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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 → 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 et al., STOC 2021], this implies that these problems are PPAD ∩ PLS-complete. As a corollary, we also obtain the following equivalence of complexity classes: CCLS = PPAD ⋂ PLS.

Settling the complexity of Nash equilibrium in congestion games