# Strategyproof and fair matching mechanism for union of symmetric m-convex constraints

## TL;DR

This paper introduces a new matching mechanism called QRDA that is fair and strategyproof for complex constraints in two-sided matching scenarios.

## Contribution

The novel Quota Reduction Deferred Acceptance (QRDA) mechanism handles a new class of distributional constraints.

## Key findings

- QRDA is fair and strategyproof for a union of symmetric M-convex sets.
- QRDA produces better matchings for students than the ACDA baseline mechanism.
- QRDA outperforms ACDA in terms of nonwastefulness in experiments.

## Abstract

We identify a new class of distributional constraints defined as a union of symmetric M-convex sets, which can represent a wide range of real-life constraints in two-sided matching settings. Since M-convexity is not closed under union, a union of symmetric M-convex sets does not belong to this well-behaved class of constraints. Consequently, devising a fair and strategyproof mechanism to handle this new class is challenging. We present a novel mechanism for it called Quota Reduction Deferred Acceptance (QRDA), which repeatedly applies the standard Deferred Acceptance mechanism by sequentially reducing artificially introduced maximum quotas. We show that QRDA is fair and strategyproof when handling a union of symmetric M-convex sets, which extends previous results obtained for a subclass of the union of symmetric M-convex sets: ratio constraints. QRDA always yields a weakly better matching for students than a baseline mechanism called Artificial Cap Deferred Acceptance (ACDA). We also experimentally show that QRDA outperforms ACDA in terms of nonwastefulness.