2020/05/25 by Alekos Cecchin, Cecchin, Alekos, François Delarue +1 · 2 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #35K65 #35L40 #35Q89 #49L25 #49N80 #60F05 #91A16 #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.AP #math.OC #math.PR #msc:35K65 #msc:35L40 #msc:35Q89 #msc:49L25 #msc:49N80 #msc:60F05 #msc:91A16
paper · pdf · doi:10.48550/arxiv.2005.12153
arxiv created 2020/05/25 · openalex publication_date 2020/05/25 · arxiv updated 2020/05/26 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
The goal of this paper is to provide a selection principle for potential mean field games on a finite state space and, in this respect, to show that equilibria that do not minimize the corresponding mean field control problem should be ruled out. Our strategy is a tailored-made version of the vanishing viscosity method for partial differential equations. Here, the viscosity has to be understood as the square intensity of a common noise that is inserted in the mean field game or, equivalently, as the diffusivity parameter in the related parabolic version of the master equation. As established recently, the randomly forced mean field game becomes indeed uniquely solvable for a relevant choice of a Wright-Fisher common noise, the counterpart of which in the master equation is a Kimura operator on the simplex. We here elaborate on this idea to make the mean field game with common noise both uniquely solvable and potential, meaning that its unique solution is in fact equal to the unique minimizer of a suitable stochastic mean field control problem. Taking the limit as the intensity of the common noise vanishes, we obtain a rigorous proof of the aforementioned selection principle. As a byproduct, we get that the classical solution to the viscous master equation associated with the mean field game with common noise converges to the gradient of the value function of the mean field control problem without common noise; we hence select a particular weak solution of the master equation of the original mean field game. Lastly, we establish an intrinsic uniqueness criterion for this solution within a suitable class.