Unexpected thresholds from independence of irrelevant alternatives in fuzzy arrow theorems

It is well known Arrow Theorem and its impact into Social Choice. It states that under an apparently mild set of conditions no rule fusing individual preferences into a social one is possible. In order to solve this situation, a possibility is to skip from dichotomic preferences to fuzzy ones. All conditions imposed to aggregation rules should be adapted to the fuzzy setting and due to the existence of different generalizations for each condition, depending on the chosen combination, a possibility or an impossibility result arises. In addition, in case we find a reasonable fuzzy aggregation rule, in most situations dichotomic decisions have to be taken at the end of the day, so the use of thresholds over fuzzy preferences is compulsory to make any decision. Surprisingly, independence of irrelevant alternatives axioms induce different thresholds which, besides they can be used on discrete and dichotomic decision making, transform fuzzy spaces of preferences and its aggregation functions into discrete ones allowing the application of new techniques to their study.