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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).\n  The default rule, the arithmetic mean of the $\\theta^{(i)}$, has been shown to be severely majoritarian; more gener","title":"The End Justifies the Mean: A Linear Ranking Rule for Proportional Sequential Decisions","url":"https://arxiv.org/abs/2605.12717","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.12717v1 Announce Type: cross \nAbstract: 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).\n  The default rule, the arithmetic mean of the $\\theta^{(i)}$, has been shown to be severely majoritarian; more gener","title":"The End Justifies the Mean: A Linear Ranking Rule for Proportional Sequential Decisions","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-14T04:43:36Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.12717"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:ed4a627a237a200b45bee536a8c421614648922a18afa2f089266781d239e16a31f4c15aedf33c1174bdd4aced76cae64adc00ae148bb341adaf0723acc21f07","signer":"crovia.substrate","subject":{"observed_at":"2026-05-14T04:43:36Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.12717"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"5b722e57f9afd76c6403ce2f5a34982a0a9b28f5cabd03b523d5d5adf53d157e","leaf_index":132530,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"700cb97aaf7e52cccecd47321cefb49803f1b7a3a143497bf68602f45e2d53b9","side":"right"},{"sibling":"006972a33a81f8e25e13711dccc0ad04cc5d29cbc4f4700ed43702142665c033","side":"left"},{"sibling":"03da6ae968edf1f1e717f612980e9e4f0a70ee32bd0c579e6ac82c75464390cc","side":"right"},{"sibling":"c6e3da853a3d65776fdd860284b0643067c467d86b5c45079d8624b4af9c3dca","side":"right"},{"sibling":"56a91ef12db06249e819653353d7cb2f25e335a4beb3dd34fe0fdfdec5f7d268","side":"left"},{"sibling":"a8e004cebfbccd83f7bc54c05605283a0d70229221861946eef503a2e83669a3","side":"left"},{"sibling":"59a072228323deebace497ba94751992cacb1b239c38d6f6fe53aa4448667365","side":"right"},{"sibling":"5481966fbd92dc66ec8b3fff9fc4dcddbff0f35b3e0cafc74158f753d7858d6b","side":"left"},{"sibling":"ed110cac74740161e005dcd47c35637ac99ecf93121b3961837570c3af56caa0","side":"left"},{"sibling":"2270188e7d95b52d2bade0ab7da2d0816592f30757dcf71636f21b8f46eb1317","side":"right"},{"sibling":"c03f0a468f574a08ffe8b17e1a17bd88216e1359f0e33a54447e160cd8675da0","side":"left"},{"sibling":"038ff12da6f55509125ef0d96e1e57dda29a2fe63bf03fba2af4cf7cbcd88b36","side":"right"},{"sibling":"b0419206fe62ef216df3900ca93cffd44df267435a4643dda354c1310079cf91","side":"right"},{"sibling":"0d4a9c03674f9d0ce64df15c15e9f656a54b41f93c428aac8e615fda26291956","side":"right"},{"sibling":"7856d920f3f1f1d2194c1ed7351bf0d674440df3cb423a6911f89cb3e9578c0b","side":"right"},{"sibling":"6ac6396bdd2e9df315427a46155531476e7f9012b4bd962e0d2d6d1209b11723","side":"right"},{"sibling":"7c8dc85cbfe43e19ac759ad176cfa11dba2467ae17927c471d5c55663c4f490d","side":"right"},{"sibling":"d841ad93efda0869e5eb97678f348f03f5caab4353e05ff4bf18f47fb945b822","side":"left"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":134292,"merkle_root":"77fc9c28fae777b81da5b495b3115474df6592dfac590333213d3bdf8b94a9b3","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260515T023701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-15T02:37:25Z","sig_algorithm":"ed25519","signature":"68107a834b00b24f5d4501e5ec727445311f132a486567ecc4c72a4e6dff24c8c21f2de3105293353ba5fdbe370d032819af6aa70f694e2e39b6af6737507009","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_99e6f16a753b30554cdb89172fb6372117eba292ccf03e2fbe4b4119f2dadcdd"}}