Matching with Generalized Lexicographic Choice Rules
aa r X i v : . [ ec on . T H ] A p r Matching with Generalized Lexicographic Choice Rules ∗ Orhan Aygün † and Bertan Turhan ‡ Abstract
Motivated by the need for real-world matching problems, this paper formulates alarge class of practical choice rules,
Generalized Lexicographic Choice Rules (GLCR),for institutions that consist of multiple divisions. Institutions fill their divisions se-quentially, and each division is endowed with a sub-choice rule that satisfies classical substitutability and size monotonicity in conjunction with a new property that we in-troduce, quota monotonicity . We allow rich interactions between divisions in the formof capacity transfers . The overall choice rule of an institution is defined as the union ofthe sub-choices of its divisions. The cumulative offer mechanism (COM) with respectto GLCR is the unique stable and strategy-proof mechanism. We define a choice-basedimprovement notion and show that the COM respects improvements . We employ thetheory developed in this paper in our companion paper,
Aygün and Turhan (2020),to design satisfactory matching mechanisms for India with comprehensive affirmativeaction constraints.
Keywords : Matching with contracts, lexicographic choice, dynamic capacity, sta-bility, strategy-proofness, respecting improvements.
JEL Codes : C78, D47. ∗ Date:
First version: July 2019, this version: April 2020.Some of the ideas in this paper were originally part of the first version of our earlier working paper “DynamicReserves in Matching Markets: Theory and Applications.” The authors thank the participants at theEconomics of Matching Conference at Boğaziçi University, as well as Aysun Aygün, Jenna Blochowicz,Selman Erol, Guillaume Hearinger, Ravi Jagadeesan, Onur Kesten, Scott Duke Kominers, and Rakesh Vohra.Turhan gratefully acknowledges the hospitality of Carnegie Mellon University and Boğaziçi University. † [email protected]; Boğaziçi University, Department of Economics, Natuk Birkan Building, Be-bek, Istanbul 34342, Turkey. ‡ [email protected]; Iowa State University, Department of Economics, Heady Hall, 518 Farm HouseLane, Ames, IA 50011, USA. Introduction
Many real-life problems involve matching agents to institutions that are composed of divisionssuch as firms, hospitals, or schools. These divisions are endowed with their own choicerules. That is, each division has a well-defined choice rule that selects from any given setof alternatives and a capacity. In some applications, choice rules are induced from a strictranking of alternatives. In many others, the choices are more complicated and cannot begenerated so easily. To make it precise, consider a choice rule that selects applicants onthe basis of their merit scores but also requires that the division select a minimum numberof disabled applicants whenever possible. Such a choice rule is obviously not a responsivechoice rule.Moreover, in many real-world institutions, there are cross-division constraints, in thesense that the number of available positions in a division might depend on the numberof applicants hired by other divisions. For example, consider a business school whosedepartments—economics, finance, marketing, etc.— are in the job market to hire new fac-ulty. Suppose that the finance group hires first, followed by the economics department, whichis then followed by the marketing department. The number of available positions for themarketing group might be increased if either the finance or the economics department hirefewer new faculty than what they initially planned, provided that the total budget of thebusiness school is not exceeded.In this paper, we study a many-to-one matching with contracts that incorporates a theoryof choice of institutions that are composed of divisions. We formulate a new and practicalfamily of choice rules, Generalized Lexicographic Choice Rules (GLCR), for institutions.Each institution has a total capacity, i.e., a number of available positions, and a pre-specifiedlinear order at which it fills its divisions. Each division is endowed with a choice rule, i.e.,a sub-choice rule . Each sub-choice rule has two inputs: the set of available options and its(dynamic) capacity. For each division, both of the inputs depend on the choices made bythe divisions that precede it. The set of available options can be thought of as the set ofremaining options from the choices of divisions that precede it. The dynamic capacity ofeach division is a function of the number of unfilled seats of the divisions that precede itgiven by an exogenously specified capacity transfer function. The overall choice rule of aninstitution is then defined as the union its divisions’ sub-choice rules. The collection of thesesub-choice rules and the capacity transfer scheme identify an overall choice rule.We impose three conditions on sub-choice rules:
Substitutability, size monotonicity and These are choice rules for which there is a rational strict preference relation that always selects the q − best elements whenever available. Such choice rules are referred to as “ q-responsive .” uota monotonicity. A choice rule is substitutable if no two alternatives x and y are com-plementary, in the sense that gaining access to x makes y desirable. A choice rule is sizemonotonic if it chooses weakly more alternatives whenever the set of available alternativesexpands. A choice rule satisfies quota monotonicity if the following conditions hold: (1) Ifthere is an increase in capacity, then the choice rule selects every alternative it chose be-forehand, given any set of alternatives, and (2)
If the capacity of a division increases by κ ,then the difference between the number of alternatives chosen under increased capacity andthe initial capacity cannot exceed κ . There are many non-responsive choice rules that arecrucial for real-world applications and satisfy these three conditions . One such family ofchoice rules in GLCR is introduced in our companion paper, Aygün and Turhan (2020), inthe context of matching problems in India with complex reservation constraints.We impose a mild condition, monotonicity, on capacity transfer functions, à la
Westkamp (2013). The monotonicity condition requires that (1) whenever weakly more seats are leftunfilled in every division preceding the j th division, weakly more slots should be availablefor the j th division, and (2) an institution cannot decrease its total capacity in responseto increased demand in some divisions. Our companion paper, Aygün and Turhan (2020),introduces several such monotonic capacity transfer functions in the context of comprehensiveaffirmative action constraints in college admissions and public sector job hiring in India.We show how markets with generalized lexicographic choice rules can be cleared by thecumulative offer mechanism (COM). To illustrate this, we borrowed the novel observabilitytheory of
Hatfield, Kominers and Westkamp (2019), henceforth
HKW (2019). We prove thatwhen sub-choice rules satisfy substitutability, size monotonicity, and quota monotonicity,and when the capacity transfer functions are monotonic, then the overall choice rules ofinstitutions satisfies: the irrelevance of rejected contracts condition (Proposition 1), theobservable substitutability (Proposition 2), the observable size monotonicity (Proposition3), and the non-manipulation via contractual terms (Proposition 4).Our first main result, Theorem 1, follows by the characterization result of
HKW (2019).The authors show that when choice rules of institutions are observably substitutable, ob-servably size monotonic and non-manipulable via contractual terms, the COM is the uniquemechanism which is stable and strategy-proof (for agents). Therefore, in marketplaces inwhich institutions’ choice rules can be modeled, as in the GLCR family, the COM is theuniquely stable and strategy-proof mechanism.We define a choice-based notion of improvement. We say that a choice rule of a division is an improvement over another choice rule for an individual if (1) whenever the latter Note that the responsiveness of a choice rule implies substitutability, size monotonicity, and quotamonotonicity.
Aygün and Turhan (2020) in the context of comprehensive affirmative action in India foradmission to publicly funded educational institutions and government sponsored jobs. How-ever, the theory we develop in this paper might be invoked to design centralized marketplacesbeyond the Indian case, such as in the implementation of the 2015 School inclusion Law inChile (
Correa et al.,
Aygün and Bó,
Practical Application: College Admission and Government Job Re-cruitment in India India has been using one of the most comprehensive affirmative action policies in the world fordecades. This policy is embedded in its constitution. There are two types of reservations inIndia: vertical (also called social ) and horizontal (also called special ) reservations . Verticalreservations have been provided as a level playing field for historically disadvantaged castesand tribes. At each institution, certain fractions of available seats are reserved for peoplefrom Scheduled Castes (SC),
Scheduled Tribes (ST) and
Other Backward Classes (OBC). Theremaining members of society are collectively categorized under the
General Category (GC) .Within each vertical category, horizontal reservations are implemented for specific groups,such as disabled people , women , people from hill areas , etc. For each horizontal reservationcategory, a certain minimum number of such individuals must be admitted within each Two recents papers,
Sönmez and Yenmez (2019a&b), study affirmative action in India with compre-hensive affirmative action. The authors formulate the complex Indian affirmative action constraints, i.e., “vertical” and “horizontal” reservations constraints.
Sönmez and Yenmez (2019a) analyze shortcomings ofthe choice procedure given in the Supreme Court judgement in
Anil Kumar Gupta vs. State of U. P. (1995).The authors provide an alternative choice rule to resolve its shortcomings.
Sönmez and Yenmez (2019b)criticize the widespread practice of tentative allocation of the open positions and propose a new design. Aygün and Turhan (2018) consider vertical reservations only and assume away horizontal reservationsfor simplicity. See also
Aygün and Turhan (2017). In 2019, the Union Government of India approved 10 percent reservation in government jobs and publiclyfunded educational institutions for the
Economically Weaker Section (EWS) in the GC. The EWS is asubcategory of people belonging to the GC having an annual family income less than a certain amount.
Aygün and Turhan (2020) formulates a sub-choice function for divisions (i.e., verticalreserve categories) that respects meritocracy while also satisfying horizontal reservations. Weshow that these non-responsive sub-choice rules are substitutable , size monotonic , and quotamonotonic . We design different monotonic capacity transfer functions depending on theapplication in India. Therefore, following the theory developed in this paper, we show thatthe COM can satisfactorily clear several different two-sided many-to-one matching marketsin India with comprehensive affirmative action constraints. Our approach in Aygün andTurhan (2020) is different than
Sönmez and Yenmez ’s (2019a&b) approach. In particular,we consider • applicants’ preferences not only over institution but also over vertical category theyare admitted under , and • reverting unfilled OBC seats to open-category seats, which is also called OBC de-reservation.Therefore, the theory we developed in this paper allows us to formulate in the Indian affir-mative action problem in its full generality. Related Literature
The matching problem with generalized lexicographic choice rules is a special case of thematching with contracts model of
Fleiner (2003) and
Hatfield and Milgrom (2005).
HKW (2019) characterize when stable and strategy-proof matching is possible in many-to-onematching setting with contracts. The authors introduce three novel conditions—observablesubstitutability, observable size monotonicity, and non-manipulability via contractual terms—and show that when these conditions are satisfied, the COM is the unique mechanism that isstable and strategy-proof (for agents). We invoke their observability theory in the followingsense: Each choice rule in the GLCR class satisfies observable substitutability, observablesize monotonicity, and non-manipulation via contractual terms conditions of
HKW (2019),as well as the irrelevance of rejected contracts in
Aygün and Sönmez (2013). Hence, followingtheir characterization result, the COM is the unique stable and strategy-proof mechanismin our environment. It is optional for SC, ST and OBC applicants to report the vertical category they belong. Manycandidates from these groups do not reveal their caste and tribe membership and utilize its benefits.
Aygünand Turhan (2020) provide further evidence on why candidates have preferences also over the categorythrough which they are admitted.
Hatfield, Kominers and Westkamp (2017). Theauthors introduce a model of institutional choice in which each institution has a set ofdivisions, as in our setting, and flexible allotment capacities that vary as a function of theset of contracts available. Each institution is modeled as having an allotment function thatdetermines how many positions are allocated to each division, given the set of availablecontracts. Our capacity transfer functions are different than their allotment functions withrespect to their very definitions and the assumptions imposed.
Aygün and Turhan (2018) study a model of dynamic reserves that is similar to the lexico-graphic choice rules considered in this paper. However, the sub-choice rules are q-responsiv ein their setting, whereas in the GLCR family the sub-choices might not be q-responsive.Hence, the model of
Aygün and Turhan (2018) cannot accommodate the matching problemsin India with both vertical and horizontal reservations, even though it can accommodate thematching problems with vertical reservations only.Our paper is related to
Afacan (2017), which is the first to define a choice-based im-provement notion. We also define a choice-based improvement notion in the absence ofpriorities. The difference between his notion and ours is critical, because he assumes thatoverall choice rules satisfy unilateral substitutability, size monotonicity, and the irrelevanceof rejected contracts conditions. Both the unilateral substitutability and size monotonicityof overall choice rules might be violated in our setting. His analysis relies on the existenceof the agent-optimal stable matching while ours—our proof for Theorem 2—does not.This paper also contributes to the literature on matching with distributional constraints.A very incomplete list includes
Abdulkadiroğlu and Sönmez (2003),
Kojima (2012),
Hafalıret al. (2013),
Echenique and Yenmez (2015),
Kamada and Kojima (2015, 2017 and 2018),
Kojima et al. (2018), and
Goto et al. (2017).We study lexicographic choice rules. Other important work studying lexicographic choiceincludes
Alva (2016),
Kominers and Sönmez (2016) , Chambers and Yenmez (2018),
Doğanet al. (2018), and
Avataneo and Turhan (2020) among others.
There is a finite set of agents I = { i , ..., i n } and a finite set of institutions S = { s , ..., s m } .There is a finite set of contracts X . Each contract x ∈ X is associated with an agent i ( x ) and an institution s ( x ) . There may be many contracts for each agent-institution pair. Wecall a set of contracts X ⊆ X an outcome , with i ( X ) ≡ ∪ x ∈ X { i ( x ) } and s ( X ) ≡ ∪ x ∈ X { s ( x ) } .For any i ∈ I ∪ S , we let X i ≡ { x ∈ X | i ∈ { i ( x ) , s ( x ) }} . An outcome X ⊆ X is feasible if | X i |≤ for all i ∈ I . 6ach agent i ∈ I has unit demand over contracts in X i and an outside option ∅ i . Thestrict preference of agent i over X i ∪ {∅ i } is denoted by P i . A contract x ∈ X i is acceptablefor i (with respect to P i ) if xP i ∅ i . Agent preferences over contracts are extended preferencesover outcomes in the natural way.Each institution s ∈ S has multi-unit demand and is endowed with a choice rule C s that describes how s would choose from any offered set of contracts. We let q s denote thephysical capacity of institution s . We assume throughout that for all X ⊆ X and for all s ∈ S , the choice rule C s (1) only selects contracts to which s is a party, i.e., C s ( X ) ⊆ X s ,and (2) selects at most one contract with any given agent and selects at most q s contracts,i.e., C s ( X ) is feasible.For any X ⊆ X and s ∈ S , we denote by R s ( X ) ≡ X \ C s ( X ) the set of contracts that s rejects from X . Stability
A feasible outcome Y ⊆ X is stable if it is1. Individually rational: C s ( Y ) = Y s for all s ∈ S , and Y i R i ∅ i for all i ∈ I .2. Unblocked:
There does not exist a nonempty Z ⊆ ( X \ Y ) , such that Z s ⊆ C s ( Y ∪ Z ) for all s ∈ s ( Z ) and ZP i Y for all i ∈ i ( Z ) .Stability requires that neither agents nor institutions wish to unilaterally walk away fromtheir assignments, and that agents and institutions cannot benefit by recontracting outsideof the match. Mechanisms A mechanism M ( · ; C ) maps preference profiles P = ( P i ) i ∈I to outcomes, given a profile ofinstitutional choice rules C = ( C s ) s ∈S . Unless otherwise stated, we assume that the choicerules of the institutions are fixed and write M ( P ) in place of M ( P ; C ) .A mechanism M is stable if M ( P ) is a stable outcome for every preference profile P . Amechanism M is strategy-proof if for every preference profile P and for each agent i ∈ I ,there is no e P i , such that M ( e P i , P − i ) P i M ( P ) . Cumulative offer mechanisms constitute a particularly important class of mechanisms.In a cumulative offer mechanism, C ⊢ , agents propose contracts according to a strict ordering ⊢ of the elements of X . In every step, some agent who does not currently have a contractheld by any institution proposes his most preferred contract that has not yet been proposed.7hen, each institution chooses its most preferred set of contracts according to its choicerule and holds this set until the next step. When multiple agents are able to propose in thesame step, the agent who actually proposes is determined by the ordering ⊢ . The mechanismterminates when no agent is able to propose; at that point, each institution is assigned the setof contracts it is holding. (We describe cumulative offer mechanisms formally in AppendixB.) In their novel analysis,
HKW (2019) characterized the conditions for institutional choicerules to guarantee the existence of stable and strategy-proof mechanisms. They also showedthat when stable and strategy-proof matching is possible, the outcome of any such mech-anism coincides with that of a cumulative offer mechanism. Moreover, the outcomes of allcumulative offer mechanisms coincide.In their seminal work,
Hatfield and Milgrom (2005) showed that the substitutes and size monotonicity of institutions’ choice rules are sufficient for stable and strategy-proofmatching in many-to-one matching settings with contracts. A choice rule is substitutable ifno two contracts x and y are “complementary” in the sense that gaining access to x makes y desirable. Formally, a choice rule C s satisfies substitutability if for all x, y ∈ X and X ⊆ X , y / ∈ C s ( X ∪ { y } ) implies y / ∈ C s ( X ∪ { x, y } ) . Substitutability is the monotonicity of the rejection function: C s is substitutable if and only if we have R s ( X ) ⊆ R s ( Y ) for allsets of contracts X and Y such that X ⊆ Y .The choice rule of institution s ∈ S is size monotonic if s chooses weakly more contractswhenever the set of available contracts expands. That is, C s is size monotonic if for all con-tracts x ∈ X and sets of contracts X ⊆ X , we have | C s ( X ) |≤| C s ( X ∪ { x } | . Substitutableand size monotonic choice rules satisfy the IRC condition. A choice rule C s satisfies the IRCif for all X ⊂ X and x ∈ X \ X , x / ∈ C s ( X ∪ { x } ) implies C s ( X ) = C s ( X ∪ { x } ) . HKW (2019) showed that for a cumulative offer mechanism to be stable and strategy-proof, substitutability and size monotonicity need only hold during the running of the mecha-nism itself. However, in that case, we also need to rule out intra-institutional manipulation.Following their terminology, we will give definitions for three properties that, if satisfied, The analysis of
Hatfield and Milgrom (2005) implicitly assumes the irrelevance of rejected contracts (IRC) condition. Please see
Aygün and Sönmez (2013) for details.
Alkan (2002) refers to the same conditionas consistency . Hatfield and Milgrom (2005) call size monotonicity the law of aggregate demand . offer process for a given institution s ∈ S , with choice rule C s , is afinite sequence of distinct contracts ( x , x , ..., x M ) , such that for all m = 1 , ..., M , x m ∈ X s .We say that an offer process ( x , x , ..., x M ) for s is observable if, for all m = 1 , ..., M , i ( x m ) / ∈ i ( C s ( { x , ..., x m − } )) , i.e, an observable offer process for institution s is a sequenceof contract offers proposed by agents, such that an agent can propose x m only if that agentis rejected by s when this institution has access to { x , x , ..., x m − } . Definition 1.
A choice rule C s of an institution s ∈ S is observable substitutable if theredoes not exist an observable offer process ( x , ...x M ) for s such that x t / ∈ C s ( { x , ..., x t , ...x M − } but x t ∈ C s ({ x ,..., x t ,..., x M }).In other words, if ( x , ..., x m ) is an observable offer process, choice rule C s satisfies ob-servable substitutability if in an economy where s is the only institution no contract that isrejected at step m − of the cumulative offer process is accepted at step m . This conditionweakens the usual substitutability condition by requiring the set of rejected contracts toexpand only at sets of contracts that can be observed in the cumulative offer process.Proposition 3 of HKW (2019) indicates that if the choice function of every institution isobservably substitutable, then for every preference profile P and any two orderings ⊢ and ⊢ ′ , C ⊢ ( P ) = C ⊢ ′ ( P ) . This implies that all cumulative offer mechanisms are equivalent, i.e.,the cumulative offer process is equivalent to the deferred acceptance mechanism describedby Gale and Shapley (1962).Another important observation of
HKW (2019) is that if choice rule of every institutionis observably substitutable and mechanism M is stable and strategy-proof, then for anypreference profile P , M is equivalent to the cumulative offer mechanism. This implies thatif the choice rules of all institutions satisfy observable substitutability, and we want to provethat a strategy-proof mechanism exists, it is enough to focus on the cumulative offer process,as it is the only candidate. The authors also find that if the choice function of each institutionis observably substitutable, then for any preference profile P , the cumulative offer process isstable.Another desirable property of the cumulative offer mechanism is strategy-proofness foragents. Without it, it is a complicated mechanism for participating agents. Even if themechanism is stable with respect to the reported preferences, it may no longer be stablefor the true preferences. However, observable substitutability is not enough to guaranteestrategy-proofness—for this, two other properties, which are defined below, are required. Definition 2.
A choice rule C s of an institution s ∈ S satisfies observable size monotonicity
9f there does not exist an offer process ( x , ..., x M ) for s such that | C b ( { x , ..., x M } ) | < | C b ( { x , ..., x M − } ) | . In other words, observable size monotonicity requires that the size of the accepted set ofcontracts weakly increases along observable offer processes. This condition weakens the usual size monotonicity condition in that it only has to be satisfied by observable offer processes.
Definition 3.
A choice rule C s of an institution s ∈ S is non-manipulable via contractualterms if there does not exist an ordering ⊢ and preference profile P for agents I , under whichonly contracts with s are acceptable, with some agent i ∈ I , with a preference relation e P i ,for which only contracts with s are acceptable, such that C ⊢ ( e P i , P − i ) P i C ⊢ ( P i , P − i ) . In other words, a choice rule C s satisfies the non-manipulability via contractual termscondition if no agent i ∈ I can profit from reporting a non-truthful preference relation thatonly finds contracts with s acceptable. Cumulative Offer Algorithm
The cumulative offer algorithm, which is the generalization of the agent-proposing deferredacceptance algorithm of
Gale and Shapley (1962), is the central allocation mechanism usedin the matching with contracts framework. We now introduce the cumulative offer processfor matching with contracts. Here, we provide an intuitive description of this algorithm; wegive a more technical statement in Appendix B.In the cumulative offer process , agents propose contracts to institutions in a sequenceof steps l = 1 , , ... : Step 1:
Some agent i ∈ I proposes his most-preferred contract, x ∈ X i . Institution 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 institutions at thebeginning of Step 2. Step 2:
Some agent i ∈ I , for whom no contract is currently held by any institution,proposes his most-preferred contract that has not yet been rejected, x ∈ X i . Institution s ( x ) holds the contract in C s ( x ) ( A s ( x ) ∪{ x } ) and rejects all other contracts in A s ( x ) ∪{ x } ;10nstitutions s ′ = s ( x ) continue to 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 agent i l ∈ I , for whom no contract is currently held by any institution,proposes his most-preferred contract that has not yet been rejected, x l ∈ X i l . Institution s ( x l ) holds the contract in C s ( x l ) ( A ls ( x l ) ∪ { x l } ) and rejects all other contracts in A ls ( x l ) ∪ { x l } ;institutions s ′ = s ( x l ) continue to hold all contracts they held at the end of Step l − . Set A l +1 s ( x l ) = A ls ( x l ) ∪ { x l } and set A l +1 s ′ = A ls ′ for each s ′ = s ( x l ) .If at any time no agent is able to propose a new contract—that is, if all agents forwhom no contracts are on hold have proposed all contracts they find acceptable—then thealgorithm terminates. The outcome of the cumulative offer process is the set of contractsheld by institutions at the end of the last step before termination.In the cumulative offer process, agents propose contracts sequentially. Institutions accu-mulate offers, choosing a set of contracts at each step (according to C s ) to hold from the setof all previous offers. The process terminates when no agent wishes to propose a contract. We now introduce a model of institutional choice in which each institution has a set of divisions and dynamic capacities for those divisions that vary as a function of the numberof unused slots in the preceding divisions. We model an institution s as having a set ofdivisions K s = { , ..., K s } . Each division k ∈ K s has an associated sub-choice rule C sk :2 X × Z ≥ −→ X s that specifies the contracts division k chooses given a set of offers and adynamic capacity to fill them. We require that each division k never chooses more contractsthan its dynamic capacity, i.e., for a set of contracts Y ⊆ X s and a dynamic capacity of κ ,we must have | C sk ( Y ; κ ) |≤ κ .We are now ready to describe GLCR in the course of an algorithm. Algorithm
Given a set contracts Y ≡ Y ⊆ X , a capacity q s for institution s , and a capacity q s fordivision 1, we compute the chosen set C s ( Y ; q s ) in K s steps where division 1 chooses in step1, division 2 chooses in step 2, and so on and so forth. Step 1
Given Y and q s , division 1 chooses C s ( Y ; q s ) . Let r = q s − | C s ( Y ; q s ) | . Let Y ≡ Y \ { x ∈ Y | i ( x ) ∈ i [ C s ( Y ; q s )] } . 11 tep k ( ≤ k ≤ K s ) : Given the set of remaining contracts Y k and its dynamic capac-ity q sk ( r , ..., r k − ) , division k chooses C sk ( Y k ; q sk ( r , ..., r k − )) . Let r k = q sk ( r , ..., r k − ) − | C sk ( Y k ; q sk ( r , ..., r k − )) | . Let Y k +1 ≡ Y k \ { x ∈ Y k | i ( x ) ∈ i [ C sk ( Y k ; q sk ( r , ..., r k − ))] } .The union of divisions’ choices is the institution’s chosen set, i.e., C s ( Y ; q s ) ≡ C s ( Y ; q s ) ∪ K s ∪ k =2 C sk ( Y k ; q sk ( r , ..., r k − )) . A more intuitive description of the above procedure is as follows: Given a set of contracts Y ≡ Y ⊆ X and a capacity q s for division 1, C s ( Y ; q s ) denotes the set of chosen contractsby division 1. We let r = q s − | C s ( Y ; q s ) | be the number of remaining slots in division1. The dynamic capacity of division 2 is then defined as q s = q s ( r ) . We remove everyagent’s contract that was chosen by division 1 for the rest of the procedure. Given the setof remaining contracts Y and its dynamic capacity q s , division 2 chooses C s ( Y ; q s ( r )) .We let r = q s ( r ) − | C s ( Y ; q s ) | be the number of vacant slots from division 2. Ingeneral, given the number of vacant slots r , r , ..., r k − , the dynamic capacity of division k is given by q sk ( r , ..., r k − ) . Given the set remaining contracts Y k and its dynamic capacity q sk ( r , ..., r k − ) , division k chooses C sk ( Y k ; q sk ( r , ..., r k − )) . We let r k = q sk ( r , ..., r k − ) − | C sk ( Y k ; q sk ( r , ..., r k − )) | be the number of vacant slots from division k . All the remainingcontracts of agents whose contracts chosen by division k is removed from Y k for the rest ofthe procedure. Capacity transfer scheme
Given an initial capacity of the first division q s , a capacity transfer scheme of institution s is a sequence of capacity functions q s = ( q s , ( q sk ) K s k =2 ) , where q sk : Z k − −→ Z + for all k ∈ K s and such that q s + q s (0) + q s (0 ,
0) + · · · + q sK (0 , ...,
0) = q s . We also impose a mild condition on capacity transfer functions, à la
Westkamp (2013). Acapacity transfer scheme q s is monotonic if, for all j ∈ { , ..., K s } and all pairs of 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 X m =2 [ q ms (˜ r , ..., e r m − ) − q ms ( r , ..., r m − )] ≤ j − X m =1 [ e r m − r m ] . Monotonicity of capacity transfer schemes requires that (1) whenever weakly more seats12re left unfilled in every division preceding the j th division, weakly more slots should beavailable for the j th division, and (2) an institution cannot decrease total capacity in responseto increased demand in some divisions.The tuple (cid:16) I , S , P, ( C sk ( · , · ) , q s ) s ∈S ,k ∈K s (cid:17) denotes a problem. Note that the collection ofsub-choice rules together with a capacity transfer function fully identify the overall choicerule and are hence regarded as the primitives of the model. Conditions on Divisions’ Sub-Choice Rules
We impose three conditions on divisions’ sub-choice rules:
Substitutability (S), size mono-tonicity (SM), and quota monotonicity (QM).We already defined substitutability and size monotonicity in Section 3. We now introducequota monotonicity.
Definition 4.
A sub-choice function C sk ( · ; · ) satisfies quota monotonicity if for any q, q ′ ∈ Z + such that q < q ′ , for all Y ⊆ X C sk ( Y, q ) ⊆ C sk ( Y, q ′ ) , and | C sk ( Y, q ′ ) | − | C sk ( Y, q ) |≤ q ′ − q. QM requires choice rules to satisfy two conditions. First, given any set of contracts, ifthere is an increase in the capacity, we require the choice rule to select every contract it waschoosing before increasing its capacity. It might choose some additional contracts. Second, ifthe capacity of a division increases by κ , then the difference between the number of contractschosen with the increased capacity and the initial capacity cannot exceed κ .Our first result relates the three conditions imposed on the sub-choice rules to the IRCthat is satisfied by the overall choice rule in the GLCR family. Proposition 1.
Suppose that every division’s sub-choice rule satisfies S, SM, and QM. Then,the institution’s overall choice rule satisfies the IRC condition.
Note that Proposition 1 does not refer to the monotonicity of the capacity transfer func-tions. That is, Proposition 1 holds even when the capacity transfer functions fail the mono-tonicity property. In the proof of Proposition 1, it can be seen that sub-choice rules thatsatisfy the IRC condition are sufficient for the overall choice rule to satisfy the IRC underany given capacity transfer function. Notice that the substitutability and size monotonicityof a choice rule imply the IRC condition (
Aygün and Sönme z, 2013).13ur next result, Proposition 2, relates three conditions imposed on the sub-choice rulestogether with monotonicity of the capacity transfer functions to the observable substitutabil-ity of the overall choice rule.
Proposition 2.
Suppose that every division’s sub-choice rule satisfies S, SM, and QM. If thecapacity transfer scheme is monotonic, then the institution’s overall choice rule is observablysubstitutable.
From the results of
HKW (2019) we know that the COM is stable when the choice ruleof every institution satisfies the observable substitutability. Another desirable property ofallocation mechanisms is strategy-proofness for agents. However, observable substitutabilityis not sufficient to guarantee strategy-proofness of the cumulative offer mechanism. For this,we need to show that choice rules in the GLCR family satisfy observable size monotonicityand non-manipulation via contractual terms.
Proposition 3.
Suppose that every division’s sub-choice rule satisfies S, SM, and QM. If thecapacity transfer scheme is monotonic, then the institution’s overall choice rule is observablysize monotonic.
Our last result in this section relates conditions imposed on sub-choice rules together withthe monotonicity of the capacity transfer functions to the non-manipulation via contractualterms condition of overall choice rules.
Proposition 4.
Suppose that every division’s sub-choice rule satisfies S, SM, and QM. Ifthe capacity transfer scheme is monotonic, then the institution’s overall choice rule is notmanipulable via contractual terms.
We are now ready to present our first main result.
Theorem 1.
Suppose that every division’s sub-choice rule satisfies S, SM, and QM at eachinstitution. If the capacity transfer scheme of each institution is monotonic, then the COMis stable and strategy-proof.
Theorem 1 is useful for practical real-life applications, as it states that a stable andstrategy-proof matching mechanism is possible under the GLCR family of institutionalchoice. The design of different mechanisms for matching problems in India in
Aygün andTurhan (2020), both for college admissions and job matching in the government sector viavertical and horizontal reservations, utilizes Theorem 1.14
Respect for Improvements
Respect for improvement is an attractive property of matching mechanisms. In some settings,especially in meritocratic systems, it is rather crucial. The property is first defined by
Balinski and Sönmez (1999) in a priority-based setting. The authors showed that deferredacceptance respects improvements in the sense that making one student more highly rankedin schools’ priority rankings improves their deferred acceptance outcome. In the matchingwith contracts setting,
Sönmez and Switzer (2013),
Sönmez (2013), and
Kominers andSönmez (2016) introduce choice rule specific improvement notions in the presence of rankinglists. In our setting, we may not have ranking lists. As opposed to these papers, we definea notion of improvement over choice rules regardless of the presence of a ranking list. Thischoice-based improvement notion was first introduced by
Afacan (2017) for overall choicerules.
Definition 5.
A sub-choice rule of division k at institution s e C sk ( · ; · ) is an improvement over C sk ( · ; · ) for agent i if, for any set of contracts X ⊆ X and for any integer κ ∈ Z + , thefollowing hold:1. if x ∈ C sk ( X ; κ ) such that i ( x ) = i , then y ∈ e C sk ( X ; κ ) for some y ∈ X such that i ( y ) = i ;2. if i / ∈ i [ C sk ( X ; κ ) ∪ e C sk ( X ; κ )] , then C sk ( X ; κ ) = e C sk ( X ; κ ) .The first condition states that if a contract of agent i is chosen from a given set under thesub-choice rule C sk , then a contract of the same agent (not necessarily the same one) mustbe chosen under e C sk given that division k has the same capacity under both sub-choice rules.It is important to note here that it is not a problem if agent i prefers x over y . As thecumulative offer algorithm is run, agents make offers in decreasing order of their preferences.If agent i offers y at some point, it means that x was rejected in earlier steps. HKW (2019)show that renegotiation does not take place during a cumulative offer process if institutions’choice rules satisfy observable substitutability. We assume that sub-choice rules satisfy S,SM, and QM. We also assume that capacity transfer functions are monotonic. Thus, by ourProposition 2, institutional overall choice rules are observably substitutable. The secondcondition states that if no contract of agent i is chosen from a given set under both C sk and e C sk , then the chosen sets are the same, given that division k has the same capacity underchoice rules C sk and e C sk .This improvement notion in conjunction with QM imply the following: If x ∈ C sk ( X ; κ ) such that i ( x ) = i , then y ∈ e C sk ( X ; κ ′ ) for some y ∈ X , such that i ( y ) = i for any κ ′ ≥ κ .15ote that QM requires that when the capacity increases, the choice rule selects a supersetof the set it was selecting beforehand.Consider two overall choice rules e C s and C s for institution s . Each rule takes the samemonotonic capacity transfer function q s as input and both of their sub-choice rules satisfyproperties S, SM, and QM. We say that an overall choice rule e C s is an improvement over C s for agent i if e C sk ( · ; · ) is an improvement over C sk ( · ; · ) for agent i at each division k = 1 , ..., K s .Finally, we say that e C ≡ ( e C s ) s ∈ S is an improvement over C ≡ ( C s ) s ∈ S for agent i if e C s isan improvement over C s for agent i at each institution s ∈ S . Definition 6.
Mechanism ϕ respects improvements if for any problem ( P, C ) and e C suchthat e C = ( e C sk ( · ; · ) , q s ) s ∈ S,k ∈K s is an improvement over C = ( C sk ( · ; · ) , q s ) s ∈ S,k ∈K s for agent i where sub-choice rules ( e C sk ( · ; · )) K s k =1 and ( C sk ( · ; · )) K s k =1 satisfy S, SM, and QM at each institution s ∈ S and capacity transfer schemes ( q s ) s ∈ S are all monotonic, ϕ ( P, e C ) R i ϕ ( P, C ) . We are now ready to present our second main result.
Theorem 2.
Suppose that at each institution divisions’ sub-choice rules satisfy S, SM, andQM and capacity transfer function is monotonic. Then, the COM respects improvements.
Theorem 2 has significant implications in real-world applications where a meritocraticsystem is integrated with affirmative action constraints, such as the matching problems inIndia with vertical and horizontal reservations.
Aygün and Turhan (2020) design choice rulesfor divisions for matching problems in India with these constraints by taking meritocraticcomponent into account. The meritocratic sub-choice rule that takes horizontal reservationsinto account we propose is not q-responsive. Yet, we show that it satisfies S, SM, andQM. Moreover, the capacity transfer functions in India—for transferring otherwise vacantOBC slots to others—is shown to be monotonic in our setting. Hence, the COM respectsimprovements in regards to our design.An important implication of respecting improvements in the case of Indian college admis-sions and job matching problems is that it incentivizes applicants to declare all horizontalreservation types they have.
This paper introduces a new and practical family of choice rules motivated by real-life insti-tutional allocation and choice problems. Institutions are divided into divisions where each16ivision is endowed with a choice rule that satisfies S,SM, and QM. Interaction betweendivisions, in the sense of capacity transfers, are allowed. The capacity transfer functions areassumed to be monotonic. The overall choice rule of an institution is then defined as theunion of its divisions’ sub-choices. In many-to-one matching frameworks, the COM, withrespect to such overall choice rules, is stable and strategy-proof. We define a choice-basednotion of improvement and show that the COM respects improvements.Our results can be used to design practical marketplaces. One such example is shownin our companion paper,
Aygün and Turhan (2020), in the context of matching problems inIndia that implement comprehensive affirmative action constraints, i.e., vertical and horizon-tal affirmative action constraints. In our companion paper, we define a sub-choice rule thatsatisfies S, SM, and QM. The sub-choice rule we designed considers additional policy goalsthat are specific to India. The design also includes monotonic capacity transfer functions,so that the COM appears as the unique stable and strategy-proof matching mechanism.Our construction can be used not only for college admissions under constraints in India butalso in job matching processes for government-sponsored job recruitments. The COM alsorespects improvements which has significant implications for the Indian case. We believethe theory we developed in this paper will find other attractive real-life applications beyondIndia.
A. Formal Description of the Cumulative Offer Process
The cumulative offer process associated with proposal order Γ is the following algo-rithm1. Let l = 0 . For each s ∈ S , let D s ≡ ∅ , and A s ≡ ∅ .2. For each l = 1 , , ... Let i be the Γ l − maximal agent i ∈ I , such that i / ∈ i ( ∪ s ∈ S D l − s ) and max P i ( X \ ( ∪ s ∈ S A ls )) i = ∅ i — that is, the first agent in the proposal order who wants to propose a new contract—if such an agent exists. (If no such agent exists, then proceed to Step 3, below.) ( a ) Let x = max P i ( X \ ( ∪ s ∈ S A ls )) i be i ’s most preferred contract that has not been pro-posed. ( b ) Let s = s ( x ) . Set D ls = C s ( A ls ∪ { x } ) and set A l +1 s = A ls ∪ { x } . For each s ′ = s ,set D ls ′ = D l − s ′ and A l +1 s ′ = A ls ′ . 17. Return the outcome Y ≡ ( ∪ s ∈ S D l − s ) = ( ∪ s ∈ S C s ( A ls )) , which consists of contracts held by institutions at the point when no agents want topropose additional contract.Here, the sets D l − s and A ls denote the set of contracts held by and available to institution s at the beginning of the cumulative offer process step l . We say that a contract z is rejected during the cumulative offer process if z ∈ A ls ( z ) but z / ∈ D l − s ( z ) for some l . B. Proofs
Before we prove the results, we first introduce some notation: • If X M = { x , ..., x M } is an observable offer process, we say X m = { x , .., x m } , i.e., X m are the contracts proposed up to step m of the observable offer process X M . • H k ( X m ) denotes the set of contracts available to division k in the computation of C s ( X m ) . • F k ( X m ) = ∪ n ≤ m H k ( X n ) , i.e., F k ( X m ) is the set of all contracts that were available todivision k at some point of offer process X m = { x , ..., x m } . • q m − k ( r , ..., r k − ) is the dynamic capacity of division k at step m − of the observableoffer process and q mk ( e r , ..., e r k − ) is the dynamic capacity of division k at step m of theobservable offer process. • R k ( X m ; q mk ( e r , ..., e r k − )) is the set of contracts rejected by division k at step m of theobservable offer process X m .We first prove the following lemma which will be key for proving our results. Lemma 1.
For all divisions k ∈ { , ..., K s } and for all m ∈ { , ..., M } where M is the laststep of observable offer process X M = { x , ..., x M } :1. C k ( H k ( X m − ); q m − k ( r , ..., r k − )) ⊆ H k ( X m ) .2. C k ( F k ( X m ); q mk ( e r , ..., e r k − )) ⊆ C k ( H k ( X m − ); q m − k ( r , ..., r k − )) ∪ [ H ( X m ) \ H k ( X m − )] .3. C k ( H k ( X m ); q mk ( e r , ..., e r k − )) = C k ( F k ( X m ); q mk ( e r , ..., e r k − )) .4. q m − k ( r , ..., r k − ) ≥ q mk ( e r , ..., e r k − ) .5. R k ( F k ( X m − ); q m − k ( r , ..., r k − )) ⊆ R k ( F k ( X m ); q mk ( e r , ..., e r k − )) . roof of Lemma 1 We use mathematical induction on pairs ( m, k ) ordered in the fol-lowing way: (1 , , (1 , , ..., (1 , K ) , (2 , , (2 , , ..., (2 , K ) , ..., ( M, , ( M, , ..., ( M, K ) . Initial Step:
Consider m = 1 and any division k = 1 , ..., K . Note that X m − = X = ∅ and X = { x } . Since H k ( X ) = H k ( ∅ ) = ∅ , condition (1) holds trivially because ∅ ⊆ H k ( X ) for all k = 1 , ..., K . Condition (2) also holds because it reduces to C k ( F k ( X ) ⊆ H k ( X ) = F k ( X ) . Condition (3) also holds trivially since H k ( X ) = F k ( X ) . Condition (4)holds at the pair (1 , as for the first division the initial capacity is given exogenously, i.e., q s . Condition (5) reduces to R k ( ∅ ) = ∅ ⊆ R k ( F k ( X )) and it trivially holds. Inductive assumption:
Assume that conditions (1)-(5) hold for • every ( m ′ , k ) with m ′ < m and k = 1 , ..., K , • every ( m, k ′ ) with k ′ < k .We need to show that conditions (1)-(5) hold for the pair ( m, k ) . We start with condition(1). (1) Take z ∈ C k ( H k ( X m − ); q m − k ( r , ..., r k − )) . If z is chosen by division k , then itmust have been rejected by all divisions that precede it. Hence, we have ( { x , ..., x m − } ) i ( z ) ⊆ ∩ k ′ Take a set of contracts Y ⊆ X and a contract z ∈ X \ Y , suchthat z / ∈ C s ( Y ∪ { z } ; q s ) . We need to prove that C s ( Y ; q s ) = C s ( Y ∪ { z } ; q s ) . Before startingour proof, note that the substitutability and size monotonicity of divisions’ sub-choice rulesimply they satisfy the IRC.Consider two different choice processes for institution s : one starts with Y and one startswith Y ∪ { z } . Let Y j and e Y j denote the set of contracts division j receives under choiceprocesses starting with Y and Y ∪ { z } , respectively. Note that Y ≡ Y and e Y ≡ Y ∪ { z } .22et r j and e r j denote the number of vacant seats at division k in the choice processes startingwith Y and Y ∪ { z } , respectively.Consider division . Under both choice processes, the given capacity of division is thesame, i.e., q s . Since z / ∈ C s ( Y ∪ { z } ; q s ) we know that z / ∈ C s ( Y ∪ { z } ; q s ) . Since thesub-choice rule of division 1 satisfies the IRC, we have that C s ( Y ∪ { z } ; q s ) = C s ( Y ; q s ) . Wealso have that r = e r since | C s ( Y ∪ { z } ; q s ) | = | C s ( Y ; q s ) | . Inductive assumption: Suppose that for all divisions j = 1 , ..., k − we have that C sj ( Y j ; q sj ( r , ..., r j − )) = C sj ( e Y j ; q sj ( e r , ..., e r j − )) .We will now prove that for division k we have that C sk ( Y k ; q sk ( r , ..., r k − )) = C sk ( e Y k ; q sk ( e r , ..., e r k − )) . The inductive assumption implies that for all j = 1 , ..., k − we have r j = e r j . Hence, wehave that q sk ( r , ..., r k − ) = q sk ( e r , ..., e r k − ) . Since z / ∈ C s ( Y ∪ { z } ; q s ) , we have that z / ∈ C sk ( e Y k ; q sk ( e r , ..., e r k − )) . Note that the inductive assumption also implies that e Y k ≡ Y k ∪ { z } . By the IRC propertyof the sub-choice rules, we have that C sk ( Y k ; q sk ( r , ..., r k − )) = C sk ( e Y k ; q sk ( e r , ..., e r k − )) . Hence, we have C s ( Y ; q s ) = C s ( Y ∪ { z } ; q s ) . Proof of Proposition 2 Consider an institution s ∈ S and observable offer process X ≡ { x , ..., x M } for s . Let X M − be the offer process { x , ..., x M − } . Suppose that y ∈ R s ( X M − ) . Since y is rejected by institution s when s faces the offer process X M − , it mustbe rejected by all divisions k = 1 , ..., K . Let r k and e r k denote the number of unfilled seatsof division k for the choice processes starting with X M − and X M , respectively. Hence, forall k = 1 , ..., K , we have that y ∈ R sk ( H k ( X M − ); q M − k ( r , ..., r k − )) . y ∈ R sk ( F k ( X M − ); q M − k ( r , ..., r k − )) . for all k = 1 , ..., K . By (5) of Lemma 1, we have that R k ( F k ( X M − ); q M − k ( r , ..., r k − )) ⊆ R k ( F k ( X M ); q Mk ( e r , ..., e r k − )) for each division k = 1 , ..., K . This implies that y ∈ R sk ( F k ( X M ); q Mk ( e r , ..., e r k − )) for each division k = 1 , ..., K . Therefore, y / ∈ C sk ( F k ( X M ); q Mk ( e r , ..., e r k − )) . By (3) of Lemma 1, we have that C sk ( H k ( X M ); q Mk ( e r , ..., e r k − )) = C sk ( F k ( X M ); q mk ( e r , ..., e r k − )) . Hence, we have that y / ∈ C sk ( H k ( X M ); q Mk ( e r , ..., e r k − )) for all k = 1 , ..., K . Thus, under the choice procedure that defines C s , we have that y / ∈ R s ( X M ) , as desired. Proof of Proposition 3 Consider an institution s ∈ S and observable offer processes X M − ≡ { x , ..., x M − } and X M ≡ { x , ..., x M } for s . Let r k and e r k be the number ofunfilled slots of division k under choice procedures starting with the observable offer processes X M − and X M , respectively. Let H k ( X M − ) and H k ( X M ) denote the sets of contractsdivision k faces under the choice procedures starting with X M − and X M , respectively.Let F k ( X M − ) = ∪ n ≤ M − H k ( X n ) and F k ( X M ) = ∪ n ≤ M H k ( X n ) , i.e., F k ( X M − ) and F k ( X M ) are the set of all contracts available to division k at some point of offer process X M − = { x , ..., x M − } and X M = { x , ..., x M } , respectively. For the ease of notation, we let r j =( r , ..., r j − ) and e r j = ( e r , ..., r j − ) . Similarly, let r k = ( r , ..., r k − ) and e r k = ( e r , ..., r k − ) .Note that by (3) of Lemma 1 we can replace H sets with F sets as follows: C sj ( H j ( X M − ); q M − j ( r j ) = C sj ( F ( X M − ); q M − j ( r j )) C sj ( H j ( X M ); q Mj ( e r j )) = C sj ( F j ( X M ); q Mj ( e r j )) . By (4) of Lemma 1, we know that q M − k ( r k ) ≥ q Mk ( e r k ) . By the second condition of monotonic capacity transfer functions, we have k X j =1 ( q M − j ( r j ) − q Mj ( e r j )) ≤ k − X j =1 [ r j − e r j ] . Replacing r j = q M − j ( r j ) − | C sj ( F j ( X M − ); q M − j ( r j )) | and e r j = q Mj ( e r j ) − | C sj ( F j ( X M ); q Mj ( e r j )) | gives us the following: k X j =1 [ q M − j ( r j ) − q Mj ( e r j )] ≤ k − X j =1 [ q M − j ( r j ) − | C sj ( F j ( X M − ); q M − j ( r j )) | − q Mj ( e r j )+ | C sj ( F j ( X M ); q Mj ( e r j )) | ] . Readjusting the terms on the right and left sides gives us ≤ q M − k ( r j ) − q Mk ( e r j ) ≤ k − X j =1 [ C sj ( F k ( X M ); q Mk ( e r j )) − C sj ( F j ( X M − ); q M − j ( r j ))] . By QM of sub-choice rules, we have the following: | C sk ( F k ( X M − ); q M − k ( r k )) | − | C sk ( F k ( X M − ); q Mk ( e r k )) |≤ q M − k ( r j ) − q Mk ( e r k ) ≤ k − X j =1 [ C sj ( F k ( X M ); q Mk ( e r j ) − C sj ( F j ( X M − ); q M − j ( r j ))] . 25y SM of the sub-choice rules, we also have the following: | C sk ( F k ( X M − ); q M − k ( r j )) |≤| C sk ( F k ( X M ); q M − k ( r j )) | since, by definition, F k ( X M − ) ⊆ F k ( X M ) . Combining the inequalities above, we obtain | C sk ( F k ( X M − ); q M − k ( r k )) | − | C sk ( F k ( X M ); q Mk ( e r k ) |≤ q M − k ( r j ) − q Mk ( e r j ) ≤ k − X j =1 [ C sj ( F j ( X M ); q Mj ( e r j )) − C sj ( F j ( X M − ); q M − j ( r j ))] . Hence, we have k X j =1 | C sj ( F j ( X M − ); q M − j ( r j )) |≤ k X j =1 | C sj ( F j ( X M ); q Mj ( e r j )) | . Applying (3) of Lemma 1 again gives us k X j =1 | C sj ( H j ( X M − ); q M − j ( r j )) |≤ k X j =1 | C sj ( H j ( X M ); q Mj ( e r j )) | . Since the inequality above holds for all k = 1 , ..., K , we have K X j =1 | C sj ( H j ( X M − ); q M − j ( r j ) |≤ K X j =1 | C sj ( H j ( X M ); q Mj ( e r j )) | , which is the desired conclusion. Proposition 2 of HKW (2019) Suppose that C s is a choice rule for institution s ∈ S that is observably substitutable andmanipulable by agent i ∈ I via contractual terms. In this case, there exists a preferenceprofile P and preferences e P i under which only contracts with s are acceptable, with P i ofthe form P i : z P i · · · P i z M , and e P i of the form e P i : z e P i z e P i · · · e P i z M , such that either 26. C i ( P i , P − i ) = ∅ while C i ( e P i , P − i ) P i ∅ , or2. C i ( e P i , P − i ) = ∅ while C i ( P i , P − i ) e P i ∅ .Before starting to prove Proposition 4 we will first prove a lemma and then introduce someextra notation to ease our proof. Lemma 2. Let X m = { x , ..., x m } and Y n = { y , ..., y n } be two observable offer processes,such that X m ⊆ Y n . Then, for all divisions k = 1 , ..., K , F k ( X m ) ⊆ F k ( Y n ) , and2. R k ( F k ( X m ); q mk ( r , ..., r k − )) ⊆ R k ( F k ( Y n ); q nk ( e r , ..., e r k − )) , and3. q mk ( r , ..., r k − ) ≥ q nk ( e r , ..., e r k − ) , where ( r , ..., r k ) and ( e r , ..., e r k − ) are the vector ofthe number of vacant seats in choice procedures starting with offer sets X m and Y n ,respectively. Proof of Lemma 2 We proceed by mathematical induction on divisions k = 1 , ..., K .For the first division, i.e., k = 1 , F ( X m ) ≡ X m and F ( Y n ) ≡ Y n by definition. Hence,we have by our assumption F ( X m ) ⊆ F ( Y n ) . Thus, (1) is satisfied. Since the capacityof the first division is given, regardless of the offer set, statement (3) is trivially satisfied.The substitutability of the sub-choice rules implies that R ( F ( X m ); q s ) ⊆ R ( F ( Y n ); q s ) .Therefore, (2) is also satisfied for k = 1 . Inductive assumption: Suppose that (1)-(3) are satisfied for all divisions j < k .We now need to show that (1)-(3) hold for division k .We start by showing that (3) holds for k . By our inductive assumption (3) and QM, forall j < k , | C sj ( F j ( X m ); q mj ( r , ..., r j − )) | − | C sj ( F j ( X m ); q nj ( e r , ..., e r j − )) |≤ q mj ( r , ..., r j − ) − q nj ( e r , ..., e r j − ) . Rearranging the terms gives us q mj ( r , ..., r j − ) − | C sj ( F j ( X m ); q mj ( r , ..., r j − )) |≥ q nj ( e r , ..., e r j − ) − | C sj ( F j ( X m ); q nj ( e r , ..., e r j − )) | . By our inductive assumption (1) and the size monotonicity, we have q mj ( r , ..., r j − ) − | C sj ( F j ( X m ); q mj ( r , ..., r j − )) |≥ q nj ( e r , ..., e r j − ) − | C sj ( F j ( Y n ); q nj ( e r , ..., e r j − )) | , r j ≥ e r j for all j < k . Then, by monotonicity of capacity transfer functions, q mk ( r , ..., r k − ) ≥ q nk ( e r , ..., e r k − ) . Hence, (3) holds for division k .To show (1), consider z ∈ F k ( X m ) . There are two cases to consider: i ( z ) / ∈ i [ C j ( F j ( X m ); q mj ( r , ..., r j − ))] , for all j < k .In this case, we know that all contracts of agent i ( z ) are rejected by all divisions thatprecede division k , i.e., ( X m ) i ( z ) ⊆ ∩ j Consider an arbitrary agent i , and let z , z , ..., z L be an arbitrary sequence of contracts in X i . Fix a profile of preferences P − i for all other agents, and let P i and e P i be given by P i : z P i · · · P i z L , e P i : z e P i z e P i · · · e P i z L . We first fix an ordering ⊢ over the set of contracts X . We let X N = { x , ..., x N } be theobservable offer process induced by the COM with ordering ⊢ under the preferences ( P i , P − i ) when institution s is the only institution available. Suppose also that b X = { b x , ..., b x b N } is theobservable offer process induced by the COM with ordering ⊢ under the preferences ( e P i , P − i ) when institution s is the only institution available. Lemma 3. If z / ∈ C s ( b X b N ; q s ) , then R s ( X N ; q s ) ⊆ R s ( b X b N ; q s ) and for all divisions k =1 , ..., K we have that F k ( X N ) ⊆ F k ( b X b N ) . Proof of Lemma 3 We proceed by mathematical induction on pairs ( m, k ) in the followingorder: (1 , , (1 , , ..., (1 , K ) , (2 , , (2 , , ..., (2 , K ) , ..., ( N, , ( N, , ..., ( N, K ) , and at each step we show the following: F k ( X m ) ⊆ F k ( b X b N ) k ( F k ( X m ); q mk ( r , ..., r k − )) ⊆ R k ( F k ( b X b N ); q b Nk (ˆ r , ..., ˆ r k − )) , where q mk ( r , ..., r k − ) and q b Nk (ˆ r , ..., ˆ r k − ) are the dynamic capacity of division k in the choiceprocesses starting with X m and b X b N , respectively.For the base case (1 , , it must be that x is either the highest-ranked contract of someagent i ( x ) = i or x = z . In the former case, x must be offered at some step of the offerprocess b X b N , as it is the best contract agent i ( x ) want to offer. In the latter case, since z isrejected by our assumption, i must offer her second-best contract under e P i , z = x , at somestep in the offer process b X b N . Hence, in both cases, x ∈ b X b N . Since F ( b X b N ) = b X b N , we havethat F ( { x } ) ⊆ F ( b X b N ) . Then, by substitutability of the sub-choice rules, we have that R ( F ( X ); q s ) ⊆ R ( F ( b X b N ); q s ) . We now show that both inclusion relations hold for ( m, k ) if they both hold for • every pair ( m ′ , k ) , such that m ′ < m and k = 1 , ..., K , and • every pair ( m, k ′ ) with k ′ < k .We first show that they hold for ( m, , given that they are satisfied for every pair ( m ′ , k ) ,such that m ′ < m and k = 1 , ..., K . By the inductive assumption for pairs ( m − , k ) , wehave R k ( F k ( X m − ); q m − k ( r , ..., r k − )) ⊆ R k ( F k ( b X b N ); q b Nk (ˆ r , ..., ˆ r k − )) for all k = 1 , ..., K . By the observability of X m we have ( { x , ..., x m − } ) i ( x m ) ⊆ R k ( H k ( X m − ); q m − k ( r , ..., r k − )) for all k = 1 , ..., K . Moreover, by the substitutability of the sub-choice rules, we have ( { x , ..., x m − } ) i ( x m ) ⊆ R k ( H k ( X m − ); q m − k ( r , ..., r k − )) ⊆ R k ( F k ( X m − ); q m − k ( r , ..., r k − )) . Hence, we have R k ( H k ( X m − ); q m − k ( r , ..., r k − )) ⊆ R k ( F k ( X m − ); q m − k ( r , ..., r k − )) . Therefore, given that R k ( F k ( X m − ); q m − k ( r , ..., r k − )) ⊆ R k ( F k ( b X b N ); q b Nk (ˆ r , ..., ˆ r k − )) forall k = 1 , ..., K , we find ( { x , ..., x m − } ) i ( x m ) ⊆ R k ( F k ( b X b N ); q b Nk (ˆ r , ..., ˆ r k − )) , k = 1 , ..., K .By (3) of Lemma 1, there exists an N ′ ≤ b N such that ( { x , ..., x m − } ) i ( x m ) ⊆ R k ( F k ( b X N ′ ); q N ′ k (ˆ r , ..., ˆ r k − )) , for all k = 1 , ..., K . By the observability of the offer process b X b N and the fact that itrepresents all the offers made under the cumulative offer process for ( e P i , P − i ) , there mustexist some step e n at which x m is proposed in b X b N . Recall that b X b N ≡ F ( b X b N ) . Therefore, wehave that x m ∈ b X b N ≡ F ( b X b N ) . Also, by the inductive assumption for the pair ( m − , , wehave that X m − ≡ F ( X m − ) ⊆ F ( b X b N ) . Moreover, since we know that F ( X m ) ≡ X m = { x m } ∪ X m − , we have F ( X m ) ⊆ F ( b X b N ) , which is the first condition we want to show for the pair ( m, . Then, by substitutability ofthe sub-choice rules, we can conclude that R ( F ( X m ); q s ) ⊆ R ( F ( b X b N ; q s ) , which ends our proof for the pair ( m, .We now show that they hold for the pair ( m, k ) when k > , given that they hold forevery pair ( m ′ , k ) such that m ′ < m and k = 1 , ..., K , and every pair ( m, k ′ ) with k ′ < k . Wewill first show that F k ( X m ) ⊆ F k ( b X b N ) . By our inductive assumption on the pair ( m − , k ) ,it is sufficient to show that H k ( X m ) ⊆ F k ( b X b N ) , since F k ( X m ) ≡ F k ( X m − ) ∪ H k ( X m ) . Take z ∈ H k ( X m ) . Since z ∈ H k ( X m ) , all contracts of agent i ( z ) must have been rejected by alldivisions that preceded division k , i.e., z ∈ ∩ k ′ 31y (3) of Lemma 1, we have that C sk ′ ( H k ′ ( b X b N ); q b Nk ′ (ˆ r , ..., ˆ r k ′ − )) = C sk ′ ( F k ′ ( b X b N ); q b Nk ′ (ˆ r , ..., ˆ r k ′ − )) , for all k ′ < k . Therefore, if there were a k ′ < k , such that z ∈ C sk ′ ( H k ′ ( b X b N ); q b Nk ′ (ˆ r , ..., ˆ r k ′ − )) ,that would imply z ∈ C sk ′ ( F k ′ ( b X b N ); q b Nk ′ (ˆ r , ..., ˆ r k ′ − )) , which contradicts z ∈ ∩ k ′ Proof of Proposition 4 By Proposition 2 of HKW (2019), it is sufficient to show thatwhen s is the only institution, the following two conditions hold:1. If i [ C ( P i , P − i )] = ∅ , then either i [ C ( e P i , P − i )] = ∅ or i [ C ( e P i , P − i )] = { z } , and2. If i [ C ( e P i , P − i )] = ∅ , then i [ C ( P i , P − i )] = ∅ .To show (1), note that Lemma 3 implies that if i [ C ( P i , P − i )] = ∅ and z / ∈ C ( e P i , P − i ) , then R s ( X N ) ⊆ R s ( b X b N ) . Moreover, if i [ C ( P i , P − i )] = ∅ , then { z , ..., z L } ⊆ R s ( X N ) and hence { z , ..., z L } ⊆ R s ( b X b N ) . Thus, i [ C ( e P i , P − i )] = ∅ .To show (2) , first notice that by Proposition 1 of HKW (2019) the COM is order in-dependent, since our overall choice rules are observably substitutable and observably sizemonotonic. Hence, we can consider X N and b X b N to be generated by the cumulative offerprocess with respect to the same proposal ordering ⊢ in which all of the agents’ contractsother than i precede all of the contracts associated with i . Under this choice of ⊢ , theremust exist an λ such that1. x m = ˆ x m for all m < λ , 32. x λ = z , and3. ˆ x λ = z .That is, λ is the first step of each cumulative offer process with respect to the order ⊢ atwhich agent i proposes. The offer process b X b N ends with the rejection of the contract z L ,since z L follows all contracts with agents other than i according to our specific ordering ⊢ and the fact that i [ C ( e P i , P − i )] = ∅ . At each step after λ , exactly one contract is newlyrejected, since the overall choice rule of the institution is observable substitutable and sizemonotonic. Formally, the following holds:1. | R s ( b X ˜ m ) \ R s ( b X ˜ m − ) | = 1 for all ˜ m = λ, λ + 1 , ..., b N , and2. z L ∈ R s ( b X b N ) \ R s ( b X b N − ) .For the offer process X N , we must have | R s ( X m ) \ R s ( X m − ) | = 1 for all m = λ, λ +1 , ..., N − .Notice that X λ − = b X λ − . Hence, we have | C s ( X λ − ; q s ) | = | C s ( b X λ − ; q s ) | . Moreover,since | R s ( b X ˜ m ) \ R s ( b X ˜ m − ) | = 1 for all ˜ m = λ, λ + 1 , ..., b N we must have that | C s ( X λ − ; q s ) | = | C s ( b X b N ; q s ) | . Similarly, since | R s ( X m ) \ R s ( X m − ) | = 1 for all m = λ, λ + 1 , ..., N − , we have | C s ( X N − ; q s ) | = | C s ( b X b N ; q s ) | . But, since X N ⊆ b X b N , the observable size monotonicity of C s implies that | C s ( X N ; q s ) |≤| C s ( b X b N ; q s ) | . Therefore, we must have R s ( X N ; q s ) \ R s ( X N − ; q s ) = ∅ . Toward a contradiction, suppose that y ∈ R s ( X N ; q s ) \ R s ( X N − ; q s ) = ∅ and y = z L .Note that y is the least-preferred acceptable contract of agent i ( y ) with respect to P i ( y ) where i ( y ) = i . Then, Lemma 3 implies that there is some step m ∗ ≥ λ , such that y ∈ R s ( b X m ∗ ) \ R s ( b X m ∗ − ) . But, since | R s ( b X m ∗ ) \ R s ( b X m ∗ − ) | = 1 and y is the least preferredacceptable contract for i ( y ) , the cumulative offer process for ( e P i , P − i ) would end at step m ∗ with the rejection of y . This contradicts the fact that the cumulative offer process for ( e P i , P − i ) ends with the rejection of z L . 33 roof of Theorem 1. By Propositions (1)-(4), the overall choice rule of each institutionsatisfies the IRC condition, observable substitutability, observable size monotonicity, andnon-manipulation via contractual terms. Then, by Theorem 4 ( H KW, 2019), the COM isthe unique stable and strategy-proof mechanism. Proof of Theorem 2. Let Φ denote the COM. The contract agent i receives for theproblem ( P, C ) is denoted by Φ i ( P, C ) . Consider a problem ( P, C ) and e C , which is animprovement over C for agent i such that each sub-choice rule satisfies S, SM, and QMunder both C and e C . Moreover, each institution’s capacity transfer function is monotonicunder both e C and C . Let x and y be the contracts agent i receives under the cumulativeoffer processes with regards to C and e C , respectively. That is, Φ i ( P, C ) = x and Φ i ( P, e C ) = y. Toward a contradiction, assume that xP i y . Note that yR i ∅ (with the possibility that y = ∅ )because the cumulative offer algorithm returns an individually rational match for agents.Consider the false preference, e P i , for agent i such that x is the only contract agent i . Thatis, e P i : x − ∅ . We will first show that Φ i ( e P i , P − i , e C ) = x . We prove this claim in two steps. In thefirst step, we will show that Φ i ( e P i , P − i , C ) = x . Toward a contradiction, suppose that Φ i ( e P i , P − i , C ) = ∅ . Recall that Φ i ( P, C ) = x . Hence, agent i can report P i instead of e P i andand obtain x . That means the COM is manipulable at the preference profile ( e P i , P − i ) . How-ever, we established in Theorem 1 that the COM is strategy-proof. This is a contradiction.Therefore, Φ i ( e P i , P − i , C ) = x must hold.In the second step, we will show that Φ i ( e P i , P − i , e C ) = x . In the first step, we showed that Φ i ( e P i , P − i , C ) = x . Recall that e C is an improvement over C for agent i . Let s ( x ) = s . Weknow that the cumulative offer process is order independent. Let agent i be the last agent topropose contracts. Then, by the definition of improvements, the cumulative offer processesunder choice profiles e C and C are identical without agent i . Since e C s is an improvementover C s for agent i , the division in institution s ( x ) that selects x at ( e P i , P − i , C ) selects x at ( e P i , P − i , e C ) , as well. Hence, we have Φ i ( e P i , P − i , e C ) = x .Therefore, agent i has an incentive to report e P i at problem ( P, e C ) . This contradicts thefact that the COM is strategy-proof under our assumptions. 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