Convergence and variational structure of a staggered scheme for mean field games with individual noise on graphs

Abstract: We propose and analyze a time-staggered numerical scheme for mean field game (MFG) systems with individual noise on finite graphs. Numerically solving such coupled forward--backward systems is delicate because the density evolves in the open probability simplex and the coefficients may degenerate at its boundary. The scheme preserves mass and satisfies a discrete fundamental identity compatible with the Lasry--Lions monotonicity argument, leading to uniqueness of the numerical solution. By establishing a timestep-uniform positive lower bound for the density and uniform bounds for the value variable, we prove first-order convergence for every interior discrete solution. For potential MFGs, we establish a variational characterization by identifying the scheme with the KKT system of a convex discrete action, yielding existence of the discrete solution and an optimization-based realization. The resulting optimization problem is solved by a feasible primal--dual Newton method in mass-preserving coordinates. Numerical experiments confirm the predicted convergence rate and illustrate topology-dependent transport and congestion-driven route choice.