Abstract
We give exponential lower bounds on the Price of Stability (PoS) of weighted congestion games with polynomial cost functions. In particular, for any positive integer d we construct rather simple games with cost functions of degree at most d which have a PoS of at least Omega(Phi_d)^{d+1}, where Phi_d ~ d/ln d is the unique positive root of equation x^{d+1}=(x+1)^d. This essentially closes the huge gap between Theta(d) and Phi_d^{d+1} and asymptotically matches the Price of Anarchy upper bound. We further show that the PoS remains exponential even for singleton games. More generally, we also provide a lower bound of Omega((1+1/alpha)^d/d) on the PoS of alphaapproximate Nash equilibria, even for singleton games. All our lower bounds extend to network congestion games, and hold for mixed and correlated equilibria as well.
On the positive side, we give a general upper bound on the PoS of alphaapproximate Nash equilibria, which is sensitive to the range W of the player weights and the approximation parameter alpha. We do this by explicitly constructing a novel approximate potential function, based on Faulhaber's formula, that generalizes Rosenthal's potential in a continuous, analytic way. From the general theorem, we deduce two interesting corollaries. First, we derive the existence of an approximate pure Nash equilibrium with PoS at most (d+3)/2; the equilibrium's approximation parameter ranges from Theta(1) to d+1 in a smooth way with respect to W. Secondly, we show that for unweighted congestion games, the PoS of alphaapproximate Nash equilibria is at most (d+1)/alpha.
BibTeX  Entry
@InProceedings{christodoulou_et_al:LIPIcs:2018:9154,
author = {George Christodoulou and Martin Gairing and Yiannis Giannakopoulos and Paul G. Spirakis},
title = {{The Price of Stability of Weighted Congestion Games}},
booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
pages = {150:1150:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770767},
ISSN = {18688969},
year = {2018},
volume = {107},
editor = {Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9154},
URN = {urn:nbn:de:0030drops91541},
doi = {10.4230/LIPIcs.ICALP.2018.150},
annote = {Keywords: Congestion games, price of stability, Nash equilibrium, approximate equilibrium, potential games}
}
Keywords: 

Congestion games, price of stability, Nash equilibrium, approximate equilibrium, potential games 
Collection: 

45th International Colloquium on Automata, Languages, and Programming (ICALP 2018) 
Issue Date: 

2018 
Date of publication: 

04.07.2018 