Time-fractional mean-field games

Develop a rigorous theory for mean-field games with time-fractional (e.g., Caputo) dynamics, including well-posedness, equilibrium characterization, and numerical schemes consistent with memory effects.

Background

The paper notes that mean-field games with spatial fractional diffusion (fractional Laplacian) are well developed, while the time-fractional case remains underexplored.

Memory in time complicates the value functional’s path dependence and the structure of associated HJB–FPK couplings, motivating foundational work for time-fractional mean-field models.

References

In large–population limits, mean–field games with spatially fractional diffusion (fractional Laplacian) are well developed, whereas time–fractional mean–field formulations remain largely open.

— Fractional Calculus in Optimal Control and Game Theory: Theory, Numerics, and Applications -- A Survey (2512.12111 - Mojahed et al., 13 Dec 2025) in Section 4, Subsection “Fractional Differential Games (FDGs)”, paragraph “Mean–field and distributed variants”