Phase transitions of the price-of-anarchy function in multi-commodity routing games

• Published in: Transportation research. Part B: methodological Vol. 182; p. 102922
• Main Authors: Cominetti, Roberto, Dose, Valerio, Scarsini, Marco
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01-04-2024
• Subjects: Network flows; Traffic demand; Wardrop equilibrium; 91A43; 91A07; Traffic demand; Network flows; 91A14; Wardrop equilibrium
• ISSN: 0191-2615; 1879-2367
• DOI: 10.1016/j.trb.2024.102922

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.
•Smoothness of the price of anarchy in nonatomic multi-commodity routing games as a function of the traffic demand.•Sufficient conditions for smoothness when the support of the Wardrop equilibrium is locally constant.•Proof of a monotonicity conjecture by O’Hare et al. (2016) for the side derivatives at breakpoints, under proportional demand.•Counterexamples for this conjecture when the demand does not vary proportionally.