Bi-directional dominance for measure modeled uncertainty

Abstract

Our objective here is to provide a formulation for a dominance relationship between variables having uncertain values, represented by measures, which can be used for ordering these uncertain values. We first discuss the use of monotonic set functions, fuzzy measures, for the representation of uncertain information. We consider the issue of comparing and ordering variables whose values are uncertain and represented via a measure and we suggest the use of bi-directional dominance, which can be seen as a generalization of stochastic dominance that is used to order probability distributions. We note the establishment of a dominance relationship between uncertain values is often not possible. In order to circumvent this we introduce the idea of surrogates for bi-directional dominance, here we associate with each uncertain value, measure, a scalar value that provides an ordering between all uncertain values. These surrogates are defined so that they maintain any relationship between measures that can be established with the use of bi-directional dominance.