aa r X i v : . [ ec on . T H ] M a y Dynamic Reserves in Matching Markets ∗ Orhan Aygün † and Bertan Turhan ‡ March, 2020
Abstract
We study a school choice problem under affirmative action policies where authorities re-serve a certain fraction of the slots at each school for specific student groups, and wherestudents have preferences not only over the schools they are matched to but also the type ofslots they receive. Such reservation policies might cause waste in instances of low demand fromsome student groups. To propose a solution to this issue, we construct a family of choice func-tions, dynamic reserves choice functions, for schools that respect within-group fairness andallow the transfer of otherwise vacant slots from low-demand groups to high-demand groups.We propose the cumulative offer mechanism (COM) as an allocation rule where each schooluses a dynamic reserves choice function and show that it is stable with respect to schools’choice functions, is strategy-proof , and respects improvements . Furthermore, we show thattransferring more of the otherwise vacant slots leads to strategy-proof Pareto improvementunder the COM.
JEL Classification : C78, D47, D61, D63 ∗ First version: September, 2016. This version: March, 2020.We are grateful to the editor, the associate editor, and two anonymous referees, as well as Péter Biró, Fuhito Kojima,Scott Duke Kominers, Utku Ünver, Rakesh Vohra, and especially Bumin Yenmez, whose detailed suggestions lead tosignificant improvements in the paper. We also thank Dilip Abreu, Oğuz Afacan, Nikhil Agarwal, Eduardo Azevedo,Jenna M. Blochowicz, Rahul Deb, Federico Echenique, Isa Hafalır, Andrei Gomberg, John W. Hatfield, YuichiroKamada, Navin Kartik, Onur Kesten, Parag Pathak, Ran Shorrer, Tayfun Sönmez, and Alexander Westkamp forhelpful comments. Finally, we thank the seminar audience at Boston College, University of St. Andrews, ITAM,2017 MATCH-UP Conference, 2017 AEA Meeting, 2016 North American Meeting of Econometric Society, 2016GAMES, the 13th Meeting of Social Choice and Welfare, 2015 Conference on Economic Design, 9th Meeting ofMatching in Practice, and Workshop on Market Design in memory of Dila Meryem Hafalır at ITAM. † [email protected]; Boğaziçi University, Department of Economics, Natuk Birkan Building, Bebek, Is-tanbul 34342, Turkey. ‡ [email protected]; Iowa State University, Department of Economics, Heady Hall, 518 Farm House Lane, Ames,IA 50011, USA. Introduction
The theory of two-sided matching and its applications has been studied since the seminal workof
Gale and Shapley (1962). Nevertheless, many real-life matching markets are subject to variousconstraints, such as affirmative action in school choice. Economists and policy makers are oftenfaced with new challenges from such constraints. Admission policies in school choice systems oftenuse reserves to grant applicants from certain backgrounds higher priority for some available slots.
Reservation in India is such a process of setting aside a certain percentage of slots in governmentinstitutions for members of underrepresented communities, defined primarily by castes and tribes.We present engineering school admissions in India as an unprecedented matching problem withaffirmative action in which students care about the category through which they are admitted.In engineering school admissions in India, students from different backgrounds (namely, sched-uled castes (SC), scheduled tribes (ST), other backward classes (OBC), and general category (GC)) are treated with different criteria. Schools reserve a certain fraction of their slots for students fromSC, ST, and OBC categories. The remaining slots at each school, which are called general category (GC) slots, are open to competition. It is optional for SC, ST, and OBC students to declare theirbackground information. Those who do declare their background information are considered forthe reserved slots in their respective category, as well as for the GC slots. Students who do notbelong to SC, ST, or OBC categories are considered only for GC slots. Students belonging toSC, ST, and OBC communities who do not reveal their background information are only consid-ered for GC slots. Aygün and Turhan (2017) documented that students from SC, ST, and OBCcategories have preferences not only for schools but also for the category through which they areadmitted. Hence, students from these communities may prefer not to declare their caste and tribeinformation in the application process. Besides this strategic calculation burden on students, thecurrent admission procedure suffers from a crucial market failure: The assignment procedure failsto transfer some unfilled slots reserved for under-privileged castes and tribes to the use of remainingstudents. Hence, it is quite wasteful.We address real-life applications as follows: There are schools and students to be matched.Each school initially reserves a certain number of its slots for different privilege groups (or studenttypes). A given student may possibly match with a given school under more than one type. Eachschool has a pre-specified sequence in which different sets of slots are considered, and where eachset accepts students in a single privilege type. Different schools might have different orders. Sincea student might have more than one privilege type, the set of students cannot be partitioned into Students who do not belong to SC, ST, and OBC categories are called general category (GC) applicants. Admission to the Indian Institute of Technologies (IITs) and its matching-theoretical shortcomings are explainedin detail in
Aygün and Turhan (2017). We will call this sequence a precedence sequence, which is different than the precedence order from
Kominersand Sönmez (2016). Precedence order is a linear order over the set of student types. Precedence sequence, onthe other hand, is more general in the sense that a given student type might appear multiple times. A technicaldefinition will be given in the model section. . If there is less demand from at leastone privilege type, schools are given the opportunity to utilize vacant slots by transferring themover to other privilege types. Authorities might require a certain capacity transfer scheme so thateach school has a complete plan where they state how they want to redistribute these slots. Thus,we take capacity transfer schemes exogenously given. The only mild condition imposed on thecapacity transfer scheme is monotonicity, which requires that (1) if more slots are left from oneor more sets, the capacity of the sets considered later in the precedence sequence must be weaklyhigher, and (2) a school cannot decrease the total capacity in response to increased demand forsome sets of slots.We design choice functions for schools that allow them to transfer capacities from low-demandprivilege types to high-demand privilege types. Each school respects an exogenously given prece-dence sequence between different sets of slots when it fills its slots. Each school has a strict priorityordering (possibly different than the other schools’) over all students. For each school, priorityorderings for different privilege types are straightforwardly derived from the school’s priority or-dering. There is an associated choice function, which we call a “sub-choice function ,” for each setof slots. In Indian engineering school admissions, sub-choice functions are q-responsive. That is, asub-choice function always selects the q-best students with respect to the priority ordering of theassociated privilege type at that school, where q denotes the capacity.The school starts filling its first set of slots according to its precedence sequence. Given theinitial capacity of the first set of slots and a contract set, the sub-choice function associated withthe first set selects contracts. The school then moves to the second set according to its precedencesequence. The (dynamic) capacity of the second set is a function of the number of unfilled slotsin the first set. The exogenous capacity transfer function of the school specifies the capacity ofthe second set. The set of available contracts for the second set of slots is computed as follows:If a student has one of her contracts chosen by the first set, then all of her contracts are removedfor the rest of the choice process. Given the set of remaining contracts and the capacity, thesub-choice function associated with the second set selects contracts. In general, the (dynamic)capacity of set k is a function of the number of vacant slots of the k − sets that precede it.The set of contracts available to the set of slots k is computed as follows: If a student has oneof her contracts chosen by one of the k − sets of slots that precede the k th set, then all of hercontracts are removed. Given the set of remaining contracts for the set of slots k and its capacity,the sub-choice function associated with the set k selects contracts. The (overall) choice of a schoolis the union of sub-choices of its sets of slots. Hard bounds and soft bounds are analyzed in detail in
Hafalır et al. (2013) and
Ehlers et al. (2014). Westkamp (2013) introduces this monotonicity condition on capacity transfer schemes.
3e propose a remedy for the Indian engineering school admissions problem through a matchingwith contracts model that has the ability to utilize vacant slots of certain types for other students.We have three design objectives: stability, strategy-proofness and respect for improvements.
Sta-bility ensures that (1) no student is matched with an unacceptable school-slot category pair, (2) schools’ dynamic reserves choices are respected, and (3) no student desires a slot at which she hasa justified claim under the priority and precedence structure. Strategy-proofness guarantees thatstudents can never game the allocation mechanism via preference manipulation. In our frame-work, it also relieves students of the strategic manipulation burden, which involves whether or notstudents declare their background. Respect for improvements is an essential property in merito-cratic systems. In allocation mechanisms that respect improvements, students have no incentiveto lower their standings in schools’ priority rankings.We propose the cumulative offer mechanism (COM) as an allocation rule. We prove that theCOM is stable with respect to schools’ dynamic reserve choice functions (Theorem 1) , is (weakly)group strategy-proof (Theorem 2) , and respects improvements (Theorem 3). The main result ofthe paper (Theorem 4) states that when a single school’s choice function becomes “more flexible,” while those of the other schools remain unchanged, the outcome of the COM under the former(weakly) Pareto dominates the outcome under the latter. Theorem 4 is of particular importancebecause it describes a strategy-proof Pareto improvement. Finally, we investigate the relationshipbetween families of dynamic reserves choice rules and Kominers and Sönmez’s (2016) slot-specificpriorities choice rules. We show that for every slot-specific priorities choice rule, there is an outcomeequivalent dynamic reserves choice rule (Theorem 5). Moreover, we give an example of a dynamicreserves choice rule for which there is no outcome equivalent slot-specific priorities choice rule(Example 1). Related Literature
The school choice problem was first introduced by the seminal paper of
Abdulkadiroğlu and Sön-mez (2003). The authors introduced a simple affirmative action policy with type-specific quotas.
Kojima (2012) showed that the minority students who purported to be the beneficiaries might in-stead be made worse off under this type of affirmative action. To circumvent inefficiencies causedby majority quotas,
Hafalır et al. (2013) offer minority reserves . Westkamp (2013) introduced amodel of matching with complex constraints. His model permits priorities to vary across slots.In his model, students are considered to be indifferent between different slots of a given school. Strategy-proofness ensures that it is a weakly dominant strategy for each student to report their caste and tribeinformation. See
Kominers (2019) for detailed discussion of respect for improvements in matching markets. We define “more flexible” criterion to compare two monotonic capacity transfer schemes given a precedencesequence. We say that a monotonic capacity transfer scheme e q is more flexible than monotonic capacity transferscheme q if e q transfers at least as many otherwise vacant slots as q at every instance. There must also be an instancewhere e q transfers strictly more otherwise vacant slots than q does. Westkamp (2013). Moreover, our comparativestatics result on transfer schemes does not have a counterpart in
Westkamp (2013).
Kominers and Sönmez (2016) introduce another prominent family of choice functions—slot-specific priorities choice functions—to implement diversity objectives in many-to-one settings. Weshow that dynamic reserves choice rules nest slot-specific priorities choice rules. Moreover, weprovide an example of a dynamic reserves choice rule that cannot be generated by a slot-specificpriorities choice rule.In a related work,
Biró et al. (2010) analyze a college admission model with common and upperquotas in the context of Hungarian college admissions. They use choice functions for colleges thatallow them to select multiple contracts of the same applicant. They show that a stable assignmentexists. The completions of dynamic reserves choice functions, discussed in Appendix 7.2, satisfythe properties they impose. Hence, their result also implies the existence of a stable allocationin our framework. However, our main focus is different as we aim to show strategy-proof Paretoimprovement by making capacity transfer function more flexible.The matching problem with dynamic reserves choice functions is a special case of the matchingwith contracts model of
Fleiner (2003) and Hatfield and Milgrom (2005). The analysis andresults of
Hatfield and Kominers (2019) are the technical backbone of our results regarding stableand strategy-proof mechanism design. We show that every dynamic reserves choice function hasa completion that satisfies the irrelevance of rejected contracts condition of
Aygün and Sönmez (2013), in conjunction with substitutability and the law of aggregate demand.
Hatfield et al. (2017) introduce a model of hospital choice in which each hospital has a set ofdivisions and flexible allotment of capacities to those divisions that vary as a function of the setof contracts available. These authors define choice functions that nest dynamic reserves choicefunctions while continuing to obtain stability and strategy-proofness for the COM. Our Theorems3 and 4 do not have a counterpart in
Hatfield et al. (2017).Our work is also related with the research agenda on matching with constraints that is studiedin a series of papers:
Kamada and Kojima (2015), (2017),
Kojima et al. (2018), and
Goto et al. (2017). In these papers, constraints are imposed on subsets of institutions as a joint restriction,as opposed to at each individual institution. Our main results distinguish our work from thesepapers. We discuss the relationship between our stability notion and that of
Kamada and Kojima (2017) in Section 3.Another related paper is
Echenique and Yenmez (2015). Dynamic reserves choice functionsmight seem similar to the family of choice functions the authors analyze: choice rules generated Fleiner’s results cover these of
Hatfield and Milgrom (2005) regarding stability. However,
Fleiner (2003) doesnot analyze incentives. Echenique (2012) has shown that under the substitutes condition, which is thoroughly assumed in
Hatfield andMilgrom (2005), the matching with contracts model can be embedded within the
Kelso and Crawford (1982) labormarket model.
Kelso and Crawford (1982) built on the analysis of
Crawford and Knoer (1981).
5y reserves . However, dynamic reserves choice functions choose contracts whereas choice rulesgenerated by reserves choose students.Two recent papers,
Sönmez and Yenmez (2019a,b) , study affirmative action in India from amatching-theoretical perspective. The authors consider both vertical and horizontal reservations while we consider only vertical reservations for simplicity. Even though they consider more generalreserve structure than ours, the authors consider agents’ preferences only over institutions anddo not take agents’ preferences over vertical categories they are admitted under into account.Moreover, they assume away capacity transfers between vertical categories. Therefore, their modeldoes not contain our model and vice versa. There is a finite set of students I = { i , ..., i n } , a finite set of schools S = { s , ..., s m } , and afinite set of student privileges (types) T = { t , ..., t p } . We call T i ⊆ T the set of privilegesthat student i can claim and T = ( T i ) i ∈ I the profile of types that students can claim. We define X i = { i } × S × T i as the set of all contracts associated with student i ∈ I . We let X = ∪ i ∈ I X i be theset of all contracts. Each contract x ∈ X is between a student i ( x ) and a school s ( x ) and specifiesa privilege t ( x ) ∈ T i ( x ) . There may be many contracts for each student-school pair. We extendthe notations i ( · ) , s ( · ) and t ( · ) to the set of contracts for any Y ⊆ X by setting i ( Y ) ≡ ∪ y ∈ Y { i ( y ) } , s ( Y ) ≡ ∪ y ∈ Y { s ( y ) } and t ( Y ) ≡ ∪ y ∈ Y { t ( y ) } . For Y ⊆ X , we denote Y i ≡ { y ∈ Y | i ( y ) = i } ;analogously, we denote Y s ≡ { y ∈ Y | s ( y ) = s } and Y t ≡ { y ∈ Y | t ( y ) = t } .Each student i ∈ I has a (linear) preference order P i over contracts in X i = { x ∈ X | i ( x ) = i } and an outside option ∅ which represents remaining unmatched. A contract x ∈ X i is acceptable for i (with respect to P i ) if xP i ∅ . We use the convention that ∅ P i x if x ∈ X \ X i .We say that the contracts x ∈ X for which ∅ P i x are unacceptable to i . The at-least-as-well relation R i is obtained from P i as follows: xR i x ′ if and only if either xP i x ′ or x = x ′ . Let P i denote the set of all preferences over X i ∪ {∅} . A preference profile of students is denoted by P = ( P i , ..., P i n ) ∈ × i ∈ I P i . A preference profile of all students except student i l is denoted by P − i l = ( P i , ..., P i l − , P i l +1 , ..., P i n ) ∈ × i = i l P i .Students have unit demand , that is, they choose at most one contract from a set of contractoffers. We assume that students always choose the best available contract, so that the choice C i ( Y ) of a student i ∈ I from contract set Y ⊆ X is the P i -maximal element of Y (or the outside optionif ∅ P i y for all y ∈ Y ). Caste-based reservations for SC, ST, and OBC categories are called vertical reservations, also referred to as socialreservations. Horizontal reservations, also referred to as special reservations, are intended for other disadvantagedgroups of citizens, such as disabled persons, and women. Horizontal reservations are implemented within eachvertical category. See
Sönmez and Yenmez (2019a,b) for details. We use the terms “type” and “privilege” interchangeably. To simplify our notation, the individual contracts are treated as interchangeable with singleton contract sets. s ∈ S , q s denotes the physical capacity of school s ∈ S . We call q = ( q s ..., q s m ) the vector of school capacities. Each school s ∈ S has a priority order π s , which is a linear orderover I ∪ {∅} . Let
Π = ( π s , ..., π s m ) denote the priority profile of schools. For each school s ∈ S ,the priority ordering for students who can claim the privilege t ∈ T , denoted by π st , is obtainedfrom π s as follows: • for i, j ∈ I such that t ∈ T i \ T j , iπ s ∅ , and jπ s ∅ , iπ st ∅ π st j , • for any other i, j ∈ I , iπ st j if and only if iπ s j .An allocation Y ⊆ X is a set of contracts such that each student appears in at most one contractand no school appears in more contracts than its capacity allows. Let X denote the set of allallocations. Given a student i and an allocation Y , we refer to the pair ( s ( x ) , t ( x )) such that i ( x ) = i as the assignment of student i under allocation Y . We extend student preferences overcontracts to preferences over outcomes in the natural way. We say that an allocation Y ⊆ X Pareto dominates allocation Z ⊆ X if Y i R i Z i for all i ∈ I and Y i P i Z i for at least one i ∈ I . Each school s ∈ S has multi-unit demand, and is endowed with a choice function C s ( · ) thatdescribes how s would choose from any offered set of contracts. Throughout the paper, we assumethat for all Y ⊆ X and for all s ∈ S , the choice function C s ( · ) :1. only selects contracts to which s is a party, i.e., C s ( Y ) ⊆ Y s , and2. selects at most one contract with any given student.For any Y ⊆ X and s ∈ S , we denote R s ( Y ) ≡ Y \ C s ( Y ) as the set of contracts that s rejects from Y .We now introduce a model of dynamic reserves choice functions in which each school s ∈ S has λ s groups of slots . School s fills its groups of slots according to a precedence sequence, which isa surjective function f s : { , ..., λ s } −→ T . The interpretation of f s is that school s fills the firstgroup of slots with f s (1) -type students, the second group of slots with f s (2) -type students, andso on. School s ∈ S has a target distribution of its slots across different types ( q t s , ..., q t p s ) , whichmeans that it has q t s slots to be reserved for privilege t , q t s slots to be reserved for privilege t , This priority order is often determined by performance on an admission exam, by a random lottery, or dictatedby law. In engineering school admissions in India, each school ranks students according to test scores. Differentschools might have different test score rankings because they use different weighted averages of math, physics,chemistry, and biology scores depending on the school. It is important to note that students whose test scores areunder a certain threshold are deemed as unacceptable for each school. ∅ π st j means student j is unacceptable for privilege t at school s . We take precedence sequences to be exogenously given. However,
Dur et al. (2018) show that precedencesequences might have significant effects on distributional objectives in the context of Boston’s school choice system. s fills its slots according to the initiallyset capacities for each group of slots ( q s , q s , ..., q λ s s ) such that P j ∈ ( f s ) − ( t ) q js = q ts for all t ∈ T . If thetarget distribution cannot be achieved because too few students from one or more privileges apply,then school s use an exogenously given capacity transfer scheme that specifies how its capacity isto be redistributed. Technically, a capacity transfer scheme is defined as follows: Definition 1.
Given a precedence sequence f s and a capacity of the first group of slots q s , a capacity transfer scheme of school s is a sequence of capacity functions q s = ( q s , ( q ks ) λ s k =2 ) ,where q ks : Z k − −→ Z + such that q ks (0 , ...,
0) = q ks for all k ∈ { , ..., λ s } .We impose a mild condition, à la Westkamp (2013), on capacity transfer functions.
Definition 2.
A capacity transfer scheme q s is monotonic if, for all j ∈ { , ..., λ s } and all pairsof sequences ( r l , e r l ) such that e r l ≥ r l for all l ≤ j − , • q js ( e r , ..., e r j − ) ≥ q js ( r , ..., r j − ) , and • j P m =2 [ q ms (˜ r , ..., e r m − ) − q ms ( r , ..., r m − )] ≤ j − P m =1 [ e r m − r m ] .Monotonicity of capacity transfer schemes requires that (1) whenever weakly more slots are leftunfilled in every groups of slots preceding the j th group of slots, weakly more slots should beavailable for the j th group, and (2) a school cannot decrease its total capacity in response toincreased demand for some groups of slots. Sub-choice functions
For each group of slots at school s ∈ S , there is an associated sub-choice function c s : 2 X × Z + × T −→ X . Given a set of contracts Y ⊆ X , a nonnegative integer κ ∈ Z + , and a privilege t ∈ T , c s ( Y, κ, t ) denotes the set of chosen contracts that name privilege t up to the capacity κ from theset of contracts Y . We require sub-choice functions to be q-responsive given the ranking π st . Definition 3. A sub-choice function c s ( · , κ, t ) of a group of slots at school s for privilege type t is q-responsive if there exists a strict priority ordering π st on the set of contracts naming privilegetype t and a positive integer κ , such that for any Y ⊆ ( X s ∩ X t ) , c s ( Y, κ, t ) = κ ∪ i =1 { y ∗ i } where y ∗ i is defined as y ∗ = max π st Y and, for ≤ i ≤ κ , y ∗ i = max π st Y \ { y ∗ , ..., y ∗ i − } . We adapt this definition from
Chambers and Yenmez (2017).
8n other words, a sub-choice function is q-responsive if there is a strict priority ordering overstudents who have privilege t for which the sub-choice function always selects the highest-rankedavailable students in privilege t up to the capacity. Remark . Since our main real-life application is engineering school admissions in India, we shallassume that at each school s ∈ S , and for each group of slots reserved for privilege t ∈ T , theassociated sub-choice function c s ( · , · , t ) is q-responsive and obtained from π st . Overall choice functions
The overall choice function of school s , C s ( · , f s , q s ) : 2 X −→ X , runs its sub-choice functionsin an orderly fashion given the precedence sequence f s and capacity transfer scheme q s . Given aset of contracts Y ⊆ X , C s ( Y, f s , q s ) denotes the set of chosen contracts from the set of contracts Y and is determined as follows: • Given q s and Y = Y ⊆ X , let Y ≡ c s ( Y , q s , f s (1)) be the set of chosen contracts withprivilege f s (1) from Y . Let r = q s − | Y | be the number of vacant slots. Define e Y ≡{ y ∈ Y | i ( y ) ∈ i ( Y ) } as the set of all contracts of students whose contracts are chosenby sub-choice function c s ( · , q s , f s (1)) . If a contract of a student is chosen, then all of thecontracts naming that student shall be removed from the set of available contracts for therest of the procedure. The set of remaining contracts is then Y = Y \ e Y . • In general, let Y k = c sk ( Y k − , q ks , f s ( k )) be the set of chosen contracts with privilege f s ( k ) from the set of available contracts Y k − , where q ks = q ks ( r , ..., r k − ) is the dynamic capacityof group of slots k as a function of the vector of the number of unfilled slots ( r , ..., r k − ) .Let r k = q ks − | Y k | be the number of vacant slots. Define e Y k = { y ∈ Y k − | i ( y ) ∈ i ( Y k ) } .The set of remaining contracts is then Y k = Y k − \ e Y k . • Given Y = Y ⊆ X and the capacity of the first group of slots q s , we define the overall choicefunction of school s as C s ( Y, f s , q s ) = c s ( Y , q s , f s (1)) ∪ ( λ s ∪ k =2 c sk ( Y k − , q ks ( r , ..., r k − ) , f s ( k ))) .The primitives of the overall choices for each school s ∈ S are the precedence sequence f s , the capac-ity transfer scheme q s , and the priority order π s . Since an overall choice is computed by using theseprimitives, it is not one of the primitives in our model. The list ( I, S, T , X, P, Π , ( f s , q s , π s ) s ∈ S ) denotes a problem. These types of sub-choice functions are often used in real-life applications. For example, in the cadet branchmatching processes in the USMA and ROTC, each sub-choice function is induced from a strict ranking of studentsaccording to test scores. See
Sönmez and Switzer (2013) and
Sönmez (2013) for further details. Stability Concept
Stability has emerged as the key to a successful matching market design. We follow the
Gale andShapley (1962) tradition in focusing on outcomes that are stable . In the matching with contractsframework,
Hatfield and Milgrom (2005) define stability as follows: An outcome Y ⊆ X is stable if 1. Y i R i ∅ for all i ∈ I ,2. C s ( Y ) = Y s for all s ∈ S , and3. there does not exist a school s ∈ S and a blocking set Z = C s ( Y ) such that Z s ⊆ C s ( Y ∪ Z ) and Z i = C i ( Y ∪ Z ) for all i ∈ i ( Z ) .If the first requirement ( individual rationality for students ) fails, then there is a student whoprefers to reject a contract that involves her (or, equivalently, there is a student who is given anunacceptable contract). In our context, the second condition ( individual rationality for schools )requires that the schools’ choice functions are respected. If the third condition ( unblockedness )fails, then there is an alternative set of contracts that a school and students associated with acontract in that set strictly prefers. Remark . Our stability notion is related to the weak stability notion of
Kamada and Kojima (2017). The authors define the feasibility constraint as a map φ : Z | H | + −→ { , } , such that φ ( w ) ≥ φ ( w ′ ) whenever w ≤ w ′ . Their interpretation is that each coordinate in w corresponds toa hospital and the number in that coordinate represents the number of doctors matched to thathospital. φ ( w ) = 1 means that w is feasible and φ ( w ) = 0 means it is not. They say that matching µ is feasible if and only if φ ( w ( µ )) = 1 , where w ( µ ) := ( | µ h | ) h ∈ H is a vector of nonnegative integersindexed by hospitals whose coordinates corresponding to h are | µ h | . Capacity transfer functionsin our setting can be represented by the feasibility constraint map from their paper. Condition 2in our stability definition takes into account not only dynamic capacities of groups of seats in eachschool but also their precedence sequences. It is a feasibility condition. Westkamp (2013) definesa similar condition in his “procedural stability” definition in a simpler matching model withoutcontracts. A direct mechanism is a mechanism where the strategy space is the set of preferences P for eachstudent i ∈ I , i.e., a function ψ : P n −→ X that selects an allocation for each preference profile.We propose the COM as our allocation function. Given the student preferences and schools’ overall10hoice functions, the outcome of the COM is computed by the cumulative offer algorithm . Thisis the generalization of the agent-proposing deferred acceptance algorithm of Gale and Shapley (1962). We now introduce the cumulative offer process (COP) for matching with contracts.Here, we provide an intuitive description of this algorithm; we give a more technical description inAppendix 7.1. Definition 4.
In the COP, students propose contracts to schools in a sequence of steps l = 1 , . . . : Step 1 : Some student i ∈ I proposes his most-preferred contract, x ∈ X i . School s ( x ) holds x if x ∈ C s ( x ) ( { x } ) , and rejects x otherwise. Set A s ( x ) = { x } , and set A s ′ = ∅ for each s ′ = s ( x ) ; these are the sets of contracts available to schools at the beginning of Step 2. Step 2 : Some student i ∈ I , for whom no school currently holds a contract, proposes hismost-preferred contact that has not yet been rejected, x ∈ X i . School s ( x ) holds the contractin C s ( x ) (cid:16) A s ( x ) ∪ { x } (cid:17) and rejects all other contracts in A s ( x ) ∪{ x } ; schools s ′ = s ( x ) continueto hold all contracts they held at the end of Step 1. Set A s ( x ) = A s ( x ) ∪ { x } , and set A s ′ = A s ′ for each s ′ = s ( x ) . Step l : Some student i l ∈ I , for whom no school currently holds a contract, proposes hismost-preferred contact that has not yet been rejected, x l ∈ X i l . School s (cid:0) x l (cid:1) holds the contract in C s ( x l ) (cid:18) A ls ( x l ) ∪ (cid:8) x l (cid:9)(cid:19) and rejects all other contracts in A ls ( x l ) ∪ (cid:8) x l (cid:9) ; schools s ′ = s (cid:0) x l (cid:1) continueto hold all contracts they held at the end of Step l −
1. Set A l +1 s ( x l ) = A ls ( x l ) ∪ (cid:8) x l (cid:9) , and set A l +1 s ′ = A ls ′ for each s ′ = s (cid:0) x l (cid:1) .If at any time no student is able to propose a new contract—that is, if all students for whomno contracts are on hold have proposed all contract they find acceptable—then the algorithmterminates. The outcome of the COP is the set of contracts held by schools at the end of the laststep before termination.In the COP, students propose contracts sequentially. Schools accumulate offers, choosing ateach step (according to their choice functions) a set of contracts to hold from the set of all previousoffers. The process terminates when no student wishes to propose a contract.Given a preference profile of students P = ( P i ) i ∈ I and a profile of choice functions for schools C = ( C s ) s ∈ S , let Φ (
P, C ) denote the outcome of the COM. Let Φ i ( P, C ) denote the assignmentof student i ∈ I and Φ s ( P, C ) denote the assignment of school s ∈ S . Remark . We do not explicitly specify the order in which students make proposals.
Hirata andKasuya (2014) show that in the matching with contracts model, the outcome of the COP is order-independent if the overall choice function of every school satisfies the bilateral substitutability(BLS) and the irrelevance of rejected contracts (IRC) conditions. Dynamic reserves choice functionssatisfy BLS and IRC. Hence, the order-independence of the COP holds. See
Hatfield and Milgrom (2005) for more details.
11 mechanism ϕ is stable if for every preference profile P ∈ P | I | the outcome ϕ ( P ) is stable withrespect to the schools’ overall choice functions. Since the COP gives a stable outcome for everyinput if each school’s capacity transfer scheme is monotonic, the COM is a stable mechanism. Theorem 1.
The cumulative offer mechanism is stable with respect to dynamic reserves choicefunctions.
Proof.
See Appendix 7.3.To analyze the incentive properties of the COM when schools use dynamic reserves choice func-tions, we first define standard strategy-proofness and (weak) group strategy-proofness in relationto a direct mechanism.
Definition 5.
A direct mechanism ϕ is said to be strategy-proof if there does not exist apreference profile P , a student i ∈ I , and preferences P ′ i of student i such that ϕ i ( P ′ i , P − i ) P i ϕ i ( P ) . That is, no matter which student we consider, no matter what her true preferences P i are,no matter what other preferences P − i other students report (true or not), and no matter whichpotential “misrepresentation” P ′ i student i considers, a truthful preference revelation is in her bestinterest. Hence, students can never benefit from gaming the mechanism ϕ . Definition 6.
A direct mechanism ϕ is said to be weakly group strategy-proof if there is nopreference profile P , a subset of students I ′ ⊆ I , and a preference profile ( P i ) i ∈ I ′ of students in I ′ such that ϕ i (cid:16) ( P ′ i ) i ∈ I ′ , ( P j ) j ∈ I \ I ′ (cid:17) P i ϕ i ( P ) for all i ∈ I ′ .That is, no subset of students can jointly misreport their preferences to receive a strictlypreferred outcome for every member of the coalition. Hatfield and Kominers (2019) show that if schools’ choice functions have substitutable comple-tions so that these completions satisfy the LAD, then the COP becomes weakly group strategy-proof.
Theorem 2.
Suppose that each school uses a dynamic reserves choice function. Then, the cumu-lative offer mechanism is weakly group strategy-proof.
Proof.
See Appendix 7.3. 12 espect for Unambiguous Improvements
We say that priority profile Π is an unambiguous improvement over priority profile Π for student i ∈ I if, for all schools s ∈ S , the following conditions hold:1. For all x ∈ X i and y ∈ (cid:0) X I \{ i } ∪ {∅} (cid:1) , if xπ s y then xπ s y .2. For all y, z ∈ X I \{ i } , yπ s z if and only if yπ s z .That is, Π is an unambiguous improvement over priority profile Π for student i if Π is obtainedfrom Π by increasing the priority of some of i ’s contracts while leaving the relative priority of otherstudents’ contracts unchanged. Definition 7.
A mechanism ϕ respects unambiguous improvements for i ∈ I if for anypreference profile P ∈ × i ∈ I P i ϕ i ( P ; Π) R i ϕ i ( P ; Π) whenever Π is an unambiguous improvement over Π for i . We say that ϕ respects unambiguousimprovement s if it respects unambiguous improvements for each student i ∈ I .Respect for improvements is essential in settings like ours where it implies that students neverwant to intentionally decrease their test scores and, in turn, their rankings. Similarly, it is alsoimportant in cadet-branch matching where cadets can influence their priority rankings directly. Sönmez (2013) argues that cadets take perverse steps to lower their priorities because the mech-anism used by the Reserve Officer Training Corps (ROTC) to match its cadets to branches failsthe respecting improvements property.
Theorem 3.
The cumulative offer mechanism with respect to dynamic reserves choice functionsrespects unambiguous improvements.
Proof.
See Appendix 7.3.
In this section, we first define a comparison criteria between two monotone capacity transferschemes. Consider a school s ∈ S with a given precedence sequence f s and target distribution q s = ( q s , ..., q λ s s ) . Let q s and e q s be two monotone capacity transfer schemes: given a vector ofunused slots from group of slots to j − , ( r , ..., r j − ) ∈ Z j − , the dynamic capacity of the j th group under capacity transfer schemes q s and e q s are q js = q js ( r , ..., r j − ) and e q js = e q js ( r , ..., r j − ) ,respectively, for all j ≥ and, q s = e q s = q s .Let q s and e q s be two monotone capacity transfer schemes that are compatible with the prece-dence sequence f s and target capacity vector q s of school s ∈ S . We say that the monotonecapacity transfer scheme e q s is more flexible than the monotone capacity transfer scheme q s if13. there exists l ∈ { , ..., λ s } and (ˆ r , ..., ˆ r l − ) ∈ Z l − such that e q ls (ˆ r , ..., ˆ r l − ) > q ls (ˆ r , ..., ˆ r l − ) ,and2. for all j ∈ { , ..., λ s } and ( r , ..., r j − ) ∈ Z j − , if j = l or ( r , ..., r j − ) = (ˆ r, ..., ˆ r l − ) , then e q js ( r , ..., r j − ) ≥ q js ( r , ..., r j − ) .The definition states that one monotonic capacity transfer scheme is more flexible than anotherif it transfers at least as many vacant slots as the other at every instance (i.e., the vectors of thenumber of unused slots). There must also be an instance where the first one transfers strictly morevacant slots than the second one to the next group of slots according to the precedence sequence.Also, both of the monotonic capacity transfer schemes take the capacity of the first group of slotswith respect to the precedence sequence equal to its target capacity. Holding all else constant,when the capacity transfer scheme becomes more flexible, it defines a particular choice functionexpansion. Expanding the overall choice function of a single school leads to Pareto improvement for stu-dents under the COM. Theorem 4.
Let C = ( C s , ..., C s m ) be the profile of schools’ overall choice functions. Fix a school s ∈ S . Suppose that e C s takes a capacity transfer scheme that is more flexible than that of C s ,holding all else constant. Then, the outcome of the cumulative offer mechanism with respect to ( e C s , C − s ) weakly Pareto dominates the outcome of the cumulative offer mechanism with respect to C . Proof.
See Appendix 7.3.Theorem 4 is of particular importance because it indicates that increasing the transferabilityof capacity from low-demand to high-demand groups leads to strategy-proof Pareto improvementwith the cumulative offer algorithm. This result provides a normative foundation for recommendinga more flexible interpretation of type-specific quotas. This result establishes that to maximizestudents’ welfare, schools’ choice functions should be expanded as much as possible.It is important to note that when more than one school’s capacity transfer scheme become moreflexible, a simple iteration of Theorem 4, one school at a time, ensures (weak) Pareto improvement.Therefore, a more flexible capacity transfer profile of schools implies that the COM with the newcapacity transfer scheme (weakly) Pareto improves the original transfer scheme. The type of choice function expansion here is different than the one
Chambers and Yenmez (2017) define . Their notion of expansion is in the sense of set inclusion while ours is not. They say that a choice function C ′ is anexpansion of another choice function C if for every offer set Y , C ( Y ) ⊆ C ′ ( Y ) . According to the expansion via amore flexible capacity transfer scheme, when a choice function C expands to C ′ it is possible to have C ( Y ) * C ′ ( Y ) for some Y . This result does not contradict the findings of
Alva and Manjunath (2019), because increasing flexibility of thecapacity transfers changes the choice functions, and therefore the set of contracts that are feasible in their context.Theorem 4 achieves the improvement by considering a dominating mechanism that is infeasible under the originaltransfer scheme. Relationship Between Slot-specific Priorities and DynamicReserves Choice Rules
In this section, we investigate the relationship between the families of slot-specific priorities choicerules and dynamic reserves choice rules. To do so, we first describe slot-specific priorities choicerules.Each school s ∈ S has a set of slots B s . Each slot can be assigned at most one contract in X s .Slots b ∈ B s have linear priority orders π b over contracts in X s . Each slot b ranks a null contract ∅ b that represents remaining unassigned. Schools s ∈ S may be assigned as many as | B s | contractsfrom an offer set Y ⊆ X —one for each slot in B s — but may hold no more than one contract with agiven student. The slots in B s are ordered according to a linear order of precedence ⊲ s . We denote B s ≡ { b , ..., b q s } with | B s | = q s . The interpretation of ⊲ s is that if b l ⊲ s b l +1 , then—wheneverpossible—school s fills slot b l before filling slot b l +1 . Formally, the choice C s ( Y ) of a school s ∈ S from contract set Y ⊆ X is defined as follows: • First, slot b is assigned the contract y that is π b -maximal among contracts in Y . • Then, slot b is assigned the contract y that is π b -maximal among contracts in the set Y \ Y i ( y ) of contracts in Y with agents other than i ( y ) . • This process continues in sequence, with each slot b l being assigned to the contract y l thatis π b l -maximal among contracts in the set Y \ Y i ( { y ,...,y l − } ) .If no contract is assigned to a slot b l ∈ B s in the computation of C s ( Y ) , then b l is assigned thenull contract ∅ b l .We first give an example of a dynamic reserves choice rule that cannot be generated by aslot-specific priorities choice rule. Example 1.
Consider I = { i, j, k, l } , S = { s } with q s = 2 , and Θ = { t , t , t } . Student i only hastype t and a single contract x . Student j only has type t and a single contract y . Student k hastypes t and t , and two contracts related to these types z and z , respectively. Finally, student l has types t and t , and two contracts related to these types w and w , respectively. The setof contracts for this problem is X = { x , y , z , z , w , w } . Students are ordered with respect totheir exam scores from highest to lowest as follows: i − j − k − l .The school reserves the first seat for type t , and the second seat for type t . If either the firstseat or the second seat cannot be filled with the students they are reserved for, they are filled witha type t student(s). The precedence order is such that the first seat is filled first with a type t student if possible, and then the second seat is filled with a type t student, if possible. If any ofthese seats cannot be filled with the intended student types, all of the vacant seats are filled withtype t students at the very end, if possible. 15e can represent the distributional objective described above by capacity-transfers as follows:Initially q t = q t = 1 and q t = 0 . The dynamic capacity of the third seat is given by q t = r + r ,where r , r ∈ { , } . Some of the choice situations under the capacity-transfer described aboveare given below: Y C ( Y ) { x , y , z , z , w , w } { x , y }{ y , z , z } { y , z }{ x , z , z } { x , z }{ y , w , w } { y , w }{ x , w , w } { x , w }{ z , z } { z }{ w , w } { w } In order to implement the choices above with slot-specific priorities, we need to find a strictranking of the contracts in X for both of the slots. Note that { x , y } is chosen from { x , y , z , z , w , w } .Then, x must be chosen for one of the slots and y must be chosen for the other. There are twocases to consider. Case 1: x is chosen from slot 1 and y is chosen from slot 2. Then, x is the highest prioritycontract in slot 1. We have C ( { x , z , z } ) = { x , z } . Then, z must have higher priority than z in the strict priority ranking of slot 2 because x will be chosen from the first slot. Noticethat both z and z must have lower priority than y in the strict ranking of slot 2. Also, since C ( { y , z , z } ) = { y , z } , then it must be the case that z has higher priority than z in the strictpriority of the first slot. Notice that z cannot be chosen from the second slot as z has higherpriority. However, C ( { z , z } ) = { z } . This is a contradiction. Case 2: y is chosen from slot 1 and x is chosen from slot 2. Then, y has the highest priorityin slot 1. We have C ( { y , w , w } ) = { y , w } . Therefore, in the ranking of slot 2, w must havehigher priority than w . Also, since C ( { x , w , w } ) = { x , w } , it follows that in the ranking ofslot 1 w must have higher priority than w . This is because w cannot be chosen from slot 2 asit has a lower priority than w there. However, C ( { w , w } ) = { w } . This is a contradiction.Hence, we cannot find a strict rankings of the contracts in X for these two slots that generatethe dynamic reserves choice rule defined above.Our last result states that the family of dynamic reserves choice rules nests the family ofslot-specific priorities choice rules. Theorem 5.
Every slot-specific priorities choice rule can be generated by a dynamic reserves choicerule. roof. See Appendix 7.3.
This paper studies a school choice problem with distributional objectives where students care aboutboth the school they are matched with as well as the category through which they are admitted.Each school can be thought of as union of different groups of slots, where each group is associatedwith exactly one category. Schools have target distributions over their groups of slots in the form ofreserves. If these reserves are considered to be hard bounds, then some slots will remain empty ininstances where demand for particular categories is less than their target capacities. To overcomethis problem and to increase efficiency, we design a family of dynamic reserves choice functions.We do so by allowing monotonic capacity transfers across groups of slots when one or more of thegroups is not able to fill to its target capacity. The capacity transfer scheme is exogenously givenfor each school and governs the dynamic capacities of groups, each of which has a q-responsivesub-choice function. The overall choice function of a school can be thought of as the union ofchoices with these sub-choice functions of its groups.We offer the COM with respect to dynamic reserves choice functions as an allocation rule.We show that the COM is stable and strategy-proof in our framework. Moreover, the COM re-spects improvements. We introduce a comparison criteria between two monotonic capacity transferschemes. If a monotone capacity transfer scheme transfers at least as many vacancies in every con-tingency compared to another monotone capacity transfer scheme, we say that the first is moreflexible than the second. We show that when capacity transfer scheme of a school becomes moreflexible, while other school choice functions remain unchanged, the outcome of the COM underthe modified profile of choice functions Pareto dominates the outcome of the COM under the orig-inal profile. This result is the main message of our paper, as it describes a strategy-proof Paretoimprovement by making capacity transfers more flexible.
Cumulative Offer Process (COP):
Consider the outcome the COM as denoted by Φ Γ ( P, C ) .For any preference profile P of students, profile of choice functions of schools C , and an ordering Γ of the elements of X , the outcome is determined by the COP with respect to Γ , P and C asfollows: Step 0:
Initialize the set of contracts available to the schools as A = ∅ .17 tep t ≥ : Consider the set U t ≡ (cid:8) x ∈ X \ A t − : i ( x ) / ∈ i (cid:0) C S ( A t − ) (cid:1) and ∄ z ∈ (cid:0) X i ( x ) \ A t − (cid:1) ∪ {∅} such that zP i ( x ) x (cid:9) . If U t is empty, then the algorithm terminates and the outcome is given by C S ( A t − ) . Other-wise, letting y t be the highest-ranked element of U t according to Γ , we say that y t is proposed andset A t = A t − ∪ { y t } and proceed to step t + 1 .A COP begins with no contracts available to the schools (i.e., A = ∅ ). Then, at each step t ,we construct U t , the set of contracts that (1) have not yet been proposed, (2) are not associated tostudents with contracts chosen by schools from the currently available set of contracts, and (3) areboth acceptable and the most-preferred by their associated students among all contracts not yetproposed. If U t is empty, then every student i either has some associated contract chosen by someschool, i.e., i ∈ i (cid:0) C S ( A t − ) (cid:1) , or has no acceptable contracts left to propose, and so the COP ends.Otherwise, the contract in U t that is highest-ranked according to Γ is proposed by its associatedstudent, and the process proceeds to the next step. Note that at some step this process must endas the number of contracts is finite.Letting T denote the last step of the COP, we call A T the set of contracts observed in the COPwith respect to Γ , P , and C . Definition 8.
A choice function C s ( · ) satisfies the irrelevance of rejected contracts (IRC)condition if for all Y ⊂ X , for all z ∈ X \ Y , and z / ∈ C s ( Y ∪ { z } ) = ⇒ C s ( Y ) = C s ( Y ∪ { z } ) . Hatfield and Milgrom (2005) introduce the substitutability condition, which generalizes theearlier gross substitutes condition of
Kelso and Crawford (1982).
Definition 9.
A choice function C s ( · ) satisfies substitutability if for all z, z ′ ∈ X , and Y ⊆ X , z / ∈ C s ( Y ∪ { z } ) = ⇒ z / ∈ C s ( Y ∪ { z, z ′ } ) . Definition 10.
A choice function C s ( · ) satisfies the law of aggregate demand (LAD) if Y ⊆ Y ′ = ⇒| C s ( Y ) | ≤ | C s ( Y ′ ) | .The following definitions are from Hatfield and Kominers (2019). A completion of a many-to-one choice function C s ( · ) of school s ∈ S is a choice function C s ( · ) , such that for all Y ⊆ X , either C s ( Y ) = C s ( Y ) or there exists a distinct z, z ′ ∈ C s ( Y ) such that i ( z ) = i ( z ′ ) . If a choice function C s ( · ) has a completion that satisfies the substitutability and IRC condition, then we say that C s ( · ) is substitutably completable. If every choice function in a profile C = ( C s ( · )) s ∈ S is substitutablycompletable, then we say that C is substitutably completable. We denote by C S ( Y ) ≡ ∪ s ∈ S C s ( Y ) the set of contracts chosen by the set of schools from a set of contracts Y ⊆ X . C s ( · , f s , q s ) be a dynamic reserve choice function given the precedence sequence f s andthe capacity transfer scheme q s . We define a related choice function C s ( · , f s , q s ) . Given a set ofcontracts Y ⊆ X , C s ( Y, f s , q s ) denotes the set of chosen contracts from set Y and is determinedas follows: • Given q s and Y = Y ⊆ X , let Y ≡ c s ( Y , q s , f s (1)) be the set of chosen contracts withprivilege f s (1) from Y . Let r = q s − | Y | be the number of vacant slots. The set ofremaining contracts is then Y = Y \ Y . • In general, let Y k = c sk ( Y k − , q ks , f s ( k )) be the set of chosen contracts with privilege f s ( k ) from the set of available contracts Y k − , where q ks = q ks ( r , ..., r k − ) is the dynamic capacityof group of slots k as a function of the vector of the number of unfilled slots ( r , ..., r k − ) .Let r k = q ks − | Y k | be the number of vacant slots. The set of remaining contracts is then Y k = Y k − \ Y k − . • Given Y = Y ⊆ X and the capacity of the first group of slots q s , we define C s ( Y, f s , q s ) = c s ( Y , q s , f s (1)) ∪ ( λ s ∪ k =2 c sk ( Y k − , q ks ( r , ..., r k − ) , f s ( k ))) .The difference between C s ( · ) and C s ( · ) is as follows: In the computation of C s ( · ) , if a contract ofa student is chosen by some group of slots then his/her other contracts are removed for the rest ofthe choice procedure. However, in the computation of C s ( · ) this is not the case. According to thechoice procedure C s ( · ) , if a contract of a student is chosen, say, by group of slots k , then his/herother contracts will still be available for the following groups of slots.The following proposition shows that C s ( · ) defined above is the completion of the dynamicreserves choice function C s ( · ) . Proposition 1. C s ( · ) is a completion of C s ( · ) . Proof.
Let f s and q s be the precedence sequence and capacity transfer scheme of school s ∈ S ,respectively. Take an offer set Y = Y ⊆ X and assume there is no pair of contracts z, z ′ ∈ Y such that i ( z ) = i ( z ′ ) and z, z ′ ∈ C s ( Y, f s , q s ) . We want to show that C s ( Y, f s , q s ) = C s ( Y, f s , q s ) . Let Y j be the set of contracts chosen by group of slots j and let Y j be the set of contractsthat remains in the choice procedure after group j selects according to dynamic reserve choicefunction C ( · ) . Similarly, let Y j be the set of contracts chosen by group of slots j and let Y j be the set of contracts that remains in the choice procedure after group j selects according tothe completion C ( · ) . Notice that Y = Y . Let r j and r j be the number of vacant slots ingroup of slots j in the choice procedures C s ( Y, f s , q s ) and C s ( Y, f s , q s ) , respectively. Also, let19 js ( r , ..., r j − ) and q js ( r , ..., r j − ) denote the dynamic capacities of group of slots j under choiceprocedures C s ( Y, f s , q s ) and C s ( Y, f s , q s ) , respectively.Given ¯ q s and Y = Y , we have Y = c s ( Y , ¯ q s , f s (1)) = Y by the construction of C s .Moreover, r = r and q s ( r ) = q s ( r ) .Suppose that for all j ∈ { , ..., k − } we have Y j = Y j . We need to show that it holds for groupof slots k , i.e., Y k = Y k . Since the chosen set is the same in every group from to k − under C ( · ) and C ( · ) , the number of remaining slots in each group is the same as well. Then, the dynamiccapacity of the group of slots k are the same under choice procedures C s ( Y, f s , q s ) and C s ( Y, f s , q s ) ,i.e., q ks ( r , ..., r k − ) = q ks ( r , ..., r k − ) . Since there are no two contracts of an agent chosen by C s ( Y, f s , q s ) , one can deduce that all of the remaining contracts of agents, whose contracts werechosen by previous sub-choice functions, are rejected by c sk ( Y k − , q ks ( r , ..., r k − ) , f s ( k )) . Therefore,the IRC of the sub-choice function implies that c sk ( Y k − , q ks ( r , ..., r k − ) , f s ( k )) = c sk ( Y k − , q ks ( r , ..., r k − ) , f s ( k )) . Hence, we have Y k = Y k , r k = r k , and q k +1 s ( r , ..., r k ) = q k +1 s ( r , ..., r k ) .Since in each group of slots the same sets of contracts are chosen by the dynamic reserve choicefunction and its completion, the result follows. Proposition 2. C s ( · ) satisfies the IRC. Proof.
For any Y ⊆ X such that Y = C s ( Y, f s , q s ) , let x be one of the rejected contracts, i.e., x ∈ Y \ C s ( Y, f s , q s ) . To show that the IRC is satisfied, we need to prove that C s ( Y, f s , q s ) = C s ( Y \ { x } , f s , q s ) . Let ˜ Y = Y \ { x } . Let ( Y j , ¯ r j , Y j ) be the sequence of the set of chosen contracts, the numberof vacant slots, and the remaining set of contracts for group j = 1 , ..., λ s from Y under C ( · ) .Similarly, let ( ˜ Y j , ˜ r j , ˜ Y j ) be the sequence of the set of chosen contracts, the number of vacant slots,and the remaining set of contracts for group j = 1 , ..., λ s from ˜ Y under C ( · ) .For the first group of slots, since the sub-choice functions satisfy the IRC, we have Y = ˜ Y .Moreover, ¯ r = ˜ r and Y \ { x } = ˜ Y . By induction, for each j = 2 , ..., k − , assume that Y j = ˜ Y j , ¯ r j = ˜ r j , and Y j \ { x } = ˜ Y j . We need to show that the above equalities hold for j = k . Since , x / ∈ C s ( Y, f s , q s ) and thesub-choice functions satisfy the IRC condition we have c sk ( Y k − , q ks ( r , ..., r k − ) , f s ( k )) = c sk ( ˜ Y k − , q ks (˜ r , ..., ˜ r k − ) , f s ( k )) . k in the choice processes beginning with Y and Y ∪ { x } , respectively. By our inductive assumption that ¯ r j = ˜ r j for each j = 2 , ..., k − , thedynamic capacity of group k is the same under both choice processes. The number of remainingslots is the same as well, i.e., ¯ r k = ˜ r k . Finally, we know that x is chosen from the set ˜ Y k − ∪ { x } ,then we have Y k = ˜ Y k ∪ { x } . Since for all j ∈ { , ..., λ s } , Y j = ˜ Y j , we have C s ( Y, f s , q s ) = C s ( ˜ Y , f s , q s ) . Hence, C s ( · , f s , q s ) satisfies the IRC. Proposition 3. C s ( · ) satisfies the substitutability. Proof.
Consider an offer set Y ⊆ X such that Y = C s ( Y, f s , q s ) . Let x be one of the rejectedcontracts, i.e., x ∈ Y \ C s ( Y, f s , q s ) , and let z be an arbitrary contract in X \ Y . To showsubstitutability, we need to show that x / ∈ C s ( Y ∪ { z } , f s , q s ) . Consider ˜ Y = Y ∪ { z } . Let ( Y j , r j , Y j ) be the sequence of the set chosen contracts, the number ofvacant slots, and the set of remaining contracts for group of slots j = 1 , ..., λ s from Y under C ( · ) .Similarly, let ( ˜ Y j , ˜ r j , ˜ Y j ) be the sequence of the set chosen contracts, the number of vacant slots,and the set of remaining contracts for group of slots j = 1 , ..., λ s from ˜ Y under C ( · ) . There aretwo cases to consider: Case 1 z ∈ ˜ Y \ C s ( ˜ Y , f s , q s ) .In this case, the IRC of C s implies C s ( ˜ Y , f s , q s ) = C s ( Y, f s , q s ) . Therefore, x / ∈ C s ( ˜ Y , f s , q s ) . Case 2 z ∈ C s ( ˜ Y , f s , q s ) .Let j be the group of slots such that z ∈ ˜ Y j . By the IRC of sub-choice functions, x / ∈ ˜ Y j = Y j ,for all j ′ = 1 , ..., j − . Moreover, ˜ Y j ′ − = Y j ′ − ∪ { z } and ˜ r j ′ = r j ′ , for all j ′ = 1 , ..., j − .First note that the dynamic capacity of group j is the same under choice procedures beginningwith Y = Y and Y ∪ { z } = ˜ Y , respectively. This is because the number of unused slots fromgroups to j − are the same under the two choice procedures. We know that z is chosen exactlyat group j in the process beginning with ˜ Y . There are two cases here: (a) The dynamic capacity of group j is exhausted in the process beginning with Y . In thiscase, by choosing z from ˜ Y another contract, we say that say y ∈ ˜ Y is rejected even though y was chosen at group j in the process beginning with Y .21 b) The dynamic capacity of group j is not exhausted in the choice process beginning with Y .In this case, z is chosen at group j in the process beginning with ˜ Y without rejecting any contractthat was chosen in the process beginning with Y at group j .In the case of ( a ) , | c sj ( Y j − , q js ( r , ..., r j − ) , f s ( j )) | = q js ( r , ..., r j − ) and z ∈ c sj ( ˜ Y j − , q js ( r , ..., r j − ) , f s ( j )) implies that there exists a contract y such that y ∈ c sj ( Y j − , q js ( r , ..., r j − ) , f s ( j )) \ c sj ( ˜ Y j − , q js (˜ r , ..., ˜ r j − ) , f s ( j )) . This implies that ˜ Y j = Y j ∪ { y } . Since the capacity of group j is exhausted under both choiceprocesses, the number of vacant slots for group j will be in both choice processes. Thus, thecapacity will be the same for group j + 1 under both.Notice that x / ∈ Y j = ⇒ x / ∈ ˜ Y j because c sj ( Y j − , q js ( r , ..., r j − ) , f s ( j )) ∪ { z } \ { y } = c sj ( ˜ Y j − , q js (˜ r , ..., ˜ r j − ) , f s ( j )) . In case ( b ) , we have | c sj ( Y j − , q js ( r , ..., r j − ) , f s ( j )) | < q js ( r , ..., r j − ) . Hence, r j > . Then, since the sub-choice functions are responsive, we have c sj ( ˜ Y j − , q js (˜ r , ..., ˜ r j − ) , f s ( j )) = { z } ∪ c sj ( Y j − , q js ( r , ..., r j − ) , f s ( j )) . Therefore, x / ∈ Y j = ⇒ x / ∈ ˜ Y j . We also have r j = ˜ r j + 1 . Moreover, the set of remaining contracts under both choice processes willbe the same, i.e., ˜ Y j = Y j . The facts r j ′ = ˜ r j ′ for all j ′ = 1 , ..., j − and r j = ˜ r j + 1 implies—bythe monotonicity of capacity transfer schemes—that either q j +1 s ( r , ..., r j ) = q j +1 s (˜ r , ..., ˜ r j ) q j +1 s ( r , ..., r j ) = 1 + q j +1 s (˜ r , ..., ˜ r j ) hold.Suppose now that for all γ = j, ..., k − we have that either h ˜ Y γ = Y γ ∪ { e y } f or some e y and q γ +1 s (˜ r , ..., ˜ r γ ) = q γ +1 s ( r , ..., r γ ) i or h ˜ Y γ = Y γ and q γ +1 s (˜ r , ..., ˜ r γ ) ≤ q γ +1 s ( r , ..., r γ ) ≤ q γ +1 s (˜ r , ..., ˜ r γ ) i . We have already shown that it holds for γ = j and we will now show that it also holds for γ = k .We will first analyze the former case. By inductive assumption, we have ˜ Y k − = Y k − ∪ { e y } for some contract e y . If e y is not chosen from the set e Y k − then exactly the same set of contractswill be chosen from Y k − and ˜ Y k − since the capacities of group k are the same under both choiceprocesses and the sub-choice function satisfies the IRC condition. Then, we will have ˜ Y k = Y k ∪{ e y } .Moreover, since the number of vacant slots at group k will be the same under both processes, wewill have q k +1 s ( r , ..., r j ) = q k +1 s (˜ r , ..., ˜ r j ) . If e y is chosen from the set ˜ Y k − , we have two sub-cases,depending on if the dynamic capacity of group k is exhausted under the choice process beginningwith Y . If it is not exhausted, then we will have c sk ( ˜ Y k − , q ks (˜ r , ..., ˜ r k − ) , f s ( k )) = { e y } ∪ c sk ( Y k − , q ks ( r , ..., r k − ) , f s ( k )) , which implies that ˜ Y k = Y k . Moreover, we will have r k = ˜ r k + 1 . The monotonicity of capacitytransfer scheme implies that q k +1 s (˜ r , ..., ˜ r k ) ≤ q k +1 s ( r , ..., r k ) ≤ q k +1 s (˜ r , ..., ˜ r k ) . The first inequality follows from the fact that ˜ r i ≤ r i for all i = 1 , ..., k . The second inequalityfollows from the second condition of the monotonicity of the capacity transfer schemes.On the other hand, if the dynamic capacity of group k is exhausted in the choice procedurebeginning with Y , then choosing e y from the set ˜ Y k − implies that there exists a contract y that ischosen from Y k − but rejected from ˜ Y k − . Then, we will have ˜ Y k = Y k ∪ { y } since the sub-choicefunction is q-responsive and group k ’s capacities are the same under both choice processes. In thiscase, we will have r k = ˜ r k = 0 . Since ˜ r i ≤ r i for all i = 1 , ..., k , we will have q k +1 s (˜ r , ..., ˜ r k ) ≤ q k +1 s ( r , ..., r k ) from the first condition of the monotonicity of the capacity transfer scheme. Since q ks (˜ r , ..., ˜ r k − ) = q ks ( r , ..., r k − ) and ˜ r k = r k , we will have q k +1 s (˜ r , ..., ˜ r k ) ≥ q k +1 s ( r , ..., r k ) by thesecond condition of the monotonicity of capacity transfer schemes. In the second condition of the monotonicity of the capacity transfer schemes, if the number of vacant slots is
23e will now analyze the latter case in which we have ˜ Y k − = Y k − and either q ks ( r , ..., r k − ) = q ks (˜ r , ..., ˜ r k − ) or q ks ( r , ..., r k − ) = 1 + q ks (˜ r , ..., ˜ r k − ) .If q ks ( r , ..., r k − ) = q ks (˜ r , ..., ˜ r k − ) , then given that ˜ Y k − = Y k − , we will have ˜ Y k = Y k . Thisalso implies r k = ˜ r k . Moreover, we obtain q k +1 s ( r , ..., r k ) = q k +1 s (˜ r , ..., ˜ r k ) by the monotonic-ity of capacity transfer scheme. Note that ˜ r i ≤ r i for all i = 1 , ..., k implies q k +1 s ( r , ..., r k ) ≥ q k +1 s (˜ r , ..., ˜ r k ) by the first condition of the monotonicity of capacity transfers. The second condi-tion of the monotonicity of capacity transfers implies q k +1 s ( r , ..., r k ) ≤ q k +1 s (˜ r , ..., ˜ r k ) .If q ks ( r , ..., r k − ) = 1 + q ks (˜ r , ..., ˜ r k − ) , then given ˜ Y k − = Y k − , we have two sub-cases here. Sub-case 1. If c sk ( ˜ Y k − , q ks (˜ r , ..., ˜ r k − ) , f s ( k )) = c sk ( Y k − , q ks ( r , ..., r k − ) , f s ( k )) , then we will have ˜ Y k = Y k . Also, the monotonicity of capacity transfer scheme implies that q k +1 s (˜ r , ..., ˜ r k ) ≤ q k +1 s ( r , ..., r k ) ≤ q k +1 s (˜ r , ..., ˜ r k ) . Sub-case 2. If c sk ( ˜ Y k − , q ks (˜ r , ..., ˜ r k − ) , f s ( k )) ∪ { y ∗ } = c sk ( Y k − , q ks ( r , ..., r k − ) , f s ( k )) for some y ∗ , then we will have ˜ Y k = Y k ∪ { y ∗ } . Moreover, the monotonicity of capacity transferschemes in this case implies that q k +1 s ( r , ..., r k ) = q k +1 s (˜ r , ..., ˜ r k ) . This is because given ˜ r i ≤ r i for all i = 1 , ..., k the first condition of the monotonicity of the capacitytransfers implies that q k +1 s ( r , ..., r k ) ≥ q k +1 s (˜ r , ..., ˜ r k ) . On the other hand, the second condition ofthe monotonicity of the capacity transfers implies that q k +1 s ( r , ..., r k ) ≤ q k +1 s (˜ r , ..., ˜ r k ) .Since x / ∈ Y k , we will have x / ∈ ˜ Y k for all k = 1 , ..., λ s . Thus, we can conclude that x / ∈ C s ( Y ∪ { z } , f s , q s ) , which tells us that the completion C s satisfies the substitutability condition. Proposition 4. C s ( · ) satisfies the LAD. written as the dynamic capacity of the group minus the number of chosen contracts then we will have the following:the dynamic capacity of the group k + 1 in the choice process beginning with Y minus the dynamic capacity of thegroup k + 1 in the choice process beginning with Y ∪ { z } = ˜ Y must be less than or equal to the summation of thedifference of the number of chosen contracts from group to group k , which is 0 in this specific case. roof. Consider two sets of contracts Y and ˜ Y such that Y ⊆ ˜ Y ⊆ X . Let f s and q s be theprecedence sequence and the capacity transfer scheme of school s ∈ S . We want to show that | C s ( Y, f s , q s ) |≤| C s ( ˜ Y , f s , q s ) | . Let ( Y j , r j , Y j ) be the sequences of sets of chosen contracts, numbers of vacant slots and setsof remaining contracts for groups j = 1 , ..., λ s under choice processes beginning with Y = Y .Similarly, let ( ˜ Y j , ˜ r j , ˜ Y j ) be the sequences of sets of chosen contracts, numbers of vacant slots andsets of remaining contracts for groups j = 1 , ..., λ s under choice processes beginning with ˜ Y = ˜ Y .For the first group with capacity q s , since the sub-choice function is q-responsive (and thusimplies the LAD), we have | Y | = | c s ( Y , q s , f s (1)) |≤| c s ( ˜ Y , q s , f s (1)) | = | ˜ Y | . Then, it implies that r = q s − | Y |≥ ˜ r = q s − | ˜ Y | . Moreover, we have Y ⊆ ˜ Y . To see this,consider a y ∈ Y . It means that y / ∈ Y . If y is not chosen from a smaller set Y , then it cannotbe chosen from a larger set ˜ Y because sub-choice function is q-responsive (hence, substitutable).Suppose that ˜ r j ≤ r j and Y j ⊆ ˜ Y j hold for all j = 1 , ..., k − . We need to show that both ofthem hold for group k .Given that ˜ r j ≤ r j for all j = 1 , ..., k − , the first condition of the monotonicity implies that q ks ( r , ..., r k − ) ≥ q ks (˜ r , ..., ˜ r k − ) . The second condition of the monotonicity puts an upper boundfor the difference between q ks ( r , ..., r k − ) and q ks (˜ r , ..., ˜ r k − ) . For group k | Y k | − | ˜ Y k |≤| Y k | − | c sk ( Y k − , q ks (˜ r , ..., ˜ r k − , f s ( k )) | because | ˜ Y k | = | c sk ( ˜ Y k − , q ks (˜ r , ..., ˜ r k − ) , f s ( k )) |≥| c sk ( Y k − , q ks (˜ r , ..., ˜ r k − ) , f s ( k )) | by the q-responsiveness of the sub-choice function. We then have | Y k | − | c sk ( Y k − , q ks (˜ r , ..., ˜ r k − , f s ( k )) |≤ q ks ( r , ..., r k − ) − q ks (˜ r , ..., ˜ r k − ) . This follows from q-responsiveness because | Y k | − | c sk ( Y k − , q ks (˜ r , ..., ˜ r k − , f s ( k )) | is the dif-ference between the number of chosen contracts when the capacity is (weakly) increased from q ks (˜ r , ..., ˜ r k − ) to q ks ( r , ..., r k − ) . Hence, the difference | Y k | − | c sk ( Y k − , q ks (˜ r , ..., ˜ r k − , f s ( k )) | cannot exceed the increase in the capacity which is q ks ( r , ..., r k − ) − q ks (˜ r , ..., ˜ r k − ) . Therefore, nowwe have | Y k | − | ˜ Y k |≤ q ks ( r , ..., r k − ) − q ks (˜ r , ..., ˜ r k − ) . q ks (˜ r , ..., ˜ r k − ) − | ˜ Y k |≤ q ks ( r , ..., r k − ) − | Y k | , which is ˜ r k ≤ r k .Given that Y k − ⊆ ˜ Y k − and q ks ( r , ..., r k − ) ≥ q ks (˜ r , ..., ˜ r k − ) , we will have Y k ⊆ ˜ Y k . For anexplanation, consider a contract x ∈ Y k . That means that x ∈ Y k − but x is not chosen from Y k − when the capacity is q ks ( r , ..., r k − ) , i.e., x / ∈ c sk ( Y k − , q ks ( r , ..., r k − ) , f s ( x )) . When the capacity isreduced to q ks (˜ r , ..., ˜ r k − ) and the set Y k − is expanded to ˜ Y k − , x cannot be chosen because thesub-choice function is q-responsive. Hence, it must be the case that x ∈ ˜ Y k .Now let η j = r j − ˜ r j . As we just proved above, η j ≥ for all j = 1 , ..., λ s . Plugging r j = q js ( r , ..., r j − ) − | Y j | and ˜ r j = q ks (˜ r , ..., ˜ r k − ) − | ˜ Y j | in η j = r j − ˜ r j gives us | ˜ Y j | = q js ( r , ..., r j − ) − q js (˜ r , ..., ˜ r j − )+ | Y j | + η j . Summing both the right and left hand sides for j = 1 , ..., λ s yields λ s X j =1 | ˜ Y j | = λ s X j =1 | Y j | + λ s X j =2 (cid:2) q js ( r , ..., r j − ) − q js (˜ r , ..., ˜ r j − ) (cid:3) + λ s X j =1 η j . Since each η j ≥ , we have λ s X j =1 | ˜ Y j |≥ λ s X j =1 | Y j | + λ s X j =2 (cid:2) q js ( r , ..., r j − ) − q js (˜ r , ..., ˜ r j − ) (cid:3) . Also, we know that q js ( r , ..., r j − ) ≥ q js (˜ r , ..., ˜ r j − ) for all j = 2 , ..., λ s by the first condition of themonotonicity of the capacity transfer scheme as, r i ≥ ˜ r i for all i = 1 , ..., j − (Notice that for j = 1 , the capacity is fixed to q s under both processes.) Therefore, we have λ s X j =1 | ˜ Y j |≥ λ s X j =1 | Y j | , which means | C s ( Y, f s , q s ) |≤| C s ( ˜ Y , f s , q s ) | . Proof of Theorem 1
In Proposition 1 we showed that each dynamic reserve choice function has a completion. Propo-sitions 2 and 3 show that the completion satisfies the IRC and substitutability conditions, respec-tively. Then, by Theorem 2 of
Hatfield and Kominers (2019), there exists a stable outcome withrespect to the profile of schools’ choice functions.26 roof of Theorem 2
In Proposition 4 we showed that the substitutable completion satisfies the LAD. Then, by theTheorem 3 of
Hatfield and Kominers (2019), the COM is (weakly) group strategy-proof for stu-dents.
Proof of Theorem 3
Assume, toward a contradiction, that the COM does not respect unambiguous improvements.Then, there exists a student i ∈ I , a preference profile of students P ∈ × i ∈ I P i , and priorityprofiles Π and Π such that Π is an unambiguous improvement over Π for student i and ϕ i ( P ; Π) P i ϕ i ( P ; Π) . Let ϕ i ( P ; Π) = x and ϕ i ( P ; Π) = x . Consider a preference e P i of student i according to which theonly acceptable contract is x , i.e., e P i : x − ∅ i . Let e P = ( e P i , P − i ) . We will first prove the followingclaim: Claim: ϕ i ( e P ; Π) = x = ⇒ ϕ i ( e P ; Π) = x .Proof of the Claim: Consider the outcome of the COM under priority profile Π given thepreference profile of students e P . Recall that the order in which students make offers has no impacton the outcome of the COP. We can thus completely ignore student i and run the COP until itstops. Let Y be the resulting set of contracts. At this point, student i makes an offer for his onlycontract x . This might create a chain of rejections, but it does not reach student i . So, his contract x is chosen by s ( x ) by, say, the group k with respect to the precedence sequence f s ( x ) of school s ( x ) . Now consider the COP under priority profile Π . Again, we completely ignore student i andrun the COP until it stops. The same outcome Y is obtained, because the only difference betweenthe two COPs is student i ’s position in the priority rankings. At this point, student i makes anoffer for his only contract x . If x is chosen by the same group k , then the same rejection chain (ifthere was one in the COP under the priority profile Π ) will occur and it does not reach student i ;otherwise, we would have a contradiction with the case under priority profile Π . The only otherpossibility is the following: since student i ’s ranking is now (weakly) better under π s ( x ) comparedto π s ( x ) , his contract x might be chosen by group l < k . Then, it must be the case that r l = 0 in the COP under both priority profiles Π and Π . Therefore, by selecting x , the group l mustreject some other contract. Let us call this contract y . If no contract of student i ( y ) = j is chosenbetween groups l and k , then, by the q-responsiveness of sub-choice functions, the groups’ chosensets between l and k under both priority profiles are the same. Hence, the number of remainingslots would be the same. In this case, y is chosen in the group k . Thus, if a rejection chain starts,27t will not reach student i ; otherwise, we could have a contradiction due to the fact that x waschosen at the end of the COP under priority profile Π . A different contract of student j cannotbe chosen between groups l and k ; otherwise, the observable substitutability of dynamic reserveschoice function of school s ( x ) would be violated. Therefore, if any contract of student j is chosenby these groups between l and k , it must be y . If y is chosen by a group that precedes k , then itmust replace a contract—we call it z . By the same reasoning, no other contract of student i ( z ) can be chosen before group k ; otherwise, we would violate the observable substitutability of thedynamic reserve choice function of school s ( x ) . Proceeding in this fashion leads the same contractin group k to be rejected and initiates the same rejection chain that occurs under priority profile Π . Since the same rejection chain does not reach student i under priority profile Π , it will notreach student i under priority profile Π , which ends our proof for the claim.Since ϕ i ( P ; Π) = x and ϕ i ( P ; Π) = x such that xP i x , if student i misreports and submits e P i under priority profile Π , then she can successfully manipulate the COM. This is a contradictionbecause we have already established that the COM is strategy-proof. Proof of Theorem 4
Consider school s ∈ S with a precedence sequence f s and a target capacity vector ( q s , ..., q λ s s ) . Let e q s and q s be two capacity transfer schemes that are compatible with the precedence sequence f s and the target capacity vector ( q s , ..., q λ s s ) . Suppose that the following two conditions hold: • there exists l ∈ { , ..., λ s } and (ˆ r , ..., ˆ r l − ) ∈ Z l − , such that e q ls (ˆ r , ..., ˆ r l − ) = 1+ q ls (ˆ r , ..., ˆ r l − ) ,and • for all j ∈ { , ..., λ s } and ( r , ..., r j − ) ∈ Z j − , if j = l or ( r , ..., r j − ) = (ˆ r , ..., ˆ r l − ) , then e q js ( r , ..., r j − ) = q js ( r , ..., r j − ) .Let e C s and C s be dynamic reserves choice functions e C s ( · , f s , e q s ) and C s ( · , f s , q s ) , respectively. Let e C = (cid:16) e C s , C − s (cid:17) and C = ( C s , C − s ) . Let the outcomes of the cumulative offer algorithm at (cid:16) P, e C (cid:17) and ( P, C ) be e Z and Z , respectively. If e Z = Z , then there is nothing to prove because it meansthe capacity flexibility of school s does not bite.Suppose that e Z = Z . That is, the capacity flexibility of school s bites, which means that thereis a student who was rejected under C s who is no longer rejected under e C s . We now define an improvement chains algorithm that starts with outcome Z . Since the capacity flexibility bites, thevector (ˆ r , ..., ˆ r l − ) must occur in the choice procedure of school s . Dynamic reserves choice functions satisfy observable substitutability condition of
Hatfield et al. (2019). Werefer readers to
Hatfield et al. (2019) for the definitions of observable offer processes and observable substitutability.Since dynamic reserves choice functions have substitutable completion that satisfies the size monotonicity, it satisfiesobservable substitutability. tep 1: Consider students who prefer ( s, f s ( l )) to their assignments under Z , i.e., e I ( s,f s ( l ))1 = { i ∈ I | ( s, f s ( l )) P i Z i } . We choose π s -maximal student in e I ( s,f s ( l ))1 (if any), call her e i , and assign her e x = ( e i , s, f s ( l )) .Update the outcome to e Z = Z ∪ { e x } \ z where z is the contract student e i receives under Z .If ( s ( z ) , t ( z )) = ∅ , then the improvement process ends and we have e Z = e Z = Z ∪ { e x } .Otherwise, we move to Step 2 because by assigning e i to ( s, f s ( l )) we create a vacancy in school s ( z ) within the privilege t ( z ) .If e I ( s,f s ( l ))1 = ∅ , then the number of vacant slots at the last group accepting students in type f s ( l ) will increase by one. When the capacity transfer scheme of school s does not transfer thisextra vacancy to any other group following the last group in type f s ( l ) in the computation of C s ( Z s , f s , e q s ) , the improvement chain process ends and we have e Z = Z . If the extra slot is trans-ferred to the group l ′ that follows the last group in type f s ( l ) in the computation of C s ( Z s , f s , e q s ) ,then we consider students who prefer ( s, f s ( l ′ )) over their assignments under Z , i.e., I ( s,f s ( l ′ ))1 = { i ∈ I | ( s, f s ( l ′ )) P i Z i } . We choose π s -maximal student in I ( s,f s ( l ′ ))1 (if there is any), call her e i , and assign her e x =( e i , s, f s ( l ′ )) . Update the outcome to e Z = Z ∪ { e x } \ z where z is the contract e i receives under Z . If ( s ( z ) , t ( z )) = ∅ , then the improvement process ends and we have e Z = e Z = Z ∪ { e x } .Otherwise, we move to Step 2. Because assigning e i to ( s, f s ( l ′ )) creates a vacancy in school s ( z ) within the privilege t ( z ) .If e I ( s,f s ( l ′ ))1 = ∅ , then the number of vacant slots at the last group that accepts students in type f s ( l ′ ) will increase by one. If the capacity transfer scheme of school s does not transfer this extravacancy to any other group following the last group that accepts students of type f s ( l ′ ) in thecomputation of C s ( Z s , f s , e q s ) , then the improvement chain process ends and we have e Z = Z . Ifthe extra slot is transferred to the group l ′′ that follows the last group that accepts students intype f s ( l ′ ) in the computation of C s ( Z s , f s , e q s ) , then we consider students who prefer ( s, f s ( l ′′ )) over their assignments under Z , and so on.Since school s has finitely many groups, Step 1 ends in finitely many iterations. If no extrastudent is assigned to school s by the end of Step 1, then the improvement chains algorithm endsand we have e Z = Z . If an extra student is assigned to school s by the end of Step 1, then we moveon to Step 2. Step t>1:
Consider students who prefer ( s ( z t − ) , t ( z t − )) to their assignments under e Z t − , i.e., e I ( s ( z t − ) , t ( z t − )) t = { i ∈ I | ( s ( z t − ) , t ( z t − )) P i ( e Z t − ) i } .
29e choose π s ( z t − ) -maximal student in e I ( s ( z t − ) , t ( z t − )) t , call her e i t , and assign her e x t = ( e i t , s ( z t − ) , t ( z t − )) .Update the outcome to e Z t = e Z t − ∪ { e x t } \ z t where z t is the contract student e i t receives under e Z t − .If ( s ( z t − ) , t ( z t − )) = ∅ , then the improvement algorithm ends and we have e Z = e Z t = e Z t − ∪{ e x t } . Otherwise, we move to Step t + 1 . Because assigning e i t to ( s ( z t − ) , t ( z t − )) creates a vacancyin school s ( z t ) within type t ( z t ) .If e I ( s ( z t − ) , t ( z t − )) t = ∅ , then the number of vacant slots at the last group that accepts studentsin type f s ( z t − ) will increase by one. If the capacity transfer scheme of school s ( z t − ) does nottransfer this extra capacity to any other group following the last group that accepts studentsin type t ( z t − ) in the computation of C s ( z t − ) (( e Z t − ) s ( z t − ) , f s ( z t − ) , q s ( z t − ) ) , then the improvementchains process ends and we have e Z = e Z t − . If the extra slot is transferred to the group ofslot m that follows the last group that accepts students in type t ( z t − ) in the computation of C s ( z t − ) (( e Z t − ) s ( z t − ) , f s ( z t − ) , q s ( z t − ) ) , then we consider students who prefer ( s ( z t − ) , f s ( z t − ) ( m )) over their assignments under e Z t − , i.e., e I ( s ( z t − ) ,f s ( zt − ) ( m )) t = { i ∈ I | ( s ( z t − ) , f s ( z t − ) ( m )) P i ( e Z t − ) i } . We choose π s ( z t − ) -maximal student in e I ( s ( z t − ) ,f s ( zt − ( m )) t , call her e i t , and assign her e x t = ( e i t , s ( z t − ) , f s ( z t − ) ( m )) .Update the outcome to e Z t = e Z t − ∪ { e x t } \ z t where z t is the contract student e i t receives under e Z t − .If ( s ( z t − ) , f s ( z t − ) ( m )) = ∅ , then the improvement algorithm ends and we have e Z = e Z t = e Z t − ∪ { e x t } . Otherwise, we move to Step t + 1 . Because assigning e i t to ( s ( z t − ) , f s ( z t − ) ( m )) createsa vacancy in school s ( z t ) within type t ( z t ) .If e I ( s ( z t − ) , t ( z t − )) t = ∅ , then the number of vacant slots at the last group that accepts studentsin type f s ( z t − ) will increase by one. If the capacity transfer scheme of school s ( z t − ) does nottransfer this extra capacity to any other group following the last group that accepts students intype f s ( z t − ) ( m ) in the computation of C s ( z t − ) (( e Z t − ) s ( z t − ) , f s ( z t − ) , q s ( z t − ) ) , then the improvementchains process ends and we have e Z = e Z t − . If the extra slot is transferred to the group of slot m ′ that follows the last group that accepts students in type f s ( z t − ) ( m ) in the computation of C s ( z t − ) (( e Z t − ) s ( z t − ) , f s ( z t − ) , q s ( z t − ) ) , then we consider students who prefer ( s ( z t − ) , f s ( z t − ) ( m ′ )) over their assignments under e Z t − , and so on.Since school s ( z t − ) has finitely many groups , Step t ends in finitely many iterations. If noextra student is assigned to school s ( z t − ) by the end of Step t , then the improvement chainsalgorithm ends and we have e Z = e Z t − . If an extra student is assigned to school s ( z t − ) by the endof Step t , then we move on to Step t + 1 .This process ends in finitely many iterations because there are finitely many contracts andwhen we move to the next step it means a student is made strictly better off. Also, notice that nostudent is worse off during the execution of the improvement chains algorithm. The improvementalgorithm, by construction, starts with the outcome Φ( P, C ) and ends at Φ( P, e C ) . Hence, we have Φ i ( P, e C ) R i Φ i ( P, C ) for all i ∈ I . 30e define the sequence of capacity transfer schemes and dynamic reserve choice functionsfor school s ∈ S : (( q s ) , ( q s ) , ... ) and ( C s ( Y, f s , ( q s ) ) , C s ( Y, f s , ( q s ) ) , ... ) . Let the sequence Φ( P, C ) , Φ( P, C ) ,... denote the outcomes of the COPs at profiles ( P, ( C s ( · , f s , ( q s ) ) , C − s )) and ( P, ( C s ( · , f s , ( q s ) ) , C − s )) ,..., respectively. Hence, by construction, we have Φ i ( P, C a +1 ) R i Φ i ( P, C a ) for all i ∈ I and a ≥ . By the transitivity of weak preferences, we have Φ i ( P, e C ) R i Φ( P, C ) for all i ∈ I . Proof of Theorem 5
Our proof is constructive. We first define an associated type space. Let X be the set of allcontracts. We define a distinct “type” for each contract in X . Let g : X → T = { τ , ..., τ | X | } bea bijective function. The interpretation of the g function is that the artificial type of a contract x ∈ X is g ( x ) ∈ { τ , ..., τ | X | } . Therefore, each contract in X is associated with a distinct (artificial)type.Consider a slot b l ∈ B s with priority order π b l . Let | π b l | denote the number of contracts thatthe slot b l finds acceptable, i.e., ranks higher than the null contract which corresponds to remainingunassigned. Let x l , x l ,..., x | π bl | l be the acceptable contracts for slot b l such that x l π b l x l π b l · · · π b l x | π bl | l . For the slot b l in school s in the true market, we create a sequence of slots — | π b l | many slots— inthe associated market, i.e., { b l , ...b | π bl | l } . The initial capacity of b l is , i.e., q b l = 1 , and the initialcapacities of b l , b l , ..., b | π bl | l are , i.e., q b kl = 0 for all k = 2 , ..., | π b l | . Define r b kl such that r b kl = 0 if slot b kl is filled and r b kl = 1 if slot b kl remains vacant. The dynamic capacity of the slot b kl , for all k = 2 , ..., | π b l | , is defined as q b kl ( r b l , ..., r b k − l ) = r b k − l . That is, if the slot b k − l remains vacant, thenthe capacity of the slot b kl becomes 1. Note that if a slot b k − l is filled, then the dynamic capacityof slots that come after b k − l become .Each slot b kl is associated with a sub-choice rule c sb kl ( · , q b kl , · ) that is defined as follows: Thesub-choice rule c sb kl ( · , q b kl , · ) can only considers contracts with artificial type g − ( x kl ) , therefore onlythe contract x kl . Given a set of contracts Y ⊆ X , c sb kl ( Y, q b kl , g − ( x kl )) = { x kl }∅ if x kl ∈ Y and q b kl = 1 ,otherwise . Note that c sb kl is a q-responsive choice function. We now describe a dynamic reserves choice rule e C s ( · ) that is outcome equivalent to the slot-specific choice rule C s ( · ) . Let Y ⊆ X be a set ofcontracts. Step 1
Consider slots { b , b , ..., b | π b | } in this step.31 tep 1.1 Apply the sub-choice function c sb . If a contract is chosen, then end Step 1, andmove to Step 2 due to the capacity transfer rule described above. Otherwise, move to Step 1.2. Step 1.2
Apply the sub-choice function c sb . If a contract is chosen, then end Step 1, andmove to Step 2 due to the capacity transfer rule described above. Otherwise, move to Step 1.3.This process continues in sequence. If a contract chosen in Step 1, then all of the contractsassociated with the student whose contract is chosen is removed for the rest of the procedure. Let y be the chosen contract in this step. Then, the set of remaining contracts is Y \ Y i ( y ) . Step n ≥ Consider slots { b n , b n , ..., b | π bn | n } in this step. Step n.1
Apply the sub-choice function c sb n . If a contract is chosen, then end Step n, andmove to Step ( n + 1) due to the capacity transfer rule described above. Otherwise, move to Stepn.2. Step n.2
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