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We extend the study of the complexity of finding an $\eps$-approximate Nashequilibrium in congestion games from the case of positive delay functions todelays of arbitrary sign. We first prove that in symmetric games with$\alpha$-bounded jump the $\eps$-Nash dynamic converges in polynomial time whenall delay functions are negative, similarly to the case of positive delays. Wethen establish a hardness result for symmetric games with $\alpha$-bounded jumpand with arbitrary delay functions: in that case finding an $\eps$-Nashequilibrium becomes $\PLS$-complete.

The complexity of approximate Nash equilibrium in congestion games with negative delays