Robustness to manipulations in school choice

Abstract

We study the school choice problem and propose a new criterion for comparing non-strategy-proof mechanisms: robustness to manipulations. Mechanism A is more robust than mechanism B if each student (given any preferences of this student and any profile of schools’ priorities) can potentially access a smaller set of schools via a profitable manipulation under mechanism A than under mechanism B. This criterion strengthens the two independent criteria proposed by Bonkoungou and Nesterov (Theor Econ 16(3):881–909, 2021) and Decerf and Van der Linden (J Econ Theory 197:105313, 2021). We then show that all results obtained with these two criteria, as well as with the original criterion proposed by Pathak and Sönmez (Am Econ Rev 103(1):80–106, 2013), can also be obtained using robustness. Our results provide a stronger rationalization for a wide range of reforms in school choice and college admissions system.

Appendix: Proofs

Let us first introduce some useful notation. For convenience, we enumerate schools with lowercase indices according to the preference relation of student j: $s_1 \ P_j\ s_2 \ P_j\ \ldots \ P_j\ s_k \ P_j\ \ldots$

In many proofs we work with preference profiles which use similar preference relations. For convenience, we place these preference relations here and refer to them throughout the proofs.

$$\begin{aligned} \begin{array}{ccc} P_j &{}\quad P_j^{'} &{}\quad P_{i \ (\forall i\ne j)} \\ \hline s_1 &{}\quad s_1 &{}\quad s_1 \\ \vdots &{}\quad \vdots &{}\quad \vdots \\ s_{k} &{}\quad s_{k+1} &{}\quad s_{k} \\ s_{k+1} &{}\quad s_{k} &{}\quad s_{k+1} \\ \emptyset &{}\quad \emptyset &{}\quad \emptyset \\ \end{array} \end{aligned}$$

(1)

$$\begin{aligned} \begin{array}{ccc} P_j &{}\quad P_j^{'} &{}\quad P_{i \ (\forall i\ne j)} \\ \hline s_1 &{}\quad s_1 &{}\quad s_1 \\ \vdots &{}\quad \vdots &{}\quad \vdots \\ s_{k-1} &{}\quad s_{k} &{}\quad s_{k-1} \\ s_k &{}\quad s_{k-1} &{}\quad s_k \\ \emptyset &{}\quad \emptyset &{} \emptyset \end{array} \end{aligned}$$

(2)

Furthermore, in most proofs the schools’ priorities are such that student j is the least preferred:

$$\begin{aligned} \begin{array}{c} \succ _{s \ (\forall s \in S)} \\ \hline 1 \\ 2 \\ \vdots \\ j \\ \end{array} \end{aligned}$$

(3)

Proof of Theorem 1

First, we show that $DA ^{k+1}$ is at least as robust as $DA ^k$. Second, we show that $DA ^{k+1}$ is more robust than $DA ^k$.

Suppose that school s is strategically accessible to student j at preference relation $P_j$ via mechanism $DA ^{k+1}$. By definition, there exist a manipulation $P'_{j}$ and a list of preferences $P_{-j}$ such that $DA ^{k+1}_j(P'_{j}, P_{-j})=s \ P_j \ DA ^{k+1}_j(P_j, P_{-j})$. Since $DA ^{k+1}(P) = DA(P^{k+1})$¹² is strategy-proof, school s is ranked in the preference relation $P_j$ as the $(k+2)$th choice or with a higher number.¹³ Hence, $DA ^{k+1}_j(P_j, P_{-j}) = \emptyset$. Since $DA ^{k+1}_j(P'_{j}, P_{-j})=s$, in manipulation $P'_{j}$ school s is reported as one of $\{1,\ldots , k+1\}$ first choices. We next use the outcomes of $DA ^{k+1}(P_j, P_{-j})$ and construct a list of preferences $P^*_{-j}$ the following way:

• For each student i (not j) such that $DA ^{k+1}_i(P_j, P_{-j}) \ne \emptyset$ fill their first row with the outcome $\mu _i= DA ^{k+1}_i(P_j, P_{-j})$. Fill their second row with the outside option ($\emptyset$).

• For each student h (not j) who were unassigned, $DA ^{k+1}_h(P_j, P_{-j}) = \emptyset$, keep their preferences the same as in $P_{-j}$.

We claim that given such preference list $P^*_{-j}$, school s must also be strategically accessible to student j at preference relation $P_j$ via mechanism $DA ^k$:

• If student j reports his true preference relation $P_j$, he remains unassigned via mechanism $DA ^k$, given $P^*_{-j}$. Since $DA ^{k+1}_j(P_j, P_{-j}) = \emptyset$, then schools $s_1$, $s_2$, ..., $s_{k+1}$ fill their full capacities via $DA ^{k+1}(P_j,P_{-j})$. Hence, via $DA ^k(P_j, P^*_{-j})$ these schools fill their full capacities with the same students in the first round. Hence, $DA ^k_j(P_j,P^*_{-j})=\emptyset$.

• If student j reports school s as one of the $\{1,\ldots ,k\}$ first choices, he obtains s for certain. Since $DA ^{k+1}_j(P'_{j},P_{-j})=s$, by the end of the algorithm there were no more than $q_s-1$ students with higher priorities than j who were accepted to this school. Thus, no more than $q_s-1$ students with higher priority than j put s in their first row in $P^*_{-j}$. So in $DA ^k$ there always exist a place for student j in school s, given $P^*_{-j}$. Thus, there exists a manipulation $P''_j$ of student j such that $DA ^k_j(P''_j,P^*_{-j})=s$.

Hence, school s is strategically accessible to student j at preference relation $P_j$ via mechanism $DA ^k$. This proves that mechanism $DA ^{k+1}$ is at least as robust as mechanism $DA ^k$.

Now, suppose there are $k+1$ students ($k>0$) and $k+1$ schools with capacities equal to 1. Consider the preference profiles with preferences described in 1 and the priority relation 3.

Then, $DA ^{k}_j(P) = \emptyset$, but $DA ^{k}_j(P^{'}_j, P_{-j}) = s_{k+1}$, so $s_{k+1}$ is strategically accessible to student j at $P_j$ via $DA ^{k}$. At the same time, there is no strategically accessible school to student j at $P_j$ via $DA ^{k+1}$, which is equivalent to unconstrained version of the DA, because this mechanism is strategy proof. Thus, school $s_{k+1}$ is strategically accessible to student j at preference relation $P_j$ via $DA ^{k}$, but not via $DA ^{k+1}$.

Then, by induction, for each $k>l$: $\forall j \in J, \ P_j\in {\mathscr {P}}{:}\, S_j(P_j, DA ^k) \subseteq S_j(P_j, DA ^l) \ \& \ \exists i \in J, \ P_i\in {\mathscr {P}}{:}\,S_i(P_i, DA ^k) \subsetneq S_i(P_i, DA ^l)$, so $DA ^k$ is more robust than $DA ^l$. $\square$

Proof of Theorem 2

First, we show that $\beta ^{k+1}$ is at least as robust as $\beta ^k$. Second, we show that $\beta ^{k+1}$ is more robust than $\beta ^k$.

Suppose school s is strategically accessible to student j at preference relation $P_j$ via $\beta ^{k+1}$. By definition, there exist a manipulation $P'_{j}$ and a list of preferences $P_{-j}$ such that $\beta ^{k+1}_j(P'_{j}, P_{-j})=s \ P_j \ \beta ^{k+1}_j(P_j, P_{-j})$. Without loss of generality assume that school s is ranked as the first choice in the report $P'_{j}$. Since $\beta ^{k+1}_j(P'_{j}, P_{-j}) = s$, no more than $q_s-1$ students with higher priority than student j ranked school s as their first choice in $P_{-j}$. Hence, $\beta ^k_j(P'_{j}, P_{-j})=s$, as school s has a place for student j in the first round. Moreover, since $s \ P_j\ \beta ^{k+1}_j(P_j, P_{-j})$, we have two cases to consider:

• If $\beta ^{k+1}_j(P_j, P_{-j})=\emptyset$, then schools $s_1$, $s_2$, ..., $s_{k}$ fill their full capacities via $\beta ^{k+1}(P_j,P_{-j})$. Hence, these schools fill their full capacities via $\beta ^k(P_j,P_{-j})$. So, $\beta ^k_j(P_j,P_{-j})=\emptyset$.

• If $\beta ^{k+1}_j(P_j, P_{-j})=s_m$, then $s \ P_j\ s_m \ P_j\ s_{k+2}$. All schools $s_1$, ..., $s_{m-1}$ fill their full capacities via $\beta ^{k+1}(P_j,P_{-j})$. Hence, these schools fill their full capacities via $\beta ^k(P_j,P_{-j})$. So, either $\beta ^k_j(P_j,P_{-j})=s_m$ or $\beta ^k_j(P_j,P_{-j})=\emptyset$.

In both cases, $\beta ^k_j(P'_{j},P_{-j})=s \ P_j\ \beta ^k_j(P_j, P_{-j})$, so school s is strategically accessible to student j at preference relation $P_j$ via mechanism $\beta ^k$. Thus, mechanism $\beta ^{k+1}$ is at least as robust as mechanism $\beta ^k$.

Now, suppose there are $k+1$ students ($k>0$) and $k+1$ schools with capacities equal to 1. Again consider the preference relations 1 and the priority relation 3.

Then, $\beta ^{k}_j(P) = \emptyset$, but $\beta ^{k}_j(P_j^{'}, P_{-j}) = s_{k+1}$, so $s_{k+1}$ is strategically accessible to student j at $P_j$ via $\beta ^{k}$. Note that $\beta ^{k+1}_j(P)\ne \emptyset$. Thus, it can not be that $s_{k+1} \ P_j \ \beta ^{k+1}_j(P)$, so school $s_{k+1}$ is not strategically accessible for student j at preference relation $P_j$ via mechanism $\beta ^{k+1}$.

Then, by induction, for each $k>l$: $\forall j \in J, \ P_j\in {\mathscr {P}}{:}\,S_j(P_j, \beta ^k) \subseteq S_j(P_j, \beta ^l) \ \& \ \exists i \in J, \ P_i\in {\mathscr {P}}{:}\,S_i(P_i, \beta ^k) \subsetneq S_i(P_i, \beta ^l)$, so $\beta ^k$ is more robust than $\beta ^l$. $\square$

Proof of Theorem 3

First, we show that $DA ^k$ is at least as robust as $FPF ^{k}$. Second, we show that $DA ^k$ is more robust than $FPF ^{k}$.

Suppose that school s is strategically accessible to student j at preference relation $P_j$ via mechanism $DA ^k$. By definition, for a given preference relation $P_j$ there exist $P'_{j}$, $P_{-j}$ such that $DA ^k_j(P'_{j}, P_{-j})=s \ P_j \ DA ^k_j(P_j, P_{-j})$. Since $DA ^k(P) = DA(P^{k})$ is strategy-proof, school s is ranked in the preference relation $P_j$ as the $(k+1)$th choice or with a higher number. Hence, $DA ^k_j(P_j, P_{-j}) = \emptyset$. Since $DA ^k_j(P'_{j}, P_{-j})=s$, in manipulation $P'_{j}$ school s is reported as one of the $\{1,\ldots , k\}$ first choices. Based on the outcomes of $DA ^k(P_j, P_{-j})$, we construct the following profile $P^*_{-j}$ (keeping $P_j$ fixed):

• For each student i (not j) such that $DA ^k_i(P_j, P_{-j}) \ne \emptyset$ fill their first row with the outcome $\mu _i= DA ^k_i(P_j, P_{-j})$. Fill their second row with the outside option ($\emptyset$).

• For each student h (not j) who were unassigned, $DA ^k_h(P_j, P_{-j}) = \emptyset$, keep their preferences the same as in $P_{-j}$.

Next, let a manipulation $P''_j$ be such that s is the only acceptable school. We claim that $FPF ^{k}_j(P''_j,P^*_{-j})=s \ P_j \ FPF ^{k}_j(P_j,P^*_{-j})$.

• (1) Since $DA ^k_j(P_j,P_{-j})=\emptyset$, then schools $s_1$, ..., $s_k$ fill their full capacities via $DA ^k(P_j, P_{-j})$. Hence, all these schools fill their full capacities via $FPF ^{k}(P_j, P^*_{-j})$ with the same students in the first round, as all these students put the outcomes of $DA ^k(P_j,P_{-j})$ in their first rows in $P^*_{-j}$.

• (2) Since $DA ^k_j(P'_{j},P_{-j})=s$, then by the end of $DA ^k(P'_{j},P_{-j})$ no more than $q_s-1$ students had higher priority to school s than student j, and listed them as their first k choices. Then, no more than $q_s-1$ students with higher priority than student j put s in $P^*_{-j}$ in the first k rows.

Now consider $FPF ^{k}(P''_j, P^*_{-j})$:

Case 1 School s—a first-preference-first school. Due to (1), if student j reports his true preferences, he remains unassigned. $FPF ^{k}_j(P_j,P^*_{-j})=\emptyset$. Due to (2), no more than $q_s-1$ students with higher scores put school s as the first choice in $P^*_{-j}$. Thus, in the first round student j obtains school s with $P''_j$ for certain.

Case 2 School s—an equal preference school. Due to (1), if student j reports his true preferences, he remains unassigned. $FPF ^{k}_j(P_j,P^*_{-j})=\emptyset$. Due to (2), no more than $q_s-1$ students with higher scores put school s in $P^*_{-j}$. Thus, during the algorithm student j must get it for certain with $P''_j$.

Thus, $FPF ^{k}_j(P_j,P^*_{-j}) = \emptyset$, $FPF ^{k}_j(P''_j,P^*_{-j})=s$, which proves that school s is strategically accessible to student j at $P_j$ via $FPF ^{k}$. So, mechanism $DA ^k$ is at least as robust as mechanism $FPF ^{k}$.

Now, suppose there are $k>1$ schools with capacities equal to 1 and $k+1$ students. Also assume that at least one school, for example $s_k$, is first-preference-first. Consider preferences in 2 and priorities 3.

Then, $FPF _j^k(P)=\emptyset$, all schools reach their full capacity. However, $FPF _j^{k}(P^{'}_j, P_{-j})=s_{k}$, which means that $s_k$ is strategically accessible to student j at $P_j$ via $FPF ^{k}$.

At the same time, school $s_k$ is not strategically accessible to student j at $P_j$ via $DA ^k$, which is equivalent to unconstrained $DA$, because this mechanism is strategy proof.

We get that $\forall j \in J, \ P_j\in {\mathscr {P}}{:}\,S_j(P_j, DA ^k) \subseteq S_j(P_j, FPF ^{k}) \ \& \ \exists i \in J, \ P_i\in {\mathscr {P}}{:}\,S_i(P_i, DA ^k) \subsetneq S_i(P_i, FPF ^{k})$. $\square$

Proof of Theorem 4

Lemma 1. (Chen and Kesten, 2017). Let k be given. Let P be a preference profile, j —a student, s—a school and $P_j^{s}$ —a preference relation in which student j has ranked school s first.

• (a) Suppose that student j is matched to school s under $Ch^{(k)}(P_j,P_{-j})$. Then he is also matched to school s under $Ch^{(k)}(P_j^{s},P_{-j})$.

• (b) Suppose that student j prefers school s to his matching under $Ch^{(k)}(P_j,P_{-j})$ and has ranked it among his top k schools under P. Then he can not obtain a seat at school s by misrepresenting his preferences.

First, we show that $Ch^{(k+1)}$ is at least as robust as $Ch^{(k)}$. Second, we show that $Ch^{(k+1)}$ is more robust than $Ch^{(k)}$.

Suppose that school s is strategically accessible to student j at $P_j$ via $Ch^{(k+1)}$ at $P_j$, i.e. for a given preference relation $P_j$ there exist $P'_{j}$ and $P_{-j}$ such that $Ch^{(k+1)}_j (P'_{j},P_{-j})=s \ P_j \ Ch^{(k+1)}_j (P_j,P_{-j})=\mu _j$.

By Lemma 1, school s is ranked in preference relation $P_j$ as the $(k+2)$th choice or with a higher number. Since school, obtained by student j without manipulation, $\mu _j$ (or the outside option), is less preferred by student j than s, this school (or the outside option) must be ranked lower in his preference relation than s. By Lemma 1, we can let $P'_{j}= P^s_j$. There are two cases how student j could be admitted to school s with the report $P'_{j}$:

Case 1 At least one student among those who were matched to school s under $Ch^{(k+1)}(P_j,P_{-j})$ has lower priority than student j under $\succ _{s}$. Based on the outcomes of $Ch^{(k+1)}(P_j, P_{-j})$, we construct a list of preferences $P^*_{-j}$ the following way:

• For each student i (not j) such that $Ch^{(k+1)}_i(P_j, P_{-j}) \ne \emptyset$ fill their first row with the outcome $\mu _i=Ch^{(k+1)}_i(P_j, P_{-j})$. Fill their second row with the outside option ($\emptyset$).

• For each student h (not j) who were unassigned, $Ch^{(k+1)}_h(P_j, P_{-j}) = \emptyset$, keep their preferences the same as in $P_{-j}$.

We claim that given such preference list $P^*_{-j}$, school s must also be strategically accessible to student j at preference relation $P_j$ via mechanism $Ch^{(k)}$:

• If student j reports his true preference relation $P_j$, he remains with $\mu _j$ via mechanism $Ch^{(k)}$, given $P^*_{-j}$. Since $Ch^{(k+1)}_j(P_j, P_{-j}) = \mu _j = s_m$, then schools $s_1$, $s_2$, ..., $s_{k+1}, \ldots , s_{m-1}$ fill their full capacities via $Ch^{(k+1)}(P_j,P_{-j})$. Hence, via $Ch^{(k)}(P_j, P^*_{-j})$ these schools fill their full capacities with the same students in the first round. Hence, $s \ P_j\ Ch^{(k)}_j(P_j,P^*_{-j})$.

• If student j reports $P'_{j}=P^s_j$, he obtains school s for certain. Since $Ch^{(k+1)}_j (P'_{j},P_{-j})=s$, by the end of the algorithm there were no more than $q_s-1$ students with higher priorities than j who were accepted to this school. Thus, no more than $q_s-1$ students with higher priorities than j put s in their first row in $P^*_{-j}$. So in $Ch^{(k)}$ there always exists a place for student j in school s, given $P^*_{-j}$. Thus, $Ch^{(k)}_j(P^s_j,P^*_{-j})=s$.

Case 2 Every student among those who were matched to school s under $Ch^{(k+1)} (P_j,P_{-j})$ has higher priority than j under $\succ _{s}$. Then, at least one such student has ranked school s as his $(k+2)$th choice or with a higher number. Based on the outcomes of $Ch^{(k+1)}(P_j, P_{-j})$, we construct a list of preferences $P^*_{-j}$ in the following way:

• For each student i (not j) such that either $Ch^{(k+1)}_i(P_j, P_{-j}) \ne \emptyset$ and $Ch^{(k+1)}_i(P_j, P_{-j}) \ne s$, or $Ch^{(k+1)}_i(P_j, P_{-j}) = s$ and s is ranked among first $k+1$ schools in $P_i$, fill their first row with the outcome $\mu _i = Ch^{(k+1)}(P_j, P_{-j})$. Fill their second row with the outside option ($\emptyset$).

• For each other student h (not j), keep their preferences the same as in $P_{-j}$. There exist some student $h'$ that keeps ranking school s as $(k+2)$th choice or with a higher number (and who has higher priority than j to s). Denote such student $h'$.

We claim that given such preference list $P^*_{-j}$, school s must also be strategically accessible to student j at preference relation $P_j$ via mechanism $Ch^{(k)}$:

• If student j reports his true preference relation $P_j$, he remains with $\mu _j=s_m$ via mechanism $Ch^{(k)}$, given $P^*_{-j}$. Since $Ch^{(k+1)}_j (P_j, P_{-j}) = s_m$, then schools $s_1$, $s_2$, ..., $s_{k+1}, \ldots , s_{m-1}$ fill their full capacities via $Ch^{(k+1)}(P_j,P_{-j})$. Hence, via $Ch^{(k)}(P_j, P^*_{-j})$ these schools fill their full capacities with the same students in the first round. Hence, $s \ P_j\ Ch^{(k)}_j(P_j,P^*_{-j})$.

• If student j reports $P'_{j}=P^s_j$, he obtains school s for certain. Since $Ch^{(k+1)}_j(P'_{j},P_{-j})=s$, by the end of the algorithm there were no more than $q_s-1$ students with higher priorities than j who were accepted to this school. Thus, no more than $q_s-1$ students with higher priorities than j put s in their first row in $P^*_{-j}$. So, in $Ch^{(k)}$ there exists a place for student j in school s, given $P^*_{-j}$, in the first k rounds, since students $h'$ with higher priority under $\succ _s$ than j put this school as their $(k+2)$th first choice or higher, so they will not be considered by school s in the first k rounds. Thus, $Ch^{(k)}_j(P^s_j,P^*_{-j})=s$.

In both cases school s is strategically accessible to student j at preference relation $P_j$ via $Ch^{(k)}$. Hence, $Ch^{(k+1)}$ is at least as robust as $Ch^{(k)}$.

Now, suppose there are $k+2$ students ($k>0$) and $k+1$ schools with capacity equal to 1. Let us consider the preferences 1 and schools’ priorities 3:

Then, $Ch^{(k)}_j(P) = \emptyset$, while $Ch^{(k)}_j(P_j^{'},P_{-j}) = s_{k+1}$. At the same time, $s_{k+1}$ is not strategically accessible to student j at $P_j$ via $Ch^{(k+1)}$, because in this case the mechanism is equivalent to $DA$ which is strategy-proof. So, $s_{k+1}$ is not strategically accessible to student j at $P_j$ under $Ch^{(k+1)}$ but under $Ch^{(k)}$. Then, by induction, for each $k>l$: $\forall j \in J, \ P_j\in {\mathscr {P}}{:}\, S_j(P_j, Ch^{(k)}) \subseteq S_j(P_j, Ch^{(l)}) \ \& \ \exists i \in J, \ P_i \in {\mathscr {P}}{:}\,S_i(P_i, Ch^{(k)}) \subsetneq S_i(P_i, Ch^{(l)})$, so $Ch^{(k)}$ is more robust than $Ch^{(l)}$. $\square$

Proof of Theorem 5

First, we show that mechanism $Ch^{(k)}$ is at least as robust as $DA ^k$. Second, we show that $Ch^{(k)}$ is more robust than $DA ^k$.

Suppose that school s is strategically accessible to j at $P_j$ via $Ch^{(k)}$. This means that there exists a manipulation $P'_{j}$ and a list of preferences $P_{-j}$ such that

$$\begin{aligned} Ch^{(k)}_j(P'_{j}, P_{-j})=s \ P_j \ Ch^{(k)}_j(P_j, P_{-j}) = \mu _j. \end{aligned}$$

The true preference relation $P_j$ is such that school s is not listed among first k schools, because otherwise the student could not obtain s via a manipulation, as in the first k rounds $Ch^{(k)}$ is equivalent to $DA ^k=DA(P^k)$ which is strategy-proof. Thus, school s is listed by student j as his $(k+1)$th choice or with a higher number in $P_j$.

Since at some profile $P = (P_j, P_{-j})$ student j can profitably manipulate to obtain s, he can do it by listing s the first. If he can not do this, then, school s will reach its full capacity at the first round with at least one student that has higher priority than student j. Then, student j can not obtain school s with another manipulations. Thus, we assume further that manipulation $P'_{j}$ such that s is listed the first and $Ch^{(k)}_j(P'_{j}, P_{-j}) = s$.

Now we construct a list of preferences $P^*_{-j}$ based on the outcomes of $Ch^{(k)}(P_j, P_{-j})$:

• Every student who is assigned to school s via $Ch^{(k)}(P_j, P_{-j})$ and every student who is unassigned via $Ch^{(k)}(P_j, P_{-j})$ lists school $s_1$ as his first choice, and the outside option $\emptyset$ as the second choice.

• Every student i who is assigned to some school $s_i \ne s$ via $Ch^{(k)}(P_j, P_{-j})$ lists this school $s_i$ as his first choice, and the outside option $\emptyset$ as the second choice.

Next, we pick this profile $P^* = (P_j, P^*_{-j})$, conduct the $DA ^k$ mechanism, and show that school s is also strategically accessible to student j at $P_j$.

Case 1 student j reports his true preferences.

Round 1: all students who were assigned to $s_1$ via $Ch^{(k)}(P_j, P_{-j})$ put $s_1$ as their first choice in $P^*_{-j}$ and either obtain it via $DA ^k(P_j, P^*_{-j})$ or remain unassigned (e.g. in case a person assigned to s in via $Ch^{(k)}(P)$ put $s_1$ and has a higher priority). Since $Ch^{(k)}_j(P_j, P_{-j}) \ne s_1$, in $Ch^{(k)}(P_j, P_{-j})$ during the first $DA ^k$ there are more students with higher priority to $s_1$ than j who are assigned to this school so that there is no remaining place to student j. Hence, in the first round of $DA ^k(P_j, P^*_{-j})$ the same students apply to $s_1$, so student j is rejected by $s_1$. Round 2: the only applicant is student j. He is rejected by $s_2$—his second choice, because by the second round of $Ch^{(k)}(P_j, P_{-j})$ this school has reached its full capacity with more priority students, and in $P^*$ these students listed $s_2$ as their first choices.

...

Round k: the only applicant is student j. He is rejected by $s_k$—his kth choice, because by the second round of $Ch^{(k)}(P_j, P_{-j})$ this school has reached its full capacity with students of higher priority, and in $P^*$ these students listed $s_k$ as their first choices.

Thus, with true preferences student j becomes unassigned via $DA ^k$ at the profile $P^*$: $DA ^k(P_j, P^*_{-j}) = \emptyset$.

Case 2 Student j reports the manipulation $P'_{j}$.

Round 1: student j sends an application to school s and obtains it, as nobody listed this school the first.

Rounds 2-k: no new applicants.

Thus, via the manipulation $P_{-j}$ student j obtains s under $DA ^k$ at the profile $P^*$: $DA ^k(P'_{j}, P^*_{-j}) = s$.

Case 1 and Case 2 imply that school s is strategically accessible to student j at $P_j$ given mechanism $DA ^k$. Hence, $Ch^{(k)}$ is at least as robust as $DA ^k$.

Now, we show that schools that are not strategically accessible to student j at $P_j$ via $Ch^{(k)}$ can be strategically accessible to j at $P_j$ via $DA ^k$ (when $k<|S|$, otherwise the two mechanisms are equivalent). To illustrate this, we provide an example.

Suppose there are $k+1$ students ($k>0$) and $k+1$ schools with capacities equal to 1. Consider preferences given by 1 and the priorities 3.

Then, $DA ^{k}_j(P) = \emptyset$, but $DA ^{k}_j(P_j^{'}, P_{-j}) = s_{k+1}$, so $s_{k+1}$ is strategically accessible to student j at $P_j$ via $DA ^{k}$. Then, note that $Ch^{(k)}_j(P)\ne \emptyset$ (Otherwise, there exists a school without student, which means that no student including j has applied to this school. This contradicts to preference relation of student j who applied to all schools). Besides, $s_{k+1}$ can be strategically accessible to student j only in the case $s_{k+1} \ P_j \ Ch^{(k)}_j(P)$. The last is true only when $Ch^{(k)}_j(P) = \emptyset$. This contradicts to the previous result. Hence, $s_{k+1}$ is not strategically accessible to student j at $P_j$ via $Ch^{(k)}$, but via $DA ^{k}$. $\square$

Proof of Proposition 1

• (1) By Remark 1 and Theorem 3, $FPF ^{k}$ is at least as AM-manipulable as $DA ^k$. It is left to show that the opposite is not true. Suppose there are $k+1$ students and k schools with capacities equal to 1. Let $s_k$ be a first-preference-first school. Consider preferences given by 2 and the priorities 3. School $s_k$ is strategically accessible to student j at $P_j$ via $FPF ^k$ (e.g. with manipulation $P'_{j}$). At the same time, there is no school that is strategically accessible to student j at $P_j$ via $DA ^k$ because this mechanism is equivalent to the unconstrained version of the Deferred Acceptance mechanism, which is strategy proof. Hence, student j has a truthful dominant strategy via $DA ^k$, but not via $FPF ^k$.

• (2) Immediately follows from Proposition 1.1, and the fact that $DA ^k$ is more AM-manipulable than $Ch^{(k)}$ if there are at least $k>1$ schools (Proposition 7, Decerf and Van der Linden 2021). $\square$

Proof of Proposition 2

• (1) By Remark 2 and Theorem 2, $\beta ^k$ is at least as BN-manipulable as $\beta ^{k+1}$. Let us show that the opposite is also true. In other words, $\beta ^k$ and $\beta ^{k+1}$ are equivalent in terms of BN-manipulability for $k>1$ (the number of schools is at least 3). Assume school s is strategically accessible to student j at some $P_j$ via $\beta ^{k}$, so there exist $P'_{j}, P_{-j}$ such that $\beta ^k_j(P_j^{'}, P_{-j})=s \ P_j \ \beta ^k_j(P_j, P_{-j})$. Without loss of generality assume that $P'_{j}$ is such that school s is listed first. Notice that if $\beta ^k_j(P_j^{'}, P_{-j})=s$, then $\beta ^{k+1}_j(P_j^{'}, P_{-j})=s$, since in the first round of Boston mechanism there are no more than $q_s-1$ applicants to school s that have higher priority under $\succ _s$ than student j. In case of truthful report $P_j$ (and given the same $P_{-j}$), the student either has the same assignments under both mechanisms, or student j is unassigned under $\beta ^k(P)$, but assigned to school $s_{k+1}$ under $\beta ^{k+1}(P)$. Let us consider the two cases separately. Case 1 Suppose $\beta _j^{k+1}(P) = \beta _j^k(P) = \mu _j$, where $\mu _j \in \{s_3, \ldots , s_k, \emptyset \}$. Note that $\mu _j \ne s_1$, because otherwise s will not be strategically accessible to j at $P_j$. Also $\mu _j \ne s_2$, as it is impossible to obtain $s_1 \ P_j\ s_2$ with any manipulation. Since $\beta ^k_j(P_j^{'}, P_{-j})=s \ P_j \ \mu _j$, we also have $\beta ^{k+1}_j(P_j^{'}, P_{-j})=s \ P_j \ \beta ^{k+1}_j(P)=\mu _j$. Hence, s is also strategically accessible to j at $P_j$ under $\beta ^{k+1}$. This is true for every $j\in J$ and $P_j\in {\mathscr {P}}$. Case 2 Let $\beta ^{k+1}_j(P)=s_{k+1} \ P_j \ \beta ^k_j(P)= \emptyset$. Here we impose the additional assumption: no two schools can accept all students. We construct a new preference profile $P^* = (P^*_j, P^*_{-j})$ in the following way:

  • $P^*_j$ is the following: school $s_1$ is ranked first, school $s_{k+1}$ is ranked second, the outside option is ranked third.

  • Take the priority relation of school $s_1$, and choose the top $q_{s_1}$ students under $\succ _{s_1}$. Every such student i put school $s_1$ as the first choice, school $s_{k+1}$ as the second choice in $P^*_{-j}$.

  • Denote by $\succ '_{s_{k+1}}$ the priority relation of school $s_{k+1}$, $\succ _{s_{k+1}}$, without top $q_{s_1}$ students under $\succ _{s_1}$. Let T denote the set of the top $q_{s_{k+1}}$ students under $\succ '_{s_{k+1}}$. There are two cases:

    • (1) Student j is in T ($j \in T$). Every student $h \in T \backslash \{j\}$ put school $s_{k+1}$ as their first choice. One student g, who has not been considered yet, put school $s_{k+1}$ as their first choice. Note that such student g always exists, because of the additional assumption. $\begin{array}{cccc} P^*_j &{}\quad P^*_{i} &{}\quad P^*_{h} &{}\quad P^*_{g} \\ \hline s_1 &{}\quad s_1 &{}\quad s_{k+1} &{}\quad s_{k+1} \\ s_{k+1} &{}\quad s_{k+1} &{}\quad \vdots &{}\quad \vdots \\ \emptyset &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots \\ \end{array}$$\begin{array}{cc} \succ _{s_1} &{}\quad \succ _{s_{k+1}} \\ \hline \vdots &{}\quad \vdots \\ i &{}\quad j \\ \vdots &{}\quad \vdots \\ j &{}\quad g \\ \vdots &{}\quad \vdots \end{array}$

    • (2) Student j is not in T ($j \notin T$). Every student $h \in T$ put school $s_{1}$ as their first choice, school $s_{k+1}$ as their second choice. $\begin{array}{ccc} P^*_j &{}\quad P^*_{i} &{}\quad P^*_{h} \\ \hline s_1 &{}\quad s_1 &{}\quad s_{1} \\ s_{k+1} &{}\quad s_{k+1} &{}\quad s_{k+1} \\ \emptyset &{}\quad \vdots &{}\quad \vdots \\ \end{array}$$\begin{array}{cc} \succ _{s_1} &{}\quad \succ _{s_{k+1}} \\ \hline \vdots &{}\quad \vdots \\ i &{}\quad h \\ \vdots &{}\quad \vdots \\ j &{}\quad j \\ \vdots &{}\quad \vdots \end{array}$

  • All other students (if any) put $s_1$ as the first choice, the outside option as the second choice.

  We claim that given such profile $P^*$, school $s_{k+1}$ is also strategically accessible to student j via $\beta ^{k+1}$. Consider a manipulation of student j such that $s_{k+1}$ is ranked first. $P''_j$: $s_{k+1}$, ... If (1) follows, then student j is unassigned if he reports his true preferences: $\beta ^{k+1}_{j}(P^*_j, P^*_{-j}) = \emptyset$. Indeed, school $s_1$ fills its full capacity in the first round with students that have the highest priority under $\succ _{s_1}$. Student j is not among these students, because he was rejected by $s_1$ in the first round of $\beta ^k(P)$. So, in the first round j is rejected by $s_1$. School $s_{k+1}$ fills its full capacity in the first round, as $q_{s_{k+1}}$ students put it the first. Hence, in the second round student j is rejected by $s_{k+1}$. So, $\beta ^{k+1}_{j}(P^*_j, P^*_{-j}) = \emptyset$. If student j reports $P''_j$, he is accepted to school $s_{k+1}$ in the first round, as $j\in T$. Hence, $\beta ^{k+1}_{j}(P''_j, P^*_{-j}) = s_{k+1} \ P^*_j \ \emptyset$. If (2) follows, then student j is unassigned if he reports his true preferences: $\beta ^{k+1}_{j}(P^*_j, P^*_{-j}) = \emptyset$. Indeed, school $s_1$ fills its full capacity in the first round with students that have the highest priority under $\succ _{s_1}$. Student j is not among these students, because he was rejected by $s_1$ in the first round of $\beta ^k(P)$. So, in the first round j is rejected by $s_1$. School $s_{k+1}$ fills its full capacity in the second round, accepting all students $h\in T$. Since $j\notin T$, student j is rejected by $s_{k+1}$. So, $\beta ^{k+1}_{j}(P^*_j, P^*_{-j}) = \emptyset$. If student j reports $P''_j$, he is accepted to school $s_{k+1}$ in the first round, as he is the only applicant to this school. Hence, $\beta ^{k+1}_{j}(P''_j, P^*_{-j}) = s_{k+1} \ P^*_j \ \emptyset$. So, $s_{k+1}$ is strategically accessible to j at $P^*_j$ via $\beta ^{k+1}$. Thus, mechanism $\beta ^{k+1}$ is at least as BN-manipulable as $\beta ^k$.

• (2) By Remark 2 and Theorem 5, $DA ^k$ is at least as BN-manipulable as $Ch^{(k)}$. It is left to show that the opposite is not true. Let us refer to the example from the proof of Theorem 5. Suppose there are $k+1$ ($k>0$) students and $k+1$ schools with capacity 1 each. Consider the preferences given by 1 and the priorities 3. Recall that school $s_{k+1}$ is strategically accessible to student j at $P_j$ via $DA ^{k}$, hence it is strategically accessible to j. We show that j has no strategically accessible school via $Ch^{(k)}$. Consider any preference relation of student j. Then, the first k most preferred schools are not strategically accessible to student j at this preference relation via $Ch^{(k)}$, because during the first k rounds the mechanism works as $DA ^{k} = DA(P^k)$, which makes it impossible to manipulate by the first k preferences. Furthermore, the less preferred school in this preference relation is also not strategically accessible to j via $Ch^{(k)}$. Assume the opposite, then, by definition, student j should receive strictly worse outcome without manipulation, that is no school $\emptyset$. However, since there are $k+1$ rounds in mechanism, $k+1$ students and $k+1$ schools with capacity 1, student j can not end up unassigned, because in his preference relation all schools are preferred to the outside option. So, no school is strategically accessible to j at arbitrarily preference relation via $Ch^{k}$. Hence, there are no strategically accessible schools to j via $Ch^{k}$, but there are some via $DA ^{k}$.

• (3) Immediately follows from Proposition 2.2 and the fact that $FPF ^k$ is more BN-manipulable than $DA ^k$. $\square$

Proof of Proposition 3

• (1) To prove that $DA ^k$ and $\beta ^l$ ($l>k>1$) are not comparable according to robustness, we provide environments, in which: (1) school s is strategically accessible to student j at $P_j$ under $DA ^k$, but not under $\beta ^l$; (2) school s is strategically accessible to student j at $P_j$ under $\beta ^l$, but not under $DA ^k$.

  • (1) Let there be l schools, each with capacity 1, and l students. Consider the following preference relations and schools’ priorities:

    $$\begin{aligned} \begin{array}{ccc} P_j &{}\quad P_j^{'} &{}\quad P_{i \ (\forall i\ne j)} \\ \hline s_1 &{}\quad s_l &{}\quad s_1 \\ \vdots &{}\quad \vdots &{}\quad \vdots \\ s_{k} &{}\quad &{}\quad s_{k} \\ \vdots &{}\quad &{}\quad \vdots \\ s_{l} &{}\quad &{}\quad s_{l}\\ \emptyset &{}\quad &{}\quad \emptyset \end{array} \ \ \ \ \ \ \ \begin{array}{c} \succ _{s \ (\forall s \in S)} \\ \hline 1 \\ 2 \\ \vdots \\ j \end{array} \end{aligned}$$

    Then $DA ^k_j(P_j, P_{-j}) = \emptyset$, but $DA ^k_j(P'_{j}, P_{-j}) = s_l$, so school $s_l$ is strategically accessible to j at $P_j$ under $DA ^k$. Under $\beta ^l$ school $s_l$ cannot be strategically accessible to student j at $P_j$ since via $\beta ^l$ with l students (acceptable to all schools) every student must be assigned to some school, so j receives at least $s_l$ if he reports truthfully.

  • (2) Consider the following preference relations and schools’ priorities:

    $$\begin{aligned} \begin{array}{cccc} P_j &{} P_j^{'} &{} P_i &{} P_{h \ (\forall h\ne j,i)} \\ \hline s_1 &{} s_2 &{} s_1 &{} s_2 \\ s_{2} &{} \vdots &{} \vdots &{} s_{1} \\ \vdots &{} &{} &{} \vdots \\ s_l &{} &{} &{} \vdots \\ \end{array} \ \ \ \ \ \ \ \begin{array}{cc} \succ _{s \ (\forall s \in S \backslash s_2)} &{} s_2 \\ \hline 1 &{} j \\ \vdots &{} \vdots \\ j &{} \end{array} \end{aligned}$$

    Then $\beta ^l(P'_{j},P_{-j})=s_2 \ P_j\ \beta ^l(P_j, P_{-j})$, so $s_2$ is strategically accessible to j at $P_j$. School $s_2$ cannot be strategically accessible to j at $P_j$ because $DA ^k(P)= DA ^k(P^k)$ (and $s_2$ is in k first preferences) makes it impossible to manipulate within first k preferences.

• (2) Suppose that school s is strategically accessible to student j under $DA ^k$. We want to show that it is also strategically accessible to j under $\beta ^l$. First, since s is not strategically accessible to j and for each school each student is better than $\emptyset$ (by assumption), there should be at least k students. Also, school s is not listed among top k preferences of student j. Denote by $s_1$ student j’s most preferred school. Let us construct the following preference profiles:

  • Student j: $s_1$ is the most preferred school, s is the second, $s_2$ is the third.

  • $q_{s_1}$ students, who have higher priorities in $s_1$ than j (they always exist since s is strategically accessible to j under $DA ^k$): $s_1$ is the most preferred school.

  • Among the remaining students (there are at least $q_s$ remaining students due to the assumption that no two schools have seats for all students), choose $q_s-1$ students with the least priorities to s. All these students put s as their top choice.

  • Among the remaining students: choose one student h with the highest priority to s. If school s prefers j to h, then put s as h’s first choice. Otherwise, construct the h’s preferences as follows: $s_1$ is the first, s is the second.

  • All the students who are left put $s_1$ as their first choice, $\emptyset$ as the second.

  Under $\beta ^l$ student j is assigned to $s_2$ if he reports his preferences truthfully. Indeed, $s_1$ fills all the seats in the first round with students having higher priority to $s_1$ than student j. School s fills all the places either in the first or in the second round. By construction, student j cannot be accepted to s. If he misreports his preferences by putting s as his first choice then student j is accepted to s, either because there is one vacant seat in the first round or by supplanting a student with lower priority to s. Therefore school s is strategically accessible to student j under $\beta ^l$. Now, we provide an example where some school s is strategically accessible to some student j under $\beta ^l$ but not under $DA ^k$. Suppose there are l schools, each with capacity 1 and l students. Consider the following preferences and priority relations as in the proof of Proposition 3 (1), 2). Then, $\beta ^{l}_j(P'_{j}, P_{-j}) = s_2 \ P_j \ \beta ^l_j(P_j, P_{-j})$, so $s_2$ is strategically accessible to j under $\beta ^l$. School $s_2$ is not strategically accessible to j under $DA ^k$, because whenever $s_2$ is not in the first k j’s preferences, student j obtains some school from the top k preferences under $DA ^k$. $\square$

Proof of Proposition 4

• (1) Assume that some profile P is vulnerable under mechanism $Ch^{(k)}$, that is, there exists student j, school s, and a manipulation $P_j^{'}$ such that $Ch^{(k)}_j(P^{'}_j, P_{-j}) = s \ P_{j} \ Ch^{(k)}_j(P_j, P_{-j})$. We need to show that this profile P is also vulnerable under mechanism $DA ^{k}$. In particular, we will show that for the same student j there exists a manipulation $P_j^{''}$ such that $DA ^k_j(P^{''}_j, P_{-j}) \ P_{j} \ DA ^k_j(P_j, P_{-j})$. Since $Ch^{(k)}(P)$ during the first k rounds is equivalent to $DA ^k(P)=DA(P^k)$, which is strategy-proof, then school s is ranked in $P_j$ as the $(k+1)$th top choice or with a higher number. Now, consider $DA ^k(P)$. If j reports $P_j$, he remains unassigned, since schools $s_1$, $s_2$, ..., $s_k$ rejected student j during the first k rounds of $Ch^{(k)}(P)$. The first k rounds in $DA ^k$ are the same. So, $DA ^k_j(P)=\emptyset$. Consider $P''_j$ such that school s is ranked the first. Since $Ch^{(k)}_j(P'_{j}, P_{-j})=s$, then no more than $q_{s}-1$ students with higher priority than j under $\succ _s$ ranked s among their first k top choices. Hence, $DA ^k_j(P''_j, P_{-j})=s \ P_j\ \emptyset$. So, we have shown that if profile P is vulnerable under mechanism $Ch^{(k)}$, then it is also vulnerable under $DA ^{k}$. Now we provide an example, where the opposite is not true. Suppose, there are $k+1$ students (enumerated as 1,2,...,$k+1$) and $k+1$ schools. The preference profile and school priorities are the following: $\begin{array}{ccc} P_{k+1} &{}\quad P_{k+1}^{'} &{}\quad P_{i\ne k+1} \\ \hline s_1 &{}\quad s_{k+1} &{}\quad s_i \\ \vdots &{}\quad \vdots &{}\quad \vdots \\ s_{k} &{}\quad &{}\quad \\ s_{k+1} &{}\quad &{}\quad \\ \emptyset &{}\quad &{}\quad \\ \end{array}$$\begin{array}{c} \succ _{s \ (\forall s \in S)} \\ \hline 1 \\ 2 \\ \vdots \\ k+1 \\ \end{array}$ We obtain the following allocations at profile P under mechanisms $Ch^{(k)}$ and $DA ^{k}$:

  $$\begin{aligned} DA ^{k}(P) = \begin{pmatrix} i &{}\quad k+1\\ s_i &{}\quad \emptyset \end{pmatrix} \quad Ch^{(k)}(P) = \begin{pmatrix} i &{}\quad k+1\\ s_i &{}\quad s_{k+1} \end{pmatrix} \end{aligned}$$

  Notice that all students, except student $k+1$, obtain his most preferred school under both mechanisms, so they can not benefit by misrepresenting his preferences. As for the student $k+1$, he obtains school $s_{k+1}$ via $Ch^{(k)}$ without manipulation. In other words, he remains unmatched after k rounds. Notice that since the first k rounds of $Ch^{(k)}$ are equivalent to $DA ^{k}$, the agent does not have a manipulation that gives him a strictly better outcome than $s_{k+1}$. So, profile P is not vulnerable under $Ch^{(k)}$. Now assume that student $k+1$ reports preferences $P^{'}_{k+1}$ instead of $P_{k+1}$:

  $$\begin{aligned} DA ^{k}(P_{k+1}^{'}, P_{-(k+1)}) = \begin{pmatrix} i &{}\quad k+1\\ s_i &{}\quad s_{k+1} \end{pmatrix} \end{aligned}$$

  Notice that student $k+1$ was unmatched via $DA ^{k}$ at the true preference relation, but obtained school $s_{k+1}$ at manipulation $P_{k+1}^{'}$. Hence, profile P is vulnerable to manipulations via $DA ^{k}$ but not via $Ch^{(k)}$.

• (2) To prove that $Ch^{(k)}$ and $FPF ^k$ are not comparable in general we provide two environments, at which: (1) some profile is vulnerable under $FPF ^{1}$, but not under $Ch^{(1)}$; (2) some profile is vulnerable under $Ch^{(3)}$, but not under $FPF ^3$.

  • (1) There are 2 students and 2 schools with capacity 1. Both schools are first-preference-first. The preference profile and school priorities are the following: $\begin{array}{ccc} P_1 &{}\quad P_2 &{}\quad P_2^{'} \\ \hline s_1 &{}\quad s_1 &{}\quad s_2 \\ s_2 &{}\quad s_2 &{}\quad s_1 \\ \emptyset &{}\quad \emptyset &{}\quad \emptyset \\ \end{array}$$\begin{array}{c} \succ _{s \ (\forall s \in S)} \\ \hline 1 \\ 2 \\ \end{array}$ We obtain the following allocations at profile P under mechanisms $Ch^{(1)}$ and $FPF ^1$:

    $$\begin{aligned} FPF ^1(P) = \begin{pmatrix} 1 &{}\quad 2\\ s_1 &{}\quad \emptyset \end{pmatrix} \\ Ch^{(1)}(P) = \begin{pmatrix} 1 &{}\quad 2\\ s_1 &{}\quad s_2 \end{pmatrix} \end{aligned}$$

    Notice that student 1 obtains his most preferred school under both mechanisms, so he can not benefit by misrepresenting his preferences. As for student 2, he obtains school $s_2$ via $Ch^{(1)}$ without manipulation. The only school that student 2 prefers to $s_2$ is $s_1$. However, no manipulation gives school $s_1$ to student 2 because the first priority of $s_1$ is student 1 who reported it as most preferred. Hence, profile P is not vulnerable under $Ch^{(1)}$. Now assume that student 2 reports preferences $P^{'}_2$ instead of $P_2$:

    $$\begin{aligned} FPF ^1(P_2^{'}, P_{-2}) = \begin{pmatrix} 1 &{}\quad 2\\ s_1 &{}\quad s_2 \end{pmatrix} \end{aligned}$$

    Notice that student 2 was unmatched via $FPF ^1$ at true preference relation, but obtained school $s_2$ at manipulation $P_2^{'}$. Hence, profile P is vulnerable to manipulations via $FPF ^1$ but not via $Ch^{(1)}$.

  • (2) Now let us refer to the Example 1 (Bonkoungou and Nesterov 2021). There are 7 students and 7 schools with capacity 1. The preference profile and school priorities are the following: $\begin{array}{cccccccc} P_1^{'} &{}\quad P_1 &{}\quad P_2 &{}\quad P_3 &{}\quad P_4 &{}\quad P_5 &{}\quad P_6 &{}\quad P_7 \\ \hline s_4 &{}\quad s_1 &{}\quad s_1 &{}\quad s_2 &{}\quad s_3 &{}\quad s_5 &{}\quad s_4 &{}\quad s_6 \\ \emptyset &{}\quad s_2 &{}\quad \emptyset &{}\quad \emptyset &{}\quad \emptyset &{}\quad s_7 &{}\quad s_5 &{}\quad s_4 \\ &{}\quad s_3 &{}\quad &{}\quad &{}\quad &{}\quad \emptyset &{}\quad s_6 &{}\quad \emptyset \\ &{}\quad s_4 &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \emptyset &{}\quad \\ &{}\quad \emptyset &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ \end{array} \ \ \ \ \ \begin{array}{ccccccc} \succ _{s_1} &{}\quad \succ _{s_2} &{}\quad \succ _{s_3}&{}\quad \succ _{s_4}&{}\quad \succ _{s_5}&{}\quad \succ _{s_6} &{}\quad \succ _{s_7} \\ \hline 2 &{}\quad 3 &{}\quad 4 &{}\quad 7 &{}\quad 6 &{}\quad 6 &{}\quad 5 \\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad 1 &{}\quad 5 &{}\quad 7 &{}\quad \vdots \\ &{}\quad &{}\quad &{}\quad 6 &{}\quad \vdots &{}\quad \vdots &{}\quad \\ &{}\quad &{}\quad &{}\quad \vdots &{}\quad &{}\quad &{}\quad \\ \end{array}$ Assume that school $s_5$ is the only first-preference-first school. It was shown that profile P is not vulnerable under $FPF ^3$. Now let us check whether it is vulnerable under $Ch^{(3)}$. We obtain the following allocation at this preference profile under $Ch^{(3)}$:

    $$\begin{aligned} Ch^{(3)}(P) = \begin{pmatrix} 1 &{}\quad 2 &{}\quad 3 &{}\quad 4 &{}\quad 5 &{}\quad 6 &{}\quad 7\\ \emptyset &{}\quad s_1 &{}\quad s_2 &{}\quad s_3 &{}\quad s_5 &{}\quad s_4 &{}\quad s_6 \end{pmatrix} \end{aligned}$$

    So, each student except student 1 obtains his most preferred school. As a result, these students can not benefit by misrepresenting their preferences. As for student 1, he ends up without school at preference relation $P_1$ via $Ch^{(3)}$. Now, assume that he reports preferences $P_1'$ instead of $P_1$:

    $$\begin{aligned} Ch^{(3)} (P^{'}_1, P_{-1}) = \begin{pmatrix} 1 &{}\quad 2 &{}\quad 3 &{}\quad 4 &{}\quad 5 &{}\quad 6 &{}\quad 7\\ s_4 &{}\quad s_1 &{}\quad s_2 &{}\quad s_3 &{}\quad s_7 &{}\quad s_5 &{}\quad s_6 \end{pmatrix} \end{aligned}$$

    Thus, student 1 obtains $s_4$ by a manipulation $P_1^{'}$. Hence, profile P is vulnerable under $Ch^{(3)}$, but not under $FPF ^3$.

$\square$