Sensitivity analysis of the perturbed utility stochastic traffic equilibrium

TL;DR

This paper develops an analytical sensitivity framework for the perturbed utility route choice (PURC) model and its flow-based stochastic traffic equilibrium, deriving closed-form Jacobians for PURC flows and for equilibrium link flows with respect to link costs and network parameters. It handles inactive links via a projection-based approach, proves differentiability almost everywhere outside a small activation boundary, and shows how to exploit sparsity for scalable, parallel sensitivity computations. The results enable rapid welfare and uncertainty analyses and provide a practical tool for bilevel transportation design and pricing problems, including large-scale networks demonstrated on a Copenhagen-area case. A key finding is that PURC can exhibit complementarity between routes, contrasting with standard discrete-choice substitutions, which has implications for network design and policy analysis. Overall, the framework bridges theoretical PURC/traffic equilibrium models with real-world, large-scale applications while offering computational gains over repeated equilibrium solves.

Abstract

This paper develops a sensitivity analysis framework for the perturbed utility route choice (PURC) model and the accompanying stochastic traffic equilibrium model. We derive analytical sensitivity expressions for the Jacobian of the individual optimal PURC flow and equilibrium link flows with respect to link cost parameters under general assumptions. This allows us to determine the marginal change in link flows following a marginal change in link costs across the network. We show how to implement these results while exploiting the sparsity generated by the PURC model. Numerical examples illustrate the use of our method for estimating equilibrium link flows after link cost shifts, identifying critical design parameters, and quantifying uncertainty in performance predictions. Finally, we demonstrate the method in a large-scale example. The findings have implications for network design, pricing strategies, and policy analysis in transportation planning and economics, providing a bridge between theoretical models and real-world applications.

Figures (5)

• Figure 1: Example network
• Figure 2: Simple network example, substitution pattern of PURC.
• Figure 3: Approximate link flow changes versus exact links flow changes [cars per day].
• Figure 4: (a) Approximate flow change, 10% increase in free-flow travel time on southbound Langebro; (b) Exact flow change, 10% increase in free-flow travel time on southbound Langebro.
• Figure 5: (a) Approximate flow change, 10% increase in free-flow travel time on southbound Langebro, zoom; (b) Exact flow change, 10% increase in free-flow travel time on southbound Langebro, zoom.

Theorems & Definitions (12)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Lemma 1
• proof : Proof of Lemma \ref{['lem:P_hat']}
• Lemma 2
• proof