Entropic mean-field min-max problems via Best Response flow

We investigate the convergence properties of a continuous-time optimization method, the $\mathit{\text{Mean-Field}}\mathit{\text{Best}}\mathit{\text{Response}}$ flow, for solving convex-concave min-max games with entropy regularization. We introduce suitable Lyapunov functions to establish exponential convergence to the unique mixed Nash equilibrium. Additionally, we demonstrate the convergence of the fictitious play flow as a by-product of our analysis.