Detecting and Interpreting Median Constrained Bucket Orders Within the Kemeny Axiomatic Framework

Preference rankings are data expressing preferences of individuals over a set of available alternatives. Suppose to have a set of n items to be ranked by m judges. When a judge gives a complete and strict precedence ranking of the items, producing in fact a permutation of the first n integers, the resulting ranking is defined complete (or full). Sometimes some individuals assign the same integer to two or more items, producing a tied ranking. Sometimes tied rankings are called bucket orders, in the sense of a set of items ranked in a tie at a given location. One of the most common problems in dealing with rank data is the identification of the so-called consensus ranking, namely that ranking which best synthesizes the consensus opinion. Depending on the reference framework, this problem is known as social choice problem, rank aggregation problem, median ranking, central ranking, Kemeny problem. This (NP-hard) problem has gained increasing importance over the years both as a main research topic and as an essential starting point for other types of analysis. It has been tackled with several different approaches. In general, except for methods based on counting such as Borda or Condorcet-like methods, the detection the consensus ranking is based on the minimization of a distance measure suitably defined for preference rankings. We are adopting Kemeny's axiomatic framework. We assume that the consensus ranking is the median ranking defined as that/those ranking/s that minimize/s the sum of the Kemeny distances between itself, and the rankings expressed by a set of m judges. This presentation concerns the median constrained bucket order, namely a specific solution of the rank aggregation problem in which the median ranking is forced to have a pre-specified number of buckets. Our proposal is motivated by the attempt to find a solution for some real data problems, one of them concerning the evaluation of a set of nurses within the so-called triage prioritization.