Ordered Contracts in Matching Markets: Properties and Stability, Study Guides, Projects, Research of Gastroenterology

Download Ordered Contracts in Matching Markets: Properties and Stability and more Study Guides, Projects, Research Gastroenterology in PDF only on Docsity! 1957 In response to a recently dismissed antitrust lawsuit against the National Residency Matching Program (NRMP), Jeremy Bulow and Jonathan Levin (2006) show that when firms set imper- sonal salaries simultaneously, before matching with workers, then such a match leads to lower aggregate wages compared to any competitive outcome. Vincent P. Crawford (forthcoming) shows that this concern can be addressed by incorporating flexible salaries in the centralized match, that is, the possibility for each position to have more than one potential salary, with the final salary to be determined together with the worker-firm pairing. This builds on earlier work that shows that a match in which each position can have a large number of contracts, which I will call a multiple contract match, allows for competitive outcomes (Crawford and Else Marie Knoer 1981; Alexander S. Kelso and Crawford 1982; Alvin E. Roth 1984b; and John Hatfield and Paul Milgrom 2005). Here, I observe that the NRMP has a feature that I will call ordered contracts, which destroys the low-wage equilibrium of Bulow and Levin and in fact allows for competitive outcomes. In a match with ordered contracts, firms can specify several possible contracts, just as in a match with flexible contracts. However, firms can also spec- ify the order in which they prefer to fill these contracts. Specifically, at any point in the match, only one type of contract is available, which is in contrast to a match with multiple contracts. This additional control of firms changes which stable outcomes are reached through appropri- ately modified deferred acceptance algorithms. Most importantly, the set of contracts reached through either a firm or worker proposing modi- fied deferred acceptance algorithm is the same, Competitive Wages in a Match with Ordered Contracts By Muriel Niederle* which in general is not the case in a multiple contract match. The results of this paper will cast the lesson we learned from Bulow and Levin in a new light. In particular, the relative lack of personal wages in the match appears to be an outcome of the market, not a constraint of the match. This also becomes apparent when we look at salaries in medical specialties that do, and do not, use a match. Since 1951, the market for medical residents has been organized through a centralized match- ing procedure that assigns medical students to residency programs using a variant of a deferred acceptance algorithm.1 In 1998, Roth and Elliot Peranson introduced a new algorithm, which switched from a hospital proposing to a student proposing deferred acceptance algorithm. The new algorithm also incorporates several special features, such as accommodating couples who want two jobs. A second special feature is to allow for ordered contracts, or “reverting posi- tions.” Programs that try to fill a position under a certain contract can, in case they do not find a suitable candidate, change (or revert) that con- tract to a position with a different contract (see Roth and Peranson 1999). This allows programs to effectively have more than one contract for any position, while being able to control the order in which they want to try to fill those positions. In the 1990s, about 7 percent of the three to four thousand programs that participate each year have contracts that could revert to other contracts if they remain unfilled (accounting for almost 6 percent of the total quota of posi- tions). Roth and Peranson (1999) note that such reversions typically occur when, for example, a director of a second-year postgraduate program arranges with the director of a first-year prereq- uisite program for his residents to spend their first year in that prerequisite program. If the 1 The match was introduced to ensure a uniform appoint- ment date, control the unraveling of hiring decisions, and reduce congestion problems of other, decentralized plans that tried to promote late and uniform hiring (see Roth 2003). * Department of Economics, Stanford University, Stan- ford, CA 94305, and NBER (e-mail: niederle@stanford.edu). I thank Georgy Artemov, Jonathan Levin, Eric Maskin, Al Roth, and Michael Schwarz for helpful discussions, the National Science Foundation and the Sloan Foundation for generous support, and the Institute for Advanced Study at Princeton and the Economics Department at Harvard Uni- versity for their hospitality. DECEMBER 20071958 THE AMERICAN ECONOMIC REVIEW second-year program fails to fill all its positions, then the vacancies can “revert” to the first-year program to be filled by other applicants. More recently, in the reinstitution of the fellowship match for gastroenterologists, this feature is especially advertised to, for example, allow pro- grams to try to fill a slot first with a research fellow, and in case no suitable research fellow can be attracted, the program can decide to fill this position with a more clinically oriented fel- low instead (see Niederle, Deborah D. Proctor, and Roth 2006). While ordered contracts have been used in practice in the NRMP, theoreti- cal properties have not been analyzed before.2 In Section IV, I’ll fill this theoretical gap and explore the additional options that ordered con- tracts give to market designers. In 2002, an antitrust lawsuit was filed, charg- ing that the main effect of the match is to sup- press wages of medical residents. Bulow and Levin develop a stylized model to analyze the effects on wages of using a centralized match such as the NRMP. In their simple model, firms simultaneously announce a wage at which they are willing to hire any worker. Workers care only about salaries and form preferences over firms after the announcement. Then an assortative matching occurs, which matches more produc- tive workers with firms that offer a higher wage. The main result of the paper is that this model yields wage compression, sub-competitive aver- age wages, and higher profits for firms compared to any competitive outcome. The intuition for the result is that, compared to a competitive market, firms cannot change their salaries depending on the worker they end up hiring.3 2 Roth and Peranson (1999) show how applicants who want two kinds of contracts, namely a first-year contract and a second-year contract, as well as couples who want two jobs, present complementarities that may make the set of stable matchings empty. They fail to note that ordered contracts, or reversions of positions in themselves, are not a source of complementarities, rather substitutes, and hence do not pose any problems for the existence of stable match- ings and are amenable to theoretical analysis. 3 The fact that firms cannot change their salary when they try to hire different workers becomes important when firms are asymmetric. For example, some firms are more productive than others. In this case, a competitive outcome calls for gaps in the wages paid to different workers, which are not reproduced by equilibrium strategies when firms simultaneously set one wage for their position independent of the worker they end up hiring. As such, the paper can be seen as providing support for the contention that a centralized match, such as the NRMP, may indeed be used to reduce wages. However, the NRMP allows for ordered contracts, which allow for wage com- petition that can restore competitive outcomes. In a stylized model of the NRMP with ordered contracts, each firm i, instead of advertising only one position at one contract (wage) pi, can create a second contract pi * for the same posi- tion, and decide which workers are eligible for each contract. Firm i first tries to fill the posi- tion at contract pi *. If it fails to fill the position, then the contract is changed (or reverted) to a new contract pi and firm i tries to fill the posi- tion at this new contract with a new set of eli- gible workers. Ordered contracts have two main effects on wages. First, when other firms play the mixed strate- gies in the wage-setting equilibrium of Bulow and Levin, every firm has a strict incentive to use ordered contracts, and can strictly increase its expected payoff. Second, if all firms use ordered contracts, there exists an equilibrium in which wages are competitive. The actual NRMP algorithm is therefore able to achieve competi- tive outcomes in the model of Bulow and Levin with the use of ordered contracts. Empirically, Niederle, Proctor, and Roth (2006) show that in the labor market for gastroenterol- ogy fellows, after their market operated with- out a match for nearly a decade, most programs offered the same wage to all their fellows (i.e., impersonal wages). That is, wage data show less differentiation than a competitive model in which gastroenterology fellows care only about wages would predict.4 In that case, we would not necessarily expect wages to be very dif- ferent in a market with and without a match, even if the match would work as described by Bulow and Levin. This implies that we may not observe every program submitting two or more contracts, in a model in which, on the one hand, workers do not care most about wages but care, rather, for the firm where they work, and, on the other hand, multiple contracts serve only to discipline potential wage manipulations by competing firms. Hence, this paper does not 4 In Section V, I will discuss the effects of eliminating a match, such as the NRMP, on the competitiveness of a market. VOL. 97 NO. 5 1961NIEDERLE: COMpETITIVE WAgEs IN A MATCH WITH ORDERED CONTRACTs The process continues until the behavior of firms 2, . . , N is specified. If D1 5 D2, then firm 1’s behavior is also specified. Otherwise, firm 1 offers zero with its remaining offer probability, namely g1102 5 1 2 g n q11n 2 · 1pn11 2 pn 2 . THEOREM 1 (Bulow and Levin 2006, 659):  There is a unique price equilibrium. Let qn 1·2 and p1, . . , pN11 and g1102 be defined as above. Then, for each firm n, and each nonempty interval 3pm, pm114 , gn 1 p 2 5 qn 1m 2 for all p [ 1 pm, pm114 . First, I show that the equilibrium of Bulow and Levin does not survive the introduction of ordered contracts (proofs of theorems are in the Appendix).6 THEOREM 2: suppose all firms have only one position, and offer wages pi according to the Bulow and Levin equilibrium. If some firm i can offer two wages, pi *—for which it can restrict which workers are acceptable—and pi —for which any worker is eligible—then the firm makes strict positive gains from using that possibility. Furthermore, if every firm has an ordered star-position contract, then competitive wages are an equilibrium outcome. Let ci be the lowest competitive wage of worker i; then c1 5 0, ci11 5 ci 1 Di. THEOREM 3 (Competitive Equilibrium Wages): The following strategies form a Nash equilib- rium. Every firm announces pi 5 ci for 1 # i # N and pN * 5 cN with only worker wN being eligible, and pj * 5 cj11 for j , N; and the work- ers being eligible for pj * are workers wj11 and higher. The workers report their preferences truthfully, that is, they rank all contracts such that they prefer higher wages to lower wages, 6 Because ordered contracts have not been discussed and theoretically analyzed before, the last section of the paper provides the algorithm for a match with ordered contracts and explores theoretical properties, most notably the fact that the algorithm yields a stable matching, that is, an out- come in which every firm employs at most one worker, and each worker works at most for one firm, such that no firm- worker pair (including the option of unemployment and an empty position) can be found that would rather be matched together than with their current partner. and for a given wage they prefer more produc- tive to less productive firms. Therefore a match with ordered contracts, which provides a description of the actual possi- bilities offered by the NRMP algorithm, allows for competitive outcomes, and does not neces- sarily result in lower wages. III.  Wages of Medical Fellows In the 1990s, about 7 percent of the three to four thousand programs that participate each year had positions with contracts that could revert to other contracts if they remain unfilled (accounting for almost 6 percent of the total quota of positions).7 In the reinstitution of the fellowship match for gastroenterology fellows, this feature is advertised in a way to allow pro- grams to try to fill a slot first with a research fel- low, and in case no suitable research fellow can be attracted, the program can decide, instead, to fill this position with a more clinically oriented fellow (see Niederle, Proctor, and Roth 2006). Do we observe price competition through ordered contracts in the NRMP? There are two main empirical questions to solve before it is clear how contracts in the NRMP should look to result in competitive wages. First, note that firm 1, in order to discipline the wage offer of firm 2, needs to offer the star- contract only with probability 1/2.8 Similarly, firm 2 has to offer the star-contract only with probability 2/3, to effectively discipline firm 3’s wage offer. Second, several firms and workers may be of identical or very similar quality. In that case, a competitive wage equilibrium with ordered contracts can be achieved even if only a subset 7 This feature is also used by the Internship Matching Program sponsored and supervised by the Association of Psychology Postdoctoral and Internship Centers (APPIC). 8 Firm 2 prefers to announce the contract p2 5 1 versus p2 5 0, if firm 1 offers p1* 5 1 with probability q $ 1/2. Furthermore, firm 2 cannot use his star-contract to exploit the fact that firm 1 offers p1* 5 1 with probability q, and whenever firm 1 does not offer a star-contract, try to recruit worker 2 with p2 * 5 0 (where only worker 2 is eligible), and in case the position is not filled, that is, whenever firm 1 does offer the star-contract, revert the contract to p2 5 1. The reason this is not robust is that worker 2 can simply announce that the low-wage star-contract is unacceptable, and so always ensure a high salary of 1. DECEMBER 20071962 THE AMERICAN ECONOMIC REVIEW of firms offers a second contract. Consider the initial example, only now assume that there are four firms and four workers, of “quality” 1, 2, 2, 3 each, where the profits of firm m from hiring worker n at wage p are mn 2 p. Then, a competi- tive wage equilibrium can be sustained even if only firm 1 and one firm 2 offer a second con- tract. This is true even as we increase the number of firms of quality 1, 2, or 3. That is, in general the number of firms that need to offer two con- tracts is highly dependent on the quality distribu- tions of hospitals, residents, and fellows. More generally, it is not clear that wages are the major determinant of hospital choice of resi- dents and fellows, rather than, say, the actual hospital at which the resident or fellow receives his education. That is, it is not clear how com- petitive wages would actually look. This is obvi- ously hard to explore directly with wage data obtained through the match, when the question is whether the match does or does not affect wage distributions. However, not all fellow- ship markets operate with a match, allowing for direct evidence on the importance of impersonal wages. Furthermore, we can compare wages in markets with and without a match. Niederle, Proctor, and Roth (2006) studied the market for gastroenterology fellows when a centralized match was not used. A survey of gastroenterology program directors reveals that only a few program directors (6 percent of respondents) did not offer the same wage to all their incoming fellows, and they all responded that wages were not adjusted to outside offers. Similar results hold for other terms of fellow- ship contracts, such as hours on call. It does not appear that in a market without a match wages are highly personalized. The survey results are supported by data on the internal medicine fellowship market. One can compare wages of fellows whose specialty par- ticipates in a match with wages of fellows whose specialty hires in a decentralized way. In internal medicine, of all subspecialties that require three years of prior residency, in the years between 2002 and 2004 four specialties used the Medical Specialties Matching Program (MSMP), while ten did not. Niederle and Roth (2003a) com- pare wages of all programs that report positive wages, excluding those from Puerto Rico, using the data from the Graduate Medical Education Library 2002–2003. A simple regression of the wage on a match dummy (which is one when the specialty uses the match) reveals no significant effect of the match. Similarly, a comparison of wages within hospitals for specialties that use a match and that do not yields a small positive, sig- nificant (but not economically significant) effect of a match on wages. Similar results were found for the next year, using the Graduate Medical Education Library 2003–2004 data (Niederle and Roth 2004). It is not clear that a match compresses or low- ers wages, because, for one thing, the ordered contracts match used by the NRMP allows for wage competition. Furthermore, the market of internal medicine subspecialties fellows strongly suggests that wages are not different for special- ties that use a match compared to those that do not. While the NRMP does not, in fact, force wages to be impersonal, we still observe a lot of impersonal wages, both in centralized and decentralized markets. In this respect, residents or first-year fellows may not be unique. For example, in many economics departments, first- year salaries of junior faculty hired in the same year are often the same. As such, the lessons of Bulow and Levin may still apply: wages are more compressed than if each worker were paid their marginal productivity; however, the match does not seem to be the cause of impersonal wages. IV.  Matching with Ordered Contracts In this final section, I analyze matching with ordered contracts. I show the existence of sta- ble matchings and how deferred acceptance algorithms have to be modified to account for ordered contracts. I point out similarities and differences between a match with ordered con- tracts and a match with multiple contracts, as introduced by Crawford and Knoer (1981) and Kelso and Crawford (1982). (For an overview, see Roth and Marilda Sotomayor (1990).) In a matching with ordered contracts, every firm can have several contracts, for each of which the firm specifies a strict preference order- ing over the set of workers eligible for this con- tract (formal definitions are in the Appendix).9 Furthermore, each firm has a strict ordering over 9 For example, a firm may decide that a specific worker may be eligible for the first contract but not for the second contract. VOL. 97 NO. 5 1963NIEDERLE: COMpETITIVE WAgEs IN A MATCH WITH ORDERED CONTRACTs which contract should be filled first. Firms have preferences over worker-contract pairs and not filling their position at all, while workers have preferences over firm-contract pairs or remain- ing unemployed. In a matching, each worker can work for at most one firm at a specific contract, and every firm can employ at most one worker and so can have at most one of its contracts accepted. A matching is stable if there exists no firm, worker, or contract triplet such that the firm would rather fill its position with that worker at that contract, and the worker would rather accept that contract from that firm, than stay with their current match. First, I show that a stable matching always exists, by providing a modified deferred accep- tance (MDA) algorithm whose outcome is always a stable match. There are two versions of MDA, worker and firm proposing. The Firm proposing MDA is basically a succession of standard deferred acceptances (DA) (see David Gale and Lloyd S. Shapley 1962). First, in MDA step 1, all firms have only their first contract available. Then comes the DA part. In DA step 1, firms make offers to their most preferred worker. Workers collect all their offers, keep their most preferred acceptable offer, and reject any other offers. More generally, in DA step k, firms whose offer was rejected in step k 2 1 make an offer to their next most desirable worker. Workers col- lect all their offers, keep their most preferred acceptable offer, and reject any other offers. The DA subalgorithm ends when either no firm has its offer rejected, or all rejected firms have no more workers they want to make an offer to, at the current contract. In the general MDA step k, any firm i that has its position at the j-th contract p i j unfilled changes the contract to p i j11 in case it has another contract to revert to. Then, the algo- rithm continues with a DA subalgorithm, where all previous offers are cancelled and have to be remade.10 The algorithm ends when all firms 10 Note that when some firm f reverts its position to a new contract, the deferred acceptance algorithm part can simply continue at whichever offers are held right now, instead of canceling all offers and restarting the whole pro- cess anew. The reason is that no firm that has not changed a contract can gain by remaking an offer that was rejected, as it was rejected because the worker either finds the offer unacceptable, or has a better offer in hand, and will have so once more as the algorithm unfolds, now that there are even more desirable contracts than before. that have no offer held by an applicant have no more contracts to change or revert to. Workers who hold an offer from a firm at a contract are matched to that firm at that contract; remaining firms and workers are unmatched. A worker proposing MDA is actually a sim- plified version of the current algorithm used by the NRMP, which has, however, not been fully described (or analyzed) in this respect before.11 In a Worker proposing MDA in MDA step 1, all firms have only their first contract available. Then comes the DA step, just as before. At the general MDA step k any firm i that has its position at j-th contract p i j unfilled reverts the contract to p i j11 in case it has another contract to change to. Then the algorithm continues with the DA steps, where all previous offers are annulled and have to be remade. The algorithm ends when all the firms that have no offer have no more position to revert to. At this point, any worker whose offer is held by a firm at a specific contract is matched to that firm at that contract. Unlike in the firm proposing algorithm, in the worker proposing MDA, interim offers have to be annulled, because some worker, who has an offer held by a firm, may prefer one of the new contracts that are introduced when some other firm changed its contract. THEOREM 4 (Stability): Whenever firms have a strict ordering over a finite number of con- tracts, which are ordered contracts, and for each contract a strict preference ordering over the workers, and workers have a strict ordering over firm-contract pairs, then both the firm and worker proposing MDA yield a stable outcome. THEOREM 5 (Firm-Optimal Stable Match): The firm proposing MDA yields the firm-optimal stable match. I now describe properties that are different from other models of matchings with multiple contracts, in which firms cannot enforce the order in which contracts should be filled (see Crawford and Knoer 1981; Kelso and Crawford 1982; Roth 1984b; and Hatfield and Milgrom 11 The algorithm designed by Roth and Peranson, and used by the NRMP, works like a worker proposing MDA (private communication from Alvin Roth) (see Roth and Peranson 1999 for other aspects of the algorithm). DECEMBER 20071966 THE AMERICAN ECONOMIC REVIEW have already been hired (for example, by pro- grams that have not tried to fill the position first under another contract), then a program may actually lose some potential candidates simply by having tried to fill a position first under a dif- ferent contract. And this is exactly how ordered contracts are used in the NRMP, and how they have been advertised for use in the new gastro- enterology fellowship match. Some of the success of economic theory comes from the ability to make vastly simplified models that capture some essential, general properties of markets. On the other hand, some of the recent successes in the market design lit- erature reflect an attention to the details of par- ticular markets.18 And sometimes case studies on details that are important in specific markets can lead to new general insights and theory. 18 For an overview on the importance of the attention to details in market design, and its successes, see Roth (2002). Appendix PROOF OF THEOREM 2: Every firm i has a highest wage interval that it offers with a constant density, let it be 3 pi L, pi H 4 , on which it competes for several workers, the highest being wH and the lowest wL. The highest (and lowest) worker are easily determined by determining the highest (or respectively, lowest) firm that is offering a wage on this highest wage interval of firm i. Suppose that all other firms use the mixed strategies from before; then the following strategy makes firm i in expectation strictly better off. Let pi s 5 1 pi H 2 pi L 2/2 and the only worker eligible for that wage be wH. It is easy to see that firm i is strictly better off with this strategy than foregoing the possibility to use a pi s job at all. The reason is that firm i is indifferent among all wages it offers in the Bulow and Levin equilibrium, so its profit is determined by, for example, offering wage pi 5 pi H and hiring any of the workers wH, … , wL with (different) positive probability. So, trying to hire worker wH at a lower wage first, with the use of the star-contract, which is successful with positive probability, strictly increases expected payoffs. PROOF OF THEOREM 3: First I show that a firm i cannot gain by deviating. Without firm i, resulting wages would be cj for worker wj, and workers wj with j . i work for firm j, while workers wj with j # i work for firm j 2 1. If firm i submits the strategies suggested by the theorem, firm i hires worker wi at the lowest competi- tive wage for wi (displacing firm i 2 1). Firm i cannot hire any workers wj with j . i, unless firm i is willing to pay e more than competitive wages, and may hire workers wj with j # i at competitive wages. So, given the definition of lowest competitive wages, firm i cannot make higher profits than hiring worker wi at ci. Now I show that workers cannot gain by deviating either. Given the strategies of firms, and workers j . i, worker i is the highest ranked worker of the standard contract of firm i at ci, the star-contract of firm i 2 1 at ci, and contracts at wages lower than ci. Any higher-wage contract is not achievable for worker wi. Hence, worker wi, by reporting truthfully, receives the highest wage he can receive given the strategies of other firms and workers. In a model with ordered contracts, every firm i can have up to K contracts pi 1, … , pi K. Let pi be the set of contracts and Ki the number of contracts of firm i. For each contract pk i, firm i specifies a strict preference ordering over the set of workers eligible for this contract W ki # W. Furthermore, firm i has a strict ordering over which contract should be filled first. Let the first contract be p1 i, and only if firm i cannot fill the position at p1 i will firm i try to recruit workers at p2 i , and so on. Firm f has preferences over 5 f 6 < pi 3 W, where, by definition, for any k, j such that k 1 j , Ki, 5w [ Wi k, 5w9 [ Wi k1j, 1 pk f , w 2 sf 1 pf k1j, w92 . Let pF be the total set of contracts, where pf [ pF qt f [ F and p is a contract that f offers, that is, pF 5 <fi[Fpi. A worker w has preferences over 5w6 < pF. A matching is a function m : pF < W S pF < W such that 5w, pf 1i 2 Zm 1w 2 Z 5 Zm 1 pf2 Z 5 1, 1ii 2 m 1w 2 [ pF < 5w6 and m 1 pf2 [ W < 5 pf6 and 1iii 2 m 1w 2 5 pf 3 m 1 pf2 5 w and 1iv 2 5f : Z 5 pf : m 1 pf2 [ W 6 Z # 1. VOL. 97 NO. 5 1967NIEDERLE: COMpETITIVE WAgEs IN A MATCH WITH ORDERED CONTRACTs For any matching m, in slight abuse of notation, let m 1 f 2 be the contract worker pair in case f has one of its contracts filled, and otherwise let m 1 f 2 be f. A matching is stable if 1i 2 5w, pf , f : If m 1w 2 5 pf then m 1w 2 sw w, and m 1 f 2 sf f ; and 1ii 2 E/ f , pf, w such that pf sw m 1w 2 and 1pf , w 2 sf m 1 f 2 . PROOF OF THEOREM 4 (Stability): Stability in the case of firm proposing MDA is trivial, as any firm makes an offer to any worker it prefers more, and gets rejected by that worker (which implies the worker has a better offer in hand). To show stability in the worker proposing MDA outcome, note that for a given set of contracts used in the final MDA step, the outcome is stable to deviations that use only these contracts, because the DA yields stable outcomes (see Gale and Shapley 1962). Therefore, I need only show that no worker prefers a position pi j that got reverted to pi j11. Suppose that at some step in the MDA, at the end of the DA part, a position pi j is unfilled. Let the interim matching be m, where m is the worker optimal stable match given the contracts available. Then, at m, for any worker w eligible for pi j, m 1w 2 sw pi j. Now pi j gets reverted into pi j11. Technically, this is equivalent to adding a new firm to an existing market. By Gale and Sotomayor (1985), adding a firm implies that the new worker optimal stable match m9 satisfies for any worker w: m91w 2 sw m 1w 2 . That is, every worker eligible for pi j still has m91w 2 sw pi j. This is true for any reversion, that is, no worker would accept a contract that was reverted into another contract. PROOF OF THEOREM 5 (Firm-Optimal Stable Match): For a firm f, define a worker contract pair 1pf, w 2 to be achievable if there exists a stable matching at which firm f is matched to worker w at contract pf . I show, by induction, that the stable outcome produced by the firm proposing MDA matches every firm to their most preferred achievable worker contract pair, and is therefore the (unique) firm optimal stable matching. Assume that up to a given step in the procedure, no firm has yet been rejected at a contract by a worker who is achievable. At this step, suppose that worker w rejects firm f at contract pf . If worker w rejects firm f at contract pf as unacceptable (i.e., w sw pf ), then this worker is unachievable at this contract and I’m done. If worker w rejects firm f at contract pf in favor of a firm g at contract pg, I show that w is not achievable for firm f at contract pf . Firm g prefers w at pg to any other worker contract pair except for those workers who have already rejected firm g at contract pg and at any contracts in place before the contract got reverted into pg , and hence (by the inductive assumption) are unachievable to firm g. Consider a hypothetical matching m that matches firm f to worker w at contract pf and everyone else to an achievable worker contract pair. Then firm g prefers w at contract pg to the achievable worker-contract pair at m. So, the matching m is unstable, since it is blocked by 1g, pg, w 2 who prefer each other to their match at m. Therefore, there is no stable matching that matches f to w at pf , and so worker w is not achievable to firm f at contract pf , which completes the proof. PROOF OF THEOREM 6: The worker proposing and firm proposing MDA follow the same steps, the only difference being that the interim matching is reached by either a firm proposing or a worker proposing DA. However, D. G. McVitie and L. B. Wilson (1970) and Roth (1984a) showed that for a given set of workers and firms, all stable matchings have the same workers and positions matched and share the set of unmatched workers and positions. This implies that at any interim match at the end of a DA step, in both MDA algorithms, the same positions are unfilled and get reverted into the same set of new contracts. I have already made the argument that the DA part of firms can as easily be thought of as one in which all former offers are annulled and remade. PROOF OF COROLLARy 1: By adding undesirable contracts at the top of the preference list (or scratching them), a firm does not influence the set of stable matchings. Hence, a firm does not influence the set of contracts that are the outcome of both the firm and worker proposing MDA. DECEMBER 20071968 THE AMERICAN ECONOMIC REVIEW REFERENCES Anantham,  Siva,  and  Jennifer  Stack.  2006. “Unraveling and Wage Formation in Entry- Level Markets.” Unpublished. Artemov,  Georgy.  2006. “Matching and Price Competition: Would Personalized Prices Help?” http://www.econ.brown.edu/students/Georgy_ Artemov/artemov_personalized_mistakes_ IJGT_f.pdf. Avery,  Christopher,  Andrew  Fairbanks,  and  Richard Zeckhauser. 2003. The Early Admis- sions game: Joining the Elite. Cambridge, MA: Harvard University Press. Bulow,  Jeremy,  and  Jonathan  Levin.  2006. “Matching and Price Competition.” American Economic Review, 96(3): 652–68. Coles, Peter. 2005. “Strategic Behavior in Match- ing Markets.” Unpublished. Crawford,  Vincent  P.  Forthcoming. “The Flex- ible-Salary Match: A Proposal to Increase the Salary Flexibility of the National Resident Matching Program.” Journal of Economic Behavior and Organization. Crawford,  Vincent  P.,  and  Elsie  Marie  Knoer.  1981. “Job Matching with Heterogeneous Firms and Workers.” Econometrica, 49(2): 437–50. Gale, David, and Lloyd S. Shapley. 1962. “College Admissions and the Stability of Marriage.” American Mathematical Monthly, 69(1): 9–15. Gale, David, and Marilda Sotomayor. 1985. “Some Remarks on the Stable Matching Problem.” Discrete Applied Mathematics, 11: 223–32. Hatfield,  John  William,  and  Paul  R.  Milgrom.  2005. “Matching with Contracts.” American Economic Review, 95(4): 913–35. Kelso, Alexander S., Jr., and Vincent P. Crawford.  1982. “Job Matching, Coalition Formation, and Gross Substitutes.” Econometrica, 50(6): 1483–1504. Kojima,  Fuhito.  2007. “Matching and Price Competition: Comment.” American Economic Review, 97(3): 1027–31. In the worker proposing MDA, no firm can benefit from delaying its reversion of positions, as the more steps of the MDA pass, the more desirable the competing positions become. Since the DA step restarts whenever there is a change in a contract, delaying to revert a position, that is, having a round in which a position by a firm is unfilled has no effect. In the firm proposing algorithm, the statement is equivalent to the statement that, in a regular DA algorithm, some firms may start making offers only after others firms have already made offers. This does not affect the outcome of a DA algorithm. Kojima,  Fuhito,  and  Parag  A.  Pathak.  2006. “Incentives and Stability in Large Two-Sided Matching Markets.” http://www.people.fas. harvard.edu/~kojima/largematchingmarket. pdf. Konishi,  Hideo,  and  Margarita  Sapozhnikov.  2006. “Decentralized Matching Markets with Endogeneous Salaries.” http://www2.bc.edu/ ~konishih/dcmatch4.pdf. McKinney,  C.  Nicholas,  Muriel  Niederle,  and  Alvin  E.  Roth.  2005. “The Collapse of a Medical Labor Clearinghouse (and Why Such Failures Are Rare).” American Economic Review, 95(3): 878–89. McVitie,  D.  G.,  and  L.  B.  Wilson. 1970. “Stable Marriage Assignments for Unequal Sets.” BIT Numerical Mathematics, 10: 295–309. Niederle, Muriel, Deborah D. Proctor, and Alvin  E.  Roth.  2006. “What Will Be Needed for the New GI Fellowship Match to Succeed?” gastroenterology, 130: 218–24. Niederle,  Muriel,  and  Alvin  E.  Roth.  2003a. “Relationship between Wages and Presence of a Match in Medical Fellowships.” Journal of the American Medical Association, 290(9): 1153–54. Niederle,  Muriel,  and  Alvin  E.  Roth.  2003b. “Unraveling Reduces Mobility in a Labor Market: Gastroenterology with and without a Centralized Match.” Journal of political Economy, 111(6): 1342–52. Niederle, Muriel, and Alvin E. Roth. 2004. “The Gastroenterology Fellowship Match: How It Failed, and Why It Could Succeed Once Again.” gastroenterology, 127: 658–66. Niederle, Muriel, and Alvin E. Roth. 2005. “The Gastroenterology Fellowship Market: Should There Be a Match?” American Economic Review, 95(2): 372–75. Niederle, Muriel, and Alvin E. Roth. 2007. “Mak- ing Markets Thick: Designing Rules for Offers and Acceptances.” http://www.stanford.edu/ ~niederle/MakingMarketsThick.May2007.pdf. Roth, Alvin E. 1984a. “The Evolution of the Labor Market for Medical Interns and Residents: A