# Local Nash Equilibria are Isolated, Strict Local Nash Equilibria in `Almost All' Zero-Sum Continuous Games

arXiv: https://arxiv.org/abs/2002.01007 · HTML: https://mlsanity.com/paper/2002.01007/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Feb 2020.

## Built on by (most cited first, 2 of 2)

- [Solving Min-Max Optimization with Hidden Structure via Gradient Descent Ascent](https://mlsanity.com/paper/2101.05248/) (2021)
- [Large deviations for interacting particle dynamics for finding mixed equilibria in zero-sum games](https://mlsanity.com/paper/2206.15177/) (2022)

## Abstract (arXiv)

We prove that differential Nash equilibria are generic amongst local Nash equilibria in continuous zero-sum games. That is, there exists an open-dense subset of zero-sum games for which local Nash equilibria are non-degenerate differential Nash equilibria. The result extends previous results to the zero-sum setting, where we obtain even stronger results; in particular, we show that local Nash equilibria are generically hyperbolic critical points. We further show that differential Nash equilibria of zero-sum games are structurally stable. The purpose for presenting these extensions is the recent renewed interest in zero-sum games within machine learning and optimization. Adversarial learning and generative adversarial network approaches are touted to be more robust than the alternative. Zero-sum games are at the heart of such approaches. Many works proceed under the assumption of hyperbolicity of critical points. Our results justify this assumption by showing `almost all' zero-sum games admit local Nash equilibria that are hyperbolic.