The Law of Large Numbers for Large Stable Matchings
aa r X i v : . [ ec on . E M ] J a n T H E L AW OF L ARG E N U MBERS FOR L ARG E S TABL E M ATCH I NG S
Jacob Schwartz and Kyungchul Song
University of Haifa and University of British Columbia A BSTRACT . A stable matching between two sets of agents (such as students andcolleges) has received a great deal of attention in the literature. For an empiricalstudy of matching markets, the main statistics of interest are the matching frequen-cies, which show the fraction of students of a given type who match with collegesof a given type. We study the concentration-of-measure phenomenon for match-ing frequencies when the observed matching is generated from a Deferred Accep-tance algorithm. We introduce a notion of partial homogeneity of preferences toexpress correlated priorities of colleges over students, and demonstrate the relationbetween partial homogeneity and concentration of measure. Our concentration in-equality immediately yields the rate of convergence for the law of large numbersfor matching frequencies.K
EY WORDS . Two-sided matching, concentration inequality, stable matching, de-ferred acceptance algorithm, law of large numbers, correlated preferencesJEL C
LASSIFICATION : C13, C78
1. Introduction
In economics, many types of matching outcomes arise in the context of a two-sided market, such as in marriage markets, college-admissions markets, hospital
Date : January 5, 2021.We thank the participants at the Econometrics Lunch Seminar at UBC and the Econometrics work-shop at Western University for valuable comments and questions. All errors are ours. Song ac-knowledges that this research was supported by Social Sciences and Humanities Research Councilof Canada. Corresponding address: Jacob Schwartz, Department of Economics, University of Haifa,Mount Carmel, Haifa, 3498838, Israel. Email address: [email protected]. residency markets, and labor markets. These matchings are two-sided in the sensethat both sides of the market have preferences and any agent can walk out of thematching when there is an incentive to do so, such as when there is a preferredalternative agent on the other side who also prefers to match with the agent than hiscurrent match. Hence a reasonable prediction from such a matching market shouldbe one that focuses on outcomes that do not permit such deviations. A matchingamong agents that satisfies such restrictions is formalized as a stable matching inthe literature.In analyzing data from matching markets, a quantity of fundamental importanceis the matching frequency which refers to the fraction of matched pairs of individ-uals with given types. For example, let N ( x ) be the set of students with observedcharacteristic x and M ( z ) be the set of colleges with observed characteristic z . Thenone of the statistics most commonly used for empirical analysis of matching data isof the following form: MF ( x, z ) = { ( i, j ) ∈ N ( x ) × M ( z ) : i and j are matched } , (1.1)which represents the proportion of college-student pairs from characteristics x and z that are matched. For example, by studying such quantities, one can explorepositive assortative matching along certain type categories.In this paper we are interested in the following type of a concentration-of-measurephenomenon: P (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) MF ( x, z ) nm − E [ MF ( x, z ) |F ] nm (cid:12)(cid:12)(cid:12)(cid:12) ≥ t |F (cid:27) ≤ B ( t ) , (1.2)for some bound B ( t ) , where n denotes the total number of students in the marketand m the total number of colleges. Here F is a σ -field that contains the colleges’priorities over students. The bound B ( t ) measures how much the distribution ofthe normalized matching frequencies MF ( x, z ) / ( nm ) concentrates around the mean E [ MF ( x, z ) |F ] / ( nm ) .It is far from trivial to study the statistical properties of matching frequencieswhen the matching realizes from a stable matching mechanism. As in many match-ing models in economics, we assume that the randomness of the matching comesprimarily from the randomness of the preferences. In any matching among a finitenumber of agents, the preference of each agent may affect the matching outcomes of the other agents. Hence the matching result of every agent is potentially corre-lated with that of every other agent. Such a cross-sectional dependence pattern ofa matching outcome in a large stable matching market is not conformable with theexisting dependence notions in probability and statistics such as time series depen-dence, spatial dependence, or global dependence through common shocks. In fact,to the best of our knowledge, there is no formal result of the law of large numbersfor stable two-sided many-to-one matching outcomes in the literature. In this paper, we focus on a stable matching that arises from a Deferred Ac-ceptance (DA) mechanism, which is one of the most studied centralized matchingmechanisms since Gale and Shapley (1962). This mechanism, when it begins withstudents applying for colleges, leads to a two-sided matching outcome that is op-timal for the students among all matching outcomes that are stable. Thus we callsuch matchings student-optimal matchings . This algorithm and its variants havebeen widely used in practice, for instance in the matching of medical residentswith hospitals, and students with schools. (See Section 5.4 in Roth and Sotomayor(1990) for a centralized matching in labor markets for medical interns. See alsoAbdulkadiro˘glu, Che, and Yasuda (2011) for a comparison between two centralizedmatching mechanisms: DA mechanism and Boston mechanism.) The DA mecha-nism has also been employed as an econometric model of a large matching in theliterature. (See e.g. Boyd, Lankford, Loeb, and Wyckoff (2013).)One of the innovative elements we introduce is our association of correlationamong colleges’ preferences over students with similarity between students and astudy of how such an approach can be used to study the concentration of measurephenomenon for matching frequencies. For example, consider a situation of collegeadmissions in which two students each apply to one or more colleges. Then, thecolleges’ ranking of the two students is more likely to be the same when measures ofthe students’ observed ability (such as their SAT scores) are very different than whenthey are similar. To formalize this notion, we introduce a graph among the agentswhose geodesic distance represents the degree of similarity between agents, i.e., thecloser two agents on the one side of the market are in the graph, the more similarthey look to agents on the other side of the market. In our model of preference, It is worth noting that Menzel (2015) developed an econometric model of a large stable one-to-onematching market and in doing so, he proved the Law of Large Numbers for the matching frequencies.We will discuss his work in relation to ours later. it is only when two agents are similar enough (i.e., within a threshold distance inthe similarity graph) that the other sides’ preferences for them are permitted to beheterogeneous. In such a setting, we call these preferences partially homogeneous .Furthermore, our setting permits a quite flexible form of preferences. Most im-portantly, we do not assume that an agent’s payoff can be additively or multiplica-tively decomposed into a systematic component and a match-specific idiosyncraticcomponent that is i.i.d. across potential matches. Such a payoff structure has beenfrequently used in the econometrics literature of large matching with either trans-ferable utility or non-transferable utility (Dagsvik (2000), Choo and Siow (2006),Menzel (2015) and Fox (2018)). In our set-up, the preference of an each agent is arandom permutation over the set of agents on the other side. The only assumptionwe require is that the students’ random permutations are conditionally independentgiven the colleges’ priorities over the students.Our main result is a concentration inequality for a general statistic on observedmatching outcomes. An important special case is when the statistic is a normalizedmatching frequency, as introduced above. In particular, our concentration bound isshown to increase as colleges’ preferences over students get more heterogeneous.This seems to be from the extensive dependence between matching outcomes aris-ing from the correlated preferences. We demonstrate this phenomenon using oursimulations.The main method we rely on for the concentration inequality is to use a con-ditional version of McDiarmid’s inequality (McDiarmid (1989)). This inequality isvery useful in our context, as it gives us a general tool to derive a bound for theconcentration-of-measure for statistics that involve independent random variablesin a complex form. The concentration bound is essentially determined by the shapeof nonlinear dependence, more specifically, by the bounded difference property thatshows how sensitively the statistic responds when one of the input random variablesis changed. Thus, our main innovation is to establish this bounded difference con-dition for matching outcomes.In deriving the bounded difference condition for the large matching, we drawheavily on the machineries of economic theory. Roth and Vande Vate (1990) pro-posed a random process through which an arbitrary matching converges to a stablematching with probability one. Blum, Roth, and Rothblum (1997) and Blum and Rothblum(2002) developed a re-stabilization operator that takes an arbitrary matching and produces a stable matching after a finite number of iterations. Similarly, we relyon a re-stabilization operator that transforms a matching into a stable matching,which we use to obtain a bound for the number of the students affected by one stu-dent’s change of preference. Our bounding the number of such students is relatedto the rejection chain method that Kojima and Pathak (2009) used in their study ofstrategic proofness of a large student optimal stable matching mechanism.A result more closely related to ours is found in Menzel (2015) who proved theconvergence of matching frequencies as part of his development of an econometricmodel of a large one-to-one matching market. There are a few major differencesbetween his paper and ours. First, our paper focuses on a many-to-one matchingmarket, where the number of students can be much greater than colleges. Thisimbalance between the two sides creates a fundamental difference from the settingMenzel (2015) studied. Second, as mentioned before, Menzel (2015) used a payoffstructure that is multiplicatively separable into systematic components and idiosyn-cratic comonents, while our setting does not. The work of Diamond and Agarwal(2017) is also related to our paper. They studied identification and asymptotic in-ference in many-to-one matching markets - in particular, discovering the value ofmany-to-one matchings as opposed to one-to-one matchings in identifying the pay-off parameters. The main difference is that they focused on a case of homogeneouspreferences on both sides of the market, where the number of students is fixed inproportion to the number of colleges. Furthermore, their asymptotic inference re-lied on a sampling process where the observed variables are drawn independentlyfrom the conditional distribution given utilities. In our setting, the number of thestudents is allowed to be much larger than the number of colleges, the preferencesof students are permitted to be fully heterogeneous, and the colleges’ preferencesare partially homogeneous. Having said that, the goal of our paper, as compared toMenzel (2015) and Diamond and Agarwal (2017), is rather modest, focusing on thelaw of large numbers for matching frequencies instead of developing a full econo-metric model for large two-sided matching markets. (See Chiappori and Salani´e To the best of our knowledge, it is a far from trivial to develop a method of asymptotic inferencethat takes into account the cross-sectional dependence structure of matching outcomes created bystrategic interdependence in a large matching market. Menzel (2015) used a random sampling (ora stratefied sampling) scheme for inference from observed matchings. The sampling process usedby Diamond and Agarwal (2017) does not preserve the entire cross-sectional dependence structureof the large matching. (2016) for a survey of various econometric models of large two-sided matchingmarkets.)The remainder of the paper is organized as follows. In the next section, we for-mally introduce a two-sided matching market drawing on the analogy of a collegeadmissions model, where colleges’ preferences are partially homogeneous. In Sec-tion 3, we provide a general concentration inequality on the Lipschitz functionalsof a large matching. Section 4 provides some numerical illustrations that showthe relation between homogeneity of preferences and concentration of measure. InSection 5, we conclude.
2. A College Admissions Model
Throughout the paper, we follow Roth and Sotomayor (1990) and refer to ageneric many-to-one matching model as a college admissions model , calling one side students and the other side colleges . While the analogy of a colleges admissionsmodel eases the exposition of our model, the reader need not assume our model istargeted to reflect all the details specific to this matching environment in reality.A college admissions model consists of the set N = { , ..., n } of students and theset M = { , ..., m } of colleges. In many situations, colleges are capacity-constrained.For each college j ∈ M , let q j be a positive integer that represents the quota of col-lege j . To accommodate the possibility of unmatched students and colleges withunfilled positions, we denote N ′ = N ∪ { } and M ′ = M ∪ { } , so that an un-matched student or an unfilled position at a college is viewed as being matched to . (When we need to view N ′ and M ′ as an ordered set, according to the ordering ofnatural numbers, we take to be the last element of N ′ and M ′ .) A (many-to-one) matching under the capacity constraint q = ( q j ) j ∈ N is defined as a (point-valued)map µ : N ′ → M ′ such that | µ − ( j ) ∩ N | ≤ q j for each j ∈ M . That is, the matching, µ , is such that the number of students assigned to each college does not exceed thecapacity of the college.In our college admissions model, we let v i represent student i ’s preference or-dering over colleges , ..., m , which is represented by a random permution of the colleges. Similarly, we let w j represent college j ’s preference ordering over stu-dents, , ..., n , which is represented by a random permutation of the students. Forany two colleges j , j , we write j ≻ i j if and only if v i ( j ) < v i ( j ) , and for anytwo students i , i , we write i ≻ j i if and only if w j ( i ) < w j ( i ) . If ≻ j i , thismeans that college j prefers to be unmatched by any student than to be matchedwith student i , and in this case we say that student i is unacceptable to college j . Let V consist of preference profiles v = ( v , ..., v n ) of students, and W the setof preference profiles w = ( w , ..., w m ) of colleges. The collection of preferenceprofiles, u = ( v , w ) is given by U = V × W , where v ∈ V and w ∈ W .In modeling the predicted outcomes of matching in economics, it is standard inthe literature to focus on (pairwise) stable matchings (Roth and Sotomayor (1990)).We call a map µ : N × U → M ′ a matching mechanism . Each u ∈ U induces a matching µ ( · ; u ) . We say that a matching mechanism µ is stable if for each u ∈ U , µ ( · ; u ) is stable in the sense that:(1) (Individual Rationality) There is no i ∈ N or j ∈ M such that ≻ i µ ( i ; u ) and ≻ j i ′ for some i ′ ∈ µ − ( j ; u ) .(2) (Incentive Compatibility) There is no ( i, j ) ∈ N × M such that both j ≻ i µ ( i ; u ) and i ≻ j i ′ for some i ′ ∈ µ − ( j ; u ) .Hence, a matching is stable if each agent prefers to be matched with their currentmatch than to be unmatched, and there is no student-college pair not currentlymatched who can improve over their current match by matching with one another.(Such a pair is called a blocking pair to the matching.) We call any triple ( N, M, u ) a matching market . A stable matching µ of a market ( N, M, u ) is called the student-optimal stable matching for that market if for every i ∈ N , µ ( i ) ≻ i µ ′ ( i ) or µ ( i ) = µ ′ ( i ) , for every other stable matching µ ′ of ( N, M, u ) . Astudent optimal matching is unique, and can be realized through the DA mechanism A college j ’s preference over groups of students is said to be responsive to its individual preferences ≻ j over students if for any two assignments differing in only one student, the college prefers theassignment containing the preferred student according to ≻ j . (See Definition 5.2. in Section 5 ofRoth and Sotomayor (1990).) Throughout the paper, we also maintain the assumption that collegepreferences are responsive. Our notation here leaves the quotas of colleges implicit, referring to them only when required. proposed by Gale and Shapley (1962). We present the description of the algorithmtaken from Kojima and Pathak (2009), p.612, below.
Step 1:
Each student applies to her first-choice college. Each college rejects the lowest rankingstudents in excess of its capacity and all unacceptable students among those who applied to it,keeping the rest of the students temporarily. Students not rejected at this step may be rejected inlater steps.
Step t in general : Each student who was rejected in Step (t-1) applies to her next highest choice(if any). Each college considers these students and students who are temporarily held from theprevious step together, and rejects the lowest-ranking students in excess of its capacity and all unac-ceptable students, keeping the rest of the students temporarily (so students not rejected at this stepmay be rejected in later steps).
As mentioned in the introduction, the DA algorithm has been one of the moststudied centralized matching mechanisms in the literature. In this paper, we focuson the student optimal matching mechanism.
The preferences of agents in a two-sided matching market are often correlatedin the sense that agents on one side of the market may agree upon the rankings ofagents on the other side of the market. In the case that preferences are perfectlycorrelated, all agents on one side of the market share the same ranking over theagents on the other side of the market. We refer to preferences in such a case as homogeneous . In this paper, we express the strength of correlation across collegepreferences in terms of the extent to which they are homogeneous. The main ideais that when two students are “dissimilar enough” (such as in terms of SAT scores),their rankings are commonly agreed upon by colleges whereas rankings over stu-dents who are similar are permitted to be heterogeneous across colleges. We callsuch preferences partially homogeneous in the paper.To formalize partial homogeneity of preferences in our model, let ≻ N be a partialorder over the students. We say that the partial order ≻ N is of degree k , if forany two students i , i ∈ N , the rank of i in any linear extension of ≻ N is higherthan i by more than k if and only if i ≻ N i . We assume that all the colleges’preferences over the students are linear extensions of the partial order ≻ N . A largerdegree k reflects a higher extent of heterogeneity in the colleges’ preferences overthe students. We can explicitly associate the degree of preference heterogeneity with similaritybetween students such that preferences are heterogeneous only if the students aresimilar enough. More specifically, suppose that there is a similarity graph G =( N, E ) on the set of students, such that the geodesic distance d G (i.e., the length ofthe shortest path) between any two students i and i represents how similar theyare. We assume that colleges’ preference profile w is k -consistent with the similaritygraph G , in the sense that if i ≻ N i , d G ( i , i ) > k . Thus, a denser similarity graphis associated with a high degree of homogeneity in colleges’ preferences.Throughout the paper, we envisage random preferences , where the preferenceprofiles are randomly drawn from a distribution, and then the students preferences v are drawn from a conditional distribuiton given w that satisfies the followingassumption. Assumption 2.1. (i) The colleges’ preference profile w is drawn from a distributionunder the restriction that w is k -consistent with a similarity graph G .(ii) The individual preferences v i of the students’ preference profile v are condi-tionally independent given w .Assumption 2.1 is the only condition we impose on preferences in this paper.We do not introduce a further structure, such as a specification where the payoffof each agent is additively separable into a systematic component or idiosyncraticcomponent. In particular, the payoffs of an agent across potential matches withagents on the other side can be arbitrarily correlated.
3. The Law of Large Numbers for Stable Matchings
Suppose that a matching mechanism µ ( u ) = [ µ ( i ; u )] i ∈ N is given. As we see be-low, statistics that summarize matching patterns typically take the form of ϕ ( µ ( u ); u ) for a functional ϕ . Our focus is on the phenomenon of a concentration-of-measureof ϕ ( µ ( u ); u ) around its conditional mean given w .While the students’ preferences u are generated independently, the matching µ ( u ) depends on the entire preference profile v nonlinearly, and hence we can-not apply standard law of large numbers results that have been established in the literature of probability and statistics. Instead, we rely on a concentration inequal-ity called McDiarmid’s inequality (McDiarmid (1989)), and prove the concentrationof ϕ ( µ ( u ); u ) around its conditional mean given w . A conditional version of McDi-armid’s inequality is given as follows. Lemma 3.1 (McDiarmid’s Inequality) . Suppose that X i ’s are random elements takingvalues in a space X which are conditionally independent given a σ -field, F . Let g : X n → R be a map such that for all i = 1 , ..., n , and all x , ...., x m and x ′ i ∈ X , | g ( x , ..., x i − , x i , x i +1 , ..., x n ) − g ( x , ..., x i − , x ′ i , x i +1 , ..., x n ) | ≤ c i , (3.1) for some c i > . Then, for all t > , P {| g ( X , ..., X n ) − E [ g ( X , ..., X n ) |F ] | ≥ t |F } ≤ (cid:18) − t P ni =1 c i (cid:19) . (3.2)A key step in applying McDiarmid’s inequality involves establishing that the match-ing mechanism of interest obeys the bounded difference condition in (3.1). Theinequality shows that the distribution of g ( X , ..., X n ) concentrates more around itsconditional mean if the bounds c i in (3.1) are small.Let us present a bounded difference result for a student-optimal stable matchingbelow. Let D ⊂ U × U be the set of pairs of profiles ( u , u ′ ) such that u , u ′ ∈ U differ by the preference of one student. Lemma 3.2.
Let µ be the student-optimal stable matching mechanism. Then for any ( u ′ , u ) ∈ D , |{ i ∈ N : µ ( i ; u ) = µ ( i ; u ′ ) }| ≤ m ( k + 1) + 2) . (3.3)In any two student-optimal stable matchings such that the preferences differ byonly one student’s preference, there are no more than m ( k + 1) + 2) studentswho are matched to different colleges between the two matchings. This bounddepends on the number m of the colleges in the market and the degree k of partialhomogeneity of colleges’ preferences. When k is larger, colleges’ preferences areallowed to be more heterogeneous, and this leads to a larger bound for the boundeddifferences.The role of partial homogeneity in the preferences is crucial for obtaining thebound in Lemma 3.2. As we show in the example below, when the disagreementamong colleges is extensive enough, a change in preference by a single student canalter the match of every student. T ABLE
1. Preferences of Agents for Example 3.1Student Preferences College Preferences i ( j , j , j ) j ( i , i , i , i , i ) i ( j , j , j ) j ( i , i , i , i , i ) i ( j , j , j ) j ( i , i , i , i , i ) i ( j , j , j ) i ( j , j , j ) Example 3.1.
Consider a college admissions market with five students. That is, N = { i , i , i , i , i } , and three colleges, M = { j , j , j } with q = 1 , q = q = 2 .Suppose that the preferences u = ( v , w ) are given by Table 1. For example, thepreference of student i is such that the student considers college j the best, j thesecond best, and j the worst. The student-optimal stable matching under the givenpreferences is ( µ ( i ) , µ ( i ) , µ ( i ) , µ ( i ) , µ ( i )) = ( j , j , j , j , j ) . Next, suppose that the preferences of agents are instead given by u ′ = ( v ′ , w ) ,where v ′ is defined to be identical to v , except that we replace the preferencesof student i with the ordering ( j , j , j ) . The student-optimal matching under u ′ = ( v ′ , w ) is µ ′ = ( j , j , j , j , j ) . Thus, as students move from mathing µ to matching µ ′ due to one student’s prefer-ence change, all of them end up being matched with a different college. (cid:3) It is worth noting that the bound in Lemma 3.2 can be tighter when there arevacancies at the colleges. Indeed, Example 3.2 below illustrates how the effectsof extensive college-preference heterogeneity can be mitigated by the presence ofvacancies at the colleges. Since the bounded difference condition must accountfor a ‘worst-case scenario’ in which such vacancies are absent at the colleges, thisexample suggests that the bound in Lemma 3.2 can be conservative in practice whenat least some colleges have vacancies.
Example 3.2.
Consider again the college admissions market introduced in Example3.1. This time, however, suppose that q = 1 , q = 3 , q = 2 . That is, college j has an additional position. As before, the student optimal matching under u is µ = ( j , j , j , j , j ) . However, the fact that college j has a vacanct position at µ implies that the change of preferences from v to v ′ (as defined in Example 3.1)would lead to only student i changing colleges. That is, the presence of a vacancyat college j prevents the ‘cascade’ of changes that occured in the previous example.By the same logic, if every college at µ had one vacant position, the change inpreferences from u to any profile u ′ that differed in the preference of one studentwould lead to at most one student changing college. (cid:3) Let us turn to the concentration inequality for a statistic ϕ ( µ ( u ); u ) . As for thefunctional ϕ in the statistic, we introduce the following condition. Assumption 3.1.
The functional ϕ ( · , · ) : ( M ′ ) n × U → R d satisfies that there ex-ist constants K > , and c i > , i ∈ N , such that for all u ∈ U , w ∈ W , anypermutations v i , v ′ i of M ′ , and any two matchings µ , µ : N → M ′ , k ϕ ( µ ; u ) − ϕ ( µ ; u ) k ≤ K k µ − µ k H , and(3.4) k ϕ ( µ ; v − i , v i , w ) − ϕ ( µ ; v − i , v ′ i , w ) k ≤ c i , where || · || H denote the Hamming norm, and µ ℓ = [ µ ℓ ( i )] i ∈ N , ℓ = 1 , .The assumption encompasses a wide range of functionals ϕ . We will discuss anexample below. If we let g ϕ ( u ) = ϕ ( µ ( u ); u ) , observe that the inequalities (3.4)and Lemma 3.2 give us the following bounded difference condition k g ϕ ( v − i , v i , w ) − g ϕ ( v − i , v ′ i , w ) k ≤ K k µ ( v i , v − i , w ) − µ ( v ′ i , v − i , w ) k H + c i (3.5) ≤ K ( m ( k + 1) + 2) + c i . Thus by McDiarmid’s inequality the concentration inequality immediately follows,as presented in the theorem below.
Theorem 3.1.
Let µ be the student-optimal stable matching mechanism and Assump-tions 2.1 and 3.1 hold. Then, for all t > : P {| ϕ ( µ ( u ); u ) − E [ ϕ ( µ ( u ); u ) | w ] | ≥ t | w } (3.6) ≤ − t K n ( m ( k + 1) + 2) + P ni =1 c i ! . For two vectors a, b ∈ R n , || a − b || H ≡ P ni =1 { a i = b i } . To appreciate the meaning of Theorem 3.1, let us consider an example of thefunctional ϕ . Suppose that X i and Z j denote the observed characteristics of student i and college j . We let U ij = ( X i , Z j ) and permit that U ij ’s are involved in thepreferences of students and colleges. For example, w j ( i ) < w j ( i ) , if and only if ϕ s ( U i ,j , ε i ,j ) > ϕ s ( U i ,j , ε i ,j ) , and(3.7) v i ( j ) < v i ( j ) , if and only if ϕ c ( U i,j , η i,j ) > ϕ c ( U i,j , η i,j ) , where ε i,j ’s and η i,j ’s are unobserved preference components, and ϕ s and ϕ c arenonstochastic maps. Hence for example, college j prefers student i to i if andonly if ϕ s ( U i ,j , ε i ,j ) > ϕ s ( U i ,j , ε i ,j ) . The results of this paper do not require anyrestrictions on the maps ϕ s and ϕ c or on the relation between the observed andunobserved components of the preferences.The statistic commonly used in the literature of matching takes the followingform: ϕ ( µ ( u ); u ) = 1 nm X i ∈ N X j ∈ M { µ ( i ; u ) = j } h ( X i , Z j ) , (3.8)for some map h . This gives an aggregate matching pattern between students andcolleges according their observed characteristics X i ’s and Z j ’s. For example, let usassume that they are discrete, taking values from X and Z respectively, and for ( x, z ) ∈ X × Z , we take h ( X i , Z j ) = 1 { X i = x, Z j = z } . (3.9)Then the quantity ϕ ( µ ( u ); u ) , after the preferences are realized, gives the proba-bility that a randomly chosen student-college pair are matched under µ ( · ; u ) andat the same time they have characteristics x and z respectively. For example, thesestatsitics are used to measure the extent of positive assortative matching betweenstudents and colleges based on their characteristics.Let us define the population counterpart of ϕ ( µ ( u ); u ) as its conditional mean E [ ϕ ( µ ( u ); u ) | w ] . (3.10)Theorem 3.1 then gives the following result. Corollary 3.1.
Let µ be the student-optimal stable matching mechanism and Assump-tion 2.1 holds. Furthermore, suppose that ϕ takes the form in (3.8) with map h bounded by . Then, P {| ϕ ( µ ( u ); u ) − E [ ϕ ( µ ( u ); u ) | w ] | ≥ t | w } (3.11) ≤ (cid:18) − nm t / m ( k + 1) + 2) + 1 (cid:19) . The corollary shows that regardless of whether the number of colleges m is fixedor increases to infinity, the rate of convergence is determined by n . Indeed, we havethe law of large numbers as long as n → ∞ . The result also shows if the partialhomogeneity degree k is such that k/ √ n → , (3.12)as n → ∞ , the law of large numbers holds. While the concentration-of-measure isdriven by the increasing number of students, it permits m to increase at the samerate as n .By the Borel-Cantelli Lemma, we obtain the following strong law of large num-bers result from our concentration inequality. Corollary 3.2.
Suppose that the conditions of Corollary 3.1 hold, and that there existconstants
C > and < b < / such that k ≤ Cn b for all n ≥ . Then, P n lim n →∞ | ϕ ( µ ( u ); u ) − E [ ϕ ( µ ( u ); u ) | w ] | = 0 o = 1 . (3.13)Thus the aggregate matching patterns are not sensitive to the realizations of stu-dent’s preferences when n is large. In other words, for any two students’ preferencesequences v and v ′ , and u = ( v , w ) and u ′ = ( v ′ , w ) , where v and v ′ are drawnfrom the same distribution that is conditionally independent across students given w , we have lim n →∞ | ϕ ( µ ( u ); u ) − ϕ ( µ ( u ′ ); u ′ ) | = 0 , (3.14)with probability one. Hence the aggregate matching patterns are robust to a partic-ular realization of the students’ preferences in a large matching. Let us give a sketch of the proof of the bounded difference result in Lemma 3.2.For each student i ∈ N , let µ i ( · ; u − i ) be the student optimal matching in a mar-ket with student i eliminated, and let g i ( µ i ( · ; u − i )) be the matching obtained after adding back student i yet leaving the student unmatched. Then by modifying there-stabilization algorithm in the literature, we define an operator T ∗ which assignsthe student-optimal matching µ ( · ; u ) to g i ( µ i ( · ; u − i )) : µ ( · ; u ) = T ∗ ( g i ( µ i ( · ; u − i ))) . The operator T ∗ reveals a relation between the two matchings µ and µ − i . Morespecifically, for each j ∈ M , define N j, = µ − i ( j ; u − i ) \ µ − ( j ; u ) , and N j, = µ − ( j ; u ) \ µ − i ( j ; u − i ) , so that N j, is the set of students displaced from college j after the re-stabilizationprocess, and N j, the set of students newly matched with college j after the process.The properties of T ∗ guarantee that N j, is empty for every college j with a vacancyunder the original matching. Further, we also have that either N j, is empty forevery j with a vacancy under the original matching, or that N j, empty for everysuch j but one, for whom the set is singleton. For every college with no vacancyunder the original matching, the operator T ∗ guarantees that(a) | N j, | = | N j, | , and(b) | N j, ∪ N j, | ≤ k + 1) − .To obtain the second result, we draw on results from the theory of ordered sets(Schr¨oder (2016)). Since N j, and N j, are disjoint, we obtain that | N j, | ≤ k + 1 , and | N j, | ≤ k + 1 . From this, we find the following bound X i ′ ∈ N { µ ( i ′ ; u ) = µ i ( i ′ ; u − i ) , i = i ′ } (3.15) ≤ X j ∈ M | µ − ( j ; u ) \ µ − i ( j ; u − i ) | + 1 ≤ m ( k + 1) + 1 . (3.16)(The additive is due to the possibility of a student being unmatched as a resultof the process. See the proof for details.) Now, to obtain the bounded difference condition, let u and u ′ differ by student i . Then, |{ i ′ ∈ N : µ ( i ′ ; u ) = µ ( i ′ ; u ′ ) }| = X i ′ ∈ N { µ ( i ′ ; u ) = µ ( i ′ ; u ′ ) } (3.17) ≤ X i ′ ∈ N { µ ( i ′ ; u ) = µ ( i ′ ; u ′ ) , i ′ = i } + 1 ≤ X i ′ ∈ N { µ ( i ′ ; u ) = µ i ( i ′ ; u − i ) , i ′ = i } + X i ′ ∈ N { µ i ( i ′ ; u − i ) = µ ( i ′ ; u ′ ) , i ′ = i } + 1 . Then, since u − i = u ′− i , we use the bound from (3.16) to bound each term on theright of (3.17). This gives the desired bound in Lemma 3.2. We consider n students and a small number m of colleges. We draw i.i.d. prefer-ences as follows: v s,i,j = β Z j + ε ij , and v ∗ c,i = γ X i + η i , where η i ’s and ε ij ’s are N (0 , and Z j ’s and X i ’s are Bernoulli random taking values1 and 0 with equal probability. The partial order of the colleges’ preferences overstudents can be constructed by discretizing v ∗ c,i , i = 1 ..., n , into B bins, and colleges’spreferences are different by randomly permuting the order of students in the samebin. We consider the following statistic: S = 1 n n X i =1 { X i = 1 , Z µ ( i ) = 1 } , (3.18)where µ ( i ) is a matching obtained by using the student-optimal Deferred Accep-tance Algorithm. For example, this is the sample moment used by Diamond and Agarwal(2017). Now, we would like to see the variance of S as we vary n , m and B .Throughout the simulations, we assume that colleges have equal capacities, set-ting q j = n/m for all colleges. For a fixed choice of n , m and B , we generate therandom variables X i ’s, Z j ’s and η i ’s outside the simulation loop, with only ε ij ’s andthe college’s preferences allowed to differ across the iterations used to compute S .We also set β = γ = 1 . T ABLE
2. Simulated Mean and Standard Deviation of S Standard Deviation Mean m n = 500 n = 1000 n = 2000 n = 500 n = 1000 n = 200010 B = 1 B = 2 B = 5 B = 500 B = 1 B = 2 B = 5 B = 500 B = 1 B = 2 B = 5 B = 500 Notes: The table reports the simulated mean and variance of S for different values of n , m , and B . Alarger B implies more homogeneity in college preferences. The variance of S falls in (1) n , for fixed m and B , (2) m for fixed n and B , and B for fixed n and m . In our simulation design, the mean value of S is also increasing in B , illustrating a positive relationship between the level of preference homogeneityand the degree of assortative matching. The simulation number is , . Table 2 reports the simulated mean and variance of S for a variety of values forvarious values of n , m , and B . As expected, the effect of an increase in the numberof students when the number of colleges and M is held fixed is a fall in the varianceof S . Likewise, the variance of S also decreases in n for fixed m and B . Lastly, forfixed m and n the general effect of an increase in M is a fall in the variance and anincrease in the mean of S . Thus, under the above simulation design, an increasein college preference homogeneity is associated with concentration of measure andstronger positive assortative matching.
4. Conclusion
This paper considers a large two-sided matching market, where a matching be-tween the two sides is realized through a centralized algorithm such as a DA mech-anism. In such a situation, it is a nontrivial matter to establish limit theorems forstatistics including matching frequencies as the number of the participants grows.Using the re-equilibration arguments from economic theory and a concentration inequality, we derive a concentration inequality for matching frequencies in thisenvironment.We believe that our result can potentially be useful for developing inference onlarge matching markets under a wide range of sampling processes. To developinference robust to sampling processes (such as random sampling), one needs totake into account the dependence structure of the observations explicitly. As men-tioned before, this is challenging in large matching markets. Our work addressesthis challenge and reveals the role of partial homogeneity of preferences in theconcentration-of-measure phenomenon for matching frequencies. It is left to futureresearch to establish limit distribution theory for the large matching setting, wherea complex dependence structure arises naturally due to the strategic interdepen-dence in the underlying market.
5. Appendix: Proofs
The crucial result for our concentration inequality is the bounded difference resultin Lemma 3.2. To establish this, we introduce a re-stabilization operator thattransforms an unstable matching in one market into a stable matching in another,by repeatedly satisfying a blocking pair . Roth and Vande Vate (1990) showed thatgiven an arbitrary matching, a sequence of matchings in which each is obtainedfrom the previous matching by satisfying blocking pairs is guaranteed to convergeto a stable matching when blocking pairs may be chosen randomly at each stepin the sequence. Rather than re-stabilizing arbitrary matchings, we focus on smallperturbations to the outcomes of stable matching mechanisms caused by a changein the preferences of a single student. Therefore, we will consider a re-stabilizationoperator that takes in a matching that is already ‘close’ to be stable. The definition in the one-to-one matching case is well known (e.g., Blum, Roth, and Rothblum(1997)): given an unstable matching µ ′ with a blocking pair ( i, j ) we say that a matching µ isobtained from µ ′ by satisfying the blocking pair ( i, j ) if i and j are matched to each other in µ , their mates (if any) in µ ′ are unmatched in µ and the status of the remaining matched agentsis unchanged. In our many-to-one setup, the notion of satisfying a blocking pair relevant for ourpurposes will be made precise later when we define a special operator (see Definition 5.2). We begin with some key definitions that we use repeatedly later. Definition 5.1.
Given a matching µ , we say that(1) µ is individually rational if there is no i ∈ N or j ∈ M such that ≻ i µ ( i ) and ≻ j i ′ for some i ′ ∈ µ − ( j ) .(2) µ is envy-free if it is individually rational and any blocking pair, if it exists,involves an unmatched student in µ .(3) µ is envy-free with respect to one student if it is (i) envy free and (ii) either µ is stable or there exists one and only student i ∈ N who belongs to everyblocking pair of µ .Given a a market ( N, M, u ) , let S ( N, M, u ) be the set of stable matchings, and let E ( N, M, u ) be the set of matchings that are envy-free with respect to one student.With each map µ i : N \{ i } → M ′ , we associate a map g i ( µ i )( · ) : N → M ′ defined by g i ( µ i )( i ′ ) = ( µ i ( i ′ ) , if i ′ = i, , if i ′ = i. (5.1)The map g i ( µ i ) is a matching on N constructed from µ i by matching student i to .The following remark follows in a straightforward way from our previous defini-tions. Remark 5.1.
For every u ∈ U and i ∈ N , µ i ∈ S ( N \{ i } , M, u − i ) if and only if g i ( µ i ) ∈ E ( N, M, u ) . Given a matching µ , a pair ( i, j ) is a student-maximal blocking pair for µ if ( i, j ) is a blocking pair for µ and j is the student’s most preferred college amongthose with whom he can form a blocking pair for µ . Our re-stablization operatoriteratively satisfies student-maximal blocking pairs. Closely related definitions have been used. In particular, the notion of doctor quasi-stable matchingsdiscussed in Wu and Roth (2018). The matchings in the second item of the definition can be viewedas a many-to-one version of the matchings studied by Blum and Rothblum (2002) in the one-to-onecase. The operator we propose is a straightforward adaptation of the correcting procedures describedin Blum and Rothblum (2002) (itself a special case of Blum, Roth, and Rothblum (1997)), adaptedto a many-to-one setup. The approach is also similar to Bir´o, Cechl´arov´a, and Fleiner (2008), whodiscuss algorithms for stabilizing matching markets when a single agent is added to a market thatis presumed stable in the absence of the additional agent. Note that Wu and Roth (2018) showedthat in many-to-one markets in which no students have justified envy, stable matchings can beobtained as fixed points of a lattice operator that generalizes the college-optimal deferred acceptance Before defining the operator, we introduce some further notation. Define B µ tobe the set of all blocking pairs to a matching µ for the market ( N, M, u ) . Let the setof student-maximal blocking pairs for a matching µ be B ′ µ = { ( i ′ , j ′ ) ∈ B µ : j ′ ≻ i ′ j ′′ , for all ( i ′ , j ′′ ) ∈ B µ } . Note that B ′ µ = ∅ if and only if B µ = ∅ . In the case that µ ∈ E ( N, M, u ) , B ′ µ iseither empty, or contains exactly one blocking pair. In the case that B µ = ∅ , wedefine j ∗ ( i ) to be such that ( i, j ∗ ( i )) ∈ B ′ µ . Definition 5.2.
For any µ ∈ E ( N, M, u ) , the operator T : E ( N, M, u ) → E ( N, M, u ) is defined as follows.Suppose B µ = ∅ . Then we take T ( µ ) = µ .Suppose B µ = ∅ . Then B ′ µ is a singleton, say, { ( i, j ∗ ( i )) } and we set T ( µ )( i ) = j ∗ ( i ) . For each i ′ = i , let us denote its matched college by j ′ (i.e., µ ( i ′ ) = j ′ ). We set T ( µ )( i ′ ) as follows:Case 1: j ′ = j ∗ ( i ) . Then T ( µ )( i ′ ) = j ′ .Case 2: j ′ = j ∗ ( i ) . Then, T ( µ )( i ′ ) = j ′ , if | µ − ( j ′ ) | < q j ′ ,j ′ , if | µ − ( j ′ ) | = q j ′ , and i ′′ ≺ j ′ i ′ for some i ′′ ∈ µ − ( j ′ )0 , if | µ − ( j ′ ) | = q j ′ , and i ′ ≺ j ′ i ′′ for all i ′′ ∈ µ − ( j ′ ) \ { i ′ } . (5.2)Note that from the definition of T , it is clear that any µ ∈ S ( N, M, u ) satisfies T ( µ ) = µ ; i.e., any stable matching is a fixed point of T . It is clear that T mapsfrom E ( N, M, u ) to itself, since for any µ ∈ E ( N, M, u ) , T ( µ ) is either stable orblockable by at most one student, i ′ .The next result, Lemma 5.1, shows that repeated iterations of the operator T yield a stable matching when the input is an envy-free matching with respect toone student. We introduce a partial order % over matchings. First, we require thefollowing straightforward definition. algorithm. Note that the “college” envy-free matchings in our context (i.e., the second item ofDefinition 5.1) is not a lattice with respect to the common preferences of colleges for the samereason that doctor quasi-stable matchings is not a lattice with respect to the common preference ofhospitals, as explained in Section 5 of Wu and Roth (2018). The sets B µ , B ′ µ , and the operator T certainly depend on u , but we will often suppress these fromour notation for simplicity. Definition 5.3.
We say that a pair of matchings µ and µ differ in the assignmentof at most one college by at most two students if either µ = µ or there is some i ∈ N with µ ( i ) = 0 and j ∈ M such that µ − ( j ′ ) = µ − ( j ′ ) for all j ′ ∈ M \{ j } andeither µ − ( j ) = µ − ( j ) ∪ { i } , or µ − ( j ) = µ − ( j ) ∪ { i }\{ i } for some i ∈ µ − ( j ) . Suppose that ≻ ∗ j is college j ’s preference ordering over groups of students. Weassume that ≻ ∗ j is responsive to ≻ j . Definition 5.4.
The preference relation of a college j , ≻ ∗ j , over sets of students is responsive to the preference over individual students if, whenever µ − ( j ) = µ − ( j ) ∪{ i }\{ i } for i ∈ µ − ( j ) and i / ∈ µ − ( j ) , then µ − ( j ) ≻ j µ − ( j ) if and only if i ≻ j i .For any pair of matchings µ and µ differing in the assignment of at most onecollege by at most two students, we write µ % µ if and only if for all j ∈ M , either µ − ( j ) ≻ ∗ j µ − ( j ) or µ − ( j ) = µ − ( j ) . Note that the preferences of colleges overgroups of students in the definition of % are well-defined under our assumptionthat college preferences are responsive. Finally, we define a partial ordering % overmatchings as the transitive closure of % . That is, for any two matchings µ ′ and µ we write µ ′ % µ if and only if there is a sequence µ = µ, µ , µ , ..., µ r with µ r = µ ′ such that µ r % µ r − % ... % µ % µ . Lemma 5.1.
The operator T satisfies T ( µ ) % µ for each µ ∈ E ( N, M, u ) . Moreover,for any µ ∈ E ( N, M, u ) there is a finite sequence µ = µ, µ , µ , ..., µ r with µ r ∈S ( N, M, u ) where µ r ′ = T ( µ r ′ − ) for each r ′ = 1 , ..., r . Proof: If B ′ µ is empty, the matching is stable and the operator defined above delivers T ( µ ) = µ. If B ′ µ is not empty, it contains exactly one blocking pair, ( i, j ∗ ( i )) , and T assigns student i to college j ∗ ( i ) ; in the case that j ∗ ( i ) has no vacancies under µ ,its worst student under µ , say, i ′ , is made unmatched. The status of all remainingstudents and colleges from µ are retained in T ( µ ) . For any µ ∈ E ( N, M, u ) wealso have that T ( µ ) % µ , since T either affects no colleges, or leaves exactly one See Definition 5.2. on page 128 of Roth and Sotomayor (1990). college, j ∗ ( i ) , strictly better off while leaving the remaining colleges unaffected. Since there are finite number of student-college pairs, repeated iterations of T fromany µ ∈ E ( N, M, u ) are guaranteed to converge to a stable matching, after finiteiterations. (cid:4) Lemma 5.1 shows that for any µ ∈ E ( N, M, u ) , repeated iterations of T lead toa stable matching of the market ( N, M, u ) , in finite iterations. Furthermore, theoutput of repeated iterations of T is uniquely determined by the given choice of µ ∈ E ( N, M, u ) , since there is at most one student-maximal blocking pair after eachiteration of the operator. It is convenient to develop notation for the stable output ofrepeated iterations of T in terms of an input matching. Given any µ ∈ E ( N, M, u ) ,we denote T ∗ ( µ ) ≡ µ r where µ = µ, µ , µ , ..., µ r is the finite sequence of matchingswith µ r ∈ S ( N, M, u ) , where µ r ′ = T ( µ r ′ − ) for each r ′ = 1 , ..., r . The next result, Lemma 5.4, allows us to refine the previous result to a situationin which the outcome of iterations of T happens to be not just a stable matching, butthe student-optimal stable matching. The result is essentially a many-to-one versionof Theorem 5.2 of Blum, Roth, and Rothblum (1997) adapted to our setup. Toproceed, we introduce a notion of one-to-one matching markets that are analogousto the many-to-one matching markets we have dealt with so far. Our definitions fol-low Section 5.2 of Roth and Sotomayor (1990), but we summarize the main detailshere for convenience.Given a many-to-one market ( N, M, u ) , we define the corresponding one-to-onemarket ( N, ¯ M , ¯ u ) as follows. First, the set of colleges ¯ M in the one-to-one marketis obtained “splitting” each college j ∈ M into q j positions, c j, , ..., c j,q j , where eachposition of j has the same preferences over students as college j . Students’ pref-erences over the positions in the one-to-one market are such that each student i prefers a position of college j to a position of college j ′ in the one-to-one market ifand only if i prefers college j to college j ′ in the many-to-one market. Moreover, By strictness and responsiveness of college preferences and the fact that ( i, j ∗ ( i )) is a blockingpair for µ , we have either (i) | µ − ( j ∗ ( i )) | = q j and j ∗ ( i ) strictly prefers µ − ( j ∗ ( i )) ∪ { i }\{ i ′ } to µ − ( j ∗ ( i )) for some i ′ ∈ µ − ( j ∗ ( i )) or (ii) | µ − ( j ∗ ( i )) | < q j and j ∗ ( i ) strictly prefers µ − ( j ∗ ( i )) ∪ { i } to µ − ( j ∗ ( i )) . We will sometimes write T ( · ; u ) and T ∗ ( · ; u ) when the distinction between the preferences in theunderlying market is important, but we will often suppress these from our notation for simplicity. A special case of the result also appears as the second item of Theorem 2.3 in Blum and Rothblum(2002). when comparing any two positions of the same college j , each student is assumedto simply to prefer the position with the smaller index. Thus, each student consid-ers c j, the best position of college j , c j, to be the second best position of j , and soon. Under the assumption of strict preferences, we obtain the following one-to-onecorrespondence between matchings for the market ( N, M, u ) and matchings for themarket ( N, ¯ M , ¯ u ) : a matching µ for market ( N, M, u ) which matches college j ∈ M with students µ − ( j ) , corresponds to a matching ¯ µ for market ( N, ¯ M , ¯ u ) in whichthe students in µ − ( j ) are matched in the order they occur in the college’s prefer-ences, with the ordered positions of j that appear in ¯ M . Thus, if i is college j ’s mostpreferred student in µ − ( j ) then ¯ µ ( i ) = c j, , and so on. Next, the notion of simplematchings (Sotomayor (1996)) are useful. Definition 5.5.
A matching ¯ µ is simple in market ( N, ¯ M , ¯ u ) if(i) there is no i ∈ N or j ∈ ¯ M such that ≻ i ¯ µ ( i ) and ≻ j ¯ µ − ( j ) , and(ii) Any ( i, j ) ∈ N × ¯ M such that j ≻ i ¯ µ ( i ) and i ≻ j ¯ µ − ( j ) satisfies that ¯ µ ( i ) = 0 .That is, ¯ µ is simple if it is (i) individually rational and (ii) any blocking pairof ¯ µ involves an unmatched student. The following result and proof is similar toProposition 2.2 of Wu and Roth (2018). Lemma 5.2.
Let µ be an envy-free matching in ( N, M, u ) (in the sense of Definition5.1). Then its corresponding matching ¯ µ in ( N, ¯ M , ¯ u ) is simple. Proof:
Let µ be envy-free in ( N, M, u ) . Suppose by contradiction that its corre-sponding matching ¯ µ is not simple in ( N, ¯ M , ¯ u ) . We assume that ¯ µ is individuallyrational, as otherwise the contradiction is immediate. This implies that there is ablocking pair ( i, j ) ∈ N × ¯ M for ¯ µ with ¯ µ ( i ) = j ′ = 0 . Suppose that j and j ′ arepositions of distinct colleges in M . Then we obtain the contradiction that µ is notenvy-free, since ( i, j ) is a blocking pair for µ , yet i is matched under µ . (cid:4) The following lemma is a many-to-one version of Theorem A6 of Blum and Rothblum(2002).
Lemma 5.3.
For any µ ′ ∈ E ( N, M, u ) , and µ = T ∗ ( µ ′ ) is the college-worst stablematching in ( N, M, u ) among those that colleges weakly prefer to µ ′ . Note that for such a blocking pair to exist, j and j ′ cannot be positions of the same college under M ; if j is the better position, then the college fills it with a preferred student; if j is the worseposition then i doesn’t prefer j to j ′ . Proof:
By Definition 5.2, each many-to-one matching in the sequence producedby repeated iterations of T from µ ′ is envy-free. Hence, by Lemma 5.2, the cor-responding one-to-one matching of each many-to-one matching in this sequence issimple. Moreover, since each many-to-one matching in the sequence is constructedby satisfying the student-maximal blocking pair of an envy-free matching, it fol-lows that each corresponding matching in the sequence of one-to-one matchings inthe related market is constructed by satisfying the student-maximal blocking pairof a simple matching. In other words, iterations of T perform the greedy correct-ing procedure of Blum and Rothblum (2002) in the related market. Let ¯ µ ′ be theone-to-one matching corresponding to µ ′ . By Theorem A6 of Blum and Rothblum(2002), for any simple matching ¯ µ ′ , the matching ¯ µ = T ∗ (¯ µ ′ ) is the college-worststable matching weakly preferred by the colleges to ¯ µ ′ .The remainder of the argument uses the following notation. Given two many-to-one matchings µ , µ with corresponding one-to-one matchings ¯ µ , ¯ µ , we use ¯ µ ≥ ¯ µ to denote ¯ µ − ( j ) ≻ j ¯ µ − ( j ) or ¯ µ − ( j ) = ¯ µ − ( j ) for all colleges j . Wenow argue that µ is the college-worst stable matching weakly preferred by collegesto µ ′ . Suppose by contradiction that there is a stable many-to-one matching µ ′′ satisfying µ % µ ′′ % µ ′ with µ ′′ = µ . Let ¯ µ ′′ denote the stable one-to-one match-ing corresponding to µ ′′ . Since µ and µ ′′ are stable it follows by Theorem 5.27 ofRoth and Sotomayor (1990) that ¯ µ ≥ ¯ µ ′′ . Next, the claim that µ ′′ % µ ′ implies that ¯ µ ′′ ≥ ¯ µ ′ by Lemma 5.25 of Roth and Sotomayor (1990). Hence, the previous twoimplications yields that ¯ µ ≥ ¯ µ ′′ ≥ ¯ µ ′ , contradicting the fact that ¯ µ is the college-worst stable matching weakly preferred to ¯ µ ′ . Hence we conclude that µ is thecollege-worst matching weakly preferred by colleges to µ ′ , as required. (cid:4) Lemma 5.4.
For any u ∈ U and i ∈ N , let µ i ≡ µ i ( · ; u − i ) be the student-optimalstable matching for market ( N \{ i } , M, u − i ) and µ ( · ; u ) the student-optimal stable Thus ¯ µ ≥ ¯ µ represents the statement that colleges weakly prefer ¯ µ to ¯ µ . This notation issimilar to that introduced in Roth and Sotomayor (1990). In fact, we use the following straight-forward adaptation of Lemma 5.25 of Roth and Sotomayor(1990) with µ ′′ ∈ S ( N, M, u ) and µ ′ ∈ E ( N, M, u ) . First, note that since ¯ µ ′′ and ¯ µ = T ∗ (¯ µ ′ ) arestable for the same market, the set of matched students and filled positions is the same in both match-ings by Theorem 5.12 of Roth and Sotomayor (1990). This implies that the set of filled positionsunder ¯ µ ′′ is either the same as under ¯ µ ′ or is increased by one. Thus, the claim in Lemma 5.25 holdsfor the college that fills a vacancy (if such a college exists). For all other colleges, the argumentis the same as in the proof of Lemma 5.25. Note that since ¯ µ ′′ is stable and ¯ µ ′ is simple, the re-quired version of the decomposition lemma is provided by Lemma A.2 of Blum, Roth, and Rothblum(1997). matching for market ( N, M, u ) . Then we have µ ( · ; u ) = T ∗ ( g i ( µ i ); u ) . The proof of Lemma 5.4 draws on Corollary A7 of Blum and Rothblum (2002).See also Theorem 4.3 of Blum, Roth, and Rothblum (1997), and Theorem 3.12 ofWu and Roth (2018).
Proof:
Let µ ′′ be any stable matching such that µ ′′ % g i ( µ i ) . Note that we have µ ′′ % µ ( · ; u ) since µ ( · ; u ) is college-worst matching by Corollary 5.30 of Roth and Sotomayor(1990). Note that µ i , the student-optimal stable matching in the market without i ,is equivalent to the student-optimal stable matching in the same market where i isadded but declares all colleges unnacceptable, g i ( µ i ) . Hence by Theorem 5.34 ofRoth and Sotomayor (1990), we have that µ ( · ; u ) % g i ( µ i ) . Therefore, µ ( · ; u ) is thecollege-worst stable matching among those that colleges weakly prefer to g i ( µ i ) . ByLemma 5.3, we conclude that µ ( · ; u ) = T ∗ ( g i ( µ i )) . (cid:4) Lemma 5.5.
Let µ be the student-optimal stable matching mechanism. Then for each j ∈ M and i ∈ N , we have that for any u ∈ U , | µ − ( j ; u ) \ µ − i ( j ; u − i ) | ≤ k + 1 , and | µ − i ( j ; u − i ) \ µ − ( j ; u ) | ≤ k + 1 , where µ i ( · ; u − i ) is the student-optimal matching for market ( N \{ i } , M, u − i ) . Proof : Let µ , µ , ..., µ r ′ with µ = g i ( µ i ( · ; u − i )) , where µ r = T ( µ r − ; u ) for each r = 1 , ..., r ′ . By Lemma 5.1, r ′ is finite and µ r ′ = T ∗ ( µ ; u ) ∈ S ( N, M, u ) . However,by Lemma 5.4, we also have that T ∗ ( µ ; u )( · ) = µ ( · ; u ) . Note that by the definitionof T , any j with | µ − i ( j ; u − i ) | < q j can be involved in at most one iteration of T .Hence the first bound holds trivially with the value of one for any college with avacancy under µ i ( · ; u − i ) . The second bound holds with the value of zero for anysuch college. Therefore, for the remainder of the argument, we consider only j ∈ M with | µ − i ( j ; u − i ) | = q j . We now argue that for every j ∈ M with | µ − i ( j ; u − i ) | = q j , | N j, | ≤ k + 1 and | N j, | ≤ k + 1 , (5.3)where N j, ≡ µ − ( j ; u ) \ µ − i ( j ; u − i ) and N j, = µ − i ( j ; u − i ) \ µ − ( j ; u ) . By Lemma 5.1, | µ − i ( j ; u − i ) | = q j implies that | µ − ( j ; u ) | = q j , since all students in µ − i ( j ; u − i ) areacceptable to j under preferences u (i.e., a college j with no vacancy unmatches a student at an iteration of the operator if and only if college j forms a student-maximal blocking pair with some other student). Thus, by Lemma 5.1, we have thefollowing equality for any j with | µ − i ( j ; u − i ) | = q j , | N j, | = | N j, | . (5.4)The following implication from Lemma 5.1 will also come in handy:(5.5) i ≻ j i ′ , for all ( i, i ′ ) ∈ N j, × N j, . We now show (5.3) using the above facts. The argument is by contradiction; i.e.,suppose that either | N j, | ≥ t or | N j, | ≥ t for some integer t > k + 1 . Let us denotethe students in N j, by i , ..., i t , where we write i t ≻ j ... ≻ j i ≻ j i . Similarly, wedenote the students in N j, by i t +1 , ..., i t , where i t ≻ j ... ≻ j i t +2 ≻ j i t +1 . The factthat the number of students in each set is the same is from (5.4). Next, by (5.5)and the above labeling convention it is obvious that i t ≻ j i t − ... ≻ j i ≻ j i . (5.6)Thus, by (5.6) and partial homogeneity of college preferences, it follows that theset ¯ N j ≡ N j, ∪ N j, satisfies the following property: i ≺ N i t +1 , i t +1 ≻ N i , i ≺ N i t +2 , i t +2 ≻ N i , ..., i t ≺ N i t . In this case, the set ¯ N j is called a fence on ( N, ≻ N ) . (See Definition 2.37 on page37 of Schr¨oder (2016).) For the required contradiction, we will use the order-theoretic fact that the size of the largest fence of any poset must be less than orequal to twice the size of largest antichain of the poset minus one. (Proposition2.50 on page 41 of Schr¨oder (2016).) Let ¯ S ⊆ N denote the largest antichain of ( N, ≻ N ) . It is straightforward to argue that | ¯ S | ≤ k + 1 under partial homogeneity. Therefore, since ¯ N j is a fence of ( N, ≻ N ) , we obtain the desired contradiction fromthe following: | ¯ N j | = 2 t > S − k + 1) − for all integers t > k + 1 . (5.7) Recall that S ⊆ N is an antichain of ( N, ≻ N ) if and only if there is no i, i ′ ∈ S with i = i ′ , suchthat i ≻ N i ′ or i ′ ≻ N i . To see that ¯ S satisfies | ¯ S | ≤ k +1 , suppose by contradiction that there exists some antichain S ′ ⊆ N of ( N, ≻ N ) with | S ′ | > k + 1 . Then, it is immediate that there is some pair of students i, i ′ ∈ S ′ whose ranks differ by more than k in some linear extension of ≻ N , which in turn implies that either i ′ ≻ N i or i ≻ N i ′ from our definitions. Hence partial homogeneity implies that | ¯ S | ≤ k + 1 . Thus, we conclude that | N j, | ≤ k + 1 and | N j, | ≤ k + 1 as required. (cid:4) Corollary 5.1.
Let µ be the student-optimal stable matching mechanism. For each i ∈ N , let µ i ( · ; u − i ) be the student-optimal matching in market ( N \{ i } , M, u − i ) . Thenfor each i ∈ N and u ∈ U , we have |{ i ′ ∈ N \{ i } : µ ( i ′ ; u ) = µ i ( i ′ ; u − i ) }| ≤ m ( k + 1) + 1 . Proof:
Observe that at most one student who is matched to a college under µ i ( · ; u − i ) can be left unmatched as a result of the iterations of T process. That is, the set A i = (cid:8) i ′ ∈ N : µ ( i ′ ; u ) = 0 and i ′ ∈ µ − i ( j ; u − i ) for some j ∈ M (cid:9) , is either a singleton, say i ∗ , for some i ∗ ∈ N \{ i } , or an empty set. A student i ′ ∈ N \{ i, i ∗ } satisfies µ ( i ′ ; u ) = µ i ( i ′ ; u − i ) , if and only if, for exactly one college, j he is a member of µ − ( j ; u ) but not µ − i ( j ; u − i ) . Thus for each student i ′ ∈ N \{ i } we have that { µ ( i ′ ; u ) = µ i ( i ′ ; u − i ) } = X j ∈ M { i ′ ∈ µ − ( j ; u ) \ µ − i ( j ; u − i ) } + 1 { i ′ = i ∗ } . Therefore, summing over i ′ ∈ N \{ i } , we have |{ i ′ ∈ N \{ i } : µ ( i ′ ; u ) = µ i ( i ′ ; u − i ) }| = X j ∈ M | µ − ( j ; u ) \ µ − i ( j ; u − i ) | + X i ′ ∈ N \{ i } { i ′ = i ∗ } ≤ m ( k + 1) + 1 , where we used Lemma 5.5 for the last bound. (cid:4) Proof of Lemma 3.2 : Choose ( u ′ , u ) ∈ D so that u ′ and u differ by one student’spreference, say, the preference of student i . Then note that |{ i ′ ∈ N : µ ( i ′ ; u ) = µ ( i ′ ; u ′ ) }| = X i ′ ∈ N { µ ( i ′ ; u ) = µ ( i ′ ; u ′ ) } (5.8) ≤ X i ′ ∈ N { µ ( i ′ ; u ) = µ ( i ′ ; u ′ ) , i ′ = i } + 1 ≤ X i ′ ∈ N { µ ( i ′ ; u ) = µ i ( i ′ ; u − i ) , i ′ = i } + X i ′ ∈ N { µ i ( i ′ ; u − i ) = µ ( i ′ ; u ′ ) , i ′ = i } + 1 . The additive in the first bound is due to the possibility that µ ( i ; u ) = 0 . ByCorollary 5.1, we can bound the first term of (5.8) by X i ′ ∈ N { µ ( i ′ ; u ) = µ i ( i ′ ; u − i ) , i ′ = i } ≤ |{ i ′ ∈ N \{ i } : µ ( i ′ ; u ) = µ i ( i ′ ; u − i ) }|≤ m ( k + 1) + 1 . Since the above bound is uniform over u ∈ U and u − i = u ′− i , an identical boundalso holds for the second term on the right hand side of (5.8). Thus we concludethat |{ i ′ ∈ N : µ ( i ′ ; u ) = µ ( i ′ ; u ′ ) }| ≤ m ( k + 1) + 2) , as required. (cid:4) Proof of Corollary 3.1:
We prove Assumption 3.1. Let a, b ∈ { , , ..., m } n suchthat P mj =0 { a i = j } = 1 and P mj =0 { b i = j } = 1 for all i = 1 , ..., n . Observe that nm | ϕ ( a ; u ) − ϕ ( b ; u ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ∈ M X i ∈ N h ( X i , Z j ) (1 { a i = j } − { b i = j } ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ∈ M X i ∈ N h ( X i , Z j )1 { a i = j } (1 { a i = j } − { b i = j } ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ∈ M X i ∈ N h ( X i , Z j )1 { b i = j } (1 { a i = j } − { b i = j } ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Let us focus on the leading term on the right hand side. We bound it by X i ∈ N X j ∈ M { a i = j } | { a i = j } − { b i = j }|≤ X i ∈ N max j ∈ M | { a i = j } − { b i = j }|≤ X i ∈ N { a i = b i } ≤ k a − b k H , where the second inequality follows because | h ( X i , Z j ) | ≤ and the third inequalityfollows because P j ∈ M { a i = j } ≤ . Hence the first inequality in (3.4) holds with K = 1 / ( nm ) . We prove the second inequality in (3.4). Let ˜ X = ( ˜ X i ) i ∈ N and ˜ Z = ( ˜ Z j ) j ∈ M bethe altered versions of X = ( X i ) i ∈ N and Z = ( Z j ) j ∈ M as we move from v i to v ′ i .Then, | ϕ ( a ; v − i , v i , w ) − ϕ ( a ; v − i , v ′ i , w ) | is bounded by nm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ∈ M { a i = j } (cid:16) h ( X i , Z j ) − h ( ˜ X i , ˜ Z j ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ nm X j ∈ M { a i = j } ≤ nm . (5.9)Hence the functional ϕ satisfies Assumption 3.1 with constants K = 1 / ( nm ) , and c i = 2 / ( nm ) . Therefore, the desired bound comes from Theorem 3.1. (cid:4) References A BDULKADIRO ˘ GLU , A., Y.-K. C HE , AND
Y. Y
ASUDA (2011): “Resolving ConflictingPreferences in School Choice: The “Boston Mechanism” Reconsidered,”
AmericanEconomic Review , 101, 399–410.B IR ´ O , P., K. C ECHL ´ AROV ´ A , AND
T. F
LEINER (2008): “The dynamics of stable match-ings and half-matchings for the stable marriage and roommates problems,”
Inter-national Journal of Game Theory , 36(3-4), 333–352.B
LUM , Y., A. E. R
OTH , AND
U. G. R
OTHBLUM (1997): “Vacancy chains and equilibra-tion in senior-level labor markets,”
Journal of Economic Theory , 76(2), 362–411.B
LUM , Y.,
AND
U. G. R
OTHBLUM (2002): ““Timing is everything” and marital bliss,”
Journal of Economic Theory , 103(2), 429–443.B
OYD , D., H. L
ANKFORD , S. L
OEB , AND
J. W
YCKOFF (2013): “Analyzing the De-terminants of the Matching of Public School Teachers to Jobs: Disentangling thePreferences of Teachers and Employers,”
Journal of Labor Economics , 31, 83–117.C
HIAPPORI , P.-A.,
AND
B. S
ALANI ´ E (2016): “The Econometrics of Matching Models,” Journal of Economic Literature , 54, 832–862.C
HOO , E.,
AND
A. S
IOW (2006): “Who Marries Whom and Why,”
Journal of PoliticalEconomy , 114, 175–201.D
AGSVIK , J. K. (2000): “Aggregation in Matching Markets,”
International EconomicReview , 41, 27–58.D
IAMOND , W.,
AND
N. A
GARWAL (2017): “Latent indices in assortative matchingmodels,”
Quantitative Economics , 8(3), 685–728.F OX , J. T. (2018): “Estimating Matching Games with Transfers,” Quantitative Eco-nomics , 9, 1–38.G
ALE , D.,
AND
L. S. S
HAPLEY (1962): “College admissions and the stability of mar-riage,”
The American Mathematical Monthly , 69(1), 9–15.K
OJIMA , F.,
AND
P. A. P
ATHAK (2009): “Incentives and Stability in Large Two-SidedMatching Markets,”
American Economic Review , 99, 608–627.M C D IARMID , C. (1989): “On the method of bounded differences,”
Surveys in com-binatorics , 141(1), 148–188.M
ENZEL , K. (2015): “Large matching markets as two-sided demand systems,”
Econometrica , 83(3), 897–941. R OTH , A. E.,
AND
M. S
OTOMAYOR (1990):
Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis . Econometric Society Monograph Series, Cam-bridge University Press, Cambridge.R
OTH , A. E.,
AND
J. H. V
ANDE V ATE (1990): “Random paths to stability in two-sidedmatching,”
Econometrica: Journal of the Econometric Society , pp. 1475–1480.S
CHR ¨ ODER , B. (2016):
Ordered Sets: An Introduction with Connections from Combi-natorics to Topology . Springer International Publishing.S
OTOMAYOR , M. (1996): “A non-constructive elementary proof of the existence ofstable marriages,”
Games and Economic Behavior , 13(1), 135–137.W U , Q., AND
A. E. R
OTH (2018): “The lattice of envy-free matchings,”