3. How Alike Is It? versus How Likely Is It?: A Disjunction Fallacy in Probability Judgments

FINAL COMMENTS

The goal of the heuristics and biases program in its early days was to understand intuitive judgment under uncertainty, not develop a unified theory of it. Judgment heuristics were described as a collection of disparate cognitive procedures that are bound together by their common function in a particular domain. In contrast, we argue here that heuristics share a common process of attribute substitution and are not limited to questions about uncertain events. Our treatment is otherwise firmly anchored in previous work. The substitution of one question for another, the representation of categories by prototypes, the view of erroneous intuitions as easy to override but almost impossible to eradicate – all these ideas are quite old (Kahneman, Slovic, & Tversky, 1982). We show here that the same ideas apply to a diverse class of difficult judgments, including retrospective evaluations of colonoscopies and decisions about saving birds from drowning in oil.

Ironically, the original research program might have been less interesting, and less influential, if the scope of its central concept had been appreciated and made explicit at the outset. The attention that the program attracted was due in no small part to an expository style in which illustrative examples embedded in the text turn readers into observers. A more general treatment could not have exploited this style because broad categories, in the classic Roschian sense, do not have ‘good’ exemplars. Just as a robin is a good bird but a mediocre animal, the Linda problem is a compelling illustration of a conjunction fallacy, but a less obvious example of a violation of dominance. The restriction of the original treatment of judgment heuristics to the domain of uncertainty was therefore as fortunate as it was arbitrary. We believe that the interpretation of representativeness and of other heuristics that we have offered here is both more complete and more general than earlier versions. But the original concepts, at least for us, had a charm that this more elaborate analysis cannot match.

* We thank Maya Bar-Hillel, Tom Gilovich, Dale Griffin, Ralph Hertwig, Denis Hilton, David Krantz, Barbara Mellers, Greg Pogarsky, Ilana Ritov, Norbert Schwarz, and Philip Tetlock for helpful comments.

3. How Alike Is It? versus How Likely Is It?: A Disjunction Fallacy in Probability Judgments *

Maya Bar-Hillel and Efrat Neter

The extension rule in probability theory states that if A is a subset of B, then the probability of A cannot exceed that of B. A special case of the extension rule is the conjunction rule, which states that the probability of A&B can exceed the probability of neither A nor B because it is contained in both.

Tversky and Kahneman (1983) demonstrated that under certain circumstances, people predictably and systematically violate the conjunction rule. In one study, they gave subjects the following description:

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations (p. 297).

This was followed by a list of eight possible outcomes, each describing possible activities of Linda at the present time (her job, her interests, or both). Subjects were asked to rank order the outcomes by the probability that they describe Linda’s current activities. Of the eight, one was representative of Linda (“Linda is active in the feminist movement”), one was unrepresentative of Linda (“Linda is a bank teller”), and one was a conjunction of these two (“Linda is a bank teller and is active in the feminist movement”). A large majority of the subjects (85%) rated the conjunctive outcome, “Linda is a bank teller and is active in the feminist movement,” more probable than “Linda is a bank teller.”

This result was predicted from the representativeness hypothesis: “Representativeness is an assessment of the degree of correspondence between a sample and a population, an instance and a category, an act and an actor or, more generally, between an outcome and a model” (Tversky & Kahneman, 1983, p. 295). Kahneman and Tversky (1972, 1973) provided much evidence that people often judge the probability of an outcome given a model by the extent to which the outcome represents the model. In addition, Tversky (1977) showed that adding to an outcome (O) a feature (F) that matches a model (M) enhances the match between the outcome and the model. In other words, the match of O&F to M can be greater than the match of O to M. Hence, insofar as people judge the probability of outcomes by their representativeness, being a bank teller and active in the feminist movement would be judged more likely an outcome for Linda than being a bank teller, due to the addition of a feature that is representative of Linda (feminism) to her unrepresentative job. Whereas there is nothing logically wrong with the judgment that being a feminist bank teller is more representative of Linda than being a bank teller, judging the conjunctive outcome to be more probable than its constituent violates the logically necessary conjunction rule.

Another special case of the extension rule is the disjunction rule, according to which the probability of A-or-B can be smaller than neither the probability of A nor the probability of B because it contains both. Formally speaking, there is no difference between the three rules (conjunction, disjunction, and extension) because for any pair of events A and B in which B is a subset of A, A can always be represented as a disjunction, one of whose constituents is B, and B can always be represented as a conjunction, one of whose constituents is A. For example, one can argue that the set of bank tellers is a disjunction – of bank tellers who are active feminists with bank tellers who are not. Viewed in this way, Tversky and Kahneman’s (1983) results could just as well have been labeled the disjunction fallacy. Why, then, are they regarded as a conjunction fallacy? Is this just a matter of arbitrary choice?

Formally speaking, the answer is yes, but psychological considerations favor one view over another. Consider the category parent and its subcategory mother. One can just as well choose to view mother as the conjunction of parent with female as to view parent as the disjunction of mother with father. In contrast, the category bank teller does not naturally evoke a representation as a union, and certainly not as a union of bank tellers who are active feminists with bank tellers who are not. At the same time, the subcategory of bank tellers who are active feminists can hardly be described except by resort to the conjunction of these two constituents. Indeed, the language does not even contain a single word label to designate this category. In that sense, the categories bank teller and bank teller and active feminist are more naturally viewed, respectively, as a unitary category and a conjunction of two categories than as a disjunction of two categories and a unitary category.

How, then, might one create a category that would be naturally viewed as disjunctive? The simplest possibility to come to mind is to replace the connective and used to create conjunctive categories with the connective or. This idea must be implemented with caution, however, because the English words and and or do not always quite correspond to the logical connectives and and or. First, the English or is often understood in its exclusive sense of “A or B, but not both,” as in, “The party will take place next week or the week after.” Second, the English and can be used to create a union as well as an intersection – the sentences, “She invited colleagues or relatives” and “She invited colleagues and relatives” could be used to describe the same guest list. Third, and most pertinent to present concerns, not all categories that can be joined meaningfully by one of these connectives lend themselves to as meaningful a joining by the other. For example, whereas putting and between bank teller and active in the feminist movement creates a meaningful category, putting or between these two category names creates a rather odd one. Similarly, whereas the question, “Is Linda more likely to be a bank teller, or a bank teller and active in the feminist movement?” makes some sense, the question, “Is Linda more likely to be a bank teller, or a bank teller or active in the feminist movement?” sounds to us rather confusing. . . .