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. Utility function V_i = −0.075 AT_i − 0.05 WT_i − 0.04 RT_i − 0.002 TC_i, with times in minutes and cost in cents, and the mode attributes below.

Find. The four mode shares under the multinomial logit model; the revised shares after the bike path cuts cycling riding time from 60 to 45 min; and an explanation of the IIA property together with the practical remedies for it.

Approach. Evaluate the systematic utility of each mode, exponentiate, and take each mode's share as its exponential divided by the sum over all four; then repeat with the single changed attribute and compare the two solutions to expose IIA numerically.

1. Part (a) — evaluate the systematic utilities. Substituting each row of the attribute table into the utility function, for automobile $$\begin{aligned}V_{auto} &= -0.075(6)-0.05(1)-0.04(25)-0.002(300) \\ &= -0.450-0.050-1.000-0.600=-2.100\end{aligned}$$ and in the same way V_bus = −3.220, V_rail = −2.375 and V_bike = −2.475. Only differences in utility matter, so the negative signs carry no meaning of their own; bus is the least attractive mode because it is penalised on all three time components at once.
2. Apply the multinomial logit share formula. The probability of choosing mode i from the set of four is $$P_i=\frac{e^{V_i}}{\sum_{j} e^{V_j}}$$ Exponentiating gives e^V = 0.122456 (auto), 0.039955 (bus), 0.093014 (rail) and 0.084163 (bike), summing to 0.339589. Dividing each by that total, $$\boxed{P_{auto}=36.06\ \%,\quad P_{bus}=11.77\ \%,\quad P_{rail}=27.39\ \%,\quad P_{bike}=24.78\ \%}$$ which sums to 100.00 per cent as it must.
3. Part (b) — re-evaluate the one utility that changes. The bike path reduces cycling riding time from 60 to 45 min, and nothing else changes, so $$V_{bike}'=-0.075(1)-0.05(0)-0.04(45)-0.002(0)=-0.075-1.800=-1.875$$ an improvement of 0.60 utility units, which raises the bike exponential from 0.084163 to e^−1.875 = 0.153355 — a factor of 1.822.
4. Recompute the shares with the new denominator. The other three exponentials are unchanged, so the sum becomes 0.122456 + 0.039955 + 0.093014 + 0.153355 = 0.408781 and $$\boxed{P_{auto}'=29.96\ \%,\quad P_{bus}'=9.77\ \%,\quad P_{rail}'=22.75\ \%,\quad P_{bike}'=37.52\ \%}$$ Cycling gains 12.73 percentage points, taken from automobile (−6.10), rail (−4.64) and bus (−1.99).

Part (c) — the IIA property, demonstrated on this paper's own numbers, and the remedies

What IIA says. The independence of irrelevant alternatives property states that for any two alternatives the ratio of their choice probabilities depends only on the attributes of those two alternatives and is completely unaffected by the presence, absence or attributes of any third alternative. It follows directly from the logit form, because the common denominator cancels:

This paper hands the demonstration to the candidate for nothing, and quoting the two numbers is far more convincing than describing the property in words. Between parts (a) and (b) only the bike utility changes, so the odds of automobile against rail must be identical before and after:

Equivalently, every mode whose utility did not change keeps its exponential and has its share rescaled by one common factor, 0.339589 / 0.408781 = 0.8307 — automobile from 36.06 to 29.96 per cent, bus from 11.77 to 9.77 and rail from 27.39 to 22.75, each an identical 16.93 per cent relative reduction. The model draws the new cyclists from the other three modes strictly in proportion to their existing shares.

Why that is unrealistic here. Nothing in the corridor's behaviour justifies proportional substitution. A 45-minute bike ride competes most directly with the 30-minute rail trip and the 40-minute bus trip — short, non-car, physically active journeys made by people who already accept an unprotected door-to-door time — and much less directly with the 25-minute automobile trip, whose users have already paid for a vehicle and revealed a preference for its comfort and flexibility. Behaviourally the unobserved components of utility for bus, rail and bike are correlated (all are affected by weather, personal security, the ability to carry goods, and the absence of a car in the household), while automobile's is not. The classic statement of the failure is the red-bus / blue-bus paradox: introducing a bus identical to an existing one except for its colour should split the bus share, but multinomial logit gives the new bus the same share as every other mode's proportional draw, inflating total bus patronage. Here the model predicts the bike path takes 6.10 points from the car, which almost certainly overstates the car diversion and understates the transit diversion — a serious matter if the project is being justified on greenhouse-gas reduction, since the emissions saving depends entirely on how much of the new cycling comes out of cars.

How to account for the limitation. Five practical routes, roughly in order of effort:

• Nested logit. Group the correlated alternatives into a nest — here a "non-auto" or "sustainable modes" nest containing bus, rail and bike, with automobile alone in its own nest — and estimate a logsum (inclusive-value) parameter for the nest. Substitution is then stronger within the nest than across it, so improving cycling draws mostly from bus and rail. The estimated logsum coefficient must lie between 0 and 1 for the model to be consistent with random-utility maximisation, and testing whether it differs significantly from 1 is itself the test of whether the nest is needed. This is the standard remedy in Canadian regional models.
• Cross-nested or paired combinatorial logit, when an alternative belongs partly to more than one group — for instance a park-and-ride mode that is both auto and rail.
• Multinomial probit or mixed (random-parameters) logit, which allow a full covariance matrix among the error terms or let the taste coefficients vary randomly across the population. These impose no nesting structure at all and are the most general remedy, at the cost of simulation-based estimation.
• Market segmentation. Estimate separate models by car availability, income, trip purpose and distance band. Much apparent IIA violation is unobserved heterogeneity: a zero-car household never substitutes between car and bike, and once the population is segmented the within-segment IIA assumption becomes far more defensible.
• Test before trusting. The Hausman–McFadden specification test re-estimates the model on a restricted choice set; if the coefficients shift significantly when an alternative is removed, IIA is rejected and a nested or mixed structure is required. Reporting that test is what distinguishes a defensible forecast from an assumed one.

This question is the numerical counterpart of Question 1(b) — the bike path is the travel demand management strategy discussed there — and its nested-logit remedy is the same idea as the stochastic-user-equilibrium and path-size-logit correction proposed in Question 6(c), where route overlap plays the role that mode correlation plays here.