{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:YAITI4NKBRY34IJS7FUGOP64K2","short_pith_number":"pith:YAITI4NK","canonical_record":{"source":{"id":"1211.2405","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2012-11-11T11:55:27Z","cross_cats_sorted":[],"title_canon_sha256":"ebf58db8b601f73b74ab6d40354fe10ae21d35080d27b9bc2bc06f66a130b8ce","abstract_canon_sha256":"e2a74d6fe3725a082d1423980834b654c27dcb3a2f668705338e1b8ea35f335a"},"schema_version":"1.0"},"canonical_sha256":"c0113471aa0c71be2132f968673fdc56915e4708978235da6b434d7b3332f729","source":{"kind":"arxiv","id":"1211.2405","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.2405","created_at":"2026-05-18T03:41:03Z"},{"alias_kind":"arxiv_version","alias_value":"1211.2405v1","created_at":"2026-05-18T03:41:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.2405","created_at":"2026-05-18T03:41:03Z"},{"alias_kind":"pith_short_12","alias_value":"YAITI4NKBRY3","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"YAITI4NKBRY34IJS","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"YAITI4NK","created_at":"2026-05-18T12:27:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:YAITI4NKBRY34IJS7FUGOP64K2","target":"record","payload":{"canonical_record":{"source":{"id":"1211.2405","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2012-11-11T11:55:27Z","cross_cats_sorted":[],"title_canon_sha256":"ebf58db8b601f73b74ab6d40354fe10ae21d35080d27b9bc2bc06f66a130b8ce","abstract_canon_sha256":"e2a74d6fe3725a082d1423980834b654c27dcb3a2f668705338e1b8ea35f335a"},"schema_version":"1.0"},"canonical_sha256":"c0113471aa0c71be2132f968673fdc56915e4708978235da6b434d7b3332f729","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:03.430642Z","signature_b64":"u0dWoynQ2S2cyPOTFNAewtyiL59J8jgbDUsjO4oKUD03qm4haJDf3ZcNHaaLygrpsi7OExTRZ2HMRVVCWfquBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c0113471aa0c71be2132f968673fdc56915e4708978235da6b434d7b3332f729","last_reissued_at":"2026-05-18T03:41:03.429866Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:03.429866Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.2405","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:41:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ivPJJeXvjD2SYb5sZLJ3g1+1Ec6SIyMwtFNoHEG1DYa2NstxDNsO2gsvesc98LZ8jRTeNMYJtohvydIiUTnyCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T06:52:05.902558Z"},"content_sha256":"9b7eee9d4d7c24d83803f2c7a77f97a832d6d5a073454deaf87dada38b927e65","schema_version":"1.0","event_id":"sha256:9b7eee9d4d7c24d83803f2c7a77f97a832d6d5a073454deaf87dada38b927e65"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:YAITI4NKBRY34IJS7FUGOP64K2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Rank-1 Games With Exponentially Many Nash Equilibria","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.GT","authors_text":"Bernhard von Stengel","submitted_at":"2012-11-11T11:55:27Z","abstract_excerpt":"The rank of a bimatrix game (A,B) is the rank of the matrix A+B. We give a construction of rank-1 games with exponentially many equilibria, which answers an open problem by Kannan and Theobald (2010)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2405","kind":"arxiv","version":1},"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":"2026-05-18T03:41:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d4OvKG1bGks5H7SvF3mN6vdIiJbLMUJklTOlvno6PeM1/l+W2LiZjg+xoiSo3aUjGw0lSUzjGm7GA7WafV5QBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T06:52:05.902898Z"},"content_sha256":"243ea663ebdd0ff74feae18d59075f301a5c2bb3f0fedb3ebd604d5ccc245664","schema_version":"1.0","event_id":"sha256:243ea663ebdd0ff74feae18d59075f301a5c2bb3f0fedb3ebd604d5ccc245664"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YAITI4NKBRY34IJS7FUGOP64K2/bundle.json","state_url":"https://pith.science/pith/YAITI4NKBRY34IJS7FUGOP64K2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YAITI4NKBRY34IJS7FUGOP64K2/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":"2026-06-24T06:52:05Z","links":{"resolver":"https://pith.science/pith/YAITI4NKBRY34IJS7FUGOP64K2","bundle":"https://pith.science/pith/YAITI4NKBRY34IJS7FUGOP64K2/bundle.json","state":"https://pith.science/pith/YAITI4NKBRY34IJS7FUGOP64K2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YAITI4NKBRY34IJS7FUGOP64K2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:YAITI4NKBRY34IJS7FUGOP64K2","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":"e2a74d6fe3725a082d1423980834b654c27dcb3a2f668705338e1b8ea35f335a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2012-11-11T11:55:27Z","title_canon_sha256":"ebf58db8b601f73b74ab6d40354fe10ae21d35080d27b9bc2bc06f66a130b8ce"},"schema_version":"1.0","source":{"id":"1211.2405","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.2405","created_at":"2026-05-18T03:41:03Z"},{"alias_kind":"arxiv_version","alias_value":"1211.2405v1","created_at":"2026-05-18T03:41:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.2405","created_at":"2026-05-18T03:41:03Z"},{"alias_kind":"pith_short_12","alias_value":"YAITI4NKBRY3","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"YAITI4NKBRY34IJS","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"YAITI4NK","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:243ea663ebdd0ff74feae18d59075f301a5c2bb3f0fedb3ebd604d5ccc245664","target":"graph","created_at":"2026-05-18T03:41:03Z","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 rank of a bimatrix game (A,B) is the rank of the matrix A+B. We give a construction of rank-1 games with exponentially many equilibria, which answers an open problem by Kannan and Theobald (2010).","authors_text":"Bernhard von Stengel","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2012-11-11T11:55:27Z","title":"Rank-1 Games With Exponentially Many Nash Equilibria"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2405","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":"sha256:9b7eee9d4d7c24d83803f2c7a77f97a832d6d5a073454deaf87dada38b927e65","target":"record","created_at":"2026-05-18T03:41:03Z","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":"e2a74d6fe3725a082d1423980834b654c27dcb3a2f668705338e1b8ea35f335a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2012-11-11T11:55:27Z","title_canon_sha256":"ebf58db8b601f73b74ab6d40354fe10ae21d35080d27b9bc2bc06f66a130b8ce"},"schema_version":"1.0","source":{"id":"1211.2405","kind":"arxiv","version":1}},"canonical_sha256":"c0113471aa0c71be2132f968673fdc56915e4708978235da6b434d7b3332f729","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c0113471aa0c71be2132f968673fdc56915e4708978235da6b434d7b3332f729","first_computed_at":"2026-05-18T03:41:03.429866Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:41:03.429866Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u0dWoynQ2S2cyPOTFNAewtyiL59J8jgbDUsjO4oKUD03qm4haJDf3ZcNHaaLygrpsi7OExTRZ2HMRVVCWfquBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:41:03.430642Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.2405","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9b7eee9d4d7c24d83803f2c7a77f97a832d6d5a073454deaf87dada38b927e65","sha256:243ea663ebdd0ff74feae18d59075f301a5c2bb3f0fedb3ebd604d5ccc245664"],"state_sha256":"613f83339c35c11bb872605f00081027c2d765f8991f49b095ca9e447d835475"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YwY6QTQ9Qn5i9Do423kGY85zSV1i2rI83xNuy+NWEFYysEumKfNCo0Sd8fNbQS9+6JJneKnGfmwVlH6eLZ5vDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T06:52:05.904943Z","bundle_sha256":"606b2ac9f29ba62f7b612d767dbaf0c49f69a44941ff1db5a603d3e066c83f6d"}}