Incentives and Manipulation in Large Market Matching with Substitutes

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1 Incentives and Manipulation in Large Market Matching with Substitutes Evan Storms May 2013 Abstract The analysis of large two-sided many-to-one matching markets available to date focuses on the class of responsive preferences. This paper gives the rst thorough analysis of large market matching under the wider class of substitutable preferences satisfying the Law of Aggregate Demand. We show that the central results of Kojima and Pathak (2009) establishing vanishing incentives to misreport hold under these more general preferences. In doing so, we apply the characterization of stable matchings developed in Hateld and Milgrom (2005) to extend the foundational large market matching theorems not only to more general preferences, but also to arbitrary stable mechanisms. This analysis culminates in the rst proof that truthful reporting constitutes an epsilon Nash equilibrium in large markets for all agents under any stable mechanism. Stanford Department of Economics: Undergraduate Honors Thesis. I would like to thank Paul Milgrom and Fuhito Kojima for providing invaluable guidance, both on this paper in particular and on how to approach theoretical economics in general.

2 The economic theory of matching mechanisms has found applications in a number of real markets, leading to improvements in the design of centralized labor markets, school assignment systems, and housing allocation programs. In a sense, the reverse is also true: the subtleties of these real-world markets have necessitated the development of a more robust, more exible theory of matching mechanisms. Two important directions in that development are the study of large market matching and the generalizations of agent preferences. Each represents an important break from the framework of initial research in matching theory. The seminal work in matching theory focused on deterministic properties of matching mechanisms holding independent of many of the specics of a particular market. Gale and Shapley's famous study of the marriage problem, for instance, required no distinction between a market with a dozen participants and one with millions. In contrast, large market matching theory examines how the properties of a matching mechanism depend on the size of the market to which it is applied. In particular, the theory is concerned with the "asymptotic" properties of matching mechanisms - properties which hold with high probability in the limit as the size of the market approaches innity. Similarly, much of the original work in matching theory focuses on a relatively simple class of preferences which Roth terms "responsive" preferences. Roughly speaking, responsive preferences are those that can be represented by a capacity and a simple rank-order list. In many-to-one matching, responsive preferences do not allow an agent on the "one" side of the market to express how the number of partners from the "many" side he prefers to be matched with depends on which partners are available, or to express that only certain combinations of partners are acceptable. The study of more general classes of preferences considers how matching mechanisms perform when agents are allowed to express complex preferences over subsets of other agents. This paper lies at the intersection of these two lines of work. Our main result generalizes one of the central results of large market matching theory, showing that the fraction of agents with incentives to unilaterally misreport in large markets under a one-sided optimal stable mechanism converges to 0 (Kojima and Pathak, 2009), to the class of substitutable preferences satisfying a condition Hatled and Milgrom (2005) refer to as the Law of Aggregate Demand. Our generalization uses the characterization of stable matchings in terms of optimal choice functions over accumulated sets of oers to simplify several of the proof of the original result, as well as to give smaller coecients of 2

3 convergence. The second part of the paper extends the main result to arbitrary stable mechanisms, and shows that under an additional market thickness condition, in the limit truthful reporting constitutes an epsilon equilibrium for all agents in the game induced by any stable mechanism. Finally, we discuss how certain regularity assumptions used in obtaining the result can be reformulated to match other previous work on large market matching. The results obtained constitute one of the most general existing analyses of incentives in large matching markets. 1 Related Literature Our paper continues an investigation that began with a conjecture by Roth and Peranson (1999). After conducting a series of simulations on data from the National Residency Matching Program, they observed that the number of hospitals which could gain by misreporting their preferences, if all other agents continued to report truthfully 1, was only a small fraction of the market. This lead them to conjecture that the number of agents who could gain by misreporting would be small in large markets. The rst theoretical treatment of the question was given in Immorlica and Mahdian (2005), which shows that in one-to-one matching markets the fraction of agents with incentives to misreport under a stable mechanism converges to 0 as the number of agents approaches innity. In probably the most inuential work on large market matching, Kojima and Pathak (2009) show that under certain regularity conditions, this result extends to agents in many-to-one matching markets. The structure of our paper most closely resembles the work by Kojima and Pathak (2009). We use a number of important concepts from their paper, in particular the rejection chain algorithm, in the proof of our main result. That said, our paper achieves some improvements and simplications of their techniques by incorporating the general theory of matching with substitutable preferences developed in Hateld and Milgrom (2005). In particular, we make use of their characterization of stable matchings as mutually optimal selection by both sides from the lattice of cumulative oers to show that all potentially gainful misreporting can be represented by a null report which simply 1 This relies on the assumption that agents' reports were truthful, which Roth and Peranson argue is reasonable in this context. 3

4 rejects all possible matching partners. Our paper also extends the study of incentives in large two-sided matching markets in a number of new directions other than generalizing agents' preferences. While Kojima and Pathak (2009) focus on incentives under the student optimal stable mechanism, we show that our results extend to arbitrary stable mechanisms, including the National Intern Matching Program used to match hospitals to medical interns before it was replaced by Roth's National Residency Matching Program. Since arbitrary stable mechanisms need not make it a weakly dominant strategy for students to report truthfully, our paper undertakes a novel study of student incentives under arbitrary stable mechanisms in large markets. In particular, we provide the rst proof that truthful reporting constitutes an epsilon-nash equilibrium for all agents in the limit as the market size approaches innity. Outside of the direct line of work on large market matching, there are a number of important connections between the study of general preferences and work on quota systems, including the study of capacity constraints as in Abdulkadiroglu and Sonmez (2003) and Kojima(2010), and minority reserves as in Hafalir, Yenmez, and Yildirim(2012). We show in Section 3 that all of these quota systems fall under class of substitutable preferences satisfying the Law of Aggregate Demand, and that our results therefore hold in matching markets using such systems. Finally, our results add to the growing body of literature on strategy proofness in large markets for general mechanisms. Beginning with Roberts and Postlewaite (1976), which showed that traders' incentives to misreport preferences in exchange economies vanishes as the market size approaches innity, a number of papers have examined incentives under direct mechanisms as a feature of market size. Azevedo and Budish (2013) connect these results with their characterization of "strategy-proofness in the large" in terms of truthful reporting giving an optimal selection from a xed choice set. Our results establish that, when preference satisfy substitutability and the Law of Aggregate Demand, any stable matching mechanism satises this characterization. 2 2 Technically, Azevedo and Budish's denition requires that the games be restricted to nite type and actions spaces, so that to make this claim precise we would have to introduce some nite classication scheme over innite sets of students and colleges. 4

5 2 General Framework 2.1 Preliminary Denitions We follow the convention of describing a two-sided many-to-one matching setting in terms of students and colleges (of course, we could equally well use doctors and hospitals, or workers and rms). A many-to-one matching market, henceforth a market, consists of a tuple (S; C; P S ; P C ), where S is a set of students, C disjoint from S is a set of colleges, P S = (P s ) s2s is a set of strict preference relations over elements of C [ f;g, and P C = (P c ) c2c is a set of strict preference relations over subsets of S. Here ; represents being unmatched. We write that student s prefers college c to c 0 as either cp s c 0 or c s c 0, and likewise for college preferences. If c s ;, we say that college c is acceptable to student s, and if for A S, A c ;, we that the set of students A is acceptable to college c. In the case A = fsg, we say that student s is acceptable to college c. Since only rankings over acceptable partners matters in our analysis, when representing preference lists we truncate at ;. For example: P s = c 1 ; c 2 means that college c 1 is student s's most preferred college, c 2 is his next most preferred college, and all other colleges are unacceptable. To reduce notation, we also sometimes use P = P C [ P S to refer all of the preferences in the market. A matching is a function from C [ S to C [ P(S) [ f;g such that: (1) j(s)j = 1 and (s) = ; if (s) 62 C for every s 2 S, (2)(s) = c if and only if s 2 (c). Condition (1) means that no student is matched to more than one college, and that if she is not matched to a college she is unmatched. Condition (2) means that the if a student is matched to a college, then that college is matched to a subset containing that student. A matching is individually rational if for all s 2 S, (c) s ;, and for all c 2 C, there is no subset of (c) which college c prefers to (c). A student-college pair (s; c) is said to block the matching if: (1) c s (s) and (2) s 2 Ch Pc ((c) [ fsg). In words, (s; c) block if s prefers c to whichever college he is matched with under, and c would prefer some subset of the students it is matched with under and student s to just those students it is matched with under. A matching is stable if it is individually rational and there is no student-college pair which blocks it. 5

6 2.2 Substitutes We have introduced college's preferences as a an ordering over subsets of students. Alternatively, we can represent a college's preferences by means a function mapping any given subset of students to a college's most preferred subset of students contained in that subset. Formally, preferences P c over subsets of S are said to induce the choice function Ch Pc : P(S) 7! P(S), where for all X S, Ch Pc (X) = argmax Pc;Y X Y = Y X j 8Z X Y P cz The choice function allows a simple denition of the important concept of substitutability. Denition 1. Students are substitutes under preferences P c if for all X S and distinct s; s 0 2 X, s 2 Ch Pc (X) implies s 2 Ch Pc (X fs 0 g). Substitutability is the primary condition on colleges' preferences we use throughout the paper. The condition requires that if a college chooses to be matched with students s when it can choose from some set X, then it will continue to choose to be matched with s when it can choose from some Y X containing s. Informally, students are substitutes when having more students to choose from never makes a college want some particular student more. Note that since students have preferences over individual colleges, as opposed to subsets of colleges, no similar condition for student preferences is needed. Colleges are necessarily substitutes for a student because the choice of one college requires not being paired with all other colleges. We said above that much of the early work in many-to-one matching theory used a subclass of substitutable preferences known as responsive preferences. Formally, a colleges preferences P c are responsive with quota q c if for any students s; s 0, fsg [ X c fs 0 g [ X for all X S such that jxj < q c, and ; c Y for all Y S such that jy j > q c. Responsive preferences correspond to a simple rank-order list of students, so that a college prefers students according to their place in this list regardless of which other students it is matched to. If we represent preferences in terms of utility functions, substitutable preferences correspond to submodular utility functions over students, 6

7 where responsive preferences correspond to the special case of additive utility functions. Below we give an example of substitutable, non-responsive preferences: Example 1. Suppose there are four students s 1 ; s 2 ; s 3 ; s 4, and a college c with preferences: P c = fs 1 ; s 3 g; fs 1 ; s 4 g; fs 2 ; s 3 g; fs 2 ; s 4 g; fs 1 g; fs 2 g; fs 3 g; fs 4 g Then the preferences P c are non-responsive since, given that c is matched with s 1, its most preferred additional students (to be matched alongside s 1 ) is s 3, while given that it is matched to s 2, its most preferred additional student is s 4. To make this example more concrete, we can imagine that students s 1 and s 2 are women and students s 3 and s 4 are men. College c's preferences then express that c prefers to be matched with one man and one woman whenever possible, and that any matching with two students of the same sex is unacceptable. 2.3 Stable Mechanisms The substitutes condition is related in an important way to the concept of stability. It is possible that for some market (S; C; P ), no stable matching exists, as the next example demonstrates: Example 2. Consider the market (S; C; P ) for which S = fs 1 ; s 2 g, C = fc 1 ; c 2 g, and P is given by: P c1 P c2 = fs 1 ; s 2 g; fs 1 g = fs 2 g; fs 1 g P s1 = c 2 ; c 1 P s2 = c 1 ; c 2 Suppose is a stable matching for this market. If s 2 2 (c 1 ), then since c 1 only nds s 2 acceptable as a pair with s 1, (c 1 ) = fs 1 ; s 2 g. But then (s 1 ; c 2 ) blocks. If s 2 62 (c 1 ), then s 2 must be matched to c 2 lest (s 2 ; c 2 ) block, and so s 1 must be matched to c 1. But then (s 2 ; c 1 ) blocks. Therefore no such stable matching exists. Notice that in Example 2, students 1 and 2 are complements for college c 1 : c 1 only wants to be matched to s 2 if she is paired with s 1. Substitutability precludes complementarities 7

8 among students. That is, it rules out the possibility that a college might nd prefer some student more when he is matched along with certain other students. It turns out that this is sucient to insure the existence of a stable match. Remark 1 (Roth). Suppose students are substitutes for every college in the market M = (S; C; P ). Then there exists a stable matching in M. The proof of this fact uses the Student-Proposing Deferred Acceptance Algorithm, rst introduced in Gale and Shapley(1982). Algorithm 1 (Student-Proposing Deferred Acceptance Algorithm (SPDA)). 1) Round 1: Each student applies to his rst choice college. Let E 1 c be the set of students who apply to college c at this step. College c holds all students in H 1 c = Ch P c (E 1 ). c 2) Round t 2: Each student not in H t 1 c for some c who has not yet applied to every acceptable college applies to his most preferred college to which he has not applied at a previous round. Let E t c be the set of students who apply to college c at this step. College c holds the students in H t c = Ch P c (H t 1 c [ E t c ). Terminate when no new applications are made. Colleges are matched with the students they are holding at the terminal round. The SPDA terminates in a nite number of rounds K, as there are only nitely many students and nitely many colleges a student can nd acceptable. When the algorithm terminates, either every student is matched to some college, or every student who is unmatched has applied to all acceptable colleges. Let A k c = [ ike k be the set of student who have applied to college c at steps up to and including round k. The SPDA where colleges have substitutable preferences has the following important property: 8

9 Lemma 1. H k c = Ch P c (A k ). c Proof. Let X Y be any increasing subsets of students. We show that with substitutable preferences, the choice function is path independent, or: Ch Pc (Y ) = Ch Pc (Ch Pc (X) [ (Y X)) (1) Suppose x 2 Ch Pc (Y ). If x 2 X, then by substitutability and the fact that X Y and Ch Pc (X) [ (Y X) Y, x 2 Ch Pc (X) and x 2 Ch Pc (Ch Pc (X) [ (Y X)). Thus Ch Pc (Y ) Ch Pc (Ch Pc (X) [ (Y X)). Now note that Ch Pc (Y ) is the maximal subset of Y which is also a xed point of the choice function. Since Ch Pc (Ch Pc (X) [ (Y X)) is a xed point of Ch Pc and contains Y, we must have Ch Pc (Ch Pc (X) [ (Y X)) = Ch Pc (Y ). The result then follows by applying (1) recursively to the sets A k c. Lemma 1 allows an easy proof or Remark 1. Proof of Remark 1. We show that the matching obtained at the end of the SPDA algorithm is stable. Clearly it is individually rational, since students only apply to acceptable colleges, and colleges hold only those students in the most preferred subset of currently held and newly applying students. Consider a potential blocking pair (s; c), where c s (s). Since s ranks c above (s) (including the possibility that (s) = ;), s applies to c at some step of the SPDA algorithm. Thus s 2 A K c but ). But then college c strictly prefers (c) to (c) [ fsg, and so s 62 (c) = H K c = Ch Pc (A K c s 62 Ch Pc ((c) [ fsg). Thus (s; c) does not form a blocking pair, and since this holds for any s and c such that c s (s), no student-college pair blocks. The SPDA algorithm is one means of implementing a stable mechanism. Generally, a stable mechanism for a given set of students and colleges is a function ' from the set of their possible preferences orderings to the set of stable matchings over S [ C. So long as we restrict attention to markets with substitutable preferences, we are assured that there is at least one such stable matching. The matching produced by the SPDA algorithm has the special property that it is unanimously weakly preferred by all students to any other stable matching. For this 9

10 reason, the stable mechanism which produces this matching is called the student optimal stable mechanism (SOSM), which we denote S. We can dene a counterpart of the SPDA in which colleges instead of students make applications, and in this case the outcome is a stable match unanimously weakly preferred by all colleges to any other stable matching. We call the stable mechanism which implements this matching the college optimal stable mechanism (COSM), and we denote it C. Letting ' be a stable mechanism, we dene college c's choice set under mechanism ' and preference report P, denoted X ' c (P ), to be the subset of those students s such that '(P )(s) s c. 3 That is, X ' c (P 0 ) is the set of students who weakly prefer college c to the college they are matched to under ' when colleges submit preferences P 0. If ' = S is the SOSM, then in terms of the DA algorithm, X ' c (P 0 ) consists of those students which apply to college c at some stage of the algorithm. The following lemma generalizes Lemma 1 to arbitrary stable mechanisms: Lemma 2. Let ' be any stable mechanism and (S; C; P ) a market where P are any substitutable preferences. Then for all c 2 C, '(P )(c) = Ch Pc (X ' c (P )). Proof. For convenience, we suppress the ' and P in our notation. Consider s 2 '(c). The stability of ' implies that there is no set of students Y such that for each s 0 2 Y, '(s 0 ) s 0 c, and s 62 Ch Pc ('(c) [ Y ). But since X c is precisely the set of students s 0 for which '(s 0 ) s 0 c, and since '(c) X c by denition, taking Y = X c gives s 2 Ch Pc ('(c) [ X c ) = Ch Pc (X c ). Thus '(c) Ch Pc (X c ). For the reverse inclusion, consider s 2 Ch Pc (X c ). Substitutability of preferences implies that for any subset Y X c containing s, s 2 Ch Pc (Y ). In particular, then, s 2 Ch Pc ('(c) [ fsg). Since s 2 X c, we must have s 2 '(c), lest (c; s) block '. Therefore Ch Pc (X c ) '(c). 2.4 Incentives and Manipulation Suppose we have a market M = (S; C; P ) and a stable mechanism ', and we wish to apply ' to M. Though it is simple mathematically to dene the object '(P ), in reality agents' preferences are private information which they must make known in order to implement the mechanism. This introduces the possibility of manipulations agents submitting preferences other than their true preferences in an attempt to achieve a more 3 Where it is clear from the context that ' = S, we suppress the ' in the notation. 10

11 preferred matching. Recall that a stable mechanism for a market species a stable matching for any preferences P 0 students and colleges might report. From the perspective of the mechanism ', it makes no dierence whether these preferences are the true, underlying market preferences or not. If it is the case that P 0 i = P i for agent i, we say that agent i is reporting truthfully. A successful manipulation for a college c, when other colleges make reports P 0 C c and students report P 0 S, is some report P 0 c 6= P c such that '(P 0 c ; P 0 C c ; P 0 s )(c) c '(P c ; P 0 C c ; P 0 s )(c). It is important to note that the comparison of the two outcomes is made under college c's true preferences P c, since these are what determine whether the college is actually better o. One of the central questions of matching theory is, when is it in agents' interests to report truthfully? Roth (1982) showed that there is no stable mechanism such that, under that mechanism, in all markets, all agents can do no better than to report truthfully, given that other agents do so. That is, there is no stable mechanism such that truthful reporting is a Nash Equilibrium for all markets. Indeed, even though the SOSM gives students their most preferred stable match, truthful reporting need not be an equilibrium for students under the SOSM. The next example illustrates. Example 3 (from Hateld and Milgrom (2005)). Consider a market (S; C; P ) with two colleges and three students, and preferences given by: P c1 P c2 = fs 3 g; fs 1 ; s 2 g; fs 1 g; fs 2 g = fs 1 g; fs 2 g; fs 3 g P s1 = c 1 ; c 2 P s2 = c 2 ; c 1 P s3 = c 2 ; c 1 It is easy to check that the only stable matching in this market, and therefore the student optimal stable match, is f(s 1 ; c 2 ); (s 3 ; c 1 )g. However, if s 2 instead reports P 0 s 2 : c 1 ; c 2, then S (P C ; P S s2 ; P 0 s 2 ) = f(s 1 ; c 1 ); (s 2 ; c 1 ); (s 3 ; c 2 )g. Therefore s 2 can manipulate successfully in this market under the SOSM. 11

12 Notice that in the above example, c 1 's preferences have the feature that they rank a subset of size one above one of size two. As a consequence, s 2 can increase the number of students c 1 accepts by applying before s 3, which opens up the chance for her to manipulate. It turns out that if we rule out preferences in this class, truthful reporting becomes a weakly dominant strategy for students. Denition 2. A college's preferences P c satisfy the Law of Aggregate Demand (LoAD) if for all any subsets X Y S, jch Pc (X)j jch Pc (Y )j. The Law of Aggregate Demand means that given a weakly larger set of students to choose from, a college will always choose to be matched with weakly more students. It thus precludes any preferences which rank a smaller (in size) subset above a larger one, such as those in Example 2. Remark 2 (Hateld and Milgrom(2005)). Let (S; C; P ) be a market such that college's preferences satisfy substitutability and LoAD. Then xing other agents' reports, no student can manipulate successfully under the SOSM, i.e., truthful reporting is a weakly dominant strategy for students under the SOSM. Remark 2 allows us to focus on manipulations by colleges when we work with the SOSM, which will be important in proving the main result. 3 A Note on Generalized Preferences Most of the theoretical work in this paper is done generalizing large-market results for responsive preferences to the more general case of substitutable, LoAD preferences. It is important, therefore, to understand exactly how much this expands the scope for the application of these results. We will argue that it expands the scope signicantly. Responsive preferences are highly restrictive in the sense that they preclude any interdependence among students' (or workers, etc.) value to colleges (rms). But in reality, it is quite common that these interdependencies should arise. In particular, having students or workers of dierent backgrounds, races, genders, skill sets, and so on, is often of great importance in matching markets. A college might well prefer a female student more when its current student population is heavily male; having a full team of cooks changes a restaurant's preferences over hiring another cook or its rst waiter. 12

13 One particular extension of responsive preferences that has received attention in the literature is responsive preferences with some form of quota. There are several varieties of quota systems, and we will show that the corresponding preferences all fall under the scope of substitutable, LoAD preferences. As formalized in Abdulkadiroglu and Sonmez (2003), responsive preferences with capacity constraints are a form of maximum quota in which students are divided into a number of types and colleges can accept only a maximum number q i students of type i. For example, a racial quota system might divide students into a "racial majority" type and a "racial minority" type, with colleges then permitted to have at most some maximum fraction of their total capacity consist of any one type. Preferences over students are represented by these capacity constraints along with a maximum total capacity and a rank-ordering over all students. We can describe a college's selection process under maximum quotas as follows: Suppose a college c has a set X of students to choose from. It selects the highest ranking students one by one until it reaches one of its capacity constraints, say for students of type i. It then rejects all students of type i it has not yet accepted. Next it selects the highest ranking students, among those not yet accepted or rejected, one by one, until it reaches its next capacity constraint, say for students of type j. It then rejects all students of type j it has not yet accepted. The college continues accepting students in this manner until it reaches its maximum total capacity. It is easy to see that this selection process corresponds to substitutable, LoAD preferences. Suppose c accepts student s from set X. Then at some point in the selection process s is the highest ranked student of the types for which c has not yet reached its quota. It follows that c will accept s from X fs 0 g for any s 0 2 X; s 0 6= s: X fs 0 g contains no more students of any type, and in particular of student s's type, ranked above s than X, so that s is still the highest ranked student with an acceptable type at some point in the selection process. Further, c always admits acceptable students of a type for which it has capacity, so that it never prefers a smaller subset to a larger one. Another form of quota is minority reserves, studied in Hafalir, Yenmez, and Yildirim (2012). Under this system, students are again broken down into types 4, though now 4 Hafalir, Yenmez, and Yildirim(2012) focuses on the case in which there are only two types, a "minority" and a "majority." We generalize to account for the possibility of more than two types. 13

14 types as well as students are rank-ordered. Each school has a set reserve r i for students of a particular type i such that, so long as it has fewer than r i students of type i, it prefers any student of type i to any student of type j such that either j > i or j < i but the college has already selected r j students. Again, each college c also has a maximum total capacity q c. We can describe a college c's selection process under minority reserves as follows: Suppose a college c has a set X of students to choose from. It chooses its most preferred students of type 1 one by one until it either it has selected r 1 such students or it has chosen all students of type 1 in X. Generally, given that c has selected r i or all available students for each type i < k, it selects its most preferred students of type k one by one until it has a total of q c students, has accepted all students of type k, or has accepted r k students of type k. If c has accepted its reserve of or exhausted all types and has fewer than q c students, it lls the remaining positions with the highest ranked remaining students of any type. This selection process likewise corresponds to substitutable, LoAD preferences. Suppose c accepts s of type i from X. Then either s is one of the r i highest ranked students of type i, or P i max(r i ; jx i j) < q c and s is among the highest q P c i max(r i ; jx i j) ranked students of those not among the highest r k for their type k. Since this is clearly still true if we remove some student s 0 6= s from X, s is accepted by c from X fs 0 g. Thus students are substitutes for college c. Since c will always accept acceptable students when below capacity, c again never prefers a smaller subset of students to a larger one. Thus these quota systems fall within the class of substitutable, LoAD preferences, and all of our results on incentives in large markets apply. That said, the advantage of considering this broader class is not just that it subsumes these and other quota systems but also that it subsumes cases in which preferences cannot be described in terms of strict market-wide quota systems. Our model allows colleges to have dierent quota systems and dierent categorizations of students. It also allows for "fuzzy" quotas, in which colleges might make exceptions to the quota system in certain cases. 4 Main Result In this section, we show that under certain conditions the expected number of colleges which can manipulate successfully under the SOSM given that other colleges report 14

15 truthfully converges to 0 as the size of the market approaches innity. The proof consists of three steps. First, we show that all protable manipulations can be recreated via a type of manipulation we term a dropping strategy, so that it is sucient to consider these types of manipulations. Second, we show that a college c can manipulate via a dropping strategy only if, starting from the student optimal stable matching, it can induce a series of rejections that include a new application to c. Third, we show that in the limit, the number of colleges who can induce such a series of rejections converges to Dropping Strategies In any sizable market, the number of reports a college can submit is quite large; the number of reports satisfying responsiveness is alone jsj!. This might seem to pose a challenge to conning the scope of potential manipulations. Fortunately, any successful manipulation can be recreated by a report in the much smaller class of dropping strategies. Denition 3. Suppose college c has true preferences P c. The report P 0 c is a dropping strategy if there exists a subset (possibly empty) of students D such that: (1)fsgP c ; but ;P 0 c S for all S containing s, for all s 2 D, and (2)SP 0 c S0 if and only if SP c S 0 for all S; S 0 such that (S [ S 0 ) \ D = ;. If s 2 D for P 0 c, we say that P 0 c drops students s. In words, a dropping strategy is the result of declaring all subsets containing certain students unacceptable, without otherwise changing preferences over still acceptable students. Notice that if the true preferences are substitutable and satisfy LOAD, then any dropping strategy P 0 represents preferences which are also substitutable and satisfy LOAD. (Substitutability follows from the fact that a dropping strategy declares all subsets containing student s 2 D unacceptable by part (1) of the denition; LOAD holds because if P 0 to contain a sub partial order violating the condition, then P would as well by part (2) of the denition.) When preferences are responsive, the above denition coincides with the denition in Kojima and Pathak (2009). (Note: With general substitutable preferences, the denition of a dropping strategy from Kojima and Pathak (2009) could potentially be generalized to take into account that colleges can declare a subset of students unacceptable without doing the same for each individual student in the subset. However, we show that this is unnecessary, because a college can replicate any protable misreporting by simply declaring any subset containing some particular students unacceptable, as in our denition of dropping 15

16 strategies.) The next lemma shows that show that dropping strategies are exhaustive, in the sense that any protable manipulation can be recreated via a dropping strategy. Lemma 3. Assume colleges have substitutable preferences P C, and x student preferences P S. Then for any preference prole P ~ c resulting in matching S ( P ~ c ; P c), there exists a dropping strategy P 0 c such that: (1) X S c ( P ~ c ; P c) X S c (P 0 c ; P c), and (2) S ( P ~ c ; P c)(c) c S (P 0 c ; P c)(c). Proof. See appendix. We call the largest choice set that a college can induce, holding the reports of other colleges and students xed at P c, its maximal choice set. It follows from Lemma 3 that this choice set can be induced via a dropping strategy. In fact, it turns out that the maximal choice set is simply the choice set which results when a college drops all acceptable students. This is a consequence of the following lemma. Lemma 4. Let P 0 c be the dropping strategy for college c which drops students in D 0, let ^P c be the dropping strategy which drops students in ^D, and suppose ^D D 0. Then for all k and all c i, A k c i ( ^P ) A k c i (P 0 ), which implies X ci ( ^P ) X ci (P 0 ). In particular, X c ( ^P ) X c (P 0 ). Proof. See appendix. Together, Lemmas 2 and 3 give us a simple characterization of any successful manipulation by a college under the SOSM: a successful manipulation occurs when a college rejects some student out of accordance with its true preferences, in order to make some other student worse o and thereby add him to its choice set. By Lemma 4, if a college cannot expand its choice set by declaring all student unacceptable, then it can never expand its choice set. This simplies the problem enormously, as now instead of considering the at least jsj possible reports a college might submit, we need only consider the consequences of a single report. 4.2 Rejection Chains To determine those consequences, we introduce the concept of a rejection chain, an approach adopted from Kojima and Pathak (2009). We leave the technical denition and 16

17 all corresponding proofs to the appendix; in the main text we settle for an intuitive explanation. Suppose we modify the SPDA algorithm by allowing college c to reject some subset of its students B 0 c after the algorithm has reached the student optimal stable match, and then continuing the SPDA algorithm from this point. If they have not yet applied to all acceptable schools, those students will make new applications to other schools with three possibilities: each student is either is rejected, is accepted to his next-choice school and some students are rejected as a consequence of his application, or is accepted and no students are rejected as a consequence. If the rst or second occurs for any student, this gives rise to a new set of displaced students, who can make new applications to any acceptable school they have not yet applied to. This process of rejections and applications terminates at a some new matching in a nite number of steps, since each student can make only nitely many applications. We refer to the process of generating this new matching from the student optimal stable matching a rejection chain with input B 0 c. By Lemma 1, the set of students college c is matched with when SPDA algorithm reaches the student optimal stable match is its preferred subset of all students who have applied. Thus college c can only be better o at the end of the rejection chain if some new student applies to c, or equivalently, if some student is added to X c because he is rejected from the college he ranks just above c. If this occurs, we say that the rejection chain returns to college c. To make matters concrete, the next example illustrates one rejection chain for a particular market. Example 4. Consider a market with three colleges and four students, in which college c 1 has preferences as in Example 1, and the other colleges' preferences are given by: P c2 P c3 = fs 2 g; fs 1 g; fs 3 g; fs 4 g = fs 4 g; fs 1 g; fs 3 g; fs 2 g 17

18 And student preferences are: P s1 = c 2 ; c 1 ; c 3 P s2 = c 1 ; c 2 ; c 3 P s3 = c 3 ; c 1 ; c 3 P s4 = c 1 ; c 3 ; c 2 The student optimal stable matching in this market is S = f(c 1 ; fs 2 ; s 4 g); (c 2 ; fs 1 g); (c 3 ; fs 3 g)g. Now let's describe the rejection chain starting with input B 0 c 1 = fs 2 ; s 4 g. Let c 1 reject s 2 and s 4. Then s 2 applies to her next choice school c 2. Since Ch (fs Pc 2 2; s 1 g) = fs 2 g, c 2 accepts s 2 and rejects s 1. Student s 1 then applies to his next choice school, c 1, and c 1 accepts him. Now student s 4 applies to his next choice school, c 3. Since Ch (fs Pc 3 3; s 4 g) = fs 4 g, c 3 accepts s 4 and rejects s 3. Student s 3 then applies to his next choice school c 1, and since Ch (fs Pc 1 1; s 3 g) = fs 1 ; s 3 g, c 1 accepts s 3. Since there are no more unmatched students, the rejection chain then terminates, and the resulting matchings is = f(c 1 ; fs 1 ; s 3 g); (c 2 ; fs 2 g); (c 3 ; fs 4 g)g. In this example, the rejection chain returned to college c 1. (To see that rejection chains need not always return, consider the above with each student's preference list truncated to just his rst choice college.) Note that in Example 4, the same matching is obtained from the SOSM if college c 1 simply drops students s 2 and s 4 from its preference list. This is not a coincidence: As the next lemma demonstrates, rejection chains provide a means of evaluating how a college's choice set changes when it plays a particular dropping strategy. In particular, we will show that if the maximal rejection chain does not return to college c, then c cannot expand its choice set by misreporting its preferences. Lemma 5. Let P 0 c be a dropping strategy which drops students in subset D, and let be the matching obtained by Algorithm 2 with input B 0 c = D \ S(P )(c). Then for all c, X S c (P 0 ) X c. Proof. See appendix. Since the rejection chain not returning to c is equivalent to X c and 5 imply the following corollary. = X S c (P ), Lemmas 3, 4, 18

19 Corollary 1. If the rejection chain does not return to college c for B 0 c = S(P )(c), then for any preference prole ~ P c, X S c ( P ~ c ; P c) X S c (P c ; P c). 4.3 The Fraction of Rejection Chains which Return Vanishes in Large Markets Thus far, we have treated markets as deterministic objects, so that a college either can or cannot manipulate successfully. To translate this into the probability that a college can manipulate successfully, we need to dene the notion of a random market. A random market is a tuple = (S; C; P C ; D; k), where S is a set of students, C disjoint from S is a set of colleges, P C = (P c ) c2c is a set of college's preference relations over subsets of students, D = (p c ) c2c is a probability distribution on C [ f0g, and k is a positive integer. A random market induces a market by the realization of student preferences P S = (P s ) s2s as follows: Algorithm 3 (Student Preference Generation). Initialize: Give an arbitrary ordering over students. Let s = s 1 and k = 1. Round k: 1) Step 1: Select a college c (possibly ;) from distribution D. Set P s(1) = c. If c = ;, terminate the round. 2) Step 2 t k: Select colleges from distribution D until a college c is selected so that c 6= P s(k) for any k < t. Let P s(t) = c. If c = ;, terminate the round. End of Round: Terminate if k = jsj. Else, let s = s k+1, and proceed to Round k+1. Student s k nds all colleges not drawn in the course of Round k of Algorithm 2 to be unacceptable, and ranks those colleges drawn according to P s as constructed in the algorithm. Algorithm 2 results in each student having a preference list of length at most k, so that k is the bound on the length of student preference lists. (Dening D as a probability distribution over C [ f0g allows for the possibility that student preference lists might be shorter than k). This bound will play an important part in proving the main result. 19

20 Next we make precise the notion of market size approaching innity. A sequence of random markets is written f n g, where n = (C n ; S n ; P n C ; kn ; D n ) such that jc n j = n. We say that a property holds in the limit for a sequence of random markets if there is some N such that for all n > N, the market n satises P. The proof that the fraction of rejection chains which return vanishes in the limit relies on the following regularity assumptions: Denition 4. A sequence of random markets f n g is regular if there exist some positive integers k and q such that for all n: 1. k n = k, 2. For all c 2 C n, no subset of more than q students is acceptable, 3. jc n j = n and js n j qn, 4. For all c 2 C n, every individual student is acceptable. Condition (1) requires that the length of student preference list be uniformly bounded across markets. Condition (2) requires that the maximum number of students colleges can acceptwhich we can think of as their capacitiesis likewise uniformly bounded across markets. Condition (3) requires that the number of students grows at most linearly in the number of colleges. Condition (4) requires that if a vacant college receives some new applications, it accepts at least one. All of the conditions will be used in our proof of the main result. See section 7 for a discussion of alternative conditions under which the result still holds. We are now in a position to present the main result. Theorem 1. If f n g is a regular sequence of random markets in which college preferences satisfy substitutes and LoAD, the expected fraction of colleges which can gain by misreporting preferences under the SOSM, given that other colleges report truthfully, converges to 0 as n approaches innity. The proof of Theorem 1 is given through a series of technical lemmas in the appendix. Here we give an intuitive explanation. The key step of the proof involves showing that when student preference lists are bounded, a non-vanishing fraction of colleges receives no applicants in the course of the SPDA algorithm. In particular, for any suciently 20

21 unpopular college c, the expected number of colleges more popular than c (p 0 c p c ) who receive no applicants is lower bounded by a positive constant. By condition (4), these vacant colleges will accept any student who proposes to them as part of a rejection chain, and since they have no students to reject in the applicants place, no new applications can come about as a result. Now, because colleges' preferences satisfy LoAD, a college receiving one new application will never reject more than one of its former students. And since college c can reject no more than q students in initiating a rejection chain, this implies that in the course of a rejection chain, there are never more than q unmatched students who have not exhausted their preference lists. It follows that once j q applications have been made to vacant colleges more popular than c, the rejection chain terminates. Then, since these colleges are more popular than c, with high probability any rejection chain starting with c terminates with j applications to these colleges before any new application is made to c. Thus by Corollary 1, a suciently unpopular college cannot add to its choice set by misreporting its preferences. The nal step of the argument involves showing that that in the limit, almost all colleges are suciently unpopular. This follows from the fact that the sum of the probability of applications to a college is upper bounded by 1, so that as the number of colleges grows to innity, the fraction of colleges whose popularity falls below any xed > 0 grows to innity. 5 Arbitrary Stable Mechanisms and Student Incentives One might expect that, if colleges cannot unilaterally manipulate under a mechanism that gives them their least preferred stable matching, then they should not be able to manipulate under a mechanism which gives them a more preferred stable matching. In this section, we show that this is indeed the case by generalizing Theorem 1 to all stable matching mechanisms. We also show that strategy-proofness holds on the student side of the market as well. 21

22 5.1 Arbitrary Stable Mechanism We will need the following result from Hateld and Milgrom (2005), which we state without proof. Remark 3. Suppose college's preferences satisfy substitutability and LoAD. Then the number of students matched to each college and the set of unmatched students is the same for all stable matchings. We will also need the following generalization of Lemma 3, which shows that for any stable mechanism, any manipulation can be recreated via a dropping strategy. Lemma 6. Assume colleges' preferences under P satisfy the substitutes and LoAD conditions. Let ' be any stable matching mechanism, and suppose there is some preference list ~ P c such that '( ~ P c ; P c) =. Then there exists a dropping strategy P 0 c such that '(P 0 c ; P c)(c) = Ch Pc ((c)). Proof. See appendix. The main lemma in this section is the following, which establishes that if a college can attain a set of students under one stable mechanism for some preference report, then there is a preference report which will give it that same set of students under any stable mechanism. Lemma 7. Let ' be any stable mechanism, and let ~ P c be any list of substitutable preferences by college c. Then there is some report P 0 c of substitutable and LoAD preferences such that (P 0 c ; P c)(c) = Ch Pc ('( ~ P c ; P c)(c)) for any stable mechanism. Proof. See appendix. We now oer a formal statement and proof of Theorem 2. Theorem 2. Let ' be any stable mechanism, and let f n g be a regular sequence of random markets. Dene: M ' = fc 2 C j '(P 0 c ; P c)(c) c '(P )(c) for some P 0 c, in market (S n ; C n ; P )g and: ' (n) = E h jm' j n i Then: ' (n) lim n!0 n = 0 22

23 Proof. By Lemma 7, if a college can obtain a set of students it prefers to that resulting from truthful reporting under the mechanism ', then it can obtain that set of students under any stable mechanism, and in particular under S. Since S is college-pessimal, if some c prefers a matching to '(P )(c), then it must prefer that matching to S (P )(c). Therefore M ' M S, so that ' S. By Theorem 1, lim n S (n) n lim n '(n) n = Student Incentives = 0. Thus Recall that by Remark 2, under the SOSM, if colleges preferences satisfy substitutability and LoAD, then it is a weakly dominant strategy for students to report their true preferences. This need not be the case for other stable matching mechanisms, however, even if colleges are restricted to responsive preferences. (Consider the simple example in which there are two students and two colleges, with P ci = s i ; s i and P si = c i; c i. If student i reports truthfully under the COSM, he is matched with c i, but if he instead reports P 0 s i = c i, he is matched with c i.) Our next result shows that as the size of the market approaches innity, the number of such potential manipulations approaches 0. Corollary 2. Let ' be an arbitrary stable mechanism, and let f n g be a sequence of regular markets in which colleges' preferences satisfy substitutability and LoAD. Dene: and: Then: N ' = fs 2 S j '(P 0 s ; P s; P c )(s) s '(P )(s) for some P 0 s, in market (S n ; C n ; P )g ' (n) = E ' (n) lim n!0 n h jn' j n i Proof. An immediate consequence of Theorem 2 is that as the size of the market approaches innity in a regular sequence of random markets, the fraction of colleges matched to the same students across all stable matchings converges to 1. Since colleges quotas are bounded, this in turn implies that the fraction of students who are matched = 0 to the same college in all stable matches converges to 1. Since truthful reporting is a weakly dominant strategy for students under the SOSM, the most preferred college a student can obtain by submitting any preference list at any 23

24 stable matching is S (P )(s). By the above reasoning, the fraction of students who attain S (P )(s) at all stable matchings converges to 1. It follows that the number of students who can obtain some school c 0 under ' such that c 0 s '(P )(s) converges to 0. 6 Equilibrium Analysis Thus far we have established that the fraction of agents with incentives to deviate unilaterally from truthful reporting vanishes in large markets. Our analysis relied on the assumption that all agents other than one under consideration report truthfully, which allowed us to consider agents' chances to manipulate one by one. In this section, we drop that assumption and allow all agents to behave strategically simultaneously in order to develop a theory of equilibrium in large markets. We construct the analysis in two stages: the rst treating student reports as xed and considering the resulting game among colleges, and the second considering the general matching market game in which all agents can behave strategically. 6.1 Strategic Play by Colleges Theorem 1 establishes that under the SOSM, when colleges' preferences satisfy substitutability and LoAD, the fraction of colleges which can gain by misreporting unilaterally converges to 0 in any regular sequence of markets. This not sucient, however, to conclude that truthful reporting is an equilibrium of the matching market when we allow colleges to behave strategically, since a vanishing but non-zero fraction of colleges can have persistent incentives to misreport. Indeed, Kojima and Pathak (2009) show that there exists a regular sequence of random markets so that a particular college has incentives to manipulate even as the size of the market approaches innity. Recall that in the proof of Theorem 1, we showed that incentives to manipulate disappear for suciently unpopular colleges, and that in the limit an arbitrarily large fraction of colleges is suciently unpopular. Equilibrium, though, requires that no college be able to gain "too much" by misreporting. It follows that what is required for equilibrium is some condition ruling out the possibility of a college being "too popular." Indeed, it turns out that exactly such a condition is sucient for truthful reporting to constitute an -Nash Equilibrium in the limit for colleges under the SOSM, and in fact 24