Verifiable sufficient conditions for Stackelberg equilibria in fractional differential games

Develop verifiable sufficient conditions ensuring the existence (and, where possible, uniqueness) of Stackelberg (leader–follower) equilibria in fractional differential games whose dynamics are governed by Caputo fractional derivatives, including conditions that account for memory and path dependence.

Background

The paper reviews bilevel and best-response formulations for leader–follower structures and notes that right-sided fractional adjoints arise in fractional PMP-based characterizations.

However, memory effects complicate verification, motivating the need for tractable sufficient conditions to certify Stackelberg equilibria in the fractional setting.

References

Open questions. Refinements are still needed on: (a) Isaacs–type conditions under memory; (b) existence/uniqueness for general–sum FO games; (c) verifiable sufficient conditions for Stackelberg equilibria in FDGs; and (d) scalable algorithms with a priori/a posteriori certificates that couple fractional–approximation error with game–theoretic suboptimality.

— 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)”, Open questions paragraph