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Kemeny–Young calculations are usually done in two steps. The first step is to create a matrix or table that counts pairwise voter preferences. The second step is to test all possible rankings, calculate a score for each such ranking, and compare the scores. Each ranking score equals the sum of the pairwise counts that apply to that ranking.
Plurality voting is the most common voting system, and has been in widespread use since the earliest democracies.As plurality voting has exhibited weaknesses from its start, especially as soon as a third party joins the race, some individuals turned to transferable votes (facilitated by contingent ranked ballots) to reduce the incidence of wasted votes and unrepresentative election results.
The high rank matrix completion in general is NP-Hard. However, with certain assumptions, some incomplete high rank matrix or even full rank matrix can be completed. Eriksson, Balzano and Nowak [10] have considered the problem of completing a matrix with the assumption that the columns of the matrix belong to a union of multiple low-rank subspaces.
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The Kendall tau rank distance is a metric (distance function) that counts the number of pairwise disagreements between two ranking lists. The larger the distance, the more dissimilar the two lists are.
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In practice, we can construct one specific rank factorization as follows: we can compute , the reduced row echelon form of .Then is obtained by removing from all non-pivot columns (which can be determined by looking for columns in which do not contain a pivot), and is obtained by eliminating any all-zero rows of .