Stability of Lagrangian Generalized Nash Equilibriums

TL;DR

This paper analyzes the stability properties of Lagrangian generalized Nash equilibriums (LGNEs), providing characterizations of their solution mapping behaviors under various conic constraints using advanced variational analysis tools.

Contribution

It offers new characterizations of the Aubin property and isolated calmness of LGNE solution mappings for general and specialized conic-constrained GNEPs, including explicit criteria.

Findings

Characterizations of Aubin property and isolated calmness using coderivative and graph derivative.

Explicit conditions for Lipschitz continuous single-valued localization.

Analysis of stability properties for classical conically constrained Nash equilibrium problems.

Abstract

Lagrangian generalized Nash equilibriums (LGNEs) were introduced by Rockafellar (2024) for a class of generalized Nash equilibrium problems (GNEPs) in which each player's strategy is subject to conic constraints. This paper investigates the stability properties of the LGNE solution set, specifically focusing on the Aubin property, isolated calmness, and Lipschitz continuous single-valued localization. For general conically constrained GNEPs, characterizations of the Aubin property and isolated calmness of the LGNE solution mapping under canonical perturbations are established. These characterizations are formulated using the coderivative and graph derivative of normal cone mappings. Subsequently, these general results are specialized to GNEPs with equality and inequality constraints, yielding explicit characterizations for both the Lipschitz continuous single-valued localization and isolated calmness of the corresponding LGNE solution mapping, which are described by nonsingularity of linear complementarity sytems. For GNEPs with shared conic constraints, the Aubin property and isolated calmness of the consensus LGNE solution mapping--where identical Lagrange multipliers are assigned to the shared constraint--are first characterized. We further analyze the case when the conic constraints are specialized as equalities and inequalities. Finally, for classical conically constrained Nash equilibrium problems, the Aubin property and isolated calmness of the Lagrangian Nash equilibrium solution mapping are also analyzed.