Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distributed Stackelberg Equilibrium Seeking for Networked Multi-Leader Multi-Follower Games with A Clustered Information Structure

Yue Chen, Peng Yi

TL;DR

The paper addresses distributed SE seeking in networked $MLMF$ (multi-leader multi-follower) games under a clustered information structure. It proposes a distributed algorithm that combines implicit gradient estimation with network consensus to approximate the followers' global best responses using only local communications. Convergence is established under diminishing step sizes (and under constant step sizes with strong monotonicity), and the follower constraints are handled via an interior-point barrier extension. Numerical simulations in microgrid and cellular-network contexts validate effectiveness and scalability to large populations.

Abstract

The Stackelberg game depicts a leader-follower relationship wherein decisions are made sequentially, and the Stackelberg equilibrium represents an expected optimal solution when the leader can anticipate the rational response of the follower. Motivated by control of network systems with two levels of decision-making hierarchy, such as the management of energy networks and power coordination at cellular networks, a networked multi-leaders and multi-followers Stackelberg game is proposed. Due to the constraint of limited information interaction among players, a clustered information structure is assumed that each leader can only communicate with a portion of overall followers, namely its subordinated followers, and also only with its local neighboring leaders. In this case, the leaders cannot fully anticipate the collective rational response of all followers with its local information. To address Stackelberg equilibrium seeking under this partial information structure, we propose a distributed seeking algorithm based on implicit gradient estimation and network consensus mechanisms. We rigorously prove the convergence of the algorithm for both diminishing and constant step sizes under strict and strong monotonicity conditions, respectively. Furthermore, the model and the algorithm can also incorporate linear equality and inequality constraints into the followers' optimization problems, with the approach of the interior point barrier function. Finally, we present numerical simulations in applications to corroborate our claims on the proposed framework.

Distributed Stackelberg Equilibrium Seeking for Networked Multi-Leader Multi-Follower Games with A Clustered Information Structure

TL;DR

The paper addresses distributed SE seeking in networked (multi-leader multi-follower) games under a clustered information structure. It proposes a distributed algorithm that combines implicit gradient estimation with network consensus to approximate the followers' global best responses using only local communications. Convergence is established under diminishing step sizes (and under constant step sizes with strong monotonicity), and the follower constraints are handled via an interior-point barrier extension. Numerical simulations in microgrid and cellular-network contexts validate effectiveness and scalability to large populations.

Abstract

The Stackelberg game depicts a leader-follower relationship wherein decisions are made sequentially, and the Stackelberg equilibrium represents an expected optimal solution when the leader can anticipate the rational response of the follower. Motivated by control of network systems with two levels of decision-making hierarchy, such as the management of energy networks and power coordination at cellular networks, a networked multi-leaders and multi-followers Stackelberg game is proposed. Due to the constraint of limited information interaction among players, a clustered information structure is assumed that each leader can only communicate with a portion of overall followers, namely its subordinated followers, and also only with its local neighboring leaders. In this case, the leaders cannot fully anticipate the collective rational response of all followers with its local information. To address Stackelberg equilibrium seeking under this partial information structure, we propose a distributed seeking algorithm based on implicit gradient estimation and network consensus mechanisms. We rigorously prove the convergence of the algorithm for both diminishing and constant step sizes under strict and strong monotonicity conditions, respectively. Furthermore, the model and the algorithm can also incorporate linear equality and inequality constraints into the followers' optimization problems, with the approach of the interior point barrier function. Finally, we present numerical simulations in applications to corroborate our claims on the proposed framework.
Paper Structure (35 sections, 14 theorems, 76 equations, 7 figures, 2 tables, 2 algorithms)

This paper contains 35 sections, 14 theorems, 76 equations, 7 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. PROBLEM FORMULATION
  4. MLMF Stackelberg Games
  5. Networked MLMF Stackelberg Games
  6. Examples:
  7. Algorithm Design
  8. The Implicit Differential Approximation Method of An MLMF Stackelberg Game
  9. Notations for Global Information
  10. Four Questions to Final Results
  11. Main Results
  12. Approximate Analysis
  13. Convergence Analysis
  14. Diminishing Step-size
  15. Constant Step-size
  16. ...and 20 more sections

Key Result

Lemma 1

Suppose Assumptions assumptionoffunction and assumption_lipschitz hold. For any $j,h\in \mathcal{I}_L$, $i\in \mathcal{I}_F$, any $x_k\in \mathbb{R}^q$, $y_{T,k} \in \mathbb{R}^{p}$, where $E^j =[I_{q_j}, \dots, I_{q_j}, \dots, I_{q_j}] \in \mathbb{R}^{q_j\times Nq_j}$, $J^{j,*}_{\mathcal{P}_h,k} := \text{diag}(\nabla_{x^j}\nabla_{z}s^i(z ,x_k)_{i\in \mathcal{P}_h})|_{z=y^{i,*}(x_k)}$, and $H^

Figures (7)

  • Figure 1: Clustered Information Structure.
  • Figure 2: Schematic of a microgrid management system.
  • Figure 3: Schematic of a heterogeneous cellular networked system.
  • Figure 4: The flowchart of Algorithm 1.
  • Figure 5: The trajectories of $\frac{\| x_k - x^\diamond \|}{ \|x_0 - x^\diamond\|}$ with different diminishing steps $\beta_k$ (left) and constant steps $\beta$ (right).
  • ...and 2 more figures

Theorems & Definitions (16)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3: The hyper-pseudo-gradient
  • Lemma 4: Best responses approximation
  • Lemma 5: Local J-H-I approximation
  • Lemma 6: Global followers information approximation
  • Lemma 7: Distance to equilibrium point
  • Proposition 1
  • Remark 1
  • ...and 6 more