The Price of Stability of Weighted Congestion Games

We give exponential lower bounds on the Price of Stability (PoS) of weighted congestiongames with polynomial cost functions. In particular, for any positive integer d we constructrather simple games with cost functions of degree at most d which have a PoS of at leastΩ(Φd), where Φd ∼ d/ ln d is the unique positive root of equation xd+1 = (x + 1).This asymptotically closes the huge gap between Θ(d) and Φd+1d and matches the Price ofAnarchy upper bound. We further show that the PoS remains exponential even for singletongames. More generally, we also provide a lower bound of Ω((1 + 1/α)/d) on the PoS ofα-approximate Nash equilibria. 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 approximate Nashequilibria, which is sensitive to the range W of the player weights. We do this by explicitlyconstructing a novel approximate potential function, based on Faulhaber’s formula, thatgeneralizes 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 pureNash equilibrium with PoS at most (d + 3)/2; the equilibrium’s approximation parameterranges from Θ(1) to d+ 1 in a smooth way with respect to W . Secondly, we show that forunweighted congestion games, the PoS of α-approximate Nash equilibria is at most (d+1)/α.