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16-Civ-A6 Highway Design, Construction, and Maintenance · May 2015

Question 7 of 7: Logit Mode Choice and the IIA Property

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National Examination — 98-Civ-A6 Transportation Planning & Engineering, May 2015. Closed book (one two-sided aid sheet), 3 hours. Seven questions of equal value (20 marks); any five constitute a complete paper. All seven are solved here as a study resource.

Reference texts (subject):

Question 7: 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. Linear-in-attributes utilities with an alternative-specific constant of 0.35 for the car alternative and all-negative time and cost coefficients (time and cost reduce utility).

Given data
Mode$TT$ (min)$ET$ (min)$TC$ ($)
Car2551.40
Bus4080.50
Car-passenger (b)2550.70

Find. (a) binary logit shares; (b) three-mode multinomial shares; (c) IIA interpretation and remedy.

Approach. Compute each mode's systematic utility, then apply the logit share $P_i = e^{V_i}/\sum_j e^{V_j}$.

  1. (a) Utilities. $$V_{\text{car}}=0.35-0.025(25)-0.042(5)-0.01(1.40)=-0.499,$$ $$V_{\text{bus}}=-0.025(40)-0.042(8)-0.01(0.50)=-1.341.$$
  2. (a) Binary logit shares. $$P_{\text{car}}=\frac{e^{-0.499}}{e^{-0.499}+e^{-1.341}}=\frac{1}{1+e^{-0.842}}=0.699,\qquad P_{\text{bus}}=0.301.$$ $$\boxed{P_{\text{car}}=0.699,\quad P_{\text{bus}}=0.301.}$$
  3. (b) Split car into car-drive and car-passenger. Both keep the car constant and $TT=25$, $ET=5$; only cost differs: $$V_{\text{cd}}=-0.499,\qquad V_{\text{cp}}=0.35-0.625-0.210-0.01(0.70)=-0.492,\qquad V_{\text{bus}}=-1.341.$$
  4. (b) Multinomial shares. With $e^{V_{\text{cd}}}=0.607$, $e^{V_{\text{cp}}}=0.611$, $e^{V_{\text{bus}}}=0.262$ (sum 1.480): $$P_{\text{cd}}=0.410,\quad P_{\text{cp}}=0.413,\quad P_{\text{bus}}=0.177.$$ $$\boxed{P_{\text{car-drive}}=0.410,\ P_{\text{car-pass}}=0.413,\ P_{\text{bus}}=0.177.}$$
Question 7 — Mode shares
Mode(a) binary(b) three-mode MNL
Car (total)0.6990.823
  — car-drive—0.410
  — car-passenger—0.413
Bus0.3010.177

(c) Does (b) make intuitive sense? IIA and its remedy

No — the result is not intuitively reasonable. Merely re-labelling the single car option as two near-identical sub-options (car-drive and car-passenger) should not change the total attractiveness of travelling by car relative to the bus. Yet the combined car share jumps from 0.699 to 0.410 + 0.413 = 0.823, and the bus share falls from 0.301 to 0.177, even though nothing about the bus changed.

This is the classic symptom of the Independence of Irrelevant Alternatives (IIA) property of the multinomial logit: the ratio of any two modes' probabilities depends only on those two modes' utilities and is unaffected by the presence of others. Here $P_{\text{cd}}/P_{\text{cp}}$ is fixed, but adding the correlated "extra car" alternative steals share from the bus in proportion to its own share — the well-known red-bus/blue-bus problem. Car-drive and car-passenger share unobserved attributes (they are both "car"), violating the logit's assumption of independent random error terms.

Remedy. Use a model that allows correlation among similar alternatives: a nested logit that places car-drive and car-passenger in a common "car" nest (with the bus in its own nest) so that splitting the nest does not draw share from the bus; or a more general cross-nested / mixed (random-parameters) logit or probit that relaxes IIA directly. Practically, nesting the two car sub-modes restores the sensible result that total car share stays near 0.70.

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