G-2024-45
Mean field social optimization: Feedback person-by-person optimality and the dynamic programming equation
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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 \(\epsilon\)
-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.
Published August 2024 , 43 pages
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