The core in the matching model with quota restriction *

Transcription

1 h cor in h maching mol wih uoa rsricion * Absrac: In his papr w suy h cor in h maching mol wih uoa rsricion whr an insiuion has o hir a s of pairs of complmnary workrs an has a uoa ha is h maximum numbr of cania pair posiions o b fill W fin as naural boh h concp of blocking by coaliion an h concp of cor nr h rsricion of h insiuion s rsponsiv prfrncs h xisnc of cor is guaran an is characrizaion is obain Kywors: aching; abiliy; Cor JL classicaion: C78; C7; 79 lfina mnia ** * his work is parially suppor by h Consjo Nacional Invsigacions Cinficas y écnicas CONIC hrough Gran PIC-04 by h Agncia Nacional Promoción Cinfica y écnica hrough Gran by h nivrsia Nacional an Luis hrough Gran 3950 an by h Insiuo Cincias Básicas nivrsia Nacional an Juan hrough Gran /776 ** nivrsia Nacional an Juan an Juan Argnina -mail: lfinafmnia@ spycomar R Bras co mp 00; 0: 37-49

2 38 h cor in h maching mol wih uoa rsricion Inroucion R Bras co mp 00; 0: On-o-on maching mols hav bn usful for suying assignmn problms wih h isinciv faur ha agns can b ivi from h vry bginning ino wo isjoin subss of complmnary workrs: h s of workrs of yp I an h s of workrs of yp II h naur of h assignmn problm consiss of maching ach agn workrs of yp I wih on agn from h ohr si of h mark workrs of yp II h funamnal usion of his assignmn problm consiss of maching ach workr wih a on workr from h ohr si Roh an 99 ongll an Roh 99 Roh an Xing 994 an Romro-ina 997 ar xampls of rsarch work which invsiga paricular maching problms lik nry-lvl profssional labor marks an sun amission a collgs h agns hav prfrncs on h ponial parnrs abiliy has bn consir h main propry o b saisfi wih any snsibl maching A maching is call sabl all h agns ar mach o an accpabl parnr an hr is no mach pair of workrs ha woul prfr anohr parnr o hir currn on Whn h mols of bilaral assignmn ar sui h hory of coopraiv gams is rlvan sinc h soluion in hs mols has ypical faurs of soluion in such gams In coopraiv gams h problm is cnr on h analysis of h possibl soluions an is sabiliy In hs gams som playrs agns may rach bining agrmns for in hm h rsuls ha ach of h coaliions of playrs ha can b form ar sui Cor is a soluion of a coopraiv gam ha is no block by any coaliion of agns an will b spci larh frnc bwn h finiion of cor an ha of a group of sabl machings for a bilaral gam lis in h facs ha h cor is fin via a coaliion of agns an ha all coaliions play a rol whil h s of sabl machings is fin only wih rspc o a crain yp of coaliion ha is o say a rsul is in h cor i is no block by an agn or by any coaliion of agns h numbr of agns ha inrvn in h coaliion no bing consir om rsuls rfrring o h non-mpy cor in a family of gnralizaions in h assignmn mark which inclus h mark of marriag ar foun in Quinzii 984 Curil an ijs 985 among ohrs Gal an haply 96 show ha h s of sabl machings an h cor coinci A varian of h mol of bilaral assignmn is h assignmn mol wih uoa rsricion prsn by mnia ar Nm an Ovio 008 whr an insiuion has o hir a s of pairs of complmnary workrs an has a uoa uoa which is h maximum numbr of pair posiions o b fill his insiuion has a prfrnc on his ponial s of pairs h propry of sabiliy in his mol pns on h prfrncs xprss by h paricipans an on h prfrncs of h insiuion; ha is why h propry of -sabiliy is fin nr h rsricion of h insiuion s rsponsiv prfrnc h xisnc of h s of -sabl machings is guaran an is characrizaion is obain In his rsarch work h cor in h assignmn mol wih uoa rricion is sui or his aim h insiuion s rol in h concp of blocking by coaliion is rconsir an h concps of -block by coaliion an -cor ar fin nr h rsricion of h insiuion s rsponsiv prfrncs h xisnc of h -cor is prov an vn hough in his mol h s of -sabl machings an h -cor o no coinci a compl characrizaion of h -cor is obain h papr is organiz as follows cion prsns a br rvision of h horical concps of h maching mol an a rsul is sa ha links h s of sabl machings o h cor h mos imporan finiions of h assignmn mol wih uoa rsricion an h rsuls ha guaran h xisnc of h s of maching -sabl an is characrizaion ar sa as wll In cion 3 w fin h concps of -block

3 mnia 39 by coaliion an -cor In cion 4 w show a rsul which links h s of -sabl machings o h -cor as wll as h characrizaion of his s In cion 5 unr h rsricion of h insiuion s rsponsiv prfrncs h xisnc of h -cor is guaran cion 6 prsns his papr s conclusions Prliminaris h maching mol I consiss of wo isjoin ss of agns i h s of n workrs of yp I an h s of m workrs of yp II no by { n } { m } rspcivly ach workr has a sric prfrnc rlaion P ovr {&} an ach workr has a sric prfrnc rlaion P ovr {&} Noic ha only sric prfrncs ar bing consir imilar rsuls may b obain infrnc is allow Prfrnc profils ar n+m-upls of prfrnc rlaion rprsn by P P ; P P P P P n m Givn a prfrnc profil P h sanar maching mark is no by P Givn a prfrnc rlaion P h subs of workrs prfrr o h mpy s by ar call accpabl imilarly givn a prfrnc rlaion P h subss of workrs prfrr o h mpy s by ar call accpabl h assignmn problm consiss of maching workrs of yp I wih workrs of yp II h naur of h rlaionship bing kp an hr bing h possibiliy for boh yps of workrs o rmain unmach ormally: finiion A maching m is a mapping from h s ino h s {&} such ha for all an : ihr m or m & ihr m or m & 3 m an only m L b h s of all possibl maching m Givn a maching mark P a maching m is block by a singl agn f & P f m f W say ha a maching is iniviually raional i is no block by any singl agn A maching m is block by a pair of workrs P m an P m finiion A maching m is sabl i is no block by any iniviual agn or by any pair of workrs Givn a maching mark P nos h s of sabl machings Noic ha Gal an haply 96 [] hav prov ha h s of sabl machings is non-mpy & Givn a maching mark on an P a maching m is block by a coaliion hr xiss m such ha: an x P x for all x x finiion 3 h cor of P is h s of machings of i is no block by any coaliion Givn a maching mark P C nos h cor of R Bras co mp 00; 0: 37-49

4 40 h cor in h maching mol wih uoa rsricion On of h cnral rsuls prov by Roh an oomayor n 990 [7] which links h cor wih h s of sabl machings is ha hy coinci: C h maching mol wih uoa rsricion I consiss of wo isjoin ss of agns h s of n workrs of yp I an h s of m workrs of yp II no by { n } { n } rspcivly an an insiuion no by Insiuion has a mflxiv ransiiv anisymmric an compl binary rlaion R ovr h s of all possibl machings h mpy maching inclu As usual l P an I no h sric an infrn prfrnc rlaions inuc by R rspcivly h pair of workrs will work for insiuion an his has a maximum numbr of posiions uoa min{nm} o b fill; hn only h machings whos carinaliy is smallr or ual o may b accpabl h insiuion may choos som machings of accoring o hir prfrnc P an hir uoa rsricion W no {m : # m } his nw maching markr is no ; R Noic ha min{mn} h s of all h machings may b accpabl i A maching m is accpabl for insiuion accoring o hir prfrncs m an mr m & whr m & is h maching such ha m & x& for vry x Givn min{nm} a maching m is iniviually raional # m mp & an m f P f & for vry workr f such ha m f & A maching m is block by pair of workrs P m P m an ihr a m an m or b m is iniviually raional an m R m o m fin by: f f f f f ohrwis Noic ha m hn m m finiion 4 A maching m is sabl i is iniviually raional an is no block by any pair of workrs Givn a maching mark ; R nos h s of -sabl machings Noic ha mnia al 008 [] hav prov ha unr h rsricion of h insiuion s rsponsiv prfrncs h s of -sabl machings is non-mpy hy also obain a characrizaion of such a s as: No: h finiion of h insiuion s rsponsiv prfrncs an h ss an ar ma xplici in ail in h Appnix R Bras co mp 00; 0: 37-49

5 mnia 4 3 h -cor Our objiv in his scion is o xn h concp of blocking for coaliion o h maching mol wih rsricion of h capaciy nlik wha happns in mol in mol h insiuion has prfrncs ovr h s of machings ; ha is why i is ncssary o rconsir h insiuion s rol in h concp of blocking by coaliion W will consir ha whn h coaliion ha blocks a maching is form by agns alray conrac by h insiuion ha is o say no singl agns h insiuion os no rais any objcion in is formaion bcaus i consirs h blocking o b inrnal On h ohr han xrnal agns blong o h coaliion ha is o say singl agns in h original maching h insiuion imposs h nw maching o b prfrr o h original on ormally: finiion 5 A maching m is -block by a coaliion xiss m such ha: m xp x mx for all x 3 m x mx for all x m 4 If x such ha mx & hn m R m In mol h cor was fin as h s of machings which ar no block by any coaliion W xn h concp of cor o h mol in h following way: finiion 6 Givn a maching mark whih uoa rsricion h -cor of is h s of machings of which ar no -block by any coaliion W no C h -cor of Proposiion L R b a maching mark wih uoa rsricion; C hn m s -iniviually raional Proof Assum ha m is no -iniviually raional hn hr xiss f such ha &P f m f W consir h coaliion { f } an h maching m such ha x x x xf f f f hy saisfy h coniions of -block by coaliion conraicing h fac ha C h following proposiion sablishs ha is a subs of -cor Proposiion If R is a maching mark wih uoa rsricion hn C h proof o his proposiion is vlop in h Appnix as Proposiion A7 h following xampl shows ha no always is a subs of C xampl L P b h maching mark such ha { } an { } ar h wo ss of workrs wih h prfrnc profil P P whr: P P P P 3 an an h following prfrncs ovr an : y R Bras co mp 00; 0: 37-49

6 4 h cor in h maching mol wih uoa rsricion consiss of h following maching: W consir h coaliion { } By im vii of h rsponsiv xnsion I whr As B B by im vi of h rsponsiv xnsion P hn P an C W fin h following subs of as follows: R { : hr xiss such as an P P P an P } W show ha vry maching ha is in R is no in h -cor Proposiion 3 L R b a maching mark an m r hn C Proof L m r hn hr xiss such ha m m & P P P an P W consir h coaliion { } an h maching x x x x x W hav ha an x x for vry x his h coniions an of -block for coaliion ar saisfi Also coniions 3 of -block for coaliion ar saisfi sinc m m & P P P an P o monsra coniion 4 w consir h maching x x x x x By ms vii of rsponsiv xnsion n by ms vi of rsponsiv xnsion P which implis ha for ransiiv propry of R u P hn h maching m is -block for coaliion an C W fin h s \ R an w will prov ha i is a subs of -cor Proposiion 4 If R is a maching mark wih uoa rsricion hn C h proof o his proposiion is vlop in h Appnix as Proposiion A8 4 h -cor an h -sabl h following rsul sablishs ha vry maching in h -cor is also a -sabl of Proposiion 5 If R is a maching mark wih uoa rsricion hn C Proof Assum ha C an rom C by proposiion 3 m is -iniviually raional As hn hr xiss ha -block o m R Bras co mp 00; 0: 37-49

7 mnia 43 L {} an l h maching whr x x x x x ohrwis Coniions an 3 of -block by coaliion ar saisfi by finiion of m As m coniion of -block by pair implis coniions of -block by coaliion o prov Coniion 4 w consir m& or m& hn Coniions of -block by coaliion imply ha R ; which conraics ha C h rsuls obain unil now allow us o giv a compl characrizaion of h -cor in rms of ss an horm If R is a maching mark wih uoa rsricion hn C Proof rom Proposiion 4 an Proposiion 6 w obain: C L C hn by Proposiion 5 As w obain: or If #m by w hav ha m L #m an l us assum ha which implis by ha m an \ R; hn m R an by Proposiion 5 C which conraics C W obain: C 3 an 3 imply ha C 5 xisnc of h -cor In orr o guaran ha C w prov ha givn a mol of mark ss an ar no mpy simulanously Lmma If hn ~ for vry ~ Proof L m which implis ha hr xiss N such ha an #m Wihou loss of gnraliy w assum ha max{ : } which implis ha ~ inc w hav ha inc m an m ~ ar sabl in an rspcivly by Lmma A in Appnix # ~ # # ~ which implis ha # ~ o # ~ If # ~ hn ~ an hrfor If # ~ hr xiss ~ such ha # ~ # ~ # ~ hn # ~ o # ~ Wnohcarinaliyofamaching h of a m by m # #{ : } #{ : } R Bras co mp 00; 0: 37-49

8 44 h cor in h maching mol wih uoa rsricion ~ k inc is fini hr xiss k an such ha # ~ hn By rpaing his procss h rsul ha follows is obain: ~ for vry ~ L us prov now ha s C is no mpy horm If ; R is a maching mark wih uoa rsricion hn C Proof L by Proposiion C L us assum an l m an # Noic ha an hn C for vry 4 Now w will show ha: C C for > 5 uppos ha C y C wih > inc C hn which implis ha # an for vry coaliion hr is no maching ha saisfis Coniions y of -block by coaliion his conraics C By 5 an C i is impli ha C for vry > 6 rom 4 an 6 w conclu ha: C for vry 7 6 Conclusions In his rsarch work h concp of cor is fin in h maching mol wih uoa rsricion an is call -cor or his purpos h insiuion s rol in h concp of blocking by coaliion in h bilaral assignaion mol was rconsir h xisnc of h -cor unr h insiuion s rsponsiv prfrnc rsricion is prov By mans of xampl i is sa ha unlik wha happns in h bilaral mol h -sabls an h -cor may no coinci; nvrhlss horm shows how a characrizaion of h -cor can b obain in rms of subss of h s of -sabls Appnix h rsricion of rom now on w will no {} an c {} such ha { c }{} an f f will no a gnric workr Givn w no h rsricion of P o by P C Givn P C w no h rsricion of o by P P or h sak of simpliciy C C w no P whr w hav o unrsan ha P P P C R Bras co mp 00; 0: 37-49

9 mnia 45 Lmma A mnia arí Ovio an Nm 008 Givn P an l m an b h sabl machings for an rspcivly hn # # # # \ h insiuion s rsponsiv prfrnc Givn a maching mark an a uoa min{ n m} w no A h s of all -sabl machings W will assum ha h insiuion has an iniviual prfrnc ovr h s an an iniviual prfrnc ovr h s an is prfrnc ovr machings ar ircly connc wih is prfrncs ovr workrs An insiuion s prfrnc is call rsponsiv o is iniviual prfrncs for any maching ha frs in only on workr h insiuion prfrs h maching ha has h mos prfrabl workr accoring o h iniviual prfrncs In orr o formaliz h insiuion s rsponsiv prfrnc w inrouc h noaions ha follow or vry maching m consir B { : } or vry f : f si f { } f f Noic ha finiion A A prfrnc rlaion R is a rsponsiv xnsion of prfrncs an ovr an rspcivly such ha i saisfis h following coniions: i ii iii iv P an only an P an only P P P an only an only for vry B v or vry m such ha # # an B B \ { } { } P an only P : vi or vry m such ha B B an P hn P vii or vry m such ha an hn I W consir a prfrnc R o b rsponsiv hr ar wo iniviual prfrncs an ovr {} an {} rspcivly such ha R is a rsponsiv xnsion Rmark A3 Givn wo prfrncs an ovr {} an {} rspcivly w can consruc a rsponsiv prfrnc rlaion R ovr h s of all machings ; morovr his xnsion is no uniu h ss an Now w will consir h mol whr R is a rsponsiv prfrnc Wihou loss of gnraliy an in orr o avoi h aiion of noaional complxiy o h mol w assum ha all h agns of ss an ar accpabl for h insiuion i for vry an w hav ha an or vry N w can fin h following subs such ha # an for vry f an f w hav ha l f f No ha whr # l Givn ss {# } an { # } ; for vry w no h rsricion of o an P i R Bras co mp 00; 0: 37-49

10 h cor in h maching mol wih uoa rsricion 46 R Bras co mp 00; 0: Givn an h following ss of machings: ohrwis vry for # an } such ha : { Proposiion A4 Givn R hr xiss K such ha K Givn an h following ss of sabl machings: \ an \ vry for or ihr :# P P an } such ha : { Proposiion A5 Givn R hr xiss ^ K such ha ˆ K Rmark A6 h ss K an ^ K on h prvious proposiions ar givn by: such ha : { N K } an such ha : { ^ N K } Proposiion A7 If R is a maching mark wih uoa rsricion hn C Proof W assum ha an; hn # an hr xis a coaliion an a maching ha saisfy h coniions of -block by coaliion Noic ha m is -block by h coaliion hn m is block by h coaliion inc hr xiss N such ha an as C C hn m is no -block by any coaliion ; hus which implis ha such ha xiss hr x x 8 inc w can wri hr bing an W consir h following ss: \ \ 9 W can wri y : 0

11 mnia 47 Now w will show ha y L us assum ha an l us consir ; hn Howvr an is a coaliion ha blocks m hn w hav ha P an P which implis ha is a -bloching pair of m his conraics imilary w can prov ha By an Coniion of -block by coaliion w obain # # # an # # # L \ an \ \ \ 3 W xprss #m y #m in rms of h ss fin in 9 # #{ \ : } # \ #{ : } 4 L now { \ : } { : } 5 By coniion of -block by coaliion w hav ha { : } an \ by 0 i is impli ha { : } rom \ an 5 w can fin a s A such ha # #{ \ : } # A# # 6 an by Coniion 3 of -block by coaliion w hav ha: # #{ \ : } # A# # 7 As # an # an by 4 an 7 w hav ha: # A # # #{ : } # \ 8 By # { : } 7 an 8 w hav ha: # # which implis ha: { # : } an # \ # # \ Noic ha for vry an ; his implis ha w can fin h following sunc of machings 0 r such ha k k inc k k no ha as an ; an his implis ha m k is fini wll # k # k an B B \ As R is a rsponsiv xnsion k k k P k an by ransiiviy of R P 9 r ymmrically by consiring # y # w obain # \ # # { : } \ an his implis ha for vry an an now w can fin h sunc of maching r r such ha k By ransiiviy k of R w hav ha r P 0 R Bras co mp 00; 0: 37-49

12 48 h cor in h maching mol wih uoa rsricion By 9 0 an ransiiviy of R P inc by 8 hr xiss x such ha x Coniion conraics h fac ha m is -block by coaliion inally C Proposiion A8 If R is a maching mark wih uoa rsricion hn C Proof W Assum ha an C hn # an hr xis a coaliion an a maching ha saisfy h coniions of -block by coaliion Noic ha m is -block by h coaliion hn m is block by h coaliion inc hr xiss N such ha an as C C hn m is no -block by any coaliion ; hus which implis ha hr xiss x suchha x inc w can wri hr bing an W consir h following ss: \ W hav ha y : An \ 3 4 Now w will show ha \ an \ 5 W assum ha hr xiss \ By coniion of -block by coaliion which impls ha or L us assumha hn As an is a coaliion ha blocks m w hav ha P an P which implis ha is a -blocking pair of m I conraics If hn an sinc \ As is a coaliion ha blocks m w hav ha P an P which implis ha \ \ I conraics hn \ imilary w can prov ha \ Now w show ha y 6 L us assum ha an By coniion of -blocks by coaliion inc P y an P hn is a blocking pair of m which conraics imilary w can prov ha inc for vry an for vry an ar no muually accpabl; hn an ; by Coniion of -blocks by coaliion an wih which h proof o 6 is compl inc R for vry an for vry hn: y 7 R Bras co mp 00; 0: 37-49

13 mnia 49 By 6 an Coniion of -blocks by coaliion # # By 5 an 7 for vry an Bsis w hav ha by Coniion of -blocks by coaliion an consunly his implis ha w can fin h following sunc of machings 0 r such ha k k inc k no ha as an i is impli ha k k is a maching such ha # k # k an B B \{ } { } As R is a rsponsiv xnsion k k k P k an by ransiiviy of R P 8 r ymmrically by consiring \ an by Coniion of -block by coaliion w hav ha for vry y which implis ha w can fin h sunc of maching r r such ha k By ransiiviy of R w hav ha k r P 9 By 8 9 an ransiiviy of R P 30 inc by hr xiss x such ha x coniion 30 conraics h fac ha m is -block by coaliion inally C Bibliography [] NIA ; ARÍ ; N A al abl soluions on machings mols wih uoa rsricions In rviw 008 [] GAL ; HAPLY L Collg amissions an h sabiliy of marriag Amrican ahmaical onhly v 69 p [3] GAL ; OOAYOR om rmarks on h sabl maching problm Amrican ahmaical onhly v p [4] QINZII Cor an compiiv uilibria wih inivisibiliis Inrnaional Journal of Gam v 3 p [5] RORO-INA A Implmnaion of abl oluions in a Rsric aching mark Rviw of conomic sign v 3 p [6] ROH A Nw Physicians: a naural xprimn in mark organizaion cinc v 50 p [7] ROH A; OOAYOR wo-si aching: A uy in Gam-horic oling an Analysis Cambrig nivrsiy Prss Cambrig nglan [conomrica ociy onographs No 8] 990 [8] HAPLY Ll On balanc ss an cors Naval Rsarch Logosic Quarrly v 4 p [9] HAPLY Ll; CAR H On cors an inivisibiliy Journal of ahmaical conomics v p [0] HAPLY Ll; hubik h assignmn gam I: h cor Inrnaional Journal of Gam hory v p R Bras co mp 00; 0: 37-49