16-Civ-A6 Highway Design, Construction, and Maintenance · December 2014
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
National Examination — 98-Civ-A6 Transportation Planning & Engineering, December 2014. Closed book (one two-sided aid sheet), 3 hours. Seven questions of equal value (20 marks); any five constitute a complete paper. All seven are solved here as a study resource.
Given. Three modes with the utility coefficients above (all time/cost coefficients negative) and the level-of-service data.
Given data — level of service by mode
Mode
IVTT (min)
OVTT (min)
TC ($)
Automobile
12
7
4.00
Bus
20
12
2.50
Light rail
18
10
3.00
Find. Choice probabilities before (a) and after (b) the bus improvement, and an interpretation via the IIA property (c).
Approach. Evaluate each mode's deterministic utility, then apply the multinomial logit $P_i=e^{V_i}/\sum_j e^{V_j}$; recompute for the improved bus and interpret through IIA.
(a) Utilities. Substituting the data (costs read in dollars):
$$V_a=0.3-0.04(12)-0.2(7)-0.06(4)=-1.82,$$
$$V_b=0.5-0.06(20)-0.2(12)-0.06(2.5)=-3.25,\qquad V_r=-0.06(18)-0.2(10)-0.06(3)=-3.26.$$
(a) Logit probabilities. With $e^{-1.82}=0.1620$, $e^{-3.25}=0.0388$, $e^{-3.26}=0.0384$ (sum $0.2392$):
$$\boxed{P_a=0.677,\quad P_b=0.162,\quad P_r=0.160.}$$
Auto dominates because its utility is far higher despite its cost.
(b) Improved bus. Only the bus changes (IVTT 15, OVTT 8):
$$V_b'=0.5-0.06(15)-0.2(8)-0.06(2.5)=-2.15,\qquad e^{-2.15}=0.1165.$$
Re-normalizing with $V_a,V_r$ unchanged (sum $0.3168$):
$$\boxed{P_a=0.511,\quad P_b=0.368,\quad P_r=0.121.}$$
The bus share more than doubles, drawing riders from both auto and rail.
(c) IIA and its limitation. Directionally the result is sensible — a faster bus should win share. But the multinomial logit obeys independence of irrelevant alternatives: the ratio $P_a/P_r=0.677/0.160=4.22$ is identical before and after ($0.511/0.121=4.22$), so the improved bus pulls proportionally from auto and rail. That is unrealistic: bus and light rail are close substitutes (both transit, similar access/wait characteristics), so a better bus should cannibalize rail far more than auto. This is the classic "red-bus/blue-bus" defect of IIA. It is overcome by relaxing the equal-cross-substitution assumption — most directly with a nested logit model that groups bus and light rail in a common "transit" nest (so improvements shift choice mainly within the nest), or with a cross-nested/mixed (random-parameters) logit or a probit model that permits correlated error terms among the transit alternatives.
