On the Rationality of Some Crisp Choice Functions Based on Strongly Complete Fuzzy Pre-Orders

Barrett, Pattanaik and Salles [Fuzzy Sets and Systems 34 (1990) 197–212] introduced nine alternative rules for generating exact choice sets from fuzzy weak preference relations (FWPR) and four rationality properties. They showed that, when preferences are fuzzy pre-orders, most of these alternative rules (preference-based choice functions or PCFs) violate at least two rationality properties. Following in the same direction, Fotso and Fono [New Mathematics and Natural Computation 8 (2012) 257–272] characterized these PCFs and analyzed their consistency in the cases of strongly complete fuzzy pre-orders and crisp complete pre-orders. In this paper, based on results of the two previous papers, we determine to what extend the PCFs are rational with respect to the structure of the underlying relation. More specifically, for each of the nine alternative rules violating a given rationality property, we determine respectively crisp pre-orders and strongly complete fuzzy pre-orders for which the PCF satisfies the property.