Subadditivity Effect: Why the Parts Add Up to More Than the Whole

In probability theory, an event's likelihood does not depend on how it is described: the chance that a death was from natural causes is the chance that it was from heart disease, or cancer, or any other natural cause, because those are the same event. Amos Tversky and Derek Koehler's support theory, published in 1994, starts from the observation that judged probability does not behave this way. People assign probability to descriptions of events, which the theory calls hypotheses, and different descriptions of the same event receive different numbers.

The theory's central quantity is support: the strength of the evidence a person can bring to mind for a hypothesis. A one-word description such as natural causes is judged from whatever comes to mind as a typical case, and much of the category never gets considered. Spelling the category out, as heart disease, cancer, stroke and so on, brings more of it to mind and increases its support. Two predictions follow. Unpacking a hypothesis into its parts raises its judged probability, and unpacking the alternative lowers it. And when the parts are judged separately, their probabilities add up to more than the probability given to the whole, which is the subadditivity the page is named for. In the binary case, where a hypothesis and its complement are judged, the two still add to about one; it is the splitting of one side into several that breaks additivity.

The effect is not confined to laypeople. Redelmeier, Koehler, Liberman and Tversky presented physicians with brief clinical scenarios in 1995 and found that they discounted possibilities that were not spelled out, as the theory predicts. Fischhoff, Slovic and Lichtenstein had found the same shape in 1978 with fault trees for why a car might not start: when branches were pruned from the tree, the probability assigned to the catch-all branch, all other problems, rose far less than the pruned branches had carried, so the missing causes were effectively forgotten.

Tversky and Koehler also found subadditivity more pronounced for probability judgments than for frequency judgments: asking how many out of a hundred deaths, rather than how likely a single death, brought the parts closer to the whole, which is one reason forecasters and clinicians are advised to work in frequencies.

Tversky and Koehler published "Support theory: A nonextensional representation of subjective probability" in Psychological Review in 1994. The paper combines a formal theory with experiments designed to test its predictions, of which the causes-of-death studies are the best known. Participants judged the probability that a person selected at random from those who had died in the previous year had died from each cause in a list. Each participant saw one of two kinds of list: an implicit disjunction, a category named in a word or two, such as death from natural causes, or the same category spelled out as an explicit disjunction, death from heart disease, cancer, or some other natural cause, or split into its parts to be judged one by one.

The mean probability given to the implicit description was lower than the sum of the mean probabilities given to its parts, and the degree of subadditivity increased with the number of parts the category was divided into. The same pattern held for unnatural causes, split into accident, homicide and other unnatural causes. Because each participant judged only one kind of list, the comparison was made between groups, so no one was asked to hold two inconsistent numbers in mind at once; the inconsistency appears only when the groups are set side by side.

Other experiments tested the theory's other predictions. Judgments of a hypothesis and its complement added to approximately one, as the theory requires for the binary case, and unpacking a hypothesis raised its judged probability while unpacking the alternative lowered it. The authors argued that no theory assigning probability to events rather than descriptions could capture these results.

The demo on this page departs from the design in one respect that matters. Because it compares one visitor's own answers, you judged both the one-word description and the spelled-out one, where the 1994 participants saw one or the other, and both sat on the same screen, so you could set them to match. An answer that matches is the careful response, not a failure of the demo. A gap between your two answers is consistent with the effect; the between-groups comparison in the paper is what shows that the description, not the person, moved the number. The recorded shares drawn behind your answers are a benchmark for the estimates, not a test of the effect, which concerns the relation between your answers rather than their accuracy.

The rows below are illustrations of where a category estimated as a word differs from the same category estimated as a list, not findings from the studies cited on this page.

The number follows the description, so the counters change the description or check one description against another.

• Estimate the whole and the parts both ways, then reconcile. If the parts add to more than the whole, one set of numbers is wrong, and the exercise shows which cases you had left out.
• Always include a line for everything else, and estimate it first. The catch-all is where unconsidered cases live, and in the 1978 fault-tree study it was the line that failed to absorb what had been pruned.
• Normalize when a list is complete: force exhaustive, exclusive options to sum to 100 percent and notice which ones shrink.
• Use frequencies. Ask how many out of 100 rather than how likely; Tversky and Koehler found subadditivity weaker for frequency judgments.
• Be suspicious of the detailed scenario. A vividly described possibility borrows probability from the vague remainder.
• When someone else supplies the list, ask who chose the items. A list is an argument about where the probability should go.

The subadditivity effect is one of several ways judged probability departs from the mathematics, and its neighbors are easy to confuse with it.

• It is not the conjunction fallacy, in which a detailed event is judged more likely than a general event that contains it, such as a bank teller who is active in the feminist movement against a bank teller. That concerns a single conjunction judged more probable than its parent; subadditivity concerns the parts of a category together outweighing the category.
• It is not neglect of probability, where probabilities are ignored in favor of outcomes. Subadditivity happens while people are attending carefully to probabilities.
• It is not the availability heuristic, though it is related. Availability is the use of ease of recall as a cue to frequency; support theory generalizes the idea to the evidence a description brings to mind, and predicts the specific pattern of unpacking and subadditivity that availability alone does not.
• It is not the base rate fallacy, which concerns ignoring prior probabilities when evidence arrives. Subadditivity needs no evidence and no updating; it appears in a single judgment.