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Distributionally Robust Nash Equilibria via Variational Inequalities

Zeinab Alizadeh, Azadeh Farsi, Afrooz Jalilzadeh

TL;DR

The paper tackles finding Distributionally Robust Nash Equilibria in $n$-player noncooperative games under distributional ambiguity, formulating the problem as a minimax objective over strategies $x_i\in K_i$ and distributions $p_i\in P_i$. It recasts DRNE as a variational inequality with a monotone (possibly set-valued) operator and introduces a Minty VI gap to quantify progress in nonsmooth settings. A gradient descent–ascent algorithm with minibatch subgradients (GDA-DRNE) and projection steps is proposed, with proven convergence guarantees: an $\mathcal{O}\left(\frac{1}{\epsilon^{2}}\log^{2}\left(\frac{1}{\epsilon}\right)\right)$ oracle complexity and almost sure convergence to a DRNE. Theoretical results rely on a robust VI framework and Robbins–Siegmund techniques, while numerical experiments on risk-averse RaNE with CVaR illustrate robustness and scalability in practice.

Abstract

Nash Equilibrium and its robust counterpart, Distributionally Robust Nash Equilibrium (DRNE), are fundamental problems in game theory with applications in economics, engineering, and machine learning. This paper addresses the problem of DRNE, where multiple players engage in a noncooperative game under uncertainty. Each player aims to minimize their objective against the worst-case distribution within an ambiguity set, resulting in a minimax structure. We reformulate the DRNE problem as a Variational Inequality (VI) problem, providing a unified framework for analysis and algorithm development. We propose a gradient descent-ascent type algorithm with convergence guarantee that effectively addresses the computational challenges of high-dimensional and nonsmooth objectives.

Distributionally Robust Nash Equilibria via Variational Inequalities

TL;DR

The paper tackles finding Distributionally Robust Nash Equilibria in -player noncooperative games under distributional ambiguity, formulating the problem as a minimax objective over strategies and distributions . It recasts DRNE as a variational inequality with a monotone (possibly set-valued) operator and introduces a Minty VI gap to quantify progress in nonsmooth settings. A gradient descent–ascent algorithm with minibatch subgradients (GDA-DRNE) and projection steps is proposed, with proven convergence guarantees: an oracle complexity and almost sure convergence to a DRNE. Theoretical results rely on a robust VI framework and Robbins–Siegmund techniques, while numerical experiments on risk-averse RaNE with CVaR illustrate robustness and scalability in practice.

Abstract

Nash Equilibrium and its robust counterpart, Distributionally Robust Nash Equilibrium (DRNE), are fundamental problems in game theory with applications in economics, engineering, and machine learning. This paper addresses the problem of DRNE, where multiple players engage in a noncooperative game under uncertainty. Each player aims to minimize their objective against the worst-case distribution within an ambiguity set, resulting in a minimax structure. We reformulate the DRNE problem as a Variational Inequality (VI) problem, providing a unified framework for analysis and algorithm development. We propose a gradient descent-ascent type algorithm with convergence guarantee that effectively addresses the computational challenges of high-dimensional and nonsmooth objectives.
Paper Structure (7 sections, 4 theorems, 31 equations, 1 figure, 1 algorithm)

This paper contains 7 sections, 4 theorems, 31 equations, 1 figure, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Related Works
  3. Variational Inequality Reformulation of DRNE
  4. Preliminaries
  5. Proposed Method
  6. Convergence Analysis
  7. Numerical Experiments

Key Result

Theorem 5

Suppose Assumptions assump:F, assum:samples, and ass:bounded_operator hold. Let $\{x^t,p^t\}_{t}$ be the sequence generated by Algorithm alg1, where $x=[x_i]_{i=1}^n$ and $p=[p_i]_{i=1}^n$. Choose the stepsizes as $\lambda_t = \gamma_t=\frac{1}{\sqrt{1+t}\ log(t+2)}$ and define $\bar{x}^T\triangleq Consequently, the oracle complexity required to achieve an $\epsilon$-gap is $\mathcal{O}\!\left(\t

Figures (1)

  • Figure 1: Performance of GDA-DRNE averaged over 5 simulations for different batch sizes.

Theorems & Definitions (7)

  • Remark 4
  • Theorem 5: Convergence rate
  • Remark 6
  • Theorem 7: Almost Sure Convergence
  • Lemma 8
  • Definition 9: Stochastic errors
  • Lemma 10