{"paper":{"title":"The End Justifies the Mean: A Linear Ranking Rule for Proportional Sequential Decisions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The angular mean of voter vectors satisfies long-run individual proportionality for repeated linear rankings.","cross_cats":["cs.AI"],"primary_cat":"cs.GT","authors_text":"Bailey Flanigan, Carmel Baharav, Maximilian T. Wittmann, Niclas Boehmer","submitted_at":"2026-05-12T20:26:37Z","abstract_excerpt":"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 $\\theta^*$ "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That long-run average proportionality is acceptable even when per-batch outcomes can deviate substantially, and that voter preferences remain fixed across batches.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The angular mean of voter scoring vectors satisfies long-run individual proportionality for sequential linear ranking decisions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The angular mean of voter vectors satisfies long-run individual proportionality for repeated linear rankings.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c8a0a0cf58106f9c53006213f855fd3b6aca16c14a34bcadcccf662a3b1d7e15"},"source":{"id":"2605.12717","kind":"arxiv","version":1},"verdict":{"id":"ce602db4-2870-4808-94e8-ad1673f33dcb","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:52:24.096293Z","strongest_claim":"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.","one_line_summary":"The angular mean of voter scoring vectors satisfies long-run individual proportionality for sequential linear ranking decisions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That long-run average proportionality is acceptable even when per-batch outcomes can deviate substantially, and that voter preferences remain fixed across batches.","pith_extraction_headline":"The angular mean of voter vectors satisfies long-run individual proportionality for repeated linear rankings."},"references":{"count":50,"sample":[{"doi":"","year":2013,"title":"On the convergence of gradient descent for finding the Riemannian center of mass","work_id":"63fa1ffd-f1c8-440a-ad4c-688565a2a94d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"The moral machine experiment","work_id":"f3db6ff3-741e-4e73-9145-61daa7f87012","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"On the stability of moral preferences: A problem with computational elicitation methods","work_id":"2aa99587-1a15-4b27-ace5-d9eff8d192c7","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Optimal budget aggregation with single-peaked preferences","work_id":"6fe369b5-00d3-4d1b-9767-8e6df37f8ff6","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"Justified representation for perpetual voting","work_id":"3a0ce6bc-8d7b-46eb-aa02-c68e86208775","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":50,"snapshot_sha256":"bf3a5ee429ad6fde8e84e430e1590463b2bd60f14745e807fca9c078b49f6696","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"57ef7755063a80e809e2cef5b1a1a2f6623d8549d7389426ed99fa04dc846fa6"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}