Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distributed Nash Equilibrium Seeking over Time-Varying Directed Communication Networks

Duong Thuy Anh Nguyen, Duong Tung Nguyen, Angelia Nedić

TL;DR

The paper addresses distributed Nash equilibrium seeking for non-cooperative convex games under partial-decision information on time-varying directed networks. It proposes a distributed algorithm that combines projected gradient steps with consensus updates and introduces a novel contraction property for time-varying row-stochastic mixing matrices, enabling explicit geometric convergence guarantees. Key contributions include a new contraction lemma for weighted dispersion, a convergence analysis that avoids augmented mappings and PF-eigenvector computations, and applicability to static and time-varying network settings, demonstrated through a Nash-Cournot example. Together, these results offer a scalable, fully decentralized NE seeking framework robust to network dynamics with practical implications for distributed markets and networked decision systems.

Abstract

This paper proposes a distributed algorithm to find the Nash equilibrium in a class of non-cooperative convex games with partial-decision information. Our method employs a distributed projected gradient play approach alongside consensus dynamics, with individual agents minimizing their local costs through gradient steps and local information exchange with neighbors via a time-varying directed communication network. Addressing time-varying directed graphs presents significant challenges. Existing methods often circumvent this by focusing on static graphs or specific types of directed graphs or by requiring the stepsizes to scale with the Perron-Frobenius eigenvectors. In contrast, we establish novel results that provide a contraction property for the mixing terms associated with time-varying row-stochastic weight matrices. Our approach explicitly expresses the contraction coefficient based on the characteristics of the weight matrices and graph connectivity structures, rather than implicitly through the second-largest singular value of the weight matrix as in prior studies. The established results facilitate proving geometric convergence of the proposed algorithm and advance convergence analysis for distributed algorithms in time-varying directed communication networks. Numerical results on a Nash-Cournot game demonstrate the efficacy of the proposed method.

Distributed Nash Equilibrium Seeking over Time-Varying Directed Communication Networks

TL;DR

The paper addresses distributed Nash equilibrium seeking for non-cooperative convex games under partial-decision information on time-varying directed networks. It proposes a distributed algorithm that combines projected gradient steps with consensus updates and introduces a novel contraction property for time-varying row-stochastic mixing matrices, enabling explicit geometric convergence guarantees. Key contributions include a new contraction lemma for weighted dispersion, a convergence analysis that avoids augmented mappings and PF-eigenvector computations, and applicability to static and time-varying network settings, demonstrated through a Nash-Cournot example. Together, these results offer a scalable, fully decentralized NE seeking framework robust to network dynamics with practical implications for distributed markets and networked decision systems.

Abstract

This paper proposes a distributed algorithm to find the Nash equilibrium in a class of non-cooperative convex games with partial-decision information. Our method employs a distributed projected gradient play approach alongside consensus dynamics, with individual agents minimizing their local costs through gradient steps and local information exchange with neighbors via a time-varying directed communication network. Addressing time-varying directed graphs presents significant challenges. Existing methods often circumvent this by focusing on static graphs or specific types of directed graphs or by requiring the stepsizes to scale with the Perron-Frobenius eigenvectors. In contrast, we establish novel results that provide a contraction property for the mixing terms associated with time-varying row-stochastic weight matrices. Our approach explicitly expresses the contraction coefficient based on the characteristics of the weight matrices and graph connectivity structures, rather than implicitly through the second-largest singular value of the weight matrix as in prior studies. The established results facilitate proving geometric convergence of the proposed algorithm and advance convergence analysis for distributed algorithms in time-varying directed communication networks. Numerical results on a Nash-Cournot game demonstrate the efficacy of the proposed method.
Paper Structure (21 sections, 10 theorems, 88 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 21 sections, 10 theorems, 88 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations and Terminologies
  3. Problem Formulation
  4. Graph-constrained Agents' Interactions
  5. Partial-Decision Information Scenario
  6. Distributed Algorithm
  7. Preliminary results
  8. Linear Combinations and Graphs
  9. Implications of the Stochastic Nature of Matrix $W_k$
  10. Contraction Property of Gradient Method
  11. Convergence Analysis
  12. Contraction Property of Weighted Dispersion
  13. Convergence of Algorithm 1
  14. Numerical results
  15. Conclusion
  16. ...and 6 more sections

Key Result

Lemma 1

Let $\{u_i, \, i\in[m]\}\subset{\mathbb R}^n$ be a collection of $m$ vectors and $\{\gamma_i,\, i\in[m]\}$ be a collection of $m$ scalars. (a) We have: (b) If $\sum_{i=1}^m \gamma_i=1$ holds, then for all $u\in {\mathbb R}^n$ we have:

Figures (5)

  • Figure 1: Network Nash-Cournot game: An edge from $i$ to $M_h$ on this graph implies that firm $i$ participates in market $M_h$.
  • Figure 2: Communication network topologies.
  • Figure 3: Comparison across different static network topologies
  • Figure 4: Comparing static and time-varying directed networks.
  • Figure 5: Comparing the proposed algorithm with Bianchi2020NashES.

Theorems & Definitions (40)

  • Definition 1: Graph Connectivity
  • Definition 2: Graph Diameter
  • Definition 3: Shortest-Path Graph Covering
  • Definition 4: Maximal Edge-Utility
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • ...and 30 more