Matching with Contracts
By JOHN WILLIAM HATFIELD AND PAUL R. MILGROM*
We develop a model of matching with contracts which incorporates, as special cases, the college admissions problem, the Kelso-Crawford labor market matching model, and ascending package auctions. We introduce a new “law of aggregate demand” for the case of discrete heterogeneous workers and show that, when workers are substitutes, this law is satisfied by profit-maximizing firms. When workers are substitutes and the law is satisfied, truthful reporting is a dominant strategy for workers in a worker-offering auction/matching algorithm. We also parameterize a large class of preferences satisfying the two conditions. (JEL C78, D44)
Since the pioneering U.S. spectrum auctions Matching algorithms based on economic the- of 1994 and 1995, related ascending multi-item ory also have important practical applications. auctions have been used with much fanfare on Alvin E. Roth and Elliott Peranson (1999) ex- six continents for sales of radio spectrum and plain how a certain two-sided matching proce- electricity supply contracts.1 Package bidding, dure, which is similar to the college admissions in which bidders can place bids not just for algorithm introduced by David Gale and Lloyd individual lots but also for bundles of lots Shapley (1962), has been adapted to match (“packages”), has found increasing use in pro- 20,000 doctors per year to medical residency curement applications (Milgrom, 2004; Peter programs. After Atila Abdulkadirog˘lu and Tay- Cramton et al., 2005). Recent proposals in the fun So¨nmez (2003) advocated a variation of the United States to allow package bidding for same algorithm for use by school choice pro- spectrum licenses and for airport landing rights grams, a similar centralized match was adopted incorporate ideas suggested by Lawrence Aus- by the New York City schools (Abdulkadirog˘lu ubel and Milgrom (2002) and by David Porter et et al., 2005b) and another is being evaluated by al. (2003) (see Ausubel et al., 2005). the Boston schools (Abdulkadirog˘lu et al., 2005a). This paper identifies and explores certain similarities among all of these auction and * Hatfield: Graduate School of Business, Stanford Uni- versity, Stanford, CA 94305 (e-mail: hatfield@stanford. matching mechanisms. To illustrate one simi- edu); Milgrom: Department of Economics, Stanford larity, consider the labor market auction model University, Stanford, CA 94305 (e-mail: paul@milgrom. of Alexander Kelso and Vincent Crawford net). We thank Atila Abdulkadirog˘lu, Federico Echenique, (1982), in which firms bid for workers in simul- Daniel Lehmann, Jon Levin, and Alvin Roth for helpful discussions. This research is supported by National Science taneous ascending auctions. The Kelso-Crawford Foundation Grant No. SES-0239910. John Hatfield has also model assumes that workers have preferences been supported by the Burt and Deedee McMurtry Stanford over firm-wage pairs and that all wage offers are Graduate Fellowship. drawn from a prespecified finite set. If that set 1 For example, a New York Times article about a spec- includes only one wage, then all that is left for trum auction in the United States was headlined “The Great- est Auction Ever” (New York Times, March 16, 1995, p. the auction to determine is the match of workers A17). The scientific community has also been enthusiastic. to firms, so the auction is effectively trans- In its fiftieth anniversary self-review, the U.S. National formed into a matching algorithm. The auction Science Foundation reported that “from a financial stand- algorithm begins with each firm proposing em- point, the big payoff for NSF’s longstanding support [of auction theory research] came in 1995 ... [when t]he Federal ployment to its most preferred set of workers at Communications Commission established a system for us- the one possible wage. When some workers turn ing auctions.” See Milgrom (2004). it down, the firm makes offers to other workers
913 914 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005 to fill its remaining openings. This procedure is possibility of monetary transfers is excluded precisely the hospital-offering version of the from those formulations. As discussed below, Gale-Shapley matching algorithm. Hence, the the quasi-linearity assumption combines with Gale-Shapley matching algorithm is a special the substitutes assumption in a subtle and re- case of the Kelso-Crawford procedure. strictive way. The possibility of extending the National This paper presents a new model that sub- Resident Matching Program (the “Match”) to sumes, unifies, and extends the models cited permit wage competition is an important con- above. The basic unit of analysis in the new sideration in assessing public policy toward the model is the contract. To reproduce the Gale- Match, particularly because work by Jeremy Shapley college admissions problem, we spec- Bulow and Jonathan Levin (2003) lends some ify that a contract is fully identified by the theoretical support for the position that the student and college; other terms of the relation- Match may compress and reduce doctors’ ship depend only on the parties’ identities. To wages relative to a perfectly competitive stan- reproduce the Kelso-Crawford model of firms dard. The practical possibility of such an exten- bidding for workers, we specify that a contract sion depends on many details, including, is fully identified by the firm, the worker, and importantly, the form in which doctors and hos- the wage. Finally, to reproduce the Ausubel- pitals would have to report their preferences for Milgrom model of package bidding, we specify use in the Match. In its current incarnation, the that a contract is identified by the bidder, the Match can accommodate hospital preferences package of items that the bidder will acquire, that encompass affirmative action constraints and the price to be paid for that package. Sev- and a subtle relationship between internal med- eral additional variations can be encompassed icine and its subspecialties, so it will be impor- by the model. For example, Roth (1984, 1985b) tant for any replacement algorithm to allow allows that a contract might specify the partic- similar preferences to be expressed. In Section ular responsibilities that a worker will have IV, we introduce a parameterized family of val- within the firm. uations that accomplishes that. Our analysis of matching models emphasizes A second important similarity is between the two conditions that restrict the preferences of Gale-Shapley doctor-offering algorithm and the the firms/hospitals/colleges: a substitutes condi- Ausubel-Milgrom proxy auction. Explaining tion and an law of aggregate demand condition. this relationship requires restating the algorithm We find that these two conditions are implied by in a different form from the one used for the the assumptions of earlier analyses, so our uni- preceding comparison. We show that if the hos- fied treatment implies the central results of pitals in the Match consider doctors to be sub- those theories as special cases. stitutes, then the doctor-offering algorithm is In the tradition of demand theory, we define equivalent to a certain cumulative offer process substitutes by a comparative statics condition. in which the hospitals at each round can choose In demand theory, the exogenous parameter from all the offers they have received at any change is a price decrease, so the challenge is to round, current or past. In a different environ- extend the definition to models in which there ment, where there is but a single “hospital” or may be no price that is allowed to change. In auctioneer with unrestricted preferences and our contracts model, a price reduction corre- general contract terms, this cumulative offer sponds formally to expanding the firm’s oppor- process coincides exactly with the Ausubel- tunity set, i.e., to making the set of feasible Milgrom proxy auction. contracts larger. Our substitutes condition as- Despite the close connections among these serts that when the firm chooses from an ex- mechanisms, previous analyses have treated panded set of contracts, the set of contracts it them separately. In particular, analyses of auc- rejects also expands (weakly). As we will show, tions typically assume that bidders’ payoffs are this abstract substitutes condition coincides ex- quasi-linear. No corresponding assumption is actly with the demand theory condition for stan- made in analyzing the medical match or the dard models with prices. It also coincides college admissions problem; indeed, the very exactly with the “substitutable preferences” VOL. 95 NO. 4 HATFIELD AND MILGROM: MATCHING WITH CONTRACTS 915 condition for the college admissions problem prove some new results for the class of auction (Roth and Marilda Sotomayor, 1990). and matching models that satisfy this law. The law of aggregate demand, which is new The paper is organized as follows. Section I in this paper, is similarly defined by a compar- introduces the matching-with-contracts nota- ative static. It is the condition that when a tion, treats an allocation as a set of contracts, college or firm chooses from an expanded set, it and characterizes the stable allocations in terms admits at least as many students or hires at least of the solution of a certain system of two as many workers.2 equations. The term “law of aggregate demand” is mo- Section II introduces the substitutes condition tivated by the relation of this condition to the and uses it to prove that the set of stable allo- law of demand in producer theory. According to cations is a nonempty lattice, and that a certain producer theory, a profit-maximizing firm de- generalization of the Gale-Shapley algorithm mands (weakly) more of any input as its price identifies its maximum and minimum elements. falls. For the matching model with prices, the These two extreme points are characterized as a law of aggregate demand requires that when doctor-optimal/hospital-pessimal point, which any input price falls, the aggregate quantity is a point that is the unanimously most preferred demanded, which includes the quantities de- stable allocation for the doctors and the unani- manded of that input and all of its substitutes, mously least preferred stable allocation for the rises (weakly). Notice that it is tricky even to hospitals; and a hospital-optimal/doctor-pessimal state such a law in producer theory with divis- point with the reverse attributes. ible inputs, because there is no general aggre- Section II also proves several related results. gate quantity measure when divisible inputs are First, if there are at least two hospitals and if diverse. In the present model with indivisible some hospital has preferences that do not satisfy workers, we measure the aggregate quantity of the substitutes condition, then even if all other workers demanded or hired by the total number hospitals have just a single opening, there exists of such workers. a profile of preferences for the students and A key step in our analysis is to prove a new colleges such that no stable allocation exists. result in demand theory: if workers are substi- This result is important for the construction of tutes, then a profit-maximizing firm’s employ- matching algorithms. It means that any match- ment choices satisfy the law of aggregate ing procedure that permits students and colleges demand. Since firms are profit maximizers and to report preferences that do not satisfy the regard workers as substitutes in the Kelso- substitutes condition cannot be guaranteed al- Crawford model, it follows that the law of ag- ways to select a stable allocation with respect to gregate demand holds for that model. Thus, one the reported preferences. implication of the standard quasi-linearity as- Another result concerns vacancy chain dy- sumption of auction theory is that the bidders’ namics, which traces the dynamic adjustment of preferences satisfy the law of aggregate de- the labor market when a worker retires or a new mand. We find that when preferences are re- worker enters the market and the dynamics are sponsive as originally and still commonly represented by the operator we have described. assumed in matching theory analyses, they sat- The analysis extends the results reported by isfy the law of aggregate demand.3 We also Yosef Blum et al. (1997) and foreshadowed by Kelso and Crawford (1982). We find that, start- ing from a stable allocation, the vacancy adjust- ment process converges to a new stable 2 In their study of a model of “schedule matching,” Ahmet Alkan and Gale (2003) independently introduced a allocation. similar notion, which they call “size monotonicity.” Section III contains the most novel results of 3 The original matching formulation of Gale and Shapley the paper. It introduces the law of aggregate (1962) assumed that colleges simply have a rank order listing of students. Roth (1985a) dubbed this “responsive- ness” and sought to relax this condition. Similarly, in a model of matching with wages, Crawford and E. M. Knoer separable. That restriction was relaxed by Kelso and Craw- (1981) assumed that firm preferences over workers were ford (1982). 916 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005 demand, verifies that it holds for a profit- algorithm of the previous sections. Since the maximizing firm when inputs are substitutes, Ausubel-Milgrom proxy auction is also a cumu- and explores its consequences. When both the lative offer process, our dominant strategy con- substitutes and the law of aggregate demand clusion of Section III implies an extension of conditions are satisfied, then (a) the set of work- the Ausubel-Milgrom dominant strategy result. ers employed and the set of jobs filled is the When contracts are not substitutes, the cumula- same at every stable collection of contracts; and tive offer process can converge to an infeasible (b) it is a dominant strategy for doctors (or allocation. We show that if there is a single workers or students) to report their preferences hospital/auctioneer, however, the cumulative truthfully in the doctor-offering version of the offer process converges to a feasible, stable extended Gale-Shapley algorithm. We also allocation, without restrictions on the hospital’s demonstrate the necessity of a weaker version preferences. Section VI concludes. All proofs of the law of aggregate demand for these are in the Appendix. conclusions. Our conclusion about this dominant strategy I. Stable Collections of Contracts property substantially extends earlier findings about incentives in matching. The first such The matching model without transfers has results, due to Lester Dubins and David Freed- many applications, of which the best known man (1981) and Roth (1982), established the among economists is the match of doctors to dominant strategy property for the marriage hospital residency programs in the United problem, which is a one-to-one matching prob- States. For the remainder of the paper, we de- lem that is a special case of the college admis- scribe the match participants as “doctors” and sions problem. Similarly, Gabrielle Demange “hospitals.” These groups play the same respec- and Gale (1985) establish the dominant strategy tive roles as students and colleges in the college property for the worker-firm matching problem admissions problem and workers and firms in in which each firm has singleton preferences. the Kelso-Crawford labor market model. These results generalize to the case of respon- sive preferences, that is, to the case where each A. Notation hospital (or college or firm) behaves just the same as a collection of smaller hospitals with The sets of doctors and hospitals are denoted one opening each. For the college admissions by D and H, respectively, and the set of con- problem, Abdulkadirog˘lu (2003) has shown that tracts is denoted by X. We assume only that the dominant strategy property also holds when each contract x ʦ X is bilateral, so that it is ʦ colleges have responsive preferences with ca- associated with one “doctor” xD D and one ʦ pacity constraints, where the constraints limit “hospital” xH H. When all terms of employ- the number of workers of a particular type that ment are fixed and exogenous, the set of con- can be hired. All of these models with a domi- tracts is just the set of doctor-hospital pairs: X ϵ nant strategy property satisfy our substitutes D ϫ H. For the Kelso-Crawford model, a con- and law of aggregate demand conditions, so the tract specifies a firm, a worker, and a wage, X ϵ earlier dominant strategy results are all sub- D ϫ H ϫ W. sumed by our new result. Each doctor d can sign only one contract. Her In Section IV, we introduce a new parame- preferences over possible contracts, including terized family of preferences—the endowed as- the null contract A, are described by the total ՝ signment preferences—and show that they order d. The null contract represents unem- subsume certain previously identified classes ployment, and contracts are acceptable or un- and satisfy both the substitutes and law of ag- acceptable according to whether they are more gregate demand conditions. preferred than A. When we write preferences as ՝ ՝ Section V introduces cumulative offer pro- Pd : x d y d z, we mean that Pd names the cesses as an alternative auction/matching algo- preference order of d and that the listed con- rithm and shows that when contracts are tracts (in this case, x, y, and z) are the only substitutes, it coincides with the doctor-offering acceptable ones. VOL. 95 NO. 4 HATFIELD AND MILGROM: MATCHING WITH CONTRACTS 917
Given a set of contracts XЈ ʚ X offered in the We formalize the notion of stable allocations Ј market, doctor d’s chosen set Cd(X ) is either as follows: the null set, if no acceptable contracts are of- fered, or the singleton set consisting of the most DEFINITION: A set of contracts XЈ ʚ Xisa preferred contract. We formalize this as follows: stable allocation if
(i) C (XЈ) ϭ C (XЈ) ϭ XЈ and (1) C ͑XЈ͒ D H d (ii) there exists no hospital h and set of con- Љ Ј tracts X Ch(X ) such that A if ͕x ʦ XЈ͉xD ϭ d, x ՝d A͖ ϭ A ϭ ͕ͭ ͕ ʦ Ј͉ ϭ ͖͖ max՝d x X xD d otherwise. XЉ ϭ Ch ͑XЈ ഫ XЉ͒ ʚ CD ͑XЈ ഫ XЉ͒.
The choices of a hospital h are more compli- If condition (i) fails, then some doctor or ՝ cated, because it has preferences h over sets of hospital prefers to reject some contract; that doctors. Its chosen set is a subset of the con- doctor or hospital then blocks the allocation. If Ј ʚ ʦ tracts that name it, that is, Ch(X ) {x condition (ii) fails, then there is an alternative Ј͉ ϭ X xH h}. In addition, a hospital can sign only set of contracts that a hospital strictly prefers one contract with any given doctor: and that its corresponding doctors weakly prefer. The first theorem states that a set of contracts (2) ͑ ᭙ h ʦ H͒͑ ᭙ XЈ ʚ X͒͑ ᭙ x, xЈ ʦ Ch ͑XЈ͒͒x is stable if any alternative contract would be rejected by some doctor or some hospital from xЈ f xD xDЈ . its suitably defined opportunity set. In the for- mulas below, think of the doctors’ opportunity Ј ϭ ഫ Ј Let CD(X ) dʦD Cd(X ) denote the set of set as XD and the hospitals’ opportunity set as Ј Ј contracts chosen by some doctor from set X . XH.IfX is the corresponding stable set, then XD The remaining offers in XЈ are in the rejected must include, in addition to XЈ, all contracts that Ј ϭ ЈϪ Ј set: RD(X ) X CD(X ). Similarly, the hos- would not be rejected by the hospitals, and XH pitals’ chosen and rejected sets are denoted by must similarly include XЈ and all contracts that Ј ϭ ഫ Ј Ј ϭ ЈϪ Ј CH(X ) hʦH Ch(X ) and RH(X ) X would not be rejected by the doctors. If X is Ј CH(X ). stable, then every alternative contract is rejected ϭ ഫ by somebody, so X XH XD. This logic is B. Stable Allocations: Stable Sets of summarized in the first theorem. Contracts ʚ 2 THEOREM 1: If (XD, XH) X is a solution to In our model, an allocation is a collection of the system of equations contracts, as that determines the payoffs to the participants. We study allocations such that (3) X ϭ X Ϫ R ͑X ͒ there is no alternative allocation that is strictly D H H preferred by some hospital and weakly pre- ferred by all of the doctors that it hires, and such and that no doctor strictly prefers to reject his con- tract. It is a standard observation in matching XH ϭ X Ϫ RD ͑XD ͒, theory that such an allocation is a core alloca- പ tion in the sense that no coalition of hospitals then XH XD is a stable set of contracts and പ ϭ ϭ and doctors can find another allocation, feasible XH XD CD(XD) CH(XH). Conversely, for for them, that all weakly prefer and some any stable collection of contracts XЈ, there ex- strictly prefer. This allocation is also a stable ists some pair (XD, XH) satisfying (3) such that Јϭ പ allocation in the sense that there is no coalition X XH XD. that can deviate profitably, even if the deviating coalition assumes that outsiders will remain Theorem 1 is formulated to apply to general willing to accept the same contracts. sets of contracts. It is the basis of our analysis of 918 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005 stable sets of contracts in the entire set of mod- doctors and W ϭ {wគ , ... , w } is a finite set of els treated in this paper. possible wages. Assume that the set of wages W is such that the hospitals’ preferences are al- II. Substitutes ways strict. Suppose that w ϭ max W is a prohibitively high wage, so that no hospital ever In this section, we introduce our first restric- hires a doctor at wage w . tion on hospital preferences, which is the re- In demand theory, it is standard to represent striction that contracts are substitutes. We use the hospital’s market opportunities by a vector ʦ D the restriction to prove the existence of a stable w W that specifies a wage wd at which set of contracts and to study an algorithm that each doctor can be hired. We can extend the identifies those contracts. domain of the choice function C to allow market Our substitutes condition generalizes the opportunities to be expressed by wage vectors, Roth-Sotomayor substitutable preferences con- as follows: dition to preferences over contracts. In words, D the substitutable preferences condition states (4) w ʦ W f C͑w͒ ϵ C͕͑͑d, wd ͉͒d ʦ D͖͒. that if a contract is chosen by a hospital from some set of available contracts, then that con- Formula (4) associates with any wage vector w ͉ ʦ tract will still be chosen from any smaller set the set of contracts {(d, wd) d D} and defines that includes it. Our substitutes condition is C(w) to be the choice from that set. similarly defined as follows: With the choice function extended this way, we can now describe the traditional demand DEFINITION: Elements of X are substitutes theory substitutes condition. The condition as- for hospital h if for all subsets XЈ ʚ XЉ ʚ Xwe serts that increasing the wage of doctor d from Ј ʚ Љ 4 Ј have Rh(X ) Rh(X ). wd to wd (weakly) increases demand for any other doctor dЈ. In the language of lattice theory, which we use below, elements of X are substitutes for DEFINITION: C satisfies the demand-theory Ј Ј hospital h exactly when the function Rh is substitutes condition if (i) d d ,(ii)(d , ʦ Ј Ͼ Ј isotone. wdЈ) C(w) and (iii) wd wd imply that (d , ʦ Ј In demand theory, substitutes is defined by a wdЈ) C(wd, wϪd). comparative static that uses prices. It says that, limiting attention to the domain of wage vectors To compare the two conditions, we need to at which there is a unique optimum for the be able to assign a vector of wages to each set hospital, the hospital’s demand for any doctor d of contracts XЈ. It is possible that, in XЈ, some is nondecreasing in the wage of each other doctor is unavailable at any wage or is available at doctor dЈ. several different wages. For a profit-maximizing Our next result verifies that for resource al- hospital, the doctor’s relevant wage is the low- location problems involving prices, our defini- est wage, if any, at which she is available. tion of substitutes coincides with the standard Moreover, such a hospital does not distinguish demand theory definition. For simplicity, we between a doctor who is unavailable and one focus on a single hospital and suppress its iden- who is available only at a prohibitively high tifier h from our notation. Thus, imagine that a wage. Thus, from the perspective of a profit- hospital chooses doctors’ contracts from a sub- maximizing hospital, having contracts XЈ avail- set of X ϭ D ϫ W, where D is a finite set of able is equivalent to facing a wage vector Wˆ (XЈ) specified as follows:
4 This condition is identical in form to the “condition (5) Wˆ d ͑XЈ͒ ϭ min͕s͉s ϭ w or ͑d, s͒ ʦ XЈ͖. alpha” used in the study of social choice, for example by Amartya Sen (1970), but the meanings of the choice sets are different. In Sen’s model, the choice set is a collection of In view of the preceding discussion, a profit- mutually exclusive choices, whereas in the matching prob- maximizing hospital’s choices must obey the lems, the set describes the choice itself. following identity: VOL. 95 NO. 4 HATFIELD AND MILGROM: MATCHING WITH CONTRACTS 919
(6) C͑XЈ͒ ϭ C͑Wˆ ͑XЈ͒͒. applications of F, starting from the minimum and maximum points of X ϫ X, converge mono- THEOREM 2: Suppose that X ϭ D ϫ Wisa tonically to a fixed point of F.5 We summarize finite set of doctor-wage pairs and that (6) the particular application here with the follow- holds. Then C satisfies the demand theory sub- ing theorem.6 stitutes condition if and only if its contracts are substitutes. THEOREM 3: Suppose contracts are substi- tutes for the hospitals. Then: In particular, this shows that the Kelso- Crawford “gross substitutes” condition is sub- (a) The set of fixed points of F on X ϫ Xisa sumed by our substitutes condition. nonempty finite lattice, and in particular includes a smallest element (Xគ , Xគ ) and a D H A. A Generalized Gale-Shapley Algorithm largest element (XD, XH); ϭ A (b) Starting at (XD, XH) (X, ), the general- We now introduce a monotonic algorithm ized Gale-Shapley algorithm converges that will be shown to coincide with the Gale- monotonically to the highest fixed point Shapley algorithm on its original domain. To describe the monotonicity that is found in the ͑XD , XH ͒ algorithm, let us define an order on X ϫ X as follows: ϭ sup͕͑XЈ, XЉ͉͒F͑XЈ, XЉ͒ Ն ͑XЈ, XЉ͖͒; and
(7) ͑͑X , X ͒ Ն ͑XЈ , XЈ ͒͒ ϭ A D H D H (c) Starting at (XD, XH) ( , X), the general- ized Gale-Shapley algorithm converges N ͑ ʛ Ј ʚ Ј ͒ XD XD and XH XH . monotonically to the lowest fixed point ϫ Ն With this definition, (X X, ) is a finite ͑Xគ D , Xគ H ͒ lattice. The generalized Gale-Shapley algorithm is ϭ inf͕͑XЈ, XЉ͉͒F͑XЈ, XЉ͒ Յ ͑XЈ, XЉ͖͒. defined as the iterated applications of a certain ϫ 3 ϫ function F : X X X X, as defined below. The facts that (XD, XH) is the highest fixed គ គ point of F and that (XD, XH) is the lowest in the (8) F ͑XЈ͒ ϭ X Ϫ R ͑XЈ͒ specified order mean that for any other fixed 1 H គ ʚ ʚ ʚ ʚ point (XD, XH), XD XD XD and XH XH គ F ͑XЈ͒ ϭ X Ϫ R ͑XЈ͒ XH. Because doctors are better off when they 2 D can choose from a larger set of contracts, it ͑ ͒ ϭ ͑ ͑ ͒ ͑ ͑ ͒͒͒ F XD , XH F1 XH , F2 F1 XH .
As we have previously observed, since the doc- 5 This special case of Tarski’s fixed point theorem can be simply proved as follows: Let Z be a finite lattice with tors’ choices are singletons, a revealed prefer- ϭ ϭ ϭ maximum point z. Let z0 z, z1 F(z0), ... , zn F(znϪ1). ence argument establishes that the function R : Յ ϭ Յ D Plainly, z1 z0 and, since F is isotone, z2 F(z1) 3 ϭ Յ X X is isotone. If the contracts are substitutes F(z0) z1 and similarly znϩ1 zn for all n. So, the 3 decreasing sequence {z } converges in a finite number of for the hospitals, then the function RH : X X n steps to a point zˆ with F(zˆ) ϭ zˆ. Moreover, for any fixed is isotone as well. When both are isotone, the Յ ϭ n n ϫ Ն 3 ϫ Ն point z˜, since z˜ z, z˜ F (z˜) Յ F (z) ϭ zˆ for n large, so function F :(X X, ) (X X, )is zˆ is the maximum fixed point. A similar argument applies Ն also isotone, that is, it satisfies ((XD, XH) for the minimum fixed point. Ј Ј f Ն Ј Ј 6 (X D, X H)) (F(XD, XH) F(X D, X H)). Hiroyuki Adachi (2000) and Federico Echenique and Thus, F : X ϫ X 3 X ϫ X is an isotone Jorge Oviedo (2004) have also characterized the Gale- Shapley algorithm for the marriage problem and the college function from a finite lattice into itself. Using admissions problem, respectively, as iterated applications of fixed point theory for finite lattices, the set of a certain operator on a lattice. See also Tama´s Fleiner fixed points is a nonempty lattice, and iterated (2003). 920 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005 follows that the doctors unanimously weakly (9) X ͑t͒ ϭ X Ϫ R ͑X ͑t Ϫ 1͒͒ គ D H H prefer CD(XD)toCD(XD)toCD(XD) and simi- larly that the hospitals unanimously prefer X ͑t͒ ϭ X Ϫ R ͑X ͑t͒͒. គ H D D CH(XH)toCH(XH)toCH(XH). Notice, by ϭ ϭ പ Theorem 1, that CD(XD) CH(XH) XD គ ϭ គ ϭ គ പ គ Ϫ XH and CD(XD) CH(XH) XD XH,so After offers have been made in iteration t 1, Ϫ we have the following welfare conclusion. the hospital’s cumulative set of offers is XH(t 1). Each hospital h holds onto the nh best offers THEOREM 4: Suppose contracts are substi- it has received at any iteration, provided that tutes for the hospitals. Then, the stable set of many acceptable offers have been made; other- പ contracts XD XH is the unanimously most wise, it holds all acceptable offers that have preferred stable set for the doctors and the been made. Thus, the accumulated set of re- Ϫ unanimously least preferred stable set for the jected offers is RH(XH(t 1)) and the unre- គ പ គ Ϫ Ϫ ϭ hospitals. Similarly, the stable set XD XH is jected offers are those in X RH(XH(t 1)) the unanimously most preferred stable set for XD(t). At round t, if a doctor’s contract is being the hospitals and the unanimously least pre- held, then the last offer the doctor made was its ferred stable set for the doctors. best contract in XD(t). If a doctor’s last offer was rejected, then its new offer is its best con- Theorems 3 and 4 duplicate and extend fa- tract in XD(t). The contracts that doctors have miliar conclusions about stable matches in the not offered at this round or any earlier one are Gale-Shapley matching problem and a similar therefore those in RD(XD(t)). So, the accumu- conclusion about equilibrium prices in the lated set of offers doctors have made are those Ϫ ϭ Kelso-Crawford labor market model. These in X RD(XD(t)) XH(t). Once a fixed point of new theorems encompass both these older mod- this process is found, we have, by Theorem 1, a പ els, and additional ones with general contract stable set of contracts XH(t) XD(t). terms. To illustrate the algorithm, consider a simple To see how the Gale-Shapley algorithm is example with two doctors and two hospitals, encompassed, consider the doctor-offering al- where X ϵ D ϫ H and agents have the follow- gorithm. As in the original formulation, we sup- ing preferences: pose that hospitals have a ranking of doctors ՝ that is independent of the other doctors they will (10) Pd1 : h1 h2 hire, so hospital h just chooses its nh most preferred doctors (from among those who are ͕ ͖ ՝ ͕ ͖ ՝ A Ph1 : d1 d2 acceptable and have proposed to hospital h). Let X (t) be the cumulative set of contracts ՝ H Pd : h1 h2 offered by the doctors to the hospitals through 2 iteration t, and let X (t) be the set of contracts D P ͕d d ͖ ՝ ͕d ͖ ՝ ͕d ͖ ՝ A that have not yet been rejected by the hospitals h2 : 1 , 2 1 2 . through iteration t. Then, the contracts “held” at ϭ the end of the iteration are precisely those that The algorithm is initialized with XD(0) ϭ have been offered but not rejected, which are X {(d1, h1), (d1, h2), (d2, h1), (d2, h2)} and പ ϭ A those in XD(t) XH(t). The process initiates XH(0) . Table 1 applies the operator given with no offers having been made or rejected, so in (9) to illustrate the algorithm. ϭ ϭ A ϭ ϭ XD(0) X and XH(0) . For t 1, starting from XD(1) X, both Iterated applications of the operator F de- doctors choose to work for the hospital h1,so scribed above define a monotonic process, in they reject their contracts with hospital h2: thus, ϭ which the set of doctors makes an ever-larger RD(XD(1)) {(d1, h2), (d2, h2)}. Next, XH(1) is (accumulated) set of offers and the set of unre- calculated as the complement of RD(XD(1)), so ϭ jected offers grows smaller round by round. XH(1) {(d1, h1), (d2, h1)}. From XH(1), the Using the specification of F and starting from hospitals choose (d1, h1) and reject (d2, h1), the extreme point (X, A), we have completing the row for t ϭ 1. VOL. 95 NO. 4 HATFIELD AND MILGROM: MATCHING WITH CONTRACTS 921
TABLE 1—ILLUSTRATION OF THE ALGORITHM
XD(t) RD(XD(t)) XH(t) RH(XH(t)) ϭ AA t 0 X {(d1, h2), (d2, h2)} ϭ t 1 X {(d1, h2), (d2, h2)} {(d1, h1), (d2, h1)} {(d2, h1)} ϭ t 2{(d1, h1), (d1, h2), (d2, h2)} {(d1, h2)} {(d1, h1), (d2, h1), (d2, h2)} {(d2, h1)} ϭ t 3{(d1, h1), (d1, h2), (d2, h2)} {(d1, h2)} {(d1, h1), (d2, h1), (d2, h2)} {(d2, h1)}
ϭ For t 2, we first compute XD(2) as the by a doctor may find that its next most preferred complement of RH(XH(1)). The doctors choose contract to the same doctor is at a higher wage. {(d1, h1), (d2, h2)}, so they reject {(d1, h2)}. The hospitals must then choose from the comple- B. When Contracts Are Not Substitutes ϭ Ϫ ment XH(2) X {(d1, h2)}. The hospitals choose (d1, h1) and (d2, h2) and reject (d2, h1). In It is clear from the preceding analysis that ϭ the third round, XD(3) XD(2) and the process that definition of substitutes is just sufficient to has reached a fixed point. allow our mathematical tools to be applied. In The example illustrates the monotonicity of this section, we establish more. We show that the algorithm: XH(t) grows larger step by step, unless contracts are substitutes for every hospi- while XD(t) grows smaller. At termination of the tal, the very existence of a stable set of contracts algorithm, the intersection of the choice sets is cannot be guaranteed.7 പ ϭ the stable set of contracts XH(3) XD(3) {(d1, h1), (d2, h2)}. Moreover, when XD(t) and THEOREM 5: Suppose H contains at least two Ј XH(t) are interpreted as suggested above, the hospitals, which we denote by h and h . Further process {XD(t), XH(t)} described by (9) with the suppose that Rh is not isotone, that is, contracts ϭ ϭ A initial conditions XD(0) X and XH(0) are not substitutes for h. Then there exist pref- coincides with the doctor-offering Gale- erence orderings for the doctors in set D, a Shapley algorithm. preference ordering for a hospital hЈ with a For the hospital-offering algorithm, a similar single job opening such that, regardless of the analysis applies, but with a different interpreta- preferences of the other hospitals, no stable set tion of the sets and a different initial condition. of contracts exists. Let XD(t) be the cumulative set of contracts offered by the hospitals to the doctors before Together, Theorems 3 and 5 characterize the iteration t, and let XH(t) be the set of contracts set of preferences that can be allowed as inputs that have not yet been rejected by the hospitals into a matching algorithm if we wish to guar- up to and including iteration t. Then, the con- antee that the outcome of the algorithm is a tracts “held” at the end of iteration t are pre- stable set of contracts with respect to the re- cisely those that have been offered but not ported preferences. According to Theorem 3, ϩ പ rejected, which are those in XD(t 1) XH(t). we can allow all preferences that satisfy substi- With this interpretation, the analysis is identical tutes and still reach an outcome that is a stable to the one above. The Gale-Shapley hospital collection of contracts. According to Theorem offering algorithm is characterized by (9) and 5, if we allow any preference that does not ϭ A ϭ the initial conditions XD(0) and XH(0) X. satisfy the substitutes condition, then there is The same logic applies to the Kelso-Crawford some profile of singleton preferences for the model, provided one extends their original treat- other parties such that no stable collection of ment to include a version in which the workers contracts exists. make offers in addition to the treatment in which firms make offers. The words of the preceding paragraphs apply exactly, but a con- 7 Kelso and Crawford (1982) provide a particular exam- tract offer now includes a wage, so, for exam- ple in which preferences do not satisfy substitutes and there ple, a hospital whose contract offer is rejected does not exist a stable allocation. 922 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005
This theory also reaffirms and extends the at hospitals in a stable match. Then suppose one close connection between the substitutes condi- doctor retires. Imagine a process in which a tion and other concepts that has been estab- hospital seeks to replace its retired doctor by lished in the recent auctions literature with raiding other hospitals to hire additional doc- quasi-linear preferences. Milgrom (2000) stud- tors. If the hospital makes an offer that would ies an auction model with discrete goods in succeed in hiring a doctor away from another which the set of possible bidder values is re- hospital, the affected hospital has three options: quired to include all the additive functions.8 He it may make an offer to another doctor (or shows that if goods are substitutes, then a com- several), improve the terms for its current doc- petitive equilibrium exists. If, however, there tor, or leave the position vacant. Suppose it are at least three bidders and if there is any makes whatever contract offer would best serve allowed value such that the goods are not all its purposes. substitutes, then there is some profile of values In models with a fixed number of positions such that no competitive equilibrium exists. and no contracts, this process in which doctors Faruk Gul and Ennio Stacchetti (1999) establish and vacancies move from one hospital to an- the same positive existence result. They also other has been called vacancy chain dynamics show that if preferences include all values in (Blum et al., 1997). Kelso and Crawford (1982) which a bidder wants only one particular good consider similar dynamics in the context of their as well as any one for which goods are not all model. substitutes, and if the number of bidders is The formal results for our extended model are sufficiently large, then there is some profile of similar to those of the older theories. Starting Ј Ј preferences for which no competitive equilib- with a stable collection of contracts X , let XH rium exists. Ausubel and Milgrom (2002) estab- be the set of contracts that some doctor weakly lish that if (a) there is some bidder for whom prefers to her current contract in XЈ, and let Ј ϭ Ј ഫ Ϫ Ј preferences are not demand theory substitutes, XD X (X XH). As in the proof of Ј Ј ϭ Ј Ј (b) values may be any additive function, and (c) Theorem 1, we have F(XD, XH) (XD, XH) and Јϭ Ј പ Ј there are at least three bidders in total, then X XD XH. there is some profile of preferences such that the To study the dynamic adjustment that results Vickrey outcome is not stable and the core from the retirement of doctor d, we suppose the imputations do not form a lattice. Conversely, if process starts from the initial state (XD(0), ϭ Ј Ј all bidders have preferences that are demand XH(0)) (XD, XH). This means that the employ- theory substitutes, then the Vickrey outcome is ees start by considering only offers that are at in the core and the core imputations do form a least as good as their current positions and that lattice. Taken together, these results establish a hospitals remember which employees have re- close connection between the substitutes condi- jected them in the past. The doctors’ rejection tion, the cooperative concept of the core, the function is changed by the retirement of doctor ˆ ˆ Љ ϭ Љ ഫ ʦ noncooperative concepts of Vickrey outcomes, d to RD, where RD(X ) RD(X ) {x Љ͉ ϭ and competitive equilibrium. X xD d}; that is, in addition to the old rejec- tions, all contract offers addressed to the retired doctor are rejected. C. “Vacancy Chain” Dynamics To synchronize the timing with our earlier notation, let us imagine that hospitals make Suppose that a labor market has reached equi- offers at round t Ϫ 1 and doctors accept or reject librium, with all interested doctors placed them at round t. Hospitals consider as poten- Ϫ ϭ Ϫ tially available the doctors in XH(t 1) X ˆ Ϫ RD(XD(t 1)) and the doctors then reject all but the best offers, so the cumulative set of offers 8 A valuation function is additive if the value of any set received is X (t) ϭ X Ϫ R (X (t Ϫ 1)). Define of items is the sum of the separate values of the elements. D H H Such a function v is also described as “modular.” Additivity A B is equivalent to the requirement that for all sets and , ˆ ͑ ͒ ϭ ͑ Ϫ ͑ ͒ Ϫ ˆ ͑ ͒͒ v(A ഫ B) ϩ v(A പ B) ϭ v(A) ϩ v(B). (11) F XD , XH X RH XH , X RD XD . VOL. 95 NO. 4 HATFIELD AND MILGROM: MATCHING WITH CONTRACTS 923
If contracts are substitutes for the hospitals, as the price falls, agents should demand more of ˆ Ј Ј ϭ Ј then F is isotone and, since F(XD, XH) (XD, a good. Here, price falls correspond to more Ј ˆ Ј Ј Ն Ј Ј XH), it follows that F(XD, XH) (XD, XH). contracts being available, and more demand ϭ Ј Ј Then, since (XD(0), XH(0)) (XD, XH), we have corresponds to taking on (weakly) more con- ϭ ˆ Ն (XD(1), XH(1)) F(XD(0), XH(0)) (XD(0), tracts. We formalize this intuition with the fol- ϭ ˆ Ϫ XH(0)). Iterating, (XD(n), XH(n)) F(XD(n lowing definition. Ϫ Ն Ϫ Ϫ 1), XH(n 1)) (XD(n 1), XH(n 1)): the doctors accumulate offers and the contracts that DEFINITION: The preferences of hospital h ʦ are potentially available to the hospitals shrink. H satisfy the law of aggregate demand if for all Ј ʚ Љ ͉ Ј ͉ Յ ͉ Љ ͉ A fixed point is reached and, by Theorem 1, it X X , Ch(X ) Ch(X ) . corresponds to a stable collection of contracts. According to this definition, if the set of THEOREM 6: Suppose that contracts are sub- possible contracts expands (analogous to a de- Ј Ј stitutes and that (XD, XH) is a stable set of crease in some doctors’ wages), then the total contracts. Suppose that a doctor retires and that number of contracts chosen by hospital h either the ensuing adjustment process is described by rises or stays the same. The corresponding prop- ϭ Ј Ј ϭ (XD(0), XH(0)) (XD, XH) and (XD(t), XH(t)) erty for doctor preferences is implied by re- ˆ Ϫ Ϫ F(XD(t 1), XH(t 1)). Then, the sequence vealed preference, because each doctor chooses {(XD(t), XH(t))} converges to a stable collection at most one contract. Just as for the substitutes of contracts at which all the unretired doctors condition, when wages are endogenous, we in- are weakly better off and all the hospitals are terpret the definition as applying to the domain Ј weakly worse off than at the initial state (XD, of wage vectors for which the hospital’s opti- Ј 9 XH). mum is unique. The next theorem shows the important rela- The sequence of contract offers and job tionship between profit maximization, substi- moves described by iterated applications of Fˆ tutes, and the law of aggregate demand. includes possibly complex adjustments. Hospi- tals that lose a doctor may seek several replace- THEOREM 7: If hospital h’s preferences are ments. Hospitals whose doctors receive contract quasi-linear and satisfy the substitutes condi- offers may retain those doctors by offering bet- tion, then they satisfy the law of aggregate ter terms or may hire a different doctor and later demand. rehire the original doctor at a new contract. All along the way, the doctors find themselves Below, we use the law of aggregate demand choosing from more and better options, and the to characterize both necessary and sufficient hospitals find themselves marching down their conditions for the rural hospitals’ property and preference lists by offering costlier terms, pay- ensure that truthful revelation is a dominant ing higher wages, or making offers to other strategy for the doctors. Previously, in matching doctors whom they had earlier rejected. models without money, the dominant strategy result was known only for responsive prefer- III. Law of Aggregate Demand ences with capacity constraints (Abdulkadiro- g˘lu, 2003). We subsume that result with our We now introduce a second restriction on theorem. preferences that allows us to prove the next two results about the structure of the set of stable A. Rural Hospitals Theorem matches. We call this restriction the law of aggregate demand. Roughly, this law states that In the match between doctors and hospitals, certain rural hospitals often had trouble filling all their positions, raising the question of 9 The theorem does not claim, and it is not generally true, whether there are other core matches at which that this new point must be the new doctor-best stable set of the rural hospitals might do better. Roth (1986) contracts. analyzed this question for the case of X ϭ D ϫ 924 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005
͉ ͉ ϭ ͉ ͉ H and responsive preferences, and found that and Ch(XH) Ch(XH) . That is, every doctor the answer is no: every hospital that has unfilled and hospital signs the same number of contracts positions at some stable match is assigned ex- at every stable collection of contracts. actly the same doctors at every stable match. In particular, every hospital hires the same number The next theorem verifies that the counterex- of doctors at every stable match. amples developed above can always be gener- In this section, we show by an example that alized whenever any hospital’s preferences this last conclusion does not generalize to the violate the law of aggregate demand. full set of environments in which contracts are substitutes.10 We then prove that if preferences THEOREM 9: If there exists a hospital h, sets Ј ʚ Љ ʚ ͉ Ј ͉ Ͼ ͉ Љ ͉ satisfy the law of aggregate demand and substi- X X X such that Ch(X ) Ch(X ) , and tutes, then the last conclusion of Roth’s theorem at least one other hospital, then there exist holds: every hospital signs exactly the same singleton preferences for the other hospitals number of contracts at every point in the core, and doctors such that the number of doctors although the doctors assigned and the terms of employed by h is different for two stable employment can vary. Finally, we show that matches. any violation of the law of aggregate demand implies that preferences exist such that the Theorem 9 establishes that the law of aggre- above conclusion does not hold. gate demand is not only a sufficient condition ϭ ϭ Suppose that H {h1, h2} and D {d1, d2, for the rural hospitals result of Theorem 8 but, d3}. For hospital h1, suppose its choices maxi- in a particular sense, a necessary one as well. ՝ ՝ ՝ ՝ mize , where {d3} {d1, d2} {d1} ՝ A ՝ ՝ {d2} {d1, d3} {d2, d3}. This prefer- B. Truthful Revelation as a Dominant 11 ence satisfies substitutes. Suppose h2 has one Strategy ՝ position, with its preferences given by {d1} ՝ ՝ A {d2} {d3} . Finally, suppose d1 and d2 The main result of this section concerns doc- prefer h1 to h2 while d3 has the reverse prefer- tors’ incentives to report their preferences truth- Јϭ ence. Then, the matches X {(h1, d3), (h2, d1)} fully. For the doctor-offering algorithm, if Љϭ and X {(h1, d1), (h1, d2), (h2, d3)} are both hospital preferences satisfy the law of aggregate stable but hospital h1 employs a different num- demand and the substitutes condition, then it is ber of doctors and the set of doctors assigned dominant strategy for doctors to reveal truth- differs between the two matches. fully their preferences over contracts.12 We fur- This example involves a failure of the law of ther show that both preference conditions play aggregate demand, because as the set of con- essential roles in the conclusion. tracts available to h1 expands by the addition of We will show the positive incentive result for (h1, d3) to the set {(h1, d1), (h1, d2)}, the number the doctor-offering algorithm in two steps of doctors demanded declines from two to one. which highlight the different roles of the two When the law of aggregate demand holds, how- preference assumptions. First, we show that the ever, we have the following result. substitutes condition, by itself, guarantees that doctors cannot benefit by exaggerating the rank- THEOREM 8: If hospital preferences satisfy ing of an unattainable contract. More precisely, substitutes and the law of aggregate demand, if there exists a preferences list for a doctor d then for every stable allocation (XD, XH) and such that d obtains contract x by submitting this ʦ ʦ ͉ ͉ ϭ ͉ ͉ every d D and h H, Cd(XD) Cd(XD) list, then d can also obtain x by submitting a preference list that includes only contract x. Second, we will show that adding the law of 10 A similar example appears in Ruth Martı´nez et al. (2000). 11 These preferences, however, do not display the “single 12 It is, of course, not a dominant strategy for hospitals to improvement property” that Gul and Stacchetti (1999) in- truthfully reveal; nor would it be so even if we considered troduce and show is characteristic of substitutes preferences the hospital-offering algorithm. For further discussion of in models with quasi-linear utility. this point, see Roth and Sotomayor (1990). VOL. 95 NO. 4 HATFIELD AND MILGROM: MATCHING WITH CONTRACTS 925 aggregate demand guarantees that a doctor does With these preferences, the only stable match at least as well as reporting truthfully as by is {(d1, h2), (d3, h1)}, which leaves d2 unem- reporting any singleton. Together, these are the ployed. However, if d2 were to reverse her dominant strategy result. ranking of the two hospitals, then {(d1, h1), (d2, To understand why submitting unattained h1), (d3, h2)} would be chosen by the doctor- contracts cannot help a doctor d, consider the offering algorithm, leaving d2 better off. Essen- following. Let x be the most-preferred contract tially, by offering a contract to h2, d2 has that d can obtain by submitting any preference changed the number of positions available. list (holding all other submitted preferences When the preferences of the hospitals satisfy fixed). Note that all that d accomplishes when the law of aggregate demand, however, mak- reporting that certain contracts are preferred to x ing more offers to the hospitals (weakly) in- is to make it easier for some coalition to block creases the number of contracts the hospitals outcomes involving x. Thus, if x is attainable accept. with any report, it is attainable with the report Pd : x that ranks x as the only acceptable con- THEOREM 11: Let hospitals’ preferences sat- tract. This intuition is captured in the following isfy substitutes and the law of aggregate de- theorem. mand, and let the matching algorithm produce the doctor-optimal match. Then, fixing the pref- THEOREM 10: Let hospitals’ preferences sat- erences of the other doctors and of all the isfy the substitutes condition and let the match- hospitals, let x be the contract that d obtains by ՝ ing algorithm produce the doctor-optimal submitting the set of preferences Pd : z1 d ՝ ... ՝ ՝ Љ match. Fixing the preferences of hospitals and z2 d d zn d x. Then the preferences Pd : ՝ ՝ ... ՝ ՝ ՝ ՝ ... ՝ of doctors besides d, let x be the outcome that d y1 y2 yn x ynϩ1 yN ՝ Љ obtains by reporting preferences Pd : z1 d obtain a contract that is Pd-preferred or indif- ՝ ... ՝ ՝ z2 d d zn d x. Then, the outcome that d ferent to x. Ј obtains by reporting preferences Pd : x is also x. According to this theorem, when a doctor’s Љ Some other doctors may be strictly better off true preferences are Pd, the doctor can never do when d submits her shorter preference list; there better according to these true preferences than are fewer collections of contracts that d now by reporting the preferences truthfully. objects to, so the core may become larger, and In fact, the law of aggregate demand is “al- the doctor-optimal point of the enlarged core most” a necessary condition as well. The excep- makes all doctors weakly better off and may tions can arise because certain violations of the make some strictly better off. law of aggregate demand are unobservable from Without the law of aggregate demand, how- the choice data of the algorithm, and these can- ever, it may still be in a doctor’s interest to not affect incentives. Thus, consider an example conceal her preferences for unattainable posi- where terms t are included in the contract, and tions. To see this, consider the case with two where a hospital h has preferences {(d1, h, ՝ ˜ ՝ ˜ ՝ hospitals and three doctors, where contracts are t1)} {(d1, h, t1), (d2, h, t2)} {(d1, h, t1)} ϫ simply elements of D H, and let preferences be: {(d2, h, t2)}. Although these preferences violate the law of aggregate demand, the algorithm will (12) P : h ՝ h never “see” the violation, as either (d , h, t ) ՝ d1 1 2 1 1 d1 (d , h, ˜t )or(d , h, ˜t ) ՝ (d , h, t ). Thus, 1 1 1 1 d1 1 1 ͕ ͖ ՝ ͕ ͖ ՝ ͕ ͖ ՝ ͕ ͖ whichever terms that d first offers will deter- Ph1 : d3 d1 , d2 d1 d2 1 mine a conditional set of preferences for the ՝ hospital that does satisfy the law of aggregate Pd : h2 h1 2 demand. (The hospital will never reject an offer of either (d , h, t )or(d , h, ˜t ).) P ͕d ͖ ՝ ͕d ͖ ՝ ͕d ͖ 1 1 1 1 h2 : 1 2 3 The next theorem says that if some hospital’s preferences violate the law of aggregate demand ՝ Pd3 : h2 h1 . in a way that can even potentially be observed 926 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005 from the hospital’s choices, then there exist of jobs to fill and an existing endowment of preferences for the other agents such that it is doctors, while doctors’ productivities vary not a dominant strategy for doctors to report among jobs. A hospital values new doctors ac- truthfully, even when the other assumptions we cording to the their incremental value in the have used are satisfied. assignment problem. From a general job assignment perspective, the key restrictions of endowed assignment val- THEOREM 12: Let hospital h have prefer- uations are that each doctor can do one and only ͉ ͉ Ͼ ͉ ഫ ͉ ences such that Ch(X) Ch(X {x}) and let one job and that the hospital’s output is the sum there exist two contracts y, z such that yD of outputs of its various jobs. A hospital cannot ʦ ഫ Ϫ zD xD yD and y, z Rh(X {x}) Rh(X). use one doctor for two jobs, nor can it combine Then if another hospital hЈ exists, there exist the skills of two doctors in the same job. Also, singleton preferences for the hospitals besides h there is no interaction in the productivity of and preferences for the doctors such that it is doctors in different jobs. Given this mathemat- not a dominant strategy for all doctors to reveal ical structure, if a doctor is lost, it will always be their preferences truthfully. optimal for the hospital to retain all of its other doctors, although it may choose to reassign Thus, to the extent that the law of aggregate some of the retained doctors to fill vacated demand for hospital preferences has observable positions. consequences for the progress of the doctor- Another way to describe the set of endowed offering algorithm, it is an indispensable condi- assignment valuations, VEA, is to build it up tion to ensure the dominant strategy property for from three properties, as follows. First, a sin- doctors. gleton valuation is one of the form v(S) ϭ ␣ ␣ maxdʦS d for some nonnegative vector . Sin- gleton valuations represent the possible valua- IV. Classes of Conforming Preferences tions by a hospital with just one opening, and VEA includes all singleton valuations. Second, if Although the class of substitutes valuations is a hospital is composed of two units, j ϭ 1, 2, 13 ʦ quite limited, it is broader than the set of each of which has a valuation vj VEA, then the responsive preferences in useful ways. The hospital’s maximum value from assigning ٚ substitutes valuations accommodate all the af- workers between its units, denoted by (v1 ϵ ϩ Ϫ firmative action and subspecialty constraints de- v2)(S) maxRʚSv1(R) v2(S R), is also in scribed above and allow a doctor’s marginal the set. We call this property closure under product to depend on which other doctors the aggregation. Third, if a hospital’s value is de- hospital attracts. rived from a value in VEA by endowing the In this section, we introduce a parameterized hospital with a set of doctors T, then the hospi- class of quasi-linear substitutes valuations for tal’s incremental value for extra doctors ͉ ϵ ഫ Ϫ hospitals evaluating sets of new hires. The val- v(S T) v(S T) v(T) is also in the set VEA. uations are based on an endowed assignment We call this property closure under endowment. model, according to which hospitals have a set Finally, two valuations v1 and v2 are called equivalent if they differ by an additive con- stant. For example, for any fixed set of doctors T, v( ⅐ ͉T) is equivalent to v( ⅐ ഫ T). 13 Hans Reijnierse et al. (2002) (see also Yuan-Chuan Lien and Jun Yan, 2003) have characterized the valuation THEOREM 13: Let VEA denote the smallest functions that satisfy the substitutes condition as follows. family of valuations of sets of doctors that in- Let v(S) be the hospital’s value of doctor set S and let cludes all the singleton valuations and is closed ϭ ഫ Ϫ vS(T) v(S T) v(S) denote the incremental value of the under aggregation and endowment. Then, for set of doctors T. Then, doctors are substitutes if for every set ʦ each v VEA, there exists a set of jobs, J, a set S and three other doctors d1, d2 and d3, there is no unique ϩ ϩ of doctors T, and a (͉D͉ ϩ ͉T͉) ϫ ͉J͉-matrix [␣ ], maximum in {vS(d1) vS(d2, d3), vS(d2) vS(d1, d3), ij ϩ vS(d3) vS(d1, d2)}. such that v is equivalent to the following: VOL. 95 NO. 4 HATFIELD AND MILGROM: MATCHING WITH CONTRACTS 927
͑ ͉ ␣ ͒ ϵ ␣ most preferred acceptable doctors, if that many (13) v S J, , T max dj zdj subject to z d ʦ S are available, and otherwise hires all of the acceptable doctors. Յ ʦ To map responsive preferences into the as- zdj 1 for j J d ʦ S signment problem framework, define a utility 3 ޒ ՝ function uh : D to represent h and set the 1 for d ʦ S ഫ T minimum acceptable utility to zero. Formally, Յ ͭ A zdj Ј ʦ ՝ Ј N Ͼ Ј 0 for d ԫ S ഫ T (1) if d, d Dh, then d h d uh(d) uh(d ) j ʦ J ʦ A N Ͼ and (2) d Dh uh(d) 0. Using this utility, ʦ ͕ ͖ we specify an assignment problem as follows: zdj 0, 1 for all d, j. ϭ ␣ ϭ ϩ J {1, ... , nh}, and dj uh(d) 1, and fix the wage at 1. Using a positive wage is a device to Conversely, for every such J, T, and ␣, v( ⅐ ͉J, ensure that the hospital strictly prefers not to ␣ ʦ , T) VEA. hire a doctor it plans not to assign to any job. It We dub this family of valuations the endowed is evident that with this valuation preference, assignment valuations because of the form of the hospital most prefers to hire the set of avail- optimization (13). The family VEA is an exten- able, acceptable doctors with the highest rank- ՝ sion of the assignment valuations derived by ing according to h. Lloyd Shapley (1962), used by Roth (1984, Among the extensions of responsive prefer- 1985b) in models of matching with wages and ences that have been most important in practice explicit job assignments, and emphasized in a are ones to accommodate quotas of various study of package bidding by Lehmann et al. kinds. For example, one use of quotas is to (2001). The endowed assignment valuations are manage subspecialties of internal medicine, in quite useful for applications. They also fit nicely which a certain number of positions are to be with the theory reported in previous sections for filled by doctors planning that subspecialty if the following reason. enough acceptable doctors are available. One can reserve ms jobs for subspecialty s by pro- THEOREM 14: Every endowed assignment viding that there are ms jobs in which doctors in ⅐ ͉ ␣ ʦ valuation v( J, , T) VEA is a substitutes that subspecialty have their productivity in- valuation. creased by a large constant M. Then, if ms doctors in subspecialty s are available, at least To the best of our knowledge, all of the that many will be demanded at the optimal substitutes valuations that have been used or solution to (13). If that many qualified doctors proposed for practical applications are included are not available, the jobs will be filled by other among the endowed assignment valuations. In- doctors, according to their productivities. deed, the question of whether all substitutes Another use of quotas is for an affirmative valuations are endowed assignment valuations action policy that absolutely reserves jobs for is an open one. Here, we review some of the members of certain target groups. For each af- most popular valuations to show how they fit firmative action group G, define a correspond- ϭ ഫ into the class. ing set of jobs JG and let J G JG. Specify ʦ ʦ ␣ ϭ The most commonly studied class of prefer- that for any doctor d G and job j JG, dj ␣ ϭ ences for matching problems like the Match, for uh(d) and otherwise dj 0. One can also which employment terms are exogenously introduce an unrestricted category of jobs jЈ that ␣ ϭ fixed, are the responsive preferences. These any doctor can fill with productivity djЈ specify that each hospital h has a fixed number uh(d). This obviously ensures that a certain A ʚ of openings nh, a set of acceptable doctors Dh number of jobs are reserved for each target ՝ D, and a strict ordering h of the acceptable group. doctors. When a doctor is unacceptable, that A recent treatment of affirmative action by means that hiring the doctor is always worse Roth (1991), extended by Abdulkadirog˘lu than leaving her position unfilled. Given any set (2003), uses the class of responsive preferences of available doctors, the hospital hires its nh with capacity constraints. This class requires 928 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005 that the set of doctors be partitioned into a finite point exists. In this section, we offer a different number of affirmative action groups and that for characterization of the Gale-Shapley doctor- each group G the hospital has a capacity mG that offering algorithm that will prove especially limits the number of doctors who can be hired well suited to situations in which contracts may from that group. In addition, at most nh doctors not be substitutes, but in which there is just one in total can be hired. “hospital”: the auctioneer. For now, we allow To represent such preferences using endowed the possibility that there are several hospitals. assignment valuations, let us specify that there The alternative representation is constructed ¥ are mG jobs of type G and G mG jobs in total. by replacing the system of equations (9) by the Ϫ ¥ The hospital is endowed with nh G mG following system: doctors. The endowed doctors have productivity ␣ ϭ ͑ ͒ ϭ Ϫ ͑ ͑ Ϫ ͒͒ ij M in every job, where M is a large (14) XD t X RH XH t 1 number. This ensures that at most nh new doc- tors will be hired at any optimum. For the XH ͑t͒ ϭ XH ͑t Ϫ 1͒ ഫ CD ͑XD ͑t͒͒. nonendowed doctors, we calculate productivi- ϭ ties uh(d) in the same fashion as for the respon- We call the algorithm that begins with XD(0) ␣ ϭ ϭ A sive preferences and set dj uh(d) for jobs j of X and XH(0) and obeys (14) a cumulative ␣ ϭ type G, and dj 0 for all other jobs. This offer process, because the formalism captures ensures that at most mG doctors will be hired the idea that hospitals accumulate offers from from group G. By inspection, this specification doctors in the set XH(t) and hold their best represents Abdulkadirog˘lu’s affirmative action choices CH(XH(t)) from the accumulated set. preferences. There is no assumption of consistency imposed Unlike the specifications used for the current on the algorithm, so it is possible that several National Resident Matching Program algo- hospitals are “holding” contract offers from the rithm, the endowed assignment valuations per- same doctor. The corresponding allocation is, of mit overlapping affirmative action categories. course, infeasible, since each doctor can ulti- For example, suppose that a small hospital has a mately accept just one contract. target of hiring one female and one minority At each round t of the cumulative offer pro- doctor. In the endowed assignment structure, cess, all doctors make their best offers from the the hospital may assign a minority female doc- set of not-yet-rejected choices XD(t), but any tor to fill either a minority slot or a female slot, doctor d for whom a contract is being held but not both. This is necessary to retain the simply repeats one of its earlier offers. Thus, substitutes demand structure, for otherwise new offers are made only by doctors who have making a minority female doctor available been rejected. ʦ could lead the hospital to hire a nonminority To see why this is so, let x CH(XH(t)) be a male doctor whom it would otherwise reject. contract that is being held, and consider the ϭ Endowed assignment valuations thus provide corresponding doctor xD d. By revealed pref- a flexible, parameterized way for hospitals to erence, doctor d strictly prefers x to any contract represent their preferences for matching with or that she has not yet offered, since those con- without wages using a class of quasi-linear pref- tracts were available to offer at the time that x erences that satisfies the substitutes condition was offered. Since d’s most preferred contract and the law of aggregate demand. in Xd(t) at the current time must be weakly preferred to x, it must be coincide with one of V. Cumulative Offer Processes and Auctions d’s earlier offers. The second equation of system (14) is the one The algorithms described by the system (9) that distinguishes the cumulative offer process with different starting points have the property from the system in (9). In the cumulative offer that they can terminate only at a stable set of process, without any assumptions about hospi- contracts. Nevertheless, unless preferences sat- tals’ preferences, XH(t) grows monotonically isfy the substitutes condition, the system is not from round to round, so the sequence of sets guaranteed to converge at all, even when a fixed converges. In contrast, the earlier process was VOL. 95 NO. 4 HATFIELD AND MILGROM: MATCHING WITH CONTRACTS 929 guaranteed to converge only when contracts are trains travel at a uniform speed along the track. substitutes for the hospitals. In this setting, the contract terms must include When contracts are substitutes, the two sys- the direction and the two times, and the seller is tems of equations are equivalent. constrained to hold only combinations of bids such that trains maintain safe distances from THEOREM 15: Suppose that contracts are one another at all times. ϭ substitutes for the hospitals and that XD(0) X A second example of the generalized proxy ϭ A and XH(0) . Then, the sequences of pairs process is a procurement auction in which the {(XD(t), XH(t))} generated by the two laws of buyer scores suppliers on the basis of such motion (9) and (14) are identical. factors as quality, excess capacity, credit rating, and historical reliability, as well as price, and in ϭ Even with the initial condition XD(0) X and which the buyer prefers to set aside some ϭ A XH(0) , the algorithms described by (9) and amount of its purchase for minority contractors (14) may differ when contracts are not substi- or to maintain geographic diversity of supply to tutes. In that case, by inspection of the system reduce the chance of supply disruptions. In an (14), the cumulative offer process still con- asset sale, the seller may weigh the probability verges, because XH(t) is bounded by the finite that the sale will be completed, for example due set X and grows monotonically from round to to financing contingencies or because a union or round. What is at issue is whether the hospital’s antitrust regulators must approve the sale. choice from its final search set in the cumulative The cumulative offer process model with offer process is a feasible and stable set of general contracts accommodates all of these contracts. possibilities. The auctioneer in the model cor- We will find below that when there is a single responds to a single “hospital”—hereafter the hospital, the outcome is indeed a feasible and auctioneer—with a choice function, CH, that stable set of contracts. In that case, the cumu- selects her most preferred collection of contract lative offer process coincides with the general- proposals. We have the following result (which ized proxy auction of Ausubel and Milgrom is first stated using different notation than in the (2002). Those authors analyze in detail the case Ausubel-Milgrom paper): when a bid consists of a price and a subset of the set of goods that the bidder wishes to buy. At THEOREM 16: When the doctor-offering cu- each round, the seller “holds” the collection of mulative offer process with a single hospital bids that maximizes its total revenues, subject to terminates at time t with outcome (XD(t), XH(t)), the constraint that each good can be sold only the hospital’s choice CH(XH(t)) is a stable col- once. The generalized proxy auction, however, lection of contracts. is not limited to the sale of goods and, in fact, is identical in scope to our present model of Cumulative offer processes connect the the- matching with contracts. In particular, the auc- ory of matching with contracts to the emerging tioneer may impose a variety of constraints on theory of package auctions and auctions with the feasible collections of bids and may weigh complex constraints. nonprice factors, either exclusively or in com- bination with prices, to decide which collection of bids to hold. Bidders, for their part, may VI. Conclusion make bids that include factors besides price, and may not include price at all. We have introduced a general model of To illustrate the role of general contracts in matching with bilateral contracts that encom- this auction setting, consider the auction design passes and extends two-sided matching models suggested by Paul Brewer and Charles Plott with and without money and certain auction (1996), in which bidders seek to buy access to a models. The new formulation allows some con- railroad track. In that application, a bid specifies tract terms to be exogenously fixed and others to a train’s direction of travel, departure and ar- be endogenous, in any combinations. In this rival times, and price offered. It is assumed that very general framework, we characterize stable 930 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005 collections of contracts in terms of the solution aggregate demand fails in any potentially ob- to a certain system of equations. servable way, then the preceding dominant The key to the analysis is to extend two strategy property does not hold. concepts of demand theory to models with or For these results to be useful for practical without prices. The first concept to be extended mechanism design, one needs to account for is the notion of substitutes. Our definition ap- how preferences, especially hospital prefer- plies essentially the Roth-Sotomayor substitut- ences, are to be reported to the mechanism. able preferences condition to a more general There needs to be a convenient way for hospi- class of contracts: contracts are substitutes if, tals to express a rich array of preferences, and whenever the set of feasible bilateral contracts one needs to know whether the preferences be- expands, the set of contracts that the firm rejects ing reported actually satisfy the conditions of also expands. We show that (a) our definition the various theorems. Toward that end, we in- coincides with the usual demand theory condi- troduce a parametric form that we call extended tion when both apply, (b) when contracts are assignment valuations, which strictly general- substitutes, a stable collection of contracts ex- izes several existing specifications and which ists, and (c) if any hospital or firm has prefer- always satisfies both the substitutes and law of ences that are not substitutes, then there are aggregate demand conditions. preferences with single openings for each other Finally, we introduce an alternative treatment firm such that no stable allocation exists. We of the doctor-offering algorithm: the cumulative further show that when the substitute condition offer process. We show that when contracts are applies, (a) both the doctor-offering and hospi- substitutes, the previously characterized doctor- tal-offering Gale-Shapley algorithms can be offering algorithm coincides exactly with a cu- represented as iterated operations of the same mulative offer process. When contracts are not operator (starting from different initial condi- substitutes, but there is just one hospital (the tions), and (b) starting at a stable allocation “auctioneer”), the cumulative offer process co- from which a doctor retires, a natural market incides with the Ausubel-Milgrom ascending dynamic mimics the Gale-Shapley process to proxy auction. This identity clarifies the con- find a new stable allocation. nection between these algorithms and, com- The second relevant demand theory concept bined with the dominant strategy theorem is the law of demand, which we extend both to reported above, generalizes the Ausubel-Milgrom include heterogeneous inputs and to encompass dominant strategy theorem for the proxy models with or without prices. The law of ag- auctions. gregate demand condition holds that when the Our new approach reveals deep similarities set of feasible contracts expands, the number of among several of the most successful auction contracts that the firm chooses to sign weakly and matching designs in current use and among increases. In terms of traditional demand theory, the environmental conditions in which, theoret- this means that, for example, when the wages of ically, the mechanisms should perform at their some of a heterogeneous group of workers falls, best. Understanding these similarities can help if the workers are substitutes, then the total us to understand the limitations of these mech- number of workers employed rises. We show anisms, paving the way for new designs. that (a) when inputs are substitutes, the choices of a profit-maximizing firm/hospital satisfy the APPENDIX law of aggregate demand. Moreover, when the choices of every hospital/firm satisfy the law of PROOF OF THEOREM 1: ʚ 2 aggregate demand and the substitutes condition, Let (XD, XH) X be a solution to (3). Then, പ ϭ Ϫ ϭ then (b) the set of workers/doctors employed is XD XH XD RD(XD) CD(XD) and പ ϭ Ϫ ϭ the same at every stable allocation, (c) the num- similarly XH XD XH RH(XH) CH(XH), പ ϭ ϭ ber employed by each firm/hospital is also the so XH XD CH(XH) CD(XD). Јϵ പ same, and (d) truthful reporting is a dominant Next, we show that X XH XD is a stable Јϭ ϭ strategy for doctors in the doctor-offering algo- set of contracts. Since X CH(XH) CD(XD), rithm. Moreover, we prove (e) that if the law of it follows by revealed preference that XЈϭ VOL. 95 NO. 4 HATFIELD AND MILGROM: MATCHING WITH CONTRACTS 931
Ј ϭ Ј CH(X ) CD(X ), so condition (i) of the defi- the doctors find the other hospitals to be nition of stability is satisfied. unacceptable. Consider any hospital h and set of contracts Suppose Rh is not isotone. Then, there exists Љ ʚ Ј ഫ Љ Ј ʦ Ј ʚ ʦ X CD(X X ) naming hospital h. Since X some x, y X and X X such that for all x ϭ Ј ϭ ʦ Ј Ϫ Ј ഫ CD(XD), revealed preference of the doctors X , xH h and such that x Rh(X ) Rh(X Љ പ ʚ ʦ Ј ഫ implies that X XD CD(XD) and hence that {y}). By construction, since x, y Ch(X Љ പ ϭ Љ പ പ ʚ X RD(XD) X XD RD(XD) CD(XD) {y}), contracts x and y specify different doctors, പ ϭ A Љ ʚ Ϫ ϭ ϵ ϵ Ј Ј RD(XD) . Thus, X X RD(XD) XH say, d1 xD yD d2. Let x and y denote the Љ Ј by (3). So if X Ch(X ), then by the revealed corresponding contracts for doctors d1 and d2 in Љ ՞ ϭ Ј preference of hospital h, X h Ch(XH) which hospital h is substituted for h. Ј Љ Ј ഫ Љ Ch(X ). Hence, X Ch(X X ), so condition We specify preferences as follows: first, for Ј Ј Ј ՝ Ј ՝ A (ii) is satisfied. So, the set of contracts X is hospital h , we take {x } hЈ {y } hЈ and all unblocked. other contracts are unacceptable. Second, doc- Ј Ј ഫ Ј ഫ Ϫ For the converse, suppose that X is a stable tors in xD(CH(X ) CH(X {y})) {d1, d2} Ј ϭ Ј Ј ഫ Ј ഫ collection of contracts. Then, CD(X ) X . Let prefer their elements of CH(X ) CH(X {y}) ՝ XH be the set of contracts that the doctors to any other contract. Third, d1 has {x} d Ј Ј 1 weakly prefer to X and XD the set of contracts {x } and ranks all other contracts lower. Fourth, Ј ՝ that doctors weakly disprefer. By construction, d2 has {y } d {y} and ranks all other con- Ј Ϫ Ј Ϫ Ј 2 {X , XH X , XD X } is a partition of X. tracts lower. Finally, the remaining doctors find To obtain (3), observe first that if there is all contracts from hospitals h and hЈ to be Ј Љϭ some h such that Ch(XH) Xh, then X unacceptable. Љ Ch(XH) violates stability condition (ii). So, Consider a feasible, acceptable allocation X ϭ Ј ϭ Ј Ј ʦ Љ Ј Ch(XH) Xh for all h, that is, CH(XH) X .It such that y X . Since h and d2 can have only Ϫ ϭ Ϫ Ϫ Ј ϭ Љ Ј ԫ Љ follows that X RH(XH) X (XH X ) one contract in X , x , y X . Then, h’s con- Ϫ Ϫ ϭ Ϫ ϭ Љ Ј X (X XD) XD and that X RD(XD) tracts in X form a subset of X ,sox is not Ϫ Ϫ ϭ Ϫ Ϫ Ј ϭ X (XD CD(XD)) X (XD X ) XH. included and d1 has a contract less preferred പ Ј Ј Ј Hence, (XD, XH) satisfies (3) and XH than x . Then, the deviation by (d1, h )tox ϭ Ј Љ XD X . blocks X . Consider a feasible, acceptable allocation XЉ PROOF OF THEOREM 2: such that yЈ ԫ XЉ. Then, either x, y ʦ XЉ or XЉ ʦ Ј Ͼ Let i j,(j, wj) C(w) and wi wi. Define is blocked by a coalition including h, d1 and d2 ϵ ͉ Ն Ј ʚ ʦ Z(w) {(j, w˜ j) w˜ j wj}. Then, Z(wi, wϪi) using the contracts x and y. However, if x, y Ј Љ Ј Ј Z(w). If contracts are substitutes, then R(Z(wi, X , then a deviation by (d2, h ) to contract y ʚ ʦ Љ wϪi)) R(Z(w)). By (6), since (j, wj) C(w), blocks X . ʦ ԫ it follows that (j, wj) C(Z(w)), so (j, wj) Since all feasible allocations are blocked, ԫ Ј R(Z(w)). Hence, (j, wj) R(Z(wi, wϪi)). So, (j, there exists no stable set of contracts. ʦ Ј Ϫ Ј ϭ Ј wj) Z(wi, wϪi) R(Z(wi, wϪi)) C(Z(wi, ʦ Ј wϪi)) and thus (j, wj) C(wi, wϪi) by assumption PROOF OF THEOREM 7: (6). Thus, C satisfies demand theory substitutes. Suppose X ϭ D ϫ H ϫ W and let Z(w) ϵ {(j, ͉ Ն Conversely, suppose contracts are not substi- w˜ j) w˜ j wj}. Then, the law of aggregate de- tutes. Then, there exists a set XЈ, an element (i, mand is the statement that for any wage vectors ԫ Ј ʦ Ј ԫ Ն wi) X , and (j, wj) R(X ) such that (j, wj) w, wˆ satisfying w wˆ such that the choices sets Љ Љϭ Ј ഫ ͉ ͉ Յ ͉ ͉ R(X ), where X X {(i, wi)}. Using (6), are singletons, C(Z(w)) C(Z(wˆ )) . The proof Ͻ ˆ Ј Љϭ ˆ Љ Ј ϭ ˆ Ј wi Wi(X ). Let w W(X ) and wi Wi(X ). is by contradiction. Suppose the law of aggre- ЈϾ Љ ʦ Љ ԫ Then, wi wi and (j, wj) C(w ), but (j, wj) gate demand does not hold. Then there exists a Љ Ј C(wϪi , wi), so C does not satisfy the demand wage vector w and a doctor d such that for some Ͼ ͉ ϩ ͉ Ͼ theory substitutes condition. (and hence all) 0, C(Z(wd , wϪd)) ͉ Ϫ ͉ C(Z(wd , wϪd)) . Since h’s preferences are PROOF OF THEOREM 5: quasi-linear, changing doctor d’s wage can af- We may limit attention to the case with fect the hiring of other doctors only if it affects exactly two hospitals by specifying that the hiring of doctor d. It follows that there are 932 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005 exactly two optimal choices for the hospital at In principle, there are three cases. Ϫ ϭ ՝ wage vector w; these are C(Z(wd , wϪd)) at If xD yD, then let Px : y x and Pz : z. ϩ D D ഫ which doctor d is hired and C(Z(wd , wϪd)) Then, there exist two stable matches, Ch(Y at which d is not hired but such that two other {x}) and Ch(Y), with zD employed in the first doctors are hired, that is, there exist doctors dЈ, match but not in the second. Љ ʦ ϩ Ϫ Ϫ ϭ d C(Z(wd , wϪd)) C(Z(wd , wϪd)). The case xD zD is symmetric. Ј Ј Ј Let the corresponding payoff for the hospital Finally, if yD xD zD, then let x , y , z (when faced with wage vector w)be. denote contracts with hospital hЈ where the doc- Јϭ Ϫ Consider the wage vector w (wd , tors (and any other terms) are the same as in x, Ϫ wdЈ 2 , wϪ{d,dЈ}). For positive and suffi- y, z, respectively. Specify the remaining prefer- ϭ Ј ՝ ϭ ՝ Ј ϭ ՝ ciently small, the hospital’s payoff at wage vec- ences by Px x x, Py y y , Pz z Ј ϩ ϩ Ј ϭD Ј ՝ ЈD ՝ A ՝ Ј D tor w is 2 if it chooses C(Z(wd , wϪd)) z , and PhЈ {y } {x } {z }. Then, ϩ Ϫ Ј ഫ and it is if it chooses C(Z(wd , wϪd)), there exist two different stable matches, {x } Ј ഫ ഫ and one of these choices must be optimal. So, Ch(Y) and {y } Ch(Y {x}), with zD em- Ј ϭ ϩ C(w ) C(Z(wd , wϪd)). But then, raising ployed in the first match but not in the second. Ј Ϫ the wage of doctor d from wdЈ 2 to wdЈ while Ј holding the other wages at wϪdЈ reduces the de- PROOF OF THEOREM 10: mand for doctor dЉ from one to zero, in violation Let XЈ denote the collection of contracts cho- of the demand theory substitutes condition. sen by the algorithm when doctor d submits preference Pd. If this collection, which is stable under the reported preferences, is not stable Ј PROOF OF THEOREM 8: under Pd, then there exists a blocking coalition. ʚ By definition, XD XD, so by revealed pref- This blocking coalition must contain d,asno ͉ ͉ Ն ͉ ͉ ʚ erence, Cd(XD) Cd(XD) . Also, XH XH,so other doctor’s preferences have changed, but ͉ ͉ Յ by the law of aggregate demand, Ch(XH) that is impossible, since x is d’s favorite con- ͉ ͉ ϭ Ј Ј Ch(XH) . By Theorem 1, CD(XD) CH(XH) tract according to the preferences Pd. Since X is ϭ ¥ ͉ ͉ ϭ Ј and CD(XD) CH(XH), so dʦD Cd(XD) stable under Pd, the doctor-optimal stable match ¥ ͉ ͉ ¥ ͉ ͉ ϭ ¥ ͉ ͉ Ј hʦH Ch(XH) and dʦD Cd(XD) hʦH Ch(XH) . under Pd (the existence of which is guaranteed ¥ ͉ ͉ Ն Combining these leads to dʦD Cd(XD) by Theorem 2) must make every doctor ¥ ͉ ͉ ϭ¥ ͉ ͉ Ն¥ ͉ ͉ ϭ Ј dʦD Cd(XD) hʦH Ch(XH) hʦH Ch(XH) (weakly) better off than at X . In particular, ¥ ͉ ͉ dʦD Cd(XD) , which begins and ends with the same doctor d must obtain x. sum. Hence, none of the inequalities can be strict. PROOF OF THEOREM 11: Ј From Theorem 10, Pd : x also obtains x. Hence, by the rural hospitals theorem, d is em- Ј PROOF OF THEOREM 9: ployed at every point in the core when Pd : x is ͉ Ј ͉ Ͼ ͉ Љ ͉ Ј Since Ch(X ) Ch(X ) , there exists some submitted. So, every allocation X at which d is set Y, XЈ ʚ Y ʚ XЉ and contract x such that unemployed is blocked by some coalition and ͉ ͉ Ͼ ͉ ഫ ͉ ʦ ഫ Ј Ch(Y) Ch(Y {x}) . Since x Ch(Y {x}) set of contracts when d submits Pd. Conse- ϭ ഫ ՝ (as otherwise Ch(Y) Ch(Y {x}) for the quently, if d submits the preferences P*d : y1 ՝ ... ՝ ՝ preferences to be rationalizable) there must ex- y2 yn x, then every allocation at which ʦ ഫ Ϫ ist two contracts y, z Rh(Y {x}) Rh(Y), d is unemployed is still blocked, by the same such that y x z y. Moreover, since y, z ʦ coalition and set of contracts. Since the doctor-
Ch(Y), yD zD. best stable allocation is one at which d gets a Ј Denoting by h the second hospital whose exis- P*d-acceptable contract, that allocation is weakly tence is hypothesized by the theorem, we specify P*d-preferred to x. Finally, the doctor-optimal preferences as follows. Let all the doctors with con- match when P*d is submitted is still the doctor- Љ tracts in Y have those contracts be their most favored, optimal match when Pd is submitted, as d’s and let all other doctors find any contract with h preferences over contracts less preferred than x unacceptable. Let all doctors find any contract not cannot be used to block a match where she involving hospital h or hЈ to be unacceptable. receives a contract weakly preferred to x. VOL. 95 NO. 4 HATFIELD AND MILGROM: MATCHING WITH CONTRACTS 933
PROOF OF THEOREM 12: valuation without endowments v( ⅐ ͉J, ␣, A)isa Consider contracts xЈ, yЈ, and zЈ such that substitutes valuation. Ausubel and Milgrom ϭ Ј ϭ Ј ϭ Ј Ј ϭ Ј ϭ Ј ϭ xD xD, yD yD, zD zD, and xH yH zH (2002) prove that a valuation v is a substitutes hЈ. Let the preferences of the three identified be valuation if and only if the corresponding indi- Ј ՝ ՝ Ј Ј ՝ ϭ Ϫ ¥ Px : x x, Py : y y , and Pz : z z, and rect profit function ( (p) v(S) dʦS pd)is D Ј D Ј ՝ ЈD ՝ Ј let those of h be PhЈ :{y } {z } {x }. For submodular. Let ˆ be the indirect profit func- ʦ ϵ ⅐ ͉ ␣ A the other contracts xˆ Y Ch(X), let xˆ be xˆD’s tion corresponding to v( J, , ). Then the most favored contract. For the remaining doctors, indirect profit function corresponding to v( ⅐ ͉J, Ј ␣ ϭ let any contract with h or h be unacceptable. , T)is (p1, ... , pN) ˆ (p1, ... , pN, 0, ... , 0). With the preceding preferences, the only sta- Since ˆ is the profit function of the substitutes Ј ble allocation includes contracts x and y , leav- valuation vˆh, it is submodular, and hence is ing doctor zD unemployed. If, however, zD submodular as well. misrepresents her preferences and reports PЈ : zD z ՝ zЈ, then the doctor-optimal stable match PROOF OF THEOREM 15: includes z, leaving doctor zD better off. Suppose that contracts are substitutes for the hospitals. We proceed by induction. The initial PROOF OF THEOREM 13: condition specifies that the sequences are iden- Setting ͉J͉ ϭ 1 and T ϭ A, it is clear by tical through time t ϭ 0. Denote the sequence inspection that this family includes all singleton corresponding to the cumulative offer process valuations. by a superscript C and denote the alternative The family is closed under endowment, be- process defined by (9) with no superscript. As- cause given any such valuation v( ⅐ ͉J, ␣, T), sume the inductive hypothesis that the se- endowing a set of doctors TЈ simply creates the quences are the same up to round t Ϫ 1 and valuation v( ⅐ ͉J, ␣, T ഫ TЈ)). suppress the corresponding superscripts for the ϭ Ϫ To verify that the family is closed under values at that round. Then, XD(t) X ⅐ ͉ Ϫ ϭ C aggregation, consider any two valuations v( J, RH(XH(t 1)) XD(t). This also implies that ␣, T) and v( ⅐ ͉JЈ, ␣Ј, TЈ). Since J and T are just index sets, we may assume without loss of X ϭ X ͑t Ϫ ͒ ഫ X ͑t͒ generality that T പ TЈϭJ പ JЈϭA. Define (16) H 1 D . TЉϭT ഫ TЈ and JЉϭJ ഫ JЈ. Let To complete the proof, we must show that ␣ ʦ ഫ ʦ C ϭ Ϫ ഫ ij if i D T, j J XH(t) XH(t) or, equivalently, that XH(t 1) ␣ Љ ϵ ␣Ј ʦ ഫ Ј ʦ Ј ϭ Ϫ (15) ij ͭ ij if i D T , j J CD(XD(t)) X RD(XD(t)). Ն C Ϫ ʚ C Ϫ 0 otherwise. For t 2, XH(t 2) XH(t 1) by con- Ϫ struction, so by the inductive hypothesis XH(t ⅐ ʚ Ϫ By inspection of the optimization problems, v( 2) XH(t 1). Since contracts are substitutes, ٚ ⅐ ͉ Ј ␣Ј Ј ϭ ⅐ ͉ Љ ␣Љ Љ Ϫ Ϫ ʚ Ϫ ␣ ͉ J, , T) v( J , , T ) v( J , , T ). RH is isotone, so X RH(XH(t 1)) X ⅐ Ϫ ʚ Ϫ For the converse, consider the valuation v( RH(XH(t 2)) and hence XD(t) XD(t 1). ͉ ␣ ϭ ʚ Ϫ J, , T) given by (13). We derive it construc- For t 1, the inclusion XD(t) XD(t 1) is ϭ tively from the singleton valuations, aggrega- implied by the initial condition XD(0) X. tion, and endowment, as follows. Let vj be the Recall that, by revealed preference, RD is ʚ Ϫ singleton valuation associated with row j in isotone. It follows that RD(XD(t)) RD(XD(t ␣ ϭ ͉ ͉ Ϫ ϭ Ϫ matrix [ ij] for j 1, ... , J . Since VEA is 1)) and hence, using (9), that XH(t 1) X ٚ ... ٚ ϭ ⅐ Ϫ ʚ Ϫ ϭ closed under aggregation, v1 v͉J͉ v( RD(XD(t 1)) X RD(XD(t)) XH(t). Thus, ͉ ␣ A ʦ Ϫ ϭ Ϫ പ Ϫ ϭ J, , ) VEA. Since VEA is also closed under XH(t 1) XH(t 1) (X RD(XD(t))) Ϫ Ϫ Ϫ ഫ endowment, we may endow the doctors in T to XH(t 1) RD(XD(t)). So, XH(t 1) ⅐ ͉ ␣ ʦ ϭ Ϫ ഫ Ϫ obtain v( J, , T) VEA. CD(XD(t)) XH(t 1) (XD(t) ϭ Ϫ Ϫ ഫ RD(XD(t))) (XH(t 1) RD(XD(t))) Ϫ ϭ Ϫ ഫ Ϫ PROOF OF THEOREM 14: (XD(t) RD(XD(t))) (XH(t 1) XD(t)) ϭ Ϫ L. Shapley (1962) (see also Lehmann et al., RD(XD(t)) X RD(XD(t)), where the last step 2001) had already proved that every assignment equality follows from (16). 934 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005
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