Cominimum Additive Operators

Abstract

This paper proposes a class of weak additivity concepts for an operator on the set of real valued functions on a finite state space omega, which include additivity and comonotonic additivity as extreme cases. Let epsilon be a collection of subsets of omega. Two functions x and y on omega are epsilon-cominimum if, for each E subseteq epsilon, the set of minimizers of x restricted on E and that of y have a common element. An operator I on the set of functions on is E- cominimum additive if I(x+y) = I(x)+I(y) whenever x and y are epsilon-cominimum. The main result characterizes homogeneous E-cominimum additive operators in terms of the Choquet integrals and the corresponding non-additive signed measures. As applications, this paper gives an alternative proof for the characterization of the E-capacity expected utility model of Eichberger and Kelsey (1999) and that of the multi-period decision model of Gilboa (1989).