Basic type dominance and its use in measure-modeled uncertainty in decision making

We introduce the concept of a fuzzy measure μ 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, μ(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 μ, denoted \(\hat{\mu }\), and note in this context \(\hat{\mu }\)(A) models the anticipation of not finding the value of V in not A. We emphasize that both μ(A) and \(\hat{\mu }\)(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 μ(A) and \(\hat{\mu }\)(A) to provide a formulation for our new concept basic measure-based dominance.