Bidirectional possibilistic dominance in uncertain decision making

Our concern is with comparing variables whose values are possibility distributions on an ordered space Y. We introduce the idea of a possibilistic exceedance function, PED, a function that indicates the possibility that the value of variable is at least as large a given value in Y. We use this to define the idea of possibilistic dominance. We say one possibility distribution possibilistically dominates another distribution if its PED has at least as large value for all elements in Y. We suggest that possibilistic dominance can be used to order possibility distributions. We introduce surrogates for possibilistic dominance. We then point out that if one distribution has a PED at least large as another distribution for all elements in Y then it has a bigger possibility of attaining both large and small values in Y. In order to circumvent this we introduce the idea of bi-directional possibilistic dominance, here we require the dominant distribution to have a larger PED value for the higher values in Y and a smaller PED for the lower values in Y.