A Bayesian optimization approach to find Nash equilibria

TL;DR

This paper introduces a Bayesian optimization method using Gaussian processes to efficiently find Nash equilibria in expensive black-box game scenarios, outperforming traditional derivative-based algorithms in cost and reliability.

Contribution

A novel Gaussian-process based Bayesian optimization approach for finding Nash equilibria in derivative-free, black-box game settings, with strategies for scalability and reduced computation.

Findings

Reliable equilibrium detection with fewer evaluations

Effective in high-dimensional decision spaces

Outperforms classical algorithms in cost efficiency

Abstract

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.