  This paper is devoted to studying the stability properties of the
Karush-Kuhn-Tucker (KKT) solution mapping $S_{\rm KKT}$ for Nash equilibrium
problems (NEPs) with canonical perturbations. Firstly, we obtain an exact
characterization of the strong regularity of $S_{\rm KKT}$ and a sufficient
condition that is easy to verify. Secondly, we propose equivalent conditions
for the continuously differentiable single-valued localization of $S_{\rm
KKT}$. Thirdly, the isolated calmness of $S_{\rm KKT}$ 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 $S_{\rm KKT}$ 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.
