Optimized Distortion in Linear Social Choice

TL;DR

This paper studies how to select candidates optimally when voters' utilities are linear functions of candidate features, providing bounds on distortion and algorithms for minimizing it, with applications in recommendation systems and opinion surveys.

Contribution

It introduces the first analysis of distortion for linear utility functions in social choice, with bounds depending on embedding dimension and new instance-optimal algorithms.

Findings

Bounds on distortion depend only on embedding dimension.

Poly-time algorithms for minimizing distortion are developed.

Empirical evaluation shows improved performance in real-world domains.

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-\`a-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.