Modelling discontinuous choices

The final complication to consider regarding discrete choice model is the range of methods that have been proposed for considering discontinuous or lexicographic preferences. As discussed above, the potential for lexicographic preferences is a concern for SP research, in particular because they would violate a key axiom of neoclassical choice theory. In this section, a number of approaches to modelling non-compensatory choices are considered. The first point to make is that lexicographic strategies for choices are compatible with utility maximisation (V. Foster & Mourato, 2002; Plott, 1987). If continuity is not assumed, then a consumer’s preferences could be such that one preference must be satisfied before the next

most important preference can be considered. In that case, choosing the alternative with the highest value on the most important attribute maximises the consumer’s utility.

Lexicographic strategies are a subset of combinatory strategies, which also include conjunctive and disjunctive strategies (Einhorn, 1970). These strategies differ from linear utility functions because they are not simply weighted sums of the attributes, but combine their assessments of the attributes in more complex ways. Conjunctive strategies require the chosen option to meet minimum levels or thresholds for all attributes; disjunctive strategies require it to be the best option on one of the attributes; and lexicographic strategies evaluate options using an ordered set of attributes (Camerer, 1995; Earl, 1983). All of these strategies are non-compensatory: if an alternative is not good enough with regard to one attribute, no combination of other attributes can compensate for this failure (Earl, 1983, 1986; Einhorn, 1970; Swait, 2001b). They are thus inconsistent with the assumption of continuity.

Furthermore, they are inconsistent with the assumption of preference separability: non- compensatory strategies rely on interactions between choice attributes in the utility function (Einhorn, 1970).

Kurauchi & Morikawa (2001) noted that non-compensatory strategies have been considerably theorised, but they found few empirical applications in the literature. Furthermore, the

empirical applications are not a literature in the sense of a coherent, interrelated body of knowledge, but are a few largely isolated attempts to deal with lexicographic preferences. What follows is a review of several empirical studies employing non-compensatory modelling.

Swait (2001b) developed an approach to non-compensatory modelling of CE data that allowed respondents to state threshold values for specific choice attributes. He surveyed consumers on rental car preferences and specifically asked whether they would rent cars of

certain sizes or would rent cars from certain companies. What respondents said they would not do was modelled as thresholds. He added these thresholds to a standard logit model to create a penalised utility function. Individuals could make choices that violated their stated ‘requirements’, but with a cost to their utility. If the estimated penalty for violation was sufficiently high, then the threshold would never be violated. If the penalty for violation is not very high and the benefits were sufficient, such as a promotion being run by a rental company with which one would prefer not to do business, then violations could occur. As a result, Swait was able to estimate the ‘value’ to respondents of their stated requirements: what was it worth to drive a car of the wrong size? This penalised utility function incorporates the idea of thresholds, which is often how lexicographic preferences are viewed (Fishburn, 1974), but in a standard CE framework that allows for both compensatory and non-compensatory decisions. There have been two main criticisms of this model. First, it is essentially a compensatory model; the thresholds ‘merely serve to locate points of nonlinearity in an attribute value function that is compensatory’ (Elrod et al., 2004; see also Gilbride & Allenby, 2004). Swait (2001b), however, argued that this treatment was realistic: thresholds are ‘fuzzy’ and

decision-makers do violate them. A second criticism is that this model relies on self-reports of what attribute levels are unacceptable (Gilbride & Allenby, 2004). Self-reporting on decision processes can interfere with the decision process by causing more careful processing,

influencing decision criteria, and causing information overload (Elrod et al., 2004; Gladwell, 2005). Other research on discrete choices has had some success in avoiding these two issues. An attempt to address the two issues that Swait (2001b) encountered is a choice model

developed by Elrod, et al. (2004). They replaced the linear function in a standard MNL with a general nonrectangular hyperbola (GNH). They argued that this functional form allows a fully non-compensatory modelling of decision-making, and can empirically distinguish

It also allows for combinations of compensatory and non-compensatory decision-making, as in the semi-lexicographic model. This model thus represents an alternative to standard MNL models and to an approach that relies on verbal protocols to determine use of decision thresholds or cut-offs.

Two aspects of the model in Elrod, et al. (2004) could be considered further. First, the model may be estimated by maximum likelihood, so that standard hypothesis tests can be used to assess model fit. However, one issue with the maximum likelihood estimation arises from the authors’ statement that the model estimates any probability on the closed interval [0,1]. If the model is estimated via maximum likelihood and if the loglikelihood statistic is used to assess model fit, then the loglikelihood should be defined for every alternative. In a fully non- compensatory model, some alternatives would be completely excluded; the probability of choosing them would be nil. However, ln(0) is undefined, so it is likely that in practice these alternatives are treated as having very small but non-zero probabilities (McFadden, 1974). Although the probabilities may be small, the positive probabilities do result in a theoretically compensatory model. The second aspect of the model that could be extended is that it was developed for binary data – whether an applicant was or was not accepted. The model could potentially be modified to account for a choice made from several options.

Gilbride & Allenby (2004) also developed a model that was non-compensatory and that did not require the respondents to identify the attribute levels that were unacceptable. Choice was modelled as a two-step approach, in which consumers first decided which products (advanced cameras, in this case) were in the choice set and then decided in a compensatory way amongst them. The first step, a screening rule, was modelled as an indicator function that could

accommodate both conjunctive and disjunctive rules for screening out unacceptable products. One finding from the research was that the two-step model was an improvement on a standard compensatory model.

Two issues arise with the model in Gilbride & Allenby (2004), however. First, whilst the authors found that they achieved better model fit ‘despite the large increase in the number of parameters’ (Gilbride & Allenby, 2004), this better fit could be due to the increase in the number of parameters. It would be interesting to assess whether the fit statistics were affected by an adjustment for the number of parameters. The second issue with the modelling for this research was the criteria for accepting or rejecting specific choice models. The researchers found that 92% of respondents were modelled as using the conjunctive rule plus

compensatory process for making decisions. In addition, 58% of respondents were found to be screening choice sets based on one attribute only. It may have been difficult to distinguish compensatory decision making from a conjunctive screening process in the absence of information about the processes that respondents used to make their choices. In this research, this type of information was not available. It may also have been difficult to distinguish a one- reason conjunctive screening process from a lexicographic screening process. Thus, it may be possible to extend this research by combining the model from this research with an expanded survey method to collect not only choice information but also information on the decision process.

Researchers in Japan did consider lexicographic decision-making, and compared it to a compensatory model in the choice of whether to drive into the central business district (CBD) or use a public transportation park-and-ride facility. They were interested in determining the impact of dynamic road signs that displayed real-time information about the level of

congestion in the CBD and the estimated travel time. One research question was whether the decision process was compensatory or non-compensatory, because new information from the signs would have different impacts depending on the decision process. In two publications (Kurauchi & Morikawa, 2001; Yamamoto, Kurauchi, & Morikawa, 2002), they assessed three different models: a standard compensatory model, a semi-lexicographic model, and a

decision-tree derived from a data-mining tool. Their models also included a latent class approach that allowed respondents to have different hierarchies of attributes. They found that the semi-lexicographic model, which included a non-compensatory decision on the most important factor and a compensatory process for the remaining factors, had the best fit. The standard model was not sufficiently sensitive to the possibility of commuters being captive to certain transport modes, and the data-mining technique did not improve the analysis of commuters’ choices. One important issue they discovered was that the theoretical analysis required richer data than the researchers actually had. While complex effects of threshold values and attribute hierarchies could be theorised, the practical modelling could examine only a limited range of non-compensatory effects.

Another dataset was the basis of research comparing five compensatory and non-

compensatory models (Lee & Geistfeld, 1998). The researchers collected SP data on washing machines and analysed respondents’ choices to determine which of five models best

represented each person’s decision-making. Importantly, they used a full factorial

experimental design. By including all possible combinations of factors in their design, they had a dataset from which they could estimate the interactions of the product attributes, which is essential for identifying non-compensatory decision-making (Einhorn, 1970). The general compensatory model, the basis of MNL, was used least. A better compensatory model was the simple additive model, in which each attribute was equally weighted. The most-used model was conjunctive, and many respondents also used a general non-compensatory model. Two important lessons can be drawn from this research. First, non-compensatory decision-making may be more prevalent than compensatory decision-making – the researchers found that 64% of respondents used a non-compensatory model. Secondly, the research demonstrated a method for applying the conjunctive and disjunctive valuation function from Einhorn (1970) to choice modelling research. Unfortunately, this research modelled choices in a different way

than other choice modelling research. The dependent variable was not which choice from choice set was made, but was whether or not each alternative was chosen. Because the fundamental choice problem is to determine which alternative from a set is preferred, CE analyses each chosen option in the context of its particular choice set. This analysis thus did not approach the choice situation in the same way as discrete choice analysis or CE research. Another example of non-compensatory decision-making based on utility maximising is Sloss (1995). The proposed model assumed that parents selecting child-care facilities made a lexicographic decision based on one of three attributes of the facilities. Whichever facility ranked the best on the attribute that the parents valued most was the facility selected. In this way, the model was completely non-compensatory. However, because the model was entirely theoretical, it would need to be combined with empirical data in order to assess whether the proposed model did represent actual choice behaviour. In addition, by construction, all the facilities in the choice set had met certain minimum criteria of acceptability. The research did not include a discussion of how this process of identifying a consideration set had occurred. An alternative approach to modelling non-compensatory choices as maximisation was

introduced by Recker & Golob (1979). They developed a model in which decision-makers use a hierarchy of attributes and critical thresholds to make choices amongst alternatives, a model later used by Kurauchi & Morikawa (2001). However, rather than maximising the likelihood that the observed choices would be made, they constrained the model to predict the actual choices made, and then adjusted the distribution of the threshold values around a mean to simulate those choices. Importantly, they found that decisions could be modelled in this way. They created a hierarchy of attributes and threshold distributions that mimicked the actual data. As they pointed out, however, they could not address the question of whether a completely non-compensatory model was any more ‘realistic’ than a completely compensatory one.

Although these examples of modelling non-compensatory choices are not numerous, they raise several issues. They did find that non-compensatory models could describe actual choices, and they found that a variety of functional forms were useful. Some of the successful models were the semi-lexicographic, the GNH, and Einhorn’s (1970) conjunctive and

disjunctive functions. This research also compared the results of different models using standard statistics that were described earlier: prediction success percentages and likelihood- based statistics. The different models were more or less successful in part because of the data available; in order to test the assumption of compensatory decision-making, richer data seems necessary. Arising out of this research are two main issues. First, the extent to which the alternative models truly are non-compensatory is open to challenge. Secondly, the most successful non-compensatory research modelled binary choice: whether an alternative was chosen or not. Further research may be able to extend these models to choice situations with more than two alternatives.

One avenue of possible work on lexicographic preferences that has been discussed

theoretically is the use of alternative distribution assumptions in a random parameter logit (RPL) model. Lexicographic preferences are discontinuous, and estimating RPL models with Bayesian techniques may allow the use of distributions that are discontinuous or multi-

dimensional, or that represent point-masses at specific values (Bhat, 2003). Whether these can be made to mimic lexicographic preferences is an open question, but they are certainly able to model more than binary choice. It should be remembered, however, that Bayesian techniques have drawbacks, including complexity of model estimation and the unavailability of classical, likelihood-based hypothesis tests (Elrod et al., 2004).