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. A calibrated multinomial logit model with mode-specific constants and generic coefficients on the four level-of-service attributes, together with the attribute values for each mode.

Find. The two-mode split, the three-mode split after light rail is introduced, and an assessment of whether the predicted redistribution is behaviourally credible.

Approach. Evaluate each observable utility, exponentiate, and apply the logit share formula; then compare the auto-to-bus odds before and after the new mode is added to demonstrate the IIA property numerically before discussing it.

1. Part (a) — evaluate the observable utility of each mode. Substituting the attributes into the calibrated functions, $$V_{Auto}=-0.33-0.10(0)-0.13(20)-0.12(5)-0.0045(225)$$ $$V_{Auto}=-0.33-2.60-0.60-1.0125=\boxed{-4.5425}$$ $$V_{Bus}=-0.27-0.10(10)-0.13(35)-0.12(0)-0.0045(100)$$ $$V_{Bus}=-0.27-1.00-4.55-0.45=\boxed{-6.2700}$$ Auto is the more attractive alternative by 1.7275 utility units, driven mainly by its much shorter in-vehicle time, which outweighs its higher out-of-pocket cost and its parking time.
2. Apply the multinomial logit share formula. The probability of choosing mode $i$ from a choice set $C$ is $$P(i)=\frac{e^{V_i}}{\displaystyle\sum_{j\in C}e^{V_j}}$$ Exponentiating, $e^{V_{Auto}}=0.010647$ and $e^{V_{Bus}}=0.001892$, so the denominator is 0.012539 and $$\begin{aligned}P(\text{Auto})&=\frac{0.010647}{0.012539}\\ P(\text{Bus})&=\frac{0.001892}{0.012539}\end{aligned}$$ $$\boxed{P(\text{Auto})=84.91 \text{ per cent}, \qquad P(\text{Bus})=15.09 \text{ per cent}}$$ Only the utility difference matters, so the same answer follows from the binary form $P(\text{Auto})=1/(1+e^{-1.7275})$.
3. Part (b) — evaluate the utility of light rail. Light rail has no mode-specific constant, so $$V_{Rail}=-0.10(15)-0.13(25)-0.12(0)-0.0045(150)$$ $$V_{Rail}=-1.50-3.25-0.675=\boxed{-5.4250}$$ It sits between auto and bus: worse than the car because of its long access and waiting time, but far better than the bus because it saves ten minutes of in-vehicle time.
4. Recompute the shares over the enlarged choice set. With $e^{V_{Rail}}=0.004405$, the denominator becomes $0.010647+0.001892+0.004405=0.016944$, and $$\begin{aligned}P(\text{Auto})&=\frac{0.010647}{0.016944}\\ P(\text{Bus})&=\frac{0.001892}{0.016944}\\ P(\text{Rail})&=\frac{0.004405}{0.016944}\end{aligned}$$ $$\boxed{P(\text{Auto})=62.83\%, \quad P(\text{Bus})=11.17\%, \quad P(\text{Rail})=26.00\%}$$ The three shares sum to 100 per cent, as they must.
5. Part (c) — demonstrate the IIA property numerically before describing it. The introduction of light rail scaled the auto share by $62.83/84.91 = 0.7400$ and the bus share by $11.17/15.09 = 0.7400$ — the identical factor. Equivalently, the auto-to-bus odds ratio is unchanged to four decimal places: $$\frac{P(\text{Auto})}{P(\text{Bus})}=\frac{e^{V_{Auto}}}{e^{V_{Bus}}}=5.6266 \text{ both before and after}$$ This is the independence of irrelevant alternatives: in a multinomial logit model the ratio of the probabilities of any two alternatives depends only on those two alternatives' utilities, so adding or removing a third alternative rescales every remaining share by a common constant and leaves every pairwise ratio untouched.
6. Say plainly whether the result is behaviourally credible. It is not. Light rail draws 22.07 percentage points from auto and only 3.92 points from bus — it takes 5.63 times as much traffic from the car as from the bus, exactly the pre-existing share ratio. Behaviourally one expects the opposite emphasis: light rail and bus are both public transit, they share unobserved attributes (no need to own a vehicle, exposure to weather while waiting, dependence on a schedule, the availability of a transit pass), so a traveller already willing to use the bus is far more likely to switch to rail than a committed car commuter is. This is the classic red-bus/blue-bus failure: because the multinomial logit model assumes the random components of utility are independent and identically distributed across alternatives, it cannot recognise that two alternatives are close substitutes, and it treats a near-duplicate of an existing mode as though it were an entirely fresh choice.
7. Set out the remedies in order of practical usefulness. The direct fix is a nested logit model with a transit nest containing bus and light rail and the car in its own nest; the nesting (logsum) parameter, estimated from data and constrained to lie between 0 and 1, controls how strongly the two transit alternatives compete with each other relative to the car, and a value near zero would concentrate the rail patronage among former bus users, which is the expected behaviour. Where alternatives belong to more than one grouping — a light rail line that is also a park-and-ride facility, for instance — a cross-nested logit allows partial membership of several nests. More general still are the multinomial probit model, which allows an unrestricted covariance matrix for the error terms, and the mixed (random-parameter) logit, which induces correlation across alternatives by letting taste coefficients vary across the population; both relax IIA fully at the cost of simulation-based estimation.
8. Complete the answer with the model-building measures that go with those specifications. Two further steps belong in a real study. First, market segmentation: estimate separate models for captive riders, choice riders and car-owning households, since much of the apparent IIA violation is unobserved heterogeneity in car availability. Second, better specification: add the alternative-specific attributes that actually distinguish rail from bus — service frequency, reliability, comfort, transfers required — so that less of the substitution pattern has to be carried by the error structure. The IIA assumption should also be tested rather than assumed away, using the Hausman–McFadden specification test, which re-estimates the model on a restricted choice set and checks whether the coefficients change significantly. For this corridor the practical conclusion is that the 26 per cent rail share is plausible as a total but the source of that patronage is not: a nested specification would predict a similar rail share drawn far more heavily from the bus, with a correspondingly smaller reduction in car use — and it is the reduction in car use, not the rail ridership, that a Canadian transit business case is normally built on.