A new distance between rankings

This paper analyzes the behavior of the well-known Spearman's footrule distance (F-distance) to measure the distance between two rankings over the same set of objects. We show that F-distance is not invariant to labeling, and therefore, it suffers from a serious drawback for its use in applications. To circumvent this problem, we propose a new distance between rankings which is invariant under indexing (i.e., labeling) and appears as a good alternative to the direct use of F-distance between rankings, and also the invariant-under-indexing Kemeny's distance as well. We also show how our new distance can work with importance weights. Some simple examples are given to show the interest of our method with respect to the classical one based on F-distance and Kemeny's distance.