## Abstract

We study school choice markets where the non-strategy-proof Boston mechanism is used to assign students to schools. Inspired by previous field and experimental evidence, we analyze a type of behavior called priority-driven: students have a common ranking over the schools and then give a bonus in their submitted preferences to those schools for which they have high priority. We first prove that under this behavior, there is a unique stable and efficient matching, which is the outcome of the Boston mechanism. Second, we show that the three most prominent mechanisms on school choice (Boston, deferred acceptance, and top trading cycles) coincide when students’ submitted preferences are priority-driven. Finally, we run some computational simulations to show that the assumption of priority-driven preferences can be relaxed by introducing an idiosyncratic preference component, and our qualitative results carry over to a more general model of preferences.

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## Notes

- 1.
An assignment is not stable if there exists a student who prefers a school over her assignment, and with priority at that school over one of the assigned students.

- 2.
This family of mechanisms includes, as a particular case, the Boston and the Deferred Acceptance mechanisms.

- 3.
With the same experimental design but in a constrained school choice environment, Calsamiglia et al. (2010) also find evidence of these misrepresentations.

- 4.
Additionally, the authors find that when markets have multiple stable matchings, approximately 71 % of the stable outcomes are the receiver-optimal stable matchings (priority-optimal stable matchings in our model). Moreover, this last result is not explained by the use of truncation strategies because it is not observed in the experiment substantial deviations from straightforward play in the receiving side of the market.

- 5.
We will also use the notation \(i \succ _s j\) to indicate that

*i*has higher priority than*j*at*s*. - 6.
Throughout the paper we will mention if we consider true or submitted preferences when discussing stability or efficiency.

- 7.
In our model, the hypothesis that students submit priority-driven preferences is a behavioral assumption. However, Calsamiglia and Miralles (2016) using a theoretical analysis, show that in presence of a bad school, and under some conditions on the distribution of capacities for schools, the unique Nash equilibrium is such that each student applies and is assigned to her safe school which Calsamiglia and Miralles (2016) call the neighborhood school.

- 8.
If \(S^F_i=\emptyset \), define \({\tilde{s}}_i=i\).

- 9.
We are grateful to an anonymous referee for suggesting us the equivalence with the application-rejection mechanisms.

- 10.
When preferences are priority-driven, it is never the case that a student applies to a school with no empty seats and where an assigned student has lower priority than her. Thus, the Boston mechanism is also equivalent to an alternative version by Dur (2015), Harless (2015), and Mennle and Seuken (2015).

- 11.
- 12.
Note that the draw of \(\alpha _s\) determines the order and the preference intensity for schools.

- 13.
Simulation code is available on request.

- 14.
Note that the figures do not represent the fraction of students who top rank their safe school as this is only one case where students’ submitted preferences are priority-driven.

## References

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## Author information

### Affiliations

### Corresponding author

## Additional information

We are grateful to Federico Echenique for suggesting us the computational simulations. We also thank to Estelle Cantillon, Li Chen, Alvaro Forteza, Antonio Miralles, Gilles Grandjean and Wouter Vergote for their comments and suggestions, as well as participants at the 7th Workshop Matching in Practice, Economics Department—FCS Uruguay and CEREC—Facultés universitaires Saint-Louis. Finally, we are very grateful to the Editor, Atila Abdulkadiroğlu, and two anonymous referees for their suggestions, which have led to a much improved version of the paper. Pereyra gratefully acknowledges financial support from ERC Grant 208535.

## Appendix

### Appendix

### Proof of Proposition 1

### Proof

Note first that when *Q* is priority-driven, at any stable matching with respect to *Q*, every student is assigned to a school which is weakly preferred to the most preferred safe school for the student according to *Q*. For future use, we summarize this observation in the following remark: \(\square \)

### Remark 1

If \(\mu \) is a stable matching with respect to *Q* and *Q* is priority-driven, then \(\mu (i) R_i s\) where \(s = max_{Q_i} S^F_i\).

We will construct a stable matching, and prove by induction that for all students their assignment at any stable matching is the same as in the constructed matching.

*Induction basis* Consider the first choice submitted by students. We can divide the set of students into those who top rank a safe school, and those who top rank \(s_1\). Note that those whose safe school is \(s_1\) but do not top rank it, have another safe school which is declared as most preferred. Given Remark 1, at every stable matching, students in the first set are assigned to the safe school that they top rank. So, assign these students to their top choices, and adjust school capacities accordingly. For the students in the second set, assign students to \(s_1\) following its priority and the adjusted capacity.

### Claim 1

Each of the assigned students in the second set receives the same school (\(s_1\)) at every stable matching.

### Proof

Consider a student *i* assigned to \(s_1\), but for whom \(s_1\) is not safe. If *i* receives another school in a stable matching \(\mu \), then there is a student *j* ranked higher than *i* at \(s_1\) who is assigned to \(s_1\) at \(\mu \) but not in the procedure we are considering (otherwise, \(\mu \) cannot be stable). Note that *j* first submitted choice is not \(s_1\), because in that case she is assigned to \(s_1\) under the procedure we are describing. Therefore, *j*’s first choice is one of her safe schools, and then she is in the first set previously mentioned. Therefore, she is assigned to the same school at every stable matching, which is a contradiction. \(\square \)

*Induction hypothesis* All students assigned to their \(1,\ldots ,k-1\) choices receive the same assignment at every stable matching.

*Induction step* For those students who are not assigned, consider their *k*-th choice. These students do not submit any of their safe schools in the first \(k-1\) positions, so the first \(k-1\) positions of their submitted preferences are \((s_1,\ldots ,s_{k-1})\). Thus, a this point schools \(s_1,\ldots ,s_{k-1}\) have filled their capacities with students who receive the same assignment at every stable matching (by the induction hypothesis). As before, divide students into those whose *k*-th choice is one of their safe schools, and those whose *k*-th choice is \(s_k\). For the students in the first set, given Remark 1, the school declared as their *k*-th choice is their most preferred safe school, and then they are assigned to it at every stable matching. So, assign these students to their *k*-th choices, and adjust school capacities accordingly. For the remaining students, assign them following schools’ priorities and adjusted capacities.

### Claim 2

Each of the assigned students in the second set receives the same school (\(s_k\)) at every stable matching.

### Proof

Consider a student *i* assigned to \(s_k\), but for whom \(s_k\) is not safe. If *i* receives another school in a stable matching \(\mu \), then there is a student *j* ranked higher than *i* at \(s_k\) who is assigned to \(s_k\) at \(\mu \) but not in the procedure we are considering (otherwise, \(\mu \) cannot be stable). Note that *j* was not assigned previously at this step and that her *k*-th submitted choice is not \(s_k\), because in that case she is assigned to \(s_k\) under the procedure we are describing. Therefore, *j* ranks one of her safe schools at position *k*, and then she is in the first set previously mentioned. Therefore, she is assigned to the same school at every stable matching, which is a contradiction. \(\square \)

Note that the last reasoning also applies to DA. In particular, no tentatively accepted student is subsequently rejected. That is, it is never the case that a student *i* who was tentatively assigned to a school *s* at step *t*, is rejected at a later step \(t'>t\). To see this, assume there is a student *j* who applies to *s* at a \(t'>t\), causing the rejection of *i* form *s*. Clearly, *s* is not a safe school of *i*, and neither of *j*. Given that *Q* is priority-driven, *j* declares *s* in a lower position in \(Q_j\) than *i* does in \(Q_i\). This is only possible, if *j* declares one of her safe schools higher than *s* in \(Q_j\). Thus, before applying to *s*, *j* applied to one of her safe schools and was rejected, which is a contradiction. This implies that the outcome of the Boston mechanism is the unique stable matching under *Q* and, in particular, that this matching is efficient.

### Proof of Proposition 2

### Proof

The proof follows directly from noting that the logic of Proposition 1 applies to all application-rejection mechanisms. Indeed, since no tentatively accepted students are subsequently rejected during the execution of the Deferred Acceptance mechanism, neither are they during the execution of any other mechanism in the family. \(\square \)

### Proof of Proposition 3

### Proof

We will prove that the TTC outcome coincides with the outcome of the DA.

During the execution of the TTC every student is assigned to her safe school. Indeed, suppose there is a cycle at the first step of the algorithm longer than 2. Without loss of generality assume the length is 4, that is, there is a cycle \((i,s,j,s')\). Then, *s* is the safe school of student *j*. But *i* declares *s* in her submitted preferences in a higher position than *j* does, and *s* is not a safe school for her (given that \(s'\) is the safe school of student *i*). This contradicts that preferences are priority-driven. So, in the first step all cycles are of length 2, and each student who leaves the market is assigned to her safe school.

Suppose that until step *k* all students that left the market were assigned to their safe school. We claim that every school which is still in the market at step *k*, points to a student for whom the school is safe. If this not the case, then there is a school such that one of the student for whom the school is safe was assigned previously to another school. But this contradicts the assumption that students that left the marker were assigned to their safe school (which is unique by assumption). As before, suppose there is a cycle \((i,s,j,s')\). Then, *s* is the safe school of student *j*, and \(s'\) of *i*. If *s* is preferred to \(s'\) in the common ranking of schools, \(s O s'\), then it cannot be that *j* prefers \(s'\) to *s* in her submitted preferences. If \(s' O s\), then it cannot be that *i* prefers *s* to \(s'\) in her submitted preferences. Thus, every cycle at step *k* is of length 2.

Given that each student is assigned with TTC to her safe school, the matching coincides with DA (and is stable) by Proposition 2. \(\square \)

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### Cite this article

Cantala, D., Pereyra, J.S. Priority-driven behaviors under the Boston mechanism.
*Rev Econ Design* **21, **49–63 (2017). https://doi.org/10.1007/s10058-017-0197-5

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### Keywords

- Two-sided many-to-one matching
- School choice
- Boston algorithm
- Manipulation strategies
- Deferred acceptance algorithm

### JEL Classification

- C72
- D47
- D78
- D82