We study the computation of static mean-field equilibria on a compact state space by formu-
lating the equilibrium condition as a variational inequality over probability measures. We propose
an entropic variant of Korpelevich’s extragradient algorithm—the Kullback–Leibler Mirror-Prox
method—in which Euclidean projections are replaced by relative-entropy proximal steps. Each
half-step is therefore an explicit exponential reweighting of the current measure, implemented on a
finite state-space discretization. Under Lasry–Lions monotonicity and continuity assumptions, we
prove convergence of mesh-refined ergodic averages and obtain finite-iteration Minty-residual and
approximate-equilibrium bounds that jointly quantify iteration and discretization errors. Under
strong monotonicity, we derive metric convergence rates for the last, best, and averaged iterates.
We also develop a KL-type Tikhonov regularization that selects the equilibrium minimizing relative
entropy with respect to a reference measure. The framework applies to potential and nonpotential
cost operators and does not require differentiability or convexity of the cost in the individual state.

Joint work with Erhan Bayraktar, Ibrahim Ekren and Lu Vy.