Transitivity analysis of preference relations under the impact of indifferent voters

Voting is a common way to reach consensus between decision makers with incompatible opinions in a practical situation. Under the situation with indifferent voters, the pairwise comparison matrix originating from a voting profile could be determined as a preference relation without additively reciprocal property, which is recalled additive non-reciprocal preference relations (ANrPRs). In this paper, some transitivity properties of ANrPRs are addressed involving additive transitivity, weak transitivity, moderate transitivity, and strong transitivity. The considered transitivity properties of ANrPRs are measured by constructing a quantification index. The simple majority rule and a novel optimization-based method are proposed to determine the winner. It is observed that the transitivity properties of ANrPRs are closely related to the occurrence of some voting paradoxes.