{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:TB5OYHWYC7ST4GD5ZFTHCBFDF5","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"5c4bbbf8f7544747dccfe71c02d8504349000ecaee6eae6d4926cc4f8d1f226b","cross_cats_sorted":["cs.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-01-12T12:51:45Z","title_canon_sha256":"c876d467ad447eae36772599977a7086bb05abd83250f3488ae73a64affaa080"},"schema_version":"1.0","source":{"id":"1601.02831","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.02831","created_at":"2026-05-18T01:22:59Z"},{"alias_kind":"arxiv_version","alias_value":"1601.02831v1","created_at":"2026-05-18T01:22:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.02831","created_at":"2026-05-18T01:22:59Z"},{"alias_kind":"pith_short_12","alias_value":"TB5OYHWYC7ST","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"TB5OYHWYC7ST4GD5","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"TB5OYHWY","created_at":"2026-05-18T12:30:44Z"}],"graph_snapshots":[{"event_id":"sha256:c1adc2a64d8c8ceaad6ccbc83423ddf3cd80ffe24ff3180dbdcf89545cfb9d7a","target":"graph","created_at":"2026-05-18T01:22:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Many important values for cooperative games are known to arise from least square optimization problems. The present investigation develops an optimization framework to explain and clarify this phenomenon in a general setting. The main result shows that every linear value results from some least square approximation problem and that, conversely, every least square approximation problem with linear constraints yields a linear value.\n  This approach includes and extends previous results on so-called least square values and semivalues in the literature. In particular, is it demonstrated how known ","authors_text":"Michel Grabisch, Ulrich Faigle","cross_cats":["cs.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-01-12T12:51:45Z","title":"Least Square Approximations and Linear Values of Cooperative Games"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.02831","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c2db05e903eb7b5a9ff6b2564636ee6af377d0555c3089622867d8b2181e96f7","target":"record","created_at":"2026-05-18T01:22:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"5c4bbbf8f7544747dccfe71c02d8504349000ecaee6eae6d4926cc4f8d1f226b","cross_cats_sorted":["cs.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-01-12T12:51:45Z","title_canon_sha256":"c876d467ad447eae36772599977a7086bb05abd83250f3488ae73a64affaa080"},"schema_version":"1.0","source":{"id":"1601.02831","kind":"arxiv","version":1}},"canonical_sha256":"987aec1ed817e53e187dc9667104a32f601145d2fb26b83a9e31158c9b667b0a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"987aec1ed817e53e187dc9667104a32f601145d2fb26b83a9e31158c9b667b0a","first_computed_at":"2026-05-18T01:22:59.505902Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:59.505902Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gVQQHSLTvjdtnFR4Mpl8o1TOM8AfsXDmibWkPHeS33HuSiTs5yceo2utJf2KvkmrjrKDcf4aKTnWtYHdCiFSDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:59.506557Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.02831","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c2db05e903eb7b5a9ff6b2564636ee6af377d0555c3089622867d8b2181e96f7","sha256:c1adc2a64d8c8ceaad6ccbc83423ddf3cd80ffe24ff3180dbdcf89545cfb9d7a"],"state_sha256":"e52b10d54118513fb99c0fed6d968ed6b464d436fcd0d8fbaa6473882d7e7447"}