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Machine-Learned Prediction Equilibrium for Dynamic Traffic Assignment

Lukas Graf, Tobias Harks, Kostas Kollias, Michael Markl

TL;DR

This work formulate a mathematically concise model and derive properties of the predictors that ensure a dynamic prediction equilibrium exists, and demonstrates the versatility of the 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.

Abstract

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.

Machine-Learned Prediction Equilibrium for Dynamic Traffic Assignment

TL;DR

This work formulate a mathematically concise model and derive properties of the predictors that ensure a dynamic prediction equilibrium exists, and demonstrates the versatility of the 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.

Abstract

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 -approximate DPE wherein no agent can improve their predicted travel time by more than 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.

Paper Structure

This paper contains 28 sections, 12 theorems, 80 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Existence Results.
  4. Computing Approximate Dynamic Prediction Equilibria.
  5. Computational Study.
  6. Comparison to Conference Version.
  7. Related Work
  8. The Flow Model
  9. Instantaneous Dynamic Equilibrium
  10. Dynamic Nash Equilibrium
  11. Dynamic Prediction Equilibria
  12. Existence of Dynamic Prediction Equilibria
  13. Existence of DPE Using a Variational Inequality
  14. A Sufficient Regularity Condition
  15. Application Predictors
  16. ...and 13 more sections

Key Result

Theorem 7

Let $X$ be a reflexive Banach space and let $\mathcal{A}: K\to X'$ be a sequentially weak-strong continuous map defined on a non-empty, closed, bounded and convex set $K\subseteq X$. Then there exists a solution $u\in X$ to the variational inequality

Figures (5)

  • Figure 1: A network where the use of a non-continuous predictor can result in a situation where no dynamic prediction equilibrium exists. Edges are labeled with $(\tau_e, \nu_e)$.
  • Figure 2: A schematic overview of the computation of a $\delta$-routed DPE.
  • Figure 3: A network with source $s$ and sink $t$. Edges are labeled with $(\tau_e, \nu_e)$.
  • Figure 4: The experiment results of the synthetic network.
  • Figure 5: The experiment results of the Anaheim II network.

Theorems & Definitions (47)

  • Definition 1
  • Definition 2
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 7
  • Definition 8
  • Definition 9
  • Lemma 10
  • Lemma 11
  • ...and 37 more