conclusiones

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Here we also briefly present other important non-EUT that potentially will be adopted in future research.

2.5.5.1 Theory of Disappointment (TD)

Allais (1979) and Hagen (1979) are the earliest attempts to address common consequence and common ratio effects. They proposed the moments of utility in which the utility of a prospect is not only dependent on expected utility (the first moment), but also on the variance of utility regarding the mean (the second moment), and skewness (the third moment). The third moment of utility is expressed as ∑ [ ̅ ], where is the utility of the consequence , and ̅ is the expected utility of the prospect . Bell (1985) and Loomes and Sugden (1986) developed the theory of disappointment which is closely related to moments of utility. Under this theory, the preference over prospects can be represented as:

∑ [ ̅ ] (2.21)

where is a non-decreasing function with , and ̅ is a measure of the ‘prior expectation’ of the utility from prospect . If the outcome utility is worse than the expected utility (i.e., ̅ ), a sense of disappointment is generated; if the outcome of the prospect is better than expected ( ̅ ), the consequence would produce elation. In a triangle diagram, TD also implies a fanning-out effect since individuals are assumed to be

‘disappointment averse’ ( ) and ‘elation prone’ ( ). It should be noted that this representation is less axiomatic compared to EUT, while it provides psychological insights in its intuitive interpretation (Loomes, 2010).

2.5.5.2 Theory of Disappointment Aversion (DA)

The psychological concept of disappointment has also been applied in the theory of disappointment aversion. This behavioural theory, introduced by Gul (1991), is not only analytically tractable but also parsimonious, with only one more parameter in addition to those required by EUT. This extra parameter carries the intuitive meaning of individuals’

disappointment aversion. From this intuitive point of view, DA is based on the individuals’ ex ante evaluation incorporating ex post disappointment or elation. The feeling of disappointment is distinguished from elation depending on whether the actual consequence is worse or better than the individual’s anticipation.

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Let the certainty equivalent of prospect is . If we assume that for all , there exist some such that for all (disappointment), and for all (elation). Hence, we can decompose the prospect into two prospects and ( ), i.e., , where represents a probability with the form ∑ . In this case, can be expressed by an elation/disappointment decomposition (EDD) with the form . Gul proposed the following functional form:

∑ ∑ (2.22)

where is the increasing transformation of probabilitya with and (decision weights). Gul (1991) proved that his theory is validated only when has the following form:

=

(2.23)

where the disappointment aversion parameter should be estimated according to the individual’s actual choice. Gul observed that if , which implies disappointment aversion; it is elation loving if . Notice that this representation reduces to EUT if . Given that is satisfied if , at most steps are required to calculate EDD and the utility of prospect .

2.5.5.3 Prospective Reference Theory (PR)

Viscusi (1989) proposed prospective reference theory in which preference is assumed to be expected-utility-maximizing with subjective posterior probabilities rather than only objective probabilities. This subjective posterior probability consists of two components, namely objective probability and prior probability. From an intuitive point of view, the prior probability can be interpreted as the ex ante judgement of the ex post likelihood of the state of the world, while objective probability, in the Bayesian sense, provides the information to update their priors. Viscusi assumed that all prior probabilities (he calls these reference risk

Elation Disappointment

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levels) are for K possible consequences. Then the subjective posterior probability can be written as:

(2.24)

where represents the relative weight which an individual gives to the objective probability. Notice that PR reduces to EUT if . The objective probability is revised to be larger if , while it is revised to be smaller if . This feature is consistent with the fact that individuals tend to over-weight extremely low probabilities and under-weight high ones (Aliev et al., 2012).

2.5.5.4 Weighted Utility Theory (WUT)

Chew and MacCrimmon (1979) introduced the weighted utility theory which has been further developed by Chew (1989) and Chew (1983). The formulation of WUT can be expressed by the following:

∑ [_∑

] (2.25)

where is the utility of consequence for prospect ; the real-value function assigns a positive weight to each consequence. The component in square brackets can be regarded as the weight associated with the consequence . It is for this reason that this theory is called weighted utility theory. Fishburn (1982) extended WUT to a more general form called skew symmetric bilinear utility theory, while Dekel (1986) proposed the implicit weighted utility theory as another generalization of WUT. The common feature of the WUT family is the use of weights on consequences. The weight not only depends on the consequence per se, but also on the whole prospect. As a result, fanning-out and fanning-in effects are captured by WUT, and the extreme outcomes with small probabilities can be measured differently compared to the EUT method. For instance, provided , the weight for the extremely good outcome is the utility of divided by the expected utility of . Thus, the extremely good outcome is over-weighted comparing with the outcomes with relatively small utility.

Chew (1989) subsequently derived this theory from three axioms, namely ordering, continuity, and the weakened form of independence. The latter can be interpreted as: if , then for each , there exist some such that

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. With respect to WUT, the indifference curves are linear and fanning out and, therefore, there must be a point at which all the indifference curves cross. Note that if is constant for all consequences WUT reduces to the EUT.

2.5.5.5 Regret Theory (RT)

In the regret theory (RT), the term regret, serving as the counterpart of utility, is generally referred to as the induced emotion when the chosen alternative turns out to be worse than the other alternatives (Bell, 1982, Fishburn, 1982, Loomes and Sugden, 1982). The intuition behind this theory is based on two assumptions: first, decision makers are aware of the fact that the chosen prospect may turn out to be less attractive than the other prospects, and such

‘mistakes’ can result in the negative emotion of regret; second, decision makers have a basic knowledge regarding the distribution of possible regrets and aim to minimize the expected regret (Chorus et al., 2006b).

Several unique features of RT distinguish it from the family of utility-based theories.

Firstly, unlike most behavioural theories, utility is replaced by the scalar of regret. Moreover, individuals make judgements based on the comparison between the attributes of alternatives, rather than within the considered alternative, as in EUT. That is, RT is capable of accounting for non-compensatory choice behaviour, i.e., the decrease of one attribute does not necessarily offset the increase of another attribute of the same prospect. Finally, the decision rule for the choice behaviour is no longer utility maximization, instead it is regret minimization. These behavioural theories have been widely used in different areas, such as psychology (Crawford et al., 2002), healthcare (Smith, 1996) and finance (Stoltz and Lugosi, 2005). In transport, RT has been applied recently to travellers’ responses to information within the context of Advanced Traveller Information Services (Chorus, 2011, Chorus et al., 2006a, Chorus et al., 2007).

Notice that several experimental tests have observed violations of monotonicity and transitivity under the RT framework (Loomes et al., 1991, Tsalatsanis et al., 2010).

Economists seem to reject the non-transitive situation in that it violates the whole theory of preference, although subsequent experimental results have revealed that these violations are largely due to experimental control, if one takes so called event-splitting effects into account (Starmer and Sugden, 1993).