Aggregating Ordinal Values Using A Measure Based Median

We introduce the concept of a fuzzy measure mu on a set X. We discuss some of the properties of a fuzzy measure. We provide some notable examples of fuzzy measures. We discuss the important application of using fuzzy measure to provide information about an uncertain variable V. Here the measure of a subset A indicates the anticipation of finding the value of V in A. Our interest here is in finding the best course of action, alternative, in this uncertain environment where each alternative is modeled as a payoff function that associates with an element in X a payoff. Here because of the uncertainty associated with the value of V our concept of best course of action is captured for each alternative by an average like value of its payoff function with respect to the measure mu. Our particular concern here is with the case where the payoffs for each alternative are drawn from a scale, which rather then being numeric, is just ordinal. Here then we become interested in finding an aggregated value of a collection of uncertain ordinal values where the uncertainty is modeled by a measure.

Automated summary

This article introduces the concept and properties of fuzzy measures, providing examples and discussing their application in determining the best course of action in uncertain environments, particularly when payoffs are ordinal rather than numeric.