Scores of hesitant fuzzy elements revisited: “Was sind und was sollen”

This paper revolves around the notion of score for hesitant fuzzy elements, the constituent parts of hesitant fuzzy sets. Scores allow us to reduce the level of uncertainty of hesitant fuzzy sets to classical fuzzy sets, or to rank alternatives characterized by hesitant fuzzy information. We propose a rigorous and normative definition capable of encapsulating the characteristics of the most important scores introduced in the literature. We systematically analyse different types of scores, with a focus on coherence properties based on cardinality and monotonicity. The hesitant fuzzy elements considered in this analysis are unrestricted. The inspection of the infinite case is especially novel. In particular, special attention will be paid to the analysis of hesitant fuzzy elements that are intervals. •We revise the definition and properties of scores for hesitant sets.•New scores are introduced for some classes of infinite subsets of the unit interval.•A score avoiding ties on closed non-degenerated intervals is built as a by-product.•Hierarchies and compatibility among properties desired for scores are analyzed.