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Nash equilibrium has long been a desired solution concept in multi-playergames, especially for those on continuous strategy spaces, which have attracteda rapidly growing amount of interests due to advances in research applicationssuch as the generative adversarial networks. Despite the fact that several deeplearning based approaches are designed to obtain pure strategy Nashequilibrium, it is rather luxurious to assume the existence of such anequilibrium. In this paper, we present a new method to approximate mixedstrategy Nash equilibria in multi-player continuous games, which always existand include the pure ones as a special case. We remedy the pure strategyweakness by adopting the pushforward measure technique to represent a mixedstrategy in continuous spaces. That allows us to generalize the Gradient-basedNikaido-Isoda (GNI) function to measure the distance between the players' jointstrategy profile and a Nash equilibrium. Applying the gradient descentalgorithm, our approach is shown to converge to a stationary Nash equilibriumunder the convexity assumption on payoff functions, the same popular setting asin previous studies. In numerical experiments, our method consistently andsignificantly outperforms recent works on approximating Nash equilibrium forquadratic games, general blotto games, and GAMUT games.

Finding Mixed Strategy Nash Equilibrium for Continuous Games through Deep Learning