Nash Flows Over Time with Tolls

Shaul Rosner; Marc Schröder; Laura Vargas Koch

doi:10.1007/978-3-032-18660-7_12

Nash Flows Over Time with Tolls

Shaul Rosner^*
, Marc Schröder
, Laura Vargas Koch

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceeding › Academic › peer-review

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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.

Original language	English
Title of host publication	Web and Internet Economics - 21st International Conference, WINE 2025, Proceedings
Editors	Vittorio Bilò, Yang Cai, Zhiyi Huang
Publisher	Springer Verlag
Pages	236-253
Number of pages	18
ISBN (Electronic)	978-3-032-18660-7
ISBN (Print)	9783032186591
DOIs	https://doi.org/10.1007/978-3-032-18660-7_12
Publication status	Published - 1 Jan 2026
Event	21st International Conference on Web and Internet Economics, WINE 2025 - New Brunswick, United States Duration: 8 Dec 2025 → 11 Dec 2025 https://wine2025.cs.rutgers.edu/

Publication series

Series	Lecture Notes in Computer Science
Volume	16266
ISSN	0302-9743

Conference

Conference	21st International Conference on Web and Internet Economics, WINE 2025
Abbreviated title	WINE 2025
Country/Territory	United States
City	New Brunswick
Period	8/12/25 → 11/12/25
Internet address	https://wine2025.cs.rutgers.edu/

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title = "Nash Flows Over Time with Tolls",

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.",

author = "Shaul Rosner and Marc Schr{\"o}der and \{Vargas Koch\}, Laura",

note = "Funding Information: The authors thank Wenzel Manegold for finding the example of Theorem 3.1, and Lukas Graf and Neil Olver for fruitful discussion on the topic of this work. We also thank the participants of the Dagstuhl Seminar 24281 [5] for helpful discussions and questions. Publisher Copyright: {\textcopyright} The Author(s), under exclusive license to Springer Nature Switzerland AG 2026.; 21st International Conference on Web and Internet Economics, WINE 2025, WINE 2025 ; Conference date: 08-12-2025 Through 11-12-2025",

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language = "English",

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editor = "Vittorio Bil{\`o} and Yang Cai and Zhiyi Huang",

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Rosner, S, Schröder, M & Vargas Koch, L 2026, Nash Flows Over Time with Tolls. in V Bilò, Y Cai & Z Huang (eds), Web and Internet Economics - 21st International Conference, WINE 2025, Proceedings. Springer Verlag, Lecture Notes in Computer Science, vol. 16266, pp. 236-253, 21st International Conference on Web and Internet Economics, WINE 2025, New Brunswick, United States, 8/12/25. https://doi.org/10.1007/978-3-032-18660-7_12

Nash Flows Over Time with Tolls. / Rosner, Shaul; Schröder, Marc; Vargas Koch, Laura.
Web and Internet Economics - 21st International Conference, WINE 2025, Proceedings. ed. / Vittorio Bilò; Yang Cai; Zhiyi Huang. Springer Verlag, 2026. p. 236-253 (Lecture Notes in Computer Science, Vol. 16266).

Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceeding › Academic › peer-review

TY - GEN

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AU - Rosner, Shaul

AU - Schröder, Marc

AU - Vargas Koch, Laura

N1 - Funding Information: The authors thank Wenzel Manegold for finding the example of Theorem 3.1, and Lukas Graf and Neil Olver for fruitful discussion on the topic of this work. We also thank the participants of the Dagstuhl Seminar 24281 [5] for helpful discussions and questions. Publisher Copyright: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2026.

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N2 - 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.

AB - 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.

U2 - 10.1007/978-3-032-18660-7_12

DO - 10.1007/978-3-032-18660-7_12

M3 - Conference article in proceeding

SN - 9783032186591

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BT - Web and Internet Economics - 21st International Conference, WINE 2025, Proceedings

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T2 - 21st International Conference on Web and Internet Economics, WINE 2025

Y2 - 8 December 2025 through 11 December 2025

ER -

Nash Flows Over Time with Tolls

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