The notion of an optimal choice is usually associated with a trade-off between attribute values. Some specific combination of attribute values generates the maximum overall value judgments across all choice alternatives. can be viewed a vector that positions choice alternative i in the cognitive space of individual n with xnik representing the projections of i on the attribute dimension k. transforms the cognitive space into utility space. An optimal choice can be viewed as finding the location in this space that maximizes the utility, that is the highest point in this multidimensional space. Any other point in this space that would be chosen represents a non-optimal choice, according to this definition.

Several non-optimal decision mechanisms have been suggested in the literature. A conjunctive decision rule involves a choice mechanism in which an individual sets minimum threshold values for each attribute utility. If an alternative fails to meet these minimum threshold values, it will not be chosen/deemed viable. Fundamentally, it implies that an individual judges a choice alternative on its least performing attribute. If the choice alternative passes this judgment, it constitutes a viable alternative; if it does not, it fails. Note that higher evaluation scores of one or more of the remaining attributes cannot compensate for any low judgment of an attribute. A conjunctive decision rule thus describes a case of non-compensatory decision-making. It implies an attribute-by-attribute processing of the choice alternatives. If the utility of one or more of the attributes that do not meet the threshold value would be positive, the choice under a conjunctive decision rule will not be the optimal, utility-maximizing choice. In case of multiple choice alternatives, conjunctive decision rules do not necessarily lead to a single choice. Rather, they lead to a partitioning of an individual's choice set into subsets of viable and non-viable (satisfying/non-satisfying) choices, each of which, but not both, may be empty. Conjunctive choice behaviour may be expressed as:

(1.1)

where μnk is an individual minimum utility threshold value, defined for each attribute k to make choice alternative i acceptable.

A pure conjunctive rule would require an individual to set a threshold utility value for each attribute and judge all attributes to decide whether it is an acceptable or non-acceptable choice. However, from a process perspective, as soon as any attribute fails to meet the corresponding threshold value, the alternative can be rejected. Wright (1975) called this the sequential elimination decision rule. It involves not only the minimum thresholds, but also a specific order in which the attributes are judged. This decision rule implies that the thresholds are endogenously defined. Simon's theory of aspiration levels and satisficing states that individuals seek to achieve a minimum acceptable utility and stop their search for better alternatives once this minimum has been achieved (Simon, 1955). It should be noted, however, that this principle is not necessarily confined to conjunctive decision rules.

Eq. (1.1) defines acceptance in utility space. Conjunctive rules may also be expressed in objective or subjective, cognitive space. In that case, the threshold parameters should be redefined. In case of a continuous monotonically increasing attribute, with a monotonically increasing underlying utility function, a minimum attribute value is defined (e.g. minimum number of parking places). Correspondingly, if the underlying utility function is monotonically decreasing, a maximum attribute value is defined (e.g. maximum acceptable travel time). If the attribute is monotonically decreasing and the underlying utility is monotonically increasing, a maximum attribute value is defined; otherwise with a monotonically decreasing utility, a minimum value is defined.

If the attribute is defined in terms of a set of ordered values, the same principles apply, except that a threshold level as opposed to a value is used. Finally, if the attributes are binomial or multinomial, thresholds are replaced with disjoint subsets, which split attribute categories into a subset of acceptable categories and a subset of non-acceptable categories.

Some studies have relied on explicit measurement of thresholds, whereas other studies have estimated thresholds, typically assuming some distribution for the aggregate level.

A second non-optimal decision rule is the disjunctive rule. It asserts that choice alternatives are evaluated on the basis of their maximum rather than their minimum utility values. A choice alternative that satisfies a maximum threshold value on at least one attribute is considered viable, regardless of its utility values on any other attribute. Thus, as the conjunctive rule, disjunctive rules partition choice sets into subsets of acceptable and non-acceptable choice alternatives. Disjunctive choice behaviour may be expressed as: where gnk is an individual's maximum utility threshold value, defined for each attribute k, at least one of which should be met to make choice alternative i acceptable.

Eq. (1.2) defines the conditions to make a choice alternative acceptable in utility space. Disjunctive rules may, however, also be expressed in objective or cognitive space. When defined in objective or cognitive space, the threshold parameters represent maximum attribute values in case of a continuous monotonically increasing attribute, and a monotonically increasing underlying utility function. Similarly, if the underlying utility function is monotonically decreasing, a minimum attribute value is defined. If the attribute is monotonically decreasing and the underlying utility is monotonically increasing, again a minimum attribute value is defined.

If the attribute is defined in terms of a set of ordered values, the same principles apply, except that a threshold level rather than a value is used. Finally, if the attributes are nominal or categorical, disjoint subsets, which split attribute categories into a subset of acceptable categories and a subset of non-acceptable categories is used.

Similar to the conjunctive decision rule, some studies have relied on explicit measurement of thresholds, whereas other studies have estimated thresholds, typically assuming some distribution for the aggregate level.

Einhorn (1970, 1971) suggested non-linear approximations of the conjunctive and disjunctive choice rules. The conjunctive model can be approximated as:

Note that Eq. (1.4) represents a parabolic response curve, implying that low attribute values cannot be compensated by higher value on one or more of the other attributes.

A non-linear approximation of the disjunctive model is given by:

(1.2)

(1.3)

where Uni is the judgment (utility) about alternative i by individual n and βίζ is a weight parameter. By taking the log, the following model is estimated:

(1.4)

(1.5)

By taking the log, this approximation can be estimated as:

(1.6)

is a parameter and aik is a reference point above the asymptotic value, for example,. Note that Eq. (1.6) represents a hyperbolic response curve.

The conjunctive and disjunctive models are based on the assumption that individuals define a set of endogenous reference points or thresholds and that the choice alternatives are compared against these reference points in utility or attribute space. Another set of decision mechanisms is based on exogenous defined reference points. Examples include the maximin, maximax and minimax regret models. Usually, these models are based on explicit measurements of utilities/satisfaction.

The maximin model assumes that individuals identify the least satisfactory attribute of each choice alternative, and then choose the alternative with the highest of these minimum utility values. In contrast, the maximax model calls for the identification of the most satisfactory attribute of each choice alternative, and assumes that an individual will choose the alternative with the highest of these maximum values. The minimax regret model assumes that an individual identifies the attribute with the largest utility difference and chooses the alternative with the highest utility on this attribute, irrespective of its utility on the other attributes. Mathematically, the maximin model can be expressed as follows. Let ,;. Then,

(1.7)

Similarly, the maximax model can be expressed as follows. Let ,

(1.8)

Finally, let , ; . Then, the minimax regret model is defined as follows:

(1.9)

These equations assume that all alternatives can be perfectly ordered on all attributes. If ties exist, that is choice alternatives cannot be perfectly ordered on one or more attributes, the choice probabilities should be adjusted for such ties.

Models of strong rational behaviour assume that preferences are invariant across context. If that would not be the case, a different model would need to be estimated for every different situation. Nevertheless, the contention and empirical evidence that choice behaviour depends on the context has received some attention in the choice literature. While conventional utility-maximizing models of rational behaviour assume that utility is derived directly from the attribute levels of a choice alternative, independent of context, several other context-dependent choice models often based on different behavioural principles have been assumed. The multinomial logit model, the most commonly used utility-maximizing discrete choice model, is characterized by the so-called IIA or Independence from Irrelevant Alternatives property, which states that the odds of choosing an alternative over another alternative is independent of any other alternative and its attributes in the choice set. To avoid this potential limitation, most scholars have allowed varying variances and/or covariances in the error terms of the utility function, reflecting the notion that choice alternatives may share some unobserved variables and these unobserved terms could have had different expected values (see Timmermans & Golledge, 1990 for an overview of these models). Others have suggested to represent the context explicitly in the specification of the utility function, for example, by including measures of similarity of choice alternatives or by estimating the effects of the existence and the attributes values of competing alternatives on the utility of the considered choice alternative. While the mathematical specification of the utility function thus differs from the conventional additive linear-in-parameters specification, the utility function is still maximized and in that sense one may argue these context-dependent choice models still represent rational behaviour.

Classic utility-maximizing models exhibit particular properties such as transitivity (if A is preferred to B and B is preferred to C, then A is by definition preferred to C) and regularity (the fact that the introduction of a new choice alternative logically can only reduce the choice probabilities of the existing choice alternatives before the introduction of the new choice alternative). Some context-dependent models violate these properties that may be viewed as essential evidence of rational behaviour. Behaviourally, these context-dependent models allow for the possibility that choice alternatives are evaluated relative to the position of the other choice alternatives in attribute space. One may assume that if individuals only have weak preferences, they do not necessarily choose extreme choice alternatives, but rather choose alternatives that are easier to justify to themselves and others (Simonson, 1989; Simonson & Tversky, 1992). Compromise alternatives would be a good example. This may be seen as evidence of bounded rationality in the sense that more extreme alternatives are less likely considered. In some cases, the model is based on different behavioural postulates such as maximization of relative advantage, maximization of relative utility or minimization of regret. On the other hand, considering the specification of these models, the processing of the information and decision rules assumes a more elaborated information processing and decision-making process and therefore a rational decision style. Thus, although we acknowledge that the question whether context-dependent models are examples of models of bounded rationality is debatable, we include a short review of the development of these models in the current chapter.

Batsell (1981) introduced a discrete choice model with an extended utility function that accounted for the similarity of choice alternatives. More specifically, his model can be expressed as:

(1.10)

Meyer and Eagle (1982) suggested a different specification by multiplying the expectation of the utility function with a measure of similarity. Borgers and Timmermans (1988) showed that if this measure of similarity is expressed as an exponential function of attribute differences and the error terms are identically and independently Gumbel distributed, the M_E model becomes equivalent to Batsell's model. Although these models do capture the effect of similarity on choice probabilities, they cannot fully capture pure competition in the sense that the market share of the perfect substitutes would be perfectly split, and the market share of the other alternatives be unaffected. Borgers and Timmermans (1988), therefore, suggested yet another specification.

(1.11)

where

(1.12)

and indicates the degree of substitution. This specification can theoretically represent the effects of perfect competition on choice probabilities for up to three choice alternatives correctly. It means that the quest for a general specification is still an unsolved problem.

If the similarity relates to the location of alternatives, spatial choice is represented and the similarity compares the average distance to competing destinations. In that case, the model resembles Fotheringham's competing destination model, except that the latter assumes a full multiplicative model and is based on a spatial interaction modelling as opposed to discrete choice modelling framework (Fotheringham, 1983, 1986).

After a period of lack of attention for context-dependent choice models, the problem recently re-emerged on the research agenda in connection with the formulation of regret-based models. These models assume that individuals minimize regret when choosing a choice alternative. Regret associated with attribute k is defined as a function of the difference between the attribute value of the considered choice alternative i and attribute valueof one or more of the remaining choice alternatives . Total (possible) regret when choosing alternative i is equal to . In line with utility-maximizing models, total regret is assumed to consist of a deterministic component and an error term: . Consequently, the probability that individual n will choose i is equal to. Assuming the error terms are identically and independently Gumbel distributed, the probabilityof individual n choosing alternative i from choice set C is given by

(1.13)

where μ is a scaling factor.

An operational regret-based model requires a specification of function g. Two different functions have been suggested in the literature. The best alternative only linear regret specification (Chorus, Arentze, & Timmermans, 2008a, 2008b) assumes that regret is defined as a linear function of the attribute difference between alternative i and the best forgone alternative: By contrast, the paired comparisons linear regret model (Chorus, 2010) assumes that regret is defined as a linear function of the attribute difference between alternative n and all other foregone alternatives in the choice set:

(1.14)

Because the discontinuity in these specifications may cause some difficulty in estimation, Chorus (2010) suggested a logarithmic transformation. In that case, the best alterative only logarithmic regret specification is defined as , while the paired comparisons logarithmic regret specification can be expressed as:

(1.15)

This new logarithmic transformation is asymptotically identical to the original specification if attribute differences are large. However, as shown by Rasouli and Timmermans (2014b), the new formulation violates the concept of regret for some attribute differences, and systematically over-estimates regret for small attribute differences.

In the chapter in this volume, Hess and Chorus (2015; see also Chorus, 2013) formulated a generalised function of the following form:

(1.16)

Note that if , Eq. (1.15) is obtained; if , Eq. (1.16) collapses into Eq. (1.14). Note that these models do not take the similarity between choice alternatives into account.

In addition to these pure regret-based models, a number of hybrid models have been formulated, combining aspects of conventional utility maximization and regret minimization behaviour. Chorus et al. (2013) suggested to divide the set of attributes be divided into subsetof attributes that are processed in a regretbased fashion and subset ; ; of attributes that are processed in a utility-maximising manner. Their hybrid utility function is then equal to:

(1.17)

Leong and Hensher's (2012) formulated a relative advantage model, originally suggested by Tversky and Simonson (1993) and Kivetz, Netzer, and Srinivasan (2004). This model differentiates between advantage and disadvantage and then calculates relative advantage as a substitute for regret. Advantage can be expressed as , with

(1.18)

Disadvantage is defined in a similar way.

(1.19)

Relative advantage is defined as:

(1.20)

The utility function is then written as a linear combination of the classic utility component and relative advantage:

(1.21)

These models have in common the use of one or more reference points to position the choice alternatives in utility space. In that sense, most of these models can be viewed as special cases of relative utility models. Zhang, Timmermans, Borgers, and Wang (2004) introduced the principle of relative utility maximization in the travel behaviour literature. It assumes that individuals will choose the alternative in their choice sets that provides the maximum relative utility. In this context, relative can mean relative to other choice alternatives, relative to previous time periods, relative to members of a social network, etc. The basic relative utility model can be expressed as:

(1.22)

where is a relative interest parameter and represents the effect of utility differences of alternative i and ϊ in the choice set.

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