Multiple Oracle Algorithm For General-Sum Continuous Games

Continuous games have compact strategy sets and continuous utility functions. Such games can have a highly complicated structure of Nash equilibria. Algorithms and numerical methods for the equilibrium computation are known only for particular classes of continuous games such as two-person polynomial games or games with pure equilibria. This contribution focuses on the computation and approximation of a mixed strategy equilibrium for the whole class of multiplayer general-sum continuous games. We extend vastly the scope of applicability of the double oracle algorithm, which was initially designed and proved to converge only for two-person zero-sum games. Specifically, we propose an iterative strategy generation technique, which splits the original problem into the master problem with only a finite subset of strategies being considered, and the subproblem in which an oracle finds the best response of each player. This simple method is guaranteed to recover an approximate equilibrium in finitely many iterations. Further, we argue that the Wasserstein distance (the earth mover's distance) is the right metric on the space of mixed strategies for our purposes. Our main result is the convergence of this algorithm in the Wasserstein distance to an equilibrium of the original continuous game. The numerical experiments show the performance of our method on several examples of games appearing in the literature.