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Matching with Contracts 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).
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