A Day-to-Day Dynamical Approach to the Most Likely User Equilibrium Problem

TL;DR

It is established that CumLog always converges to the MEUE route flow if (i) travelers have no prior information about routes and thus, are forced to give all routes an equal initial choice probability or if (ii) all travelers gather information from the same source such that the general proportionality condition is satisfied.

Abstract

The lack of a unique user equilibrium (UE) route flow in traffic assignment has posed a significant challenge to many transportation applications. The maximum-entropy principle, which advocates for the consistent selection of the most likely solution as a representative, is often used to address the challenge. Built on a recently proposed day-to-day (DTD) discrete-time dynamical model called cumulative logit (CULO), this study provides a new behavioral underpinning for the maximum-entropy UE (MEUE) route flow. It has been proven that CULO can reach a UE state without presuming travelers are perfectly rational. Here, we further establish that CULO always converges to the MEUE route flow if (i) travelers have zero prior information about routes and thus are forced to give all routes an equal choice probability, or (ii) all travelers gather information from the same source such that the so-called general proportionality condition is satisfied. Thus, CULO may be used as a practical solution algorithm for the MEUE problem. To put this idea into practice, we propose to eliminate the route enumeration requirement of the original CULO model through an iterative route discovery scheme. We also examine the discrete-time versions of four popular continuous-time dynamical models and compare them to CULO. The analysis shows that the replicator dynamic is the only one that has the potential to reach the MEUE solution with some regularity. The analytical results are confirmed through numerical experiments.

Figures (6)

• Figure 1: A three-node-four-link (3N4L) network.
• Figure 2: Distribution of $\lambda$ corresponding to UE strategies of the 3N4L network, obtained from 5000 different initial points by CULO. The red line highlights the $\lambda$ value corresponding to the equal-distribution initial point.
• Figure 3: Initial entropy v.s. limiting entropy for 250 samples of the 3N4L network. The red point highlights the pair corresponding to the equal-distribution initial point (i.e., ${\bm{s}}^0 = {\bm{0}}$); the black dashed line is the 45-degree line.
• Figure 4: Convergence patterns of CULO for the Sioux-Falls network in four scenarios. Scenario (A): CULO with predetermined routes. Scenario (B): Algorithm \ref{['alg:route-based']}. Scenario (C): Algorithm \ref{['alg:link-based']}. Scenario (D): Algorithm \ref{['alg:link-based']} with exploration noises. In each column, plot (a) reports the relative equilibrium gap, (b) reports the difference in entropy between the CULO solution and the baseline solution, normalized by the number of OD pairs and plotted in symlog scale (where the blue dashed line corresponds to a gap of zero); and (c) reports the number of routes actively used by travelers (where the red dashed line corresponds to the number of routes contained in the benchmark solution).
• Figure 5: The relationship between the limiting points of different models with respect to their step sizes. In each column, plot (a) reports the value of $\lambda$ corresponding to the UE strategy reached by the model (the red dashed line highlights the corresponding value of the MEUE); plot (b) reports the number of iterations required for convergence.

Theorems & Definitions (20)

• Definition 2.3: UE strategy
• Proposition 2.4: dafermos1980traffic
• Proposition 2.5
• Proposition 2.6
• Definition 2.7: Maximum-entropy user equilibrium, or MEUE
• Definition 2.8: General proportionality condition
• Proposition 2.9: borchers2015traffic, Theorem 3.3
• Proposition 2.10: li2023wardrop, Theorem 5.4
• Remark 2.11: Relation with classical DTD models
• Lemma 3.1