School choice with controlled choice constraints: Hard bounds versus soft bounds

Abstract

Controlled choice over public schools attempts giving options to parents while maintaining diversity, often enforced by setting feasibility constraints with hard upper and lower bounds for each student type. We demonstrate that there might not exist assignments that satisfy standard fairness and non-wastefulness properties; whereas constrained non-wasteful assignments which are fair for same type students always exist.We introduce a "controlled" version of the deferred acceptance algorithm with an improvement stage (CDAAI) that finds a Pareto optimal assignment among such assignments. To achieve fair (across all types) and non-wasteful assignments, we propose the control constraints to be interpreted as soft bounds-flexible limits that regulate school priorities. In this setting, a modified version of the deferred acceptance algorithm (DAASB) finds an assignment that is Pareto optimal among fair assignments while eliciting true preferences. CDAAI and DAASB provide two alternative practical solutions depending on the interpretation of the control constraints.JEL C78, D61, D78, I20. * An earlier version (Ehlers, 2010) of this paper emerged from a joint project of the first author with Atila Abdulkadiroglu. We are grateful for his extensive comments and contribution to that paper. Ehlers acknowledges financial support from the SSHRC (Canada).

Matching with floor constraints

“…To the best of our knowledge, the study of hard floor constraints in matching markets was pioneered by Ehlers (2010). An example developed by Ehlers et al (2014) (which supersedes Ehlers 2010 show that an immediate extension of the stability notion to one that accommodates floor constraints may result in the nonexistence of a stable matching. 2 We show that this nonexistence problem disappears when one imposes a condition on the reassignment of doctors outside the coalition.…”

“…Ehlers (2014) shows that an efficient allocation may fail to exist when we allow for indifferences in agents' preferences, but its existence may be guaranteed by imposing restrictions on the reassignment of objects. 3 Fragiadakis et al (2016) take the impossibility result of Ehlers et al (2014) and propose two mechanisms that offer different trade-offs between fairness and nonwastefulness. Tomoeda (2018) adopts the stability notion introduced by Ehlers et al (2014) and finds sufficient conditions on hospital preferences such that a stable matching exists.…”

“…3 Fragiadakis et al (2016) take the impossibility result of Ehlers et al (2014) and propose two mechanisms that offer different trade-offs between fairness and nonwastefulness. Tomoeda (2018) adopts the stability notion introduced by Ehlers et al (2014) and finds sufficient conditions on hospital preferences such that a stable matching exists. Kamada and Kojima (2017b) propose stability notions for a broad class of matching markets under ceiling constraints.…”

“…Condition (iv) in Definition 1 is critical to guarantee the existence of floor respecting stable matchings. Ehlers et al (2014) adopt a stability notion similar to ours, but they allow coalitions to reassign doctors who are not in the coalition to different hospitals. They show that a stable matching may not exist.…”

“…The key element of this definition is the specification that a doctor outside the coalition cannot be reassigned to another hospital without her consent (although a doctor can be fired without her consent). This condition is absent from the stability notion introduced previously in the literature (Ehlers et al 2014) and is the key element of our definition that allows us to guarantee the existence of stable matchings.…”