Nash equilibrium seeking over digraphs with row-stochastic matrices and network-independent step-sizes

TL;DR

This paper introduces a distributed algorithm for Nash equilibrium seeking in non-cooperative convex games over directed networks, requiring only local information and providing convergence guarantees under a local diagonal dominance condition.

Contribution

It presents a novel NE seeking algorithm that works with row-stochastic matrices and network-independent step-sizes, relaxing previous monotonicity assumptions.

Findings

Convergence to NE under diagonal dominance condition.

Explicit step-size bounds independent of communication topology.

Validated effectiveness through numerical simulations on optical network power control.

Abstract

In this paper, we address the challenge of Nash equilibrium (NE) seeking in non-cooperative convex games with partial-decision information. We propose a distributed algorithm, where each agent refines its strategy through projected-gradient steps and an averaging procedure. Each agent uses estimates of competitors' actions obtained solely from local neighbor interactions, in a directed communication network. Unlike previous approaches that rely on (strong) monotonicity assumptions, this work establishes the convergence towards a NE under a diagonal dominance property of the pseudo-gradient mapping, that can be checked locally by the agents. Further, this condition is physically interpretable and of relevance for many applications, as it suggests that an agent's objective function is primarily influenced by its individual strategic decisions, rather than by the actions of its competitors. In virtue of a novel block-infinity norm convergence argument, we provide explicit bounds for constant step-size that are independent of the communication structure, and can be computed in a totally decentralized way. Numerical simulations on an optical network's power control problem validate the algorithm's effectiveness.