Convergence under mesh refinement

Establish convergence under mesh refinement of the Riesz-map preconditioned projected extragradient method for the discrete price-formation mean field game, rather than only convergence to a variational-inequality solution on each fixed grid.

Background

The paper proves convergence of the exact metric-projection extragradient iteration to a solution of the discrete variational inequality on each fixed grid. The admissible step-size threshold can be chosen independently of the spatial and temporal mesh sizes when uniform density, slope, and price bounds are imposed.

However, the analysis does not establish that solutions or iterates of the discrete problems converge to a solution of the underlying continuous price-formation mean field game as the mesh is refined. Numerical refinement experiments provide empirical evidence of decreasing errors but do not resolve this analytical issue.

References

Convergence under mesh refinement is not proved.

— A Riesz-Map Preconditioned Extragradient Method for Price Formation in Mean Field Games  (2610.13934 - Gevorgyan et al., 9 Oct 2026) in Abstract; Section 1, Introduction

The main remaining problems are therefore to analyze this variant or develop an efficient exact metric projection, derive a priori mesh-independent slope bounds for the clipped iteration, and establish convergence under mesh refinement.

— A Riesz-Map Preconditioned Extragradient Method for Price Formation in Mean Field Games  (2610.13934 - Gevorgyan et al., 9 Oct 2026) in Section 6, Conclusion and Future Work