Existence and Complexity of Approximate Equilibria in Weighted Congestion Games
Abstract
We study the existence of approximate pure Nash equilibria ($\alpha$PNE) in weighted atomic congestion games with polynomial cost functions of maximum degree $d$. Previously it was known that $d$approximate equilibria always exist, while nonexistence was established only for small constants, namely for $1.153$PNE. We improve significantly upon this gap, proving that such games in general do not have $\tilde{\Theta}(\sqrt{d})$approximate PNE, which provides the first superconstant lower bound. Furthermore, we provide a blackbox gapintroducing method of combining such nonexistence results with a specific circuit gadget, in order to derive NPcompleteness of the decision version of the problem. In particular, deploying this technique we are able to show that deciding whether a weighted congestion game has an $\tilde{O}(\sqrt{d})$PNE is NPcomplete. Previous hardness results were known only for the special case of exact equilibria and arbitrary cost functions. The circuit gadget is of independent interest and it allows us to also prove hardness for a variety of problems related to the complexity of PNE in congestion games. For example, we demonstrate that the question of existence of $\alpha$PNE in which a certain set of players plays a specific strategy profile is NPhard for any $\alpha < 3^{d/2}$, even for unweighted congestion games. Finally, we study the existence of approximate equilibria in weighted congestion games with general (nondecreasing) costs, as a function of the number of players $n$. We show that $n$PNE always exist, matched by an almost tight nonexistence bound of $\tilde\Theta(n)$ which we can again transform into an NPcompleteness proof for the decision problem.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 arXiv:
 arXiv:2002.07466
 Bibcode:
 2020arXiv200207466C
 Keywords:

 Computer Science  Computer Science and Game Theory;
 Computer Science  Computational Complexity