Abstract
We experimentally investigate in the laboratory prominent mechanisms that are employed 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 the Boston mechanism independently of whether individuals 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” strategies 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.
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Notes
That is, the mechanism employed in Boston before it was replaced by the Gale–Shapley mechanism.
The literature has also studied other mechanisms. Abdulkadiroğlu and Sönmez (2003) proposed a mechanism 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. Boston. For further recent developments on school choice we refer to Al Roth’s blog on market design.
Miralles (2008) drew a similar conclusion based on his analytical results and simulations.
On the other hand, since the market we consider in the second phase is small, the results may not scale up to very large reallife matching markets.
Coarse school priorities are a common feature of many school choice environments. Then, in order to apply the assignment mechanisms, random tiebreaking rules are 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 aversion behave differently because of strategic uncertainty or because of the random tiebreaking. In other words, we assume that the schools’ priority orders are strict in order to study whether the behavioral effect of risk aversion is associated with strategic uncertainty. For the very same reason, we also assume that the induced game is common knowledge even though in practice individuals are likely to have incomplete information regarding the other participants’ preferences.
Loosely speaking, a subject plays a protective strategy if she protects herself from the worst eventuality to the extent possible. Consequently, a protective strategy is a maximin strategy.
A rational individual may always choose lottery B, in which case the switching point is equal to 1.
We “framed” the school choice problem from the point of view of teachers who are looking for jobs because this presentation provides a natural environment that is easy to understand. For example, material payoffs can be directly interpreted as salaries (see Pais and Pintér 2008).
If teachers had to list only one school, the two constrained 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.
Since the payoff of the second most preferred school varies for all subjects, subjects face different kinds of opponents in different games. In one alternative design to possibly overcome 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 to the change in the preference intensities to which they respond by adapting their behavior as well, etc. The elicitation of beliefs would certainly provide important information regarding the individual motives but would, at the same time, further complicate the design. Also, if we only changed the preferences of one subject the data to be collected would triple (to a total of 654 subjects).
In each treatment using the Boston algorithm, we had one student left that could not be matched with other participants. These two students took decisions without knowing that they remained unmatched. Finally, we paid them as if they were assigned a place at their most preferred school.
It is well known (Dubins and Freedman 1981 and Roth 1982) that teachers have incentives to report their ordinal preferences truthfully in treatment GS _{ u }, in which case the induced matching would be stable and efficient with respect to the teachers’ true preferences. However, to put all treatments at the same level, these incentives were neither directly revealed in the instructions nor were they indirectly taught by going over several examples. Otherwise, a convincing argument in favor of truthtelling in GS _{ u } would render the comparison between GS _{ u } and the other mechanisms rather obvious. Also, explicit advice would only increase the (observed) efficiency and stability gap between GS _{ u } and BOS _{ u }, i.e., strengthen our results.
In Chen and Sönmez (2006), in their “random” and “designed” treatments of GS _{ u }, 56 % and 72 % of the subjects, respectively, submitted their true preferences. The numbers are 58 % and 57 % in Calsamiglia et al. (2009). Our numbers seem to be slightly lower but a real comparison is not possible due to the very different environments.
Using χ ^{2} tests for homogeneity one verifies that for all cardinal payoff constellations, (a) the distribution of submitted rankings in treatment GS _{ u } (BOS _{ u }) is significantly different from the one in treatment GS _{ c } (BOS _{ c }) and (b) the distributions of submitted rankings in treatments GS _{ u } and BOS _{ u } (GS _{ c } and BOS _{ c }) are not significantly different from each other. The second finding might create the impression that subjects perceive the Gale–Shapley matching algorithm in the same way as the Boston algorithm. Results 1 and 2 presented below, however, will reveal that this is not the case.
We only considered data from subjects who behaved rationally in the first phase of the experiment, omitting those that switch from lottery B to lottery A.
The slightly cumbersome calculations are available from the authors upon request.
The common switching point has not been chosen arbitrarily. According to our data, the average switching point is 6.47 in GS _{ u }, 5.98 in GS _{ c }, 6.70 in BOS _{ u }, and 6.55 in treatment BOS _{ c } so that the difference in the group sizes is minimal if the seventh decision situation is taken as the dividing line.
In theory, the two unconstrained mechanisms should yield stable matchings if subjects recognize that telling the truth is weakly dominant (in the case of GS _{ u }) and do not fail to play Nash equilibria (in the case of BOS _{ u }, see Ergin and Sönmez 2006).
The only two exceptions are found in the efficiency levels for the full subject pool and the low risk aversion group when school 2 has a value of 13 ECU.
Note that the analysis performed in this paper may still apply to a “large” market where agents do not have complete information, but may still have a good idea of how preference distributions look like.
Barberà and Dutta (1995) showed that under GS _{ u } truthtelling is the unique protective strategy for all participants on both sides of a twosided matching market.
References
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We are very grateful for comments and suggestions from the Editor, two referees, Eyal Ert, Bettina Klaus, Muriel Niederle, Al Roth, and the seminar audiences in Alicante, Braga, Évora, Maastricht, and Málaga. F. Klijn gratefully acknowledges a research fellowship from Harvard Business School for academic year 2009–2010 when he was visiting HBS and the first draft of the paper was written. He also gratefully acknowledges support from Plan Nacional I+D+i (ECO200804784 and ECO201129847), Generalitat de Catalunya (SGR200901142), and the ConsoliderIngenio 2010 (CSD200600016) program. J. Pais gratefully acknowledges financial support from Fundação para a Ciência e a Tecnologia under project reference no. PTDC/EGEECO/113403/2009. M. Vorsatz gratefully acknowledges financial support from the Spanish Ministry of Education and Science through the project ECO200907530.
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Appendices
Appendix A: Holt and Laury (2002)
Appendix B: Protective strategies
Consider the game \(G =[I,\mathbb{A},S,g,u]\), where I={1,2,…,n} is the set of players, \(\mathbb{A}\) is the set of outcomes, S=S _{1}×⋯×S _{ n } and S _{ i } is the set of strategies of player i, \(g:S \rightarrow \mathbb{A}\) is an outcome function, and u=(u _{1},…,u _{ n }) denotes a vector of utility functions \(u_{i}:\mathbb{A} \rightarrow \mathbb{R}\), where i=1,2,…,n. Take any number k∈ℝ, any i∈I, and s _{ i }∈S _{ i }. Let c(k,s _{ i })={s _{−i }∈S _{−i }:u _{ i }(g(s _{ i },s _{−i }))=k}.
Definition 1
(Barberà and Dutta 1995)
For any i∈I and \(s_{i},s'_{i} \in S_{i}\), s _{ i } protectively dominates \(s'_{i}\), if there exists k∈ℝ such that

P1.
\(c(r,s_{i}) \cap c(r',s'_{i})= \emptyset\) for all r≤k and r<r′, and

P2.
\(c(k,s_{i}) \subset c(k,s'_{i})\).
It follows from the definition that if s _{ i } protectively dominates \(s'_{i}\), then \(s'_{i}\) does not protectively dominate s _{ i }.
Definition 2
A protective strategy is a strategy that is not protectively dominated.
Let us now apply the above definition to our school choice problem. Take, for instance the mechanism BOS _{ u } and the payoff structure 20 ECU. They define a game \(G =[I,\mathbb{A},S,\mathit{BOS}_{u},u]\), where I={1,2,3} is the set of teachers; \(\mathbb{A}\) is the set of matchings; S=S _{1}×S _{2}×S _{3}, where S _{ i }={(X,Y,Z),(X,Z,Y),(Y,X,Z),(Y,Z,X),(Z,X,Y),(Z,Y,X)} is the set of rankings over schools of teacher i, i∈I; and u=(u _{1},u _{2},u _{3}) is a vector of utility functions. To define player i’s utility function u _{ i }, note that i is indifferent between matchings that deliver the same partner, but has strict preferences over matchings that deliver different partners; four situations have to be considered: i may end up unmatched and receive a level of utility of 0, matched to the school ranked third in her preferences and receive a utility of 10, matched to the school ranked second and receive 20, and matched to the school ranked first, receiving a utility of 30.
Now let us consider teacher 1’s problem. The other teachers’ problems are similar. Note that every strategy guarantees that teacher 1 is matched, so that c(k,(×,×,×))=∅ for all k<10, implying that P2 is never satisfied for k in this range. Therefore, let us compute for each strategy of teacher 1 the set of complementary strategy profiles that match teacher 1 with school Z, with a corresponding utility of 10:
Let us start by comparing strategies (X,Y,Z) and (X,Z,Y). Since c(10,(X,Y,Z))⊂c(10,(X,Z,Y)), P2 is fulfilled for k=10. Moreover, P1 is fulfilled for r=10. Since c(r,(X,Y,Z))=∅ for all r<10, P1 is also fulfilled for r<10. It follows that strategy (X,Y,Z) protectively dominates (X,Z,Y) (and (X,Z,Y) does not protectively dominate (X,Y,Z)).
On the other hand, c(10,(Y,X,Z))⊂c(10,(Y,Z,X)) and c(r,(Y,X,Z))=∅ for all r<10 guarantee that (Y,X,Z) protectively dominates (Y,Z,X) (and (Y,Z,X) does not protectively dominate (Y,X,Z)). Furthermore, since c(10,(Z,×,×))=S _{2}×S _{3}, the strategies (Z,×,×) are protectively dominated by the other four strategies (and do not protectively dominate any of them).
Comparing c(10,(X,Y,Z)) and c(10,(Y,X,Z)), P2 is not verified for k=10. To make sure none of these strategies protectively dominates the other, we have to check what happens for higher levels of k. Computing c(20,(Y,X,Z)), it is easy to show that c(10,(X,Y,Z))∩c(20,(Y,X,Z))≠∅, so that P1 fails to hold for k>10 (with r=10 and r′=20) and (X,Y,Z) does not protectively dominate (Y,X,Z). On the other hand, (Y,X,Z) does not protectively dominate (X,Y,Z) as c(10,(Y,X,Z))∩c(30,(X,Y,Z))≠∅ and P1 fails to hold for k>10 (with r=10 and r′=30).
To ensure (X,Y,Z) is not protectively dominated, we still have to compare it with (Y,Z,X). Note that P2 is not verified for k=10. As for k>10, it can easily be shown that c(10,(Y,Z,X))∩c(30,(X,Y,Z))≠∅, so that P1 fails (with r=10 and r′=30). Similarly, (X,Z,Y) does not protectively dominate (Y,X,Z) as P2 is not verified for k=10 and c(10,(X,Z,Y))∩c(20,(Y,X,Z))≠∅, invalidating P1 for k>10 (with r=10 and r′=20).
Therefore, strategies (X,Y,Z) and (Y,X,Z) are not protectively dominated. The set of protective strategies of teacher 1 in BOS _{ u20}—in fact, in any game induced by BOS _{ u }—is {(X,Y,Z),(Y,X,Z)}.
Protective strategies can readily be calculated for the other mechanisms. In fact, following the informal description of protective strategies in Barberà and Dutta (1995, p. 289), in our school choice problem protective behavior means the following. For any distribution over the others’ strategy profiles: First, choosing a strategy that guarantees access to a school; second, among these, if possible, one that maximizes the probability of obtaining the best or the second best schools; and finally, within this set of strategies and whenever possible, picking one that maximizes the probability of being matched to the best school.
As such, since under GS _{ u } telling the truth never hurts and, for some strategy profiles of the others, leads to a better school slot, truthtelling is the unique protective strategy under this mechanism.^{Footnote 23} In what constrained mechanisms are concerned, protective behavior ensures in the first place that a subject is not left unassigned for any profile of complementary strategies. This implies using a strategy where the least preferred school is ranked first under BOS _{ c }—the unique protective strategy under this mechanism—and, given that acceptance is deferred in GS _{ c }, ranking the least preferred school first or second in the list under this mechanism. Moreover, given that ranking the least preferred school second increases the chances of being assigned to a better school both (X,Z,Y) and (Y,Z,X) are protective strategies for teacher 1 in GS _{ c }.
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Klijn, F., Pais, J. & Vorsatz, M. Preference intensities and risk aversion in school choice: a laboratory experiment. Exp Econ 16, 1–22 (2013). https://doi.org/10.1007/s1068301293295
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DOI: https://doi.org/10.1007/s1068301293295
Keywords
 School choice
 Risk aversion
 Preference intensities
 Laboratory experiment
 Gale–Shapley mechanism
 Boston mechanism
 Efficiency
 Stability
 Constrained choice