Adaptation and Natural Selection revisited

In Adaptation and Natural Selection, George C. Williams linked the distinction between group and individual adaptation with the distinction between group and individual selection. Williams' Principle, as we will call it, says that adaptation at a level requires selection at that level. This is a necessary but not a sufficient condition; for example, group adaptation requires group selection, but the fact that group selection influences a trait's evolution does not suffice for the resulting trait frequency to be a group adaptation. What more is required? In this paper, we describe an answer to this question that has been developed in multilevel selection theory. We also discuss an alternative framework for defining units of adaptation that violates Williams' Principle.