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Models for School Choice and Empirical Analysis in Boston

With education being an important mechanism for social mobility and career opportunity, school choice remains an important consideration for parents when choosing an area to reside while raising a child. Some school choice mechanisms have been more effective than others, and an important recent example that incorporates many of the matching principles discussed in class is the transition in the Boston Public Schools (BPS) assignment mechanism for kindergarten, middle, and high schools that occurred in 2006.


The Boston school choice transition differs from kidney exchange in that the traditional allocation mechanism appeared an effective “market” on the surface, but in actuality contained significant underlying efficient losses masked by strategic manipulation of preferences that produced politically-attractive outcomes where many students were reported as having received their “first choice.”


The previous mechanism, which is one of the most commonly used for school choice in the United States [2], worked by maximizing the number of students who get their first choice school. This traditional mechanism operated as follows (simplified):


Setup: Let the set of schools be X and the set of students S. Each school x (in X) has a priority order over students of S based on student characteristics such as sibling enrollment or residence within walking distance. A random lottery number is assigned to each student s such that x has strict and complete preferences over all students. Each student s provides a strict preference ranking for up to five schools.


Note that although both students and schools have priority orderings, the school choice model still has many normative qualities of a one-sided market as student outcomes are taken much more seriously relative to school outcomes in the related literature.


Stage i: In this step we only consider the ith best choices of unassigned students. For each school x, consider every unassigned student s for which x is the s’ ith preferred school. Consider qualifying students in order of x’s strict preferences. For each s, if school x still has spots available then assign s to x permanently. Else, x is full and no more students may be assigned to x.


The traditional mechanism follows stages from i=1 to i=5. Empirically, students that remain unassigned are placed into the closest open school by geography. These remaining open schools may be of extremely poor quality.


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School choice received under traditional model. Note the non-trivial number of unassigned students [1].


Although approximately 75% percent of students received their first choice, the traditional mechanism suffers from a considerable drawback that we have discussed regarding mechanisms in class. The traditional mechanism is not Strategy-Proof (SP). We define Stragey-Proofness as inability of any student (agent) to acquire a more preferred assignment in any scenario by deviating his or her reported preferences from his or her true preferences.


Abdulkadiroglu et al. were able to tease out empirical traces of strategic behavior in student self-reported preferences, but more telling was BPS parent groups’ recommendation of certain strategies for improved school choice [1]. In certain scenario, students may report a second or third-choice school as first choice or report a “safety school” as a second choice due to overcrowding in highly desirable schools [1]. This result was troublesome to the BPS due to a lack of “fairness” — differential outcomes between students who could strategize well and those who could not [2] — in addition to the value of acquiring true preference data for determining which schools are truly demanded, and to formulate education policy. The result also suggests significant losses in efficiency [1]. This is not to say the matching produced is not necessarily Pareto-efficient for students given true preferences, but that the matching produced by the traditional mechanism may be Pareto-dominated by some other matching if students lie about their true preferences.


(Example: Suppose we have four schools A,B,C,D, each with one open spot and students S1,S2,S3. S1 has priority order A,B,C,D, S2: A,B,C,D, and S3: A, C, B, D. Suppose S2 and S3 anticipate overcrowding and each put their third choice as a second “safety” choice, as is suggested by the actual BPS data. Tiebreak by alphabetical order. Then we assign S1-A, S2-C, and S3-B. But S2 and S3 can switch and each be better off. Thus, efficiency gains could be had with a system that incentivizes the reporting of true preferences).


Harkening back to content from lecture, a top-trading cycles (TTC) mechanism was strongly considered as a new mechanism for school choice. Recall that in a one-sided market with initial endowments and strict (and complete) preferences, the TTC mechanism produces a unique core matching, implying Pareto-efficiency and Individual Rationality. The TTC mechanism is also SP. For empirically analysis, the TTC algorithm had each student point to their favorite remaining school (and, implicitly, being assigned some initial endowment such that no schools were initially over capacity). The TTC outcome assigned slightly fewer students to their top-choice school, but had more students assigned to second and third-choice schools. If TTC were to be implemented, students would be able to rank an unlimited number of acceptable schools in some strict order for more complete preferences (and strict preferences outside the top five). Abdulkadiroglu et al. found that even with strategic behavior, TTC would still be about as efficient as the traditional mechanism in assignment [1].


This is a scenario, however, where we must take into consideration the intersection of theory and application. The TTC mechanism raised concerns due to difficulty of understanding among the public that could impact the perception of the mechanism’s fairness. The then-SCP superintendent states:


‘[The] Top Trading Cycles Mechanism presents the opportunity for the priority for one student at a given school to be “traded” for the priority of a student at another school, assuming each student has listed the other’s school as a higher choice than the one to which he/she would have been assigned. There may be advantages to this approach, particularly if two lesser choices can be “traded” for two higher choices. It may be argued, however, that certain priorities – e.g., sibling priority – apply only to students for particular schools and should not be traded away. Moreover, Top Trading Cycles is less transparent– and therefore more difficult to explain to parents – because of the trading feature executed by the algorithm, which may perpetuate the need or perceived need to “game the system.”’ [1]


As a consequence, the new mechanism used for BPS was a Deferred Acceptance (DA) mechanism rooted in the theory of Gale-Shapley stable matching that was initially intended for two-sided markets. The DA mechanism functions as a many-to-one matching, and is also Strategy-Proof for students (but not for schools. Again, however, school preferences for students are determined formulaically and whether schools acquire their top-preferred students is not as high of a normative priority in the literature. Thus we can ignore school model concerns).


A simplified description of DA involves each student proposing to his or her top remaining school. Schools may tentatively accept a student, but in a proceeding round may reject a previously accepted student if new students that are more preferred by the school x, but had a lower preference for x and so were considered at a later time, are now considered. Continue until all students are assigned. We have yet to discuss the matching model thoroughly, but the DA mechanism provides an enticing improvement over the traditional model in that it is stable: “no student who loses a seat to a lower priority student and receives a less-preferred assignment” [1]. Thus, the DA mechanism corrects for the traditional model’s additional drawback of possible justified envy.


Although the DA mechanism is not necessarily Pareto-efficient for students in this contexts [1], it produces comparable empirical outcomes to the TTC mechanism and is simpler to understand and justify politically. Further, the DA mechanism corrects for the traditional model’s potentially largest drawback, lack of Strategy-Proofness.


The DA mechanism was implemented in 2006. The case of BPS school assignment is an insightful look at the practical considerations that need to be considered when applying our in-class models, and an important reminder that “fairness” of a mechanism and how “good” a matching is may not always be easily defined. While we may think of PE and IR as the two most important qualities of a mechanism, in this situation Strategy-Proofness was extremely relevant for a variety of policy considerations.


Even when we study optimal mechanisms for matching, however, we must be prepared to understand and defend them. BPS was warned that the implementation of the DA mechanism would produce fewer reported top-choice assignments than the previous top-choice maximizing traditional model, which could come at political cost. In this case, the researchers were able to give compelling reasons to implement TTC or DA based on Strategy-Proofness properties and the mechanisms’ assigning fewer students to an extremely undesirable school if students could not manipulate their reported preferences well.


The recent 2014 announcement of a new school choice plan in Boston provides opportunity for new theoretical and empirical assessment (‘Improving School Choice’, 2015). Among the additions to the school choice mechanism is a wider range of schools from which to pick from — sets of schools had previously been clustered in three non-overlapping regions. This change alone suggest possible efficiency gains to be had, but more research could be conducted on any relevant alterations to the mechanisms and studying the market as a two-sided market, with the perspective and outcomes of individual schools being considered in addition to student outcomes.




[1] A. Abdulkadiroglu et al., ‘Changing the Boston School Choice Mechanism’, NBER Working Paper 11965, 2005.

[2] P. Pathak and T. Sönmez, ‘Leveling the Playing Field: Sincere and Sophisticated Players in the Boston Mechanism’, American Economic Review, vol. 98, no. 4, pp. 1636-1652, 2008.

[3] Improving School Choice, ‘Improving School Choice’, 2015. [Online]. Available:



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