On the exact convergence to Nash equilibrium in monotone regimes under partial-information

Abstract

In this paper, we consider distributed Nash equilibrium seeking in monotone and hypomonotone games. We first assume that each player has knowledge of the opponents' decisions and propose a passivity-based modification of the standard gradient-play dynamics, that we call "Heavy Anchor". We prove that Heavy Anchor allows a relaxation of strict monotonicity of the pseudo-gradient, needed for gradient-play dynamics, and can ensure exact asymptotic convergence in merely monotone regimes. We extend these results to the setting where each player has only partial information of the opponents' decisions. Each player maintains a local decision variable and an auxiliary state estimate, and communicates with their neighbours to learn the opponents' actions. We modify Heavy Anchor via a distributed Laplacian feedback and show how we can exploit equilibrium-independent passivity properties to achieve convergence to a Nash equilibrium in hypomonotone regimes. N j=1 w ij . Assume that W = W T so the weighted Laplacian of G is L = Deg − W . When G is connected and undirected, 0 is a simple eigenvalue of L, L1 N = 0, 1 T N L = 0 T , and all other eigenvalues are positive, 0 < λ 2 (L) ≤ • • • ≤ λ N (L).

Learning generalized Nash equilibria in monotone games: A hybrid adaptive extremum seeking control approach

“…The algorithm in [13] proves convergence to a NE for a monotone pseudogradient by combining an additional filtering dynamics with the standard NE seeking one.…”

“…The authors in [13] propose a passivity framework for the convergence of their algorithm. Instead, we offer a different intuition for the convergence.…”

“…Recently, the golden ratio algorithm has been proven to converge in the monotone case with only one pseudogradient computation [31]. There also exist continuoustime versions of the aforementioned algorithms, like the forward-backward-forward algorithm [4] and the golden ratio algorithm [13], albeit without projections in the latter case, rendering it unusable for GNEPs, as projections are essential for the dual dynamics. To the best of our knowledge, in the merely monotone case, there currently exist no continuous-time GNEP algorithm that can be paired with extremum seeking.…”

“…• We propose a novel, semi-decentralized, single pseudogradient computation generalized Nash equilibrium (GNE) seeking algorithm for solving monotone GNEPs, inspired by the golden ratio algorithm in [31], [13]. • We propose a novel dual variable gain adaptation scheme using the framework of hybrid dynamical systems in order to alleviate the problem of improper scaling of the cost and constraint functions.…”

“…Comparison with [22] and [23]: We emphasize that since here we assume non-strong monotonicity of the pseudogradient mapping, the methodology in [23] based on the forward-backward splitting is not applicable -see [13,Equ. 4] for an example of non-convergence.…”