The lattice of envy-free many-to-many matchings with contracts

TL;DR

This paper investigates envy-free allocations in a many-to-many matching model with contracts, establishing a lattice structure and a dynamic process to reach stable allocations from envy-free states.

Contribution

It introduces a lattice framework for envy-free allocations and models a dynamic process to converge to stable matchings using a Tarski operator.

Findings

Envy-free allocations form a lattice structure.

A Tarski operator models the dynamic process to reach stability.

Starting from any envy-free allocation, a stable one can be reached.

Abstract

We study envy-free allocations in a many-to-many matching model with contracts in which agents on one side of the market (doctors) are endowed with substitutable choice functions and agents on the other side of the market (hospitals) are endowed with responsive preferences. Envy-freeness is a weakening of stability that allows blocking contracts involving a hospital with a vacant position and a doctor that does not envy any of the doctors that the hospital currently employs. We show that the set of envy-free allocations has a lattice structure. Furthermore, we define a Tarski operator on this lattice and use it to model a vacancy chain dynamic process by which, starting from any envy-free allocation, a stable one is reached.