Condorcet Dimension and Pareto Optimality for Matchings and Beyond

TL;DR

This paper explores the concept of Condorcet-winning sets in matching problems, revealing their connection to Pareto optimality, and investigates their existence, size, and computational complexity under various preference structures.

Contribution

It establishes a link between Condorcet-winning sets and Pareto optimality, and analyzes their properties and computational complexity under different preference orderings and constraints.

Findings

Any Pareto-optimal set of two matchings is a Condorcet-winning set.

Existence of small Condorcet-winning sets varies with preference structures.

Deciding the existence of a Condorcet-winning set of fixed size is NP-hard.

Abstract

We study matching problems in which agents form one side of a bipartite graph and have preferences over objects on the other side. A central solution concept in this setting is popularity: a matching is popular if it is a (weak) Condorcet winner, meaning that no other matching is preferred by a strict majority of agents. It is well known, however, that Condorcet winners need not exist. We therefore turn to a natural and prominent relaxation. A set of matchings is a Condorcet-winning set if, for every competing matching, a majority of agents prefers their favorite matching in the set over the competitor. The Condorcet dimension is the smallest cardinality of a Condorcet-winning set. Our main results reveal a connection between Condorcet-winning sets and Pareto optimality. We show that any Pareto-optimal set of two matchings is, in particular, a Condorcet-winning set. This implication continues to hold when we impose matroid constraints on the set of matched objects, and even when agents' valuations are given as partial orders. The existence picture, however, changes sharply with partial orders. While for weak orders a Pareto-optimal set of two matchings always exists, this is -- surprisingly -- not the case under partial orders. Consequently, although the Condorcet dimension for matchings is 2 under weak orders (even under matroid constraints), this guarantee fails for partial orders: we prove that the Condorcet dimension is $\Theta(\sqrt{n})$, and rises further to $\Theta(n)$ when matroid constraints are added. On the computational side, we show that, under partial orders, deciding whether there exists a Condorcet -- winning set of a given fixed size is NP-hard. The same holds for deciding the existence of a Pareto-optimal matching, which we believe to be of independent interest. Finally, we also show that the Condorcet dimension for a related problem on arborescences is also 2.