Sensitivity of Wardrop Equilibria

• Authors: Matthias Englert;Thomas Franke;Lars Olbrich
• Venue: SAGT '08 Proceedings of the 1st International Symposium on Algorithmic Game Theory
• Year: 2008

We study the sensitivity of equilibria in the well-known gametheoretic traffic model due to Wardrop. We mostly considersingle-commodity networks. Suppose, given a unit demand flow atWardrop equilibrium, one increases the demand by εor removes an edge carrying only an ε-fraction offlow. We study how the equilibrium responds to such anε-change.Our first surprising finding is that, even for linear latencyfunctions, for every ε 0, there are networks inwhich an ε-change causes every agent to change itspath in order to recover equilibrium. Nevertheless, we can provethat, for general latency functions, the flow increase or decreaseon every edge is at most ε.Examining the latency at equilibrium, we concentrate onpolynomial latency functions of degree at most pwithnonnegative coefficients. We show that, even though the relativeincrease in the latency of an edge due to anε-change in the demand can be unbounded, the pathlatency at equilibrium increases at most by a factor of (1 +ε)p. The increase of the priceof anarchyis shown to be upper bounded by the same factor.Both bounds are shown to be tight.Let us remark that all our bounds are tight. For themulti-commodity case, we present examples showing that neither thechange in edge flows nor the change in the path latency can bebounded.