Finding anonymous tolls to realize target flow pattern in networks with continuously distributed value of time

Considering user heterogeneity in terms of a continuously distributed value of time, we investigate, on the base of so-called Wardrop's third principle, the time-versus-cost network equilibrium and system optimum problem with fixed demand. To realize a target flow pattern through imposing anonymous path tolls, we propose a time difference-based pricing in which every user is charged a toll computed from the time difference between adjacent paths ordered in the decreasing order of travel times. We find that, whenever travel disutility is measured in cost or time units, the anonymous marginal cost pricing does not exist. We further show that, if and only if the coefficient matrix of a linear system corresponding to the derived optimal path toll pattern and its augmented matrix have the same rank, an anonymous link toll pattern exists.