1. Closed-loop Equilibria for Mean-Field Games in Randomly Switching Environments with General Discounting Costs
- Author
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Mei, Hongwei, Nguyen, Son Luu, and Yin, George
- Subjects
Mathematics - Optimization and Control ,60H10, 91A16, 93E03, 34K50 - Abstract
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.
- Published
- 2024