LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY

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1 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY MARCIN PĘSKI Absrac. We analyze a large roommae problem.e., marrage machng n whch he marrage s no resrced solely o machngs beween men and women) wh non-ransferable uly. I s well known ha whle a roommae problem may no have a sable proper machng, each roommae problem does have an sable mproper machng. In a random uly model wh ypes from Dagsvk 000) and Menzel 05), we show ha all mproper sable machngs are asympocally close o beng a proper sable machng. Moreover, he dsrbuon of ypes n sable machngs proper or no) converges o he unque maxmzer of an expresson ha s a sum of wo erms: he average welfare of he machng and he Shannon enropy of he dsrbuon. In he noseless lm, when he random componen of he uly s reduced o zero, he dsrbuon of ypes of mached pars converges o he oucome of he ransferable uly model.. Inroducon We analyze sable machngs n he large roommae model Gale and Shapley 96)) wh random and non-ransferable uly. We assume ha he uly of ndvdual machng wh ndvdual j s drawn from a dsrbuon U j F j,θ {,j} ) ha depends on he ypes of boh of ndvduals for example, ncome, beauy, ec.), and an unobservable mach-specfc shock θ {,j}. The mach-specfc shock s drawn from a dsrbuon ha depends on he ypes of he ndvduals. The key assumpon s ha dsrbuons F s,θ are absoluely connuous wh respec o each oher a he op end of her suppors. The assumpon ensures ha he ules are subjec o a non-rval randomness ha remans even afer learnng he ypes and mach-specfc Dae: 0/06/5.

2 MARCIN PĘSKI shock of wo ndvduals. A specal case s he random uly specfcaon of Dagsvk 000): U j = υ j + γε j, ) where υ j s he deermnsc par of he uly, he random shock ε j s ndependenly dsrbued wh exreme value ype I dsrbuon, and γ > 0 s a parameer. The roommae problem s a generalzaon of he classc marrage machng model n whch machng s no longer resrced solely o members of wo dfferen sdes of he marke. In parcular, hs generalzaon allows for he possbly of same-sex marrages, or parnershps beween workers n worker-frm machng. I s well-known ha he roommae problem may no have a sable proper machng. For hs reason, we use an approxmae concep of mproper) machng where some ndvduals are badly) mached wh wo, nsead of one, oher ndvduals. The concep was nroduced n Tan 99) who esablshed he exsence of a sable bu, possbly, mproper) machng. Our man resul esablshes wo asympoc properes of all sable and, possbly, mproper) machngs when he populaon grows large and ules are derved from he random uly model ). Frs, we show ha, wh a probably convergng o, he fracon of badly mached ndvduals n any sable machng converges o 0. I follows ha all sable machngs are asympocally proper. Second, we characerze he lm of he probably masses d T { } T { }) over he pars of ypes of mached ndvduals. Here, mach wh corresponds o sayng alone. As he sze of he populaon grows o nfny, he dsrbuons over he mached ypes n any sable machng converge o he unque soluon of he concave maxmzaon problem max d Welfare d; F.. ) + Enropy d) s. d sasfes feasbly resrcons. 3) γ The welfare erm of formula 3) s equal o he average sum of deermnsc ules n he sable mach, where he uly coeffcens are derved from he op end denses of dsrbuons ). The second erm does no depend on he properes of he dsrbuons ) and s a verson of a well-known measure of dsorder from hermodynamcs and also bears resemblance o he Shannon measure of nformaonal conen). As n physcs, he enropy here has a counng nerpreaon and

3 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 3 s equal modulo a consan) o he logarhm of he number of mcro-confguraons.e., ndvdual machngs) ha are conssen wh a gven macro-sae.e, dsrbuon d). In parcular, he larger he enropy, he more mprecse he nformaon abou he ndvdual machngs. Fnally, he mos mporan feasbly resrcon s ha he margnals of dsrbuon d mach he dsrbuon of ypes n he populaon. Oher feasbly resrcons may by requred o reflec specal addonal feaures of he envronmen. For example, n case of he marrage machng, male ypes are resrced from marryng oher male ypes, ec. Togeher, our resul says ha he sable machng balances he welfare maxmzaon wh an enropy cos. The random uly model ) was nroduced n Dagsvk 000) for he case of he marrage machng and exreme value ype dsrbuon of shocks). Dagsvk 000) reles on a heursc o argue ha he lm of sable dsrbuons mus be he unque soluon o a sysem of demand-supply ype of equaons. Menzel 05) provdes a formal proof of he clam n Dagsvk 000). The ngenous argumen follows he demand-supply logc and reles on varous probably bounds obaned from a careful analyss of he rejecon chans n he Gale-Shapley algorhm. Ths paper conrbues o he leraure n he followng ways. Frs, we work wh a more general roommae problem where, as Gale and Shapley 96) have already observed, he deferred accepance algorhm does no apply. For hs reason, our proof s necessarly very dfferen from Menzel 05). Insead, our argumen s drec and consss of wo pars. In he frs par, we esmae a combnaoral upper bound on he number of all no necessarly sable) machngs ha nduce a gven dsrbuon d. I urns ou ha hs number s ncreasng wh enropy. In he second par, we esmae an upper bound on he probably ha a gven machng s sable. We show ha hs probably s monooncally ncreasng wh he average welfare erm of 3). The esmae reles on approxmaons from he large devaons heory see Ells 005)). By mulplyng boh esmaes, we oban an upper bound on he probably of a sable machng wh ype dsrbuon d. We show ha f d does no maxmze expresson 3), hen he probably of a sable machng wh a ype d converges o 0 a an exponenal rae. Thus, he dsrbuon ha s nduced by all sable machng mus be a maxmzer of 3).

4 4 MARCIN PĘSKI Second, Menzel 05) assumes ha he uly shocks are ndependen across parners n he mach. However, uly correlaons are common n a ypcal machng suaon. ) Examples of common shocks nclude: random evens ha affec boh pares smulaneously weaher durng a frs dae, marke condons durng a job nervew), common hsory aendng he same unversy, common frends), unobserved relaonshp characerscs par-me vs full-me, possbly of workng from home, ec.) ha jonly affec he preferences of boh machng pares. I s mporan o undersand how he correlaed shocks affec he predcons of he model. For hs reason, we allow for any common unobservable mach-specfc shocks. Our proof s smple and flexble enough so ha he mach correlaons do no cause any exra complcaon. Thrd, Dagsvk 000) and Menzel 05) show ha he unque lm dsrbuon s a soluon o a sysem of demand-supply ype of equaons. Our resul s conssen wh her fndngs, as he unque maxmzer of 3) solves he same se of equaons. We are no aware of any oher, earler use of formula 3) o descrbe he socal oucomes. Boh approaches are complemenary and shed lgh on dfferen aspecs of he problem. In parcular, our approach emphaszes he fac ha he lkelhood of a ceran socal oucome s drecly relaed o s randomness, or he cardnaly of mcro-confguraons ha are conssen wh he observed oucome. Fourh, formula 3) leads o a surprsng connecon beween large populaon machng wh non-ransferable and random uly ) and machng models wh ransferable uly wh coeffcens υs. We consder a noseless lm of he model ) where γ 0. In he noseless lm, expresson 3) becomes domnaed by he frs erm and he sable machng dsrbuon maxmzes average welfare. The laer s also he unque dsrbuon obaned n machng wh preferences gven by υs and ransferable uly. A he same me, he lm s ypcally no equal o he dsrbuon obaned when uly s non-ransferable and γ = 0. We nerpre he resul as a non-ransferable foundaon for he ransferable uly model. Dagsvk 000) allows for correlaons n hs model, bu hs argumen remans heursc.

5 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 5 Choo and Sow 006) nroduced a dfferen verson of marrage machng model ) wh ransferable uly and where he random shock ε j = ε j s he same for all ndvduals j of he same ype. Workng wh ha model, Galchon and Salane 0) show ha, wh he ransferable uly, he lm dsrbuon over ypes n sable machng wh ransfers.e., welfare maxmzng machng) converges o he soluon of a verson of 3), bu where Enropy d) s replaced by a generalzed enropy. The generalzed enropy s defned relave o he dsrbuon of he vecor of he shocks ε ṭ ) T and s equal o he value of a ceran sochasc allocaon problem. The man dfference wh our paper s ha he enropy n 3) does no depend on he dsrbuon of he shocks and he enre mpac of he dsrbuon s capured by he average welfare erm. Secon descrbes a general roommae problem. Secons 3 and 4 nroduce, respecvely, he models of ypes and random uly. Secon 5 presens he man resul and descrbes he man deas of he proof.. Roommae problem In hs secon, we dscuss he roommae problem. There are N ndvduals I. An ndvdual may form a mach wh anoher ndvdual j, n whch case receves uly U j. Alernavely, ndvdual may decde o reman alone wh uly U. We allow for he possbly ha some maches are no feasble. Le R I I be he se of resrced maches. We assume ha he resrcon s symmerc: f, j) R, hen j, ) R, and ha he opon of sayng alone s no resrced:, ) / R. For example, n he marrage machng model of Gale and Shapley 96), he populaon s dvded no wo sexes, and only oppose-sex maches are allowed. A proper machng s a bjecon µ : I I such ha µ µ )) = for each and such ha, µ )) / R. Proper machng µ s sable f,j)/ R eher U j U µ) or U j U µj) j 4) An alernave way of modelng mach resrcons s o assume ha he uly from resrced maches are, or a leas smaller han he uly from sayng alone. We ake he explc approach because s more convenen when we presen he random uly model.

6 6 MARCIN PĘSKI a b d m n c e j o f h g l k Fgure. An mproper) machng. In oher words, a machng s sable f here s no unresrced blockng mach. I s well-known ha a roommae model may no have a sable proper machng and, n he parcular, ha he Gale-Shapley algorhm does no apply). Insead, we use a relaxed soluon proposed n Tan 99). An mproper) machng s a bjecon µ : I I such ha, µ )) / R. 3 If µ µ )) =, hen, we say ha s properly mached; oherwse, s badly mached. Because µ s a bjecon, all badly mached ndvduals can be dvded no cycles, µ ), µ ),..., µ n ) =, where each ndvdual s mached wh a predecessor and successor n he cycle and all members of he cycle are badly mached. An example of a machng s drawn on Fgure. An arrow from ndvdual o j ndcaes ha µ ) = j. Thus, ndvduals a, b, f, g, m, n are properly mached, ndvduals c, d, h,, j, k, l are badly mached, and ndvdual o remans sngle. For each machng µ, defne a mappng c µ : I {s, p, b} ha assgns each ndvdual wh a caegory of s mach: s, f µ ) =.e., f s sngle), c µ ) = p, f µ ), µ µ )) =, e., f s properly mached), b, oherwse. 3 Tan 99) refers o µ as a paron.

7 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 7 Improper) machng µ s sable f 4) holds and, addonally, for each, U µ) U µ ). The frs requremen s obvous. The second requremen bes only for he badly mached ndvduals and plays he role of sorng: each badly ndvdual s mached wh wo oher ndvduals, and µ ) s he worse of wo maches. Assumpon. Src Preferences) For each and j j, U j U j. The assumpon s sasfed wh probably n he random uly model descrbed laer n he paper. Theorem. Tan 99)) Assume Src Preferences. Then, here exss a sable possbly, mproper) machng µ Types Followng Dagsvk 000) and Choo and Sow 006), we are neresed n he dsrbuon of ypes among he mached ndvduals. A ype s any observable characersc, ncludng gender, beauy, ncome, educaon. A ype of ndvdual s denoed as T, where T s a fne se. 5 Le α = N { : = } denoe he fracon of ndvduals wh ype. We assume ha he se of resrcons R s ype-measurable: e., here s R T T such ha for each j,, j) R f and only f, j ) R. There are wo specal cases: no-resrcon case, n whch R = Ø, and 4 Theorem s closely relaed o Chappor, Galchon, and Salane 04) who consder he roommae problem wh ransferable uly. They show ha f each ndvdual can be cloned no wo dencal copes wh he same preferences, and reaed n he same way by oher players), hen here exss a proper sable machng. 5 Ths assumpon can be relaxed o a suaon where T s compac. The dea s o dvde dscreze ) he space of ypes T no fnely many small areas. The formal argumen would no add much o he dffculy of he curren proof, bu would complcae he exposon. We wll provde he deals for an neresed reader.

8 8 MARCIN PĘSKI he Gale-Shapley marrage machng, where he se of ypes T = M W s a dsjon unon of men ypes M and women ypes W and R = M M W W. A ype dsrbuon s a probably measure d T T {s, p, b}). Each machng µ nduces a machng ype d µ so ha for each,, c), d µ,, c) = N { : =, µ) =, c µ ) = c }. In oher words, d µ s a probably dsrbuon of an ordered uple of random varables T, T, C), where T s a ype of a randomly drawn ndvdual n he populaon, T s he ype of µ ), and C s he caegory of he mach, µ )). We wre d c) =, d,, c) for each c C. Le D N α) denoe he se of ype dsrbuons d µ nduced from some machng µ., : Any nduced ype dsrbuon d = d µ mus sasfy he followng equales for each d,, c) = α,,c d,, c) = 0 f, ) R for each c, 5a) 5b) d,, s) = 0 f, 5c) d,, p) = d,, p) for each, 5d) d,, b) = d,, b) =: β d), 5e) Ideny 5a) ensures ha all ype ndvduals are couned. Ideny 5b) reflecs he resrcons. Ideny 5c) s abou sngle mached ndvduals. Ideny 5d) s mpled by he fac ha all proper maches are symmerc. Ideny 5e) follows from he fac ha each bad mach of ndvdual wh µ ) corresponds o exacly one bad mach wh ndvdual µ ). We defne β d) as he fracon of badly mached ndvduals of ype. For each α T and β ) [0, ] T, le D α, β )) be he se of machng ypes ha sasfy 5a)-5e) and such ha β d) = β for each. Fnally, le D α) = D α, 0) be he subse of machng ypes of proper machngs e., ds such ha d b) = 0). Somemes, we shall use he followng assumpon: Assumpon. Se {d D α) : d s) = 0} s non-empy.

9 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 9 If he assumpon s sasfed, hen one can always dvde he ndvduals no pars wh he possble excepon of nsgnfcan fracons of he populaon). The assumpon s always sasfed n he no-resrcon case. In he marrage machng case, he assumpon s sasfed f and only f he number of men and women are equal, M α = W α. For each machng ype d, defne Enropy d) = d,, s) log d,, s), d,, b) log d,, b) + )., d,, p) log d,, p) + ) 6) We refer o Enropy d) as a per capa enropy of d. The coeffcen n fron of he proper parwse erms comes from he fac ha each such mach nvolves wo people. Because each badly mached ndvdual belongs o wo maches, here s no analogous coeffcen n fron of he enropy of bad maches. The per-capa enropy s a normalzed verson of he Shannon enropy of random varable T, T ), where T s a ype of randomly drawn ndvdual and T s he ype of hs or her mach. The Shannon enropy s ypcally nerpreed as a measure of randomness or unpredcably) of dsrbuon d. In hs paper, he role of he enropy comes from a closely relaed fac: serves as a bound on he number of machngs of a gven ype. For each machng marke of sze N, le M N d) = {µ : d µ = d}. Lemma. For each machng ype d D α, β )) N log M N d) 7) ) d p) + d b) log N + α log α + β log β + Enropy d) + o ) The remnder erm o ) 6 does no depend on d. The proof can be found n Appendx A. 7 6 Recall ha o ) denoes a sequence a N 0 as N. 7 Alhough s no necessary for our purposes, one can show ha he bound 7) s asympocally gh.

10 0 MARCIN PĘSKI 4. Random uly For each j, he uly U j s condonally ndependenly drawn from dsrbuon ), ha depends on he ype of boh of he ndvduals, and an unobservable machspecfc shock θ {,j} Θ. For each, uly U s drawn from dsrbuon F. Followng Menzel 05), we assume ha uly U s equal o he maxmum among J N..d draws from dsrbuon F and ha lm N / J N = j 0. We allow ha j = 0. If j > 0, hs assumpon ensures ha, as N, he fracon of he populaon ha remans sngle does no converge o 0. If j = 0 for nsance, because J N = for each N), each ndvdual receves many more uly draws from parwse maches han from sayng alone, whch leads o asympocally small fracons of populaons who prefer o say sngle. If J N grows a rae hgher han N /, hen almos all ndvduals reman sngle n large populaons. The mach-specfc shock s chosen ndependenly for each mach from probably dsrbuon Φ, over fne se Θ. Is role s o generae unobservable) correlaons beween he ules of mach parners. The exsence of such correlaons s naural n almos all machng suaons. Defnon. The random uly model F = { } F,θ, F s regular f each dsrbuon F F a) s aomless, b) has suppor resrced o [0, ], c) has connuous densy f wh respec o he Lebesgue measure on he nerval, and d) he op end densy s srcly posve f ) > 0. Par d) s he key resrcon: ensures ha, a leas a he op of he suppor of he dsrbuons, even afer learnng he ype and he mach specfc shocks, he uly s subjec o non-rval remanng randomness. Our characerzaon of machng ype n sable machngs depends on he dsrbuon of ules only hrough he op end dervaves. Because he noon of sably s purely ordnal, he regulary assumpon can be relaxed n he followng way. The model s essenally regular f here exs srcly ncreasng funcons φ : R R for each ype such ha he random uly model F = { } F,θ φ, F φ s regular. Example below llusraes hs defnon. From now on, we assume ha he model s regular.

11 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY For any wo ypes,, θ, le V,θ := f,θ ) > 0,V, = θ Φ, θ) V,θ V,θ, and υ, = log V,, 8) We refer o υ, V := f ) > 0, and υ = log V + log j. as he welfare of a mach of ndvdual ype wh ype and υ as he welfare of sayng sngle. The laer s modfed by he normalzed number of draws j ha s used o deermne he uly of beng sngle. We use he erm welfare for wo reasons. Frs, υ, has a naural nerpreaon of deermnsc or, observable) welfare n some applcaons see Example below). Second, consan υ, each machng d, le Welfare d; υ..) = plays he role of mach welfare n he followng formula. For d,, s) υ + d,, p) υ, + d,, b) υ,. 9),, We nerpre Welfare d; υ..) as a average welfare of machng ype d. The coeffcen n fron of he average welfare of proper parwse maches comes from he fac ha each such mach nvolves wo people. Because each badly mached ndvdual belongs o wo maches, here s no analogous coeffcen n fron of average welfare n bad maches. By convenon, he value of 9) s well-defned whenever j > 0 or d s) = 0; oherwse, s equal o. Example. Suppose ha υ he sum are consans and suppose ha he random uly s u j = υ j of a ype-dependen deermnsc normalzed) uly υ j, and an dosyncrac random shock ε j 0 ha s drawn..d from dsrbuon G. Furher, suppose ha he + ε j suppor of he dsrbuon G s unbounded, and for each a, he lm G x + a) lm x G x) > 0 0) exss and s srcly posve. Examples nclude he exreme value ype I dsrbuon used n Dagsvk 000) and he exponenal dsrbuon.) Ths model s essenally regular wh welfare consans equal o υ,s = υ s +υ s. Indeed, le U j = G u j ). Then,

12 MARCIN PĘSKI he cumulave dsrbuon funcon F j of U j s equal o F j x) = G G x) υ ) j. 8 Moreover, one can show ha he logarhm of he densy a s equal o 9 for some consan C G > 0. log f j ) = C G υ j 5. Man resul The man resul characerzes he machng ype of sable machngs. Consder a maxmzaon problem max Welfare d; υ) + Enropy d). ) d Dα) Problem ) has a soluon f and only f he objecve funcon s well-defned over a leas some par of he feasble se, or when eher j > 0, or Assumpon s sasfed. Moreover, because Welfare d; υ) s lnear and Enropy d) s srcly concave, f he soluon exss, s necessarly unque. We denoe he soluon as d α, υ). The consran d D α) ensures ha he soluon respecs he dsrbuon of ypes, feasbly resrcons R, and, mporanly, assgns probably o proper maches. In parcular, because d b) = 0 for each d D α), problems 3) and ) are equvalen. Theorem. Fx a ype space T, a measure α T, a mach shock space Θ, dsrbuons Φ, of mach shocks, and a regular random uly model { F,θ, F }. Le υ. be derved as n 8). Suppose ha eher j > 0, or Assumpon holds. Then, here ) ) ) ) ) 8 F j x) = P U j < x = P G u j < x = P u j < G x) = P ε j < G x) υ j = ) G G x) υ j. 9 Le L υ) denoe he value of he lm n 0). I mus be ha L υ + υ ) = L υ) L υ ), or L υ) = e C Gυ for some consan C G. Because he lm s decreasng, mus be ha C G > 0. By L Hospal s rule, f j ) = lm y d )) g G G y) υ j dy = lm x x υ j ) g x) ) G x υ j = lm = e C Gυ j. x G x)

13 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 3 exss a unque soluon d α, υ) o problem 3) and lm Prob N d µ d α, υ) < ε for each sable machng µ) =. N The proof of Theorem can be found below. The heorem says ha for suffcenly large regular markes, n all random sable machngs, he machng ype converges o he unque soluon of he maxmzaon problem ). In parcular, he fracons of badly mached ndvduals n all sable machngs converge o 0 and all sable machngs are asympocally proper. The hypohess of Theorem requres ha eher Assumpon holds and here exss a machng n whch no ndvdual says sngle, or j > 0. Assumpon s always sasfed n he unresrced roommae problem. In he marrage machng case, he Assumpon holds as long as he number of men and women are equal. In such a case, he hess of Theorem holds even when j = 0. 0 I follows from he proof of he Theorem, ha he ules of asympocally) all ndvduals converge o,.e., o he op end of he suppor of he dsrbuon of ules. One could nerpre hs observaon as a saemen abou he effcency of he sable machngs. A he same me, s worh rememberng ha our model of sochasc uly s purely ordnal and he comparson of ndvdual ules hrough aggregae welfare mgh no always be jusfed. 0 In Menzel 05), J N = N / and j =. The proofs n Menzel 05) rely n mulple ways on he assumpon ha j > 0. Our proof demonsraes ha hs assumpon s no necessary a leas, as long as Assumpon holds). In cases where cardnal nerpreaon and he nerpersonal comparson of ules are warraned, he above observaon confrms a recen resul of Lee and Yarv 04) who sudy a closely relaed random uly model, bu wh ransferable uly.

14 4 MARCIN PĘSKI 5.. Relaon o Dagsvk 000) and Menzel 05). The frs-order condons of problem ) lead o he followng sysem of equaons log d,, p) = υ, + λ + λ for, ) / R, ) log d,, s) = υ + λ, where λ are Lagrange mulplers assocaed wh consrans 5a). In parcular, he soluon can be represened as a funcon of Lagrange mulplers λ. Afer subsung o he feasbly consrans, usng he defnons 8), we oban α e λ = V + s α s V,s α s e λs ). Furher, afer leng Γ = α e λ V, we oban Γ = s α s V,s V + Γ s. 3) Equaon 3) corresponds o equaons 3.5) from Menzel 05) recall ha n ha paper, j V =.) Menzel 05) refers o erms Γ as an nclusve value of ype ndvduals. Dagsvk 000) and Menzel 05) esablsh he unqueness of he soluon o 3) usng he conracon properes of a ceran operaor. Here, we show Consder frs he non-resrced case. Because of 5d), we can ake d,, p) = d {, }, p). The Lagrangan assocaed wh problem ) can be wren as The FOC are: L d, λ) = d,, s) υ log d,, s) ) + d {}, p) υ, log d {}, p) ) + ) d {, }, p) υ, log d,, p) + {, } λ d,, s) + d {, }, p) ) log d,, s) = + λ + υ, log d {}, p) = + λ + υ,, log d,, p) = + λ + λ + υ, for. By changng he varable λ λ, we oban )..

15 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 5 ha equaon 3) descrbes a soluon o he srcly concave maxmzaon problem. Such a soluon s, necessarly, unque. 5.. Proof of Theorem. The probably ha he machng ype of some sable machng s ε-away from d α, υ) s bounded from above by Prob d µ d α, υ) < ε for each sable machng µ) 4) =Prob d µ d α, υ) > ε for some sable machng µ) Prob µ s a sable machng) = = µ, d µ d α,υ) >ε d D N α), d d α,υ) >ε µ M N d) d D N α), d d α,υ) >ε p µ) M N d) p N d) where we denoe p µ) as he probably ha he realzaon of ndvdual ules makes machng µ sable, and p N d) = M N d) µ M N d) p µ) s he average probably of a sable machng among machngs ha nduce d. Below, we show he followng clam: 4) converges o 0 a an exponenal rae. Noce ha N log Prob dµ d α, V ) < ε for each sable machng µ)) N log D N α) + N log M N d) + N log p N d)). max d D α), d d α,v ) >ε The number of ype dsrbuons, D N α), s polynomal n N. 3 Thus, N log D N α) 0 as N ncreases, and n order o prove our clam, s enough o show ha he second erm of he las lne above s srcly negave. Le j N = N / J N, and le υ N d) = d,, s) log j N V ) + d,, p) υ, + d,, b) υ,,, 3 Noce ha for each d D N α), d,,, c) { 0, N,..., } for each,, c. Thus, D N α) N 3 T.

16 6 MARCIN PĘSKI be a verson of he welfare where j s replaced by he approxmang sequence j N j. In Appendx B, we esablsh he followng bound. Lemma. There exss a connuous funcon g : R R + such ha g x) > 0 for x > 0 and such ha for each machng ype d D α, β)), N log p N d) υ N d) max [υ N δ) + Enropy δ) + g δ b))] δ Dα,β)) d p) + d b) ) log N The remnder erm o ) does no depend on ype d. α log α β log β + o ). Recall ha Lemma derves a bound on he number of machngs wh ype d. Togeher wh Lemma, we have for each d D α, β )), N log M N d) + N log p N d) υ N d) + Enropy d) + g d b)) max [υ N δ) + Enropy δ) + g δ b))] g d b)) + o ). 5) δ Dα,β)) As N, he rgh hand sde of nequaly 5) s negave f eher d b) > 0, or d b) = 0 and he value of he erm n he frs square bracke s srcly smaller han he value of he opmzaon problem max [υ N δ) + Enropy δ)]. δ Dα,β)) If j > 0 or Assumpon holds, hen he value of he above opmzaon problem converges o he value of ). The resul follows. 6. Noseless lm In hs secon, we consder a paramerzed verson of Example wh he mach uly gven by U j = υ j + γε j, where γ > 0 s a consan parameer, and ε j are..d. drawn from he dsrbuon G ha sasfes 0). As a normalzaon, we assume ha C G =. Such a model s

17 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 7 essenally regular see Example ). By Theorem, as N, he dsrbuon over he mached ypes n all sable machngs converges o he unque soluon d α, ) γ υ = arg max d; Welfare γ ) υ + Enropy d) 6) d Dα) = arg max d Dα) Welfare d; υ) + γenropy d) Indeed, a monoonc ransformaon ranslaes he model no s regular form wh he welfare coeffcens equal o γ υ... The second lne follows from he fac ha funcon Welfare d;.) s lnear n he welfare coeffcens υ...) As he nose dsappears, γ 0, he maxmzaon problem 6) becomes domnaed by he frs welfare) erm, and he lm dsrbuon over ypes n he sable maches converges o he soluons) of he welfare maxmzaon problem max d Dα) Welfare d; υ). 7) Ths leads o a surprsng dsconnuy. As s well-known, problem 7) characerzes oucomes ha can be obaned n sable maches of he ransferable uly TU) model wh no nose, γ = 0. 4 Typcally, such oucomes do no arse as sable machngs n he non-ransferable NTU) uly model. As he nose converges o γ 0, he large populaon lm of he NTU random uly model converges o he TU oucome of he model wh no nose, and no necessarly o he NTU oucome. 5 We llusrae hs observaon wh a smple example. Example. Suppose ha he populaon s dvded equally no wo ypes, H and L. The deermnsc uly υ s descrbed n Table. Le d γ be he unque soluon of 3), or he lm of ype dsrbuons n sable machngs. As γ 0, d γ converges o he dsrbuon d ha assgns probably o mxed pars, d H, L)+d L, H) =. When γ = 0, d s he unque oucome of he TU model, and emphacally no he 4 The classc resul from Shapley and Shubk 97) saes ha any sable machng n he marrage marke maxmzes welfare. I s easy o exend he observaon o he roommae problem. The converse holds as well see Chappor, Galchon, and Salane 04)). 5 The order of he lms s mporan. If we ake he nose lm γ 0 frs, and he large populaon lm N second, hen he sable machngs rvally converge o he NTU case.

18 8 MARCIN PĘSKI υ = H = L = H 0 = L 0 0 Table. oucome of he NTU model. In he laer, he sable machng s assorave.e., assgns a mass of zero o he mxed pars). In order o develop he nuon for hs observaon, s useful o hnk abou ndvduals as radng off he payoffs from he deermnsc uly υ and he random shocks ε. The H ndvduals are wllng o mach wh an L ndvdual f he random par of he uly s suffcenly hgh. On he oher hand, because he dfference n he deermnsc values for L ndvduals s much hgher, he laer are wllng o gve up on he random shock n order o secure he larger deermnsc value. In a sense, he random shocks n he random non-ransferable uly model play a role of ransfers. Ths observaon can be nerpreed as a non-ransferable foundaon for he ransferable uly model. 7. Conclusons In hs paper, we consder he large random uly machng model nroduced n Dagsvk 000). We exend he resul from Menzel 05) o he roommae problem. Snce he Gale-Shapley algorhm does no apply o he roommae problem, our proof echnque s necessarly very dfferen. Insead, our proof s based on he esmaes of he number of dfferen machngs and an esmaes of he probably for each machng beng sable. The proof leads o a succnc characerzaon of ype dsrbuons obaned n sable machngs as maxmzers of a welfare plus enropy formula. The formula leads o a surprsng connecon beween a noseless lm of he non-ransferable model wh large populaon. I s naural o ask wheher he model and convergence resuls can be exended furher, for nsance, o many-o-many machng. One dffculy s ha our resuls rely on Tan 99) s characerzaon of approxmae sable machng n he roommae problem. As far as we know, here are no appropraely gh approxmae soluons

19 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 9 conceps for he general many-o-many machng models ha can be used n he probably esmaes as n he Lemma. Some of he mos recen resuls on he exsence of approxmae soluons can be found n Che, Km, and Kojma 05).) An alernave approach s o abandon classc machng heory and analyze manyo-many machng n he game-heorec framework of search heory. The advanage of such approach s ha he exsence of equlbrum s assured by sandard argumens. We follow hs approach n companon paper Pesk 05), where we consder a searchbased model of many-o-many machng wh he uly exendng ). We show ha he equlbrum dsrbuon over he mach ypes can be characerzed as crcal pons of a welfare plus enropy formula. In he specal case of he roommae problem wh one-o-one machng, he characerzaons from hs paper and from Pesk 05) concde. Appendx A. Number of machngs: proof of Lemma For each mproper) machng µ, defne ses A µ,, c) = { : =, µ) =, c µ ) = c }. Ses A µ.,.,.) are dsjon and A µ,, c) = Nd µ,, c). Se A µ,, c) consss of ype ndvduals who are mached wh ype ndvduals and he caegory of her mach s c. From now on, we fx he ype d D α, β )). To shoren he noaon, for each, T and c {s, p, b}, we wre d c) nsead of, d,, c), d,., c) nsead of d,, c) and d.,, c) nsead of d,, c). The counng of machngs wh ype d s dvded no he followng seps. Dfferen choces of ses A µ. The number of dfferen ways of dvdng he populaon no ses A µ,, c) s equal o ) α N)!. 8a),c Nd,, c))! Proper machngs beween ndvduals n ses A µ,, p) and A µ,, p) for. Any wo parwse machngs beween wo dsjon ses wh he same cardnaly can be obaned from each oher by he permuaon of he elemens of one of he ses.

20 0 MARCIN PĘSKI For hs reason, he number of parwse machngs beween A µ,, p) and A µ,, p) s equal o Nd,, p))!. The number of proper machngs ha can be formed by he ndvduals n se Aµ,, p) so ha ndvduals n A µ,, p) mach wh A µ,, p) s equal o Nd,, p))! = / Nd,, p))!. 8b) {, }:, ): The square roo on he rgh-hand sde comes from he fac ha each wo-elemen subse n he produc on he lef-hand sde corresponds o exacly wo ordered pars on he rgh-hand sde. Proper machngs beween ndvduals n ses A µ,, p). If se A µ,, p) has an odd number of elemens, hen here are no parwse machngs wh se A µ,, p). Oherwse, for each, he number of parwse machngs whn populaon of ype ndvduals s no larger o he number of ways of choosng Nd,, p) ndsungushable pars from Nd,, p)-elemen ses, or Nd,, p))!) Nd,,p) Nd,, p))! ) = 3... Nd,, p) ) Nd,, p))!) /. The number of proper machngs ha can be formed by he ndvduals n se A µ,, p) so ha each ype ndvdual s mached wh anoher ype s no larger han / Nd,, p))!). 8c) Bad machngs. In order o coun he number of bad machngs, noce ha each bad machng can be unquely represened as a dreced graph wh he se of nodes equal o he ses of wo copes, lef and rgh, of badly mached ndvduals, and he egdes connecng a lef copy of o he rgh copy of j f and only f j = µ ). Because µ s a bjecon, each lef node has exacly one ougong edge, and each rgh node has exacly one ncomng edge. Each lef copy of an ndvdual n se A µ,, b) s conneced o a a rgh node of ype ndvdual ha belongs o se B µ,, b) = { j : j =, µ j) =, c µ j) = b }.

21 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY The number of ndvduals n se B µ,, b) s equal o d µ,, b). The number of ways of dvdng he se of rgh nodes of ype ndvduals no ses B µ,, b) s equal o Nd.,, b))! Nd,, b))!. Gven he defnon of ses B µ.), each dreced graph corresponds o a proper) machng beween dsjon ses gven ha a lef node n se A µ,, b) mus be mached wh a rgh node n se B µ,, b). The number of such dreced graphs s equal o, Nd,, b))!. Thus, he oal number of bad machngs beween all badly mached ndvduals s equal o Nd.,, b))! Nd,, b))! = Nd,, c))!, Nd.,, b))! = Nβ d))! 8d) The las equaly follows from 5e). All machngs. The number of all mproper machngs wh ype d s no larger han he produc of 8a)-8d): M d) ) α N)! / Nd,, p))! Nβ,c Nd, d))!, c))!, = ) ) / α N)! Nβ d))!) Nd,, s))!, Nd,., p))! Nd,,, b))! We use he fac ha d,, s) = 0 for and he convenon ha 0! =.) The Srlng approxmaon mples ha for each a [0, ], log an)! = a log a + a log N ) + o ), N

22 MARCIN PĘSKI where o ) does no depend on a [0, ]. Thus, log M d) N α log α + log N + β d) log β d) + d b) log N ) d,, s) log d,, s) d s) log N ) d,, p) log d,, p) d p) log N ),, d,, b) log d,, b) d b) log N ) + o ). The resul follows from he defnon of enropy 6) and he fac ha d s) + d p) + d b) =. Appendx B. Proof of Lemma B.. Prelmnary observaon. Le F = { F,θ, F be a regular random uly },,θ model. For each F F, le V F = df ) denoe he assocaed welfare consan. du We use he followng observaon abou regular model F: Lemma 3. There exs connuous, srcly ncreasng and connuously dfferenable funcons H, h : [0, ] R + such ha h ), and, for each F F, h 0) = H 0) = 0, h 0) = H 0) =, h x) x H x) for each x, V F h u) F u) V F H u), and df du u) V H H u). Proof. Le G be a collecon of funcons G x) = V F F x)) for each F F. Each G G s connuous, connuously dfferenable, G 0) = 0, and G 0) =. Because se G s fne, follows ha here exss funcons funcons H, h : [0, ]

23 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 3 R + ha have all he requred properes saed n he Lemma and such ha for each G G, h x) G x) H x), and G x) H x). Le H = max x [0,], h x) H x) ) { } <. Le V = max F V, V,θ. From now on, fx a machng ype d D α, β )). Le β = β. Le S = {{, j} :, j I, j,, j) / R} be he se of all feasble) mach pars and le S = S I be he se of all possble maches, ha ncludes mach pars as well as he possbly of sayng alone. Le S µ) = {{, µ )} : I} be he se of maches ha occur n a machng µ. Le B µ) = { : µ µ )) } be he se of badly mached ndvduals. B.. Overvew of he proof. We descrbe he man seps of he proof, leavng deals for subsequen secons. In he frs sep, we derve a bound on he probably p µ) ha a gven machng µ s sable. The dea s o consder frs he condonal probably ha µ s sable, condonally on he realzaon of he mach ules U µ) as well as he mach specfc shocks θ {,j}. Such a probably s a produc of he probables of ndependen evens: for each such ha µ ), he even ha J N..d. draws from dsrbuon F s no larger han U µ), for each unmached par of ndvduals {, j} S \ S µ), he complemen of he even ha boh U j s larger han U µ) and Uj s larger han U µ) j, for each badly mached ndvdual B µ), he even ha he uly from he beer mach U µ ) s larger han he uly from he worse mach U µ). Usng approxmaons from Lemma 3, negrang over he normalzed ules and mach specfc shocks, and changng he varables x = h U µ) ) µ) U

24 4 MARCIN PĘSKI where he approxmaon s good for uly values close o he op end of he suppor U µ) ), we oban he followng bound: Lemma 4. There exss a connuous funcon a x) such ha a 0) = 0 and for each N, each machng µ, p µ) J ds)n N V, j x + a x ))... 9) {,j} Sµ) Bµ) 0 x... V x ) J ) N + V, j x x j dx. + {,j} S \Sµ) We wre x) + = max x, 0).) The expresson n he square bracke can be reaed as a funcon of N ndependen and unformly dsrbued varables X and he negral s he expecaon of such a funcon. In he second sep, we fnd a bound on he average probably p N d) of a sable machng wh ype d. The bound has he form of an nergal of an expresson ha depends on he average of normalzed) ules across he populaon. For each x : I [0, ], each ype, le x = α N : = x. Le µ m be he probably dsrbuon of he average m= X m, where X are ndependen and unformly dsrbued on he nerval [0, ]. We use bound 9), an nequaly, x ) + e x and some algebra o show

25 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 5 Lemma 5. There exss a connuous funcon A x) such ha A 0) = 0 and for each N, each machng ype d p N d) e Nυ N d)+ ds) log N) exp N 3/ j N V α x N 0 x exp N )) β log x + NA α x, ) R V, dµ αn x ) α x α x 0) The goal of he hrd sep s o esmae he convergence rae of he negral 0) as N. We rely on wo deas from he large devaon heory. Frs, we use he fac ha for large m, he followng approxmaon of measure µ m can be used for x 0: dµ m x) e m+log x). Second, Varadhan s Lemma Ells 005)) says ha for large m, he value of he negral grows a he rae of he larges exponen of he negrand: ˆ m log e mf x) dx max F x). x Usng a careful verson of) he above approxmaons, we can bound he convergence rae of negral 0) wh he soluon o a ceran maxmzaon problem: Lemma 6. When N, N log p N d) υ N d) + d s) log N + + o ) ) + max x. [0,] T N / j N V α x N, ) R V, α x α x + α + β ) log x + A α x ) In he fourh sep, we fnd an upper bound on he value of he maxmzaon problem n )..

26 6 MARCIN PĘSKI Lemma 7. There exss a connuous funcon g : R R + such ha g x) > 0 for x > 0 and such ha as N, N / j N V α x lm sup max N, ) R V, α x α x N x. [0,] T + α + β ) log x + A α x ) α log α β log β d s) + ) d p) + d b) log N max [υ N δ) + Enropy δ) + g δ b))] + o ). δ Dα,β)) Lemma follows from Lemmas 6 and 7 B.3. Proof of Lemma 4: Probably ha machng µ s sable. Le Θ : S Θ be a realzaon of mach-specfc shocks. realzaons of ules U µ) Θ are clear from he conex, we wre F j V j, Θ{,j} for each, j. Smlarly, le Ũ : I R denoe he of agens n her worse) maches. When he values of nsead of F j, Θ{,j}, and Ṽ j nsead of B.3.. Condonal probably gven Ũ and Θ. Frs, we esmae he condonal probably ha µ s a sable machng gven he realzaon of mach-specfc Θ and ules Ũ. By condonal ndependence assumpons, he condonal probably p µ Θ, Ũ ) s equal o he produc of he probably ha he ndvdual raonaly s preserved for all ndvduals who are alone: :µ) F Ũ )) JN, he probably ha no unmached par {, j} blocks an exsng mach: {,j} S \Sµ) F j Ũ )) F j Ũj ))), and he probably ha he uly of badly mached ndvduals n her beer mach s no smaller han he uly n he worse mach: Bµ) )) µ F ) Ũ.

27 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 7 B.3.. Condonal probably gven Θ. The condonal probably p µ Θ ) ha µ s sable gven Θ s equal o he above produc negraed over he realzaon of ules Ũ. Condonally on he realzaon of mach-specfc shock, he varables Ũ are ndependenly dsrbued wh densy d F j Ũ ) f µ ), and J N F Ũ )) JN df Ũ ) f µ ) =. Thus, he condonal probably p µ Θ ) ha µ s sable gven Θ s equal o p µ Θ ) = :µ) Ũ, I {,j} S \Sµ) :µ)= F Ũ )) JN F j Bµ) J N F Ũ )) JN df Ũ ) )) µ F ) Ũ Ũ )) F j Ũj ))) :µ) d ) µ) F Ũ. ) B.3.3. Uncondonal probably. Le y = Ũ. Lemma 3, he fac ha F Ũ ), and equaon ) mply p µ; Θ ) J ds)n N V h y )) J N Bµ) Ũ, I H y ) Ṽ µ ) {,j} S \Sµ) V :µ)= :µ) Because he mach shocks are ndependen, and V, j p µ) J ds)n N 0 y, I Bµ) V V, j :µ)= {,j} S Sµ) Ṽ µ) Ṽ j Ṽj h y ) h y j ) ) H y ) dy. = θ Φ, j θ) Ṽ j Ṽ j j, we oban V h y )) J N V, j h y ) h y j ) ) {,j} S \Sµ) H y ) H y ) dy.

28 8 MARCIN PĘSKI By anoher change of varables x = h y ), he value of he negral from he above expresson s no larger han V x ) J N Le 0 x h), I max, H h x )) x a x) = max ) {,j} S \Sµ) H h ) x )) dx h h. x )), H h x)) x V, j x x j ) )) H h x)) h h x)) Bµ) for x > 0, x 3) and a 0) = 0. By properes of funcons h and H lsed n Lemma 3, a x) s connuous. Lemma 4 follows from he fac ha h ). B.4. Proof of Lemma 5: Average probably ha a machng of ype d s sable. The proof follows from 9) and he subsequen observaons. Frs, observe ha ds)n log JN N V V, j :µ)= {,j} S Sµ) =d s) log J N + log V + log V,j N Sµ)\S {,j} Sµ) S =d s) log j N + ) log N + d,, s) log V + d,, p) υ, + d,, b) υ,,, =υ N d) + d s) log N. Lemma 8. For each d, M d) x µ Md) Bµ) x βn. Proof. Le B be a collecon of subses B { : = } of ype ndvduals such ha B = β. Symmery mples ha x = ) x β d)n x. M d) µ Md) Bµ) B B B B α N : = The las nequaly follows from he Newon mulnomal formula.

29 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 9 Lemma 9. There exs a consan K <, such ha for all N, for all x [0, ] N log max V x ) J ) N V, j x x j, 0 {,j} S \Sµ) J N V α x N V, α x α x + KN α x., ) R Proof. Frs, noce ha log x) x. Thus, [ ] log V x ) J N J N ) V x J N V α x + V α x, and ) log V, j x x j V, j x x j V,j x x j + V,j x x j. {,j} S \Sµ) {,j} S {,j} Sµ) S {,j} S \Sµ) Second, observe ha V,j x x j = {,j} S V, Nα x ) Nα x ) + V, x., ) R : = Thrd, because x [0, ] for all, whch mples ha x x µ) x, we have V,j x x j V x x j V x x µ) {,j} Sµ) S {,j} Sµ) S V x = V Nα x. Smlarly, we show ha V x : = V V x : = α x. The Lemma follows from he above nequales. Lemma 0. There exss a connuous ncreasng funcon b : [0, ] R + such ha b 0) = 0 and for each N, each )) + a x )) exp Nb α x.

30 30 MARCIN PĘSKI Proof. We assume w.l.o.g. ha a ) >. For each y > 0, le b y) be defned as a unque soluon b a ) y o he equaon a ) a b )x = b. Because he lef-hand sde s srcly decreasng n b and he rgh-hand sde s srcly ncreasng, s clear ha he soluon exss. Moreover, one shows ha he soluon s connuous n y, and ha lm y 0 b 0) = 0. Fx x,..., x N [0, ], and le x = N x. Observe ha for each c, c { : x c} x, whch mples ha I follows ha N a N ) { b x) a x ) b x) b x) : x a b x) )} x = x. N + { )} b x) N a ) : x a + a ) a bx) )x = b x). The resul follows from he fac ha + a x )) exp N a x ). Fnally, le A x) = Kx + b x). Funcon A.) s connuous and A 0) = 0 by Lemma 0. B.5. Proof of Lemma 6: Large devaon bound. The resul follows from equaon 0) and he followng resul. Lemma. For any connuous funcon h : [0, ] k R of k < varables, ˆ log h x,..., x k ) dµ m x )...dµ m k x k ) k 0 x,...,x k log m k + k m k + max 0 x,...,x k log h x,..., x k ) + k ) m k ) log x k.

31 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 3 Lemma s a verson of he Varadhan s Lemma from large devaons heory see for example, Theorem II.7.. n Ells 005)). The Varadhan s Lemma esablshes asympoc upper and lower bound on he values of ceran sequences of negrals. Here, we need a proper and no jus asympoc) upper bound. Proof. Assume frs ha k =. We are gong o show ha ˆ 0 x h x) dµ m x) me m max ) h x) x m. 0 x For each probably measure µ on [0, ], any funcon f : [0, ] R, le E µ f = f x) dµ x) denoe he expeced value of f wh respec o measure µ. Noce ha 0 µ m x x 0 ) = µ m e mx/x 0 e m) e m E µ me mx/x 0 4) = e m E λ e x/x 0) m e m x m 0, 5) where λ s he Lebesgue measure on he nerval [0, ]. The frs nequaly s a verson of Markov s nequaly. The second equaly comes from he fac ha µ m s a probably dsrbuon of he average x = x of m..d. varables. The las m equaly comes from he fac ha E λ e x/x 0 = 0 e x x 0 dx x 0.) Take any funcon h ) and le h x) = max y x h y) be he smalles decreasng funcon, no smaller

32 3 MARCIN PĘSKI han h.). By negraon by pars, E µ mh x) E µ mh x) = = ˆ 0 ˆ 0 ˆ e m h x) dµ m x) h ) y) ) µ m x y) dy + h ) 0 ˆ = e m m 0 e m m max x h ) x) ) x m dx + h ) h x) x m dx e m ) h ) h x) x m = e m m max h x) x m. x The mul-varable case k follows from nducon on k and a repeaed applcaon of sngle-varable case k = : = ˆ 0 x,...,x k ˆ 0 x k k l= k = = l= k l= h x,..., x k ) dµ m x )...dµ m k x k ) ˆ 0 x,...,x k m l ) e k l= m l h x,..., x k, x k ) dµ m x )...dµ m k x ) dµm k x k ) ˆ 0 x k m l ) e k l= m l m k ) e m k max 0 x,...,x k m l ) e k l= m l max 0 x,...,x k max 0 x k h x,..., x k ) h x,..., x k, x k ) max 0 x,...,x k ) k x m l l. l= k l= h x,..., x k ) ))) x m l l dµ m k x k ) k l= ) )) x m l l x m k k

33 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 33 B.6. Proof of Lemma 7: Maxmzaon. Frs, noce ha by changng varables x = N / y, he maxmzaon problem n ) urns no max y 0 j N V α y, ) R V, α y α y + α + β ) log y + A ) N α y α + β ) log N. 6) Noe ha α + β ) = d s) + d p) + d b). Second, because A s a connuous funcon such ha A 0) = 0, noce ha as N, he value of he maxmzaon problem n 6) converges o max y 0 j N V α y V, α y α y +, ) R α + β ) log y. 7) Thrd, we show ha Lemma. For each δ D α, β )), he value of problem 7) s no larger han α + β ) log α [ υ N δ) + E 0 δ) + δ b) ], where E 0 d) = d,, s) log d,, s) d,, p) + d,, b)) log d,, p) + d,, b)) + )., ) R Proof. Observe frs ha for each δ D α, β )), each ype δ,, s)+ δ,, p) + δ,, b)) = δ,, s)+ δ,, p) + δ,, b)) = α +β. :, ) R :, ) R 8)

34 34 MARCIN PĘSKI Thus, j N V α y V, α α y y +, ) R α + β ) log y = j N V α y V, α α y y + δ,, s) log y +, ) R + δ,, p) + δ,, b)) log y + δ,, p) + δ,, b)) log y, ) R, ) R = j N V α y + δ,, s) log y V, α α y y + δ,, p) + δ,, b)) log y y, ) R, ) R = δ,, s) j ) NV α δ,, s) y + log y + ) ) δ,, p) + δ, α α V,, b) δ,, ) R, p) + δ,, b) y y + log y y )) jn V α δ,, s) log δ,, s) + ) )) δ,, p) + δ, α α V,, b) log, δ,, p) + δ,, b), ) R where he las lne follows from he fac ha ax + log x log a. for any a, x > 0. By he deny 8) and he defnon of funcons υ N and E 0, he laer s equal o = δ s) + ) δ p) + δ b) υ N δ) + δ,, s) log δ,, s) + δ,, p) + δ,, b)) log δ,, p) + δ,, b)), ) R δ,, s) log α δ,, p) + δ,, b)) log α + log α )., ) R The expresson n he bracke n he frs lne s equal o + δ p) due o 5a). The las lne s equal o α + β ) log α due o 5d), 5e) and 5a). The resul follows.

35 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 35 Fourh, we show ha Lemma 3. There exss a connuous funcon g : R R + such ha g x) > 0 for x > 0 and for each δ D α, β )), E 0 δ) Enropy δ) + δ b) + β log α β log β g δ b)). Proof. For each χ 0, le l χ) = χ log χ. Then, l.) s a concave funcon such ha l ) = 0. For fuure use, noce ha for each a, χ 0, Also, defne l aχ) = χl a) + al χ). 9) f ξ) = l ξ) l + ξ) ξ 30) Observe ha f 0) = 0 and f.) s srcly decreasng and srcly concave. For each,, defne Then, for each δ D α, β )), a, ) = δ,, p) + δ,, b), ξ, ) = a, ) δ,, b). Enropy δ) = δ p) δ b) d,, s) log d,, s) + l δ,, p)) + l δ,, b))), ) R = δ p) δ b) d,, s) log d,, s) + l a, ) ξ, ))) + l a, ) ξ, ))),, ) R

36 36 MARCIN PĘSKI and E 0 δ) = δ p) δ b) d,, s) log d,, s) + l δ,, p) + δ,, b))), ) R = δ p) δ b) d,, s) log d,, s) Moreover, I follows ha Enropy δ) E 0 δ) +, ) R l a, ) + ξ, ))). β = a, ) ξ, )., ) R β = a, ) f ξ, )) + a, ) l ξ, ))., ) R, ) R Usng 9), 30), he concavy of f and l, and he fac ha :, ) R a, ) α ξ, ) = β α, we oban ha he laer s no larger han a, δ p) + δ b)) ) δ p) + δ b) f ξ, )) + a, ) l ξ, )), ) R, ) R δ p) + δ b)) f δ b)) + δ p) + δ b)) f δ b)) + = δ p) + δ b)) f δ b)) δ b) f δ b)) + β log α a, α ) l ξ, )) α :, ) R ) β α l α ) β β α log α α β log β. The las equaly follows from he fac ha f δ b)) 0. Le g x) = xf x). References Che, Y.-K., J. Km, and F. Kojma 05): Sable Machng n Large Economes,. 7

37 LARGE ROOMMATE PROBLEM WITH NON-TRANSFERABLE RANDOM UTILITY 37 Chappor, P.-A., A. Galchon, and B. Salane 04): The Roommae Problem - Is More Sable Than You Thnk, Games and Economc Behavor. 4, 4 Choo, E., and A. Sow 006): Who Marres Whom and Why, Journal of Polcal Economy, 4), 75 0., 3 Dagsvk, J. K. 000): Aggregaon n Machng Markes, Inernaonal Economc Revew, 4), documen),,,, 3,, 5., 5., 7 Ells, R. 005): Enropy, Large Devaons, and Sascal Mechancs. Sprnger, Berln ; New York, reprn of he s ed. sprnger-verlag new york 985 edon edn., B., B.5 Gale, D., and L. Shapley 96): College Admssons and he Sably of Marrage, Amercan Mahemacal Monhly, 69), 9 5.,, Galchon, A., and B. Salane 0): Cupd s Invsble Hand: Socal Surplus and Idenfcaon n Machng Models, SSRN Scholarly Paper ID 80463, Socal Scence Research Nework, Rocheser, NY. Lee, S., and L. Yarv 04): On he Effcency of Sable Machngs n Large Markes, SSRN Scholarly Paper ID 46440, Socal Scence Research Nework, Rocheser, NY. Menzel, K. 05): Large Machng Markes as Two-Sded Demand Sysems, Economerca. documen),, 4, 0, 5., 5., 7 Pesk, M. 05): Welfare plus Enropy n Many-o-many Machng Models, Dscusson paper, Unversy of Torono. 7 Shapley, L., and M. Shubk 97): The assgnmen game I: The core, Inernaonal Journal of Game Theory, ), Tan, J. J. M. 99): A necessary and suffcen condon for he exsence of a complee sable machng, Journal of Algorhms, ), ,, 3,, 7