This talk presents a primal-dual formulation for continuous-time mean field games (MFGs) and establishes 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 {\it 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 \emph{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.

Compared with existing approaches for MFGs, the primal-dual formulation and its NE characterization require neither the convexity of the associated Hamiltonian nor the  uniqueness of its optimizer, and remain applicable when the HJB equation lacks classical or even continuous solutions.

Short Bio: Anran Hu is an assistant professor at Columbia University, IEOR. She works at the intersection of stochastic control, game theory, optimization and machine learning. Her primary research areas are mean-field games, continuous-time stochastic control and reinforcement learning. She is also interested in FinTech and applying machine learning and reinforcement learning to finance. Before coming to Columbia University, Hu was a Hooke Research Fellow in the Mathematical Institute at the University of Oxford. She completed her Ph.D. in Berkeley IEOR, advised by Prof. Xin Guo.