Hierarchical Variational Inequality Problem for Noncooperative Game-Theoretic Selection of Generalized Nash Equilibrium

Abstract The equilibrium selection problem in the variational Generalized Nash Equilibrium Problem (v-GNEP) has been reported as an optimization problem defined over the solution set of v-GNEP, called in this paper the lower-level v-GNEP. However, to make such a selection fair for every player, we have to rely on an unrealistic assumption, that is, the availability of a trusted center that does not induce any bias among players. In this paper, to ensure fairness for every player even in the process of equilibrium selection, we propose a new equilibrium selection problem, named the upper-level v-GNEP. The proposed upper-level v-GNEP is formulated as a v-GNEP defined over the solution set of the lower-level v-GNEP. We also present an iterative algorithm, of guaranteed convergence to a solution of the upper-level v-GNEP, as an application of the hybrid steepest descent method to a fixed point set characterization of the solution of the lower-level v-GNEP. Numerical experiments illustrate the proposed equilibrium selection and algorithm.