# Pure Nash Equilibria and Best-Response Dynamics in Random Games

arXiv: https://arxiv.org/abs/1905.10758 · HTML: https://mlsanity.com/paper/1905.10758/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 May 2019.

## Built on by (most cited first, 4 of 4)

- [Nash Equilibria and Pitfalls of Adversarial Training in Adversarial Robustness Games](https://mlsanity.com/paper/2210.12606/) (2022)
- [From Leiden to Pleasure Island: The Constant Potts Model for Community Detection as a Hedonic Game](https://mlsanity.com/paper/2509.03834/) (2025)
- [Learning Recommender Mechanisms for Bayesian Stochastic Games](https://mlsanity.com/paper/2505.22979/) (2025)
- [Decision Making under Imperfect Recall: Algorithms and Benchmarks](https://mlsanity.com/paper/2602.15252/) (2026)

## Abstract (arXiv)

In finite games mixed Nash equilibria always exist, but pure equilibria may fail to exist. To assess the relevance of this nonexistence, we consider games where the payoffs are drawn at random. In particular, we focus on games where a large number of players can each choose one of two possible strategies, and the payoffs are i.i.d. with the possibility of ties. We provide asymptotic results about the random number of pure Nash equilibria, such as fast growth and a central limit theorem, with bounds for the approximation error. Moreover, by using a new link between percolation models and game theory, we describe in detail the geometry of Nash equilibria and show that, when the probability of ties is small, a best-response dynamics reaches a Nash equilibrium with a probability that quickly approaches one as the number of players grows. We show that a multitude of phase transitions depend only on a single parameter of the model, that is, the probability of having ties.