General convergence of Nash-equilibrium computation

Characterize convergence guarantees for numerical methods that compute feedback Nash equilibria in general nonzero-sum stochastic differential games, beyond the local, equilibrium-dependent conditions established for the sequential policy-iteration method.

Background

The paper situates policy iteration for stochastic differential games within a broader numerical-computation problem. Unlike single-player optimization, coupled nonzero-sum equilibrium conditions form a fixed-point problem and do not generally inherit the monotonicity properties used to prove convergence of classical policy-iteration methods.

The authors emphasize that multiple stabilizing feedback Nash equilibria may exist, even in deterministic special cases. Consequently, global convergence to a distinguished equilibrium cannot generally be expected without additional assumptions, and the convergence behavior depends on the selected equilibrium. The paper resolves a narrower problem by deriving local convergence criteria for its sequential policy-iteration scheme, leaving convergence for the general computation of Nash equilibria open.

References

Since only viscosity solutions can be expected in full generality, relaxing the $C1$/$C2$ requirements and establishing convergence of policy iteration in that framework remains a challenging open problem, which we identify as another interesting direction for future work.

— Policy iteration for Hamilton-Jacobi-Isaacs equations with control constraints and comparison with Hamilton-Jacobi-Bellman equations  (2609.29368 - Kundu et al., 24 Sep 2026) in Concluding remarks

Assessing convergence remains an open question for the computation of Nash equilibria in general.

— Policy Iteration for Linear-Quadratic Stochastic Differential Games with State- and Control-Dependent Noise  (2608.17940 - Handwerker et al., 18 Aug 2026) in Section 1, Introduction