1.1 Related work

Our mechanism achieves strategyproofness and fairness through a sequential reduction of artificial quotas combined with the application of DA. The idea of sequentially reducing maximum quotas is also used by Fragiadakis and Troyan [23], albeit for a different objective. In their model, students are categorized into different types, and the objective is to fulfill type-specific minimum and maximum quotas.

Several papers have investigated strategyproofness in matching models with quotas constraints, although they consider a setting distinct from ours [24–27]. These papers address a student-course setting where only students express preferences over courses, and both students and courses have quotas or capacities. In contrast, our setting involves preferences from both students and schools, with only schools having constraints. A study closer to our setting is conducted by Kamada and Kojima [28], who investigate student-school matchings and characterize the constraints referred to as general upper bounds, ensuring the existence of a student-optimal and fair matching. However, they consider constraints imposed on individual schools whereas our constraints are symmetric over the set of schools. Moreover, they allow constraints to depend on the identity of students, whereas our constraints are solely based on the number of students.

Two streams of work exist on matching with distributional constraints. One stream scrutinizes constraints that arise from real-life applications, such as regional maximum quotas [2], individual/regional minimum quotas [4, 5], affirmative actions [9, 11], and ratio constraints [21]. The other stream mathematically investigates an abstract and general class of constraints, exemplified by those represented by a substitute choice function [29], an M-convex set [20], and a general upper bound [28]. The results derived from the first stream are more accessible due to their practical grounding and direct applicability to real-life scenarios. In contrast, the results obtained from the second stream are rigorously organized mathematically and have broader applicability.

Our paper belongs to the second stream. In particular, it builds upon our previous work [21], which belongs to the first stream focusing on ratio constraints (Definition 15). These constraints specify the acceptable minimum ratio between the least and most popular schools (Definition 15). The class of distributional constraints we study is a strict generalization of ratio constraints, and it is defined based on a well-known concept in discrete convex analysis: M-convex sets. Although our QRDA mechanism uses a similar idea as a mechanism for ratio constraints, such as the sequential reduction of quotas, the proof techniques for our main results, notably strategyproofness with Theorem 2, differ significantly. This distinction arises due to the reliance of proof techniques for ratio constraints on specific constraint structures, rendering them inapplicable to our setting. Consequently, these proof techniques need to undergo adaptations to suit our setting.

1.2 Outline

Section 2 introduces the model used to study the proposed matching problem and defines the desirable properties. In section 3, the family of M-convex sets is introduced, and we go beyond this family by defying the union of symmetric M-convex constraints. Subsequently, the main matching mechanism, QRDA, is described in Section 4, accompanied by proofs of its properties. Additionally, QRDA is theoretically compared with a baseline mechanism, ACDA. In Section 5, we generalize the symmetric constraints to the asymmetric space, demonstrating QRDA’s adaptability. Empirical comparisons of several matching mechanisms are conducted in Section 6. Finally, this paper concludes in Section 7.

All technical notations used in this paper are listed in Table 1.

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Table 1. Notations used in this paper.

https://doi.org/10.1371/journal.pone.0289965.t001

3.1 M-convex sets

We first present the well-studied class of M-convex constraints. Assume distributional constraints are defined by a set of vectors , i.e., is school-feasible if . A set of distributional constraints is said to be M-convex if it can be defined as an M-convex set.

Definition 9 (M-convex Set). Let χ_i denote an m-element unit vector, where its i-th element is 1 and all other elements are 0. A set of m-element vectors forms an M-convex set, if for all , for all i such that , exists such that and hold.

M-convexity is a discrete analogue of a convex set in a continuous domain. The concept of an M^♮-convex set [38] is closely related to a convex set. An M-convex set represents a set of maximum elements within an M^♮-convex set. Intuitively, an M-convex set does not contain any hollow. Definition 9 states that for any two vectors, , we can find another element of by moving one step from ν towards ν′ (i.e., ), as well as by moving one step from ν′ toward ν (i.e., ). For instance, consider a standard school choice market comprising two schools and five students, with distributional constraints forming a feasible set denoted as . Upon examining two feasible vectors ν¹ = (1, 4) and ν² = (4, 1), we observe that does not satisfy the definition of M-convexity (Definition 9), since and , but is not in . However, by adding (3, 2) to , we obtain a school-feasible set {(1, 4), (4, 1), (2, 3), (3, 2)} which is M-convex. The new school-feasible vector set corresponds to the distributional constraints where each school’s maximum and minimum quotas are four and one.

Kojima et al. [20] show that when the set of feasible matchings has a matroid structure [39] and the schools’ soft preferences (which aggregate the preference of each school over the students) satisfy certain easy-to-verify conditions (e.g., they can be represented as a sum of values associated with individual contracts), the Generalized Deferred Acceptance mechanism is strategyproof and fair. In our model, distributional constraints are defined on the allocation vector of , denoted as , rather than . This distinction implies that, for each school c, all contracts related to it are equivalent in terms of hard distributional constraints. In contrast to Kojima et al. [20], the preference of each school over students is represented as a soft preference, which is separated from hard distributional constraints. Our model mandates that each student must be assigned to a school, and all schools are acceptable to all students, and vice versa. Consequently, the set of feasible matchings (specifically, the set of feasible contracts and their subsets) forming a matroid structure is tantamount to the set of feasible vectors being an M-convex set.

3.2 Symmetric M-convex sets

The concept of symmetry is motivated by settings where schools exhibit comparable sizes, resulting in similar constraints for all schools. Thus, the set of distributional constraints is symmetric over the set of schools.

Definition 10 (Symmetry). Set is symmetric if for all , any permutation of (ν₁, …, ν_m) also belongs to .

We identify several distributional constraints that can be represented by a symmetric M-convex set. Uniform min/max quotas constraints impose minimum quotas on schools, which guarantee a minimum number of students in each school, as well as the capacity limits of schools that are generally imposed in traditional matching theory.

Definition 11 (Uniform Min/Max Quotas Constraints). Let p and q respectively be the minimum and maximum quotas, i.e., the number of students assigned in each school must be between p and q. Then the feasible set is given as .

It is easy to verify that the general individual min/max quotas constraints can be represented by an M-convex set. When all the schools have identical min/max quotas, becomes symmetric.

Introducing another class of constraints that can be represented by a symmetric M-convex set, namely, symmetric distance constraints, requires us to define the most balanced vectors.

Definition 12 (Most Balanced Vectors). Vector is most balanced if for each i ∈ [m], ν_i is either ⌊n/m⌋ or ⌈n/m⌉. Let denote the set of all the most balanced vectors.

Definition 13 (Symmetric Distance Constraints). Let be the set of the most balanced vectors. Given a distance threshold , the feasible set of symmetric distance constraints is given as , where δ(ν) is given as either (i) Manhattan distance (L¹ norm), i.e., , or (ii) Chebyshev distance (L^∞ norm), i.e., .

It is easy to verify that for a single vector ν, a set of vectors whose distance from ν is within a given threshold can be represented by an M-convex set. Since we consider distance constraints from the set of all the most balanced vectors (which is symmetric), is also symmetric. This class of constraints is appropriate when a policymaker believes that the most balanced vectors are ideal, although she can accept any matching that is not too far from them.

3.3 Union of symmetric M-convex sets

In this section, we explore the class of constraints that can be represented by a union of symmetric M-convex sets.

3.4 Theoretical properties

Now we present two properties that are satisfied by a union of symmetric M-convex constraints. They are crucial for analyzing our mechanism.

Lemma 1. If where is a union of symmetric M-convex sets and ν_i > ν_j holds, then vector ν′ = (ν₁, …, ν_i − 1, …, ν_j + 1, …, ν_m) and all of its permutations also belong to .

Proof. From symmetry, vector ν′′, obtained by exchanging the i-th and j-th elements of ν, is also feasible. Then by the definition of an M-convex set, since , , and for all k ≠ i, j, vector ν − χ_i + χ_j = ν′ must be in .

Intuitively, Lemma 1 states that when transforming a feasible vector ν into a more balanced vector ν′, feasibility is preserved. This idea leads to the following lemma about most balanced vectors.

Lemma 2. If , where , is a union of symmetric M-convex set, it contains all of the most balanced vectors.

Proof. Assume first that is a symmetric M-convex set, and let ν be a vector in . If ν is most balanced, then from symmetry, all of its permutations must belong to . Thus, contains all of the most balanced vectors. If ν is not most balanced, from Lemma 1, we can move one student from a popular school to a less popular school in ν and obtain another vector included in . By repeatedly applying such modification, we eventually obtain a most balanced vector that belongs to .

Therefore, if is a union of symmetric M-convex sets, it includes all the most balanced vectors.

4.1 Mechanism description

We first introduce the standard Deferred Acceptance mechanism (DA), which is a component of our mechanism. Standard DA is defined with a standard market. A standard market is a tuple (S, C, ≻_S, ≻_C, q_C), whose definition resembles a market with distributional constraints. The only difference is that its constraints are given as a profile of individual maximum quotas: q_C = (q_c)_{c ∈ C}. Matching is school-feasible if for all c ∈ C, holds. Standard DA is defined as follows.

Mechanism 1 (Standard DA)

1. Step 1 Each student s applies to her favorite school, according to ≻_s, among the schools that have not rejected her so far.
2. Step 2 Each school c provisionally accepts the top q_c students from the applying students based on ≻_c and rejects the rest of them without any distinctions between newly applying and already provisionally accepted students.
3. Step 3 Barring any rejected students, return the current matching. Otherwise, go to Step 1.

To develop a DA-based mechanism on distributional constraints of union of symmetric M-convex sets, the crux is to specify the appropriate maximum quotas, by using the standard DA which can always be guaranteed to output feasible matchings. We explore this idea by utilizing flexible maximum quotas, such that we can enhance efficiency while guaranteeing feasibility and other desired properties. We also compare our proposed mechanism with another, which uses a fixed way to determine the maximum quotas, later in Section 4.3.

Informally, QRDA produces an initial standard market from a market with distributional constraints and then, at each stage, it iteratively (i) restricts the constraints on this market (i.e., reduces the maximum quotas) and (ii) applies DA to it until the matching returned by DA is also feasible with respect to the distributional constraints.

Which school’s maximum quota is reduced at the beginning of each stage is defined by σ, a sequence of schools based on the round-robin order c₁, c₂, …, c_m. Let σ(k) denote the k-th school in σ, i.e., σ(k) = c_j, where . For simplicity, although we assume σ is based on a fixed round-robin order, all our results hold for any balanced sequence σ. A sequence σ is said to be balanced if for each , σ(mℓ + 1), σ(mℓ + 2), …, σ(mℓ + m) is a permutation of c₁, c₂, …, c_m. The balanced sequence requirement is crucial to guarantee the strategyproofness of QRDA, as Example 7 later demonstrates.

QRDA is defined with respect to a specific quota reduction sequence σ. However, in the following, for simplicity, we assume that σ is based on the round-robin order c₁, c₂, …, c_m. When necessary, we specify σ and denote QRDA based on σ by QRDA^σ.

Let ν^max be a value that satisfies . Also, let denote the quota of school c at stage k of QRDA. Given σ and ν^max, QRDA is defined.

Mechanism 2 (QRDA)

1. Initialization:
   For all c ∈ C, .
2. Initial Stage:
   1. Step 1 Run standard DA in market and obtain matching .
   2. Step 2 If is school-feasible, then return . Otherwise, go to Stage 2.
3. Stage k (≥2):
   1. Step 1 For school c′ = σ(k − 1), , and for c ≠ c′, .
   2. Step 2 Run standard DA in market and obtain matching .
   3. Step 3 If is school-feasible, then return . Otherwise, go to Stagek + 1.

Note that in the following arguments, especially in Theorem 7, we assume that it only costs constant time to verify whether a matching is feasible or not. Also, note that the complexity/cost for finding depends on the representation of feasible vectors , which can be high. However, we only require that ν^max is not smaller than . Thus, we can always choose ν^max = n. The choice of ν^max does not influence the complexity result in Theorem 7.

4.2 Mechanism properties

First we show that QRDA always returns a feasible and fair matching by Theorem 1.

Theorem 1. QRDA returns a feasible and fair matching when is a union of symmetric M-convex sets.

Proof. If QRDA returns a matching, it is clearly feasible. According to Lemma 2, all the most balanced vectors must be included in . Eventually, at some stage k′, the quota of each school equals either ⌊n/m⌋ or ⌈n/m⌉, and . At this stage, standard DA returns a feasible matching. Thus, QRDA must terminate at an earlier stage k (≤ k′) by returning a feasible matching. The matching returned at stage k is identical to the one returned by standard DA for the market . Therefore, since standard DA is fair [18], QRDA is also fair.

Actually, this theorem holds under a much weaker condition; the fact that most balanced vectors are included in is sufficient.

Now we show that QRDA is strategyproof. Although using DA iteratively, QRDA does not trivially inherit strategyproofness from DA. Since the quotas of schools are decreasing, a student might have an incentive to terminate the mechanism early to secure the school seat, which might not be available at later stages. Therefore, the quotas have to be reduced carefully to preserve strategyproofness, i.e., the quota reduction sequence has to be balanced.

Moreover, under general non-M-convex constraints, a balanced quota reduction sequence may not be sufficient to preserve the strategyproofness of iterative DA mechanisms. Yahiro et al. [21] provided an example illustrating this fact, which we reproduce with Example 6 for the sake of completeness. In this example, initial quotas are equal to the largest number of students in a school in any feasible matching and the quota reduction sequence is balanced.

Example 6 ([21]). Consider S = {s₁, s₂, s₃, s₄, s₅, s₆}, C = {c₁, c₂, c₃}, σ: c₁, c₂, c₃, and . The following are the preferences of the students and the schools:

The initial maximum quotas are . At stage 1, all the students are assigned to their favorite school, and the matching is not feasible. The mechanism proceeds by reducing the quota of schools c₁ and c₂ by one at stages 2 and 3, and the matching remains the same. At stage 4, the quota of school c₃ is decreased by one. Student s₆ is rejected and then applies to c₁, which also rejects her. Hence, s₆ is assigned to c₂, and the matching becomes feasible. However, if s₆ misreports her preference with such that c₁ is her favorite school, she is assigned to c₁ at stage 1 and the matching is feasible. Thus, s₆ can successfully manipulate the mechanism.

Hence, QRDA’s strategyproofness is not trivial, and to prove it, we develop novel proof techniques, which require several lemmas.

The first is called the Scenario lemma and it requires two definitions. A scenario is a sequence of schools to which a student plans to apply. For example, suppose that student s has scenario , defined as c₁, c₂, c₃. This scenario means that student s first applies to school c₁, and if she is rejected, then she applies to school c₂, and then to c₃ if she is rejected again. A scenario is not necessarily exhaustive. A rejection chain , based on scenario , is the sequence of all the students and the schools’ actions (applications and rejections) that follow when student s enters the market with scenario , starting when she applies to the first school of this scenario. A rejection chain ends when (i) student s is rejected by the last school in or (ii) the mechanism terminates. A simple example of a rejection chain is presented in Table 2.

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Table 2. Example of a rejection chain.

https://doi.org/10.1371/journal.pone.0289965.t002

Given these definitions, we can define the Scenario lemma, which is inspired by the original Scenario lemma [40]. It was proven in the context of ratio constraints [21], and the proof is trivially extended to constraints represented by a union of symmetric M-convex sets.

Lemma 3 (Scenario Lemma). Consider two scenarios, and , of student s starting from the same stage of QRDA. If (1) each school that appears in also appears in (the order is irrelevant), (2) student s applies to all the schools in , and (3) all the actions of happen at the same stage, then all the actions in also happen in .

Proof. The first action in is “student s applies to school c,” where c is the first school that appears in . Since c also appears in , and s applies to all the schools in , also includes this action. For an inductive step, assume the first i − 1 actions in also happen in , and consider the i-th action of . The i-th action in must be either (i) “student s′ applies to school c′” or (ii) “school c′ rejects student s′.”

In case (i) with s′ = s, since school c′ must appear in and s applies to all the schools in , also includes this action. In case (i) with s′ ≠ s, there must be a previous action, “school c′′ rejects student s′,” in . From the inductive assumption, this action also happens in . Thus, the action “student s′ applies to school c′” also happens in .

In case (ii), let be the set of students who applied to c′ before the i-th action in , and let S_c′ be the set of all the students applying to c′ until all actions in are executed. Here, holds since every application before the i-th action in also appears in . Since in the i-th action of , s′ is rejected by school c′, she is not among c′’s favorite students in set . Since the quotas of schools are non-increasing as QRDA continues, in some stage k′ (≥k), student s′ must not be among the favorite students in S_c′. Thus, the action “school c′ rejects student s′” eventually occurs in .

Another essential lemma for QRDA’s strategyproofness in our setting is Lemma 4. Before introducing it, we introduce the following notation and concepts. Given a matching , consider that some school c_i’s student number is decreased by one and another school c_j’s student number is increased by one. We denote by η the corresponding operation, i.e., the operation that transforms vector into vector . Moreover, at stage k of QRDA, we denote by η_k the operation that transforms vector into vector since at most one student number changes in a QRDA’s stage. Finally, at stage k of QRDA, a school c is said to be full if , and maximum if . Note that c_i is always full and maximum, and c_j is neither full nor maximum in QRDA since σ is a balanced sequence.

Intuitively, Lemma 4 means that, at some stage of QRDA, if operation η exists that makes the current QRDA’s matching feasible, at any subsequent stage of QRDA, this operation (more specifically, any operation that resembles η, as defined in Lemma 4) also makes the current QRDA’s matching feasible.

Lemma 4. Consider an infeasible matching obtained at stage k of QRDA and an operation η that leads to a feasible matching when applied to . Assume that operation η increases school c_j’s student number, and let k′ denote the first stage of QRDA after k (<k′) where c_j’s student number increases. Then at any stage such that and for any maximum school c_max at stage , applying an operation that reduces c_max’s student number and increases c_j’s student number to matching leads to a feasible matching.

Proof. Consider an infeasible matching obtained at stage k of QRDA and an operation η such that imposing η on leads to a feasible vector. Assume that operation η reduces school c_i’s student number by one and increases school c_j’s student number by one. The proof is done by induction.

For , vector is feasible by assumption. If school c_i is the unique maximum school at stage k − 1, then the claim is satisfied for . Hence, consider a maximum school c_max ≠ c_i at stage k − 1. By definition, , which implies , and thus with Lemma 1 on , vector is feasible.

Consider some stage such that and assume that the claim is true at stage . We show that the claim is also true for . Assume that operation reduces school c_a’s student number and increases school c_b’s student number. Since school c_a is maximum at stage and the claim is true at stage , vector is feasible. Note that if c_b = c_j then is feasible and QRDA terminates strictly before k′, which is a contradiction. Hence, in the following, we assume c_b ≠ c_j. Two cases are possible.

If , then vector is a permutation of vector . Hence, since c_b ≠ c_j, for any maximum school c_max at , vector is a permutation of vector . Hence, by symmetry, vector is feasible.

Otherwise, . By Lemma 1 on vector , we obtain that vector is feasible, i.e., vector is feasible. If school c_a is the unique maximum school at stage , then the claim is satisfied at . Hence, consider a maximum school c_max ≠ c_a at . By definition, , which implies , and thus with Lemma 1 on , vector is feasible, which concludes the proof.

Now we are ready to show that QRDA is strategyproof under constraints represented by a union of symmetric M-convex sets.

Theorem 2. QRDA is strategyproof when is a union of symmetric M-convex sets.

Proof. By contradiction, assume student s is assigned to a better school when she misreports. Without loss of generality, we assume that her true preference is c₁ ≻_s c₂ ≻_s …≻_sc _m, and that s is assigned to school c_j in stage k when misreporting, but s is assigned to c_i at stage k′ under s’s true preference, where j < i.

First, we show that if k′ ≤ k, student s cannot benefit from misreporting. The standard DA satisfies a property called resource monotonicity, i.e., DA’s outcome is weakly less preferred by each student if the quotas decrease [41]. This property implies that when student s truthfully reports her preference at both stages k and k′, her assignment is (weakly) worse in k than in k′. Furthermore, DA is strategyproof [40, 42]. Hence, at stage k, student s’s assignment is worse when she misreports than when she truthfully reports. Therefore, s’s assignment is (weakly) worse when she misreports in k than when she truthfully reports in k′, and thus, s cannot benefit from misreporting if k′ ≤ k. Hence, in the following, k < k′ holds.

At stage k, consider an alternative (and equivalent) DA mechanism where we run a standard DA for S \ {s} under the quotas of stage k and add s to the market. The matching obtained in this way is identical to the matching obtained by applying DA when all the students enter the market simultaneously [40]. When s is added to the market with a misreported preference, s is assigned to school c_j and a feasible matching is obtained; when s is added with a true preference, s is assigned to school c_o, and an infeasible matching is obtained.

Now we define two scenarios: (a) , based on the true preference of s, and so the last action in must be “school c_i−1 rejects student s,” and (b) , based on the misreported preferences in which the last school is c_j. For , the following two cases are possible: (i) c_j is truly the least preferred school for s within or (ii) contains at least one school that is less preferred by s than c_j. We start by proving case (i); case (ii) is proven in the last paragraph of the proof. A key point in case (i) is that each school c that appears in also appears in . Thus by Lemma 3, it implies that action “student s applies to school c_j” appears in rejection chain .

At stage k, when s enters the market (after all the other students) with true preferences, assume that school c_e gains one student, and when s misreports, another school c_d gains one student. We show the proof for a particular case when c_e = c_o and c_d = c_j; a similar argument can be developed when c_e ≠ c_o or c_d ≠ c_j by replacing c_o to c_e, and c_j to c_d, respectively, in the following argument.

Note that , , and for all other schools c (≠ c_o, c_j), . Therefore, the vector of feasible matching verifies .

It follows that if holds, then by Lemma 1 on , is also feasible, which is a contradiction. Thus we obtain the following equation: (1)

The rest of the proof for case (i) is composed of three arguments: (A), (B), and (C).

(A) First, we show that under scenario , school c_j’s student number does not increase strictly between k and k′. By contradiction, assume that c_j’s student number increases at some stage (). Let c_⋆ denote a school whose student number is decreased by operation . Since school c_⋆ is maximum at stage , by Lemma 4 with , vector is feasible, i.e., vector is feasible. Hence, QRDA terminates strictly before k′, which is a contradiction.

(B) Recall that under scenario , action “student s applies to school c_j” must occur; however, in this scenario student s’s final assignment is c_i (≠ c_j). Thus, school c_j rejects student s at some stage of QRDA. Notice that if school c_j is not full at stage k′ − 1, then c_j cannot reject any students at stage k′ or at any previous stage. Hence, school c_j is full at stage k′ − 1, i.e., , which implies that for any school c, . Since school c_j’s student number does not increase strictly between k and k′, the following equation holds for any school c: (2)

Let denote the first stage after k such that for any school c. With Eqs 1 and 2, we know that . Assume that operation decreases c_a’s student number and increases c_b. By the definition of stage , and hold.

(C) Finally, we show that vector is feasible. Since school c_a is maximum at stage and , by Lemma 4, vector is feasible. Recall that school c_j’s student number does not increase strictly between k and k′. Moreover, since , school c_j cannot be full at stage , which implies that school c_j’s student number does not decrease between k and . Therefore, we obtain . Since , it leads to . Thus, by Lemma 1 on vector , vector is also feasible, i.e., vector is feasible. Hence, QRDA terminates strictly before k′, which is a contradiction.

Last, for case (ii), we can create a new scenario, , by removing all the schools that are less desired than c_j based on ≻_s from . If s is assigned to c_j in , we obtain the same contradiction as for case (i) by comparing and . Thus, action “school c_j rejects student s” must appear in . By Lemma 3, this action also appears in , but this also leads to a contradiction.

QRDA’s strategyproofness is heavily based on the fact that σ is balanced. Under ratio constraints, Yahiro et al. [21] provided an example to show that QRDA is not strategyproof if σ is not balanced. To be self-contained, we reproduce this example with Example 7.

Example 7. Consider S = {s₁, …, s₉}, C = {c₁, c₂, c₃}, α = 1/4 (the ratio constraints), and σ starting with c₂, c₂, c₁. The following are the preferences of the students and the schools:

First, QRDA^σ sets , for all c ∈ C. At stage 1, all students are assigned to their favorite schools, but this is not feasible. At both stages 2 and 3, the quota of c₂ is decreased by one, although the matching remains the same. Then at stage 4, the quota of c₁ is decreased by one and s₅ is rejected from c₁. Student s₅ applies to c₂, which also rejects her. She is finally assigned to c₃ and the corresponding matching is feasible. However, if student s₅ misreports by declaring that c₂ is her most preferred school, QRDA stops at stage 1 by returning a feasible matching where she is assigned to school c₂. Hence, student s₅ manipulated the mechanism to get a better result (s₅, c₂).

Under ratio constraints, QRDA has been proven to hold additional gratifying properties [21], most of which can be trivially extended to our setting. Their proofs for weak non-bossiness, weak Maskin monotonicity, weak group strategyproofness, and non-domination rely only on QRDA’s strategyproofness under ratio constraints and DA’s properties. Since Theorem 2 shows that QRDA is strategyproof in our setting, Theorems 3 and 4 trivially hold.

Theorem 3. QRDA is weakly non-bossy, weakly Maskin monotone, and weakly group strategyproof.

Theorem 4. Given a balanced σ, no strategyproof mechanism exists that dominates QRDA^σ.

However, weak Pareto optimality does not trivially extend to a union of symmetric M-convex sets since its proof under ratio constraints strongly relies on the structure of the ratio constraints. With Theorem 5, we show that QRDA remains weakly Pareto optimal in our setting.

Theorem 5. QRDA is weakly Pareto optimal.

Proof. Toward a contradiction, assume a matching that strongly dominates the matching returned by QRDA, denoted by , i.e., holds for all s ∈ S. Let k denote the last stage of QRDA, i.e., . The proof is conducted by showing that matching is feasible, i.e., the matching returned by QRDA at stage k − 1 is feasible, which contradicts that QRDA terminates at stage k. We develop the proof through arguments (A), (B), and (C).

(A) This argument shows that for all c ∈ C such that , it holds .

First, we show that for all c ∈ C, . Assume c′ ∈ C exists such that . The definition of QRDA’s quota reduction sequence implies that for all c ∈ C, both and hold. Therefore, for any school c ∈ C, both and hold. Hence, vector is a permutation of vector and thus, by symmetry, is feasible, which contradicts that QRDA terminates at stage k.

Then consider a school c′ ∈ C such that . Since for all c ∈ C, , no student applies to c′ during QRDA, implying that for all s ∈ S, . If , then for any student s assigned to c′ in , it holds that , which is a contradiction.

(B) The next argument shows that any school c such that is full at stage k − 1 of QRDA, i.e., .

Consider a school c′, which is not full at stage k − 1, implying that it rejects no student during QRDA. Toward a contradiction, assume that , and consider a student s′ who is assigned to c′ in . Since strongly dominates , holds. Hence, during QRDA, student s′ has to be rejected by school c′ before being allowed to apply to , which contradicts that c′ rejects no student during QRDA.

(C) Finally, we show that we can transform matching into matching by applying permutations and operations that satisfy the conditions of Lemma 1.

Denote and thus . Let matching be one of the matchings such that (i) for all c ∈ C\C′, , and (ii) for all c ∈ C′, , i.e., is one of the most balanced matchings when we restrict the setting to schools in C′. With a similar argument as proof of Lemma 2, we can show that any matching can be transformed into a most balanced matching only by permutations and operations satisfying the conditions of Lemma 1. Hence, since is one of the most balanced vectors when we restrict schools to C′, we can transform matching into only by permutations and operations satisfying the conditions of Lemma 1. Therefore, since is feasible when considering all schools C, is also feasible when considering all schools C.

By definition of , holds, and thus also holds. Furthermore, by argument (B), all schools c ∈ C′ are full at stage k − 1, i.e., . With QRDA’s quota reduction sequence, it implies that for all c ∈ C, . Therefore, in addition to satisfying (i) and (ii), matching can be chosen such that it also satisfies (iii) for all c ∈ C′, . Now note that holds, and, with condition (iii), holds. It intuitively means that the surplus of students in the schools of C′ when comparing and equals the student number in the schools of C\C′ in . To transform into , we can transfer the surplus of students from a school in C′ to a school in C\C′ with operations that satisfy the conditions of Lemma 1 and with permutations.

is feasible by Lemma 1 and symmetry, which is a contradiction. Hence, no matching exists that strongly dominates .

As Proposition 2 shows, fairness and nonwastefulness are incompatible. Since QRDA is fair, it cannot be nonwasteful. Nevertheless, QRDA satisfies the following weaker version of nonwastefulness.

Definition 18 (Weak Nonwastefulness). In matching , where , student s strongly claims an empty seat in c′ if (s, c′) ≻_s (s, c), is school-feasible, and . Matching is weakly nonwasteful if no student strongly claims an empty seat in . A mechanism is weakly nonwasteful if it always outputs a weakly nonwasteful matching.

Intuitively, a student strongly claims an empty seat if, by unilaterally moving the student to her more preferred school, the result allocation vector is not only school-feasible but also more balanced than the original allocation vector; if we move s from school c to school c′, the student number in c′ is still fewer than or equal to that in c.

We have the following result in terms of this weaker version of nonwastefulness.

Theorem 6. QRDA is weakly nonwasteful when is a union of symmetric M-convex sets.

Proof. We prove this theorem towards a contradiction. Assume that QRDA terminates at stage k, outputting feasible matching . Student s exists such that and s strongly claims an empty seat in school c′ in matching . According to Definition 18, s prefers c′ over c and holds. Since s prefers c′ over c, s must have applied to c′ and been rejected. Thus, c′ must be full, i.e., holds. Also, must hold, since school c’s student number cannot exceed its quota. By the fact that holds, we obtain . However, this contradicts the fact that the quota reduction sequence is balanced, that is, the difference between the maximum quotas of any pair of schools is at most one.

Lastly, we briefly delve into the time complexity of QRDA. At each stage k of QRDA, we do not have to run DA from the beginning (that is, each student starts by applying to the first-ranked school). Instead, at stage k of QRDA, each student first applies to her most preferred school among those that have not rejected her at all previous stages. This modification ensures that the resulting matching remains unaffected, as the execution order in DA can be flexible [40].

Theorem 7. The time complexity of QRDA is O(mn), assuming that verifying the school-feasibility of a vector can be done in constant time.

QRDA repeatedly applies the standard DA. By the above modification, a student is rejected by each school at most once during the whole QRDA’s execution. Thus, each step in DA is executed at most mn times in total. Hence, the time complexity of QRDA is O(mn).

4.3 Theoretical comparison with baseline mechanism

To the best of our knowledge, no fair and strategyproof mechanism exists in the literature that can handle the union of symmetric M-convex constraints except QRDA. An alternative approach to address such distributional constraints is by transforming them into artificial maximum quotas, thereby ensuring that standard DA returns a matching that satisfies these constraints. This method, called Artificial Cap Deferred Acceptance (ACDA), inherits DA’s properties while losing some of the flexibility of the original constraints due to the restriction of the set of feasible matchings. ACDA is employed in Japanese medical resident matching programs [2] and serves as a benchmark mechanism in various works focused on distributional constraints [4, 5, 30].

If prior information were available regarding the popularity of schools, it might be possible to set q_C to meet distributional constraints and maximize student welfare. Otherwise, a pragmatic and rational approach to determine suitable q_C is by utilizing a most balanced vector. Similar to QRDA, ACDA is defined with respect to a specific sequence σ, denoted as ACDA^σ. We assume that σ is the round-robin order c₁, c₂, …, c_m, unless otherwise specified. ACDA is defined as follows:

Mechanism 3 (ACDA (based on a most balanced vector))

1. Initialization:
   For each i where , , and for each i where , .
2. Execution:
   Run standard DA in market (S,C,≻_S,≻_C,q_C).

As stated above, ACDA inherits the properties of DA like fairness and strategyproofness.

Theorem 8. ACDA is strategyproof and returns a feasible and fair matching.

Proof. Since all the most balanced vectors are included in , the matching obtained under q_C is clearly feasible. Since standard DA is fair [18], the matching obtained by ACDA is also fair. Furthermore, since q_C is determined independently from ≻_S and the standard DA is strategyproof, ACDA is strategyproof.

The next result shows that QRDA is overwhelmingly favored by stating that all students weakly prefer the school that they obtain in QRDA to the one in ACDA.

Theorem 9. Given a balanced sequence σ, all students weakly prefer the matching obtained by QRDA^σ over that by ACDA^σ.

Proof. Let k′ denote the stage of QRDA^σ where becomes identical to the q_C used in ACDA^σ. This situation can happen since ACDA^σ generates the most balanced vector based on the quota reduction sequence σ. As shown in the proof of Theorem 1, QRDA^σ terminates at stage k where k ≤ k′. Since holds for all c ∈ C and DA satisfies resource monotonicity (described in the proof of Theorem 2), each student weakly prefers the matching obtained by QRDA^σ over that by ACDA^σ.

Since QRDA always obtains a (weakly) better matching for students than ACDA, one may assume that QRDA is always less wasteful than ACDA, i.e., more students claim empty seats in ACDA compared to QRDA. However, we cannot guarantee that QRDA is less wasteful than ACDA. Yahiro et al. [21] identified a case where the number of students who claim empty seats in QRDA exceeds that of ACDA under ratio constraints.

Theorem 10 (based on Theorem 11 by Yahiro et al. [21]). Given a balanced σ, a case exists where the number of students who claim empty seats in QRDA^σ exceeds that of ACDA^σ.

Theorem 10 implies that there are cases where QRDA still has room for amelioration by mitigating students’ claims compared to ACDA. Under ratio constraints, this QRDA’s drawback is compensated by the fact that when ACDA returns a nonwasteful matching, QRDA returns the same nonwasteful matching [21]. Theorem 11 shows that this fact extends to a union of symmetric M-convex sets.

Theorem 11. Given a balanced σ, when ACDA^σ returns a nonwasteful matching, QRDA^σ and ACDA^σ return the same matching.

Proof. In market , assume that QRDA^σ returns matching and that ACDA^σ returns matching that differs from . By contradiction assume also that is nonwasteful.

Consider a procedure that starts with matching and keeps applying QRDA’s stages, i.e., reducing quotas and applying DA (even though is feasible) until the quotas reach the same quotas as in ACDA^σ, and thus the procedure returns . During this procedure, since differs from , some rejection chains must occur and let r denote the last of these rejection chains. Assume rejection chain r decreases the student number in school c′ and increases c′′. Denote by the matching returned by this procedure before rejection chain r happens, i.e., . Since is feasible, by Lemma 1, any matching returned during this procedure is feasible, in particular is feasible. In the following, we show that one student involved in the rejection chain r is a claiming student in .

Notice that each school that rejects a student during rejection chain r has to be full in matching , and can be either maximum or not in , i.e., its student number can be or .

First, we assume that rejection chain r involves a rejection from a school that is maximum in . Consider the first maximum school in that rejects a student in r, denoted as c_b, and then . Assume that school c_b accepts student s_b (and then rejects another one), and that s_b was initially rejected by school c_a. We have two cases to consider: either c_a = c′ or c_a ≠ c′.

If c_a = c′, consider matching . Then note that . By assumption, , and since and , vector is a permutation of . By symmetry, matching is feasible, and thus student s_b is a claiming student in .

Otherwise, c_a ≠ c′, and consider matching . Note that . By definition of c_b, school c_a is not maximum in , and thus holds. In addition with , it implies that vector is a permutation of . By symmetry, matching is feasible, and thus student s_b is a claiming student in .

Therefore, assume no rejection exists in r that involves a school that is maximum in . Consider the last rejection in rejection chain r, and assume that school c_d rejects student s_d who is then accepted by school c′′ (and rejection chain r stops). Note that . By assumption, since school c_d is not maximum in , . Then vector is a permutation of . By symmetry, matching is feasible, and thus student s_d is a claiming student in .

5.1 Union of symmetric M-convex constraints with an offset

A union of symmetric M-convex constraints with an offset is defined as follows.

Definition 19 (Union of Symmetric M-convex Sets with an Offset). Set is a union of symmetric M-convex sets with an offset if offset vector and exist such that is a union of symmetric M-convex sets and is given as .

In short, a union of symmetric M-convex sets with an offset is derived from a union of symmetric M-convex sets by adding offset vector ν* to each element in . Note that offset vector ν* may contain negative elements, as long as each element in is non-negative.

One practical example of a union of symmetric M-convex sets with an offset is when each school requires a specific number of students to operate, represented as its minimum quota. More specifically, each school c has its own minimum quota p_c, which is defined analogously to Definition 11. However, these minimum quotas may vary based on the size of the schools. We apply the union of symmetric M-convex constraints to the number of surplus students beyond p_c for all c ∈ C, rather than the number of overall students. The market is denoted by a tuple . In this market, the constraints are represented as follows.

1. The individual minimum quotas, , which is an m-dimensional vector and each p_c is the minimum quota of school c. We require ∑_c∈C p_c < n.
2. The constraints on the number of surplus students for each school beyond p_C are given as a union of m-dimensional symmetric M-convex sets .
3. Matching is feasible if holds.

5.2 QRDA for union of symmetric M-convex constraints with an offset

To handle this generalized class of constraints, we slightly modify our proposed mechanism, QRDA, with different initial individual maximum quotas to handle the asymmetry introduced by the offset vector. Recall that under the constraints in Section 3.3, QRDA commences with uniform maximum quota ν^max, which satisfies for each school, and reduces the schools’ quotas based on a given sequence σ. On the other hand, for any union of symmetric M-convex sets with an offset , we define as a value that satisfies . For each school c, is defined as . In short, is the sum of the common upper-bound and the offset of c. The modified QRDA sets to , signifying that the initial maximum quota for each school is an upper-bound for any possible value in . Intuitively, in the generalized constraints handled in this section, a set of “most balanced vectors” still exists, although they have been shifted by the offset vector. The constraints are therefore symmetric and M-convex around the most balanced vectors. Therefore, the modified QRDA ensures that any returned matching still satisfies all the properties described above. Applying similar arguments in the previous sections, we deduce the following theorem.

Theorem 12. In any market under a union of symmetric M-convex constraints with an offset, the modified QRDA is feasible, fair, and strategyproof.