New 'Angular Mean' Rule Solves AI Proportionality Problem
Researchers find a simple fix for democratic AI ranking that satisfies all voter groups over time.
A new paper from Carmel Baharav, Niclas Boehmer, Bailey Flanigan, and Maximilian Wittmann tackles a core AI alignment challenge: how to collectively choose a decision rule when multiple human voters have conflicting preference vectors (scoring vectors). The authors study linear ranking rules that repeatedly score items within batches. The default arithmetic mean of voters' vectors is shown to be severely majoritarian, overrepresenting larger groups.
Their key contribution is the angular mean—the spherical average of voting vectors—which satisfies long-run individual proportionality (IP): every voter type agrees with the resulting rankings proportionally over time. The team proves that exact per-batch IP is impossible for fixed linear rules, but the gap between per-batch and long-run IP shrinks quickly with batch size. Real-world tests on three preference datasets show the angular mean substantially improves proportionality when voter preferences are diverse.
- Angular mean achieves long-run individual proportionality (IP) for sequential ranking decisions, unlike the arithmetic mean which is majoritarian.
- Exact per-batch proportionality is provably impossible for fixed linear rules, but the gap decreases rapidly with batch size.
- Experiments on three real-world preference datasets show angular mean performs similarly in homogeneous groups but improves fairness significantly in high-disagreement scenarios.
Why It Matters
A practical mathematical tool to democratize AI systems that make repeated ranking decisions, from content moderation to resource allocation.