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

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  • Cominetti, Roberto
  • Dose, Valerio
  • Scarsini, Marco

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.

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  • Cominetti, Roberto & Dose, Valerio & Scarsini, Marco, 2024. "Phase transitions of the price-of-anarchy function in multi-commodity routing games," Transportation Research Part B: Methodological, Elsevier, vol. 182(C).
    • Handle: RePEc:eee:transb:v:182:y:2024:i:c:s0191261524000468
    DOI: 10.1016/j.trb.2024.102922
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    References listed on IDEAS

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