# Stability-Preserving, Time-Efficient Mechanisms for School Choice in Two Rounds

## TL;DR

This paper develops efficient algorithms for dynamic school choice in two rounds, ensuring stability and minimal re-allocations, and explores the computational complexity and incentive issues involved.

## Contribution

It introduces polynomial algorithms for stable matchings in two-round school choice scenarios and analyzes the structure and complexity of re-allocation minimization.

## Key findings

- Polynomial algorithms for stable matchings after changes
- Re-allocation minimizing matchings form a sublattice
- NP-hardness results for certain mechanisms

## Abstract

We address the following dynamic version of the school choice question: a city, named City, admits students in two temporally-separated rounds, denoted $\mathcal{R}_1$ and $\mathcal{R}_2$. In round $\mathcal{R}_1$, the capacity of each school is fixed and mechanism $\mathcal{M}_1$ finds a student optimal stable matching. In round $\mathcal{R}_2$, certain parameters change, e.g., new students move into the City or the City is happy to allocate extra seats to specific schools. We study a number of Settings of this kind and give polynomial time algorithms for obtaining a stable matching for the new situations. It is well established that switching the school of a student midway, unsynchronized with her classmates, can cause traumatic effects. This fact guides us to two types of results, the first simply disallows any re-allocations in round $\mathcal{R}_2$, and the second asks for a stable matching that minimizes the number of re-allocations. For the latter, we prove that the stable matchings which minimize the number of re-allocations form a sublattice of the lattice of stable matchings. Observations about incentive compatibility are woven into these results. We also give a third type of results, namely proofs of NP-hardness for a mechanism for round $\mathcal{R}_2$ under certain settings.