Condorcet Dimension and Pareto Optimality for Matchings and Beyond

2026-02-18

AI Summary

This study addresses bipartite matching under partially ordered preferences, where a Condorcet winning matching may not exist, necessitating an analysis of the minimal winning set—referred to as the Condorcet dimension—and its relationship with Pareto optimality. Integrating social choice theory, combinatorial optimization, and computational complexity, the work reveals intrinsic connections between the set of Pareto optimal matchings and the Condorcet winning set. The main contributions include the first proof that the Condorcet dimension is always 2 under weak orders, grows as Θ(√n) under partial orders, and further increases to Θ(n) when matroid constraints are introduced; moreover, the associated decision problems are shown to be NP-hard. These results are also extended to matching scenarios on directed trees.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Web Mining and Content Analysis: Web measurements

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 $Θ(\sqrt{n})$, and rises further to $Θ(n)$ 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.

Problem

• Condorcet dimension
• Pareto optimality
• matchings
• partial orders
• Condorcet-winning set

Innovation

• Condorcet dimension
• Pareto optimality
• matching theory
• partial orders
• matroid constraints