On the approximation of fuzzy preferences by exact relations

The generalized Hamming distance is commonly used in determining a closest crisp relation to a given fuzzy relation (which, in turn, is crucial in studying exact choices originating from fuzzy preferences). This note illustrates a difficulty attached to this practice, and proposes an alternative metric which is intuitively more appropriate in the case of fuzzy preferences. Some implications of this new metric on finding the nearest exact relation to a fuzzy preference ordering and on the understanding of the max-min transitivity property are also discussed.