Nash Flows Over Time with Tolls

2025-10-15

AI Summary

This study investigates the cost properties and steady-state behavior of nonatomic equilibria in dynamic routing games under the Vickrey bottleneck model with tolls, challenging the conventional assumption that dynamic equilibria converge uniquely to a steady state. Method: We integrate dynamic network flow modeling, nonatomic game-theoretic equilibrium analysis, and differential game theory, explicitly coupling queueing dynamics with path-cost-based optimal route choice. Contribution/Results: We propose the first systematic, computationally tractable algorithm for determining steady-state solutions of such toll-inclusive dynamic routing models. We rigorously prove the existence of steady-state equilibria and develop an efficient numerical implementation. Our framework provides both theoretical foundations and practical computational tools for modeling traffic equilibria with pricing mechanisms, advancing the understanding of convergence, uniqueness, and stability in dynamic toll-based transportation systems.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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.

Problem

• Modeling dynamic routing games with tolls and traffic congestion effects
• Analyzing non-unique equilibria and cost variations in tolled networks
• Developing computational methods to determine steady states with tolls

Innovation

• Introducing tolls to Vickrey bottleneck model
• Analyzing non-unique equilibria with toll costs
• Computing steady states in tolled routing games