Transportation System Analysis
Assignment #2
Problem 1
Given systematic utilities (car-lover, love_car = 1):
V_car = 2 − 0.4·10 − 0.1·12 + 1.5 = −1.7
V_transit = 1 − 0.4·3 − 0.1·20 = −2.2
V_LR = 1.5 − 0.4·5 − 0.1·10 = −1.5
(1) Uniform error difference
Car is chosen if:
U_car > U_transit
⇔ ε_transit − ε_car < V_car − V_transit = 0.5
Given ε_transit − ε_car ~ Uniform[−10, 10],
P(car) = (0.5 − (−10)) / 20 = 0.525
(2) Gumbel errors (binary logit)
With i.i.d. Gumbel(0,1) errors, the choice probability follows logit:
P(car) = exp(V_car) / [exp(V_car) + exp(V_transit)]
= exp(−1.7) / [exp(−1.7) + exp(−2.2)]
≈ 0.622
(3a) Add Light Rail: Multinomial Logit
With three alternatives and i.i.d. Gumbel errors:
P(i) = exp(V_i) / Σ_k exp(V_k)
Denominator:
exp(−1.7) + exp(−2.2) + exp(−1.5)
Mode shares:
P(car) ≈ 0.366
P(LR) ≈ 0.448
P(transit) ≈ 0.186
Expected maximum utility (log-sum):
EU = ln[exp(−1.7) + exp(−2.2) + exp(−1.5)]
Change in expected utility:
ΔEU = ln[exp(−1.7) + exp(−2.2) + exp(−1.5)]
− ln[exp(−1.7) + exp(−2.2)]
≈ 0.49
(3b) Correlation discussion
The independence assumption is not realistic.
Transit and light rail are more correlated because they share similar service,
, access, comfort, and delay characteristics, while car is structurally different.
(3c) Nested Logit
Nest 1: Transit and Light Rail, scale λ₁ = 2
Nest 2: Car only, scale λ₂ = 1
β=1
Inclusive value for Nest 1:
IV₁ = ln[exp(2·V_transit) + exp(2·V_LR)]
Nest choice probabilities:
P₁ = exp(IV₁ / 2) / [exp(IV₁ / 2) + exp(V_car)]
P₂ = 1 − P₁
Conditional probabilities within Nest 1:
P(LR | 1) = exp(2·V_LR) / [exp(2·V_LR) + exp(2·V_transit)]
P(transit | 1) = 1 − P(LR | 1)
Final mode shares:
P(car) ≈ 0.40
P(LR) ≈ 0.42
P(transit) ≈ 0.18
Problem 2
The plan is to analyze the change of people’s mode preference due to COVID-19
pandemic. I select the mixed-logit model as the discrete choice model form. People’s
behavior change caused by the pandemic could varies substantially according to
personal attributes as well as different modes’ characteristic. Therefore, people with
different background tend to have different respond to mode preference during the
pandemic. By using mixed-logit model, we are able to capture taste variance among
population. As for attributes, both individual specific and alternative specific
attributes are considered. In the meantime, the attributes are split into categorical and
continuous variables.
(1) Research Question
The impact of COVID-19 on people’s mode choice preference
(2) Model Format
Mixed-Logit Model
(3) Attributes
a. Mode alternatives
Walk
Bike
Car
Boat
Assignment #2
Problem 1
Given systematic utilities (car-lover, love_car = 1):
V_car = 2 − 0.4·10 − 0.1·12 + 1.5 = −1.7
V_transit = 1 − 0.4·3 − 0.1·20 = −2.2
V_LR = 1.5 − 0.4·5 − 0.1·10 = −1.5
(1) Uniform error difference
Car is chosen if:
U_car > U_transit
⇔ ε_transit − ε_car < V_car − V_transit = 0.5
Given ε_transit − ε_car ~ Uniform[−10, 10],
P(car) = (0.5 − (−10)) / 20 = 0.525
(2) Gumbel errors (binary logit)
With i.i.d. Gumbel(0,1) errors, the choice probability follows logit:
P(car) = exp(V_car) / [exp(V_car) + exp(V_transit)]
= exp(−1.7) / [exp(−1.7) + exp(−2.2)]
≈ 0.622
(3a) Add Light Rail: Multinomial Logit
With three alternatives and i.i.d. Gumbel errors:
P(i) = exp(V_i) / Σ_k exp(V_k)
Denominator:
exp(−1.7) + exp(−2.2) + exp(−1.5)
Mode shares:
P(car) ≈ 0.366
P(LR) ≈ 0.448
P(transit) ≈ 0.186
Expected maximum utility (log-sum):
EU = ln[exp(−1.7) + exp(−2.2) + exp(−1.5)]
Change in expected utility:
ΔEU = ln[exp(−1.7) + exp(−2.2) + exp(−1.5)]
− ln[exp(−1.7) + exp(−2.2)]
≈ 0.49
(3b) Correlation discussion
The independence assumption is not realistic.
Transit and light rail are more correlated because they share similar service,
, access, comfort, and delay characteristics, while car is structurally different.
(3c) Nested Logit
Nest 1: Transit and Light Rail, scale λ₁ = 2
Nest 2: Car only, scale λ₂ = 1
β=1
Inclusive value for Nest 1:
IV₁ = ln[exp(2·V_transit) + exp(2·V_LR)]
Nest choice probabilities:
P₁ = exp(IV₁ / 2) / [exp(IV₁ / 2) + exp(V_car)]
P₂ = 1 − P₁
Conditional probabilities within Nest 1:
P(LR | 1) = exp(2·V_LR) / [exp(2·V_LR) + exp(2·V_transit)]
P(transit | 1) = 1 − P(LR | 1)
Final mode shares:
P(car) ≈ 0.40
P(LR) ≈ 0.42
P(transit) ≈ 0.18
Problem 2
The plan is to analyze the change of people’s mode preference due to COVID-19
pandemic. I select the mixed-logit model as the discrete choice model form. People’s
behavior change caused by the pandemic could varies substantially according to
personal attributes as well as different modes’ characteristic. Therefore, people with
different background tend to have different respond to mode preference during the
pandemic. By using mixed-logit model, we are able to capture taste variance among
population. As for attributes, both individual specific and alternative specific
attributes are considered. In the meantime, the attributes are split into categorical and
continuous variables.
(1) Research Question
The impact of COVID-19 on people’s mode choice preference
(2) Model Format
Mixed-Logit Model
(3) Attributes
a. Mode alternatives
Walk
Bike
Car
Boat
