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1.5. Considering a Subset of Choice Alternatives

Another simplifying decision approach is to restrict the number of considered choice alternatives, and focus on that subset only. Sometimes, this subset has been called the consideration set, whereas the (universal) choice set is comprised of all available choice alternatives (e.g. Hauser & Wernerfelt, 1990; Shocker, Ben-Akiva, Boccara, & Nedungadi, 1991). Models that are based on the assumption that individuals simplify the choice task and in that sense evidence bounded rationality have typically been developed in the context of the so-called choice set formation problem. It assumes that individuals apply a two-stage decision process, during which they first select a subset of choice alternatives from the universal choice set, and then choose the alternative they like best from this reduced subset. Different ideas have been used to reduce the choice set. Individuals may not consider all available options because some cannot be reached within a reasonable time, or given space – time prism. Another reason may be that individuals first apply thresholds on the attributes of the choice alternatives so that a smaller set of alternatives remains. Another consideration is that individuals may not be aware of particular alternatives or that some may not belong to their activity space. Manski (1977) suggested the following formulation:

(1.29)

where C* is the consideration set and C+ is the set of all possible consideration sets. Note that if choice set C+ consists of I choice alternatives, there are 21 – 1 possible consideration sets. Thus, this approach becomes problematic in case of a large number of choice alternatives.

To avoid this problem, several simplifying approaches have been suggested in the literature. Morikawa (1995) compared alternatives in pairs and derived a computationally tractable form even for a large number of alternatives. Most approaches, however, suggest applying screening heuristics to define the consideration sets. Choice probabilities then become equal to

(1.30)

whereis the deterministic utility value of choice alternative i andis

an indicator function with generic argument and condition . Together, these define whether choice alternative i will be considered. may be defined in attribute space, cognitive space or utility space. may be a given value, a set of categories or a parameter to be estimated from the data.

Gilbride and Allenby (2004) discussed some examples of screening rules. A compensatory screening rule states that the deterministic part of the utility of the choice alternative must exceed a threshold value to be acceptable. Thus, Similarly, a conjunctive screening heuristic can be represented as

(1.31)

Consideration sets formed by a conjunctive decision rule require that a choice alternative to be included is acceptable on all relevant attributes. Hence, the multiplication of the indicator variables should be equal to one to satisfy this condition. A disjunctive screening heuristic stating that at least one attribute should be acceptable for a choice alternative to be included in the consideration set can be represented as

(1.32)

Seminal work on choice set composition involved deterministic thresholds. Recker, Mc.Nally, and Root (1983) suggested applying constrained combinatorial scheduling algorithms to generate a set of feasible activity-travel patterns. Next, pattern recognition procedures were applied to identify clearly distinct patterns. After eliminating all inferior patterns, the consideration set is derived. Ben-Akiva, Bergman, Daly, and Ramaswamy (1984), in the context of route choice behaviour, launched the idea of defining a set of labelled paths, found by applying different choice criteria. By identifying these paths, the choice set is dramatically reduced.

It goes without saying that the validity of such deterministic thresholds or approaches in general may be questioned (e.g. Huber & Klein, 1991). Thresholds are likely stochastic because individuals have varying perceptions, or differ in terms of their judgments of what is considered acceptable. Moreover, researchers have limited knowledge about the choice set generating process (Swait & Ben-Akiva, 1987). Finally, the urban and transportation system themselves are stochastic (Rasouli & Timmermans, 2012).

Several authors have developed models that allow for such stochastic thresholds. Assume that the screening process requires that a choice alternative will only be included in a choice set if its attributes are equal to or larger than a set of thresholds. Let denote the vectors of thresholds of individual n. Then, the probability space is represented by joint density function with mean and covariance matrix Σ. The probability that choice alternative i will be included in the choice set of individual n is then equal to:

(1.33)

The probability that choice alternative i will not be included in the choice set of individual n equals:

(1.34)

The probability that an individual's choice set is empty is equal to the probability that none of the choice alternatives satisfies the inclusion criteria:

(1.35)

Thus, where , with . This equation includes the probability of an empty choice set, which would imply a non-choice. If one wishes to correct for this, expression (1.22) can be reformulated as:

(1.36)

(1.37)

Assume that the choice process in the second stage is a utility-maximizing process based on the multinomial logit model, then choice probabilities are equal to:

(1.38)

In principle, both stages can be substituted with alternative screening and choice models. It is no surprise, therefore, that various authors suggested replacing the simple multinomial logit model with more complex choice models. For example, Kaplan, Bekhor, and Shiftan (2009) allowed for correlated ordered-response thresholds, while Kaplan, Bekhor, and Shiftan (2010a, 2010b) assumed multinomial thresholds. Kaplan and Prato (2010) examined the use of hazard-based thresholds, while Kaplan, Bekhor, and Shiftan (2011a, 2011b) developed a model allowing for the selection of multiple independent ordered-response thresholds related to individuals' characteristics. All these models assumed IID error choice alternatives at the choice stage. To relax this assumption, Kaplan, Shiftan, and Bekhor (2012) incorporated a nested correlation pattern across choice alternatives and random taste variation across the population. Technically, their model jointly represents the conjunctive heuristic with a multidimensional mixed ordered-response probit model and the choice mechanism with an error components logit model.

This most general function is very demanding. To some extent, these formulations reflect increasing computing power and the formulation of more complex discrete choice models in general. In earlier work, several simplifying assumptions have therefore been introduced to formulate less computationally demanding models. For example, Borgers, Timmermans, and Veldhuisen (1986) and Swait and Ben-Akiva (1987) assumed that the thresholds were independent, so that

(1.39)

If the thresholds are normally distributed with mean and variance , the probability that alternative i meets the selection criteria on attribute k is given by:

(1.40)

where Ф(*) is the standard cumulative normal distribution function.

Swait and Ben-Akiva (1987) also considered the case where choice is limited to a subgroup of choice alternatives and identified the set of assumptions under which this leads to the logit model (Gaudry & Dagenais, 1979). Swait (2001a, 2001b) included in the utility function a linear penalty function for violating thresholds points with lower and upper bounds. Ceteris paribus, with an increasing penalty, the probability of choosing a choice alternative that violates one or more of the thresholds is reduced. Swait and Ben-Akiva (1987) incorporated random constraints

in their model of choice set generation. According to their model an individual will include a choice alternative in his/her choice set if and only if all its attributes are within their respective thresholds.

Several applications have used exogenous covariates to explain choice set formation. Thresholds are assumed to vary as a function of socio-demographic and economic profiles of individuals. However, as argued by Horowitz and Louviere (1995), choice sets may be just another expression of underlying utilities as opposed to separate constructs. Swait (2001a, 2001b), therefore, suggested a generalized extreme value model in which inclusion in the consideration set is a function of the expected maximum utility derived from the alternatives in the choice set. His approach has the advantage that as the expected utility of an alternative is increasing, all choice sets including that alternative have a higher probability of being chosen. Cascetta and Papola (2001) extended the utility function by including a cutoff factor. Martinez, Aguila, and Hurtubia (2009) proposed the Constrained Multinomial Logit model (CMNL), which extends previous models by imposing multiple thresholds.

 
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