On the Sampling-based Computation of Nash Equilibria under Uncertainty via the Nikaido-Isoda Function

TL;DR

This paper introduces a sampling-based projected gradient method for computing Nash equilibria in stochastic games with convex, Lipschitz continuous objectives, without requiring monotonicity or potentiality, and provides convergence guarantees.

Contribution

It develops a novel sampling-enabled gradient approach for stochastic Nash equilibria that relaxes traditional assumptions and offers convergence and complexity analysis.

Findings

Convergence guarantees for the proposed method.

Rate and complexity bounds based on sampling and inexactness.

Applicability to games with convex, Lipschitz continuous objectives.

Abstract

We consider the computation of an equilibrium of a stochastic Nash equilibrium problem, where the player objectives are assumed to be $L_0$-Lipschitz continuous and convex given rival decisions with convex and closed player-specific feasibility sets. To address this problem, we consider minimizing a suitably defined value function associated with the Nikaido-Isoda function. Such an avenue does not necessitate either monotonicity properties of the concatenated gradient map or potentiality requirements on the game but does require a suitable regularity requirement under which a stationary point is a Nash equilibrium. We design and analyze a sampling-enabled projected gradient descent-type method, reliant on inexact resolution of a player-level best-response subproblem. By deriving suitable Lipschitzian guarantees on the value function, we derive both asymptotic guarantees for the sequence of iterates as well as rate and complexity guarantees for computing a stationary point by appropriate choices of the sampling rate and inexactness sequence.