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 Ω , which include additivity and comonotonic additivity as extreme cases. Let E ⊆ 2 Ω be a collection of subsets of Ω . Two functions x and y on Ω are E -cominimum if, for each E ∈ E , 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 Ω is E -cominimum additive if I ( x + y ) = I ( x ) + I ( y ) whenever x and y are E -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 [Eichberger, J., Kelsey, D., 1999. E-capacities and the Ellsberg paradox. Theory and Decision 46, 107–140] and that of the multiperiod decision model of Gilboa [Gilboa, I., 1989. Expectation and variation in multiperiod decisions. Econometrica 57, 1153–1169].