antiadditivity

Antiadditivity is a mathematical property of functions or measures that contrasts with additivity. In an additive system, the value associated with a combined entity equals the sum of the values of its components: f(A ∪ B)=f(A)+f(B) when A and B are disjoint. For an antiadditive function, the opposite inequality holds for all disjoint sets A and B: f(A ∪ B) ≥ f(A)+f(B). This relationship is sometimes called superadditivity. Conversely, when f(A ∪ B) ≤ f(A)+f(B) the function is subadditive, which is distinct from antiadditivity.

In economics antiadditivity is used to model increasing returns to scale, where the benefit of combining inputs exceeds the sum of benefits from each input. In set theory, the cardinality of a union of disjoint finite sets is additive, but cardinalities of infinite sets can exhibit antiadditive behaviour under certain constructions. In information theory, mutual information can display antiadditive characteristics when considering overlapping sources. Antiadditivity also appears in probability theory; non-additive probability measures (e.g., capacities) assign weights that satisfy an antiadditive inequality, reflecting uncertainty or ambiguity about events.

The concept is useful in modeling phenomena where collective performance or risk is greater than the sum of individual contributions. It has implications for cooperative game theory, where the value of a coalition can be superadditive, encouraging collective action. Antiadditivity is contrasted with additivity, subadditivity and superadditivity, providing a framework for studying systems with synergistic or cumulative effects.