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

Luke Marrinan, Farzad Yousefian, Uday V. Shanbhag

TL;DR

The paper addresses computing Nash equilibria in stochastic NEPs with convex, Lipschitz objectives by reformulating the problem with the Nikaido-Isoda (NI) function and its regularized variant $V_\alpha$. It develops a sampling-enabled, inexact projected gradient method that handles inexact best-responses and uses stochastic gradient estimates to minimize $V_\alpha$, proving almost-sure convergence to stationary points of $V_\alpha$ and deriving explicit rate and complexity guarantees. Crucially, under a suitable regularity condition, stationary points of $V_\alpha$ correspond to Nash equilibria, enabling equilibrium computation without requiring gradient map monotonicity or game potentiality. The results yield concrete projection and sampling complexities: with a fixed stepsize, $\mathcal{O}(\varepsilon^{-2})$ projections and $\mathcal{O}(\varepsilon^{-4})$ samples; with diminishing stepsizes, $\mathcal{O}(\varepsilon^{-4})$ projections and $\mathcal{O}(\varepsilon^{-6})$ samples, offering practical guarantees for stochastic equilibrium computation in uncertain multi-agent settings.

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.

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

TL;DR

The paper addresses computing Nash equilibria in stochastic NEPs with convex, Lipschitz objectives by reformulating the problem with the Nikaido-Isoda (NI) function and its regularized variant . It develops a sampling-enabled, inexact projected gradient method that handles inexact best-responses and uses stochastic gradient estimates to minimize , proving almost-sure convergence to stationary points of and deriving explicit rate and complexity guarantees. Crucially, under a suitable regularity condition, stationary points of correspond to Nash equilibria, enabling equilibrium computation without requiring gradient map monotonicity or game potentiality. The results yield concrete projection and sampling complexities: with a fixed stepsize, projections and samples; with diminishing stepsizes, projections and samples, offering practical guarantees for stochastic equilibrium computation in uncertain multi-agent settings.

Abstract

We consider the computation of an equilibrium of a stochastic Nash equilibrium problem, where the player objectives are assumed to be -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.

Paper Structure

This paper contains 11 sections, 15 theorems, 90 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Prior research on stochastic Nash equilibrium problems
  3. Motivation, contributions, and outline
  4. Background and Preliminaries
  5. Equilibrium Conditions
  6. The Regularized NI Function
  7. An Inexact Algorithm For Computing Equilibria
  8. Inexact computation of yᵛᵅ(x)
  9. Algorithm description
  10. Convergence and rate analysis
  11. Concluding remarks

Key Result

Theorem 2.3

Consider the NEP$(\Theta,{\bf X})$. Suppose Assumption ass:ass-1 holds. Then a Nash equilibrium exists if one of the following holds. (a) $\mathcal{X}^{\nu}$ is bounded for every $\nu \in \{1, \dots, N\}.$ (b) There exists a vector ${\mathbf{x}}^{\rm ref} \, \in \, \mathcal{X}$ such that

Theorems & Definitions (29)

  • Definition 2.1: Nash Equilibrium Problem
  • Theorem 2.3: Existence of NE
  • Theorem 2.4: Uniqueness of NE
  • Proposition 2.5: Properties of the regularized NI function $V_{\alpha}{(\bullet)}$
  • Lemma 2.6: Danskin's Theorem
  • Lemma 2.7
  • Proof 1
  • Proposition 2.8: Smoothness and Lipschitzian properties of $V_{\alpha}{(\bullet)}$
  • Proof 2
  • Proposition 3.1: A Sufficient Condition For A Stationary Point to be an NEP kanzow
  • ...and 19 more