The Price of Stability of Weighted Congestion Games
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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 (Φ_d)^d+1, where Φ_d∼ d/ d is the unique positive root of equation x^d+1=(x+1)^d. This asymptotically closes the huge gap between (d) and Φ_d^d+1 and 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 ((1+1/α)^d/d) on the PoS of α-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 Nash equilibria, which is sensitive to the range W of the player weights. 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 (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.
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