## Discrete Choice Models – Introduction

Discrete choice models, which belong to random utility maximization (RUM) models, are widely used to analyze individual choice behavior (see Ben-Akiva and Lerman, 1985; Train, 1986 and 2003; McFadden, 2000 and 2001; Hensher et al., 2005). In the discrete choice framework, a decision maker facing a mutually exclusive and collective exhaustive set of finite number of alternatives obtains utility from each alternative and chooses the one with the highest utility.

Discrete choice models thus deal with discrete or qualitative^{[1]} outcomes involving a behavioral choice such as choice of occupation or mode of travel^{[2]}. The discrete outcomes can be ordered, e.g., number of cars a household owns or a respondent’s level of agreement to a statement in a Likert scale^{[3]}, or unordered, that is, the ordering of the outcomes has no effect on the choice process, e.g., choice of travel mode or type of housing.

The discrete choice models based on the RUM framework are important tools for analysis of individual choice behavior and have successfully been applied in diverse fields, including, transportation (c.f. McFadden, 2000; Hess, 2005; Bhatta, 2010)^{[4]}, consumer behavior (Train et al., 1987; Ashok et al., 2002), education (DesJardins et al., 1999), political science (Glasgow, 2001), economics (Herriges and Phaneuf, 2002) and peace and conflict (Barros and Proenca, 2005) to name only a few. Discrete choice analysis is an increasingly popular tool to model individual choice behavior.

Transportation is the most important field of research and application of discrete choice models. Initially, the discrete choice models in transportation were used in analyses of binary choice of travel modes in the 1960s (e.g., Warner, 1962; Lave, 1969; Lisco, 1967; Quarmby, 1967; cited in Ben-Akiva and Lerman, 1985). The models were used in estimating a value of travel time savings and predicting the market shares of alternative travel modes. The choice models have been extensively applied in transportation for nearly five decades^{[5]}. Travel demand modeling is one of the well-researched topics. There is an extensive and lively body of literature on the topic (see Domencich and McFadden, 1975; Ben-Akiva and Lerman, 1985; Train, 1986 and 2003; McFadden, 2000; Hess, 2005; Bhatta, 2010 and relevant references therein). Highly advanced models such as complex generalized extreme value (GEV) models (e.g. GEV models allowing for cross-nesting, multi-level GEV models, recursive GEV models, and so on) and models with mixed distributions (e.g. mixed logit) are developed in the RUM framework of travel demand (see Train, 2003; Hess et al., 2007; Hess, 2005; McFadden, 2000; and relevant references therein).

The choice models are based on discrete choice theory, which typically involves the following elements in the choice process, concerns the behavioral choice of discrete alternatives (c.f. Ben-Akiva and Lerman, 1985).

__The decision makers__ are the individual persons or households or firms or governments or any other decision making units that possess preferences or tastes over alternatives. For example, travelers are the decision makers for a trip to work.

__The characteristics of the decision makers__ are income, age, sex, and so on the decision makers.

__The alternatives__ are competing products, course of action, or any other option or items over which a decision must be made. The alternatives form the choice set The choice set in the RUM framework of discrete choice analysis exhibits three characteristics, viz., mutually exclusive, collectively exhaustive and finite (c.f., e.g., Train, 2003). The choice set that includes all the alternatives from the perspective of population (or an analyst) is called the universal choice set The set of alternatives that is viable for a decision maker is the feasible choice set The feasible choice set is thus the subset of the universal choice set.

__The attributes__ are something that make the alternatives useful (or just opposite) for a decision maker. For example, travel time, travel cost, comfort and so on are the attributes of a travel mode.

__The decision rule__ is the criteria followed by the decision maker to come up with the actual choice. Discrete choice theory uses random utility maximization as the decision rule.

As utility is the fundamental concept in economic theory (c.f., e.g., Varian, 1992; Silberberg and Suen, 2001) and utility maximization is the decision rule in discrete choice analysis, the discrete choice models are based on the economic theory of utility maximization (Train, 2003; Ben-Akiva and Lerman, 1985). The utilities are, however, latent variables. The actual choice, which we can observe as analysts, is a manifestation of the underlying utilities of the alternatives.

Since choice behavior of a decision maker is probabilistic from the perspective of an analyst, the discrete choice theory incorporates this probabilistic behavior through the concept of random utility. According to random utility theory, a concept first proposed by Thurstone (1927) and subsequently developed by Luce (1959) and Marschak (1960), the utility of an alternative is a random variable which consists of observable and unobservable parts from the perspective of analyst. The observable part of utility is assumed to be a function of attributes of alternatives (following Lancaster, 1966^{[6]}) and characteristics of the decision makers. The characteristics of decision makers are included in utility function in order to capture heterogeneity across decision makers since all the decision makers are not alike. The final component of the utility is a random term introduced in order to account for uncertainty due to analyst’s incomplete information about the choice process. The random utility maximization is thus the basic principle in discrete choice theory which directly follows from microeconomic consumer theory.

Footnotes:

[1] They are nominal scale variables which simply denote categories, so they are also called categorical variables. Conceptually, we can classify the discrete outcomes as those involving a behavioral choice (e.g. choice of occupation or mode of travel) or those simply describing discrete outcomes of a physical event (e.g. type of an accident) (Washington et al., 2003).

[2] There are numerous examples on discrete outcomes involving a behavioral choice in diverse fields, including transportation for choice of travel mode, route, destination, car brand, type of a vehicle to own, and so on; economics for choice of technology, market participation, production plant/plan, etc; sociology for choice of marital status (single-married-living together), marriage partner, etc; housing for choice of residential location, type of housing (rent-single own-company own), etc; marketing for choice of brand, menu, ad media (radio-TV-newspaper), etc; business for choice of costumer, portfolio, securities, etc; and education for choice of college, degree, subject/discipline choice, etc., to name only a few.

[3] A Likert scale is a psychometric scale widely used in a survey research in order to know a respondent’s attitude and level of agreement (or disagreement) to a statement. The scale is named after its discoverer Rensis Likert (1932). The Likert scale is normally used to describe items with five or seven ordered response options. For example, the response options in a typical five-point Likert scale are: (1) strongly disagree, (2) disagree, (3) neutral, (4) agree, and (5) strongly agree.

[4] There are enumerable applications in transportation.

[5] McFadden (2000) excellently reviews a historical account of the development of the state-of-the-art in the field of travel behavior research and its connection to RUM models.

[6] Lancaster proposed a new approach to consumer theory. According to the approach, it is not a good itself but its attributes that determine its utility. Utility is therefore a function of attributes of the goods.

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*Bharat P. Bhatta, Ph.D.*

*Head, Research Department
King’s College*