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Proportionality in Thumbs Up and Down Voting

Sonja Kraiczy, Georgios Papasotiropoulos, Grzegorz Pierczyński, Piotr Skowron

TL;DR

This work examines proportionality when voters may both approve and disapprove candidates under a cap $k$, motivated by applications like constitutional AI and participatory budgeting. It develops two core approaches: a symmetric utility model combining positive and negative votes, and an asymmetric model separating positive representation from veto costs via opposition taxes. It proves that adapted versions of Phragmén's rule and PAV provide strong guarantees in the symmetric setting (Base PJR and near-BEJR), while in the asymmetric setting EJPR and Group Veto are addressed through Tax-MES and Tax-Phragmén, with EJPR satisfied by the former and PJPR by the latter; generalized Thiele rules are shown incompatible with these negative guarantees. The results clarify fundamental trade-offs between representation and veto control, guiding the design of proportional rules for systems with both up and down votes and informing practical deployment in PB, constitutional AI, and digital governance platforms.

Abstract

Consider the decision-making setting where agents elect a panel by expressing both positive and negative preferences. Prominently, in constitutional AI, citizens democratically select a slate of ethical preferences on which a foundation model is to be trained. There, in practice, agents may both approve and disapprove of different ethical principles. Proportionality has been well-studied in computational social choice for approval ballots, but its meaning remains unclear when negative sentiments are also considered. In this work, we propose two conceptually distinct approaches to interpret proportionality in the presence of up and down votes. The first approach treats the satisfaction from electing candidates and the impact of vetoing them as comparable, leading to combined proportionality guarantees. The second approach considers veto power separately, introducing guarantees distinct from traditional proportionality. We formalize axioms for each perspective and examine their satisfiability by suitable adaptations of Phragmén's rule, Proportional Approval Voting rule and the Method of Equal Shares.

Proportionality in Thumbs Up and Down Voting

TL;DR

This work examines proportionality when voters may both approve and disapprove candidates under a cap , motivated by applications like constitutional AI and participatory budgeting. It develops two core approaches: a symmetric utility model combining positive and negative votes, and an asymmetric model separating positive representation from veto costs via opposition taxes. It proves that adapted versions of Phragmén's rule and PAV provide strong guarantees in the symmetric setting (Base PJR and near-BEJR), while in the asymmetric setting EJPR and Group Veto are addressed through Tax-MES and Tax-Phragmén, with EJPR satisfied by the former and PJPR by the latter; generalized Thiele rules are shown incompatible with these negative guarantees. The results clarify fundamental trade-offs between representation and veto control, guiding the design of proportional rules for systems with both up and down votes and informing practical deployment in PB, constitutional AI, and digital governance platforms.

Abstract

Consider the decision-making setting where agents elect a panel by expressing both positive and negative preferences. Prominently, in constitutional AI, citizens democratically select a slate of ethical preferences on which a foundation model is to be trained. There, in practice, agents may both approve and disapprove of different ethical principles. Proportionality has been well-studied in computational social choice for approval ballots, but its meaning remains unclear when negative sentiments are also considered. In this work, we propose two conceptually distinct approaches to interpret proportionality in the presence of up and down votes. The first approach treats the satisfaction from electing candidates and the impact of vetoing them as comparable, leading to combined proportionality guarantees. The second approach considers veto power separately, introducing guarantees distinct from traditional proportionality. We formalize axioms for each perspective and examine their satisfiability by suitable adaptations of Phragmén's rule, Proportional Approval Voting rule and the Method of Equal Shares.

Paper Structure

This paper contains 13 sections, 14 theorems, 50 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Symmetric Utility Model
  4. The Base Extended Justified Representation Axiom
  5. Phragmén's Rule
  6. Proportional Approval Voting Rule
  7. The Asymmetric Utility Model
  8. The Extended Justified Positive Representation and the Group Veto Axioms
  9. Method of Equal Shares and Phragmén with Opposition Tax
  10. Generalized Thiele Rules
  11. Conclusion & Discussion
  12. Proofs Omitted from the Main Text
  13. Negative Guarantee for Exhaustive Priceable Rules

Key Result

Lemma 0

An outcome $W$ satisfies Base EJR if and only if for every set of voters $S$ there exists a voter $i \in S$ with $|A_i \cap W| + |D_i \cap (C \setminus W)| \geqslant {{\mathrm{claim}}}(S)$, where:

Figures (3)

  • Figure 1: Illustration of sets and parameters involved in the proof of \ref{['thm:pavpositive']}. We denote by $\text{dom}(\phi)$ the domain of an injection $\phi$ and by $\text{co-dom}(\phi)$ its co-domain.
  • Figure 2: Illustration of the example presented in the proof of \ref{['thm:pavpositive']}. The table on the left depicts the primary swaps associated with $\phi$. The right part of the figure illustrates the operation of merging two swaps.
  • Figure :

Theorems & Definitions (35)

  • Example 1
  • Definition 1: Base Extended Justified Representation (Base EJR)
  • Definition 2: Base Proportional Justified Representation (Base PJR)
  • Lemma 0: $\hbox{$\spadesuit$}$
  • proof : Proof sketch
  • Lemma 0: $\hbox{$\spadesuit$}$
  • proof
  • Lemma 1
  • proof
  • Lemma 1
  • ...and 25 more