Convergence conditions for sequential best-response iteration

Establish conditions guaranteeing convergence of the sequential damped best-response iteration used to compute open-loop generalized Nash equilibrium candidates for the library’s constrained dynamic games.

Background

The solver computes generalized Nash equilibrium candidates by sequentially updating each player’s control through a best-response optimal-control problem, optionally applying damping to each update. The paper notes that convergence of the complete sequence would imply convergence to an equilibrium, but that the relevant sufficient conditions are generally unavailable or highly restrictive, particularly when generalized constraints couple the players’ feasible sets. Consequently, the implemented iteration has no general convergence guarantee, and establishing such conditions remains an unresolved methodological problem for this class of dynamic games.

References

If the whole sequence of profiles of such a method converges, its limit is an equilibrium, but conditions that guarantee convergence are typically not known or very restrictive, and generalized problems require additional properties of the constraints \citep{FacchineiKanzow2010}.

— CorleoneGame: A library of biological dynamic games with a companion Julia solver  (2610.06307 - Sager et al., 5 Oct 2026) in Section “Solving the games with CorleoneGame.jl,” subsection “Sequential response iteration and refinement”