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

TL;DR

This paper introduces a distributed algorithm for Nash equilibrium seeking in aggregative games that guarantees linear convergence by modeling the process as a singularly perturbed system, applicable to smart grid voltage control.

Contribution

It proposes a novel gradient-based distributed scheme with a tracking mechanism, analyzed through the singular perturbations framework, for strongly monotone games with local constraints.

Findings

The algorithm converges linearly to the Nash equilibrium.

It effectively reconstructs the aggregative variable locally.

Demonstrated success in a smart grid voltage support case study.

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.