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 L2 norms for density and price directions and a space-time H1 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)=p2/2, 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.
Related
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BibTeX
@misc{gevorgyan2026rieszmap,
title = {{A Riesz-Map Preconditioned Extragradient Method for Price Formation in Mean Field Games}},
author = {Yeva Gevorgyan and Diogo Gomes},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13934},
eprint = {2610.13934},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13934},
}
RIS
TY - UNPB
TI - A Riesz-Map Preconditioned Extragradient Method for Price Formation in Mean Field Games
AU - Yeva Gevorgyan
AU - Diogo Gomes
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13934
UR - https://arxiv.org/abs/2610.13934
AB - 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^2 norms for density and price directions and a space-time H^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^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.
ER -