{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:ZGUHVHJ63OEC4H4GOIP2QEMZ37","short_pith_number":"pith:ZGUHVHJ6","canonical_record":{"source":{"id":"1707.09090","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-07-28T02:39:39Z","cross_cats_sorted":[],"title_canon_sha256":"f02b277afba5553ffd75142f8dab0c97421e5b6a6cec0ede04196e970eb52200","abstract_canon_sha256":"17111194f77875f3901c926844dbee5268108581f01e290940e2f76254c61871"},"schema_version":"1.0"},"canonical_sha256":"c9a87a9d3edb882e1f86721fa81199dfe1c76d65c4378768c1d53e732f5fb7c1","source":{"kind":"arxiv","id":"1707.09090","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.09090","created_at":"2026-05-18T00:39:17Z"},{"alias_kind":"arxiv_version","alias_value":"1707.09090v1","created_at":"2026-05-18T00:39:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.09090","created_at":"2026-05-18T00:39:17Z"},{"alias_kind":"pith_short_12","alias_value":"ZGUHVHJ63OEC","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZGUHVHJ63OEC4H4G","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZGUHVHJ6","created_at":"2026-05-18T12:31:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:ZGUHVHJ63OEC4H4GOIP2QEMZ37","target":"record","payload":{"canonical_record":{"source":{"id":"1707.09090","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-07-28T02:39:39Z","cross_cats_sorted":[],"title_canon_sha256":"f02b277afba5553ffd75142f8dab0c97421e5b6a6cec0ede04196e970eb52200","abstract_canon_sha256":"17111194f77875f3901c926844dbee5268108581f01e290940e2f76254c61871"},"schema_version":"1.0"},"canonical_sha256":"c9a87a9d3edb882e1f86721fa81199dfe1c76d65c4378768c1d53e732f5fb7c1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:17.138007Z","signature_b64":"XsOnYEFYLzx87VsfFraIyG/X3Ybd7x/Mj6fS3Q5ps1iBjheTEDDHBj6I89XuBPDWGdj3jQ6OB11TxtuD5fXmBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9a87a9d3edb882e1f86721fa81199dfe1c76d65c4378768c1d53e732f5fb7c1","last_reissued_at":"2026-05-18T00:39:17.137387Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:17.137387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1707.09090","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":"2026-05-18T00:39:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LYusWmf9va3MgdMpBDwvGKVGR628FJU7FdQ/TtEpJN9R5LnXYhVIOXJEGGJnqLFeXzZNTy9vqjnHs/uo8FYOCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:36:55.734042Z"},"content_sha256":"49c0058aa16bc7e7c2f044e3d524063895fd9bdcebea0bbf3652282c34c00078","schema_version":"1.0","event_id":"sha256:49c0058aa16bc7e7c2f044e3d524063895fd9bdcebea0bbf3652282c34c00078"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:ZGUHVHJ63OEC4H4GOIP2QEMZ37","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotic Analysis of Mean Field Games with Small Common Noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Saran Ahuja, Tzu-Wei Yang, Weiluo Ren","submitted_at":"2017-07-28T02:39:39Z","abstract_excerpt":"In this paper, we consider a mean field game (MFG) model perturbed by small common noise. Our goal is to give an approximation of the Nash equilibrium strategy of this game using a solution from the original no common noise MFG whose solution can be obtained through a coupled system of partial differential equations. We characterize the first order approximation via linear mean-field forward-backward stochastic differential equations whose solution is a centered Gaussian process with respect to the common noise. The first order approximate strategy can be described as follows: at time $t \\in ["},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.09090","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":"2026-05-18T00:39:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uHoR+Z+1N+R+lVrQcB7XTNQDqPC/60elBCVqFVUS76rUk8qoKz/3TviATy/DfSiX3RLkoNofFyKVs+bhsSypCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:36:55.734639Z"},"content_sha256":"d69f4799762d1492c95b9e13ba41093376d5f4326a6878b4face0f9d0c2095ee","schema_version":"1.0","event_id":"sha256:d69f4799762d1492c95b9e13ba41093376d5f4326a6878b4face0f9d0c2095ee"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZGUHVHJ63OEC4H4GOIP2QEMZ37/bundle.json","state_url":"https://pith.science/pith/ZGUHVHJ63OEC4H4GOIP2QEMZ37/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZGUHVHJ63OEC4H4GOIP2QEMZ37/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":"2026-06-10T23:36:55Z","links":{"resolver":"https://pith.science/pith/ZGUHVHJ63OEC4H4GOIP2QEMZ37","bundle":"https://pith.science/pith/ZGUHVHJ63OEC4H4GOIP2QEMZ37/bundle.json","state":"https://pith.science/pith/ZGUHVHJ63OEC4H4GOIP2QEMZ37/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZGUHVHJ63OEC4H4GOIP2QEMZ37/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:ZGUHVHJ63OEC4H4GOIP2QEMZ37","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":"17111194f77875f3901c926844dbee5268108581f01e290940e2f76254c61871","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-07-28T02:39:39Z","title_canon_sha256":"f02b277afba5553ffd75142f8dab0c97421e5b6a6cec0ede04196e970eb52200"},"schema_version":"1.0","source":{"id":"1707.09090","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.09090","created_at":"2026-05-18T00:39:17Z"},{"alias_kind":"arxiv_version","alias_value":"1707.09090v1","created_at":"2026-05-18T00:39:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.09090","created_at":"2026-05-18T00:39:17Z"},{"alias_kind":"pith_short_12","alias_value":"ZGUHVHJ63OEC","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZGUHVHJ63OEC4H4G","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZGUHVHJ6","created_at":"2026-05-18T12:31:59Z"}],"graph_snapshots":[{"event_id":"sha256:d69f4799762d1492c95b9e13ba41093376d5f4326a6878b4face0f9d0c2095ee","target":"graph","created_at":"2026-05-18T00:39:17Z","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 consider a mean field game (MFG) model perturbed by small common noise. Our goal is to give an approximation of the Nash equilibrium strategy of this game using a solution from the original no common noise MFG whose solution can be obtained through a coupled system of partial differential equations. We characterize the first order approximation via linear mean-field forward-backward stochastic differential equations whose solution is a centered Gaussian process with respect to the common noise. The first order approximate strategy can be described as follows: at time $t \\in [","authors_text":"Saran Ahuja, Tzu-Wei Yang, Weiluo Ren","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-07-28T02:39:39Z","title":"Asymptotic Analysis of Mean Field Games with Small Common Noise"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.09090","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":"sha256:49c0058aa16bc7e7c2f044e3d524063895fd9bdcebea0bbf3652282c34c00078","target":"record","created_at":"2026-05-18T00:39:17Z","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":"17111194f77875f3901c926844dbee5268108581f01e290940e2f76254c61871","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-07-28T02:39:39Z","title_canon_sha256":"f02b277afba5553ffd75142f8dab0c97421e5b6a6cec0ede04196e970eb52200"},"schema_version":"1.0","source":{"id":"1707.09090","kind":"arxiv","version":1}},"canonical_sha256":"c9a87a9d3edb882e1f86721fa81199dfe1c76d65c4378768c1d53e732f5fb7c1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c9a87a9d3edb882e1f86721fa81199dfe1c76d65c4378768c1d53e732f5fb7c1","first_computed_at":"2026-05-18T00:39:17.137387Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:39:17.137387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XsOnYEFYLzx87VsfFraIyG/X3Ybd7x/Mj6fS3Q5ps1iBjheTEDDHBj6I89XuBPDWGdj3jQ6OB11TxtuD5fXmBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:39:17.138007Z","signed_message":"canonical_sha256_bytes"},"source_id":"1707.09090","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:49c0058aa16bc7e7c2f044e3d524063895fd9bdcebea0bbf3652282c34c00078","sha256:d69f4799762d1492c95b9e13ba41093376d5f4326a6878b4face0f9d0c2095ee"],"state_sha256":"ce2874289648b29bd703fae5bed588407df46a5fa0be181356443844cb298ba2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NvSru37wSkwvDgwKuCfoVJ3CV3v00qHCp9TzbzAT/+UGEsAHOGDdpWdjxTIIZOMqRij4Z9YXP4bi8xVxBMjbDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T23:36:55.738258Z","bundle_sha256":"0d491841ae1db50fedd1128d3d3d5811e43e5ceb54cdb75c08a92761367ffc4b"}}