Introduction (I)
The table below shows transit’s mode share in the urban US (all trip
purposes) and the 10 urban areas where it is most popular (2008 data):
Modern Mode Choice Analysis
Area
Share (%)
Urban US
Philip A. Viton
New York
San Francisco
Washington DC
Chicago
Honolulu
Boston
Seattle
Philadelphia
Portland
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Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
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Introduction (II)
CRP 4110 —()Mode Choice
11.0
5.0
4.5
3.9
3.8
3.3
2.8
2.7
2.3
April 24, 2015
2 / 52
Introduction (III)
Clearly, one major problem facing transit (and transportation
planners) is that not many people use it.
Can anything be done about this? Is there a way to make transit
more attractive to potential users?
To answer this question we need to look at what determines people’s
choices of transportation modes, for a speci…c trip purpose.
This is the mode choice analysis problem.
We want to look at the determinants of choice as a function of things
that we as planners can potentially change: it’s not helpful to say
that people choose their cars because of deep psychological
structures, because, if that’s so, it’s not likely that the transport
planner can do very much to change it.
Philip A. Viton
Philip A. Viton
1.6
CRP 4110 —()Mode Choice
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Until relatively recently, mode choice studies were dominated by
aggregate approaches: researchers would try to explain the fraction
(share) of people in a city who chose a given mode (for example, the
share using transit) based on average observed modal characteristics
in the city, average disposable personal income, etc.
But two developments have changed this: we now have:
1. Extensive surveys of individuals.
2. The computing power needed to analyze this individual data.
So nowadays it is usual to focus on modeling and understanding
behavior at the individual level.
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
4 / 52
The Setting
Discrete Choice
Consider one individual making a single trip between an origin and a
destination for a given purpose (say, work).
She has available a variety of modes on which she could make her
trip: drive alone, carpool, bus, walk, etc. These are her Choice Set.
Though our interest here is in mode choice, the discrete choice setting is
applicable in a wide variety of decision-making contexts. Examples:
Choice of a principal place (location) to live, or for a vacation home.
Choice of a vacation destination.
From her list of available modes (her choice set) she must select
exactly one to use for her trip. (Technically, this may involve
rede…nitions of the modes, in order to have the trip made on a single
mode).
Choice of a car/major appliance model to purchase.
Choice of a university (from among those where you’ve been
accepted) to attend.
This setting — select exactly one from a list of options — de…nes the
discrete choice setting.
Appellate judge: vote to a¢ rm or reverse the lower court decision.
Discrete choice is contrasted with the more usual setting (called
continuous choice) in which individuals can make multiple selections,
and in fractional quantities, from the alternatives facing them.
Government agency: decide where to build a new freeway, or locate a
waste dump or a library branch, from among a list of alternative
possibilities.
Philip A. Viton
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
5 / 52
Individual Behavior
City council member: vote for or against a project/plan.
CRP 4110 —()Mode Choice
April 24, 2015
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Structure of Utility
We study the behavior of a utility-maximizing, price-taking individual
i.
An individual’s utility depends on:
The individual faces J discrete alternatives j = 1, 2, . . . , J.
Factors observable to the analyst (eg, prices of alternatives).
Conditional utility: if individual i selects alternative j he/she gets
utility uij .
Factors unobservable to the analyst, but known to the decision-maker
(eg, whether the individual’s last experience with a particular
alternative was good/bad).
Alternative interpretation: uij represents i’s overall summary of the
attractiveness of mode j (to him or her) for this trip purpose.
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
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Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
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The Unobservable Part of Utility
Detailed Structure of Utility
To do this, we now assume: for individual i, alternative j’s total utility:
Consider an individual making repeated choices in a setting where:
Has an observable (systematic) component vij .
the choice set does not change
the observable factors do not change
Are we surprised if the individual makes di¤erent choices each time?
So (total) utility uij is represented as:
Clearly not: the reason is the in‡uence of the unobservables, which
may be changing even though we can’t observe them.
But conditional on the observables, the individual’s decisions vary for
no observable reason: they appear to an observer/analyst to have an
element of randomness to them.
We would like our models to re‡ect this seeming randomness.
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
9 / 52
Implications for the Study of Choice
The random variable (η ij ) represents the analyst’s ignorance of some
of the factors in‡uencing individual i: it does not imply that the
individual behaves randomly (from his/her own perspective).
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
10 / 52
We therefore de…ne:
Pij = Pr[individual i chooses alternative j]
Next, because of the random component η ij , we note that the total
utility uij is also a random variable.
This means that we can analyze only the probability that total utility
takes on a given value.
Hence, we as analysts can study only an individual’s choice
probability: the probability that the individual makes a given choice.
CRP 4110 —()Mode Choice
uij = vij + η ij
Choice Probabilities
It is the random component in utility that allows the analyst to
account for the apparent randomness exhibited by individuals in
situations where the observables do not change.
Philip A. Viton
Has an unobservable (idiosyncratic) component that we represent by
a random variable η ij distributed independently of vij .
April 24, 2015
11 / 52
By utility maximization, i will select alternative j if it yields the
highest utility.
So we have, equivalently:
Pij = Pr[ alternative j has highest utility, for individual i]
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
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Choice Probabilities — The Binary Case (I)
Choice Probabilities — The Binary Case (II)
It may be easier to see what is going on if we begin with the binary
case, where the individual has only two modes, say auto (mode 1)
and transit (mode 2).
In this case the mode-1 choice probability is:
Pi 1 = Pr[individual i chooses alternative 1]
Now expand the utilities into their systematic and random
components:
Pi 1 = Pr[vi 1 + η i 1 vi 2 + η i 2 ]
Reverse the direction of the inequality:
= Pr[alternative 1 has highest utility, for individual i ]
(and of course we know that Pi 2 = 1
Pi 1 = Pr[vi 2 + η i 2 < vi 1 + η i 1 ]
Pi 1 ) .
and collect the random terms on the left, and the systematic terms on
the right:
Pi 1 = Pr[η i 2 η i 1 vi 1 vi 2 ]
Write Pi 1 in terms of the utilities:
Pi 1 = Pr[ui 1
ui 2 ]
(where ties in the utilities are assumed to be settled in favor of mode
1).
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
13 / 52
Choice Probabilities — The Binary Case (III)
Philip A. Viton
ηi 1
vi 1
Pi1
vi 2 ]
G(ηi2−ηi1)
We recognize this as the cumulative distribution function of the
compound random variable η i 2 η i 1 evaluated at the “point”
vi 1 vi 2 .
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
14 / 52
Choice Probabilities — The Binary Case (IV)
We see that in binary choice, individual i’s choice probability for
mode 1 can be written as:
Pi 1 = Pr[η i 2
CRP 4110 —()Mode Choice
April 24, 2015
vi1-vi2
15 / 52
Philip A. Viton
Geometrically, individual i’s
mode-1 choice probability Pij is
the cumulative distribution
function G (η i 2 η i 1 ) of the
compound random variable
η i 2 η i 1 evaluated at the
“point” vi 1 vi 2 .
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April 24, 2015
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Choice Probabilities — The General Case (I)
Choice Probabilities — The General Case (II)
The general case is done in the same way, though the expressions are
more complicated, since there are now J modes/alternatives.
Alternative j is better than alternative k if uij
So
uik .
alternative j is better than alternative 1 AND
alternative j is better than alternative 2 AND
. . . AND . . .
alternative j is better than the last alternative J
(this is J
= Pr[individual i chooses alternative j]
= Pr[alternative j has the highest utility for individual i]
= Pr[uij ui 1 & uij ui 2 & . . . & uij uiJ ]
Pij
Alternative j is best if:
(J
1 conditions).
1 conditions).
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
17 / 52
Choice Probabilities — The General Case (III)
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
18 / 52
Choice Probabilities — The General Case (IV)
Now expand each utility into its systematic and random components:
Pij = Pr[vij + η ij
vi 1 + η i 1 , . . . , , vij + η ij
viJ + η iJ ]
So we have:
Reverse the direction of each of the inequalities:
Pij = Pr[vi 1 + η i 1
vij + η ij , . . . , viJ + η iJ
Pij = Pr[η i 1
vij + η ij ]
(J
η ij
vij
vi 1 , . . . , η iJ
η ij
vij
vij
vi 1 , . . . , η iJ
η ij
vij
viJ ]
We recognize this as saying that the choice probabilities are the joint
cumulative distribution function of the J 1 random variables
η ik η ij (k 6= j ) evaluated at the “points” vij vik .
and collect up terms (random on the left) in each inequality:
Pij = Pr[η i 1
η ij
viJ ]
1 terms in each expression; decide ties in favor of alternative j).
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
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Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
20 / 52
The Logit Model (I)
The Logit Model (II)
Any assumption about the joint distribution of the random variables
(the η ij ) will in principle give rise to a parametric choice model.
For our binary choice case (2 alternatives/modes) the logit model is:
Pi 1 =
But one model has dominated the literature.
Suppose the η ij are independently and identically distributed (i.i.d)
random variables with Type-1-Extreme-Value distributions (see
Appendix).
Then the choice probabilities are given by:
Pij =
e vij
∑Jk =1
In the numerator we have exp[vi 1 ] for the mode (here, mode 1) whose
choice probability we’re focussing on.
In the denominator we have the sum of the exp[vim ] for all the modes
(here, 2 of them) in the choice set.
The general case (J modes) is exactly the same, so we have a sum of
J terms in the denominator.
e vik
The important point here is that once you know the vij is is very easy
to calculate the choice probabilities Pij . So we turn now to the vij .
known as the Logit model of discrete choice.
Philip A. Viton
e vi 1
e vi 1 + e vi 2
CRP 4110 —()Mode Choice
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Structure of Systematic Utility
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
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Data (I)
Suppose alternative j has K observable characteristics (as perceived
by individual i): (xij 1 , xij 2 , . . . , xijK ) = xij
We want to estimate β from data on a sample of individuals.
It is conventional to assume that systematic utility vij is a
linear-in-parameters function of the xij :
vij
= xij 1 β1 + xij 2 β2 +
= xij β
The simplest assumption is that we have a random sample.
For each individual we observe the complete set of characteristics (the
xij ) of the alternatives, whether chosen or not.
+ xijK βK
We also observe a choice indicator, de…ned as:
So the logit choice probabilities become:
Pij =
yij =
e xij β
1 if individual i chooses alternative j
0 otherwise
∑Jk =1 e xik β
in which the only unknown is the weighting vector β.
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
23 / 52
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
24 / 52
Data (II)
Estimation
Indiv
1
2
..
.
Philip A. Viton
Choice
y
y11
y12
x1
x111
x121
J
1
2
..
.
y1J
y21
y22
x1J 1
x211
x221
x1J 2
x212
x222
...
...
...
x1JK
x21K
x22K
J
..
.
y2J
..
.
x2J 1
..
.
x2J 2
..
.
...
x2JK
..
.
Mode
1
2
..
.
The model is usually estimated by maximum likelihood: choose β to
maximize:
Characteristics
x2
. . . xK
x112 . . . x11K
x122 . . . x12K
CRP 4110 —()Mode Choice
...
April 24, 2015
I
y
i =1 j =1
where I is the sample size. (Or, to avoid numerical issues: maximize
log likelihood, which is equivalent).
Properties: in large samples, the MLE of β is
consistent (ie converges to true — but unknown — parameter
value).
asymptotically normal (so we can use standard tests, like t-test).
asymptotically e¢ cient (ie has minimum variance among all
consistent estimators).
Software: any stand-alone statistics package almost certainly includes
estimation of the discrete-choice logit model. Examples include
SYSTAT, LIMDEP, R, Stata, SAS, SPSS.
25 / 52
Example (I)
Philip A. Viton
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Example (II)
Individuals face a discrete choice between 4-modes: auto-alone ; bus
+ walk access ; bus + auto access ; carpool.
Choice model: model 12 from McFadden + Talvitie (1978), estimated
for San Francisco Bay Area (SFBA) work trips.
Structure of utility: naive model in which decisions are explained by
only a few factors:
Cost (and the individual’s post-tax wage).
In-vehicle time.
Excess time: this is the time to access the mode, plus wait time
for transit modes.
Dummys: these take the value 1 for the mode in question, 0
otherwise. Also known as alternative-speci…c constants. They
act like the constants in an ordinary regression equation.
Philip A. Viton
J
L = ∏ ∏ Pij ij
CRP 4110 —()Mode Choice
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27 / 52
Results of estimating the logit model:
Indep var.
Cost post-tax-wage (c/
In-vehicle time (mins)
Excess time (mins)
Auto dummy
Bus+Auto dummy
Carpool dummy
Estd. Coe¤
t-stat.
0.0412
0.0201
0.0531
0.892
1.78
2.15
7.63
2.78
7.54
3.38
7.52
8.56
c//min)
This model correctly predicted 58.5% of the choices actually made by the
sample, which is considered good for these disaggregate models.
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
28 / 52
Example — Assumed Data
Example — Systematic Utilities (I)
We assume that for a certain individual, the modal characteristics are:
Cost post-tax wage
In vehicle time (mins)
Excess time (mins)
Auto dummy
Bus+Auto dummy
Carpool dummy
Auto
Bus+Walk
Bus+Auto
Carpool
2.448
20
0
1
0
0
3.06
30
10
0
0
0
3.06
30
5
0
1
0
1.224
20
10
0
0
1
100
125
125
50
Raw costs, fares (c/)
Mode 1 (auto-drive-alone). Cost = 100c/ ; post-tax wage =
40.8479c/, so [cost
post-tax wage] is 100 40.8479 = 2. 448 .
Then:
vi 1 = xi 1 β = (2.448
=
+(0
+(0
1.39486
( 0.0412)) + (20 ( 0.0201))
( 0.0531)) + (1 ( 0.892))
( 1.78)) + (0 ( 2.15))
(Assume that car-pooling involves 2 people and they split the auto costs).
Post-tax income: $50,000 per year = 40.8479 c//min
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
29 / 52
Example — Systematic Utilities (II)
Value
Estd.Coe¤
Cost wage
IVTT
Excess time
Auto dummy
Bus+Auto dummy
Carpool dummy
2.448
20
0
1
0
0
0.0412
0.0201
0.0531
0.892
1.78
2.15
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Example — Predicted Probabilities
It may be easier to see what is going on in this calculation if we arrange
the data in a di¤erent array, where Product = Value
Estd Coe¤ :
Characteristic
Philip A. Viton
Product
0.100 86
0.402
0
0.892
0
0
Sum of products
Here are the details for computing the four mode choice probabilities
based on the example data, where the …rst row follows the same logic as in
the previous slides:
Mode:
Auto
Bus+Walk
Bus+Auto
Carpool
xij β
e xij β
∑j e xij β
Pij
1.39486
0.24787
0.637 44
0.38885
1.26007
0.283 63
0.637 44
0.44495
2.77457
0.06237
0.637 44
0.09785
3.13343
0.04357
0.637 44
0.06835
1.39486
Calculations for the other modes follow the same logic.
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
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Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
32 / 52
Marginal Own-E¤ects
Marginal Cross-E¤ects
How do the choice probabilities change as we change one of the
observable characteristic? The answer depends on whether we change
an own-characteristic or a cross-characteristic.
For an own-characteristic we ask: how does Pij (i’s choice probability
for mode j) change if we change the k-th characteristic xijk of mode j
itself?
For a cross-characteristic we ask: how does Pij change if we change
the k-th characteristic xirk of some other mode r ?
Example: how does the probability of choosing COTA change if we
change the cost of using the automobile?
In this case we have:
∂Pij
=
∂xirk
For example: how does the probability of choosing COTA change if
we change COTA’s fare?
Note that in both cases the impact depends on the starting
(pre-change) choice probabilities.
In this case we have:
∂Pij
= βk Pij (1
∂xijk
Philip A. Viton
βk Pij Pir
Pij )
CRP 4110 —()Mode Choice
April 24, 2015
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Example — Marginal E¤ects
Philip A. Viton
CRP 4110 —()Mode Choice
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The Remainder of These Notes
The impact on individual i of a 1-unit (here, a 1-cent) increase in the
bus+walk-mode’s (mode 2) bus fare (characteristic 1, which is an
own-characteristic for this mode) is (remembering that in the
speci…cation of utility we divide by the post-tax wage):
∂Pi 2
∂xi 21
=
=
=
β1
Pi 2 ( 1 Pi 2 )
wi
0.0412
0.46935 (1
40.8479
0.0025
The remainder of these notes discuss several additional topics in mode
choice analysis which will not be needed for our discussion. These topics
are covered in more detail in CRP 5700.
0.46935)
The impact of the same change on the auto (mode 1, now a
crpss-e¤ect) probability is:
∂Pi 1
∂xi 21
=
=
Philip A. Viton
β1
Pi 1 Pi 2
wi
0.0412
0.36354 0.46935 = 0.0017
40.8479
CRP 4110 —()Mode Choice
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Philip A. Viton
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Value of Time (I)
Value of Time (II)
What does this story tell us?
Consider the following story:
An individual has bus transit available at a price of $1.75/trip; and
her trips take 20 minutes.
We conclude that (because her behavior is unchanged) the change
∆C = $2.10 $175 = $0.35 in cost exactly balances the change
∆T = 20 15 = 5 mins in travel time.
So for her, a 5 minute reduction in travel time is worth $0.35.
Suppose that the transit provider reduces the travel time to 15
mins/trip; but at the same time, raises the fare to $2.10.
We say that her (unit) value of time is:
And suppose that we see that our individual makes exactly the same
number of transit trips in the two situations.
wT =
∆C
0.35
=
= $0.07 / minute
∆T
5
60 = $4. 20 per hour.
ie 7c/ per minute or 0.07
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
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Value of Time (III)
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
38 / 52
Value of Time (IV)
Let’s translate this story into symbols.
The change in utility is:
Assume that we can write the systematic utility as:
∆vij
vij = βC Cij + βT Tij + βE xijE
where:
Cij is the cost to i of alternative j
Tij is the time taken, if alternative j is chosen by i (or more
generally, any observed characteristic of alternative j)
xijE is everything else
In order for behavior to be unchanged, we must have ∆vij = 0. Now
solve for the value of time: we …nd:
wT
At the original cost Cij0 and time Tij0 , her utility is:
Now let cost change by ∆C and travel time by ∆T . Then her new
utility is:
April 24, 2015
∆C
∆T
βT
βC
and note that this can be read o¤ directly from the results of our logit
model, where we estimate the β coe¢ cients.
vij1 = βC (Cij0 + ∆C ) + βT (Tij0 + ∆C ) + βE xijE
CRP 4110 —()Mode Choice
=
=
vij0 = βC Cij0 + βT Tij0 + βE xijE
Philip A. Viton
= vij1 vij0
= βC ∆C + βT ∆T
39 / 52
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
40 / 52
Value of Time (V)
Value of Time — Example (I)
In general, for any formulation of the systematic utility function vij
(including linear-in-parameters), i’s value of time is given by:
Our example empirical speci…cation was:
vij = βC
wT =
∂vij /∂Cij
∂vij /∂Tij
where mi is i’s post-tax wage. This is slightly di¤erent from the model
in slide 39. So the …rst thing to do is to re-write it to match slide 39.
This represents i’s willingness to trade time savings for cost savings,
while keeping utility (ie behavior) constant.
We have:
vij
Values of time represent one easy way to assess the impact on
individuals of changes in travel time associated with usage of transit
modes.
CRP 4110 —()Mode Choice
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Value of Time — Example (II)
This expresses the value of time wT as a proportion of the wage (mi ).
For the empirical model, we compute the value of in-vehicle time as:
=
=
=
βT
mi
βC
0.0201
mi
0.0412
0.487 86 mi
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CRP 4110 —()Mode Choice
For the logit model, a simple calculation shows:
Pij
e vij
= v
Pik
e ik
We say: the odds of i’s choosing alternative j over alternative k
(Pij /Pik ) depends only on the observed characteristics of alternatives
j and k (ie only on vij and vik ), and not on the characteristics of any
other ‘irrelevant’alternatives.
Another interpretation: Pij /Pik is the same, no matter what other
alternatives are present in individual i’s choice set.
In the mode-choice context it is conventional to ignore the minus sign
and say that the value of in-vehicle time is (about) 49% of the
post-tax wage.
Philip A. Viton
Philip A. Viton
IIA Property
Then, applying our earlier result, we immediately see that the value of
time in this model is:
βT
βT
wT =
mi
0 =
βC
βC
wT
βC
Cij + βT Tij + βE xijE
mi
= βC0 Cij + βT Tij + βE xijE
=
where βC0 = βC /mi (that is we have a coe¢ cient βC0 multiplying
Cij ). In this form, it matches slide 39.
Of course, exactly the same idea can be applied to any other
characteristic in a utility function.
Philip A. Viton
Cij
+ βT Tij + βE xijE
mi
April 24, 2015
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This is the Independence of Irrelevant Alternatives (IIA) property.
Philip A. Viton
CRP 4110 —()Mode Choice
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Red Bus / Blue Bus (I)
Red Bus / Blue Bus (II)
Consider a situation where a population faces 2 travel alternatives,
say auto (A) and a transit bus system whose vehicles are painted red:
call it the Red-Bus (RB) system.
We will interpret the choice probabilities as aggregate modal shares:
suppose that PA = 0.70 while PRB = 0.30.
Now suppose we add a new mode: a second transit system identical
to the …rst in every respect except that its buses are painted blue
(alternatively: paint half the red buses blue). Call this the Blue Bus
(BB) system.
What do we expect for the modal shares? Obviously, the two bus
systems will split the transit-using population, so:
PA0
0
PRB
0
PBB
Philip A. Viton
00
00
= PBB
PRB
00
00
PA
0.7
=
= 2.333
PRB
0.3
=
00
PA + PRB + PBB
= 1
(by IIA)
(identical bus systems)
(probabilities must sum to 1)
Hence we predict:
00
PRB
00
PBB
00
1
= 0.230 79
4.333
= 0.230 79
=
= 0.538 42
which we know to be incorrect. Conclusion: We should not rely on
the logit model in cases where the alternatives are very similar (here,
identical).
= 0.15
= 0.15
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Testing for IIA
Philip A. Viton
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Models Without IIA
Before using the logit model it is important to know: are the choices
made by the individuals consistent with the IIA property?
If so, then it is safe to estimate a logit model and use it for
prediction.
If not, then the logit model may give wrong predictions.
CRP 4110 —()Mode Choice
April 24, 2015
In advanced treatments we can formulate models that do not involve the
IIA property, and hence will not mis-predict in the Red-Bus / Blue-Bus (or
similar) cases. Some of these are:
Nested logit.
Heteroskedastic logit.
There are several tests in the literature to decide whether observed
choices satisfy IIA.
Examples include the Hausmann-McFadden and the Small-Hsiao
tests. (See advanced texts for references).
Philip A. Viton
00
PA
00
PRB
PA
= 0.70
CRP 4110 —()Mode Choice
The logit model does not agree. It says:
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Mixed logit : this is the most recent development, in which the
coe¢ cients (the β’s) of systematic utility themselves become random
variables, that is, vary over the population.
Modern computational techniques also allow estimation of models in which
the idiosyncratic elements of utility (the η’s) are (non-independently)
Normally distributed: this is the Probit family of models.
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
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Elasticity Interpretation (I)
Elasticity Interpretation (II)
Consider the elasticity of i’s choice probability for mode j with respect
to the k-th characteristic of some other mode r : The elasticity is
de…ned as
% change in Pij
Eijrk =
% change in xirk
Eijrk
To see why this could be a problem for the logit model, suppose
mode 1 is COTA, mode 2 is a cheap Yugo subcompact, and mode 3 a
high-end Lexus.
Suppose COTA raises its fare by 10%.
And for the logit model we have:
∂Pij xirk
=
=
∂xirk Pij
What is interesting about the elasticity expression is that it is independent
of mode j, or putting it di¤erently, will take the same value for any mode j
other than r .
xirk
βk Pij Pir
=
Pij
Then the equation implies that this will have an equal impact on the
choice probability for the Yugo and for the Lexus.
βk Pir xirk
(using the expression for the marginal cross-e¤ect previously derived).
This is surely implausible: one would expect the impact of a COTA
fare change to be major for the Yugo and relatively minor for the
Lexus.
But the logit model predicts di¤erently. This is another way of stating
the IIA problem with logit.
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
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Appendix — T1EV
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
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CRP 4110 —()Mode Choice
April 24, 2015
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References
A random variable η has a Type-1-Extreme-Value distribution (sometimes
known as Weibull or Gnedenko distribution) if its probability density
(frequency) function is
η
g (η ) = e η e e
and its cumulative distribution function is
G (a) = Pr(η
a) = e
e
a
This (plus independence) is the distribution of the random elements (η ij )
underlying the logit model.
Philip A. Viton
CRP 4110 —()Mode Choice
April 24, 2015
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Philip A. Viton