{"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"}