On the Robustness of the Approximate Price of Anarchy in Generalized Congestion Games

TL;DR

This paper extends the understanding of the robustness of the price of anarchy in generalized congestion games, showing that for a broader class of social functions and latency conditions, the worst-case inefficiency remains consistent across approximate equilibria.

Contribution

It proves that the worst-case price of anarchy for approximate equilibria matches that of pure equilibria in more general congestion game settings, beyond previous smoothness-based results.

Findings

Worst-case price of anarchy matches for approximate and pure equilibria.

Results apply to broader classes of social functions and latency functions.

Primal-dual method can determine the price of anarchy in these settings.

Abstract

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.