Weighted Merging Operators: Product, Utility-based Operators and Egalitarianism

We propose new operators for weighted propositional belief merging operators. We introduce distance-based operators that use the product as aggregation function. In social choice theory, the product, called the Nash welfare function, is known to be a more equitable social welfare function than the classical utilitarian welfare function (based on a sum). We study which properties are satisfied by the obtained corresponding weighted merging operators. In particular, we show that, unlike the Nash welfare function, distance-based operators using the product do not satisfy the Pigou-Dalton property. Then, we introduce a new family of weighted merging operators, which we call utility-based weighted merging operators, where the utility is roughly the converse of a distance for distance-based operators. For most well-known distance-based operators, it is easy to find the corresponding utility-based merging operators. But an interesting result is that the utility-based weighted merging operator based on the product does not correspond to any standard distance-based weighted merging operator, and this operator satisfies the Pigou-Dalton property.