{"ID":2880856,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.14236","arxiv_id":"2508.14236","title":"Mean field social optimization: feedback person-by-person optimality and the dynamic programming equation","abstract":"We consider mean field social optimization in nonlinear diffusion models. By dynamic programming with a representative agent employing cooperative optimizer selection, we derive a new Hamilton--Jacobi--Bellman (HJB) equation to be called the master equation of the value function. Under some regularity conditions, we establish $ε$-person-by-person optimality of the master equation-based control laws, which may be viewed as a necessary condition for nearly attaining the social optimum. A major challenge in the analysis is to obtain tight estimates, within an error of $O(1/N)$, of the social cost having order $O(N)$. This will be accomplished by multi-scale analysis via constructing two auxiliary master equations. We illustrate explicit solutions of the master equations for the linear-quadratic (LQ) case, and give an application to systemic risk.","short_abstract":"We consider mean field social optimization in nonlinear diffusion models. By dynamic programming with a representative agent employing cooperative optimizer selection, we derive a new Hamilton--Jacobi--Bellman (HJB) equation to be called the master equation of the value function. Under some regularity conditions, we es...","url_abs":"https://arxiv.org/abs/2508.14236","url_pdf":"https://arxiv.org/pdf/2508.14236v2","authors":"[\"Minyi Huang\",\"Shuenn-Jyi Sheu\",\"Li-Hsien Sun\"]","published":"2025-08-19T19:54:28Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[\"Diffusion Model\",\"Large Language Model\"]","has_code":false}