In the preceding chapter we saw how rank dependence can be used to model pessimism and optimism. Another important component of probability weighting, orthogonal to the optimism–pessimism component, and cognitive rather than motivational, concerns likelihood sensitivity. This component is introduced informally in the following section, and is presented formally in §7.7. Several other extensions and applications of rank dependence are given.

Likelihood insensitivity and pessimism as two components of probabilistic risk attitudes

Figures 7.1.1–3 illustrate how two kinds of deviations from additive probabilities combine to create the probability weighting functions commonly found. Fig. 7.1.1a depicts traditional EU with probabilities weighted linearly; i.e., w(p) = p. Fig. 1b depicts pessimism as discussed in the preceding chapter.

Fig. 2a shows another psychological phenomenon. It reflects “diminishing sensitivity” for probabilities, which we will call likelihood insensitivity. Relative to EU, the weighting function is too shallow in the middle region, and too steep near both endpoints. An extreme case is shown in Fig. 3a. Here w is extremely steep at 0 and 1, and completely shallow in the middle. Such behavior is typically found if people distinguish only between “sure to happen,” “sure not to happen,” and “don't know.” An example of such a crude distinction is in Shackle (1949b p. 8). The expression 50–50 is commonly used to express such crude perceptions of uncertainty. “Either it happens or it won't; you can't say more about it.” is another way of expressing such beliefs. No distinction is made between different levels of likelihood.