Settling the complexity of Nash equilibrium in congestion games

TL;DR

This paper proves the computational equivalence and complexity classification of finding Nash equilibria in congestion games and fixed points of gradient descent dynamics, establishing their PPAD∩PLS-completeness.

Contribution

It demonstrates the equivalence of Nash equilibrium computation and fixed point finding in smooth functions, and classifies their complexity as PPAD∩PLS-complete.

Findings

Nash equilibrium in congestion games is PPAD∩PLS-complete.

Fixed points of gradient descent dynamics are PPAD∩PLS-complete.

CCLS equals PPAD∩PLS in complexity classes.

Abstract

We consider (i) the problem of finding a (possibly mixed) Nash equilibrium in congestion games, and (ii) the problem of finding an (exponential precision) fixed point of the gradient descent dynamics of a smooth function $f:[0,1]^n \rightarrow \mathbb{R}$. We prove that these problems are equivalent. Our result holds for various explicit descriptions of $f$, ranging from (almost general) arithmetic circuits, to degree-$5$ polynomials. By a very recent result of [Fearnley, Goldberg, Hollender, Savani '20] this implies that these problems are PPAD$\cap$PLS-complete. As a corollary, we also obtain the following equivalence of complexity classes: CCLS = PPAD$\cap$PLS.