A Bayesian optimization approach to find Nash equilibria
This addresses a gap in algorithmic solutions for game theory applications in engineering and machine learning where derivative information is unavailable or costly to obtain.
The authors tackled the problem of finding Nash equilibria in derivative-free, expensive black-box games by proposing a Gaussian-process based Bayesian optimization approach. They showed that equilibria can be found reliably for a fraction of the cost compared to classical derivative-based algorithms, though specific numerical results are not provided.
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.