  Nash equilibrium has long been a desired solution concept in multi-player
games, especially for those on continuous strategy spaces, which have attracted
a rapidly growing amount of interests due to advances in research applications
such as the generative adversarial networks. Despite the fact that several deep
learning based approaches are designed to obtain pure strategy Nash
equilibrium, it is rather luxurious to assume the existence of such an
equilibrium. In this paper, we present a new method to approximate mixed
strategy Nash equilibria in multi-player continuous games, which always exist
and include the pure ones as a special case. We remedy the pure strategy
weakness by adopting the pushforward measure technique to represent a mixed
strategy in continuous spaces. That allows us to generalize the Gradient-based
Nikaido-Isoda (GNI) function to measure the distance between the players' joint
strategy profile and a Nash equilibrium. Applying the gradient descent
algorithm, our approach is shown to converge to a stationary Nash equilibrium
under the convexity assumption on payoff functions, the same popular setting as
in previous studies. In numerical experiments, our method consistently and
significantly outperforms recent works on approximating Nash equilibrium for
quadratic games, general blotto games, and GAMUT games.
