{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:TAOMUDRTHRTOVUTILCGX32ROFG","short_pith_number":"pith:TAOMUDRT","canonical_record":{"source":{"id":"1603.06312","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-03-21T02:26:10Z","cross_cats_sorted":["math.OC","q-fin.MF"],"title_canon_sha256":"2a9432dedc0db7a5d7f819f1f05d00bbd5783a1f4bd50a5836db0e972a49d499","abstract_canon_sha256":"7ca676a91fd063c94f427fb681a37ab76e58e227fca5148d988934a335b1cc27"},"schema_version":"1.0"},"canonical_sha256":"981cca0e333c66ead268588d7dea2e29a2003f34850beeb943a9da33286e2a28","source":{"kind":"arxiv","id":"1603.06312","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.06312","created_at":"2026-05-18T01:02:12Z"},{"alias_kind":"arxiv_version","alias_value":"1603.06312v3","created_at":"2026-05-18T01:02:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.06312","created_at":"2026-05-18T01:02:12Z"},{"alias_kind":"pith_short_12","alias_value":"TAOMUDRTHRTO","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"TAOMUDRTHRTOVUTI","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"TAOMUDRT","created_at":"2026-05-18T12:30:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:TAOMUDRTHRTOVUTILCGX32ROFG","target":"record","payload":{"canonical_record":{"source":{"id":"1603.06312","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-03-21T02:26:10Z","cross_cats_sorted":["math.OC","q-fin.MF"],"title_canon_sha256":"2a9432dedc0db7a5d7f819f1f05d00bbd5783a1f4bd50a5836db0e972a49d499","abstract_canon_sha256":"7ca676a91fd063c94f427fb681a37ab76e58e227fca5148d988934a335b1cc27"},"schema_version":"1.0"},"canonical_sha256":"981cca0e333c66ead268588d7dea2e29a2003f34850beeb943a9da33286e2a28","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:12.996271Z","signature_b64":"2Lfkl4fTaX4j4AVefdlsJl5woDj1coY4jvehhiCEJP8goXSjBy5ACHi1SZRAPX6SBXzuFwiO3bRJdJm0m2cpBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"981cca0e333c66ead268588d7dea2e29a2003f34850beeb943a9da33286e2a28","last_reissued_at":"2026-05-18T01:02:12.995535Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:12.995535Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.06312","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":"2026-05-18T01:02:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uwRClA184lx6pIGxeZ4R769NFks4+xStvlDjTBonmxxKakoevrmeGbG67k6/m1EwQrwMOIXQBVy3oa6qGzbXCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T22:55:02.540914Z"},"content_sha256":"4c67b9fca14318c47e35d54f12ed5eb0bb65426e253c50e408da7ea44c7c8332","schema_version":"1.0","event_id":"sha256:4c67b9fca14318c47e35d54f12ed5eb0bb65426e253c50e408da7ea44c7c8332"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:TAOMUDRTHRTOVUTILCGX32ROFG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A rank based mean field game in the strong formulation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC","q-fin.MF"],"primary_cat":"math.PR","authors_text":"Erhan Bayraktar, Yuchong Zhang","submitted_at":"2016-03-21T02:26:10Z","abstract_excerpt":"We discuss a natural game of competition and solve the corresponding mean field game with \\emph{common noise} when agents' rewards are \\emph{rank dependent}. We use this solution to provide an approximate Nash equilibrium for the finite player game and obtain the rate of convergence."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06312","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":"2026-05-18T01:02:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0YKtredcp1Ciqox+SaoejmILZW7CfimqhX4mRsizSq3rnjaVUY/MhDAc1dKrzy81HjLS4AVY3W1zAxTe0+jeDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T22:55:02.541308Z"},"content_sha256":"27a3bd1322408aac6076dd740a3e1674bf94f0a6efd1620efddb96410e349798","schema_version":"1.0","event_id":"sha256:27a3bd1322408aac6076dd740a3e1674bf94f0a6efd1620efddb96410e349798"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TAOMUDRTHRTOVUTILCGX32ROFG/bundle.json","state_url":"https://pith.science/pith/TAOMUDRTHRTOVUTILCGX32ROFG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TAOMUDRTHRTOVUTILCGX32ROFG/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":"2026-05-20T22:55:02Z","links":{"resolver":"https://pith.science/pith/TAOMUDRTHRTOVUTILCGX32ROFG","bundle":"https://pith.science/pith/TAOMUDRTHRTOVUTILCGX32ROFG/bundle.json","state":"https://pith.science/pith/TAOMUDRTHRTOVUTILCGX32ROFG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TAOMUDRTHRTOVUTILCGX32ROFG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:TAOMUDRTHRTOVUTILCGX32ROFG","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":"7ca676a91fd063c94f427fb681a37ab76e58e227fca5148d988934a335b1cc27","cross_cats_sorted":["math.OC","q-fin.MF"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-03-21T02:26:10Z","title_canon_sha256":"2a9432dedc0db7a5d7f819f1f05d00bbd5783a1f4bd50a5836db0e972a49d499"},"schema_version":"1.0","source":{"id":"1603.06312","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.06312","created_at":"2026-05-18T01:02:12Z"},{"alias_kind":"arxiv_version","alias_value":"1603.06312v3","created_at":"2026-05-18T01:02:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.06312","created_at":"2026-05-18T01:02:12Z"},{"alias_kind":"pith_short_12","alias_value":"TAOMUDRTHRTO","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"TAOMUDRTHRTOVUTI","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"TAOMUDRT","created_at":"2026-05-18T12:30:44Z"}],"graph_snapshots":[{"event_id":"sha256:27a3bd1322408aac6076dd740a3e1674bf94f0a6efd1620efddb96410e349798","target":"graph","created_at":"2026-05-18T01:02:12Z","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":"We discuss a natural game of competition and solve the corresponding mean field game with \\emph{common noise} when agents' rewards are \\emph{rank dependent}. We use this solution to provide an approximate Nash equilibrium for the finite player game and obtain the rate of convergence.","authors_text":"Erhan Bayraktar, Yuchong Zhang","cross_cats":["math.OC","q-fin.MF"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-03-21T02:26:10Z","title":"A rank based mean field game in the strong formulation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06312","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":"sha256:4c67b9fca14318c47e35d54f12ed5eb0bb65426e253c50e408da7ea44c7c8332","target":"record","created_at":"2026-05-18T01:02:12Z","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":"7ca676a91fd063c94f427fb681a37ab76e58e227fca5148d988934a335b1cc27","cross_cats_sorted":["math.OC","q-fin.MF"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-03-21T02:26:10Z","title_canon_sha256":"2a9432dedc0db7a5d7f819f1f05d00bbd5783a1f4bd50a5836db0e972a49d499"},"schema_version":"1.0","source":{"id":"1603.06312","kind":"arxiv","version":3}},"canonical_sha256":"981cca0e333c66ead268588d7dea2e29a2003f34850beeb943a9da33286e2a28","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"981cca0e333c66ead268588d7dea2e29a2003f34850beeb943a9da33286e2a28","first_computed_at":"2026-05-18T01:02:12.995535Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:02:12.995535Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2Lfkl4fTaX4j4AVefdlsJl5woDj1coY4jvehhiCEJP8goXSjBy5ACHi1SZRAPX6SBXzuFwiO3bRJdJm0m2cpBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:02:12.996271Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.06312","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4c67b9fca14318c47e35d54f12ed5eb0bb65426e253c50e408da7ea44c7c8332","sha256:27a3bd1322408aac6076dd740a3e1674bf94f0a6efd1620efddb96410e349798"],"state_sha256":"9eec0e31c232585b3e38a81ce0e4e5f510ccf6d8f8fea4f7a36666719876880f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KyrHroPdj+ayNWFn9ZU6BQwEKO2N6ezrW8NfmQDRjFxcBqtLB438eOhT1kOjf5o33xswtxLR7ind9iuzvhxOCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-20T22:55:02.544602Z","bundle_sha256":"43c4766dad69252cc65672654c396118e9959f026164947a92af3b6c2e8b89fb"}}