Hierarchical Nash Equilibrium over Variational Equilibria via Fixed-point Set Expression of Quasi-nonexpansive Operator
Shota Matsuo, Keita Kume, Isao Yamada · 2025
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.