{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:DZQMZLWKTKZ4QYFB7QV7G53UZG","short_pith_number":"pith:DZQMZLWK","schema_version":"1.0","canonical_sha256":"1e60ccaeca9ab3c860a1fc2bf37774c9aa80f52102dbb6fcb49aed8829bcdd26","source":{"kind":"arxiv","id":"1103.0868","version":2},"attestation_state":"computed","paper":{"title":"On minimum integer representations of weighted games","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.GT"],"primary_cat":"math.CO","authors_text":"Josep Freixas, Sascha Kurz","submitted_at":"2011-03-04T11:23:36Z","abstract_excerpt":"We study minimum integer representations of weighted games, i.e., representations where the weights are integers and every other integer representation is at least as large in each component. Those minimum integer representations, if the exist at all, are linked with some solution concepts in game theory. Closing existing gaps in the literature, we prove that each weighted game with two types of voters admits a (unique) minimum integer representation, and give new examples for more than two types of voters without a minimum integer representation. We characterize the possible weights in minimu"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1103.0868","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-04T11:23:36Z","cross_cats_sorted":["cs.GT"],"title_canon_sha256":"abfc0805231592e63edbf5a582c2ea1a19616e81cdf26ca61f766e19b04210ad","abstract_canon_sha256":"da6adc9bbb2729e12ed4c8577a1e229d3f2447bf38c03a8cf0692345855b4db7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:32.995890Z","signature_b64":"KcN5T6OFyirUbhFbQ4cwaCUoDcQAdEInj2uQXhQKz8wd6BiOKJkorS8RED/1v8hbkexyjS6iyKfXCR1T0505Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1e60ccaeca9ab3c860a1fc2bf37774c9aa80f52102dbb6fcb49aed8829bcdd26","last_reissued_at":"2026-05-18T03:06:32.995282Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:32.995282Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On minimum integer representations of weighted games","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.GT"],"primary_cat":"math.CO","authors_text":"Josep Freixas, Sascha Kurz","submitted_at":"2011-03-04T11:23:36Z","abstract_excerpt":"We study minimum integer representations of weighted games, i.e., representations where the weights are integers and every other integer representation is at least as large in each component. Those minimum integer representations, if the exist at all, are linked with some solution concepts in game theory. Closing existing gaps in the literature, we prove that each weighted game with two types of voters admits a (unique) minimum integer representation, and give new examples for more than two types of voters without a minimum integer representation. We characterize the possible weights in minimu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.0868","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1103.0868","created_at":"2026-05-18T03:06:32.995393+00:00"},{"alias_kind":"arxiv_version","alias_value":"1103.0868v2","created_at":"2026-05-18T03:06:32.995393+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.0868","created_at":"2026-05-18T03:06:32.995393+00:00"},{"alias_kind":"pith_short_12","alias_value":"DZQMZLWKTKZ4","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_16","alias_value":"DZQMZLWKTKZ4QYFB","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_8","alias_value":"DZQMZLWK","created_at":"2026-05-18T12:26:26.731475+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DZQMZLWKTKZ4QYFB7QV7G53UZG","json":"https://pith.science/pith/DZQMZLWKTKZ4QYFB7QV7G53UZG.json","graph_json":"https://pith.science/api/pith-number/DZQMZLWKTKZ4QYFB7QV7G53UZG/graph.json","events_json":"https://pith.science/api/pith-number/DZQMZLWKTKZ4QYFB7QV7G53UZG/events.json","paper":"https://pith.science/paper/DZQMZLWK"},"agent_actions":{"view_html":"https://pith.science/pith/DZQMZLWKTKZ4QYFB7QV7G53UZG","download_json":"https://pith.science/pith/DZQMZLWKTKZ4QYFB7QV7G53UZG.json","view_paper":"https://pith.science/paper/DZQMZLWK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1103.0868&json=true","fetch_graph":"https://pith.science/api/pith-number/DZQMZLWKTKZ4QYFB7QV7G53UZG/graph.json","fetch_events":"https://pith.science/api/pith-number/DZQMZLWKTKZ4QYFB7QV7G53UZG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DZQMZLWKTKZ4QYFB7QV7G53UZG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DZQMZLWKTKZ4QYFB7QV7G53UZG/action/storage_attestation","attest_author":"https://pith.science/pith/DZQMZLWKTKZ4QYFB7QV7G53UZG/action/author_attestation","sign_citation":"https://pith.science/pith/DZQMZLWKTKZ4QYFB7QV7G53UZG/action/citation_signature","submit_replication":"https://pith.science/pith/DZQMZLWKTKZ4QYFB7QV7G53UZG/action/replication_record"}},"created_at":"2026-05-18T03:06:32.995393+00:00","updated_at":"2026-05-18T03:06:32.995393+00:00"}