Assume that when people compare two alternatives, it is not their absolute values but their value ranks that are compared. The probability that alternative i is considered better than alternative / is:

(5.24)

where is the rank of alternative value within the overall value set is the rank difference between alternative i and /. Thus the number of rank differences as well as the value ofis limited to K, the maximum number of overall values. It implies that the decision-maker sets a discriminant threshold to judge whether an alternative has enough advantage over the other, which is more behaviourally plausible than discriminating arbitrarily small utility differences. Note thatis not equivalent to the probability that alternative i is chosen over alternative /, which is defined as:

(5.25)

That means that alternative i is chosen when it has enough rank advantage over l. In addition, it is assumed that a uniform random choice is applied when neither

alternative is significantly better than the counterpart. Similarly, the expected choice probability considering latent preference structures turns out to be:

(5.26)

where is the probability of choosing alternative i over l based on discriminant threshold

The same specification as in Eq. (5.16) can be used to model the selection of comparison heuristics, with extra modifications on the mental effort, risk perception and expected outcome. The stopping rule for attribute search under a heuristics depends on whether the two alternatives can or cannot be discriminated when subsequent possible value ranks are considered. Based on Eqs. (5.17)-(5.21), expected effort is defined as:

(5.27)

(5.28)

(5.29)

In Eq. (5.27), because by definition processing one attribute implies attributes of both alternatives are processed in this context, the major difference is that the probability beliefs of attribute states of both alternatives have to be included in order to form a joint probability. Identity function is 0 when afteris processed, all subsequent possible value rank differences, ( means regardless of compare sequences between alternatives), are all smaller than the discriminant threshold, implying that alternatives cannot be discriminated or are all no smaller than the discriminant threshold, meaning alternatives can be discriminated. The process can stop here. Otherwise, attribute search will continue and the effort for processing the next attribute must be invested. The same definition logic applies to

For the specification of risk perception, only needs to be modified as:

(5.30)

where is the probability of alternative i having the overall value , and is the rank difference between the overall value ranks of the two alternatives. The specification of the expected outcome changes accordingly, withrepresenting the situation that the two alternatives can be discriminated under , and representing the situation that they cannot be discriminated.

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