Generalized Weak Transitivity of Preference

Decision making processes are often based on (pairwise) preference relations. An important property of preference relations is transitivity. Many types of transitivity have been proposed in the literature, such as max–min and max–max transitivity, restricted max–min and max–max transitivity, additive and multiplicative transitivity, or Łukasiewicz transitivity. This paper focuses on weak transitivity. Weak transitivity has been defined for additive preference relations. We extend this definition to multiplicative preference relations and further introduce a generalized version called generalized weak transitivity. We show that for reciprocal additive and multiplicative preference relations weak transitivity is equivalent to generalized weak transitivity, and we also illustrate generalized weak transitivity for preference relations that are neither additive nor multiplicative. Finally, we show how a total order (ranking of the options) can be constructed for any generalized weak transitive preference relation.