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Robust equilibria in continuous games: From strategic to dynamic robustness

Kyriakos Lotidis, Panayotis Mertikopoulos, Nicholas Bambos, Jose Blanchet

TL;DR

The paper analyzes robustness of Nash equilibria in continuous games from static (strategic) and dynamic (learning-based) perspectives. It defines strategic robustness via a gradient-field distance and provides a geometric-variational characterization, showing robust equilibria occur at boundary points where the gradient lies in the interior of the polar cone. It then proves that strategic robustness implies dynamic robustness under follow-the-regularized-leader learning with both gradient and payoff-based feedback, and shows the necessity of strategic robustness for achieving dynamic robustness, along with geometric convergence rates (geometric under entropy regularization) in affinely constrained spaces. The results illuminate a deep connection between equilibrium stability under perturbations and learning dynamics, with practical relevance for robust decision-making in ML systems employing continuous action spaces.

Abstract

In this paper, we examine the robustness of Nash equilibria in continuous games, under both strategic and dynamic uncertainty. Starting with the former, we introduce the notion of a robust equilibrium as those equilibria that remain invariant to small -- but otherwise arbitrary -- perturbations to the game's payoff structure, and we provide a crisp geometric characterization thereof. Subsequently, we turn to the question of dynamic robustness, and we examine which equilibria may arise as stable limit points of the dynamics of "follow the regularized leader" (FTRL) in the presence of randomness and uncertainty. Despite their very distinct origins, we establish a structural correspondence between these two notions of robustness: strategic robustness implies dynamic robustness, and, conversely, the requirement of strategic robustness cannot be relaxed if dynamic robustness is to be maintained. Finally, we examine the rate of convergence to robust equilibria as a function of the underlying regularizer, and we show that entropically regularized learning converges at a geometric rate in games with affinely constrained action spaces.

Robust equilibria in continuous games: From strategic to dynamic robustness

TL;DR

The paper analyzes robustness of Nash equilibria in continuous games from static (strategic) and dynamic (learning-based) perspectives. It defines strategic robustness via a gradient-field distance and provides a geometric-variational characterization, showing robust equilibria occur at boundary points where the gradient lies in the interior of the polar cone. It then proves that strategic robustness implies dynamic robustness under follow-the-regularized-leader learning with both gradient and payoff-based feedback, and shows the necessity of strategic robustness for achieving dynamic robustness, along with geometric convergence rates (geometric under entropy regularization) in affinely constrained spaces. The results illuminate a deep connection between equilibrium stability under perturbations and learning dynamics, with practical relevance for robust decision-making in ML systems employing continuous action spaces.

Abstract

In this paper, we examine the robustness of Nash equilibria in continuous games, under both strategic and dynamic uncertainty. Starting with the former, we introduce the notion of a robust equilibrium as those equilibria that remain invariant to small -- but otherwise arbitrary -- perturbations to the game's payoff structure, and we provide a crisp geometric characterization thereof. Subsequently, we turn to the question of dynamic robustness, and we examine which equilibria may arise as stable limit points of the dynamics of "follow the regularized leader" (FTRL) in the presence of randomness and uncertainty. Despite their very distinct origins, we establish a structural correspondence between these two notions of robustness: strategic robustness implies dynamic robustness, and, conversely, the requirement of strategic robustness cannot be relaxed if dynamic robustness is to be maintained. Finally, we examine the rate of convergence to robust equilibria as a function of the underlying regularizer, and we show that entropically regularized learning converges at a geometric rate in games with affinely constrained action spaces.

Paper Structure

This paper contains 32 sections, 28 theorems, 141 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our contributions in the context of related work.
  3. Preliminaries
  4. The game-theoretic framework
  5. NE.
  6. Regularized learning in games
  7. The algorithm's step-size.
  8. The mirror map.
  9. The feedback process.
  10. Stochastic gradient feedback.
  11. Payoff-based feedback.
  12. Strategic robustness: Geometric and variational characterization
  13. A first approach and insights
  14. Defining strategic robustness
  15. Geometric characterization.
  16. ...and 17 more sections

Key Result

Proposition 1

For any game $\mathcal{G}$ and any equilibrium $x^{\ast}\in\mathcal{X}$ of $\mathcal{G}$, there exists a perturbed game $\tilde{\mathcal{G}}$, arbitrarily close to $\mathcal{G}$ in the uniform metric eq:temp-dist such that $x^{\ast} \in \mathcal{X}$ is not an equilibrium of $\tilde{\mathcal{G}}$.

Figures (2)

  • Figure 1: Different equilibrium configurations: an interior equilibrium ($v(x^{\ast}) = 0$); a boundary, non-extreme equilibrium (normal cone with empty topological interior); an extreme, non-robust equilibrium ($v(x^{\ast})$ on the boundary of the normal cone); a robust equilibrium ($v(x^{\ast})$ in the interior of the normal cone). Only the robust equilibrium remains invariant under strategic perturbations of the underlying game.
  • Figure 2: Convergence and non-convergence to different type of equilibria. Only robust equilibria are stochastically asymptotically stable under \ref{['eq:FTRL']}.

Theorems & Definitions (58)

  • Example 2.1: Euclidean projection
  • Example 2.2: Exponential weights
  • Proposition 1
  • Example 3.1
  • Example 3.2
  • Remark 1
  • Definition 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • ...and 48 more