superadditivität

Superadditivität is a property of functions where the value of the function for the sum of two arguments is greater than or equal to the sum of the function values for each argument individually. Formally, a function f is superadditive if for all x and y in its domain, f(x + y) >= f(x) + f(y). This concept appears in various fields of mathematics and economics.

In cooperative game theory, superadditivity is a crucial property for characteristic functions. A game is superadditive if the value of the union of two disjoint coalitions is at least the sum of their individual values. This means that players are better off by cooperating and forming a larger group than by staying in separate smaller groups. This property simplifies the analysis of games and ensures that adding more players to a coalition never decreases its overall value.

The concept also finds application in areas like optimization and analysis. For instance, certain types of growth models might exhibit superadditive behavior, where the rate of increase accelerates with larger inputs. In functional analysis, superadditive functions are studied in relation to other properties like convexity or concavity. The inverse property, where f(x + y) <= f(x) + f(y), is called subadditivität. A function that is both superadditive and subadditive is additive, meaning f(x + y) = f(x) + f(y). Understanding superadditivität helps in predicting outcomes and understanding the behavior of systems where combined effects are at least as large as the sum of individual effects.