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

Shota Matsuo, Keita Kume, Isao Yamada

TL;DR

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 and the proposed algorithm is established by applying the hybrid steepest descent method to a variational inequality defined over the fixed point set of a quasi-nonexpansive operator.

Abstract

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.

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

TL;DR

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 and the proposed algorithm is established by applying the hybrid steepest descent method to a variational inequality defined over the fixed point set of a quasi-nonexpansive operator.

Abstract

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.
Paper Structure (8 sections, 3 theorems, 20 equations, 3 figures, 1 algorithm)

This paper contains 8 sections, 3 theorems, 20 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proposed algorithm for solving HNEP
  4. Application of the hierarchical Nash equilibrium problem to linearly-coupled aggregative game
  5. Linearly-coupled aggregative game
  6. Designing each player's upper-level cost function
  7. Numerical experiments
  8. Conclusion

Key Result

Proposition 2.2

Under $\bm{\mathcal{V}}\neq\varnothing$ and Assumption asm:monotone, $\bm{\mathcal{V}}$ is closed convex. In addition, by defining $\bm{T}_{\mathrm{FBF}}:\bm{\mathcal{H}} \times\mathcal{G} \rightarrow \bm{\mathcal{H}} \times\mathcal{G}$, with $\gamma \in (0, 1/(\kappa_{\bm{G}} + \|\bm{L}\|_{\mathrm{ and its $\alpha(\in(0, 1))$-averaged operator where we have $\mathrm{Fix}(\bm{T}_{\mathrm{FBF}})

Figures (3)

  • Figure 1: Conceptional comparison of two models for equilibrium selections (existing models $\llbracket$left$\rrbracket$ and proposed model $\llbracket$right$\rrbracket$) over $\bm{\mathcal{V}}$ of the lower-level non-cooperative game among all players $i \in \mathcal{I}$. $\llbracket$Left$\rrbracket$ The existing models g.scutariEquilibriumSelectionPower2012g.scutariRealComplexMonotone2014e.benenatiOptimalSelectionTracking2023w.heDistributedOptimalVariational2024a have been formulated to choose a special variational equilibrium by minimizing a single upper-level cost function $\bm{\mathfrak{f}}_{{}}^{\langle \mathrm{u} \rangle}$, designed hopefully by a trusted center. $\llbracket$Right$\rrbracket$ The proposed model is formulated to choose a special variational equilibrium, but in a different sense from the existing models, i.e., as an upper-level generalized Nash equilibrium (GNE) of a new non-cooperative game among all players $i\in\mathcal{I}$ with upper-level cost functions $\bm{\mathfrak{f}}_{{i}}^{\langle \mathrm{u} \rangle}$ designed by each player $i \in \mathcal{I}$.
  • Figure 2: Approximation error $\|(P_{\overline{B}(0, r)} \circ \bm{T}_{\alpha})(\bm{\xi}_n) - \bm{\xi}_n\|_2$ of variational equilibria for \ref{['eq:Cournot-Nash']}
  • Figure 3: Values of each player's upper-level cost function $\bm{\mathfrak{f}}_{{i}}^{\langle \mathrm{u} \rangle} \ (i\in\mathcal{I})$

Theorems & Definitions (5)

  • Remark 1.3: On the HNEP \ref{['eq:HierarchicalGame_0']}
  • Proposition 2.2: Fixed point expression of $\bm{\mathcal{V}}$
  • Lemma 3.1
  • Theorem 3.3: Convergence of Algorithm \ref{['alg:HSDM']}
  • Remark 3.4: Elimination of possible concerns regarding Assumption \ref{['asm:paramonotone']}