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Optimized Distortion in Linear Social Choice

Luise Ge, Gregory Kehne, Yevgeniy Vorobeychik

TL;DR

This paper introduces a comprehensive study of distortion in linear social choice where voters and candidates are embedded as nonnegative vectors in $\mathbb{R}^d$ and utilities are inner products $u_v(c)=v^T c$. It derives dimension-dependent bounds for both deterministic and randomized rules (e.g., MCP with $\mathrm{D}=O(d^3)$ and Linear Stable Lottery achieving $\mathrm{D}=\Theta(\sqrt{d})$ when embeddings are known, with PSLR and $O(d)$ guarantees for unknown embeddings) and develops an LP-based framework to compute instance-optimal distortion-minimizing rules. The authors also provide empirical validation on real-world embeddings (MovieLens and abortion opinions) showing that instance-optimal mechanisms outperform standard rules and that distortion behavior can be governed by dimension rather than candidate count. The results offer robust, scalable approaches for alignment and recommendation settings where linear utility structure and high-dimensional embeddings are natural, and they open paths for future exploration of alternative welfare notions and robustness to learning errors.

Abstract

Social choice theory offers a wealth of approaches for selecting a candidate on behalf of voters based on their reported preference rankings over options. When voters have underlying utilities for these options, however, using preference rankings may lead to suboptimal outcomes vis-à-vis utilitarian social welfare. Distortion is a measure of this suboptimality, and provides a worst-case approach for developing and analyzing voting rules when utilities have minimal structure. However in many settings, such as common paradigms for value alignment, alternatives admit a vector representation, and it is natural to suppose that utilities are parametric functions thereof. We undertake the first study of distortion for linear utility functions. Specifically, we investigate the distortion of linear social choice for deterministic and randomized voting rules. We obtain bounds that depend only on the dimension of the candidate embedding, and are independent of the numbers of candidates or voters. Additionally, we introduce poly-time instance-optimal algorithms for minimizing distortion given a collection of candidates and votes. We empirically evaluate these in two real-world domains: recommendation systems using collaborative filtering embeddings, and opinion surveys utilizing language model embeddings, benchmarking several standard rules against our instance-optimal algorithms.

Optimized Distortion in Linear Social Choice

TL;DR

This paper introduces a comprehensive study of distortion in linear social choice where voters and candidates are embedded as nonnegative vectors in and utilities are inner products . It derives dimension-dependent bounds for both deterministic and randomized rules (e.g., MCP with and Linear Stable Lottery achieving when embeddings are known, with PSLR and guarantees for unknown embeddings) and develops an LP-based framework to compute instance-optimal distortion-minimizing rules. The authors also provide empirical validation on real-world embeddings (MovieLens and abortion opinions) showing that instance-optimal mechanisms outperform standard rules and that distortion behavior can be governed by dimension rather than candidate count. The results offer robust, scalable approaches for alignment and recommendation settings where linear utility structure and high-dimensional embeddings are natural, and they open paths for future exploration of alternative welfare notions and robustness to learning errors.

Abstract

Social choice theory offers a wealth of approaches for selecting a candidate on behalf of voters based on their reported preference rankings over options. When voters have underlying utilities for these options, however, using preference rankings may lead to suboptimal outcomes vis-à-vis utilitarian social welfare. Distortion is a measure of this suboptimality, and provides a worst-case approach for developing and analyzing voting rules when utilities have minimal structure. However in many settings, such as common paradigms for value alignment, alternatives admit a vector representation, and it is natural to suppose that utilities are parametric functions thereof. We undertake the first study of distortion for linear utility functions. Specifically, we investigate the distortion of linear social choice for deterministic and randomized voting rules. We obtain bounds that depend only on the dimension of the candidate embedding, and are independent of the numbers of candidates or voters. Additionally, we introduce poly-time instance-optimal algorithms for minimizing distortion given a collection of candidates and votes. We empirically evaluate these in two real-world domains: recommendation systems using collaborative filtering embeddings, and opinion surveys utilizing language model embeddings, benchmarking several standard rules against our instance-optimal algorithms.
Paper Structure (19 sections, 24 theorems, 53 equations, 6 figures, 2 tables, 3 algorithms)

This paper contains 19 sections, 24 theorems, 53 equations, 6 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Summary of contributions.
  3. Further related work.
  4. Model and Preliminaries
  5. Special case: unit-sum utilities.
  6. Deterministic Rules
  7. Randomized Rules
  8. Known Candidate Embeddings
  9. Unknown Candidate Embeddings
  10. Optimizing Distortion
  11. Experiments
  12. Conclusion
  13. Omitted Proofs
  14. Pseudocode for Implementations of Instance-Optimal Rules
  15. Results Under $\ell^2$ Normalization
  16. ...and 4 more sections

Key Result

Lemma 1

For any $v$ and any candidates ${C=\{c_j\}_{j \in [m]}}$, the maximum utility of $v$ is at least $\max_{c \in C} u_v(c) \geq \frac{1}{d}$.

Figures (6)

  • Figure 1: Instance distortion $\mathrm{D}(f,\vec{\sigma})$. Left: MovieLens; Right: Abortion Opinion Survey
  • Figure 2: MovieLens Varying d (n=100,m=25). Left: Empirical Distortion; Right: Running Time.
  • Figure 3: MovieLens Varying n (m=12, d=10). Left: Instance Distortion; Right: Empirical Distortion.
  • Figure 4: MovieLens Varying m (n=100,d=10). Left: Instance Distortion; Right: Empirical Distortion.
  • Figure 5: MovieLens Running Time. Left: Varying n (m=12, d=10); Right: Varying m (n=100, d=10).
  • ...and 1 more figures

Theorems & Definitions (54)

  • Example 1
  • Definition 1
  • Definition 2: Stable Lotteries
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • Definition 3: Maximum Coordinate Plurality
  • Theorem 3
  • ...and 44 more