DRIVEN BY PRIORITIES MANIPULATIONS UNDER THE BOSTON MECHANISM ´ PEREYRA DAVID CANTALA AND JUAN SEBASTIAN

Abstract. Inspired by real-life manipulations used when the Boston mechanism is in place, we study school choice markets where students submit preferences driven by priorities; that is, when students declare among the most preferred those schools for which they have high priority. Under this behavior assumption, we prove that the outcome of the Boston mechanism is the school-optimal stable matching. Moreover, the condition is necessary: if the outcome of the Boston mechanism is the school-optimal stable matching, then preferences are driven by priorities. Thus, under these manipulations, the final allocation of students may be purely shaped by schools’ priorities. Additionally, we run some computational simulations to show that the assumption of driven by priorities preferences can be relaxed by introducing an idiosyncratic preference component, and our main results hold for almost all students.

December 14, 2014 Keywords: Two-sided many-to-one matching; school choice; Boston algorithm, manipulation strategies, Deferred Acceptance algorithm, Top trading cycles. JEL Classification: C72; D47; D78; D82. 1. Introduction Centralized school choice programs are aimed at expanding the capacity of families to choose the school their children will attend. Before allocating students to schools, families can express their preferences by submitting a rank order list of schools to the central clearinghouse, and when a school is overdemanded, priorities are used to resolve We are grateful to Federico Echenique for suggesting the computational simulations. We also thank to Estelle Cantillon, Li Chen, Alvaro Forteza and Antonio Miralles for their comments and suggestions, as well as participants at the 7th Workshop Matching in Practice and dECON-Uruguay. Pereyra gratefully acknowledges financial support from ERC grant 208535. Cantala is affiliated with the Centro de Estudios Econ´ omicos at El Colegio de M´exico; Pereyra is affiliated with ECARES-Universit`e Libre de Bruxelles and F.R.S.-FNRS, emails [email protected], [email protected]. 1

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ties. An important decision faced by participants is about the preferences they will submit. For example, when the Boston mechanism is in place, their best strategy may not be to submit truthfully but manipulate the mechanism with unfaithful preferences. Previous studies (see Section 1.1 for a detailed description) have shown that many families misrepresent their preferences using “district school bias”: they declare those schools where they have high priority in a higher position than in the true preference. In this paper we show that under these preferences misrepresentations the final allocation may be purely shaped by schools’ priorities. By doing so, we give new theoretical insights into regularities that have been found in previous empirical and experimental papers. Given a profile of schools’ priorities, we study a class of preferences called driven by priorities. To define this domain of preferences we consider the safe school of each student, that is, the school where the position of the student in the school priority order is lower than the quota of the school. Then, if a student declares a safe school as her first option, she will be assigned to it. Submitted preferences are driven by priorities if every student declares her safe school as acceptable, and whenever a student submits her safe school in a lower position in her preferences than another student for whom the school is not safe, the second student declares as preferred her safe school over the previous one. We analyze those markets where the Boston mechanism is used and students’ submitted preferences are driven by priorities. In particular, we assume that each student has only one safe school which can be considered as the student’s district school. Although we are aware that this assumption does not hold for all school choice markets, there are many cases where it does, for example, the Charlotte-Mecklenburg School Public School District in North Carolina (Hastings, Kane, and Staiger (2008)), Denver Public Schools, and the secondary school choice market in Scotland (see the online appendix for a discussion on the extensions of our results to the general case).1 Moreover, the assumption is used in many experiments (see, for example, Chen and S¨onmez (2006), Calsamiglia, Haeringer, and Klijn (2010), and Klijn, Pais, and Vorsatz (2013)). In this framework, we present two main results. 1

Manlove, David (2012), Matching Practices for Primary and Secondary Schools- Scotland, matchingin-practice.eu, accessed 18.11.2014

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First, we prove that when students submit preferences driven by priorities the outcome of the Boston mechanism is the best stable matching for schools (the school-optimal stable matching). Moreover, the condition is necessary: if the outcome of the Boston mechanism is the school-optimal stable matching, then preferences are driven by priorities. It is well known that the assignment of the Boston mechanism may not be stable under the reported preferences (that is, it may exist a student that prefers a school over her assignment, and with priority at that school over one of the assigned students). Our first result shows that if students manipulate the Boston mechanism by submitting preferences driven by priorities, the matching found by the mechanism is stable. Thus, in this case students receive their “worst” stable assignment. However, given that the outcome is stable, families may not have complains about the “fairness” of the allocation. Second, we analyze a situation where the Boston mechanism is replaced by a stable mechanism (like the student-proposing Deferred Acceptance or Deferred Acceptance for short), or by the top trading cycles mechanism. We observe that, even though in general each student interacts only once with the school choice mechanism, experiences and resulting lessons derived from previous interactions are very likely to spread among families over time. This situation holds specially when the mechanism is not strategyproof: strategies that have been proved “successful” in the past, may be replicated by new families when they submit their preferences over schools.2 In particular, this is one of the main findings of Ding and Schotter (2014) that experimentally shows the impact of intergenerational advice that is passed from one year cohort to the next.3 Then, it is unlikely that when a strategy-proof mechanism replaces another that participants were used to manipulate, agents change instantaneously and begin to submit their true preferences. Our second result shows that when students’ submitted preferences are driven by priorities, then the three mentioned mechanisms (Boston, Deferred Acceptance, and top trading cycles) coincide in the allocation of students to schools. Thus, if there exists a

2

For example, there are parents groups in the US aimed at giving advices to participants who enter the market for the first time, based on their own experiences. This was the case of the West Zone Parent Group in Boston (Abdulkadiroglu, Pathak, Roth, and Sonmez (2006)). 3 See also Ding and Schotter (2013) for the effects of chatting between families on the chosen strategies.

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persistent behavior, and students use the strategies indicated by the folk understanding, the change of the mechanism will not produce large modifications of the assignment.4 Our last finding gives a theoretical explanation to some empirical results found by Abdulkadiroglu, Pathak, Roth, and Sonmez (2006). These authors compare the outcomes of the Boston, Deferred Acceptance and top trading cycles mechanisms, considering the reported preferences under the Boston mechanism during the period 2001-02. They show that there are no significant differences between the three mechanisms, and conclude: “Thus if stated preferences change only slowly following the change to a strategy-proof mechanism, we should not anticipate large changes to the assignment in the meantime.”5 Finally, we investigate the extent to which our results hold when we relax the definition of preferences driven by priorities. In particular, we show that as submitted preferences tend to be driven by priorities (in a precise sense that we will define), the difference between the Boston matching and the school-optimal stable matching tends to zero. The same is true when we compare the Boston and the DA matchings. Moreover, computational simulations show that we can relax the definition by allowing non trivial amounts of idiosyncratic shocks in students’ preferences, and our main results hold for almost all students.

1.1. Related Literature. In the remaining of this section we present empirical and experimental papers that show that schools’ priorities are the main driving force behind manipulation strategies. For example, before 2005 some groups of parents in Boston gave the following advice to families: identify one non-popular school (that is, one with low demand in previous periods) which you like, and declare it as their first option, or put a popular school as the first choice, and then submit a safe school as you second option, that is a non-popular school where you are very likely to be accepted (Pathak

4

A related question is explored by Kagel and Roth (2000) in a laboratory experiment that studies the transition between a decentralized and a centralized market. In particular, they show that agents do not respond automatically to changes in the organization of the market, and present some inertia respect to their previously behavior. 5 Abdulkadiroglu, Pathak, Roth, and Sonmez (2006), page 17.

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and S¨onmez (2008)). Abdulkadiroglu, Pathak, Roth, and Sonmez (2006) based on microlevel datasets from the Boston Public Schools, find that some families submitted their preferences strategically following the previous advises. Calsamiglia and G¨ uell (2014) conduct an empirical investigation in Barcelona, where the Boston mechanism is used. The paper exploits an unexpected change in the neighborhood definition in the school choice system. They find that many families apply to the school that they consider to be safe, and more precisely, families declare as their most preferred school the one where they have the highest priority. Thus, safety plays a crucial role in students’ submitted preferences. Chen and S¨onmez (2006) conduct an experiment to analyze agents’ behavior when the Boston mechanism is used. Their findings show that two-third of the agents misrepresent their preferences using “district school bias”, which means that each participant declares her district school (where she has high priority) into a higher position than that in the true preference order. With the same experimental design but in a constrained school choice environment, Calsamiglia, Haeringer, and Klijn (2010) also find evidence of these misrepresentations. In a different experiment, Pais and Pint´er (2008) find the same manipulations, and moreover, when each participant only knows her own preferences, schools’ capacities, and the favorite candidates of all schools, up to their capacities, a substantial proportion of the subjects misrepresent their preferences using “district school bias”, both under the Deferred Acceptance and the Boston mechanisms. Recently, Chen, Jiang, Kesten, Robin, and Zhu (2013) conduct a large scale school choice experiment, and show that a high proportion of agents (more than one half) use the district school bias type of manipulations. Chen and Kesten (2013) find similar evidence of these manipulations. Finally, Echenique, Wilson, and Yariv (2013) experimentally study the Deferred Acceptance mechanism and find that participants instead of acting truthfully, “skip” down their true preferences. That is, when making a proposal decision, participants take into consideration how participants on the other side of the market perceive them. In our framework, this behavior implies that students look at their priority at each school to decide about the preferences they will submit. Additionally, these authors find that when

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markets have multiple stable matchings, approximately 71% of the stable outcomes are the receiver-optimal stable matchings (school-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. As Proposition 1 shows, this last finding can be explained by the fact that the proposer side of the market submits preferences which are driven by priorities. The organization of the paper is as follows. The next section introduces the model and the mechanisms. Section 3 includes the main results, and Section 4 presents the robustness checks of our main results. We conclude with some final remarks in Section 5. 2. The Model We consider a school choice problem (Abdulkadiro˘glu and S¨onmez (2003)), where students have to be assigned a seat at one school. Let S be a finite set of schools and I a finite set of students, a generic school is denoted s and a generic student by i. Each school s has a finite number of available seats (capacity) denoted by qs , and let q = (qs )s∈S be the vector of capacities. Since attendance to school is compulsory, we assume that P schools’ total capacity is equal to the number of students (that is, s∈S qs = |I|). For each school there is a strict priority order (a complete, transitive, and antisymmetric relation) of all students. Denote by s the strict priority order of school s; the relation i s j means that student i has priority over student j for school s. A priority profile, that specifies a strict priority order for each school, is denoted by = (s )s∈S . Let Pi be the preferences of student i defined over S ∪ {i}, and Ri be the at-leastas-good-as relation associated with Pi . The relation sPi s0 means that student i prefers attending school s over school s0 , and sPi i that the student prefers a seat at s to be unassigned (if iPi s we say that school s is unacceptable for student i). P−i denotes the preferences of all students different from i, and let P = (Pi )i∈I denote a profile of students’ preferences. For every student i and every school s acceptable for i, Pi (s) is the position of school s in the preferences of student i. For example, if school s is the first option of student

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i, then Pi (s) = 1. Symmetrically, s (i) is the position of student i in the priority order of school s. A safe school for student i is a school s such that s (i) ≤ qs . As we discussed in the Introduction, we assume that each student has only one safe school, which can be considered as the student’s district school. Thus, for each student i there exists one and only one school s such that s (i) ≤ qs .6 When a centralized mechanism is used to assign students to schools, students have to report their preferences to the central clearinghouse. We denote by Q = (Q)i∈I the profile of preferences submitted by students, and by Qi (s) the position of the acceptable school s in the preferences submitted by student i. A school choice problem is a tuple (I, S, P, , q). We fix throughout this paper I, S and q, thus a school choice problem is described by (P, ). A matching is a function µ : I → S ∪ I such that, if µ(i) 6∈ S then µ(i) = i, and |{i ∈ I, µ(i) = s}| ≤ qs for every s. Let M be the set of all possible matchings. A matching is stable if no student prefers being unassigned to her assigned school, and whenever a student prefers another school to her own, she has lower priority at that school than the assigned students, and there is no empty seat at that school. Formally, a matching µ is stable if: (1) µ(i)Ri i for every i ∈ I, (2) there is no pair (s, i) ∈ S × I such that sPi µ(i) and i s j for some j such that µ(j) = s, and (3) if a student i is such that sPi µ(i) for some s, then |{j ∈ I, µ(j) = s}| = qs . A matching is efficient if there is no other matching such that all students are weakly better off, with one of them being strictly better off: µ is efficient if there is no other matching υ ∈ M such that υ(i)Ri µ(i) for all i, and υ(j)Pj µ(j) for at least one student j. We have defined stability and efficiency considering true students’ preferences, but the same concepts can be defined respect to submitted preferences.7 6In

the online appendix we present a discussion on the extensions of our results to the general case. the paper we will mention if we consider true or submitted preferences when discussing about stability or efficiency. 7Throughout

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A mechanism is a function φ such that for each profile of submitted preferences Q, and priority orders , φ(Q, ) ∈ M. We denote by φi (Q, ) ∈ S ∪ {i} the assignment of student i when mechanism φ is used. A stable (efficient) mechanism is a mechanism that associates a stable (efficient) matching (with respect to submitted preferences) for every profile of submitted preferences and priorities. A mechanism is strategy-proof if it is a weakly dominant strategy for students to report their true preferences; formally, φ is strategy-proof if for all i ∈ I, Pi , Qi , and Q−i , φi (Pi , Q−i , )Ri φi (Qi , Q−i , ). In the remaining of this section we introduce the mechanisms that we study. 2.1. Mechanisms. All the properties we mention in this section are defined considering the submitted preferences. The first mechanism we consider is based on the student-proposing Deferred Acceptance algorithm (Gale and Shapley (1962)). The algorithm runs as follows: Step 1: Each student proposes to her top-choice school. When a school s receives more than qs proposals, it tentatively accepts proposers one at a time following the priority order up to its quota. The other proposals are rejected. In general, Step k: Any student rejected in the previous step makes a new proposal to her most preferred school that has not yet rejected her. Each school considers the students it has been holding and the new proposers. Then, it tentatively accepts proposers one at a time following the priority order up to its quota. The other proposals are rejected. The algorithm finishes when no rejections are made, the final outcome is the matching where each school accepts the students whose proposals it is “holding.” The Deferred Acceptance (DA) mechanism denoted by φDA associates each profile of submitted preferences and priorities to the outcome of the student-proposing Deferred Acceptance algorithm. The outcome of the DA algorithm is stable, and it is at least as good as any other stable matching for all students though it may not be efficient. Moreover, the mechanism is strategy-proof (Roth (1985)).

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By reversing the roles of students and schools (proposals are made by schools to students), we obtain a stable and potentially different matching, which is the best stable matching for schools, and the worst stable matching for students (called the schooloptimal stable matching). The second mechanism we consider is based on the Boston algorithm ( Abdulkadiro˘glu and S¨onmez (2003)), and runs as follows: Step 1: Only the top choice of the students are considered. For each school, consider the students who have listed it as their top choice and assign seats of the school to these students, one at a time, following school’s priority order until either there are no seats left or there is no student left who has listed it as her top choice. In general, Step k: Consider the remaining students. For each school still with available seats, consider the students who have listed it as their kth choice and assign the remaining seats to these students one at a time following school’s priority order until either there are no seats left or there is no student left who has listed it as her kth choice. The algorithm terminates when each student is assigned a seat or all submitted choices are considered. The Boston mechanism denoted by φB associates each profile of submitted preferences and priorities to the outcome of the Boston algorithm. The major drawbacks of this mechanism are that it is not stable, nor strategy-proof, so students have incentives to manipulate the mechanism. Nonetheless, the mechanism is efficient. The last mechanism is based on the top trading cycles algorithm which was introduced by Abdulkadiro˘glu and S¨onmez (2003) as an extension of Gale’s top trading cycles procedure described originally in Shapley and Scarf (1974). The mechanism runs as follows: Step 1: Assign a counter for each school that keeps track of the number of seats that are still available at the school. Initially set the counters equal to the capacities of the schools. Consider the following directed graph in which the set of nodes is the set of students and schools. The set of edges is defined as follows. Each student points to her

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favorite school. Each school points to the student who has the highest priority. There is at least one cycle in this graph. (A cycle is an ordered list of distinct schools and distinct students (s1 , i1 , s2 , . . . , sk , ik ) where s1 points to i1 , i1 points to s2 ,. . . , sk points to ik , and ik points to s1 ). Every student in a cycle is assigned a seat at the school she points to and is removed from the market. The counter of each school in a cycle is reduced by one and if it reduces to zero, the school is also removed. In general, Step k: Each remaining student points to her favorite school among the remaining schools and each remaining school points to the student with highest priority among the remaining students. There is at least one cycle. Every student in a cycle is assigned a seat at the school she points to and is removed. The counter of each school in a cycle is reduced by one and the school is also removed when it reduces to zero.8 The algorithm ends when all students are assigned a school or students’ preferences have been considered. The top trading cycles (TTC) mechanism denoted by φT T C associates each profile of submitted preferences and priorities to the outcome of the TTC algorithm. As Abdulkadiro˘glu and S¨onmez (2003) show, the TTC mechanism is efficient and strategyproof. 3. Results The Boston mechanism induces a game where each student has to submit some preferences over schools, and the mechanism computes the final allocation of students. The information available to students includes the priority order and the quota of each school. Truthful revelation is not always an equilibrium strategy in this revelation game, so students may have incentives to manipulate the mechanism by submitting preferences different from their true preferences. The evidence presented in the Introduction suggests that in some school choice markets students declare their safe schools in a higher position than that in the true preferences. The following definition captures this idea.9 8If

at some step a student prefers to being unassigned over all remaining schools, she points to herself and forms a cycle of length 1. 9In our model, the hypothesis that students submit preferences driven by priorities is a behavioral assumption. However, Calsamiglia and Miralles (2012) using a theoretical analysis, show that in presence

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Definition 1. Consider a priority profile . A profile of preferences Q is driven by priorities if for all i, j ∈ I and s, s0 ∈ S, such that s is a safe school for i and s0 for j, with s 6= s0 : (1) s Qi i, and (2) if Qi (s) > Qj (s), then Qj (s0 ) < Qj (s). The first condition requires that no student declares as unacceptable her safe school. As for the second condition, no student submits her safe school, say s, in a position lower than another student for whom s is not safe, otherwise the second student submits her safe school in a position higher than the position of s. Then, when a school is/is not the safe school of two students, there is no restriction on its position in their submitted preferences. Note that the definition relates submitted preferences with priorities and leaves room for other elements that may also influence students’ preferences. For example, two students may consider a school as the best one and report it as their first option, independently of the fact that for only one of them the school is safe. 3.1. The effects of driven by priorities manipulations. There are many Nash equilibria in the revelation game induced by the Boston mechanism. Since Ergin and S¨onmez (2006) we know that the set of Nash equilibrium outcomes of the revelation game induced by the Boston mechanism is equal to the set of stable matchings under the true preferences. Although a preference profile driven by priorities may not be a Nash equilibrium, in this section we show that when students manipulate the mechanism by reporting preferences driven by priorities, the matching selected by the mechanism is the school-optimal stable matching under the true preferences.10 The result could explain why the Boston mechanism continues to be widely used in many school districts: its outcome is stable, so, even though it may produce the worst 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 (2012) call the neighborhood school. 10Given that we are assuming that no student has more that one safe school, the school-optimal stable matching does not depend on students’ preferences. Then, it is the same matching either we consider the true or the submitted preferences.

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stable matching for students, families will consider it as a fair matching, and then they will not find strong arguments to complain about it. Proposition 1. Consider a priority profile  and a profile of preferences Q such that for every student her safe school is acceptable. Then, Q is driven by priorities if, and only if, φB (Q, ) is the school-optimal stable matching. Proof. See Appendix for a proof.



In the last proposition we need to include the assumption that every student declares her safe school as acceptable, otherwise the “if” part of the result does not hold (see Example 2 of the online appendix). Finally, the next example illustrates that the condition that each student has no more than one safe school, cannot be discarded in the last proposition. Example 1. There are four students {1, 2, 3, 4}, and four schools {s1 , s2 , s3 , s4 }, each having only one seat to fill. Students’ submitted preferences and priorities are: 



Q1 Q2 Q3 Q4

  s  2   s  1   

s4

s4

s2

s1

s1



 s4        

s1 s2 s3 s4

  1    2  3

s3

1

2 3

The outcome of the Boston mechanism is:  µB = 

1

2

3

4

s2 s3 s1 s4

 ,

which is different from the school-optimal stable matching:  µSOSM =  

1

2

3

4

s2 s1 s3 s4

 .



 4     

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As it is shown in the Appendix, the following result is a direct corollary of Proposition 1.

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Corollary 1. Consider a priority profile  and a profile of preferences Q driven by priorities. Then, there exists a unique stable matching. 3.2. Changing the Boston mechanism. Recently, important theoretical and empirical results have lead in some markets to switch from one mechanism to another. For example, in 2005 the Boston Public Schools System replaced the mechanism that had been used (the Boston mechanism) by a new mechanism: the student-proposing Deferred Acceptance mechanism. As it is noted by Abdulkadiroglu, Pathak, Roth, and Sonmez (2006), the transition from a non strategy-proof mechanism to another where it is safe for students to state their true preferences, may not produce an immediate response in the behavior of participants. Indeed, it may take some periods before students start to behave truthfully, and during this transition it is very likely that they will try to manipulate the new mechanism as other students did previously. Also, as it is suggested by Ding and Schotter (2014), the advice of previous generations may reinforce non-truthful behaviors even when a strategy-proof mechanism is in place. Moreover, as Echenique, Wilson, and Yariv (2013) show, there is evidence that although the DA mechanism is strategy-proof, agents misreport by submitting preferences which take priorities into account. Similar manipulations have been found also when the TTC mechanism is in place (Guillen and Hing (2014)). The following proposition studies the performance of any stable mechanism and the TTC mechanism, when students follow the same manipulation strategies that were used 11Eeckhout

(2000) presents a sufficient condition for uniqueness of stable matching in the one-to-one matching model called the marriage market. It can be shown that a problem where each school has only one seat and preferences are driven by priorities, satisfies the Eeckhout (2000) condition. However, the other direction is not true. Recently, Romero-Medina and Triossi (2013) introduce a sufficient condition for the existence of singleton cores in matching markets. These authors show that if there are no simultaneous cycles in men and women preferences, there exists a unique stable matching. It is straightforward to show that the existence of simultaneous cycles implies that preferences are not driven by priorities. See also Akahoshi (2013).

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previously with the Boston mechanism. In line with the empirical findings of Abdulkadiroglu, Pathak, Roth, and Sonmez (2006), we prove that in that case the outcome of the three mechanisms coincide. Proposition 2. Consider a priority profile  and a profile of preferences Q driven by priorities. Then: (1) The Boston mechanism is equivalent to any stable mechanism, that is: φB (Q, ) = ψ(Q, ), for any stable mechanism ψ. (2) The Boston mechanism is equivalent to the TTC mechanism, that is: φB (Q, ) = φT T C (Q, ). Proof. See Appendix for a proof.



Finally, it is worth noting that, although learning is an important issue in market design, it has not been emphasized in the research agenda of school choice. One exception is Erev and Roth (2014) that highlights the crucial role of experiences in changing school choice system. As these authors stress, it is important to describe and make explicit the good properties when a new mechanism is implemented. When families are used to manipulate the mechanism, the change to a strategy-proof mechanism may not produce a instantaneous change in their behavior. In that cases, as Proposition 2 shows, the allocation of students may not vary. 4. Robustness analysis When students’ submitted preferences are driven by priorities, the matching found by the Boston mechanism is the school-optimal stable matching, and the Boston mechanism is equivalent to the DA mechanism. In this section, we investigate the extent to which these results hold when this behavioral assumption is relaxed.12 In particular, we simulate markets where submitted preferences tend towards driven by priorities preferences, 12Given

our main motivation, and the real-life case we are interested in, we focus in the section on the equivalence between the Boston and the DA mechanism.

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and we show that the difference between the Boston matching and the school-optimal stable matching tends to zero. The same is true when we compare the Boston outcome and the DA matching. Moreover, the simulations show that we can relax our assumption by allowing an idiosyncratic preference component, and our main results hold for almost all students. For the remaining of this section assume that each student always declares her safe school as acceptable. We based the computational simulations on the following result. Proposition 3. Consider a priority profile  and a profile of preferences Q. Partition the set of students based on the position in which they submit their safe school; specifically, ˜ = (Q ˜ 1, . . . , Q ˜ k ) where each Q ˜ i is a subset of preferences defined as follows: Q ˜i define Q is the set of preferences of those students who declare their safe school in a lower position in their preferences than the position of the safe school of the students with preferences ˜ j for j < i. Then, Q is driven by priorities if, and only if, Q ˜ is such that: in Q ˜ 1 submits her safe school as the most preferred. (1) Each student i such that Qi ∈ Q ˜ 2 , the schools preferred over her safe school, (2) For each student i such that Qi ∈ Q ˜ 1. is the safe school of a student with preferences in Q ˜ 3 , the schools preferred over her safe school, (3) For each student i such that Qi ∈ Q ˜1 ∪ Q ˜ 2. is the safe school of a student with preferences in Q In general: ˜ j , the schools preferred over her safe school, (4) For each student i such that Qi ∈ Q ˜j . is the safe school of a student with preferences in ∪k


˜ may arise in a market where stuA profile of submitted preferences ordered as in Q dents rank all schools according to an objective criterion, such as the performance of students that have previously attended the school, but differ in the position where they submit their safe school. Then, the last Proposition allows us to construct a family of preferences profiles and to control how close each profile is from a driven by priorities profile. Specifically, we model the utility of each student for attending each school with

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three components. The first component is a common ranking of schools. The second component gives more utility to the safe school of the student. Finally, component 3 is an idiosyncratic component. The value of each component is drawn independently from the uniform distribution over [0, 1]. We can construct a profile of preferences driven by priorities by giving positive weight only to the first two components of the utility (Proposition 3). In that cases, we know that the outcome of the Boston mechanism is the school-optimal stable matching, and that the Boston mechanism is equivalent to the DA mechanism. We will analyze how these results vary as we increase the weight of the third component and decrease the weight of the other two components. Precisely, we construct the preferences of students and priorities as follows. First we order students such that for the first q1 students, school s1 is their safe school, for students indexed by q1 + 1, . . . , q2 , s2 is their safe school, and so forth for the rest of students. Then, for each school there are two sets of students, those for whom the school is safe, and the others students. Within each set, students are randomly ordered. As we have mentioned, preferences are modeled by three components. The first component is a common ranking of schools which is described by a vector (α1 , . . . , α|S| ) such that α1 > . . . > α|S| , where αi is the utility derived from attending school si . The second component is βis which is positive only if s is the safe school of i, and zero for the rest of schools. Finally, the third component depends on the student and the school, and it is denoted by γis . Then, we have the following definition of the utility that student i has from attending school s: uis = λ1 αs + λ2 βis + (1 − λ1 − λ2 )γis where λ1 > 0, λ2 > 0, λ1 + λ2 ≤ 1, βis > 0 if, and only if, s is the safe school of student i, and αi , βis and γis are drawn independently from the uniform distribution over [0, 1], for i ∈ I and j ∈ S. We analyze a market with 10,000 students (40 schools, each one with 250 seats). First, we draw a profile of priorities and utilities for each student and component, and then we consider 861 possible combinations of values for the weights of the components. For

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each preferences profile defined by a pair of weights for the first two components, we run the Boston mechanism, and so as to compare its output with the school-optimal stable matching, and with the DA matching, we compute the number of students that receive a different assignment in each matching. Finally, we simulate 100 markets and we take the average over all these markets.13 4.1. Proposition 1: The outcome of the Boston mechanism is the schooloptimal stable matching. Figure 1 presents the results of the simulations. In the left plot, each curve represents a value of the weight of the first component, and for each value we vary the weight of the second component. The picture we obtain by exchanging the roles of λ1 and λ2 is presented in the right plot. In both plots the first curve from the right corresponds to the lowest value of the parameter, and as we increase its value the curve shifts to the left.

Figure 1. Percentage of students with different assignment in the Boston than in the school-optimal stable matching Two main conclusions arise from the simulations. First, when λ1 and λ2 tend to zero, which means that preferences tend to be fully idiosyncratic, the school assigned 13Simulation

code is available on request.

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to almost all the students by the Boston mechanism is different from the one in the school-optimal stable matching. On the contrary, when the sum of λ1 and λ2 tends to 1, the percentage of students with a different assignment tends to zero. Second, for other combinations of the parameters, we have, for example, that when the three components have approximately the same weight, 44% of the students receive the same school in each of the matchings. When λ1 = 0.15 and λ2 = 0.65, the number of students with the same allocation is 84%. Third, if we fix the weight of one of the first two components, and we increase the other, then the percentage of students with different assignments tends monotonically to zero. When we compare the relative effect of λ1 and λ2 , Figure 1 shows clearly that the component that reflects the driven by priorities part of preferences (λ2 ) has a stronger effect than the other component. To see this, compare the percentage for each possible combination of (λ1 , λ2 ) of the form (x, y) and (y, x). The percentage is lower when the value of λ2 is higher. Thus, the main driving force behind our result is the fact that students submit their preferences emphasizing their safe school, and that they do not heavily rely on the homogeneous preferences assumption. Nevertheless, the homogeneous component of preferences amplifies the first effect. Additionally, we assess the effects of varying the number of students. With this aim we increase the number of schools and the quota of each school, and we compare the allocation of students by the Boston mechanism and under the school-optimal stable matching in each market. Simulations show that as the size of the market increases, for the same combination of values of λ1 and λ2 , the percentage of students with different assignment increases. Then, we can allow for stronger idiosyncratic shocks in those cases when the market is not very populated (see the online appendix for more details). 4.2. Proposition 2: The Boston mechanism is equivalent to the DA mechanism. We run the same simulations as before to study the extent to which our second result holds when we relax the assumption that submitted preferences are driven by priorities. Figure 2 shows the results. As before, in the right plot the first curve from the right corresponds to the lowest value of λ2 , and as we increase its value the curve

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shifts down. In the left plot, the first curve that corresponds to λ1 = 0 is the one that is approximately 10% for λ2 = 0, as λ1 increases the curve becomes steeper.

Figure 2. Percentage of students with different assignment in the Boston than in DA outcome Consider Proposition 2 as a benchmark to compare the results of the simulations. Note first that the situation studied in the proposition corresponds to the case where the sum of λ1 and λ2 is 1. In all these cases, the difference between the matchings is zero. Second, when all the components have approximately the same weight, 66% of the students receive the same school in each of the matchings. Moreover, when λ1 = 0.15 and λ2 = 0.65, the number of students with the same allocation is 88%. This means that we can relax our assumption by allowing an idiosyncratic component of preferences, and the main message of our second result still holds. Third, if we fix the value of λ1 , and we increase the one of λ2 (left plot), the percentage of students with different assignment tends monotonically to zero. The picture when we vary λ1 is different (right plot): when the weight of the component of the homogenous part increases, the difference first increases, and then decreases for high values of λ1 . The reason behind this result is the following: when the weight of the component of the

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homogeneous part of preferences increases, a small perturbation of the common ranking of schools in each student preferences due to idiosyncratic shocks, has larger consequences on the final assignment. Remember that the Boston mechanism strongly depends on the position of each school in each student’s preferences, thus if a student changes the position of one school in her preferences, she may be assigned to that school by the Boston mechanism, but not by the DA mechanism. Note that this effect is stronger as the weight of the second component decreases (see Section 2.2 in the online appendix for a detailed explanation of this effect). Finally, the second component, the weight of each student’s safe school, has a stronger effect on the result than the first component. Thus, as we commented before, the main driving force behind our result seems to be the fact that students put more weight on their safe school.14

5. Concluding Remarks Previous studies on school choice problems have emphasized the importance of students’ preferences to compute the final assignment, because priorities are only used to accept or reject students when a school is overdemanded. Moreover, in general schools do not define their priorities, which are imposed by the school district, based on State and local laws. In this paper, we argue that priorities may be crucial in some school choices markets, in particular, in those markets where students have incentives to submit preferences nontruthfully. An important information that students look at to decide which preferences submit are schools’ priorities. As we have shown, when students prioritize those schools where they have high chances to be admitted, the final outcome may be purely defined

14In

Figure 2 when preferences are fully idiosyncratic (λ1 = λ2 = 0), the percentage of students with different assignment is 10%. This low percentage is explained by the fact that the number of schools is not very large, and each school has a large quota. When considering markets with the same number of students but a larger number of schools with lower quotas, this percentage increases significantly. Nevertheless the same general conclusions hold also in that cases, that is, the difference between the Boston matching and the DA matching tends to zero as preferences tend to be driven by priorities. See Section 2.3 of the online appendix for details.

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by schools’ priorities, and the choice between the classic school choice mechanisms may not have large effects on the final assignment.15 6. Appendix: Proofs Before presenting the main results, we will prove two lemmas. Lemma 1. Consider a priority profile  and a profile of preferences Q driven by priorities. Then φDA (Q, ) is the school-optimal stable matching. Proof. Given that each student has no more than one safe school which she declares as acceptable, the school-optimal stable matching does not depend on students’ preferences. That is, when we run the DA mechanism with schools making the proposals, it stops in the first step. Suppose that φDA (Q, ) is different from the school-optimal stable. So, we have DA a student i is such that s (i) ≤ qs , and φDA i (Q, ) 6= s. Then, φi (Q, ) 6= i,

otherwise the assignment of i at φDA i (Q, ) and at the school-optimal stable matching coincide because by (Roth, 1986), we know that if a student is assigned in one stable matching, then she is assigned in all stable matchings. Thus, there exists a school s1 such that φDA i (Q, ) = s1 , s1 (i) > qs1 . Let student 1 be such that s1 (1) ≤ qs1 and φDA 1 (Q, ) = s2 6= s1 . Student 1 exists, otherwise if all students for whom s1 is a safe school are assigned to that school, it cannot be that i is also assigned to s1 . Given that φDA (Q, ) is stable, then Qi (s) > Qi (s1 ). Also, Q1 (s1 ) ≤ Qi (s1 ) because preferences are driven by priorities (otherwise we should have Qi (s) < Qi (s1 )). Thus, Qi (s) > Qi (s1 ) ≥ Q1 (s1 ). Thus, we have another student 2 such that s2 (2) ≤ qs2 and φDA 2 (Q) = s3 6= s2 . As before, we have that Q2 (s3 ) < Q2 (s2 ) ≤ Q1 (s2 ) < Q1 (s1 ) ≤ Qi (s1 ) < Qi (s). Continuing with this reasoning we construct a list of schools (s, s1 , s2 , . . . , s0 ) such that s0 is one of the schools that has already appeared in the list. Suppose, without loss of generality, 15

A similar message, but with very different arguments, is presented by Calsamiglia and G¨ uell (2014) and Calsamiglia and Miralles (2012).

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that s0 = s. So we have a cycle (s0 ≡ s, i0 ≡ i, s1 , 1, s2 , . . . , sk , k) such that, sl is a safe school of l, and l is assigned to sl+1 , for l = 0, . . . , k (mod k). Also, we have that: Qi (s) > Qi (s1 ) ≥ Q1 (s1 ) > Q1 (s2 ) ≥ . . . ≥ Qk (sk ) > Qk (s) Thus, given that Qi (s) > Qk (s) and that preferences are driven by priorities, we have that k s i. But then both s and sk are safe schools for student k, which is a contradiction.



Under the assumption that submitted preferences are driven by priorities and no student has more than one safe school, the school-optimal stable matching does not depend on students’ true preferences, and then it is stable under both submitted and true preferences. So, we have the following corollary. Corollary 2. Consider a priority profile  and a profile of preferences Q driven by priorities. Then φDA (Q, ) is stable under the true preferences. In the online appendix we present an example (Example 1) that illustrates that a preference profile not driven by priorities can be such that the outcome of the DA mechanism is the school-optimal stable matching. Lemma 2. Consider a priority profile  and a profile of preferences Q driven by priorities. Then, the Boston and DA mechanisms coincide: φDA (Q, ) = φB (Q, ). In particular, the Boston mechanism is stable under Q. Proof. Fix a priority profile . We will show that if φDA (Q, ) 6= φB (Q, ), then submitted preferences Q are not driven by priorities. Thus, consider the first step, t, when the two algorithms differ. Clearly, this step is not the first one. Then, we have two students i, j, and a school s such that, with the Boston mechanism i is rejected from s, and j is accepted, and with the DA, i is accepted from s, and j is rejected. This implies that j made her proposal at an early step t0 < t, and i did it at step t (otherwise, if both

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students apply to the school at t, the outcome of the mechanism coincides). Also, i has a higher priority than j at school s, because she causes j to be rejected from school s under the DA, and j declared s in her preferences in a higher position than i did (because she proposed to s at an early step). That is, we have that s (j) > qs (otherwise, j cannot be rejected from s when we apply the DA), and Qs (j) < Qs (i). Moreover, there is no school s0 such that s0 (j) ≤ qs0 and Qj (s0 ) < Qj (s) (otherwise, j would not make a proposal to s under the DA). If s (i) ≤ qs , submitted preferences are not driven by priorities, and we finish the proof. Suppose s (i) > qs . Given that j is assigned to s when we apply the Boston mechanism (and s is not a safe school for j), there is another student h such that s (h) ≤ qs and she did not apply to s or if she did, she was rejected (under the Boston mechanism). In the last case, submitted preferences are not driven by priorities, and we finish the proof. In the first case, student h is assigned to a non-safe school s0 . So we should have another student to whom s0 is a safe school, and she did not apply to s0 , or if she did it, she was rejected. In the last case, we conclude that submitted preferences are not driven by priorities. In the first case, we repeat the reasoning, and since the set of students is finite, we will find a student who applied to her safe school but she was rejected, and then submitted preferences are not driven by priorities.



Now, we have all the elements to prove Proposition 1. Proof of Proposition 1 Proof. We can restrict to the if part, because the other implication is a direct consequence of the previous two lemmas. Suppose Q is such that φB (Q, ) is the school-optimal stable matching, and let i, j ∈ I and s ∈ S be such that s (i) ≤ qs <s (j), and Qi (s) > Qj (s). Then, j does not apply to s during the execution of the Boston mechanism, otherwise, she will be assigned to s because when i proposes to s which would be in a later step, there are empty seats. This implies that j is assigned to a school which she declares in a higher position than s; but that school should be her safe school (because φB (Q, ) is the school-optimal stable

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matching). Finally, there exists a school s0 such that j ≤ qs0 and Qj (s0 ) < Qj (s), and then preferences are driven by priorities.



Corollary 3. Consider a priority profile  and a profile of preferences Q driven by priorities. Then φB (Q) is stable under P . Proof of Proposition 2 Proof. To prove the equivalence between Boston and any stable mechanism, just note that since there exists a unique stable matching, any stable mechanism will select this matching which it is also the outcome of the Boston mechanism. Regarding the equivalence between Boston and TTC, we will prove that the TTC outcome coincides with the outcome of the DA. First note that during the execution of the TTC all cycles are of length 2, that is, they are always formed by a student that points to a school and by the school pointing to the student. 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, s1 , j, s2 ). Then, s1 is the safe school of student j, and we have that Qi (s1 ) < Qj (s1 ), which contradicts that preferences are driven by priorities. So, in the first steps all cycles are of length 2, and each student who leaves the market is assigned to her safe school. Note that, in the second step, each school that has no filled all its seats, will point to a student to whom the school is safe. The same argument can be applied to the second step, and son on for all the steps of the algorithm. Given that all cycles are of length 2 in the execution of the TTC algorithm, student do not exchange their priorities for schools, and thus the outcome of TTC algorithm is stable. As long as the outcome of the TTC is efficient, then it must be equal to the DA matching.



Finally, we have to note that there is no relation between the fact that the DA and Boston mechanisms coincide and the existence of a unique stable matching. One can easily find an example where there is a unique stable matching but the outcomes of

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the two mechanisms are different. Also, Example 2.4 (page 22) in Roth and Sotomayor (1990) presents a problem for which both mechanisms find the same matching, but there exist multiple stable matchings. Proof of Proposition 3 ˜ verifies the Proof. Suppose first that Q is driven by priorities, we will show that Q property enunciated before. Note that there must exist a student i and a school s such that s is the safe school of i and Qi (s) = 1. Otherwise, consider a student and the school she declares as most preferred. Given our assumptions, that school is the safe school of another student who does not declare it as her first choice; but then Q is not driven by priorities. Now consider a student i with submitted preferences in Q˜j , and let s denote her safe school. If s0 is such that Qi (s0 ) < Qi (s), then those students whose safe school is s0 declare it in a higher position than i does, implying that their submitted preferences belong to a set Q˜k with k < j. To prove the converse, consider two students i, j and be s the safe school of i, such that Qj (s) < Qi (s). Suppose that j declares her safe school s0 in a lower position than s, that is, Qj (s) < Qj (s0 ). This implies that the submitted preferences of i, Qi , belongs ˜ k with k ≤ i, and then, Qi (s) ≤ Qj (s), which is a contradiction. to a set Q  References Abdulkadiroglu, A., P. Pathak, A. E. Roth, and T. Sonmez (2006): “Changing the Boston school choice mechanism: Strategy-proofness as Equal Access,” Discussion paper, National Bureau of Economic Research. ˘ lu, A., and T. So ¨ nmez (2003): “School choice: A mechanism design Abdulkadirog approach,” American Economic Review, 93(3), 729–747. Akahoshi, T. (2013): “Singleton core in many-to-one matching problems,” Manuscript, Waseda University. ¨ ell (2014): “The Illusion of School Choice: Empirical Calsamiglia, C., and M. Gu Evidence from Barcelona,” Manuscript, UAB.

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