Single-Winner Voting on Matchings

TL;DR

This paper explores voting on matchings where voters have preferences over entire matchings, analyzing the computational complexity of selecting socially desirable matchings and identifying conditions for tractability.

Contribution

It provides a complete complexity landscape for social welfare maximization, Pareto optimality, and Condorcet winners in voting on matchings, highlighting sharp boundaries between tractable and intractable cases.

Findings

Complexity results vary with utility models and solution concepts.

Certain cases allow polynomial-time algorithms for finding optimal matchings.

Other cases are proven to be computationally intractable.

Abstract

We introduce a single-winner perspective on voting on matchings, in which voters have preferences over possible matchings in a graph, and the goal is to select a single collectively desirable matching. Unlike in classical matching problems, voters in our model are not part of the graph; instead, they have preferences over the entire matching. In the resulting election, the candidate space consists of all feasible matchings, whose exponential size renders standard algorithms for identifying socially desirable outcomes computationally infeasible. We study whether the computational tractability of finding such outcomes can be regained by exploiting the matching structure of the candidate space. Specifically, we provide a complete complexity landscape for questions concerning the maximization of social welfare, the construction and verification of Pareto optimal outcomes, and the existence and verification of Condorcet winners under one affine and two approval-based utility models. Our results consist of a mix of algorithmic and intractability results, revealing sharp boundaries between tractable and intractable cases, with complexity jumps arising from subtle changes in the utility model or solution concept.