Fairness and efficiency trade-off in two-sided matching

The theory of two-sided matching has been developed and has been applied to many real-life application domains (see Roth and Sotomayor (1990) for a comprehensive survey in this literature). It has attracted considerable attention from AI researchers Aziz et al. (2022c); Aziz et al. (2019); Hosseini et al. (2015); Ismaili et al. (2019); Kawase and Iwasaki (2017); Yahiro et al. (2020); Suzuki et al. (2023). As the theory has been applied to increasingly diverse types of environments, researchers and practitioners have encountered various forms of distributional constraints (see Aziz et al. (2022b) for a comprehensive survey on various distributional constraints). There exist three representative classes of constraints. First, the standard model of two-sided matching considers only the maximum quota of each individual college Roth and Sotomayor (1990), which we call maximum quotas constraints.¹¹1Although our paper is described in the context of a student-college matching problem, the obtained result is applicable to matching problems in general.

More general classes of constraints are hereditary constraints Aziz et al. (2022a); Goto et al. (2017); Kamada and Kojima (2017) and hereditary M^♮-convex set constraints Kojima et al. (2018). An M^♮-convex set is a discrete counterpart of a convex set in a continuous domain. Hereditary constraints require that if a matching between students and colleges is feasible, then any matching that places weakly fewer students at each college is also feasible.

As a mechanism can handle a more general class of constraints, we can incorporate more complex constraints required for real-life application domains. Also, we obtain more flexibility in assigning students to colleges. As a result, we can expect that students’ welfare can be increased in the obtained matching. Furthermore, maximum quotas constraints can be considered to be too restrictive. In a real-life situation, it is common that some flexibility exists in determining the capacity of each college, i.e., we can increase the maximum quota of a college if it turns out to be very popular (say, by assigning additional resources). Such flexibility can be modeled naturally using a more general class of constraints.

In this paper, we focus our attention on strategyproof mechanisms, which guarantee that students have no incentive to misreport their preference over colleges. From a theoretical standpoint, if we are interested in a property achieved in dominant strategies, strategyproof mechanisms can be exclusively considered without any loss of generality, as supported by the well-known revelation principle Gibbard (1973). This principle states that if a certain property is satisfied in a dominant strategy equilibrium using a mechanism, it can also be achieved through a strategyproof mechanism. Strategyproof mechanisms are not only theoretically significant but also practically beneficial, as students do not need to speculate about the actions of others to achieve desirable outcomes; they only need to report their preferences truthfully.

Most existing works in two-sided matching require that the obtained matching must be fair, i.e., no student has justified envy. However, just requiring fairness is not sufficient since the matching that no student is assigned to any college is fair; we should achieve some requirement on students’ welfare (which is referred to as efficiency in economics) in conjunction with fairness. In the standard maximum quotas model, the renowned Deferred Acceptance mechanism (DA) Gale and Shapley (1962) can achieve an efficiency property called nonwastefulness in conjunction with fairness. A matching satisfying fairness and nonwastefulness together is called stable.

However, when some distributional constraints are imposed, there exists a trade-off between fairness and efficiency/students’ welfare. In particular, Cho et al. (2022) show that no strategyproof mechanism satisfies fairness and a weaker efficiency property called weak nonwastefulness under hereditary constraints.

The first goal of this paper is to clarify the tight boundaries on whether a strategyproof and fair mechanism can satisfy certain efficiency properties for each class of constraints (see Table 1 in Section 4). In particular, we show that under hereditary constraints, no strategyproof mechanism can simultaneously satisfy fairness and a very weak efficiency requirement called no vacant college property.

This impossibility result illustrates a dilemma: we are expanding/generalizing the classes of constraints in the hope that we can improve students’ welfare. However, if we require strict fairness, we cannot guarantee a very weak requirement of students’ welfare under general hereditary constraints. Given this dilemma, our next goal is to establish a weaker fairness requirement. In this paper, we propose a novel concept called envy-freeness up to $k$ peers (EF-$k$). This concept is inspired by a criterion called envy-freeness up to $k$ items, which is commonly used in the fair division of indivisible items Budish (2011). EF-$k$ guarantees that each student has justified envy towards at most $k$ students. By varying $k$, EF-$k$ can represent different levels of fairness. On one hand, EF-$0$ is equivalent to standard fairness. On the other hand, any matching satisfies EF-($n-1$), where $n$ is the number of students. To the best of our knowledge, this paper is the first to address the relaxed notion of fairness in two-sided, many-to-one matching.

We show that there exists a case that no matching is nonwasteful and EF-$k$ for any $k<n-1$, and checking whether a nonwasteful and EF-$k$ matching exists or not is NP-complete. Then, we develop two contrasting strategyproof mechanisms that work for general hereditary constraints. One is based on the Serial Dictatorship mechanism (SD) Goto et al. (2017), which utilizes an optimal master-list (where students are assigned in its order) that minimize $k$ based on colleges’ preferences, such that the obtained matching is guaranteed to satisfy EF-$k$. Although $k=n-1$ holds in the worst case, we experimentally show that $k$ tends to be much smaller when colleges’ preferences are similar. The other one is based on the Sample and Deferred Acceptance mechanism (SDA) Liu et al. (2023), which is developed for a special case of hereditary constraints called student-project-resource matching-allocation problem. This mechanism satisfies EF-$k$ for any given $0\leq k<n-1$. We extend SDA such that the obtained matching satisfies no vacant college property under a mild assumption. We experimentally show that this mechanism can significantly improve students’ welfare compared to a fair (EF-$0$) mechanism even when $k$ is very small.

A matching market under distributional constraints is given by $I=(S,C,X,\succ_{S},\succ_{C},{f})$. The meaning of each element is as follows.

• •

  $S=\{s_{1},\ldots,s_{n}\}$ is a finite set of students. Let $N$ denote $\{1,2,\ldots,n\}.$

• •

  $C=\{c_{1},\ldots,c_{m}\}$ is a finite set of colleges. Let $M$ denote $\{1,2,\ldots,m\}$.

• •

  $X\subseteq S\times C$ is a finite set of contracts. Contract $x=(s,c)\in X$ represents the matching between student $s$ and college $c$.

• •

  For any $Y\subseteq X$, let $Y_{s}:=\{(s,c)\in Y\mid c\in C\}$ and $Y_{c}:=\{(s,c)\in Y\mid s\in S\}$ denote the sets of contracts in $Y$ that involve $s$ and $c$, respectively.

• •

  $\succ_{S}=(\succ_{s_{1}},\ldots,\succ_{s_{n}})$ is a profile of the students’ preferences. For each student $s$, $\succ_{s}$ represents the preference of $s$ over $X_{s}\cup\{(s,\emptyset)\}$, where $(s,\emptyset)$ represents an outcome such that $s$ is unmatched. We assume $\succ_{s}$ is strict for each $s$. We say contract $(s,c)$ is acceptable for $s$ if $(s,c)\succ_{s}(s,\emptyset)$ holds. We sometimes use notations like $c\succ_{s}c^{\prime}$ instead of $(s,c)\succ_{s}(s,c^{\prime})$.

• •

  $\succ_{C}=(\succ_{c_{1}},\ldots,\succ_{c_{m}})$ is a profile of the colleges’ preferences. For each college $c$, $\succ_{c}$ represents the preference of $c$ over $X_{c}\cup\{(\emptyset,c)\}$, where $(\emptyset,c)$ represents an outcome such that $c$ is unmatched. We assume $\succ_{c}$ is strict for each $c$. We say contract $(s,c)$ is acceptable for $c$ if $(s,c)\succ_{c}(\emptyset,c)$ holds. We sometimes write $s\succ_{c}s^{\prime}$ instead of $(s,c)\succ_{c}(s^{\prime},c)$.

• •

  ${f}:{\mathbf{Z}}_{+}^{m}\rightarrow\{-\infty,0\}$ is a function that represents distributional constraints, where $m$ is the number of colleges and ${\mathbf{Z}}_{+}^{m}$ is the set of vectors of $m$ non-negative integers. For $f$, we call a family of vectors $F=\{\nu\in{\mathbf{Z}}_{+}^{m}\mid f(\nu)=0\}$ induced vectors of $f$.

We assume each contract $x$ in $X_{c}$ is acceptable for $c$. This is without loss of generality because if some contract is unacceptable for a college, we can assume it is not included in $X$.

We say $Y\subseteq X$ is a matching, if for each $s\in S$, either (i) $Y_{s}=\{x\}$ and $x$ is acceptable for $s$, or (ii) $Y_{s}=\emptyset$ holds.

For two $m$-element vectors $\nu,\nu^{\prime}\in\mathbf{Z}_{+}^{m}$, we say $\nu\leq\nu^{\prime}$ if for all $i\in M$, $\nu_{i}\leq\nu^{\prime}_{i}$ holds. We say $\nu<\nu^{\prime}$ if $\nu\leq\nu^{\prime}$ and $\nu\neq\nu^{\prime}$ hold. Also, let $|\nu|$ denote the $L_{1}$ norm of $\nu$, i.e., $|\nu|=\sum_{i\in M}\nu_{i}$.

We assume $F$ is bounded, i.e., $|F|$ is finite. This is without loss of generality because we can assume each college $c_{i}$ can accept at most $|X_{c_{i}}|$ students, i.e., $f(\nu)=-\infty$ holds when $\exists i\in M,\nu_{i}>|X_{c_{i}}|$.

Let us first introduce a very general class of constraints called hereditary constraints. Intuitively, heredity means that if $Y$ is feasible in $f$, then any subset $Y^{\prime}\subset Y$ is also feasible in $f$. Let $e_{i}$ denote an $m$-element unit vector, where its $i$-th element is $1$ and all other elements are $0$. Let $e_{0}$ denote an $m$-element zero vector $(0,\ldots,0)$.

Kojima et al. (2018) show that when $f$ is hereditary, and its induced vectors satisfy one additional condition called M^♮-convexity, there exists a general mechanism called Generalized Deferred Acceptance mechanism (GDA), which satisfies several desirable properties.²²2To be more precise, Kojima et al. (2018) show that to apply their framework, it is necessary that the family of feasible matchings forms a matroid. When distributional constraints are defined on $\nu(Y)$ rather than on contracts $Y$, the fact that the family of feasible contracts forms a matroid corresponds to the fact that (i) the family of feasible vectors forms an M^♮-convex set, and (ii) it is hereditary Murota and Shioura (1999).

Let us formally define an M^♮-convex set.

An M^♮-convex set can be considered as a discrete counterpart of a convex set in a continuous domain. Intuitively, Definition 2 means that for two feasible vectors $\nu$ and $\nu^{\prime}$, there exists another feasible vector, which is one step closer starting from $\nu$ toward $\nu^{\prime}$, and vice versa. An M^♮-convex set has been studied extensively in discrete convex analysis, a branch of discrete mathematics. Recent advances in discrete convex analysis have found many applications in economics (see the survey paper by Murota (2016)). Note that heredity and M^♮-convexity are independent properties.

Kojima et al. (2018) show that various real-life distributional constraints can be represented as a hereditary M^♮-convex set. The list of applications includes matching markets with regional maximum quotas Kamada and Kojima (2015), individual/regional minimum quotas Fragiadakis et al. (2015); Goto et al. (2017), diversity requirements in school choice Ehlers et al. (2014); Kurata et al. (2017), distance constraints Kojima et al. (2018), and so on. However, M^♮-convexity can be easily violated by introducing some additional constraints.

Let us introduce the most basic model where only distributional constraints are colleges’ maximum quotas.

If $f$ is given as colleges’ maximum quotas, then $f$ is a hereditary M^♮-convex set, but not vice versa.

With a slight abuse of notation, for two sets of contracts $Y$ and $Y^{\prime}$, we denote $Y_{s}\succ_{s}Y^{\prime}_{s}$ if either (i) $Y_{s}=\{x\}$, $Y^{\prime}_{s}=\{x^{\prime}\}$, and $x\succ_{s}x^{\prime}$ for some $x,x^{\prime}\in X_{s}$, or (ii) $Y_{s}=\{x\}$ for some $x\in X_{s}$ that is acceptable for $s$ and $Y^{\prime}_{s}=\emptyset$. Furthermore, we denote $Y_{s}\succeq_{s}Y^{\prime}_{s}$ if either $Y_{s}\succ_{s}Y^{\prime}_{s}$ or $Y_{s}=Y^{\prime}_{s}$. Also, we use notations like $x\succ_{s}Y_{s}$ or $c\succ_{s}Y_{s}$, where $x$ is a contract, $Y$ is a matching, and $c$ is a college.

Let us introduce several desirable properties of a matching and a mechanism. We say a mechanism satisfies property A if the mechanism produces a matching that satisfies property A in every possible matching market.

First, we define fairness.

Fairness implies that if student $s$ is not assigned to college $c$ (although she hopes to be assigned), then $c$ prefers all students assigned to it over $s$.

Next, we define a series of properties on students’ welfare (efficiency).

In short, feasible matching $Y$ is Pareto efficient if there exists no other feasible matching $Y^{\prime}$ such that all students weakly prefer $Y^{\prime}$ over $Y$, and at least one student strictly prefers $Y^{\prime}$ over $Y$.

Intuitively, nonwastefulness means that we cannot improve the matching of one student without affecting other students.

When additional distributional constraints (besides colleges’ maximum quotas) are imposed, fairness and nonwastefulness become incompatible in general. One way to address the incompatibility is weakening the requirement of nonwastefulness. Aziz et al. (2022a) introduce a weaker efficiency concept called cut-off nonwastefulness.

Intuitively, we consider the claim of student $s$ to move her to college $c$ from her current match is not considered legitimate if by doing so, another student $s^{\prime}$ would have justified envy toward $s$. Aziz et al. (2022a) show that a fair and cut-off nonwasteful matching always exists under hereditary constraints. This result carries over to less general hereditary and M^♮-convex set constraints, as well as weaker efficiency requirements described below. Note that the existence of a fair and cut-off nonwasteful matching does not guarantee the existence of a strategyproof mechanism for obtaining it, as shown in Section 4.

Kamada and Kojima (2017) propose another weaker version of the nonwastefulness concept, which we refer to as weak nonwastefulness.

Student $s$ can strongly claim an empty seat of $c$ only when $Y\cup\{(s,c)\}$, i.e., the matching obtained by adding her to college $c$ (without removing her from her current college), is feasible.

Let us define two more weaker efficiency properties.

Intuitively, no vacant college property means that the claim of student $s$ to move her to college $c$ from her current match is considered legitimate only when $s$ is not matched to any college and no student is assigned to $c$.

Note that this series of efficiency properties becomes monotonically weaker in this order as long as distributional constraints are hereditary. More specifically, Pareto efficiency implies nonwastefulness, but not vice versa, nonwastefulness implies cut-off nonwastefulness, but not vice versa, and so on. Pareto efficiency means that we cannot improve the matching of a set of students without hurting other students, while nonwastefulness means that we cannot improve the matching of one student without affecting other students. Thus, Pareto efficiency implies nonwastefulness. If $Y$ is nonwasteful, no student can claim an empty seat. If $Y$ is cut-off nonwasteful, a student can claim an empty seat in some cases. Thus, cut-off nonwastefulness is weaker than nonwastefulness. Next, we show that cut-off nonwastefulness implies weak nonwastefulness by showing its contraposition. More specifically, we assume student $s$ strongly claims an empty seat of college $c$ in $Y$. Then, we show that $Y$ cannot be cut-off nonwasteful. The fact that $s$ strongly claims an empty seat of $c$ implies that $s$ also claims an empty seat of $c$ since if $Y\cup\{(s,c)\}$ is feasible, $(Y\setminus Y_{s})\cup\{(s,c)\}$ is also feasible. Assume there exists another student $s^{\prime}$, where $c\succ_{s^{\prime}}Y_{s^{\prime}}$ and $s^{\prime}\succ_{c}s$ hold. Then, since $Y\cup\{(s,c)\}$ is feasible, $(Y\setminus Y_{s^{\prime}})\cup\{(s^{\prime},c)\}$ is also feasible. Thus, $Y$ cannot be cut-off nonwasteful.

Next, we show that weak nonwastefulness implies no vacant college property by showing its contraposition. More specifically, no vacant college property means that the claim of student $s$ to move her from the current matching to $c$ is considered legitimate only when $s$ is not matched to any college and no student is assigned to $c$. Let us assume $Y$ does not satisfy no vacant college property, i.e., there exists student $s$ who claims an empty seat of $c$ when $Y_{s}=\emptyset$ and $Y_{c}=\emptyset$. Then, we show $s$ strongly claims an empty seat of $c$ in $Y$. Since $Y_{s}=\emptyset$, the fact that $(Y\setminus Y_{s})\cup\{(s,c)\}$ is feasible implies that $Y\cup\{(s,c)\}$ is feasible. Thus, $s$ also strongly claims an empty seat of $c$. Finally, we show that no vacant college property implies no empty matching property by showing its contraposition. More specifically, we assume student $s$ very strongly claims an empty seat of $c$ in matching $Y$. Then, we show that $Y$ does not satisfy no vacant college property. The fact that student $s$ very strongly claims an empty seat of $c$ implies $c\succ_{s}\emptyset$, $Y_{s}=\emptyset$, and $Y_{c}=\emptyset$ hold. Thus, $Y$ does not satisfy no vacant college property.

Next, we introduce strategyproofness.

Here, we consider strategic manipulations only by students. It is well-known that even in the most basic model of one-to-one matching Gale and Shapley (1962), satisfying strategyproofness (as well as basic fairness and efficiency requirements) for both sides is impossible Roth (1982). One rationale for ignoring the college side would be that the preference of a college must be presented in an objective way and cannot be skewed arbitrarily.

In this section, we briefly introduce existing mechanisms, which are strategyproof for a given class of constraints. First, let us introduce Generalized Deferred Acceptance mechanism (GDA), which works under hereditary M^♮-convex set constraints Hatfield and Milgrom (2005). As its name shows, it is a generalized version of the Deferred Acceptance mechanism Gale and Shapley (1962). To define GDA, we first introduce choice functions of students and colleges.

As long as ${f}$ induces a hereditary M^♮-convex set, a unique subset $Y^{\prime}$ exists that maximizes the above formula. Furthermore, such a subset can be efficiently computed in the following greedy way. Let $Y^{\prime}$ denote the set of chosen contracts, which is initially $\emptyset$. Then, sort $Y$ in the decreasing order of their weights. Then, choose contract $x$ from $Y$ one by one and add it to $Y^{\prime}$, as long as $Y^{\prime}\cup\{x\}$ is feasible.

Using $Ch_{S}$ and $Ch_{C}$, GDA is defined as Mechanism 1.

Note that we describe the mechanism using terms like ”student $s$ offers” to make the description more intuitive. In reality, GDA is a direct-revelation mechanism, where the mechanism first collects the preference of each student, and the mechanism chooses a contract on behalf of each student.

Kojima et al. (2018) show that when $f$ induces a hereditary M^♮-convex set, GDA is strategyproof, the obtained matching $Y$ satisfies a property called Hatfield-Milgrom stability (HM-stability), and $Y$ is the student-optimal matching within all HM-stable matchings (i.e., all students weakly prefer $Y$ over any other HM-stable matching).

Intuitively, HM-stability means there exists no contract in $X\setminus Y$ that is mutually preferred by students and colleges. Note that HM-stability implies fairness. If student $s$ has justified envy in matching $Y$, there exists $(s,c)\in X\setminus Y$, $(s^{\prime},c)\in Y$, s.t. $(s,c)\succ_{s}Y_{s}$ and $w((s,c))>w((s^{\prime},c))$ holds. Then, $(s,c)\in Ch_{S}(Y\cup\{(s,c)\})$ and $(s,c)\in Ch_{C}(Y\cup\{(s,c)\})$ hold, i.e., $Y$ is not HM-stable.

For standard maximum quotas constraints, the only distributional constraints are $(q_{c})_{c\in C}$, i.e., ${f}(Y)=0$ iff for each $c\in C$, $|Y_{c}|\leq q_{c}$ holds. Then, $Ch_{C}(Y)$ is defined as $\bigcup_{c\in C}Ch_{c}(Y_{c})$, where $Ch_{c}$ is the choice function of each college $c$, which chooses top $q_{c}$ contracts from $Y_{c}$ based on $\succ_{c}$. When $Ch_{C}$ is defined this way, GDA becomes equivalent to the standard DA.

Next, we introduce two mechanisms that work for hereditary constraints. The Serial Dictatorship (SD) mechanism Goto et al. (2017) is parameterized by an exogenous serial order over the students called a master-list. We denote the fact that $s$ is placed in a higher/earlier position than student $s^{\prime}$ in master-list $L$ as $s\succ_{L}s^{\prime}$. Students are assigned sequentially according to the master-list. In our context with constraints, student $s$ is assigned to her most preferred college $c$, where $c$ considers her acceptable (i.e., $(s,c)\in X$ holds) and assigning $s$ to $c$ does not cause any constraint violation. More specifically, assume the obtained matching for students placed higher than $s$ in $L$ is $Y$. Then, $s$ can be assigned to $c$ when $f(\nu(Y\cup\{(s,c)\}))=0$ holds. SD is strategyproof and achieves Pareto efficiency.

The Artificial Cap Deferred Acceptance mechanism (ACDA) is defined as follows. First, we choose one vector $\nu^{*}$ s.t. $f(\nu^{*})=0$, and there exists no $\nu^{\prime}>\nu^{*}$ where $f(\nu^{\prime})=0$, i.e., a maximal feasible vector. Note that $\nu^{*}$ must be chosen independently from students’ preferences $\succ_{S}$ to guarantee strategyproofness. Then, we apply standard DA, where maximum quota $q_{c_{i}}$ for each college $c_{i}$ is given as $\nu^{*}_{i}$. Intuitively, in ACDA, the set of feasible vectors $F$ is artificially reduced to a hyper-rectangle, where $\nu$ is feasible iff $\nu\leq\nu^{*}$. ACDA is strategyproof and fair, assuming $\nu^{*}$ is chosen independently from students’ preferences.

In this section, we examine whether a fair and strategyproof mechanism exists under a given class of distributional constraints in conjunction with some efficiency property. The classes of constraints we consider are: maximum quotas constraints, hereditary and M^♮-convex set constraints, and hereditary constraints.

First, we list known results.

• •

  For maximum quotas constraints, fairness, nonwastefulness, and strategyproofness are compatible, i.e., the standard DA satisfies these properties Roth (1985). On the other hand, fairness and Pareto efficiency are incompatible, i.e., even without strategyproofness, a matching that satisfies Pareto efficiency and fairness may not exist Roth (1982).

• •

  For hereditary and M^♮-convex set constraints, fairness, weak nonwastefulness, and strategyproofness are compatible, i.e., Generalized DA satisfies these properties Kimura et al. (2023). On the other hand, fairness and nonwastefulness are incompatible Kamada and Kojima (2017).

• •

  For hereditary constraints, fairness, weak nonwastefulness, and strategyproofness are incompatible Cho et al. (2022)

Given these known results, the remaining open questions are as follows.

1. (1)

   Under hereditary and M^♮-convex set constraints, does a strategyproof, fair, and cut-off nonwasteful mechanism exist?

2. (2)

   Under hereditary constraints, can a strategyproof and fair mechanism satisfy any property weaker than weak nonwastefulness?

For question (1), we obtain a negative answer, as shown in Theorem 4. For question (2), we obtain a stronger result than Cho et al. (2022), i.e., Theorem 4 shows that no mechanism simultaneously satisfies strategyproofness, fairness, and no vacant college property. Then, we show a simple mechanism that satisfies strategyproofness, fairness, and no empty matching property (Theorem 4). In summary, we obtain tight boundaries (at least in the granularity of efficiency properties we consider in this paper) on whether a strategyproof and fair mechanism can satisfy certain efficiency properties for each class of constraints (Table 1).

No mechanism can simultaneously satisfy fairness, strategyproofness, and cut-off nonwastefulness under hereditary M^♮-convex set constraints.

No mechanism can simultaneously satisfy fairness, strategyproofness, and no vacant college property under hereditary constraints.

Next, we show that there exists a mechanism that satisfies fairness, strategyproofness, and no empty matching property under hereditary constraints. This mechanism utilizes GDA. More specifically, for given $f$, which is hereditary, we construct a set of vectors $F^{\prime}$ such that $\forall\nu\in F^{\prime}$, $f(\nu)=0$ holds (i.e., $F^{\prime}$ is a subset of vectors induced by $f$), and $F^{\prime}$ is a hereditary M^♮-convex set. Then, we apply GDA by using $f^{\prime}$ (where $f^{\prime}(\nu)=0$ iff $\nu\in F^{\prime}$) instead of $f$. $F^{\prime}$ is constructed as follows. We initialize $F^{\prime}\leftarrow\{e_{0}\}$. Then, for each $i\in M$, if $f(e_{i})=0$, we add $e_{i}$ to $F^{\prime}$. Clearly, $F^{\prime}$ is an M^♮-convex set; it contains only $e_{0}$ and $e_{i}$ ($i\in M$).

Under hereditary constraints, GDA using $f^{\prime}$ is fair, strategyproof, and satisfies no empty matching property.

In this section, we introduce a weaker fairness concept called envy-free up to $k$ peers (EF-$k$). For matching $Y$ and student $s$, let $Ev(Y,s)$ denote $\{s^{\prime}\mid s^{\prime}\in S,s\text{ has }\text{justified envy toward }s^{\prime}\text{ in }Y\}$.

EF-$0$ is equivalent to fairness. Any matching is EF-$(n-1)$, where $n=|S|$.

There are other ways to relax fairness than EF-$k$. One straightforward way is to minimize the total number of justified envies. However, this criterion can be unfair among students, e.g., one student has many envies while others have only a few. Our definition of EF-$k$ is more egalitarian; it minimizes the envies of the worst student. Other egalitarian criteria are also possible. For example, instead of counting the number of students to whom each student has envy, we can count the colleges at which each student has envy. Also, we can count the number of students by whom each student is envied. Which concept is socially acceptable is difficult to tell. This work is a first step that brings up new research directions in two-sided matching, i.e., how to relax the fairness concept in a socially acceptable way.

We use the following example to show that nonwastefulness and EF-$k$ are incompatible for any $k<n-1$ under hereditary M^♮-convex set constraints.

Under hereditary M^♮-convex set constraints, there exists a case that no matching is nonwasteful and EF-$k$ for any $k<n-1$.

Given Theorem 5, a natural question is the complexity of checking the existence of a nonwasteful and EF-$k$ matching (for $k<n-1$). Let us assume $f$ can be computed in a constant time. To examine this complexity, we utilize the following lemma.

Checking whether an EF-$k$ ($k<n-1$) and nonwasteful matching exists or not is NP-complete, even when distributional constraints form a hereditary M^♮-convex set.

In this section, we introduce two contrasting strategyproof mechanisms that work for general hereditary constraints. The first one (called SD^∗) satisfies the strongest efficiency property, i.e., Pareto efficiency, while it cannot guarantee EF-$k$ for any fixed $k<n-1$. The second one (called SDA with reserved quotas) satisfies EF-$k$ for any fixed $k<n-1$, while it can only guarantee a rather weak efficiency property. In the next section, we experimentally show that SD^∗ can guarantee EF-$k$ where $k$ is much smaller than $n-1$ when colleges’ preferences are similar. Furthermore, we experimentally show that SDA with reserved quotas can significantly improve students’ welfare compared to a fair (EF-$0$) mechanism even when $k$ is very small.