DESIGNING RANDOM ALLOCATION MECHANISMS: THEORY AND APPLICATIONS

Transcription

1 DESIGNING RANDOM ALLOCATION MECHANISMS: THEORY AND APPLICATIONS ERIC BUDISH, YEON-KOO CHE, FUHITO KOJIMA, AND PAUL MILGROM Abstract. Randomzaton s an mportant feature of resource allocaton when the resources assgned are ndvsble and monetary transfers are lmted. We expand the theory of effcent random assgnment to accommodate mult-unt demand and supply, and certan real-world features such as group-specfc quotas ( controlled choce ) and endogenous capactes n school choce and house allocaton, and schedulng and currculum constrants n course allocaton. We develop new mechansms that are ex ante far and effcent n these respectve problems. Our method can also be appled to certan two-sded matchng problems to produce far matchups n nterleague games and speed datng. Keywords: Market Desgn, Random Assgnment, Brkhoff-von Neumann Theorem, Probablstc Seral, Pseudo-Market, Utlty Guarantee, Assgnment Messages. Date: February 17, Budsh: Unversty of Chcago Booth School of Busness, 5807 S Woodlawn Ave, Chcago IL 60637, erc.budsh@chcagobooth.edu. Che: Department of Economcs, Columba Unversty, 420 West 118th Street, 1016 IAB New York, NY 10027, and YERI, Yonse Unversty, yeonkooche@gmal.com. Kojma: Department of Economcs, Stanford Unversty, CA 94305, fuhtokojma1979@gmal.com. Mlgrom: Department of Economcs, Stanford Unversty, CA 94305, paul@mlgrom.net. We are grateful to Emr Kamenca, Sebasten Lahae, Mha Manea, Herve Mouln, Tayfun Sönmez, Al Roth, Utku Ünver, and semnar partcpants at Boston College, Gakushun, Johns Hopkns, Prnceton, Tokyo, UC San Dego, Waseda, Yahoo! Research, and Yale for helpful comments. We especally thank Tomom Matsu, Akhsa Tamura, Jay Sethuraman, and Rakesh Vohra. Yeon-Koo Che s grateful to Korea Research Foundaton for World Class Unversty Grant, R Paul Mlgrom s research was supported by NSF grant SES

2 2 ERIC BUDISH, YEON-KOO CHE, FUHITO KOJIMA, AND PAUL MILGROM Randomzaton s commonplace n everyday resource allocaton. It s used to break tes among students applyng for overdemanded publc schools and for popular after-school programs, to raton offces, parkng spaces, and tasks among employees, to allocate courses and unversty dormtores among college students, and to assgn jury and mltary dutes among ctzens. 1 The ubqutous frst-come frst-served method, or queung, s often only a less apparent way to nclude random elements n settng a prorty order. Randomzaton s sensble n these examples and many others because the objects to be assgned are ndvsble and monetary transfers are lmted or unavalable. 2 In these crcumstances, any non-random assgnment of resources s lkely to be asymmetrc and, wthout compensatng monetary transfers, unfar. Randomzaton can sometmes restore ex ante symmetry and the percepton that the mechansm s far. To fnd a desrable random allocaton, t s often helpful to vew agents as consumng lotteres over objects, subject to a jont producton constrant and to evaluate ncentves and welfare on that bass. 3 Snce a random assgnment the lotteres over objects receved by agents s dvsble n probablty unts, one can then apply the classcal frameworks developed for dvsble objects. Hylland and Zeckhauser (1979) ( HZ ) were the frst to apply ths perspectve to market desgn. Ther pseudo-market mechansm endows agents wth equal fxed budgets n a fcttous currency, allows them to use the currency to buy probablty shares of alternatve objects, and fnds a compettve equlbrum by solvng for prces (per unt probablty of obtanng each object) that clear the market. The resultng allocaton s then ex ante effcent and envy-free. Bogomolnaa and Mouln (2001) ( BM ) also adopted the random assgnment approach to fnd an allocaton that s effcent n an ordnal sense and envy-free. 1 Lotteres played hstorcal roles n assgnng publc lands to homesteaders (Oklahoma Land Lottery of 1901), and rado spectra to broadcastng companes (FCC assgnment of rado frequences durng ). Lotteres are also used annually to select 50,000 wnners of the US permanent resdency vsas ( green cards ) from those qualfed n the DoJ s mmgraton dversty program. 2 The lmtaton of monetary transfers arses from moral objecton to commodtzng objects such as human organs and from farness consderaton (cf. Roth (2007)). Assgnment of resources based on prces often favor those best endowed wth money rather than those most deservng, and can be regarded as unfar for many goods and servces. See Che and Gale (2008) for an argument makng ths pont based on utltaran effcency. 3 An more common alternatve perspectve evaluates the effcency of agents ex post assgnment of objects, rather than of the ex ante lotteres they receve. In that vew, a mechansm dentfes ex post desrable allocatons and uses randomzaton only to ensure ex ante farness. For example, the random seral dctatorshp (RSD), wdely used n practce, s often analyzed that way. In RSD, the agents are randomly ordered and, followng that order, each agent s assgned hs/her most preferred object not yet assgned. The resultng pure assgnment s ex post effcent and ex ante envy-free, but may ental ex ante neffcences even n an ordnal sense (see Bogomolnaa and Mouln (2001)). Extensons of other well known mechansms, such as Gale and Shapley s deferred acceptance and Gale s top tradng cycles, suffer smlar ex ante neffcences when prortes are set randomly and used to break tes.

3 DESIGNING RANDOM ALLOCATION MECHANISMS 3 The purpose of the current paper s to broaden the random assgnment methodology (ncludng HZ and BM) to enhance ts practcal applcablty. To be appled to problems ncludng school choce and course allocaton, the random assgnment model must be extended n two ways. Frst, the model needs to account for varous polcy constrants. A case n pont s the so-called controlled-choce n school assgnment. Schools often seek to balance ther student bodes n terms of gender, ethncty, race, test scores, and the geographc locaton of students resdence. For nstance, publc schools n Massachusetts are dscouraged by the Racal Imbalance Law from havng student enrollments that are more than 50% mnorty. Mam-Dade County Publc Schools control for the socoeconomc status of students n order to dmnsh concentratons of low-ncome students at certan schools. In New York Cty, Educatonal Opton (EdOpt) schools must balance ther student bodes n terms of students test scores. 4 Publc schools n Seoul restrct the number of seats for those students resdng n dstant school dstrcts, n order to allevate mornng commutes. In a course allocaton problem, a student may wsh to enroll n no more than a certan number of courses n a gven subject (currculum constrants) or n a gven tme slot (schedulng constrants). The precedng examples are ones n whch we add constrants to the ones already present n the HZ and BM models, but some applcatons also nvolve removng or relaxng the tradtonal constrants. For nstance, when schools assgn ther students to dfferent foregn language programs, the exact composton may be adjustable to a degree determned by the avalable staff and resource. Second, n order to accommodate applcatons lke course allocaton, n whch a sngle course may accept multple students and a sngle student may take multple courses, we need to drop the one-to-one restrcton. We do that by usng the matrx of expected assgnments to generalze the random assgnment matrx of one-to-one matchng. Ths expected assgnment formulaton s most useful when the agents and mechansm desgner s partes payoffs are at least approxmately lnear functons of the pure assgnment over the support of the relevant lottery. Part of our analyss below wll focus on mplementng usng lotteres wth supports for whch ths restrcton mght reasonably apply. Even for one-to-one matchng, descrbng the ndvdual agent s consumpton outcome n terms of ther ndvdual random assgnments poses a techncal challenge. For even f each agent s assgned just one tem n expectaton and each tem s assgned just once n expectaton, mplementaton stll requres fndng a jont lottery over feasble pure assgnments wth the rght margnal dstrbutons for each agent. For HZ and BM, the only feasblty constrants on the pure assgnments are those of one-to-one matchng: each agent s assgned 4 In partcular, 16 percent of students that attend an EdOpt school must score above grade level on the standardzed Englsh Language Arts test, 68 percent must score at grade level, and the remanng 16 percent must score below grade level (Abdulkadroglu, Pathak, and Roth, 2005).

4 4 ERIC BUDISH, YEON-KOO CHE, FUHITO KOJIMA, AND PAUL MILGROM exactly one tem and each tem s assgned exactly once. Those papers were then able to solve ther techncal challenge by appealng to the celebrated celebrated Brkhoff-von Neumann theorem (Brkhoff, 1946; von Neumann, 1953), whch asserts that every random assgnment matrx, that s, every non-negatve matrx wth all the row sums and column sums equal to one, s a convex combnaton of pure assgnment matrces. Consequently, every random assgnment matrx can be mplemented by some lottery over pure assgnments. Is there an extenson of the BvN theorem avalable when there are addtonal constrants, such as the ones descrbed above? Does the extenson apply as well to many-to-many matchng problems? The frst part of the paper answers these questons by dentfyng a maxmal generalzaton of the Brkhoff-von Neumann theorem. Appealng to results n the combnatoral optmzaton lterature, we show that a certan underlyng structure of constrants s suffcent for an expected assgnment to be always mplementable by a lottery over feasble pure assgnments. Then we demonstrate that the same condton s not only suffcent but also necessary n canoncal two-sded envronments. The second part of the paper apples the expected assgnment methodology to specfc market desgn contexts. One applcaton s a generalzaton of BM s Probablstc Seral mechansm that accommodates new knds of supply-sde constrants, whch may arse n unt-demand applcatons such as school choce and dormtory assgnment. We show how to modfy BM s algorthm to accommodate constrants such as controlled choce and adjustable capactes and prove that the attractve propertes of BM s algorthm extend to ths more general envronment. Our second applcaton s a generalzaton of HZ s pseudo-market mechansm, to accommodate new knds of demand-sde constrants that may express mportant aspects of partcpants preferences n mult-unt demand applcatons such as course allocaton and assgnment of shfts to nterchangeable workers. In ths secton, we also relax the assumpton of addtve preferences. We adapt the assgnment messages developed by Mlgrom (2009), to accommodate some schedulng and currcular constrants ( I want just one course n fnance and at most one course before noon ) arsng n course allocaton, and to express nonlnear preferences such as dmnshng margnal utltes for an tem or category ( the second fnance course s worth less to me than the frst ). We then utlze these enrched messages to develop a generalzed multunt pseudo-market mechansm, establsh exstence of compettve equlbrum prces n the pseudo-market, and nvoke our earler suffcency result to ensure mplementablty of the expected assgnment that results from ths compettve equlbrum. We fnally show that ths generalzed mechansm nherts the attractve effcency and farness propertes of the

5 DESIGNING RANDOM ALLOCATION MECHANISMS 5 orgnal one-to-one mechansm. Ths extended mechansm may be useful for practce, especally because several mult-unt assgnment mechansms currently n use have been shown to allow outcomes that are neffcent and ex post qute unfar (Sönmez and Ünver, 2008; Budsh and Cantllon, 2009). Fnally, our mplementaton result has an unexpected applcaton for promotng ex post farness n mult-unt resource allocaton. When agents demand multple objects, there can be many ways to mplement a gven expected assgnment. For nstance, suppose there are two agents, 1 and 2, dvdng four objects, a, b, c, and d, whch they prefer n the order lsted. An ex ante far expected assgnment may assgn each object to each agent wth probablty 0.5. One way to mplement ths expected assgnment s to assgn a and b to 1 and c and d to 2 wth probablty one half and a and b to 2 and c and d to 1 wth the remanng probablty one half. If utlty s addtve over objects, then ths allocaton may be ex ante effcent and far, but ex post unfar, snce one agent always gets the two best and the other gets the two worst. There s another mplementaton of the same expected assgnment that s more far ex post: whenever one agent gets one of the two best objects, he must also get one of the two worst objects. It turns out our mplementaton result can be utlzed to avod the former unfar mplementaton, and more generally to ensure that pure assgnments used n the mplementng lottery have a small varaton n utlty. The method also ensures that every pure assgnment approxmates the orgnal expected assgnment n terms of expected utltes. Ths procedure can be appled n the context of course allocaton, for nstance n conjuncton wth our generalzaton of HZ, or n other mult-unt demand envronments such as task assgnment and far dvson of estates. Ths utlty guarantee method can also be adapted to a two-sded matchng problem, n whch both sdes of the market are agents. Startng wth any expected matchng, we can ntroduce ex post utlty guarantees on both sdes, ensurng ex post utlty levels that are close to the promsed ex ante levels. Ths method can be used, for example, to desgn a far schedule of nter-league sports matchups or a far speed-datng mechansm. The rest of the paper s organzed as follows. Secton 1 presents the model. Secton 2 presents the suffcency and necessty results for mplementng expected assgnments. Secton 3 presents the generalzaton of Bogomolnaa and Mouln s (2001) Probablstc Seral mechansm, for applcatons such as school choce. Secton 4 presents the generalzaton of Hylland and Zeckhauser s (1979) Pseudo-market mechansm, for applcatons such as course allocaton. Secton 5 presents the utlty guarantee results, ncludng the applcaton to two-sded matchng. Secton 6 collects some negatve results for non-blateral matchng envronments. Secton 7 concludes.

6 6 ERIC BUDISH, YEON-KOO CHE, FUHITO KOJIMA, AND PAUL MILGROM 1. Setup Consder a problem n whch a fnte set N of agents are assgned to a fnte set O of objects (as wll be clear, our framework works wth no loss f N are objects or O are agents). A (pure) assgnment s descrbed as a matrx X = [x a ] ndexed by all agents and objects, where each entry x a s the nteger quantty of object a that agent receves. Note that we allow for assgnng more than one unt of an object and even for assgnng negatve quanttes. Negatve quanttes can be nterpreted as supply oblgatons. The requrement that the matrx be nteger-valued captures the ndvsblty of the assgnment. We ntroduce a general class of constrants. A constrant structure H s a collecton of subsets of N O (that s, H 2 N O ) that ncludes all sngletons (.e., sets of the form {(, a)}). That s, each element S H, called a constrant set, s a set of agent-object pars. A vector q = (q S, q S ) S H of ntegers are the quotas assocated wth each set n H, where ntegers q S and q S are the floor and the celng constrants on the total amount assgned n S, respectvely. The constrant structure H and the quotas q restrct the set of assgnments. We say that a pure assgnment X s feasble under q f (1.1) q S x a q S for each S H. (,a) S For nstance, the constrant structure H of the smple one-to-one matchng settng conssts of all sngleton sets, all rows ({(, a) a O} for each N), and all columns ({(, a) N} for each a O), and the quotas satsfy q S = q S = 1 for every non-sngleton constrant set S H. The constrants say that each agent s assgned one object and each object s assgned to one agent. Gven a constrant structure H and assocated quotas q, a random allocaton can be descrbed as a lottery over pure assgnments each of whch s feasble under quotas q. As wth HZ and BM, however, our approach s to focus drectly on an expected assgnment namely, the expected numbers of each tem assgned to each agent as a basc unt of analyss. Formally, an expected assgnment s a matrx X = [x a ] ndexed by agents and objects where x a (, ) for every N and a O. In contrast to pure assgnment matrces, an expected assgnment allows for fractonal allocatons of objects. A natural queston s: when can an expected assgnment be mplemented by some lottery over pure assgnments? Defnton 1. Gven a constrant structure H, an expected assgnment X s mplementable (as a lottery over feasble pure assgnments) under quotas q f there exst postve numbers {λ k } K k=1 that sum up to one and (pure) assgnments {Xk } K k=1, each of whch s feasble under

7 DESIGNING RANDOM ALLOCATION MECHANISMS 7 q, such that (1.2) X = K λ k X k. k=1 In words, gven quotas, an expected assgnment s mplementable f t can be expressed as a lottery over feasble assgnments each of whch satsfes the quotas. If an expected assgnment X s mplementable under quotas q, then the expected assgnment wll trvally satsfy the quotas q: (1.3) q S x a q S for each S H. (,a) S The more challengng queston s the reverse: when s an expected assgnment mplementable? More specfcally, our nterest s to dentfy the constrant structures for whch any quotas are mplementable. Our characterzaton wll be provded n terms of the constrant structure, so the followng defnton wll prove useful. Defnton 2. Constrant structure H s unversally mplementable f, for any quotas q = (q S, q S ) S H, every expected assgnment satsfyng q s mplementable under q. If a constrant structure H s unversally mplementable, then every expected assgnment satsfyng any quotas defned on H can be expressed as a convex combnaton of pure assgnments that are feasble under the gven quotas. In other words, for any gven quotas, any expected assgnment satsfyng (1.3) can be mplemented as a lottery over feasble pure assgnments. Unversal mplementablty ams to capture the sort of nformaton that s lkely avalable to a planner when the mechansm s beng desgned. For example, n a school choce problem, the planner may consder whether to apply certan prncpled geographc and ethnc composton constrants that s, what the constrant structure wll be before knowng the exact numbers of spaces n each school or the precse preferences of the students. By studyng unversal mplementablty, we characterze the knds of constrant structures that are robust to these numercal detals and certan to be mplementable. 2. Implementng Expected Assgnments Ths secton provdes a condton whch guarantees that a constrant structure s unversally mplementable. To do so, we ntroduce concepts that wll be useful for our characterzaton. A constrant structure H s a herarchy f, for every par of elements S and S n

8 8 ERIC BUDISH, YEON-KOO CHE, FUHITO KOJIMA, AND PAUL MILGROM H, we have S S or S S or S S =. 5 That s, H s a herarchy f, for any two of ts elements, one of them s a subset of the other or they are dsjont. The followng concept proves to be central throughout the paper. Defnton 3. A constrant structure H s a bherarchy f there exst herarches H 1 and H 2 such that H = H 1 H 2 and H 1 H 2 =. A bherarchy s a constrant structure that can be expressed as a unon of two dsjont herarches. Note that the partton of H nto H 1 and H 2 need not be unque. For nstance, sngleton sets can be put nto the two famles n any arbtrary fashon. The result offered below follows readly from the combnatoral optmzaton lterature, establshng that the bherarchy condton s suffcent for unversal mplementablty. Theorem 1. (Suffcency) If a contrant structure s a bherarchy, then t s unversally mplementable. Proof. Suppose that constrant structure H s a bherarchy. Consder any expected assgnment X that satsfes q gven constrant structure H. Snce q S and q S are ntegers for each S H, we must have q S x S x S x S q S, where x S := (,a) S x a, x S s the largest nteger no greater than x S, and x S s the smallest nteger no less than x S. Hence, X must belong to the set (2.1) X = [x a] x S The set (2.1) forms a bounded polytope. (,a) S x a x S, S H. Hence, any pont of t, ncludng X, can be wrtten as a convex combnaton of ts vertces. To prove the result, therefore, t suffces to show that the vertces of (2.1) are nteger-valued. Hoffman and Kruskal (1956) show that the vertces of (2.1) are nteger-valued f and only f the ncdence matrx M = [m (,a),s ], (, a) N O, S H, where m (,a),s = 1 f (, a) S and m (,a),s = 0 f (, a) S, s totally unmodular. 6 The total unmodularty of matrx M n turn follows from Edmonds (1970). A fuller self-contaned proof s avalable n Web Appendx C. As s clear from the proof, the pure assgnments used n the mplementng lottery n Theorem 1 are not just feasble under the gven quotas; rather, the mplementaton ensures that each of the resultng pure assgnments s rounded up or down to the nearest nteger for each constrant set, whch s a stronger property. 5 Herarches are usually called lamnar famles n the combnatoral optmzaton lterature. 6 A zero-one matrx s totally unmodular f the determnant of every square submatrx s 0, 1 or +1.

9 DESIGNING RANDOM ALLOCATION MECHANISMS 9 For practcal purposes, knowng smply that an expected assgnment s mplementable s not satsfactory; mplementaton must be computable, preferably by a fast algorthm. Fortunately, there exsts an algorthm, formally descrbed n Web Appendx D wth an llustratve example, that mplements expected assgnments n polynomal tme. 7 At each step of the algorthm, an expected assgnment X satsfyng gven quotas s decomposed nto a convex combnaton γx + (1 γ)x of two expected assgnments, each of whch satsfes the quotas and has at least one more nteger-valued constrant set than X. Then, a random number s generated and wth probablty γ the algorthm contnues by smlarly decomposng X, whle wth probablty 1 γ the algorthm contnues by decomposng X. The algorthm stops when t reaches a pure assgnment. As argued more formally n the appendx, ths process has a run tme polynomal n H Examples of Bherarchy. We show here that a number of constrants llustrated n the Introducton satsfy the bherarchy condton One-to-one assgnment and the Brkhoff-von Neumann theorem. Suppose n agents are to be assgned to n objects, one for each. Notce that any assgnment s descrbed as an n n permutaton matrx; namely, each entry s zero or one, each row sums to one, and each column sums to one. Any expected assgnment s n turn represented as an n n bstochastc matrx,.e., a matrx wth entres n [0, 1], satsfyng the same row-sum and column-sum constrants. The Brkhoff-von Neumann theorem states that any bstochastc matrx can be expressed as a convex combnaton of permutaton matrces. Clearly, ths result follows from Theorem 1; all rows are dsjont and thus form a herarchy, and all columns form another. (Sngletons can be added arbtrarly to ether herarchy). Corollary 1. (Brkhoff, 1946; von Neumann, 1953) Any bstochastc matrx can be wrtten as a convex combnaton of permutaton matrces Endogenous Capactes. Consder a school choce problem n whch the school authorty wshes to run several educaton programs wthn one school buldng. An assgnment n such an envronment can be descrbed as a matrx n whch rows correspond to students and columns correspond to educaton programs (rather than school buldngs). Wth ths representaton, a school buldng corresponds to multple columns. Formally, we can let H 1 nclude all rows, whch correspond to students, whle H 2 ncludes sets of the form N O where O s a subset of educatonal programs: A sngleton set O represents an ndvdual 7 We thank Tomom Matsu and Akhsa Tamura for suggestng the algorthm. An earler draft of ths paper ncluded an alternatve algorthm generalzng the steppng-stones algorthm descrbed by Hylland and Zeckhauser (1979).

10 10 ERIC BUDISH, YEON-KOO CHE, FUHITO KOJIMA, AND PAUL MILGROM educatonal program whle a larger set O corresponds to a school buldng wth multple programs. The celng q N O then descrbes the total number of students who can be admtted wthn O, whch may apply to a sngle program or a set of programs n the same buldng. If the sum of celngs a O q N {a} s larger than the celng q N O, then that means the szes of programs wthn the same school buldng O can be adjusted, subject to the (say physcal) capacty of the school buldng. For nstance, n Fgure 1, columns b and c represent two programs (wthn a school) each of whch s subject to a quota, and there s a school wde quota mpngng on b and c, together. Note that a constrant structure composed of H 1 and H 2 descrbed above s a bherarchy, thus t s unversally mplementable. Notce also that the herarchcal structure H 2 allows for nested constrants on program szes. Fgure 1 about here Group-specfc Quotas. Affrmatve acton polces are sometmes mplemented as quotas on students fttng specfc gender, racal, or economc profles. 8 A smlar mathematcal structure results from New York Cty s Educatonal Opton programs, whch acheve a mx of students by mposng quotas on students wth test scores (Abdulkadroğlu, Pathak and Roth, 2005). Quotas may be based on the resdence of applcants as well: The school choce program of Seoul, Korea, lmts the percentage of seats allocated to applcants from outsde the dstrct. 9 resdental areas as well. A number of school choce programs n Japan have smlar quotas based on Agan constrants pertanng to ndvdual students (.e., rows ) can be organzed as a herarchy H 1. All constrants pertanng to schools capactes are organzed as a separate herarchy H 2. Group-specfc quotas can be handled by ncludng sets of the form N {a} for a O and N N n the second herarchy H 2. The celng q N {a} then determnes the maxmum number of agents school a can admt from group N. Quotas on multple groups can be mposed for each a wthout volatng the herarchy structure as long as they do not overlap wth each other. 10 Moreover, a nested seres of constrants can be accommodated. 8 Abdulkadroğlu and Sönmez (2003b) and Abdulkadroğlu (2005) analyze assgnment mechansms under affrmatve acton constrants. 9 See Students Hgh School Choce n Seoul Outlned, Dgtal Chosun Ilbo, October 16, 2008 ( 10 In fact, an overlap of constrant sets can be accommodated wth a small error. Suppose a school has maxmal quotas for whte and male at 60 and 55, respectvely. Suppose an expected assgnment assgns 40.5 whte male, 14.5 black male, and 19.5 whte female students to that school. Notce both celngs are bndng at ths expected assgnment. Ths expected assgnment can be mplemented recognzng only whte and male, whte and female, and male as the constrant sets, whch then forms a herarchy. Implementng wth ths modfed constrant structure may volate the maxmal quota for whtes, snce the constrant set for whte s not ncluded n the structure; for nstance, the school may get 41 whte male, 14 black male and 20 whte female students. However, the volaton s by only one student. In fact, the degree of volaton s at most one when there s only one overlap of constrant sets. Such a small volaton can often be tolerated n realstc

11 DESIGNING RANDOM ALLOCATION MECHANISMS 11 For nstance, a school system can requre that a school admt at most 50 students from dstrct one, at most 50 students from dstrct two, and at most 80 students from ether dstrct one or two. It s also possble to accommodate both flexble-capacty constrants and group-specfc quota constrants wthn the same herarchy H 2. Flexble-capacty constrants are defned on multple columns of an expected assgnment matrx X, whereas group-specfc quota constrants are defned on subsets of sngle columns of X. Any subset of a sngle column wll be a subset of or dsjont from any set of multple columns Course Allocaton. In course allocaton, each student may enroll n multple courses, but cannot receve more than one seat n any sngle course. Moreover, each student may have preference or feasblty constrants that lmt the number of courses taken from certan sets. For example, schedulng constrants prohbt any student from takng two courses that meet durng the same tme slot. Or, a student mght prefer to take at most two courses on fnance, at most three on marketng, and at most four on fnance or marketng n total. Many such restrctons can be modeled usng a bherarchy, wth H 1 ncludng all rows and H 2 ncludng all columns. Settng q {(,a)} = 1 and q {} O > 1 for each N and a O ensures that each student can enroll n multple courses but be assgned to at most one seat n each course. Lettng F and M be the sets of fnance courses and marketng courses, respectvely, f H 1 contans {} F,{} M and {} (F M), then we can express the constrants student can take at most q {} F courses n fnance, q {} M courses n marketng, and q {} (F M) n fnance and marketng combned. Schedulng constrants are handled smlarly; for nstance, F and M are sets of classes offered at dfferent tmes (e.g., Frday mornng and Monday mornng). It may be mpossble, however, to express both subject and schedulng constrants whle stll mantanng a bherarchy constrant structure. Note that the flexble producton and group-specfc quota constrants descrbed n Sectons can also be ncorporated nto the course allocaton problem wthout jeopardzng the bherarchcal structure. These constrants can be ncluded n H 2, whle the preference and schedulng constrants descrbed above can be ncluded n H 1. So long as H 1 and H 2 are both herarches, H = H 1 H 2 s a bherarchy Interleague Play Matchup Desgn. Some professonal sports assocatons, ncludng Major League Baseball (MLB) and the Natonal Football League (NFL), have two separate leagues. In MLB, teams n the Amercan League (AL) and Natonal League (NL) had tradtonally played aganst teams only wthn ther own league durng the regular season, controlled-choce envronments. In case the quotas are rgd, the quota can be set more conservatvely; for nstance n the above example, the quota for whtes can be set at 59 nstead of 60.

12 12 ERIC BUDISH, YEON-KOO CHE, FUHITO KOJIMA, AND PAUL MILGROM but play across the AL and NL, called nterleague play, was ntroduced n Unlke the ntraleague games, the number of nterleague games s relatvely small, and ths can make the ndvsblty problem partcularly dffcult to deal wth n desgnng the matchups. For example, suppose there are two leagues, N and O, each wth 9 teams. Suppose each team must play 15 games aganst teams n the other league. Ignorng ndvsblty, a far matchup wll requre each team n a league to play every team n the other league the same number of tmes, that s, 15/ tmes. Of course, ths fractonal matchup tself s nfeasble, but one can mplement ths expected matchup as a convex combnaton of feasble matchups. In dong so, one can also specfy addtonal constrants: e.g., each team n N has a geographc rval n O, and they must play twce; teams n each league must face opponents n the other league of smlar dffculty. Specfcally, one could requre each team to play at least 4 games wth the top 3 teams, 4 games wth the mddle 3 teams and 4 games wth the bottom 3 teams of the other league. It s not dffcult to see that the resultng constrant structure forms a bherarchy. Our approach can produce a feasble matchup that mplements the unform expected assgnment satsfyng these addtonal constrants Necessty of a bherarchcal constrant structure. Theorem 1 shows that bherarchy s suffcent for unversal mplementablty. Ths secton dentfes a sense n whch t s also necessary. Dong so also provdes an ntuton about the role bherarchy plays for mplementaton of expected assgnments. We begn wth an example of a non-bherarchcal constrant structure that s not unversally mplementable. Example 1. Consder the followng envronment wth two objects a, b and two agents 1, 2, and the constrant structure H that conssts of the frst row {(1, a), (1, b)}, the frst column {(1, a), (2, a)}, and a dagonal set {(1, b), (2, a)} (n addton to all sngleton sets). Clearly, ths constrant structure s not a bherarchy as there s no way to partton t nto two herarches. Suppose each constrant set n H has a common floor and celng quota of one. The followng expected assgnment ( ) X = cannot be mplemented as a lottery over feasble pure assgnments. 12 To see ths, frst observe that the lottery mplementng X must choose wth postve probablty a pure assgnment X n whch x 1a = 1. Snce the frst row has a quota of one, t follows that x 1b = 0. Snce the 11 See Interleague play, Wkpeda ( 12 Notatonally, the conventon throughout the paper s that the th row of the expected assgnment matrx from the top corresponds to agent whle the frst column from the left corresponds to object a, the second column corresponds to object b, and so on.

13 DESIGNING RANDOM ALLOCATION MECHANISMS 13 dagonal set has a quota of one, t follows that x 2a = 1. Ths s a contradcton because the quota for the frst column s volated at X snce x {(1,a),(2,a)} = x 1a + x 2a = 2. Example 1 suggests that the falure of mplementablty s caused by a cycle formed by an odd number of constrant sets. In the above example, for nstance, a cycle formed by three constrant sets (the frst row, the frst column, and the dagonal set {(1, b), (2, a)}) leads to a stuaton where at least one of the constrants s volated. Generalzng ths dea, we say that a sequence of constrant sets (S 1,..., S l ) n H s an odd cycle f l s odd and there exsts a sequence of agent-object pars (s 1,..., s l ) n N O such that for each = 1,..., l, we have s S S +1 and x / S j for any j, + 1, where subscrpt l + 1 s understood to be 1. An argument generalzng the above example yelds the followng (a formal proof s n the Appendx). Lemma 1. (Odd Cycles) If a constrant structure contans an odd cycle, then t s not unversally mplementable. An mportant role of the bherarchy s to rule out odd cycles. To see ths, suppose for contradcton that a bherarchy H = H 1 H 2 contans an odd cycle (S 1,..., S l ). Assume S 1 H 1 wthout loss of generalty. Then, S 2 must belong to H 2 snce S 1 S 2 and nether s a subset of the other (snce s 1 S 1 S 2, s 2 S 2 \ S 1 and s l S 1 \ S 2 ). Argung n the same fashon, S 3 must be n H 1, S 4 n H 2,..., and S l must be n H 1 snce l s an odd number. But S l S 1 and nether s a subset of the other. So H 1 cannot be a herarchy, and H cannot be a bherarchy, a contradcton. Is bherarchy necessary for unversal mplementaton? It turns out ths s not the case; nor s t the case that rulng out odd cycles s suffcent for unversal mplementaton. 13 Fgure 2 depcts how the sets of constrant structures satsfyng dfferent propertes relate to 13 To see that bherarchy s not necessary, consder an envronment wth 2 objects and 2 agents as before, but let H = {{(1, a), (1, b)}, {(1, a), (2, a)}, {(1, a), (2, b)}}, and the floor and celng quotas for each constrant set be one. Note H s not a bherachy. Yet, any expected assgnment ( ) s t X =, t t wth s + t = 1, can be decomposed by a convex combnaton of pure assgnments as ( ) ( ) ( ) s t X = = s + t. t t Next, we show that an absence of odd cycles s not suffcent for unversal mplementablty. Consder H = {{(1, a), (1, b)}, {(1, a), (2, a)}, {(1, a), (2, b)}, {(1, a), (1, b), (2, a), (2, b)}}. Ths structure does not contan an odd cycle (and t s not a bherarchy). Assume the quota for each of the frst three sets s one and the quota for the last set s two. Now consder the expected assgnment X of Example 1. Even though X satsfes the quotas, t s not mplementable.

14 14 ERIC BUDISH, YEON-KOO CHE, FUHITO KOJIMA, AND PAUL MILGROM each other. In partcular, there are gaps between the set of constrant structures contanng no odd cycles, the set of those unversally mplementable, and the set of those that are bherarchcal. No Odd Cycles Unversal Implementaton Canoncal Two-sded Assgnment Bherarchy Fgure 2: Constrant structures satsfyng dfferent propertes We now show that these gaps dsappear for an mportant class of constrant structures (depcted on the rght sde of Fgure 2). In a two-sded assgnment problem, there are often quotas for each ndvdual agent and quotas for each object. We say that H s a canoncal two-sded constrant structure f H contans all rows (.e., sets of the form {(, a) a O} for each N) and all columns (.e., sets of the form {(, a) N} for each a O). The next result demonstrates that bherarchy s necessary for unversal mplementablty for such constrant structures. 14 Theorem 2. (Necessty) If a canoncal two-sded constrant structure s not a bherarchy, then t s not unversally mplementable. The formal proof of Theorem 2 s n the Appendx. The basc strategy of the proof s to show that there exsts an odd cycle whenever a canoncal two-sded constrant structure s not a bherarchy. 14 One may wonder f the restrcton to canoncal two-sded constrant structures has any real bte n lght of the fact that one can seemngly convert any H nto one contanng each row and column smply by mposng non-bndng quotas, e.g., q S = and q S =. Ths cosmetc converson does not alter the fact that no constrant s ever bndng for the added set, however. Recall that the noton of unversal mplementablty requres the ablty to mplement an expected assgnment subject to the tghtest possble constrants on each set n H. Hence, the sets wth non-bndng constrants cannot be added to a constrant structure n ths manner.

15 DESIGNING RANDOM ALLOCATION MECHANISMS 15 Thus, n the context of typcal two-sded assgnment and matchng problems, bherarchy s both necessary and suffcent for unversal mplementaton. We now turn to applcatons. 3. A Generalzaton of The Probablstc Seral Mechansm for Assgnment wth Sngle-unt Demand In ths secton, we consder a problem of assgnng ndvsble objects to agents who can consume at most one object each. Examples nclude unversty housng allocaton, publc housng allocaton, offce assgnment, and student placement n publc schools. A common method for allocatng objects n such a settng s the random prorty mechansm. In ths mechansm, every agent reports preference rankngs of the objects. The mechansm desgner then randomly orders the agents wth equal probablty. The frst agent n the realzed order receves her stated favorte (the most preferred) object, the next agent receves hs stated favorte object among the remanng ones, and so on. Random prorty s strategy-proof, that s, reportng ordnal preferences truthfully s a weakly domnant strategy for every agent. Moreover, random prorty s ex-post effcent, that s, every pure assgnment that occurs wth postve probablty under the mechansm s Pareto effcent. Despte ts many advantages, the random prorty mechansm may ental unambguous effcency loss ex ante. Adaptng an example by Bogomolnaa and Mouln (2001), suppose that there are two types of objects a and b wth one copy each and the null object representng the outsde opton. There are four agents 1, 2, 3 and 4, where agents 1 and 2 prefer a to b to whle agents 3 and 4 prefer b to a to. By calculaton, the random prorty mechansm results n the expected assgnment 5/12 1/12 1/2 5/12 1/12 1/2 X = 1/12 5/12 1/2. 1/12 5/12 1/2 Ths assgnment entals an unambguous effcency loss. Notce frst that every agent consumes the less preferred of the two proper objects wth postve probablty. Ths happens snce two agents of the same preference (e.g., agents 1 and 2) are chosen wth postve probablty to the be frst two n the seral order, n whch case the second agent wll clam the less preferred of the two proper objects. Clearly, t wll beneft all f agents 1 and 2 can trade off 1/12 share of b for the same share of a wth agents 3 and 4. In other words, every agent

16 16 ERIC BUDISH, YEON-KOO CHE, FUHITO KOJIMA, AND PAUL MILGROM prefers an alternatve expected assgnment, 1/2 0 1/2 X 1/2 0 1/2 = 0 1/2 1/2. 0 1/2 1/2 An expected assgnment s sad to be ordnally effcent f t s not frst-order stochastcally domnated for all agents by some other expected assgnment. random prorty may result n an ordnally neffcent expected assgnment. The example mples that The probablstc seral mechansm, ntroduced by Bogomolnaa and Mouln (2001) n the smple one-to-one assgnment settng, elmnates ths form of neffcency. Imagne that each ndvsble object s a dvsble object of probablty shares: If an agent receves fracton p of an object, we nterpret that she receves the object wth probablty p. Gven reported preferences, consder the followng eatng algorthm. Tme runs contnuously from 0 to 1. At every pont n tme, each agent eats her reported favorte object wth speed one among those that have not been completely eaten up. At tme t = 1, each agent s endowed wth probablty shares of objects. The probablstc seral assgnment s defned as the resultng probablty shares. In the above example, agents 1 and 2 start eatng a and agents 3 and 4 start eatng b at t = 0 n the eatng algorthm. Snce two agents are consumng one unt of each object, both a and b are eaten away at tme t = 1. As no (proper) object remans, agents consume 2 the null object between t = 1 and t = 1. Thus the resultng probablstc seral assgnment 2 s gven by X defned above. In partcular, the probablstc seral mechansm elmnates the neffcency that was present under random prorty. More generally, Bogomolnaa and Mouln (2001) show that the probablstc seral random assgnment s ordnally effcent wth respect to any reported preferences. 15 The man goal of ths secton s to generalze the probablstc seral mechansm to accommodate constrants absent n the smple settng. To begn, we consder our basc setup wth 15 The contrbuton of Bogomolnaa and Mouln has led to much subsequent work on random assgnment mechansms for sngle-unt assgnment problems. The probablstc seral mechansm s generalzed to allow for weak preferences and exstng property rghts by Katta and Sethuraman (2006) and Ylmaz (2009). Kesten (2007) defnes two random assgnment mechansms and shows that these mechansms are equvalent to the probablstc seral mechansm. Ordnal effcency s characterzed by Abdulkadroğlu and Sönmez (2003a), McLennan (2002) and Manea (2006). Behavor of the random prorty and probablstc seral mechansms n large markets s studed by Kojma and Manea (2008), Manea (2009) and Che and Kojma (2008). In the schedulng problem (a specal case of the current envronment), Crès and Mouln (2001) show that the probablstc seral mechansm s group strategy-proof and frst-order stochastcally domnates the random prorty mechansm, and Bogomolnaa and Mouln (2002) gve two characterzatons of the probablstc seral mechansm.

17 DESIGNING RANDOM ALLOCATION MECHANISMS 17 agents N and objects O, where O now contans a null object wth unlmted supply. 16 We then consder a bherarchy constrant structure H = H 1 H 2 such that H 1 s composed of all rows whle H 2 ncludes (but s not restrcted to) all columns. We assume that q {} O = q {} O = 1 for all N, that s, each agent obtans exactly one object, rather than at most one. Ths s wthout loss of generalty snce O contans. We assume that q S = 0 for any S H that s not a row, that s, there are no other floor constrants. The celng quota for each object a, q N {a}, can be an arbtrary nonnegatve nteger. Recall that an expected assgnment X s sad to satsfy quotas q f q S (,a) S x a q S for each S H. Each agent has a strct preference over the set of objects. We wrte a b f ether a b or a = b holds. We wrte for ( ) N and for ( j ) j N\{}. As mentoned earler, the bherarchy structure n ths secton accommodates a range of practcal stuatons faced by a mechansm desgner. Frst, the objects may be produced endogenously based on the reported preferences of the agents, as n the case of school choce wth flexble capacty (Secton 2.1.2). Second, a mechansm desgner may need to treat dfferent groups of agents dfferently, as n the case of school choce wth group-specfc quotas (Secton 2.1.3). Now we ntroduce the generalzed probablstc seral mechansm. As n BM, the basc dea s to regard each ndvsble object as a dvsble object of probablty shares. More specfcally, the algorthm s descrbed as follows: Tme runs contnuously from 0 to 1. At every pont n tme, each agent eats her reported favorte object wth speed one among those that are avalable at that nstance, and the probablstc seral assgnment s defned as the probablty shares eaten by each agent by tme 1. In order to obtan an mplementable expected assgnment n the presence of addtonal constrants, however, we modfy the defnton of the algorthm. More specfcally, we say that object a s avalable to agent f and only f, for every constrant set S such that (, a) S, the total amount of probablty shares eaten away wthn S (the sum, over every agent-object par (j, b) S, of shares of b eaten by j) s less than ts celng quota q S. Ths algorthm s formally defned n Appendx A. Gven reported preferences, the generalzed probablstc seral assgnment s denoted P S( ). Note that a few modfcatons are made n the defnton of the algorthm from the verson of Bogomolnaa and Mouln (2001). Frst, we specfy avalablty of objects wth respect to both agents and objects n order to accommodate complex constrants such as affrmatve acton. Second, we need to keep track of multple constrants for each agent-object par (, a) 16 Formally, we assume that q S = + for each constrant set S that s not a row and has a nonempty ntersecton wth N { }.

18 18 ERIC BUDISH, YEON-KOO CHE, FUHITO KOJIMA, AND PAUL MILGROM durng the algorthm, snce there are potentally multple constrants that would make the consumpton of the object a by the agent no longer feasble. Recall that the constrant structure n ths secton s a bherarchy. Buldng on ths observaton and our analyss from the prevous secton, we pont out that any expected assgnment produced by the generalzed probablstc seral mechansm s mplementable. Corollary 2. For any preference profle, the generalzed probablstc seral assgnment P S( ) s mplementable. Proof. The result follows mmedately from Theorem 1, because the constrant structure under consderaton forms a bherarchy and P S( ) satsfes all quotas assocated wth that constrant structure by constructon. In ths sense, the mechansm s well-defned as a random assgnment mechansm n the current settng Propertes of The Generalzed Probablstc Seral Mechansm. We ntroduce the ordnal effcency concept n our setup. A lottery x = [x a ] a O for an agent (frst-order) stochastcally domnates another lottery x = [x a] a O at f b a x b b a for every object a O, and x strctly stochastcally domnates x f the former stochastcally domnates the latter and x x. An expected assgnment X = [x ] N ordnally domnates another expected assgnment X = [x ] N at f X X, and, for each, x stochastcally domnates x at. If X ordnally domnates X at, then every agent prefers x to x accordng to any expected utlty functon wth utlty ndex consstent wth. An expected assgnment that satsfes q s ordnally effcent at f t s not ordnally domnated at by any other expected assgnment that satsfes q. Note that our model allows for a varety of constrants, so the current noton has the flavor of constraned effcency n that the effcency s defned wthn x b, the set of expected assgnments satsfyng the quota constrants. Bogomolnaa and Mouln (2001) show that the probablstc seral mechansm results n an ordnally effcent expected assgnment n ther settng. Ther result can be generalzed to our settng although ts proof requres new arguments In BM s envronment, ordnal effcency s equvalent to the nonexstence of a Pareto-mprovng trade n probablty shares among agents (n the sense of leadng to a ordnally domnatng expected assgnment). Ths enables a characterzaton of ordnal effcency n terms of a certan bnary relaton over objects, whch s crucal n BM s proof of ther mechansm s ordnal effcency. There are two man dffcultes for generalzng

19 DESIGNING RANDOM ALLOCATION MECHANISMS 19 Theorem 3. For any preference profle, the generalzed probablstc seral expected assgnment P S( ) s ordnally effcent at. Bogomolnaa and Mouln (2001) also show that the probablstc seral mechansm s far n a specfc sense n ther settng. Formally, an expected assgnment X = [x ] N s envyfree at f x stochastcally domnates x j at for every, j N. It turns out that the generalzed probablstc seral assgnment s not necessarly envy-free n our envronment. To see ths pont, consder the followng envronment: there are three agents 1, 2, and 3, and two unts of object a plus a null object. Even though there are two copes of a, only one of them can be assgned to to 1 or 2. Suppose all agents prefer a over the null object. Then, by a smple calculaton the generalzed probablstc seral expected assgnment P S( ) s gven by P S( ) = Note that P S( ) s not envy-free snce P S 3 ( ) s not stochastcally domnated by P S 1 ( ) wth respect to 1 (ndeed, P S 3 ( ) strctly stochastcally domnates P S 1 ( ) n ths example). However, there s a sense n whch the above expected assgnment should not be consdered unfar despte the exstence of envy. To see ths pont, note that t s nfeasble to assgn object a to agent 1 wth hgher probablty smply by movng some probablty share of a from agent 3 to agent 1, because such a change would volate the celng quota on {1, 2} {a}. In that sense the envy s based on a desre of agent 1 that cannot be feasbly accommodated. Motvated by the above example, we ntroduce the followng concept. Expected assgnment X entals no feasble envy at f, whenever x does not stochastcally domnate x j at, t s mpossble to feasbly reassgn agent to x j whle keepng ntact expected assgnments of all agents except possbly of agent j;.e., f no expected assgnment Y defned by y = x j and y k = x k for all k, j, satsfes q. 18 Thus the above defnton requres that ether does not envy j, or an alternatve expected assgnment n whch receves j s lottery volates the result to our settng. Frst, not all trades n probablty shares are feasble because the new expected assgnment may volate constrants such as group-specfc quotas. Second, the nonexstence of a Paretomprovng trade does not mply ordnal effcency because, thanks to flexble capacty, a dfferent aggregate supply of objects may exst that makes every agent better off. These dfferences make t mpossble to drectly apply BM s technque. We address these complcatons by defnng a new bnary relaton over agent-object pars. See Appendx F for detals. 18 Note we do not put any restrcton on agent j s lottery n consderng feasble reassgnment. Not puttng any restrcton on j s lottery can only weaken the condton for a feasble reassgnment, and thus can only strengthen the noton of envy-freeness, n comparson wth mposng restrctons such as assgnng x to agent j n the new assgnment.

20 20 ERIC BUDISH, YEON-KOO CHE, FUHITO KOJIMA, AND PAUL MILGROM quotas. In the above example, P S( ) entals no feasble envy. Ths property turns out to hold generally, as stated below. Theorem 4. For any preference profle, the generalzed probablstc seral expected assgnment P S( ) entals no feasble envy at. Nether ordnal effcency nor no feasble envy s satsfed by random prorty even n the smplest settng of one-to-one matchng (Bogomolnaa and Mouln, 2001). Unfortunately the probablstc seral mechansm s not strategy-proof, that s, there are stuatons n whch an agent s made better off by msreportng her preferences. However, Bogomolnaa and Mouln (2001) show that the probablstc seral mechansm s weakly strategy-proof n ther settng, that s, an agent cannot msstate hs preferences and obtan an expected assgnment that strctly stochastcally domnates the one obtaned under truthtellng. Formally, we clam that the generalzed probablstc seral mechansm s weakly strategy-proof, that s, there exst no, N and such that P S (, ) strctly stochastcally domnates P S ( ) at n our more general envronment. 19 Theorem 5. The generalzed probablstc seral mechansm s weakly strategy-proof. Proof. The proof s an adaptaton of Proposton 1 of Bogomolnaa and Mouln (2001) and we omt the proof. One lmtaton of our generalzaton s that the algorthm s defned only for cases wth maxmum quotas: The mnmum quota for each group must be zero. In the context of school choce, ths precludes the admnstrator from requrng that at least a certan number of students from a group attend a partcular school. Despte ths lmtaton, admnstratve goals can often be suffcently represented usng maxmum quotas alone. For nstance, f there are two groups of students, rch and poor, a requrement that at least a certan number of poor students attend some hghly desrable school mght be adequately replaced by a maxmum quota on the number of rch students who attend. 4. A Generalzaton of The Pseudo-market Mechansm for Assgnment wth Mult-Unt Demand In a semnal paper, Hylland and Zeckhauser (1979) propose an ex-ante effcent mechansm for the problem of assgnng n objects amongst n agents wth sngle-unt demand. Based 19 Kojma and Manea (2008) show that truthtellng becomes a domnant strategy for a suffcently large market under the probablstc seral mechansm n a smpler envronment than the current one. Showng a smlar clam n our envronment s beyond the scope of ths paper, but we conjecture that the argument can be extended.