Abstract
We introduce the concept of a fuzzy measure mu on a finite space X and discuss its use in providing information about the value of an uncertain variable V. Here, the measure of a subset A of X, mu(A), models the anticipation of finding the value of V in A. We are here in a situation in which the values in X are ordered and in particular, the problem is one of determining which of two measures on X can be considered as bigger. To address this problem, we make use of the idea of stochastic dominance from probability theory and generalize it to the concept of basic measure-based dominance. To define our concept of basic measure-based dominance, we introduce the idea of the dual of mu, denoted (mu) over cap, and note in this context (mu) over cap (A) models the anticipation of not finding the value of V in not A. We emphasize that both mu(A) and (mu) over cap (A) are getting at the same idea. We point out that only for a special class of measures, called self-dual are they exactly equal. Here, we use the average of mu(A) and (mu) over cap (A) to provide a formulation for our new concept basic measure-based dominance.