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Fast Distributed Algorithm for Aggregative Games in Malicious Environment

Kai-Yuan Guo, Yan-Wu Wang, Xiao-Kang Liu, Zhi-Wei Liu

TL;DR

The paper tackles distributed NE seeking in aggregative games under malicious environments. It introduces a heterogeneous trustworthiness probabilistic framework based on stochastic trust observations and integrates it with a fast NE algorithm that uses a multi-round communication scheme on unbalanced directed networks. A resilient variant combines trust broadcast and neighbor isolation to maintain convergence toward the NE despite adversaries, with theoretical guarantees showing linear convergence to a neighborhood (or sublinear exact convergence with diminishing steps). Simulations on a PHEV energy-consumption game validate improved convergence speed, accuracy, and robustness against malicious agents, highlighting practical resilience for networked multi-agent systems.

Abstract

This paper addresses the distributed Nash Equilibrium seeking problem for aggregative games, where legitimate players' decisions are affected by potential malicious players. To describe players' behavior, we introduce a novel heterogeneous trustworthiness probabilistic framework by employing stochastic trust observations. To mitigate the waste of communication and gradient computation, we utilize a compressible unbalanced network information matrix and a multi-round communication mechanism to develop a fast Nash equilibrium seeking algorithm for aggregative games with unbalanced directed networks. By integrating the multi-round communication mechanism and a trustworthiness broadcast mechanism, we embed our fast convergence algorithm into the heterogeneous trustworthiness probabilistic framework, yielding a resilient fast Nash equilibrium seeking algorithm. Theoretical analysis confirms the convergence of the algorithm. Comparative simulations verify the accuracy of our fast convergence algorithm, and validation simulations verify the resilience of the algorithm.

Fast Distributed Algorithm for Aggregative Games in Malicious Environment

TL;DR

The paper tackles distributed NE seeking in aggregative games under malicious environments. It introduces a heterogeneous trustworthiness probabilistic framework based on stochastic trust observations and integrates it with a fast NE algorithm that uses a multi-round communication scheme on unbalanced directed networks. A resilient variant combines trust broadcast and neighbor isolation to maintain convergence toward the NE despite adversaries, with theoretical guarantees showing linear convergence to a neighborhood (or sublinear exact convergence with diminishing steps). Simulations on a PHEV energy-consumption game validate improved convergence speed, accuracy, and robustness against malicious agents, highlighting practical resilience for networked multi-agent systems.

Abstract

This paper addresses the distributed Nash Equilibrium seeking problem for aggregative games, where legitimate players' decisions are affected by potential malicious players. To describe players' behavior, we introduce a novel heterogeneous trustworthiness probabilistic framework by employing stochastic trust observations. To mitigate the waste of communication and gradient computation, we utilize a compressible unbalanced network information matrix and a multi-round communication mechanism to develop a fast Nash equilibrium seeking algorithm for aggregative games with unbalanced directed networks. By integrating the multi-round communication mechanism and a trustworthiness broadcast mechanism, we embed our fast convergence algorithm into the heterogeneous trustworthiness probabilistic framework, yielding a resilient fast Nash equilibrium seeking algorithm. Theoretical analysis confirms the convergence of the algorithm. Comparative simulations verify the accuracy of our fast convergence algorithm, and validation simulations verify the resilience of the algorithm.

Paper Structure

This paper contains 13 sections, 9 theorems, 53 equations, 8 figures, 4 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES and PROBLEM FORMULATION
  3. Aggregative Games with malicious players
  4. Trustworthiness values
  5. Fast NE seeking algorithm
  6. Resilient NE seeking Algorithm
  7. The classification and isolation of players
  8. The trustworthiness broadcast mechanism
  9. Convergence Analysis
  10. Simulations
  11. Comparison for the fast NE seeking algorithm
  12. Verification for the resilient fast NE seeking algorithm
  13. conclusion

Key Result

Lemma 1

Suppose that Assumption AssumptionCommunicationGraph holds and the weight matrix $A$ is row-stochastic. Denote $V_{k+1} = A^k$, $V_{\infty} = \lim\limits_{k \to \infty}A^{k}$, $\hat{V}_{k} = diag(V_{k})$ and $\hat{V}_{\infty} = diag(V_{\infty})$. There exist constants $\gamma>0$ and $\theta$ satisfy where $\rho = \sup_{k}\|\hat{V}_{k}^{-1}\|$,.

Figures (8)

  • Figure 1: The directed communication network with legitimate players $1,2,3,5$ and a malicious player $4$.
  • Figure 2: The directed communication network.
  • Figure 3: The squared errors of different algorithms with same gradient computation times.
  • Figure 4: The squared errors by Algorithm \ref{['Algorithm 2']} with constant step-size.
  • Figure 5: The squared errors by Algorithm \ref{['Algorithm 2']} with diminishing step-size.
  • ...and 3 more figures

Theorems & Definitions (22)

  • Definition 1: 04bacsar1998dynamic
  • Definition 2
  • Lemma 1: xi2018linear
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • proof
  • ...and 12 more