Preference Intensities and Risk Aversion
in
School Choice: A Laboratory Experiment
∗
Flip Klijn
†
Joana Pais
‡
Marc Vorsatz
§
April 2012
Abstract
We
experimentally investigate in the laboratory prominent mechanisms that are em-
ployed in school choice programs to assign students to public schools and study how
individual behavior is influenced by preference intensities and risk aversion. Our
main results show that (a) the Gale–Shapley mechanism is more robust to changes
in cardinal preferences than th e Boston mechan ism independently of whether indi-
viduals can submit a complete or only a restricted ranking of the schools and (b)
subjects with a higher degree of risk aversion are more likely to play “safer” strate-
gies under the Gale–Shapley but not under the Boston mechanism. Both results
have important implications for enrollment planning and the possible protection
risk averse agents seek.
Keywords: school choice, risk aversion, preference intensities, laboratory experi-
ment, Gale–Shapley mechanism, Boston mechanism, efficiency, stability, constrained
choice.
JEL–Numbers: C78, C91, C92, D78, I20.
∗
W
e are very grateful for comments and suggestions from the Editor, two referees, Eyal Ert, Bettina
Klaus, Muriel Niederle, Al Roth, and the se minar audiences in Alicante, Braga,
´
Evora, Maastricht, and
M´alaga.
†
Institute for Economic Analysis (CSIC) and Barcelona GSE, Barcelona, Spain; e-mail:
flip.klijn@iae.csic.es. He gratefully acknowledges a rese arch fellowship from Harvard Business
School for academic year 2009-2010 when he was visiting HBS and the fir st dr aft of the paper was writ-
ten. He also gratefully acknowledges support from Plan Nacional I+D+i (ECO2008–04784 and E CO2011–
29847), Generalitat de Catalunya (SGR200 9–01142), and the Consolider-Ingenio 2010 (CSD2006–00016)
program.
‡
Corresponding author. ISEG/Technical University of Lisbon and UECE–Research Unit on Complex-
ity and Economics, Rua Miguel Lupi, 20, 1249-078, Lisbo a, Portugal; e-mail: jpais@iseg.utl.pt. She
gratefully acknowledges financial support from Funda¸c˜ao para a Ciˆencia e a Tecnologia under project
reference no. PTDC/EGE-ECO/113403/2009.
§
Departamento de An´alisis Econ´omico II, UNED, Paseo Senda del Rey 11, 2804 0 Madrid, Spain and
Fundaci´on de Estudios de Econom´ıa Aplicada (FEDEA), Calle Jorge Juan 46, 28001 Madrid, Spain; e-
mail: mvorsatz@cee.uned.es. He gratefully acknowledges financial support from the Spanish Ministry
of Education and Science through the project ECO2009–07530.
1
Post-print of: Experimental Economics, 16(1), 2013, 1-22.
1 Introducti on
In school choice programs parent s can express their preferences regarding the assignment
of their children to public schoo ls. Abdulkadiro˘glu and S¨onmez [5] showed that promi-
nent assignment mechanisms in the US lacked efficiency, were manipulable, and/or had
other serious shortcomings that often led to lawsuits by unsatisfied parents. To overcome
these critical issues, Abdulkadiro˘glu and S¨onmez [5] took a mechanism design approach
and employed matching theory to propose alternative school choice mechanisms. Their
seminal paper triggered a rapidly growing literature that has looked into the design and
performance of assignment mechanisms. Simultaneously, several economists were invited
to meetings with the school district authorities o f New York City and Boston to explore
possible ways to redesign the assignment procedures. It was decided to adopt variants
of the so–called deferred acceptance mechanism due to Ga le and Shapley [14] (aka the
Gale–Shapley mechanism) in New Yo r k City and Boston as of 2004 and 2006, respec-
tively.
1
Since many other US school districts still use variants of what was baptized the
“Boston” mechanism,
2
it is not unlikely that these first redesign decisions will lead to
similar adoptions elsewhere.
3
Chen and S¨onmez [9] turned to cont rolled labo ratory experiments and showed that
the Gale–Shapley mechanism outperforms the Boston mechanism in terms of efficiency if
subjects are allowed to rank all schools. Since parents are only allowed to submit a list
containing a limited number of schools in many real–life instances, Calsamiglia, Haeringer,
and Klijn [8] experimentally analyzed the impact of imposing such a constraint . They
find that manipulation is drastically increased and both efficiency and stability of the
final allocations are negatively affected. Another important issue concerns the level of
information agents hold on the preferences of the others. Pais and Pint´er [19] focused
on this comparing environments where subjects, while aware of their own preferences,
have no info rmation at all about the preferences of their peers. A different approach was
taken in Featherstone and Niederle [12], where subjects may not know the preferences
of the others, but are aware of their underlying distribution. Both papers studied how
strategic behavior is affected by the level of information subjects hold. Featherstone and
Niederle [12] found that truth–telling rates of the two mechanisms are very similar. In
Pais and Pint´er [19], truth–telling is higher under Gale–Shapley only when information
is substantia l, so that the Gale–Shapley mechanism outp erforms the Boston mechanism
only in some informational settings.
The need of reassessing the school choice mechanisms is reinforced by the recent the-
oretical findings in Abdulkadiro˘glu, Che, and Yasuda [1] who showed that the Boston
1
Abdulkadir o˘glu, Pathak, and Roth [2, 3] and Abdulkadiro˘glu, Pathak, Roth, and S¨onmez [4] re ported
in more detail on their assistance and the key issues in the redesign for New York City and Boston,
respectively.
2
That is, the mechanism employed in Bosto n before it was replaced by the Ga le –Shapley mechanism.
3
The literature has also studied other mechanisms. Abdulkadiro˘glu and S¨onmez [5] proposed a mech-
anism based on Gale’s top trading cycles algorithm as a second alternative for the Boston mechanism.
However, we are not aware of school districts that employ this other alternative. More importantly, since
in Boston and New York the Boston mechanism was replaced by Gale–Shapley, our study focuses on the
ongoing debate on Gale–Shapley vs. B oston. For further recent developments on school choice we refer
to Al Roth’s blog on market design.
2
mechanism Pareto dominates the Gale–Shapley mechanism in ex ante welfare in certain
school choice environments. This happens because the Boston mechanism induces par-
ticipants to reveal their cardinal preferences (i.e., their relative preference intensities),
whereas the Gale–Shapley mechanism does not. In view of this and other results, Ab-
dulkadiro˘glu et al. [1] cautioned aga inst a hasty rejection of the Boston mechanism in
favor of mechanisms such as the Gale–Shapley mechanism.
4
Theoretically, whereas t he Gale–Shapley mechanism is strategy–proof (that is, agents
have incentives to report their ordinal preferences truthfully), a student can increase
the likelihood of being assigned a given school by ranking it higher under the Boston
mechanism. That is, the Boston mechanism is manipulable and therefore sensitive to
underlying cardinal preferences and attitudes towards risk. Motivated by these findings,
we experimentally investigate how individual behavior in the Gale–Shapley and Boston
mechanisms is influenced by preference intensities and risk aversion and whether this
affects the performance of the two mechanisms. We opt for a stylized experimental design
that has several importa nt advantages. First, by letting subjects par ticipate repeatedly in
the same market with varying payoffs, we are able to investigate the impact of preference
intensities on individual behavior and welfare. Second, a special feature of our laboratory
experiment is that befo r e subjects participate in the matching markets, they go through
a first phase in which they have to make lottery choices. This allows us to see whether
subjects with different degrees of risk aversion behave differently in the matching market.
Third, the complete information a nd the simple preference structure form an environment
that can b e thought through by the subjects, so that clear theoretical predictions about
how preference intensities a nd risk aversion should affect behavior can be made.
5
Finally,
our setup purposely does not include coarse school priorities in order to avoid possible
problems in entangling the causes of observed behavior.
6
Our main results are as follows. Subjects tend to list a school higher up (lower down)
in the submitted ranking if the payoff of that particular school is increased (decreased)
everything else equal. Moreover, the Gale–Shapley mechanism is more robust to chang es
in cardinal preferences than the Boston mechanism (Result 1). This finding has policy
appeal as robustness implies predictability, a valuable asset in enrollment planning. We
also find that subjects with a higher degree of risk aversion are more likely to play protec-
tive strategies
7
under the Gale–Shapley but not under the Boston mechanism (Result 2).
Ease in recognizing protective strategies may make risk averse agents more comfortable
4
Miralles [17] drew a similar conclusion based on his analy tical results and simulations.
5
On the other hand, since the market we consider in the second phase is small, the results may not
scale up to very large real–life matching markets.
6
Coarse school priorities are a common feature of many scho ol choice environments. Then, in order to
apply the assignment mechanisms, random tie–breaking rules ar e often used. However, the incorporation
of such rules in our design would make it very hard to see whether individuals with different degrees of
risk avers ion behave differently because of strategic uncertainty or because of the random tie–breaking.
In o ther words, we assume that the schools’ priority orders are strict in order to study whether the
behavioral effect of ris k aversion is associated with strategic uncertainty. For the very same r e ason, we
also assume that the induced game is co mmon knowledge even though in practice individuals are likely
to have incomplete infor mation regarding the other participants’ preferences.
7
Loosely speaking, a subject plays a protective strategy if she protects herself from the worst eventu-
ality to the extent possible. Consequently, a protective stra tegy is a maximin strategy.
3
with the Gale–Shapley mechanism.
The remainder of the paper is organized a s f ollows. The experimental design is ex-
plained in Section 2. In Section 3, we derive hypotheses regarding the effect of relative
preference intensities and risk aversion on strategic behavior. In Section 4, we analyze
the impact of changes in cardinal preferences, how risk aversion affects behavior in the
matching market, and the implications of two variables for the welfare pro perties of the
mechanisms. In Section 5, we conclude with some possible po licy implications.
2 Exper i mental D esign and Procedures
Our experimental study comprises four different treatments. Each treatment is divided
into two phases.
In the first phase, which is identical for all treatments, we elicit the subjects’ degree
of risk aversion using the paired lottery choice design introduced by Holt and Laury [16].
Subj ects are presented with a list of ten different choices between two lotteries (see Ta-
ble 10 in Appendix A). Lottery A is less risky than lottery B for the first nine choices, but
lottery B first–order stochastically dominates lottery A for the tenth choice. A rational
individual may choose A at the top of the list, but always chooses B at the bottom,
implying some switching point in between. The switching point, corresponding to the
first time lottery B is chosen, roughly determines the number of safe choices and, in turn,
provides a measure of the degree of risk aversion.
8
In the second phase, subjects face the following stylized school choice problem: There
are three teachers, denoted by the natural numbers 1, 2, and 3, and three schools, denoted
by the capital letters X, Y , and Z, with one open t eaching position each.
9
The preferences
of the teachers over schoo ls and the priority orderings of schools over teachers, both
commonly known to all participants, are presented in Ta ble 1.
Preferences Priorities
Teacher 1 Teacher 2 Teacher 3 School X School Y School Z
Best match X Y Z 2 3 1
Second best match Y Z X 3 1 2
Worst m atch Z X Y 1 2 3
Ta ble 1: Preferences of teachers over schools (left) and priority orderings of schools over teachers (right).
It can be seen in Table 1 that the preferences of the teachers form a Condorcet cycle.
The priority orderings of the schools form another Condorcet cycle in such a way that
every teacher is ranked la st in her most preferred schoo l, second in her second most
preferred school, and first in her least preferred school. The setup is competitive, so that
risk aversion may have a bite, and symmetric to simplify the data analysis.
8
A rational individual may always choose lottery B, in which case the switching point is equal to 1.
9
We “framed” the school choice problem from the point of view of teachers who are lo oking for jobs
because this presentation provides a natural environment that is easy to understand. Fo r ex ample,
material payoffs can be directly interpreted as salaries (see Pais a nd Pint´er [19]).
4
A 2 × 2 between–subjects design is obtained from two treatment variables that ar e
known to be empirically relevant in this type of market. The first treatment variable refers
to the restrictions on the ra nkings teachers can submit. We consider the unconstrained
and one constrained setting. In the unconstrained setting (u), teachers have to report a
ranking over all three schools. In the constrained setting (c), t hey have to report the two
schools they want to list first and second. The second treatment variable refers to how
reported rankings are used by the central clearinghouse to assign teachers to schools. We
apply here both Gale–Shapley’s teacher–proposing deferred acceptance algorithm (GS)
and the Boston algorithm (BOS). For the particular school choice problem at hand, they
are as follows:
Step 1. Each teacher sends an application to the school she listed first.
Step 2. Each school retains the applicant with the highest priority and rejects all other
applicants.
Step 3. If a teacher is rejected a t a school, she applies to the next highest listed school.
Step 4. (The two algorithms only differ in t his step.)
GS: Whenever a school receives new applications, these applications are considered to-
gether with the previously retained application (if any). Among the retained and
the new applicants, the teacher with the highest priority is retained and all other
applicants are rejected.
BOS: Whenever a school receives new applications, all of them are rejected in case the
school already retained an application before. If the school did not retain an appli-
cation so far, it retains among all applicants the one with the highest priority and
rejects all other applicants.
Step 5. The procedure described in Steps 3 and 4 is repeated until no more applications
can be rejected. Each teacher is finally assigned to t he school that retains her application
at the end of the process. In case none of a teacher’s applications are retained at the
end of the process, which can only happen in the constrained mechanisms, she remains
unemployed and gets 0 ECU.
10
Each subject fa ces one of the four treatments and plays the role of a teacher (schools
are not strategic players). The task is to submit a ranking over schools (not necessarily the
true preferences) to be used by a central clearinghouse to assign teachers to schools. This
is done three times, in three games with payoff structures that differ only in the payoff
of the second most preferred school: A subject always receives 30 ECU for her most
and 10 ECU for her least preferred schools, but in the first, second, and third games,
a subject receives 20 ECU, 13 ECU, and 2 7 ECU, respectively, if she obtains a job at
her second most preferred school.
11
To maintain the not ation as simple as possible, we
10
If teachers had to list only one school, the two c onstrained mechanisms would be identical; that
is, for all profiles of submitted (degenerate) rankings, the same matching would be obtained under the
Gale–Shapley and Boston algorithms.
11
Since the payoff of the second most preferred school varies for all subjects, subjects face different
kinds of o pponents in different games. In one alternative design to possibly ove rcome this drawback the
payoff for only one subject (in each group of three subjects) varies. Yet, in this alternative approach, the
subjects with fixed preferences would probably believe that the third subject modifies her strategy due
5
