Developing the Kemeny's Weighted Median for the Rank Aggregation Problem

Abstract

A coordinated ranking as the opinion of an expert group usually can be represented by the well-known Kemeny's median. The Kemeny's median is the least different ranking from other rankings and is free of known contradictions of the majority rule problem. As a mathematical principle, the Kemeny's median gives a decision in any case, in particular, for conflicting experts’ decisions in ordinal scales. In practice, competing opinions are usually modified by special approval procedures to achieve the required level of consensus. The known approach consists in assigning weights to experts’ opinions. In this paper, the problem to find the median for a linear combination of experts’ rankings is investigated using the well-known locally optimal Kemeny's algorithm. It is proposed to use the weighted loss matrix in it.