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

Roberto Cominetti, Valerio Dose and Marco Scarsini

Transportation Research Part B: Methodological, 2024, vol. 182, issue C

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.

Keywords: Wardrop equilibrium; Network flows; Traffic demand (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1016/j.trb.2024.102922

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