso e otfsm odealrsde e - 1 Fore xamplien:a markewti tehq naulllllloufie uresl lticaia nsuie Jlclnetrixuc nayie n;:o fs ome homogeneous inJg.oiovdio,.u sticuoimeei slt lh ec oraer ec ltaracteLryia z seidn gplrei cbee,t ween ther eservatpiroinco eft heb uyerasn ut hes elieartws b,i cahl tlr adaerse J IIauTeh.eu uyeorp timal COroeu tcomeisst heo nea tw hicht hep riciesa sb igha sp ossibil.ee,a. t t heN eller reservation pricaen,d t hes elloeprt imaclor oel ltcoimste h eo nea tw hicthh ep riciesa sl owa sp ossible. RevidsetE ac onomet1r6(i 2aN) o vem1b9r9o6 3 Two-sdiedm atching rivientd h sea mwea yf rotmh sea maes sumpTthieao pnp.r opriate assumptioount b tetus oru nbbstu lstei mptlhere:e suilnqt use stion alflo lflroowtm h aes sumptthiatothnc e o rceo incwiidttehhsc e o re definbeywd e adko minaItnti homena .r rimaogdeet lha,es sumption ofs trpircetf erceanucsteehtssew soe ttosc oincwihdiielnt,e h ceo n­ tinumooudset lhste w soe ctosi nbceicdaeau gseenh tasvc eo ntinuous preferaennpdcr eiscc aebnse a djusctoendtu isnluyo2. Ino ufro rmtarle atmweedn etv,e al colpa osfms o dewlhsi ch generbaoltithzh meea rriaangcdeo ntinmuoodueIsln st .hm iosd el, the cormemsoufnlo ttrhs e tsweco l asosfme osd ewlisfl oll flroow m thaes sumptthiaoutnt si lairtiein ecsr eianms oinnegty h,at th see t ofi ndivirdautailtolrnyaa nls ifsbe orusn daenddt, h atth ceo ries non-emapntdcy l osteodg,e twhiettrhh aes sumptthiatothn ce o re coincwiidttehhs ce o re dbeyfiw neeaddko mina(tAiloonlft. h ese assumpetxicoefnpostt rh lea asrtse a tibsyfit ehdme a rrimaogdee l evewnh epnr eferaernneoc stet sr ict.) Two-sided mmoadteaclrhsnei o nmtge reolfty h eoretical inter­ estth;e ayra el spoo werdfeuslc rmiopdteiDlvises. c mroedteehl asv e provteobd e approprwiiatwthhe i tcthooa onlaslm yasrek ient s which cwaabnge et sr eaatspe adro tft hjeo dbe scrisputcaihso n, thmea rkfeotmrse diicnatlei rntn hsUe . S-R.o t(h1 9a8,14 9 8a6n)d thUe. K-R.o t(h1 919909,o1 rn) o,n -wmaagtec hpirnogc essesee s, MongaenldRl o t(h1 9a9n1dS) o toma(y1o9cr9) I6.ts eelmisk ely 2 (1989) Anan alogouunsi ficawast ioanc ocmplishienR do tahn dS otorrnyor concertnhien g rrnrriagrmde el (whiicnhV OlovnESe -torrnt-cohnien ga)n dt hec olladmiesgsei onrsmde l (which involmmvye;- Io-onerrnt ching)T.h at psapehro wtehda tth ceo roeft hceo lladmisesige onsgarre shartESh per opertiESi ntf horrnreur nida gtroed ele vewnh ecno llemagyES b ei ndiffbeer""te"nnt diffeernetnetr cliasnsesg ,so lonasg thehya vset ripcrte fereonvceiesrn divisdtuuadle nTthse. reasoins t hawth enc ollehgaESv set ripcrte fereonvcesESrt udentthse,wy i lnlo tb ei ndifferent bewetena nyt wo entercilnagss tESh ecany beas siguedi nt hec oree,v etnh ougthh emayy be indiffebreetnwteeso nrre paiorfse ntercilnagssES .A ndt hreES ulftostr h rrnre riage trodelf ollow frotmh ew eakaessru mptiotnh anto a genitsi ndiffebreewetnetn a nyt worrnte; whocan be assiguetod himi nt hceo re. 4 RevisdteaE coonmetria 16( 2)N ovembro19 96 AlvEi.nR ot&h M ariSlodtao mayor thacto ntinmuooduemsla syb em orues effouarln alymsairnkge ts inw hicwha gaersde e termeinndeodg enTohuecs llayos.fms o dels introdhuecraeeld l owwasgte osb et reaetietdwh aeyri, . eei.t ahse r parotft hjeo dbe scrioprat sia onen n,d ogendoeutselrymp ianretd oft hmea rkoeutt come. 2.T he MathematicaMlo del. Theraert ew fion idties,j soeiotnfspt l ayPe r{=sI ..,.i ., .m}. andQ = {l,,.,.· n}j·.,T hep layienPr asn dQ wiblelc alPl­ed playaenrQds- plaryeesrpse cTthiesv eeotlfa y l.pl e rmistsria­bnlse febrest wteewpnol ayiesars se otfn umbewrhsi wcihbl eld enoted byS .F omra themactoincvaelnw ieew niacllell aop wl ayteomr a ke at ranosffs e0r=t oh imsFeulrft.h eirifmt orraen sStf oej tr hse n wes atyh ajtt rans-fteoir .s s Thaitsw ,ew iblela ssumtihnagt 1r0 E S ; 12-E S -8E S . s = Thes trucotfpu rreef eriesgn icvebesynt huet ilfiutnyc tions: Ui:j S anVd.i . ; :T hSuU si .iAtss)h u et itloiii t ny of -+ R -+ R. P beimnagt chweidtj ha nrde ceiavm ionnge tpaaryym eonfstf rom j,i sf 2: 0,or t ransf8et rojr i,isf n :s; g O. A nalogVoiujsi(ls8ty )h e utiloifjti ynQ fobre imnagt chweidti h. Ther eservuattiilofinott rih epesl ayiea rnsdj are gbiyv en Ti= Uii(aOn)Sd j = Vi.ir(eOs)p,e c.Tt hiemv aerlkygeatm iest h en givbeyn( PQ,,S ,u ,V ). A transffreorim t oj isi ndividually s ratiioUfni ail( 2:- T8i)a ndVij( s2:) SiW'e w iblela ssumtihnagt foarl ili nP anadl jli nQ : A)-Uia.nidV i ajr set riicntclrye iants hivena gr iSa;bl e A2-thsee otfa liln divirdautailtolrnyaa nls tfheacrtas bn e m ade betweia ennjd i sb ound(eImdta. y b ee mp)t.y RevidsetE ac onomet1r6(i 2aN) o vem1b9r9o6 5 Two-smiadtecdh ing Lacktihnigs assiutwm oputlbideop no,s sfiobarlp el ayteor regasrodmm ea tcahs" infingiotoe,dil "nyt hsee ntsheaa tp layer woubldep repatrope aday n fyi naimtoeu tnota chiietvS.ep ecifi­ calalnyiy,n divirdautailmolanyta cblhe tw·iea enn.di c ourledm ain possniobm laet theorwl artghere e servuattiilooif.int . y Definitio1n. A matchiisanb gi j{'e cftrioPom nU Q ontPo U Q ofo rdtewr(o t hias/Lt,2 (=x x))s ,u ch it"fhC airf:t) Q t heIn" (=i i) anidIf (.i" ) rf: P theI(.in" ) = . i. Thaits /L,( =i. )im eantsh aattt hmea tchWiiin gsm atchteojd , wherIe"a(=si i m) e aniis ss ingle. Definitio2n. Ano 'Utciosamm ea tchIi"an ngda p aiorfv ectors m ('Uv,)c altlhepeda yowff'i,t hE IR anvdE IR nA.n o utcowmiel l 'U bed enobtye( d'Uv ,;I ")' Definitio3n. Theo utco('mUv,e ;I "i)sf ae siibtflh ee irsaes eotf numbesriEsjS ,d efionnelidyIf i" )( =s ujc h that i)'U i= Uii(-.S,)a nVd i= V .ii(SiiIf." 1C=)i j,) j and ii')iU= 7'ii/Lf (=ii) iii�) r i,v i� S:i( indivriadtuiaoln ality). ·Ui. Thep ayo(ff' Ui,sfva e)s iib(fl' eU I,"vi);sa f easoiubtlceof moer some maIt"cI.hnt i hncigas s wees atyh aIti" sc ompawtiittbhhl ee feaspiabyl(oe'Uffv , ) . Definitio4n. Thef easoiubtlceCo umv,e;I "i)ss t aibtflh eei rnseo pai(ri ,EjP ) x Q wit/Lh( irf:).ia nSd E sSu cthh aUti -,Sj)>( 'U i anVdi :i>(vS i') 6 RevidseEt cao nomet1r6i( a2N )o vem1b9r9o6 AlvEi.nR ot&h M ariSlodtao mayor Thiisse quivtaolr eenqtu itrhianntogf easoiubtlceoC muve''; , ' )1" exissutct hh aI"t') ( ija nd = , Viv i' < Similwaedr elfiyan s et able payoff. Thedne fini4ts iaoytnsh a(tv) i ss taibiflit esn odto minated u, bya nyf easpiabyloveffi s ao mpea i(rij ,).( Suac pha i(rij ,)i s callaeb dl ockinWge t paaktiehr re)u .lo efts h gea mteo b et hat anpya i(rij ,)i n xQ isf rteoem atcthoo naen otahtae nrpy r ice P si ft hebyo tahg reaen,da nyi ndiviisd uaftlorr eeem asiinn gle. Thetnh pea i(rijs,)a rteh oen leys senctoiaallia tnidaop nasy,o ff iss tabialfne od n liyif ti isn t hceo rweh,i wcihbl eld enobtyeC d. (Thciosi ncibdeetnwcteehes ene otfs tapbalyeo affnstd h ceo rdeo es nostu rvtihvgeee neralizationm aftrcohtmio mn oagnn ey--ttoo-­one manmya tchsienReg. :o atnhSd o tomayor (199(01)9,9 2S otomayor an1d9 9a6, b)). Definitio5n. Thes taboluet co(um,V e;/ 1i)ss triscttalibyfl e theirsne o p ai(rij ,) xQ wit/1h( i§t) a jn ds sucthh at E P E S Uij(-'Usia) n Vd ii>(V sio)r U i.i>(U -ias n)Vd ij(Vsi)' = = Similwa.edr elfiyan s et riscttaplbayly eo ff. Defin5is taiytoshn a (tv )i ss triscttalibyifl t ie ss tabalnde n, itis n owte adkolmyi nabtyae ndfy e aspiabylvoeffi s ao mceo alition (ij.,)T hep ai(rij ,)i sc alalw eeda bkl ocpkaiinrg. Thes eotfs triscttalpbyal yeo cffosi ncwiidttehhsc e o rdee fined byw eadko minawthiioicnsah,s ubsoefCt a ndw ibleld enobtye d C. Thef ollomwoidneaglr swe e lkln owsnp ecciaasoleft s h meo del wea rter eahteirneg: RevidsetE ac onomet1r6(i 2aN) o vem1b9r9o6 7 Two-smiadtecdh ing The Marrigae Model HerSe = {O}S.i nSc ei sf initthepe,l aycearnls i tshte ir acceptpaabrlteni enor rsd,ei rna, f initoefp rleifsetr eLnecte s. L(id)e notthele i osftp referoefpn lcaeyseT rh·uiLs.( i)( j,0), = (Ic0,)" , (·0- ,)i .".,m eantsh aUti ,>;(U Oi)/ ,� ,,(e Ot)cT .h ef act "; thadto cnso ptl aayn ryo ilnet hgiasm iem pltiheaastn oyu tcome s ('(u);,/ )v1i cs omplestpeelcyib fyt ihemeda tchfitSn.ug c ahm atch­ inigsc alslteadib tflh ece o rrespoountdcionmsget aibTslh eai.ts , fitas s table miatfth ceihrnsieo pn agi Or, i.nPi lQx w itfOht) 01 j sucthh aUti.i (>O)U ;,'a(niVd)i ('O>;)V ( ,O,)( ;).;(O). Thu/si",sa s tambaltec hiitfnh ge irsne o p ai(rij ,)i nP Qx, witchn j01,s ucthh aiat n jdp reefaecroh t hteotr h eriers pective /" matuensd /e1r . Whetnh per eferaernscct ertsih crete sulmtoidneiglst hwee ll knowmna rrimaogdeei ln,t rodbuyGc aeladen Sdh ap(l1e9y6 2). The Assginment Game Int hicsa se andt hearrene u mbeariasjn db if:oiar,l l S = R paiCrijs,)i nP Qx,s ucthh at Ui:ia(i8+i) 1/"(b.8 +) = s 1.1, = 1.1., s. Ifw ed efinCtei .im ax{aib:iOij} ,t, h eanf easoiubtlceo me = + (u(,);v/ )1i sst aibfl ea nidtf h oenrlneyop aii(sr i i,nPj )Qx ,w ith ft01( ji,)s ucthh aUti+ viCt ij· < Thimso dieeslq uivtaotl heaens ts igngmaemonefSt h apalnedy Shub(i1k9 7a2nid)ti sa speccaisoaeftl h meo deblel ow. The GeneralizAesds ginmentG ame Int hmiosd el antdh uet ifluintcyt airsoetn rsi ctly increas­ S = R inagn don to( ancdo nsequceonnttlinyIu toiu sts ha)ed. d itional R assumptthiatothnu e t ility afroucnn tcott ihoahntas ts h ceo nse­ R quence thpalta yceeavrbne e mr ayd ien diffbeertewnaetne pyna ir 8 RevidseEt cao nomet1r6i(a 2N )o vem1b9r9o6 AlvEi.nR ot&h M ariSlodtao mayor ofm atebysa na ppropardijautseto mfpe rnitcT ehsef. a ctth afto r eveprayi( rji ),t h eirsse o mter anstfheamrta keiis n diffteor ent 8 beiunngm atc(ahnesddo moet hterra ns' tfheamrta ke.jis n differ­ 8 en)ti mpltiheatsth see otfi ndivirdautailtolrnyaa nls tfheacrtas n bem adbee twieae nnjd i sb ounded. ] ] Ifw ed efifni.ei= Ujiangdi =j V ii,t heanf easoiubtlceo me (( v)J.l;)i sst aibafln eod n liytf h eirnse op ai(rji ),in P Qx,w ith u, J.li(f) j ,s ucthh afti( jU i9)i (iv i0). + < Thimso deilse quivtaolte hnmeto deolfD emanagnedG ale (198F5o)Dr.e managneGd a ltehc eo ntionftu hiuett yi fluintcyt ions isar equiroeftm hemeno td el. Wew icloln cenotnmr aartkefe otersi thieasrn sye otfi nteger S numbesrast is(fhyo ir1n 2go) r Wew irlelf teoar m arket S = R. whicsha tiassfiseusm pAt]ia onndAs2 a sd iscorrec toen tinuous, accortdowi hnegt hiesars eotfi ntenguemrb eorrs S S = R. There suolftt hnsee xste ctmioosntd leyp eonntd h aes sumption that C.T hicso ndiitssi aotni sifife idsd iscarnedta el l C = S preferaernsect ersiI ctit sa. l ssoa tiisnfit ehdce o ntincuaosouefs DemanagneGd a lfeo,tr h etmh uet ilUiita.inieVd i.s ia rceo ntinuous. Withotuhtoe p posaistseu mptthiaoptnr so dutchceeo mmon resuflodtris s carnecdto en tinmuooduetslh so rsees umlatyfs a tiol holadst, h eex istoefanQ c-eo ptsitmaaoblul tec owmhei,ica sht l east weakplrye febryar leQld- platyoae nroyst hsetra obultec oSmuec.h ano utcoemxeii sntt hse mwoidtfiehnl i ntuem boefpr a yments when preferaernsect reaisnc dit n,t hmeo dewli tcho ntinduiovuissliyb le paymewnhtestn h uet ilfiutnyc tairooenn st os ot haatn ayg ent R cabne m adien diffbeertewnaetne tynw moa tches. Ift hper eferaernneoc stet sr iintc htme a rrimaogdeet lh,ee x is­ tenocfae m ano ptimmaatlc h(oirna wg o maonp timmaatlc h)ii sn g nogtu aranatsie ctea dbn,e s eeinnR otahn Sdo toma(y1o9rE9 x0:, 2.1p5.3, 4 )O.n t hoet hhearn idtf,h erneo s iasla trhya mta kes 8 aw orkienrd iffbeertewnbetee inen mgp loaynebdde iunnge mployed RevidsetE ac onomet1r6(i 2aN) o vem1b9r9o6 9 Two-smiadtecdh ing themraey b en os taboluet cotmheai tso ptifmoatrlh weo rkers. Thee xambpelleio lwl ustthrreao tplelesa ybeyid n diffeirnte hnec e discarnecdto en tinmuooduesl s. Tos eteh reo pllea ybeyid n diffewrheennc e constihdee r S = R, follotwwieonx ga mpilnwe hsi tchhe mraeyb en ow agweh imcahk es aw orkperre ciselyb eitnwdteiwefdonfi effreeernmetpn tl oyompe­nt portunIintt hiee sefi.xr asmtpt lhueet ilairtneio ectso ntiniuno us; thsee coenxda mtphlueet ilairtceio enst inuous. Counterexampl1e. Consitdhceear s oefo nfier ma ntdw woo rk­ eris.,Pe . U}a ndQ {w]',tV 2L}e.t anlde tth uet ility = = S = R funcotfit ohfiner mb e U(lO(1+S S) = Urr(O)0 = anlde tth uet ilityf oferua ncwchot rikoe(nr )b e w 'tv = WI'tV,2 foSr� 0 S for 0,a nd s < VlO'W(OO.) = Theirsne os alatrhya mta keasw orkienrd iffbeertewnbetee inn g s emploaynedbd e iunnge mploAynedtd h.e irsne o s taboluet come thaitso ptifmoatrlh weo rkeTrhse.ar ree ex acttwlsoyt aobulet ­ C0111eS, ",cn= w ];11.r = 1,V I= 1,V 2= 0, and /.,= '' tV(2uf;f'= ) 1 ,v� = 0,v S= 1, an'tvd] pre",f etro",s' w hi'tVl2eh atsh oep popsrietfee rence. Int hiesx amptlhceeo riesn oatl attnincdete hrpe r eferences oft hweo rkbeercsa (use' ')i sn ostt abCloen.t rtaor y 11 /\ ,11 .V vV 10 RevidseEt ac onomet1r6i( a2N )o vem1b9r9o6

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de dais lados qne hlcllli modelos discreto:; (tais COUlO 0 lllodeia do casamento de Gale Vve presellt a tmified treatmellt of a class of two-sided lllatcliillg models that iu .. based on the continuity of the utility functions and on the assump .. medical interns and residents: a case study in gam
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