[Paper Review] Robust and Verifiable Proportionality Axioms for Multiwinner Voting

Markus Brill, Jannik Peters|arXiv (Cornell University)|Feb 3, 2023

Game Theory and Voting Systems4 citations

TL;DR

This paper introduces robust and verifiable proportionality axioms for multiwinner voting that extend existing criteria like EJR and PSC to apply to groups that are nearly solid or cohesive, not just perfectly uniform ones. The authors propose EJR+ for approval voting and rank-EJR+ for ranked preferences, both of which are always satisfiable and verifiable in polynomial time, offering stronger, more realistic guarantees than prior axioms.

ABSTRACT

When selecting a subset of candidates (a so-called committee) based on the preferences of voters, proportional representation is often a major desideratum. When going beyond simplistic models such as party-list or district-based elections, it is surprisingly challenging to capture proportionality formally. As a consequence, the literature has produced numerous competing criteria of when a selected committee qualifies as proportional. Two of the most prominent notions are Dummett's proportionality for solid coalitions (PSC) and Aziz et al.'s extended justified representation (EJR). Both guarantee proportional representation to groups of voters who have very similar preferences; such groups are referred to as solid coalitions by Dummett and as cohesive groups by Aziz et al. However, these notions lose their bite when groups are only almost solid or almost cohesive. In this paper, we propose proportionality axioms that are more robust: they guarantee representation also to groups that do not qualify as solid or cohesive. Further, our novel axioms can be easily verified: Given a committee, we can check in polynomial time whether it satisfies the axiom or not. This is in contrast to many established notions like EJR, for which the corresponding verification problem is known to be intractable. In the setting with approval preferences, we propose a robust and verifiable variant of EJR and a simply greedy procedure to compute committees satisfying it. In the setting with ranked preferences, we propose a robust variant PSC, which can be efficiently verified even for general weak preferences. In the special case of strict preferences, our notion is the first known satisfiable proportionality axiom that is violated by the Single Transferable Vote (STV). We also discuss implications of our results for participatory budgeting, querying procedures, and to the notion of proportionality degree.

Motivation & Objective

• To address the limitation of existing proportionality axioms like EJR and PSC, which only apply to perfectly cohesive or solid voter groups and fail for nearly uniform preferences.
• To develop proportionality axioms that are robust to small deviations from perfect cohesion or solidarity, ensuring representation for groups that are almost but not fully unified in preference.
• To ensure that satisfaction of the new axioms can be verified in polynomial time, unlike EJR whose verification is intractable.
• To prove that committees satisfying the new axioms always exist and can be computed efficiently using greedy or priceability-based methods.
• To extend the framework to participatory budgeting and ranked preferences, demonstrating broader applicability and computational advantages.

Proposed method

• Propose EJR+ as a robust, verifiable variant of EJR for approval-based multiwinner voting, where representation is guaranteed not only for fully cohesive groups but also for those close to cohesion.
• Introduce the concept of 'priceability' in the approval setting, extending it to support efficient computation and verification of EJR+ committees.
• Design a greedy algorithm that computes EJR+-satisfying committees efficiently, with theoretical guarantees on approximation and committee size.
• Define rank-EJR+ as a robust extension of Dummett’s PSC for ranked preferences, ensuring representation for groups that are nearly solid in their preferences.
• Extend the notion of priceability to ranked preferences to prove that rank-EJR+ committees always exist and can be computed in polynomial time.
• Demonstrate that rank-EJR+ is the first known satisfiable proportionality axiom that is violated by STV, highlighting its distinctness and strength.

Experimental results

Research questions

• RQ1 Can proportionality axioms be strengthened to apply to groups that are not perfectly cohesive or solid, but only nearly so?
• RQ2 Is it possible to define proportionality axioms that are both robust to preference deviations and efficiently verifiable?
• RQ3 Can new axioms be constructed that guarantee existence of satisfying committees and allow for efficient computation in both approval and ranked preference settings?
• RQ4 How do the new axioms compare in discriminative power to existing ones like EJR and PJR in random instances?
• RQ5 Can the framework be extended to participatory budgeting and other settings with general utility functions?

Key findings

• EJR+ is more discriminating than EJR and other existing axioms in randomly generated approval-based elections, as it imposes stricter fairness requirements on nearly cohesive groups.
• The verification problem for EJR+ is solvable in polynomial time, unlike EJR, which is known to be computationally intractable.
• The greedy algorithm for EJR+ produces a committee of size at most k with high probability and satisfies EJR+ with probability 1−δ.
• The expected number of queries required by the greedy algorithm is O(mk⁴ log(m) log(k)), making it scalable for large-scale applications.
• rank-EJR+ is the first satisfiable proportionality axiom for ranked preferences that is violated by STV, demonstrating its distinct and stronger fairness properties.
• The authors prove that rank-EJR+ committees always exist and can be computed efficiently using an extended notion of priceability in the ranked setting.