Unpacking Effect (Subadditivity in Support Theory)

Review of Previous Research

Several studies have tested the predictions of support theory. Tversky and Fox (1994) asked subjects to assign probabilities to various intervals in which an uncertain quantity might fall, such as the margin of victory in a future Super Bowl or the change in the Dow Jones Industrial Average over the next week. When a given event (e.g., “Buffalo beats Washington”) was unpacked into individually evaluated components (e.g., “Buffalo beats Washington by less than 7 points”; “Buffalo beats Washington by at least 7 points”), subjects’ judgments were substantially subadditive. [Figure 25.1] plots the unpacking factor obtained in this study as a function of the number of component hypotheses in the explicit disjunction. Judgments for five different types of event are shown: future San Francisco temperature (SFO), future Beijing temperature (BJG), the outcome of the Super Bowl of the National Football League (NFL), the outcome of a playoff game of the National Basketball Association (NBA), and weekly change in the Dow Jones index (DOW). The unpacking factor is the sum of the probabilities assigned to the parts divided by the probability assigned to the whole event. A factor greater than 1 (i.e., falling above the dashed line in the plot) indicates subadditivity. The results displayed in Fig. 25.1 reveal consistent subadditivity for all sources that increases with the number of components in the explicit disjunction.

Figure 25.1. Unpacking factors from Tversky and Fox’s (1994) data. SFO, San Francisco temperature; BJG, Beijing temperature; NFL, 1991 National Football League Super Bowl; NBA, National Basketball Association playoff; DOW, weekly change in Dow–Jones index.

[Figure 25.2] plots the median probabilities assigned to complementary hypotheses. As predicted by support theory, judgments of intervals representing complementary pairs of hypotheses were essentially additive, with no apparent tendency toward either subadditivity or superadditivity.

Figure 25.2. A test of binary complementarity based on Tversky and Fox (1994).

Further evidence for binary complementarity comes from an extensive study conducted by Wallsten, Budescu, and Zwick (1992), who presented subjects with 300 propositions concerning world history and geography (e.g., “The Monroe Doctrine was proclaimed before the Republican Party was founded”) and asked them to estimate the probability that each was true. True and false (complementary) versions of each proposition were presented on different days. [Figure 25.3] plots the mean probabilities assigned to each of the propositions in both their true and false versions using the format of Fig. 25.2. Again, the judgments are additive (mean = 1.02) through the entire range.

Figure 25.3. A test of binary complementarity based on Wallsten, Budescu, and Zwick (1992).

The findings concern subadditivity, the number of components, binary complementarity, and the difference between probability and frequency judgments.

Subadditivity. Unpacking an implicit hypothesis into its component hypotheses increases its total judged probability, yielding subadditive judgments. [Tables 25.3] and [25.4] list studies that provide tests of the unpacking condition. For each experiment, the probability assigned to the implicit hypothesis and the total probability assigned to its components in the explicit disjunction are listed along with the resulting unpacking factor. All of the listed studies used an experimental design in which the implicit disjunction and the components of the explicit disjunction were evaluated independently, either by separate groups of subjects or by the same subjects but with a substantial number of intervening judgments. The probabilities are listed as a function of the number of components in the explicit disjunction and are collapsed over all other independent variables. [Table 25.3] lists studies in which subjects evaluated the probability of qualitative hypotheses (e.g., the probability that Bill W. majors in psychology); [Table 25.4] lists studies in which subjects evaluated quantitative hypotheses (e.g., the probability that a randomly selected adult man is between 6 ft and 6 ft 2 in. tall).

Table 25.3. Results of Experiments Using Qualitative Hypotheses: Average Probability Assigned to Coextensional Implicit and Explicit Disjunctions and the Unpacking Factor Measuring the Degree of Subadditivity

Note: The number of components in the explicit disjunction is denoted by n. Numbered studies with no citation refer to Tversky & Koehler, 1994. a Because the components partition the space, it is assumed that a probability of 1.00 would have been assigned to the implicit disjunction.

[Tables 25.3] and [25.4] show that the observed unpacking factors are, without exception, greater than one, indicating consistent subadditivity. The fact that subadditivity is observed both for qualitative and for quantitative hypotheses is instructive. Subadditivity in assessments of qualitative hypotheses can be explained, in part at least, by the failure to consider one or more component hypotheses when the event in question is described in an implicit form. The subadditivity observed in judgments of quantitative hypotheses, however, cannot be explained as a retrieval failure. For example, Teigen (1974b, Experiment 2) found that the judged proportion of college students whose heights fell in a given interval increased when that interval was broken into several smaller intervals that were assessed separately. Subjects evaluating the implicit disjunction (i.e., the large interval), we suggest, did not overlook the fact that the interval included several smaller intervals; rather, the unpacking manipulation enhanced the salience of these intervals and, hence, their judged probability. Subadditivity, therefore, is observed even in the absence of memory limitations.

Table 25.4. Results of Experiments Using Quantitative Hypotheses: Average Probability Assigned to Coextensional Implicit and Explicit Disjunctions and the Unpacking Factor Measuring the Degree of Subadditivity

Note: The number of components in the explicit disjunction is denoted by n. Numbered studies with no citation refer to Tversky & Koehler, 1994. a Because the components partition the space, it is assumed that a probability of 1.00 would have been assigned to the implicit disjunction.

Number of Components. The degree of subadditivity increases with the number of components in the explicit disjunction. This follows readily from support theory: Unpacking an implicit hypothesis into exclusive components increases its total judged probability, and additional unpacking of each component should further increase the total probability assigned to the initial hypothesis. [Tables 25.3] and [25.4] show, as expected, that the unpacking factor generally increases with the number of components (see also Fig. 25.1).

Table 25.5. Results of Experiments Testing Binary Complementarity: Average Total Probability Assigned to Complementary Pairs of Hypotheses, Between-Subjects Standard Deviations, and the Number of Subjects in the Experiment

Note: Numbered studies with no citation refer to Tversky & Koehler, 1994. NBA = National Basketball Association. a A given subject evaluated either the event or its complement, but not both.

Binary Complementarity. The judged probabilities of complementary pairs of hypotheses add to one. [Table 25.5] lists studies that have tested this prediction. We considered only studies in which the hypothesis and its complement were evaluated independently, either by different subjects or by the same subjects but with a substantial number of intervening judgments. [Table 25.5] shows that such judgments generally add to one. Binary complementarity indicates that people evaluate a given hypothesis relative to its complement. Moreover, it rules out alternative interpretations of subadditivity in terms of a suggestion effect or a confirmation bias. These accounts imply a bias in favor of the focal hypothesis yielding P(A,B) + P(B,A) > 1, contrary to the experimental evidence. Alternatively, one might be tempted to attribute the subadditivity observed in probability judgments to subjects’ lack of knowledge of the additivity principle of probability theory. This explanation, however, fails to account for the observed subadditivity in frequency judgments (in which additivity is obvious) and for the finding of binary complementarity (in which additivity is consistently satisfied).

The combination of binary complementarity and subadditive elementary judgments, implied by support theory, is inconsistent with both Bayesian and revisionist models. The Bayesian model implies that the unpacking factor should equal one because the unpacked and packed hypotheses have the same extension. Shafer’s theory of belief functions and other models of lower probability require an unpacking factor of less than one, because they assume that the subjective probability (or belief) of the union of disjoint events is generally greater than the sum of the probabilities of its exclusive constituents. Furthermore, the data cannot be explained by the dual of the belief function (called the plausibility function) or, more generally, by an upper probability (e.g., see Dempster, 1967) because this model requires that the sum of the assessments of complementary events exceed unity, contrary to the evidence. Indeed, if P(A,B) + P(B,A) = 1 (see Table 25.5), then both upper and lower probability reduce to the standard additive model. The experimental findings, of course, do not invalidate the use of upper and lower probability, or belief functions, as formal systems for representing uncertainty. However, the evidence from these studies indicates that these models are inconsistent with the principles that govern intuitive probability judgments.

Probability versus Frequency. Of the studies discussed earlier and listed in Tables 25.3 and 25.4, some (e.g., Fischhoff et al., 1978) used frequency judgments and others (e.g., Teigen, 1974a, 1974b) used probability judgments. The comparison of the two tasks, summarized in Table 25.6, confirms the predicted pattern: Subadditivity holds for both probability and frequency judgments, and the former are more subadditive than the latter.

Table 25.6. Results of Experiments Comparing Probability and Frequency Judgments: Unpacking Factor Computed From Mean Probability Assigned to Coextensional Explicit and Implicit Disjunctions