The consensus ranking problem has received much attention in the statistical literature. Given m rankings of n objects the objective is to determine a consensus ranking. The input ranking may contain ties, be incomplete, and may be weighted. Two solution concepts are discussed, the first maximizing the average weighted rank correlation of the solution ranking with the input rankings and the second minimizing the average weighted Kemeny–Snell distance. A new rank correlation coefficient called τx is presented which is shown to be the unique rank correlation coefficient which is equivalent to the Kemeny–Snell distance metric. The new rank correlation coefficient is closely related to Kendall's tau but differs from it in the way ties are handled. It will be demonstrated that Kendall's τb is flawed as a measure of agreement between weak orderings and should no longer be used as a rank correlation coefficient. The use of τx in the consensus ranking problem provides a more mathematically tractable solution than the Kemeny–Snell distance metric because all the ranking information can be summarized in a single matrix. The methods described in this paper allow analysts to accommodate the fully general consensus ranking problem with weights, ties, and partial inputs.
