A Riesz-Map Preconditioned Extragradient Method for Price Formation in Mean Field Games

Abstract: We develop and analyze a projected extragradient method, preconditioned by a Riesz map, for a first-order price-formation mean field game in one space dimension. The model couples Hamilton--Jacobi--Bellman and transport equations through a global market-clearing constraint. Finite propagation justifies truncating the first-order transport equation to a bounded interval. A monotone weak formulation incorporates the mixed initial--terminal data. Its Riesz map uses $L<sup>2$ norms for density and price directions and a space-time $H<sup>1$ norm for value-function directions, preconditioning the residual without adding higher-order regularization terms to the MFG equations. A density-only argument rules out cocoercivity on the localized admissible set in every equivalent Hilbert metric, motivating extragradient updates. For $H(x,p)=p<sup>2/2$, we use a monotone upwind discretization with adjoint-consistent transport and a state-constraint boundary closure. We establish a mesh-independent Lipschitz bound on bounded convex admissible sets with uniform density, slope, and price bounds. With exact metric projection onto these sets, the iteration converges to a solution of the discrete variational inequality on each fixed grid, with an admissible step-size threshold independent of both mesh sizes. Convergence under mesh refinement is not proved. On a quadratic benchmark, all reported runs reach the stopping tolerance with one fixed step size across a factor-of-16 reduction in both mesh sizes. Price errors and value-function errors measured away from low-density regions decay approximately at first order under refinement, while density errors decrease more slowly.