On the welfare effects of affirmative actions in school choice

Abstract

Recent evidence, from both academia and practice, indicates that implementing affirmative action policies in school choice problems may induce substantial welfare losses on the intended beneficiaries. This paper addresses the following two questions: what are the causes of such perverse consequences, and when we can effectively implement affirmative action policies without unsatisfactory outcomes. Using the minority reserve policy in the student optimal stable mechanism as an example, I show that two acyclicity conditions, type-specific acyclicity and strongly type-specific acyclicity, are crucial for the effective implementations of minority reserve policies. I further illustrate how restrictive these two acyclicity conditions are, and the intrinsic difficulty of embedding diversity goals into stable matching mechanisms. Under some regularity conditions, I demonstrate that the minority reserve policy is very unlikely to cause welfare losses on any minority students when the number of schools is sufficiently large.

Appendices

Appendix A: Proofs for Sect. 3

Notations Let $\mu$ be the matching outcome of SOSM-R in market $\Gamma = (C, P, \succ , (q, r^m))$ and $\tilde{\mu }$ be the matching after a stronger minority reserve $\tilde{r}^m$. Denote the market after implementing $\tilde{r}^m$ by $\tilde{\Gamma } = (C, P, \succ , (q, \tilde{r}^m))$. $\Gamma ^m = (C^m, P', (\succ ^o, \succ ^r), (q, r^m))$ and $\tilde{\Gamma }^m = (C^m, P', (\succ ^o, \succ ^r), (q, \tilde{r}^m))$ are the two respective auxiliary markets of $\Gamma$ and $\tilde{\Gamma }$ after splitting each school into the original sub-school ($c^o$) and the reserve sub-school ($c^r$).

1.1 A.1: Proof of Theorem 1

I first give the following Lemma which is identical with Proposition 3 of Doğan (2016), except that in the proof of Lemma 1 I introduce the reserve parameter with the corresponding original sub-schools and reserve sub-schools.

Lemma 1

For a market $\Gamma = (C, P, \succ , (q, r^m))$ and its corresponding $\Gamma ^m = (C^m, P', (\succ ^o, \succ ^r), (q, r^m))$, let $\mu$ and $\tilde{\mu }$ be the matching outcomes of SOSM-R before and after a stronger minority reserve $\tilde{r}^m$. If $\tilde{\mu }(s)$ is Pareto dominated by $\mu (s)$ for all $s \in S^m$, then $\tilde{\mu }(s)$ is Pareto dominated by $\mu (s)$ for all $s \in S$.

Proof

I will argue by contradiction. Suppose that if $\tilde{\mu } (s)$ is Pareto dominated by $\mu (s)$ for all $s \in S^m$, there has at least one majority student $s_0 \in S^M$ who prefers $\tilde{\mu }$ to $\mu$ in $\tilde{\Gamma }^m$, $\tilde{\mu } (s_0) P_{s_0} \mu (s_0)$. Let $\tilde{\mu } (s_0) = c_1$ and $\mu (s_0) = c_0$. Since $s_0$ is a majority student, she is rejected by a reserve sub-school. Because $s_0$ prefers $c_1$ to $c_0$, she must have been rejected by $c_1$ in $\Gamma ^m$ (which leads to the matching $\mu$) at an earlier step before $s_0$ applies to $c_0$. Let step l be the step when $s_0$ is rejected by $c_1$ in $\Gamma ^m$ of the SOSM algorithm. Clearly, $c_1$ must have exhausted its capacity $|\mu (c_1)| = q_{c_1}$, and $s \succ _{c_1}^x s_0$, $x= o,r$, for all s tentatively accepted by $c_1$ at step l. Since $\tilde{\mu } (s_0) = c_1$, there must have another student, denoted by $s_1$, such that $s_1$ is tentatively accepted by $c_1$ at step l in $\Gamma ^m$ but matches with another school in $\tilde{\Gamma }^m$. Recall that at step l in $\Gamma ^m$, $s_0$ is rejected by $c_1$, it implies that $s_1 \succ _{c_1}^r s_0$.

I first show that $s_1$ must be a majority student who has applied to $c_1$ at a step earlier than l in $\Gamma ^m$. Suppose that $s_1 \in S^m$, because $\tilde{\mu }$ is Pareto dominated by $\mu$ for all $s \in S^m$, while $\mu (s_1) \ne \tilde{\mu } (s_1)$, it implies that $\mu (s_1) P_{s_1} \tilde{\mu }(s_1)$. Also, recall that $s_1$ is tentatively accepted by $c_1$ before her final match in $\Gamma ^m$, $c_1 R_{s_1} \mu (s_1)$, we have $c_1 P_{s_1} \tilde{\mu } (s_1)$. Since the stronger minority reserve $\tilde{r}^m$ only increases the capacity of some $c^r$, $(s_1, c_1)$ forms a blocking pair in $\tilde{\Gamma }^m$, which contradicts the stability of $\tilde{\mu }$. Thus, $s_1 \in S^M$. Obviously, $s_1$ applies to $c_1$ at a step earlier than l.

Next, since $\tilde{\mu }(s_1) \ne c_1$, while $s_1 \succ _{c_1}^r s_0$ and $s_0, s_1 \in S^M$, it implies that $\tilde{\mu } (s_1) P_{s_1} c_1$. Otherwise, $(s_1, c_1)$ is a blocking pair in $\tilde{\Gamma }^m$. Combine with $c_1 R_{s_1} \mu (s_1)$, we have $\tilde{\mu } (s_1) P_{s_1} \mu (s_1)$. Denote $\tilde{\mu } (s_1) = c_2$. Recall that in $\Gamma ^m$, $s_1$ applies to $c_1$ before step l. Without loss of generality, denote this step by $l-1$. I can repeat the arguments for $s_0$ and $s_1$, and construct a set of l majority students who are all better-off in $\tilde{\Gamma }^m$. That is, $\tilde{\mu } (s_i) P_{s_i} c_i R_{s_i} \mu (s_i)$, $i = \{0, \ldots , l-1 \}$, $s_i \in S^M$, and $c_i$ belongs to a set of l schools such that for each $s_i$ she is tentatively accepted at step $l-i$. In particular, let step 1 be the step that initiates the rejection and acceptance chain in market $\Gamma ^m$ when $s_{l-1}$ applies to $c_{l-1}$. Because $s_{l-1}$ applies to $c_{l-1}$ at the first step, it implies that $c_{l-1} P_{s_{l-1}} c$, for all $c \in C \backslash c_{l-1}$. Recall that $c_{l-1} \ne \tilde{\mu } (s_{l-1})$, which contradicts to $\tilde{\mu } (s_{l-1}) P_{s_{l-1}} c_{l-1}$. $\square$

Lemma 1 indicates that when all minorities prefer the previous matching outcome without the stronger minority reserve $\tilde{r}^m$, then there is no majority who is strictly better-off after implementing $\tilde{r}^m$. In other words, no students benefit from $\tilde{r}^m$. With Lemma 1, I am now ready to prove Theorem 1.

(i) Type-specific acyclicity $\Longrightarrow$ Respect the spirit of a stronger minority reserve. I will prove the contrapositive. That is, if $\mu (s) R_s \tilde{\mu }(s)$ for all $s \in S^m$, and $\mu (s) P_s \tilde{\mu } (s)$ for at least one $s \in S^m$, then the priority structure $((\succ ^o, \succ ^r), (q, \tilde{r}^m))$ of market $\tilde{\Gamma }^m$ must contain a type-specific cycle with at least two schools and three students.

Lemma 1 indicates that if $\tilde{\mu }(s)$ is Pareto dominated by $\mu (s)$ for all $s \in S^m$, then there has at least one $s' \in S$, $\mu (s') P_{s'} \tilde{\mu }(s')$. Denote $\tilde{S} = \{ s \in S | \mu (s) P_s \tilde{\mu }(s) \}$ be the set of students strictly prefer the matching $\mu$. Because $\mu (s) R_s \tilde{\mu }(s)$ for all $s \in S \backslash \tilde{S}$, for those who are not strictly worse off after implementing $\tilde{r}^m$, they are matched with the same school under $\mu$, i.e., $S \backslash \tilde{S} = \{ s \in S | \mu (s) = \tilde{\mu }(s) \}$.

Choose a set of students from $\tilde{S}$, $\tilde{S'} \subseteq \tilde{S}$, such that for all $s \in \tilde{S'}$, $\tilde{\mu }(s) \ne s$. $\tilde{S'}$ is nonempty. Otherwise, there has at least one minority student $s \in \tilde{S}$ and $\tilde{\mu }(s) = s$, such that s and $\mu (s)$ form a blocking pair after the stronger minority reserve $\tilde{r}^m$. Further, $\tilde{S'}$ contains at least one minority student and one majority student. Because if all $s' \in S^M \cap \tilde{S}$, $\tilde{\mu }(s') = s'$, then for some $s \in S^m \cap \tilde{S}$, s and $\mu (s)$ form a blocking pair after implementing $\tilde{r}^m$.

Without loss of generality, let $s_j \in S^M \cap \tilde{S'}$ be the majority student who is directly affected by $\tilde{r}^m$,³³ $\mu (s_j) = c_0$. Since $c_0 P_{s_j} \tilde{\mu }(s_j)$, and $\tilde{\mu }$ is stable (Hafalir et al. 2013), this implies that $|\tilde{\mu } (c_0)| = q_{c_0}$, $|\tilde{\mu } (c_0) \cap S^m| = \tilde{r}_{c_0}^m$, and for all s who are tentatively accepted by $c_0^x$, $s \succ _{c_0}^x s_j$, $x = o, r$. Since $s_j$ is a majority student who is directly affected by the stronger minority reserve $\tilde{r}^m$, there has a minority student tentatively accepted by $c_0$, denoted by $s_i$, such that $c_0 P_{s_i} \mu (s_i)$ and $s_j \succ _{c_0}^o s_i$. It implies that $s_i \in {c_0^r}$ in $\tilde{\Gamma }^m$ (because of $s_i \succ _{c_0}^r s_j$). Otherwise, $(s_j, c_0)$ forms a blocking pair. However, $\tilde{\mu }(s_i) \ne c_0$ by assumption (otherwise $\tilde{\mu } (s_i) P_{s_i} \mu (s_i)$). Since $s_i$ cannot be rejected by a majority student from $c_0^r$, there must have another minority student, denoted by $s_k$, such that $s_k \in S^m \cap \tilde{S'}$, $s_k \in \tilde{\mu } (c_0) \backslash \mu (c_0)$ and $s_k \succ _{c_0}^r s_i$. Thus, we have

$$\begin{aligned} s_k \succ _{c_0}^r s_i \succ _{c_0}^r s_j, \quad s_i, s_k \in S^m, \quad s_j \in S^M \end{aligned}$$

(1)

Denote $\mu (s_k) = c_k$. Because $c_k P_{s_k} c_0$ and $\tilde{\mu }$ is stable, it implies that $|\tilde{\mu } (c_k)| = q_{c_k}$, and there is another student in $\tilde{S'}$, denoted by $s_{k-1}$, such that

$$\begin{aligned} s_{k-1} \in \tilde{\mu }(c_k) \backslash \mu (c_k), \quad s_{k-1} \succ _{c_k}^o s_k \end{aligned}$$

(2)

Otherwise, $(s_k, c_k)$ forms a blocking pair in $\tilde{\Gamma }^m$. Apply similar arguments of $s_{k-1}, s_k$ and $c_0, c_k$ for each student in $\tilde{S'}$ iteratively. Because the set of students in $\tilde{S'}$ is finite, let $\{ s_0, s_1, \ldots , s_{k-2}, s_{k-1} \} \in \tilde{S'} \backslash \{s_k\}$, I can construct a finite sequence of schools $c_1, c_2, \ldots , c_{k-1}, c_k$ such that for each $l = \{0, 1, 2, \ldots , k-1\}$

$$\begin{aligned}&s_l \in \tilde{\mu } (c_{l+1}) \backslash \mu (c_{l+1}), \quad \mu (s_l) = c_l, \quad c_l P_{s_l} c_{l+1} \end{aligned}$$

(3)

$$\begin{aligned}&|\tilde{\mu } (c_l)| = q_{c_l}, \quad s \succ _{c_l}^x s_l, \; x = o, r, \quad \text {for each}\; s \in \tilde{\mu } (c_l) \end{aligned}$$

(4)

In particular, I have

$$\begin{aligned} {\left\{ \begin{array}{ll} &{} s_l \succ _{c_{l+1}}^o s_{l+1} \quad \text {if}\; s_{l+1} \in S^m \\ &{} s_l \succ _{c_{l+1}}^r s_{l+1} \quad \text {if}\; s_{l+1} \in S^M, \quad l = \{0, 1, 2, \ldots , k-1\} \end{array}\right. } \end{aligned}$$

(5)

It is not difficult to see that $s_0 \equiv s_j$ by the preceding arguments. Combining (1) and (5) gives the cycle condition. The scarcity condition is satisfied by (2), (3) and (4), and the stability of the SOSM.

(ii) Respect the spirit of a stronger minority reserve $\Longrightarrow$ Type-specific acyclicity. I will prove the contrapositive. Suppose that the priority structure $((\succ ^o, \succ ^r), (q, \tilde{r}^m))$ has a type-specific cycle with: (1) $k+1$ schools $c_0, c_1, \ldots , c_k$, (2) $k+2$ students $s_i, s_j, s_k, {\mathbf {s}}_l$, where $s_i, s_k \in S^m$, $s_j \in S^M$, ${\mathbf {s}}_l = \{s_1, s_2, \ldots , s_{k-1} \} \in S$, and (3) $k+1$ disjoint sets of students $S_{c_0}, S_{c_1}, \ldots , S_{c_k} \subset S \backslash \{ s_i, s_j, s_k, s_1, s_2, \ldots , s_{k-1} \}$, as in Definition 3. Let: (1) $c_0 P_{s_i} s_i$,  $c_k P_{s_k} c_0 P_{s_k} s_k$,  $c_0 P_{s_j} c_1 P_{s_j} s_j$, (2) $c_x P_{s_x} c_{x+1} P_{s_x} s_x$, for all $s_x \in {\mathbf {s}}_l$, and (3) for each of the students belongs to $S_{c_y}$, $y=0, 1,\ldots ,k$, she ranks $c_y$ as her top choice. Also, let all students outside $\{s_i, s_j, s_k, {\mathbf {s}}_l\} \cup S_{c_0} \cup S_{c_1} \cup \ldots \cup S_{c_k}$ prefer being unmatched to being matched with any school.

Suppose that the stronger minority reserve $\tilde{r}^m$ only requires school $c_0$ to reserve one more seat for minorities. That is, $\tilde{r}_{c_0}^m = r_{c_0}^m +1$, and $\tilde{r}_{c'}^m = r_{c'}^m$, for all $c' \in C \backslash c$. In the unique matching produced by SOSM before implementing $\tilde{r}^m$ (i.e., in market $\Gamma ^m$), we have $\mu (c_k) = s_k \cup S_{c_k}$, $\mu (c_0) = s_j \cup S_{c_0}$, $\mu (c_x) = s_x \cup S_{c_x}$, $x=1,2,\ldots ,k-1$. In the unique matching produced by SOSM after implementing $\tilde{r}^m$ (i.e., in market $\tilde{\Gamma }^m$), we have $\tilde{\mu }(c_k) = s_{k-1} \cup S_{c_k}$, $\tilde{\mu }(c_0) = s_k \cup S_{c_0}$, $\tilde{\mu }(c_1) = s_j \cup S_{c_0}$, $\tilde{\mu }(c_x) = s_{x-1} \cup S_{c_x}$, $x=2,3,\ldots ,k-1$. Clearly, $\mu (s) P_{s} \tilde{\mu }(s)$ for all $s \in \{s_k, s_1, s_2, \ldots , s_{k-1} \} \cap S^m$, and $\mu (s) R_{s} \tilde{\mu }(s)$ for all $s \in S^m \backslash \{s_k, s_1, s_2, \ldots , s_{k-1} \}$, as we need. $\square$

1.2 A.2: Proof of Theorem 2

I first give the following Lemma.

Lemma 2

For a market $\Gamma = (C, P, \succ , (q, r^m))$ and its corresponding $\Gamma ^m = (C^m, P', (\succ ^o, \succ ^r), (q, r^m))$, let $\mu$ and $\tilde{\mu }$ be the matching outcomes of SOSM-R before and after a stronger minority reserve $\tilde{r}^m$. If there is at least one $s \in S^m$ who is strictly worse off in $\tilde{\mu }$ than in $\mu$, then there must have at least one majority student who is strictly worse off in $\tilde{\mu }$ than in $\mu$.

Proof

I prove the Lemma by contradiction. Suppose at least one of the minority students is strictly worse off in $\tilde{\Gamma }^m$ compared to in $\Gamma ^m$, but no majority students are strictly worse off after implementing $\tilde{r}^m$. Let $\tilde{S}^m$ denote the set of minority students who are strictly worse off after $\tilde{r}^m$, $\mu (s) P_s \tilde{\mu }(s)$, for all $s \in \tilde{S}^m \subset S^m$. And for all $s' \in S^m \backslash \tilde{S}^m$, either $\tilde{\mu }(s') R_{s'} \mu (s')$ or $\tilde{\mu }(s') P_{s'} \mu (s')$.

Suppose that a minority student, denoted by $s_0$, is strictly worse off in $\tilde{\Gamma }^m$ compared to in $\Gamma ^m$. Let $\mu (s_0) = c_1$. Since $c_1 P_{s_0} \tilde{\mu }(s_0)$, the capacity of $c_1$ is full at the step when $s_0$ is rejected by $c_1$ in $\tilde{\Gamma }^m$, there must be another student, denoted by $s_1$, who is tentatively accepted by $c_1$ when $s_0$ is rejected. Let step l be the step when $s_1$ applies to $c_1$ (or equivalently, $s_0$ is rejected by $c_1$) in $\tilde{\Gamma }^m$ of the SOSM algorithm.

I first show that if $s_1$ is a majority student, then there has a group of minority students, denoted by $\tilde{S}_1^m$, who are strictly worse off in $\tilde{\Gamma }^m$ compared to in $\Gamma ^m$, i.e., $\tilde{S}_1^m \in \tilde{S}^m$, and apply to $c_1$ at a step earlier than l in $\tilde{\Gamma }^m$. Since $s_1 \in S^M$, and no majority students are strictly worse off after implementing $\tilde{r}^m$ by assumption, we know that either $c_1 P_{s_1} \mu (s_1)$ or $c_1 R_{s_1} \mu (s_1)$. Recall that $c_1 P_{s_0} \tilde{\mu }(s_0)$, $s_0 \in S^m$, and all minorities have higher priorities than any majorities in all reserve sub-schools $c^r$, we have $s_1 \succ _{c_1}^o s_0$, $s_0 \in \mu (c_1^r)$ but $s_0 \in \tilde{\mu }(c_1^o)$. Otherwise $(s_0, c_1)$ would form a blocking pair in $\tilde{\Gamma }^m$. Therefore, there must have a group of minority students, denoted by $\tilde{S}_1^m$, who have applied to and been tentatively accepted by $c_1$ at a step earlier than l (when $c_1$ rejects $s_0$) in $\tilde{\Gamma }^m$, but do not apply to $c_1$ in $\Gamma ^m$. $s \succ _{c_1}^o s_0$ for all $s \in \tilde{S}_1^m$,³⁴ but $\mu (s) P_s c_1$ for all $s \in \tilde{S}_1^m$. Otherwise, $(s, c_1)$ are blocking pairs in $\Gamma ^m$ for all $s \in \tilde{S}_1^m$. Without losing of generality, denote the least preferred student in $\tilde{S}_1^m$ be $s_2$ (if $|\tilde{S}_1^m| =1$, then $\tilde{S}_1^m \equiv s_2$), such that $s \succ _{c_1}^o s_2 \succ _{c_1}^o s_0$ for all $s \in \tilde{S}_1^m \backslash s_2$.

If $s_1$ is a minority student, we know that $s_1$ must be strictly worse off in $\tilde{\Gamma }^m$ compared to in $\Gamma ^m$, $\mu (s_1) P_{s_1} c_1$. Otherwise, $(s_1, c_1)$ forms a blocking pair in $\Gamma ^m$. Thus, $s_1 \in \tilde{S}^m$, and $s_1$ is rejected by $\mu (s_1)$ at a step earlier than l by another student, denoted by $\dot{s}$. Since $s_0$ is a random minority student who is strictly worse off after $\tilde{r}^m$, I can equivalently treat $s_1$ as $s_0$ when $s_1 \in S^m$. Therefore, (i) if $\dot{s} \in S^m$, repeat the same arguments in this paragraph, we know that $\dot{s} \in \tilde{S}^m$, rewrite $\dot{s}$ as $s_2$; (ii) if $\dot{s} \in S^M$, apply the arguments in the previous paragraph (i.e., equivalently treat $\dot{s}$ as $s_1$ when $s_1 \in S^M$), and I have another set of minority students, denoted by $\tilde{S}_2^m$, who are strictly worse off in $\tilde{\Gamma }^m$ compared to in $\Gamma ^m$, write the least preferred minority student in $\tilde{S}_2^m$ as $s_2$.

Hence, if there is one minority student, $s_0$, who is strictly worse off in $\tilde{\Gamma }^m$ compared to in $\Gamma ^m$, there must have another minority student, $s_2$, who is also strictly worse off in $\tilde{\Gamma }^m$ and is rejected by $\mu (s_2)$ at a step earlier than l in $\tilde{\Gamma }^m$. Repeat the preceding arguments I can construct a set of l minority students, denoted by $\tilde{S}_l$, such that $c_{i+1} P_{s_i} c_i$, $i = \{1, \ldots , l \}$, $s_i \in \tilde{S}_l \subset \tilde{S}^m$, where $c_{i+1} = \mu (s_i)$, and $c_i$ belongs to a set of l schools in which $s_i$ is tentatively accepted by $c_i$ at step $l-i+1$ of the SOSM algorithm in $\tilde{\Gamma }^m$. In particular, $s_l \in \tilde{S}_l$ applies to and is tentatively accepted by $c_l$ at step 1. It implies that $c_l P_{s_l} c$, for all $c \in C \backslash c_l$. Recall that $\mu (s_l) \ne c_l$, which contradicts to $\mu (s_l) P_{s_l} c_l$. $\square$

Compared to Lemma 1, Lemma 2 allows situations where some minorities may benefit from a stronger minority reserve policy. I am now ready to prove Theorem 2.

(i) Strongly type-specific acyclicity $\Longrightarrow$ Respect the improvement of a stronger minority reserve. Suppose if $\mu (s) P_s \tilde{\mu }(s)$ for at least one $s \in S^m$, I show that $\tilde{\Gamma }^m$ must have a quasi type-specific cycle with two schools and three students.

Lemma 2 implies that if $\mu (s) P_s \tilde{\mu }(s)$ for at least one $s \in S^m$, then $\mu (s') P_{s'} \tilde{\mu }(s')$ for at least one $s' \in S^M$. Let $s_0$ denote a minority student who is strictly worse off after the stronger minority reserve $\tilde{r}^m$. Let $\mu (s_0) = c_0$, and step k be the step of the SOSM algorithm when $s_0$ is rejected by $c_0$ in $\tilde{\Gamma }^m$. Without loss of generality, I can construct a set of $k-1$ students, $\mathbf {s_l} = \{s_1, s_2, \ldots , s_{k-1} \} \in S$, such that $\mu (s_l) P_{s_l} \tilde{\mu } (s_l) \ne s_l$, $\mu (s_l) = c_l$, $l = \{1, \ldots , k-1\}$, $k \ge 2$. Let $k-l$ be the step when $s_l$ is rejected by $\mu (s_l)$ in $\tilde{\Gamma }^m$. $s_l$ applies to $c_{l-1}$ at step $k-l+1$. In particular, we have $s_1$ who is rejected by $\mu (s_1) = c_1$ at step $k-1$ and applies to $c_0$ at step k. Thus,

(i.a.) if all students in $\mathbf {s_l}$ except $s_{k-1}$ are minorities. By my construction of $\mathbf {s_l}$, $s_{k-1}$ is rejected by $\mu (s_{k-1}) = c_{k-1}$ at step 1 of the SOSM algorithm in $\tilde{\Gamma }^m$, and applies to $c_{k-2}$ in the next step. Obviously, $s_{k-1}$ is directly affected by $\tilde{r}^m$ (recall Footnote 33), and $s_{k-1} \in S^M$. Thus, there must have another minority student, denoted by $\dot{s}$, $\dot{s} \in S^m \backslash \mathbf {s_l}$, who prefers $c_{k-1}$ to all the rest schools but is rejected by $c_{k-1}$ in $\Gamma ^m$ (i.e., before the stronger minority reserve $\tilde{r}^m$). That is, $c_{k-1} P_{\dot{s}} \,c$, for all $c \in C \backslash c_{k-1}$, $s_{k-1} \succ _{c_{k-1}}^o \dot{s}$ but $\dot{s} \succ _{c_{k-1}}^r s_{k-1}$. Otherwise, $(s_{k-1}, c_{k-1})$ forms a blocking pair in $\tilde{\Gamma }^m$. In addition, we know that $s_{k-2}$ is rejected by $\mu (s_{k-2}) = c_{k-2}$ at step 2 when $s_{k-1}$ applies to $c_{k-2}$. As $s_{k-1} \in S^M$ and $s_{k-2} \in S^m$, we have $s_{k-1} \succ _{c_{k-2}}^o s_{k-2}$. Thus, $\dot{s} \succ _{c_{k-1}}^r s_{k-1} \succ _{c_{k-2}}^o s_{k-2}$.

(i.b.) if $s_1 \in S^m$, and there is at least one student in $\mathbf {s_l}$ besides $s_{k-1}$ is a majority student. Let $s_l$ be a minority student in $\mathbf {s_l}\backslash \{s_1\}$, who is rejected by $\mu (s_l) = c_l$ in $\tilde{\Gamma }^m$ when a majority student in $\mathbf {s_l}$ applies to $c_l$. Denote this majority student as $s_{l-1}$ and $\mu (s_{l-1}) = c_{l-1}$. Thus, $s_{l-1} \succ _{c_l}^o s_l$ (a minority student can only be rejected by a majority student from an original sub-school). By my construction of $\mathbf {s_l}$, there is another student $s_{l-2} \in \mathbf {s_l}$, who is tentatively accepted by $c_{l-1}$ at the step when $s_{l-1}$ is rejected by $c_{l-1}$. Thus, $s_{l-2} \succ _{c_{l-1}}^r s_{l-1}$ (a majority student can only be rejected by another student from a reserve sub-school). With $s_{l-2} \in S$, $s_{l-1} \in S^M$, and $s_l \in S^m$, we have $s_{l-2} \succ _{c_{l-1}}^r s_{l-1} \succ _{c_l}^o s_l$.

(i.c.) if $s_1 \in S^M$. Since $s_0 \in s^m$, $c_0$ rejects $s_0$ at step k of the SOSM algorithm when $s_1$ applies to $c_0$, we have $s_1 \succ _{c_0}^o s_0$. Similar to the previous cases, by my construction of $\mathbf {s_l}$, $s_1$ is rejected by $\mu (s_1) = c_1$ at step $k-1$ when $s_2 \in \mathbf {s_l}$ applies and is tentatively accepted by $c_1$. Thus, $s_2 \succ _{c_1}^r s_1$, and we have $s_2 \succ _{c_1}^r s_1 \succ _{c_0}^o s_0$, with $s_2 \in S$, $s_1 \in S^M$ and $s_0 \in S^m$.

Condition (S’) is trivially satisfied through the preceding arguments and the stability of the SOSM in all three cases.

(ii) Respect the improvement of a stronger minority reserve $\Longrightarrow$ Strongly type-specific acyclicity. I will prove the contrapositive. Suppose that the priority structure $((\succ ^o, \succ ^r), (q, \tilde{r}^m))$ has a quasi type-specific cycle with: (1) two schools $c, c'$, (2) three students $s_i, s_j, s_k$, where $s_i \in S$, $s_k \in S^m$, and $s_j \in S^M$, and (3) two disjoint sets of students $S_c, S_{c'} \subset S \backslash \{ s_i, s_j, s_k \}$, as in Definition 4. Let $c P_{s_i} s_i$,  $c P_{s_j} c' P_{s_j} s_j$,  $c' P_{s_k} s_k$, and for each of the students belongs to $S_x$, $x=c, c'$, she ranks school x as her top choice. Also, let all students outside $\{s_i, s_j, s_k\} \cup S_{c} \cup S_{c'}$ prefer being unmatched to being matched with any school.

Suppose the stronger minority reserve $\tilde{r}^m$ only requires school c to reserve one more seat for minorities. That is, $\tilde{r}_{c}^m = r_{c}^m +1$, and $\tilde{r}_{c''}^m = r_{c''}^m$, for all $c'' \in C \backslash c$. In the unique matching produced by SOSM before implementing $\tilde{r}^m$ (i.e., in market $\Gamma ^m$), we have $\mu (c) = s_j \cup S_{c}$, and $\mu (c') = s_k \cup S_{c'}$. In the unique matching produced by SOSM after implementing $\tilde{r}^m$ (i.e., in market $\tilde{\Gamma }^m$), we have $\tilde{\mu }(c) = s_i \cup S_c$, $\tilde{\mu }(c') = s_j \cup S_{c'}$, and $\tilde{\mu }(s_k) = s_k$. Clearly, $\mu (s_k) P_{s_k} \tilde{\mu }(s_k)$, as we need. $\square$

1.3 A.3: Proof of Theorem 3

For a given $\Gamma ^m$, let $|S^m| = m$ and $s_j$ be a random majority student. Choose two schools $c, c' \in C$ and relabel the minority students with the lowest priority and second lowest priority in c as $i_{m-1}$ and $i_m$, and in $c'$ as $k_{m-1}$ and $k_m$. I prove the contrapositive for both of the two parts.

Part (i) Suppose that $s_j$ ranks higher than two different minority students in $c^o$ and $c'^o$, I will show that $\Gamma ^m$ contains a type-specific cycle.

Case (i.a.) $i_m \ne k_m$. Because $s \succ _c^r i_m$, for all $s \in S^m \backslash i_m$, we have $k_m \succ _c^r i_m$, and there are other $m-2$ minority students who have higher priorities than $k_m$ in c (recall that the pointwise priority orders of minorities remain unchanged within each of $c^r$ and $c^o$). As I assume $q_c + q_{c'} \le m$, it implies that $q_c -1 \le m - 2$. Thus, we can find a set of $q_c -1$ minority students who have higher priority than $i_m$ in c from $S^m \backslash \{ i_m, k_m\}$, denoted by $S_c$. For school $c'$, because $m - 2 - (q_c -1) \ge m - 2 - (m - q_{c'} -1) = q_{c'} -1$, we can find another set of minority students that are distinct from $i_m, k_m$ and $S_c$ who are ranked higher than $k_m$ in $c'$, denoted by $S_{c'}$. Condition (C) is satisfied by $k_m \succ _c^r i_m \succ _c^r s_j \succ _c^o k_m$. $S_c$ and $S_{c'}$ suffices Condition (S).

Case (i.b.) $i_m = k_m$. Without loss of generality, suppose that $s_j \succ _{c'}^o k_{m-1}$. Since there are $m-2$ minority students who have higher priorities than $k_{m-1}$ in $c'$, with similar arguments in Case (i.a.), we can find a set of $q_{c'} -1$ minority students that are distinct from $k_m, k_{m-1}$, denoted by $S'_{c'}$, and another set of $q_c -1$ minority students that are distinct from $k_m, k_{m-1}$ and $S'_{c'}$, denoted by $S'_c$. Condition (C) is satisfied by $k_{m-1} \succ _c^r i_m \succ _c^r s_j \succ _{c'}^o k_{m-1}$. $S'_{c'}$ and $S'_c$ suffices Condition (S).

Part (ii) I have already shown in Part (i) that when $s_j$ ranks higher than two different minority students in two schools, there is a type-specific cycle. Recall Remark 2, if $(\succ , (q, r^m))$ has a type-specific cycle, then it has a quasi type-specific cycle. Thus, I only need to discuss the situation when there is only one minority student ranked lower than $s_j$ in one (original sub-)school. Without loss of generality, suppose that $s_j \succ _{c'}^o k_m$.

Case (ii.a.) $i_m \ne k_m$, since $i_m \succ _c^r s_j$, $i_m, s_j$ and $k_m$ suffice Condition (C$'$). Case (ii.b.) $i_m = k_m$, Condition (C$'$) is given by $i_{m-1} \succ _c^r s_j \succ _{c'}^o k_m$. Condition (S$'$) is satisfied in both of the two cases with the same arguments in (i.a.), and is omitted. $\square$

Appendix B: Proof of Theorem 4

The proof involves a few steps. In brief, I first define a stochastic version of the Sequential SOSM-R in which all students draw their preferences iteratively from the set of schools (Step 1). I then show that when the number of schools becomes large, at the end of Loop 1 of the algorithm, many of them either are not listed on any minorities’ preference orders, or even if they have been listed by some minority students but are not required to implement minority reserve policies (Step 2). Last, under the regularity conditions (Definition 6), the probability that no majority students apply to these schools converges to one when the number of schools goes to infinity (Step 3). Step 2 and 3 closely follow the asymptotic techniques used in the proof of Theorem 1 of Kojima et al. (2013).

Step 1: Define the stochastic sequential SOSM-R algorithm

Notations First, relabel the minority student $s^m$ and the majority student $s^M$ by s(m) and s(M), respectively. S(m) and S(M) are the respective sets of minority students and majority students. Also, relabel the original sub-school $c^o$ and reserve sub-school $c^r$ by c(o) and c(r), respectively. Use $A_s$ and $D_s$ to record the respective sets of schools that s(m) and s(M) has already drawn from ${\mathcal {P}}^n$. When $|A_s| = k$ is reached, $A_s$ is the set of schools that is acceptable to s. Also, denote $B^i$ (resp. $E^j$) the set of rejected minorities (resp. majorities) from Loop 2 (resp. Loop 1).

Algorithm 1. Stochastic sequential SOSM-R algorithm

• (1) Initialization: Let $l(m)=1$ and $l(M)=1$. For every $s(m) \in S(m)$ (resp. $s(M) \in S(M)$), let $A_s = \emptyset$ (resp, $D_s = \emptyset$). Order all majorities and minorities in their respective arbitrarily fixed manner. Set $B^0 = \emptyset$, $E^0 = \emptyset$, $i=0$.

• (2) Loop 1:

  • (a) If $B^i = \emptyset$, then go to Step (2b). Otherwise, pick some minority s(m) in $B^i$. Let $B^{i+1} = B^i \backslash s(m)$, increment i by one and go to Step (2c).

  • (b) Choosing the applicant:

    • (i) If $l(m) \le |S(m)|$, then let s(m) be the l(m)th student and increment l(m) by one.

    • (ii) Otherwise, go to Step (3).

  • (c) Choosing the applied:

    • (i) If $|A_s| \ge k$, then go back to Step (2a).

    • (ii) If not, select c randomly from the distribution ${\mathcal {P}}^n$ until $c \notin A_s$. Add c to $A_s$. Split each c listed by students into two corresponding sub-schools, the original sub-school c(o) and the reserve sub-school c(r), according to the procedure defined in Sect. 2.2.

  • (d) Acceptance and/or rejection:

    Each s(m) first applies to the reserve sub-school c(r) of her most preferred school c that she has not applied yet.

    • (i) If c(r) prefers each of its current mates to s(m) and there is no empty seat, c(r) rejects s(m). s(m) then applies to the corresponding original sub-school c(o) of c. If c(o) prefers each of its current mates to s(m) and there is no empty seat, c(o) rejects s(m). Go back to Step (2c).

    • (ii.) If c(r) has no empty seat but it prefers s(m) to one of its current mates, c(r) rejects the matched student with the lowest priority. If the rejected student is a majority, add her to $E^j$, and go back to Step (2a). If the rejected student is a minority, let this student be s(m) and let her apply to the corresponding original sub-school c(o) of c.

      • A. If c(o) prefers each of its current mates to s(m) and there is no empty seat, then c(o) rejects s(m), and go back to Step (2c).

      • B. If c(o) prefers s(m) to one of her current mates, s(m) is tentatively accepted. If the rejected student is a majority, add her to $E^j$, and go back to Step (2a). If the rejected student is a minority, let this student be s(m), and go back to Step (2c).

    • (iii.) If c(r) prefers each of its current mates to s(m) and there is no empty seat, then c(r) rejects s(m). If the corresponding c(o) of c also has no empty seat but it prefers s(m) to one of its current mates, then c(o) rejects the matched student with the lowest priority. If the rejected student is a majority, add her to $E^j$, and go back to Step (2a). If the rejected student is a minority, let this student be s(m) and go back to Step (2c).

    • (iv.) If either c(r) or its corresponding c(o) has an empty seat, then s(m) is tentatively accepted. Go back to Step (2a).

• (3) Loop 2 (“Round j”):

  • (a) If $E^j = \emptyset$, then go to Step (3c).

  • (b) Otherwise, pick some majority s(M) in $E^j$. Let $E^{j+1} = E^j \backslash s(M)$, increment j by one and go to Step (3d).

  • (c) Choosing the applicant:

    • (i) If $l(M) \le |S(M)|$, then let s(M) be the l(M)th student and increment l(M) by one.

    • (ii) Otherwise,

      • (A) If $B^i \ne \emptyset$, then go back to Step (2).

      • (B) Otherwise, terminate the algorithm.

  • (d) Choosing the applied:

    • (i) If $|D_s| \ge k$, then go back to Step (3a).

    • (ii) If not, select c randomly from the distribution ${\mathcal {P}}^n$ until $c \notin D_s$. Add c to $D_s$. Split each c listed by any student into two corresponding sub-schools, the original sub-school c(o) and the reserve sub-school c(r), according to the procedure defined in Sect. 2.2.

  • (e) Acceptance and/or rejection:

    • (i) If s(M) applies to a school c which has reserves for minorities ($|c(r)| > 0$) and has tentatively accepted some minorities, then there are four cases:

      • (A) Each s(M) first applies to the original school c(o) of her most preferred school c. If c(o) prefers each of its current mates to s(M) and there is no empty seat, c(o) rejects s(M). s(M) then applies to the corresponding reserve sub-school c(r) of c. If c(r) prefers each of its current mates to S(M) and there is no empty seat, then c(r) rejects s(M). Go back to Step (3d).

      • (B) If c(o) has no empty seat but it prefers s(M) to one of its current mates, then c(o) rejects the matched student with the lowest priority. If the rejected student is a minority, add her to $B^i$ and go back to Step (3a). If the rejected student is a majority, let this student be s(M) and let her apply to the corresponding reserve sub-school c(r) of c:

        • If c(r) prefers each of its current mates to S(M) and there is no empty seat, then c(r) rejects s(M). Go back to Step (3d).

        • If c(r) prefers s(M) to one of its current matched majority, s(M) is tentatively accepted. Let the rejected majority student be s(M), go back to Step (3d).

      • (C) If c(o) prefers each of its current mates to s(M) and there is no empty seat, then c(o) rejects s(M). If the corresponding c(r) of c also has no empty seat but it prefers s(m) to one of current matched majority student, then c(r) rejects the least preferred majority tentatively accepted. Let the rejected majority student be s(M), go back to Step (3d).

      • (D) If either c(o) or its corresponding c(r) has an empty seat, then s(M) is tentatively accepted. Go back to Step (3a).

    • (ii) If s(M) applies to a school c which either has no reserves for minorities ($|c(r)| = 0$) or even if has reserves but has not been applied by any minorities yet, then there are three cases:

      • (A) If school c prefers each of its current mates to s(M) and there is no vacant position, then c rejects s(M). Go back to Step (3d).

      • (B) If c prefers s(M) to one of its current mates, then c rejects the least preferred student tentatively accepted. If the rejected student is a majority, let this rejected student be s(M), go back to Step (3d). If the rejected student is a minority, add her to $B^i$ and go back to Step (3a).

      • (C) If c has an empty seat, then s(M) is tentatively accepted. Go back to Step (3a).

The Stochastic Sequential SOSM-R terminates at Step 3(c)iiB. The probability that the above algorithm arrives at any steps is identical regardless of whether the random preferences are drawn at once in the beginning (i.e., the Sequential SOSM-R case) or drawn iteratively during the execution of the algorithm (i.e., the Stochastic Sequential SOSM-R case). This argument is called the principle of deferred decisions (see, for example, Motwani and Raghavan (1996)).

Step 2: The market is strongly type-specific acyclic with a high probability

Let $V_n$ be a random set of schools that are either not listed in any minorities’ preference orders at the end of Loop 1 of the Stochastic Sequential SOSM-R, or listed by some minority students but are not required to implement minority reserve policies. Let $X_n = |V_n|$ be a random variable counts the number of schools in $V_n$.³⁵ I first state the following result which provides a lower bound of $X_n$ at the beginning of Loop 2. Since it is almost identical to Lemma 2 of Kojima et al. (2013), the proof is omitted.

Lemma 3

For any $n > 4k$

$$\begin{aligned} E[X_n] \ge \frac{n}{2} e^{-16 \lambda k} \end{aligned}$$

Kojima et al. (2013) write their result based on the set of hospitals (denoted by $Y_n$) that are not listed by any single doctors at the end of their Stochastic Doctor-Proposing Deferred Acceptance Algorithm, and prove that $E[|Y_n|] \ge \frac{n}{2} e^{-16 \lambda k}$. The Stochastic Doctor-Proposing Deferred Acceptance Algorithm runs analogously to the Loop 1 of the Stochastic Sequential SOSM-R, in which single doctors and hospitals can be treated equivalently to minority students and schools respectively. Denote $Y'_n$ the set of schools that are not listed in any minorities’ preference orders at the end of Loop 1 of the Stochastic Sequential SOSM-R. Clearly,

$$\begin{aligned} E[|Y'_n|] \ge \frac{n}{2} e^{-16 \lambda k} \end{aligned}$$

Recall that $V_n$ is the set of schools which includes $Y'_n$ and an additional set of schools that have been listed by some minorities but without seats reserved for minorities (i.e. the capacity of its sub-school c(r) is zero). Clearly, $E[X_n] \ge E[|Y'_n|]$.

Let $Prob \,(\hat{\Gamma }^{n, stsc})$ be the probability that the corresponding priority structure in a random market $\hat{\Gamma }^n$ is strongly type-specific acyclic. Also, let $\bar{R} = bn^a$ be the upper bound on the number of seats reserved for minority students in the random market $\hat{\Gamma }^n$. The following lemma states that when the market is sufficiently large (and conditional on $X_n > \frac{E[X_n]}{2}$), it becomes strongly type-specific acyclic with a high probability.

Lemma 4

For any sufficiently large n,

$$\begin{aligned} Prob \left( \hat{\Gamma }^{n, stsc} \ \Big | \ X_n > \dfrac{E[X_n]}{2} \right) \ \ge \ \left( 1 - \dfrac{\bar{R}}{E[X_n] / 4r} \right) ^{\bar{R}} \end{aligned}$$

(6)

if the conditioning event has a strictly positive probability.

Proof

First, note that there are at most $\bar{R}$ seats reserved for minorities, which also implies the maximum number of schools implemented with minority reserve policy. Let $C_1$ be the set of schools with minority reserves and are listed by a minority student in her preference order at the end of Loop 1. Recall the condition (4) of Definition 6, which can be rewritten as

$$\begin{aligned} \sum _{c \in C_1} p_c \le r \bar{R} \cdot \min _{c \in C} \{p_c\} \end{aligned}$$

Also, we have

$$\begin{aligned} \sum _{c \in V_n} p_c \ge X_n \cdot \min _{c \in C} \{p_c\} \end{aligned}$$

I am interested in computing the probability that Round 1 of Loop 2 arrives at Step 3(e)ii, which is the probability when a majority student applies to some school not in $C_1$. It is bounded below by:

$$\begin{aligned} 1 - \dfrac{\sum _{c \in C_1} p_c}{\sum _{c \in V_n} p_c + \sum _{c \in C_1} p_c} \ge 1- \dfrac{\bar{R}}{\frac{X_n}{r} + \bar{R}} > 1 - \dfrac{\bar{R}}{\frac{E[X_n]/2}{r} + \bar{R}} \end{aligned}$$

Now assume that at all Rounds $1,\ldots ,j - 1$, no majority matches to the schools in $C_1$. Then there are still at least $X_n - (j - 1)$ schools which are either not listed by any minorities, or matched with some minorities but without minority reserved seat(s). This follows since at most $j - 1$ schools have had their seats filled at Rounds $1, \ldots , j - 1$ from the set of schools in $V_n$. Similarly, I can compute that at Round j, the probability that the Stochastic Sequential SOSM-R produces the same matching before and after implementing a (stronger) minority reserve policy is at least,

$$\begin{aligned} 1- \dfrac{\bar{R}}{\frac{X_n - (j-1)}{r} + \bar{R}} > 1 - \dfrac{\bar{R}}{\frac{E[X_n]/2 - (j-1)}{r} + \bar{R}} \end{aligned}$$

Since there are at most $\bar{R}$ minority reserved seats which may cause some majorities to be rejected from their previously matched schools before the (stronger) minority reserve, the probability that Algorithm 1 produces the same matching after a (stronger) minority reserve policy (conditional on $X_n > \frac{E[X_n]}{2}$) is at least,

$$\begin{aligned} \prod _{j=1}^{\bar{R}} \bigg ( 1 - \dfrac{\bar{R}}{\frac{E[X_n]/2 -(j-1)}{r} + \bar{R}} \bigg ) \ge \bigg ( 1 - \dfrac{\bar{R}}{\frac{E[X_n]/2 - (\bar{R}-1)}{r} + \bar{R}} \bigg )^{\bar{R}} \ge \bigg ( 1 - \dfrac{\bar{R}}{E[X_n]/4r} \bigg )^{\bar{R}} \end{aligned}$$

where the first inequality follows as $j \le \bar{R}$, $j \in \{1,\ldots ,\bar{R}\}$. The second inequality holds since $E[X_n]/2 - \bar{R} + 1 \ge E[X_n]/4 > 0$, which follows from Lemma 3 and the assumption that n is sufficiently large. $\square$

Step 3: Proof of Theorem 4

The last step is to show that the unconditional probability of strongly type-specific acyclicity converges to one as the market becomes large, which can be verified through the following inequalities.

$$\begin{aligned} \begin{aligned} Prob \left( \hat{\Gamma }^{n, stsc} \right)&\ge Prob \left( X_n > \dfrac{E[X_n]}{2} \right) \cdot \left( 1 - \dfrac{\bar{R}}{E[X_n] / 4r} \right) ^{\bar{R}} \\&\ge \left( 1 - \dfrac{4}{E[X_n]} \right) \cdot \left( 1 - \dfrac{\bar{R}}{E[X_n] / 4r} \right) ^{\bar{R}}\\&\ge \left( 1- \dfrac{8 e^{16 \lambda k}}{n} \right) \cdot \left( 1 - \dfrac{8r\bar{R}e^{16 \lambda k}}{n} \right) ^{\bar{R}} \end{aligned} \end{aligned}$$

The first inequality is given by Lemma 4. The second inequality follows the result by Kojima et al. (2013),³⁶ and the last inequality is given by Lemma 3.

For the two items in the last inequality, it is obvious that the first item converges to one as $n \rightarrow \infty$. For the second item, recall that there exists $b > 0$, such that $\bar{R} < bn^a$, for any n (condition (3) of Definition 6). Therefore,

$$\begin{aligned} \begin{aligned} \left( 1 - \dfrac{8r\bar{R}e^{16 \lambda k}}{n} \right) ^{\bar{R}}&> \left( 1 - \dfrac{8r bn^a e^{16 \lambda k}}{n} \right) ^{bn^a} \\&= \left( 1 - \dfrac{8r b e^{16 \lambda k}}{n^{1-a}} \right) ^{n^{1-a} b n^{2a-1}} \ge (e^{8r b e^{-16 \lambda k}})^{b n^{2a-1}} \end{aligned} \end{aligned}$$

where the last inequality follows as $(1-\frac{\beta }{x})^x \ge e^{-\beta }$, when $\beta , x > 0$. Since I assume $a \in [0, \frac{1}{2})$, the term $n^{2a-1}$ converges to zero as $n \rightarrow \infty$. Thus, $(e^{8r b e^{-16 \lambda k}})^{b n^{2a-1}}$ converges to one as $n \rightarrow \infty$. This completes the proof of Theorem 4, given the (material) equivalence between strongly type-specific acyclicity and respect of the improvement of a stronger minority reserve policy (Theorem 2). $\square$