Generalized Nash Equilibrium from a Robustness Perspective in Variational Analysis

Generalized Nash equilibrium is updated by a formulation in terms of extended-real-valued functions being minimized by the agents. Tools of variational analysis are employed to investigate how a local equilibrium, focused on local solutions in the minimization, might respond to perturbations of the parameters on which the agents’ problems depend. Stability properties in this setting of competitive multiple-agent optimization are developed and contrasted with those already known for single-agent optimization in understanding solution robustness. However, the purpose of the investigation goes beyond these results, in themselves, to the deeper issue of whether the concept of Nash equilibrium is adequate for the constructive modeling of agent interactions. To be meaningful, an equilibrium needs to be well-posed in reflecting natural tendencies and circumstances instead of something fragile and ephemeral. Unfinished business is indicated with unanswered questions, and a mathematical landscape is thereby revealed where much more remains to be explored.