$\Upsilon$-Values: Power Indices à La Orness for Nonadditive Measures

Fuzzy measures, also known as capacities, nonadditive measures, and monotonic games, are increasingly used in all kind of applications. Fuzzy measures are set functions. So, for a given set
$X$
, we need to define
$2^{|X|} - 2$
parameters (excluding the measure on the empty set and on
$X$
itself). Because of that they are difficult to visualize, and indices and metrics have been defined. The Shapley value is an example. It permits us to determine weights of importance of each element in
$X$
. In this article, we introduce an alternative index. We call it
$\Upsilon$
-values. We provide an axiomatic characterization. These values are inspired on the Shapley values, and they are associated to set size, or position (order statistics) in a chain. Thus, also position when the measure is used in combination with a fuzzy integral. Andness and orness are measures that permit to evaluate the degree of simultaneity (conjunction) and substitutability (disjunction) of an aggregation function. We show the connection between our value and these concepts. In a way,
$\Upsilon$
-values define a power index à la orness.