Nash equilibria: Complexity, symmetries, and approximation

We survey recent joint work with Christos Papadimitriou and Paul Goldberg on the computational complexity of Nash equilibria. We show that finding a Nash equilibrium in normal form games is computationally intractable, but in an unusual way. It does belong to the class NP; but Nash’s theorem, showing that a Nash equilibrium always exists, makes the possibility that it is also NP-complete rather unlikely. We show instead that the problem is as hard computationally as finding Brouwer fixed points, in a precise technical sense, giving rise to a new complexity class called PPAD. The existence of the Nash equilibrium was established via Brouwer’s fixed-point theorem; hence, we provide a computational converse to Nash’s theorem. To alleviate the negative implications of this result for the predictive power of the Nash equilibrium, it seems natural to study the complexity of approximate equilibria: an efficient approximation scheme would imply that players could in principle come arbitrarily close to a Nash equilibrium given enough time. We review recent work on computing approximate equilibria and conclude by studying how symmetries may affect the structure and approximation of Nash equilibria. Nash showed that every symmetric game has a symmetric equilibrium. We complement this theorem with a rich set of structural results for a broader, and more interesting class of games with symmetries, called anonymous games.