Subadditivity of Support

Determinants of Subadditivity of Support

Tversky and Koehler (1994) introduced subadditivity as a general feature of the support scale: the total support for a disjunctive hypothesis was assumed to be less than or equal to the sum of the support values for its components. The amount of subadditivity, however, was essentially treated as a free parameter, likely to be influenced by various characteristics of the judgment context. Subsequent research has examined the relationship between various features of judgment tasks and the degree of subadditivity of support.

Original Demonstrations of Variations in Subadditivity. Tversky and Koehler (1994) initially proposed two influences on the degree of subadditivity. The first is the number of components in an unpacked hypothesis. They predicted greater subadditivity would be found for unpacked hypotheses involving more components, and indeed found that hypotheses unpacked into seven components yielded more subadditivity than hypotheses unpacked into three components.

A second influence on subadditivity proposed by Tversky and Koehler concerned whether the judged probability is interpreted as a propensity of an individual case or interpreted in terms of a relative frequency applied to a collection of cases. As an illustration of this distinction, consider a physician evaluating either the probability that an individual patient will survive or evaluating the percentage of patients sharing common symptoms who will survive. Tversky and Koehler (1994) predicted more subadditivity for judged probabilities than for judged percentages because evaluating collections of instances is more likely to invoke considerations of inclusion and exclusion relations among hypotheses. They indeed found greater subadditivity for judgments of probability than for judgments of relative frequency.

Rottenstreich and Tversky (1997) considered two separate sources of subadditivity. Support for an implicit disjunction was assumed to be less than or equal to support for a corresponding explicit disjunction, and the latter was assumed to be less than or equal to the sum of supports for the disjunction’s components. The former is labeled implicit subadditivity; the latter explicit subadditivity. In several experiments, Rottenstreich and Tversky (1997) examined how the similarity between the components of a disjunction influences the degree of explicit subadditivity. They found that explicit disjunctions involving similar components, which are more easily “repacked” into a composite hypothesis, yielded greater explicit subadditivity than disjunctions involving dissimilar components.

Similarity between Hypotheses and Evidence. Rottenstreich, Brenner, and Sood (1999) examined another aspect of similarity as a determinant of subadditivity. Support for a hypothesis may often be closely related to the similarity or conceptual match between a hypothesis and the background evidence (as when the representativeness heuristic is used to assess likelihood). For example, in a criminal trial, the likelihood of a defendant’s guilt may be judged by the extent to which the evidence introduced at trial matches aspects of the scenario of guilt proposed by the prosecution. How the target hypothesis is described may influence the perceived degree of similarity or match with the evidence. The hypothesis “The defendant murdered Mr. Smith” may match the evidence less well than an elaborated and detailed hypothesis, such as, “The defendant used his key to enter the locked office and murdered Mr. Smith using the decorative bust on the mantle.” Such elaborated, more restrictive hypotheses may seem more likely because they appear to match the evidence better than broader hypotheses, a pattern termed the conjunction fallacy (Tversky & Kahneman, 1983).

To address the role of similarity in subadditivity of support, Rottenstreich et al. (1999) contrasted judgments of disjunctive hypotheses in the presence of specific evidence (case judgments) with those made in the absence of specific evidence (class judgments). In case judgments, the perceived similarity between hypotheses and evidence is likely to dominate evaluation of support, but in class judgments the similarity between hypotheses and evidence is less readily evaluated, and other mechanisms (such as availability) presumably drive the assessment of support.

In one study, subjects read a description of a character Linda, whose background evidence closely matched the hypothesis journalist. Subjects then estimated the probability that Linda works as either a journalist or realtor. Whereas the description of Linda may appear highly similar to journalist, it may in fact appear less similar to the disjunction journalist or realtor, because the nonmatching component realtor dilutes the strong similarity with journalist. Such dilution is unlikely to occur in class judgments, where similarity does not form the basis of the judgments. As a result, Rottenstreich et al. (1999) predicted more subadditivity when judgments of likelihood were based on similarity, rather than other factors such as availability. Consistent with their predictions, judgment based on similarity yielded extreme forms of subadditivity. Judgments based on similarity often showed nonmonotonicity of support, in which support for a disjunction is less than the support for one of its components – a pattern analogous to the conjunction fallacy, but evaluated at the level of support rather than probability (cf. Tversky & Kahneman, 1983).

Subadditivity for Residual Hypotheses. Another stream of research has examined the determinants of subadditivity for residual hypotheses, defined as the complement of a particular hypothesis (e.g., “A team other than the New York Yankees will win the World Series”; “The patient suffers from something other than pneumonia”).

Koehler et al. (1997) proposed that subadditivity would be greater for residual hypotheses defined by strong focal hypotheses. Specifically, they tested a linear discounting model in which the subadditivity weight w_Ā for a residual hypothesis decreases with the support for the focal hypothesis that defines the residual:

The subadditivity weight wĀ represents the degree of subadditivity for the residual hypothesis Ā (read as “not A” or “the complement of A”). For example, if there are four mutually exclusive and exhaustive hypotheses A, B, C, and D, then wĀ = s(Ā)/[s(B) + s(C) + s(D)]. Smaller values of w represent more subadditivity. The linear discounting model was intended to capture the intuition that when evidence for the focal possibility is strong, the residual hypothesis (i.e., the set of all other possibilities) is less likely to be spontaneously unpacked into its components, compared to when the focal hypothesis is weak (which might yield more careful scrutiny of alternative possibilities). Koehler et al. (1997) found consistent support for both the qualitative and quantitative predictions of the linear discounting model.

Brenner and Koehler (1999) examined subadditivity of residual hypotheses in greater detail. Rather than measuring subadditivity for a residual hypothesis with the global weight w_Ā, Brenner and Koehler examined local weights that measure the specific contribution of each component included within the residual. Using local weights, the total support for a residual hypothesis can be expressed as a weighted sum of its components’ support values:

In this formulation, the local weight w(B, Ā) is interpreted as, “the weight of component B within the residual hypothesis Ā.”

Brenner and Koehler (1999) proposed and tested several ordinal properties of local weights. Properties analogous to subadditivity and enhancement were found to hold for local weights as well as global weights. A generalization of the linear discounting model accounted for most of the variability in these local weights, giving a compact model of the “microstructure” of support for residual hypotheses.

Evidential Influences on Subadditivity. Koehler (2000) examined the effects of several evidential features on the degree of subadditivity. In a simulated medical diagnosis task, participants judged the likelihood of a designated diagnosis (flu strain 1, 2, or 3) on the basis of a pattern of symptoms (e.g., cough, sore throat, chills). Over many trials, participants judged the likelihood of each of the three possible flu strains for a given pattern of symptoms; the total of these judged probabilities reflects the degree of subadditivity of the residual hypothesis. Subadditivity was greater when symptoms were in conflict with each other – for example when a pattern of symptoms included both cough (implicating flu strain 1) and sore throat (implicating flu strain 2). The effect of cue conflict was largely accounted for by the linear discounting model of Koehler et al. (1997). According to this interpretation, cue conflict leads to greater perceived support for the focal hypothesis, which in turn produces greater discounting of the residual hypothesis.

Summary of Factors Influencing Subadditivity. The studies reviewed here found that the degree of subadditivity varied with several factors, including the number of components in a hypothesis, the similarity of those components to each other and to the evidence, the strength of the focal hypothesis, the conflict among evidential cues, and whether the judgment is elicited in terms of probability or relative frequency. These seemingly unrelated factors all influence either (1) the ease with which a hypothesis is naturally or spontaneously broken into pieces, or (2) the extent to which compelling evidence is seen to support the components of a disjunctive hypothesis. The more easily evidence is seen to support individual components of a “packed” hypothesis, the less subadditivity is expected. Put another way: When a hypothesis is unpacked into components, the unpacking effect should be greatest when the act of unpacking recruits evidence for the unpacked components that would not be brought to mind by the packed hypothesis. Thus, evidence that encourages spontaneous unpacking of a hypothesis and supports individual components should yield little or no subadditivity.