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

2026-05-12

AI Summary

This study addresses the challenge of achieving individual proportionality (IP)—ensuring that the preferences of distinct groups are reflected in aggregate rankings in proportion to their population sizes—under repeatedly applied linear scoring rules. Focusing on batch ranking aggregation with a fixed linear scoring vector, the work proposes using the angular mean from spherical geometry as the aggregation rule. Theoretical analysis establishes, for the first time, that the angular mean satisfies long-term individual proportionality, while also revealing that exact proportionality per batch is unattainable under any fixed linear rule; however, fairness rapidly converges to the ideal as batch size increases. Empirical evaluation on real-world preference data with high inter-group disagreement demonstrates that the angular mean substantially outperforms conventional approaches such as the arithmetic mean, significantly enhancing proportional fairness without compromising ranking utility.

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 $\langle θ^*,x_j\rangle$ according to a fixed scoring vector $θ^*$. Given voters' preferred scoring vectors $θ^{(1)},\dots,θ^{(n)}$ and their population fractions $α^{(1)},\dots,α^{(n)}$, we ask how to choose a collective vector $θ^*$ satisfying individual proportionality (IP): every voter type $i$ should agree with the resulting rankings to an $α^{(i)}$-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 $θ^{(i)}$, 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.

Problem

• linear ranking rules
• individual proportionality
• collective decision-making
• AI alignment
• participatory design

Innovation

• angular mean
• individual proportionality
• linear ranking rules
• collective decision-making
• AI alignment