Subadditivity

We extend the theory by providing a more detailed representation of subadditivity. Let A be an implicit hypothesis with the same extension as the explicit disjunction of the elementary hypotheses A1, . . ., A**n; that is, A′ = (A1 ∨ . . . ∨ A**n)′. Assume that any two elementary hypotheses, B and C, with the same extension have the same support; that is, B′, C′ ⊆ T and B′ = C′ implies s(B) = s(C). It follows that, under this assumption we can write:

[ s(A) = \sum_{i=1}^n w_i^A s(A_i) ]

In this representation, the support of each elementary hypothesis is "discounted" by its respective weight, which reflects the degree to which the judge attends to the hypothesis in question. If w_i^A = 1 for all i, then s(A) is the sum of the support of its elementary hypotheses, as in an explicit disjunction. However, w_j^A = 0 for some j indicates that A_j is effectively ignored. Finally, if the weights add to one, then s(A) is a weighted average of the s(A_i), 1 ≤ i ≤ n. This equation should not be interpreted as a process of deliberate discounting in which the judge assesses the support of an implicit disjunction by discounting the assessed support of the corresponding explicit disjunction. Instead, the weights are meant to represent the result of an assessment process in which the judge evaluates A without explicitly unpacking it into its elementary components. It should also be kept in mind that elementary hypotheses are defined relative to a given sample space. Such hypotheses may be broken down further by refining the level of description.

Note that whereas the support function is unique, except for a unit of measurement, the "local" weights w_i^A are not uniquely determined by the observed probability judgments. These data, however, determine the "global" weights w^A defined by:

[ w^A = \frac{s(A)}{s(A_1 \lor \ldots \lor A_n)} ]

The global weight w^A, which is the ratio of the support of the corresponding implicit (A) and explicit (A1 ∨ . . . ∨ A**n) disjunctions, provides a convenient measure of the degree of subadditivity induced by A. The degree of subadditivity, we propose, is influenced by several factors, one of which is the interpretation of the probability scale. Specifically, subadditivity is expected to be more pronounced when probability is interpreted as a propensity of an individual case than when it is equated with, or estimated by, relative frequency. Kahneman and Tversky (1979, 1982) referred to these modes of judgment as singular and distributional, respectively, and argued that the latter is usually more accurate than the former (see also Reeves & Lockhart, 1993). Although many events of interest cannot be interpreted in frequentistic terms, there are questions that can be framed in either a distributional or a singular mode. For example, people may be asked to assess the probability that an individual, selected at random from the general population, will die as a result of an accident. Alternatively, people may be asked to assess the percentage (or relative frequency) of the population that will die as a result of an accident. We propose that the implicit disjunction "accident" is more readily unpacked into its components (e.g., car accidents, plane crashes, fire, drowning, poisoning) when the judge considers the entire population rather than a single person. The various causes of death are all represented in the population's mortality statistics, but not in the death of a single person. More generally, we propose that the tendency to unpack an implicit disjunction is stronger in the distributional than in the singular mode. Hence, a frequentistic formulation is expected to produce less discounting (i.e., higher w s) than a formulation that refers to an individual case.