## On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract) (2007)

Venue: | IN PROC. FOCS |

Citations: | 37 - 4 self |

### BibTeX

@INPROCEEDINGS{Etessami07onthe,

author = {Kousha Etessami and Mihalis Yannakakis},

title = {On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract)},

booktitle = {IN PROC. FOCS},

year = {2007},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 ɛ> 0, compute an approximation within ɛ of some (actual) Nash equilibrium. We show that approximation of an actual Nash Equilibrium, even to within any non-trivial constant additive factor ɛ < 1/2 in just one desired coordinate, is at least as hard as the long standing square-root sum problem, as well as a more general arithmetic circuit decision problem that characterizes P-time in a unit-cost model of computation with arbitrary precision rational arithmetic; thus placing the approximation problem in P, or even NP, would resolve major open problems in the complexity of numerical computation. We show similar results for market equilibria: it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium, where the economy is given by explicit algebraic formulas for the excess demand functions. We define a class, FIXP, which captures search problems that can be cast as fixed point