# Learning Generalized Nash Equilibria in a Class of Convex Games

## TL;DR

This paper introduces a distributed payoff-based algorithm for learning Nash equilibria in convex games with coupling constraints, proving convergence under various conditions and analyzing the convergence rate.

## Contribution

It presents a novel distributed algorithm that uses only local information and proves its convergence to Nash equilibria in convex games with and without coupling constraints.

## Key findings

- Algorithm converges to Nash equilibrium under potential function assumption.
- Convergence is guaranteed with weaker assumptions in absence of coupling constraints.
- Convergence rate is derived for strongly monotone game maps.

## Abstract

We consider multi-agent decision making where each agent optimizes its convex cost function subject to individual and coupling constraints. The constraint sets are compact convex subsets of a Euclidean space. To learn Nash equilibria, we propose a novel distributed payoff-based algorithm, such that each agent uses information only about its cost function values and the constraint function values with their associated dual multiplier. We prove convergence of this algorithm to a Nash equilibrium, under the assumption that the game admits a strictly convex potential function. In the absence of coupling constraints, we prove convergence to Nash equilibria under significantly weaker assumptions, not requiring a potential function. Namely, strict monotonicity of the game mapping is sufficient for convergence. We also derive the convergence rate of the algorithm for strongly monotone game maps.