Matching with Contracts

Transcription

1 Matcing wit Contracts Jon William atfield and Paul Milgrom * January 13, 2005 Abstract. We develop a model of matcing wit contracts wic incorporates, as special cases, te college admissions problem, te Kelso-Crawford labor market matcing model, and ascending package auctions. We introduce a new law of aggregate demand for te case of discrete eterogeneous workers and sow tat, wen workers are substitutes, tis law is satisfied by profit maximizing firms. Wen workers are substitutes and te law is satisfied, trutful reporting is a dominant strategy for workers in a workeroffering auction/matcing algoritm. We also parameterize a large class of preferences satisfying te two conditions. JEL Classification: C78, 44 Since te pioneering US spectrum auctions of 1994 and 1995, related ascending multiitem auctions ave been used wit muc fanfare on six continents for sales of radio spectrum and electricity supply contracts. 1 Package bidding, in wic bidders can place bids not just for individual lots but also for bundles of lots ( packages ), as found increasing use in procurement applications (Paul Milgrom (2004), Peter Cramton et al. (2005)). Recent proposals in te US to allow package bidding for spectrum licenses and for airport landing rigts incorporate ideas suggested by Lawrence Ausubel and Paul Milgrom (2002) and by avid Porter et al. (2003) (see Lawrence Ausubel et al. (2005)). 1

2 Matcing algoritms based on economic teory also ave important practical applications. Alvin E. Rot and Elliott Peranson (1999) explain ow a certain two-sided matcing procedure, wic is similar to te college admissions algoritm introduced by avid Gale and Lloyd Sapley (1962), as been adapted to matc 20,000 doctors per year to medical residency programs. After Atila Abdulkadiroğlu and Tayfun Sönmez (2003) advocated a variation of te same algoritm for use by scool coice programs, a similar centralized matc was adopted by te New York City scools (Atila Abdulkadiroğlu et al. (2005b)) and anoter is being evaluated by te Boston scools (Atila Abdulkadiroğlu et al. (2005a)). Tis paper identifies and explores certain similarities among all of tese auction and matcing mecanisms. To illustrate one similarity, consider te labor market auction model of Alexander Kelso and Vincent Crawford (1982), in wic firms bid for workers in simultaneous ascending auctions. Te Kelso-Crawford model assumes tat workers ave preferences over firm-wage pairs and tat all wage offers are drawn from a pre-specified finite set. If tat set includes only one wage, ten all tat is left for te auction to determine is te matc of workers to firms, so te auction is effectively transformed into a matcing algoritm. Te auction algoritm begins wit eac firm proposing employment to its most preferred set of workers at te one possible wage. Wen some workers turn it down, te firm makes offers to oter workers to fill its remaining openings. Tis procedure is precisely te ospital-offering version of te Gale-Sapley matcing algoritm. ence, te Gale-Sapley matcing algoritm is a special case of te Kelso-Crawford procedure. Te possibility of extending te National Resident Matcing Program (te Matc ) to permit wage competition is an important consideration in assessing public policy towards te Matc, particularly because work by Jeremy Bulow and Jonatan Levin (2003) lends some 2

3 teoretical support for te position tat te Matc may compress and reduce doctors wages relative to a perfectly competitive standard. Te practical possibility of suc an extension depends on many details, including importantly te form in wic doctors and ospitals would ave to report teir preferences for use in te Matc. In its current incarnation, te matc can accommodate ospital preferences tat encompass affirmative action constraints and a subtle relationsip between internal medicine and its subspecialties, so it will be important for any replacement algoritm to allow similar preferences to be expressed. We introduce a parameterized family of valuations tat accomplises tat in section IV. A second important similarity is between te Gale-Sapley doctor-offering algoritm and te Ausubel-Milgrom proxy auction. Explaining tis relationsip requires restating te algoritm in a different form from te one used for te preceding comparison. We sow tat if te ospitals in te Matc consider doctors to be substitutes, ten te doctor-offering algoritm is equivalent to a certain cumulative offer process in wic te ospitals at eac round can coose from all te offers tey ave received at any round, current or past. In a different environment, were tere is but a single ospital or auctioneer wit unrestricted preferences and general contract terms, tis cumulative offer process coincides exactly wit te Ausubel-Milgrom proxy auction. espite te close connections among tese mecanisms, previous analyses ave treated tem separately. In particular, analyses of auctions typically assume tat bidders payoffs are quasi-linear. No corresponding assumption is made in analyzing te medical matc or te college admissions problem; indeed, te very possibility of monetary transfers is excluded from tose formulations. As discussed below, te quasi-linearity assumption combines wit te substitutes assumption in a subtle and restrictive way. 3

4 Tis paper presents a new model tat subsumes, unifies and extends te models cited above. Te basic unit of analysis in te new model is te contract. To reproduce te Gale- Sapley college admissions problem, we specify tat a contract is fully identified by te student and college; oter terms of te relationsip depend only on te parties identities. To reproduce te Kelso-Crawford model of firms bidding for workers, we specify tat a contract is fully identified by te firm, te worker, and te wage. Finally, to reproduce te Ausubel-Milgrom model of package bidding, we specify tat a contract is identified by te bidder, te package of items tat te bidder will acquire, and te price to be paid for tat package. Several additional variations can be encompassed by te model. For example, Alvin E. Rot (1985b, (1984) allows tat a contract migt specify te particular responsibilities tat a worker will ave witin te firm. Our analysis of matcing models empasizes two conditions tat restrict te preferences of te firms/ospitals/colleges: a substitutes condition and an law of aggregate demand condition. We find tat tese two conditions are implied by te assumptions of earlier analyses, so our unified treatment implies te central results of tose teories as special cases. In te tradition of demand teory, we define substitutes by a comparative statics condition. In demand teory, te exogenous parameter cange is a price decrease, so te callenge is to extend te definition to models in wic tere may be no price tat is allowed to cange. In our contracts model, a price reduction corresponds formally to expanding te firm s opportunity set, i.e., to making te set of feasible contracts larger. Our substitutes condition asserts tat wen te firm cooses from an expanded set of contracts, te set of contracts it rejects also expands (weakly). As we will sow, tis abstract substitutes condition coincides exactly wit te demand teory condition for standard models wit prices. It also coincides 4

5 exactly wit te substitutable preferences condition for te college admissions problem (Alvin E. Rot and Marilda Sotomayor (1990)). Te law of aggregate demand, wic is new in tis paper, is similarly defined by a comparative static. It is te condition tat wen a college or firm cooses from an expanded set, it admits at least as many students or ires at least as many workers. 2 Te term law of aggregate demand is motivated by te relation of tis condition to te law of demand in producer teory. According to producer teory, a profit-maximizing firm demands (weakly) more of any input as its price falls. For te matcing model wit prices, te law of aggregate demand requires tat wen any input price falls, te aggregate quantity demanded, wic includes te quantities demanded of tat input and all of its substitutes, rises (weakly). Notice tat it is tricky even to state suc a law in producer teory wit divisible inputs, because tere is no general aggregate quantity measure wen divisible inputs are diverse. In te present model wit indivisible workers, we measure te aggregate quantity of workers demanded or ired by te total number of suc workers. A key step in our analysis is to prove a new result in demand teory: if workers are substitutes, ten a profit maximizing firm s employment coices satisfy te law of aggregate demand. Since firms are profit maximizers and regard workers as substitutes in te Kelso- Crawford model, it follows tat te law of aggregate demand olds for tat model. Tus, one implication of te standard quasi-linearity assumption of auction teory is tat te bidders preferences satisfy te law of aggregate demand. We find tat wen preferences are responsive as originally and still commonly assumed in matcing teory analyses, tey satisfy te law of aggregate demand. 3 We also prove some new results for te class of auction and matcing models tat satisfy tis law. 5

6 Te paper is organized as follows. Section I introduces te matcing-wit-contracts notation, treats an allocation as a set of contracts, and caracterizes te stable allocations in terms of te solution of a certain system of two equations. Section II introduces te substitutes condition and uses it to prove tat te set of stable allocations is a non-empty lattice, and tat a certain generalization of te Gale-Sapley algoritm identifies its maximum and minimum elements. Tese two extreme points are caracterized as a doctor-optimal/ospital-pessimal point, wic is a point tat is te unanimously most preferred stable allocation for te doctors and te unanimously least preferred stable allocation for te ospitals, and a ospital-optimal/doctor-pessimal point wit te reverse attributes. Section II also proves several related results. First, if tere are at least two ospitals and if some ospital as preferences tat do not satisfy te substitutes condition, ten even if all oter ospitals ave just a single opening, tere exists a profile of preferences for te students and colleges suc tat no stable allocation exists. Tis result is important for te construction of matcing algoritms. It means tat any matcing procedure tat permits students and colleges to report preferences tat do not satisfy te substitutes condition cannot be guaranteed always to select a stable allocation wit respect to te reported preferences. Anoter result concerns vacancy cain dynamics, wic traces te dynamic adjustment of te labor market wen a worker retires or a new worker enters te market and te dynamics are represented by te operator we ave described. Te analysis extends te results reported by Yosef Blum et al. (1997) and foresadowed by Alexander Kelso and Vincent Crawford (1982). We find tat, starting from a stable allocation, te vacancy adjustment process converges to a new stable allocation. 6

7 Section III contains te most novel results of te paper. It introduces te law of aggregate demand, verifies tat it olds for a profit-maximizing firm wen inputs are substitutes, and explores its consequences. Wen bot te substitutes and te law of aggregate demand conditions are satisfied, ten (1) te set of workers employed and te set of jobs filled is te same at every stable collection of contracts and (2) it is a dominant strategy for doctors (or workers or students) to report teir preferences trutfully in te doctor-offering version of te extended Gale-Sapley algoritm. We also demonstrate te necessity of a weaker version of te law of aggregate demand for tese conclusions. Our conclusion about tis dominant strategy property substantially extends earlier findings about incentives in matcing. Te first suc results, due to Lester ubins and avid Freedman (1981) and Alvin E. Rot (1982), establised te dominant strategy property for te marriage problem, wic is a one-to-one matcing problem tat is a special case of te college admissions problem. Similarly, Gabrielle emange and avid Gale (1985) establis te dominant strategy property for te worker-firm matcing problem in wic eac firm as singleton preferences. Tese results generalize to te case of responsive preferences, tat is, to te case were eac ospital (or college or firm) beaves just te same as a collection of smaller ospitals wit one opening eac. For te college admissions problem, Atila Abdulkadiroğlu (2003) as sown tat te dominant strategy property also olds wen colleges ave responsive preferences wit capacity constraints, were te constraints limit te number of workers of a particular type tat can be ired. All of tese models wit a dominant strategy property satisfy our substitutes and law of aggregate demand conditions, so te earlier dominant strategy results are all subsumed by our new result. 7

8 In section IV, we introduce a new parameterized family of preferences te endowed assignment preferences and sow tat tey subsume certain previously identified classes and satisfy bot te substitutes and law of aggregate demand conditions. Section V introduces cumulative offer processes as an alternative auction/matcing algoritm and sows tat wen contracts are substitutes, it coincides wit te doctor-offering algoritm of te previous sections. Since te Ausubel-Milgrom proxy auction is also a cumulative offer process, our dominant strategy conclusion of section III implies an extension of te Ausubel-Milgrom dominant strategy result. Wen contracts are not substitutes, te cumulative offer process can converge to an infeasible allocation. We sow tat if tere is a single ospital/auctioneer, owever, te cumulative offer process converges to a feasible, stable allocation, witout restrictions on te ospital s preferences. Section VI concludes. All proofs are in te appendix. I. Stable Collections of Contracts Te matcing model witout transfers as many applications, of wic te best known among economists is te matc of doctors to ospital residency programs in te United States. For te remainder of te paper, we describe te matc participants as doctors and ospitals. Tese groups play te same respective roles as students and colleges in te college admissions problem and workers and firms in te Kelso-Crawford labor market model. A. Notation Te sets of doctors and ospitals are denoted by and, respectively, and te set of contracts is denoted by X. We assume only tat eac contract x X is bilateral, so tat it is 8

9 associated wit one doctor x and one ospital x. Wen all terms of employment are fixed and exogenous, te set of contracts is just te set of doctor-ospital pairs: X. For te Kelso-Crawford model, a contract specifies a firm, a worker and a wage, X W. Eac doctor d can sign only one contract. er preferences over possible contracts, including te null contract, are described by te total order d. Te null contract represents unemployment, and contracts are acceptable or unacceptable according to weter tey are more preferred tan. Wen we write preferences as Pd : x d y d z, we mean tat P d names te preference order of d and tat te listed contracts (in tis case, x, y and z) are te only acceptable ones. Given a set of contracts X X offered in te market, doctor d s cosen set C ( X ) is eiter te null set, if no acceptable contracts are offered, or te singleton set consisting of te most preferred contract. We formalize tis as follows: d (1) { x X x d x d } { x X x = d} if =, = Cd ( X ) = { max } oterwise d Te coices of a ospital are more complicated, because it as preferences over sets of doctors. Its cosen set is a subset of te contracts tat name it, tat is, C ( X ) { x X x = }. In addition, a ospital can sign only one contract wit any given doctor. X X x x C X x x x x. (2) ( )( )(, ( )) 9

10 Let C ( X ) = C ( X ) denote te set of contracts cosen by some doctor from set d d X. Te remaining offers in X are in te rejected set: R ( X ) = X C ( X ). Similarly, te ospitals cosen and rejected sets are denoted by C ( X ) = C ( X ) and R ( X ) = X C ( X ). B. Stable allocations: Stable Sets of Contracts In our model, an allocation is a collection of contracts, as tat determines te payoffs to te participants. We study allocations suc tat tere is no alternative allocation tat is strictly preferred by some ospital and weakly preferred by all of te of doctors tat it ires and suc tat no doctor strictly prefers to reject is contract. It is a standard observation in matcing teory tat suc an allocation is a core allocation in te sense tat no coalition of ospitals and doctors can find anoter allocation, feasible for tem, tat all weakly prefer and some strictly prefer. Tis allocation is also a stable allocation in te sense tat tere is no coalition tat can deviate profitably, even if te deviating coalition assumes tat outsiders will remain willing to accept te same contracts. We formalize te notion of stable allocations as follows: efinition. A set of contracts X X is a stable allocation if (i) C ( X ) = C ( X ) = X and (ii) tere exists no ospital and set of contracts X C ( X ) suc tat X = C ( X X ) C ( X X ). 10

11 If condition (i) fails, ten some doctor or ospital prefers to reject some contract; tat doctor or ospital ten blocks te allocation. If condition (ii) fails, ten tere is an alternative set of contracts tat a ospital strictly prefers and tat its corresponding doctors weakly prefer. Te first teorem states tat a set of contracts is stable if any alternative contract would be rejected by some doctor or some ospital from its suitably defined opportunity set. In te formulas below, tink of te doctors opportunity set as X and te ospitals opportunity set as X. If X te corresponding stable set, ten X must include, in addition to X, all contracts tat would not be rejected by te ospitals and X must similarly include X and all contracts tat would not be rejected by te doctors. If X is stable, ten every alternative contract is rejected by somebody, so X = X X. Tis logic is summarized in te first teorem. Teorem 1. If ( X, X ) X 2 is a solution to te system of equations (3) X = X R ( X ) and X = X R ( X ), ten X X is a stable set of contracts and X X = C( X) = C( X). Conversely, for any stable collection of contracts X, tere exists some pair ( X, X ) satisfying (3) suc tat X = X X. Teorem 1 is formulated to apply to general sets of contracts. It is te basis of our analysis of stable sets of contracts in te entire set of models treated in tis paper. 11

12 II. Substitutes In tis section, we introduce our first restriction on ospital preferences, wic is te restriction tat contracts are substitutes. We use te restriction to prove te existence of a stable set of contracts and to study an algoritm tat identifies tose contracts. Our substitutes condition generalizes te Rot-Sotomayor substitutable preferences condition to preferences over contracts. In words, te substitutable preferences condition states tat if a contract is cosen by a ospital from some set of available contracts, ten tat contract will still be cosen from any smaller set tat includes it. Our substitutes condition is similarly defined as follows: efinition. Elements of X are substitutes for ospital if for all subsets X X X we ave R ( X ) R ( X ). 4 In te language of lattice teory, wic we use below, elements of X are substitutes for ospital exactly wen te function R is isotone. In demand teory, substitutes is defined by a comparative static tat uses prices. It says tat, limiting attention to te domain of wage vectors at wic tere is a unique optimum for te ospital, te ospital s demand for any doctor d is non-decreasing in te wage of eac oter doctor d. Our next result verifies tat for resource allocation problems involving prices, our definition of substitutes coincides wit te standard demand teory definition. For simplicity, we focus on a single ospital and suppress its identifier from our notation. Tus, imagine tat a ospital cooses doctors contracts from a subset of X = W, were is a finite set of doctors and W = { w,..., w} is a finite set of possible wages. Assume tat te set of wages W is 12

13 suc tat te ospitals preferences are always strict. Suppose tat w= maxw is a proibitively ig wage, so tat no ospital ever ires a doctor at wage w. In demand teory, it is standard to represent te ospital s market opportunities by a vector w W tat specifies a wage d w at wic eac doctor can be ired. We can extend te domain of te coice function C to allow market opportunities to be expressed by wage vectors, as follows: ( d ) (4) w W C( w) C {( d, w ) d } Formula (4) associates wit any wage vector w te set of contracts {( dw, ) d } and defines Cw ( ) to be te coice from tat set. d Wit te coice function extended tis way, we can now describe te traditional demand teory substitutes condition. Te condition asserts tat increasing te wage of doctor d from to w d (weakly) increases demand for any oter doctor d. w d efinition. C satisfies te demand-teory substitutes condition if (i) d d, (ii) ( d, wd ) C( w) and (iii) wd wd > imply tat ( d, w ) C( w, w ). d d d To compare te two conditions, we need to be able to assign a vector of wages to eac set of contracts X. It is possible tat, in X, some doctor is unavailable at any wage or is available at several different wages. For a profit-maximizing ospital, te doctor s relevant wage is te lowest wage, if any, at wic se is available. Moreover, suc a ospital does not distinguis between a doctor wo is unavailable and one wo is available only at a proibitively ig wage. Tus, from te perspective of a profit-maximizing ospital, aving contracts X available is equivalent to facing a wage vector Wˆ ( X ) specified as follows: 13

14 (5) Wˆ ( X ) = min{ s s = w or ( d, s ) X }. d In view of te preceding discussion, a profit-maximizing ospital s coices must obey te following identity: (6) C( X ) = C( Wˆ ( X )). Teorem 2. Suppose tat X = W is a finite set of doctor-wage pairs and tat (6) olds. Ten C satisfies te demand teory substitutes condition if and only if it contracts are substitutes. In particular, tis sows tat te Kelso-Crawford gross substitutes condition is subsumed by our substitutes condition. A. A Generalized Gale-Sapley Algoritm We now introduce a monotonic algoritm tat will be sown to coincide wit te Gale- Sapley algoritm on its original domain. To describe te monotonicity tat is found in te algoritm, let us define an order on X X as follows: (7) (( X, X) ( X, X ) ) ( X X and X X ) Wit tis definition, ( X X, ) is a finite lattice.. Te generalized Gale-Sapley algoritm is defined as te iterated applications of a certain function F : X X X X, as defined below. (8) F1 ( X ) = X R ( X ) F2 ( X ) = X R ( X ) (, ) = ( ), ( ) ( 1 2( 1 )) F X X F X F F X 14

15 As we ave previously observed, since te doctors coices are singletons, a revealed preference argument establises tat te function R : X X is isotone. If te contracts are substitutes for te ospitals, ten te function R : X X is isotone as well. Wen bot are isotone, te function F :( X X, ) ( X X, ) is also isotone, tat is, it satisfies ( ) (( X, X) ( X, X )) F( X, X) F( X, X ). Tus, F : X X X X is an isotone function from a finite lattice into itself. Using fixed point teory for finite lattices, te set of fixed points is a non-empty lattice and iterated applications of F, starting from te minimum and maximum points of X X, converge monotonically to a fixed point of F. 5 We summarize te particular application ere wit te following teorem. 6 Teorem 3. Suppose contracts are substitutes for te ospitals. Ten, 1. te set of fixed points of F on X X is a non-empty finite lattice, and in particular includes a smallest element ( X, X ) and a largest element ( X, X ), 2. starting at ( X, X ) ( X, ) monotonically to te igest fixed point =, te generalized Gale-Sapley algoritm converges {( ) ( ) ( )} ( X, X ) = sup X, X F X, X X, X, and 3. starting at ( X, X ) (, X) monotonically to te lowest fixed point =, te generalized Gale-Sapley algoritm converges {( ) ( ) ( )} ( X, X ) = inf X, X F X, X X, X. 15

16 Te facts tat ( X, X ) is te igest fixed point of F and tat ( X, X ) is te lowest in te specified order mean tat for any oter fixed point ( X, X ), X X X and X X X. Because doctors are better off wen tey can coose from a larger set of contracts, it follows tat te doctors unanimously weakly prefer C ( X ) to C ( X ) to C ( X ) and similarly tat te ospitals unanimously prefer C ( X ) to C ( X ) to C ( X ). Notice, by teorem 1, tat C ( X ) = C ( X ) = X X and C ( X ) = C ( X ) = X X, so we ave te following welfare conclusion. Teorem 4. Suppose contracts are substitutes for te ospitals. Ten, te stable set of contracts X X is te unanimously most preferred stable set for te doctors and te unanimously least preferred stable set for te ospitals. Similarly, te stable set X X is te unanimously most preferred stable set for te ospitals and te unanimously least preferred stable set for te doctors. Teorems 3 and 4 duplicate and extend familiar conclusions about stable matces in te Gale-Sapley matcing problem and a similar conclusion about equilibrium prices in te Kelso- Crawford labor market model. Tese new teorems encompass bot tese older models, and additional ones wit general contract terms. To see ow te Gale-Sapley algoritm is encompassed, consider te doctor-offering algoritm. As in te original formulation, we suppose tat ospitals ave a ranking of doctors tat is independent of te oter doctors tey will ire, so ospital just cooses its n most preferred doctors (from among tose wo are acceptable and ave proposed to it). 16

17 Let X ( t ) be te cumulative set of contracts offered by te doctors to te ospitals troug iteration t, and let X ( t ) be te set of contracts tat ave not yet been rejected by te ospitals troug iteration t. Ten, te contracts eld at te end of te iteration are precisely tose tat ave been offered but not rejected, wic are tose in X ( t) X ( t). Te process initiates wit no offers aving been made or rejected so, X (0) = X and X (0) =. Iterated applications of te operator F described above define a monotonic process, in wic te set of doctors make an ever-larger (accumulated) set of offers and te set of unrejected offers grows smaller round by round. Using te specification of F and starting from te extreme point ( X, ), we ave: (9) ( ) ( ) X () t = X R X ( t 1) X () t = X R X () t After offers ave been made in iteration t 1, te ospital s cumulative set of offers is X ( 1) t. Eac ospital old onto te n best offers it as received at any iteration provided tat many acceptable offers ave been made; oterwise it olds all acceptable offers tat ave been made. Tus, te accumulated set of rejected offers is R ( X ( 1) t ) and te unrejected offers are tose in ( ) X R X ( t 1) = X ( t). At round t, if a doctor s contract is being eld, ten te last offer te doctor made was its best contract in X ( t ). If a doctor s last offer was rejected, ten its new offer is its best contract in X ( t ). Te contracts tat doctors ave not offered at tis round or any earlier one are terefore tose in R( X () t ). So, te accumulated set of offers doctors ave made are tose in ( ) X R X () t = X () t. Once a fixed point is found of tis process is found, we ave, by Teorem 1, a stable set of contracts X ( t) X ( t). 17

18 To illustrate te algoritm, consider a simple example wit two doctors and two ospitals, were X and agents ave te following preferences: (10) d { } { } { } { } { } P : P : d d P : P : d, d d d d X (0) Te algoritm is initialized wit X (0) X {( d, 1 1), ( d, 1 2), ( d, 2 1)(, d, 2 2) } = = and =. Table 1 applies te operator given in (9) to illustrate te algoritm. ==Table 1 ere== For t = 1, starting from X (1) = X, bot doctors coose to work for te ospital 1, so tey reject teir contracts wit ospital 2 : Tus, R( X (1) ) = {( 1, 2),( 2, 2) } calculated as te complement of R( X (1) ), so X (1) {( d1, 1),( d2, 1) } ospitals coose ( d1, 1) and reject ( d2, 1), completing te row for t = 1. d d. Next, X (1) is =. From X (1), te For t = 2, we first compute X (2) as te complement of R ( X (1)). Te doctors coose {( d, ),( d, )}, so tey reject {( d, )} Te ospitals must ten coose from te complement = { }. Te ospitals coose ( d, ) and ( d, ) and reject (, ) X X d (2) (, ) round, X (3) = X (2) and te process as reaced a fixed point. d. In te 3 rd 2 1 Te example illustrates te monotonicity of te algoritm: X ( t ) grows larger step-by-step wile X ( t ) grows smaller. At termination of te algoritm, te intersection of te coice sets is te stable set of contracts X (3) X (3) {( d, ),( d, )} =. Moreover, wen X ( t ) and X () t are interpreted as suggested above, te process { X ( t), X( t )} described by (9) wit te 18

19 initial conditions X (0) algoritm. = X and X (0) = coincides wit te doctor-offering Gale-Sapley For te ospital offering algoritm, a similar analysis applies but wit a different interpretation of te sets and a different initial condition. Let X ( t ) be te cumulative set of contracts offered by te ospitals to te doctors before iteration t and X ( t ) be te set of contracts tat ave not yet been rejected by te ospitals up to and including iteration t. Ten, te contracts eld at te end of iteration t are precisely tose tat ave been offered but not rejected, wic are tose in X ( t+ 1) X ( t). Wit tis interpretation, te analysis is identical to te one above. Te Gale-Sapley ospital offering algoritm is caracterized by (9) and te initial conditions (0) X = and X (0) = X. Te same logic applies to te Kelso-Crawford model, provided one extends teir original treatment to include a version in wic te workers make offers in addition to te treatment in wic firms make offers. Te words of te preceding paragraps apply exactly, but a contract offer now includes a wage so, for example, a ospital wose contract offer is rejected by a doctor may find tat its next most preferred contract is at a iger wage to te same doctor. B. Wen Contracts are Not Substitutes It is clear from te preceding analysis tat tat definition of substitutes is just sufficient to allow our matematical tools to be applied. In tis section, we establis more. We sow tat unless contracts are substitutes for every ospital, te very existence of a stable set of contracts cannot be guaranteed. 7 19

20 Teorem 5. Suppose contains at least two ospitals, wic we denote by and. Furter suppose tat R is not isotone, tat is, contracts are not substitutes for. Ten, tere exist preference orderings for te doctors in set, a preference ordering for a ospital wit a single job opening suc tat, regardless of te preferences of te oter ospitals, no stable set of contracts exists. Togeter, teorems 3 and 5 caracterize te set of preferences tat can be allowed as inputs into a matcing algoritm if we wis to guarantee tat te outcome of te algoritm is a stable set of contracts wit respect to te reported preferences. According to teorem 3, we can allow all preferences tat satisfy substitutes and still reac an outcome tat is a stable collection of contracts. According to teorem 5, if we allow any preference tat does not satisfy te substitutes condition, ten tere is some profile of singleton preferences for te oter parties suc tat no stable collection of contracts exists. Tis teory also reaffirms and extends te close connection between te substitutes condition and oter concepts tat as been establised in te recent auctions literature wit quasi-linear preferences. Paul Milgrom (2000) studies an auction model wit discrete goods and transfers and in wic bidder values are allowed may be any additive function and may include oter functions as well. 8 e sows tat if goods are substitutes, ten a competitive equilibrium exists. If, owever, tere are at least tree bidders and if tere is any allowed value suc tat te goods are not all substitutes, ten tere is some profile of values suc tat no competitive equilibrium exists. Faruk Gul and Ennio Staccetti (1999) establis te same positive existence result. Tey also sow tat if preferences include all values in wic a bidder wants only one particular good as well as any one for wic goods are not all substitutes, and if te number of bidders is sufficiently large, ten tere is some profile of preferences for wic no competitive 20

21 equilibrium exists. Lawrence Ausubel and Paul Milgrom (2002) establis tat if (i) tere is some bidder for wom preferences are not demand teory substitutes, (ii) values may be any additive function and (iii) tere are at least tree bidders in total, ten tere is some profile of preferences suc tat te Vickrey outcome is not stable and te core imputations do not form a lattice. Conversely, if all bidders ave preferences tat are demand teory substitutes, ten te Vickrey outcome is in te core and te core imputations do form a lattice. Taken togeter, tese results establis a close connection between te substitutes condition, te cooperative concept of te core, te non-cooperative concepts of Vickrey outcomes, and competitive equilibrium. C. Vacancy Cain ynamics Suppose tat a labor market as reaced equilibrium, wit all interested doctors placed at ospitals in a stable matc. Suppose some doctor ten retires. Imagine a process in wic a ospital seeks to replace its retired doctor by raiding oter ospitals to ire additional doctors. If te ospital makes an offer tat would succeed in iring a doctor away from anoter ospital, te affected ospital as tree options: it may make an offer to anoter doctor (or several), improve te terms for its current doctor, or leave te position vacant. Suppose it makes watever contract offer would best serve its purposes. In models wit a fixed number of positions and no contracts, tis process in wic doctors and vacancies move from one ospital to anoter as been called vacancy cain dynamics Yosef Blum, Alvin E. Rot and Uriel Rotblum (1997). Alexander Kelso and Vincent Crawford (1982) consider similar dynamics in te context of teir model. Te formal results for our extended model are similar to tose of te older teories. Starting wit a stable collection of contracts X, let X be te set of contracts tat some doctor 21

22 weakly prefers to er current contract in X and let X X ( X X ) teorem 1, we ave F( X, X ) = ( X, X ) and X = X X. =. As in te proof of To study te dynamic adjustment tat results from te retirement of doctor d, we suppose te process starts from te initial state ( X (0), X (0)) = ( X, X ). Tis means tat te employees start by considering only offers tat are at least as good as teir current positions and tat ospitals remember wic employees ave rejected tem in te past. Te doctors rejection function is canged by te retirement of doctor d to R ˆ, were Rˆ ( X ) = R ( X ) { x X x = d }, tat is, in addition to te old rejections, all contract offers addressed to te retired doctor are rejected. To syncronize te timing wit our earlier notation, let us imagine tat ospitals make offers at round t 1 and doctors accept or reject tem at round t. ospitals consider as potentially available te doctors in X ( t 1) = X Rˆ ( X ( t 1)) and te doctors ten reject all but te best offers, so te cumulative set of offers received is X ( t) = X R ( X ( t 1)). efine: (11) Fˆ( X, X ) = ( X R ( X ), X Rˆ ( X )) If contracts are substitutes for te ospitals, ten ˆF is isotone and, since F( X, X ) = ( X, X ), it follows tat F ˆ ( X, X ) ( X, X ). Ten, since ( X (0), X (0)) = ( X, X ), we ave ( X (1), X (1)) = Fˆ ( X (0), X (0)) ( X (0), X (0)). Iterating, ( X ( n), X ( n)) = Fˆ ( X ( n 1), X ( n 1)) ( X ( n 1), X ( n 1)) : te doctors 22

23 accumulate offers and te contracts tat are potentially available to te ospitals srinks. A fixed point is reaced and, by Teorem 1, it corresponds to a stable collection of contracts. Teorem 6. Suppose tat contracts are substitutes and tat ( X, X ) is a stable set of contracts. Suppose tat a doctor retires and tat te ensuing adjustment process is described by ( X (0), X (0)) = ( X, X ) and ( X ( t), X ( t)) = Fˆ ( X ( t 1), X ( t 1)). Ten, te sequence {( ( ), ( ))} X t X t converges to a stable collection of contracts at wic all te unretired doctors are weakly better off and all te ospitals are weakly worse off tan at te initial state ( X, X ). 9 Te sequence of contract offers and job moves described by iterated applications of ˆF includes possibly complex adjustments. ospitals tat lose a doctor may seek several replacements. ospitals wose doctors receive contract offers may retain tose doctors by offering better terms or may ire a different doctor and later reire te original doctor at a new contract. All along te way, te doctors find temselves coosing from more and better options and te ospitals find temselves marcing down teir preference lists by offering costlier terms, paying iger wages, or making offers to oter doctors wom tey ad earlier rejected. III. Law of Aggregate emand We now introduce a second restriction on preferences tat allows us to prove te next two results about te structure of te set of stable matces. We call tis restriction te law of aggregate demand. Rougly, tis law states tat as te price falls, agents sould demand more of a good. ere, prices falling corresponds to more contracts being available, and demanding more 23

24 corresponds to taking on (weakly) more contracts. We formalize tis intuition wit te following definition. efinition. Te preferences of ospital satisfy te law of aggregate demand if for all X X, C ( X ) C ( X ). According to tis definition, if te set of possible contracts expands (analogous to a decrease in some doctors wages), ten te total number of contracts cosen by ospital eiter rises or stays te same. Te corresponding property for doctor preferences is implied by revealed preference, because eac doctor cooses at most one contract. Just as for te substitutes condition, wen wages are endogenous, we interpret te definition as applying to te domain of wage vectors for wic te ospital s optimum is unique. Te next teorem sows te important relationsip between profit maximization, substitutes, and te law of aggregate demand. Teorem 7. If ospital s preferences are quasilinear and satisfy te substitutes condition, ten tey satisfy te law of aggregate demand. Below, we use te law of aggregate demand to caracterize bot necessary and sufficient conditions for te rural ospitals property and ensure tat trutful revelation is a dominant strategy for te doctors. Previously, in matcing models witout money, te dominant strategy result was known only for responsive preferences wit capacity constraints Atila Abdulkadiroğlu (2003). We subsume tat result wit our teorem. 24

25 A. Rural ospitals Teorem In te matc between doctors and ospitals, certain rural ospitals often ad trouble filling all teir positions, raising te question of weter tere are oter core matces at wic te rural ospitals migt do better. Alvin E. Rot (1986) analyzed tis question for te case of X = and responsive preferences and found tat te answer is no: every ospital tat as unfilled positions at some stable matc is assigned exactly te same doctors at every stable matc. In particular, every ospital ires te same number of doctors at every stable matc. In tis section, we sow by an example tat tis last conclusion does not generalize to te full set of environments in wic contracts are substitutes. 10 We ten prove tat if preferences satisfy te law of aggregate demand and substitutes, ten te last conclusion of Rot s teorem olds: every ospital signs exactly te same number of contracts at every point in te core, altoug te doctors assigned and te terms of employment can vary. Finally, we sow tat any violation of te law of aggregate demand implies preferences exist suc tat te above conclusion does not old. Suppose tat = {, } and { d, d, d } 1 2 =. For ospital 1, suppose its coices maximize, were{ d } { d, d } { d } { d } { d, d } { d, d }. Tis preference satisfies substitutes. 11 Suppose 2 as one position, wit its preferences given by { } { } { } d d d. Finally, suppose d 1 and d 2 prefer 1 to 2 wile d 3 as te reverse preference. Ten, te matces X = {( 1, d3),( 2, d1) } and X {( 1, d1),( 1, d2)(, 2, d3) } = are bot stable but ospital 1 employs a different number of doctors and te set of doctors assigned differs between te two matces. 25

26 Tis example involves a failure of te law of aggregate demand, because as te set of contracts available to 1 expands by te addition of ( 1, d 3) to te set {( 1, 1)(, 1, 2) } d d, te number of doctors demanded declines from two to one. Wen te law of aggregate demand olds, owever, we ave te following result. Teorem 8. If ospital preferences satisfy substitutes and te law of aggregate demand, ten for every stable allocation (, ) X X and every d and, Cd ( X) = Cd ( X) and C ( X ) = C ( X ). Tat is, every doctor and ospital signs te same number of contracts at every stable collection of contracts. Te next teorem verifies tat te counterexamples developed above can always be generalized wenever any ospital s preferences violate te law of aggregate demand. Teorem 9. If tere exists a ospital, sets X X X suc tat C ( X ) C ( X ) >, and at least one oter ospital, ten tere exist singleton preferences for te oter ospitals and doctors suc tat te number of doctors employed by is different for two stable matces. Teorem 9 establises tat te law of aggregate demand is not only a sufficient condition for te rural ospitals result of teorem 8 but, in a particular sense, a necessary one as well. B. Trutful Revelation as a ominant Strategy Te main result of tis section concerns doctors incentives to report teir preferences trutfully. For te doctor-offering algoritm, if ospital preferences satisfy te law of aggregate demand and te substitutes condition, ten it is dominant strategy for doctors to trutfully reveal teir preferences over contracts. 12 We ten furter sow tat bot preference conditions play essential roles in te conclusion. 26

27 We will sow te positive incentive result for te doctor offering algoritm in two steps tat igligt te different roles of te two preference assumptions. First, we sow tat te substitutes condition, by itself, guarantees tat doctors cannot benefit by exaggerating te ranking of an unattainable contract. More precisely, if tere exists a preferences list for a doctor d suc tat d obtains contract x by submitting tis list, ten d can also obtain x by submitting a preference list tat includes only contract x. Second, we will sow tat adding te law of aggregate demand guarantees tat a doctor does at least as well as reporting trutfully as by reporting any singleton. Togeter, tese are te dominant strategy result. To understand wy submitting unattained contracts cannot elp a doctor d, consider te following. Let x be te most-preferred contract tat d can obtain by submitting any preference list (olding all oter submitted preferences fixed). Note tat all tat d accomplises wen reporting tat certain contracts are preferred to x is to make it easier for some coalition to block outcomes involving x. Tus, if x is attainable wit any report, it is attainable wit te report Pd : x tat ranks x as te only acceptable contract. Tis intuition is captured in te following teorem: Teorem 10. Let ospitals preferences satisfy te substitutes condition and let te matcing algoritm produce te doctor-optimal matc. Fixing te preferences of ospitals and of doctors besides d, let x be te outcome tat d obtains by reporting preferences P : z z... z x. Ten, te outcome tat d obtains by reporting preferences P : x is d 1 d 2 d d n d d also x. Some oter doctors may be strictly better off wen d submits er sorter preference list; tere are fewer collections of contracts tat d now objects to, so te core may become larger, and 27

28 te doctor-optimal point of te enlarged core makes all doctors weakly better off and may make some strictly better off. Witout te law of aggregate demand, owever, it may still be in a doctor s interest to conceal er preferences for unattainable positions. To see tis, consider te case wit two ospitals and tree doctors, were contracts are simply elements of, and let preferences be: (12) { } { } { } { } { } { } { } P : P : d d, d d d d P : P : d d d d P : d3 2 1 Wit tese preferences, te only stable matc is {( 1, 2),( 3, 1) } d d, wic leaves d 2 unemployed. owever, if d 2 were to reverse er ranking of te two ospitals, ten {( 1, 1),( 2, 1),( 3, 2) } d d d would be cosen by te doctor-offering algoritm, leaving d 2 better off. Essentially, by offering a contract to 2, d 2 as canged te number of positions available. owever, wen te preferences of te ospitals satisfy te law of aggregate demand, making more offers to te ospitals (weakly) increases te number of contracts te ospitals accept. Teorem 11. Let ospitals' preferences satisfy substitutes and te law of aggregate demand and let te matcing algoritm produce te doctor-optimal matc. Ten, fixing te preferences of te oter doctors and of all te ospitals, let x be te contract tat d obtains by submitting te set of preferences Pd : z1 d z2 d... d zn d x. Ten te preferences P d : y1 y2... yn x yn yn obtain a contract tat is P d -preferred or indifferent to x. 28

29 According to tis teorem, wen a doctor s true preferences are P d, te doctor can never do better according to tese true preferences tan by reporting te preferences trutfully. In fact, te law of aggregate demand is almost a necessary condition as well. Te exceptions can arise because certain violations of te law of aggregate demand are unobservable from te coice data of te algoritm, and tese cannot affect incentives. Tus, consider an example were terms t are included in te contract, and were a ospital as preferences { } {( ) } {( )} {( )} ( ) ( ) d,, t d,, t, d,, t d,, t d,, t. Altoug tese preferences violate te law of aggregate demand, te algoritm will never see te violation, as eiter ( d,, t ) ( d,, t ) or ( d,, t ) ( d,, t ) 1 1 d Tus, wicever terms tat d 1 first offers will 1 1 d1 1 1 determine a conditional set of preferences for te ospital tat do satisfy te law of aggregate demand. (Te ospital will never reject an offer of eiter ( d,, t ) or (,, ) 1 1 d t.) 1 1 Te next teorem says tat if some ospital s preferences violate te law of aggregate demand in a way tat can even potentially be observed from te ospital s coices, ten tere exist preferences for te oter agents suc tat it is not a dominant strategy for doctors to report trutfully, even wen te oter assumptions we ave used are satisfied. ( ) Teorem 12. Let ospital ave preferences suc tat C ( X) C X { x} tere exist two contracts y, z suc tat y z x y > and let and yz, R( X { x} ) R( X). Ten if anoter ospital exists, tere exist singleton preferences for te ospitals besides and preferences for te doctors suc tat it is not a dominant strategy for all doctors to reveal teir preferences trutfully. 29

30 Tus, to te extent tat te law of aggregate demand for ospital preferences as observable consequences for te progress of te doctor-offering algoritm, it is an indispensable condition to ensure te dominant strategy property for doctors. IV. Classes of Conforming Preferences Altoug te class of substitutes valuations is quite limited, 13 it is broader tan te set of responsive preferences in useful ways. Te substitutes valuations accommodate all te affirmative action and subspecialty constraints described above and allow a doctor s marginal product to depend on wic oter doctors te ospital attracts. In tis section, we introduce a parameterized class of quasi-linear substitutes valuations for ospitals evaluating sets of new ires. Te valuations are based on an endowed assignment model, according to wic ospitals ave a set of jobs to fill and an existing endowment of doctors, wile doctors productivities vary among jobs. A ospital values new doctors according to te teir incremental value in te assignment problem. From a general job assignment perspective, te key restrictions of endowed assignment valuations are tat eac doctor can do one and only one job and tat te ospital s output is te sum of outputs of its various jobs. A ospital cannot use one doctor for two jobs nor can it combine te skills of two doctors in te same job. Also, tere is no interaction in te productivity of doctors in different jobs. Given tis matematical structure, if a doctor is lost, it will always be optimal for te ospital to retain all of its oter doctors, altoug it may coose to reassign some of te retained doctors to fill vacated positions. Anoter way to describe te set of endowed assignment valuations, V EA, is to build it up from tree properties, as follows. First, a singleton valuation is one of te form 30

31 vs ( ) = max d S α d for some non-negative vector α. Singleton valuations represent te possible valuations by a ospital wit just one opening, and V EA includes all singleton valuations. Second, if a ospital is composed from two units, j=1,2, eac of wic as a valuation vj V ospital s maximum value from assigning workers between its units, denoted by EA, ten te ( v1 v2)( S) max v 1( R ) v R S 2( S R + ), is also in te set. We call tis property closure under aggregation. Tird, if a ospital s value is derived from a value in V EA by endowing te ospital wit a set of doctors T, ten te ospital s incremental value for extra doctors vs ( T) vs ( T) vt ( ) is also in te set V EA. We call tis property closure under endowment. Finally, two valuations v 1 and v 2 are called equivalent if tey differ by an additive constant. For example, for any fixed set of doctors T, v( i T) is equivalent to v( i T). Teorem 13. Let V EA denote te smallest family of valuations of sets of doctors tat includes all te singleton valuations and is closed under aggregation and endowment. Ten, for eac v V, tere exists a set of jobs, J, a set of doctors T, and a ( + T ) J -matrix [ α ], EA ij suc tat v is equivalent to te following: (13) vs ( J, α, T) max α z subject to z dj z d S d S dj z 1 for j J dj 1 for d S T z j J dj 0 for d S T {0,1} for all d, j dj Conversely, for every suc J, T, and α, v( i J, α, T) VEA. We dub tis family of valuations te endowed assignment valuations because of te form of optimization (13). Te family V EA is an extension of te assignment valuations derived by L. Sapley (1962), used by Alvin E. Rot (1985b, (1984) in models of matcing wit wages and 31