The Price of Stability of Weighted Congestion Games

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 Ω(Φ d ) d+1 , where Φ d ∼ d/ ln d is the unique positive root of equation x d+1 = (x + 1) d . This almost closes the huge gap between Θ(d) and Φ d+1 d . 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.

On approximate pure Nash equilibria in weighted congestion games with polynomial latencies

“…As a matter of fact, our potentials, despite their simplicity, provide an approximation of d instead of d + 1; although it is worth noticing that, for very small degrees, the two approaches provide the same approximation guarantee. As a corollary of Theorem 3, we also show (Corollary 4) that the social optimum of an instance of WCG(d) is always a (d + 1)-approximate pure Nash equilibrium, as it has already been observed in [7]. More importantly, Theorem 3 implies that, as state by Corollary 5, every mildly congested instance of WCG(d) always admits a e e−1 -approximate potential function, where e is the Euler's number.…”

“…Specifically, we first give bounds (Lemma 6 and Lemma 7) relating the value of the (d + δ)-approximate potential function for a given state to the social cost of that state; if we then perform a sequence of (d + δ)-improvement moves starting from an optimal state, the potential does not increase, and hence we can bound the cost of any (d + δ)-approximate pure Nash equilibrium that we reach. Notice that our bound does not depend on the range of the players' weights and significantly improves the bound provided in [7], by making use of a different and simpler potential function.…”

“…Now, let us focus on the expression Ψ γe e (P ) − Ψ γe e (P \ {i}). For every i ∈ P , let λ i (P ) = − γ e k e + 1 λ i (P ) ke+1 (12) Therefore, by combining (11) and 12, we conclude that, for |P | ≥ 2, for the expression (7) we have .…”

“….. For the 1-approximate price of stability, for the unweighted game, there exists a bound of 1.577 for linear latencies [8,3] and a bound of Θ(d) for polynomial latencies [6]. The 1-approximate price of stability for the weighted game with polynomial latencies has been recently investigated in [7]; they provided a lower bound of Ω(d/ log d) d+1 , matching the upper bound in [1]. The authors also showed bounds to the α-approximate price of stability.…”

Optimizing the strategy of activities using numerical methods for determining equilibrium

“…Authors of work [23] proposed methods for calculation of the lower and upper bounds of the price of stability for a class of balanced games in the overloaded state with polynomial delays with negative coefficients. The upper bound on the value of the price of stability is the Nash equilibrium in this case.…”