Settling the Complexity of Nash Equilibrium in Congestion Games

We consider

(i) the problem of finding a (possibly mixed) Nash equilibrium in congestion game, and

(ii) the problem of finding an (exponential precision) fixed point of the gradient descent dynamics of a smooth function over the cube. We prove that these problems are equivalent.

Our result holds for various explicit descriptions of the function, ranging from (almost general) arithmetic  circuits, to degree-5 polynomials. By a very recent result of [Fearnley, Goldberg, Hollender, Savani 2020], this implies that these problems are PPAD∩PLS-complete. As a corollary, we also obtain the following equivalence of complexity classes: CCLS=PPAD∩PLS.

Remark: The talk will not assume prior knowledge of the complexity classes (PPAD, PLS, CCLS).

Joint work with Aviad Rubinstein

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