Interdistrict School Choice: A Theory of Student Assignment
IINTERDISTRICT SCHOOL CHOICE: A THEORY OF STUDENT ASSIGNMENT † ISA E. HAFALIR, FUHITO KOJIMA, AND M. BUMIN YENMEZ ∗ Abstract. Interdistrict school choice programs—where a student can be assigned to aschool outside of her district—are widespread in the US, yet the market-design literaturehas not considered such programs. We introduce a model of interdistrict school choice andpresent two mechanisms that produce stable or efficient assignments. We consider three cate-gories of policy goals on assignments and identify when the mechanisms can achieve them.By introducing a novel framework of interdistrict school choice, we provide a new avenueof research in market design.
1. Introduction
School choice is a program that uses preferences of children and their parents over pub-lic schools to assign children to schools. It has expanded rapidly in the United States andmany other countries in the last few decades. Growing popularity and interest in schoolchoice stimulated research in market design, which has not only studied this problem inthe abstract, but also contributed to designing specific assignment mechanisms. Existing market-design research about school choice is, however, limited to intradistrict choice, where each student is assigned to a school only in her own district. In other words,the literature has not studied interdistrict choice, where a student can be assigned to aschool outside of her district. This is a severe limitation for at least two reasons. First,interdistrict school choice is widespread: some form of it is practiced in 43 U.S. states. Date : January 8, 2019, First draft: July 15, 2017.
Keywords : Interdistrict school choice, student assignment, stability, efficiency.We thank Mehmet Ekmekci, Haluk Ergin, Yuichiro Kamada, Kazuo Murota, Tayfun S ¨onmez, Utku ¨Unver,Rakesh Vohra, and the audiences at various seminars and conferences. Lukas Bolte, Ye Rin Kang, and espe-cially Kevin Li provided superb research assistance. Kojima acknowledges financial support from the Na-tional Research Foundation through its Global Research Network Grant (NRF-2016S1A2A2912564). Hafaliris affiliated with the UTS Business School, University of Technology Sydney, Sydney, Australia; Kojima iswith the Department of Economics, Stanford University, 579 Serra Mall, Stanford, CA, 94305; Yenmez is withthe Department of Economics, Boston College, 140 Commonwealth Ave, Chestnut Hill, MA, 02467. Emails: [email protected] , [email protected] , [email protected] . See Abdulkadiro ˘glu et al. (2005a,b, 2009) for details of the implementation of these new school choiceprocedures in New York and Boston. See http://ecs.force.com/mbdata/mbquest4e?rep=OE1705 , accessed on July 14, 2017. a r X i v : . [ ec on . T H ] J a n HAFALIR, KOJIMA, AND YENMEZ
Second, as we illustrate in detail below, many policy goals in school choice impose con-straints across districts in reality, but the existing literature assumes away such constraints.This omission limits our ability to analyze these policies of interest.In this paper, we propose a model of interdistrict school choice. Our paper builds uponmatching models in the tradition of Gale and Shapley (1962). We study mechanisms andinterdistrict admissions rules to assign students to schools under which a variety of policygoals can be established, an approach similar to the intradistrict school choice literature(Abdulkadiro ˘glu and S ¨onmez, 2003). In our setting, however, policy goals are defined onthe district level—or sometimes even over multiple districts—rather than the individualschool level, placing our model outside of the standard setting. To facilitate the analysisin this setting, we model the problem as matching with contracts (Hatfield and Milgrom,2005) between students and districts in which a contract specifies the particular schoolwithin the district that the student attends. Following the school choice literature, we begin our analysis by considering stability (we also consider efficiency, as explained later). To define stability in our framework, weassume that each district is endowed with an admissions rule represented by a choicefunction over sets of contracts. We focus our attention on the student-proposing deferred-acceptance mechanism (SPDA) of Gale and Shapley (1962). In our setting, this mechanismis not only stable but also strategy-proof—i.e., it renders truthtelling a weakly dominantstrategy for each student.In this context, we formalize a number of important policy goals. The first is individualrationality in the sense that every student is matched with a weakly more preferred schoolthan the school she is initially matched with (in the absence of interdistrict school choice).This is an important requirement, because if an interdistrict school choice program harmsstudents, then public opposition is expected and the program may not be sustainable.The second policy is what we call the balanced-exchange policy : The number of students thateach district receives from the other districts must be the same as the number of studentsthat it sends to the others. Balanced exchange is also highly desired by school districts inpractice. This is because each district’s funding depends on the number of students that itserves and, therefore, if the balanced-exchange policy is not satisfied, then some districtsmay lose funding, possibly making the interdistrict school choice program impossible. Foreach of these policy goals, we identify the necessary and sufficient condition for achievingthat goal under SPDA as a restriction on district admissions rules. We use the terms assignment and matching interchangeably for the rest of the paper. One might suspect that an interdistrict school choice problem can readily be reduced to an intradistrictproblem by relabeling a district as a school. This is not the case because, among other things, which schoolwithin a district a student is matched with matters for that student’s welfare.
NTERDISTRICT SCHOOL CHOICE 3
Last, but not least, we also consider a requirement that there be enough student diversityin each district. In fact, diversity appears to be the main motivation for many interdistrictschool choice programs. To put this into context, we note that the lack of diversity isprevalent under intradistrict school choice programs even though they often seek diversityby controlled-choice constraints. This is perhaps unsurprising given that only residentsof the given district can participate in intradistrict school choice and there is often severeresidential segregation. In fact, a number of studies such as Rivkin (1994) and Clotfelter(1999, 2011) attribute the majority—as high as 80 percent for some data and measure—of racial and ethnic segregation in public schools to disparities between school districtsrather than within school districts. Given this concern, many interdistrict choice programsexplicitly list achieving diversity as their main goal.A case in point is the
Achievement and Integration (AI) Program of the Minnesota Depart-ment of Education (MDE). Introduced in 2013, the AI program incentivizes school dis-tricts for integration. A district is required to participate in this program if the proportionof a racial group in the district is considerably higher than that in a neighboring district.In particular, every year the MDE commissioner analyzes fall enrollment data from everydistrict and, when a district and one of its adjoining districts have a difference of 20 percentor higher in the proportion of any group of enrolled protected students (American Indian,Asian or Pacific Islander, Hispanic, Black, not of Hispanic origin, and White, not of His-panic origin), the district with the higher percentage is required to be in the AI program. In the 2015-16 school year, more than 120 school districts participated in this program(Figure 1, taken from MDE’s website, shows school districts in the Minneapolis-Saint Paulmetro area that take part in this program).Motivated by Minnesota’s AI program, we consider a policy goal requiring that thedifference in the proportions of each student type across districts be within a given bound.Then, we provide a necessary and sufficient condition for SPDA to satisfy the diversitypolicy. The condition provided is one on district admissions rules that have a structure oftype-specific ceilings, an analogue of the class of choice rules analyzed by Abdulkadiro ˘gluand S ¨onmez (2003) and Ehlers et al. (2014) in the context of a more standard intradistrictschool-choice problem. We refer to Wells et al. (2009) for a review and discussion of interdistrict integration programs. Examples of controlled school choice include Boston before 1999, Cambridge, Columbus, and Min-neapolis. See Abdulkadiro ˘glu and S ¨onmez (2003) for details of these programs as well as analysis of con-trolled school choice. In Minnesota’s AI program, if the difference in the proportion of protected students at a school is 20percent or higher than a school in the same district, the school with the higher percentage is considereda racially identifiable school (RIS) and districts with RIS schools also need to participate in the AI program.In this paper, we focus on diversity issues across districts rather than within districts. Diversity problemswithin districts are studied in the controlled school choice literature that we discuss below.
HAFALIR, KOJIMA, AND YENMEZ
RI/RIS ARIS/A RI RIRI/RISA RI/RISA/RIS VRI RI/RISA ARI/RIS RI/ARI/RIS ARIRI A RIRI/RIS A/RISRI/RISA A ARI V AA VElk River728Buffalo-Hanover-Montrose877 Rockford 883SibleyEast2310
Minneapolis-St.Paul Int'lAirport0Minneapolis1 SouthSt.Paul6Anoka-Hennepin11 Centennial12ColumbiaHeights13
Fridley14
St.Francis15 Spring LakePark16Central108 Waconia110Watertown-Mayer111 EasternCarver County112 Burnsville-Eagan-Savage191 Farmington192Lakeville194 Randolph195Rosemount-AppleValley-Eagan196West St. Paul-MendotaHts.-Eagan197 InverGrove Heights199 Hastings200Hopkins270 Bloomington271EdenPrairie272 Edina273Minnetonka276Westonka277 Orono278 Osseo279 Richfield280Robbinsdale281
St. Anthony-NewBrighton282
St. LouisPark283Wayzata284 BrooklynCenter286 MoundsView621 North St.Paul-Maplewood622Roseville623 WhiteBear Lake624St. Paul625BellePlaine716 Jordan717 PriorLake-Savage719Shakopee720New PragueArea721 ForestLake831 Mahtomedi832SouthWashington County833 StillwaterArea834
For a listing of School Districts go to Page 2
Achievement and Integration Districts by CollaborativeMetro Area2015-2016 School Year
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Brooklyn_CenterSibleyNWSISDBloomington (RIS)Burnsville-LakevilleEast Metro Integration District (EMID)Educational Equity AllianceMinnesota River ValleyNorthwest Suburban Integration School District (NWSISD)Rosemount-Apple Valley-Eagan (RIS)SEBPNICWest Metro Education Program (WMEP)
AA ARI
Figure 1. Minnesota-Saint Paul metro area school districts participating inthe AI program. The districts with the same color are adjoining districts thatexchange students with one another.Next, we turn our attention to efficiency. Given that the distributional policy goals workas constraints on matchings, we use the concept of constrained efficiency . We say that amatching is constrained efficient if it satisfies the policy goal and is not Pareto dominatedby any matching that satisfies the same policy goal. In addition, we require individualrationality and strategy-proofness. We first demonstrate an impossibility result; when thediversity policy is given as type-specific ceilings at the district level, there is no mechanismthat satisfies the policy goal, constrained efficiency, individual rationality, and strategy-proofness. By contrast, a version of the top trading cycles mechanism (TTC) of Shapleyand Scarf (1974) satisfies these properties when the policy goal satisfies M-convexity, a Without individual rationality, all the other desired properties can be attained by a serial dictatorship.
NTERDISTRICT SCHOOL CHOICE 5 concept in discrete mathematics (Murota, 2003). We proceed to show that the balanced-exchange policy and an alternative form of diversity policy—type-specific ceilings at theindividual school level instead of at the district level—are M-convex, so TTC satisfies thedesired properties for these policies. The same conclusion holds even when both of thesepolicy goals are imposed simultaneously.We also consider the case when there is a policy function that measures how well amatching satisfies the policy goal. For example, diversity of a matching can be measuredas its distance to an ideal distribution of students. We show that TTC satisfies the samedesirable properties when the policy function satisfies pseudo M-concavity , a notion of con-cavity for discrete functions that we introduce. Furthermore, we show that there is anequivalence between two approaches based on the M-convexity of the policy set and thepseudo M-concavity of the policy function. Therefore, both results can naturally be ap-plied in different settings depending on how the policy goals are stated.
Related Literature.
Our paper is closely related to the controlled school choice literaturethat studies student diversity in schools in a given district. Abdulkadiro ˘glu and S ¨onmez(2003) introduce a policy that imposes type-specific ceilings on each school. This policyhas been analyzed by Abdulkadiro ˘glu (2005), Ergin and S ¨onmez (2006), and Kojima (2012),among others. More accommodating policies using reserves rather than type-specific ceil-ings have been proposed and analyzed by Hafalir et al. (2013) and Ehlers et al. (2014). Thelatter paper finds difficulties associated with hard floor constraints, an issue further an-alyzed by Fragiadakis et al. (2015) and Fragiadakis and Troyan (2017). In addition tosharing the motivation of achieving diversity, our paper is related to this literature in thatwe extend the type-specific reserve and ceiling constraints to district admissions rules. Incontrast to this literature, however, our policy goals are imposed on districts rather thanindividual schools, which makes our model and analysis different from the existing ones.The feature of our paper that imposes constraints on sets of schools (i.e., districts),rather than individual schools, is shared by several recent studies in matching with con-straints. Kamada and Kojima (2015) study a model where the number of doctors who canbe matched with hospitals in each region has an upper bound constraint. Variations andgeneralizations of this problem are studied by Goto et al. (2014, 2017), Biro et al. (2010),and Kamada and Kojima (2017, 2018), among others. While sharing the broad interest inconstraints, these papers are different from ours in at least two major respects. First, theydo not assume a set of hospitals is endowed with a well-defined choice function, whileeach school district has a choice function in our model. Second, the policy issues studied In addition to the works discussed above, recent studies on controlled school choice and other two-sided matching problems with diversity concerns include Westkamp (2013), Echenique and Yenmez (2015),S ¨onmez (2013), Kominers and S ¨onmez (2016), Dur et al. (2014), Dur et al. (2016), and Nguyen and Vohra(2017).
HAFALIR, KOJIMA, AND YENMEZ in these papers and those studied in ours are different given differences in the intendedapplications. These differences render our analysis distinct from those of the other papers,with none of their results implying ours and vice versa.One of the notable features of our model is that district admissions rules do not nec-essarily satisfy the standard assumptions in the literature, such as substitutability , whichguarantee the existence of a stable matching. In fact, even a seemingly reasonable districtadmissions rule may violate substitutability because a district can choose at most one con-tract associated with the same student—namely just one contract representing one schoolthat the student can attend. Rather, we make weaker assumptions following the approachof Hatfield and Kominers (2014). This issue is playing an increasingly prominent role inmatching with contracts literature; for example, in matching with constraints (Kamadaand Kojima, 2015), college admissions (Ayg ¨un and Turhan, 2016; Yenmez, 2018), and post-graduate admissions (Hassidim et al., 2017), to name just a few.Our analysis of Pareto efficient mechanisms is related to a small but rapidly growingliterature that uses discrete optimization techniques for matching problems. Closest toours is Suzuki et al. (2017), who show that a version of TTC satisfies desirable propertiesif the constraint satisfies M-convexity. Our analysis on efficiency builds upon and gen-eralizes theirs. While the use of discrete convexity concepts for studying efficient objectallocation is still rare, it has been utilized in an increasing number of matching problemssuch as two-sided matching with possibly bounded transfer (Fujishige and Tamura, 2006,2007), matching with substitutable choice functions (Murota and Yokoi, 2015), matchingwith constraints (Kojima et al., 2018a), and trading networks (Candogan et al., 2016).There is also a recent literature on segmented matching markets in a given district. Man-junath and Turhan (2016) study a setting where different clearinghouses can be coordi-nated, but not integrated in a centralized clearinghouse, and show how a stable matchingcan be achieved. In a similar setting, Dur and Kesten (2018) study sequential mechanismsand show that these mechanisms lack desired properties. In another work, Ekmekci andYenmez (2014) study the incentives of a school to join a centralized clearinghouse. In con-trast to these papers, we study which interdistrict school choice policies can be achievedwhen districts are integrated.At a high level, the present paper is part of research in resource allocation under con-straints. Real-life auction problems often feature constraints (Milgrom, 2009), and a greatdeal of attention was paid to cope with complex constraints in a recent FCC auction forspectrum allocation (Milgrom and Segal, 2014). Auction and exchange markets underconstraints are analyzed by Bing et al. (2004), Gul et al. (2018), and Kojima et al. (2018b).Handling constraints is also a subject of a series of papers on probabilistic assignment See Kurata et al. (2016) for an earlier work on TTC in a more specialized setting involving floor con-straints at individual schools.
NTERDISTRICT SCHOOL CHOICE 7 mechanisms (Budish et al., 2013; Che et al., 2013; Pycia and ¨Unver, 2015; Akbarpour andNikzad, 2017; Nguyen et al., 2016). Closer to ours are Dur and ¨Unver (2018) and Dur etal. (2015). They consider the balance of incoming and outgoing members—a requirementthat we also analyze—while modeling exchanges of members of different institutions un-der constraints. Although the differences in the model primitives and exact constraintsmake it impossible to directly compare their studies with ours, these papers and oursclearly share broad interests in designing mechanisms under constraints.The rest of the paper is organized as follows. Section 2 introduces the model. In Sections3 and 4, we study when the policy goals can be satisfied together with stability and con-strained efficiency, respectively. Section 5 concludes. Additional results, examples, andomitted proofs are presented in the Appendix.
2. Model
In this section, we introduce our concepts and notation.
There exist finite sets of students S , districts D , and schools C . Each student s and school c has a home district denoted by d ( s ) and d ( c ) , respectively.Each student s has a type τ ( s ) that can represent different aspects of the student such asthe gender, race, socioeconomic status, etc. The set of all types is finite and denoted by T .Each school c has a capacity q c , which is the maximum number of students that the schoolcan enroll. There exist at least two school districts with one or more schools. For eachdistrict d , k d is the number of students whose home district is d . In each district, schoolshave sufficiently large capacities to accommodate all students from the district, i.e., forevery district d , k d ≤ (cid:80) c : d ( c )= d q c . For each type t , k t is the number of type- t students.We model interdistrict school choice as a matching problem between students and dis-tricts. However, merely identifying the district with which a student is matched leavesthe specific school she is enrolled in unspecified. To specify which school within a dis-trict the student is matched with, we use the notion of contracts: A contract x = ( s, d, c ) specifies a student s , a district d , and a school c within this district, i.e., d ( c ) = d . For anycontract x , let s ( x ) , d ( x ) , and c ( x ) denote the student, district, and school associated withthis contract, respectively. Let X ≡ { ( s, d, c ) | d ( c ) = d } denote the set of all contracts. Forany set of contracts X , let X s denote the set of all contracts in X associated with student s , i.e., X s = { x ∈ X | s ( x ) = s } . Similarly, let X d and X c denote the sets of all contracts in X associated with district d and school c , respectively. For ease of exposition, a contract will sometimes be denoted by a pair ( s, c ) with the understandingthat the district associated with the contract is the home district of school c . HAFALIR, KOJIMA, AND YENMEZ
Each district d has an admissions rule that is represented by a choice function Ch d .Given a set of contracts X , the district chooses a subset of contracts associated with itself,i.e., Ch d ( X ) = Ch d ( X d ) ⊆ X d .Each student s has a strict preference order P s over all schools and the outside optionof being unmatched, which is denoted by ∅ . Likewise, P s is also used to rank contractsassociated with s . Furthermore, we assume that the outside option is the least preferredoutcome, so for every contract x associated with s , x P s ∅ . The corresponding weak orderis denoted by R s . More precisely, for any two contracts x, y associated with s , x R s y if x P s y or x = y .A matching is a set of contracts. A matching X is feasible for students if there existsat most one contract associated with every student in X . A matching X is feasible if itis feasible for students and the number of contracts associated with every school in X isat most its capacity, i.e., for any c ∈ C , | X c | ≤ q c . We assume that there exists a feasible initial matching ˜ X such that every student has exactly one contract. For any student s ,if ˜ X s = { ( s, d, c ) } for some district d and school c , then c is called the initial school of s .A problem is a tuple ( S , D , C , T , { d ( s ) , τ ( s ) , P s } s ∈S , { Ch d } d ∈D , { d ( c ) , q c } c ∈C , ˜ X ) . In whatfollows, we assume that all the components of a problem are publicly known except forstudent preferences. Therefore, we sometimes refer to a problem by the student preferenceprofile which we denote as P S . The preference profile of a subset of students S ⊆ S isdenoted by P S . A district admissions rule Ch d is feasible if it al-ways chooses a feasible matching. It is acceptant if, for any contract x associated withdistrict d and matching X that is feasible for students; and if x is rejected from X , then at Ch d ( X ) , either • the number of students assigned to school c ( x ) is equal to q c ( x ) , or • the number of students assigned to district d is at least k d .In words, when a district admissions rule is acceptant, a contract x = ( s, d, c ) can berejected by district d from a set which is feasible for students only if either the capacity ofschool c is filled or district d has accepted at least k d students. Equivalently, if neither ofthese two conditions is satisfied, then the district has to accept the student. Throughoutthe paper, we assume that admissions rules are feasible and acceptant. A district admissions rule satisfies substitutability if, whenever a contract is chosenfrom a set, it is also chosen from any subset containing that contract (Kelso and Crawford, In Appendix A.1, we also consider the case when the initial matching for each district is constructedusing student preferences and district admissions rules. In Section 3.3, we assume a weaker notion of acceptance when the admissions rule limits the numberof students of each type that the district can accept.
NTERDISTRICT SCHOOL CHOICE 9 Ch d satisfies substitutability if,for every x ∈ X ⊆ Y ⊆ X with x ∈ Ch d ( Y ) , it must be that x ∈ Ch d ( X ) . A district admis-sions rule satisfies the law of aggregate demand (LAD) if the number of contracts chosenfrom a set is weakly greater than that of any of its subsets (Hatfield and Milgrom, 2005).Mathematically, a district admissions rule Ch d satisfies LAD if, for every X ⊆ Y ⊆ X , | Ch d ( X ) | ≤ | Ch d ( Y ) | . A completion of a district admissions rule Ch d is another admis-sions rule Ch (cid:48) d such that for every matching X either Ch (cid:48) d ( X ) is equal to Ch d ( X ) or it isnot feasible for students (Hatfield and Kominers, 2014). Throughout the paper, we assumethat district admissions rules have completions that satisfy substitutability and LAD. InAppendix B, we provide classes of district admissions rules that satisfy our assumptions.
A feasible matching X satis-fies individual rationality if every student weakly prefers her outcome in X to her initialschool, i.e., for every student s , X s R s ˜ X s .A distribution ξ ∈ Z |C|×|T | + is a vector such that the entry for school c and type t isdenoted by ξ tc . The entry ξ tc is interpreted as the number of type- t students in school c at ξ . Furthermore, let ξ td ≡ (cid:80) c : d ( c )= d ξ tc , which is interpreted as the number of type- t studentsin district d at ξ . Likewise, for any feasible matching X , the distribution associated with X is ξ ( X ) whose c, t entry ξ tc ( X ) is the number of type- t students assigned to school c at X . Similarly, ξ td ( X ) denotes the number of type- t students assigned to district d at X .We represent a distributional policy goal Ξ as a set of distributions. The policy thateach student is matched without assigning any school more students than its capacityis denoted by Ξ , i.e., Ξ ≡ { ξ | (cid:80) c,t ξ tc = (cid:80) d k d and q c ≥ (cid:80) t ξ tc for all c } . A matching X satisfies the policy goal Ξ if the distribution associated with X is in Ξ .A feasible matching X Pareto dominates another feasible matching Y if every studentweakly prefers her outcome in X to her outcome in Y and at least one student strictlyprefers the former to the latter. Given a distributional policy goal, a feasible matching X that satisfies the policy goal satisfies constrained efficiency if there exists no feasiblematching that satisfies the policy goal and Pareto dominates X .A matching X is stable if it is feasible and • districts would choose all contracts assigned to them, i.e., Ch d ( X ) = X d for everydistrict d , and • there exist no student s and no district d who would like to match with each other,i.e., there exists no contract x = ( s, d, c ) / ∈ X such that x P s X s and x ∈ Ch d ( X ∪{ x } ) . Alkan (2002) and Alkan and Gale (2003) introduce related monotonicity conditions. Hatfield and Kojima (2010) introduce other notions of weak substitutability.
Stability was introduced by Gale and Shapley (1962) for the college admissions problem.In the context of assigning students to public schools, it is viewed as a fairness notion(Abdulkadiro ˘glu and S ¨onmez, 2003).A mechanism φ takes a profile of student preferences as input and produces a feasi-ble matching. The outcome for student s at the reported preference profile P S undermechanism φ is denoted as φ s ( P S ) . A mechanism φ satisfies strategy-proofness if no stu-dent can misreport her preferences and get a strictly more preferred contract. More for-mally, for every student s and preference profile P S , there exists no preference P (cid:48) s such that φ s ( P (cid:48) s , P S\{ s } ) P s φ s ( P S ) . For any property on matchings, a mechanism satisfies the prop-erty if, for every preference profile, the matching produced by the mechanism satisfies theproperty.
3. Achieving Policy Goals with Stable Outcomes
To achieve stable matchings with desirable properties, we use a generalization of thedeferred-acceptance algorithm of Gale and Shapley (1962).
Student-Proposing Deferred Acceptance Algorithm . Step 1:
Each student s proposes a contract ( s, d, c ) to district d where c is her mostpreferred school. Let X d denote the set of contracts proposed to district d . District d tentatively accepts contracts in Ch d ( X d ) and permanently rejects the rest. If thereare no rejections, then stop and return ∪ d ∈D Ch d ( X d ) as the outcome. Step n ( n > ): Each student s whose contract was rejected in Step n − proposesa contract ( s, d, c ) to district d where c is her next preferred school. If there is nosuch school, then the student does not make any proposals. Let X nd denote theunion of the set of contracts that were tentatively accepted by district d in Step n − and the set of contracts that were proposed to district d in Step n . District d tentatively accepts contracts in Ch d ( X nd ) and permanently rejects the rest. If thereare no rejections, then stop and return ∪ d ∈D Ch d ( X nd ) .The student-proposing deferred acceptance mechanism (SPDA) takes a profile of stu-dent preferences as input and produces the outcome of this algorithm at the reported stu-dent preference profile. When district admissions rules have completions that satisfy sub-stitutability and LAD, SPDA is stable and strategy-proof (Hatfield and Kominers, 2014).Therefore, when we analyze SPDA, we assume that students report their preferences truth-fully.We illustrate SPDA using the following example. We come back to this example later tostudy the effects of interdistrict school choice. NTERDISTRICT SCHOOL CHOICE 11
Example . Consider a problem with two school districts, d and d . District d has school c with capacity one and school c with capacity two. District d has school c with capacitytwo. There are four students: students s and s are from district d , whereas students s and s are from district d . The initial matching is { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } .Given any set of contacts, district d chooses students who have contracts with school c first and then chooses from the remaining students who have contracts with school c .For school c , the district prioritizes students in the order s (cid:31) s (cid:31) s (cid:31) s and choosesone applicant if there is any. For school c , the district prioritizes students according tothe order s (cid:31) s (cid:31) s (cid:31) s and chooses as many applicants as possible without goingover the school’s capacity while ignoring the contracts of the students who have alreadybeen accepted at school c . Likewise, district d prioritizes students according to the order s (cid:31) s (cid:31) s (cid:31) s and chooses as many applicants as possible without going over thecapacity of school c . These admissions rules are feasible and acceptant, and they havecompletions that satisfy substitutability and LAD. In addition, student preferences aregiven by the following table, P s P s P s P s c c c c c c c c c c c c which means that, for instance, student s prefers c to c to c .In this problem, SPDA runs as follows. At the first step, student s proposes to dis-trict d with contract ( s , c ) , student s proposes to district d with contract ( s , c ) , stu-dent s proposes to district d with contract ( s , c ) , and student s proposes to district d with contract ( s , c ) . District d first considers contracts associated with school c , ( s , c ) and ( s , c ) , and tentatively accepts ( s , c ) while rejecting ( s , c ) because student s hasa higher priority than student s at school c . Then district d considers contracts of theremaining students associated with school c . In this case, there is only one such contract, ( s , c ) , which is tentatively accepted. District d considers contract ( s , c ) and tentativelyaccepts it. The tentative matching is { ( s , c ) , ( s , c ) , ( s , c ) } . Since there is a rejection, thealgorithm proceeds to the next step.At the second step, student s proposes to district d with contract ( s , c ) . District d firstconsiders contract ( s , c ) and tentatively accepts it. Then district d considers contracts ( s , c ) and ( s , c ) and tentatively accepts them both. District d does not have any new In Appendix B.1, we provide a general class of admissions rules, including this one as a special case.We show that these admissions rules are feasible and acceptant, and they have completions that satisfysubstitutability and LAD. contracts, so tentatively accepts ( s , c ) . Since there is no rejection, the algorithm stops.The outcome of SPDA is { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } . (cid:3) In the rest of this section, we formalize three policy goals and characterize conditionsunder which SPDA satisfies them.
In our context, individual rationality requires that every stu-dent is matched with a weakly more preferred school than her initial school. As a result,SPDA does not necessarily satisfy individual rationality even though each student is eitherunmatched or matched with a school that is more preferred than being unmatched.If individual rationality is violated so that some students prefer their initial schools tothe outcome of SPDA, then there may be public opposition that harm interdistrict schoolchoice efforts. For this reason, individual rationality is a desirable property for policymak-ers. The following condition proves to play a crucial role for achieving this property.
Definition 1.
A district admissions rule Ch d respects the initial matching if, for any student s whose initial school c is in district d and matching X that is feasible for students, ( s, d, c ) ∈ X implies ( s, d, c ) ∈ Ch d ( X ) . When a district’s admissions rule respects the initial matching, it has to admit thosecontracts associated with itself in which students apply to their initial schools from everymatching that is feasible for students. The following result shows that this is exactly thecondition for SPDA to satisfy individual rationality.
Theorem 1.
SPDA satisfies individual rationality if, and only if, each district’s admissions rulerespects the initial matching.
The intuition for the “if” part of this theorem is simple. When district admissions rulesrespect the initial matching, no student is matched with a school which is strictly lesspreferred than her initial school under SPDA because she is guaranteed to be accepted bythat school if she applies to it. For the “only if” part of the theorem, we construct a specificstudent preference profile such that SPDA assigns one student a strictly less preferredschool than her initial school whenever there exists one district with an admissions rulethat does not respect the initial matching.In the next example, we illustrate SPDA with district admissions rules that respect theinitial matching.
Example . Consider the problem in Example 1. Recall that in this problem, the outcomeof SPDA is { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } . This matching is not individually rational be-cause student s prefers her initial school c to school c that she is matched with. Thisobservation is consistent with Theorem 1 because the admissions rule of district d does NTERDISTRICT SCHOOL CHOICE 13 not respect the initial matching. In particular, Ch d ( { ( s , c ) , ( s , c ) } ) = { ( s , c ) } , so stu-dent s is rejected from a matching that is feasible for students and includes the contractwith her initial school.Now modify the priority ranking of district d at school c so that s (cid:31) s (cid:31) s (cid:31) s but, otherwise, keep the construction of the district admissions rules and student prefer-ences the same as before. With this change, district admissions rules respect the initialmatching because each student is accepted when she applies to the district with her initialschool. In particular, the proposal of student s to district d with her initial school c isalways accepted. With this modification, it is easy to check that the outcome of SPDA is { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } . This matching satisfies individual rationality. (cid:3) In some school districts, each student gets a priority at her neighborhood school, as inthis example. In the absence of other types of priorities, neighborhood priority guaranteesthat SPDA satisfies individual rationality.
For interdistrict school choice, maintaining a balance of studentsincoming from and outgoing to other districts is important. To formalize this idea, we saythat a mechanism satisfies the balanced-exchange policy if the number of students that adistrict gets from the other districts and the number of students that the district sends tothe others are the same for every district and for every profile of student preferences. Sincedistrict choice rules are acceptant and students prefer every school to the outside optionof being unmatched, every student is matched with a school under SPDA. Therefore, forSPDA, this policy is equivalent to the requirement that the number of students assignedto a district must be equal to the number of students from that district.The balanced-exchange policy is important because the funding that a district gets de-pends on the number of students it serves. Therefore, an interdistrict school choice pro-gram may not be sustainable if SPDA does not satisfy the balanced-exchange policy. Forachieving this policy goal, the following condition on admissions rules proves important.
Definition 2.
A matching X is rationed if, for every district d , it does not assign strictly morestudents to the district than the number of students whose home district is d . A district admissionsrule is rationed if it chooses a rationed matching from any matching that is feasible for students. When a district admissions rule is rationed, the district does not accept strictly morestudents than the number of students from the district at any matching that is feasiblefor students. The result below establishes that this property is exactly the condition toguarantee that SPDA satisfies the balanced-exchange policy. In Appendix B.2, we construct a class of district admissions rules that includes this admissions rule asa special case. These admissions rules are feasible and acceptant, and have completions that satisfy substi-tutability and LAD. Furthermore, they also respect the initial matching.
Theorem 2.
SPDA satisfies the balanced-exchange policy if, and only if, each district’s admissionsrule is rationed.
To obtain the intuition for this result, consider a student. Acceptance requires that adistrict can reject all contracts of this student only when the number of students assignedto the district is at least as large as the number of students from that district. As a result,all students are guaranteed to be matched. In addition, when district admissions rulesare rationed, a district cannot accept more students than the number of students from thedistrict. These two facts together imply that the number of students assigned to a districtin SPDA is equal to the number of students from that district. Therefore, SPDA satisfiesthe balanced-exchange policy when each district’s admissions rule is rationed. Conversely,when there exists one district with an admissions rule that fails to be rationed, then we canconstruct student preferences such that this district is matched with strictly more studentsthan the number of students from the district in SPDA, which means that the outcome doesnot satisfy the balanced-exchange policy.Now we illustrate SPDA when district admissions rules are rationed.
Example . Consider the problem in Example 1. Recall that in this problem, the SPDAoutcome is { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } . Since there are three students matched withdistrict d and there are only two students from that district, SPDA does not satisfythe balanced-exchange policy. This is consistent with Theorem 2 because the admis-sions rule of district d is not rationed. In particular, Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) , ( s , c ) , ( s , c ) } , so district d accepts strictly more students than the number ofstudents from there given a matching that is feasible for students.Suppose that we modify the admissions rule of district d as follows. If the dis-trict chooses a contract associated with school c , then at most one contract associatedwith school c is chosen. Therefore, the district never chooses more than two con-tracts, which is the number of students from there. Therefore, the updated admis-sions rule is rationed. With this change, it is easy to check that the SPDA outcome is { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } , which satisfies the balanced-exchange policy. (cid:3) An implication of Theorems 1 and 2 is that SPDA is guaranteed to satisfy individualrationality and the balanced-exchange policy if, and only if, each district’s admissions rulerespects the initial matching and is rationed.
The third policy goal we consider is that of diversity. More specifically,we are interested in how to ensure that there is enough diversity across districts so that In Appendix B.3, we construct a class of rationed district admissions rules that includes this admissionsrule as a special case. These admissions rules are feasible and acceptant, and they have completions thatsatisfy substitutability and LAD.
NTERDISTRICT SCHOOL CHOICE 15 the student composition in terms of demographics does not vary too much from districtto district.We are mainly motivated by a program that is used in the state of Minnesota. State lawin Minnesota identifies racially isolated (relative to one of their neighbors) school districtsand requires them to be in the
Achievement and Integration (AI) Program . The goal is to in-crease the racial parity between neighboring school districts. We first introduce a diversitypolicy in the spirit of this program: Given a constant α ∈ [0 , , we say that a mechanismsatisfies the α -diversity policy if for all preferences, districts d and d (cid:48) , and type t , the dif-ference between the ratios of type- t students in districts d and d (cid:48) is not more than α . Weinterpret α to be the maximum ratio difference tolerated under the diversity policy; forinstance, α = 0 . for Minnesota.We study admissions rules such that SPDA satisfies the α -diversity policy when thereis interdistrict school choice. Since this policy restricts the number of students across dis-tricts, a natural starting point is to have type-specific ceilings at the district level. However,it turns out that type-specific ceilings at the district level may yield district admissionsrules resulting in no stable matchings (see Theorem 9 in Appendix A.2).Since there is an incompatibility between district-level type-specific ceilings and the ex-istence of a stable matching, we impose type-specific ceilings at the school level as follows. Definition 3.
A district admissions rule Ch d has a school-level type-specific ceiling of q tc atschool c for type- t students if the number of type- t students admitted cannot exceed this ceiling.More formally, for any matching X that is feasible for students, |{ x ∈ Ch d ( X ) | τ ( s ( x )) = t, c ( x ) = c }| ≤ q tc . Note that district admissions rules typically violate acceptance once school-level type-specific ceilings are imposed. This is because a student can be rejected from a set that isfeasible for students even when the number of applicants to each school is smaller than itscapacity and the number of applicants to the district is smaller than the number of studentsfrom that district. Given this, we define a weaker version of the acceptance assumption asfollows.
Definition 4.
A district admissions rule Ch d that has school-level type-specific ceilings is weaklyacceptant if, for any contract x associated with a type- t student and district d and matching X that is feasible for students, if x is rejected from X , then at Ch d ( X ) , • the number of students assigned to school c ( x ) is equal to q c ( x ) , or • the number of students assigned to district d is at least k d , or • the number of type- t students assigned to school c ( x ) is at least q tc . In other words, a student can be rejected from a set that is feasible for students onlywhen one of these three conditions is satisfied.
In SPDA, a student may be left unassigned due to school-level type-specific ceilings evenwhen district admissions rules are weakly acceptant. To make sure that every student ismatched, we make the following assumption.
Definition 5.
A profile of district admissions rules ( Ch d ) d ∈D accommodates unmatched stu-dents if for any student s and feasible matching X in which student s is unmatched, there exists x = ( s, d, c ) ∈ X such that x ∈ Ch d ( X ∪ { x } ) . When a profile of district admissions rules accommodates unmatched students, for anyfeasible matching in which a student is unmatched, there exists a school such that the dis-trict associated with the school would admit that student if she applies to that school. Forexample, when each admissions rule respects the initial matching, the profile of districtadmissions rules accommodates unmatched students because an unmatched student’s ap-plication to her initial school is always accepted. Lemma 2 in Appendix D shows that whena profile of district admissions rules accommodates unmatched students, every student ismatched to a school in SPDA.In general, accommodation of unmatched students may be in conflict with type-specificceilings because there may not be enough space for a student type when ceilings are smallfor this type. To avoid this, we assume that type-specific ceilings are high enough so that ( Ch d ) d ∈D accommodates unmatched students. Our assumptions on district admissions rules allow us to control the distribution of theSPDA outcome. In particular, the SPDA outcome satisfies the following conditions: (i) (cid:80) t ξ td ( X ) = k d for all d ∈ D , (ii) (cid:80) c ∈C ξ tc ( X ) = k t for all t ∈ T , (iii) (cid:80) t ∈T ξ tc ( X ) ≤ q c for all c ∈ C , and (iv) ξ tc ( X ) ≤ q tc for all t ∈ T and c ∈ C . We call any matching X satisfying theseconditions legitimate .In this framework, type- t ceilings of schools in district d may result in a floor of anothertype t (cid:48) in this district in the sense that the number of type- t (cid:48) students in the district shouldbe at least a certain number. Moreover, this may further impose a ceiling for type t (cid:48) inanother district d (cid:48) . To see this, suppose, for example, that (i) there are two districts d and d (cid:48) , (ii) in each district, there is one school and 100 students, (iii) 100 students are of type t and 100 students are of another type t (cid:48) , and (iv) each school has a type- t ceiling of 60 and atype- t (cid:48) ceiling of 70. In a legitimate matching, each district needs to have at least 40 type- t (cid:48) students (because, otherwise, the number of type- t students in that district would have tobe more than 60). Moreover, this would mean that there cannot be more than 60 type- t (cid:48) students in any district (because, otherwise, there would need to be more than 40 type- t (cid:48) students in the other district, contradicting the floor we just calculated). Hence, in this For instance, ignoring integer problems, q td ≥ k d k t (cid:80) t (cid:48)∈T k t (cid:48) for all t, d , would make ceilings compatiblewith this property as it would be possible to assign the same percentage of students of each type to alldistricts. NTERDISTRICT SCHOOL CHOICE 17 example, in effect we have a floor of 40 and a (further restricted) ceiling of 60 for type- t (cid:48) students for each district.Faced with this complication, our approach is to find the tightest lower and upperbounds induced by these constraints. For this purpose, a certain optimization problemproves useful. More specifically, consider a linear-programming problem where for eachtype t and district d , we seek the minimum and maximum values of (cid:80) c : d ( c )= d y tc subject to(i) (cid:80) t (cid:48) ∈T (cid:80) c : d ( c )= d (cid:48) y t (cid:48) c = k d (cid:48) for all d (cid:48) ∈ D , (ii) (cid:80) c ∈C y t (cid:48) c = k t (cid:48) for all t (cid:48) ∈ T , (iii) (cid:80) t (cid:48) ∈T y t (cid:48) c ≤ q c for all c ∈ C , and (vi) y t (cid:48) c ≤ q t (cid:48) c for all t (cid:48) ∈ T and c ∈ C . Let ˆ p td and ˆ q td be the solutions to theminimization and maximization problems, respectively.Both of these optimization problems belong to a special class of linear-programmingproblems called a minimum-cost flow problem, and many computationally efficient algo-rithms to solve it are known in the literature. A straightforward but important observa-tion is that ˆ p td (resp. ˆ q td ) is exactly the lowest (resp. highest) number of type- t students whocan be matched to district d in a legitimate matching (Lemma 3 in Appendix D). Giventhis observation, we call ˆ p td the implied floor and ˆ q td the implied ceiling .Now we are ready to state the main result of this section. Theorem 3.
Suppose that each district admissions rule has school-level type-specific ceilings andis rationed and weakly acceptant. Moreover, suppose that the district admissions rule profile ac-commodates unmatched students. Then, SPDA satisfies the α -diversity policy if, and only if, ˆ q td /k d − ˆ p td (cid:48) /k d (cid:48) ≤ α for every type t and districts d, d (cid:48) such that d (cid:54) = d (cid:48) . The proof of this theorem, given in Appendix D, is based on a number of steps. First,as mentioned above, we note that ˆ p td and ˆ q td are the lower and upper bounds, respectively,of the number of type- t students who can be matched with district d in any legitimatematching. This observation immediately establishes the “if” part of the theorem. Then,we further establish that the implied floors and ceilings can be achieved simultaneouslyin the sense that, for any pair of districts d and d (cid:48) with d (cid:54) = d (cid:48) , there exists a legitimatematching that assigns exactly ˆ q td type- t students to district d and exactly ˆ p td (cid:48) type- t studentsto district d (cid:48) (Lemma 4). In other words, we establish that the implied ceiling and floorare achieved in two different districts, and they are achieved at one legitimate matching si-multaneously. We complete the proof of the theorem by constructing student preferencessuch that the outcome of SPDA achieves these bounds. In Appendix C, we provide an To see that our problem is a minimum-cost flow problem, note that we can take ( k d ) d ∈D as the “supply,” ( k t ) t ∈T as the “demand,” ( q td ) d ∈D ,t ∈T as the “arc capacity bounds,” and the objective functions for ˆ p td and ˆ q td to be min y td and min − y td , respectively. These problems have an “integrality property” so that if the supply,demand, and bounds are integers, then all the solutions are integers as well. As already mentioned, manyalgorithms have been proposed to solve different objective functions for these problems. For instance, thecapacity scaling algorithm of Edmonds and Karp (1972) gives the solutions in polynomial time. For moreinformation, see Chapter 10 of Ahuja (2017). We are grateful to Fatma Kilinc-Karzan for helpful discussions. example that illustrates Theorem 3. In Appendix B.4, we provide a fairly general class ofdistrict admissions rules that satisfies our assumptions in this result.The analysis in this section characterizes conditions under which different policy goalsare achieved under SPDA. One of the facts worth mentioning in this context is that achiev-ing multiple policies can be overly demanding. To see this point, we note that individualrationality and α -diversity policy are often incompatible with one another. For example,consider a problem such that each student’s most preferred school is her initial school anda constant α such that the initial matching does not satisfy the α -diversity policy. Indeed,in this case, no mechanism can simultaneously satisfy individual rationality and the α -diversity policy because the initial matching is the unique individually rational matching,but it fails the α -diversity policy.
4. Achieving Policy Goals with Efficient Outcomes
In this section, we turn our focus to efficiency. More specifically, we study the existenceof a mechanism that satisfies a given policy goal on the distribution of agents, constrainedefficiency, strategy-proofness, and individual rationality. We first consider a policy goalwith type-specific ceilings at the district level. In this setting, we establish an impossibilityresult.
Theorem 4.
There exist a problem and ceilings ( q td ) t ∈T ,d ∈D such that the the initial matching ˜ X satisfies the policy goal Ξ ≡ { ξ | q td ≥ ξ td for all d and t, q c ≥ (cid:80) t ξ tc for all c } , while there existsno mechanism that satisfies the policy goal Ξ , constrained efficiency, individual rationality, andstrategy-proofness. We show this result using the following example.
Example . Consider the following problem with districts d and d . District d has schools c , c , and c and district d has schools c , c , and c . All schools have a capacity of one.There are six students: students s and s have type t , students s and s have type t ,and students s and s have type t . Both districts have a ceiling of one for types t and t : q t d = q t d = 1 and q t d = q t d = 1 . Initially, student s i is matched with school c i , for i = 1 , . . . , , so the initial matching satisfies the policy goal Ξ . Student preferences are asfollows: s s s s s s c c c c c c c c c c c c ... ... c ... ... c ... ... NTERDISTRICT SCHOOL CHOICE 19 where the dots in the table mean that the corresponding parts of the preferences are arbi-trary.In this example, there are two matchings that satisfy the policy goal Ξ , constrained effi-ciency, and individual rationality: X = { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } , and X (cid:48) = { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } . If a mechanism satisfies the desired properties, then its outcome at the above studentpreference profile must be either matching X or X (cid:48) .Consider the case where the mechanism produces matching X at the above studentpreference profile. Suppose student s misreports her preference by ranking c first and c second (while the ranking of other schools is arbitrary). Under the new report, themechanism produces matching X (cid:48) because it is the only matching that satisfies the policygoal Ξ , constrained-efficiency, and individual rationality. Since student s strictly prefersher school in X (cid:48) to her school in X , she has a profitable deviation.Similarly, consider the case where the mechanism produces matching X (cid:48) at the abovestudent preference profile. Suppose student s misreports her preference by ranking c first and c second (while the ranking of the other schools is arbitrary). In this case, themechanism produces matching X because it is the only matching that satisfies the policygoal Ξ , constrained-efficiency, and individual rationality. Since student s strictly prefersher school in X to her school in X (cid:48) , she has a profitable deviation.In both cases, there exists a student who benefits from misreporting, so the desiredconclusion follows. (cid:3) This example also shows that there is no mechanism that satisfies the α -diversity policygoal for α = 0 introduced in Section 3.3, constrained efficiency, individual rationality,and strategy-proofness. Consequently, without any assumptions, a policy goal may notbe implemented with the desirable properties. To establish a positive result, we considerdistributional policy goals that satisfy the following notion of discrete convexity, which isstudied in the mathematics and operations research literatures (Murota, 2003). Definition 6.
Let χ c,t denote the distribution where there is one type- t student at school c and thereare no other students. A set of distributions Ξ is M-convex if whenever ξ, ˜ ξ ∈ Ξ and ξ tc > ˜ ξ tc forsome school c and type t then there exist school c (cid:48) and type t (cid:48) with ξ t (cid:48) c (cid:48) < ˜ ξ t (cid:48) c (cid:48) such that ξ − χ c,t + χ c (cid:48) ,t (cid:48) ∈ Ξ and ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ∈ Ξ . The letter M in the term M-convex set comes from the word matroid , a closely related and well-studiedconcept in discrete mathematics.
To illustrate this concept, suppose that a set of distributions Ξ is M-convex. Considertwo distributions ξ and ˜ ξ in this set such that there are more type- t students in school c at ξ than at ˜ ξ . Then there exist school c (cid:48) and type t (cid:48) such that there are more type- t (cid:48) students inschool c (cid:48) at ˜ ξ than ξ with the following two properties. First, removing one type- t studentfrom school c and adding one type- t (cid:48) student to school c (cid:48) in ξ produces a distribution in Ξ . Second, removing one type- t (cid:48) student from school c (cid:48) and adding one type- t studentto school c in ˜ ξ gives a distribution in Ξ (see Figure 2). Intuitively, from each of these twodistributions we can move closer to the other distribution in an incremental manner within Ξ , a property analogous to the standard convexity notion but adapted to a discrete setting.We illustrate this concept with the following example.Figure 2. Illustration of M-convexity Example . Consider the problem and the set of distributions Ξ defined in Example 4. Weshow that Ξ is not M-convex. Recall matchings X and X (cid:48) in that example. By construction,both X and X (cid:48) satisfy the policy goal Ξ . Furthermore, ξ t c ( X ) = 1 > ξ t c ( X (cid:48) ) because (i)school c is matched with student s at X , whose type is t , while (ii) school c is matchedwith student s at X (cid:48) , whose type is t (cid:54) = t . If the set of distributions Ξ is M-convex, thereexist a school c and a type t such that ξ tc ( X ) < ξ tc ( X (cid:48) ) and ξ ( X ) − χ c ,t + χ c,t is in Ξ . Becauseeach school’s capacity is one, and at matching X all schools have filled their capacities, thismeans that the only candidate for ( c, t ) satisfying the above condition is such that c = c .But the only nonzero ξ tc ( X (cid:48) ) is for t = t (because s is the unique student matched with c at X (cid:48) ), and ξ ( X ) − χ c ,t + χ c ,t does not satisfy the policy goal because district d ’s ceilingfor type t is violated (note ξ t c ( X ) = 1 because student s is matched with c at X .) NTERDISTRICT SCHOOL CHOICE 21
The above argument implies that Ξ ∩ Ξ is not M-convex either. To see this, note thatboth ξ ( X ) and ξ ( X (cid:48) ) are in Ξ ∩ Ξ because all students are matched. Because we haveshown that no distribution of the form ξ ( X ) − χ c ,t + χ c,t is in Ξ , by set inclusion relation Ξ ∩ Ξ ⊆ Ξ , there is no distribution of the form ξ ( X ) − χ c ,t + χ c,t in Ξ ∩ Ξ either. (cid:3) Now we introduce a mechanism that achieves the desirable properties whenever thepolicy goal is M-convex. To do this, we first create a hypothetical matching problem. Onone side of the market, there are school-type pairs ( c, t ) where c ∈ C and t ∈ T . On theother side, there are students from the original problem, S . Given any student s ∈ S anda preference order P s of s in the original problem, define preference order ˜ P s over school-type pairs in the hypothetical problem as follows: letting t be the type of student s and c be her initial school in the original problem, ( s, c ) P s ( s, c (cid:48) ) ⇐⇒ ( c, t ) ˜ P s ( c (cid:48) , t ) for any c, c (cid:48) ∈ C , and ( c , t ) ˜ P s ( c, t (cid:48) ) for any c ∈ C and t (cid:48) ∈ T such that t (cid:48) (cid:54) = t . That is, ˜ P s is apreference order over school-type pairs that ranks the school-type pairs in which the typeis t in the same order as in P s , while finding all school-type pairs specifying a differenttype as less preferred than the pair corresponding to her initial school. Furthermore, let ( c , t ) be the initial school-type pair for s in the hypothetical problem.Next we define a priority ordering of students that school-type pairs use to rank stu-dents. For school-type pair ( c, t ) , students initially matched with ( c, t ) have the highestpriority, and then all other students have the second highest priority. This gives us twopriority classes for students. Then, ties are broken according to a master priority list thatevery school-type pair uses.We say that a type- t student s with the initial school-type pair ( c, t ) is permissible toschool-type pair ( c (cid:48) , t (cid:48) ) at matching X if ξ ( X ) + χ c (cid:48) ,t (cid:48) − χ c,t is in Ξ . Note that a type- t student with initial school-type pair ( c, t ) is always permissible to pair ( c, t ) at matching X whenever ξ ( X ) is in Ξ .The following is a generalization of Gale’s top trading cycles mechanism (Shapley andScarf, 1974), building on its recent extension by Suzuki et al. (2017). Top Trading Cycles Algorithm . Consider a hypothetical problem.
Step 1:
Let X ≡ ˜ X . Each school-type pair points to the permissible student at match-ing X with the highest priority. If there exists no such student, remove the school-type pair from the market. Each student s points to the highest ranked remainingschool-type pair with respect to ˜ P s . Identify and execute cycles. Any student whois part of an executed cycle is assigned the school-type pair she is pointing to andis removed from the market. Step n ( n > ): Let X n denote the matching consisting of assignments in the previoussteps and initial assignments for all students who have not been processed in theprevious steps. Each remaining school-type pair points to the unassigned student who is permissible at matching X n with the highest priority. If there exists no suchstudent, remove the school-type pair from the market. Each unassigned student s points to the highest ranked remaining school-type pair with respect to ˜ P s . Identifyand execute cycles. Any student who is part of an executed cycle is assigned theschool-type pair she is pointing to and is removed from the market.This algorithm terminates in the first step such that no student remains to be processed.The outcome is defined as the matching induced by the outcome of the hypothetical prob-lem at this step. The top trading cycles mechanism (TTC) takes a profile of student prefer-ences as input and produces the outcome of this algorithm at the reported student prefer-ence profile. Note that the definition of permissibility and, hence, the definition of TTC,depend on the policy goal. Nevertheless, we do not explicitly state the policy goal underconsideration when it is clear from the context.The main result of this section is as follows. Theorem 5.
Suppose that the initial matching satisfies the policy goal Ξ . If Ξ ∩ Ξ is M-convex,then TTC satisfies the policy goal Ξ , constrained efficiency, individual rationality, and strategy-proofness. The assumption that the initial matching satisfies the policy goal is necessary for theresult: Consider student preferences such that each student’s highest-ranked school isher initial school. Then the initial matching is the unique individually rational matching.Therefore, if there exists a mechanism with the desired properties, then the outcome atthis preference profile has to be the initial matching. Hence, we need the assumption thatthe initial matching satisfies the policy goal to have such a mechanism.To see one of the implications of this theorem, suppose that the policy goal Ξ is such thatno school is matched with more students than its capacity. In that case, if Ξ is M-convex,then TTC satisfies the desirable properties. Corollary 1.
Suppose that the policy goal Ξ is such that for every ξ ∈ Ξ and c ∈ C , (cid:80) t ξ tc ≤ q c .Furthermore, suppose that the initial matching satisfies Ξ . If Ξ is M-convex, then TTC satisfies thepolicy goal Ξ , constrained efficiency, individual rationality, and strategy-proofness. In the proof of this corollary, we show that when Ξ is M-convex and no distribution in Ξ assigns more students to a school than its capacity, then Ξ ∩ Ξ is also M-convex. Therefore,the corollary follows directly from Theorem 5.Next we illustrate TTC with an example. Example . Consider a problem with two school districts, d and d . District d has school c with capacity three and school c with capacity two. District d has NTERDISTRICT SCHOOL CHOICE 23 school c with capacity two and school c with capacity one. There are seven stu-dents: students s , s , s , and s are from district d and have type t , whereas stu-dents s , s , and s are from district d and have type t . The initial matching is { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } . Student preferences are as fol-lows. P s P s P s P s P s P s P s c c c c c c c c c c c c c c ... ... ... c c c c c c c c In addition to the school capacities, there is only one additional constraint that school c cannot have more than one type- t student. As we show in the proof of Corollary 2, theset of distributions that satisfy this policy goal and the requirement that every student ismatched is an M-convex set. Therefore, by Theorem 5, TTC satisfies constrained efficiency,individual rationality, strategy-proofness, and the policy goal.To run TTC, we use a master priority list. Suppose that the master priority list ranksstudents as follows: s (cid:31) s (cid:31) s (cid:31) s (cid:31) s (cid:31) s (cid:31) s .At Step , there are eight school-type pairs. Consider ( c , t ) . Initially, students s and s are matched with it, so they are both permissible to this pair. We use the master prioritylist to rank them, so s gets the highest priority at ( c , t ) . Therefore, ( c , t ) points to s .Now consider ( c , t ) . Initially, it does not have any students because there is no type- t student assigned to c in the original problem. Furthermore, s is permissible to ( c , t ) because she can be removed from ( c , t ) and a type- t student can be assigned to ( c , t ) without violating the school capacities or the policy goal. Therefore, ( c , t ) points to s aswell, who gets a higher priority than the other permissible students because of the masterpriority list. The rest of the pairs also point to the highest-priority permissible students.Each student points to the highest ranked school-type pair of the same type as shown inFigure 3A. There is only one cycle: s → ( c , t ) → s → ( c , t ) → s . Therefore, s ismatched with ( c , t ) and s is matched with ( c , t ) .At Step , there are six remaining school-type pairs: There are no permissible studentsfor ( c , t ) and ( c , t ) because c has a capacity of one and it is already assigned to s .Each remaining school-type pair points to the highest-ranked remaining permissible stu-dent. Each student points to the highest-ranked remaining school-type pair (see Figure3B). There is only one cycle: s → ( c , t ) → s . Hence, s is assigned to ( c , t ) . (a) Step 1 of TTC (b) Step 2 of TTC(c) Step 3 of TTC (d) Step 4 of TTC Figure 3. The first four steps of TTC. In each step, there is only one cycle,which is represented by the dashed lines.The algorithm ends in five steps. Steps 3 and 4 are also shown in Figure 3. In Step 5, s points to ( c , t ) , which points back to the student. The outcome is { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } . It can be easily seen that the distribution associated with this matching satisfies thepolicy goal because no school has more students than its capacity and c has only onetype- t student. (cid:3) Sometimes it may be more convenient to describe a policy goal using a real-valued func-tion rather than a set of distributions. The interpretation is that the policy function mea-sures how satisfactory the distribution is in terms of the policy goal. To formalize this al-ternative approach let f : Z |C|×|T | + → R be a function on distributions such that f ( ξ ) ≥ f ( ξ (cid:48) ) means that distribution ξ satisfies the policy at least as well as distribution ξ (cid:48) . Let λ ∈ R bea constant. Consider the following ( f, λ ) − policy : Ξ( f, λ ) ≡ { ξ ∈ Z |C|×|T | + | f ( ξ ) ≥ λ } . Notethat the initial matching ˜ X satisfies the ( f, λ ) -policy if, and only if, f ( ξ ( ˜ X )) ≥ λ . NTERDISTRICT SCHOOL CHOICE 25
We introduce the following condition on functions, which plays a crucial role in theM-convexity of the ( f, λ ) -policy. Definition 7.
A function f is pseudo M-concave , if for every distinct ξ, ˜ ξ ∈ Ξ , there exist ( c, t ) and ( c (cid:48) , t (cid:48) ) with ξ tc > ˜ ξ tc and ξ t (cid:48) c (cid:48) < ˜ ξ t (cid:48) c (cid:48) such that min { f ( ξ − χ c,t + χ c (cid:48) ,t (cid:48) ) , f ( ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ) } ≥ min { f ( ξ ) , f ( ˜ ξ ) } . This is a notion of concavity for functions on a discrete domain. Lemma 1 shows thatpseudo M-concavity characterizes when upper contour sets are M-convex. It is strongerthan quasi M-concavity but not logically related to the M-concavity studied in the discretemathematics literature (Murota, 2003).
Lemma 1. Ξ( f, λ ) ∩ Ξ is M-convex for every λ if, and only if, f is pseudo M-concave. Therefore, we get the following result:
Theorem 6. If f is pseudo M-concave and λ is such that f ( ξ ( ˜ X )) ≥ λ , then TTC satisfies the ( f, λ ) -policy, constrained efficiency, individual rationality, and strategy-proofness. To see why this theorem holds, recall that by Lemma 1, Ξ( f, λ ) ∩ Ξ is M-convex. Fur-thermore, by assumption, the initial matching satisfies the ( f, λ ) -policy. Therefore, theresult follows from Theorem 5.Before we consider specific policy goals, we show that the set-based approach in The-orem 5 and the function-based approach in Theorem 6 are equivalent. For that purpose,note first that Lemma 1 already shows that the ( f, λ ) -policy for a pseudo M-concave func-tion f yields an M-convex policy set Ξ( f, λ ) ∩ Ξ . We establish a sense in which a converseresult holds. Theorem 7.
Suppose that Ξ is a set of distributions. If Ξ ∩ Ξ is M-convex, then there exist apseudo M-concave function f and a constant λ ∈ R such that Ξ( f, λ ) ∩ Ξ = Ξ ∩ Ξ . Now that we have established general results based on M-convexity of the policy setor pseudo M-concavity of the policy function, we proceed to apply them to a variety ofsituations. To begin, consider the set Ξ of distributions of all feasible matchings. In otherwords, consider a situation in which no policy goal is imposed other than (cid:80) t ξ tc ≤ q c foreach c . Then it is rather straightforward to show that the set Ξ ∩ Ξ is an M-convex set.This implies that when there is no policy goal, TTC is efficient, individually rational, andstrategy-proof, a standard result in the literature (Abdulkadiro ˘glu and S ¨onmez, 2003).Next we apply Theorems 5 and 6 to a variety of policy goals. These results turn out to beapplicable to many specific cases, as a wide variety of policy goals induce distributions thatsatisfy M-convexity or can be expressed by policy functions that are pseudo M-concave. To be more specific, first suppose that the policy goal Ξ sets type-specific floors and ceilingsat each school, i.e., Ξ ≡ { ξ | q tc ≥ ξ tc ≥ p tc for all c and t } where q tc is the ceiling and p tc is thefloor for type t at school c . Therefore, for each school, the number of students of a giventype must be within the ceiling and floor of this type at the school. We call a policy goal Ξ of this form a school-level diversity policy and show that Ξ ∩ Ξ is an M-convex set. Thisfinding, together with Theorem 5, implies the following positive result. Corollary 2.
Suppose that the initial matching satisfies a school-level diversity policy. Then TTCsatisfies the school-level diversity policy, constrained efficiency, individual rationality, and strategy-proofness.
We note a sharp contrast between this result and Theorem 4. The latter result demon-strates that no mechanism is guaranteed to satisfy the policy goal and other desideratasuch as constrained efficiency, individual rationality, and strategy-proofness if the floorsor ceilings are imposed at the district level. Corollary 2, in contrast, shows that a mech-anism with the desirable properties exists if the floors and ceilings are imposed at theschool level. Taken together, these results inform policy makers about what kinds of di-versity policies are compatible with the other desiderata.One possible shortcoming of Corollary 2 is that the result holds under the assumptionthat the initial matching satisfies the school-level diversity policy. This may be undesir-able given that often diversity policies are implemented because schools or districts areregarded as insufficiently diverse, as in the case of the diversity law in Minnesota. In sucha setting, a potential diversity requirement can be that the diversity should not decreaseas a result of interdistrict school choice according to a diversity measure f . Such a con-sideration can be formally described as the ( f, ξ ( ˜ X )) -policy, Ξ( f, ξ ( ˜ X )) . The next corollaryestablishes a positive result for a Ξ( f, ξ ( ˜ X )) -policy where the diversity is measured via the“Manhattan distance” to an ideal point. Corollary 3.
Let ˆ ξ ∈ Ξ be an ideal distribution and f ( ξ ) ≡ − (cid:80) c,t | ξ tc − ˆ ξ tc | be the policy function.Then TTC satisfies ( f, ξ ( ˜ X )) -policy, constrained efficiency, individual rationality, and strategy-proofness. Note that the initial matching ˜ X always satisfies ( f, ξ ( ˜ X )) -policy. Furthermore, we showthat the policy function f is pseudo M-concave. Therefore, this corollary follows from The-orem 6. More generally, when the diversity is measured by a pseudo M-concave function,then the TTC outcome is as diverse as the initial matching. Furthermore, TTC also satisfiesthe other desirable properties.Next, we study the balanced-exchange policy introduced in Section 3.2. We establishthat the balanced-exchange policy imposed on Ξ is represented by a distribution thatsatisfies M-convexity. This implies the following result. NTERDISTRICT SCHOOL CHOICE 27
Corollary 4.
TTC satisfies the balanced-exchange policy, constrained efficiency, individual ratio-nality, and strategy-proofness.
One of the advantages of our approach is that M-convexity of a set and pseudo M-concavity of a function are so general that a wide variety of policy goals satisfy them,and that it is likely to be applicable for policy goals that one may encounter in the fu-ture. To highlight this point, we consider imposing the diversity and balanced-exchangepolicies at the same time. More specifically, define a set of distributions Ξ ≡ { ξ | q tc ≥ ξ tc ≥ p tc for all c and t and (cid:80) t (cid:80) c : d ( c )= d ξ tc = k d for all d } and call it the combination of balanced-exchange and school-level diversity policies . This is the set of distributions that satisfyboth the school-level floors and ceilings and the balanced-exchange requirement. We es-tablish Ξ ∩ Ξ is M-convex, implying the following result. Corollary 5.
Suppose that the initial matching satisfies the combination of balanced exchange andschool-level diversity policies. Then TTC satisfies the combination of balanced exchange and school-level diversity policies, constrained efficiency, individual rationality, and strategy-proofness.
In general, the intersection of two M-convex sets need not be M-convex. Therefore, theproof of this result does not follow from the proofs of Corollaries 2 and 4.
5. Conclusion
Despite increasing interest in interdistrict school choice in the US, the scope of matchingtheory has been limited to intradistrict choice. In this paper, we proposed a new frame-work to study interdistrict school choice that allows for interdistrict admissions, both fromstability and efficiency perspectives. For stable mechanisms, we characterized conditionson district admissions rules that achieve a variety of important policy goals, such as stu-dent diversity across districts. For efficient mechanisms, we showed that certain types ofdiversity policies are incompatible with desirable properties such as strategy-proofness,while alternative forms of diversity policies can be achieved by a variation of the top trad-ing cycles mechanism, which is strategy-proof. Overall, our analysis suggests that inter-district school choice can help achieve desirable policy goals such as student diversity, butonly with an appropriate design of constraints, admissions rules, and placement mecha-nisms.We regard this paper as a first step toward formal analysis of interdistrict school choicebased on tools of market design. As such, we envision a variety of directions for futureresearch. For example, it may be interesting to study cases in which the conditions for ourresults are violated. Although we already know the policy goals are not guaranteed to besatisfied for our stability results (our results provide necessary and sufficient conditions), Such an example is available from the authors upon request. the seriousness of the failure of the policy goals studied in the present paper is an openquestion. Quantitative measures or an approximation argument like those used in “largematching market” studies (e.g., Roth and Peranson (1999), Kojima and Pathak (2009), Ko-jima et al. (2013), and Ashlagi et al. (2014)) may prove useful, although this is speculativeat this point and beyond the scope of the present paper.We studied policy goals that we regarded as among the most important ones, but theyare far from being exhaustive. Other important policy goals may include a diversity policyrequiring certain proportions of different student types in each district (see Nguyen andVohra (2017) for a related policy at the level of schools), as well as a balanced exchangepolicy requiring a certain bound on the difference in the numbers of students receivedfrom and sent to other districts (see Dur and ¨Unver (2018) for a related policy at the levelof schools). Given that the existing literature has not studied interdistrict school choice,we envision that many policy goals await to be studied within our framework.While our paper is primarily theoretical and aimed at proposing a general frameworkto study interdistrict school choice, the main motivation comes from applications to actualprograms such as Minnesota’s AI program. Given this motivation, it would be interestingto study interdistrict school choice empirically. For instance, evaluating how well the ex-isting programs are doing in terms of balanced exchange, student welfare, and diversity,and how much improvement could be made by a conscious design based on theories suchas the ones suggested in the present paper, are important questions left for future work. Inaddition, implementation of our designs in practice would be interesting. Doing so may,for instance, shed new light on the tradeoff between SPDA and TTC, which has been stud-ied in the intradistrict school choice from a practical perspective (e.g., Abdulkadiro ˘glu etal. (2006), Abdulkadiro ˘glu et al. (2017)). We are only beginning to learn about the inter-district school choice problem, and thus we expect that these and other questions couldbe answered as more researchers analyze it.
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Appendix A. Additional Results
In this section, we provide some additional results. Before we proceed, we introducetwo admissions rule properties. An admissions rule Ch satisfies path independence iffor every X, Y ⊆ X , Ch ( X ∪ Y ) = Ch ( X ∪ Ch ( Y )) . Path independence states that aset can be divided into not-necessarily disjoint subsets and the admissions rule can beapplied to the subsets in any order so that the chosen set of contracts is always the same.An admissions rule Ch satisfies the irrelevance of rejected contracts (IRC) if for every X ⊆ X and x / ∈ Ch ( X ) , Ch ( X \ { x } ) = Ch ( X ) . The irrelevance or rejected contractsstates that a rejected contract can be removed from a set without changing the chosen set.Path independence is equivalent to substitutability and IRC (Aizerman and Malishevski, A.1. Improving Student Welfare for Districts with Intradistct School Choice.
In Section3.1, we studied when SPDA satisfies individual rationality, which requires that, under in-terdistrict school choice, every student is matched with a school that is weakly more pre-ferred than her initial school. In this section, we consider an alternative setting where eachdistrict uses SPDA to assign its students to schools when there is no interdistrict schoolchoice. In other words, the status quo is SPDA when there is only intradistrict schoolchoice. More explicitly, each student ranks schools in their home districts (or contracts as-sociated with their home districts) and SPDA is used between a district and students fromthat district. Note that we assume each student’s ranking over contracts associated withthe home district is the same as the relative ranking in the original preferences. Impor-tantly, in this setting, we compare SPDA outcomes in interdistrict and intradistrict schoolchoice. In such a setting, we characterize district admissions rules which guarantee thatno student is hurt from interdistrict school choice.The next property of district admissions rules plays a crucial role to achieve this policy.
Definition 8.
A district admissions rule Ch d favors own students if for any matching X thatis feasible for students, Ch d ( X ) ⊇ Ch d ( { x ∈ X | d ( s ( x )) = d } ) . When a district admissions rule favors own students, any contract that is chosen from aset of contracts associated with students from a district is also chosen from a superset thatincludes additional contracts associated with students from the other districts. Roughly,this condition requires that, under interdistrict school choice, a district prioritizes its ownstudents that it used to admit over students from the other districts (even though an out-of-district student can still be admitted when a student from the district is rejected).The following result shows that this is exactly the condition which guarantees that in-terdistrict school choice weakly improves the outcome for every student.
Theorem 8.
Every student weakly prefers the SPDA outcome under interdistrict school choice tothe SPDA outcome under intradistrict school choice for all student preferences if, and only if, eachdistrict’s admissions rule favors own students.
In the proof, we show that in the intradistrict school choice the SPDA outcome can alter-natively be produced by an interdistrict school choice model where students rank contractswith all districts and districts have modified admissions rules: For any set of contracts X , See Ayg ¨un and S ¨onmez (2013) for a study of IRC and Chambers and Yenmez (2017) for a study of pathindependence in a matching context.
NTERDISTRICT SCHOOL CHOICE 35 each district d chooses the following contracts: Ch d ( { x ∈ X | d ( s ( x )) = d } ) . Since theoriginal district admissions rules favor own students, the chosen set under the modifiedadmissions rule is a subset of Ch d ( X ) when X is feasible for students. Then the conclusionthat students receive weakly more preferred outcomes in interdistrict school choice thanin intradistrict school choice follows from a comparative statics property of SPDA that weshow (Lemma 5). To show the “only if” part, when there exists a district admissions rulethat fails to favor own students, we construct student preferences such that interdistrictschool choice makes at least one student strictly worse off than intradistrict school choice.
A.2. District-level Type-specific Ceilings.
In this section, we show the incompatibilityof type-specific ceilings at the district level with the existence of a stable matching.
Definition 9.
A district admissions rule Ch d has a district-level type-specific ceiling of q td for type- t students if the number of type- t students admitted from a matching that is feasible forstudents cannot exceed this ceiling. More formally, for any matching X that is feasible for students, |{ x ∈ Ch d ( X ) | τ ( s ( x )) = t }| ≤ q td . Note that, as in the case of school-level type-specific ceilings, district admissions rulesdo not necessarily satisfy acceptance once district-level type-specific ceilings are imposed.We define a weaker version of the acceptance assumption as follows.
Definition 10.
A district admissions rule Ch d that has district-level type-specific ceilings is d-weakly acceptant if, for any contract x associated with a type- t student and district d and match-ing X that is feasible for students, if x is rejected from X , then at Ch d ( X ) , • the number of students assigned to school c ( x ) is equal to q c ( x ) , or • the number of students assigned to district d is at least k d , or • the number of type- t students assigned to district d is at least q td . This admissions rule property states that a student can be rejected only when one ofthese three conditions is satisfied.We establish that in an interdistrict school choice problem in which district admissionsrules have district-level type-specific ceilings that also satisfy some other desired proper-ties, there may exist no stable matching.
Theorem 9.
There exist districts, schools, students, and their types such that for every admissionsrule of a district with district-level type-specific ceilings that satisfies d -weak acceptance and IRC,there exist admissions rules for the other districts that satisfy substitutability and IRC and studentpreferences such that no stable matching exists. We cannot use the comparative statics result of Yenmez (2018) because in our setting Ch d ( X ) ⊇ Ch (cid:48) d ( X ) only when X is feasible for students, whereas Yenmez (2018) requires this property for all sets of contracts X . To show this result, we construct an environment such that a district admissions rulewith the desired properties cannot satisfy weak substitutability , a necessary condition toguarantee the existence of a stable matching (Hatfield and Kojima, 2008).
Appendix B. Examples of District Admissions Rules
In this section, we first provide a class of district admissions rules that are feasible andacceptant and, furthermore, have completions that satisfy substitutability and LAD. Then,based on this class, we identify admissions rules that also satisfy the properties stated inTheorems 1, 2, 3, and 8.
B.1. An Example of a District Admissions Rule.
Consider a district d with schools c , . . . , c n . Each school c i has an admissions rule Ch c i such that, for any set of contracts X , Ch c i ( X ) = Ch c i ( X c i ) ⊆ X c i . District d ’s admissions rule Ch d is defined as follows. Forany set of contracts X , Ch d ( X ) = Ch c ( X ) ∪ Ch c ( X \ Y ) ∪ . . . ∪ Ch c n ( X \ Y n − ) ,where Y i for i = 1 , . . . , n − is the set of all contracts in X associated with students whohave contracts in Ch c ( X ) ∪ . . . ∪ Ch c i ( X \ Y i − ) . In words, we order the schools andlet schools choose in that order. Furthermore, if a student is chosen by some school, weremove all contracts associated with this student for the remaining schools.We study when district admissions rule Ch d satisfies our assumptions. Claim 1.
Suppose that for every school c i and matching X , | Ch c i ( X ) | ≤ q c i . Then district admis-sions rule Ch d is feasible.Proof. Since every student-school pair uniquely defines a contract, for every matching X ,every school c i , and every student s , there is at most one contract associated with s in Ch c i ( X ) . In addition, whenever a student’s contract with a school c i is chosen, her con-tracts with the remaining schools are included in Y j for every j ≥ i by the constructionof Ch d . Hence, for every X , Ch d ( X ) is feasible for students. Furthermore, by assumption, | Ch c i ( X ) | ≤ q c i for each c i . Therefore, Ch d is feasible. (cid:3) Claim 2.
Suppose that for every school c i and matching X , | Ch c i ( X ) | = min { q c i , | X c i |} . Thendistrict admissions rule Ch d is acceptant.Proof. Suppose that matching X is feasible for students and x ∈ X d \ Ch d ( X ) . There exists i ≤ n such that c i = c ( x ) . Since X is feasible for students, x ∈ X \ Y i − where Y i − is asdefined in the construction of Ch d . Because x ∈ X d \ Ch d ( X ) , x / ∈ Ch c i ( X \ Y i − ) . Then | Ch c i ( X \ Y i − ) | = q c i by assumption, which implies that district admissions rule Ch d isacceptant. (cid:3) NTERDISTRICT SCHOOL CHOICE 37
Next we study when district admissions rule Ch d has a completion that satisfies sub-stitutability and LAD. Consider the following district admissions rule Ch (cid:48) d : For any set ofcontracts X , Ch (cid:48) d ( X ) = Ch c ( X ) ∪ . . . ∪ Ch c n ( X ) . Claim 3.
Suppose that for every school c i , Ch c i satisfies substitutability and LAD. Then districtadmissions rule Ch (cid:48) d is a completion of Ch d , and it satisfies substitutability and LAD.Proof. To show that Ch (cid:48) d is a completion of Ch d , suppose that X is a set of contracts suchthat Ch (cid:48) d ( X ) is feasible for students. By mathematical induction, we show that Ch c i ( X ) = Ch c i ( X \ Y i − ) for i = 1 , . . . , n , where Y i is defined as above for i > and Y = ∅ . The claimtrivially holds for i = 1 . Suppose that it also holds for , . . . , i − . We show the claim for i .Since Ch (cid:48) d ( X ) is feasible for students, Ch c i ( X ) and Ch c ( X ) ∪ . . . ∪ Ch c i − ( X ) do not haveany contracts associated with the same student. Therefore, Ch c i ( X ) ∩ Y i − = ∅ . Since Ch c i satisfies IRC, Ch c i ( X ) = Ch c i ( X \ Y i − ) . As a result, Ch d ( X ) = Ch (cid:48) d ( X ) , which completesthe proof that Ch (cid:48) d is a completion of Ch d .Since all school admissions rules satisfy substitutability and LAD, so does Ch (cid:48) d . (cid:3) All of the assumptions on school admissions rules stated in Claims 1, 2, and 3 are satis-fied when school admissions rules are responsive : each school has a ranking of contractsassociated with itself and, from any given set of contracts, each school chooses contractswith the highest rank until the capacity of the school is full or there are no more contractsleft. Responsive admissions rules satisfy substitutability and LAD. Furthermore, for everyschool c i , | Ch c i ( X ) | = min { q c i , | X c i |} . By the claims stated above, when school admis-sions rules are responsive, district admissions rule Ch d is feasible and acceptant, and ithas a completion that satisfies substitutability and LAD.Based on these results, we provide examples of district admissions rules that furthersatisfy the additional assumptions considered in different parts of our paper. B.2. District Admissions Rules Satisfying the Assumptions in Theorem 1.
We use thedistrict admissions rule construction above and we further specify each school’s admis-sions rule. Each school has a responsive admissions rule. If a student is initially matchedwith a school, then her contract with this school is ranked higher than contracts of stu-dents who are not initially matched with the school. As before, district admissions rule Ch d is feasible and acceptant, and it has a completion that satisfies substitutability andLAD. Claim 4.
District admissions rule Ch d respects the initial matching. See Chambers and Yenmez (2018) for a characterization of responsive admissions rules using substi-tutability.
Proof.
Let c be the initial school of student s and x = ( s, d, c ) . By construction, for anymatching X that is feasible for students, x ∈ X implies x ∈ Ch d ( X ) because c chooses x from any set of contracts and s does not have any other contract in X . Therefore, Ch d respects the initial matching. (cid:3) B.3. District Admissions Rules Satisfying the Assumptions in Theorem 2.
We modifythe district admissions rule construction in Appendix B.1. Each school has a ranking ofcontracts associated with itself. When it is the turn of a school, it accepts contracts thathave the highest rank until the capacity of the school is full, or the number of contractschosen by the district is k d , or there are no more contracts left. The remaining contracts ofa chosen student are removed.District admissions rule Ch d is feasible because no school admits more students than itscapacity and no student is admitted to more than one school. Claim 5.
District admissions rule Ch d is acceptant.Proof. To show acceptance, suppose that matching X is feasible for students and x ∈ X d \ Ch d ( X ) . There exists i ≤ n such that c i = c ( x ) . Since X is feasible for students, x ∈ X \ Y i − where Y i − is the set of all contracts in X associated with students who arechosen by schools c , . . . , c i − . Because x ∈ X d \ Ch d ( X ) , x is not chosen by c i . Then, byconstruction, either c i fills its capacity or the district admits k d students, which impliesthat Ch d is acceptant. (cid:3) Claim 6.
District admissions rule Ch d has a completion that satisfies substitutability and LAD.Proof. First, we construct a completion of Ch d . Define the following district admissionsrule: given a set of contracts X , when it is the turn of a school, it chooses from all the con-tracts in X . Each school chooses contracts using the same priority order until the schoolcapacity is full, or the district has k d contracts, or there are no more contracts left. De-note this admissions rule by Ch (cid:48) d . Suppose that Ch (cid:48) d ( X ) is feasible for students. Then, byconstruction, Ch (cid:48) d ( X ) = Ch d ( X ) . Therefore, Ch (cid:48) d is a completion of Ch d .Next, we show that Ch (cid:48) d satisfies LAD. Suppose that Y ⊇ X . Every school c i choosesweakly more contracts from Y than X unless the number of contracts chosen from Y bythe district reaches k d . Since the number of chosen contracts from X cannot exceed k d byconstruction, Ch (cid:48) d satisfies LAD.Finally, we show that Ch (cid:48) d satisfies substitutability. Suppose that x ∈ X ⊆ Y and x ∈ Ch (cid:48) d ( Y ) . Therefore, the number of contracts chosen from Y by schools preceding c ( x ) is strictly less than k d . This implies that the number of contracts chosen from X byschools preceding c ( x ) is weakly less than this number as weakly more contracts are cho-sen by schools preceding school c ( x ) in Y than X . As a result, for school c ( x ) , weakly morecontracts can be chosen from X than Y . NTERDISTRICT SCHOOL CHOICE 39
The ranking of contract x among Y in the ranking of school c ( x ) is high enough thatit is chosen from set Y . Therefore, the ranking of contract x among X in the ranking ofschool c ( x ) must be high enough to be chosen from set X because weakly more contractsare chosen from X than Y for school c ( x ) . (cid:3) Furthermore, by construction, district admissions rule Ch d never chooses more than k d students. Therefore, it is also rationed. B.4. District Admissions Rules Satisfying the Assumptions in Theorem 3.
A profileof district admissions rules can accommodate unmatched students by reserving seats fordifferent types of students:
Definition 11.
Let c be a school in district d . A district admissions rule Ch d has a reserve of r tc for type- t students at school c if, for any feasible matching X that does not have any contractassociated with type- t student s , if |{ x ∈ X c | τ ( s ( x )) = t }| < r tc , then x = ( s, d, c ) satisfies x ∈ Ch d ( X ∪ { x } ) . A reserve for a student type at a school c guarantees space for this type at school c .Therefore, when a student is unmatched at a feasible matching and the reserve for hertype is not yet filled at a school, the district will accept this student at that school if sheapplies to it. Claim 7.
Suppose that districts have admissions rules with reserves such that (cid:80) c r tc = k t for everytype t . Then the profile of district admissions rules accommodates unmatched students.Proof. Suppose that student s is unmatched at a feasible matching X . Let t be the typeof student s . Then there exists a school c such that the number of type- t students in c at X is strictly less than r tc because (cid:80) c r tc = k t . By definition of reserves, x = ( s, c ) satisfies x ∈ Ch d ( c ) ( X ∪ { x } ) . (cid:3) A district can have type-specific reserves at its schools in different ways. In the rest ofthis subsection, we use school admissions rules with reserves introduced by Hafalir etal. (2013) to construct a fairly general example in which a district has schools with type-specific reserves. Let r tc be the number of seats reserved by school c for type- t students.Suppose that the type-specific ceilings for schools are given and that they satisfy the as-sumptions in Section 3.3. Assume that, for every district d , (cid:80) c : d ( c )= d (cid:80) t r tc = k d , (cid:80) c r tc = k t and, for every type t and school c , r tc ≤ q tc . Furthermore, assume that (cid:80) t r tc ≤ q c for everyschool c .Consider the following district admissions rule for district d . Schools are ordered as c , c , . . . , c n . Each school has a ranking over contracts associated with it and a linear orderover student types. First, all schools choose contracts for their reserve seats accordingto the order c , c , . . . , c n . When it is the turn of school c i , all contracts associated with students whose contracts were previously chosen are removed. School c i chooses contractsfor its reserved seats so that, for every type, either reserved seats are filled or there are nomore contracts associated with students of that type remaining. Then all schools choosecontracts for their empty seats in order. When it is the turn of school c i , all contracts ofpreviously chosen students are removed. School c i chooses from the remaining contractsin order. When a contract of a type- t student is considered, this contract is chosen unlessthe school’s capacity is filled or its type- t ceiling is filled or the district has k d contracts.Denote this district admissions rule by Ch d .District admissions rule Ch d is feasible because a student cannot have more than onecontract and a school cannot have more contracts than its capacity at any chosen set ofcontracts. It is also weakly acceptant and rationed by construction. Furthermore, for everytype t and school c , the district cannot admit more than q tc type- t students at c , so it has aschool-level type-specific ceiling of q tc for type- t students and school c . Claim 8.
District admissions rule Ch d has a completion that satisfies substitutability and LAD.Proof. For any set of contracts X , school c , and type t , let X tc denote the set of all contractsin X that are associated with school c and type- t students.Consider the construction of Ch d above, but modify it by not removing contracts ofstudents who are chosen previously. Denote this district admissions rule by Ch (cid:48) d . To showthat Ch (cid:48) d is a completion of Ch d , consider a set of contracts X and suppose that Ch (cid:48) d ( X ) is feasible for students. Since the only difference in the constructions of Ch d and Ch (cid:48) d isthe removal of contracts of previously chosen students, it must be that Ch (cid:48) d ( X ) = Ch d ( X ) .Therefore, Ch (cid:48) d is a completion of Ch d .To prove substitutability of Ch (cid:48) d , suppose, for contradiction, that there exist sets of con-tracts X and Y with X ⊆ Y and a contract x ∈ X such that x ∈ Ch (cid:48) d ( Y ) \ Ch (cid:48) d ( X ) . Let s and c be such that x = ( s, c ) and t = τ ( s ) . First, note that | X tc | > r tc because x (cid:54)∈ Ch (cid:48) d ( X ) .Since Y ⊇ X , | Y tc | ≥ | X tc | > r tc is implied. Therefore, it is after all schools in d have chosencontracts based on their reserves in the algorithm describing Ch (cid:48) d that contract x is chosenby Ch (cid:48) d given Y . Let n ( X ) and n ( Y ) be the numbers of contracts that have been chosenby all schools before the step (call it step κ c ) at which school c chooses students beyondits reserve under X and Y , respectively. Because x ∈ Ch (cid:48) d ( Y ) , it follows that n ( Y ) < k d .Therefore, for each school c (cid:48) , the number of contracts chosen by c (cid:48) before step κ c under Y is weakly larger than those under X , which we prove as follows: • Suppose that school c (cid:48) is processed after school c in the algorithm deciding Ch d .Then, by step κ c , c (cid:48) is matched with students of each type t (cid:48) only up to its type- t (cid:48) reserve. More formally, the numbers of type- t (cid:48) students matched to c (cid:48) are equal to min { r t (cid:48) c (cid:48) , | X t (cid:48) c (cid:48) |} and min { r t (cid:48) c (cid:48) , | Y t (cid:48) c (cid:48) |} under X and Y , respectively. Obviously, the latterexpression is no smaller than the former expression. NTERDISTRICT SCHOOL CHOICE 41 • Suppose that school c (cid:48) is processed before school c in the algorithm deciding Ch d .Recall that n ( Y ) < k d . Therefore, for school c (cid:48) , it is either (i) as many as q c (cid:48) studentsare matched to c (cid:48) under Ch d ( Y ) , or (ii) for each type t (cid:48) , the number of type- t (cid:48) stu-dents matched to c (cid:48) in Y is min { q t (cid:48) c (cid:48) , | Y t (cid:48) c (cid:48) |} . In case (i), the desired conclusion followstrivially because, given any set of contracts, the number of students matched to c (cid:48) cannot exceed q c (cid:48) . For case (ii), under X , the number of type- t (cid:48) students matchedto c (cid:48) cannot exceed min { q t (cid:48) c (cid:48) , | X t (cid:48) c (cid:48) |} ≤ min { q t (cid:48) c (cid:48) , | Y t (cid:48) c (cid:48) |} . Summing up across all types,we obtain the desired conclusion.Thus n ( X ) ≤ n ( Y ) , so k d − n ( X ) ≥ k d − n ( Y ) . Now, in step κ c , school c will choose allthe applications until either the total number of contracts chosen reaches k d , or the totalnumber of contracts chosen at c reaches q c , or the number of contracts chosen at c thatare associated with type t students reaches q tc . Given the previous fact that k d − n ( X ) ≥ k d − n ( Y ) , the fact that Y ⊇ X , and the fact that x is chosen by c in step κ c under Y , ithas to be the case that x is also chosen by c under X in step κ c or before. We prove this asfollows: • If | X tc | ≤ r tc , then x is chosen in the reserve stage by construction. • Let | X tc | > r tc . First note that at step κ c under X and Y , for each type t , there arefewer contracts associated with school c and type- t students that remain to be pro-cessed under X than under Y ( X ⊆ Y , and there is no contract in X = X ∩ Y thatis processed in the reserve stage under Y but not under X ), so the subset of X thatshould be processed in κ c is a subset of the corresponding subset of Y . Moreover,the remaining number of contracts to be chosen before reaching the ceiling at c foreach type t in step κ c is weakly larger at X than at Y by the definition of the reservestage. Finally, as argued above, the total number of students in the district who canstill be chosen at κ c is weakly larger under X than at Y , so whenever x is chosenunder Y in this stage, x is chosen under X in this stage or the reserve stage.This is a contradiction to the assumption that x / ∈ Ch (cid:48) d ( X ) .To show that Ch (cid:48) d satisfies LAD, suppose, for contradiction, that there exist two setsof contracts X, Y with X ⊆ Y and | Ch (cid:48) d ( Y ) | < | Ch (cid:48) d ( X ) | . Then, because Ch (cid:48) d ( Y ) = (cid:83) c : d ( c )= d ( Ch (cid:48) d ( Y ) ∩ Y c ) and Ch (cid:48) d ( X ) = (cid:83) c : d ( c )= d ( Ch (cid:48) d ( X ) ∩ X c ) , there exists a school c with d ( c ) = d such that | Ch (cid:48) d ( Y ) ∩ Y c | < | Ch (cid:48) d ( X ) ∩ X c | . (1)Fix such c arbitrarily. Next, note that | Ch (cid:48) d ( Y ) | < | Ch (cid:48) d ( X ) | ≤ k d , | Ch (cid:48) d ( Y ) ∩ Y c | < | Ch (cid:48) d ( X ) ∩ X c | ≤ q c , where the first line follows because Ch (cid:48) d is rationed by construction, and the second linealso holds by construction of Ch (cid:48) d . Therefore, | Ch (cid:48) d ( Y ) ∩ Y tc | = min {| Y tc | , q tc }≥ min {| X tc | , q tc } = | Ch (cid:48) d ( X ) ∩ X tc | , (2)for each type t ∈ T . Because Ch (cid:48) d ( Y ) ∩ Y c = ∪ t ∈T ( Ch (cid:48) d ( Y ) ∩ Y tc ) and Ch (cid:48) d ( X ) ∩ X c = ∪ t ∈T ( Ch (cid:48) d ( X ) ∩ X tc ) , inequality (2) and the fact Y tc ∩ Y t (cid:48) c = X tc ∩ X t (cid:48) c = ∅ for any pair of types t, t (cid:48) with t (cid:54) = t (cid:48) imply | Ch (cid:48) d ( Y ) ∩ Y c | ≥ | Ch (cid:48) d ( X ) ∩ X c | , which contradicts inequality (1). (cid:3) B.5. District Admissions Rules Satisfying the Assumptions in Theorem 8.
Consider thedistrict admissions rule construction in Appendix B.1. In this example, let each school usea priority ranking in such a way that all contracts of students from district d are rankedhigher than the other contracts. Claim 9.
District admissions rule Ch d favors own students.Proof. Suppose that X is feasible for students. When it is the turn of school c i , it consid-ers X c i . Therefore, Ch d ( X ) = Ch c ( X c ) ∪ . . . ∪ Ch c k ( X c k ) . Furthermore, Ch c i ( X c i ) ⊇ Ch c i ( { x ∈ X c i | d ( s ( x )) = d } ) by construction. Taking the union of all subset inclusionsyields Ch d ( X ) ⊇ Ch d ( { x ∈ X d | d ( s ( x )) = d } ) . Therefore, Ch d favors own students. (cid:3) Appendix C. An Example for Section 3.3
In this section, we provide an example in which the conditions on the admissions rulesstated in Theorem 3 are satisfied and, therefore, SPDA satisfies the diversity policy.Consider a problem with two school districts, d and d . District d has school c withcapacity three and school c with capacity two. District d has school c with capacity twoand school c with capacity one. There are seven students: students s , s , s , and s arefrom district d , whereas students s , s , and s are from district d . Students s , s , s ,and s have type t and s , s , and s have type t . To construct district admissions rulesthat satisfy the properties stated in Theorem 3, let us first specify type-specific ceilings andcalculate implied floors and implied ceilings. Suppose that q t c = 1 , q t c = 2 , q t c = 1 , q t c = 1 ,q t c = 2 , q t c = 1 , q t c = 1 , q t c = 1 . NTERDISTRICT SCHOOL CHOICE 43
These yield the following implied floors, ˆ p t d = 1 , ˆ p t d = 2 , ˆ p t d = 2 , ˆ p t d = 0 , and implied ceilings ˆ q t d = 2 , ˆ q t d = 3 , ˆ q t d = 3 , ˆ q t d = 1 . For any type t and two districts d and d (cid:48) , denote ˆ q td /k d − ˆ p td (cid:48) /k d (cid:48) by ∆ td,d (cid:48) . Using theimplied floors and ceilings above, we get: ∆ t d ,d = 2 / − / − / , ∆ t d ,d = 3 / − / / , ∆ t d ,d = 3 / − / / , and ∆ t d ,d = 1 / − / − / . Hence, these type-specific ceilings satisfy the condition stated in Theorem 3 that ∆ td,d (cid:48) ≤ α for α = 0 . .We construct district admissions rules that have type-specific ceilings, and are rationedand weakly acceptant. Furthermore, the profile of district admissions rules accommodatesunmatched students. As in Appendix B.4, we consider type-specific reserves (as detailedbelow, we first fill in the reserves while applying the district admissions rule that usestype-specific reserves). Let us consider the reserves for schools as follows: r t c = 0 , and r tc = 1 for all other c, t. Consider the following district admissions rule. For each district, schools and studenttypes are ordered and each school has a linear order over students. First, schools choosecontracts for their reserved seats following the master priority list until the reserves arefilled or all the applicants of the relevant type are processed. Then, following the givenorder over schools and student types, schools choose from the remaining contracts fol-lowing the linear order over students in order to fill the rest of their seats until the school To see this, note that there cannot be zero type- t students in d (otherwise not all type- t studentscan be matched since there are only three spaces available for type- t students in d ). If there is one type- t student in d , there has to be three type- t students in d , which implies there cannot be any type- t studentsin d , and this implies there will be three type- t students in d . If there are two type- t students in d , therehave to be two type- t students in d , which implies there is one type- t student in d , and this implies therewill be two type- t students in d . By noting these minimum and maximum numbers, we obtain the impliedreserves and implied ceilings accordingly. These bounds are achievable because it is feasible to have (i) onetype- t student in d , three type- t students in d , zero type- t students in d , and three type- t students in d , and (ii) two type- t students in d , two type- t students in d , one type- t student in d , and two type- t students in d . capacity is filled, or the district has k d contracts, or district type-specific ceilings are filled,or there are no more remaining contracts. To give a more concrete example, suppose that the linear order over students for eachschool is as follows: s (cid:31) s (cid:31) s (cid:31) s (cid:31) s (cid:31) s (cid:31) s and schools and types are orderedfrom the lowest index to the highest. Then, for example, we have the following: Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } . Let us elaborate on how we determine the chosen set of contracts in the above case. School c considers contracts with students s , s , s , and s . Among these students, c accepts s for its reserve for type t , and s for its reserve for type t . Moreover, school c considerscontracts with students s and s . Among these students students, c accepts s for itsreserve for type t . For the remainder of seats, s is accepted by c since (i) c ’s type t ceiling is not full, (ii) c ’s capacity is not full, and (iii) district d has only three acceptedcontracts at this point. Next, s and s are rejected since d has accepted four contracts atthis point. This results in the chosen set of contracts presented above.To illustrate the SPDA outcomes, consider student preferences given by the followingtable. P s P s P s P s P s P s P s c c c c c c c ... c c c c c c ... ... c c c ... c c c SPDA results in the following outcome: { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } . District d is assigned two students of both types and district d is assigned two type- t students and one type- t student. As a result, the ratio difference for type- t students be-tween these districts is roughly . , and the ratio difference for type- t students is roughly . . This example illustrates that the actual ratio differences can be significantly lowerthan the one given by Theorem 3 ( . versus . ). Appendix D. Omitted Proofs
In this section, we provide the omitted proofs. In Appendix B.4, we provide a class of admissions rules that include the one we consider here. Theseadmissions rules satisfy all of the assumptions that we make in this section.
NTERDISTRICT SCHOOL CHOICE 45
Proof of Theorem 1.
First, to show the “if” part, suppose that all district admissions rulesrespect the initial matching. In SPDA, each student s goes down in her preference order,and either SPDA ends before student s reaches her initial school (which is a preferred out-come over the initial school), or student s reaches her initial school. In the latter case, sheis matched with her initial school because the district’s admissions rule respects the initialmatching and the district always considers a set of contracts that is feasible for studentsat any step of SPDA. From this step on, the district accepts this contract, so student s ismatched with her initial school. Therefore, SPDA satisfies individual rationality.To prove the “only if” part, suppose that there exists a district d with an admissions rulethat fails to respect the initial matching. Hence, there exists a matching X , which is feasiblefor students, that includes x = ( s, d, c ) where school c is the initial school of student s and x / ∈ Ch d ( X ) . Now, consider student preferences such that every student associated with acontract in X d prefers that contract the most and all other students prefer a contract asso-ciated with a different district the most. Then, at the first step of SPDA, district d considersmatching X d and tentatively accepts Ch d ( X d ) . Since x / ∈ Ch d ( X d ) , contract x is rejected atthe first step. Therefore, student s is matched with a strictly less preferred school than herinitial school, which implies that SPDA does not satisfy individual rationality. (cid:3) Proof of Theorem 2.
We first prove that if each district admissions rule is rationed, thenSPDA satisfies the balanced-exchange policy. Let X be the matching produced by SPDAfor a given preference profile.We begin by showing that each student must be matched with a school in X . Suppose,for contradiction, that student s is unmatched. Since X is a stable matching, every contract x = ( s, d, c ) associated with the student is rejected by the corresponding district, i.e., x / ∈ Ch d ( X ∪ { x } ) . Otherwise, student s and district d would like to match with each otherusing contract x , contradicting the stability of matching X . Since X ∪ { x } is feasible forstudents, acceptance implies that, for each district d , either every school in the district isfull or that the district has at least k d students at matching X . Both of them imply thatthe district has at least k d students in matching X since the sum of the school capacitiesin district d is at least k d . But this is a contradiction to the assumption that student s isunmatched since the existence of an unmatched student implies that there is at least onedistrict d such that the number of students in X d is less than k d . Therefore, all students arematched in X .Because X is the outcome of SPDA, it is feasible for students. Therefore, because districtadmissions rules are rationed, the number of students in district d cannot be strictly morethan k d for any district d . Furthermore, since every student is matched, the number of students in district d must be exactly k d (because, otherwise, at least one student wouldhave been unmatched.) As a result, SPDA satisfies the balanced-exchange policy.Next, we prove that if at least one district’s admissions rule fails to be rationed, thenthere exists a student preference profile under which SPDA does not satisfy the balanced-exchange policy. Suppose that there exist a district d and a matching X , which is feasiblefor students, such that | Ch d ( X ) | > k d . Consider a feasible matching X (cid:48) such that (i) allstudents are matched, (ii) X (cid:48) d = Ch d ( X ) , and (iii) for every district d (cid:48) (cid:54) = d , | X (cid:48) d (cid:48) | ≤ k d (cid:48) .The existence of such X (cid:48) is guaranteed since every district has enough capacity to serveits students (i.e., for every district d (cid:48) , (cid:80) c : d ( c )= d (cid:48) q c ≥ k d (cid:48) ), and | Ch d ( X ) | > k d . Now, considerany student preferences, where every student likes her contract in X (cid:48) the most.We show that SPDA stops in the first step. For district d (cid:48) (cid:54) = d , X (cid:48) d (cid:48) is feasible and thenumber of students matched to d (cid:48) at X (cid:48) d (cid:48) is weakly less than k d (cid:48) . Since Ch d (cid:48) is acceptant, Ch d (cid:48) ( X (cid:48) d (cid:48) ) = X (cid:48) d (cid:48) . For district d , we need to show that Ch d ( X (cid:48) d ) = X (cid:48) d , which is equivalent to Ch d ( Ch d ( X )) = Ch d ( X ) . Let Ch (cid:48) d be a completion of Ch d that satisfies path independence.Because X and Ch d ( X ) are feasible for students, Ch (cid:48) d ( X ) = Ch d ( X ) and Ch (cid:48) d ( Ch (cid:48) d ( X )) = Ch d ( Ch d ( X )) . Furthermore, since Ch (cid:48) d is path independent, Ch (cid:48) d ( Ch (cid:48) d ( X )) = Ch (cid:48) d ( X ) ,which implies Ch d ( Ch d ( X )) = Ch d ( X ) . As a result, Ch d ( X (cid:48) d ) = X (cid:48) d . Therefore, SPDAstops at the first step since no contract is rejected.Since SPDA stops at the first step, the outcome is matching X (cid:48) . But X (cid:48) fails the balanced-exchange policy because | X (cid:48) d | = | Ch d ( X ) | > k d . (cid:3) Proof of Theorem 3.
To prove this result, we provide the following lemmas.
Lemma 2.
If a profile of district admissions rules accommodates unmatched students, every stu-dent is matched to a school in SPDA.Proof of Lemma 2.
Let X be the outcome of SPDA for some preference profile. Suppose,for contradiction, that student s is unmatched. Since X is a stable matching and student s prefers any contract x = ( s, d, c ) to being unmatched, x / ∈ Ch d ( X ∪ { x } ) . But this is acontradiction to the assumption that the profile of district admissions rules accommodatesunmatched students. (cid:3) Lemma 3.
For each type t , district d , and legitimate matching X , we have ˆ q td ≥ ξ td ( X ) ≥ ˆ p td . More-over, for each type t and district d , there exist legitimate matchings X and X (cid:48) such that ξ td ( X ) = ˆ p td and ξ td ( X (cid:48) ) = ˆ q td .Proof of Lemma 3. Observe that for every legitimate matching X , the induced distributionsatisfies the constraints of the linear program. Therefore, the first part follows from thedefinition of the implied floors and ceilings. For the second part, note that there exists NTERDISTRICT SCHOOL CHOICE 47 a solution to the linear program such that the ceiling and the floor are attained. Further-more, every solution y = ( y tc ) c ∈C ,t ∈T of the linear program can be supported by a legitimatematching X such that y tc = ξ tc ( X ) for every c and t . (cid:3) Lemma 4.
For each t ∈ T and d, d (cid:48) ∈ D with d (cid:54) = d (cid:48) , there exists a legitimate matching X suchthat ξ td ( X ) = ˆ q td and ξ td (cid:48) ( X ) = ˆ p td (cid:48) .Proof of Lemma 4. Let ˆ X be a legitimate matching such that ξ td ( ˆ X ) = ˆ q td and M be the setof all legitimate matchings. Let M ≡ { X ∈ M | ξ td (cid:48) ( X ) = ˆ p td (cid:48) } . M is nonempty due to Lemma 3. Next, let M ≡ { X ∈ M | (cid:88) ˜ t, ˜ c | ξ ˜ t ˜ c ( X ) − ξ ˜ t ˜ c ( ˆ X ) |≤ (cid:88) ˜ t, ˜ c | ξ ˜ t ˜ c ( X (cid:48) ) − ξ ˜ t ˜ c ( ˆ X ) | for every X (cid:48) ∈ M } . M is nonempty because M is a finite set. We will show that for any X ∈ M , ξ td ( X ) = ξ td ( ˆ X ) = ˆ q td .To prove the above claim, assume for contradiction that there exists X ∈ M such that ξ td ( X ) (cid:54) = ξ td ( ˆ X ) . By Lemma 3, ξ td ( X ) (cid:54) = ξ td ( ˆ X ) implies that ξ td ( X ) < ξ td ( ˆ X ) . Then there exists c with d ( c ) = d such that ξ tc ( X ) < ξ tc ( ˆ X ) . Consider the following procedure. Step 0:
Initialize by setting ( t , c ) := ( t, c ) . Note that ξ t c ( X ) < ξ t c ( ˆ X ) by definitionof c . Step i ≥ : Given sequences of type-school pairs (( t j , c j )) ≤ j ≤ i and (( t j +1 , c ∗ j )) ≤ j ξ t (cid:48) c i ( ˆ X ) , then set ( t i +1 , c ∗ i ) := ( t (cid:48) , c i ) .(3) If not, then note that (cid:80) ˜ t ∈T ξ ˜ tc i ( X ) < q c i . Also note that there exists a type-school pair ( t (cid:48) , c (cid:48) ) with c (cid:48) (cid:54) = c i such that ξ t (cid:48) c (cid:48) ( X ) > ξ t (cid:48) c (cid:48) ( ˆ X ) and d ( c (cid:48) ) = d i because (cid:80) ˜ c : d (˜ c )= d i , ˜ t ∈T ξ ˜ t ˜ c ( X ) = (cid:80) ˜ c : d (˜ c )= d i , ˜ t ∈T ξ ˜ t ˜ c ( ˆ X ) = k d i .(a) If t (cid:48) = t i , then let ¯ X be a matching such that ξ ˜ t ˜ c ( ¯ X ) = ξ t i c i ( X ) + 1 for (˜ t, ˜ c ) = ( t i , c i ) ,ξ t (cid:48) c (cid:48) ( X ) − for (˜ t, ˜ c ) = ( t i , c (cid:48) ) ,ξ ˜ t ˜ c ( X ) otherwise. A proof of this fact is as follows. By an earlier argument, ξ t i c i ( X ) < ξ t i c i ( ˆ X ) . Moreover, by assumption ξ ˜ tc i ( X ) ≤ ξ ˜ tc i ( ˆ X ) for every ˜ t ∈ T . Therefore, (cid:80) ˜ t ∈T ξ ˜ tc i ( X ) < (cid:80) ˜ t ∈T ξ ˜ tc i ( ˆ X ) ≤ q c i . Note that ¯ X ∈ M . Also, by construction, (cid:80) ˜ t, ˜ c | ξ ˜ t ˜ c ( ¯ X ) − ξ ˜ t ˜ c ( ˆ X ) | = (cid:80) ˜ t, ˜ c | ξ ˜ t ˜ c ( X ) − ξ ˜ t ˜ c ( ˆ X ) | − < (cid:80) ˜ t, ˜ c | ξ ˜ t ˜ c ( X ) − ξ ˜ t ˜ c ( ˆ X ) | , which contradicts theassumption that X ∈ M .(b) Therefore, suppose that t (cid:48) (cid:54) = t i and set ( t i +1 , c ∗ i ) := ( t (cid:48) , c (cid:48) ) .(4) The pair ( t i +1 , c ∗ i ) created above satisfies ξ t i +1 c ∗ i ( X ) > ξ t i +1 c ∗ i ( ˆ X ) , so there exists c (cid:48) ∈ C such that ξ t i +1 c (cid:48) ( X ) < ξ t i +1 c (cid:48) ( ˆ X ) . Set c i +1 = c (cid:48) . Note that ξ t i +1 c i +1 ( X ) < ξ t i +1 c i +1 ( ˆ X ) .We follow the procedure above to define ( t , c ) , ( t , c ∗ ) , ( t , c ) , ( t , c ∗ ) , ( t , c ) , and soforth. Because T is a finite set, we have i and j > i with t i = t j . Consider the smallest j with this property (note that given such j , i is uniquely identified). Now, let ¯ X be amatching such that ξ ˜ t ˜ c ( ¯ X ) = ξ t k c k ( X ) + 1 for (˜ t, ˜ c ) = ( t k , c k ) for any k ∈ { i, i + 1 , . . . , j − } ,ξ t k +1 c ∗ k ( X ) − for (˜ t, ˜ c ) = ( t k +1 , c ∗ k ) for any k ∈ { i, i + 1 , . . . , j − } ,ξ ˜ t ˜ c ( X ) otherwise.We will show ¯ X ∈ M . To do so, by construction of ¯ X , first note that (cid:80) ˜ t ∈T ξ ˜ t ˜ c ( ¯ X ) ≤ (cid:80) ˜ t ∈T ξ ˜ t ˜ c ( X ) + 1 ≤ q ˜ c for any ˜ c ∈ { c i , . . . , c j − } such that (cid:80) ˜ t ∈T ξ ˜ t ˜ c ( X ) < q ˜ c . Next, byconstruction of ¯ X , (cid:80) ˜ t ∈T ξ ˜ t ˜ c ( ¯ X ) = (cid:80) ˜ t ∈T ξ ˜ t ˜ c ( X ) = q ˜ c for every ˜ c ∈ { c i , . . . , c j − } such that (cid:80) ˜ t ∈T ξ ˜ t ˜ c ( X ) = q ˜ c . Moreover, (cid:80) ˜ t ∈T ξ ˜ t ˜ c ( ¯ X ) ≤ (cid:80) ˜ t ∈T ξ ˜ t ˜ c ( X ) = q ˜ c for every ˜ c ∈ { c ∗ i , . . . , c ∗ j − } . Fi-nally, for every ˜ c ∈ C \ { c i , . . . , c j − , c ∗ i , . . . , c ∗ j − } , (cid:80) ˜ t ∈T ξ ˜ t ˜ c ( ¯ X ) = (cid:80) ˜ t ∈T ξ ˜ t ˜ c ( X ) ≤ q ˜ c . Thus,all school capacities are satisfied by ¯ X . Also by construction of ¯ X , for each ˜ d ∈ D , (cid:80) ˜ c : d (˜ c )= ˜ d ξ ˜ t ˜ c ( ¯ X ) = (cid:80) ˜ c : d (˜ c )= ˜ d ξ ˜ t ˜ c ( X ) = k ˜ d , so ¯ X is rationed. Furthermore, for every ˜ c ∈ C and ˜ t ∈ T , ξ ˜ t ˜ c ( ¯ X ) ≤ max { ξ ˜ t ˜ c ( X ) , ξ ˜ t ˜ c ( ˆ X ) } by construction, so all type-specific ceilings are sat-isfied. Moreover, by construction of ¯ X , for each ˜ t ∈ T , either ξ ˜ t ˜ c ( ¯ X ) = ξ ˜ t ˜ c ( X ) for every ˜ c ∈ C or there exists exactly one pair of schools ˜ c (cid:48) and ˜ c (cid:48)(cid:48) in C such that ξ ˜ t ˜ c (cid:48) ( ¯ X ) = ξ ˜ t ˜ c (cid:48) ( ¯ X ) + 1 , ξ ˜ t ˜ c (cid:48)(cid:48) ( ¯ X ) = ξ ˜ t ˜ c (cid:48)(cid:48) ( ¯ X ) − , and ξ ˜ t ˜ c ( ¯ X ) = ξ ˜ t ˜ c ( X ) for every ˜ c ∈ C \ { ˜ c (cid:48) , ˜ c (cid:48)(cid:48) } . Thus, ˜ t ∈ T , (cid:80) ˜ c ∈C ξ ˜ t ˜ c ( ¯ X ) = (cid:80) ˜ c ∈C ξ ˜ t ˜ c ( X ) for every ˜ t ∈ T . Therefore, ¯ X is legitimate.By construction of ¯ X , either ξ td (cid:48) ( ¯ X ) = ξ td (cid:48) ( X ) or ξ td (cid:48) ( ¯ X ) = ξ td (cid:48) ( X ) − . This implies that ¯ X ∈ M . Furthermore, (cid:80) ˜ t, ˜ c | ξ ˜ t ˜ c ( ¯ X ) − ξ ˜ t ˜ c ( ˆ X ) | < (cid:80) ˜ t, ˜ c | ξ ˜ t ˜ c ( X ) − ξ ˜ t ˜ c ( ˆ X ) | , since while creatingthe ξ ˜ t ˜ c ( ¯ X ) entries, we add to some entries of X that satisfy ξ ˜ t ˜ c ( X ) < ξ ˜ t ˜ c ( ˆ X ) and subtract from some entries of X that satisfy ξ ˜ t ˜ c ( X ) > ξ ˜ t ˜ c ( ˆ X ) . These lead to a contradiction to theassumption that X ∈ M , which completes the proof. (cid:3) A proof of this fact is as follows. Because (cid:80) ˜ t ∈T ξ ˜ tc i ( X ) < q c i , (cid:80) ˜ t ∈T ξ ˜ tc i ( ¯ X ) = (cid:80) ˜ t ∈T ξ ˜ tc i ( X )+1 ≤ q c i . Forevery ˜ c (cid:54) = c i , (cid:80) ˜ t ∈T ξ ˜ t ˜ c ( ¯ X ) ≤ (cid:80) ˜ t ∈T ξ ˜ t ˜ c ( X ) ≤ q ˜ c . Thus, all school capacities are satisfied. For all ˜ c, ˜ t , ξ ˜ t ˜ c ( ¯ X ) ≤ max { ξ ˜ t ˜ c ( X ) , ξ ˜ t ˜ c ( ˆ X ) } ≤ q ˜ t ˜ c by construction, so all type-specific ceilings are satisfied. And (cid:80) ˜ t ∈T , ˜ c ∈C ξ ˜ t ˜ c ( ¯ X ) = (cid:80) ˜ t ∈T ξ ˜ t ˜ c ( X ) by definition of ¯ X , so ¯ X is a legitimate matching. Finally, ξ ˜ t ˜ d ( ¯ X ) = ξ ˜ t ˜ d ( X ) for every ˜ t and ˜ d , so ¯ X ∈ M . NTERDISTRICT SCHOOL CHOICE 49
Now we are ready to prove the theorem. The “if” part follows from Lemmas 2 and 3.Specifically, by Lemma 2, SPDA produces a legitimate matching. Therefore, by Lemma 3,we have ˆ p td ≤ ξ td ( X ) ≤ ˆ q td for every t ∈ T and d ∈ D . For each school district d , hence, themaximum proportion of type- t students that can be admitted is ˆ q td /k d and the minimumproportion of type t students that can be admitted is ˆ p td /k d . Therefore, the ratio differenceof type- t students in any two districts is at most max d (cid:54) = d (cid:48) { ˆ q td /k d − ˆ p td (cid:48) /k d (cid:48) } . We conclude that the α -diversity policy is achieved when ˆ q td /k d − ˆ p td (cid:48) /k d (cid:48) ≤ α for every t , d , and d (cid:48) with d (cid:54) = d (cid:48) .The “only if” part of the theorem follows from Lemma 4. Suppose that ˆ q td /k d − ˆ p td (cid:48) /k d (cid:48) > α for some t , d , and d (cid:48) with d (cid:54) = d (cid:48) . From Lemma 4, we know the existence of a legitimatematching X such that ξ td ( X ) = ˆ q td and ξ td (cid:48) ( X ) = ˆ p td (cid:48) . Consider a student preference profilewhere each student prefers her contract in X the most. Then, since the admissions rulesare weakly acceptant, SPDA ends at the first step as all applications are accepted. Thus X is the outcome of SPDA and, therefore, the α -diversity policy is not satisfied. (cid:3) Proof of Theorem 5.
Suzuki et al. (2017) study a setting in which each student is initiallymatched with a school and there are no constraints associated with student types, that is,when there is just one type. In that setting, they show that if the distribution is M-convex,then their mechanism, called TTC-M, satisfies the policy goal, constrained efficiency, indi-vidual rationality, and strategy-proofness. To adapt their result to our setting, consider thehypothetical matching problem that we have introduced before the definition of TTC inwhich each student is matched with a school-type pair and each student has strict prefer-ences over all school-type pairs. It is straightforward to verify that this hypothetical prob-lem satisfies all the conditions assumed by Suzuki et al. (2017). In particular, M-convexityof Ξ ∩ Ξ holds by assumption. Therefore, TTC-M in this market satisfies the policy goal,constrained efficiency, individual rationality, and strategy-proofness.We note that the outcome of our TTC is isomorphic to the outcome of TTC-M in thehypothetical problem in the following sense. Student s is allocated to contract ( s, c ) underpreference profile P = ( P s ) s ∈S at the outcome of TTC if, and only if, student s is allocatedto the school-type pair ( c, t ) under preference profile ˜ P = ( ˜ P s ) s ∈S at TTC-M in the hypo-thetical problem. The rest of the proof is devoted to showing that our TTC satisfies thedesired properties in the original problem.The result that TTC satisfies the policy goal follows from the result in Suzuki et al. (2017)that the distribution corresponding to the TTC-M outcome is in Ξ ∩ Ξ .To show constrained efficiency, let X be the outcome of TTC and, for each student s ∈ S ,let ( s, c s ) be the contract associated with student s at matching X . Suppose, for contradic-tion, that there exists a feasible matching X (cid:48) with ξ ( X (cid:48) ) ∈ Ξ that Pareto dominates match-ing X . Denoting X (cid:48) s = ( s, c (cid:48) s ) for each student s ∈ S , this implies ( s, c (cid:48) s ) R s ( s, c s ) for every student s ∈ S , with at least one relation being strict. Then, by the construction of pref-erences ˜ R s in the hypothetical problem, we have ( c (cid:48) s , τ ( s )) ˜ R s ( c s , τ ( s )) for every student s ∈ S , with at least one relation being strict. Moreover, because matching X (cid:48) is feasible inthe original problem, Y (cid:48) = { ( c (cid:48) s , τ ( s )) | ( s, c (cid:48) s ) ∈ X (cid:48) } is feasible in the hypothetical problem,and Y = { ( c s , τ ( s )) | ( s, c s ) ∈ X } is the result of TTC-M. This is a contradiction to the resultin Suzuki et al. (2017) that TTC-M is constrained efficient.To show individual rationality, let matching X be the outcome of TTC and, for eachstudent s ∈ S , let X s = ( s, c s ) be the contract associated with student s at matching X .Additionally, let Y = { ( c s , τ ( s )) | ( s, c s ) ∈ X } be the result of TTC-M in the hypotheticalproblem. Suzuki et al. (2017) establish that TTC-M is individually rational, so ( c s , τ ( s )) ˜ R s ( c ( s ) , τ ( s )) for every s ∈ S , where c ( s ) denotes the initial school of student s . By theconstruction of ˜ R s , this relation implies ( s, c s ) R s ( s, c ( s )) for every student s ∈ S , whichmeans X is individually rational in the original problem.To show strategy-proofness, in the original problem, let s be a student, t her type, P − s thepreference profile of students other than student s , P s the true preference of student s , and P (cid:48) s a misreported preference of student s . Furthermore, let c and c (cid:48) be schools assigned tostudent s under ( P s , P − s ) and ( P (cid:48) s , P − s ) for TTC, respectively. Note that the previous argu-ment establishes that, in the hypothetical problem, student s is allocated to ( c, t ) and ( c (cid:48) , t ) under ( ˜ P s , ˜ P − s ) and ( ˜ P (cid:48) s , ˜ P (cid:48)− s ) , respectively. Because TTC-M is strategy-proof, it followsthat ( c, t ) ˜ P s ( c (cid:48) , t ) or c = c (cid:48) . By the construction of ˜ P s , this relation implies ( s, c ) P s ( s, c (cid:48) ) or ( s, c ) = ( s, c (cid:48) ) , establishing strategy-proofness of TTC in the original problem. (cid:3) Proof of Corollary 1.
Assume that (cid:80) t ξ tc ≤ q c for every ξ ∈ Ξ and c ∈ C . Under this pre-sumption, we show that when the policy goal Ξ is M-convex, so is Ξ ∩ Ξ . Then the resultfollows immediately from Theorem 5 because the initial matching satisfies Ξ .Suppose that ξ, ˜ ξ ∈ Ξ ∩ Ξ such that ξ tc > ˜ ξ tc for some school c and type t . Since Ξ is M-convex and ξ, ˜ ξ ∈ Ξ , there exist school c (cid:48) and type t (cid:48) with ξ t (cid:48) c (cid:48) < ˜ ξ t (cid:48) c (cid:48) such that ˆ ξ ≡ ξ − χ c,t + χ c (cid:48) ,t (cid:48) ∈ Ξ and ¯ ξ ≡ ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ∈ Ξ . We will show that ˆ ξ, ¯ ξ ∈ Ξ .Because ξ ∈ Ξ , (cid:80) ˜ c ∈C , ˜ t ∈T ˆ ξ ˜ t ˜ c = (cid:80) ˜ c ∈C , ˜ t ∈T ξ ˜ t ˜ c = (cid:80) d k d . Furthermore, for every ˜ c, ˜ t , bydefinition of ˆ ξ , we have ˆ ξ ˜ t ˜ c ≤ max { ξ ˜ t ˜ c , ˜ ξ ˜ t ˜ c } ≤ q ˜ t ˜ c . These two properties imply that ˆ ξ ∈ Ξ . Asimilar argument shows ¯ ξ ∈ Ξ . (cid:3) Proof of Lemma 1.
Note that Ξ( f, λ ) ∩ Ξ = { ξ ∈ Ξ | f ( ξ ) ≥ λ } . The “if” direction:
Suppose that ξ ∈ Ξ( f, λ ) ∩ Ξ and ˜ ξ ∈ Ξ( f, λ ) ∩ Ξ are distinct. There-fore, f ( ξ ) , f ( ˜ ξ ) ≥ λ . By assumption, there exist ( c, t ) and ( c (cid:48) , t (cid:48) ) with ξ tc > ˜ ξ tc and ξ t (cid:48) c (cid:48) < ˜ ξ t (cid:48) c (cid:48) such that min { f ( ξ − χ c,t + χ c (cid:48) ,t (cid:48) ) , f ( ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ) } ≥ min { f ( ξ ) , f ( ˜ ξ ) } . NTERDISTRICT SCHOOL CHOICE 51
This implies f ( ξ − χ c,t + χ c (cid:48) ,t (cid:48) ) , f ( ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ) ≥ λ . Furthermore, ξ − χ c,t + χ c (cid:48) ,t (cid:48) , ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ∈ Ξ since the sum of coordinates is equal to (cid:80) d k d and no school is assignedmore students than its capacity. Therefore, ξ − χ c,t + χ c (cid:48) ,t (cid:48) , ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ∈ Ξ( f, λ ) ∩ Ξ , so,by Theorem 4.3. of Murota (2003), Ξ( f, λ ) ∩ Ξ is M-convex. The “only if” direction:
Suppose that the function f is not pseudo M-concave, so thatthere exist distinct ξ, ˜ ξ ∈ Ξ such that for all ( c, t ) and ( c (cid:48) , t (cid:48) ) with ξ tc > ˜ ξ tc and ξ t (cid:48) c (cid:48) < ˜ ξ t (cid:48) c (cid:48) wehave min { f ( ξ − χ c,t + χ c (cid:48) ,t (cid:48) ) , f ( ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ) } < min { f ( ξ ) , f ( ˜ ξ ) } .Let λ ≡ min { f ( ξ ) , f ( ˜ ξ ) } . The above condition implies that Ξ( f, λ ) ∩ Ξ is not M-convex. (cid:3) Proof of Theorem 7.
Let f ( ξ ) = 1 when ξ ∈ Ξ ∩ Ξ and f ( ξ ) = 0 otherwise.First we show that f is pseudo M-concave. Take two distinct ξ, ˜ ξ ∈ Ξ . If min { f ( ξ ) , f ( ˜ ξ ) } = 0 , then min { f ( ξ − χ c,t + χ c (cid:48) ,t (cid:48) ) , f ( ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ) } ≥ for every ( c, t ) and ( c (cid:48) , t (cid:48) ) , so the desired inequality holds. Suppose that f ( ξ ) = f ( ˜ ξ ) = 1 . By theconstruction of f , we have ξ, ˜ ξ ∈ Ξ ∩ Ξ . Since Ξ ∩ Ξ is M-convex, there exist ( c, t ) and ( c (cid:48) , t (cid:48) ) such that ξ − χ c,t + χ c (cid:48) ,t (cid:48) , ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ∈ Ξ ∩ Ξ . By the construction of f , f ( ξ − χ c,t + χ c (cid:48) ,t (cid:48) ) = f ( ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ) = 1 . Therefore, the desired inequality also holds forthis case, so f is pseudo M-concave.Next we show that Ξ( f, λ ) ∩ Ξ = Ξ ∩ Ξ for λ = 1 . For any ξ ∈ Ξ( f, ∩ Ξ , f ( ξ ) = 1 ,which implies that ξ ∈ Ξ ∩ Ξ by the construction of f . Therefore, Ξ( f, ∩ Ξ ⊆ Ξ ∩ Ξ .Now, let ξ ∈ Ξ ∩ Ξ . Then, by the construction of f , f ( ξ ) = 1 , so ξ ∈ Ξ( f, ∩ Ξ . Therefore, Ξ ∩ Ξ ⊆ Ξ( f, ∩ Ξ . We conclude that Ξ( f, ∩ Ξ = Ξ ∩ Ξ . (cid:3) Proof of Corollary 2.
Suppose that Ξ is a school-level diversity policy. We will first showthat Ξ ∩ Ξ is an M-convex set. Recall that Ξ = { ξ |∀ c, t q tc ≥ ξ tc ≥ p tc } and Ξ = { ξ | (cid:80) c,t ξ tc = (cid:80) d k d and ∀ c q c ≥ (cid:80) t ξ tc } .Suppose that there exist ξ, ˜ ξ ∈ Ξ ∩ Ξ such that ξ tc > ˜ ξ tc . To show M-convexity, we willfind school c (cid:48) and type t (cid:48) with ξ t (cid:48) c (cid:48) < ˜ ξ t (cid:48) c (cid:48) such that (1) ˆ ξ ≡ ξ − χ c,t + χ c (cid:48) ,t (cid:48) ∈ Ξ ∩ Ξ and(2) ¯ ξ ≡ ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ∈ Ξ ∩ Ξ . To show both conditions, we look at two possible casesdepending on whether c (cid:48) = c or not. Case 1:
First consider the case in which there exists type t (cid:48) such that ξ t (cid:48) c < ˜ ξ t (cid:48) c . We prove(1) for c (cid:48) = c . First, by definition of ˆ ξ , we have (cid:80) ˜ t ∈T , ˜ c ∈C ˆ ξ ˜ t ˜ c = (cid:80) ˜ t ∈T , ˜ c ∈C ξ ˜ t ˜ c = (cid:80) d k d . Next,since (cid:80) ˜ t ∈T ˆ ξ ˜ tc = (cid:80) ˜ t ∈T ξ ˜ tc , we have (cid:80) ˜ t ∈T ˆ ξ ˜ tc ≤ q c . Therefore, ˆ ξ ∈ Ξ .Next, we have ˆ ξ tc = ξ tc − ≥ ˜ ξ tc ≥ p tc (the equality comes from the definition of ˆ ξ , thefirst inequality comes from the assumption ξ tc > ˜ ξ tc , and the second inequality comes fromthe assumption ˜ ξ ∈ Ξ ), and ˆ ξ tc = ξ tc − < ξ tc ≤ q tc (the equality comes from the definitionof ˆ ξ , the first inequality is obvious, and the second inequality comes from the assumption ξ ∈ Ξ ). Moreover, we have ˆ ξ t (cid:48) c = ξ t (cid:48) c + 1 > ξ t (cid:48) c ≥ p t (cid:48) c (the equality comes from the definitionof ˆ ξ , the first inequality is obvious, and the second inequality comes from the assumption ξ ∈ Ξ ), and ˆ ξ t (cid:48) c = ξ t (cid:48) c + 1 ≤ ˜ ξ t (cid:48) c ≤ q t (cid:48) c (the equality comes from the definition of ˆ ξ , the firstinequality comes from the assumption ξ t (cid:48) c < ˜ ξ t (cid:48) c , and the second inequality comes from theassumption ˜ ξ ∈ Ξ ). For any ˜ c, ˜ t with (˜ c, ˜ t ) (cid:54)∈ { ( c, t ) , ( c, t (cid:48) ) } , we have ˆ ξ ˜ t ˜ c = ξ ˜ t ˜ c by definition of ˆ ξ , so p ˜ t ˜ c ≤ ˆ ξ ˜ t ˜ c ≤ q ˜ t ˜ c . Therefore, ˆ ξ ∈ Ξ and hence we conclude (1).The proof that (1) is satisfied follows from the facts that ξ tc > ˜ ξ tc and ξ t (cid:48) c < ˜ ξ t (cid:48) c . By changingthe roles of t with t (cid:48) and ξ with ˜ ξ in the preceding argument, we get the implication of (1)that ¯ ξ ∈ Ξ ∩ Ξ . But this is exactly (2). Case 2:
Second, consider the case in which there exists no type t (cid:48) such that ξ t (cid:48) c < ˜ ξ t (cid:48) c .Then, ξ t (cid:48) c ≥ ˜ ξ t (cid:48) c for every t (cid:48) (cid:54) = t . This in particular implies (cid:80) ˜ t ∈T ξ ˜ tc > (cid:80) ˜ t ∈T ˜ ξ ˜ tc . Because (cid:80) ˜ t ∈T , ˜ c ∈C ξ ˜ t ˜ c = (cid:80) ˜ t ∈T , ˜ c ∈C ˜ ξ ˜ t ˜ c by the assumption that ξ, ˜ ξ ∈ Ξ , there exists a school c (cid:48) (cid:54) = c suchthat (cid:80) ˜ t ∈T ξ ˜ tc (cid:48) < (cid:80) ˜ t ∈T ˜ ξ ˜ tc (cid:48) . In particular, there exists a type t (cid:48) such that ˜ ξ t (cid:48) c (cid:48) > ξ t (cid:48) c (cid:48) .Now we proceed to show condition (1) for this case. To do so first note that, by definitionof ˆ ξ , we have (cid:80) ˜ t ∈T , ˜ c ∈C ˆ ξ ˜ t ˜ c = (cid:80) ˜ t ∈T , ˜ c ∈C ξ ˜ t ˜ c . In addition, the relation (cid:80) ˜ t ∈T ξ ˜ tc > (cid:80) ˜ t ∈T ˜ ξ ˜ tc = (cid:80) ˜ t ∈T ˆ ξ ˜ tc and the assumption ξ ∈ Ξ imply that (cid:80) ˜ t ∈T ˆ ξ ˜ tc ≤ q c . Likewise, (cid:80) ˜ t ∈T ˆ ξ ˜ tc (cid:48) = (cid:80) ˜ t ∈T ξ ˜ tc (cid:48) +1 ≤ (cid:80) ˜ t ∈T ˜ ξ ˜ tc (cid:48) ≤ q c (cid:48) . Finally, for any ˜ c, ˜ t with (˜ c, ˜ t ) (cid:54)∈ { ( c, t ) , ( c (cid:48) , t (cid:48) ) } , we have ˆ ξ ˜ t ˜ c = ξ ˜ t ˜ c bydefinition of ˆ ξ , so (cid:80) ˜ t ∈T ˆ ξ ˜ t ˜ c ≤ q ˜ c for every ˜ c (cid:54) = c, c (cid:48) . Thus, ˆ ξ ∈ Ξ .Next, ˆ ξ tc = ξ tc − ≥ ˜ ξ tc ≥ p tc (the first inequality follows from the assumption ξ tc > ˜ ξ tc and the second from ˜ ξ ∈ Ξ ), and ˆ ξ tc = ξ tc − < ξ tc ≤ q tc (the first inequality is obvious andthe second inequality follows from ξ ∈ Ξ ). Moreover, ˆ ξ t (cid:48) c (cid:48) = ξ t (cid:48) c (cid:48) + 1 > ξ t (cid:48) c (cid:48) ≥ p t (cid:48) c (cid:48) (the firstinequality is obvious and the second follows from ξ ∈ Ξ ), and ˆ ξ t (cid:48) c (cid:48) = ξ t (cid:48) c (cid:48) + 1 ≤ ˜ ξ t (cid:48) c (cid:48) ≤ q t (cid:48) c (cid:48) (thefirst inequality follows from ξ t (cid:48) c (cid:48) < ˜ ξ t (cid:48) c (cid:48) and the second follows from ˜ ξ ∈ Ξ ). For any ˜ c, ˜ t with (˜ c, ˜ t ) (cid:54)∈ { ( c, t ) , ( c (cid:48) , t (cid:48) ) } , we have ˆ ξ ˜ t ˜ c = ξ ˜ t ˜ c by definition of ˆ ξ , so p ˜ t ˜ c ≤ ˆ ξ ˜ t ˜ c ≤ q ˜ t ˜ c . Therefore, ˆ ξ ∈ Ξ and hence we conclude (1).The proof that (1) is satisfied follows from the facts that ξ tc > ˜ ξ tc , ˜ ξ t (cid:48) c (cid:48) > ξ t (cid:48) c (cid:48) , there are morestudents assigned to school c at ξ than ˜ ξ , and there are more students assigned to school c (cid:48) at ˜ ξ than ξ . If we change the roles of ξ with ˜ ξ , c with c (cid:48) , and t with t (cid:48) , then (1) would imply ¯ ξ ∈ Ξ ∩ Ξ . But this is exactly (2). Therefore, Ξ ∩ Ξ is an M-convex set.The desired conclusion then follows from the fact that Ξ ∩ Ξ is an M-convex set andTheorem 5. (cid:3) Proof of Corollary 3.
We show that f is pseudo M-concave. Let ξ, ˜ ξ ∈ Ξ be distinct.Then U ≡ { ( c, t ) | ξ ct > ˜ ξ ct } is a nonempty set. Partition this set into three subsets: U ≡ { ( c, t ) | ˆ ξ ct ≥ ξ ct > ˜ ξ ct } , U ≡ { ( c, t ) | ξ ct > ˆ ξ ct > ˜ ξ ct } , and U ≡ { ( c, t ) | ξ ct > ˜ ξ ct ≥ ˆ ξ ct } . Like-wise V ≡ { ( c (cid:48) , t (cid:48) ) | ˜ ξ c (cid:48) t (cid:48) > ξ c (cid:48) t (cid:48) } is a nonempty set that can be partitioned into three subsets: NTERDISTRICT SCHOOL CHOICE 53 V ≡ { ( c (cid:48) , t (cid:48) ) | ˆ ξ c (cid:48) t (cid:48) ≥ ˜ ξ c (cid:48) t (cid:48) > ξ c (cid:48) t (cid:48) } , V ≡ { ( c (cid:48) , t (cid:48) ) | ˜ ξ c (cid:48) t (cid:48) > ˆ ξ c (cid:48) t (cid:48) > ξ c (cid:48) t (cid:48) } , and V ≡ { ( c (cid:48) , t (cid:48) ) | ˜ ξ c (cid:48) t (cid:48) > ξ c (cid:48) t (cid:48) ≥ ˆ ξ c (cid:48) t (cid:48) } .We consider several cases. Case 1: U is nonempty. There exists ( c, t ) such that ξ ct > ˆ ξ ct > ˜ ξ ct . Since ξ, ˜ ξ ∈ Ξ , thereexists ( c (cid:48) , t (cid:48) ) such that ξ c (cid:48) t (cid:48) < ˜ ξ c (cid:48) t (cid:48) . In this case, f ( ξ − χ c,t + χ c (cid:48) ,t (cid:48) ) ≥ f ( ξ ) and f ( ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ) ≥ f ( ˜ ξ ) by definition of f . Therefore, min { f ( ξ − χ c,t + χ c (cid:48) ,t (cid:48) ) , f ( ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ) } ≥ min { f ( ξ ) , f ( ˜ ξ ) } . Case 2: V is nonempty. There exists ( c (cid:48) , t (cid:48) ) such that ˜ ξ c (cid:48) t (cid:48) > ˆ ξ c (cid:48) t (cid:48) > ξ c (cid:48) t (cid:48) . The proof of this caseis similar to the proof of Case 1. Case 3: U and V are nonempty. There exist ( c, t ) and ( c (cid:48) , t (cid:48) ) such that ˆ ξ ct ≥ ξ ct > ˜ ξ ct and ˆ ξ c (cid:48) t (cid:48) ≥ ˜ ξ c (cid:48) t (cid:48) > ξ c (cid:48) t (cid:48) . In this case, f ( ξ − χ c,t + χ c (cid:48) ,t (cid:48) ) = f ( ξ ) and f ( ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ) = f ( ˜ ξ ) bydefinition of f . Then, min { f ( ξ − χ c,t + χ c (cid:48) ,t (cid:48) ) , f ( ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ) } = min { f ( ξ ) , f ( ˜ ξ ) } . Case 4: U and V are nonempty. There exist ( c, t ) and ( c (cid:48) , t (cid:48) ) such that ξ ct > ˜ ξ ct ≥ ˆ ξ ct and ˜ ξ c (cid:48) t (cid:48) > ξ c (cid:48) t (cid:48) ≥ ˆ ξ c (cid:48) t (cid:48) . The proof is similar to the proof of Case 3. Case 5: U = U and V = V . For every ( c, t ) such that ξ ct > ˜ ξ ct , we have ˆ ξ ct ≥ ξ ct > ˜ ξ ct , andfor every ( c (cid:48) , t (cid:48) ) such that ˜ ξ c (cid:48) t (cid:48) > ξ c (cid:48) t (cid:48) , we have ˜ ξ c (cid:48) t (cid:48) > ξ c (cid:48) t (cid:48) ≥ ˆ ξ c (cid:48) t (cid:48) . In this case, f ( ˜ ξ ) + 2 ≤ f ( ξ ) ,so min { f ( ˜ ξ ) , f ( ξ ) } = f ( ˜ ξ ) . Furthermore, for any choice of ( c, t ) and ( c (cid:48) , t (cid:48) ) such that ξ tc > ˜ ξ tc and ξ t (cid:48) c (cid:48) < ˜ ξ t (cid:48) c (cid:48) , we have f ( ξ − χ c,t + χ c (cid:48) ,t (cid:48) ) = f ( ξ ) − and f ( ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ) = f ( ˜ ξ ) + 2 .Since f ( ξ ) − ≥ f ( ˜ ξ ) and f ( ˜ ξ ) + 2 > f ( ˜ ξ ) , we get the desired conclusion that min { f ( ξ − χ c,t + χ c (cid:48) ,t (cid:48) ) , f ( ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ) } ≥ f ( ˜ ξ ) = min { f ( ξ ) , f ( ˜ ξ ) } . Case 6: U = U and V = V . For every ( c, t ) such that ξ ct > ˜ ξ ct , we have ξ ct > ˜ ξ ct ≥ ˆ ξ ct , andfor every ( c (cid:48) , t (cid:48) ) such that ˜ ξ c (cid:48) t (cid:48) > ξ c (cid:48) t (cid:48) , we have ˆ ξ c (cid:48) t (cid:48) ≥ ˜ ξ c (cid:48) t (cid:48) > ξ c (cid:48) t (cid:48) . The proof is similar to the proofof Case 5.We have considered all the possible cases: If U or V are nonempty, then we are doneby Cases 1 and 2, respectively. Suppose that they are both empty. Therefore, U = U ∪ U and V = V ∪ V are both nonempty. If U and V are nonempty, then we are done by Case3. If U and V are nonempty, then we are done by Case 4. If U and V are nonempty andone of U or V is nonempty, then we are done by Cases 3 or 4. Otherwise, if U and V areempty when U and V are nonempty, then U = U and V = V , which is covered by Case5. If U and V are nonempty and one of U or V is nonempty, then we are done by Cases3 or 4. Otherwise, if U and V are empty when U and V are nonempty, then U = U and V = V , which is covered by Case 6.We conclude that f is pseudo M-concave since in all possible cases we derive the desiredinequality. Then the proof follows from Theorem 6. (cid:3) Proof of Corollary 4.
Let the balanced-exchange policy be denoted by Ξ . We first show that Ξ ∩ Ξ is M-convex.Suppose that there exist ξ, ˜ ξ ∈ Ξ ∩ Ξ such that ξ tc > ˜ ξ tc . To show M-convexity, we needto find school c (cid:48) and type t (cid:48) with ξ t (cid:48) c (cid:48) < ˜ ξ t (cid:48) c (cid:48) such that (1) ˆ ξ ≡ ξ − χ c,t + χ c (cid:48) ,t (cid:48) ∈ Ξ ∩ Ξ and (2) ¯ ξ ≡ ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ∈ Ξ ∩ Ξ .If there exists t (cid:48) such that ˜ ξ t (cid:48) c > ξ t (cid:48) c , then (cid:80) ˜ t ∈T ˆ ξ ˜ td = (cid:80) ˜ t ∈T ¯ ξ ˜ td = (cid:80) ˜ t ∈T ξ ˜ td = k d for every d and (cid:80) ˜ t ∈T ˆ ξ ˜ tc = (cid:80) ˜ t ∈T ¯ ξ ˜ tc = (cid:80) ˜ t ∈T ξ ˜ tc ≤ q c for every c ∈ C , so both (1) and (2) are satisfied.Now suppose ˜ ξ t (cid:48) c ≤ ξ t (cid:48) c for every type t (cid:48) (cid:54) = t . Therefore, (cid:80) ˜ t ∈T ˜ ξ ˜ tc < (cid:80) ˜ t ∈T ξ ˜ tc . Because (cid:80) ˜ t ∈T , ˜ c : d (˜ c )= d ˜ ξ ˜ t ˜ c = (cid:80) ˜ t ∈T , ˜ c : d (˜ c )= d ξ ˜ t ˜ c , where d ≡ d ( c ) , there exists another school c (cid:48) in district d such that (cid:80) ˜ t ∈T ˜ ξ ˜ tc (cid:48) > (cid:80) ˜ t ∈T ξ ˜ tc (cid:48) . In particular, there exists a type t (cid:48) such that ˜ ξ t (cid:48) c (cid:48) > ξ t (cid:48) c (cid:48) .We first show (1). To do so, first note that since both schools c and c (cid:48) are in district d , (cid:80) ˜ t ∈T , ˜ c : d (˜ c )= d ˆ ξ ˜ t ˜ c = (cid:80) ˜ t ∈T , ˜ c : d (˜ c )= d ξ ˜ t ˜ c = k d . Moreover, for any ˜ d (cid:54) = d , (cid:80) ˜ t ∈T , ˜ c : d (˜ c )= ˜ d ˆ ξ ˜ t ˜ c = (cid:80) ˜ t ∈T , ˜ c : d (˜ c )= ˜ d ξ ˜ t ˜ c = k ˜ d because ˆ ξ ˜ t ˜ c = ξ ˜ t ˜ c for any ˜ t and ˜ c with d (˜ c ) = ˜ d by definition of ˆ ξ . Thus, ˆ ξ ∈ Ξ . Next we show ˆ ξ ∈ Ξ . To do so, first observe that (cid:80) ˜ t ∈T ˜ ξ ˜ tc = (cid:80) ˜ t ∈T ξ ˜ tc − < q c .Moreover, (cid:80) ˜ t ∈T ˆ ξ ˜ tc (cid:48) = (cid:80) ˜ t ∈T ξ ˜ tc (cid:48) + 1 ≤ (cid:80) ˜ t ∈T ˜ ξ ˜ tc (cid:48) ≤ q c (cid:48) . Furthermore, for any ˜ c (cid:54) = c, c (cid:48) , (cid:80) ˜ t ∈T ˆ ξ ˜ t ˜ c = (cid:80) ˜ t ∈T ξ ˜ t ˜ c ≤ q ˜ c . Therefore, ˆ ξ ∈ Ξ and hence (1) holds.Note that the above argument relies on the facts ξ tc > ˜ ξ tc , ξ t (cid:48) c (cid:48) < ˜ ξ t (cid:48) c (cid:48) , and d ( c ) = d ( c (cid:48) ) . If weswitch the roles of c with c (cid:48) and ξ with ˜ ξ , the implication of (1) is (2).The result then follows from Theorem 5 because Ξ ∩ Ξ is M-convex and the initial match-ing trivially satisfies the balanced-exchange policy. (cid:3) Proof of Corollary 5.
The proof is very similar to those of Corollary 2 and Corollary 4.We first show that Ξ ∩ Ξ is an M-convex set. Recall that Ξ = { ξ |∀ c, t q tc ≥ ξ tc ≥ p tc and ∀ d (cid:80) t ξ td = k d } and Ξ = { ξ | (cid:80) c,t ξ tc = (cid:80) d k d and ∀ c q c ≥ (cid:80) t ξ tc } .Suppose that there exist ξ, ˜ ξ ∈ Ξ ∩ Ξ such that ξ tc > ˜ ξ tc . To show M-convexity, we needto find school c (cid:48) and type t (cid:48) with ξ t (cid:48) c (cid:48) < ˜ ξ t (cid:48) c (cid:48) such that (1) ˆ ξ ≡ ξ − χ c,t + χ c (cid:48) ,t (cid:48) ∈ Ξ ∩ Ξ and (2) ¯ ξ ≡ ˜ ξ + χ c,t − χ c (cid:48) ,t (cid:48) ∈ Ξ ∩ Ξ . Let d ≡ d ( c ) . To show both conditions, we look at two possiblecases depending on whether c (cid:48) = c or not. Case 1:
First consider the case in which there exists type t (cid:48) such that ξ t (cid:48) c < ˜ ξ t (cid:48) c . We prove(1) for c (cid:48) = c . First, by definition of ˆ ξ , we have (cid:80) ˜ t ∈T , ˜ c ∈C ˆ ξ ˜ t ˜ c = (cid:80) ˜ t ∈T , ˜ c ∈C ξ ˜ t ˜ c = (cid:80) d k d . Next,since (cid:80) ˜ t ∈T ˆ ξ ˜ tc = (cid:80) ˜ t ∈T ξ ˜ tc , we have (cid:80) ˜ t ∈T ˆ ξ ˜ tc ≤ q c . Therefore, ˆ ξ ∈ Ξ .Next, we have ˆ ξ tc = ξ tc − ≥ ˜ ξ tc ≥ p tc (the equality comes from the definition of ˆ ξ , thefirst inequality comes from the assumption ξ tc > ˜ ξ tc , and the second inequality comes fromthe assumption ˜ ξ ∈ Ξ ), and ˆ ξ tc = ξ tc − < ξ tc ≤ q tc (the equality comes from the definitionof ˆ ξ , the first inequality is obvious, and the second inequality comes from the assumption ξ ∈ Ξ ). Moreover, we have ˆ ξ t (cid:48) c = ξ t (cid:48) c + 1 > ξ t (cid:48) c ≥ p t (cid:48) c (the equality comes from the definitionof ˆ ξ , the first inequality is obvious, and the second inequality comes from the assumption NTERDISTRICT SCHOOL CHOICE 55 ξ ∈ Ξ ), and ˆ ξ t (cid:48) c = ξ t (cid:48) c + 1 ≤ ˜ ξ t (cid:48) c ≤ q t (cid:48) c (the equality comes from the definition of ˆ ξ , the firstinequality comes from the assumption ξ t (cid:48) c < ˜ ξ t (cid:48) c , and the second inequality comes from theassumption ˜ ξ ∈ Ξ ). For any ˜ c, ˜ t with (˜ c, ˜ t ) (cid:54)∈ { ( c, t ) , ( c, t (cid:48) ) } , we have ˆ ξ ˜ t ˜ c = ξ ˜ t ˜ c by definition of ˆ ξ , so p ˜ t ˜ c ≤ ˆ ξ ˜ t ˜ c ≤ q ˜ t ˜ c . Finally, (cid:80) ˜ t ∈T ˆ ξ ˜ td = (cid:80) ˜ t ∈T ¯ ξ ˜ td = (cid:80) ˜ t ∈T ξ ˜ td = k d for every d . Therefore, ˆ ξ ∈ Ξ and hence we conclude (1).The proof that (1) is satisfied follows from the facts that ξ tc > ˜ ξ tc and ξ t (cid:48) c < ˜ ξ t (cid:48) c . By changingthe roles of t with t (cid:48) and ξ with ˜ ξ in the preceding argument, we get the implication of (1)that ¯ ξ ∈ Ξ ∩ Ξ . But this is exactly (2). Case 2:
Second, consider the case in which there exists no type t (cid:48) such that ξ t (cid:48) c < ˜ ξ t (cid:48) c .Then, ξ t (cid:48) c ≥ ˜ ξ t (cid:48) c for every t (cid:48) (cid:54) = t . This in particular implies (cid:80) ˜ t ∈T ξ ˜ tc > (cid:80) ˜ t ∈T ˜ ξ ˜ tc . Because (cid:80) ˜ t ∈T , ˜ c : d (˜ c )= d ˜ ξ ˜ t ˜ c = (cid:80) ˜ t ∈T , ˜ c : d (˜ c )= d ξ ˜ t ˜ c , where d ≡ d ( c ) , there exists another school c (cid:48) in district d such that (cid:80) ˜ t ∈T ˜ ξ ˜ tc (cid:48) > (cid:80) ˜ t ∈T ξ ˜ tc (cid:48) . In particular, there exists a type t (cid:48) such that ˜ ξ t (cid:48) c (cid:48) > ξ t (cid:48) c (cid:48) .Now we proceed to show condition (1) for this case. To do so first note that, by definitionof ˆ ξ , we have (cid:80) ˜ t ∈T , ˜ c ∈C ˆ ξ ˜ t ˜ c = (cid:80) ˜ t ∈T , ˜ c ∈C ξ ˜ t ˜ c . In addition, the relation (cid:80) ˜ t ∈T ξ ˜ tc > (cid:80) ˜ t ∈T ˜ ξ ˜ tc = (cid:80) ˜ t ∈T ˆ ξ ˜ tc and the assumption ξ ∈ Ξ imply that (cid:80) ˜ t ∈T ˆ ξ ˜ tc ≤ q c . Likewise, (cid:80) ˜ t ∈T ˆ ξ ˜ tc (cid:48) = (cid:80) ˜ t ∈T ξ ˜ tc (cid:48) +1 ≤ (cid:80) ˜ t ∈T ˜ ξ ˜ tc (cid:48) ≤ q c (cid:48) . Finally, for any ˜ c, ˜ t with (˜ c, ˜ t ) (cid:54)∈ { ( c, t ) , ( c (cid:48) , t (cid:48) ) } , we have ˆ ξ ˜ t ˜ c = ξ ˜ t ˜ c bydefinition of ˆ ξ , so (cid:80) ˜ t ∈T ˆ ξ ˜ t ˜ c ≤ q ˜ c for every ˜ c (cid:54) = c, c (cid:48) . Thus, ˆ ξ ∈ Ξ .Next, ˆ ξ tc = ξ tc − ≥ ˜ ξ tc ≥ p tc (the first inequality follows from the assumption ξ tc > ˜ ξ tc and the second from ˜ ξ ∈ Ξ ), and ˆ ξ tc = ξ tc − < ξ tc ≤ q tc (the first inequality is obvious andthe second inequality follows from ξ ∈ Ξ ). Moreover, ˆ ξ t (cid:48) c (cid:48) = ξ t (cid:48) c (cid:48) + 1 > ξ t (cid:48) c (cid:48) ≥ p t (cid:48) c (cid:48) (the firstinequality is obvious and the second follows from ξ ∈ Ξ ), and ˆ ξ t (cid:48) c (cid:48) = ξ t (cid:48) c (cid:48) + 1 ≤ ˜ ξ t (cid:48) c (cid:48) ≤ q t (cid:48) c (cid:48) (the first inequality follows from ξ t (cid:48) c (cid:48) < ˜ ξ t (cid:48) c (cid:48) and the second follows from ˜ ξ ∈ Ξ ). For any ˜ c, ˜ t with (˜ c, ˜ t ) (cid:54)∈ { ( c, t ) , ( c (cid:48) , t (cid:48) ) } , we have ˆ ξ ˜ t ˜ c = ξ ˜ t ˜ c by definition of ˆ ξ , so p ˜ t ˜ c ≤ ˆ ξ ˜ t ˜ c ≤ q ˜ t ˜ c . Next,note that since both schools c and c (cid:48) are in district d , (cid:80) ˜ t ∈T , ˜ c : d (˜ c )= d ˆ ξ ˜ t ˜ c = (cid:80) ˜ t ∈T , ˜ c : d (˜ c )= d ξ ˜ t ˜ c = k d .Moreover, for any ˜ d (cid:54) = d , (cid:80) ˜ t ∈T , ˜ c : d (˜ c )= ˜ d ˆ ξ ˜ t ˜ c = (cid:80) ˜ t ∈T , ˜ c : d (˜ c )= ˜ d ξ ˜ t ˜ c = k ˜ d because ˆ ξ ˜ t ˜ c = ξ ˜ t ˜ c for any ˜ t and ˜ c with d (˜ c ) = ˜ d by definition of ˆ ξ . Therefore, ˆ ξ ∈ Ξ and hence we conclude (1).The proof that (1) is satisfied follows from the facts that ξ tc > ˜ ξ tc , ˜ ξ t (cid:48) c (cid:48) > ξ t (cid:48) c (cid:48) , there are morestudents assigned to school c at ξ than ˜ ξ , and there are more students assigned to school c (cid:48) at ˜ ξ than ξ . If we change the roles of ξ with ˜ ξ , c with c (cid:48) , and t with t (cid:48) , then (1) would imply ¯ ξ ∈ Ξ ∩ Ξ . But this is exactly (2). Therefore, Ξ ∩ Ξ is an M-convex set.The result then follows from Theorem 5 because Ξ ∩ Ξ is M-convex. (cid:3) Proof of Theorem 8.
Suppose that district admissions rules favor own students. Fix a stu-dent preference profile. Recall that under interdistrict school choice, students are as-signed to schools by SPDA, where each student ranks all contracts associated with her and each district d has the admissions rule Ch d . Under intradistrict school choice, stu-dents are assigned to schools by SPDA where students only rank the contracts asso-ciated with their home districts and each district d has the admissions rule Ch d . Wefirst show that the intradistrict SPDA outcome can be produced by SPDA when all dis-tricts participate simultaneously and students rank all contracts, including the ones as-sociated with the other districts, by modifying admissions rules for the districts. Let Ch (cid:48) d ( X ) ≡ Ch d ( { x ∈ X | d ( s ( x )) = d } ) be the modified admissions rule.In SPDA, if district admissions rules have completions that satisfy path independence,then SPDA outcomes are the same under the completions and the original admissionsrules because in SPDA a district always considers a set of proposals which is feasible forstudents. Furthermore, SPDA does not depend on the order of proposals when districtadmissions rules are path independent. As a result, SPDA does not depend on the orderof proposals when district admissions rules have completions that satisfy path indepen-dence. Therefore, the intradistrict SPDA outcome can be produced by SPDA when alldistricts participate simultaneously and students rank all contracts including the ones as-sociated with the other districts and each district d has the admissions rule Ch (cid:48) d . The reasonbehind this is that when each district d has admissions rule Ch (cid:48) d , a student is not admittedto a school district other than her home district. Furthermore, because Ch d favors ownstudents, the set of chosen students under Ch (cid:48) d is the same as that under Ch d for any setof contracts of the form { x ∈ X | d ( s ( x )) = d } for any set X .We next show that Ch (cid:48) d has a path-independent completion. By assumption, for everydistrict d , there exists a path-independent completion (cid:102) Ch d of Ch d . Let (cid:102) Ch (cid:48) d ( X ) ≡ (cid:102) Ch d ( { x ∈ X | d ( s ( x )) = d } ) . We show that (cid:102) Ch (cid:48) d is a path-independent completion of Ch (cid:48) d . To showthat (cid:102) Ch (cid:48) d ( X ) is a completion, consider a set X such that (cid:102) Ch (cid:48) d ( X ) is feasible for students.Let X ∗ ≡ { x ∈ X | d ( s ( x )) = d } . Then we have the following: (cid:102) Ch (cid:48) d ( X ∗ ) = (cid:102) Ch d ( X ∗ ) = Ch d ( X ∗ ) = Ch (cid:48) d ( X ∗ ) , where the first equality follows from the definition of (cid:102) Ch (cid:48) d , the second equality followsfrom the fact that (cid:102) Ch d is a completion of Ch d , and the third equality follows from thedefinition of Ch (cid:48) d . Furthermore, because (cid:102) Ch (cid:48) d ( X ) = (cid:102) Ch (cid:48) d ( X ∗ ) and Ch (cid:48) d ( X ∗ ) = Ch (cid:48) d ( X ) , weget (cid:102) Ch (cid:48) d ( X ) = Ch (cid:48) d ( X ) . Therefore, (cid:102) Ch (cid:48) d is a completion of Ch (cid:48) d .To show that (cid:102) Ch (cid:48) d is path independent, consider two sets of contracts X and Y . Let X ∗ ≡ { x ∈ X | d ( s ( x )) = d } and Y ∗ ≡ { x ∈ Y | d ( s ( x )) = d } . Then we have the following: NTERDISTRICT SCHOOL CHOICE 57 (cid:102) Ch (cid:48) d ( X ∪ (cid:102) Ch (cid:48) d ( Y )) = (cid:102) Ch (cid:48) d ( X ∪ (cid:102) Ch d ( Y ∗ ))= (cid:102) Ch d ( X ∗ ∪ (cid:102) Ch d ( Y ∗ ))= (cid:102) Ch d ( X ∗ ∪ Y ∗ )= (cid:102) Ch (cid:48) d ( X ∪ Y ) , where the first and second equalities follow from the definition of (cid:102) Ch (cid:48) d , the third equalityfollows from path independence of (cid:102) Ch d , and the last equality follows from the definitionof (cid:102) Ch (cid:48) d . Therefore, (cid:102) Ch (cid:48) d is path independent.Because Ch d favors own students, we have Ch d ( X ) ⊇ Ch (cid:48) d ( X ) for every X that is feasiblefor students. Furthermore, for any such X , (cid:102) Ch d ( X ) = Ch d ( X ) and (cid:102) Ch (cid:48) d ( X ) = Ch (cid:48) d ( X ) because (cid:102) Ch d is a completion of Ch d and (cid:102) Ch (cid:48) d is a completion of Ch (cid:48) d , respectively. Therefore,for any X that is feasible for students, (cid:102) Ch d ( X ) ⊇ (cid:102) Ch (cid:48) d ( X ) . We use this result to show thefollowing lemma. Lemma 5.
Every student weakly prefers the interdistrict SPDA outcome under ( (cid:102) Ch d ) d ∈D to theinterdistrict SPDA outcome under ( (cid:102) Ch (cid:48) d ) d ∈D .Proof. Let µ be the interdistrict SPDA outcome under ( (cid:102) Ch d ) d ∈D and µ (cid:48) be the interdistrictSPDA outcome under ( (cid:102) Ch (cid:48) d ) d ∈D . If µ (cid:48) is stable under ( (cid:102) Ch d ) d ∈D , then the conclusion followsfrom the result that µ is the student-optimal stable matching under ( (cid:102) Ch d ) d ∈D because each (cid:102) Ch d is path independent (Chambers and Yenmez, 2017).Suppose that µ (cid:48) is not stable under ( (cid:102) Ch d ) d ∈D . Since µ (cid:48) is stable under ( (cid:102) Ch (cid:48) d ) d ∈D , (cid:102) Ch (cid:48) d ( µ (cid:48) d ) = µ (cid:48) d for every district d . Furthermore, µ (cid:48) d is feasible for students, so (cid:102) Ch d ( µ (cid:48) d ) ⊇ (cid:102) Ch (cid:48) d ( µ (cid:48) d ) = µ (cid:48) d . By definition of admissions rules, µ (cid:48) d ⊇ (cid:102) Ch d ( µ (cid:48) d ) , so (cid:102) Ch d ( µ (cid:48) d ) = µ (cid:48) d . Asa result, there must exist a blocking contract for matching µ (cid:48) so that it is not stable un-der ( (cid:102) Ch d ) d ∈D . Whenever there exists a blocking pair, we consider the following algorithmto improve student welfare. Let d be a district associated with a blocking contract. Set µ ≡ µ (cid:48) . Step n ( n ≥ ): Consider the following set of contracts associated with a district d n for which there exists an associated blocking contract: X nd n ≡ { x = ( s, d n , c ) | x P s µ n − s } . District d n accepts (cid:102) Ch d n ( µ n − d ∪ X nd n ) and rejects the rest of the contracts.Let µ nd n ≡ (cid:102) Ch d n ( µ n − d n ∪ X nd n ) and µ nd ≡ µ n − d \ Y n where Y n ≡ { x ∈ µ n − |∃ y ∈ µ nd n s.t. s ( x ) = s ( y ) } for d (cid:54) = d n . If there are no blocking contracts for matching µ n under ( (cid:102) Ch d ) d ∈D , then stop and return µ n , otherwise go to Step n + 1 .We show that district d n does not reject any contract in µ n − d n by mathematical inductionon n , i.e., µ nd n ⊇ µ n − d n for every n ≥ . Consider the base case for n = 1 . Recall that µ d = (cid:102) Ch d ( µ d ∪ X d ) = (cid:102) Ch d ( µ (cid:48) d ∪ X d ) . By construction, µ d is a feasible matching. We claimthat µ (cid:48) d ∪ µ d is feasible for students. Suppose, for contradiction, that it is not feasible forstudents. Then there exists a student s who has one contract in µ (cid:48) d and one in µ d \ µ (cid:48) d . Callthe latter contract z . By construction, z P s µ (cid:48) s , and by path independence, z ∈ (cid:102) Ch d ( µ (cid:48) d ∪{ z } ) . Furthermore, since student s is matched with district d in µ (cid:48) , d ( s ) = d . Therefore, (cid:102) Ch d ( µ (cid:48) d ∪ { z } ) = (cid:102) Ch (cid:48) d ( µ (cid:48) d ∪ { z } ) by definition of (cid:102) Ch (cid:48) d and construction of µ (cid:48) . Hence, z ∈ (cid:102) Ch (cid:48) d ( µ (cid:48) d ∪ { z } ) , which contradicts the fact that µ (cid:48) is stable under ( (cid:102) Ch (cid:48) d ) d ∈D . Hence, µ (cid:48) d ∪ µ d is feasible for students. Feasibility for students implies that (cid:102) Ch d ( µ (cid:48) d ∪ µ d ) ⊇ (cid:102) Ch (cid:48) d ( µ (cid:48) d ∪ µ d ) . Path independence and construction of µ d yield µ d = (cid:102) Ch d ( µ (cid:48) d ∪ µ d ) .Furthermore, there exists no student s , such that d ( s ) = d ,who has a contract in µ d \ µ (cid:48) ,as this would contradict stability of µ (cid:48) under ( (cid:102) Ch (cid:48) d ) d ∈D . This implies, by definition of (cid:102) Ch (cid:48) d ,that (cid:102) Ch (cid:48) d ( µ (cid:48) d ∪ µ d ) = (cid:102) Ch (cid:48) d ( µ (cid:48) d ) , and, by stability of µ (cid:48) under ( (cid:102) Ch (cid:48) d ) d ∈D , (cid:102) Ch (cid:48) d ( µ (cid:48) d ) = µ (cid:48) d .Therefore, µ d = (cid:102) Ch d ( µ (cid:48) d ∪ µ d ) ⊇ (cid:102) Ch (cid:48) d ( µ (cid:48) d ∪ µ d ) = µ (cid:48) d = µ d , which means that district d does not reject any contracts.Now consider district d n where n > . There are two cases to consider. First considerthe case when d n (cid:54) = d i for every i < n . In this case, µ n − d n ⊆ µ d n = µ (cid:48) d n . We repeatthe same arguments as in the previous paragraph. Stability of µ (cid:48) under ( (cid:102) Ch (cid:48) d ) d ∈D andpath independence of (cid:102) Ch (cid:48) d n implies that µ nd n ∪ µ n − d n is feasible for students. Therefore, (cid:102) Ch d n ( µ n − d n ∪ µ nd n ) ⊇ (cid:102) Ch (cid:48) d n ( µ n − d n ∪ µ nd n ) . Furthermore, there exists no student s , such that d ( s ) = d n , who has a contract in µ nd n \ µ n − d n . As a result, by definition of (cid:102) Ch (cid:48) d n and by pathindependence, (cid:102) Ch (cid:48) d n ( µ n − d n ∪ µ nd n ) = (cid:102) Ch (cid:48) d n ( µ n − d n ) = µ n − d n . As in the previous paragraph, weconclude that µ nd n = (cid:102) Ch d n ( µ n − d n ∪ µ nd n ) ⊇ (cid:102) Ch (cid:48) d n ( µ n − d n ∪ µ nd n ) = µ n − d n .The second case is when there exists i < n such that d i = d n . Let i ∗ be the last suchstep before n . Since student welfare improves at every step before n by the mathematicalinduction hypothesis, µ i ∗ − d n ∪ X i ∗ d n ⊇ µ n − d n ∪ X nd n . By definition, µ i ∗ d n = (cid:102) Ch d n ( µ i ∗ − d n ∪ X i ∗ d n ) ,which implies by path independence that µ n − d n ⊆ (cid:102) Ch d n ( µ n − d n ∪ X nd n ) = µ nd n since µ n − d n ⊆ µ i ∗ d n .Finally, we need to show that the improvement algorithm terminates. We claim that µ nd n (cid:54) = µ n − d n . Suppose, for contradiction, that these two matchings are the same. Then,by path independence of (cid:102) Ch d n , for every x ∈ X nd n , (cid:102) Ch d n ( µ n − d n ∪ { x } ) = µ n − d n . This is acontradiction because there exists at least one blocking contract associated with district d n .Therefore, district d n gets at least one new contract at Step n . Hence, at least one studentgets a strictly more preferred contract at every step of the algorithm while every otherstudent gets a weakly more preferred contract. Since the number of contracts is finite, thealgorithm has to end in a finite number of steps. (cid:3) NTERDISTRICT SCHOOL CHOICE 59
Because the interdistrict SPDA outcome under ( Ch d ) d ∈D is the same as the interdistrictSPDA outcome under ( (cid:102) Ch d ) d ∈D and the interdistrict SPDA outcome under ( Ch (cid:48) d ) d ∈D is thesame as the interdistrict SPDA outcome under ( (cid:102) Ch (cid:48) ) d ∈D , the lemma implies that every stu-dent weakly prefers the outcome of interdistrict SPDA under ( Ch d ) d ∈D to the outcome ofintradistrict SPDA (which is the same as the interdistrict SPDA outcome under ( Ch (cid:48) d ) d ∈D ).This completes the proof of the first part.To prove the second part of the theorem, we show that if at least one district’s admissionsrule fails to favor own students, then there exists a student preference profile such that notevery student is weakly better off under interdistrict SPDA than under intradistrict SPDA.Suppose that for some district d , there exists a matching X , which is feasible for students,such that Ch d ( X ) is not a superset of Ch d ( X ∗ ) , where X ∗ ≡ { x ∈ X | d ( s ( x )) = d } . Now,consider a matching Y where (i) all students from district d are matched with schools indistrict d , (ii) Y is feasible, and (iii) Y ⊇ Ch d ( X ∗ ) . The existence of such a Y follows fromthe fact that Ch d ( X ∗ ) is feasible and k d (cid:48) ≤ (cid:80) c : d ( c )= d (cid:48) q c , for every district d (cid:48) (that is, thereare enough seats in district d (cid:48) to match all students from district d (cid:48) .) Because Y is feasibleand Ch d is acceptant, Ch d ( Y d ) = Y d .Now consider the following student preferences. First we consider students from dis-trict d . Each student s who has a contract in X ∗ ranks X ∗ s as her top choice. Note that doingso is well defined because X ∗ is feasible for students. Each student s who has a contract in X ∗ \ Ch d ( X ∗ ) ranks contract Y s as her second top choice. Note that, in this case, Y s cannotbe the same as X ∗ s because Ch d ( Y d ) = Y d and Ch d is path independent. Each student s whohas a contract in Y \ X ∗ ranks that contract as her top choice. Next we consider studentsfrom the other districts. Each student s who has a contract in X \ X ∗ ranks that contractas her top choice. Any other student ranks a contract not associated with district d as hertop choice. Complete the rest of the student preferences arbitrarily.Consider SPDA for district d in intradistrict school choice. At the first step, students whohave a contract in X ∗ propose that contract. The remaining students who have contractsin Y \ X ∗ propose the associated contracts. Because Y is feasible, Y contains Ch d ( X ∗ ) , and Ch d is acceptant, only contracts in X ∗ \ Ch d ( X ∗ ) are rejected. At the second step, thesestudents propose their contracts in Y d , and the set of proposals that the district considersis Y d . Because Ch d ( Y d ) = Y d , no contract is rejected, and SPDA stops and returns Y d . Inparticular, every student who has a contract in Ch d ( X ∗ ) has the corresponding contract atthe outcome.In interdistrict SPDA, at the first step, each student who has a contract in X proposes thatcontract and every other student proposes a contract associated with a district differentfrom d . District d considers X (or X d ), and tentatively accepts Ch d ( X ) . Because Ch d ( X ) (cid:54)⊇ Ch d ( X ∗ ) by assumption, at least one student who has a contract in Ch d ( X ∗ ) is rejected. Therefore, this student is strictly worse off under interdistrict school choice than underintradistrict school choice. (cid:3)
Proof of Theorem 9.
To show the result, we first introduce the following weakening of thesubstitutability condition (Hatfield and Kojima, 2008). A district admissions rule Ch d sat-isfies weak substitutability if, for every x ∈ X ⊆ Y ⊆ X with x ∈ Ch d ( Y ) and | Y s | ≤ foreach s ∈ S , it must be that x ∈ Ch d ( X ) .Under weak substitutability, the following result is known (the statement is slightlymodified for the present setting). Theorem 10 (Hatfield and Kojima (2008)) . Let d and d (cid:48) be two distinct districts. Suppose that Ch d satisfies IRC but violates weak substitutability. Then, there exist student preferences anda path-independent admissions rule for d (cid:48) such that, regardless of the other districts’ admissionsrules, no stable matching exists. Given this result, for our purposes it suffices to show the following.
Theorem 9’.
Let d be a district. There exist a set of students, their types, schools in d , and type-specific ceilings for d such that there is no district admissions rule of d that has district-level type-specific ceilings, is d -weakly acceptant, and satisfies IRC and weak substitutability. To show this result, consider a district d with k d = 2 . There are three schools c , c , c inthe district, each with capacity one, and four students s , s , s , s of which two are froma different district. Students s and s are of type t and students s and s are of type t .The district-level type-specific ceilings are as follows: q t d = q t d = 1 .Suppose, for contradiction, that the district admissions rule has district-level type-specific ceilings, is d -weakly acceptant, and satisfies IRC and weak substitutability.Consider Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } ) . Since types are symmetric and twostudents are symmetric within each type, without loss of generality, we can assume Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) } because q c = 1 .Next, consider Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) . Because q c = 1 and Ch d is d -weakly ac-ceptant, this is either equal to { ( s , c ) } or { ( s , c ) } (the case when it is equal to { ( s , c ) } is symmetric to the case when { ( s , c ) } . We analyze these two cases separately.(1) Suppose Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) } . Then, by IRC, weconclude that Ch d ( { ( s , c ) , ( s , c ) } ) = { ( s , c ) } . Next, we argue that Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) } . This is because the only twocases that satisfy d-weak acceptance and type-specific ceilings are { ( s , c ) } and { ( s , c ) , ( s , c ) } . The latter would violate weak substitutability since in that case ( s , c ) would be accepted in a larger set { ( s , c ) , ( s , c ) , ( s , c ) } and rejected from NTERDISTRICT SCHOOL CHOICE 61 a smaller set { ( s , c ) , ( s , c ) } . Then, by IRC, Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) } implies Ch d ( { ( s , c ) , ( s , c ) } ) = { ( s , c ) } . Then we note that Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) , ( s , c ) } since by weak substitutabil-ity ( s , c ) cannot be chosen, and therefore ( s , c ) and ( s , c ) have to bechosen due to d -weak acceptance. Next, again by weak substitutabil-ity, we note that Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) , ( s , c ) } implies Ch d ( { ( s , c ) , ( s , c ) } ) = { ( s , c ) } . Finally, we note that this contradicts Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) } and IRC.(2) Suppose Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) } . Consider Ch d ( { ( s , c ) , ( s , c ) } ) . Because q c = 1 and Ch d is d -weakly acceptant, thisis either { ( s , c ) } or { ( s , c ) } . We consider these two possible cases separately.These two subcases will follow similar arguments to Case (1) above and changethe indices appropriately in order to get a contradiction.(a) Suppose Ch d ( { ( s , c ) , ( s , c ) } ) = { ( s , c ) } . Next, we argue that Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) } . This is because the only twocases that satisfy d-weak acceptance and type-specific ceilings are { ( s , c ) } and { ( s , c ) , ( s , c ) } . The latter would violate weak substitutability since inthat case ( s , c ) would be accepted in a larger set { ( s , c ) , ( s , c ) , ( s , c ) } and rejected from a smaller set { ( s , c ) , ( s , c ) } . Then, by IRC, Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) } implies Ch d ( { ( s , c ) , ( s , c ) } ) = { ( s , c ) } . Then we note that Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) , ( s , c ) } since by weak substitutability ( s , c ) cannot to be chosen, therefore ( s , c ) and ( s , c ) have to be chosen due to d-weak acceptance. Next, again by weaksubstitutability, we note that Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) , ( s , c ) } implies Ch d ( { ( s , c ) , ( s , c ) } ) = { ( s , c ) } . Finally, we note that this contra-dicts Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) } and IRC.(b) Suppose Ch d ( { ( s , c ) , ( s , c ) } ) = { ( s , c ) } . Next, we argue that Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) } . This is because the only twocases that satisfy d-weak acceptance and type-specific ceilings are { ( s , c ) } and { ( s , c ) , ( s , c ) } . The latter would violate weak substitutability since inthat case ( s , c ) would be accepted in a larger set { ( s , c ) , ( s , c ) , ( s , c ) } and rejected from a smaller set { ( s , c ) , ( s , c ) } . Then, by IRC, Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) } implies Ch d ( { ( s , c ) , ( s , c ) } ) = { ( s , c ) } . Then we note that Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) , ( s , c ) } since by weak substitutability ( s , c ) cannot to be chosen, therefore ( s , c ) and ( s , c ) have to be chosen due to d-weak acceptance. Next, again by weaksubstitutability, we note that Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) , ( s , c ) } implies Ch d ( { ( s , c ) , ( s , c ) } ) = { ( s , c ) } . Finally, we note that this contra-dicts Ch d ( { ( s , c ) , ( s , c ) , ( s , c ) } ) = { ( s , c ) } and IRC.and IRC.