16-Civ-A6 Highway Design, Construction, and Maintenance · December 2015

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: 98-Civ-A6 Transportation Planning & Engineering, National Examination, December 2015. Seven questions, each of equal value (20 marks); any five constitute a complete examination. Closed book, one two-sided aid sheet permitted. Three hours. All seven questions are solved below as a study resource.

Reference texts. Mannering, Washburn & Kilareski, Principles of Highway Engineering and Traffic Analysis (Wiley); Papacostas & Prevedouros, Transportation Engineering and Planning (Prentice Hall); Roess, Prassas & McShane, Traffic Engineering (Pearson); Ortúzar & Willumsen, Modelling Transport (Wiley) for the demand-model chapters (trip generation, distribution, mode choice, assignment).

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. A linear-in-attributes utility with common coefficients across modes.

Given data — mode attributes
Mode	C	$X_1$ wait (min)	$X_2$ travel (min)	$X_3$ park (min)	$X_4$ cost (cents)
Auto	−0.33	0	20	5	225
Bus	−0.27	10	35	0	100
Light rail	0	15	25	0	150

Find. Binary auto/bus split (a); three-way split with light rail (b); an IIA critique and remedy (c).

Approach. Evaluate each utility $V_m$, then apply the logit share $P_m=e^{V_m}/\sum_j e^{V_j}$. Only utility differences matter, so the answer is unchanged by any common shift.

Mode utilities. Substituting each attribute set: $$V_{\text{auto}}=-0.33-0.13(20)-0.12(5)-0.0045(225)=-4.5425,$$ $$V_{\text{bus}}=-0.27-0.10(10)-0.13(35)-0.0045(100)=-6.2700.$$
(a) Binary logit split. $$P_{\text{auto}}=\frac{e^{-4.5425}}{e^{-4.5425}+e^{-6.2700}}=\frac{1}{1+e^{-1.7275}}=0.849,$$ $$\boxed{P_{\text{auto}}=84.9\%,\qquad P_{\text{bus}}=15.1\%}$$
(b) Add light rail. Its utility is $$V_{\text{LR}}=0-0.10(15)-0.13(25)-0.0045(150)=-5.4250.$$ With $\sum_j e^{V_j}=e^{-4.5425}+e^{-6.2700}+e^{-5.4250}$: $$\boxed{P_{\text{auto}}=62.8\%,\quad P_{\text{bus}}=11.2\%,\quad P_{\text{LR}}=26.0\%}$$
(c) IIA check. The auto:bus odds are $0.849/0.151=5.62$ before light rail and $0.628/0.112=5.62$ after — identical. The logit model draws the new mode’s 26 % share from auto and bus in proportion to their existing shares, exactly as IIA requires.

(c) Interpretation and remedy. The prediction is not fully intuitive. Light rail and bus are both public-transit modes and share many unobserved attributes (transfers, schedule adherence, "transit" stigma), so a new rail line should draw disproportionately from the bus, not evenly from auto and bus. This is the classic red-bus/blue-bus failure of IIA: the multinomial logit treats all alternatives as equally substitutable because its error terms are independent. Here auto loses 22.1 percentage points (84.9 % → 62.8 %) while bus loses only 3.9 (15.1 % → 11.2 %), so the model overstates the diversion from auto and understates how much light rail would cannibalise the bus. Remedies: use a nested logit that groups bus and light rail under a "transit" nest (so they compete more closely with each other than with auto), or a cross-nested/mixed (random-parameters) logit or probit model that allows correlated error terms across the similar modes. Adding a mode-specific bus–rail correlation restores the intuitive result.

Question 7 — mode-choice results
Mode	Utility $V$	(a) Share	(b) Share with LR
Auto	−4.5425	84.9%	62.8%
Bus	−6.2700	15.1%	11.2%
Light rail	−5.4250	—	26.0%

Topic: Multinomial logit mode choice and the Independence of Irrelevant Alternatives property.

Key relations: $P_m=e^{V_m}/\sum_j e^{V_j}$; only utility differences matter; IIA $\Rightarrow$ $P_a/P_b$ independent of any other alternative.

Why this works: Under the assumption of independent, identically Gumbel-distributed error terms, the choice probabilities take the closed-form logit expression, and the ratio of any two shares depends only on those two modes’ utilities. This is analytically convenient but implies every new alternative steals share proportionally — the substance of IIA.

Common pitfall: Dropping the negative signs on the time and cost coefficients (which would make more expensive, slower modes more attractive), or forgetting that the constants already absorb all unmeasured mode preferences. The deeper error is trusting the logit’s even redistribution when two alternatives are close substitutes — that is when a nested model is required.

Source: Ortúzar & Willumsen, Modelling Transport, Ch. 7–8 (discrete choice, logit and nested logit); Papacostas & Prevedouros, Ch. 12 (mode choice).

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