Pareto Optimal Matchings in Many-to-Many Markets with Ties

TL;DR

This paper characterizes Pareto-optimal matchings in complex many-to-many markets with ties, introduces a new mechanism to generate all such matchings, and discusses strategic manipulation limitations.

Contribution

It unifies and generalizes existing results by characterizing POMs and proposing the GSDT mechanism for many-to-many markets with ties.

Findings

GSDT can generate all POMs with different applicant priorities

GSDT is truthful only for certain priority orderings

Any mechanism generating all POMs in this setting is susceptible to manipulation

Abstract

We consider Pareto-optimal matchings (POMs) in a many-to-many market of applicants and courses where applicants have preferences, which may include ties, over individual courses and lexicographic preferences over sets of courses. Since this is the most general setting examined so far in the literature, our work unifies and generalizes several known results. Specifically, we characterize POMs and introduce the \emph{Generalized Serial Dictatorship Mechanism with Ties (GSDT)} that effectively handles ties via properties of network flows. We show that GSDT can generate all POMs using different priority orderings over the applicants, but it satisfies truthfulness only for certain such orderings. This shortcoming is not specific to our mechanism; we show that any mechanism generating all POMs in our setting is prone to strategic manipulation. This is in contrast to the one-to-one case (with or without ties), for which truthful mechanisms generating all POMs do exist.