Stability for Nash Equilibrium Problems

This paper is devoted to studying the stability properties of the Karush-Kuhn-Tucker (KKT) solution mapping SKKT for Nash equilibrium problems (NEPs) with canonical perturbations. Firstly, we obtain an exact characterization of the strong regularity of SKKT and a sufficient condition that is easy to verify. Secondly, we propose equivalent conditions for the continuously differentiable single-valued localization of SKKT. Thirdly, the isolated calmness of SKKT is studied based on two conditions: Property A and Property B, and Property B proves to be sufficient for the robustness of both $E(p)$ and SKKT under the convex assumptions, where $E(p)$ denotes the Nash equilibria at perturbation p. Furthermore, we establish that studying the stability properties of the NEP with canonical perturbations is equivalent to studying those of the NEP with only tilt perturbations based on the prior discussions. Finally, we provide detailed characterizations of stability for NEPs whose each individual player solves a quadratic programming (QP) problem.