Support Theory

Support Theory

Support theory is a nonextensional model of subjective probability. Let (T) be a finite set of at least two states of the world, exactly one of which obtains, though the judge does not know which. Subsets of (T) are events, while verbal descriptions of events are hypotheses. The set of hypotheses (H) is finite and includes at least one description for each event; different hypotheses may describe the same event (e.g., rolling a pair of dice: “the sum is 3” and “the product is 2” both pick out the same outcome). (H) is closed under explicit disjunction: if (A) and (B) are exclusive, their explicit disjunction (A \vee B) is in (H), with ((A \vee B)' = A' \cup B').

A central distinction is between explicit and implicit disjunctions. An implicit hypothesis is neither elementary, null, nor an explicit disjunction of exclusive non-null components. For example, “Ann majors in a natural science” is coextensional with the explicit disjunction “Ann majors in either a biological science or a physical science,” but it is implicit because it is not expressed as a disjunction.

An evaluation frame ((A, B)) consists of a focal hypothesis (A) and an alternative hypothesis (B), assumed exclusive. The judged probability (P(A, B)) is the subjective probability that (A) rather than (B) holds, given that exactly one of them obtains. In this model, judged probability depends on the descriptions (A) and (B), not only on the underlying events.

Support theory postulates a ratio-scale support function (s) on hypotheses such that for any exclusive (A, B \in H),

[ P(A, B) = \frac{s(A)}{s(A) + s(B)}. ]

If (B) and (C) are exclusive and (A) is an implicit disjunction coextensional with (B \vee C), then

[ s(A) \le s(B) + s(C). ]

Support is additive for explicit disjunctions and subadditive for implicit ones. This subadditivity reflects a basic feature of human judgment: when evaluating an implicit disjunction, people form a global impression based on the most available or representative instances, rather than exhaustively unpacking it. Unpacking an implicit category into explicit components tends to increase total support, because it reminds the judge of otherwise neglected possibilities and enhances the salience of the mentioned outcomes. For instance, support for “death from an unnatural cause” is typically less than the sum of support for “homicide,” “fatal car accidents,” “drowning,” etc.

Support is interpreted as the strength of evidence available to the judge. It may derive from objective data, from heuristic cues such as representativeness or availability (e.g., Bill’s resemblance to the stereotype of an accountant; how easily similar oil spills come to mind), or from recruited reasons and arguments. Because support is based on such impressionistic evidence, it is not monotonic with respect to set inclusion: (s(B)) may exceed (s(A)) even though (A' \supset B'), though for explicit disjunctions (s(B) \le s(B \vee C)). A well-known example: support for “a randomly selected word ends with ing” can exceed support for “has n in the next-to-last position,” but once the inclusion relation is made transparent by reformulating the latter as “ends with ing or otherwise has n in the next-to-last position,” the unpacked version receives greater support.

Direct Assessments of Support

Because support is a function of a single hypothesis, it can be elicited independently of alternatives and without explicit reference to chance. Several studies show that direct ratings of support accurately predict judged probabilities. For example:

• Koehler, Brenner, and Tversky (1997, Experiment 1) used relatedness ratings between academic fields (e.g., Economics and Statistics) to predict the probability that a student who took a Statistics course is majoring in Economics.
• Ratings of “suspiciousness” of a suspect in a crime story predicted the judged probability of guilt (Koehler et al., 1997, Experiment 5).
• Judgments of resemblance to a prototypical flu patient predicted the probability that a patient had that strain (Koehler, 2000, Experiment 5).
• Direct ratings of “team strength” predicted the judged probability that one team would defeat another (e.g., Fox & Tversky, 1998; Koehler, 1996).

These results support modeling probability judgments as comparisons between the support for a focal hypothesis and its alternative. Predicting those judgments does not establish that either the support ratings or the resulting probabilities are accurate.

Support theory remains mostly agnostic about the determinants of support; they vary with domain, expertise, and the accessibility of evidence. However, most direct support ratings primarily reflect the strength of evidence—how strongly it points toward the hypothesis—rather than its weight (reliability or predictive validity). For instance, a suspect’s failure on a lie-detector test may markedly increase her perceived suspiciousness even when the evidence is known to be unreliable, an example of strength overpowering weight. Courts attempt to guard against this by excluding evidence where prejudicial value outweighs probative value.

Consequences

The theory yields several testable properties of the observable probability judgments (P(A,B)):

• Binary complementarity: (P(A,B) + P(B,A) = 1), as long as neither hypothesis is null.
• Proportionality: For mutually exclusive (A, B, C) with (B) non-null, the odds of (A) against (B) are independent of (C): (P(A, B \vee C) / P(B, A \vee C) = P(A, B) / P(B, A)).
• Product rule: For exclusive, non-null pairs, the product of the odds for (A) against (B) and for (C) against (D) equals the product of the odds for (A) against (D) and for (C) against (B).
• Unpacking principle: If (B, C, D) are mutually exclusive and (A) is an implicit disjunction coextensional with (B \vee C), then [ P(B \vee C, D) \ge P(A, D) \quad \text{and} \quad P(D, A) \ge P(D, B \vee C). ] That is, unpacking the focal hypothesis raises its judged probability and unpacking the alternative hypothesis lowers the focal’s probability. Stated differently, subjective probability is subadditive for implicit disjunctions, in contrast to additive for explicit ones.

These conditions are also sufficient: a theorem shows that if judged probabilities satisfy binary complementarity, proportionality, the product rule, and the unpacking principle, then there exists a nonnegative ratio scale (s) on (H) that represents them via the support equation with the subadditivity property for implicit disjunctions. The unpacking principle distinguishes support theory from lower‑probability models (e.g., Shafer’s theory), which assume extensionality and superadditivity, reflecting a different intuition—that in partial ignorance people hold some belief in reserve. The studies discussed here provide evidence for the subadditivity predicted by support theory.

Extensions

Studies discussed in this account report several patterns consistent with support theory:

• Probability judgments increase when the focal hypothesis is unpacked and decrease when the alternative hypothesis is unpacked.
• Binary complementarity holds, but in the general case subjective probabilities are subadditive.
• Subadditivity is more pronounced for verbally expressed probability judgments than for judgments of frequency, and it is enhanced when the components are compatible with the focal hypothesis.
• The theory provides a method for predicting judged probability from independent assessments of evidential support.

Support theory allows judgments to violate set inclusion: a narrower hypothesis may receive more support than a broader one. It can therefore represent the conjunction effect, as in the Linda problem, when a conjunction receives more support than one of its constituents. This does not mean that the support values of the conjuncts are simply added. The additive rule in the model applies to explicit disjunctions of exclusive hypotheses. Explaining what makes a particular description persuasive requires further assumptions about the evidence people attend to.