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.
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 [1] and the number of minutes of telecommunications service a customer decides to purchase.
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 specify the probability that an individual chooses an option among a set of alternatives.
In practice, we cannot know all factors affecting individual choice decisions as their determinants are partially observed or imperfectly measured.
Note that the alternative "other" is included in order to make the choice set exhaustive.
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 (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.
This derivation is useful for three reasons: Uni is the utility (or net benefit or well-being) that person n obtains from choosing alternative i.
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.
Equivalently, adding a constant to the utilities of all the alternatives does not change the choice probabilities.
The scale of utility is often defined by the variance of the error term in discrete choice models.
Normalization of the variance therefore affects the interpretation of parameters estimated across diverse datasets.
In particular, Pn1 can also be expressed as Note that if two error terms are iid extreme value,[nb 1] their difference is distributed logistic, which is the basis for the equivalence of the two specifications.
Then the probability of taking the action is where Φ is the cumulative distribution function of standard normal.
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 alternatives that might not always be realistic in a given situation.
This pattern of substitution is often called the Independence of Irrelevant Alternatives (IIA) property of standard logit models.
[16][17] A number of models have been proposed to allow correlation over alternatives and more general substitution patterns: The following sections describe Nested Logit, GEV, Probit, and Mixed Logit models in detail.
is the identity matrix (such that there is no correlation or heteroscedasticity), the model is called independent probit.
Mixed Logit models have become increasingly popular in recent years for several reasons.
accommodates random taste variation over people and correlation across alternatives that generates flexible substitution patterns.
In addition, McFadden and Train have shown that any true choice model can be approximated, to any degree of accuracy by a mixed logit with appropriate specification of explanatory variables and distribution of coefficients.
Or, in a survey, a respondent might be asked: The models described above can be adapted to account for rankings beyond the first choice.
The most prominent model for rankings data is the exploded logit and its mixed version.
[32][33] One application is the Combes et al. paper explaining the ranking of candidates to become professor.
For instance, in the example of the helping people facing foreclosure, the person chooses for some real numbers a, b, c, d. Defining
Logistic, then the probability of each possible response is: The parameters of the model are the coefficients β and the cut-off points a − d, one of which must be normalized for identification.