Abstract
Successful algorithms have been developed for computing Nash equilibrium in a variety of finite game classes. However, solving continuous games—in which the pure strategy space is (potentially uncountably) infinite—is far more challenging. Nonetheless, many real-world domains have continuous action spaces, e.g., where actions refer to an amount of time, money, or other resource that is naturally modeled as being real-valued as opposed to integral. We present a new algorithm for approximating Nash equilibrium strategies in continuous games. In addition to two-player zero-sum games, our algorithm also applies to multiplayer games and games with imperfect information. We experiment with our algorithm on a continuous imperfect-information Blotto game, in which two players distribute resources over multiple battlefields. Blotto games have frequently been used to model national security scenarios and have also been applied to electoral competition and auction theory. Experiments show that our algorithm is able to quickly compute close approximations of Nash equilibrium strategies for this game.
Highlights
Successful algorithms have been developed for computing approximate Nash equilibrium strategies in a variety of finite game classes, even classes that are challenging from a computational complexity perspective
Even solving three-player perfect-information strategicform games is challenging from a theoretical complexity perspective; it is PPAD-hard1 to compute a Nash equilibrium in two-player general-sum and multiplayer games, and it is widely believed that no efficient algorithms exist [2,3,4]
We presented a new algorithm for computing Nash equilibrium in a broad class of continuous games
Summary
Successful algorithms have been developed for computing approximate Nash equilibrium strategies in a variety of finite game classes, even classes that are challenging from a computational complexity perspective. An algorithm that was recently applied for approximating Nash equilibrium strategies in six-player no-limit Texas hold’em poker defeated strong human professional players [1] This is an extremely large extensive-form game of imperfect information. Nash equilibrium existence and computational complexity results from strategic-form games hold for imperfect-information extensive-form games; e.g., all finite games are guaranteed to have a Nash equilibrium, two-player zerosum games can be solved in polynomial time, and equilibrium computation for other game classes is PPAD-hard. The concept of a behavioral strategy in an extensive-form game corresponds to a strategy that assigns a probability distribution over the set of possible actions at each of the player’s information sets. We can define extensive-form imperfect-information continuous games to that for finite games, with analogous definitions of mixed and behavioral strategies. The continuous Blotto game that we consider does not fit in any of these classes, and has discontinuous utility functions, so we cannot apply Theorem 1 or these algorithms
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