Nash Flows Over Time with Tolls

TL;DR

This work investigates Nash flows over time in a dynamic routing game with fixed tolls on edges of a Vickrey bottleneck model. It shows that tolls can destroy uniqueness of equilibrium costs and even prevent convergence to a steady state, unlike the toll-free case, and it advances a constructive LP-based method to compute steady states via deficit-flow initialization built on thin-flow concepts. The core contribution is a primal–dual linear program (LP) formulation that yields an LP-induced deficit flow which is practically in steady state and simultaneously a dynamic equilibrium, with the thin-flow structure mediating the linkage between edge costs and flow rates. The paper also supplies critical counterexamples illustrating non-uniqueness of costs and non-termination, highlighting the richer and more complex geometry of tolled dynamic equilibria. Overall, the results illuminate how fixed tolls interact with queue dynamics to shape long-run traffic behavior and point to several intriguing theoretical questions about existence, structure, and efficiency of tolled dynamic equilibria.

Abstract

We study a dynamic routing game motivated by traffic flows. The base model for an edge is the Vickrey bottleneck model. That is, edges are equipped with a free flow transit time and a capacity. When the inflow into an edge exceeds its capacity, a queue forms and the following particles experience a waiting time. In this paper, we enhance the model by introducing tolls, i.e., a cost each flow particle must pay for traversing an edge. In this setting we consider non-atomic equilibria, which means flows over time in which every particle is on a cheapest path, when summing up toll and travel time. We first show that unlike in the non-tolled version of this model, dynamic equilibria are not unique in terms of costs and do not necessarily reach a steady state. As a main result, we provide a procedure to compute steady states in the model with tolls.

Figures (5)

• Figure 1: Source and sink thin flows. Transit times and tolls are not given as they do not affect the thin flows.
• Figure 2: Multiple dynamic equilibria with different cost functions.
• Figure 3: A Nash flow over time that never reaches steady state.
• Figure 4: Flow division and queue lengths for different $\theta_i$ values for the example in \ref{['thm:no_steady_states_reached']}. It can be observed that queue length as well as flow rates converge rather quickly, while the explicit formulas show that the steady state is never actually reached.
• Figure 5: All tolls in the above example are 0. With a deficit flow steady states can be described in a simple way.

Theorems & Definitions (29)

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