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, \ie 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.

翻译：我们研究了非原子多商品路由博弈中价格无政府函数及均衡流随交通需求变化的行为。我们分析其光滑性，特别关注需求在特定数值处沃德罗普均衡支撑集发生相变（即最优路径集合突变）的情况。通常，当此类相变发生时，价格无政府函数会出现断点（即不可微）。我们证明：若所有商品需求按比例变化，则在断点处，价格无政府函数及均衡社会成本的最大左/右导数均与较小的均衡支撑集相关。这一结果在比例需求假设下证实了o'Hare等人（2016）通过仿真观察到的猜想。我们还提供反例表明：当需求非比例变化时（即使沿不经过原点的直线移动），这种单边导数的单调性可能失效。