Discrete choice


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 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 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 and over time 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 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:
  1. 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.
  2. 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.
  3. 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 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. 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:
where
In the mode of transport example above, the attributes of modes, such as travel time and cost, and the characteristics of consumer, 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
  • where J is the total number of alternatives.
  • where N is the number of people making the choice.
Different models 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:
  1. It gives a precise meaning to the probabilities Pni
  2. It motivates and distinguishes alternative model specifications, e.g., the choice of a functional form for G.
  3. It provides the theoretical basis for calculation of changes in consumer surplus from changes in the attributes of the alternatives.
Uni is the utility 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:
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
where
  • 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 for some numerical function z,
  • is a corresponding vector of coefficients of the observed variables, and
  • captures the impact of all unobserved factors that affect the person's choice.
The choice probability is then
Given β, the choice probability is the probability that the random terms, are below the respective quantities Different choice models arise from different distributions of εni for all i and different treatments of β.