On the robustness of the approximate price of anarchy in generalized congestion games

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 ϵ-approximate Coarse Correlated Equilibria still coincides with that of ϵ-approximate Pure Nash Equilibria, for any ϵ≥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.