We study the problem of computing an approximate Nash equilibrium of a game whose strategy space is continuous without access to gradients of the utility function. Lack of access to gradients is common in reinforcement learning settings, where the environment is treated as a black box, as well as equilibrium finding in mechanisms such as auctions, where the mechanism's payoffs are discontinuous in the players' actions. To tackle this problem, we turn to zeroth-order optimization techniques that combine pseudo-gradients with equilibrium-finding dynamics. Specifically, we introduce a new technique that requires a number of utility function evaluations per iteration that is constant rather than linear in the number of players. It achieves this by performing a single joint perturbation on all players' strategies, rather than perturbing each one individually. This is very important for many-player games, especially when the utility function is expensive to compute in terms of wall time, memory, money, or other resources. We evaluate our approach on various games, including auctions, which have important real-world applications. Our approach yields a dramatic improvement in performance in terms of the wall time required to reach an approximate Nash equilibrium.

We study the problem of computing an approximate Nash equilibrium of continuous-action game without access to gradients. Such game access is common in reinforcement learning settings, where the environment is typically treated as a black box. To tackle this problem, we apply zeroth-order optimization techniques that combine smoothed gradient estimators with equilibrium-finding dynamics. We model players' strategies using artificial neural networks. In particular, we use randomized policy networks to model mixed strategies. These take noise in addition to an observation as input and can flexibly represent arbitrary observation-dependent, continuous-action distributions. Being able to model such mixed strategies is crucial for tackling continuous-action games that lack pure-strategy equilibria. We evaluate the performance of our method using an approximation of the Nash convergence metric from game theory, which measures how much players can benefit from unilaterally changing their strategy. We apply our method to continuous Colonel Blotto games, single-item and multi-item auctions, and a visibility game. The experiments show that our method can quickly find a high-quality approximate equilibrium. Furthermore, they show that the dimensionality of the input noise is crucial for performance. To our knowledge, this paper is the first to solve general continuous-action games with unrestricted mixed strategies and without any gradient information.