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Computing Voting Rules with Improvement Feedback

Evi Micha, Vasilis Varsamis

TL;DR

The paper analyzes how improvement feedback can be used to identify winners under common voting rules when only partial, neighborhood-based feedback is available. It provides a complete characterization for learnability of positional scoring rules: for general $t$, only linear combinations of the plurality rule and a derived $t$-dependent vector $\vec{s}^*_t$ are learnable (and this extends under a uniform feedback model); Condorcet-consistent rules remain infeasible to identify in the worst case with $t$-improvement feedback. The authors construct $t$-indistinguishable preference profiles to prove information-theoretic impossibilities and show that improvement feedback does not suffice for Condorcet-consistent rules, in contrast to pairwise comparisons. Experiments on synthetic ranking models reveal nuanced practical performance, with improvement feedback outperforming pairwise queries in some distributions (e.g., Mallows) and underperforming in others (e.g., Plackett-Luce), suggesting potential benefits from hybrid feedback schemes. The work highlights fundamental limits of incremental feedback for broad classes of voting rules and informs design choices in systems relying on partial user feedback.

Abstract

Aggregating preferences under incomplete or constrained feedback is a fundamental problem in social choice and related domains. While prior work has established strong impossibility results for pairwise comparisons, this paper extends the inquiry to improvement feedback, where voters express incremental adjustments rather than complete preferences. We provide a complete characterization of the positional scoring rules that can be computed given improvement feedback. Interestingly, while plurality is learnable under improvement feedback--unlike with pairwise feedback--strong impossibility results persist for many other positional scoring rules. Furthermore, we show that improvement feedback, unlike pairwise feedback, does not suffice for the computation of any Condorcet-consistent rule. We complement our theoretical findings with experimental results, providing further insights into the practical implications of improvement feedback for preference aggregation.

Computing Voting Rules with Improvement Feedback

TL;DR

The paper analyzes how improvement feedback can be used to identify winners under common voting rules when only partial, neighborhood-based feedback is available. It provides a complete characterization for learnability of positional scoring rules: for general , only linear combinations of the plurality rule and a derived -dependent vector are learnable (and this extends under a uniform feedback model); Condorcet-consistent rules remain infeasible to identify in the worst case with -improvement feedback. The authors construct -indistinguishable preference profiles to prove information-theoretic impossibilities and show that improvement feedback does not suffice for Condorcet-consistent rules, in contrast to pairwise comparisons. Experiments on synthetic ranking models reveal nuanced practical performance, with improvement feedback outperforming pairwise queries in some distributions (e.g., Mallows) and underperforming in others (e.g., Plackett-Luce), suggesting potential benefits from hybrid feedback schemes. The work highlights fundamental limits of incremental feedback for broad classes of voting rules and informs design choices in systems relying on partial user feedback.

Abstract

Aggregating preferences under incomplete or constrained feedback is a fundamental problem in social choice and related domains. While prior work has established strong impossibility results for pairwise comparisons, this paper extends the inquiry to improvement feedback, where voters express incremental adjustments rather than complete preferences. We provide a complete characterization of the positional scoring rules that can be computed given improvement feedback. Interestingly, while plurality is learnable under improvement feedback--unlike with pairwise feedback--strong impossibility results persist for many other positional scoring rules. Furthermore, we show that improvement feedback, unlike pairwise feedback, does not suffice for the computation of any Condorcet-consistent rule. We complement our theoretical findings with experimental results, providing further insights into the practical implications of improvement feedback for preference aggregation.

Paper Structure

This paper contains 24 sections, 18 theorems, 61 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Model
  5. Voting Rules.
  6. Improvement Feedback Distribution.
  7. Improvement Feedback Queries.
  8. $t$-Indistinguishable Preference Profiles.
  9. Main t-Indistinguishable Preference Profiles
  10. Positional Scoring Rules
  11. General t-Improvement Feedback Distribution
  12. Uniform t-Improvement Feedback Distribution
  13. Condorcet Consistent Rules
  14. Uniform t-Improvement Feedback Distribution
  15. General t-Improvement Feedback Distribution
  16. ...and 9 more sections

Key Result

Lemma 3.1

Let $D_{i,j,\ell}$ be a preference profile defined with respect to two candidates $a$ and $b$, as follows: For any $m\geq 4$ and any $t \in [m-1]$, the preference profiles $D_{i,j,\ell}$ and $D^{a\leftrightarrow b}_{i,j,\ell}$ are $t$-indistinguishable when $p = \frac{P^t_{\ell}}{P^t_{i} - P^t_{j} + P^t_{\ell}}$.

Figures (3)

  • Figure 1: Approximation ratio of the optimal Borda score (top) and the optimal Copeland score (bottom) for different numbers of agents and various underlying ranking distributions, with $t = 5$ under the uniform improvement feedback distribution.
  • Figure 2: Different $t$-improvement feedback distributions
  • Figure 3: Varying values of $t$

Theorems & Definitions (33)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 23 more