Linear Social Choice with Few Queries: A Moment-Based Approach

TL;DR

This paper introduces a moment-based approach to linear social choice under extreme communication constraints, showing that a small number of pairwise or graded comparisons per voter can recover voter distribution moments for diverse social choice objectives.

Contribution

It demonstrates that only two comparisons per voter are needed to identify the entire voter-type distribution in a linear social choice model, enabling complex social choice objectives.

Findings

One comparison per voter suffices for social welfare maximization.

Two comparisons per voter identify the second moment of voter types.

Richer queries enable full recovery of voter distribution moments.

Abstract

Most social choice rules assume access to full rankings, while current alignment practice -- despite aiming for diversity -- typically treats voters as anonymous and comparisons as independent, effectively extracting only about one bit per voter. Motivated by this gap, we study social choice under an extreme communication budget in the linear social choice model, where each voter's utility is the inner product between a latent voter type and the embedding of the context and candidate. The candidate and voter spaces may be very large or even infinite. Our core idea is to model the electorate as an unknown distribution over voter types and to recover its moments as informative summary statistics for candidate selection. We show that one pairwise comparison per voter already suffices to select a candidate that maximizes social welfare, but this elicitation cannot identify the second moment and therefore cannot support objectives that account for inequality. We prove that two pairwise comparisons per voter, or alternatively a single graded comparison, identify the second moment; moreover, these richer queries suffice to identify all moments, and hence the entire voter-type distribution. These results enable principled solutions to a range of social choice objectives including inequality-aware welfare criteria such as taking into account the spread of voter utilities and choosing a representative subset.