# Representing Quasi-Nested Choice Structure

When the number of alternatives in a choice set increases, one can expect that there is a higher possibility that correlations between alternatives matter. In such a case, grouping alternatives into several nests and adopting a nested logit model is a popular approach. However, when the nested choice structure is not homogeneous across population groups, the modelling task becomes difficult. One may apply, for example, a latent class modelling approach (Wu, Zhang, & Fujiwara, 2011); however, if there are more choice nests, the latent class modelling approach may not work well either. To represent this kind of heterogeneous choice structure flexibly, one can transform Eq. (3.7) into, where *w*ng is a weight parameter for group *g* rather than for a single alternative.

Again, assuming that error term *e*ni follows an independent and identical Weibull distribution results in the following choice model.

One can see that alternatives in the group *g* are, in fact, nested under alternative *i* with the help of *w*ng This model is called a quasi-nested choice model (r_QNL model: Zhang & Fujiwara, 2004).

# Endogenous Modelling of Choice Set Generation

To date, endogenous modelling of choice set generation has been dominated by the two-stage procedure proposed by Manski (1977). The first stage treats the generation of a choice set and the second stage consists of choosing an alternative from the choice set. This approach seems logical at first glance. However, modelling practice suggests that the separate specification of choice sets provides no useful information for predicting choices beyond the information contained in the utility functions (Horowitz & Louviere, 1995). Furthermore, when the number of alternatives becomes large, the number of possible choice sets increases dramatically. Under these circumstances, model estimations become very difficult because compared to the large number of choice sets, the information available to estimate parameters, representing the different choice sets, is in fact very limited.

Motivated by this shortcoming of the two-stage model, Zhang, Fujiwara, and Kusakabe (2005) applied the concept of relative utility to represent choice set

(3.10)

(3.11)

generation endogenously within a unified modelling framework. First, transform Eq. (3.7) as, where,and

The validity of Eq. (3.12) is supported by the fact that relative interest parameters can take on any real value. Then, the choice probability can be re-written as,

If , alternative *i* is excluded from the choice set. Because , theoretically, one can useto represent the probability of an alternative being included in a choice set. Completely different from the Manski's two-stage procedure, Eq. (3.13) (called r GenMNL model) does not need any combination of alternatives. The utility of an alternative includes the information about not only the choice of the alternative but also whether the alternative is included in the choice set or not. Thus, choice set generation and choices of alternatives are represented simultaneously in a very flexible way. Especially, the new parameterstill meets the condition , suggesting that it can still play the same role as the original interest parameter. In other words, one can apply Eq. (3.13) to represent choice set generation together with context dependencies. Further details are discussed in Section 3.5.

# Reflecting the Non-linearity of Context Dependency

All above model specifications assume a linear form to define relative utility. However, this does not mean that non-linear forms cannot be used. Redefinein

Eq. (3.7) as, where is an alternative-specific attribute, is an alternative-generic attribute, is a constant term, and and are unknown parameters of and , respectively.

Substituting Eq. (3.14) into Eq. (3.7) results in the following :

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

where

It is obvious that Eq. (3.16) allows for attribute-based comparisons, as shown by . Distinguishing the sign of leads to three types of outcomes: positive outcome (also called gain or advantageous outcome), negative outcome (loss, regret or disadvantageous outcome) and indifferent outcomes. In all previous models, these outcomes are not distinguished. In other words, it is implicitly assumed that people show symmetric responses to advantageous and disadvantageous outcomes. However, behavioural economics suggests that this assumption may not always be true.

To overcome the shortcoming of relative utility models described previously, one can integrate relative utility with the concept of prospect theory (see Zhang, 2013; Zhang, Wang, Timmermans, & Fujiwara, 2010) by re-specifying in the above as,

(3.17)

Here, two dummy variables are newly introduced: is equal to 1 if is non-negative and 0 otherwise; and is equal to 1 if is negative and 0 otherwise. Thus, represents the gain from a comparison and indicates the loss. Parameters and (equal to or smaller than 1) determine the convexity/concavity of the utility function, and (equal to or larger than 1) describes the degree of loss aversion.