Existence, Uniqueness, and Trip Cost Function Properties of User Equilibrium in the Bottleneck Model with Multiple User Classes
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Bibliographic record
Abstract
Under relatively general assumptions a unique deterministic departure-time user equilibrium with a finite departure rate exists in the bottleneck model with drivers who differ in their unit costs of travel time, preferred times of arrival, and schedule delay cost functions. Existence requires that schedule delay cost functions be upper semicontinuous with respect to arrival time, and that schedule delay costs decline at a rate smaller than the unit cost of travel time. Uniqueness requires, more restrictively, that schedule delay cost functions be continuous. Several properties of equilibrium trip cost functions are derived for n groups of users with N i in group i. The trip cost of a user in group i is a nondecreasing function of each N j , but typically rises more quickly with respect to N i than N j , j≠i. Thus, users experience lower trip costs when they travel with users unlike themselves than with an equal number of users like themselves.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it