Approximating Nash Equilibria for Black-Box Games: A Bayesian Optimization Approach

Game theory has emerged as a powerful framework for modeling a large range of multi-agent scenarios. Many algorithmic solutions require discrete, finite games with payoffs that have a closed-form specification. In contrast, many real-world applications require modeling with continuous action spaces and black-box utility functions where payoff information is available only in the form of empirical (often expensive and/or noisy) observations of strategy profiles. To the best of our knowledge, few tools exist for solving the class of expensive, black-box continuous games. In this paper, we develop a method to find equilibria for such games in a sequential decision-making framework using Bayesian Optimization. The proposed approach is validated on a collection of synthetic game problems with varying degree of noise and action space dimensions. The results indicate that it is capable of improving the maximum regret in noisy and high dimensions to a greater extent than hierarchical or discretized methods.