Enhancement Effect

The Enhancement Effect

In support theory, increasing assessed support for a hypothesis need not decrease support for its competitors. Evidence compatible with every elementary hypothesis may increase support for them all. The enhancement assumption predicts that this will increase subadditivity in separate probability judgments.

Consider a case in which one of several suspects committed a murder. To simplify matters, assume that there are four suspects who, in the absence of specific evidence (low information), are considered equally likely to be guilty. Suppose further evidence is then introduced (high information) that implicates each of the suspects to roughly the same degree, so that they remain equally probable. Let L and H denote the evidence available under low and high information conditions, respectively. Let Ā denote the negation of A; that is, “Suspect A is not guilty.” Here the notation compares the first hypothesis with the second: a pair of suspects is a comparison between those suspects, not their joint guilt. Comparing a suspect with its negation gives the elementary judgment. According to the Bayesian model, then, P(A, B | H) = P(A, B | L) = 1/2, P(A, Ā| H) = P(A, Ā| L) = 1/4, and so forth.

In contrast, Shafer’s (1976) belief-function approach requires that the probabilities assigned to each of the suspects add to less than one and suggests that the total will be higher in the presence of direct evidence (i.e., in the high-information condition) than in its absence. As a consequence, 1/2 ≥ P(A,B | H) ≥ P(A,B | L), 1/4 ≥ P(A, Ā| H) ≥ P(A, Ā| L), and so forth. In other words, both the binary and the elementary judgments are expected to increase as more evidence is encountered. In the limit, when no belief is held in reserve, the binary judgments approach one-half and the elementary judgments approach one-fourth.

The enhancement assumption yields a different pattern; namely, P(A, B | H) = P(A, B | L) = 1/2, P(A, Ā| H) ≥ P(A, Ā| L) ≥ 1/4, and so forth. As in the Bayesian model, the binary judgments are one-half; in contrast to that model, however, the elementary judgments are expected to exceed one-fourth and to be greater under high- than under low-information conditions. Although both support theory and the belief-function approach yield greater elementary judgments under high- than under low-information conditions, support theory predicts that they will exceed one-fourth in both conditions, whereas Shafer’s theory requires that these probabilities be less than or equal to one-fourth.

The assumption of equally probable suspects is not essential for the analysis. Suppose that initially the suspects are not equally probable, but the new evidence does not change the binary probabilities. Here, too, the Bayesian model requires additive judgments that do not differ between low- and high-information conditions; the belief-function approach requires superadditive judgments that become less superadditive as more information is encountered; and the enhancement assumption predicts subadditive judgments that become more subadditive with the addition of (compatible) evidence.