Question 6 of 7: Four-Mode Multinomial Logit and the IIA Property

Question 6: Four-Mode Multinomial Logit 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.

with $V_i = -0.075\,AT_i - 0.05\,WT_i - 0.04\,RT_i - 0.002\,TC_i$, and in part (b) the bike riding time falls from 60 to 45 minutes.

Find. The modal split before and after the bike-path investment, and an explanation of the independence-of-irrelevant-alternatives property together with practical remedies for it.

Approach. Evaluate the deterministic utility of each mode, exponentiate, and normalise. Because only the bike attribute changes in part (b), the three other exponentials carry over unchanged, which both saves work and makes the IIA property in part (c) visible directly in the arithmetic.

1. (a) Evaluate the four utilities. Substituting each row of the attribute table: $$V_{\text{auto}} = -0.075(6) - 0.05(1) - 0.04(25) - 0.002(300) = -0.450 - 0.050 - 1.000 - 0.600 = -2.100$$ $$V_{\text{bus}} = -0.075(10) - 0.05(15) - 0.04(40) - 0.002(60) = -0.750 - 0.750 - 1.600 - 0.120 = -3.220$$ $$V_{\text{rail}} = -0.075(7) - 0.05(10) - 0.04(30) - 0.002(75) = -0.525 - 0.500 - 1.200 - 0.150 = -2.375$$ $$V_{\text{bike}} = -0.075(1) - 0.05(0) - 0.04(60) - 0.002(10) = -0.075 - 0.000 - 2.400 - 0.020 = -2.495$$ Automobile has the highest (least negative) utility despite being by far the most expensive, because it is fastest and involves almost no access or waiting time; bus is the worst on every attribute except cost.
2. Exponentiate and normalise. The multinomial logit probability is $$P_i = \frac{e^{V_i}}{\displaystyle\sum_{j\in C}e^{V_j}}$$ $$e^{V_{\text{auto}}} = e^{-2.100} = 0.122456,\quad e^{V_{\text{bus}}} = e^{-3.220} = 0.039955$$ $$e^{V_{\text{rail}}} = e^{-2.375} = 0.093014,\quad e^{V_{\text{bike}}} = e^{-2.495} = 0.082496$$ $$\sum_j e^{V_j} = 0.122456 + 0.039955 + 0.093014 + 0.082496 = 0.337922$$ $$\boxed{P_{\text{auto}} = 36.24\ \text{per cent},\quad P_{\text{bus}} = 11.82\ \text{per cent},\quad P_{\text{rail}} = 27.53\ \text{per cent},\quad P_{\text{bike}} = 24.41\ \text{per cent}}$$ The four shares sum to 100.00 per cent, as they must.
3. (b) Recompute the bike utility with the improved riding time. Only $RT_{\text{bike}}$ changes, from 60 to 45 minutes: $$V_{\text{bike}}' = -0.075(1) - 0.05(0) - 0.04(45) - 0.002(10) = -0.075 - 1.800 - 0.020 = -1.895$$ The gain is exactly $0.04 \times 15 = 0.60$ utility units. To put that in context, the coefficient ratio $0.04/0.002 = 20$ cents per minute of riding time means the improvement is worth $20 \times 15 = 300$ cents, or three dollars, per trip to a cyclist — the same as the entire out-of-pocket cost of driving.
4. Renormalise with the three unchanged exponentials. $$e^{V_{\text{bike}}'} = e^{-1.895} = 0.150319$$ $$\sum_j e^{V_j} = 0.122456 + 0.039955 + 0.093014 + 0.150318 = 0.405744$$ $$\boxed{P_{\text{auto}}' = 30.18,\quad P_{\text{bus}}' = 9.85,\quad P_{\text{rail}}' = 22.92,\quad P_{\text{bike}}' = 37.05\ \text{per cent}}$$ Bike share rises by 12.64 percentage points, from 24.41 to 37.05 per cent, and the improvement moves it from third place to first.
5. Read the structure of the loss. Where did the 12.64 points come from? Auto loses 6.06 points, rail loses 4.61 and bus loses 1.97. Each loses in strict proportion to the share it already held: $$\frac{30.18}{36.24} = \frac{9.85}{11.82} = \frac{22.92}{27.53} = 0.8328$$ Every surviving mode is scaled by the identical factor 0.8328. That is not a coincidence of these numbers — it is the IIA property, and part (c) is about it.
   Figure 6.1 — Modal shares before and after the bike-path improvement. Auto, bus and rail are each multiplied by the same factor 0.8328, which is the independence-of-irrelevant-alternatives property made visible.
6. (c) State the IIA property precisely. For any two alternatives $i$ and $k$ in the choice set, $$\frac{P_i}{P_k} = \frac{e^{V_i}/\sum_j e^{V_j}}{e^{V_k}/\sum_j e^{V_j}} = \frac{e^{V_i}}{e^{V_k}} = e^{V_i - V_k}$$ The normalising denominator cancels, so the odds ratio between any two alternatives depends only on their own two utilities. It is independent of the existence, the attributes, or the improvement of every other — "irrelevant" — alternative in the set. The results above demonstrate it numerically: the auto-to-rail odds are $$\frac{36.24}{27.53} = 1.3165\ \text{before, and}\ \frac{30.18}{22.92} = 1.3165\ \text{after}$$ identical to four decimal places, even though the bike alternative has become substantially more attractive.
7. Explain why this is a limitation. IIA forces any new or improved alternative to draw from all existing alternatives in proportion to their current shares. That is behaviourally wrong whenever some alternatives are closer substitutes for one another than others, because a new alternative should draw disproportionately from the modes it most resembles. The classic illustration is the red-bus/blue-bus paradox: introducing a bus identical to an existing bus except for its colour ought to split the bus market and leave the car share untouched, but multinomial logit halves every mode's share equally and predicts a fall in car use that will not occur. In this question the same defect is present. The model predicts that better bike paths take 6.06 points from the automobile and only 1.97 from the bus, whereas empirically a bike-path programme draws most heavily from short transit trips and from walking — travellers who are already unprotected from the weather and already accept an active access leg — and much less from the long-distance car commuter who is a poor substitute for a 45-minute cycle. The forecast benefit in vehicle-kilometres removed is therefore overstated, and the forecast loss of transit ridership understated, which are precisely the two numbers a business case rests on.
8. Set out the remedies. The property arises from the assumption that the random utility components $\varepsilon_i$ are independently and identically Gumbel-distributed, so every practical fix relaxes either the independence or the identical-distribution part of that assumption.

   Nested logit is the standard first response and the one to name first. Group the close substitutes into a nest — here, a motorised nest containing auto, bus and rail with a transit sub-nest for bus and rail, against a non-motorised nest containing bike (and walk, if present) — and estimate a logsum (nesting) parameter for each nest. Alternatives within a nest share an unobserved component, so substitution is stronger within a nest than across it, and IIA holds only within nests. The nesting parameter is estimable and testable, and a value not significantly different from one recovers the plain multinomial logit, so nothing is lost by testing for it.

   Cross-nested logit handles the common case in which an alternative belongs partly to two nests — a park-and-ride or bike-and-ride option is genuinely both motorised and not — by allocating it fractionally to several nests.

   Multinomial probit assumes multivariate-normal error terms with a full covariance matrix, so any pattern of correlation among alternatives can be represented and IIA is absent entirely. Its cost is that the choice probabilities have no closed form and must be simulated, and the covariance parameters are difficult to identify with typical sample sizes.

   Mixed (random-parameters) logit is the modern general answer: let the taste coefficients vary randomly across the population, so that two alternatives sharing an attribute become correlated through the shared random coefficient. It can approximate any random-utility model arbitrarily closely and is estimated by simulated maximum likelihood.

   Two cheaper, partial fixes are also worth naming, because they are what a consultant is most likely to do in practice. Market segmentation — estimating separate models for captive riders, choice riders, licence holders and non-holders, or by trip distance band — removes much of the offending heterogeneity, since IIA is far less damaging within a homogeneous segment. And adding alternative-specific constants and interaction terms (a distance–bike interaction, for example, so that the bike utility falls away sharply on long trips) at least prevents the model from predicting large bike shares where cycling is implausible. Whichever route is taken, the model should be tested for IIA before it is relied upon, using a Hausman–McFadden specification test: estimate the model on the full choice set and again with one alternative removed, and reject IIA if the coefficient vectors differ significantly.