As introduced in the previous section, the logit model is derived from the assumption that the error terms of the utility functions are independent and identically Gumbel distributed. These models were first introduced in the context of binary choice models, where the logistic distribution is used to derive the probability. Their generalization to more than two alternative is referred to as multinomial logit models.
If the error terms are independent and
identically Gumbel distributed, with location parameter 0 and scale parameter
, the probability that a given individual choose alternative i within
is given by
The derivation of this result is attributed to Holman and Marley by Luce and Suppes (1965). We refer the reader to Ben-Akiva and Lerman (1985) and Anderson et al. (1992) for additional details.
It is interesting to note that the multinomial logit model can also be derived
from the choice axiom defined by (6) and
(7). Indeed, defining 
and 
, we have that (11) is equivalent
to (35).
An important property of the multinomial logit model is the
Independence from Irrelevant Alternatives (IIA). This property can be
stated as follows. The ratio of the probabilities of any two
alternatives is independent from the choice set. That is,
for any choice sets 
and 
such that
, for any
alternative 
and 
in , we have
This result can be proven easily using (35). Ben-Akiva and Lerman (1985) propose an equivalent definition: The ratio of the choice probabilities of any two alternatives is entirely unaffected by the systematic utilities of any other alternatives.
The IIA property of multinomial logit models is a limitation for some practical applications. This limitation is often illustrated by the red bus/blue bus paradox (see, for example, Ben-Akiva and Lerman, 1985) in the modal choice context. We prefer here the path choice example presented in Figure 9.
Figure 9: A path choice example
The probability provided by the multinomial logit model (35) for this example are
which is not consistent with the intuitive result. This situation appears in choice problems with significantly correlated alternatives, as it is clearly the case in the example. Indeed, alternatives 2a and 2b are so similar that their utilities share many unobserved attributes of the path and, therefore, the assumption of independence of the random part of these utilities is not valid in this context.
The Nested Logit Model, presented in the next section, partly overcomes this limitation of the multinomial logit model.
