Dynamic Matching in School Choice: Efficient Seat Reassignment After Late Cancellations

Abstract

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.

Theory of and Experiments on Minimally Invasive Stability Preservation in Changing Two-Sided Matching Markets

“…Besides the work of Bredereck et al [2020], there are several other works dealing with adapting a (stable) matching to a changing agent set or changing preferences [Bhattacharya et al, 2015, Feigenbaum et al, 2020, Gajulapalli et al, 2020, Ghosal et al, 2020, Kanade et al, 2016, Nimbhorkar and Rameshwar, 2019. Closest to our work, Gajulapalli et al [2020] designed polynomialtime algorithms for two variants of an incremental version of HOSPITAL RESIDENTS where the given matching is always resident-optimal (unlike in our setting where the given matching can be an arbitrary stable matching) and in the updated instance either new residents are added or the quotas of some hospitals are modified.…”

“…Here, students are matched to schools, trying to accommodate the students' preferences over the schools as well as possible. However, due to students reallocating or deciding to visit a private school, according to Feigenbaum et al [2020], in New York typically around 10% of the students drop out after a first round of assignments, triggering some readjustments in the school-student matchings in a further round.…”

Dynamic Bipartite Matching Market with Arrivals and Departures

“…Another example is a paired kidney exchange market, which requires simultaneous transplantation between different patients [34,30]. Other applications include crowd sourcing platform [20], task allocation [11,5,33], house allocation [1,31], school choice [15], and real-time ride sharing [28].…”

A binomial decision tree to manage yield‐uncertainty in multi‐round academic admissions processes

“…Abdulkadiroglu and Sönmez (2003) highlighted the somewhat different problem of centralized public school admissions, where student preferences take precedence and school preferences relate only to government policy guidelines. There has since been a substantial body of literature on this subject too (e.g., Erdil & Ergin, 2008;Fack et al, 2019;Feigenbaum et al, 2020). However, both the college and school admissions problems are distinct from the problem of our interest-where a single educational institution is deciding the best way of handling its admission applications.…”