How to transform all multiple solutions of the Kemeny Ranking Problem into a single solution

Abstract

Preference aggregation as a problem of a single consensus ranking determination, using Kemeny rule, for m rankings, including ties, of n alternatives is considered in the paper. The Kemeny Ranking Problem (KRP) may have considerably more than one optimal solutions (strict orders or permutations of the alternatives) and, hence, special efforts to deal with this phenomenon are needed. In the paper, there is proposed an efficient formal rule for convolution of the N multiple optimal permutations, the output profile Я(N, n), into an exact single final consensus ranking, which can include ties. The convolution rule is as follows: in the final consensus ranking, alternatives are arranged in ascending order of their rank sums (total ranks) calculated for the output profile Я; some two alternatives are considered to be tolerant if they have the same rank sums in Я. The equivalent convolution rule can be also applied as follows: in the final consensus ranking, alternatives are arranged in descending order of row sums (total scores) calculated for a tournament table built for Я; some two alternatives are deemed to be tolerant if they have the same row sums. It is shown that, for any alternative, its total rank and total score are equal in sum to the output profile dimension NЧn. The convolution rules are validated using Borda count.