Stability for Nash Equilibrium Problems

Consider the stability properties of the Karush-Kuhn-Tucker (KKT) solution mapping $S_{\text{KKT}}$ for Nash equilibrium problems (NEPs) with canonical perturbations. Firstly, we obtain an exact characterization of the strong regularity of $S_{\text{KKT}}$ as well as an easily verified sufficient condition. Secondly, we propose equivalent conditions for the continuously differentiable single-valued localization of $S_{\text{KKT}}$. Thirdly, the isolated calmness of $S_{\text{KKT}}$ is studied based on the I-property. The P-property is proposed as a sufficient condition for the robust isolated calmness of $S_{\text{KKT}}$ under convex assumptions. 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 the NEPs in which each individual player solves a quadratic programming problem.

Funding: Financial support from the Strategic Priority Research Program of the Chinese Academy of Sciences [Grant XDA27010101] is gratefully acknowledged. Y.-H. Dai received financial support from the Natural Science Foundation of China [Grants 11991021, 11991020, and 12021001]. L. Zhang received financial support from the Major Program of the National Natural Science Foundation of China [Grants 72192830 and 72192831], the National Natural Science Foundation of China [Grant 12371298], and the 111 Project [Grant B16009].