16-Civ-A6 Highway Design, Construction, and Maintenance · May 2014

Question 7 of 7: Multinomial Logit Mode Choice and IIA

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

Notes on this paper

Paper format. National Examination, 98-Civ-A6 Transportation Planning & Engineering, May 2014 — 3 hours, closed book (one two-sided aid sheet). Seven questions; any five constitute a complete examination and all are of equal value (20 marks). All seven are solved below as a study resource.

Reference texts (subject).

Garber, N.J. & Hoel, L.A., Traffic and Highway Engineering, 5th ed. — traffic flow, deterministic queueing, shock waves.
Papacostas, C.S. & Prevedouros, P.D., Transportation Engineering and Planning, 3rd ed. — the four‑step urban travel‑demand model.
Ortuzar, J. de D. & Willumsen, L.G., Modelling Transport, 4th ed. — trip generation, trip distribution, discrete mode choice, traffic assignment.
Sheffi, Y., Urban Transportation Networks — user‑equilibrium assignment and Wardrop’s principles.

Question 7: Multinomial Logit Mode Choice and IIA (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. Utility functions above; mode attributes below. Choice by multinomial logit, $P_i=\dfrac{e^{V_i}}{\sum_j e^{V_j}}$.

Mode attributes and systematic utilities
Mode	TC ($)	TT (min)	Utility V
Auto	2.00	10	−2.300
Bus	0.75	18	−3.575
Light rail	1.25	15	−3.625

Find. (a) mode shares for the three modes; (b) shares after splitting auto into drive/passenger; (c) an IIA critique and remedy.

Approach. Evaluate each systematic utility, exponentiate, and normalise by the sum (multinomial logit). In part (b) add the two auto sub‑modes and re‑normalise; compare the combined auto share with part (a).

Utilities (a). The paper prints every utility function with the subscripts $\text{TC}_b$ and $\text{TT}_a$; this is a typesetting slip, so each mode is evaluated with its own cost and time ($\text{TC}_i$, $\text{TT}_i$) as the definitions below the functions state. $V_a=0.7-0.5(2.00)-0.2(10)=-2.300$; $V_b=0.4-0.5(0.75)-0.2(18)=-3.575$; $V_r=0-0.5(1.25)-0.2(15)=-3.625$.
Exponentiate and sum (a). $e^{V_a}=0.10026$, $e^{V_b}=0.02803$, $e^{V_r}=0.02667$; $\sum e^{V}=0.15496$.
Mode shares (a). $P_a=0.10026/0.15496=\boxed{0.647}$; $P_b=0.02803/0.15496=0.181$; $P_r=0.02667/0.15496=0.172$. Auto dominates because its lower time and higher constant outweigh its higher cost.
Split auto (b). Auto‑drive keeps $V=-2.300$; auto‑passenger has $V=0.7-0.5(1.00)-0.2(10)=-1.800$, so $e^{V}=0.16530$. New sum $=0.10026+0.16530+0.02803+0.02667=0.32026$.
Four‑mode shares (b). $P_{\text{drive}}=0.10026/0.32026=0.313$; $P_{\text{pass}}=0.16530/0.32026=0.516$; $P_{\text{bus}}=0.02803/0.32026=0.088$; $P_{\text{rail}}=0.02667/0.32026=0.083$. The combined auto share is $\boxed{0.313+0.516=0.829}$.
Diagnosis (c). Splitting auto into two nearly identical sub‑modes inflates total auto share from $0.647$ to $0.829$ and drains bus (0.181→0.088) and rail (0.172→0.083) roughly in proportion — the classic red‑bus/blue‑bus symptom of the IIA property. This is not intuitively sensible: merely re‑labelling auto as two options should not win auto more travellers.

(c) Why IIA fails here and how to fix it. Multinomial logit assumes independent, identically distributed error terms, so the ratio of any two modes' probabilities depends only on those two modes (independence of irrelevant alternatives). When two alternatives are close substitutes — auto‑drive and auto‑passenger are almost the same travel experience — MNL treats them as fully distinct and effectively "double counts" the auto option, stealing share from the genuinely different transit modes. The remedy is a model that recognises correlation among similar alternatives: a nested logit model that groups auto‑drive and auto‑passenger in one "auto" nest (so competition within the nest does not spill onto bus and rail), or more general forms such as cross‑nested/mixed (random‑parameter) logit or a probit specification with a full error‑covariance structure. Any of these breaks the IIA restriction and restores a sensible combined auto share close to the original 0.647.

Question 7 — mode shares
Mode	(a) 3 modes	(b) 4 modes
Auto‑drive	0.647 (auto)	0.313
Auto‑passenger	0.647 (auto)	0.516
Bus	0.181	0.088
Light rail	0.172	0.083
Combined auto	0.647	0.829

Topic. Multinomial logit mode choice and the independence‑of‑irrelevant‑alternatives (IIA) property.

Key relations. $P_i=\dfrac{e^{V_i}}{\sum_j e^{V_j}}$; IIA: $P_i/P_k=e^{V_i-V_k}$ (independent of all other alternatives).

Why this works. Only utility differences matter, so the constants and coefficients set each mode's relative attractiveness. IIA is convenient (easy estimation, simple forecasting) but assumes uncorrelated alternatives; when that fails, nesting the correlated options fixes the substitution pattern.

Common pitfall. Reading the cost/time coefficients as positive — they are negative (higher cost or time lowers utility) — and expecting plain MNL to handle close substitutes without a nest.

Source. Ortuzar & Willumsen, ch. 7 (discrete choice); Ben‑Akiva & Lerman, Discrete Choice Analysis, ch. 5.

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