“The Boston Public School Match,” and
“Changing the Boston School Choice Mechanism: Strategy-Proofness as Equal Access”
Abdulkadiroglu, Pathak, Roth, and Sonmez.
November 18th, 2008
Speaker: Yusuke Narita
In July 2005 the Boston School Committe votes to replace
the existing Boston school choice mechanism (henceforth Boston mechanism (BM))
with a deferred acceptance mechanism.
This follows 2 years of discussion between officials and researchers and
the decisive factor is whether there is the incentives to "game the system."
Why did they decide to embark on such reform, and
what lessons do we learn from it?
Background & theory
---Boston mechanism --- History & algorithm
---Two alternatives --- Deferred Acceptance & Top Trading Cycles
---Comparison & implication
---Data & summary statistics
---Strategic & unsophisticated behavior
Recommendation, consequence & lessons
---Recommendation and Consequence
---Lesson #1: Empirical difficulty in investigation of strategic play
---Lesson #2: Strategy-proofness as "equal access"
---Lesson #3: Gap in intuitions between economists and non-economists
Boston mechanism --- History
School choice in Boston has been partly shaped by desegregation,
e.g., "busing" ordered since 1974.
In 1999, Boston Public School (henceforth BPS) eliminated racial preferences
and adopted the current mechanism.
Boston mechanism --- Background
In Boston, there are over 60,000 students from grades K-12
in almost 140 schools in 3 three zones, East, North and West.
See Fig.1 in NBER ver.
Most elementary and middle schools are zone schools,
which admit students only from their zone.
Only 5 elementary and middle schools are citywide schools,
to which students from all zones can apply. All high schools are citywide.
Each school's maximum capacity is determined not by each school but by BPS
and the assignment system has 3 rounds
with the majority of students participating in the 1st.
Each winter, students at grades 1, 6, and 9 are asked to
submit a preference ranking of 3-5 schools.1
Boston mechanism --- Algorithm
For most schools, for half of the seats at a given school
the students are priority ordered as follows:
1 Students who are not in the main transition grades continue on in their current school
through guaranteed priority.
1. Students who are guaranteed a space at the school by virtue of
already attending that school or a feeder school (guaranteed priority),
2. sibling & walk zone priority,
3. only sibling priority,
4. only walk zone priority, and
5. other students in the zone. 23
Then, a random # for each student breaks ties in each category. (random tie-breaker)
For the other half of the seats, walk zone priorities do not apply, and
Priorities are based on guaranteed and sibling priority and the random tie-breaker.
Based on preferences, priorities and school capacities,
student assignments are determined by the following algorithm:
Step k: Only the kth choices of the students not previously assigned are considered.
For each school with still available seats, assign the seats to the students
who have listed it as their kth choice following their priority order,
until either there are no seats left or no student left who listed it as her kth choice.
The procedure terminates when every student is assigned a seat at a school,
or if the only students who remain unassigned listed no more.
At the end of the assignment process, if a student is not given any of his choice,
or did not return an application, BPS assigns the student to
the school closest to home that has space.
Boston mechanism --- Remark
・There are 13 special admission high schools and 3 special-education programs
that process applicants separately.
2 After the 1st registration period there is no longer a walk-zone priority.
3 For schools with walk zone priority for half of the seats, the school is treated as two
identical schools, each half the size of the original school, only one of which gives
priority to students from the walk zone.
・Not every residential location in the city has a school for which
they obtain walk-zone priority.
・The BM is one of a class of "priority mechanisms" used to
match medical graduates to positions in the British National Health Service
and eventually abandoned (Roth 1990, 1991).
・Versions are widely used in American school choice systems,
for example in Cambridge, Charlotte-Mecklenberg, Denver, Miami-Dade,
Rochester, Tampa-St. Petersburg, and White Plains.
Boston mechanism --- Problem
The BM is not strategy-proof, i.e., students (or their parents) may improve
their assignments by misrepresenting their preferences.4
It is especially the case when their top choice are overdemanded schools.
・Excellent info., e.g., about which schools are highly sought after
is available at Family Resource Centers, and on BPS websites.
・In fact, there are many signs that both the school district and
families are aware that students may not always want to rank schools truthfully.
(cf. The BPS school guide, and the West Zone Parent Group (WZPG)
More on the Claim
Q: What is essential reason for the fact that the BM not strategy proof?
A1: Endogenous "effective priorities"
4 Def. of "mispresenting" = Submitting a preference list that differs from the true
Sibling, walk zone, and random priorities: Exogenous and only utilized to tie-breaking
"Effective priorities": Endogenous in the sense that each student who ranks a school as
her kth choice is considered before each student who ranks it (k +1)st .
→Ability of students to influence the "effective priorities"
that makes the BM vulnerable to preference manipulation.
A2: "No more than 5 choices"
Is there any other mechanisms which can solve this problem?
→ YES! (Abdulkadiroglu and Sonmez (2003) )
Alternative #1 --- Student-proposing DA mechanism
For a given list of priorities, student preferences and school capacities,
this mechanism determines a student assignment as follows:
Step k : Each student who was rejected in the previous step proposes to
her next choice if one remains. Each school considers the set consisting
of the students it has been holding and its new proposers, and
tentatively assigns its seats to these students one at a time in priority order.
Any students in the set remaining after all the seats are filled are rejected.
The algorithm terminates when no student is rejected, and each student is assigned
her final tentative assignment.
In contrast with the BM, student-proposing DA mechanism is
・stable (no justified envy), i.e., there is no student who loses a seat to a lower priority
student and receives a less-preferred assignment.
・student-optimal, i.e., all students prefer their outcome to
any other stable matching (Gale and Shapley 1962), and
・strategy-proof (Roth 1985) but
・not ex post Pareto efficient.
Alternative #2 --- Top Trading Cycle (TTC) mechanism
For a given list of priorities, student preferences and school capacities
this mechanism determines a student assignment with the following algorithm:
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 1 and if it reaches 0,
the school is removed.
The procedure terminates when each student is assigned a seat or
all submitted choices have been considered.
This version of the TTC mechanism, an extension of
Gale's top trading cycles mechanism (Shapley and Scarf (1974))
introduced by Abdulkadiroglu and Sonmez (2003) is
・ex post Pareto efficient (Shapley and Scarf 1974) and
・strategy-proof (Roth 1982b).
Comparison and implication
If the intention of the school board is that priorities be "strictly enforced,"
the student proposing DA mechanism is a leading candidate.
However, if welfare considerations apply only to students,
there is tension between stability and Pareto efficiency.
(Roth 1982, Balinski and Sonmez 1999, Abdulkadiroglu and Sonmez 2003).
In addition, while TTC is a Pareto efficient when only students are considered, and
the student-proposing DA is not, the former does not Pareto dominate the latter.
To avoid the complications with the transition from the old system, we focus our
empirical analysis on
・the second year of the system, school year 2001-02 and
・students submitting preferences in round 1 in the main transition grades: K2, 6, and 9.
Table 1 presents which stated choice students received from the Boston mechanism.
See Table 1…
Table 2 shows the priority through which students are assigned.
Strategic or unsophisticated behavior?
Note that at least some families are responding to the incentives to
mispresent their preferences.
Now let’s see the pattern that those manipulations predict is also found in the data.
However, because we do not know families' true preferences,
we have to confine our attention here to a small subset of clear mispresenting.
A school is overdemanded
if (# of students who rank that school as their 1st choice) > (# of seats at the school).
No one who lists an overdemanded school as a second choice
will be assigned to it by the Boston mechanism, and
listing an overdemanded school as a second choice can only reduce
the probability of receiving schools ranked lower.
Thanks to its observability, we remove many of the difficulties involved
in assessing empirically the extent of strategic play or its absence
when the true preferences are un-known by concentrating on this strategic mistake.
Two striking examples in Panel A of Table 3 illustrate
the evidence of strategic behavior in a naive sense.
See Panel A of Table 3…
Panel B of Table 3 shows the relationship between the drop
between 1st choice and 2nd choice applicants and
proxies for whether the school was overdemanded in the previous year .
With either proxy, there is a statistically significant relationship
between the preference discontinuity and how overdemanded the school was.
See Panel B of Table 3…
On the other hand, the gap between the 1st and 2nd choices is
either not present or much smaller at overdemanded citywide schools,
suggesting strategically unsophisticated behavior
on the part of many parents interested in citywide schools in a naive sense.
e.g. At the Hernandez Elementary School which has 42 seats,
there are 115 students who rank it 1st, 84 who rank it 2nd, and 90 who rank it 3rd.
Moreover, Table 4 looks at students who did rank two overdemanded schools
as their 1st choices despite the fact that
if the student only has random priority at her 1st choice
so that the odds of receiving it are low:
a majority of all unassigned students submitted a rank order with this property.
Table 4 Panel A shows that these students are disproportionately unassigned, and
that some of them could have been assigned to one of their ranked schools
if they had simply deleted the second overdemanded school
from the preference list they submitted.5
See Table 4 Panel B…
Table 4 Panel B looks at the outcome of students who rank two overdemanded schools
as their 1st and 2nd choice and have only random priority at their 1st choice.
See Table 4 Panel B…
Recommendation and consequence
Based on our theoretical and empirical analysis of the behavior of Boston families,
the researchers think it seems likely that
・Boston families are faced with solving a complex strategic problem,
rather than just a problem of forming preferences over schools
and recommend that a strategy-proof mechanism like the TTC or the DA would
・lift this strategic burden from parents, and
makes the school choice process more transparent and fair in a sense.
・make more reliable submitted preferences as an indicator of families' true preferences.
As a result, in July 2005 the Boston School Committee votes to
replace the existing BM with a DA mechanism.
Two remaining concerns
Concern #1: How smoothly will Boston families learn it is safe to state true preferences?
5 Omitting the 2nd overdemanded choice from the list gives us a lower bound on how
many students suffered ex post from the mistake of listing 2 overdemanded schools 1st.
Concern #2: How accurate do we judge the extent to which
preferences are currently being manipulated?
According to the comparison of the outcomes of the 3 mechanisms
using the stated preferences submitted under the Boston mechanism during 2001-02,
for each year and grade, the outcomes of all 3 mechanisms are very similar
→Probably no substantial change in the performance of the assignment system
Lesson #1 = Empirical difficulty in investigation of strategic play
Because we cannot know true preferences,
investigating strategic play is empirically challenging.
So the fact that Boston school choice system was not broken in an obvious way
becomes one of the challenges.
Lesson #2 = "Starategy-proofness as equal access"
Q: Why is strategy-proofness important?
A1: For students and parents, a strategy-proof mechanism adds clarity
to the assignment process, by allowing for clear advice to parents regarding
how to rank schools. ("List your true choice.")
→A strategy-proof algorithm levels the playing field.
A2: For school officials, they will be able to use the submitted preferences
as indicators of family preferences, to determine which schools are in fact
the most highly regarded, and to estimate the effect of policy changes.
Lesson #3 = Gap in intuitions between economists and non-economists
At first, central to the theoretical discussion of TTC versus student proposing DA
was the tradeoff between stability and efficiency.
However, an additional, related concern has arised, i.e.,
the uneasiness of BPS to allow priorities to be traded, particularly
where sibling priority is involved.
It is the example of a gap in intuitions between economists and non-economists
about what kinds of things should and should not be traded.