On co-Möbius representation of fuzzy measures

The Möbius transform has facilitated great progress in both theory and practical applications of fuzzy measures. As well as simplifying a number of calculations used for interpretation, the Möbius representation of certain families like k-additive fuzzy measures significantly reduces the number of defining parameters.For some other classes, like k-interactive fuzzy measures, we can use the less well-known co-Möbius representation, however this is not as intuitive or straightforward when it comes to interpreting its values. In this contribution we propose a modification to this calculation that we call the complement co-Möbius representation, which leads to more natural expressions, useful simplifications and insights. We provide the conversion formulas between each of the representations as well as some examples, highlighting k-interactive and plausibility measures in particular.