The Price of Stability of Weighted Congestion Games

TL;DR

This paper establishes exponential lower bounds on the Price of Stability in weighted congestion games with polynomial cost functions, and provides upper bounds using a novel potential function, advancing understanding of equilibrium efficiency.

Contribution

It introduces exponential lower bounds for the PoS in weighted congestion games and develops a new potential function for upper bounds on approximate equilibria.

Findings

Exponential lower bounds on PoS for weighted congestion games.

A new potential function for analyzing approximate Nash equilibria.

Upper bounds on PoS depending on weights and approximation parameters.

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 $\varOmega(\Phi_d)^{d+1}$, where $\Phi_d\sim d/\ln d$ is the unique positive root of equation $x^{d+1}=(x+1)^d$. This almost closes the huge gap between $\varTheta(d)$ and $\Phi_d^{d+1}$. 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 $\varOmega((1+1/\alpha)^d/d)$ on the PoS of $\alpha$-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 $\alpha$-approximate 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 $\varTheta(1)$ to $d+1$ in a smooth way with respect to $W$. Secondly, we show that for unweighted congestion games, the PoS of $\alpha$-approximate Nash equilibria is at most $(d+1)/\alpha$.