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The Proportional Veto Principle for Approval Ballots

Daniel Halpern, Ariel D. Procaccia, Warut Suksompong

TL;DR

This work introduces flexible-voter representation (FVR) for approval ballots, quantifying how a large, flexible coalition can veto a poorly supported candidate and proving robust guarantees across all flexibility thresholds. It develops a complete single-winner theory: a universal lower bound $\mathrm{FVR}(R,s) \ge 1-s$, a baseline $s$-threshold rule achieving equality, and a suite of scoring rules including the optimal $R_{\text{OPT}}$ with $w(f)=\frac{1}{1-f}$ that attains FVR-optimality for every $s$. The authors extend FVR to multi-winner elections, deriving hypergeometric-based lower bounds, constructing simultaneously optimal rules via expandedInstances and a polynomial-time sequential algorithm, and showing incompatibilities with other representation notions such as JR. Collectively, the results provide a principled, tunable framework for approval-based representation that rewards voter flexibility and yields strong, theory-backed guarantees for both single and multi-winner settings. The work has practical relevance for participatory budgeting and governance contexts where broad consensus and legitimacy are paramount, and it establishes foundational connections between FVR, proportional veto, and scoring-rule design under approval-based preferences.

Abstract

The proportional veto principle, which captures the idea that a candidate vetoed by a large group of voters should not be chosen, has been studied for ranked ballots in single-winner voting. We introduce a version of this principle for approval ballots, which we call flexible-voter representation (FVR). We show that while the approval voting rule and other natural scoring rules provide the optimal FVR guarantee only for some flexibility threshold, there exists a scoring rule that is FVR-optimal for all thresholds simultaneously. We also extend our results to multi-winner voting.

The Proportional Veto Principle for Approval Ballots

TL;DR

This work introduces flexible-voter representation (FVR) for approval ballots, quantifying how a large, flexible coalition can veto a poorly supported candidate and proving robust guarantees across all flexibility thresholds. It develops a complete single-winner theory: a universal lower bound , a baseline -threshold rule achieving equality, and a suite of scoring rules including the optimal with that attains FVR-optimality for every . The authors extend FVR to multi-winner elections, deriving hypergeometric-based lower bounds, constructing simultaneously optimal rules via expandedInstances and a polynomial-time sequential algorithm, and showing incompatibilities with other representation notions such as JR. Collectively, the results provide a principled, tunable framework for approval-based representation that rewards voter flexibility and yields strong, theory-backed guarantees for both single and multi-winner settings. The work has practical relevance for participatory budgeting and governance contexts where broad consensus and legitimacy are paramount, and it establishes foundational connections between FVR, proportional veto, and scoring-rule design under approval-based preferences.

Abstract

The proportional veto principle, which captures the idea that a candidate vetoed by a large group of voters should not be chosen, has been studied for ranked ballots in single-winner voting. We introduce a version of this principle for approval ballots, which we call flexible-voter representation (FVR). We show that while the approval voting rule and other natural scoring rules provide the optimal FVR guarantee only for some flexibility threshold, there exists a scoring rule that is FVR-optimal for all thresholds simultaneously. We also extend our results to multi-winner voting.
Paper Structure (26 sections, 13 theorems, 46 equations, 1 figure, 1 algorithm)

This paper contains 26 sections, 13 theorems, 46 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Overview of Results
  3. Additional Discussion
  4. Single-Winner Voting
  5. Preliminaries
  6. Lower Bound
  7. Approval Voting Rule
  8. Power Scoring Rules
  9. Optimal Scoring Rule
  10. General Weight Function Analysis
  11. Multi-Winner Voting
  12. Preliminaries
  13. Lower Bound
  14. Simultaneously-Optimal Rules
  15. Compatibility with Other Notions of Representation
  16. ...and 11 more sections

Key Result

Theorem 2.2

For any rule $R$ and any $s\in (0,1)$, we have $\mathrm{FVR}\xspace(R,s) \ge 1-s$. Moreover, for each $s\in (0,1)$, there exists a rule $R$ such that $\mathrm{FVR}\xspace(R,s) = 1-s$.

Figures (1)

  • Figure 1: FVR guarantees of $p$-power scoring rules for flexibility thresholds $s\in (0,1)$, compared to the optimal guarantees indicated by the thick line. For $p=1, 2$, the guarantees match the optimal ones at $s = 1/2, 2/3$, respectively.

Theorems & Definitions (21)

  • Example 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • proof
  • ...and 11 more