Subadditivity in Expert Judgments
People may assign a broad event less probability than the sum they assign to its mutually exclusive subcategories when those are described separately. Studies of experts suggest that professional knowledge does not always prevent this inconsistency.
THE IDEA
Subadditivity in Expert Judgments
Is subadditivity confined to novices, or does it also hold for experts? Redelmeier, Koehler, Liberman, and Tversky (1995) explored this question in the context of medical judgments. They presented physicians at Stanford University (N = 59) with a detailed scenario concerning a woman who reported to the emergency room with abdominal pain. Half of the respondents were asked to assign probabilities to two specified diagnoses (gastroenteritis and ectopic pregnancy) and a residual category (none of the above); the other half assigned probabilities to five specified diagnoses (including the two presented in the other condition) and a residual category (none of the above). Subjects were instructed to give probabilities that summed to one because the possibilities under consideration were mutually exclusive and exhaustive. If the physicians’ judgments conform to the classical theory, then the probability assigned to the residual category in the two-diagnosis condition should equal the sum of the probabilities assigned to its unpacked components in the five-diagnosis condition. Consistent with the predictions of support theory, however, the judged probability of the residual in the two-diagnosis condition (mean = .50) was significantly lower than that of the unpacked components in the five-diagnosis condition (mean = .69), p < .005 (Mann–Whitney U test).
In a second study, physicians from Tel Aviv University (N = 52) were asked to consider several medical scenarios consisting of a one-paragraph statement including the patient’s age, gender, medical history, presenting symptoms, and the results of any tests that had been conducted. One scenario, for example, concerned a 67-year-old man who arrived in the emergency room suffering a heart attack that had begun several hours earlier. Each physician was asked to assess the probability of one of the following four hypotheses: patient dies during this hospital admission (A); patient is discharged alive but dies within 1 year (B); patient lives more than 1 but less than 10 years (C); or patient lives more than 10 years (D). The authors call these elementary judgments because they pit an elementary hypothesis against its complement, which is an implicit disjunction of all of the remaining elementary hypotheses. After assessing one of these four hypotheses, all respondents assessed P(A, B), P(B, C), and P(C, D) or the complementary set. We refer to these as binary judgments because they involve a comparison of two elementary hypotheses.
As predicted, the elementary judgments were substantially subadditive. The means of the four groups in the preceding example were 14% for A, 26% for B, 55% for C, and 69% for D, all of which overestimated the actual values reported in the medical literature. In problems like this, when individual components of a partition are evaluated against the residual, the denominator of the unpacking factor is taken to be 1; thus, the unpacking factor is simply the total probability assigned to the components (summed over different groups of subjects). In this example, the unpacking factor was 1.64. In sharp contrast, the binary judgments (produced by two different groups of physicians) exhibited near-perfect additivity, with a mean total of 100.5% assigned to complementary pairs.
Further evidence for subadditivity in expert judgment has been provided by Fox, Rogers, and Tversky (1996), who investigated 32 professional options traders at the Pacific Stock Exchange. These traders made probability judgments regarding the closing price of Microsoft stock on a given future date (e.g., it will be less than $88 per share). In the study’s setting, these traders dealt with Microsoft stock options and routinely assessed possible future prices. Nevertheless, their judgments exhibited the predicted pattern of subadditivity and binary complementarity. The average unpacking factor for a fourfold partition was 1.47, and the average sum of complementary binary events was 0.98. Subadditivity in expert judgments has been documented in other domains by Fischhoff et al. (1978), who studied auto mechanics, and by Dube-Rioux and Russo (1988), who studied restaurant managers.
When to use it
In diagnosis, compare the probability assigned to a broad residual category with the total assigned to the same mutually exclusive diagnoses when they are listed separately.
In financial forecasting, analysts who enumerate multiple possible outcomes (e.g., stock price ranges) may overestimate the total probability, affecting risk assessment and portfolio allocation.
In project planning, compare the probability of any delay with the sum assigned to a set of mutually exclusive delay scenarios. Define the scenarios carefully so they do not overlap.
In legal reasoning, check whether probabilities assigned to mutually exclusive accounts of an event are consistent with the probability assigned to their combined category.
FROM THEORY TO EVERYDAY LIFE
In practice
Medical Diagnosis Example
Physicians at Stanford were presented a scenario of a woman with abdominal pain. When only two diagnoses (gastroenteritis and ectopic pregnancy) plus a residual category were listed, the average probability assigned to the residual was 0.50. When five specific diagnoses were listed (unpacking the residual), the combined probability of the unpacked components was 0.69, demonstrating subadditivity.
Options Trading Example
Professional traders at the Pacific Stock Exchange judged probabilities for the closing price of Microsoft stock. For a fourfold partition into price ranges—illustrative ranges might be less than $88 or between $88 and $92—the sum of probabilities across components averaged 1.47, exceeding 1.0 and indicating subadditivity.
A MOMENT TO REFLECT
Put it to work.
Think of a time you estimated the likelihood of an event with several subcategories. Did the sum of the subcategory probabilities exceed the probability you assigned to the general category?
How might subadditivity affect expert decision-making in high-stakes fields like medicine or finance? Can you identify a real-world case where this bias may have had consequences?
When you unpack a general category into specific components, why do you think the total probability increases? What psychological mechanisms (e.g., availability, representativeness) might drive this effect?
Often confused with
References
- Subadditivity in medical judgments — Redelmeier, D. A., Koehler, D. J., Liberman, V., & Tversky, A. (1995)
- Subadditivity in financial judgments — Fox, C. R., Rogers, B. A., & Tversky, A. (1996)
- Subadditivity in auto mechanics — Fischhoff, B., et al. (1978)
- Subadditivity in restaurant managers — Dube-Rioux, L., & Russo, J. E. (1988)
CONTINUE THE CONVERSATION
Source texts
- Heuristics and Biases The Psychology of Intuitive Judgment (Thomas Gilovich, Dale Griffin, Daniel Kahneman)