On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract)

d Abstract) Kousha Etessami LFCS, School of Informatics University of Edinburgh Mihalis Yannakakis Department of Computer Science Columbia University We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 or more players, and given > 0, compute a vector x (a mixed strategy profile) that is within distance (say, in l∞) of some (exact) Nash equilibrium. We show that approximation of an (actual) Nash equilibrium for games with 3 players, even to within any non-trivial constant additive factor < 1/2 in just one desired coordinate, is at least as hard as the long standing square-root sum problem, as well as more general arithmetic circuit decision problems, and thus that even placing the approximation problem in NP would resolve a major ...

• Type: Conference
• Year: 2007
• Where: FOCS
• Authors: Kousha Etessami, Mihalis Yannakakis
• Keywords: FOCS 2007, Nash Equilibria, Nash Equilibrium, Theoretical Computer Science, fixed Point