{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:3LO77KM2RM5QADFTFTKWPTC2L4","short_pith_number":"pith:3LO77KM2","canonical_record":{"source":{"id":"1709.00625","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-02T20:20:56Z","cross_cats_sorted":[],"title_canon_sha256":"52811f6859e81aa2f94e4eb48406426eea53c0478a4c56e1107afd328200b217","abstract_canon_sha256":"bcae5868143edb038940f69ecb5ca0710b3b091442e7efdd24589845d7a502d4"},"schema_version":"1.0"},"canonical_sha256":"daddffa99a8b3b000cb32cd567cc5a5f16c792ee07233f87e21176f0a85f82e7","source":{"kind":"arxiv","id":"1709.00625","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.00625","created_at":"2026-05-17T23:53:52Z"},{"alias_kind":"arxiv_version","alias_value":"1709.00625v2","created_at":"2026-05-17T23:53:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.00625","created_at":"2026-05-17T23:53:52Z"},{"alias_kind":"pith_short_12","alias_value":"3LO77KM2RM5Q","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"3LO77KM2RM5QADFT","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"3LO77KM2","created_at":"2026-05-18T12:30:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:3LO77KM2RM5QADFTFTKWPTC2L4","target":"record","payload":{"canonical_record":{"source":{"id":"1709.00625","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-02T20:20:56Z","cross_cats_sorted":[],"title_canon_sha256":"52811f6859e81aa2f94e4eb48406426eea53c0478a4c56e1107afd328200b217","abstract_canon_sha256":"bcae5868143edb038940f69ecb5ca0710b3b091442e7efdd24589845d7a502d4"},"schema_version":"1.0"},"canonical_sha256":"daddffa99a8b3b000cb32cd567cc5a5f16c792ee07233f87e21176f0a85f82e7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:52.107046Z","signature_b64":"v+Ye1xN0I1aGLyZ7ID+Y8NHl4emOtB0NA9TTbvIpO/tiFW90Qs2FB+80s/L4qtEhdHiSgNz5TKbWfjQ/IysEAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"daddffa99a8b3b000cb32cd567cc5a5f16c792ee07233f87e21176f0a85f82e7","last_reissued_at":"2026-05-17T23:53:52.106342Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:52.106342Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1709.00625","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:53:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X8ljbfi/pOLJU13ZPiFanIuCbGFwyYZUCEpLuSnuNclPxUSHHCJcNAxQL1sDToiglcSydH4kaqXeN5ULTUCLCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T13:07:41.126711Z"},"content_sha256":"bc637a391b6559e3545663469b8614be60516b6d7ea38715c3d26cd4055ef0b8","schema_version":"1.0","event_id":"sha256:bc637a391b6559e3545663469b8614be60516b6d7ea38715c3d26cd4055ef0b8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:3LO77KM2RM5QADFTFTKWPTC2L4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Power index rankings in bicameral legislatures and the US legislative system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Victoria Powers","submitted_at":"2017-09-02T20:20:56Z","abstract_excerpt":"In this paper we study rankings induced by power indices of players in simple game models of bicameral legislatures. For a bicameral legislature where bills are passed with a simple majority vote in each house we give a condition involving the size of each chamber which guarantees that a member of the smaller house has more power than a member of the larger house, regardless of the power index used. The only case for which this does not apply is when the smaller house has an odd number of players, the larger house has an even number of players, and the larger house is less than twice the size "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.00625","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:53:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UIAmNypdEwbCuwtT/hPXSd91RtxQk1kPQnqJN/tiAzlqhg1ntaTDkZ8SfhWoloSX15YnhDti/EcojeNZ9oeqDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T13:07:41.127052Z"},"content_sha256":"0778f9592720a331a6a969570f768a128f0e36376ca42a799229add5bb606957","schema_version":"1.0","event_id":"sha256:0778f9592720a331a6a969570f768a128f0e36376ca42a799229add5bb606957"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3LO77KM2RM5QADFTFTKWPTC2L4/bundle.json","state_url":"https://pith.science/pith/3LO77KM2RM5QADFTFTKWPTC2L4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3LO77KM2RM5QADFTFTKWPTC2L4/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":"2026-06-02T13:07:41Z","links":{"resolver":"https://pith.science/pith/3LO77KM2RM5QADFTFTKWPTC2L4","bundle":"https://pith.science/pith/3LO77KM2RM5QADFTFTKWPTC2L4/bundle.json","state":"https://pith.science/pith/3LO77KM2RM5QADFTFTKWPTC2L4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3LO77KM2RM5QADFTFTKWPTC2L4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:3LO77KM2RM5QADFTFTKWPTC2L4","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":"bcae5868143edb038940f69ecb5ca0710b3b091442e7efdd24589845d7a502d4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-02T20:20:56Z","title_canon_sha256":"52811f6859e81aa2f94e4eb48406426eea53c0478a4c56e1107afd328200b217"},"schema_version":"1.0","source":{"id":"1709.00625","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.00625","created_at":"2026-05-17T23:53:52Z"},{"alias_kind":"arxiv_version","alias_value":"1709.00625v2","created_at":"2026-05-17T23:53:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.00625","created_at":"2026-05-17T23:53:52Z"},{"alias_kind":"pith_short_12","alias_value":"3LO77KM2RM5Q","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"3LO77KM2RM5QADFT","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"3LO77KM2","created_at":"2026-05-18T12:30:58Z"}],"graph_snapshots":[{"event_id":"sha256:0778f9592720a331a6a969570f768a128f0e36376ca42a799229add5bb606957","target":"graph","created_at":"2026-05-17T23:53:52Z","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":"In this paper we study rankings induced by power indices of players in simple game models of bicameral legislatures. For a bicameral legislature where bills are passed with a simple majority vote in each house we give a condition involving the size of each chamber which guarantees that a member of the smaller house has more power than a member of the larger house, regardless of the power index used. The only case for which this does not apply is when the smaller house has an odd number of players, the larger house has an even number of players, and the larger house is less than twice the size ","authors_text":"Victoria Powers","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-02T20:20:56Z","title":"Power index rankings in bicameral legislatures and the US legislative system"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.00625","kind":"arxiv","version":2},"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:bc637a391b6559e3545663469b8614be60516b6d7ea38715c3d26cd4055ef0b8","target":"record","created_at":"2026-05-17T23:53:52Z","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":"bcae5868143edb038940f69ecb5ca0710b3b091442e7efdd24589845d7a502d4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-02T20:20:56Z","title_canon_sha256":"52811f6859e81aa2f94e4eb48406426eea53c0478a4c56e1107afd328200b217"},"schema_version":"1.0","source":{"id":"1709.00625","kind":"arxiv","version":2}},"canonical_sha256":"daddffa99a8b3b000cb32cd567cc5a5f16c792ee07233f87e21176f0a85f82e7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"daddffa99a8b3b000cb32cd567cc5a5f16c792ee07233f87e21176f0a85f82e7","first_computed_at":"2026-05-17T23:53:52.106342Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:53:52.106342Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"v+Ye1xN0I1aGLyZ7ID+Y8NHl4emOtB0NA9TTbvIpO/tiFW90Qs2FB+80s/L4qtEhdHiSgNz5TKbWfjQ/IysEAQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:53:52.107046Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.00625","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bc637a391b6559e3545663469b8614be60516b6d7ea38715c3d26cd4055ef0b8","sha256:0778f9592720a331a6a969570f768a128f0e36376ca42a799229add5bb606957"],"state_sha256":"ab4e333d32742d57ebc346c766ba23d3e713ee2f31f8aa3f10a4026e7194fba4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ojTfb2abDAPfDrwNbEE0W+l7DiQfZdPDg0eufkPJ0Q+O90rgU62Fy9WuvPrWamZU0NP7WfeQSJAkJ62V8iw9Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T13:07:41.128887Z","bundle_sha256":"18a3bc5405ae6896674100455837508c2c28d0db207946697f7186ec437fb3d3"}}