Question 7 of 7: Multinomial logit mode choice and the IIA property

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 the utility coefficients above (all time and cost coefficients negative, as printed) and the following level‑of‑service data.

Find. (a) base‑case mode shares; (b) shares after the bus improvement; (c) an IIA critique.

Approach. Evaluate each mode’s systematic utility $V_i$, then apply the multinomial logit formula $P_i=e^{V_i}/\sum_j e^{V_j}$; only utility differences matter.

1. Base utilities (a). $$V_a=0.1-0.02(12)-0.15(7)-0.03(2.5)=-1.265,$$ $$V_b=0.2-0.03(20)-0.15(12)-0.03(0.75)=-2.223,$$ $$V_r=-0.03(18)-0.15(10)-0.03(1.2)=-2.076.$$
2. Base shares. Exponentiating, $e^{V_a}=0.2822$, $e^{V_b}=0.1083$, $e^{V_r}=0.1254$ (sum $=0.5160$), so $$\boxed{P_a=54.7\%,\ P_b=21.0\%,\ P_r=24.3\%}.$$
3. Improved bus utility (b). With $\mathrm{IVTT}_b=15$, $\mathrm{OVTT}_b=10$: $$V_b'=0.2-0.03(15)-0.15(10)-0.03(0.75)=-1.773,$$ while $V_a$ and $V_r$ are unchanged.
4. New shares. Now $e^{V_b'}=0.1699$, sum $=0.5776$, giving $$\boxed{P_a=48.9\%,\ P_b=29.4\%,\ P_r=21.7\%}.$$ The bus share rises by 8.4 percentage points, drawn from auto (54.7→48.9) and light rail (24.3→21.7).
5. IIA check. The auto‑to‑rail odds are unchanged: $P_a/P_r=0.547/0.243=2.25$ before and $0.489/0.217=2.25$ after — the improvement to bus drew from auto and rail strictly in proportion to their existing shares.

(c) Does part (b) make sense? The IIA limitation

The direction is sensible — a faster bus should gain share — but the way it gains share exposes the IIA weakness of the multinomial logit model. IIA states that the ratio of probabilities of any two alternatives depends only on those two alternatives’ utilities, so an improvement to bus must draw from auto and light rail in exact proportion to their current shares (the $P_a/P_r=2.25$ result above). That is behaviourally implausible here: light rail and bus are both transit modes and are closer substitutes for each other than either is for the car, so a better bus should cannibalise light‑rail ridership more than it cannibalises auto. This is the classic “red‑bus/blue‑bus” problem. The remedy is to abandon the assumption that the random error terms are independent across modes: a nested logit model that groups bus and light rail in a common transit nest (or a cross‑nested, mixed/random‑parameters logit, or probit model) lets the two transit modes share correlated unobserved utility, so the improved bus correctly draws proportionally more from light rail than from the automobile.