  One of the main results shown through Roughgarden's notions of smooth games
and robust price of anarchy is that, for any sum-bounded utilitarian social
function, the worst-case price of anarchy of coarse correlated equilibria
coincides with that of pure Nash equilibria in the class of weighted congestion
games with non-negative and non-decreasing latency functions and that such a
value can always be derived through the, so called, smoothness argument. We
significantly extend this result by proving that, for a variety of (even
non-sum-bounded) utilitarian and egalitarian social functions and for a broad
generalization of the class of weighted congestion games with non-negative (and
possibly decreasing) latency functions, the worst-case price of anarchy of
$\epsilon$-approximate coarse correlated equilibria still coincides with that
of $\epsilon$-approximate pure Nash equilibria, for any $\epsilon\geq 0$. As a
byproduct of our proof, it also follows that such a value can always be
determined by making use of the primal-dual method we introduced in a previous
work. It is important to note that our scenario of investigation is beyond the
scope of application of the robust price of anarchy (for as it is currently
defined), so that our result seems unlikely to be alternatively proved via the
smoothness framework.
