{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:VLTR543GU5T55BMLGUDERUQR6Z","short_pith_number":"pith:VLTR543G","canonical_record":{"source":{"id":"2605.14363","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-05-14T04:45:34Z","cross_cats_sorted":[],"title_canon_sha256":"22efcea9b2393a8d8d3774cbc78eec7ace67230d916caa2437b55508a8c2e0c7","abstract_canon_sha256":"66de6614abc3ce524599ee634ade8a44960eb370b9b9fb4670447cb4c7950ea2"},"schema_version":"1.0"},"canonical_sha256":"aae71ef366a767de858b350648d211f6481ab16e382ba004b7d1fd0f3bd6b9a5","source":{"kind":"arxiv","id":"2605.14363","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.14363","created_at":"2026-05-17T23:39:07Z"},{"alias_kind":"arxiv_version","alias_value":"2605.14363v1","created_at":"2026-05-17T23:39:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14363","created_at":"2026-05-17T23:39:07Z"},{"alias_kind":"pith_short_12","alias_value":"VLTR543GU5T5","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"VLTR543GU5T55BML","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"VLTR543G","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:VLTR543GU5T55BMLGUDERUQR6Z","target":"record","payload":{"canonical_record":{"source":{"id":"2605.14363","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-05-14T04:45:34Z","cross_cats_sorted":[],"title_canon_sha256":"22efcea9b2393a8d8d3774cbc78eec7ace67230d916caa2437b55508a8c2e0c7","abstract_canon_sha256":"66de6614abc3ce524599ee634ade8a44960eb370b9b9fb4670447cb4c7950ea2"},"schema_version":"1.0"},"canonical_sha256":"aae71ef366a767de858b350648d211f6481ab16e382ba004b7d1fd0f3bd6b9a5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:07.926702Z","signature_b64":"pDYbQUhfLC177JPGfiZL2eU2IZJwnYr2fK3ZVpQYrX9MjofbaEZgUtxjjwL41S0sAsOM8TwnYf1MlfqqIzviBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aae71ef366a767de858b350648d211f6481ab16e382ba004b7d1fd0f3bd6b9a5","last_reissued_at":"2026-05-17T23:39:07.925886Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:07.925886Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.14363","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":"2026-05-17T23:39:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aHDOPN1Bvu14qMwGjnpXZJh1LfbBKvjDj8y+nMP5wr0kuLR0Gk7/o7EHEjd5ZWYB60lG4sKSIXfUiYeraynRCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T06:24:40.141156Z"},"content_sha256":"21b79154048d192a39647f60957a50726f2b0bd5868dbdb24b1954b4aec9c48c","schema_version":"1.0","event_id":"sha256:21b79154048d192a39647f60957a50726f2b0bd5868dbdb24b1954b4aec9c48c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:VLTR543GU5T55BMLGUDERUQR6Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Equilibrium for Time-inconsistent Mean Field Games: A Systematic Analysis by Entropy Regularization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Entropy regularization establishes existence of equilibria for time-inconsistent mean field games via convergence of regularized solutions.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Erhan Bayraktar, Keyu Zhang, Xiang Yu, Zhenhua Wang","submitted_at":"2026-05-14T04:45:34Z","abstract_excerpt":"This paper studies the existence and approximation of equilibria for general time-inconsistent mean field game (MFG) problems in the continuous-time setting. To handle the intricate nonlocal equilibrium Hamilton-Jacobi-Bellman (EHJB) system arising from initial-time dependence, such as non-exponential discounting, we develop a vanishing entropy regularization approach for solving the MFG. With entropy regularization, we first characterize the regularized equilibrium via a coupled exploratory equilibrium HJB (EEHJB) equation and a law-dependent stochastic differential equation. By exploiting Sc"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"By employing compactness arguments, Young measure techniques, and a duality tool for divergence-form Fokker-Planck equations, we prove that the regularized equilibria converge, up to subsequences, to an equilibrium of the original time-inconsistent MFG.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Mild assumptions on the data for global existence of regularized equilibria; short time horizon and weak terminal interaction conditions for convergence of the policy iteration algorithm.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Entropy regularization establishes existence and convergence of equilibria for time-inconsistent mean field games via fixed-point arguments and compactness techniques.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Entropy regularization establishes existence of equilibria for time-inconsistent mean field games via convergence of regularized solutions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"79ba4fdfe19ce32963b653fa0e6c9361478fc909d5ec0e99bb0a7c8d3c72faf9"},"source":{"id":"2605.14363","kind":"arxiv","version":1},"verdict":{"id":"5e3c24d8-65ab-4e64-82d7-42f087d7d422","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:09:12.877005Z","strongest_claim":"By employing compactness arguments, Young measure techniques, and a duality tool for divergence-form Fokker-Planck equations, we prove that the regularized equilibria converge, up to subsequences, to an equilibrium of the original time-inconsistent MFG.","one_line_summary":"Entropy regularization establishes existence and convergence of equilibria for time-inconsistent mean field games via fixed-point arguments and compactness techniques.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Mild assumptions on the data for global existence of regularized equilibria; short time horizon and weak terminal interaction conditions for convergence of the policy iteration algorithm.","pith_extraction_headline":"Entropy regularization establishes existence of equilibria for time-inconsistent mean field games via convergence of regularized solutions."},"references":{"count":60,"sample":[{"doi":"","year":2025,"title":"Bayraktar, E. and Huang, Y.-J. and Wang, Z. and Zhou, Z. , title =. Mathematics of Operations Research , volume =. 2025 , pages =","work_id":"d093b4ff-5b90-4e90-9051-839b9ccefbbb","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"SIAM Journal on Financial Mathematics , volume =","work_id":"7ee26a97-4126-4b74-bb77-50acc7464b9d","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Mathematics of Operations Research , year =","work_id":"296ce239-9412-4e30-8b87-d3ce1f7ef586","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"On time-inconsistent stochastic control in continuous time , journal =","work_id":"1cdd903d-6eba-47df-a135-7a862fad626a","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Extended hjb equation for mean-variance stopping problem: Vanishing regularization method.Preprint, available at arXiv:2510.24128, 2025","work_id":"c01a2704-b761-4553-b3ed-03da4cfdc7a4","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":60,"snapshot_sha256":"a644c1afc9c1599ac47e05d40ea30b8dc0205fc5a722d54f2297f2f9f128319f","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"db3552ca9fda607a2acc66fea7b72bc1d4f5472b9274f927963cf0e719838225"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"5e3c24d8-65ab-4e64-82d7-42f087d7d422"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:39:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PizequX1tzDYxSvKIEaph9W95HaQ/sbCzH9s8SFHBgbEhAiVtMOBCTpK7MWpucnv8DQW3i3DmbOU8LULdA8JCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T06:24:40.142155Z"},"content_sha256":"310fb175de62e53a2c7f9ba54d49555df9b5fe07f8c0c52cfd68f7ca7004f93f","schema_version":"1.0","event_id":"sha256:310fb175de62e53a2c7f9ba54d49555df9b5fe07f8c0c52cfd68f7ca7004f93f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VLTR543GU5T55BMLGUDERUQR6Z/bundle.json","state_url":"https://pith.science/pith/VLTR543GU5T55BMLGUDERUQR6Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VLTR543GU5T55BMLGUDERUQR6Z/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":"2026-05-27T06:24:40Z","links":{"resolver":"https://pith.science/pith/VLTR543GU5T55BMLGUDERUQR6Z","bundle":"https://pith.science/pith/VLTR543GU5T55BMLGUDERUQR6Z/bundle.json","state":"https://pith.science/pith/VLTR543GU5T55BMLGUDERUQR6Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VLTR543GU5T55BMLGUDERUQR6Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VLTR543GU5T55BMLGUDERUQR6Z","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":"66de6614abc3ce524599ee634ade8a44960eb370b9b9fb4670447cb4c7950ea2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-05-14T04:45:34Z","title_canon_sha256":"22efcea9b2393a8d8d3774cbc78eec7ace67230d916caa2437b55508a8c2e0c7"},"schema_version":"1.0","source":{"id":"2605.14363","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.14363","created_at":"2026-05-17T23:39:07Z"},{"alias_kind":"arxiv_version","alias_value":"2605.14363v1","created_at":"2026-05-17T23:39:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14363","created_at":"2026-05-17T23:39:07Z"},{"alias_kind":"pith_short_12","alias_value":"VLTR543GU5T5","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"VLTR543GU5T55BML","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"VLTR543G","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:310fb175de62e53a2c7f9ba54d49555df9b5fe07f8c0c52cfd68f7ca7004f93f","target":"graph","created_at":"2026-05-17T23:39:07Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"By employing compactness arguments, Young measure techniques, and a duality tool for divergence-form Fokker-Planck equations, we prove that the regularized equilibria converge, up to subsequences, to an equilibrium of the original time-inconsistent MFG."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"Mild assumptions on the data for global existence of regularized equilibria; short time horizon and weak terminal interaction conditions for convergence of the policy iteration algorithm."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Entropy regularization establishes existence and convergence of equilibria for time-inconsistent mean field games via fixed-point arguments and compactness techniques."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Entropy regularization establishes existence of equilibria for time-inconsistent mean field games via convergence of regularized solutions."}],"snapshot_sha256":"79ba4fdfe19ce32963b653fa0e6c9361478fc909d5ec0e99bb0a7c8d3c72faf9"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"db3552ca9fda607a2acc66fea7b72bc1d4f5472b9274f927963cf0e719838225"},"paper":{"abstract_excerpt":"This paper studies the existence and approximation of equilibria for general time-inconsistent mean field game (MFG) problems in the continuous-time setting. To handle the intricate nonlocal equilibrium Hamilton-Jacobi-Bellman (EHJB) system arising from initial-time dependence, such as non-exponential discounting, we develop a vanishing entropy regularization approach for solving the MFG. With entropy regularization, we first characterize the regularized equilibrium via a coupled exploratory equilibrium HJB (EEHJB) equation and a law-dependent stochastic differential equation. By exploiting Sc","authors_text":"Erhan Bayraktar, Keyu Zhang, Xiang Yu, Zhenhua Wang","cross_cats":[],"headline":"Entropy regularization establishes existence of equilibria for time-inconsistent mean field games via convergence of regularized solutions.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-05-14T04:45:34Z","title":"Equilibrium for Time-inconsistent Mean Field Games: A Systematic Analysis by Entropy Regularization"},"references":{"count":60,"internal_anchors":0,"resolved_work":60,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Bayraktar, E. and Huang, Y.-J. and Wang, Z. and Zhou, Z. , title =. Mathematics of Operations Research , volume =. 2025 , pages =","work_id":"d093b4ff-5b90-4e90-9051-839b9ccefbbb","year":2025},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"SIAM Journal on Financial Mathematics , volume =","work_id":"7ee26a97-4126-4b74-bb77-50acc7464b9d","year":2023},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"Mathematics of Operations Research , year =","work_id":"296ce239-9412-4e30-8b87-d3ce1f7ef586","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"On time-inconsistent stochastic control in continuous time , journal =","work_id":"1cdd903d-6eba-47df-a135-7a862fad626a","year":2017},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Extended hjb equation for mean-variance stopping problem: Vanishing regularization method.Preprint, available at arXiv:2510.24128, 2025","work_id":"c01a2704-b761-4553-b3ed-03da4cfdc7a4","year":null}],"snapshot_sha256":"a644c1afc9c1599ac47e05d40ea30b8dc0205fc5a722d54f2297f2f9f128319f"},"source":{"id":"2605.14363","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-15T02:09:12.877005Z","id":"5e3c24d8-65ab-4e64-82d7-42f087d7d422","model_set":{"reader":"grok-4.3"},"one_line_summary":"Entropy regularization establishes existence and convergence of equilibria for time-inconsistent mean field games via fixed-point arguments and compactness techniques.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Entropy regularization establishes existence of equilibria for time-inconsistent mean field games via convergence of regularized solutions.","strongest_claim":"By employing compactness arguments, Young measure techniques, and a duality tool for divergence-form Fokker-Planck equations, we prove that the regularized equilibria converge, up to subsequences, to an equilibrium of the original time-inconsistent MFG.","weakest_assumption":"Mild assumptions on the data for global existence of regularized equilibria; short time horizon and weak terminal interaction conditions for convergence of the policy iteration algorithm."}},"verdict_id":"5e3c24d8-65ab-4e64-82d7-42f087d7d422"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:21b79154048d192a39647f60957a50726f2b0bd5868dbdb24b1954b4aec9c48c","target":"record","created_at":"2026-05-17T23:39:07Z","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":"66de6614abc3ce524599ee634ade8a44960eb370b9b9fb4670447cb4c7950ea2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-05-14T04:45:34Z","title_canon_sha256":"22efcea9b2393a8d8d3774cbc78eec7ace67230d916caa2437b55508a8c2e0c7"},"schema_version":"1.0","source":{"id":"2605.14363","kind":"arxiv","version":1}},"canonical_sha256":"aae71ef366a767de858b350648d211f6481ab16e382ba004b7d1fd0f3bd6b9a5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aae71ef366a767de858b350648d211f6481ab16e382ba004b7d1fd0f3bd6b9a5","first_computed_at":"2026-05-17T23:39:07.925886Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:07.925886Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pDYbQUhfLC177JPGfiZL2eU2IZJwnYr2fK3ZVpQYrX9MjofbaEZgUtxjjwL41S0sAsOM8TwnYf1MlfqqIzviBA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:07.926702Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.14363","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:21b79154048d192a39647f60957a50726f2b0bd5868dbdb24b1954b4aec9c48c","sha256:310fb175de62e53a2c7f9ba54d49555df9b5fe07f8c0c52cfd68f7ca7004f93f"],"state_sha256":"1ef5c47704b642c473a17735292adb5250d3ebadb580b4c27cfba270748625a0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UEVaSOBeGvpG2C7yc7lbaf1NR6jttfQvM4bjjiG1tplcdtxaySDvln0x76u47beeoVfhqmANyuCv7CBVHn/HDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T06:24:40.146390Z","bundle_sha256":"c230a872843cd9977888824140540156db13ad851c0fbada247313f7102db3ea"}}