Simple Manipulations in School Choice Mechanisms

We introduce a new criterion to compare the properties of mechanisms when the solution concept used induces multiple solutions. Our criterion generalizes previous approaches in the literature. We use our criterion to compare the stability of constrained versions of the Boston (BOS) and deferred acceptance (DA) school choice mechanisms in which students can only rank a subset of the schools they could potentially access. When students play a Nash equilibrium, we show that there is a stability cost to increasing the number of schools students can rank in DA. On the other hand, when students only play undominated strategies, increasing the number of schools students can rank increases stability. We find sim- ilar results for BOS. We also compare BOS and DA. Whatever the number of schools students can rank, we find that BOS is more stable than DA in Nash equilibrium, but less stable in undominated strategies.

Since 2001, many Chinese provinces have transitioned from a “sequential” to a “parallel” school choice or college admissions mechanism. Inspired by this natural experiment, we evaluate the sequential (immediate acceptance, IA), parallel (PA), and deferred acceptance (DA) mechanisms in the laboratory. We find that participants are most likely to reveal their preferences truthfully under DA, followed by PA and then DA. While stability comparisons also follow the same order, efficiency comparisons vary across environments. Regardless of the metrics, the performance of PA is robustly sandwiched between IA and DA. Furthermore, 53% of our subjects adopt an insurance strategy under PA, making them at least as well off as what they could guarantee themselves under IA. These results help explain the recent reforms in Chinese school choice and college admissions.

We present an experimental study of three school choice mechanisms under complete information, using the designed environment in Chen & Sonmez (2006). We find that the top trading cycles (TTC) mechanism outperforms both the Gale-Shapley deferred acceptance (DA) and the Boston immediate acceptance (BOS) mechanism in terms of truth-telling and efficiency, whereas DA is more stable than either TTC or BOS. Compared to the incomplete information setting in Chen & Sonmez (2006), the performance of both TTC and BOS improves with more information, whereas that of DA does not.

Most school choice and other matching mechanisms are based on deferred acceptance (DA) for its incentive properties. However, non-strategyproof mechanisms can dominate DA in welfare because manipulation in preference rankings can reflect the intensities of underlying cardinal preferences. In this work, we use the parallel mechanism of Chen and Kesten, which interpolates between Boston mechanism and DA, to quantify this tradeoff. While it is previously known that mechanisms that are closer to Boston mechanism are more manipulable, we show that they are also more efficient in student welfare if school priorities are weak. Our theoretical results show the inefficiency-manipulability tradeoff in the worst case, while our simulation results show the same tradeoff in the typical case.

Criteria for evaluating school choice mechanisms are first, whether truth-telling is sometimes punished and second, how efficient the match is. With common knowledge preferences, Deferred Acceptance (DA) dominates the Boston mechanism by the first criterion and is ambiguously ranked by the second. Our laboratory experiments confirm this. A new ex ante perspective, where preferences are private information, introduces new efficiency costs borne by strategy-proof mechanisms, like DA. In a symmetric environment, truth-telling can be an equilibrium under Boston, and Boston can first-order stochastically dominate DA in terms of efficiency, both in theory and in the laboratory.

In recent years, China has experienced a trend of changing from the Boston mechanism (BM) to the Chinese parallel mechanism for high school and college admissions. Using a unique data set from the high-school-assignment system in China that combines survey data eliciting students’ school preferences with administrative data that cover students’ school choices and admission records under both mechanisms, this paper compares the welfare performance of BM, the Chinese parallel mechanism, and the Deferred Acceptance (DA) mechanism. We find a non-monotonic relationship between the manipulability and efficiency of school choice mechanisms: DA yields significantly higher welfare than the Chinese parallel mechanism and BM, but BM yields higher welfare than Chinese parallel mechanism although not significantly. We also find that switching from BM to Chinese parallel mechanism hurts students regardless of their socioeconomic status, but benefits students with higher test scores. Students with lower socioeconomic status benefit more than those with higher socioeconomic status when switching from BM to DA, but when switching from Chinese parallel mechanism to DA, students with higher socioeconomic status benefit more.

Manipulability in school choice

Comparing school choice mechanisms by interim and ex-ante welfare

In recent years, China has experienced a trend of changing from the Boston mechanism (BM) to the Chinese parallel mechanism for high school and college admissions. Using a unique data set from the high-school-assignment system in China that combines survey data eliciting students' school preferences with administrative data that cover students' school choices and assignment results under both mechanisms, this paper compares the welfare performance of BM, the Chinese parallel mechanism, and the Deferred Acceptance (DA) mechanism. We find DA yields higher average welfare than the Chinese parallel mechanism, which is higher than the average utility under BM. We also find a more manipulable mechanism hurts students from low socioeconomic districts. Less manipulable mechanisms are more likely to assign students with higher score percentiles to schools with higher education quality, especially for students from higher socioeconomic districts. However, lower-scored students from higher socioeconomic districts decreased their human capital production after the mechanism was changed to the Chinese parallel mechanism.

In the school choice market, where scarce public school seats are assigned to students, a key operational issue is how to reassign seats that are vacated after an initial round of centralized assignment. Practical solutions to the reassignment problem must be simple to implement, truthful, and efficient while also alleviating costly student movement between schools. We propose and axiomatically justify a class of reassignment mechanisms, the permuted lottery deferred acceptance (PLDA) mechanisms. Our mechanisms generalize the commonly used deferred acceptance (DA) school choice mechanism to a two-round setting and retain its desirable incentive and efficiency properties. School choice systems typically run DA with a lottery number assigned to each student to break ties in school priorities. We show that under natural conditions on demand, the second-round tie-breaking lottery can be correlated arbitrarily with that of the first round without affecting allocative welfare and that reversing the lottery order between rounds minimizes reassignment among all PLDA mechanisms. Empirical investigations based on data from New York City high school admissions support our theoretical findings. This paper was accepted by Gad Allon, operations management.

In the school choice market, where scarce public school seats are assigned to students, a key operational issue is how to reassign seats that are vacated after an initial round of centralized assignment. Practical solutions to the reassignment problem must be simple to implement, truthful and efficient while also alleviating costly student movement between schools. We propose and axiomatically justify a class of reassignment mechanisms, the Permuted Lottery Deferred Acceptance (PLDA) mechanisms. Our mechanisms generalize the commonly used Deferred Acceptance (DA) school choice mechanism to a two-round setting and retain its desirable incentive and efficiency properties. School choice systems typically run DA with a lottery number assigned to each student to break ties in school priorities. We show that under natural conditions on demand, the second round tie-breaking lottery can be correlated arbitrarily with that of the first round without affecting allocative welfare, and reversing the lottery order between rounds minimizes reassignment among all PLDA mechanisms. Empirical investigations based on data from NYC high school admissions support our theoretical findings.

Level-k reasoning in school choice

In recent years, China has experienced a trend of changing from the Boston mechanism (BM) to the Chinese parallel mechanism for high school and college admissions. Using a unique data set from the high-school-assignment system in China that combines survey data eliciting students' school preferences with administrative data that cover students' school choices and admission records under both mechanisms, this paper compares the welfare performance of BM, the Chinese parallel mechanism, and the Deferred Acceptance (DA) mechanism. We find a non-monotonic relationship between the manipulability and efficiency of school choice mechanisms: DA yields significantly higher welfare than the Chinese parallel mechanism and BM, but BM yields higher welfare than Chinese parallel mechanism although not significantly. We also find that switching from BM to Chinese parallel mechanism hurts students regardless of their socioeconomic status. Students put schools they prefer more on all of their ranks under Chinese parallel mechanism than BM, and only students who are admitted by one of their first two choices benefit from the mechanism change.

In the school choice market, where scarce public school seats are assigned to students, a key issue is how to reassign seats that are vacated after an initial round of centralized assignment. Every year around 10% of students assigned a seat in the NYC public high school system eventually do not use it, and their vacated seats can be reassigned. Practical solutions to the reassignment problem must be simple to implement, truthful and efficient. I propose and axiomatically justify a class of reassignment mechanisms, the Per- muted Lottery Deferred Acceptance (PLDA) mechanisms, which generalize the commonly used Deferred Acceptance (DA) school choice mechanism to a two-round setting and retain its desirable in- centive and efficiency properties. I also provide guidance to school districts as to how to choose the appropriate mechanism in this class for their setting. Centralized admissions are typically conducted in a single round using Deferred Acceptance, with a lottery used to break ties in each school’s prioritization of students. Our proposed PLDA mechanisms reassign vacated seats using a second round of DA with a lottery based on a suitable permutation of the first-round lottery numbers. I demonstrate that under a natural order condition on aggregate student demand for schools, the second-round tie-breaking lottery can be correlated arbitrarily with that of the first round without affecting allocative welfare. I also show how the identifying char- acteristic of PLDA mechanisms, their permutation, can be chosen to control reallocation. vacated after the initial round are reassigned using decentralized waitlists that create significant student movement after the start of the school year, which is costly for both students and schools. I show that reversing the lottery order between rounds minimizes reassignment among all PLDA mechanisms, allowing us to alleviate costly student movement between schools without affecting the ef- ficiency of the final allocation. In a setting without school priorities, I also characterize PLDA mechanisms as the class of mechanisms that provide students with a guarantee at their first-round assign- ment, respect school priorities, and are strategy-proof, constrained Pareto efficient, and satisfy some mild symmetry properties. Finally, I provide simulations of the performance of different PLDA mecha- nisms in the presence of school priorities. All simulated PLDAs have similar allocative efficiency, while the PLDA based on reversing the tie-breaking lottery between rounds minimizes the number of reassigned students. These results support our theoretical findings. This is based on joint work with Itai Feigenbaum, Yash Kanoria, and Jay Sethuraman.

Matching a set of agents to a set of objects has many real applications. One well-studied framework is that of priority-based matching, in which each object is assumed to have a priority order over the agents. The Deferred Acceptance (DA) and Top-Trading-Cycle (TTC) mechanisms are the best-known strategy-proof mechanisms. However, in highly anonymous environments, the set of agents is not known a priori, and it is more natural for objects to instead have priorities over characteristics (e.g., the student's GPA or home address). In this paper, we extend the model so that each agent reports not only its preferences over objects, but also its characteristic. We derive results for various notions of strategy-proofness and false-name-proofness, corresponding to whether agents can only report weaker characteristics or also incomparable or stronger ones, and whether agents can only claim objects allocated to their true accounts or also those allocated to their fake accounts. Among other results, we show that DA and TTC satisfy a weak version of false-name-proofness. Furthermore, DA also satisfies a strong version of false-name-proofness, while TTC fails to satisfy it without an acyclicity assumption on priorities.