Strategyproof Matching with Minimum Quotas

Transcription

1 Strategyproof Matching with Minimum Quotas Daniel E. Fragiadakis Atsushi Iwasaki Peter Troyan Suguru Ueda Makoto Yokoo October 7, 2013 Abstract We study matching markets in which institutions may have minimum (in addition to the more standard maximum) quotas. Minimum quotas are important in many settings (for example, hospital residency matching, military cadet matching, and school choice) but popular mechanisms are unable to accommodate them. We introduce two new classes of strategyproof mechanisms that allow for minimum quotas as an explicit input, and show that our mechanisms improve welfare relative to current approaches. Because of an incompatibility between standard fairness and nonwastefulness axioms in the presence of minimum quotas, we introduce new second-best axioms and show that they are satisfied by our mechanisms. Last, use computer simulations to quantify (i) the number of agents who will strictly prefer our mechanisms and (ii) how far they are from the first-best axioms of fairness and nonwastefulness. Combining both the theoretical and simulation results, we argue that our mechanisms should improve the performance of matching markets with minimum quota constraints in practice. JEL Classification: C78, D61, D63, I20 Keywords: acceptance minimum quotas, school choice, strategyproofness, envy, fairness, deferred We are grateful to Brian Baisa, John Hatfield, Fuhito Kojima, Scott Kominers, Muriel Niederle, Al Roth and members of the Stanford market design reading group for helpful comments. We thank Naoyuki Hashimoto and Masahiro Goto for research assistance. Fragiadakis and Troyan: Department of Economics, Stanford University, 579 Serra Mall, Stanford, CA, Iwasaki, Ueda, and Yokoo: Graduate School of Information Science and Electrical Engineering, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, , Japan. s: 1

2 1 Introduction The theory of matching has been extensively developed for markets in which the agents (students/schools, hospitals/residents, workers/firms) have maximum quotas that cannot be exceeded. 1 However, in many real-world markets, minimum quotas are also present, and there is a lack of mechanisms that take minimum quotas into account. This paper fills this gap by providing new strategyproof mechanisms that fill all minimum quotas and satisfy other important desiderata (fairness and e ciency) as much as possible. There are many examples of matching problems with minimum quotas. School districts may need at least a certain number of students in each school in order for the school to operate, as in college admissions in Hungary (Biró, Fleiner, Irving, and Manlove (2010)). In medical residency matching markets in many countries, there is a shortage of doctors assigned to hospitals in rural areas. The United States Military Academy (USMA) solicits cadet preferences over assignments to various Army branches, and each branch has minimum manning requirements. In the context of schools, minimum quotas are important not only in assigning students across schools, but also in assigning students to classes within schools. For example, computer science students at Kyushu University must all complete a laboratory requirement. Students submit preferences over the labs, but each lab has certain minimum and maximum quotas. Matching with minimum quotas is also important to firms that want to assign employees to specific projects: for example, newly graduated doctors often must complete an intern year in which they rotate through various departments in a hospital. The hospitals consider doctor preferences when making assignments, but each department has a minimum sta ng requirement which must be satisfied above all else. 2 Because standard matching mechanisms cannot accommodate minimum quotas explicitly, many markets choose to impose artificially lower maximum quotas as a way to force agents to apply to institutions that would not have otherwise filled their minimum quotas. We call this approach artificial caps. For example, in hospital-resident markets in Japan, the Japan Residency Matching Program places regional caps on urban areas such as Tokyo that are less than the total number of positions actually o ered by hospitals in the region. They do so before running the standard deferred acceptance algorithm so that during the running of the algorithm, more doctors will be forced to apply to hospitals in rural areas. 3 An October 1, 2007 memorandum from the Army to USMA describes a similar procedure for their cadet assignment algorithm, whereby before the algorithm is run, artificial caps are imposed on popular branches to ensure that the less popular 1 See Roth and Sotomayor (1990) for a comprehensive survey of many results in this literature. 2 The Kyushu University computer science problem can also be interpreted in this context if we think of the students as employees and the department as a firm. 3 See Kamada and Kojima (2013) for more detail on the specifics of this market. 2

3 branches will meet their manning requirements. 4 Notice that imposing artificial caps will also obscure the presence of minimum quotas in markets in which they are truly present, but in which designers simply lack good tools to handle them. Though the ultimate goal is to satisfy the minimum quotas, artificial caps only does so implicitly, by eliminating positions ex-ante, without regard to agent preferences. This leads to e ciency losses, since after demand is realized, some positions in high demand will end up below their true capacity, making it possible to reassign the agents and make everyone better o. Our mechanisms recover these e ciency losses by lowering capacities at popular institutions only when it is actually necessary, which depends on the realized demand of the agents. Importantly, we show that we are able to increase e ciency without compromising any incentive or fairness properties. More specifically, we start by introducing standard axioms that have been identified as key in the matching literature without minimum quotas: fairness, 5 nonwastefulness, and strategyproofness. Unfortunately, with minimum quotas, matchings that are simultaneously fair and nonwasteful may not even exist, which means that one of these axioms must be weakened. Because di erent institutions may weigh the relative importance of each axiom di erently, we provide two mechanisms: extended-seat deferred acceptance (ESDA) will be fair, while multistage deferred acceptance (MSDA) will be nonwasteful; importantly, both mechanisms are strategyproof. 6 ESDA works by dividing the seats into two classes. Each school is given a number of regular seats equal to its minimum quota, and a number of extended seats equal to the di erence between its minimum and maximum quota. Students apply as in DA, first to the regular seats and then to the extended seats according to their preferences. By restricting the number of extended seats that can be assigned in an appropriate manner, we ensure that all of the regular seats will be filled and all minimum quotas will be satisfied. MSDA, on the other hand, runs by first reserving a number of students such that, no matter how the remaining students are assigned, we will have enough students reserved to fill any remaining minimum quotas. The students not reserved are then assigned according to standard DA, and we calculate how many minimum quota seats remain. This process is then repeated until all students are assigned. Note that both of these mechanisms take the minimum quotas as an explicit input, and use this information together with the student preferences to allocate the flexible seats (those above the minimum quota) based on 4 Sönmez and Switzer (2013) and Sönmez (2013) provide further details on military cadet matching and study other aspects of the problem unrelated to minimum quotas. 5 Fairness means that if one student envies the assignment of another, then the second student must have a higher priority at her assigned school than the first. It is also called no justified envy in the school choice literature. See Section 2 for a formal definition. 6 Our mechanisms reduce to the standard DA algorithm when the minimum quotas are 0 or are equal to the maximum quotas. 3

4 student demand, in contrast to artificial caps. We are the first paper to provide (nontrivial) strategyproof mechanisms for matching markets with minimum quota constraints. While it is impossible to simultaneously satisfy the standard axioms of fairness and nonwastefulness with minimum quotas, they are still appealing normative properties, and hence it is desirable to weaken them as little as possible. Theoretically, we provide new second-best axioms and show that they are satisfied by our mechanisms. In addition, we use computer simulations to quantify how far our mechanisms are from the first-best axioms, since an institution that opted for a nonwasteful mechanism likely would desire to at least minimize the number of priority violations (blocking pairs), and vice-versa for an institution that opted for a fair mechanism. Finally, we compare our new mechanisms to artificial caps DA (ACDA), which is currently in use in several markets (see above). Theoretically, ACDA is strategyproof and fair (as is ESDA), but it does not satisfy even the weaker definition of nonwastefulness that ESDA does. Simulations show that both ESDA and MSDA waste significantly fewer seats than ACDA, and are overwhelmingly preferred by the students, in the sense that the rank distributions of ESDA and MSDA first order stochastically dominate that of ACDA. Thus, from a policy perspective, there seems to be little theoretical or empirical justification for using ACDA. As mentioned above, if fairness is a larger concern, then ESDA should be used, while if nonwastefulness is paramount, then MSDA is the better mechanism. Let us close the introduction by emphasizing that the main goal of this paper is to provide practical mechanisms that can easily be implemented in markets in which minimum quotas must be satisfied. Any such mechanism must inevitably give up at least one of fairness or nonwastefulness, at least in their strongest form. This necessitates non-standard ways of thinking about such properties, which is an additional contribution of this paper. We follow the school choice literature in taking fairness and nonwastefulness as normative axioms, and so while one of them must be weakened, we hope to satisfy the weakened axiom as much as possible. 7 By showing that our mechanisms satisfy second-best theoretical axioms and using computer simulations to quantitatively measure the losses in fairness and nonwastefulness, we argue that while minimum quotas unfortunately lead to some impossibilities, the mechanisms we introduce do indeed satisfy desirable properties, and should be useful in practical applications where minimum quotas are an important concern. 7 Starting with Gale and Shapley (1962), the matching literature has classically combined fairness and nonwastefulness into one property known as stability, which it then interprets as a positive (rather than normative) constraint. See Section 2 for further discussion of the two viewpoints, and why the normative approach is more relevant to the current work. 4

5 Related Literature While minimum quotas seem like a natural extension to the standard matching models, there is little work on the topic, most likely because the problem becomes di cult and many of the results from the previous literature have been negative. The paper most related to this one is Ehlers, Hafalir, Yenmez, and Yildirim (2011), who study diversity constraints in school choice. They also note several impossibility results once minimum quotas are introduced, and provide mechanisms that are either manipulable or may not satisfy all minimum quotas. We instead provide strategyproof mechanisms that guarantee all minimum quotas will be satisfied. We view this as an important contribution, as the matching literature has found strategyproofness to be a key property in a wide variety of settings (see, for example, Roth (1991), Abdulkadiroğlu, Pathak, Roth, and Sönmez (2005), Ergin and Sönmez (2006), Chen and Sönmez (2006), Pathak and Sönmez (2008), and Pathak and Sönmez (2012), who discuss the negative consequences of manipulable mechanisms). Fragiadakis and Troyan (2013) consider a model similar to Ehlers, Hafalir, Yenmez, and Yildirim (2011), but provide a di erent class of mechanisms. 8 Kojima (2012) show that some types of a rmative action quotas may actually hurt the very minorities they are supposed to help. Hafalir, Yenmez, and Yildirim (2013) use minority reserves (as opposed to majority quotas) to alleviate the problem identified by Kojima (2012), while Kominers and Sönmez (2012) generalize the mechanism of Hafalir, Yenmez, and Yildirim (2013) to allow for slot-specific priorities. Westkamp (2012) analyzes complex (maximum) quota constraints in the German university admissions system, while Braun, Dwenger, Kübler, and Westkamp (2012) conducts an experimental analysis of the same system. Biró, Fleiner, Irving, and Manlove (2010) study college admissions in Hungary, in which colleges may declare minimum quotas for their programs. They study the di culty (from a computer science perspective) of finding stable matchings when minimum quotas are introduced, but do not provide explicit mechanisms or consider incentive or e ciency issues, as we do here. Hamada, Iwama, and Miyazaki (2011) also study matching with minimum quotas from a computer science perspective, showing that minimizing the number of blocking pairs is an NP-hard problem when minimum quotas are imposed. 8 The use of strategyproof mechanisms also advances the so-called Wilson Doctrine (Wilson (1987)), which argues that in order to analyze practical problems, economic models should reduce their reliance on common knowledge assumptions amongst the players. In particular, strategyproof mechanisms require no knowledge or beliefs about the preferences of others in order for students to formulate a best-response (see also Bergemann and Morris (2005) for a discussion of robustness to beliefs in general mechanism design settings). It should be noted, however, that while economists have generally advocated for the replacement of manipulable mechanisms with strategyproof ones, the benefits are not without a cost: Miralles (2009), Abdulkadiroğlu, Che, and Yasuda (2011), Featherstone and Niederle (2011), and Troyan (2012) have shown that nonstrategyproof mechanisms (the Boston mechanism in particular) may sometimes outperform strategyproof ones, at least in equilibrium. 5

6 As a final possible application, consider the medical residency market studied first by Roth (1984). In these markets, the shortage of doctors in rural areas is a well-known problem, and the so-called rural hospitals theorem suggests it is di cult to solve (see Roth (1986), Martinez, Masso, Neme, and Oviedo (2000), and Hatfield and Milgrom (2005)). Kamada and Kojima (2013) discuss one possible solution used in Japan: capping the number of residents who can be assigned to a given region. To the extent that these caps are simply an ad-hoc way to ensure some true minimum quotas are satisfied, imposing these quotas directly and using one of our mechanisms is another possible approach. Methodologically, our approach of ranking mechanisms that cannot achieve full fairness on the entire preference domain is similar to approaches taken by Pathak and Sönmez (2012) and Carroll (2011), who introduce methods for ranking nonstrategyproof school choice and voting mechanisms (respectively) by their degree of vulnerability to manipulation. The remainder of this paper is organized as follows. In Section 2, we present a basic model with minimum and maximum quotas and introduce some standard axioms. We show how these axioms become incompatible in the presence of minimum quotas, and so must be weakened in some way. Section 3 introduces the ESDA algorithm and discusses its properties, while Section 4 does the same for the MSDA algorithm. Section 5 uses computer simulations to quantitatively study ESDA, MSDA, and ACDA with respect to fairness, nonwastefulness, and student welfare. Section 6 concludes. All proofs are in the appendix, unless otherwise stated. 2 Matching with minimum quotas 2.1 Model For convenience, we use the language of matching students and schools, but our model can be applied to many other types of markets, including those mentioned in the introduction. In addition, we will sometimes speak of the goals of a school district, but in general this can be interpreted as any type of institution/authority that desires to match agents to positions and/or objects (e.g., a firm or the military). A market consists of (S, C, p, q, S, C). S = {s 1,s 2,...,s n } is a set of n students, C = {c 1,c 2,...,c m } asetofm schools ( colleges ), and p =(p c1,...,p cm ) and q =(q c1,...,q cm )are vectors of minimum and maximum quotas, respectively, for each school. We assume p c 0, q c > 0, and p c apple q c for all c 2 C and P c2c p c <n< P c2c q c to ensure a feasible matching exists. 9 Define 9 We assume both inequalities strict because if n = P c2c p c or P c2c q c there is no flexibility in the seats to be assigned and the standard DA algorithm can be used. Our model only becomes interesting when choices must be 6

7 P e = n c2c p c to be the number of excess students above the sum of the minimum quotas. To make the problem interesting, we additionally assume that m>2and p c <q c for at least two schools. 10 Each student s has a strict preference relation s over C, while each school c has a strict priority relation c over S. Vectors of such relations, one for each agent, are denoted S =( s) s2s for the students and C =( c) c2c for the schools. Let P denote the set of possible preference relations over C, and P S denote the set of all preference vectors for all students. As is standard in the school choice literature, the school priorities are fixed and known to all students (in applications, priorities are often related to such things as the distance a student lives from a school or whether or not a student has a sibling attending the school). For now, we assume all students are acceptable to all schools and vice-versa. This is a reasonable assumption in many contexts: in public school choice, school districts are legally required to assign every student a seat at some school, and school districts often assign students to schools they did not express any preference for (though they may still of course take their outside option); in military cadet matching, cadets are obligated to serve in the Army, and so must express a preference over every possible branch. 11 While obviously required for a feasible matching to be guaranteed to exist, this assumption is not practically necessary to actually run our algorithms; in Section 5, we will allow students to declare schools as unacceptable, and show that our mechanisms still can be run and outperform artificial caps. A matching is a mapping µ : S [ C! 2 S [ C that satisfies: (i) µ(s) 2 C for all s 2 S, (ii) µ(c) S for all c 2 C, and (iii) for any s 2 S and c 2 C, we have µ(s) =c if and only if s 2 µ(c). A matching is feasible if p c apple µ(c) appleq c for all c 2 C. LetM denote the set of feasible matchings. A mechanism : P S!Mis a function that takes as an input any possible preference profile of the students and gives as an output a feasible matching of students to schools. If the students submit preference profile S2 P S, then ( S) is the assigned matching, and we write i( S) for the assignment to agent i 2 S [ C. Ifi 2 S, then i ( S) 2 C is the school student i is assigned to, while if i 2 C, then i ( S) S is the set of students assigned to school i. 12 made about where to assign the flexible seats. 10 These assumptions are likely to be satisfied in any real-world market of reasonable size. The special cases where these assumptions do not hold are dealt with in Appendix F. 11 See the appendix for a more detailed discussion of markets where the assumption of complete preference listings is satisfied. 12 Since student preferences are the only private information, we only explicitly write this as a function of S; however, this function of course implicitly depends on C, p, q, and C as well. 7

8 2.2 Axioms In this section, we discuss several axioms that are important both theoretically and practically. 13 The axioms have all been used previously in the matching literature, and have been key considerations in the redesign of school choice mechanisms in many school districts. Each of the axioms is intuitively appealing, but unfortunately, it will turn out that with minimum quotas, an impossibility result will obtain. Remark 1. There is a important distinction in the matching literature as to whether the axioms should be treated as positive or normative. This paper mostly adheres to a normative viewpoint. We will discuss the positive vs. normative distinction in further detail below, after providing the axioms themselves. In order to define our first axiom (fairness), we must introduce the notion of a blocking pair. Definition 1. Given a matching µ, student-school pair (s, c) form a blocking pair if c and s c s 0 for some s 0 2 µ(c). s µ(s) In words, student s would rather be matched to school c than her current match µ(s), and she has higher priority at c than some student s 0 who is currently assigned there; thus, s has a claim on a seat at c over student s 0. In some papers, it is said that s has justified envy towards s 0, and we will sometimes use these terms interchangeably. Priorities are often based on criteria such as distance between a student and the school or test scores, and if one student justifiably envies another, she may be able to take legal action against the school district (see Abdulkadiroğlu and Sönmez (2003)). Thus, an important goal of many school districts is for a matching to contain no such blocking pairs. When this is true, we say that the matching is fair. 14 Definition 2. A matching µ is fair (or eliminates all justified envy) if no student-school pair (s, c) can form a blocking pair. The next important axiom is nonwastefulness. Say that student s claims an empty seat at school c if c s µ(s) and µ(c) <q c. 15 Definition 3. A matching µ is nonwasteful if whenever a student s claims an empty seat at school c, we have µ(µ(s)) = p µ(s). 13 This section has benefited greatly from both an editor and an anonymous referee, who helped us clarify our thinking about the proper way to understand and interpret the axioms we use. 14 For example, fairness was an extremely important criterion to administrators of the Boston school district when they were redesigning their school assignment mechanism. See Abdulkadiroğlu, Pathak, Roth, and Sönmez (2005). 15 Nonwastefulness is an intuitive e ciency requirement. A strengthening of nonwastefulness is Pareto e ciency. However, it is well-known that even without minimum quotas, Pareto e ciency is incompatible with fairness. The widely used deferred acceptance algorithm will find a matching that is nonwasteful and fair, but it will in general not be Pareto e cient, even without minimum quotas. 8

9 Put another way, this definition says that if student s prefers school c to her current assignment µ(s), school c has an empty seat, and the number of students assigned to her current school µ(s) is strictly above its minimum quota, then student s should be moved to that school. The last part is what di erentiates this definition from the standard definition of nonwastefulness; if it is not satisfied, then moving s would violate feasibility, and so the matching is nonwasteful. These properties also have counterparts for mechanisms: we say is fair (nonwasteful) if for every preference profile it produces a fair (nonwasteful) matching. Remark 2. (Positive vs. normative interpretations) Starting with Gale and Shapley (1962), most of the matching literature has combined fairness and nonwastefulness into a single axiom called stability. Classical theory treats stability as a positive concept. In hospital residency matching, for example, if a matching is not stable, a hospital and a doctor may block the match by deviating from their assignments and signing contracts with each other. In these markets, whether a blocking pair involves filling an empty seat or replacing a less preferred doctor with a more preferred one is immaterial, and so fairness and nonwastefulness can be combined to stability. The literature then focuses on stable matches because when there is no outside enforcement, blocking pairs will lead to failure of the mechanism. Recently, however, there have been many applications of matching theory that deviate from this classical paradigm and do indeed have mechanisms in place to enforce a match. In these markets, even though the market organizer may have this power, implementing a match that ignores the preferences and priorities still seems undesirable. The first paper in this vein (to our knowledge) is Balinski and Sönmez (1999), who study Turkish college admissions. In this market, all seats are controlled by a centralized authority, and so participants deviating from their assignment is not a concern. They view the school seats as merely objects to be consumed, and so interpret the schools as passive participants that have priorities (based on test scores, for example) rather than active participants with preferences (hence, these are often called one-sided matching problems). These features lead them to separate stability into fairness and nonwastefulness and discuss each as important normative axioms. Public school districts in many cities in the United States similarly control all seats and thus have the power to prevent blocking pairs, but still view fairness and nonwastefulness as important from a normative perspective (see Abdulkadiroğlu and Sönmez (2003) or Abdulkadiroğlu, Pathak, Roth, and Sönmez (2005) for examples). This normative framework can also be applied to other markets beyond school choice. Military cadet matching is an excellent example: while the Army formally has the power to assign cadets to whatever branch they wish, it is a rooted institutional feature that cadets who rank higher on the Order of Merit List deserve to receive more preferred assignments. Thus, while satisfying the 9

10 minimum quotas is paramount, fairness is still an important normative concern. 16 Social choice theory provides a useful analogy for the normative approach we use. Consider, for example, Arrow s Impossibility Theorem (Arrow (2012)). The analysis starts by proposing several axioms for social choice functions (universal domain, Pareto e ciency, independence of irrelevant alternatives, and non-dictatorship) which, upon introspection, are all appealing. While each axiom has an intuitive normative appeal, the impossibility theorem says no social choice function can satisfy them all simultaneously. In our setting, fairness and nonwastefulness are both intuitively normatively appealing to many institutions, but a similar impossibility result will obtain: no mechanism can satisfy both axioms (in fact, matchings that are simultaneously fair and nonwasteful may not even exist). Given this, one of these axioms must be weakened, and we go on to provide two mechanisms: one that is fair (and weakly nonwasteful), and one that is nonwasteful (and weakly fair). The last important axioms concern incentives for the students to report truthfully. 0 Definition 4. A mechanism is strategyproof if s( S) s s ( s, S\{s}) for all S 2P S, 0 s 2 S, and s 2P. In words, a mechanism is strategyproof if no student ever has any incentive to misreport her preferences, no matter what the other students report. As discussed in the introduction, strategyproofness has been found to be a very important property in the success of matching mechanisms, for both positive and normative reasons. All of the mechanisms we provide will be strategyproof. In addition, our mechanisms will be immune to certain types of group manipulations. Definition 5. A mechanism is group strategyproof if there does not exist a preference profile S 2P S, a group of students S 0 S, and a preference profile ( 0 s ) s2s 0 0 S 0 such that s( 0 S 0, S\S 0) s s( S) for all s 2 S 0. That is, there is no subset of students who can jointly misreport their preferences and make every member of the set strictly better o. Clearly, group strategyproofness implies strategyproofness. It is well known in the literature that without minimum quotas, DA is group strategyproof (Hatfield and Kojima (2009)). Both of our new mechanisms will also be group strategyproof. 16 There are even some seemingly two-sided markets where stability constraints actually have more natural interpretations as normative: in Japan, the Japanese Resident Matching Program prevents hospitals and doctors from forming blocking pairs if doing so violates certain regional caps. Kamada and Kojima (2013) provide normative justifications for stability constraints in this context. 10

11 2.3 Impossibility of a simultaneously fair and nonwasteful matching Unfortunately, it turns out that matchings that are simultaneously fair and nonwasteful may not even exist in the presence of minimum quotas. This is shown in the following example, which is very instructive in illustrating the main problem that arises. 17 Let there be two students s 1 and s 2 and three schools c 1,c 2, and c 3. The student preferences and school priorities and quotas are given in the following table. Note that each school has 1 seat, and school c 1 has a minimum quota of 1. All other minimum quotas are 0. c 1 c 2 c 3 s 2 s 2 s 1 s 1 s 1 s 2 p c q c s 1 s 2 c 2 c 3 c 3 c 2 c 1 c 1 The minimum quota requirement at c 1 means that one of s 1 or s 2 must be assigned there; nonwastefulness then requires that the other student be assigned his most preferred school. However, the student assigned c 1 will then justifiably envy the other student. For example, in the allocation indicated by the boxes, s 1 prefers c 3 and has higher priority than s 2 there, and thus forms a blocking pair. The case where s 2 is assigned c 1 is similar. This example shows that there is no mechanism that simultaneously satisfies all of our axioms (recall the analogy with Arrow s Theorem above). 18 Thus, in order to proceed, either fairness or nonwastefulness must be weakened. Because di erent institutions may judge the relative importance of the two properties di erently, we will provide two mechanisms: one that keeps fairness (and weakens nonwastefulness), and one that keeps nonwastefulness (and weakens fairness). Even though we do not expect weakening these axioms to result in market failure (since we interpret the axioms as normative), it is still desirable to weaken them as little as possible. In Section 5, we quantify how far our mechanisms are from ideal, based on the idea that all else equal, a school district that opted for a nonwasteful mechanism would still like to minimize the number of blocking pairs, and a school district that opted for a fair mechanism would still like to waste as few seats as possible. Besides computer simulations, we also provide new theoretical axioms to accommodate the impossibility result. In the next paragraph, we discuss weak nonwastefulness, while in Section 4, we do the same for fairness. To weaken nonwastefulness, we start by saying that a student s weakly claims a seat at school c if there exists a chain of students and schools (c 0,s 1,c 1,s 2,...,c J 1,s J ), J 1, such that 17 This example was first noted in Ehlers, Hafalir, Yenmez, and Yildirim (2011). 18 This example can be easily embedded in larger markets. 11

12 (i) µ(c 0 ) <q c 0, (ii) c j 1 s j cj, (iii) µ(s j )=c j, and (iv) c J 1 = c and s J = s. In words, this says that while there may not be an empty seat at c under matching µ for s to claim, there is a way to reassign the students so that s 1 takes an empty seat at school c 0, s 2 takes the seat he vacates, and so forth, ending with s being assigned a seat at c, and this reassignment makes every student in the chain strictly better o. Note that when J = 1, weakly claiming a seat is equivalent to claiming an empty seat. Definition 6. Given a matching µ, let Z(µ) be the set of students who weakly claim a seat at some school. If µ(µ(s)) >p µ(s) for all s 2 Z(µ), then matching µ is strongly wasteful. If matching µ is not strongly wasteful, we say it is weakly nonwasteful. We say that mechanism for every possible preference profile. is weakly nonwasteful if it produces a weakly nonwasteful matching To understand weak nonwastefulness intuitively, think of the following scenario. Upon assigning a matching µ, a school district may receive calls from parents requesting that their child be moved to an empty seat. If the school district grants these first-round requests, the seats vacated by these students will open up, and an additional group of parents may call and request that their child be moved to the now vacated seat, and so forth. The set Z(µ) captures all parents who may request an empty seat in such a way. If the school district cannot grant all requests to move while still ensuring feasibility, it may instead opt for a policy of not granting any student requests to move to any empty seat. Weak nonwastefulness gives the school district justification for not entertaining any requests to move, as this will lead to an avalanche of other requests, not all of which can be granted. Nonwastefulness is a special case of weak nonwastefulness in which only the first-round requests are considered. 19 It is important to note that this definition does have content: the artificial caps mechanism described in the Introduction (and discussed formally below) will not be even weakly nonwasteful, while our new mechanisms will be. Weak nonwastefulness captures a formal sense in which a fair mechanism does not give up on nonwastefulness entirely, even if it must do so somewhat due to the aforementioned impossibility. 19 Ehlers, Hafalir, Yenmez, and Yildirim (2011) weaken nonwastefulness in an alternative way by calling a matching constrained nonwasteful if moving a student to an empty seat he claims would result in a matching that is not fair. Unfortunately, they show that fairness and constrained nonwastefulness in their sense are incompatible with strategyproofness. Since we are searching for strategyproof mechanisms, we must use the alternative weakening of nonwastefulness given here. 12

13 2.4 Artificial caps deferred acceptance (ACDA) If we are willing to give up on nonwastefulness entirely, there is a simple mechanism that is strategyproof and fair, namely, the artificial caps deferred acceptance (ACDA) mechanism described in the introduction. 20 Artificial caps deferred acceptance proceeds by imposing artificial maximum quotas at every school. Then, deferred acceptance is run with only these maximum quotas, ignoring the minimum quotas. By imposing su ciently stringent artificial caps, we can ensure that no matter how the students are allocated, the minimum quotas will be satisfied. As an example, consider a market of n = 100 students and m = 10 schools, each with p c =5 and q c = 20. Now, imagine imposing artificial caps of qc = 10 at each school and running the standard DA algorithm. By doing so, exactly 10 students will be assigned to each school, thereby satisfying all minimum and maximum quotas. However, this may be very wasteful if, for example, there is high demand for c 1, because many students could be moved there and made better o without violating any quotas. The problem is that the mechanism has no flexibility, and eliminates seats without regard to student preferences. (Note also that to an outside observer it would appear that each school had only 10 seats with no minimum quotas. In reality, this is not the case, and only appears so because the school district has no good way of handling minimum quotas.) More formally, the school district may impose an artificial maximum quota q c at each c. By choosing the vector q =(q c 1,...,q c m ) wisely, the school district can ensure that no matter how the students are allocated, as long the artificial caps q are satisfied, the true quota vectors p and q will be satisfied as well. Definition 7. The vector q ensures a feasible matching if the following holds: µ(c) appleq c 8c 2 C =) p c apple µ(c) appleq c 8c 2 C. In words, this definition simply says that q ensures a feasible matching if, whenever matching µ satisfies the artificial caps q, it also satisfies the true minimum and maximum quotas p and q. In general, there will be many vectors q that ensure a feasible matching, and at least one such vector always exists: simply choose any feasible matching µ and set q c = µ(c). 21 After imposing artificial caps of q, a school district can then run any mechanism that has maximum quotas as an input and ensure a feasible matching. 22 Because the DA algorithm is a popular mechanism in many settings and is our focus in this paper, we will mostly be concerned 20 The deferred acceptance algorithm is well-known in the literature, and it is also a special case of the new mechanisms we define in section 3, and so we do not give its definition here. See Gale and Shapley (1962) for the original description, or Abdulkadiroğlu and Sönmez (2003) for a discussion of DA in the context of school choice. 21 A necessary and su cient condition for q to ensure a feasible matching is that the following hold for all c 2 C: (i) p c apple qc P apple q c and (ii) p c apple n d2c\{c} q d. 22 For example, imposing artificial caps and then using a serial dictatorship or the top trading cycles algorithm would also produce a feasible match. 13

14 with the ACDA mechanism. Because we are simply running DA under a di erent set of maximum quotas, ACDA will inherit some of the good properties of DA, namely strategyproofness and fairness. However, since ACDA eliminates seats ex-ante, without regard for student preferences, it will tend to waste seats. Formally, we have the following result. Theorem 3. For any q that ensures a feasible matching, the artificial caps deferred acceptance algorithm is group strategyproof and fair, and produces a feasible matching for all preference profiles of the students; however, ACDA is strongly wasteful. Feasibility follows by definition of q, while strategyproofness and fairness follow from the fact that the DA mechanism itself is strategyproof and fair. To see that ACDA may be strongly wasteful (i.e., is not weakly nonwasteful), consider the example in section 2.3, and impose an artificial cap of 0 at c 2 (e ectively eliminating the seat at c 2 to ensure that one of the two students will be assigned to c 1 ). Then, consider the following preferences of the agents: s 1 s 2 c 2 c 1 c 3 c 2 c 1 c 3 The ACDA algorithm gives the allocation shown in the boxes. Now, the only student who weakly claims an empty seat is s 1 who claims a seat at c 2 (in the language of Definition 6, Z(µ) = {s 1 }). Since we do not need to deny any other student s claim to an empty seat, the matching is strongly wasteful. The problem is that ACDA eliminates seats ex-ante, without any regard for student preferences. The new mechanisms that we introduce will correct this by allowing the extra seats to be assigned more flexibly, based on realized student demand. While weak nonwastefulness gives a formal sense in which ACDA is a deficient mechanism, the question of how wasteful is ACDA remains. In Section 5, we use computer simulations to quantify the wastefulness of the ACDA mechanism; in particular, we show that the new mechanisms that we introduce waste far fewer seats and will be preferred by the students. 3 Extended-seat DA (ESDA) Deferred acceptance is a very popular mechanism in practice, and so, developing DA-like algorithms that satisfy our axioms as much as possible seems to be a sensible approach for practical market design. This is indeed the likely inspiration for markets that use ACDA, but, we show 14

15 that, by considering the minimum quotas in a more careful manner, it is possible to develop alternative DA-based mechanisms that improve upon ACDA. Our first mechanism, extended-seat DA (ESDA), will be strategyproof, fair, and weakly nonwasteful. To define the ESDA algorithm, we take the original market (S, C, p, q, S, C) and define a corresponding extended market : (S, C, q, S, C). When extending the market, the set of students is unchanged. Consider a school c in the original market with a minimum and maximum quota of p c and q c, respectively. When extending the market, we divide c into two smaller schools: a standard school, which, with slight abuse of notation, we label c, and that has a maximum quota of q c = p c, and an extended school c, which has a maximum quota of q c = q c p c. Each school (standard and extended) uses the original priority relation of school c; formally, c = c = c. Thus, the set of schools is now C = C [ C = {c 1,...,c m,c 1,...,c m} and the vector of maximum quotas is q = { q c 0} c 0 2 C. Note that the extended market has no minimum quotas. By assigning P no more than e = n c2c p c students to extended schools, all standard schools will be filled to capacity, thereby satisfying all minimum quotas in the original market. For the students, preferences over C [ C are created by taking the original preference relation s and inserting school c j immediately after school c j. That is, preference relation s: c j s c k becomes s : c j s c j s c k s c k The main issue that arises is how to assign the extended seats when more than e students have applied to them. To do so, we fix an ordering of the schools, and let the schools accept students one-by-one in this order until e students have been accepted across all of the extended schools. The remaining students who have applied to the extended schools are rejected. Extended-seat deferred acceptance Choose a vector of target capacities ( q c ) c 2C such that q c apple q c and P c 2C q c apple e. Without loss of generality, order the extended schools in C as c 1 >c 2 > >c m.let µ be the matching produced in the extended market. 1. Begin with an empty matching, such that µ(s) =; for all s 2 S. 2. Choose a student s who is not currently tentatively matched to any school. If no such student exists, end the algorithm. 3. Let s apply to the most preferred school c 2 C according to s that has not yet rejected her. If c is a regular school, let school c choose the q c highest-ranked students according to c and reject the rest. If c is an extended school, proceed to step 4. 15

16 4. In this step, we consider all extended schools. For each extended school c 2 C, let S c be the set of students tentatively held at c (i.e., those students who have applied to c but have not yet been rejected). Let each school c 2 C choose the q c highest-ranked students in S c according to c ; denote this set S 0 c. Starting with a tentative match of S0 c for each school, let the schools choose, one-by-one, the best remaining student in their current applicant pool (i.e., those students in S c school s true capacity q c that have not yet been chosen), until either the is reached, or the total number of students assigned to extended schools reaches the cap e, or there are no more students who can be chosen without violating the true capacity of a school q c. Formally, set j = 1 and: (a) If either (i) the number of students assigned to extended seats across all extended schools is equal to e, or (ii) for each extended school c 2 C, the number of students chosen so far throughout step 4 is equal to min{ q c, S c }, then reject all remaining students not chosen by any extended school and return to step 2. (b) If not, let c j choose its most preferred student in S c j who has not yet been chosen, as long as the number of students chosen so far is strictly less than q c. Increment j by 1, and return to step 4(a). The algorithm above outputs a matching in the extended market µ. We then take this outcome and map it to an outcome in the original market in the obvious way: if µ(s) =c or µ(s) =c, then µ(s) =c, and the final output of the ESDA algorithm is the matching µ. 23 We next present an example to show how the ESDA mechanism runs. Example 1. There are five students s 1,...,s 5 and three schools c 1,c 2,c 3. The preferences, priorities and minimum and maximum quotas are shown in the table below. 23 In the running of the mechanism, a choice must be made about how to set the target capacities. One natural choice for the target capacities is q c = 0 for all c 2 C, which corresponds to a situation in which each school is first allowed to accept a number of students equal to its minimum quota, and then schools continue to choose students one-by-one. However, nonsymmetric target capacities may also be used if we expect some schools to be more popular than others ex-ante. 16

17 c 1 c 2 c 3 s 5 s 3 s 3 s 3 s 4 s 4 s 1 s 1 s 2 s 2 s 2 s 5 s 4 s 5 s 1 p q s 1 s 2 s 3 s 4 s 5 c 2 c 2 c 1 c 2 c 1 c 1 c 1 c 2 c 3 c 2 c 3 c 3 c 3 c 1 c 3 To run ESDA, our extended market uses schools C [ C = {c 1,c 2,c 3,c 1,c 2,c 3}. The maximum quotas are q c1 = q c2 = q c3 = 1, and q c 1 =1, q c 2 = 2 and q c 3 = 0. Note that there are no minimum quotas in the extended market. Recall that the ordering of the schools is c 1 >c 2 >c 3. The cap on the number of extended seats is e = 2. Set the target capacities at q c = 0 for all c 2 C. We additionally modify all students preferences by inserting school c j after school c j. For example, the modified preferences of student s 1 are as follows: s1 : c 2 s1 c 2 s1 c 1 s1 c 1 s1 c 3 s1 c 3. This leads to the following extended market, where the changes are shown in red: c1 c 1 c2 c 2 c3 c 3 s 5 s 5 s 3 s 3 s 3 s 3 s 3 s 3 s 4 s 4 s 4 s 4 s 1 s 1 s 1 s 1 s 2 s 2 s 2 s 2 s 2 s 2 s 5 s 5 s 4 s 4 s 5 s 5 s 1 s 1 q s1 s2 s3 s4 s5 c 2 c 2 c 1 c 2 c 1 c 2 c 2 c 1 c 2 c 1 c 1 c 1 c 2 c 3 c 2 c 1 c 1 c 2 c 3 c 2 c 3 c 3 c 3 c 1 c 3 c 3 c 3 c 3 c 1 c 3 The ESDA algorithm begins with s 1,s 2 and s 4 applying to school c 2 and students s 3 and s 5 apply to school c 1. Schools c 1 and c 2 tentatively accept s 5 and s 4, respectively. Everyone else is rejected. Students s 1 and s 2 then apply to c 2 and s 3 applies to c 1. Since the target capacities of all extended schools are 0, we allow the extended schools to admit students from their applicant pools one by one. First, c 1 admits its only applicant s 3, and then c 2 admits student s 1 (since s 1 c 2 s 2 ). At this point, two students have been admitted to extended schools, and so school c 2 must reject the student it was tentatively holding, s 2. 17

18 Student s 2 continues by applying to c 1, but is rejected in favor of s 5, who is currently sitting at c 1. She then applies to c 1. We again allow the extended schools to admit students from their applicant pools one by one. This begins by c 1 accepting s 3 (from its current applicant pool of {s 2,s 3 }). We then move to school c 2, which again admits s 1. Once again, at this point two students have been admitted to extended schools, and so s 2, who is tentatively sitting at c 1, is rejected. She then proceeds to apply to c 3 and is admitted, and the algorithm ends. The output in the extended market is µ = c 1 c 1 c 2 c 2 c 3 c 3 s 5 s 3 s 4 s 1 s 2 ; Mapping this back to a matching in the original market, the final matching is:!! µ = c 1 c 2 c 3 {s 3,s 5 } {s 1,s 4 } s 2 We now discuss the properties of the ESDA algorithm. 24 Theorem 4. The ESDA mechanism is (i) group strategyproof (ii) fair (iii) weakly nonwasteful. We show strategyproofness by first showing that no student can gain by misreporting her preferences in the extended market. Since this is a larger set of possible manipulations than in the original market, this means that no student can gain by misrepresenting her preferences in the original market either. In the extended market, strategyproofness is intuitively inherited from the standard deferred acceptance mechanism: a student need not give up trying for high ranked schools (whether regular or extended) because if she is rejected, she is still able to apply (and be admitted to) her lower ranked schools by rejecting some other student who is tentatively sitting there (i.e., the acceptances of the other students were deferred ). At a formal level, the proof in the appendix relates our model to that of Kamada and Kojima (2013), who prove strategyproofness using the matching with contracts model of Hatfield and 24 In a short extended abstract appearing in the Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems (Ueda, Fragiadakis, Iwasaki, Troyan, and Yokoo (2012)), we discuss an alternative extended-seat DA algorithm. Though the terminology is the same, the two algorithms run entirely di erently and satisfy di erent properties. 18

19 Milgrom (2005). E ectively, we can think of all of the extended seats as being controlled by one single umbrella school which has a capacity of e (all of the extended seats belong to one region in the Kamada and Kojima framework). A contract in the Hatfield-Milgrom terminology then specifies a student and the specific extended school to which she is assigned, and these contracts are chosen according to the choice function of our umbrella school. Hatfield and Milgrom (2005) show that if the choice functions of all schools satisfy a key substitutes condition, then the DA algorithm in the matching with contracts model is strategyproof, while Hatfield and Kojima (2009) show that it is in addition group strategyproof (see also Hatfield and Kominers (2012), who obtain the same result in a more general model). The standard schools obviously have substitutable choice functions, and so the fact that the umbrella school controlling all of the extended seat contracts also has a substitutable choice function means that the mechanism is group strategyproof. For fairness, note that if a student s is rejected from a regular school, it must be because that school is filled to its capacity with higher ranked students. Since the ranking of students assigned to a school only improves as the algorithm continues, at the end of the algorithm, s must be lower ranked than every student assigned to this school, and hence cannot form a blocking pair with any of these students. If, on the other hand, s is rejected from an extended school, it must be because either (a) that extended school is filled to capacity with higher ranked students, in which case s cannot form a blocking pair for the same reason given previously, or (b) e other students are tentatively assigned to extended schools. For case (b), when the e th student is accepted, the school s is rejected from is tentatively holding some set of students (possibly empty) who are higher ranked than s. Because of the fixed order in which extended schools are allowed to accept new students, no student ranked lower than any of the students currently held at this school will ever be admitted, and so s cannot form a blocking pair. 4 Multistage DA (MSDA) The ESDA algorithm introduced in the last section satisfies the desirable properties of strategyproofness and fairness, and while it satisfies a stronger nonwastefulness property than artificial caps DA, ESDA will still result in some wasted seats. If nonwastefulness is a bigger concern to policymakers than fairness, then ESDA may not be a desirable mechanism to use. In this section, we define an alternative mechanism, called multistage deferred acceptance (MSDA), which will be strategyproof and nonwasteful, though it will satisfy only a weaker definition of fairness. Since we do not want to give up fairness entirely, we must introduce a new way to think about the concept. The problem, of course, is that any mechanism that is strategyproof and nonwasteful 19