Mechanisms for a dynamic many-to-many school choice problem

TL;DR

This paper introduces new mechanisms for assigning teachers to schools over time, ensuring stability and efficiency in a complex dynamic many-to-many matching environment with tenured teachers.

Contribution

It proposes the TRDA and TREADA mechanisms that achieve dynamic stability, efficiency, and Pareto improvements in a novel dynamic school choice setting.

Findings

A dynamically stable matching always exists.

TREADA Pareto-dominates any dynamically stable matching.

Mechanisms can be manipulated but are non-obviously manipulable under certain conditions.

Abstract

We examine the problem of assigning teachers to public schools over time when teachers have tenured positions and can work simultaneously in multiple schools. To do this, we investigate a dynamic many-to-many school choice problem where public schools have priorities over teachers and teachers hold path-independent choice functions selecting subsets of schools. We introduce a new concept of dynamic stability that recognizes the tenured positions of teachers and we prove that a dynamically stable matching always exists. We propose the Tenure-Respecting Deferred Acceptance (TRDA) mechanism, which produces a dynamically stable matching that is constrained-efficient within the class of dynamically stable matchings and minimizes unjustified claims. To improve efficiency beyond this class, we also propose the Tenure-Respecting Efficiency-Adjusted Deferred Acceptance (TREADA) mechanism, an adaptation of the Efficiency-Adjusted Deferred Acceptance mechanism to our dynamic context. We demonstrate that the outcome of the TREADA mechanism Pareto-dominates any dynamically stable matching and achieves efficiency when all teachers consent. Additionally, we examine the issue of manipulability, showing that although the TRDA and TREADA mechanisms can be manipulated, they remain non-obviously dynamically manipulable under specific conditions on schools' priorities.