Subadditivity effect

The subadditivity effect is the tendency to judge the probability of a disjunction of mutually exclusive events as lower than the sum of the probabilities of its individual components. Put plainly, one estimate for the whole can be smaller than separate estimates for the same nonoverlapping parts.

That comparison is meaningful only when the parts are mutually exclusive (no two can occur together) and exhaustive (the list covers every possibility in the whole). If categories overlap or an outcome is missing, a mismatched total may be a problem with the list rather than evidence of this effect.

The archive alarm that changed when its causes were named

The following scene is an illustrative hypothetical. The archive, alarms, people, numbers, and outcome are invented; they do not describe a study or public incident.

A fictional archive is reviewing overnight humidity alarms. For the exercise, every alarm receives one sealed root-cause code:

• sensor drift;
• power fluctuation;
• a door left open;
• HVAC fault; or
• another cause.

Only one code can be assigned, and the five codes cover every alarm.

Before seeing that list, Mara estimates a 30% chance that the next alarm will have any non-HVAC cause. Later, colleagues assess each non-HVAC cause separately. They give sensor drift 14%, power fluctuation 12%, an open door 11%, and another cause 9%. Those fictional component judgments total 46%, even though their union is the same non-HVAC event Mara judged at 30%.

No new evidence arrived. What changed was the description. Each named cause became a focal possibility with details that were easier to retrieve, while its competitors receded into a less vivid residual category.

Why probability should add up

For mutually exclusive events, probability theory requires:

P(A or B or C) = P(A) + P(B) + P(C)

If A, B, and C also exhaust the possible outcomes, their probabilities should total 100%. Psychological experiments have found that judged probabilities for disjunctions of more than two mutually exclusive events can be systematically lower than the sum of judgments for the individual events. This is a pattern in human estimates, not a change in the mathematical rule.

The archive example gives the same event two descriptions: “any non-HVAC cause” is packed, while “sensor drift, power fluctuation, an open door, or another cause” is unpacked. A coherent probability should remain the same across those extensionally equivalent descriptions. Human judgment does not always remain invariant.

Implicit and explicit subadditivity

These terms describe two related tests, and they should not be collapsed into one.

Implicit subadditivity

Implicit subadditivity compares a packed event with an unpacked description of that same event. If “a non-HVAC cause” receives 30%, but “sensor drift, power fluctuation, an open door, or another cause” receives 40% when judged as one explicit disjunction, unpacking has raised the estimate.

Explicit subadditivity

Explicit subadditivity compares a disjunction with the sum of separate judgments for its disjoint components. The whole might receive 30%, while separately assessed parts add to 46%. In a complete partition, a related procedure asks about each possible outcome one at a time; those focal judgments can total more than 100%.

The distinction matters because unpacking a single description and summing separately elicited probabilities are different tasks. They can engage different judgment processes.

What support theory explains

Tversky and Koehler's support theory treats subjective probability as a judgment about the support for a focal hypothesis relative to its alternative. The same event can attract different support when described differently.

Naming components may retrieve possibilities or evidence that a packed label did not bring to mind. When one component is judged alone, its alternatives may be compressed into a catchall such as “anything else.” That packing can reduce how fully the competing possibilities are represented. The focal component then receives a larger share of judged probability. Repeat the procedure for every component, and the separate estimates may add to more than 100%.

Rottenstreich and Tversky extended this account by separating implicit from explicit subadditivity. They linked the former to enhanced availability and proposed repacking and anchoring accounts for the latter. These are evidence-based explanations, but no single mechanism covers every result. Category interpretation, typicality, evidence strength, probability versus frequency formats, and the alternatives a person generates can all affect the pattern.

What the experiments found, and what they do not prove

In Brenner and Koehler's 1999 Experiment 1, 186 University of Waterloo students made predictions about the five nominees for the 1998 Best Picture Oscar. The sum of judgments made with each film separately in focus was 135% (SE 6.7%), above the additive benchmark of 100%. The researchers excluded another 54 participants who did not follow the allocation instruction requiring totals to equal 100%.

That result is a clean illustration of separately focal judgments exceeding the available probability. It does not establish a universal 35-point distortion. The task concerned an award prediction, the retained sample came from one university, and exclusions were substantial enough to report alongside the result.

The direction can also reverse. Sloman and colleagues tested how the typicality of named examples affected unpacking. In one experiment, 124 undergraduates judged nine well-defined categories. The packed mean was 65%. Unpacking with typical examples produced means of 63.8% and 63.5%, close to additivity, while atypical unpackings averaged 58.9%, below the packed judgment. A later experiment involved 169 undergraduates, 129 in the probability task and 40 in the typicality task, and again found strong decreases for atypical unpackings.

This counterevidence rules out a simple promise that “listing more parts fixes the estimate.” An atypical list may narrow what a category seems to mean or draw attention toward weak examples. The number, typicality, and structure of the named components matter; adding components alone does not determine the result.

What the subadditivity effect is not

• Conjunction fallacy: This occurs when A and B is judged more likely than A or B alone. It violates the rule for an intersection. Subadditivity concerns a union or partition of mutually exclusive events.

• Disjunction fallacy: This occurs when A or B is judged less likely than A alone or B alone. Explicit subadditivity only requires the union estimate to be below the sum of component estimates; the union need not fall below either component.

• Partition dependence: This is the broader finding that judgments change with the way a possibility space is divided. Subadditivity is one specific additivity failure that can appear when packed and unpacked partitions attract different support.

• Denominator neglect: This involves underweighting the denominator in a ratio or frequency comparison. Subadditivity can occur without any displayed ratio; its diagnostic comparison is between a whole and its nonoverlapping parts.

• Probability weighting: This usually describes how stated or objective probabilities are transformed when risky options are valued. Support theory addresses how a judged probability is constructed from descriptions and evidence in the first place.

Availability can contribute by making named possibilities easier to retrieve, but the availability heuristic is a proposed mechanism and neighboring pattern, not another name for subadditivity.