@article{ref1,
title="Phase transitions of the price-of-anarchy function in multi-commodity routing games",
journal="Transportation research part B: methodological",
year="2024",
author="Cominetti, Roberto and Dose, Valerio and Scarsini, Marco",
volume="182",
number="",
pages="e102922-e102922",
abstract="We consider the behavior of the price of anarchy and equilibrium flows in nonatomic multi-commodity routing games as a function of the traffic demand. We analyze their smoothness with a special attention to specific values of the demand at which the support of the Wardrop equilibrium exhibits a phase transition with an abrupt change in the set of optimal routes. Typically, when such a phase transition occurs, the price of anarchy function has a breakpoint, i.e., is not differentiable. We prove that, if the demand varies proportionally across all commodities, then, at a breakpoint, the largest left or right derivatives of the price of anarchy and of the social cost at equilibrium, are associated with the smaller equilibrium support. This proves - under the assumption of proportional demand - a conjecture of O'Hare et al. (2016), who observed this behavior in simulations. We also provide counterexamples showing that this monotonicity of the one-sided derivatives may fail when the demand does not vary proportionally, even if it moves along a straight line not passing through the origin.<p /> <p>Language: en</p>",
language="en",
issn="0191-2615",
doi="10.1016/j.trb.2024.102922",
url="http://dx.doi.org/10.1016/j.trb.2024.102922"
}