In economics, discrete choice models, or qualitative choice models, describe, explain, and predict choices between two or more discrete alternatives, such as entering or not entering the labor market, or choosing between modes of transport. Such choices contrast with standard consumption models in which the quantity of each good consumed is assumed to be a continuous variable. In the continuous case, calculus methods (e.g. first-order conditions) can be used to determine the optimum amount chosen, and demand can be modeled empirically using regression analysis. On the other hand, discrete choice analysis examines situations in which the potential outcomes are discrete, such that the optimum is not characterized by standard first-order conditions. Thus, instead of examining "how much" as in problems with continuous choice variables, discrete choice analysis examines "which one". However, discrete choice analysis can also be used to examine the chosen quantity when only a few distinct quantities must be chosen from, such as the number of vehicles a household chooses to own and the number of minutes of telecommunications service a customer decides to purchase. Techniques such as logistic regression and probit regression can be used for empirical analysis of discrete choice. Discrete choice models theoretically or empirically model choices made by people among a finite set of alternatives. The models have been used to examine, e.g., the choice of which car to buy, where to go to college, which mode of transport (car, bus, rail) to take to work among numerous other applications. Discrete choice models are also used to examine choices by organizations, such as firms or government agencies. In the discussion below, the decision-making unit is assumed to be a person, though the concepts are applicable more generally. Daniel McFadden won the Nobel prize in 2000 for his pioneering work in developing the theoretical basis for discrete choice. Discrete choice models statistically relate the choice made by each person to the attributes of the person and the attributes of the alternatives available to the person. For example, the choice of which car a person buys is statistically related to the person's income and age as well as to price, fuel efficiency, size, and other attributes of each available car. The models estimate the probability that a person chooses a particular alternative. The models are often used to forecast how people's choices will change under changes in demographics and/or attributes of the alternatives. Discrete choice models specify the probability that an individual chooses an option among a set of alternatives. The probabilistic description of discrete choice behavior is used not to reflect individual behavior that is viewed as intrinsically probabilistic. Rather, it is the lack of information that leads us to describe choice in a probabilistic fashion. In practice, we cannot know all factors affecting individual choice decisions as their determinants are partially observed or imperfectly measured. Therefore, discrete choice models rely on stochastic assumptions and specifications to account for unobserved factors related to a) choice alternatives, b) taste variation over people (interpersonal heterogeneity) and over time (intra-individual choice dynamics), and c) heterogeneous choice sets. The different formulations have been summarized and classified into groups of models. When discrete choice model are combined with structural equation models to integrate psychological (latent) variables, they are referred as hybrid choice models.
Applications Marketing researchers use discrete choice models to study consumer demand and to predict competitive business responses, enabling choice modelers to solve a range of business problems, such as pricing, product development, and demand estimation problems. In market research, this is commonly called conjoint analysis. Transportation planners use discrete choice models to predict demand for planned transportation systems, such as which route a driver will take and whether someone will take rapid transit systems. The first applications of discrete choice models were in transportation planning, and much of the most advanced research in discrete choice models is conducted by transportation researchers. Disaster planners and engineers rely on discrete choice models to predict decision take by householders or building occupants in small-scale and large-scales evacuations, such as building fires, wildfires, hurricanes among others. These models help in the development of reliable disaster managing plans and safer design for the built environment. Energy forecasters and policymakers use discrete choice models for households' and firms' choice of heating system, appliance efficiency levels, and fuel efficiency level of vehicles. Environmental studies utilize discrete choice models to examine the recreators' choice of, e.g., fishing or skiing site and to infer the value of amenities, such as campgrounds, fish stock, and warming huts, and to estimate the value of water quality improvements. Labor economists use discrete choice models to examine participation in the work force, occupation choice, and choice of college and training programs. Ecological studies employ discrete choice models to investigate parameters that drive habitat selection in animals.
Common features of discrete choice models Discrete choice models take many forms, including: Binary Logit, Binary Probit, Multinomial Logit, Conditional Logit, Multinomial Probit, Nested Logit, Generalized Extreme Value Models, Mixed Logit, and Exploded Logit. All of these models have the features described below in common.
Choice set The choice set is the set of alternatives that are available to the person. For a discrete choice model, the choice set must meet three requirements:
The set of alternatives must be collectively exhaustive, meaning that the set includes all possible alternatives. This requirement implies that the person necessarily does choose an alternative from the set. The alternatives must be mutually exclusive, meaning that choosing one alternative means not choosing any other alternatives. This requirement implies that the person chooses only one alternative from the set. The set must contain a finite number of alternatives. This third requirement distinguishes discrete choice analysis from forms of regression analysis in which the dependent variable can (theoretically) take an infinite number of values. As an example, the choice set for a person deciding which mode of transport to take to work includes driving alone, carpooling, taking bus, etc. The choice set is complicated by the fact that a person can use multiple modes for a given trip, such as driving a car to a train station and then taking train to work. In this case, the choice set can include each possible combination of modes. Alternatively, the choice can be defined as the choice of "primary" mode, with the set consisting of car, bus, rail, and other (e.g. walking, bicycles, etc.). Note that the alternative "other" is included in order to make the choice set exhaustive. Different people may have different choice sets, depending on their circumstances. For instance, the Scion automobile was not sold in Canada as of 2009, so new car buyers in Canada faced different choice sets from those of American consumers. Such considerations are taken into account in the formulation of discrete choice models.
Defining choice probabilities A discrete choice model specifies the probability that a person chooses a particular alternative, with the probability expressed as a function of observed variables that relate to the alternatives and the person. In its general form, the probability that person n chooses alternative i is expressed as:
P n i ≡ Pr ( Person n chooses alternative i ) = G ( x n i , x n j , j ≠ i , s n , β ) , {\displaystyle P_{ni}\equiv \Pr({\text{Person }}n{\text{ chooses alternative }}i)=G(x_{ni},\;x_{nj,j\neq i},\;s_{n},\;\beta ),}
where
x n i {\displaystyle x_{ni}} is a vector of attributes of alternative i faced by person n,
x n j , j ≠ i {\displaystyle x_{nj,j\neq i}} is a vector of attributes of the other alternatives (other than i) faced by person n,
s n {\displaystyle s_{n}} is a vector of characteristics of person n, and
β {\displaystyle \beta } is a set of parameters giving the effects of variables on probabilities, which are estimated statistically. In the mode of transport example above, the attributes of modes (xni), such as travel time and cost, and the characteristics of consumer (sn), such as annual income, age, and gender, can be used to calculate choice probabilities. The attributes of the alternatives can differ over people; e.g., cost and time for travel to work by car, bus, and rail are different for each person depending on the location of home and work of that person. Properties:
Pni is between 0 and 1
∀ n : ∑ j = 1 J P n j = 1 , {\displaystyle \forall n:\;\sum _{j=1}^{J}P_{nj}=1,} where J is the total number of alternatives. (Expected fraction of people choosing i ) = 1 N ∑ n = 1 N P n i , {\displaystyle ={1 \over N}{\sum _{n=1}^{N}P_{ni}},} where N is the number of people making the choice. Different models (i.e., models using a different function G) have different properties. Prominent models are introduced below.
Consumer utility Discrete choice models can be derived from utility theory. This derivation is useful for three reasons:
It gives a precise meaning to the probabilities Pni It motivates and distinguishes alternative model specifications, e.g., the choice of a functional form for G. It provides the theoretical basis for calculation of changes in consumer surplus (compensating variation) from changes in the attributes of the alternatives. Uni is the utility (or net benefit or well-being) that person n obtains from choosing alternative i. The behavior of the person is utility-maximizing: person n chooses the alternative that provides the highest utility. The choice of the person is designated by dummy variables, yni, for each alternative:
y n i = { 1 U n i > U n j ∀ j ≠ i 0 otherwise {\displaystyle y_{ni}={\begin{cases}1&U_{ni}>U_{nj}\quad \forall j\neq i\\0&{\text{otherwise}}\end{cases}}}
Consider now the researcher who is examining the choice. The person's choice depends on many factors, some of which the researcher observes and some of which the researcher does not. The utility that the person obtains from choosing an alternative is decomposed into a part that depends on variables that the researcher observes and a part that depends on variables that the researcher does not observe. In a linear form, this decomposition is expressed as
U n i = β z n i + ε n i {\displaystyle U_{ni}=\beta z_{ni}+\varepsilon _{ni}}
where
z n i {\displaystyle z_{ni}} is a vector of observed variables relating to alternative i for person n that depends on attributes of the alternative, xni, interacted perhaps with attributes of the person, sn, such that it can be expressed as z n i = z ( x n i , s n ) {\displaystyle z_{ni}=z(x_{ni},s_{n})} for some numerical function z,
β {\displaystyle \beta } is a corresponding vector of coefficients of the observed variables, and
ε n i {\displaystyle \varepsilon _{ni}} captures the impact of all unobserved factors that affect the person's choice. The choice probability is then
P n i = Pr ( y n i = 1 ) = Pr ( ⋂ j ≠ i U n i > U n j , ) = Pr ( ⋂ j ≠ i β z n i + ε n i > β z n j + ε n j , ) = Pr ( ⋂ j ≠ i ε n j − ε n i < β z n i − β z n j , ) {\displaystyle {\begin{aligned}P_{ni}&=\Pr(y_{ni}=1)\\&=\Pr \left(\bigcap _{j\neq i}U_{ni}>U_{nj},\right)\\&=\Pr \left(\bigcap _{j\neq i}\beta z_{ni}+\varepsilon _{ni}>\beta z_{nj}+\varepsilon _{nj},\right)\\&=\Pr \left(\bigcap _{j\neq i}\varepsilon _{nj}-\varepsilon _{ni}<\beta z_{ni}-\beta z_{nj},\right)\end{aligned}}}
Given β, the choice probability is the probability that the random terms, εnj − εni (which are random from the researcher's perspective, since the researcher does not observe them) are below the respective quantities ∀ j ≠ i : β z n i − β z n j . {\displaystyle \forall j\neq i:\beta z_{ni}-\beta z_{nj}.} Different choice models (i.e. different specifications of G) arise from different distributions of εni for all i and different treatments of β.
Properties of discrete choice models implied by utility theory
Only differences matter The probability that a person chooses a particular alternative is determined by comparing the utility of choosing that alternative to the utility of choosing other alternatives:
P n i = Pr ( y n i = 1 ) = Pr ( ⋂ j ≠ i U n i > U n j ) = Pr ( ⋂ j ≠ i U n i − U n j > 0 ) {\displaystyle P_{ni}=\Pr(y_{ni}=1)=\Pr \left(\bigcap _{j\neq i}U_{ni}>U_{nj}\right)=\Pr \left(\bigcap _{j\neq i}U_{ni}-U_{nj}>0\right)}
As the last term indicates, the choice probability depends only on the difference in utilities between alternatives, not on the absolute level of utilities. Equivalently, adding a constant to the utilities of all the alternatives does not change the choice probabilities.
Scale must be normalized Since utility has no units, it is necessary to normalize the scale of utilities. The scale of utility is often defined by the variance of the error term in discrete choice models. This variance may differ depending on the characteristics of the dataset, such as when or where the data are collected. Normalization of the variance therefore affects the interpretation of parameters estimated across diverse datasets.
Prominent types of discrete choice models Discrete choice models can first be classified according to the number of available alternatives.
* Binomial choice models (dichotomous): 2 available alternatives * Multinomial choice models (polytomous): 3 or more available alternatives Multinomial choice models can further be classified according to the model specification:
* Models, such as standard logit, that assume no correlation in unobserved factors over alternatives * Models that allow correlation in unobserved factors among alternatives In addition, specific forms of the models are available for examining rankings of alternatives (i.e., first choice, second choice, third choice, etc.) and for ratings data. Details for each model are provided in the following sections.
Binary choice
A. Logit with attributes of the person but no attributes of the alternatives
Un is the utility (or net benefit) that person n obtains from taking an action (as opposed to not taking the action). The utility the person obtains from taking the action depends on the characteristics of the person, some of which are observed by the researcher and some are not. The person takes the action, yn = 1, if Un > 0. The unobserved term, εn, is assumed to have a logistic distribution. The specification is written succinctly as:
{ U n = β s n + ε n y n = { 1 U n > 0 0 U n ⩽ 0 ε ∼ Logistic ⇒ P n 1 = 1 1 + exp ( − β s n ) {\displaystyle {\begin{cases}U_{n}=\beta s_{n}+\varepsilon _{n}\\y_{n}={\begin{cases}1&U_{n}>0\\0&U_{n}\leqslant 0\end{cases}}\\\varepsilon \sim {\text{Logistic}}\end{cases}}\quad \Rightarrow \quad P_{n1}={\frac {1}{1+\exp(-\beta s_{n})}}}
B. Probit with attributes of the person but no attributes of the alternatives
The description of the model is the same as model A, except the unobserved terms are distributed standard normal instead of logistic.
{ U n = β s n + ε n y n = { 1 U n > 0 0 U n ⩽ 0 ε ∼ Standard normal ⇒ P n 1 = Φ ( β s n ) , {\displaystyle {\begin{cases}U_{n}=\beta s_{n}+\varepsilon _{n}\\y_{n}={\begin{cases}1&U_{n}>0\\0&U_{n}\leqslant 0\end{cases}}\\\varepsilon \sim {\text{Standard normal}}\end{cases}}\quad \Rightarrow \quad P_{n1}=\Phi (\beta s_{n}),}
where Φ {\displaystyle \Phi } is cumulative distribution function of standard normal.
C. Logit with variables that vary over alternatives Uni is the utility person n obtains from choosing alternative i. The utility of each alternative depends on the attributes of the alternatives interacted perhaps with the attributes of the person. The unobserved terms are assumed to have an extreme value distribution.
{ U n 1 = β z n 1 + ε n 1 U n 2 = β z n 2 + ε n 2 ε n 1 , ε n 2 ∼ iid extreme value ⇒ P n 1 = exp ( β z n 1 ) exp ( β z n 1 ) + exp ( β z n 2 ) {\displaystyle {\begin{cases}U_{n1}=\beta z_{n1}+\varepsilon _{n1}\\U_{n2}=\beta z_{n2}+\varepsilon _{n2}\\\varepsilon _{n1},\varepsilon _{n2}\sim {\text{iid extreme value}}\end{cases}}\quad \Rightarrow \quad P_{n1}={\frac {\exp(\beta z_{n1})}{\exp(\beta z_{n1})+\exp(\beta z_{n2})}}}
We can relate this specification to model A above, which is also binary logit. In particular, Pn1 can also be expressed as
P n 1 = 1 1 + exp ( − β ( z n 1 − z n 2 ) ) {\displaystyle P_{n1}={\frac {1}{1+\exp(-\beta (z_{n1}-z_{n2}))}}}
Note that if two error terms are iid extreme value, their difference is distributed logistic, which is the basis for the equivalence of the two specifications.
D. Probit with variables that vary over alternatives The description of the model is the same as model C, except the difference of the two unobserved terms are distributed standard normal instead of logistic. Then the probability of taking the action is
P n 1 = Φ ( β ( z n 1 − z n 2 ) ) , {\displaystyle P_{n1}=\Phi (\beta (z_{n1}-z_{n2})),}
where Φ is the cumulative distribution function of standard normal.
Multinomial choice without correlation among alternatives
E. Logit with attributes of the person but no attributes of the alternatives
The utility for all alternatives depends on the same variables, sn, but the coefficients are different for different alternatives:
Uni = βisn + εni, Since only differences in utility matter, it is necessary to normalize β i = 0 {\displaystyle \beta _{i}=0} for one alternative. Assuming β 1 = 0 {\displaystyle \beta _{1}=0} , εni are iid extreme value The choice probability takes the form
P n i = exp ( β i s n ) ∑ j = 1 J exp ( β j s n ) , {\displaystyle P_{ni}={\exp(\beta _{i}s_{n}) \over \sum _{j=1}^{J}\exp(\beta _{j}s_{n})},}
where J is the total number of alternatives.
F. Logit with variables that vary over alternatives (also called conditional logit)
The utility for each alternative depends on attributes of that alternative, interacted perhaps with attributes of the person:
{ U n i = β z n i + ε n i ε n i ∼ iid extreme value ⇒ P n i = exp ( β z n i ) ∑ j = 1 J exp ( β z n j ) , {\displaystyle {\begin{cases}U_{ni}=\beta z_{ni}+\varepsilon _{ni}\\\varepsilon _{ni}\sim {\text{iid extreme value}}\end{cases}}\quad \Rightarrow \quad P_{ni}={\exp(\beta z_{ni}) \over \sum _{j=1}^{J}\exp(\beta z_{nj})},}
where J is the total number of alternatives. Note that model E can be expressed in the same form as model F by appropriate respecification of variables. Define w n j k = s n δ j k {\displaystyle w_{nj}^{k}=s_{n}\delta _{jk}} where δ j k {\displaystyle \delta _{jk}} is the Kronecker delta and sn are from model E. Then, model F is obtained by using
z n j = { w n j 1 , ⋯ , w n j J } and β = { β 1 , ⋯ , β J } , {\displaystyle z_{nj}=\left\{w_{nj}^{1},\cdots ,w_{nj}^{J}\right\}\quad {\text{and}}\quad \beta =\left\{\beta _{1},\cdots ,\beta _{J}\right\},}
where J is the total number of alternatives.
Multinomial choice with correlation among alternatives A standard logit model is not always suitable, since it assumes that there is no correlation in unobserved factors over alternatives. This lack of correlation translates into a particular pattern of substitution among a