## Transitivity and fuzzy preferences.(English)Zbl 1075.91526

From the text: Let $$X$$ be a finite set of alternatives and $$R$$ be a fuzzy relation on $$X$$. $$R$$ is connected if for all $$x,y\in X$$, we have $$R(x,y)+R(y,x)\geq 1$$. The authors give eight different transitivity conditions for their connected fuzzy relations. Three criteria: (1) restrictiveness (the extent to which imposition of transitivity prevents fuzzy preferences from being ”truly fuzzy”); (2) factorizability (whether the transitivity of fuzzy weak preference can be factored into the transitivity of fuzzy strict preference and the transitivity of fuzzy indifference); and (3) normality (the degree to which transitivity prevents choice cycles, ensuring consistency of choice) are used to evaluate all the eight relations. The authors rate each fuzzy transitive relation as poor, fair, good, or excellent with respect to each of the three criteria. The result is: no fuzzy relation receives all excellent ratings but two get all poor ratings.

### MSC:

 91B08 Individual preferences
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