Deriving 2-ranks from multiplicative preference relations in a group context: An investigation with axiomatic design

Abstract

Multiplicative preference relations (MPRs) are broadly employed in the Analytic Hierarchy Process (AHP) to support group decision making (GDM). In AHP-GDM, aggregation of individual judgments (AIJ) and aggregation of individual priorities (AIP) are frequently used to acquire a full ranking of alternatives. However, in practical AHP-GDM, alternatives sometimes need to be assigned to two preference-ordered classes. In this paper, we call it the 2-rank GDM problem with MPRs and investigate it through axiomatic design. We start by proposing six axioms: Anonymity for alternatives (AA), Anonymity for individuals (AI), Positive responsiveness (PR), Non-dictatorship (ND), Independence of irrelevant alternatives (IIA), and Power invariance (PI) to identify a “perfect” 2-rank approach. Next, an impossibility theorem is presented stating that no 2-rank approach satisfies six axioms concurrently due to the inherent conflict among them. Based on AIJ and AIP, five 2-rank approaches M I, AIJ-M II, AIP-M II, AIP-M III, and AIP-M IV are developed. We then analyze and compare them based on the proposed axioms and prove that M I satisfies all axioms except for IIA, while other approaches only satisfy AA, AI, PR, and ND. Finally, simulation experiments are designed to further explore and compare the properties of these five 2-rank approaches. From the axiomatic and simulation analyses, it is found that M I is the best approach for the 2-rank GDM problem with MPRs.