Departure Time Choice with Parametric Heterogeneity: Equilibrium and Instability

Hillel Bar-Gera; Liron Ravner; Stephen D. Boyles

arxiv: 2604.13831 · v1 · submitted 2026-04-15 · 💻 cs.GT

Departure Time Choice with Parametric Heterogeneity: Equilibrium and Instability

Hillel Bar-Gera , Stephen D. Boyles , Liron Ravner This is my paper

Pith reviewed 2026-05-10 12:14 UTC · model grok-4.3

classification 💻 cs.GT

keywords departure time choicebottleneck modelday-to-day dynamicsinstabilityparametric heterogeneityVickrey modelequilibriumtraffic flow

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The pith

Continuous preference variation leaves departure-time dynamics unstable in the bottleneck model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper asks whether adding a continuous spread of travelers' preferences for early versus late arrival can eliminate the instability that appears in Vickrey's single-bottleneck model under day-to-day adjustment rules. It first establishes that an equilibrium still exists and is unique once earliness and lateness parameters are required to move together monotonically. The central result then shows that this added heterogeneity does not remove instability for any day-to-day rule that satisfies local-pressure and order-preservation conditions. If correct, the finding implies that the mismatch between model predictions and observed steady traffic patterns must come from other modeling choices rather than the lack of preference diversity.

Core claim

Under a monotonic relationship between earliness and lateness parameters, the model with continuously distributed schedule-delay preferences possesses a unique equilibrium. Every day-to-day dynamic obeying local-pressure and order-preservation conditions is nevertheless unstable at this equilibrium.

What carries the argument

The continuous parametric heterogeneity model with monotonic earliness-lateness linkage, together with the local-pressure and order-preservation restrictions on day-to-day dynamics.

If this is right

A unique equilibrium exists once the monotonicity condition is imposed.
Instability holds for the entire class of dynamics satisfying the stated conditions.
The result does not depend on the specific form of the continuous distribution, only on monotonicity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Stability in real traffic may require either non-monotonic preferences or adjustment rules that violate order preservation.
Testing the model on networks with multiple bottlenecks could reveal whether the instability survives added route-choice freedom.
Empirical work could check whether observed schedule-delay parameters are in fact monotonic.

Load-bearing premise

Earliness and lateness parameters must be monotonically related and the day-to-day rules must obey local pressure and order preservation.

What would settle it

A measured single-bottleneck setting in which travelers' earliness and lateness parameters satisfy monotonicity and the observed day-to-day departure times converge to a stable pattern would contradict the instability claim.

Figures

Figures reproduced from arXiv: 2604.13831 by Hillel Bar-Gera, Liron Ravner, Stephen D. Boyles.

**Figure 1.** Figure 1: ) That is, β(n) and γ(n) both reflect the same infinitesimal traffic fraction. Effectively, we assume that ordering travelers by increasing value of β is the same as ordering them by decreasing value of γ. We discuss the implications of this assumption at greater length at the end of this section. We will show below that this ordering — from travelers with lowest β (and highest γ) to travelers with the low… view at source ↗

**Figure 1.** Figure 1: The figure illustrates two examples of penalty functions (linear on the [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Figure 2: The cost functions Cn(t) are plotted for different traveler types. In equilibrium, the minimal cost for every type n traveler is attained at t(n). Definition 1. The functions Q(t) and t(n) are an equilibrium if Cn(t(n)) = min u∈T {Cn(u)}, (2) for all n ∈ [0, N], with Cn given in terms of Q by equation (1). The classic Vickrey bottleneck is a limiting case of our model, by taking the support of β and γ to b… view at source ↗

**Figure 3.** Figure 3: The equilibrium queue, earliness and lateness penalties of travelers as [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Equilibrium departure and arrival dynamics. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: The solid black line is the equilibrium profile [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

**Figure 6.** Figure 6: The cost functions Cn(t) after a small perturbation away from equilibrium. Travelers within the perturbation interval are pressured to move in the direction of a new local minimum. costs and face no pressure to move [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗

**Figure 7.** Figure 7: The division line. The marked areas correspond to parameter pairs [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗

read the original abstract

Vickrey's classic single-bottleneck departure time choice equilibrium model exhibits instability under many plausible day-to-day learning dynamics. Such instability is not observed in reality -- does this difference stem from the day-to-day dynamics or from one of the simplifying assumptions of the basic model? This paper explores a variant of the basic model with a continuous distribution of schedule delay parameters which we intuitively expect to have more favorable stability properties. To attain tractability we assume a monotonic relationship between earliness and lateness parameters. We first verify the existence and uniqueness of the equilibrium solution for this model. We then study a broad class of day-to-day dynamics satisfying local pressure and order preservation conditions. Our main contribution is a formal proof that, surprisingly, all such day-to-day dynamics in this context are unstable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Heterogeneity in schedule delays does not stabilize day-to-day dynamics in the Vickrey model once monotonicity is imposed for tractability.

read the letter

The main takeaway is that this paper shows instability survives the move to continuous heterogeneity in departure-time choice. They add a distribution over schedule-delay parameters to the classic single-bottleneck setup, prove the equilibrium still exists and is unique, and then show that every day-to-day adjustment process obeying local pressure and order preservation is unstable. That instability result for the heterogeneous case is the new formal piece relative to the Vickrey literature they cite. The existence-uniqueness step is careful groundwork that lets the instability argument go through cleanly. The proof itself looks like a direct extension rather than a radical departure, but it is executed without obvious gaps in the abstract framing. The monotonicity restriction between earliness and lateness parameters is stated up front as the price of tractability, and the stress-test note is right that this keeps the result inside a narrow parametric family. Drop the monotonicity and both uniqueness and the propagation of instability could fail. The class of dynamics is also restricted, so the claim does not cover arbitrary learning rules. No numerical checks or sensitivity tests are mentioned in the abstract, which leaves the practical bite of the instability open. This is for transportation modelers and game theorists who work on equilibrium stability in traffic. A reader already following Vickrey extensions will get a precise negative result on whether heterogeneity alone fixes the instability problem. It deserves peer review because the question is well-posed, the formal steps are transparent, and the modeling choices are explicit even if they limit scope.

Referee Report

2 major / 2 minor

Summary. The manuscript extends Vickrey's single-bottleneck departure time choice model to include a continuous distribution of schedule delay parameters, assuming a monotonic relationship between earliness and lateness parameters for tractability. It verifies the existence and uniqueness of the equilibrium and provides a formal proof that all day-to-day dynamics satisfying local-pressure and order-preservation conditions are unstable.

Significance. If the result holds, it is significant in showing that heterogeneity in schedule delay parameters does not mitigate the instability of day-to-day dynamics in departure time choice, contrary to initial intuition. This formal mathematical contribution strengthens the case for instability in such models and highlights the role of parametric assumptions in equilibrium analysis. The provision of a rigorous proof for both equilibrium properties and instability is a positive aspect of the work.

major comments (2)

[§2 (Model and Assumptions)] The monotonicity assumption between earliness and lateness parameters is imposed for tractability and is load-bearing for both the uniqueness of equilibrium and the propagation of instability in the dynamics proof. The paper should explicitly discuss the restrictiveness of this assumption and its implications for the generality of the instability result, as non-monotonic distributions might alter the order-preservation property.
[Instability Proof (main theorem)] The proof that all such dynamics are unstable relies on the continuous distribution and the order-preservation condition. A more detailed exposition of how the heterogeneity interacts with the local-pressure condition to ensure instability would be beneficial, including any potential boundary cases or sensitivity to the distribution parameters.

minor comments (2)

[Abstract] The abstract clearly states the main contribution but could briefly mention the monotonicity assumption to set expectations for readers.
[Notation] Ensure consistent use of notation for the schedule delay parameters throughout the manuscript to avoid confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments highlight important aspects of our modeling assumptions and proof exposition, and we plan to incorporate revisions that address these points while preserving the core contributions.

read point-by-point responses

Referee: [§2 (Model and Assumptions)] The monotonicity assumption between earliness and lateness parameters is imposed for tractability and is load-bearing for both the uniqueness of equilibrium and the propagation of instability in the dynamics proof. The paper should explicitly discuss the restrictiveness of this assumption and its implications for the generality of the instability result, as non-monotonic distributions might alter the order-preservation property.

Authors: We agree that the monotonicity assumption is central to achieving tractability and enabling the order-preservation property used in both the equilibrium uniqueness and instability proofs. In the revised manuscript, we will add an explicit discussion in Section 2 (likely as a new subsection or extended paragraph) on the restrictiveness of this assumption. This will include its implications for the generality of the instability result and acknowledge that non-monotonic distributions could potentially violate order-preservation, leading to different stability outcomes under the dynamics. We will frame this as a limitation and suggest it as an avenue for future work. revision: yes
Referee: [Instability Proof (main theorem)] The proof that all such dynamics are unstable relies on the continuous distribution and the order-preservation condition. A more detailed exposition of how the heterogeneity interacts with the local-pressure condition to ensure instability would be beneficial, including any potential boundary cases or sensitivity to the distribution parameters.

Authors: We concur that expanding the exposition of the main instability theorem would improve clarity. In the revised version, we will augment the proof section with a more detailed breakdown of how the continuous parametric heterogeneity interacts with the local-pressure condition to propagate instability across the population. This will include additional remarks on boundary cases (e.g., when the distribution approaches a degenerate case) and sensitivity to distribution parameters such as variance or support width. These additions will be integrated without altering the theorem statement or core logic. revision: yes

Circularity Check

0 steps flagged

No circularity: formal existence-uniqueness-stability proof under explicit parametric assumptions

full rationale

The paper states its monotonicity assumption on earliness/lateness parameters explicitly for tractability, then proves equilibrium existence and uniqueness, followed by a general instability result for any day-to-day dynamics obeying local-pressure and order-preservation. No step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the theorems are derived from the stated conditions without renaming known results or smuggling ansatzes. The derivation is self-contained as a mathematical proof within the restricted model.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The model rests on the monotonicity assumption linking earliness and lateness parameters and on the two abstract conditions (local pressure, order preservation) that define the class of dynamics; no free parameters or invented entities are described in the abstract.

axioms (2)

domain assumption Monotonic relationship between earliness and lateness parameters
Introduced to attain tractability for the continuous distribution
domain assumption Day-to-day dynamics satisfy local pressure and order preservation
Defines the broad class for which instability is proved

pith-pipeline@v0.9.0 · 5434 in / 1212 out tokens · 41449 ms · 2026-05-10T12:14:46.672726+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

(Here we require no ordering of departures,nis an arbitrary traveler.) 29

The departure time for each traveler,τ(n). (Here we require no ordering of departures,nis an arbitrary traveler.) 29

work page
[2]

permutation

The departure order, specified by a “permutation”ϕ: [0, N]→[0, N]. The individual departure times [1] determine this function throughϕ(n) = λ({n ′ :τ(n ′)< τ(n)}) whereλdenotes the Lebesgue measure. (As a reg- ularity condition, we assume thatτ(n) is chosen in a way that these sets are Lebesgue measurable.)

work page
[3]

We can express this in terms of the departure times [1] asνD(t) =λ({n:τ(n)≤t|})

The cumulative departures by timet,ν D(t). We can express this in terms of the departure times [1] asνD(t) =λ({n:τ(n)≤t|}). (We again impose as a regularity condition onτ(n) that these sets are Lebesgue measurable.)

work page
[4]

Cumulative arrivals can be calculated from cumulative departures [3] using the queuing dynamics, or explicitly asν A(t) = inf t′<t{νD(t′) +s(t−t ′)}

The cumulative arrivals by timet,ν A(t). Cumulative arrivals can be calculated from cumulative departures [3] using the queuing dynamics, or explicitly asν A(t) = inf t′<t{νD(t′) +s(t−t ′)}

work page
[5]

The index of the traveler arriving at timet, writtenn(t), determined from the cumulative arrival profile [4] and the ordering [2]:n(t) =ϕ −1(νA(t)), using the fact that the departure and arrival orders are the same

work page
[6]

The arrival time of the traveler of indexn, given byt(n) =n −1(t) as the inverse of [5]

work page
[7]

The queuing delay experienced by a traveler arriving at timet,Q(t) = t−τ(n(t))

work page
[8]

In particular, [1] uniquely determines all of the other values [2–8], using the expressions given in the list above

The departure time for a traveler arriving at timet,D(t) =t−Q(t). In particular, [1] uniquely determines all of the other values [2–8], using the expressions given in the list above. But it is difficult to characterize the equilibrium in terms of [1], because the queueing effects and delays are not explicit. It is more convenient to work with the traveler...

work page
[9]

Proof.We use implication arrows to express relationships among the above quantities; [i]⇒[j] means that item [i] in the list is sufficient to determine item [i], and so forth

and [7] in the list) uniquely determine all other values in the list above. Proof.We use implication arrows to express relationships among the above quantities; [i]⇒[j] means that item [i] in the list is sufficient to determine item [i], and so forth. The notation [i] + [j]⇒[k] means that items [i] and [j] together are sufficient to determine [k]. Using t...

work page
[10]

+ [7]⇒[3]: Given the cumulative arrival curve and queueing delay, con- struct the cumulative departure curve by shifting the arrival curve to the left by the queueing delay:ν D(t−Q(t)) =ν A(t)

work page
[11]

direction

+ [2]⇒[1]: Apply the permutationϕto determine where a particular trav- eler is in the departure order, and apply the inverse cumulative departure curve:τ(n) =ν −1 D (ϕ(n)). Conclusion: given [6] and [7], we can compute the values of all other quantities in the list, so no information is lost by analyzing the equilibrium in terms of t(n) andQ(t). Using thi...

work page

[1] [1]

(Here we require no ordering of departures,nis an arbitrary traveler.) 29

The departure time for each traveler,τ(n). (Here we require no ordering of departures,nis an arbitrary traveler.) 29

work page

[2] [2]

permutation

The departure order, specified by a “permutation”ϕ: [0, N]→[0, N]. The individual departure times [1] determine this function throughϕ(n) = λ({n ′ :τ(n ′)< τ(n)}) whereλdenotes the Lebesgue measure. (As a reg- ularity condition, we assume thatτ(n) is chosen in a way that these sets are Lebesgue measurable.)

work page

[3] [3]

We can express this in terms of the departure times [1] asνD(t) =λ({n:τ(n)≤t|})

The cumulative departures by timet,ν D(t). We can express this in terms of the departure times [1] asνD(t) =λ({n:τ(n)≤t|}). (We again impose as a regularity condition onτ(n) that these sets are Lebesgue measurable.)

work page

[4] [4]

Cumulative arrivals can be calculated from cumulative departures [3] using the queuing dynamics, or explicitly asν A(t) = inf t′<t{νD(t′) +s(t−t ′)}

The cumulative arrivals by timet,ν A(t). Cumulative arrivals can be calculated from cumulative departures [3] using the queuing dynamics, or explicitly asν A(t) = inf t′<t{νD(t′) +s(t−t ′)}

work page

[5] [5]

The index of the traveler arriving at timet, writtenn(t), determined from the cumulative arrival profile [4] and the ordering [2]:n(t) =ϕ −1(νA(t)), using the fact that the departure and arrival orders are the same

work page

[6] [6]

The arrival time of the traveler of indexn, given byt(n) =n −1(t) as the inverse of [5]

work page

[7] [7]

The queuing delay experienced by a traveler arriving at timet,Q(t) = t−τ(n(t))

work page

[8] [8]

In particular, [1] uniquely determines all of the other values [2–8], using the expressions given in the list above

The departure time for a traveler arriving at timet,D(t) =t−Q(t). In particular, [1] uniquely determines all of the other values [2–8], using the expressions given in the list above. But it is difficult to characterize the equilibrium in terms of [1], because the queueing effects and delays are not explicit. It is more convenient to work with the traveler...

work page

[9] [9]

Proof.We use implication arrows to express relationships among the above quantities; [i]⇒[j] means that item [i] in the list is sufficient to determine item [i], and so forth

and [7] in the list) uniquely determine all other values in the list above. Proof.We use implication arrows to express relationships among the above quantities; [i]⇒[j] means that item [i] in the list is sufficient to determine item [i], and so forth. The notation [i] + [j]⇒[k] means that items [i] and [j] together are sufficient to determine [k]. Using t...

work page

[10] [10]

+ [7]⇒[3]: Given the cumulative arrival curve and queueing delay, con- struct the cumulative departure curve by shifting the arrival curve to the left by the queueing delay:ν D(t−Q(t)) =ν A(t)

work page

[11] [11]

direction

+ [2]⇒[1]: Apply the permutationϕto determine where a particular trav- eler is in the departure order, and apply the inverse cumulative departure curve:τ(n) =ν −1 D (ϕ(n)). Conclusion: given [6] and [7], we can compute the values of all other quantities in the list, so no information is lost by analyzing the equilibrium in terms of t(n) andQ(t). Using thi...

work page