Lecture 6 - Advanced Choice Models
ENCI707: Engineering Demand and Policy Analysis
Discrete Choice: Choice Bundle
- Choice bundle, for example choice of making a shopping trip at a particular location:
- Choice of making a shopping trip compared with no trip or any other trips.
- Choosing a mode of travel to reach a destination.
- Choosing a particular location over other possible locations.
- Different substitution patterns may exist in three components of shopping trip choice.
- Overall, a single discrete choice but multiple components of choices.
- Individual component have no meaning
Discrete Choice: Choice Bundle
- Choice of Park & Ride as a mode to make commuting trip:
- Choice of Park & Ride station location
- Choice of Park & Ride over all other possible modes, e.g., drive all way, car passenger, transit all way
- Different substitution patterns may exist between choices of modes as well as between alternative Park & Ride station locations
- Overall, a single discrete choice but multiple components of choices
- Individual component have no meaning
Sequential Logit Model of Choice Bundle
- Examples: Mode (\(m\)) and Destination Choice (\(d\)):
- Choose destination
- Then choose mode, given (conditional upon) the destination choice
- Sequentially modelling the choices of alternatives step by step
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Sequential Logit Model of Choice Bundle
\[Pr(d) = \frac{\exp(V_d)}{\sum_{d' \in D} \exp(V_d')} \text{, } Pr(m|d) = \frac{\exp(V_{m|d})}{\sum_{m' \in M} \exp(V_{m'|d})}\] - Joint probability of choice \[Pr(md) = \frac{\exp(V_d)}{\sum_{d' \in D} \exp(V_d') \sum_{m' \in M} \exp(V_{m'|d})}\] \[Pr(md) = \frac{\exp(V_d + V_{m|d})}{\sum_{d' \in D} \exp(V_{d'})} \frac{\exp(V_{m|d})}{\sum_{m' \in M} \exp(V_{m'|d})}\]
Simultaneous Logit Model of Choice Bundle
- Restructure the sequential decisions into a finite number of alternative pairs
- Alternative pairs are mutually exclusive and collectively exhaustive: single discrete choice case
- The example of mode(m) and destination(d) choice:
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\[Pr(md) = \frac{\exp(V_d + V_{m|d})}{\sum_{d' \in D}\sum_{m' \in M} \exp(V_{d'} + V_{m'|d})}\]
Simultaneous Logit Model of Choice Bundle
- Unlike sequential model, simultaneous model incorporates expectations of conditional level utility at the conditioning levels:
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- Systematic utility of destination choice: \[\widetilde{V_d} = V_d + \text{{Expected Maximum Utility of choice of mode}}_d\]
- Scale parameters are same (unity) at both levels
Nested Logit Model
\[U_{m|d} = V_{m|d} + \epsilon_{m|d} \text{ (Lower-level conditional utility function)}\] \[\begin{align}
U_{d} = V_d + (V_{m|d} + \epsilon_{m|d}) + \epsilon_d = \widetilde{V_d} + \epsilon_d & \text{ (Upper-level unconditional} \\
& \text{utility function for a pair of d-m)}
\end{align}\] \[\begin{align}
\widetilde{V_d} = V_d + (V_{m|d} + \epsilon_{m|d}) & \text{ (Upper-level compounded} \\
& \text{systematic utility for a pair of d-m)}
\end{align}\] \[\epsilon_d \sim\text{IID EV Type I with scale, }\mu_1\text{; }Var(\epsilon_d)=\pi^2/6\mu^2_d\] \[\epsilon_{m|d} \sim\text{IID EV Type I with scale, }\mu_2\text{; }Var(\epsilon_{m|d})=\pi^2/6\mu^2_m\]
Nested Logit Model: Elasticities
Nested Logit Model: Elasticities
IID and Nested Logit Model
- Consider a route choice problem: In network, multiple alternative routes having common link will have correlated properties.
- Example: 2 routes with no partial overlap. Under equilibrium, both have the same average travel time: \(t_1=t_2=t\)
- Consider: systematic utility of choosing a route: \(V_r=V_1=V_2=-t\)
- Logit choice probability: \[Pr(route1) = Pr(route2) = \frac{\exp(V_1)}{\exp(V_1)+\exp(V_2)} = \frac{\exp(-t)}{\exp(-t)+\exp(-t)} = 0.5\]
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IID and Nested Logit Model
- Add a 3rd route that has partial overlap with route 2
- Under equilibrium, all have the same average travel time: \(t_1=t_2=t_3=t\)
- Consider: systematic utility of choosing a route: \(V_1=V_2=V_3=-t\)
- Logit choice probability: \[Pr(routeX) = \frac{\exp(V_X)}{\exp(V_1)+\exp(V_2)+\exp(V_3)} = \frac{\exp(-t)}{\exp(-t)+\exp(-t)+\exp(-t)} = 0.33\]
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- Route 2 and 3 are very similar!
IID and Nested Logit Model
- Let us change the utility structure \[U_1 = V_1 + \epsilon_1\] \[U_2 = V_2 + w + \epsilon_2\] \[U_3 = V_3 + w + \epsilon_3\]
- Commonalities between 2 and 3 by common \(w\) lead to correlation with respect to common \(\epsilon\).
- Further assumption:
- \(\epsilon\) are IID Type I EV, with scale \(\mu_1\)
- \(w\) are IID Type I EV, with scale \(\mu_2\)
- Normalizing \(\mu_1 = 1\) and defining \(\phi=1/\mu_2\)
Nested Logit Model for Single Choice
\[\text{For } \phi=0 \text{; }Pr(1)=\frac{1}{1+2^{\phi}} = 1/2\] \[\text{For } \phi=1 \text{; }Pr(1)=\frac{1}{1+2^{\phi}} = 1/3\] \[\text{For } 0<\phi<1 \text{; }Pr(1)=\frac{1}{1+2^{\phi}} < 1/2\]
- Nested Logit (NL) model can capture various levels of correlated properties (substitution patterns) among alternatives of a single choice (as opposed to choice bundle) through the nesting structure:
- Within a particular nest*, alternatives are IIA
- Between nests, various levels of substitution patterns are captured through nesting structures
Nested Logit to Address IIA
Variance-Covariance Matrix of Utility Functions (assuming unit scale) \[\frac{\pi^2}{6} \begin{bmatrix} 1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 1 \end{bmatrix}\text{; }Pr(j) = \frac{\exp(V_j)}{\sum_{k=1}^7 \exp(V_k)}\]
Nested Logit to Address IIA
Nested Logit to Address IIA
