The End Justifies the Mean: A Linear Ranking Rule for Proportional Sequential Decisions

TL;DR

This paper introduces a simple linear ranking rule called the angular mean that satisfies long-run proportionality among voters, addressing fairness in repeated decision-making scenarios.

Contribution

It proves the angular mean satisfies long-run individual proportionality and analyzes the limitations of fixed linear rules for per-batch proportionality.

Findings

The angular mean satisfies long-run individual proportionality.

Exact per-batch proportionality is impossible with fixed linear rules.

Angular mean improves proportionality in high-disagreement voter datasets.

Abstract

AI alignment and participatory design motivate a new democratic design problem: how to collectively choose a decision rule to use repeatedly. We study this problem for linear ranking rules, which repeatedly rank items $x_j$ within batches $X=(x_1,\dots,x_m)\in(\mathbb{R}^d)^m$, where each item's ranking is dictated by its score $\langle \theta^*,x_j\rangle$ according to a fixed scoring vector $\theta^*$. Given voters' preferred scoring vectors $\theta^{(1)},\dots,\theta^{(n)}$ and their population fractions $\alpha^{(1)},\dots,\alpha^{(n)}$, we ask how to choose a collective vector $\theta^*$ satisfying individual proportionality (IP): every voter type $i$ should agree with the resulting rankings to an $\alpha^{(i)}$-proportional degree, either on average over time (long-run IP) or even within each batch (per-batch IP). The default rule, the arithmetic mean of the $\theta^{(i)}$, has been shown to be severely majoritarian; more generally, it is not clear that any fixed linear rule can balance many voters' disparate opinions. Our main result is that, surprisingly, there is a simple rule that does satisfy long-run IP: the angular mean, the spherical analog of the arithmetic mean. We then show that exact per-batch IP is impossible for fixed linear rules, but that the gap between per-batch and long-run IP shrinks quickly with batch size. Experiments on three real-world preference datasets show that all rules perform similarly when voters' preferences are homogeneous, while the angular mean substantially improves proportionality in high-disagreement regimes.