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We study the problem of computing an approximate Nash equilibrium ofcontinuous-action game without access to gradients. Such game access is commonin reinforcement learning settings, where the environment is typically treatedas a black box. To tackle this problem, we apply zeroth-order optimizationtechniques that combine smoothed gradient estimators with equilibrium-findingdynamics. We model players' strategies using artificial neural networks. Inparticular, we use randomized policy networks to model mixed strategies. Thesetake noise in addition to an observation as input and can flexibly representarbitrary observation-dependent, continuous-action distributions. Being able tomodel such mixed strategies is crucial for tackling continuous-action gamesthat lack pure-strategy equilibria. We evaluate the performance of our methodusing an approximation of the Nash convergence metric from game theory, whichmeasures how much players can benefit from unilaterally changing theirstrategy. We apply our method to continuous Colonel Blotto games, single-itemand multi-item auctions, and a visibility game. The experiments show that ourmethod can quickly find high-quality approximate equilibria. Furthermore, theyshow that the dimensionality of the input noise is crucial for performance. Toour knowledge, this paper is the first to solve general continuous-action gameswith unrestricted mixed strategies and without any gradient information.

Finding mixed-strategy equilibria of continuous-action games without gradients using randomized policy networks