Random Matching under Priorities: Stability and No Envy Concepts

TL;DR

This paper explores various stability and fairness concepts for random matchings with preferences and priorities, providing a comprehensive taxonomy, interpretations, and transformations that preserve these concepts.

Contribution

It formalizes and compares multiple stability and no envy concepts for random matchings, including weak preferences and unacceptability, and introduces a transformation preserving these properties.

Findings

Provides a taxonomy of stability concepts for random matchings.

Establishes logical relations between different stability and fairness notions.

Introduces a transformation that preserves stability concepts across settings.

Abstract

We consider stability concepts for random matchings where agents have preferences over objects and objects have priorities for the agents. When matchings are deterministic, the standard stability concept also captures the fairness property of no (justified) envy. When matchings can be random, there are a number of natural stability / fairness concepts that coincide with stability / no envy whenever matchings are deterministic. We formalize known stability concepts for random matchings for a general setting that allows weak preferences and weak priorities, unacceptability, and an unequal number of agents and objects. We then present a clear taxonomy of the stability concepts and identify logical relations between them. Furthermore, we provide no envy / claims interpretations for some of the stability concepts that are based on a consumption process interpretation of random matchings. Finally, we present a transformation from the most general setting to the most restricted setting, and show how almost all our stability concepts are preserved by that transformation.