Selecting Matchings via Multiwinner Voting: How Structure Defeats a Large Candidate Space

TL;DR

This paper investigates how leveraging structure in approval preferences can make the computation of fair matchings feasible, transforming intractable voting rules into polynomial-time algorithms in complex multiwinner election scenarios.

Contribution

It demonstrates that exploiting preference structure allows efficient computation of approval-based multiwinner rules like PAV and seq-PAV in large candidate spaces.

Findings

PAV becomes polynomial-time computable with symmetric approvals.

seq-PAV guarantees extended justified representation.

Structural exploitation extends to other complex election models.

Abstract

Given a set of agents with approval preferences over each other, we study the task of finding $k$ matchings fairly representing everyone's preferences. We model the problem as an approval-based multiwinner election where the set of candidates consists of all possible matchings and agents' preferences over each other are lifted to preferences over matchings. Due to the exponential number of candidates in such elections, standard algorithms for classical sequential voting rules (such as those proposed by Thiele and Phragm\'en) are rendered inefficient. We show that the computational tractability of these rules can be regained by exploiting the structure of the approval preferences. Moreover, we establish algorithmic results and axiomatic guarantees that go beyond those obtainable in the general multiwinner setting. Assuming that approvals are symmetric, we show that proportional approval voting (PAV), a well-established but computationally intractable voting rule, becomes polynomial-time computable, and its sequential variant (seq-PAV), which does not provide any proportionality guarantees in general, fulfills a rather strong guarantee known as extended justified representation. Some of our positive computational results extend to other types of compactly representable elections with an exponential candidate space.