# Near-Feasible Stable Matchings with Budget Constraints

## TL;DR

This paper addresses the challenge of finding near-feasible stable matchings in a matching with contracts framework under hospital budget constraints, introducing a new compatibility condition to ensure existence and proposing mechanisms for efficient solutions.

## Contribution

It introduces the concept of compatibility to guarantee near-feasible stable matchings under budget constraints and develops mechanisms that efficiently find such matchings with optimal budget excess.

## Key findings

- Compatibility condition ensures existence of near-feasible stable matchings.
- Mechanisms achieve efficient solutions with minimal budget excess.
- Strategy-proofness is sacrificed for optimal budget excess bounds.

## Abstract

We consider the matching with contracts framework of Hatfield and Milgrom when one side (a firm or hospital) can make monetary transfers (offer wages) to the other (a worker or doctor). In a standard model, monetary transfers are not restricted. However, we assume that each hospital has a fixed budget; that is, the total amount of wages allocated by each hospital to the doctors is constrained. With this constraint, stable matchings may fail to exist and checking for the existence is hard. To deal with the nonexistence, we focus on near-feasible matchings that can exceed each hospital budget by a certain amount, and We introduce a new concept of compatibility. We show that the compatibility condition is a sufficient condition for the existence of a near-feasible stable matching in the matching with contracts framework. Under a slight restriction on hospitals' preferences, we provide mechanisms that efficiently return a near-feasible stable matching with respect to the actual amount of wages allocated by each hospital. By sacrificing strategy-proofness, the best possible bound of budget excess is achieved.