16-Civ-A6 Highway Design, Construction, and Maintenance · December 2016

Question 7 of 7: Multinomial Logit Mode Choice and the IIA Property

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. National Examination, December 2016 — 98-Civ-A6, Transportation Planning & Engineering. Three hours, closed book (one two-sided aid sheet, approved calculator). Seven questions of equal value (20 marks each); any five constitute a complete examination. All seven are solved here, because the set is a study resource rather than a sat examination.

Reference texts. Mannering, Washburn & Kilareski, Principles of Highway Engineering and Traffic Analysis (Wiley) — traffic-stream models, deterministic queueing and shock waves; Papacostas & Prevedouros, Transportation Engineering and Planning (Prentice Hall) — the land-use/transport system and the four-step demand model; Ortuzar & Willumsen, Modelling Transport (Wiley) — trip generation, distribution, mode choice and traffic assignment; Sheffi, Urban Transportation Networks (Prentice Hall) — user-equilibrium assignment; Transportation Association of Canada, Geometric Design Guide for Canadian Roads — Canadian planning and design practice.

Question 7: Multinomial Logit Mode Choice and the IIA Property (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Three modes with linear-in-parameters utilities sharing common time and cost coefficients, differing only in their alternative-specific constants.

Given data — utility specification and level of service
Mode	Constant	$TT$ (min)	$TC$ ($)
Auto	1.1	16	3.50
Bus	0.1	30	2.00 (1.00 in part b)
Light rail	0.0	25	2.50
Common coefficients: $-0.05$ per minute of travel time, $-0.25$ per dollar of travel cost

Find. The three modal shares before and after the bus fare reduction, and an explanation of the logit property that makes the predicted response questionable.

Approach. Evaluate each observable utility, exponentiate, and normalise by the sum — the multinomial logit formula — then compare the before-and-after odds between the two unchanged modes to expose the independence-of-irrelevant-alternatives property.

Evaluate the three utilities. Substituting the level-of-service values into each specification, $$V_a=1.1-0.05(16)-0.25(3.50)=1.1-0.80-0.875=-0.575$$ $$V_b=0.1-0.05(30)-0.25(2.00)=0.1-1.50-0.50=-1.900$$ $$V_r=0-0.05(25)-0.25(2.50)=-1.25-0.625=-1.875$$ Auto leads not only because of its large positive constant but because it is much faster; light rail edges out bus on utility despite costing more, because it saves five minutes.
Apply the multinomial logit formula. With $P_i=e^{V_i}\big/\sum_j e^{V_j}$, $$e^{V_a}=0.5627,\qquad e^{V_b}=0.1496,\qquad e^{V_r}=0.1533,\qquad \sum=0.8656$$ $$\boxed{P_a=0.650,\qquad P_b=0.173,\qquad P_r=0.177}$$ That is 65.0 % auto, 17.3 % bus and 17.7 % light rail; the three sum to unity as they must.
Recompute the bus utility after the fare reduction. Only $TC_b$ changes, from $2.00 to $1.00, and only through the cost coefficient: $$V_b'=0.1-0.05(30)-0.25(1.00)=-1.650,\qquad \Delta V_b = +0.25$$ so $e^{V_b'}=0.1920$ and the new denominator is $0.5627+0.1920+0.1533=0.9080$.
Predict the new shares. $$\boxed{P_a'=0.620,\qquad P_b'=0.211,\qquad P_r'=0.169}$$ The dollar off the fare raises the bus share by 3.87 percentage points, from 17.3 % to 21.1 %. Auto loses 3.05 points and light rail loses 0.82 points.
Expose the substitution pattern. Examine the ratio of the two shares that did not change: $$\frac{P_a}{P_r}=\frac{0.6501}{0.1771}=3.669 \qquad\text{and}\qquad \frac{P_a'}{P_r'}=\frac{0.6196}{0.1689}=3.669$$ The odds are identical, and each losing mode surrenders exactly the same fraction of its own share: $$1-\frac{P_a'}{P_a}=1-\frac{P_r'}{P_r}=4.68\%$$ $$\boxed{\text{Bus gains proportionally from auto and light rail alike — the IIA property.}}$$

Final results — Question 7
Quantity	(a) Base	(b) After $1.00 fare cut
$V_a$ / $V_b$ / $V_r$	−0.575 / −1.900 / −1.875	−0.575 / −1.650 / −1.875
Auto share, $P_a$	0.650 (65.0 %)	0.620 (62.0 %)
Bus share, $P_b$	0.173 (17.3 %)	0.211 (21.1 %)
Light-rail share, $P_r$	0.177 (17.7 %)	0.169 (16.9 %)
Auto : light-rail odds	3.669	3.669 (unchanged — IIA)
Share each loser sheds	4.68 % of its own share, both modes

(c) The unrealistic assumption and how to account for it

The property at work is independence of irrelevant alternatives (IIA). It follows directly from the multinomial logit derivation, in which the unobserved parts of utility are assumed to be independently and identically Gumbel-distributed across alternatives. Under that assumption the ratio of any two choice probabilities depends only on those two alternatives' own utilities, so it is unaffected by a change in a third. The consequence is proportional substitution: when bus becomes cheaper, it draws new riders from auto and from light rail in exact proportion to their existing shares, and every competitor loses the same 4.68 % of its own patronage.

That prediction is not credible here. Bus and light rail are both public transit: they share a great many unobserved attributes that neither utility function measures — the need to walk to a stop, exposure to weather, schedule adherence and waiting, transfers, crowding, luggage and child-carrying difficulty, and the absence of a parked car at the destination. A traveller who is willing to use one of them is far more likely to be willing to use the other, which means their unobserved utility components are strongly correlated, violating the independence assumption at the heart of the model. A cheaper bus fare should therefore attract mostly light-rail riders (and some new transit trips), with only a small effect on committed auto users. The model instead says that 79 % of the bus's gain comes out of auto, which would flatter a fare-reduction business case considerably — it over-predicts the mode shift away from the car and hence over-predicts the congestion and emissions benefit. This is the classic red-bus/blue-bus problem in its practical form.

The correction is to relax the independence assumption in the error structure. The standard remedy for this exact situation is a nested logit model: place bus and light rail together in a "transit" nest, with auto competing against the nest at the upper level. The nest's logsum parameter (equivalently the dissimilarity parameter, $0\lt\theta\le1$) then measures how substitutable the two transit modes are within the nest; a low value concentrates the substitution inside the nest, so the fare cut moves riders mainly from light rail to bus and only weakly from auto. The odds between auto and light rail are then free to change, as intuition demands. Where the correlation structure is more complicated than a clean tree — light rail competing with both bus and auto through different shared attributes — a cross-nested logit, mixed (error-components) logit or multinomial probit model allows a general covariance matrix, at the cost of simulation-based estimation. A partial alternative, useful when the data will not support a richer error structure, is to specify the missing attributes explicitly — adding access time, waiting time, transfer count and service frequency as separate variables with their own coefficients — since correlation in the unobserved term shrinks as more of the shared attributes are moved into the observed utility. In all cases the nesting structure must be tested rather than assumed: estimate the logsum parameter and check that it lies in the admissible range, and confirm the improvement with a likelihood-ratio test against the plain multinomial logit.

Topic. Discrete-choice mode split: the multinomial logit model, the IIA property, and nested-logit correction.

Key relations. $P_i=e^{V_i}\big/\sum_j e^{V_j}$; only utility differences matter, so one alternative-specific constant is normalised to zero (light rail here); IIA: $P_i/P_j=e^{V_i-V_j}$ is independent of every other alternative; implied value of time $=\beta_{TT}/\beta_{TC}=0.05/0.25=$ $0.20/min = $12/h. Nested logit: $P_i = P_{\text{nest}}\times P_{i\mid\text{nest}}$ with the upper level driven by the nest logsum $\theta\ln\sum_{k\in\text{nest}}e^{V_k/\theta}$.

Why this works. The logit form arises from utility maximisation with independent Gumbel errors, which is what makes the choice probabilities close-form and the model estimable on modest data. That same independence is what produces IIA, so the convenience and the defect are two faces of one assumption. Nesting fixes it by allowing the errors of alternatives within a nest to be correlated, which is exactly the structure the bus/light-rail pair requires: substitution is then concentrated where the alternatives really are close substitutes.

Common pitfall. Reading the light-rail constant of zero as an error rather than as the normalisation it is. A second pitfall is applying the cost coefficient to the fare change alone and forgetting to re-exponentiate and re-normalise: because the denominator changes too, the new shares are not obtained by adding the utility change to the old share. Third, do not report the substitution pattern as a plausible forecast — part (c) is asking you to recognise that it is an artefact of the model, and an answer that computes (b) without questioning it misses eight of the twenty marks.

Source. Ortuzar & Willumsen, Modelling Transport, Ch. 7 (discrete-choice models: MNL, IIA, nested logit) and Ch. 8 (model specification and estimation); Ben-Akiva & Lerman, Discrete Choice Analysis, Ch. 4–5 & 10; Papacostas & Prevedouros, Transportation Engineering and Planning, Ch. 6 (mode choice).

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