[Paper Review] Characterizations of Sequential Valuation Rules

Chris Dong, Patrick Lederer|arXiv (Cornell University)|Feb 23, 2023

Game Theory and Voting SystemsEconomics, Econometrics and Finance3 citations

TL;DR

This paper provides axiomatic characterizations of approval-based committee (ABC) voting rules within the class of sequential valuation rules, introducing the novel axiom of consistent committee monotonicity. It shows that proper sequential valuation rules—satisfying anonymity, neutrality, non-imposition, and continuity—are exactly step-dependent sequential scoring rules, and further characterizes sequential Thiele rules and three prominent ABC rules: sequential approval voting, sequential proportional approval voting, and sequential Chamberlin-Courant voting.

ABSTRACT

Approval-based committee (ABC) voting rules elect a fixed size subset of the candidates, a so-called committee, based on the voters' approval ballots over the candidates. While these rules have recently attracted significant attention, axiomatic characterizations are largely missing so far. We address this problem by characterizing ABC voting rules within the broad and intuitive class of sequential valuation rules. These rules compute the winning committees by sequentially adding candidates that increase the score of the chosen committee the most. In more detail, we first characterize almost the full class of sequential valuation rules based on mild standard conditions and a new axiom called consistent committee monotonicity. This axiom postulates that the winning committees of size k can be derived from those of size k-1 by only adding candidates and that these new candidates are chosen consistently. By requiring additional conditions, we derive from this result also a characterization of the prominent class of sequential Thiele rules. Finally, we refine our results to characterize three well-known ABC voting rules, namely sequential approval voting, sequential proportional approval voting, and sequential Chamberlin-Courant approval voting.

Motivation & Objective

• To address the lack of axiomatic characterizations for approval-based committee (ABC) voting rules, especially within the broad class of sequential valuation rules.
• To introduce and formalize the new axiom of consistent committee monotonicity, combining committee monotonicity and consistency to ensure stable and rational sequential extension of committees.
• To characterize the full class of proper sequential valuation rules (satisfying anonymity, neutrality, non-imposition, and continuity) as step-dependent sequential scoring rules.
• To refine the characterization to identify sequential Thiele rules and three specific ABC rules: sequential approval voting, sequential proportional approval voting, and sequential Chamberlin-Courant approval voting.
• To establish clone-proportionality as a key property distinguishing sequential proportional approval voting from other sequential Thiele rules, using axiomatic reasoning.

Proposed method

• Introduces sequential valuation rules that build winning committees incrementally by selecting candidates that maximize the total score based on a valuation function over ballot-committee pairs.
• Defines a valuation function that depends on the sizes of the ballot, the committee, and their intersection, leading to the class of step-dependent sequential scoring rules.
• Proposes the new axiom of consistent committee monotonicity, requiring that when two disjoint electorates independently extend a committee with the same candidates, the combined electorate should also extend it with those same candidates.
• Applies standard axioms—anonymous, neutral, non-imposing, and continuous—to narrow the class of rules and derive structural properties of the valuation functions.
• Uses proof by contradiction and strategic profile construction to show that only specific valuation functions (e.g., harmonic weights for seqPAV) satisfy clone-proportionality and other refined axioms.
• Employs case analysis on Thiele counting functions to prove that only sequential proportional approval voting satisfies clone-proportionality among sequential Thiele rules.

Experimental results

Research questions

• RQ1 Which sequential valuation rules satisfy the mild standard axioms of anonymity, neutrality, non-imposition, and continuity?
• RQ2 How can consistent committee monotonicity be formalized, and what structural constraints does it impose on sequential ABC voting rules?
• RQ3 Which sequential valuation rules are equivalent to step-dependent sequential scoring rules under the proposed axioms?
• RQ4 Which sequential Thiele rules are uniquely characterized by additional axioms such as clone-proportionality?
• RQ5 What distinguishes sequential proportional approval voting from other sequential Thiele rules in terms of axiomatic properties?

Key findings

• All proper sequential valuation rules—satisfying anonymity, neutrality, non-imposition, and continuity—are equivalent to step-dependent sequential scoring rules, where the valuation function depends only on the sizes of the ballot, committee, and their intersection.
• The class of proper and consistently committee monotone sequential valuation rules is exactly the class of step-dependent sequential scoring rules, establishing a strong foundation for their use.
• Sequential approval voting is the unique sequential valuation rule satisfying consistency, committee monotonicity, and non-imposition, and is characterized by the harmonic scoring function.
• Sequential proportional approval voting is the unique sequential Thiele rule satisfying clone-proportionality, distinguishing it from other sequential Thiele rules.
• Sequential Chamberlin-Courant approval voting is characterized as the unique sequential valuation rule where the valuation function depends only on whether the intersection is non-empty, and the committee is built to maximize representation diversity.
• The paper proves that any sequential Thiele rule other than sequential proportional approval voting fails clone-proportionality, by constructing profiles where the rule elects a candidate that should be prioritized due to high approval support.