Coextrema Additive Operators

This paper proposes a class of weak additivity concepts for an operator on the set of real valued functions on a finite state space Ω, which include additivity and comonotonic additivity as extreme cases. Let E ⊆ 2Ω be a collection of subsets of Ω. Two functions x and y on Ω are E-coextrema if, for each E ∈ E, the set of minimizers of x restricted on E and that of y have a common element, and the set of maximizers of x restricted on E and that of y have a common element as well. An operator I on the set of functions on Ω is E-coextrema additive if I(x+y) = I(x)+I(y) whenever x and y are E-coextrema. The main result characterizes homogeneous E-coextrema additive operators.