# A Bayesian optimization approach to find Nash equilibria

arXiv: https://arxiv.org/abs/1611.02440 · HTML: https://mlsanity.com/paper/1611.02440/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Nov 2016.

## Built on by (most cited first, 9 of 9)

- [Joint Entropy Search for Multi-objective Bayesian Optimization](https://mlsanity.com/paper/2210.02905/) (2022)
- [Derivative-free Alternating Projection Algorithms for General Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/2108.00473/) (2021)
- [Multi-objective optimisation via the R2 utilities](https://mlsanity.com/paper/2305.11774/) (2023)
- [No-Regret Learning of Nash Equilibrium for Black-Box Games via Gaussian Processes](https://mlsanity.com/paper/2405.08318/) (2024)
- [Do We Really Need to Approach the Entire Pareto Front in Many-Objective Bayesian Optimisation?](https://mlsanity.com/paper/2604.09417/) (2026)
- [An active learning method for solving competitive multi-agent decision-making and control problems](https://mlsanity.com/paper/2212.12561/) (2022)
- [Bayesian Optimization for Non-Cooperative Game-Based Radio Resource Management](https://mlsanity.com/paper/2512.01245/) (2025)
- [Converge Then Diversify: Decoupling Convergence and Diversity in Multi-Objective Bayesian Optimisation](https://mlsanity.com/paper/2609.13396/) (2026)
- [Efficient Nash Equilibrium Computation for Cybersecurity Games](https://mlsanity.com/paper/2609.19399/) (2026)

## Abstract (arXiv)

Game theory finds nowadays a broad range of applications in engineering and machine learning. However, in a derivative-free, expensive black-box context, very few algorithmic solutions are available to find game equilibria. Here, we propose a novel Gaussian-process based approach for solving games in this context. We follow a classical Bayesian optimization framework, with sequential sampling decisions based on acquisition functions. Two strategies are proposed, based either on the probability of achieving equilibrium or on the Stepwise Uncertainty Reduction paradigm. Practical and numerical aspects are discussed in order to enhance the scalability and reduce computation time. Our approach is evaluated on several synthetic game problems with varying number of players and decision space dimensions. We show that equilibria can be found reliably for a fraction of the cost (in terms of black-box evaluations) compared to classical, derivative-based algorithms. The method is available in the R package GPGame available on CRAN at https://cran.r-project.org/package=GPGame.