External-cost continuous-type wardrop equilibria in routing games

We propose a bi-criterion Wardrop equilibrium which we call an external cost continuous type Wardrop equilibrium for nonatomic routing games where agents incur some type dependent cost to have access to different sets of available routes. Rather than being anonymous, the population is described by a distribution over the type parameter (that can be supported on any compact interval of the real line). At equilibria, no member of the population can improve their type dependent cost either by switching sets of routes or by switching routes within their chosen set. From this equilibrium condition, we derive how the population mass will divide up among the various routing options. We then formulate a potential function optimization program for finding the equilibrium mass distribution. This work revisits the cost-vs-time equilibria of Leurent [1] and Marcotte [2] while specifically allowing the type parameter to be positive or negative. Applications include modeling the value of information in routing, the effect of privacy concern on congestion, and how commuters make tradeoffs between different forms of transportation.