A seminal result of [Fleischer et al., 2004], [Karakostas and Kolliopoulos, 2004] and [Yang and Huang, 2004] states that system optimal multi-commodity static network flows are always implementable as tolled Wardrop equilibrium flows even if users have heterogeneous value-of-time sensitivities. Their proof uses LP-duality to characterize the general implementability of network flows by tolls. For the much more complex setting of dynamic flows, [Graf et al., 2025] identified necessary and sufficient conditions for a dynamic s-d flow to be implementable as a tolled dynamic equilibrium. They used the machinery of (infinite-dimensional) strong duality to obtain their characterizations. Their work, however, does not answer the question of whether system optimal dynamic network flows are implementable by tolls.

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 having different valuations of travel time and money spent. As our main result, we give the first characterization of this type of implementability showing that for single-source single-destination networks and heterogeneous users, a dynamic edge flow is implementable by tolls if and only if the induced subgraph of the edge flow contains no cycle of positive length containing the destination. For the proof of this result we make several technical contributions: We formulate a novel infinite dimensional optimization problem, where the goal is to minimize the weighted travel times with respect to the fixed network loading induced by the given edge flow. Using the recently introduced concept of parameterized network loadings (cf. [23]), we prove existence of optimal solutions, strong duality, and a characterization of special optimal solutions for which an inequality is tight. These results are then all used for the proof of the above mentioned main characterization.

In this article, we study the dynamic traffic assignment problem using the general path-delay-operator form as proposed by Friesz et al. [1989] and augment this model with side-constraints. Our contribution consists of four types of results:

We study a dynamic traffic assignment model, where agents base their instantaneous routing decisions on real-time delay predictions. We formulate a mathematically concise model and define dynamic prediction equilibrium (DPE) in which no agent can at any point during their journey improve their predicted travel time by switching to a different route. We demonstrate the versatility of our framework by showing that it subsumes the well-known full information and instantaneous information models, in addition to admitting further realistic predictors as special cases. We then proceed to derive properties of the predictors that ensure a dynamic prediction equilibrium exists. Additionally, we define $\varepsilon$-approximate DPE wherein no agent can improve their predicted travel time by more than $\varepsilon$ and provide further conditions of the predictors under which such an approximate equilibrium can be computed. Finally, we complement our theoretical analysis by an experimental study, in which we systematically compare the induced average travel times of different predictors, including two machine-learning based models trained on data gained from previously computed approximate equilibrium flows, both on synthetic and real world road networks.

We study a dynamic traffic assignment model, where agents base their instantaneous routing decisions on real-time delay predictions. We formulate a mathematically concise model and derive properties of the predictors that ensure a dynamic prediction equilibrium exists. We demonstrate the versatility of our framework by showing that it subsumes the well-known full information and instantaneous information models, in addition to admitting further realistic predictors as special cases. We complement our theoretical analysis by an experimental study, in which we systematically compare the induced average travel times of different predictors, including a machine-learning model trained on data gained from previously computed equilibrium flows, both on a synthetic and a real road network.

We initiate the study of dynamic traffic assignment for electrical vehicles addressing the specific challenges such as range limitations and the possibility of battery recharge at predefined charging locations. We pose the dynamic equilibrium problem within the deterministic queueing model of Vickrey and as our main result, we establish the existence of an energy-feasible dynamic equilibrium. There are three key modeling-ingredients for obtaining this existence result: * We introduce a walk-based definition of dynamic traffic flows which allows for cyclic routing behavior as a result of recharging events en route. * We use abstract convex feasibility sets in an appropriate function space to model the energy-feasibility of used walks. * We introduce the concept of capacitated dynamic equilibrium walk-flows which generalize the former unrestricted dynamic equilibrium path-flows. Viewed in this framework, we show the existence of an energy-feasible dynamic equilibrium by applying an infinite dimensional variational inequality, which in turn requires a careful analysis of continuity properties of the network loading as a result of injecting flow into walks. We complement our theoretical results by a computational study in which we design a fixed-point algorithm computing energy-feasible dynamic equilibria. We apply the algorithm to standard real-world instances from the traffic assignment community illustrating the complex interplay of resulting travel times, energy consumption and prices paid at equilibrium.