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 . Our bound extends also to network congestion games. We further show that the PoS remains exponential even for singleton games. More generally, we provide a lower bound of Ω((1 + 1/α) d /d) on the PoS of α-approximate Nash equilibria for singleton games. All our lower bounds hold for mixed and correlated equilibria as well. On the positive side, we give a general upper bound on the PoS of α-approximate Nash equilibria, which is sensitive to the range W of the player weights and the approximation parameter α. 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 Θ(1) to d + 1 in a smooth way with respect to W . Secondly, we show that for unweighted congestion games, the PoS of α-approximate Nash equilibria is at most (d + 1)/α.
However, recently, Christodoulou et al. REF showed that polynomial weighted congestion games have an approximate analogue of a potential function, which they called Faulhaber's potential.
The Price of Stability of Weighted Congestion Games
ArXiv
ArXiv
Computer Science Mathematics