Weak Solutions for Potential Mean Field Games of Controls

Jameson Graber; Alan Mullenix; Laurent Pfeiffer

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arXiv · 2010.09912

Weak Solutions for Potential Mean Field Games of Controls

Jameson Graber·Alan Mullenix·Laurent Pfeiffer

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Abstract

We analyze a system of partial differential equations that model a potential mean field game of controls, briefly MFGC. Such a game describes the interaction of infinitely many negligible players competing to optimize a personal value function that depends in aggregate on the state and, most notably, control choice of all other players. A solution of the system corresponds to a Nash Equilibrium, a group optimal strategy for which no one player can improve by altering only their own action. We investigate the second order, possibly degenerate, case with non-strictly elliptic diffusion operator and local coupling function. The main result exploits potentiality to employ variational techniques to provide a unique weak solution to the system, with additional space and time regularity results under additional assumptions. New analytical subtleties occur in obtaining a priori estimates with the introduction of an additional coupling that depends on the state distribution as well as feedback.

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Jameson Graber, Alan Mullenix, Laurent Pfeiffer. 2020-10-26. Weak Solutions for Potential Mean Field Games of Controls. https://arxiv.org/abs/2010.09912

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