Abstract : In this talk, we will establish a primal-dual formulation for continuous-time mean field games (MFGs) and provide a complete analytical characterization of the set of all Nash equilibria (NEs). We first show that for any given mean field flow, the representative player's control problem with measurable coefficients is equivalent to a linear program over the space of occupation measures. We then establish the dual formulation of this linear program as a maximization problem over smooth subsolutions of the associated Hamilton-Jacobi-Bellman (HJB) equation, which plays a fundamental role in characterizing NEs of MFGs. Finally, a complete characterization of all NEs for MFGs is established by the strong duality between the linear program and its dual problem. This strong duality is obtained by studying the solvability of the dual problem, and in particular through analyzing the regularity of the associated HJB equation. its NE characterization do not require the convexity of the associated Hamiltonianor the uniqueness of its optimizer, and remain applicable when the HJB equation lacks classical or even continuous solutions.

Keywords : mean field game; Nash equilibrium; primal-dual characterization; occupation measure; controlled martingale; superposition principle; strong duality; Hamilton-Jacobi-Bellman equation; Fokker–Planck equation