Mode choice models

Mode Choice – Models

Philip A. Viton

October 30, 2013

Philip A. Viton ()CRP/CE 5700 – Mode Choice Models October 30, 2013 1 / 35

Introduction

We consider an individual choosing among the availabletransportation modes for a specific trip purpose.

The set of available modes is called the individual’s Choice Set.

Our task is to understand an individual’s decision (behavior) – whichmode she selects – in terms of observables: that is, determinants ofbehavior which we as analysts can observe, measure, and potentiallychange.


Discrete Choice (I)

We assume that once she has decided on a mode, she utilizes onlythat mode. We will say that she consumes 1 unit of the mode.

This means that we will need to define “compound modes” so that ifshe decides to access transit by (say) walking, this will be a differentmode than if she decides to get to the transit system by car.

This setup – the individual has a set of alternatives and must chooseexactly one – is known as Discrete Choice.

It differs from a more common situation where the individual canchoose any quantities from any of the available alternatives (think ofgrocery shopping: in the vegetable department you can choose asmany as you want; you are not restricted to buying just onevegetable).


Discrete Choice (II)

The Discrete Choice setup applies to many more settings than simplymode choice, and the methods we will be developing can be used,with only a change in terminology, to analyze a wide variety of choicesettings.

Examples:

Individual choice of a place (location) to liveIndividual choice of a vacation destinationChoice of a car or major appliance model to purchaseChoice of which university (from among those where you’ve beenaccepted) to attendAppellate judge: vote to affi rm or overrule the lower court decisionCity council member: vote for or against a project/plan

(Note that in each of these examples we will assume that you canchoose just one of the available alternatives).


Understanding Choice (I)

We now focus explicitly on the mode-choice decision.

Consider an individual making repeated mode choices in a situationwhere the observables – the observable characteristics of the modesand of the individual herself – do not change.

Do we expect that she will make the same choice every time? Thatis, would we surprised if she made different choices on different days,even if the observable determinants of her decision were unchanged?

Alternatively: suppose we knew all and only the observables. Are weconfident that we could predict her choices each and every time?


Understanding Choice (II)

Of course, the answer is No, and the reason is clear: in addition tothe determinants of her decision that we can observe, the individualalso takes into account factors that she knows, but which we (asanalysts) do not observe.

For example, we may not know that her car is in the shop for repairson one day (and that would incline her to use transit); or that she hasa doctor’s appointment after work and needs to arrive promptly (andthat would incline her to use her car).


Understanding Choice (III)

We can summarize our analytical situation in two ways:

There are determinants of her choice that we as analysts do notobserve: that is, our knowledge of the determinants of her behavior isincomplete.

From our perspective, there appears to be an element of randomnessin her decision-making: when the observables do not change herbehavior may change for no observable reason.

Note that this is not to say that our individual’s choice is random: it isjust the way things appear to us, as analysts trying to understand (explain)her decision.

Our analysis will need to take these sources of variation into account if weare to make plausible inferences about her decisions.


Choice and Probability

The natural way to do this is to employ notions of probability.

Probability is designed to handle the situation where there are severalpotential outcomes (here: choices) and we cannot say exactly whichwill occur.

By focussing on the probability that the individual makes a givenchoice we will be incorporating our lack of complete knowledge aboutout individual’s decision-making.


Individual Behavior (I)

We now describe the discrete choice setting more formally.

Individual i has J modes indexed by j (j = 1, 2, . . . , J) to choosefrom. We say that her choice set is of size J (is J-dimensional).

There is a vector of observable characteristics of mode j as applicableto individual i . Here, observable means observable to us: we cangather (or have gathered) data on these characteristics.

We assume that there are K observables and write them, forindividual i and mode j as xij = (xij1, xij2, . . . , xijK ) Note that we aresaying that all modes can be described by the same K characteristics,though some modes may have values indicating that the characteristicmay not be present or relevant.

Individual i is a characteristics-taker: her decision whether or not touse a particular mode will not alter their characteristics.


Individual Behavior (II)

We will assume that individual i makes her choice in two steps:

1. She examines all the characteristics of each mode in her choice set,and constructs a single summary value for each mode.Note that this summary involves all the modal characteristics,including the ones that are not observable to us (the analysts).

2. She chooses the mode (for this trip) that has the highest summarynumber (ie is best for her).


Individual Behavior (III)

We will say that individual i summarizes all the characteristics of eachavailable mode via a special function called a utility function. Thesummary value associated with a mode is her utility from that mode.

If individual i selects mode j she gets utility uij from it (this issometimes called her conditional utility, since it is conditional on herhaving selected mode j).

We will assume that utility function has the property that uij > uik ifand only if individual i regards mode j as better than mode k. We saythat the utility function represents her preferences.

It can be shown that under very general conditions a utility functionthat represents an individual’s preferences exists.

Individual i chooses her best (most-preferred) mode: she is autility-maximizer.


Structure of Utility

As we have argued, individual choice depends on:

Factors observable to the analyst (eg, prices or travel times of themodes).

Factors unobservable to the analyst, but known to the decision-maker(eg whether the individual’s last experience with a particular modewas good/bad).


Structure of Utility (I)

In light of this, we now assume that mode j’s utility for individual i :

Has an observable (systematic) component that depends on theobservable factors influencing her choice. We write this as vij (xij ) orvij for short.

Has an unobservable (idiosyncratic) component representingdeterminants of choice that we as analysts do not observe (but thatare known to the decision-maker).

Following our earlier suggestion that from our perspective theseunobservables induce an appearance of randomness in the individual’sdecision-making, we represent the idiosyncratic component by arandom variable ηij .


Structure of Utility (II)

So utility uij is represented from the analyst’s perspective as:

uij = vij + ηij

where vij is the systematic part of utility and ηij is the idiosyncraticpart.

The random variable ηij represents the analyst’s ignorance of allfactors influencing individual i : it does not imply that the individualbehaves randomly (from his/her own perspective).


Implications for the Study of Choice

From the analyst’s perspective:

(Conditional) utility for i if mode j is selected:

uij = vij + ηij

Because of the random component ηij , the utility that i gets if shechooses mode j , uij , is also a random variable.

Thus, the event "for individual i , the utility of mode j is highest” isalso a random variable.

This means that we can analyze only the probability that mode jmaximizes i’s utility.

This will be individual i’s choice probability for mode j .


Binary Choice

We now study what this means in more detail.

We begin with the situation where the individual faces a binary choice:her choice set has two elements (modes; say, auto and transit).

Mode 1 has utility (for individual i) of:

ui1 = vi1 + ηi1

And similarly for mode 2:

ui2 = vi2 + ηi2


Binary Choice – Probabilities

Because the individual is a utility maximizer, she will choose themode that gives her the highest total utility.

And since utility uij is a random variable, we focus on the probabilitythat mode 1 (say) is best for individual i :

Pi1 = Pr[ui1 ≥ ui2]

Pi1 is individual i’s choice probability for mode 1. Clearly, in a binarychoice setting, Pi2 = 1− Pi1.We will assume that ties (the case ui1 = ui2) are settled in favor ofmode 1. If the η’s are continuous random variables, the event of a tiehas probability zero, so this is a harmless assumption.



We now derive an expression for the choice probability Pi1 in a binarychoice setting. We have:

Pi1 = Pr[ui1 ≥ ui2]= Pr[vi1 + ηi1 ≥ vi2 + ηi2]

= Pr[vi2 + ηi2 ≤ vi1 + ηi1]

= Pr[ηi2 − ηi1 ≤ vi1 − vi2]

where we have collected the random variables (the η’s) on the left-handside of the inequality, and the non-random terms – the v’s – on theright-hand side. So in summary:

Pi1 = Pr[ηi2 − ηi1 ≤ vi1 − vi2]



We recognize Pij as the cumulative distribution function (cdf) of therandom variable ηi2 − ηi1 evaluated at the “point” vi1 − vi2. Ifηi2 − ηi1 has cdf G (·) then we can write:

Pi1 = G (vi1 − vi2)

Alternatively, if g(·) is the probability density (frequency) function(pdf) of the random variable ηi2 − ηi1 then

Pi1 =∫ vi1−vi2

−∞g(ηi2 − ηi1).

That is, the choice probability is the area under the probability densityfunction from −∞ to vi1 − vi2, the difference in the systematicutilities.


Binary Choice – Probabilities (cdf)

vi1vi2

G(ηi2−ηi1)

Pi1


Binary Choice – Probabilities (cdf)

vi1 vi2


Binary Choice – Functional Form

To make these expressions usable in practice, we now need to saysomething more specific about:

The (joint) distribution of the random variables ηi1 and ηi2.

The way in which the systematic utility (vij ) depends on theobservables (xij ).


Binary Choice Models

We begin with the distributions of the random variables. As regards this,there are two important observations:

1. Any assumption about the joint distribution of the two random(idiosyncratic) elements ηij will yield a choice model. There is usuallyno theoretical basis for preferring one distributional assumption overanother.

2. However, largely for practical (often computational) reasons, theliterature has been dominated by two assumptions.


Binary Choice – Probit Model (I)

Assume that ηi1 and ηi2 have independent N(0,12 ) distributions.

(The second parameter is the variance, not the standard deviation).

Then ηi2 − ηi1 has a N(0, 1) distribution and:

Pi1 = G (vi1 − vi2)= Φ(vi1 − vi2)

where Φ is the standard normal cumulative distribution function (cdf).

This defines the Binary Probit model of (mode) choice.


Binary Choice – Probit Model (II)

To evaluate mode choice probabilities according to this model, youneed to use tables of the normal cdf Φ, available in almost anystatistics textbook. A useful relation is Φ(−a) = 1−Φ(a).Examples: suppose vi1 − vi2 = 1.3. Then from tables of the normaldistribution:

Pi1 = Φ(1.3)= 0.9032

If vi1 − vi2 = −0.5 then:

Pi1 = Φ(−0.5)= 1−Φ(0.5)= 1− 0.69146= 0.3085


Binary Choice – Probit Model (III)

Note that these results make intuitive sense:

When vi1 − vi2 = 1.3, we are saying that the observablecharacteristics for mode 1 appear better to individual i than theobservable characteristics of mode 2.

This does not imply that she will select mode 1 (remember that herchoice will also involve the unobservables); but it does suggest thatthe probabilities favor mode 1; in our case we calculate that in factPi1 = 0.9032.

On the other hand, when vi1 − vi2 = −0.5, we are saying that theobservable characteristics for mode 2 appear better to individual ithan the observable characteristics of mode 1.

This suggests that the probabilities favor mode 2; and in fact wecalculate that here Pi1 = 0.3085 and hence thatPi2 = 1− Pi1 = 1− 0.3085 = 0.691 5.Philip A. Viton ()CRP/CE 5700 – Mode Choice Models October 30, 2013 26 / 35

Binary Choice – Logit Model (I)

Suppose on the other hand that ηi1 and ηi2 are independent randomvariates with the Type-1 Extreme Value (T1EV) distribution, whosefrequency (g) and distribution (G ) functions are:

g(ηij ) = e−ηij e−e

−ηijand G (α) = e−e

−α

(also known as the Weibull or Gnedenko distribution).

Then it can be shown (see separate optional handout) that

Pi1 =evi1

evi1 + evi2

=1

1+ e−(vi1−vi2)

This defines the Binary Logit model of (mode) choice.


Binary Choice – Logit Model (II)

Note that the logit probabilities can be evaluated without tables –all you need is a calculator.Examples: suppose vi1 − vi2 = 1.3, then:

Pi1 =1

1+ e−(1.3)

= 0.785 835 0

If vi1 − vi2 = −0.5 then:

Pi1 =1

1+ e−(−0.5)

= 0.377 540 7

Note that the probit and logit models give different results for thechoice probabilities : the probability model you select matters.


Multinomial Choice

We now extend our work to the general case of a J-dimensionalchoice set (individual i can choose from among J available modes).

This is known as multinomial choice.

As before, i will select mode j if it yields the highest utility.

Mode j is best for i if:

mode j is better than mode 1 for i ANDmode j is better than mode 2 for i AND. . . AND . . .mode j is better than mode J for i

(this is J − 1 conditions).


Multinomial Choice Probabilities (I)

Translating this into symbols:

Mode j is better than mode 1 if uij ≥ ui1, or if

vij + ηij ≥ vi1 + ηi1

Mode j is better than mode 2 if uij ≥ ui2, or if

vij + ηij ≥ vi2 + ηi2

Mode j is better than mode k if uij ≥ uik , or if

vij + ηij ≥ vik + ηik


Multinomial Choice Probabilities (II)

So

Pij = Pr[uij ≥ ui1, uij ≥ ui2, . . . , uij ≥ uiJ ]= Pr[vij + ηij ≥ vi1 + ηi1, vij + ηij ≥ vi2 + ηi2, . . . ,

vij + ηij ≥ viJ + ηiJ , ]

= Pr[vi1 + ηi1 ≤ vij + ηij , vi2 + ηi2 ≤ vij + ηij , . . . ,

viJ + ηiJ ≤ vij + ηij ]

= Pr[ηi1 − ηij ≤ vij − vi1, ηi2 − ηij ≤ vij − vi2, . . . ,

ηiJ − ηij ≤ vij − viJ ]

(J − 1 terms in each expression; decide ties in favor of mode j)


Multinomial Choice Probabilities (III)

To clean this up a bit, write the difference ηi1 − ηij as ηi1j , or moregenerally, ηik − ηij = ηikjAnd write vij − vi1 as vij1, or more generally, vij − vik = vijkThen we can write:

Pij = Pr[ηi1j ≤ vij1, ηi2j ≤ vij2 , . . . , ηijJ ≤ viJj ]

We see that the choice probability is the joint cumulative distributionfunction of the random variables (ηi1j , ηi2j , . . . , ηiJj ) evaluated at the“point” (vij1, vij2, . . . , vijJ ).

As before, any assumption about the joint distribution of the randomvariables (the ηij or the ηikj ) will give rise to a choice model.


Multinomial Probit Model

If we assume that the (ηi1, ηi2, . . . , ηiJ ) are distributed as multivariatenormal with mean vector 0 (this is a J-vector of 0’s) and a J × Jcovariance matrix Σ, this defines the multinomial probit model.The diffi culty with this is that in order to evaluate these choiceprobabilities we need to perform a J − 1-dimensional integration ofthe multivariate normal distribution.

For J greater than about 4, until very recently this has been regardedas effectively impossible.

So for moderately high-dimensional choice sets, there has been verylittle use of the multinomial probit model.


Multinomial Logit Model

On the other hand, assume that the ηij are distributed as independentT1EV random variates.

Then the choice probabilities are given by:

Pij =evij

∑Jk=1 evik

known as the (Multinomial) Logit model. (See separate handout for aderivation).

Note that, like the binary logit model, the choice probabilities can beevaluated using just a pocket calculator – no integrations necessary.

For this reason, the logit model has dominated the literature. Butnote that there is a loss of generality: the multinomial probit modelallows for correlations among the unobserved characteristics of themodes while the multinomial logit model rules that out.


Structure of Systematic Utility

Suppose mode j has K observable characteristics (as applicable toindividual i): xij = (xij1, xij2, . . . , xijK )It is conventional to assume that systematic utility vij islinear-in-parameters:

vij = xijβ

= xij1β1 + xij2β2 + · · ·+ xijK βK

So in the logit case the choice probabilities become:

Pij =exij β

∑Jk=1 exik β

in which the only unknown is the weighting vector β.

So our remaining task is to say something about β. This is anestimation problem, and we turn next to strategies to allow us to dothis.Philip A. Viton ()CRP/CE 5700 – Mode Choice Models October 30, 2013 35 / 35

Mode choice models

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