Fundamentals of Transportation/Choice Modeling

Mode choice analysis is the third step in the conventional four-step transportation forecasting model, following Trip Generation and Destination Choice but before Route Choice. While trip distribution's zonal interchange analysis yields a set of origin destination (OD) tables which tells where the trips will be made, mode choice analysis allows the modeler to determine what mode of transport will be used.

Disaggregate Travel Demand models

Travel demand theory was introduced in the appendix on traffic generation. The core of the field is the set of models developed following work by Stan Warner in 1962 (Strategic Choice of Mode in Urban Travel: A Study of Binary Choice). Using data from the CATS, Warner investigated classification techniques using models from biology and psychology. Building from Warner and other early investigators, disaggregate demand models emerged. Analysis is disaggregate in that individuals are the basic units of observation, yet aggregate because models yield a single set of parameters describing the choice behavior of the population. Behavior enters because the theory made use of consumer behavior concepts from economics and parts of choice behavior concepts from psychology. Researchers at the University of California, Berkeley (especially Daniel McFadden, who won a Nobel Prize in Economics for his efforts) and the Massachusetts Institute of Technology (Moshe Ben-Akiva) (and in MIT associated consulting firms, especially Cambridge Systematics) developed what has become known as choice models, direct demand models (DDM), Random Utility Models (RUM) or, in its most used form, the multinomial logit model (MNL).

Choice models have attracted a lot of attention and work; the Proceedings of the International Association for Travel Behavior Research chronicles the evolution of the models. The models are treated in modern transportation planning and transportation engineering textbooks.

One reason for rapid model development was a felt need. Systems were being proposed (especially transit systems) where no empirical experience of the type used in diversion curves was available. Choice models permit comparison of more than two alternatives and the importance of attributes of alternatives. There was the general desire for an analysis technique that depended less on aggregate analysis and with a greater behavioral content. And, there was attraction too, because choice models have logical and behavioral roots extended back to the 1920s as well as roots in Kelvin Lancaster’s consumer behavior theory, in utility theory, and in modern statistical methods.

The Logit Model

The Logit Model, widely used for transportation forecasting in various forms, was first theorized by Daniel McFadden. The Logit model says, the probability that a certain mode choice will be taken is proportional to $e$ raised to the utility over the sum of $e$ raised to the utility.

$P_{m}{\rm {}}={\rm {}}{\frac {e^{u_{ijm}}}{\sum {e^{u_{ijm}}}}}{\rm {}}\,\!$

For any Logit Model the sum of the probability of all modes will equal 1.

$1{\rm {}}={\rm {}}\sum {P_{m}}{\rm {}}\,\!$

The Logit Model also says that if a new mode of transportation is added to a system (or taken away) then the original modes will lose (or gain) an amount of travels proportional to their share originally.

Steps for the Logit Model:

• Compute the Utility for each OD pair and mode
• Compute Exponentiated utilities for each OD pair and mode
• Sum Exponentiated utilities for each OD pair
• Compute Probability for each mode by OD pair
• Multiply Probability for OD pair by number of trips for each OD pair

Psychological roots

Distribution of perceived weights

Early psychology work involved the typical experiment: Here are two objects with weights, $w_{1}$ and $w_{2}$, which is heavier? The finding from such an experiment would be that the greater the difference in weight, the greater the probability of choosing correctly. Graphs similar to the one on the right result.

Louis Leon Thurstone proposed (in the 1920s) that perceived weight,

$w=v+e$,

where $v$ is the true weight and $e$ is random with $E(e)=0.$

The assumption that $e$ is normally and identically distributed (NID) yields the binary probit model.

Econometric formulation

Economists deal with utility rather than physical weights, and say that

observed utility = mean utility + random term.

Utility in this context refers to the total satisfaction (or happiness) received from making a particular choice or consuming a good or service.

The characteristics of the object, x, must be considered, so we have

u(x) = v(x) + e(x).

If we follow Thurston's assumption, we again have a probit model.

An alternative is to assume that the error terms are independently and identically distributed with a Weibull, Gumbel Type I, or double exponential distribution (They are much the same, and differ slightly in their tails (thicker) from the normal distribution). This yields the multinomial logit model (MNL). Daniel McFadden argued that the Weibull had desirable properties compared to other distributions that might be used. Among other things, the error terms are normally and identically distributed. The logit model is simply a log ratio of the probability of choosing a mode to the probability of not choosing a mode.

$\log \left({\frac {P_{i}}{1-P_{i}}}\right)=v(x_{i})\,\!$

Observe the mathematical similarity between the logit model and the S-curves we estimated earlier, although here share increases with utility rather than time. With a choice model we are explaining the share of travelers using a mode (or the probability that an individual traveler uses a mode multiplied by the number of travelers).

The comparison with S-curves is suggestive that modes (or technologies) get adopted as their utility increases, which happens over time for several reasons. First, because the utility itself is a function of network effects, the more users, the more valuable the service, higher the utility associated with joining the network. Second, because utility increases as user costs drop, which happens when fixed costs can be spread over more users (another network effect). Third, technological advances, which occur over time and as the number of users increases, drive down relative cost.

An illustration of a utility expression is given:

$\log \left({\frac {P_{A}}{1-P_{A}}}\right)=\beta _{0}+\beta _{1}\left({c_{A}-c_{T}}\right)+\beta _{2}\left({t_{A}-t_{T}}\right)+\beta _{3}I+\beta _{4}N=v_{A}\,\!$

where

P_i = Probability of choosing mode i.

P_A = Probability of taking auto

c_A,c_T = cost of auto, transit

t_A,t_T = travel time of auto, transit

I = income

N = Number of travelers

With algebra, the model can be translated to its most widely used form:

${\frac {P_{A}}{1-P_{A}}}=e^{v_{A}}\,\!$

$P_{A}=e^{v_{A}}-P_{A}e^{v_{A}}\,\!$

$P_{A}\left({1+e^{v_{A}}}\right)=e^{v_{A}}\,\!$

$P_{A}={\frac {e^{v_{A}}}{1+e^{v_{A}}}}\,\!$

It is fair to make two conflicting statements about the estimation and use of this model:

1. It's a "house of cards", and
2. Used by a technically competent and thoughtful analyst, it's useful.

The "house of cards" problem largely arises from the utility theory basis of the model specification. Broadly, utility theory assumes that (1) users and suppliers have perfect information about the market; (2) they have deterministic functions (faced with the same options, they will always make the same choices); and (3) switching between alternatives is costless. These assumptions don’t fit very well with what is known about behavior. Furthermore, the aggregation of utility across the population is impossible since there is no universal utility scale.

Suppose an option has a net utility u_jk (option k, person j). We can imagine that having a systematic part v_jk that is a function of the characteristics of an object and person j, plus a random part e_jk, which represents tastes, observational errors, and a bunch of other things (it gets murky here). (An object such as a vehicle does not have utility, it is characteristics of a vehicle that have utility.) The introduction of e lets us do some aggregation. As noted above, we think of observable utility as being a function:

$v_{A}=\beta _{0}+\beta _{1}\left({c_{A}-c_{T}}\right)+\beta _{2}\left({t_{A}-t_{T}}\right)+\beta _{3}I+\beta _{4}N\,\!$

where each variable represents a characteristic of the auto trip. The value β₀ is termed an alternative specific constant. Most modelers say it represents characteristics left out of the equation (e.g., the political correctness of a mode, if I take transit I feel morally righteous, so β₀ may be negative for the automobile), but it includes whatever is needed to make error terms NID.

Econometric estimation

Figure: Likelihood Function for the Sample {1,1,1,0,1}.

Turning now to some technical matters, how do we estimate v(x)? Utility (v(x)) isn’t observable. All we can observe are choices (say, measured as 0 or 1), and we want to talk about probabilities of choices that range from 0 to 1. (If we do a regression on 0s and 1s we might measure for j a probability of 1.4 or -0.2 of taking an auto.) Further, the distribution of the error terms wouldn’t have appropriate statistical characteristics.

The MNL approach is to make a maximum likelihood estimate of this functional form. The likelihood function is:

$L^{*}=\prod _{n=1}^{N}{f\left({y_{n}\left|{x_{n},\theta }\right.}\right)}\,\!$

we solve for the estimated parameters

${\hat {\theta }}\,\!$

that max L*. This happens when:

${\frac {\partial L}{\partial {\hat {\theta }}_{N}}}=0\,\!$

The log-likelihood is easier to work with, as the products turn to sums:

$\ln L^{*}=\sum _{n=1}^{N}{\ln f\left({y_{n}\left|{x_{n},\theta }\right.}\right)}\,\!$

Consider an example adopted from John Bitzan’s Transportation Economics Notes. Let X be a binary variable that is gamma and 0 with probability (1- gamma). Then f(0) = (1- gamma) and f(1) = gamma. Suppose that we have 5 observations of X, giving the sample {1,1,1,0,1}. To find the maximum likelihood estimator of gamma examine various values of gamma, and for these values determine the probability of drawing the sample {1,1,1,0,1} If gamma takes the value 0, the probability of drawing our sample is 0. If gamma is 0.1, then the probability of getting our sample is: f(1,1,1,0,1) = f(1)f(1)f(1)f(0)f(1) = 0.1*0.1*0.1*0.9*0.1=0.00009. We can compute the probability of obtaining our sample over a range of gamma – this is our likelihood function. The likelihood function for n independent observations in a logit model is

$L^{*}=\prod _{n=1}^{N}{P_{i}^{Y_{i}}}\left({1-P_{i}}\right)^{1-Y_{i}}\,\!$

where: Y_i = 1 or 0 (choosing e.g. auto or not-auto) and Pi = the probability of observing Yi=1

The log likelihood is thus:

$\ell =\ln L^{*}=\sum _{i=1}^{n}{\left[{Y_{i}\ln P_{i}+\left({1-Y_{i}}\right)\ln \left({1-P_{i}}\right)}\right]}\,\!$

In the binomial (two alternative) logit model,

$P_{auto}={\frac {e^{v(x_{auto})}}{1+e^{v(x_{auto})}}}\,\!$, so

$\ell =\ln L^{*}=\sum _{i=1}^{n}{\left[{Y_{i}v(x_{auto})-\ln \left({1+e^{v(x_{auto})}}\right)}\right]}\,\!$

The log-likelihood function is maximized setting the partial derivatives to zero:

${\frac {\partial \ell }{\partial \beta }}=\sum _{i=1}^{n}{\left({Y_{i}-{\hat {P}}_{i}}\right)=}0\,\!$

The above gives the essence of modern MNL choice modeling.

Independence of Irrelevant Alternatives (IIA)

Independence of Irrelevant Alternatives is a property of Logit, but not all Discrete Choice models. In brief, the implication of IIA is that if you add a mode, it will draw from present modes in proportion to their existing shares. (And similarly, if you remove a mode, its users will switch to other modes in proportion to their previous share). To see why this property may cause problems, consider the following example: Imagine we have seven modes in our logit mode choice model (drive alone, carpool 2 passenger, carpool 3+ passenger, walk to transit, auto driver to transit (park and ride), auto passenger to transit (kiss and ride), and walk or bike). If we eliminated Kiss and Ride, a disproportionate number may use Park and Ride or carpool.

Consider another example. Imagine there is a mode choice between driving and taking a red bus, and currently each has 50% share. If we introduce another mode, let's call it a blue bus with identical attributes to the red bus, the logit mode choice model would give each mode 33.3% of the market, or in other words, buses will collectively have 66.7% market share. Logically, if the mode is truly identical, it would not attract any additional passengers (though one can imagine scenarios where adding capacity would increase bus mode share, particularly if the bus was capacity constrained.

There are several strategies that help with the IIA problem. Nesting of choices allows us to reduce this problem. However, there is an issue of the proper Nesting structure. Other alternatives include more complex models (e.g. Mixed Logit) which are more difficult to estimate.

Returning to roots

The discussion above is based on the economist’s utility formulation. At the time MNL modeling was developed there was some attention to psychologist's choice work (e.g., Luce’s choice axioms discussed in his Individual Choice Behavior, 1959). It has an analytic side in computational process modeling. Emphasis is on how people think when they make choices or solve problems (see Newell and Simon 1972). Put another way, in contrast to utility theory, it stresses not the choice but the way the choice was made. It provides a conceptual framework for travel choices and agendas of activities involving considerations of long and short term memory, effectors, and other aspects of thought and decision processes. It takes the form of rules dealing with the way information is searched and acted on. Although there is a lot of attention to behavioral analysis in transportation work, the best of modern psychological ideas are only beginning to enter the field. (e.g. Golledge, Kwan and Garling 1984; Garling, Kwan, and Golledge 1994).

Examples

Example 1: Mode Choice Model

TProblem

Problem:

You are given this mode choice model $U_{ijm}=-0.412(C_{c}/w)-0.0201*C_{ivt}-0.0531*C_{ovt}-0.89*D_{1}-1.78D_{3}-2.15D_{4}\,\!$

Where:

• $C_{c}/w\,\!$ = cost of mode (cents) / wage rate (in cents per minute)
• $C_{ivt}\,\!$ = travel time in-vehicle (min)
• $C_{ovt}\,\!$ = travel time out-of-vehicle (min)
• $D\,\!$ = mode specific dummies: (dummies take the value of 1 or 0)
  • $D_{1}$ = driving,
  • $D_{2}$ = transit with walk access, [base mode]
  • $D_{3}$ = transit with auto access,
  • $D_{4}$ = carpool

With these inputs:

'	Driving	Walk Connect Transit	Auto Connect Transit	Carpool
t = travel time in-vehicle (min)	10	30	15	12
t0 = travel time out-of-vehicle (min)	0	15	10	3
$D_{1}$ = driving,	1	0	0	0
$D_{2}$ = transit with walk access, [base mode]	0	1	0	0
$D_{3}$ = transit with auto access,	0	0	1	0
$D_{4}$ = carpool	0	0	0	1
COST	25	100	100	150
WAGE	60	60	60	60

What are the resultant mode shares?

Example

Solution:

Outputs	1: Driving	2: Walk-connect transit	3: Auto-connect transit	4: Carpool	Sum
Utilities	-1.26	-2.09	-3.30	-3.58
EXP(V)	0.28	0.12	0.04	0.03
P(V)	59.96%	26.31%	7.82%	5.90%	100%

Interpretation

Value of Time:

$.0411/2.24=\$0.0183/min=\$1.10/hour$

(in 1967 $, when the wage rate was about $2.85/hour)

implication, if you can improve the travel time (by more buses, less bottlenecks, e.g.) for less than $1.10/hour/person, then it is socially worthwhile.

Example 2: Mode Choice Model Interpretation

What mode would a perfectly rational, perfectly informed traveler choose in a deterministic world given these facts:

TProblem

Case 1

'	Bus	Car	Parameter
Tw	10 min	5 min	-0.147
Tt	40 min	20 min	-0.0411
C	$2	$1	-2.24

Car always wins (independent of parameters as long as all are < 0)

TProblem

Case 2

'	Bus	Car	Parameter
Tw	5 min	5 min	-0.147
Tt	40 min	20 min	-0.0411
C	$2	$4	-2.24
Results	-6.86	-10.51

Under observed parameters, bus always wins, but not necessarily under all parameters.

It is important to note that individuals differ in parameters. We could introduce socio-economic and other observable characteristics as well as a stochastic error term to make the problem more realistic.

Sample Problem

• Problem ( Solution )
• Problem ( Solution )

Additional Questions

• Homework
• Additional Questions

Variables

• $U_{ijm}$- Utility of traveling from i to j by mode m
• $D_{n}$ = mode specific dummies: (dummies take the value of 1 or 0)
• $P_{m}$ = Probability of mode m
• $C_{c}/w$ = cost of mode (cents) / wage rate (in cents per minute)
• $C_{ivt}$ = travel time in-vehicle (min)
• $C_{ovt}$ = travel time out-of-vehicle (min)
• $D_{n}$ = mode specific dummies: (dummies take the value of 1 or 0)

Abbreviations

• WCT - walk connected transit
• ADT - auto connect transit (drive alone/park and ride)
• APT - auto connect transit (auto passenger/kiss and ride)
• AU1 - auto driver (no passenger)
• AU2 - auto 2 occupants
• AU3+ - auto 3+ occupants
• WK/BK - walk/bike
• IIA - Independence of Irrelevant Alternatives

Key Terms

• Mode choice
• Logit
• Probability
• Independence of Irrelevant Alternatives (IIA)
• Dummy Variable (takes value of 1 or 0)

Videos

• Mode Choice
• Mode Choice Problem

References

• Garling, Tommy Mei Po Kwan, and Reginald G. Golledge. Household Activity Scheduling, Transportation Research, 22B, pp. 333–353. 1994.
• Golledge. Reginald G., Mei Po Kwan, and Tommy Garling, “Computational Process Modeling of Household Travel Decisions,” Papers in Regional Science, 73, pp. 99–118. 1984.
• Lancaster, K.J., A new approach to consumer theory. Journal of Political Economy, 1966. 74(2): p. 132-157.
• Luce, Duncan R. (1959). Individual choice behavior, a theoretical analysis. New York, Wiley.
• Newell, A. and Simon, H. A. (1972). Human Problem Solving. Englewood Cliffs, NJ: Prentice Hall.
• Ortuzar, Juan de Dios and L. G. Willumsen’s Modelling Transport. 3rd Edition. Wiley and Sons. 2001,
• Thurstone, L.L. (1927). A law of comparative judgement. Psychological Review, 34, 278-286.
• Warner, Stan 1962 Strategic Choice of Mode in Urban Travel: A Study of Binary Choice