Strategic schools under the Boston mechanism revisited

Abstract

Ergin and Sönmez (J Public Econ 90(1):215–237, 2006) showed that for schools it is a dominant strategy to report their preferences truthfully under the Boston mechanism, and that the Nash equilibrium outcomes in undominated strategies of the induced game are stable. We show that these results rely crucially on two assumptions. First, schools need to be restricted to reporting all students as acceptable. Second, students cannot observe the preferences reported by the schools before submitting their own preferences. We show that relaxing either assumption gives schools an incentive to manipulate their reported preferences. We provide a full characterization of undominated strategies for schools and students for the simultaneous move game induced by the Boston mechanism. Nash equilibrium outcomes in undominated strategies of that game may contain unstable matchings. Furthermore, when students observe schools’ preferences before submitting theirs, the subgame perfect Nash equilibria of the sequential game induced by the Boston mechanism may also contain unstable matchings. Finally, we show that schools may have an incentive to manipulate capacities only if students observe the schools’ strategies before submitting their own preferences.

Appendix

1.1 Proof of Lemma 1

Necessity of the conditions Suppose that a strategy ${\tilde{P}}_{s}$ is undominated. We will show that if each of the conditions above fail, there is a strategy that dominates ${\tilde{P}}_{s}$.

For (i), suppose that there is some student i such that $i{\tilde{P}}_{s}\varnothing$ and $\varnothing P_{s}i$. Consider a strategy $(P'_{s},{\tilde{q}}_{s})$ where $P'_{s}$ is the same as ${\tilde{P}}_{s}$ except that for every student j such that $\varnothing P_{s} j$ we make $\varnothing P'_{s}j$ (including i). Under the simultaneous Boston mechanism, a school can’t change the set of students who apply to that school in each period. As a result, under $P'_{s}$, the set of acceptable students who are matched to s under ${\tilde{P}}_{s}$ will still be matched to s under $P'_{s}$, but if some unacceptable student is matched to s under ${\tilde{P}}_{s}$, she will be replaced by either an acceptable student or an empty seat. By responsiveness, both cases are preferred by s to being matched with some unacceptable student. Therefore, $P'_{s}$ dominates ${\tilde{P}}_{s}$.

For (ii), suppose ${\tilde{q}}_{s}<q_{s}$ and consider a strategy $({\tilde{P}}_{s},q_{s})$. For given preferences and capacities of the other schools and for given preferences of the students, each round of the Boston mechanism will proceed identically under $({\tilde{P}}_{s},{\tilde{q}}_{s})$ and $({\tilde{P}}_{s},q_{s})$ until the capacity constraint ${\tilde{q}}_{s}$ starts to bind, say in round t. This means that in round t there are more students applying to school s than the remaining capacity. However, since $q_{s}>{\tilde{q}}_{s}$ under $({\tilde{P}}_{s},q_{s})$ not only will the same set of students be accepted as under $({\tilde{P}}_{s},{\tilde{q}}_{s})$ but, in addition, some more students will be accepted. Since preferences are responsive and since ${\tilde{P}}_{s}$ lists as acceptable only those students that are acceptable under $P_{s}$, it follows that the outcome of the Boston mechanism under $({\tilde{P}}_{s},q_{s})$ is weakly preferred to the outcome under $({\tilde{P}}_{s},{\tilde{q}}_{s})$. Hence, only strategies with ${\tilde{q}}_{s}=q_{s}$ can be undominated.

For (iii), consider two students, i, j such that $i,j{\tilde{P}}_{s}\varnothing$, $iP_{s}j$ and $j{\tilde{P}}_{s}i$. Consider the strategy $(P'_{s},{\tilde{q}}_{s})$ where $P'_{s}$ is such that $k{\tilde{P}}_{s}\varnothing \Leftrightarrow kP'_{s}\varnothing$ and for all $k,k'P'_{s}\varnothing$ we have $kP'_{s}k'\Leftrightarrow kP_{s}k'$. Since under $({\tilde{P}}_{s},{\tilde{q}}_{s})$ and $(P'_{s},{\tilde{q}}_{s})$ the same set of students is acceptable, each round of the Boston mechanism is equivalent under both strategies unless in some round there are more acceptable applicants than available capacity, for given preferences and capacities of other schools and preferences of the students. Because capacities are the same, this is the same round under both strategies. In that round the preferences over the student determine which students will be accepted. Under $(P'_{s},{\tilde{q}}_{s})$ the best students according to $P_{s}$ will be accepted. This is not the case under $({\tilde{P}}_{s},{\tilde{q}}_{s})$. Thus school s strictly prefers the outcome under $(P'_{s},{\tilde{q}}_{s})$ to the outcome under $({\tilde{P}}_{s},{\tilde{q}}_{s})$ whenever cases arise in which both i and j apply in the same round, but only j is accepted under $({\tilde{P}}_{s},{\tilde{q}}_{s})$. Otherwise the outcome under both strategies is the same. Hence, $({\tilde{P}}_{s},{\tilde{q}}_{s})$ is dominated by $(P'_{s},{\tilde{q}}_{s})$, which is a contradiction.

For (iv), let i be the most-preferred student under $P_{s}$ but $\varnothing {\tilde{P}}_{s}i$. We construct a strategy for school s, $(P'_{s},{\tilde{q}}_{s})$ that dominates $({\tilde{P}}_{s},{\tilde{q}}_{s})$. Let $P'_{s}$ be the same as ${\tilde{P}}_{s}$ except that i is now the most-preferred acceptable student. For given preferences and capacities of the other school and preferences of the students, each round of the Boston mechanism will proceed identically under both strategies until student i applies to school s. In that round, if school s still has available capacity, i will be rejected under $({\tilde{P}}_{s},{\tilde{q}}_{s})$ but not under $(P'_{s},{\tilde{q}}_{s})$ and otherwise the outcome under both strategies is the same. The rejection of i under $({\tilde{P}}_{s},{\tilde{q}}_{s})$ can at most lead to one additional application from an acceptable student, with other applications being the same in each subsequent round. This additional application occurs when student i in later rounds is matched to some other school $s'$ and thereby prevents another student $i'$ from being matched there and $i'$ likewise prevents another student from being matched to some other school. This sequence of rejections either ends with one student being matched to a school whose capacity constraint under the Boston mechanism and $(P'_{s},{\tilde{q}}_{s})$ is not binding (this includes being left unmatched) or it ends with some student $i''$ applying to school s. If $i''$ is not acceptable, the sequence of rejections continues until it either ends or some other student applies to s. If s accepts $i''$ then the assignment under both strategies will differ by only a single agent. But since i is the most-preferred student under $P_{s}$, the outcome under $(P'_{s},{\tilde{q}}_{s})$ is preferred by s. Hence, $({\tilde{P}}_{s},{\tilde{q}}_{s})$ is dominated by $(P'_{s},{\tilde{q}}_{s})$, which is a contradiction.

For (v), let there be only $l<q_{s}$ acceptable students under $({\tilde{P}}_{s},q_{s})$. Construct $(P'_{s},q_{s})$ such that it is the same as $({\tilde{P}}_{s},q_{s})$ except that one additional student i, who is acceptable under $P_{s}$ is acceptable under it. This is possible since we assumed that at least $q_{s}$ students are acceptable under each school’s true preferences. Since only l students under ${\tilde{P}}_{s}$ are acceptable, the capacity constraint will never bind in any round of the Boston mechanism under either $(P'_{s},q_{s})$ or $({\tilde{P}}_{s},q_{s})$ for any given preferences and capacities of the other schools and preferences of the students. This means that for both strategies any application by the l students, that are acceptable under $({\tilde{P}}_{s},q_{s})$, will be accepted. In addition, under $(P'_{s},q_{s})$ student i can also be accepted. Therefore, $(P'_{s},q_{s})$ never yields a worse outcome than $({\tilde{P}}_{s},q_{s})$ but may sometimes yield a preferred outcome, evaluated under the true preferences, which contradicts $({\tilde{P}}_{s},q_{s})$ being undominated.

Sufficiency of the conditions Suppose $({\tilde{P}}_{s},{\tilde{q}}_{s})$ satisfies all five conditions, so that we can replace ${\tilde{q}}_{s}$ by $q_{s}$. Without loss of generality, any strategy $(P'_{s},q'_{s})$ that is a candidate for a strategy that dominates $({\tilde{P}}_{s},q_{s})$ also needs to satisfy the five conditions. Otherwise we could replace it with a strategy that dominates it. So we have $q'_{s}=q_{s}$. Furthermore, whenever two students $i,i'$ are acceptable under ${\tilde{P}}_{s}$ and $P'_{s}$ we must have $i{\tilde{P}}_{s}i'\Leftrightarrow iP'_{s}i'$, since both satisfy the third condition. Thus, $P'_{s}$ can only meaningfully differ from ${\tilde{P}}_{s}$ in terms of the set of students who are reported to be acceptable.

Suppose first that $q_{s}$ students are acceptable under $({\tilde{P}}_{s},q_{s})$. Define the set of students that are acceptable under this strategy to be $\tilde{A}\subseteq I$. Let $i_{1}$ be the most-preferred student under $P_{s}$. We then have that $i_{1}\in \tilde{A}$. It cannot be dominated by another strategy $(P'_{s},q{}_{s})$ that declares a different set of $q_{s}$ students acceptable. To see this, suppose that students in $\tilde{A}$, and only those students, rank school s first. All other students declare s to be unacceptable. In that case, under strategy $({\tilde{P}}_{s},q_{s})$ the set $\tilde{A}$ is accepted, but only a subset of those students are accepted under $(P'_{s},q{}_{s})$, which contradicts $(P'_{s},q{}_{s})$ dominating $({\tilde{P}}_{s},q_{s})$.

Now suppose that $({\tilde{P}}_{s},q_{s})$ with $q_{s}$ acceptable students is dominated by some other strategy $(P'_{s},q{}_{s})$ that declares k students acceptable, with $q_{s}<k\le n$. Denote this set of students as $A'$. We have that $\tilde{A}\subset A'$. Otherwise, let the students in $\tilde{A}{\setminus } A'$ be the only ones who consider s acceptable and let them rank s first. Then $({\tilde{P}}_{s},q_{s})$ yields a better outcome to s than $(P'_{s},q_{s})$.

Suppose now that a subset of $q_{s}-1$ students in $\tilde{A}$ and another student $i'$ in $A'{\setminus }\tilde{A}$ rank s first. Suppose further that $i_{1}$ is the student in $\tilde{A}$ who does not rank s first, but instead ranks another school first, which does not consider $i_{1}$ acceptable. Furthermore, let s be the second choice of $i_{1}$. Then the outcome of the Boston mechanism under $({\tilde{P}}_{s},q_{s})$ differs from that under $(P'_{s},q_{s})$ only in that $i_{1}$ is matched to s in the former, while $i'$ is matched to s in the latter. Since $i_{1}P_{s}i'$, the outcome under $({\tilde{P}}_{s},q_{s})$ is preferred to that under $(P'_{s},q_{s})$. This argument can be adapted straightforwardly to show that no strategy $({\tilde{P}}_{s},q_{s})$ that considers $k\ge q_{s}$ students acceptable is dominated by another strategy $(P'_{s},q_{s})$ that considers $k'\ge k$ students acceptable.

Finally, suppose that k, with $n\ge k>q_{s}$, students are acceptable under $({\tilde{P}}_{s},q_{s})$. We will show that $({\tilde{P}}_{s},q_{s})$ cannot be dominated by any strategy $(P'_{s},q_{s})$ declaring $k'<k$ students acceptable. The associated sets of acceptable students are $\tilde{A}$ and $A'$ as before. Since fewer students are acceptable under $(P'_{s},q_{s})$ we have $\tilde{A}{\setminus } A'\ne \varnothing$. Now suppose all students in $\tilde{A}{\setminus } A'$ rank school s first and are the only students to consider i as acceptable. Then up to the $q_{s}$ most-preferred students will be accepted by s under $({\tilde{P}}_{s},q_{s})$ but no student will be matched to s under $(P'_{s},q_{s})$. Thus, $({\tilde{P}}_{s},q_{s})$ cannot be dominated by a strategy $(P'_{s},q_{s})$ that considers fewer students acceptable.

1.2 Proof of Lemma 2

Necessity of $sP_{i}\varnothing \Leftrightarrow s{\tilde{P}}_{i}\varnothing$. For a contradiction suppose that ${\tilde{P}}_{i}$ is an undominated strategy. Suppose that there exists some student i and school s such that $sP_{i}\varnothing$ but $\varnothing {\tilde{P}}_{i}s$. Let preferences $P'_{i}$ be the same as ${\tilde{P}}_{i}$ except that s is the least-preferred acceptable school. For all strategies ${\tilde{P}}_{-i}$, $({\tilde{P}}_{S},{\tilde{q}}_{s})$ of the other agents in which i is matched to some school under ${\tilde{P}}_{i}$, the assignment under $P'_{i}$ has to be the same under the Boston mechanism, since in each round i applies to the same schools. If i remains unmatched under ${\tilde{P}}_{i}$ then i also remains unmatched under $P'_{i}$, unless i is accepted by school s when applying to it. Hence, the outcome under both is the same, unless i is accepted by school s, which cannot happen under ${\tilde{P}}_{i}$. One profile of preferences and capacities that has this property is one in which all schools except s consider i to be unacceptable and school s considers i and no other student to be acceptable. For this profile of preferences, under $P'_{i}$, i is matched to s, which because $sP_{i}\varnothing$, is strictly preferred to the outcome when reporting ${\tilde{P}}_{i}$. Hence ${\tilde{P}}_{i}$ is dominated by $P'_{i}$, a contradiction.

For any strategy ${\tilde{P}}_{i}$ that declares a school s with $\varnothing P_{i}s$ to be acceptable, one can find preferences for the schools and the other students so that i is matched to s. For any such preferences, a strategy that is identical to ${\tilde{P}}_{i}$, but declares s and any other school $s'$, with $\varnothing P_{i}s'$, to be unacceptable, can at worst result in i being unmatched. This is preferred to being matched to an unacceptable school, so ${\tilde{P}}_{i}$ is a dominated strategy, which is a contradiction.

Sufficiency of $sP_{i}\varnothing \Leftrightarrow s{\tilde{P}}_{i}\varnothing$. To see the result, we need to show that any ordering of acceptable schools constitutes an undominated strategy. We show first that no strategy ${\tilde{P}}_{i}$ that ranks some acceptable school s first can be dominated by another strategy $P'_{i}$ that ranks another acceptable school $s'$ first. We assume that both schools are acceptable under $P_{i}$. Suppose that ${\tilde{P}}_{-i}$ is such that $q_{s}$ other students rank s first, but that i is the most-preferred student according to ${\tilde{P}}_{s}$. The other $q_{s}$ students are acceptable to s. Suppose in addition that no other school, including $s'$, considers i to be acceptable. In that case, under ${\tilde{P}}_{i}$ student i obtains school s. Under $P'_{i}$ student i remains unmatched. Hence, a strategy ${\tilde{P}}_{i}$ ranking school s first can only be dominated by a strategy that also ranks s first.

Suppose that the top k schools, with $2\le k<m$, under ${\tilde{P}}_{i}$ are ranked as follows: $s_{1}{\tilde{P}}_{i}s_{2}\dots s_{k-1}{\tilde{P}}_{i}s_{k}$. Suppose that $P'_{i}\ne {\tilde{P}}_{i}$ ranks the first $k-1$ schools the same way, but ranks another acceptable school $s'_{k}\ne s_{k}$ as the kth best school. The preferences over the remaining schools under both ${\tilde{P}}_{i}$ and $P'_{i}$ are arbitrary. We show that no such $P'_{i}$ can dominate ${\tilde{P}}_{i}$. We construct reported preferences for the other students and strategies for the schools as follows. Suppose school s reports a capacity of ${\tilde{q}}_{s_{k}}=1$. Let there be one other student $i'$ who reports the same preferences over the first k schools as student i. Let both i and $i'$ be acceptable to school $s_{k}$ but unacceptable to any school $s_{1},s_{2},\dots ,s_{k-1}$. Let the preferences of other students be such that none of them apply to school $s_{k}$ until after round k. Furthermore, let $i{\tilde{P}}_{s_{k}}i'$. Last, we assume that $s_{k}$ is the only school that considers i to be acceptable.

Under these (incompletely) specified preferences it follows that i is matched to $s_{k}$ when reporting ${\tilde{P}}_{i}$ but remains unmatched when reporting $P'_{i}$. Since $s_{k}$ is acceptable, it follows that $P'_{i}$ does not dominate ${\tilde{P}}_{i}$. Hence, no strategy is dominated by another strategy that is identical for the first $k-1$ schools and deviates thereafter. It also follows that for a given k and $s_{1},s_{2},\dots ,s_{k}$, the strategy ${\tilde{P}}_{i}$ can also not be dominated by any strategy that differs in the ranking of the $k-1$ most-preferred schools. It follows that no strategy ${\tilde{P}}_{i}$ that satisfies for all $s\in S$, $s{\tilde{P}}_{i}\varnothing \Leftrightarrow sP_{i}\varnothing$ is dominated by some other strategy that also satisfies this condition.

1.3 Proof of Theorem 1

To prove the theorem, we will propose undominated strategies for the students and schools and verify that these strategies constitute a Nash equilibrium. Take some matching $\mu$ that is stable with respect to the true preferences and capacities. For all $i\in I$ such that $\mu (i)=\varnothing$ let i report preferences truthfully. For all $i\in I$ such that $\mu (i)\in S$ let i report school $\mu (i)$ as the most preferred and all other acceptable schools arbitrarily ranked below. Let all schools report preferences truthfully. We will refer to this strategy-profile as $(P_{I}^{\mu },P_{S},q)$, to make clear that only students’ strategies depend on the matching $\mu$.

Claim 1

$(P_{I}^{\mu },P_{S},q)$ consists only of undominated strategies.

From Lemma 1 and Corollary 2 it follows that schools’ strategies are undominated. From Lemma 2 it follows that students’ strategies are undominated.

Claim 2

$(P_{I}^{\mu },P_{S},q)$ is a Nash equilibrium.

If all agents use these strategies, the outcome under the Boston mechanism is $\mu$. Every student i applying to $\mu (i)$ will be accepted by the school in the first round of the Boston mechanism, since $\mu$ is stable. Students unmatched under $\mu$ will be rejected by all schools to which they apply, since if they were accepted by some school, this would contradict the stability of $\mu$ given that schools are assumed to report preferences truthfully. No school s can gain from a deviation. Any student applying to a school other than s will be accepted in the first round, and so cannot be obtained by school s. The only students that s could obtain under some alternative strategy ${\tilde{P}}_{s}$ are those who apply to it in the first round and those who are unmatched under $\mu$ and who consider s acceptable. However, since $\mu$ is stable, it follows that s prefers $\mu (s)$ to obtaining some students who are unmatched under $\mu$ and who consider s acceptable. Hence, s cannot possibly gain from any type of deviation.

No student i who remains unmatched under $\mu$ can gain from a deviation. Since $\mu$ is stable and schools report preferences truthfully, any unmatched student must be less preferred by the schools than the students accepted by the school in the first round of the Boston mechanism. If a school accepts some student i who is unmatched under $\mu$, this contradicts the stability of $\mu$. Similarly, any student i who is matched to a school under $\mu$ cannot gain from a deviation. If such a student ranked another, preferred school s first then that school would not accept the application. Otherwise, if it accepted, this would contradict the stability of $\mu$, since s would have chosen i in the first round even when $\mu (s)$ was available. If some other strategy resulted in i being matched to s, then i ranking s first would also do so. Hence, no other deviation can yield student i being assigned to a school s that is preferred over $\mu (i)$. Hence, no student can gain from a deviation.

1.4 Proof of Proposition 3

Fewer students matched under $({\tilde{P}}_{I},{\tilde{P}}_{S},q)$ than in stable matchings. Consider the following true preferences and capacities:

$$\begin{aligned} \begin{array}{rl} I=\{i_{1},i_{2}\} &{}\quad S=\{s_{1},s_{2}\},\quad q_{s_{1}}=q_{s_{2}}=1\\ P_{i_{1}}:s_{1} &{}\quad P_{s_{1}}:i_{2}\ i_{1}\\ P_{i_{2}}:s_{2}\ s_{1} &{}\quad P_{s_{2}}:i_{1}\ i_{2} \end{array} \end{aligned}$$

It can be easily verified that the unique stable matching under the true preferences is $\{(s_{1},i_{1}),(s_{2},i_{2})\}$. Consider the following strategy profile (assuming capacities are reported truthfully):

$$\begin{aligned} \begin{array}{rl} {\tilde{P}}_{i_{1}}:s_{1} &{}\quad {\tilde{P}}_{s_{1}}:i_{2}\ i_{1}\\ {\tilde{P}}_{i_{2}}:s_{1}\ s_{2} &{}\quad {\tilde{P}}_{s_{2}}:i_{1} \end{array} \end{aligned}$$

Using Lemmas 1 and 2, it follows that $({\tilde{P}}_{I},{\tilde{P}}_{S},q)$ are undominated strategies. The outcome of the Boston mechanism under $({\tilde{P}}_{I},{\tilde{P}}_{S},q)$ is $\{(s_{1},i_{2}),(s_{2},\varnothing ),(\varnothing ,i_{1})\}$, which leaves $i_{1}$ unmatched. In contrast, in the stable matching, $i_{1}$ is matched to $s_{1}$.

It remains to show that $({\tilde{P}}_{I},{\tilde{P}}_{S},q)$ constitutes a Nash equilibrium. School $s_{1}$ gets its most-preferred outcome, so have no incentive to deviate. Student $i_2$ is deemed unacceptable by school $s_2$ under ${\tilde{P}}_{s_{2}}$, and therefore cannot deviate and be matched there. Since $i_{2}$ is matched to $s_{1}$ in the first round of the Boston mechanism, there is no report that $s_{2}$ could make to obtain either $i_{2}$ or $i_{1}$. Similarly, $i_{1}$ applies to $s_{1}$ in the first round and is rejected. Student $i_{1}$ has no incentive to deviate, since $s_{1}$ is the only school that is acceptable to $i_{1}$.

More students matched under $({\tilde{P}}_{I},{\tilde{P}}_{S},q)$ than in stable matchings. Consider the following true preferences and capacities:

$$\begin{aligned} \begin{array}{rl} I=\{ i_{1},i_{2}\} &{}\quad S=\{ s_{1},s_{2},s_{3}\},\quad q_{1}=2,\quad q_{2}=1\\ P_{i_{1}}:s_{1}\ s_{2} &{}\quad P_{s_{1}}:i_{1}\ i_{2}\ i_{3}\\ P_{i_{2}}:s_{1}\ s_{2} &{}\quad P_{s_{2}}:i_{1}\ i_{2}\\ P_{i_{3}}:s_{1}\ s_{2} \end{array} \end{aligned}$$

It can be easily verified that the unique stable matching under the true preferences is $\{(s_{1},(i_{1},i_{2})),(s_{2},\varnothing ),(\varnothing ,i_{3})\}$. Student $i_{3}$ is thus unassigned in the stable matching. Consider the following strategy profile (assuming capacities are reported truthfully):

$$\begin{aligned} \begin{array}{rl} {\tilde{P}}_{i_{1}}:s_{1}\ s_{2} &{}\quad {\tilde{P}}_{s_{1}}:i_{1}\ i_{3}\\ {\tilde{P}}_{i_{2}}:s_{2}\ s_{1} &{}\quad {\tilde{P}}_{s_{2}}:i_{1}\ i_{2}\\ {\tilde{P}}_{i_{3}}:s_{1}\ s_{2} \end{array} \end{aligned}$$

Using Lemmas 1 and 2, it follows that $({\tilde{P}}_{I},{\tilde{P}}_{S},q)$ are undominated strategies. The outcome under the Boston mechanism of strategy profile $({\tilde{P}}_{I},{\tilde{P}}_{S},q)$ is $\{(s_{1},(i_{1},i_{3})),(s_{2},i_{2})\}$, which has $i_{3}$ being assigned to $s_{1}$.

It remains to verify that $({\tilde{P}}_{I},{\tilde{P}}_{S},q)$ constitutes a Nash equilibrium. Students $i_{1}$ and $i_{3}$ are matched to their most-preferred school, so have no incentive to deviate. Student $i_{2}$ is declared unacceptable under ${\tilde{P}}_{s_{1}}$, so being matched to $s_{2}$ is the best $i_{2}$ can achieve. School $s_{1}$ has no incentive to deviate, since $i_{2}$ is matched to $s_{2}$ in the first round of the Boston mechanism, independent of the preferences reported by $s_{1}$. Similarly, $i_{1}$ is matched to $s_{1}$ in the first round of the Boston mechanism, so $s_{2}$ cannot obtain $i_{1}$ by reporting other preferences. Hence, $s_{2}$ also has no incentive to deviate.