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 competing modes with calibrated linear-in-parameters utility functions sharing the same time and cost coefficients, and mode-specific constants of 1.1 (auto), 0.1 (bus) and 0 (light rail, the reference mode).

Find. The three mode shares before and after a bus fare reduction of one dollar, and an explanation of the modelling assumption that makes the predicted response implausible together with the remedies available.

Approach. Evaluate each mode's systematic utility, exponentiate, and divide by the sum over all three alternatives; repeat with the reduced bus fare and compare the pattern of the changes rather than only their magnitudes.

1. (a) Evaluate the systematic utilities. Substituting the attributes into each utility function, $$V_a=1.1-0.05(16)-0.25(3.50)=1.1-0.80-0.875=-0.575$$ $$V_b=0.1-0.05(30)-0.25(2.00)=0.1-1.50-0.50=-1.900$$ $$V_r=0-0.05(25)-0.25(2.50)=-1.25-0.625=-1.875$$ Note that light rail already out-scores bus despite having no constant — it is five minutes faster, and the 0.50 dollar fare premium does not offset that.
2. Exponentiate and normalise. The multinomial logit choice probability is $$P_i=\frac{e^{V_i}}{\displaystyle\sum_{j\in C}e^{V_j}}$$ with $$e^{V_a}=0.562705,\quad e^{V_b}=0.149569,\quad e^{V_r}=0.153355,\quad \textstyle\sum_j e^{V_j}=0.865629$$
3. Report the base-case shares. Dividing each exponential by the sum, $$\boxed{P_a=0.6501\;(65.01\%),\quad P_b=0.1728\;(17.28\%),\quad P_r=0.1772\;(17.72\%)}$$ The three shares sum to 1.0000, as they must. Auto dominates because its mode-specific constant of 1.1 — capturing comfort, flexibility and privacy — more than compensates for its higher money cost.
4. (b) Recompute the bus utility at the lower fare. Only the bus cost changes, from 2.00 to 1.00 dollars: $$V_b'=0.1-0.05(30)-0.25(1.00)=0.1-1.50-0.25=-1.650$$ $$e^{V_b'}=0.192050, \qquad \textstyle\sum_j e^{V_j}=0.562705+0.192050+0.153355=0.908110$$ The fare cut is worth \(0.25(1.00)=0.25\) utility units, equivalent by the time coefficient to \(0.25/0.05=5\) minutes of travel time saved.
5. Report the post-reduction shares. $$\boxed{P_a'=0.6196\;(61.96\%),\quad P_b'=0.2115\;(21.15\%),\quad P_r'=0.1689\;(16.89\%)}$$ Again the shares sum to 1.0000. The bus gains 3.87 percentage points, a 22.4 per cent increase in its own ridership; auto loses 3.04 points and light rail loses 0.83 points.
6. Expose the structure of the change. Because only the bus attribute changed, the numerators for auto and rail are unaltered and both are divided by the same new denominator. Every unchanged mode is therefore rescaled by the identical factor $$\frac{\sum_j e^{V_j}}{\sum_j e^{V_j'}}=\frac{0.865629}{0.908110}=0.9532$$ and the odds ratio between any two unchanged modes is preserved exactly: $$\frac{P_a}{P_r}=\frac{0.6501}{0.1772}=3.6693 \qquad\text{and}\qquad \frac{P_a'}{P_r'}=\frac{0.6196}{0.1689}=3.6693$$ This numerical identity is the independence of irrelevant alternatives, and it is what part (c) asks about.

(c) The IIA assumption, why it fails here, and what to do about it

The assumption. The multinomial logit model assumes the random components of utility, \(\varepsilon_i\), are independently and identically Gumbel-distributed across alternatives. Independence means no unobserved attribute is shared by any two modes; identical distribution means every mode has the same error variance. Together they produce the independence of irrelevant alternatives: the ratio of the choice probabilities of any two alternatives depends only on their own utilities and is completely unaffected by the existence or the attributes of any third alternative.

Why the part (b) result is implausible. IIA forces the bus fare cut to draw its new riders from auto and light rail in exact proportion to their existing shares. The arithmetic above shows this literally: auto and rail are both multiplied by 0.9532, so auto surrenders 3.04 points and rail only 0.83 points — auto supplies 78.6 per cent of the bus's gain. Behaviourally this is backwards. Bus and light rail are both public transit: they share unobserved attributes (waiting and transfer disutility, exposure to weather, no door-to-door capability, dependence on a published schedule, the same fare-card system, often the same corridors and interchanges), so their error terms are strongly correlated and they are close substitutes. A one-dollar fare cut on the bus should draw predominantly from light rail — the mode a transit user can most easily switch to — and only marginally from car commuters, whose choice is anchored by a large mode-specific constant and by long-run decisions about vehicle ownership and residential location. The model as specified therefore overstates the auto diversion, understates the rail diversion, and consequently overstates the net congestion and emissions benefit of the fare policy. This is the classic red-bus / blue-bus problem in its practical form: adding or improving one member of a correlated group cannibalises that group, not the whole market.

How to account for it. Several remedies, roughly in order of practicality:

• Nested logit. Group bus and light rail into a "transit" nest with auto in its own nest. Choice is then modelled as a transit-versus-auto decision at the upper level and a bus-versus-rail decision within the nest. The nest's logsum parameter \(\theta\) (with \(0 < \theta \le 1\)) measures the correlation: a value near zero means the two transit modes are near-perfect substitutes and almost all of the bus's gain comes from rail, while \(\theta = 1\) collapses the model back to multinomial logit. IIA is retained within each nest but relaxed between nests, which is exactly the structure this problem needs. This is the standard practical fix and requires only re-estimation, not new data.
• Cross-nested or paired combinatorial logit, when an alternative belongs partly to more than one group — for example a park-and-ride option that is partly auto and partly transit.
• Mixed (random-parameters) logit, which lets the coefficients vary randomly across the population and permits any substitution pattern; it also captures taste heterogeneity, so a fare cut can matter far more to low-income travellers than to others.
• Multinomial probit, which allows a full covariance matrix on the error terms and so removes the IIA restriction entirely, at the price of simulation-based estimation and much heavier computation.
• Market segmentation — estimating separate models by income, vehicle availability, or trip length — which reduces the unobserved heterogeneity that causes the correlation in the first place, and is often the cheapest partial remedy.
• Testing before trusting. The Hausman–McFadden specification test re-estimates the model on a restricted choice set (say, dropping light rail); if the coefficients shift significantly, IIA is rejected and a nested structure is required. This test should be run as a matter of routine before any elasticity or policy forecast is quoted from a multinomial logit model.

For this corridor the recommended answer is a nested logit with a transit nest containing bus and light rail, re-estimated on the same revealed-preference data, with the fare-cut forecast re-run and the difference in auto diversion reported to the client as the measure of the specification's importance. Cross-referring to Question 1(c), the practical implication is that a bus fare reduction is a weak tool for raising average vehicle occupancy on this corridor — it mainly reshuffles existing transit riders — whereas parking pricing or a congestion charge acts directly on the auto alternative's utility, which is where the auto share actually comes from.