Distributed equilibrium seeking in aggregative games: linear convergence under singular perturbations lens

2024-12-16

IEEE Conference on Decision and Control

AI Summary

This paper addresses the challenge of computing Nash equilibria in networked aggregative games, where agents lack access to the global aggregate variable and cannot rely on centralized coordination. We propose a fully distributed algorithm wherein each agent performs local gradient updates while simultaneously tracking the aggregate variable via dynamic consensus—requiring no central node or global information. Methodologically, we model the algorithm as a singularly perturbed system, uncovering its intrinsic two-timescale (fast/slow) dynamical structure. A key innovation is the support for generalized aggregate functions beyond arithmetic averages, substantially enhancing modeling flexibility. Under strong monotonicity and constraint qualification assumptions, we establish linear convergence with constant step sizes using variational inequality and constrained optimization theory. The algorithm’s efficiency and robustness are empirically validated on a voltage-support problem in smart grids.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Search and Optimization: Distributed Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present a fully-distributed algorithm for Nash equilibrium seeking in aggregative games over networks. The proposed scheme endows each agent with a gradient-based scheme equipped with a tracking mechanism to locally reconstruct the aggregative variable, which is not available to the agents. We show that our method falls into the framework of singularly perturbed systems, as it involves the interconnection between a fast subsystem – the global information reconstruction dynamics – with a slow one concerning the optimization of the local strategies. This perspective plays a key role in analyzing the scheme with a constant stepsize, and in proving its linear convergence to the Nash equilibrium in strongly monotone games with local constraints. By exploiting the flexibility of our aggregative variable definition (not necessarily the arithmetic average of the agents’ strategy), we show the efficacy of our algorithm on a realistic voltage support case study for the smart grid.

Problem

• Distributed Nash equilibrium seeking in aggregative games
• Linear convergence via singular perturbation analysis
• Voltage support application in smart grids

Innovation

• Fully-distributed gradient-based Nash equilibrium seeking
• Singularly perturbed systems framework analysis
• Linear convergence in strongly monotone games