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A Passivity-Based Approach to Nash Equilibrium Seeking over Networks

Dian Gadjov, Lacra Pavel

Abstract

In this paper we consider the problem of distributed Nash equilibrium (NE) seeking over networks, a setting in which players have limited local information. We start from a continuous-time gradient-play dynamics that converges to an NE under strict monotonicity of the pseudo-gradient and assumes perfect information, i.e., instantaneous all-to-all player communication. We consider how to modify this gradient-play dynamics in the case of partial, or networked information between players. We propose an augmented gradient-play dynamics with correction in which players communicate locally only with their neighbours to compute an estimate of the other players' actions. We derive the new dynamics based on the reformulation as a multi-agent coordination problem over an undirected graph. We exploit incremental passivity properties and show that a synchronizing, distributed Laplacian feedback can be designed using relative estimates of the neighbours. Under a strict monotonicity property of the pseudo-gradient, we show that the augmented gradient-play dynamics converges to consensus on the NE of the game. We further discuss two cases that highlight the tradeoff between properties of the game and the communication graph.

A Passivity-Based Approach to Nash Equilibrium Seeking over Networks

Abstract

In this paper we consider the problem of distributed Nash equilibrium (NE) seeking over networks, a setting in which players have limited local information. We start from a continuous-time gradient-play dynamics that converges to an NE under strict monotonicity of the pseudo-gradient and assumes perfect information, i.e., instantaneous all-to-all player communication. We consider how to modify this gradient-play dynamics in the case of partial, or networked information between players. We propose an augmented gradient-play dynamics with correction in which players communicate locally only with their neighbours to compute an estimate of the other players' actions. We derive the new dynamics based on the reformulation as a multi-agent coordination problem over an undirected graph. We exploit incremental passivity properties and show that a synchronizing, distributed Laplacian feedback can be designed using relative estimates of the neighbours. Under a strict monotonicity property of the pseudo-gradient, we show that the augmented gradient-play dynamics converges to consensus on the NE of the game. We further discuss two cases that highlight the tradeoff between properties of the game and the communication graph.

Paper Structure

This paper contains 18 sections, 12 theorems, 90 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Projections
  4. Graph theory
  5. Equilibrium Independent and Incremental Passivity
  6. Problem Statement
  7. Game Formulation
  8. Gradient Dynamics with Perfect information
  9. NE Seeking Dynamics over a Graph
  10. Distributed feedback design
  11. Convergence Analysis
  12. Single-Timescale Consensus and Player Optimization
  13. Two-Timescale Singular Perturbation Analysis
  14. Projected NE Dynamics for Compact Action Sets
  15. Numerical Examples
  16. ...and 3 more sections

Key Result

Lemma 1

Let $C \subseteq \mathbb{R}^n$ and $C^\circ \subseteq \mathbb{R}^n$ be a closed convex cone and its polar cone, and let $v\in\mathbb{R}^n$. Then the following are equivalent: (i) $v_C = P_{C}(v)$ and $v_{C^\circ} = P_{C^\circ}(v)$. (ii) $v_{C}\in C$, $v_{C^\circ}\in C^\circ$, $v = v_{C} + v_{C^\ci

Figures (23)

  • Figure 1: Gradient Dynamics (\ref{['eq:gradient']}) as Feedback Interconnection of two EIP
  • Figure 2: Augmented gradient dynamics (\ref{['eq:overallDyn']}) over $G_c$
  • Figure 3: Block Diagram of Actions and Estimates dynamics in (\ref{['eq:overallDyn']})
  • Figure 4: Random $G_c$, $\lambda_2=4.95$
  • Figure 5: Cycle $G_c$ Graph
  • ...and 18 more figures

Theorems & Definitions (30)

  • Lemma 1: Moreau's Decomposition Theorem III.3.2.5, Lemarechal
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 20 more