[Paper Review] Best-of-Both-Worlds Fairness in Committee Voting

Haris Aziz, Xinhang Lu+3 more|arXiv (Cornell University)|Mar 7, 2023

Game Theory and Voting SystemsEconomics, Econometrics and Finance3 citations

TL;DR

This paper introduces a best-of-both-worlds fairness framework for approval-based committee voting, combining ex-ante fairness (via individual, unanimous, and group fair share) with ex-post fairness (via EJR and FJR). It presents a polynomial-time algorithm satisfying ex-post EJR, ex-ante group fair share (GFS), and strong unanimous fair share (UFS), and a non-polynomial-time algorithm achieving ex-post FJR alongside strong UFS.

ABSTRACT

The best-of-both-worlds paradigm advocates an approach that achieves desirable properties both ex-ante and ex-post. We launch a best-of-both-worlds fairness perspective for the important social choice setting of approval-based committee voting. To this end, we initiate work on ex-ante proportional representation properties in this domain and formalize a hierarchy of notions including Individual Fair Share (IFS), Unanimous Fair Share (UFS), Group Fair Share (GFS), and their stronger variants. We establish their compatibility with well-studied ex-post concepts such as extended justified representation (EJR) and fully justified representation (FJR). Our first main result is a polynomial-time algorithm that simultaneously satisfies ex-post EJR, ex-ante GFS and ex-ante Strong UFS. Subsequently, we strengthen our ex-post guarantee to FJR and present an algorithm that outputs a lottery which is ex-post FJR and ex-ante Strong UFS, but does not run in polynomial time.

Motivation & Objective

• To extend the best-of-both-worlds fairness paradigm—previously applied only to resource allocation—into the domain of social choice, specifically approval-based committee voting.
• To formalize a hierarchy of ex-ante fairness axioms, including Individual Fair Share (IFS), Unanimous Fair Share (UFS), and Group Fair Share (GFS), as extensions of single-winner fairness concepts.
• To investigate the compatibility of these new ex-ante fairness properties with established ex-post fairness guarantees such as Extended Justified Representation (EJR) and Fully Justified Representation (FJR).
• To design algorithms that simultaneously satisfy strong ex-ante and ex-post fairness properties, particularly focusing on computational efficiency and fairness guarantees.

Proposed method

• Proposes a hierarchy of ex-ante fairness axioms—positive share, IFS, UFS, GFS, and their strong variants—based on expected representation and voter coalitions.
• Introduces the Best-Weighted (BW-MES) algorithm, a polynomial-time method that achieves ex-post EJR, ex-ante GFS, and strong UFS by combining a modified method of equal shares (MES) with a greedy candidate selection process.
• Develops the Best-Weighted (BW-GCR) algorithm, which extends the GCR (Group-Consensus-Responsive) approach to ensure ex-post FJR and strong UFS, though it does not run in polynomial time.
• Uses voter-specific candidate payments $ y_{ij} $ and price systems to analyze fairness guarantees, drawing on the concept of priceability from prior work.
• Employs logical and game-theoretic reasoning to prove that the BW-MES algorithm satisfies strong UFS and GFS, even when voters have disjoint approvals.
• Provides a counterexample showing that the GCR-based algorithm does not guarantee GFS, highlighting the need for more refined payment mechanisms.

Experimental results

Research questions

• RQ1 Can ex-ante fairness properties such as GFS and strong UFS be meaningfully defined and formalized in the context of approval-based committee voting?
• RQ2 Is it possible to design a polynomial-time algorithm that simultaneously satisfies ex-post EJR and strong ex-ante fairness (GFS and strong UFS)?
• RQ3 Can ex-ante GFS be achieved in conjunction with stronger ex-post fairness guarantees such as FJR?
• RQ4 What is the computational and fairness trade-off between polynomial-time algorithms and non-polynomial-time algorithms that achieve stronger ex-post fairness?

Key findings

• The BW-MES algorithm runs in polynomial time and satisfies ex-post EJR, ex-ante GFS, and strong UFS, demonstrating the feasibility of combining strong ex-ante and ex-post fairness in a computationally efficient way.
• The BW-GCR algorithm achieves ex-post FJR and ex-ante strong UFS, but does not run in polynomial time, indicating a trade-off between fairness strength and computational efficiency.
• The paper shows that the GCR algorithm alone does not guarantee ex-ante GFS, even though it satisfies strong UFS, by constructing a counterexample with three voters and four candidates.
• A price system can be constructed for the output of GCR, but this does not suffice to guarantee a non-zero lower bound on voter contributions to approved candidates, thus failing to imply GFS.
• The paper establishes a logical hierarchy of ex-ante and ex-post fairness properties, showing that strong UFS implies GFS, and that FJR implies EJR, with implications across the fairness spectrum.
• An open question remains: whether a randomized committee exists that satisfies both ex-ante GFS and ex-post FJR, highlighting a key direction for future research.