Closed-loop Equilibria for Mean-Field Games in Randomly Switching Environments with General Discounting Costs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911787097522176 |
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| author | Mei, Hongwei Nguyen, Son Luu Yin, George |
| author_facet | Mei, Hongwei Nguyen, Son Luu Yin, George |
| contents | This work is devoted to finding the closed-loop equilibria for a class of mean-field games (MFGs) with infinitely many symmetric players in a common switching environment when the cost functional is under general discount in time. There are two key challenges in the application of the well-known Hamilton-Jacobi-Bellman and Fokker-Planck (HJB-FP) approach to our problems: the path-dependence due to the conditional mean-field interaction and the time-inconsistency due to the general discounting cost. To overcome the difficulties, a theory for a class of systems of path-dependent equilibrium Hamilton-Jacobi-Bellman equations (HJBs) is developed. Then closed-loop equilibrium strategies can be identified through a two-step verification procedure. It should be noted that the closed-loop equilibrium strategies obtained satisfy a new form of local optimality in the Nash sense. The theory obtained extends the HJB-FP approach for classical MFGs to more general conditional MFGs with general discounting costs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_00227 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Closed-loop Equilibria for Mean-Field Games in Randomly Switching Environments with General Discounting Costs Mei, Hongwei Nguyen, Son Luu Yin, George Optimization and Control 60H10, 91A16, 93E03, 34K50 This work is devoted to finding the closed-loop equilibria for a class of mean-field games (MFGs) with infinitely many symmetric players in a common switching environment when the cost functional is under general discount in time. There are two key challenges in the application of the well-known Hamilton-Jacobi-Bellman and Fokker-Planck (HJB-FP) approach to our problems: the path-dependence due to the conditional mean-field interaction and the time-inconsistency due to the general discounting cost. To overcome the difficulties, a theory for a class of systems of path-dependent equilibrium Hamilton-Jacobi-Bellman equations (HJBs) is developed. Then closed-loop equilibrium strategies can be identified through a two-step verification procedure. It should be noted that the closed-loop equilibrium strategies obtained satisfy a new form of local optimality in the Nash sense. The theory obtained extends the HJB-FP approach for classical MFGs to more general conditional MFGs with general discounting costs. |
| title | Closed-loop Equilibria for Mean-Field Games in Randomly Switching Environments with General Discounting Costs |
| topic | Optimization and Control 60H10, 91A16, 93E03, 34K50 |
| url | https://arxiv.org/abs/2403.00227 |