Robust equilibria in continuous games: From strategic to dynamic robustness

TL;DR

This paper investigates the robustness of Nash equilibria in continuous games, defining strategic and dynamic robustness, establishing their relationship, and analyzing convergence rates of regularized learning dynamics.

Contribution

It introduces a geometric characterization of strategic robustness, links it to dynamic robustness, and analyzes convergence rates of entropic regularization in constrained action spaces.

Findings

Robust equilibria are invariant to small payoff perturbations.

Strategic robustness implies dynamic robustness in continuous games.

Entropic regularization leads to geometric convergence in constrained 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.