{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:7Z6BKV2MGH5M4EM7YWPPSZRXPI","short_pith_number":"pith:7Z6BKV2M","canonical_record":{"source":{"id":"1711.06611","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-17T16:11:30Z","cross_cats_sorted":[],"title_canon_sha256":"50fb2a23a383383c04ea2fc8a5fe32ead275f144f583ce345a0e46472884666e","abstract_canon_sha256":"a69823d81e8925a8815c3c3c79443ed8c5d86dbb6dfe1ebea0543013a4e5883e"},"schema_version":"1.0"},"canonical_sha256":"fe7c15574c31face119fc59ef966377a1aa77c65799157b4ab494eae16da9498","source":{"kind":"arxiv","id":"1711.06611","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.06611","created_at":"2026-05-17T23:59:39Z"},{"alias_kind":"arxiv_version","alias_value":"1711.06611v3","created_at":"2026-05-17T23:59:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.06611","created_at":"2026-05-17T23:59:39Z"},{"alias_kind":"pith_short_12","alias_value":"7Z6BKV2MGH5M","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7Z6BKV2MGH5M4EM7","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7Z6BKV2M","created_at":"2026-05-18T12:31:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:7Z6BKV2MGH5M4EM7YWPPSZRXPI","target":"record","payload":{"canonical_record":{"source":{"id":"1711.06611","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-17T16:11:30Z","cross_cats_sorted":[],"title_canon_sha256":"50fb2a23a383383c04ea2fc8a5fe32ead275f144f583ce345a0e46472884666e","abstract_canon_sha256":"a69823d81e8925a8815c3c3c79443ed8c5d86dbb6dfe1ebea0543013a4e5883e"},"schema_version":"1.0"},"canonical_sha256":"fe7c15574c31face119fc59ef966377a1aa77c65799157b4ab494eae16da9498","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:39.446089Z","signature_b64":"2qVa92Oo+/8K52Kg6LkqA+vWNnqTH9TA+wEw1pDDc5Q+8YiyVGZknpF2PGFvJxuFKiKNGYwT+MCF5/E3/7hQDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe7c15574c31face119fc59ef966377a1aa77c65799157b4ab494eae16da9498","last_reissued_at":"2026-05-17T23:59:39.445333Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:39.445333Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1711.06611","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:59:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E72rk+Bg12u9VX+61gdf8knRk55DDdagNB/lxzec5GjlNSMRou4dnLHZS8ozCYOmpblFvlmPx7U+wXNgPskLBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T18:22:09.419488Z"},"content_sha256":"eb9073bb569198ef774f16f26434712e0a1692f822b0a28acfd13ea8f29e46c8","schema_version":"1.0","event_id":"sha256:eb9073bb569198ef774f16f26434712e0a1692f822b0a28acfd13ea8f29e46c8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:7Z6BKV2MGH5M4EM7YWPPSZRXPI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bounds for the Nakamura number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Josep Freixas, Sascha Kurz","submitted_at":"2017-11-17T16:11:30Z","abstract_excerpt":"The Nakamura number is an appropriate invariant of a simple game to study the existence of social equilibria and the possibility of cycles. For symmetric quota games its number can be obtained by an easy formula. For some subclasses of simple games the corresponding Nakamura number has also been characterized. However, in general, not much is known about lower and upper bounds depending of invariants on simple, complete or weighted games. Here, we survey such results and highlight connections with other game theoretic concepts."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.06611","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:59:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5i+MAnM+0GzoC0DIgRBRMXuyxALm3+NE8IGigMGLe0v/DwaunuXEWnC+vMJd3OpSsrs6oseB8ZuSjQX1Ygh/CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T18:22:09.420147Z"},"content_sha256":"f4feccdd1d660cf2246f4eb21281edb728760b2ea4fc585c03dcb0bcd92431d4","schema_version":"1.0","event_id":"sha256:f4feccdd1d660cf2246f4eb21281edb728760b2ea4fc585c03dcb0bcd92431d4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7Z6BKV2MGH5M4EM7YWPPSZRXPI/bundle.json","state_url":"https://pith.science/pith/7Z6BKV2MGH5M4EM7YWPPSZRXPI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7Z6BKV2MGH5M4EM7YWPPSZRXPI/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-25T18:22:09Z","links":{"resolver":"https://pith.science/pith/7Z6BKV2MGH5M4EM7YWPPSZRXPI","bundle":"https://pith.science/pith/7Z6BKV2MGH5M4EM7YWPPSZRXPI/bundle.json","state":"https://pith.science/pith/7Z6BKV2MGH5M4EM7YWPPSZRXPI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7Z6BKV2MGH5M4EM7YWPPSZRXPI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:7Z6BKV2MGH5M4EM7YWPPSZRXPI","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":"a69823d81e8925a8815c3c3c79443ed8c5d86dbb6dfe1ebea0543013a4e5883e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-17T16:11:30Z","title_canon_sha256":"50fb2a23a383383c04ea2fc8a5fe32ead275f144f583ce345a0e46472884666e"},"schema_version":"1.0","source":{"id":"1711.06611","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.06611","created_at":"2026-05-17T23:59:39Z"},{"alias_kind":"arxiv_version","alias_value":"1711.06611v3","created_at":"2026-05-17T23:59:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.06611","created_at":"2026-05-17T23:59:39Z"},{"alias_kind":"pith_short_12","alias_value":"7Z6BKV2MGH5M","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7Z6BKV2MGH5M4EM7","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7Z6BKV2M","created_at":"2026-05-18T12:31:05Z"}],"graph_snapshots":[{"event_id":"sha256:f4feccdd1d660cf2246f4eb21281edb728760b2ea4fc585c03dcb0bcd92431d4","target":"graph","created_at":"2026-05-17T23:59:39Z","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":"The Nakamura number is an appropriate invariant of a simple game to study the existence of social equilibria and the possibility of cycles. For symmetric quota games its number can be obtained by an easy formula. For some subclasses of simple games the corresponding Nakamura number has also been characterized. However, in general, not much is known about lower and upper bounds depending of invariants on simple, complete or weighted games. Here, we survey such results and highlight connections with other game theoretic concepts.","authors_text":"Josep Freixas, Sascha Kurz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-17T16:11:30Z","title":"Bounds for the Nakamura number"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.06611","kind":"arxiv","version":3},"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:eb9073bb569198ef774f16f26434712e0a1692f822b0a28acfd13ea8f29e46c8","target":"record","created_at":"2026-05-17T23:59:39Z","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":"a69823d81e8925a8815c3c3c79443ed8c5d86dbb6dfe1ebea0543013a4e5883e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-17T16:11:30Z","title_canon_sha256":"50fb2a23a383383c04ea2fc8a5fe32ead275f144f583ce345a0e46472884666e"},"schema_version":"1.0","source":{"id":"1711.06611","kind":"arxiv","version":3}},"canonical_sha256":"fe7c15574c31face119fc59ef966377a1aa77c65799157b4ab494eae16da9498","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fe7c15574c31face119fc59ef966377a1aa77c65799157b4ab494eae16da9498","first_computed_at":"2026-05-17T23:59:39.445333Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:39.445333Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2qVa92Oo+/8K52Kg6LkqA+vWNnqTH9TA+wEw1pDDc5Q+8YiyVGZknpF2PGFvJxuFKiKNGYwT+MCF5/E3/7hQDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:39.446089Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.06611","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:eb9073bb569198ef774f16f26434712e0a1692f822b0a28acfd13ea8f29e46c8","sha256:f4feccdd1d660cf2246f4eb21281edb728760b2ea4fc585c03dcb0bcd92431d4"],"state_sha256":"e9f430d3107b79deb14a7c1191b89e73cf8ced653e1ea480cd126893b0f031c8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hhSb5KxNuzCtiYtyjIR2bFhXDaAACOTNt5rpwyETzKDh8vAsPbvdtlgkyolHDEA5NwJ2bXpM/tXkUFY/5YmEAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T18:22:09.423401Z","bundle_sha256":"cc3757323fa14bbac9b55b76c1ae61fa8acfb1d1e42443e72b21e93ff0e6be9d"}}