# Robust equilibria in continuous games: From strategic to dynamic robustness

arXiv: https://arxiv.org/abs/2512.08138 · HTML: https://mlsanity.com/paper/2512.08138/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 59 references that are in the map.
- First posted on arXiv 9 Dec 2025.

## Builds on (most cited first, 17 of 17)

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
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- [Global Convergence of Multi-Agent Policy Gradient in Markov Potential Games](https://mlsanity.com/paper/2106.01969/) (2021)
- [Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains](https://mlsanity.com/paper/1707.06386/) (2017)
- [Learning in games with continuous action sets and unknown payoff functions](https://mlsanity.com/paper/1608.07310/) (2016)
- [Adaptive Learning in Continuous Games: Optimal Regret Bounds and Convergence to Nash Equilibrium](https://mlsanity.com/paper/2104.12761/) (2021)
- [The limits of min-max optimization algorithms: convergence to spurious non-critical sets](https://mlsanity.com/paper/2006.09065/) (2020)
- [No-regret learning and mixed Nash equilibria: They do not mix](https://mlsanity.com/paper/2010.09514/) (2020)
- [Fighting Bandits with a New Kind of Smoothness](https://mlsanity.com/paper/1512.04152/) (2015)
- [Bandit learning in concave N-person games](https://mlsanity.com/paper/1810.01925/) (2018)
- [Survival of the strictest: Stable and unstable equilibria under regularized learning with partial information](https://mlsanity.com/paper/2101.04667/) (2021)
- [A unified stochastic approximation framework for learning in games](https://mlsanity.com/paper/2206.03922/) (2022)
- [No-Regret Learning in Games with Noisy Feedback: Faster Rates and Adaptivity via Learning Rate Separation](https://mlsanity.com/paper/2206.06015/) (2022)
- [On the convergence of policy gradient methods to Nash equilibria in general stochastic games](https://mlsanity.com/paper/2210.08857/) (2022)
- [The equivalence of dynamic and strategic stability under regularized learning in games](https://mlsanity.com/paper/2311.02407/) (2023)
- [The impact of uncertainty on regularized learning in games](https://mlsanity.com/paper/2506.13286/) (2025)

## Abstract (arXiv)

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.