The “Boston” school-choice mechanism: an axiomatic approach

Abstract

The Boston mechanism is a popular student-placement mechanism in school-choice programs around the world. We provide two characterizations of the Boston mechanism. We introduce two new axioms; favoring higher ranks and rank-respecting invariance. A mechanism is the Boston mechanism for some priority if and only if it favors higher ranks and satisfies consistency, resource monotonicity, and rank-respecting invariance. In environments where each type of object has exactly one unit, as in house allocation, a characterization is given by favoring higher ranks, individual rationality, population monotonicity, and rank-respecting invariance.

Appendices

Appendix A: Proof of Proposition 1

For contradiction, assume that matching $\mu$ favors higher ranks but is not constrained Pareto efficient. Since $\mu$ favors higher ranks, there exists no school $c$ and student $i$ such that $i \in I_c, |\mu _{c}|<q_{c}$ and $cP_{i}\mu _{i}$. This and the assumption that $\mu$ is not constrained Pareto efficient imply that there exists a sequence of students $i_{1},i_{2},\ldots ,i_{n}$ such that $i_k \in I_{\mu _{i_{k+1}}}$, and $\mu _{i_{k+1}}P_{i_{k}}\mu _{i_{k}}$ or equivalently $P_{i_{k}}(\mu _{i_{k+1} })<P_{i_{k}}(\mu _{i_{k}})$ for all $k=1,\ldots ,n$ (with the convention that $n+1=1$). Since $\mu$ favors higher ranks, $P_{i_{k+1}}(\mu _{i_{k+1} })\le P_{i_{k}}(\mu _{i_{k+1}})$ for all $k=1,\ldots ,n$. Combining these inequalities, we obtain

$$\begin{aligned} P_{i_{1}}(\mu _{i_{1}})&\le P_{i_{n}}(\mu _{i_{1}})<P_{i_{n}}(\mu _{i_{n}})\le P_{i_{n-1}}(\mu _{i_{n}})<P_{i_{n-1}}(\mu _{i_{n-1}})\\&\le \dots \le P_{i_{1}} (\mu _{i_{2}})<P_{i_{1}}(\mu _{i_{1}}), \end{aligned}$$

a contradiction.

Appendix B: Proof of Lemma 1

Suppose that $\varphi$ satisfies resource monotonicity and consistency, and fix $P, q$, and $i$ arbitrarily. If $\varphi _{i}[P;q]=\emptyset$, then consistency implies

$$\begin{aligned} \varphi _{j}[P_{i}^{\emptyset },P_{-i};q]=\varphi _{j}[P;q]\,\quad {for all}\,j\not =i\, {and all}\,P_i^\emptyset \in \mathcal P ^\emptyset . \end{aligned}$$

Suppose $\varphi _{i}[P;q]\ne \emptyset$. Then, by consistency, $\varphi _{j}\left[ P_{i}^{\emptyset },P_{-i};q_{\varphi _{i}\left[ P;q\right] }-1;q_{-\varphi _{i}\left[ P;q\right] }\right] =\varphi _{j}\left[ P;q\right]$ for all $j\not =i$ and all $P_i^\emptyset \in \mathcal P ^\emptyset$. Hence, resource monotonicity implies that for all $j\ne i$

$$\begin{aligned} \varphi _{j}\left[ P_{i}^{\emptyset },P_{-i};q\right] R_{j}\varphi _{j}\left[ P_{i}^{\emptyset },P_{-i};q_{\varphi _{i}\left[ P;q\right] }-1,q_{-\varphi _{i}\left[ P;q\right] }\right] =\varphi _{j}\left[ P;q\right] . \end{aligned}$$

The above displayed relations show that $\varphi$ satisfies population monotonicity.

Appendix C: Proof of Theorem 1

It is straightforward to see that the Boston mechanism for an arbitrary priority profile $\succ$ satisfies all the axioms in the statement. Thus, we show the converse.

Based on Lemma 1, we invoke population monotonicity of $\varphi$ in various parts of the proof. For any $c\in C$ and $I^{\prime }\subseteq I$, let $\mathcal P ^{\left| I\right| }\left\langle c;I^{\prime }\right\rangle$ be the set of preference profiles such that all students in $I^{\prime }$ rank $c$ as the first choice and all other students rank $\emptyset$ as the first choice. We prove the theorem in two parts.

1.1 Part 1: Construction of priority profile $\succ$

For each $c\in C$, construct $\succ _{c}$ as follows: Let $q_{c}=1$ and $q_{c^{\prime }}=0$ for all $c^{\prime }\in C\setminus \{c\}$. Fix some $P^{\left( 1\right) }\in \mathcal P ^{\left| I\right| }\left\langle c,I\right\rangle$ and let the top-priority student under $\succ _{c}$, denoted by $i_{c}^{1}$, be the student in $\varphi _{c}[P^{1};q]$ if any (who is unique, if existent, since $q_{c}=1$). Iteratively continue, such that: For each $\ell \ge 2$, fix some $P^{\left( \ell \right) }\in \mathcal P ^{\left| I\right| }\left\langle c,I\setminus \{i_{c}^{1},\ldots ,i_{c}^{\ell -1}\} \right\rangle$ and let the $\ell ^{\text {th}}$ priority student under $\succ _{c}$, denoted by $i_{c}^{\ell }$, be the student in $\varphi _{c}[P^{\left( \ell \right) };q]$ if any (who is unique, if existent, since $q_{c}=1$). If $\varphi _{c}[P^{\left( \ell \right) };q]=\emptyset$ for any $\ell \ge 1$, then order all students in $I \setminus \{i_c^1,\ldots ,i_c^{\ell -1}\}$ so that $\emptyset \succ _c i$ for all $i \in I \setminus \{i_c^1,\ldots ,i_c^{\ell -1}\}$. This procedure defines a priority order $\succ _{c}$. It is an implication of the following claim that the construction of $\succ _{c}$ is independent of the choice of preference $P^{\left( \ell \right) }\in \mathcal P ^{\left| I\right| }\left\langle c,I\setminus \{i_{c}^{1},\ldots ,i_{c}^{\ell -1}\} \right\rangle$ in each step $\ell$:

Claim 1

Let $I^{\prime }\subseteq I$. Suppose $\varphi _{c}[P;q]=i$ where $P\in \mathcal P ^{\left| I\right| }\left\langle c,I^{\prime }\right\rangle$ and $q_{c}=1,q_{c^{\prime }}=0$ for all $c^{\prime }\ne c$. Then $\varphi _{c}[P^{\prime };q^{\prime }]=i$ for all $q^{\prime }$ with $q_{c}^{\prime }=1$ and $P^{\prime }\in \mathcal P ^{\left| I\right| }\left\langle c,I^{\prime \prime }\right\rangle$ with $\{i\} \subseteq I^{\prime \prime }\subseteq I^{\prime }$.

Proof

We will prove the claim in three steps:

Proof Step 1:

First, we show the claim when $q^{\prime }=q$ and $I^{\prime \prime }=I^{\prime }$. Let $P\in \mathcal P ^{\left| I\right| }\left\langle c,I^{\prime }\right\rangle$ and $\varphi _{i}[P;q]=c$. Assume $P_{j}^{\prime }\ne P_{j}$ for some $j\in I$ and $P_{k}^{\prime }=P_{k}$ for all $k\ne j$ without loss of generality. Two cases are possible for the identity of $j$, where Case 2 has also two sub-cases:

Case 1

$j=i$: Then, since $\varphi _{i}[P;q]=c$ and $c$ is top-ranked both at $P_{i}$ and $P_{i}^{\prime }, P^{\prime }\, {m.t.}\,P$ at $\varphi [P;q]$ and

$$\begin{aligned} U_{k}\left( P,\varphi [P;q]\right)&= U_{k}\left( P^{\prime },\varphi [P;q]\right) \!,\\ V_{k}\left( P,\varphi [P;q]\right)&= V_{k}\left( P^{\prime },\varphi [P;q]\right) \end{aligned}$$

for all $k\in I$. Thus, $P^{\prime }\, {r.r.m.t.}\,P$ at $\varphi [P;q]$. Since $\varphi$ satisfies rank-respecting invariance, we conclude that $\varphi [P^{\prime };q]=\varphi [P;q]$ and hence $\varphi _{i}[P^{\prime };q]=c$.

Case 2

$j\ne i$ and $\varphi _{j}[P^{\prime };q]\ne c$:

For any $P_{j}^{\emptyset }\in \mathcal P ^{\emptyset }$, since $\varphi$ satisfies population monotonicity, $\varphi _{i}[P_{j}^{\emptyset } ,P_{-j};q]R_{i}\varphi _{i} [P;q]=c.$ Since $P_{i}$ top-ranks $c$, we obtain

$$\begin{aligned} \varphi _{i}[P_{j}^{\emptyset },P_{-j};q]=c. \end{aligned}$$

(1)

Suppose for contradiction that $\varphi _{i}[P^{\prime };q]\ne c$. Then, since $\varphi$ favors higher ranks, $i \in I_c$, and $\varphi _j[P;q] \ne c$, there exists $i^{\prime }\ne i,j$ such that $\varphi _{i^{\prime }}[P^{\prime };q]=c$. Thus, $P_{i^{\prime }} =P_{i^{\prime }}^{\prime }$ top-ranks $c$ as $\varphi$ favors higher ranks (otherwise $i^{\prime }$ receives $\emptyset$ by individual rationality) and, by population monotonicity,

$$\begin{aligned} \varphi _{i^{\prime }}[P_{j}^{\emptyset },P_{-j}^{\prime };q]=c. \end{aligned}$$

(2)

Relations (1) and (2) contradict each other since $i^{\prime }\ne i, q_{c}=1,$ and $P_{-j}^{\prime }=P_{-j}$.

Case 2-(ii)

$j\ne i$ and $\varphi _{j}[P^{\prime };q]=c$:

Then, since both $P_{j}$ and $P_{j}^{\prime }$ top-rank $c$,²⁵ $P\, {m.t.}\,P^{\prime }$ at $\varphi [P^{\prime };q]$ and

$$\begin{aligned} U_{k}\left( P,\varphi [P';q]\right)&= U_{k}\left( P^{\prime },\varphi [P';q]\right) \!,\\ V_{k}\left( P,\varphi [P';q]\right)&= V_{k}\left( P^{\prime },\varphi [P';q]\right) \end{aligned}$$

for all $k\in I$. Thus, $P\, {r.r.m.t.}\,P^{\prime }$ at $\varphi [P^{\prime };q]$. Since $\varphi$ satisfies rank-respecting invariance we conclude $\varphi _{j}[P;q]=c$, a contradiction to $j\ne i, \varphi _{i}[P;q]=c$ and $q_{c}=1$.

Proof Step 2:

Given the preceding argument, population monotonicity of $\varphi$ implies that the claim holds for all cases when $q^{\prime }=q$ and $\{i\} \subseteq I^{\prime \prime }\subseteq I^{\prime }$, noting that $c$ is the top-ranked school at $P_{i}.$

Proof Step 3:

: Finally, we will show that the claim holds for all $q^{\prime }$ with $q_{c}^{\prime }=1$ and $P^{\prime }\in \mathcal P ^{\left| I\right| }\left\langle c,I^{\prime \prime }\right\rangle$ with $\{i\} \subseteq I^{\prime \prime }\subseteq I^{\prime }$. Thus, assume that $q^{\prime }$ satisfies $q_{c} ^{\prime }=1$ and $P^{\prime }\in \mathcal P ^{\left| I\right| }\left\langle c,I^{\prime \prime }\right\rangle$ with $\{i\} \subseteq I^{\prime \prime }\subseteq I^{\prime }$. By Proof Step 2,

$$\begin{aligned} \varphi _{i}[P^{\prime };q]=c. \end{aligned}$$

(3)

Since $\varphi$ satisfies resource monotonicity and $q_{c}^{\prime }\ge q_{c}$ for all $c\in C$ (as $q_{c}^{\prime }=q_{c}=1$ and $q_{c^{\prime }}^{\prime } \ge 0=q_{c^{\prime }}$ for all $c^{\prime }\ne c$),

$$\begin{aligned} \varphi _{i}[P^{\prime };q^{\prime }]R_{i}^{\prime } \varphi _{i}[P^{\prime };q]. \end{aligned}$$

(4)

Relations (3) and (4) imply

$$\begin{aligned} \varphi _{i}[P^{\prime };q^{\prime }]R_{i}^{\prime }c. \end{aligned}$$

(5)

Since $c$ is top-ranked under $P_{i}^{\prime }$ by assumption, relation (5) implies

$$\begin{aligned} \varphi _{i}[P^{\prime };q^{\prime }]=c, \end{aligned}$$

completing the proof. $\square$

1.2 Part 2: Proof that $\varphi =\psi ^{\succ }$

Let $\succ =(\succ _c)_{c \in C}$ be the priority order profile constructed in Part 1. For any given preference profile $P$ and quota profile $q$, we will show that $\varphi [P;q]=\psi ^{\succ }[P;q]$. Construct the following student sets and quotas:

For any $c\in C\cup \left\{ \emptyset \right\}$, define

$$\begin{aligned} I_{c}(0)&=\emptyset ,\\ J_{c}(0)&=\emptyset ,\\ q_{c}(0)&=q_{c}. \end{aligned}$$

For any $\ell \ge 1$, given $(I_{c}(0),J_{c}(0),q_{c}(0))_{c\in C\cup \left\{ \emptyset \right\} },\ldots ,(I_{c}(\ell -1),J_{c}(\ell -1),q_{c}(\ell -1))_{c\in C\cup \left\{ \emptyset \right\} }$, recursively define

$$\begin{aligned} I_{c}(\ell )&= \left\{ i\in I_c \setminus \left( { \bigcup \limits _{d\in C\cup \left\{ \emptyset \right\} }} \bigcup _{\ell ^{\prime }=1}^{\ell -1}J_{d}(\ell ^{\prime })\right) :P_{i}(c)=\ell \right\} ,\\ J_{c}(\ell )&= \left\{ i\in I_{c}(\ell ):\varphi _{i}[P;q]=c\right\} ,\\ q_{c}(\ell )&= q_{c}(\ell -1)-|J_{c}(\ell -1)|. \end{aligned}$$

$I_{c}(\ell )$ is the set of qualified students who rank $c$ as their $\ell ^{\text {th}}$ choice and have not received any higher-ranked school. $J_{c}(\ell )$ is the set of students in $I_{c}(\ell )$ who receive seats at $c$ under $\varphi \left[ P;q\right]$. $q_{c}(\ell )$ denotes the number of seats remaining after assigning seats to students who rank $c$ as the $(\ell -1)^{\text {st}}$ choice or higher. Let $\succ _{\emptyset }\in \Pi$ be an arbitrary priority order. Individual rationality of $\varphi$ (following from resource monotonicity of $\varphi$) implies $I_\emptyset =I$, and it suffices to show the following claim.

Claim 2

For all $\ell \ge 1,\,J_{c}(\ell )=\arg \max _{I_{c}(\ell ),q_{c}(\ell )}\succ _{c}$ for all $c\in C\cup \left\{ \emptyset \right\}$, where

$$\begin{aligned} \arg \max _{I_{c}(\ell ),q_{c}(\ell )}\succ _{c}:=\left\{ i\in I_{c}(\ell ):\left| \left\{ j\in I_{c}(\ell ):j\succeq _{c}i\right\} \right| \le q_{c}(\ell )\right\} , \end{aligned}$$

is the set of (at most) $q_{c}(\ell )$ students who have the highest priorities at $c$ among those in $I_{c}(\ell )$.

Proof

Let $c\in C\cup \left\{ \emptyset \right\} .$ Fix $\ell \ge 1$. If $|I_{c}(\ell )|\le q_{c}(\ell )$, then $\arg \max _{I_{c}(\ell ),q_{c}(\ell )}\succ _{c}=I_{c}\left( \ell \right)$. Then $J_{c}(\ell )=I_{c}(\ell )$ because $\varphi$ favors higher ranks. Hence, the conclusion $J_{c}\left( \ell \right) =\arg \max _{I_{c}(\ell ),q_{c}(\ell )}\succ _{c}$ holds.

Thus, assume $|I_{c}(\ell )|>q_{c}(\ell )$. Suppose for contradiction that the conclusion $J_{c}\left( \ell \right) =\arg \max _{I_{c}(\ell ),q_{c}(\ell )}\succ _{c}$ does not hold. Since $\varphi$ favors higher ranks, it follows that $|J_{c}(\ell )|=q_{c}(\ell )$, and hence, there exist $i\in \arg \max _{I_{c} (\ell ),q_{c}(\ell )}\succ _{c}$ such that $\varphi _{i}[P;q]\ne c$ and $j\in I_{c}(\ell )\setminus \left( \arg \max _{I_{c}(\ell ),q_{c}(\ell )}\succ _{c}\right)$ such that $\varphi _{j}[P;q]=c$.

Let $P^{\prime }=(P_{i},P_{j},P_{-i,j}^{\emptyset })$ for some $P_{-i,j} ^{\emptyset }\in \left( \mathcal P ^{\emptyset }\right) ^{\left| I\right| -2}.$ Let $q^{\prime }=(q_{c^{\prime }}^{\prime })_{{c^{\prime }}\in C}$ be defined by

$$\begin{aligned} q_{c^{\prime }}^{\prime }=q_{c^{\prime }}-\left| \{k\in I\setminus \{i,j\} : \varphi _{k}[P;q]=c^{\prime }\} \right| \end{aligned}$$

for each $c^{\prime }\in C$. By construction of $I_{c}\left( \ell \right) ,$ since $i\in I_{c}\left( \ell \right)$, she has not been matched to a higher-choice school than $c$. Thus, $cP_{i}\varphi _{i}[P;q]$. Since $\varphi$ is consistent,

$$\begin{aligned} \varphi _{j}[P^{\prime };q^{\prime }]&= \varphi _{j}[P;q]=c,\\ \varphi _{i}[P^{\prime };q^{\prime }]&= \varphi _{i}[P;q]\quad \text {and hence,}\ cP_{i}\varphi _{i}[P^{\prime };q^{\prime }]. \end{aligned}$$

Consider preference $P_{i}^{\prime \prime },P_{j}^{\prime \prime }$ such that $c$ is top-ranked at $P_{i}^{\prime \prime }$ and $P_{j}^{\prime \prime }$ and relative rankings of all other schools are unchanged from $P_{i}^{\prime }$ and $P_{j}^{\prime }$, respectively. Then $P^{\prime \prime }:=(P_{i}^{\prime \prime },P_{j}^{\prime \prime },P_{-i,j}^{\prime })\, {r.r.m.t.}\,P^{\prime }$ at $\varphi [P^{\prime };q^{\prime }]$. Thus, since $\varphi$ satisfies rank-respecting invariance, $\varphi _{i}[P^{\prime \prime };q^{\prime }]\ne c$. This is a contradiction since Claim 1 and the assumption that $i$ has a higher priority than $j$ at $c$ imply $\varphi _{i}[P^{\prime \prime };q^{\prime }]=c$. $\square$

Claim 2 completes the proof of the Theorem.

Appendix D: Proof of Theorem 2

The Boston mechanism for an arbitrary priority profile clearly satisfies all the axioms in the statement. Let $q$ be the quota vector with $q_{c}=1$ for all $c \in C$, which is fixed throughout the current analysis.

1.1 Part 1: Construction of priority profile $\succ$

Construct $\succ _{c}$ in an analogous manner to the one in the proof of Theorem 1.²⁶ It is an implication of the following claim that the construction of $\succ _{c}$ is independent of the choice of preference $P^{\left( \ell \right) }\in \mathcal P ^{\left| I\right| }\left\langle c,I\setminus \{i_{c}^{1},\ldots ,i_{c}^{\ell -1}\} \right\rangle$ in each step $\ell$:

Claim 3

Let $I^{\prime }\subseteq I$. Suppose $\varphi _{c}[P]=i$ where $P\in \mathcal P ^{\left| I\right| }\left\langle c,I^{\prime }\right\rangle .$ Then $\varphi _{c}[P^{\prime }]=i$ for all $P^{\prime } \in \mathcal P ^{\left| I\right| }\left\langle c,I^{\prime \prime }\right\rangle$ with $\{i\} \subseteq I^{\prime \prime }\subseteq I^{\prime }$.

Proof

The proof is omitted since it is identical to Steps 1 and 2 of the proof of Claim 1. $\square$

1.2 Part 2: Proof that $\varphi =\psi ^{\succ }$

Let $\succ =(\succ _c)_{c \in C}$ be the priority order profile constructed in Part 1. For any given preference profile $P$, we will show that $\varphi [P]=\psi ^{\succ }[P]$. For each $c \in C \cup \{ \emptyset \}$ and $\ell =1,2,\ldots ,$ define $I_{c}(\ell ), J_{c}(\ell )$, and $q_{c}(\ell )$ as in Part 2 of the Proof of Theorem 1. Recall that $I_{c}(\ell )$ is the set of students who rank $c$ as their $\ell ^{\text {th}}$ choice and have not received any higher-ranked school (except those who cannot be matched to $c$ under any preference profile), $J_{c}(\ell )$ is the set of students in $I_{c} (\ell )$ who receive seats at $c$ under $\varphi \left[ P \right]$, and $q_{c}(\ell ) \in \{ 0, 1 \}$ denotes the number of seats in $c$ remaining after assigning seats to students who rank $c$ as the $(\ell -1)^{\text {st}}$ choice or higher. Let $\succ _{\emptyset }\in \Pi$ be an arbitrary priority order. Individual rationality of $\varphi$ implies $I_\emptyset =I$, and it suffices to show the following claim.

Claim 4

For all $\ell \ge 1, J_{c}(\ell )=\arg \max _{I_{c}(\ell ),q_c(\ell )}\succ _{c}$ for all $c\in C\cup \left\{ \emptyset \right\}$, where

$$\begin{aligned} \arg \max _{I_{c}(\ell ),q_{c}(\ell )}\succ _{c}:=\left\{ i\in I_{c}(\ell ):\left| \left\{ j\in I_{c}(\ell ):j\succeq _{c}i\right\} \right| \le q_{c}(\ell )\right\} , \end{aligned}$$

is the set of (at most) $q_{c}(\ell )$ students who have the highest priorities at $c$ among those in $I_{c}(\ell )$.

Proof

Let $c\in C\cup \left\{ \emptyset \right\}$ . Fix $\ell \ge 1$. If $|I_{c}(\ell )|\le q_c(\ell )$, then $\arg \max _{I_{c}(\ell ),q_c(\ell )}\succ _{c}=I_{c}\left( \ell \right)$. Then $J_{c}(\ell )=I_{c}(\ell )$ because $\varphi$. Hence, the conclusion $J_{c}\left( \ell \right) =\arg \max _{I_{c}(\ell ),q_{c}(\ell )} \succ _{c}$ holds.

Thus, assume $|I_{c}(\ell )|>q_c(\ell )$. Suppose for contradiction that the conclusion does not hold. Since $\varphi$ favors higher ranks, it follows that $|J_{c}(\ell )|=1$, and hence, there exist $i\in \arg \max _{I_{c} (\ell )}\succ _{c}$ such that $\varphi _{i}[P]\ne c$ and $j\in I_{c}(\ell )\setminus \left( \arg \max _{I_{c}(\ell ),q_c(\ell )}\succ _{c}\right)$ such that $\varphi _{j}[P]=c$.

Consider preference profile $P^{\prime }$ such that (1) at $P_{k}^{\prime }$ for every student $k \in I_{c}(\ell ), c$ is top ranked and relative rankings of all other schools are unchanged from $P_{k}$, and (2) preferences of all other students are unchanged. Since $q_{c}=1$, it follows that $P^{\prime }\, {r.r.m.t.}\,P$ at $\varphi [P]$. Thus, since $\varphi$ satisfies rank-respecting invariance, $\varphi _{j}[P^{\prime }]= c$.

Let $P^{\prime \prime }=(P_{i}^{\prime },P_{j}^{\prime },P_{-i,j}^{\emptyset })$ for some $P_{-i,j}^{\emptyset }\in \left( \mathcal P ^{\emptyset }\right) ^{\left| I\right| -2}.$ Since $\varphi$ satisfies population monotonicity,

$$\begin{aligned} \varphi _{j}[P^{\prime \prime }]R_{j}^{\prime }\varphi _{j}[P^{\prime }]=c. \end{aligned}$$

This and the assumption that $c$ is top-ranked at $P_{j}^{\prime }$ imply

$$\begin{aligned} \varphi _{j}[P^{\prime \prime }]=c. \end{aligned}$$

Since $q_{c}=1$, the above relation implies

$$\begin{aligned} \varphi _{i}[P^{\prime \prime }]\ne c, \end{aligned}$$

which is a contradiction since Claim 3 and the construction of $\succ$ that gives $i$ a higher priority than $j$ at $c$ imply $\varphi _{i}[P^{\prime \prime }]=c$. $\square$

Claim 4 completes the proof of the Theorem.

Appendix E: A welfare maximization interpretation of the Boston mechanism

For each problem $\left[ P;q\right] ,$ the solution of the following linear assignment program, which maximizes the sum of the induced utilities, is equivalent to the solution of the Boston mechanism induced by a priority profile $\succ$: First, define for each $c \in C, I_c=\{i \in I : i \succ _c \emptyset \}$ as the set of acceptable students for school $c$.

$$\begin{aligned}&\max _{\left[ z_{i,c} \right] } {\sum \limits _{i\in I,c\in C\cup \left\{ \emptyset \right\} }} z_{i,c}u_{i}\left( c\right) \\&\quad \text {subject to}\\&\quad i \not \in I_c \Rightarrow z_{i,c} =0\quad \quad \forall i \in I, c \in C . \\&\quad 0 \le z_{i,c}\quad \quad \forall i\in I,c\in C\cup \left\{ \emptyset \right\} \\&\quad {\sum \limits _{c\in C\cup \left\{ \emptyset \right\} }} z_{i,c} =1\quad \quad \forall i\in I\\&\quad {\sum \limits _{i\in I}} z_{i,c} \le q_{c}\quad \quad \forall c\in C\cup \left\{ \emptyset \right\} , \end{aligned}$$

where $q_{\emptyset }=\infty$ and $\left( u_{i}\right) _{i\in I} \subseteq (\mathbb R _{+}^{\left| C\right| \cup \left\{ \emptyset \right\} })^{|I|}$ are utility functions consistent with the given preferences $P$ (i.e., for any $i \in I$ and $c,d \in C \cup \{\emptyset \}$, we have $u_i(c)>u_i(d)$ if and only if $c \, P_i \, d$) satisfying

$$\begin{aligned} \left| I\right| u_{i}\left( c\right)&< u_{j}\left( d\right) \quad \quad \forall i,j\in I\text { and }c,d\in C\cup \left\{ \emptyset \right\} \text { with }P_{j}\left( c\right) <P_{i}\left( d\right) \\ \left| I\right| u_{i}\left( c\right)&< u_{j}\left( c\right) \quad \quad \forall c\in C\cup \left\{ \emptyset \right\} \text {with } P_{i}\left( c\right) =P_{j}\left( c\right) \text { and }\succ _{c}\left( j\right) <\succ _{c}\left( i\right) \end{aligned}$$

In the solution $\left[ z_{i,c}\right] _{i\in I,c\in C\cup \left\{ \emptyset \right\} }$, which is unique and where $z_{i,c}\in \left\{ 0,1\right\}$ for all $i\in I$ and $c\in C\cup \left\{ \emptyset \right\}$, school $i$ is matched with school $c$ if and only if $z_{i,c}=1$.

Appendix F: Independence of axioms for remaining cases

The main text has presented examples showing that the axioms in the characterizations are independent for all but a few values of $|I|$ and $|C|$. This section completes the investigation by considering all other cases.

1.1 Axioms for Theorem 1

A mechanism violating only consistency: Suppose $\left| I\right| \le 2$: If $\left| I\right| = 1$, then consistency is vacuously satisfied by any mechanism. If $\left| I\right| =2$, then favoring higher ranks implies consistency. To see this first note that, for any $i \in I$ and $j \ne i, \varphi _i[P;q]$ is the most preferred school in $\{c \in C \cup \{ \emptyset \}: i \in I_{c}, q_{c}- \mathbf 1 _{\varphi _j[P;q]=c} \ge 1 \}$ by favoring higher ranks, where $\mathbf 1 _{\varphi _j[P;q]=c}=1$ if $\varphi _j[P;q]=c$ and $0$ otherwise. By inspection, favoring higher ranks implies that $\varphi _i[P_i,P^\emptyset _j;(q_{c}-\mathbf 1 _{\varphi _j[P;q]=c})_{c \in C}]$ is the most preferred school in $\{c \in C \cup \{ \emptyset \}: i \in I_{c}, q_{c}- \mathbf 1 _{\varphi _j[P;q]=c} \ge 1 \}$, showing consistency. Thus, there is no mechanism that violates consistency while favoring higher ranks.

A mechanism violating only resource monotonicity: Suppose that $|I|=1$ or $|C|=1$: If $|I|=1$, then favoring higher ranks implies resource monotonicity. To see this point observe that the unique agent, denoted $i$, receives her most preferred school in $\{c \in C: i \in I_c, q_c \ge 1\}$. Since this set is increasing in each $q_c$ in the set inclusion sense, resource monotonicity follows.

If $|C|=1$, then favoring higher ranks, consistency, and rank-respecting invariance imply resource monotonicity. To show this, first recall that favoring higher ranks implies individual rationality (since $I_\emptyset =I$ and $q_\emptyset =\infty$). Now suppose, for contradiction, a mechanism $\varphi$ satisfies favoring higher ranks, consistency, and rank-respecting invariance, while violating resource monotonicity. Then there exists a student $i \in I$, preference profile $P$, and a quota $q_c$ of the unique school $c$ such that

$$\begin{aligned}&\varphi _i[P;q_c-1]=c, \end{aligned}$$

(6)

$$\begin{aligned}&\varphi _i[P;q_c] = \emptyset ,\end{aligned}$$

(7)

$$\begin{aligned}&c \, P_i \, \emptyset . \end{aligned}$$

(8)

By relationships (6)–(8) and favoring higher ranks, $|\varphi _c[P;q_c]|=q_c$, and hence, there exists a student $j \ne i$ such that

$$\begin{aligned} \varphi _j[P;q_c]&= c,\end{aligned}$$

(9)

$$\begin{aligned} \varphi _j[P;q_c-1]&= \emptyset . \end{aligned}$$

(10)

By consistency of $\varphi$ and relation (9), it follows that

$$\begin{aligned} \varphi _i[P^\emptyset _j,P_{-j};q_c-1]=\varphi _i[P;q_c]. \end{aligned}$$

(11)

Relations (7) and (11) imply

$$\begin{aligned} \varphi _i[P^\emptyset _j,P_{-j};q_c-1]=\emptyset . \end{aligned}$$

(12)

Meanwhile relation (10) implies that $(P^\emptyset _j,P_{-j}) \, {r.r.m.t.} \, P$ at $\varphi [P;q_c-1]$. Thus, by rank-respecting invariance of $\varphi$,

$$\begin{aligned} \varphi _i[P^\emptyset _j,P_{-j};q_c-1]=\varphi _i[P;q_c-1]. \end{aligned}$$

(13)

Then, by relations (6) and (13), we obtain

$$\begin{aligned} \varphi _i[P^\emptyset _j,P_{-j};q_c-1]=c. \end{aligned}$$

(14)

Relations (12) and (14) contradict each other, showing that $\varphi$ is resource monotonic.

A mechanism violating only rank-respecting invariance: Suppose that $\left| C\right| =1$ or $|I|=1$: If $|C|=1$, then consistency, resource monotonicity, and favoring higher ranks imply rank-respecting invariance. To see this point, first recall that these properties imply individual rationality and population monotonicity. Let $\varphi$ be a mechanism for that these axioms hold and for each $i \in I$ let

$$\begin{aligned} \begin{array}[c]{ll} P'_i &{} P_i'' \\ \hline c &{} \emptyset \\ \emptyset &{} c \end{array} \end{aligned}$$

Fix a preference profile $P$ arbitrarily. Since $\varphi$ satisfies individual rationality,

$$\begin{aligned} \varphi _i[P;q]=c \Rightarrow P_i=P_i' \end{aligned}$$

(15)

Moreover, if $\varphi _i[P;q]$ is the top-ranked school for $i$ at $P_i$, then the only monotonic transformation of $P_i$ at $\varphi _i[P;q]$ is $P_i$ itself. By this fact and (15), if $\tilde{P}$ is a monotonic transformation of $P$ at $\varphi [P;q]$, then for any $i \in I$, either

• 1. $\tilde{P}_i=P_i$, or

• 2. $P_i=P_i', \varphi _i[P;q]=\emptyset$, and $\tilde{P}_i=P_i''.$

For any $i$ such that Case 1 above applies, population monotonicity of $\varphi$ and Cases 1 and 2 imply $\varphi _i[\tilde{P};q] \, R_i \varphi _i[P;q]$. So, if $\varphi _i[P;q]=c$, then $\varphi _i[\tilde{P};q]=c$. Since $\varphi$favors higher ranks, it follows that $\varphi _i[P;q]=\emptyset$ implies $\varphi _i[\tilde{P};q]=\emptyset$. Thus, we conclude $\varphi _i[P;q]=\varphi _i[\tilde{P};q]$. For any $i$ such that Case 2 above applies, individual rationality of $\varphi$ implies $\varphi _i[\tilde{P};q]=\emptyset =\varphi _i[P;q]$. Therefore, we conclude $\varphi [\tilde{P};q]=\varphi [P;q]$, showing rank-respecting invariance.

If $|I|=1$, then favoring higher ranks implies rank-respecting invariance. To show this, let $i$ be the unique student in $I$ and $C_i=\{c \in C \cup \{ \emptyset \} : i \in I_c, q_c \ge 1\}$. Since $i$ is the unique student, favoring higher ranks imply that $\varphi _i[P_i;q]$ is the school that is top-ranked by $P_i$ within $C_i$. Any monotonic transformation $P'_i$ of $P_i$ at $\varphi _i[P_i;q]$ leaves the top-ranked school in $C_i$ unchanged (namely $\varphi _i[P_i;q]$), so $\varphi _i[P_i';q]=\varphi _i[P_i;q]$ by favoring higher ranks. This shows that $\varphi$ satisfies rank-respecting invariance.

1.2 Axioms for Theorem 2

A mechanism violating only favoring higher ranks: Suppose that $|C|=1$ or $|I|=1$: If $|C|=1$, then any mechanism $\varphi$ that satisfies individual rationality and rank-respecting invariance favors higher ranks. To see this, let $C=\{c\}$ and assume for contradiction that $\varphi$ does not favor higher ranks. Then there exists $i \in I_c$ (so $\varphi _i[P]= c$ for some $P$) such that $c P'_i \varphi _i[P']$ for some $P'$ where either $\varphi _c[P']=\emptyset$ or $\varphi _c[P']=j$ with $P'_i(c)<P'_j(c)$. The latter is a contradiction to individual rationality of $\varphi$, because $P'_i(c)<P'_j(c)$ and $|C|=1$ imply $\emptyset P'_j c$. Thus, assume $\varphi _c[P']=\emptyset$. Since $q_c=1$ and $C=\{c\}$, it follows that $\varphi _j[P]=\varphi _j[P']=\emptyset$ for all $j \ne i$. Also note that $P_i=P_i'$ because $\varphi _i[P]=c$ implies $c P_i \emptyset$ by individual rationality of $\varphi , c P'_i \emptyset$ from before, and $|C|=1$. Hence, $(P_i,P_{-i}^\emptyset ) {r.r.m.t.} P$ at $\varphi [P]$ and $(P_i,P_{-i}^\emptyset ) {r.r.m.t.} P'$ at $\varphi [P']$ for any $P_{-i}^\emptyset \in \mathcal ( P^\emptyset )^{|I|-1}$. Thus, rank-respecting invariance implies $c=\varphi _i[P]=\varphi _i[P_i,P_{-i}^\emptyset ]=\varphi _i[P']=\emptyset$, a contradiction.

If $|I|=1$, then rank-respecting invariance implies favoring higher ranks. To see this point suppose for contradiction that a mechanism $\varphi$ violates favoring higher ranks while satisfying rank-respecting invariance. Then, for the unique student $i$, there exist $P_i,P_i'$ such that

$$\begin{aligned} \varphi _i[P_i'] P_i \varphi _i[P_i]. \end{aligned}$$

(16)

Then a preference $P_i''$ such that

$$\begin{aligned} \begin{array}{c} P''_{i}\\ \hline \varphi _i[P_i']\\ \varphi _i[P_i]\\ \vdots \end{array} \end{aligned}$$

satisfies $P_i'' \, {r.r.m.t.} \, P_i$ at $\varphi [P_i]$ and $P_i'' \, {r.r.m.t.} \, P_i'$ at $\varphi [P_i']$. By rank-respecting invariance, we obtain

$$\begin{aligned} \varphi [P_i]=\varphi [P_i'']=\varphi [P'_i], \end{aligned}$$

a contradiction to relation (16).

A mechanism violating only population monotonicity: Suppose that $|C|=1$ or $|I| \le 2$: If $|I|=1$, then population monotonicity is vacuously satisfied. If $|I|=2$, then individual rationality and favoring higher ranks imply population monotonicity. To see this point, let $I=\{1,2\}$ and $P_2^\emptyset \in \mathcal P ^\emptyset$. Consider an arbitrary preference profile $P$. Since $\varphi$ is individually rational, $\varphi _1[P] \, R_1 \, \emptyset$. If $\varphi _1[P]=\emptyset$, then individual rationality of $\varphi$ implies that $\varphi _1[P_1,P_2^\emptyset ] \,R_1 \,\emptyset =\varphi _1[P]$, satisfying the conclusion of population monotonicity. Thus, suppose $\varphi _1[P] \ne \emptyset$ and, for contradiction, that $\varphi _1[P] \, P_1 \, \varphi _1[P_1,P^\emptyset _2]$. Then, by favoring higher ranks, $\varphi _2[P_1,P^\emptyset _2]=\varphi _1[P]$. However, $\emptyset \,P_2^\emptyset \, \varphi _1[P]$ since $P_2^\emptyset$ top-ranks $\emptyset$, which contradicts individual rationality of $\varphi$. By a symmetric argument, the conclusion of population monotonicity holds for matchings of student $2$, showing that population monotonicity holds.

If $|C|=1$, then individual rationality and rank-respecting invariance imply population monotonicity. To see this point, first note that $\varphi _i[P] \, R_i \, \emptyset$ for all $i$ and $P$ since $\varphi$ is individually rational. Hence, if $\varphi _i[P]=\emptyset$, then $\varphi _i[P^\emptyset _j,P_{-j}] R_i \emptyset =\varphi _i[P]$ for any $i \in I, j \ne i$ and $P^\emptyset _j \in \mathcal P ^\emptyset$; thus, the conclusion of population monotonicity holds. So suppose $\varphi _i[P]=c \ne \emptyset$. Then, since $|C|=1$ and $q_c=1, \varphi _k[P]=\emptyset$ for all $k \ne i$. Thus, for any $j \ne i$ and $P^\emptyset _j \in \mathcal P ^\emptyset$, we have $(P^\emptyset _j,P_{-j}) \, {r.r.m.t.} \, P$ at $\varphi [P]$. By rank-respecting invariance of $\varphi$, we obtain $\varphi _i[P^\emptyset _j,P_{-j}]=\varphi _i[P]$, showing that the conclusion of population monotonicity holds. These arguments show the claim.

A mechanism violating only rank-respecting invariance: Suppose that $|I|=1$ or $|C|=1$: If $|I|=1$ or $|C|=1$, then the paragraph on rank-respecting invariance in Sect. F.1 shows that there exists no mechanism that satisfies favoring higher ranks, individual rationality, and population monotonicity, and yet violates rank-respecting invariance.²⁷