Hierarchical Nash Equilibrium over Variational Equilibria via Fixed-point Set Expression of Quasi-nonexpansive Operator

The equilibrium selection problem in the generalized Nash equilibrium problem (GNEP) has recently been studied as an optimization problem, defined over the set of all variational equilibria achievable through a lower-level non-cooperative game among players. 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 for every player. In this paper, we study a new equilibrium selection problem, named the hierarchical Nash equilibrium problem (HNEP), and propose an iterative algorithm for solving the HNEP. The HNEP is designed to ensure a fair selection without assuming any trusted center. More precisely, the HNEP is the GNEP for an upper-level non-cooperative game defined over the set of all variational equilibria of the lower-level non-cooperative game. The proposed algorithm for the HNEP is established by applying the hybrid steepest descent method to a variational inequality defined over the fixed point set of a quasi-nonexpansive operator. Numerical experiments show the effectiveness of the proposed equilibrium selection problem and its algorithmic solution.