One family of power means unifies minimum, mean, and maximum as substitutability endpoints.

Evidence

Note that when the exponent t = 1 this expression conveniently yields the arithmetic mean. Moreover, as t approaches ∞, m(t) approaches themaximumofai andast approaches−∞, m(t)approachestheminimum, thus defining the endpoints of the continuum (as illustrated in table 5.2). Additional discriminating markers can be added to this basic structure as needed. For example, the geometric mean used by Lemke and Reed (2001) to measure dyadic democracy serves well as a marker of quite modest substitutability situated between the partial substitutability of the arithmetic mean and the strict nonsubstitutability of the minimum value. — Gary Goertz, Social Science Concepts, 894

Categorization

• DEFINE: Gives the formal t-parameterized framework that renders weakest link, geometric mean, arithmetic mean, and best shot as points on one continuum.
• XREF: Links to Lemke and Reed’s geometric-mean democracy measure and to the t = -infinity, 1, infinity table of aggregating devices.