Sharp convergence rates for nonlinear closed-loop alpha-potential games

Establish sharp, optimal convergence rates for the potential approximation parameter \(\alpha\) in nonlinear closed-loop alpha-potential stochastic differential games.

Background

Theorem 2 provides sufficient upper bounds for α\alpha in closed-loop stochastic differential games with mean-field interactions. These bounds contain several fractional powers of the population size, and the authors emphasize that closed-loop feedback may prevent α\alpha from converging to zero even as the number of players becomes large.

The stated bounds are not claimed to be optimal. Determining whether the rates N−1/4N^{-1/4}, N−1/2N^{-1/2}, N−1N^{-1}, and N−2N^{-2} can be improved, or identifying the exact asymptotic behavior of α\alpha under nonlinear feedback, remains unresolved.

References

We stress that these are sufficient upper bounds; establishing sharp, optimal convergence rates for nonlinear closed‑loop \alpha-potential stochastic differential games remains an open direction for future investigation.

— Closed-loop $α$-Potential Stochastic Differential Games via a BSDE Approach  (2609.09756 - Li et al., 9 Sep 2026) in Remark following Theorem 2, Remark \ref{rem:scaling_constants_closedloop}, Section 4

In contrast, extending this methodology to continuous-time SDGs introduces profound analytical challenges: the infinite-dimensional nature of control spaces, the coupling between drift and diffusion terms in stochastic dynamics, and the absence of finite-dimensional sufficient statistics collectively impede the direct application of linear programming duality arguments---rendering the computation and bounding of \alpha an unresolved technical frontier.

— Closed-loop $α$-Potential Stochastic Differential Games via a BSDE Approach  (2609.09756 - Li et al., 9 Sep 2026) in Section 1, paragraph “\(\alpha\)-NE and estimation of \(\alpha\)”