CIVL2700 · Transport Systems
Discrete Choice Modelling (Logit)
Discrete choice modelling is the Week 3 demand-side engine of CIVL2700 Transport Systems at the University of Sydney: it predicts what fraction of travellers pick each mode, route or departure time from a set of alternatives. The multinomial-logit (MNL) model turns each option's systematic utility into a market share, and its defining property — Independence of Irrelevant Alternatives (IIA) — is both its strength and its limit. This chapter feeds directly into travel-demand forecasting and is examinable in In-class Test 1 and the comprehensive final exam.
What this chapter covers
- 01Random utility theory: total utility U = V + ε, with an observed part and an unobserved Gumbel error
- 02Systematic utility V written linear-in-parameters from cost, time and an alternative-specific constant
- 03Sign of the taste coefficients (β_cost, β_time < 0) and the value of time β_time/β_cost
- 04The multinomial-logit choice probability P(i) = e^V_i / Σ e^V_j
- 05Only differences in utility matter; fixing one alternative-specific constant to zero
- 06The logit S-curve: how share responds smoothly to a utility difference
- 07Deterministic (all-or-nothing) versus probabilistic (logit) mode split from the same utilities
- 08Independence of Irrelevant Alternatives (IIA) and the odds ratio e^(V_i - V_j)
- 09The red-bus / blue-bus paradox and when nested logit is needed
- 10Turning shares into headcounts and reading which split the exam wants
Binary logit mode split, then a third mode (IIA check)
- +2Systematic utilities. V(car) = 0.50 − 0.04·6 − 0.03·25 = 0.50 − 0.24 − 0.75 = −0.49 utils. V(bus) = 0 − 0.04·3 − 0.03·40 = −0.12 − 1.20 = −1.32 utils.
- +2Exponentiate and sum. e^(−0.49) = 0.6126, e^(−1.32) = 0.2671; sum = 0.8797. Divide across the whole choice set, not just the other mode.
- +2Logit shares and headcounts. P(car) = 0.6126/0.8797 = 0.696 (≈ 1393 commuters); P(bus) = 0.2671/0.8797 = 0.304 (≈ 607). Shares sum to 1.000.
- +1Deterministic split. V(car) = −0.49 > V(bus) = −1.32, so all-or-nothing sends all 2000 to the car and 0 to the bus.
- +2Add the train. e^(−0.99) = 0.3716; new sum = 0.6126 + 0.2671 + 0.3716 = 1.2513. P(car) = 0.490, P(bus) = 0.213, P(train) = 0.297.
- +1IIA check. P(car)/P(bus) = 0.490/0.213 = 2.29 = e^(−0.49−(−1.32)) = e^0.83 — identical to the two-mode odds, so IIA holds.
Key terms
- Systematic utility (V)
- The observed, measurable part of an alternative's utility, written linear-in-parameters as an alternative-specific constant plus weighted attributes, e.g. V = β0 + β_cost·cost + β_time·time. Measured in dimensionless 'utils'; only differences between alternatives affect choice.
- Random utility (ε)
- The unobserved part of utility — comfort, habit, taste — that the modeller cannot measure. Assuming these errors are independent and identically distributed Gumbel (Type I extreme-value) is exactly what produces the logit model.
- Multinomial logit (MNL)
- The model giving each alternative's choice probability as P(i) = e^V_i / Σ_j e^V_j, summed over every alternative in the choice set. Shares always lie between 0 and 1 and add to 1.
- Alternative-specific constant (ASC)
- The intercept β0 in an alternative's utility, capturing average preference for that mode beyond its measured attributes. Because only utility differences matter, one ASC is fixed to zero as the reference alternative.
- Independence of Irrelevant Alternatives (IIA)
- The logit property that the odds of choosing i over j, P(i)/P(j) = e^(V_i − V_j), depend only on i and j and not on any other alternative. Adding a mode rescales all shares but preserves every pairwise odds ratio.
- Deterministic choice
- The all-or-nothing rule: with no random error, every traveller picks the single maximum-utility alternative, so 100% of demand goes to it. A step function — simple but sensitive to tiny utility changes.
- Value of time (VoT)
- The ratio of the time coefficient to the cost coefficient, β_time/β_cost, expressed in $/min (or $/h) — how much money a traveller implicitly places on saving one minute of travel time.
- Red-bus / blue-bus paradox
- The classic failure of IIA: adding a near-identical alternative (a second bus of a different colour) makes plain logit wrongly draw share from unrelated modes such as car. Nested logit corrects this by grouping close substitutes.
Discrete Choice Modelling (Logit) FAQ
Why can systematic utility be negative in a logit model?
Because utility is measured on an arbitrary scale where only differences between alternatives matter. Adding the same constant to every V cancels in the ratio P(i) = e^V_i / Σ e^V_j, so a negative V is perfectly valid. e^V is always positive, so the shares are still genuine probabilities between 0 and 1. If a share ever falls outside that range, the exponentials were summed wrong, not a sign chosen wrong.
What is the difference between deterministic and probabilistic (logit) choice?
They use the same systematic utilities but differ in what they do with them. Deterministic (all-or-nothing) choice sends 100% of demand to the single highest-utility alternative — a step that can flip with a tiny change. Probabilistic logit spreads demand smoothly in proportion to e^V, so the better mode wins the larger share while weaker modes keep a non-zero share, reflecting unobserved differences in taste.
Can AI help me with discrete choice modelling in CIVL2700?
Yes — Sia can explain it step by step. Paste a mode-choice question and Sia will walk through writing each utility, exponentiating over the whole choice set, dividing to get the logit shares, and checking the IIA odds ratio, so you learn the method for the exam. Sia does not sit your assessments or promise a grade; it is a study aid that builds your own working, which you should confirm against your Canvas materials.
Exam move
Drill the three-move routine until it is automatic: write each alternative's systematic utility V with the correct negative cost and time coefficients, exponentiate and sum e^V across the whole choice set, then divide to get every share. Practise converting shares to headcounts and be ready to give the deterministic (all-or-nothing) split from the same utilities — exam parts often ask for both. Memorise the IIA odds ratio P(i)/P(j) = e^(V_i − V_j) and the red-bus/blue-bus limitation, since 'explain a property' questions reward it. Because a logit part is method-credit marked, always show your e^V values so an arithmetic slip costs one mark, not the whole question. The final exam runs 2.5 hours; budget time in proportion to marks (about 1.5 minutes per mark on a 100-mark paper) and confirm the exact date, room and open/closed-book status on Canvas and the University exam timetable.
Working through Discrete Choice Modelling (Logit) in CIVL2700? Sia is AskSia’s AI Engineering tutor — ask any CIVL2700 Discrete Choice Modelling (Logit) question and get a clear, step-by-step explanation grounded in how CIVL2700 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
