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We consider the behavior of the price of anarchy and equilibrium flows innonatomic multi-commodity routing games as a function of the traffic demand. Weanalyze their smoothness with a special attention to specific values of thedemand at which the support of the Wardrop equilibrium exhibits a phasetransition with an abrupt change in the set of optimal routes. Typically, whensuch a phase transition occurs, the price of anarchy function has a breakpoint,\ie is not differentiable. We prove that, if the demand varies proportionallyacross all commodities, then, at a breakpoint, the largest left or rightderivatives of the price of anarchy and of the social cost at equilibrium, areassociated with the smaller equilibrium support. This proves -- under theassumption of proportional demand -- a conjecture of o'Hare et al. (2016), whoobserved this behavior in simulations. We also provide counterexamples showingthat this monotonicity of the one-sided derivatives may fail when the demanddoes not vary proportionally, even if it moves along a straight line notpassing through the origin.

Phase Transitions of the Price-of-Anarchy Function in Multi-Commodity Routing Games