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Stochastic Prediction Equilibrium for Dynamic Traffic Assignment

Lukas Graf, Tobias Harks, Michael Markl

TL;DR

The work develops a general dynamic traffic assignment framework based on an edge-loading operator $\Phi$ and a routing operator $\mathscr{R}$ to model physical flow propagation and route choice. It proves existence of coherent flows for finite horizons via Kakutani-Fan-Glicksberg fixed-point theory and extends to infinite horizons under causality through extension properties, with uniqueness guaranteed under strict causality or Lipschitz contraction conditions. The paper introduces the stochastic prediction equilibrium (SPE), capturing noisy, ML-based predictions, and establishes existence and (under bounded density noise) uniqueness; SPE encompasses deterministic prediction equilibria, dynamic Nash, and IDE as special cases. By unifying prescriptive, descriptive, and stochastic routing in one framework, these results generalize and extend classical DTA analyses (e.g., Bayen’s prescriptive results, dynamic prediction equilibria, full-information equilibria) and offer pathways for computation via fixed-point iterations. The framework thus provides a principled basis for analyzing and designing dynamic routing under uncertainty, with applicability to standard models like Vickrey’s queueing and affine-linear volume-delay dynamics.

Abstract

Stochastic effects significantly influence the dynamics of traffic flows. Many dynamic traffic assignment (DTA) models attempt to capture these effects by prescribing a specific ratio that determines how flow splits across different routes based on the routes' costs. In this paper, we propose a new framework for DTA that incorporates the interplay between the routing decisions of each single traffic participant, the stochastic nature of predicting the future state of the network, and the physical flow dynamics. Our framework consists of an edge loading operator modeling the physical flow propagation and a routing operator modeling the routing behavior of traffic participants. The routing operator is assumed to be set-valued and capable to model complex (deterministic) equilibrium conditions as well as stochastic equilibrium conditions assuming that measurements for predicting traffic are noisy. As our main results, we derive several quite general equilibrium existence and uniqueness results which not only subsume known results from the literature but also lead to new results. Specifically, for the new stochastic prediction equilibrium, we show existence and uniqueness under natural assumptions on the probability distribution over the predictions.

Stochastic Prediction Equilibrium for Dynamic Traffic Assignment

TL;DR

The work develops a general dynamic traffic assignment framework based on an edge-loading operator and a routing operator to model physical flow propagation and route choice. It proves existence of coherent flows for finite horizons via Kakutani-Fan-Glicksberg fixed-point theory and extends to infinite horizons under causality through extension properties, with uniqueness guaranteed under strict causality or Lipschitz contraction conditions. The paper introduces the stochastic prediction equilibrium (SPE), capturing noisy, ML-based predictions, and establishes existence and (under bounded density noise) uniqueness; SPE encompasses deterministic prediction equilibria, dynamic Nash, and IDE as special cases. By unifying prescriptive, descriptive, and stochastic routing in one framework, these results generalize and extend classical DTA analyses (e.g., Bayen’s prescriptive results, dynamic prediction equilibria, full-information equilibria) and offer pathways for computation via fixed-point iterations. The framework thus provides a principled basis for analyzing and designing dynamic routing under uncertainty, with applicability to standard models like Vickrey’s queueing and affine-linear volume-delay dynamics.

Abstract

Stochastic effects significantly influence the dynamics of traffic flows. Many dynamic traffic assignment (DTA) models attempt to capture these effects by prescribing a specific ratio that determines how flow splits across different routes based on the routes' costs. In this paper, we propose a new framework for DTA that incorporates the interplay between the routing decisions of each single traffic participant, the stochastic nature of predicting the future state of the network, and the physical flow dynamics. Our framework consists of an edge loading operator modeling the physical flow propagation and a routing operator modeling the routing behavior of traffic participants. The routing operator is assumed to be set-valued and capable to model complex (deterministic) equilibrium conditions as well as stochastic equilibrium conditions assuming that measurements for predicting traffic are noisy. As our main results, we derive several quite general equilibrium existence and uniqueness results which not only subsume known results from the literature but also lead to new results. Specifically, for the new stochastic prediction equilibrium, we show existence and uniqueness under natural assumptions on the probability distribution over the predictions.
Paper Structure (25 sections, 32 theorems, 65 equations, 2 figures)

This paper contains 25 sections, 32 theorems, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Model
  5. Physical Flow Model
  6. Behavioural model
  7. Existence and Uniqueness
  8. Existence
  9. Uniqueness
  10. Stochastic Prediction Equilibrium
  11. Deterministic Prediction Equilibrium
  12. Prescriptive Routing Operators
  13. Noisy Predictions
  14. Conclusion
  15. Open Questions.
  16. ...and 10 more sections

Key Result

Theorem 3.1

Let $T\in\mathbb{R}_{\geq 0}$ and let $\Phi$ be a locally bounded edge-loading operator such that $\Phi_T$ is sequentially weak-weak-continuous and let $\mathscr{R}$ be a routing operator such that $\mathscr{R}_T$ has a sequentially weakly closed graph and non-empty and convex values. Then, there ex

Figures (2)

  • Figure 1: A network in Vickrey's model with edge labels $(\nu_e, c^0_e)$ with capacity $\nu_e$ and free-flow travel time $c^0_e$.
  • Figure 2: An example for how the auxiliary network (right) used in the proof of \ref{['lem:CharacterizationStochasticIDERouting']} is constructed for some node $v$ with outgoing edges $\delta^+(v)$ (left). The labels on the arcs of the auxiliary network indicate their capacities.

Theorems & Definitions (89)

  • Definition 2.2
  • Definition 2.3
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.10
  • Definition 2.11
  • Definition 2.12
  • Theorem 3.1
  • ...and 79 more