Dynamic Tolling for Inducing Socially Optimal Traffic Loads

How to design tolls that induce socially optimal traffic loads with dynamically arriving travelers who make selfish routing decisions? We propose a two-timescale discrete-time stochastic dynamics that adaptively adjusts the toll prices on a parallel link network while accounting for the updates of traffic loads induced by the incoming and outgoing travelers and their route choices. The updates of loads and tolls in our dynamics have three key features: (i) The total demand of incoming and outgoing travelers is stochastically realized; (ii) Travelers are myopic and selfish in that they choose routes according to a perturbed best response given the current latency and tolls on parallel links; (iii) The update of tolls is at a slower timescale as compared to the the update of loads. We show that the loads and the tolls eventually concentrate in a neighborhood of the fixed point, which corresponds to the socially optimal load and toll price. Moreover, the fixed point load is also a stochastic user equilibrium with respect to the toll price. Our results can be useful for traffic authorities to efficiently manage traffic loads in response to the arrival and departure of travelers.