Introduction I
Transits mode share in the urban US and the 10 urban areas where it is
most popular (2008 data):
Modern Mode Choice Analysis
Area Share (%)
US 1.6
Philip A. Viton New York 11.0
San Francisco 5.0
Washington DC 4.5
Chicago 3.9
May 9, 2012 Honolulu 3.8
Boston 3.3
Seattle 2.8
Philadelphia 2.7
Portland 2.3
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 1 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 2 / 39
Introduction II Introduction III
Clearly, one major problem facing transit (and transportation In this note we study the problem of mode choice: what determines
planners) is that not many people use it. peoples decisions as to which mode to use for a specic trip purpose.
Can anything be done about this? Is there a way to make transit Until recently, mode choice studies were dominated by aggregate
more attractive to potential users? approaches: researchers would try to explain the percentage (share)
To answer this question we need to look at what determines peoples of people in a city who chose a given mode (for example, the percent
choices of transportation modes. using transit) based on average modal characteristics in the city,
This is the mode choice analysis problem. average disposable personal income, etc.
Of course, we want to look at the determinants of choice as a But two developments have changed this: we now have
function of things that we as planners have some chance of being able 1. Extensive surveys of individuals.
to change: its not helpful to say that people choose their cars 2. The computing power needed to analyze this individual data.
because of deep psychological structures, because, if thats so, then
its not likely that the transport planner can do very much to change So nowadays it is usual to focus on modeling and understanding
behavior. behavior at the individual level.
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 3 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 4 / 39
The Setting Discrete Choice
Consider an individual making a single trip between an origin and a Though our interest here is in mode choice, the discrete choice setting is
destination. applicable in a wide variety of decision-making contexts. Examples:
She has available a variety of modes on which to make the trip: drive Choice of a principal place (location) to live, or for a vacation home.
alone, carpool, bus, walk, etc. These constitute her Choice Set.
Choice of a vacation destination.
From her list of available modes (her choice set) she must select
Choice of a car/major appliance model to purchase.
exactly one to use for her trip. (This may involve redenitions of the
modes, in order to have a trip made on a single mode). Choice of a university (from among those where youve been
accepted) to attend.
This setting select exactly one from a list of options denes 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 City council member: vote for or against a project/plan.
continuous choice) in which individuals can choose varying quantities State agency: decide where to build a new freeway, from among a list
of multiple options. of alternative routes.
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 5 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 6 / 39
Individual Behavior Structure of Utility
Individual choice depends on:
Utility-maximizing, price-taking individual i.
J 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 Factors unobservable to the analyst, but known to the decision-maker
utility uij . (eg, whether the individuals last experience with a particular
alternative was good/bad).
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 7 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 8 / 39
The Unobservable Part of Utility Detailed Structure of Utility
We now assume: for individual i, alternative js 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 Has an unobservable (idiosyncratic) component that we represent by
a random variable ij distributed independently of vij .
Are we surprised if the individual makes dierent choices each time?
So (total) utility uij is represented as:
Clearly not: the reason is the inuence of the unobservables.
But conditional on the observables, the individuals decisions vary for uij = vij + ij
no observable reason: they appear to an analyst to have an element
of randomness to them. The random variable represents the analysts ignorance of some of the
We would like our models to reect this seeming randomness. factors inuencing individual i: it does not imply that the individual
behaves randomly (from his/her own perspective).
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 9 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 10 / 39
Implications for the Study of Choice Choice Probabilities I
We dene
It is the random component in utility that allows the analyst to mimic
Pij = Pr[individual i chooses alternative j]
the apparent randomness exhibited by individuals in their
decision-making in situations where the observables do not change.
By utility maximization, i will select alternative j if it yields the
Because of the random component ij , the total utility uij is also a highest utility
random variable.
Alternative j is best if:
This means that we can analyze only the probability that total utility
takes on a given value. alternative j is better than alternative 1 AND
alternative j is better than alternative 2 AND
Hence, we can study only the choice probability that the individual
. . . AND . . .
makes a given choice.
alternative j is better than alternative J
(this is J 1 conditions).
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 11 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 12 / 39
Choice Probabilities II Choice Probabilities III
Alternative j is better than alternative 1 if:
uij ui 1
We have:
vij + ij vi 1 + i 1
Alternative j is better than alternative k if: Pij = Pr[ i 1 ij vij vi 1 , . . . , iJ ij vij viJ ]
uij uik We see that the choice probabilities are the joint cumulative
vij + ij vik + ik distribution function of the random variables ik ij evaluated at
the points vij vik .
So Any assumption about the joint distribution of the random variables
Pij = Pr[vij + ij vi 1 + i 1 , . . . , , vij + ij viJ + iJ ] (the ij ) will in principle give rise to a choice model.
= Pr[vi 1 + i 1 vij + ij , . . . , viJ + iJ vij + ij ] But one model has dominated the literature.
= Pr[ i 1 ij vij vi 1 , . . . , iJ ij vij viJ ]
(J 1 terms in each expression; decide ties in favor of alternative j).
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 13 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 14 / 39
The Logit Model Structure of Systematic Utility
Suppose alternative j has K observable characteristics (as perceived
by individual i): (xij 1 , xij 2 , . . . , xijK ) = xij
Suppose the ij are independently and identically distributed (i.i.d) It is conventional to assume that systematic utility vij is a
random variables with Type1-Extreme-Value distributions (see Appendix). linear-in-parameters function of the xij :
Then the choice probabilities are given by: vij = xij
e vij = xij 1 1 + xij 2 2 + + xijK K
Pij =
Jk =1 e vik
So the logit choice probabilities become
known as the Logit model of discrete choice.
e xij
Pij =
Jk =1 e xik
in which the only unknown is the weighting vector .
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 15 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 16 / 39
Data I Data II
Characteristics
Indiv Mode y x1 x2 . . . xK
We want to estimate from data on a sample of individuals.
1 1 y11 x111 x112 . . . x11K
The simplest assumption is that we have a random sample. 2 y12 x121 x122 . . . x12K
For each individual we observe the complete set of characteristics (the ..
.
xij ) of the alternatives, whether chosen or not. J y1J x1J 1 x1J 2 ... x1JK
We also observe a choice indicator, dened as 2 1 y21 x211 x212 ... x21K
2 y22 x221 x222 ... x22K
1 if individual i chooses alternative j ..
yij = .
0 otherwise
J y2J x2J 1 x2J 2 ... x2JK
.. .. .. .. .. ..
. . . . . ... .
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 17 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 18 / 39
Estimation Example I
The model is usually estimated by maximum likelihood: choose to
maximize Individual faces a choice of 4-modes: auto-alone ; bus + walk access ;
I J
L =
y bus + auto access ; carpool.
Pij ij
i =1 j =1 Demand model: model 12 from McFadden + Talvitie (1978)
where I is the sample size. (Or, to avoid numerical issues: maximize estimated for SFBA work trips.
log likelihood, which is equivalent). Structure of utility: naive model in which choices are explained by
Properties of the estimators: in large samples, the only a few factors:
maximum-likelihood estimate of is
Cost (and the post-tax wage).
consistent (ie converges to true parameter value).
In-vehicle time.
asymptotically normal (so we can use standard tests, like t-test).
Excess time: this is the time to access the mode, plus wait time
asymptotically e cient (ie has minimum variance among all
for transit modes.
consistent estimators).
Dummys: these take the value 1 for the mode in question, 0
Software: any stand-alone statistics package almost certainly includes otherwise. Also known as alternative-specic constants.
estimation of the discrete-choice logit model. Examples include
SYSTAT, LIMDEP, R, Stata, SAS, SPSS.
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 19 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 20 / 39
Example II Marginal Eects I
Results:
How do the choice probabilities change as we change one of the
Indep var. Estd. Coe t-stat. observable characteristic? The answer depends on whether we change
Cost/post-tax-wage (c/ c//min) 0.0412 7.63 on own-characteristic or a cross-characteristic.
In-vehicle time (mins) 0.0201 2.78 For an own-characteristic we ask: how does Pij change if we change
Excess time (mins) 0.0531 7.54 the k-th characteristic xijk of mode j itself? (How does the
Auto dummy 0.892 3.38 probability of choosing COTA change if we change COTAs fare?). In
Bus+Auto dummy 1.78 7.52 this case we have
Carpool dummy 2.15 8.56 Pij
= k Pij (1 Pij )
xijk
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 394 ()Mode Choice May 9, 2012 21 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 22 / 39
Marginal Eects II Value of Time I
For a cross-characteristic we ask: how does Pij change if we change Consider the following story:
the k-th characteristic xirk of some other mode r ? (How does the
probability of choosing COTA change if we change the cost of using An individual has bus transit available at a price of $1.75/trip; and
the automobile?). In this case we have her trips last 20 minutes.
Suppose that the transit agency (a) reduces her travel time to 15
Pij
= k Pij Pir mins/trip; but at the same time, raises the fare to $2.10.
xirk
And suppose that we see that our individual makes exactly the same
Note that in both cases the impact depends on the starting choice number of transit trips in the two situations.
probabilities.
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 23 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 24 / 39
Value of Time II Value of Time III
Lets translate this story into symbols.
What does this story tell us? Write the systematic utility as
vij = C Cij + T Tij + E xijE
We conclude that (because her behavior is unchanged) the
C = $2.10 $175 = $0.35 cost increase exactly balances the where:
T = 20 15 = 5 mins travel time improvement. Cij is the cost to i of alternative j
Tij is the time taken, if alternative j is chosen by i (or more
So for her, a 5 minute reduction in travel time is worth $0.35.
generally, any observed characteristic of alternative j)
We say that her (unit) value of time is xijE is everything else (a K 2-vector)
C 0.35 At the original cost Cij0 and time Tij0 her utility is
wT = = = $0.07 / minute
T 5 vij0 = C Cij0 + T Tij0 + E xijE
ie 7c/ per minute or 0.07 60 = $4. 20 per hour. Now let cost change by C and travel time by T . Then her new
utility is
vij1 = C (Cij0 + C ) + T (Tij0 + C ) + E xijE
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 25 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 26 / 39
Value of Time IV Value of Time V
The change in utility is In general, for any formulation of the systematic utility function vij
(including linear-in-parameters) we have: is value of time is given by
vij = vij1 vij0
= C C + T T
vij /Cij
wT =
In order for behavior to be unchanged, we must have vij = 0. Now vij /Tij
solve for the value of time: we nd
This represents is willingness to trade time savings for cost savings,
C
wT = while keeping utility (ie behavior) constant.
T
T Values of time represent one easy way to asses the impact on
= individuals of changes in travel time associated with usage of transit
C
modes.
and note that this can be read o directly from the results of our logit Of course, exactly the same idea can be applied to any other
model, where we estimate the coe cients. characteristic in the empirical specication of a utility function.
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 27 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 28 / 39
Value of Time Example I Value of Time Example
Our empirical model is: Then, applying our earlier result, we immediately see that the value of
time in this model is:
Cij T T
vij = C + T Tij + E xijE wT = 0 = yi
yi C C
where yi is is post-tax wage. This diers from the form we analyzed So the coe cient ratio T /C is the value of time as a proportion
above. So the rst thing to do is re-write it to match the previous of the wage (yi ). For the empirical model, we compute the value of
analysis. in-vehicle time as:
We have: T
wT = yi
C
C 0.0201
vij = Cij + T Tij + E xijE = yi
yi 0.0412
= C0 Cij + T Tij + E xijE = 0.487 86 yi
where C0 = C /yi (that is we have a coe cient C0 multiplying Cij ). In the mode-choice context it is conventional to ignore the minus sign
In this form, it matches our previous expression. and say that the value of in-vehicle time is (about) 49% of the
post-tax wage.
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 29 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 30 / 39
IIA Property Red Bus / Blue Bus I
Consider a situation where a population faces 2 travel alternatives,
say auto (A) and a transit bus system whose vehicles are painted red:
For the logit model, a simple calculation shows:
call it the Red-Bus (RB) system.
Pij e vij We interpret the choice probabilities as aggregate modal shares:
= v suppose that PA = 0.70 while PRB = 0.30.
Pik e ik
Now suppose we add a new mode: a second transit system identical
We say: the odds of is choosing alternative j over alternative k to the rst in every respect except that its buses are painted blue
(Pij /Pik ) depends only on the observed characteristics of alternatives (alternatively: paint half the red buses blue). Call this the Blue Bus
j and k (ie only on vij and vik ), and not on the characteristics of any (BB) system.
other irrelevantalternatives.
What do we expect for the modal shares? Obviously, the two bus
Another interpretation: Pij /Pik is the same, no matter what other systems will split the transit-using population, so
alternatives are present in individual is choice set.
This is the Independence of Irrelevant Alternatives (IIA) property. PA0 = 0.70
0
PRB = 0.15
0
PBB = 0.15
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 31 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 32 / 39
Red Bus / Blue Bus II Testing for IIA
The logit model does not agree. It says:
00
PA PA 0.7
00 = = = 2.333 (by IIA)
PRB PRB 0.3 Before using the logit model it is important to know: are the choices
00 00 made by the individuals consistent with the IIA property?
PRB = PBB (identical bus systems)
00 00 00 If so, then it is safe to estimate a logit model and use it for
PA + PRB + PBB = 1 (probabilities must sum to 1)
prediction.
Hence we predict If not, then the logit model may give wrong predictions.
00 1 There are several tests in the literature to decide whether observed
PRB = = 0.230 79
4.333 choices satisfy IIA.
00
PBB = 0.230 79 Examples: the Hausmann-McFadden and the Small-Hsiao tests. (See
00
PA = 0.538 42 advanced texts for references).
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).
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 33 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 34 / 39
Models Without IIA Elasticity Interpretation I
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 Consider the elasticity of is choice probability for mode j with respect
(or similar) cases. Some of these are: to the k-th characteristic of some other mode r : The elasticity is
dened as
Nested logit. % change in Pij
Eijrk =
Heteroskedastic logit. % change in xirk
Mixed logit : this is the most recent development, in which the And for the logit model we have:
coe cients (the s) of systematic utility themselves become random
variables, that is, vary over the population. Pij xirk xirk
Eijrk = = k Pij Pir = k Pir xirk
xirk Pij Pij
Modern computational techniques also allow estimation of models in which
the idiosyncratic elements of utility (the s) are (non-independently) (using the expression for the marginal cross-eect previously derived).
Normally distributed: this is the Probit family of models.
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 35 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 36 / 39
Elasticity Interpretation II Appendix T1EV
What is interesting about the elasticity expression is that it is independent
of mode j, or putting it dierently, will take the same value for any mode j
other than r . A random variable has a Type-1-Extreme-Value distribution (sometimes
known as Weibull or Gnedenko distribution) if its probability density
To see why this could be a problem for the logit model, suppose
(frequency) function is
mode 1 is COTA, mode 2 is a cheap Yugo subcompact, and mode 3 a
f ( ) = e e e
high-end Lexus.
Suppose COTA raises its fare by 10%. and its cumulative distribution function is
Then the equation implies that this will have an equal impact on the F (a) = Pr( a) = e e a
choice probability for the Yugo and for the Lexus.
This is surely implausible: one would expect the impact of a COTA This (plus independence) is the distribution of the random elements ( ij )
fare change to be signicant for the Yugo and relatively minor for the underlying the logit model.
Lexus.
But the logit model predicts dierently. This is another way of stating
the IIA problem with logit.
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 37 / 39 Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 38 / 39
References
Daniel McFadden, Antti Talvitie, et al.
Demand model estimation and validation.
Special Report UCB-ITS-SR-77-9, Institute of Transportation Studies,
University of California, Berkeley, 1977.
Kenneth E. Train.
Qualitative Choice Analysis.
MIT Press, Cambridge, MA., 1986.
Kenneth E. Train.
Discrete Choice Methods with Simulation.
Cambridge University Press, Cambridge, UK, 2002.
Philip A. Viton CRP 394 ()Mode Choice May 9, 2012 39 / 39
