Lower bounds for the closed-loop potential approximation parameter

Construct policy perturbations and quantify the asymmetry of second-order variations generated by the joint distribution of sensitivity processes and adjoint backward stochastic differential equations in order to obtain meaningful lower bounds for \(\alpha\) in closed-loop stochastic differential games.

Background

The paper develops computable upper bounds for the potential approximation parameter α\alpha by representing first- and second-order variations through forward sensitivity processes and adjoint BSDEs. These estimates show that feedback-induced sensitivity processes can produce a nonvanishing upper bound in the large-population limit.

No corresponding nontrivial lower-bound theory is established. The authors identify the need for explicit policy perturbations and a quantitative analysis of second-order asymmetry to determine when the approximation error is genuinely bounded away from zero.

References

Obtaining meaningful lower bounds for \alpha in the closed‑loop setting is also an interesting open problem. It would require constructing policy perturbations and quantifying the asymmetry of second‑order variations from the joint distribution of sensitivity processes and adjoint BSDEs, which we leave for future work.

— Closed-loop $α$-Potential Stochastic Differential Games via a BSDE Approach  (2609.09756 - Li et al., 9 Sep 2026) in Section 6, Concluding remarks