Mean field social optimization: feedback person-by-person optimality and the dynamic programming equation
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| Format: | Preprint |
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2025
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| _version_ | 1866910227105841152 |
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| author | Huang, Minyi Sheu, Shuenn-Jyi Sun, Li-Hsien |
| author_facet | Huang, Minyi Sheu, Shuenn-Jyi Sun, Li-Hsien |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_14236 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean field social optimization: feedback person-by-person optimality and the dynamic programming equation Huang, Minyi Sheu, Shuenn-Jyi Sun, Li-Hsien Optimization and Control 49L20, 49N80, 93A15, 93E20 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. |
| title | Mean field social optimization: feedback person-by-person optimality and the dynamic programming equation |
| topic | Optimization and Control 49L20, 49N80, 93A15, 93E20 |
| url | https://arxiv.org/abs/2508.14236 |