Tolls for Dynamic Equilibrium Flows

Abstract

We consider dynamic network flows and study the following question: Which dynamic edge flows can be implemented as tolled dynamic equilibrium flows? We study this question for the heterogeneous-user model, where the flow particles are partitioned into populations characterized by their own source,destination-pairs and a cost function associating with any walk and departure time some costs. As our two main results, we first provide a duality-based characterization of implementability of dynamic edge flows for the multi-source, multi-destination case. Secondly, we derive both, a combinatorial and duality-based characterization of implementability of dynamic edge flows for the multi-source, single-destination case. Both results are derived under a fairly general network loading model. For the proof, we make several technical contributions: We formulate a novel infinite dimensional optimization problem, where the goal is to minimize the aggregated costs of the particles with respect to the fixed network loading induced by the given edge flow. This requires the recently introduced concept of autonomous network loadings for which we show several new structural insights. In particular, we give an alternative (tighter) characterization of the existence of autonomous network loadings for our setting by deriving a generalization of a result of M.A. Zarecki\u{\i} on the Lusin $N^{-1}$ property of absolutely continuous monotone functions which may also be of independent interest. These insights allow us to prove the stated characterizations under the assumption of strong duality. Finally, for the case of a single-destination, we are able to provide a non-trivial proof that this assumption is always fulfilled for finitely supported edge flows with costs representing weighted travel times.