Sensitivity of Wardrop Equilibria

Abstract

We study the sensitivity of equilibria in the well-known game theoretic traffic model due to Wardrop. We mostly consider single-commodity networks. Suppose, given a unit demand flow at Wardrop equilibrium, one increases the demand by ε or removes an edge carrying only an ε-fraction of flow. We study how the equilibrium responds to such an ε-change. Our first surprising finding is that, even for linear latency functions, for every ε > 0, there are networks in which an ε-change causes every agent to change its path in order to recover equilibrium. Nevertheless, we can prove that, for general latency functions, the flow increase or decrease on every edge is at most ε. Examining the latency at equilibrium, we concentrate on polynomial latency functions of degree at most p with nonnegative coefficients. We show that, even though the relative increase in the latency of an edge due to an ε-change in the demand can be unbounded, the path latency at equilibrium increases at most by a factor of (1 + ε) p. The increase of the price of anarchy is 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 the multi-commodity case, we present examples showing that neither the change in edge flows nor the change in the path latency can be bounded.

The price of anarchy in routing games as a function of the demand

“…In a different direction, Patriksson [28] characterized the existence of directional derivatives for the equilibrium and [20] proved that equilibrium costs are always directionally differentiable, whereas equilibrium edge loads not always are. Takalloo and Kwon [14] showed that there exist single OD network games where an ε-increase in the traffic demand produces a global migration of traffic from one set of equilibrium paths to a disjoint set of paths, but, nevertheless, the load on each edge changes at most by ε. Moreover, if the cost functions are polynomials of degree at most d, then the equilibrium costs increase at most of a multiplicative factor (1 + ε) d .…”

On the Price of Anarchy of Highly Congested Nonatomic Network Games

“…[22] and [14] perform sensitivity analysis of Wardrop equilibrium to some parameters of the model. Closer to the scope of our paper, Englert et al [9] examine how the equilibrium of a congestion game changes when either the total mass of players is increased by ε or an edge that carries an ε fraction of the mass is removed. For polynomial cost functions they bound the increase of the equilibrium cost when a mass ε of players is added to the system.…”

Mechanisms that govern how the Price of Anarchy varies with travel demand

“…For example, for traffic networks with separable, polynomial link costs, Correa et al (2008) showed that tighter upper bounds than those presented by Roughgarden (2003) can be derived provided the free-flow travel cost on each network link is at least a non-zero, fixed proportion of its travel cost under a UE assignment of travel demand. Englert et al (2010) also showed that the maximum increase in the Price of Anarchy, due to an increase in demand, can be bounded for traffic networks with separable, polynomial link costs and a single origin-destination (OD) pair.…”