Abstract: Random utility maximization discrete choice models are widely used in transportation and other fields to represent the choice of one among a set of mutually exclusive alternatives. The decision maker, in each case, is assumed to choose the alternative with the highest utility to him/her.  The utility to the decision maker of each alternative is not completely known by the modeler; thus, the modeler represents the utility by a deterministic portion which is a function of the attributes of the alternative and the characteristics of the decision-maker and an additive random component which represents unknown and/or unobservable components of the decision maker's utility function.

Early development of choice models was based on the assumption that the error terms were multivariate normal or independently and identically Type I extreme value (gumbel) distributed (Johnson and Kotz, 1970).  The multivariate normal assumption leads to the multinomial probit (MNP) model (Daganzo, 1979); the independent and identical gumbel assumption leads to the multinomial logit (MNL) model (McFadden, 1973).  The probit model allows complete flexibility in the variance-covariance structure of the error terms but it's use requires numerical integration of a multi-dimensional normal distribution.  The multinomial logit probabilities can be evaluated directly but the assumption that the error terms are independently and identically distributed across alternatives and cases (individuals, households or choice repetitions) places important limitations on the competitive relationships among the alternatives.  Developments in the structure of discrete choice models have been directed at either reducing the computational burden associated with the multinomial probit model (McFadden, 1989; Hajivassiliou and McFadden, 1990; Börsch-Supan and Hajivassiliou, 1992; Keane, 1994) or increasing the flexibility of extreme value models.

Two approaches have been taken to enhance the flexibility of the MNL model.  One approach, the development of open form discrete choice models is discussed by Bhat in another chapter of this handbook. This chapter describes the development of closed form models which relax the assumption of independent and identically distributed random error terms in the multinomial logit model to provide a more realistic representation of choice probabilities.