Ranked pairs

Ranked pairs (sometimes abbreviated "RP") or the Tideman method is an electoral system developed in 1987 by Nicolaus Tideman that selects a single winner using votes that express preferences. The ranked-pairs procedure can also be used to create a sorted list of winners. If there is a candidate who is preferred over the other candidates, when compared in turn with each of the others, the ranked-pairs procedure guarantees that candidate will win. Because of this property, the ranked-pairs procedure complies with the Condorcet winner criterion (and is a Condorcet method). The ranked-pairs procedure operates as follows: Tally the vote count comparing each pair of candidates, and determine the winner of each pair (provided there is not a tie) Sort (rank) each pair, by strength of victory, from largest first to smallest last. "Lock in" each pair, starting with the one with the largest strength of victory and, continuing through the sorted pairs, add each one in turn to a graph if it does not create a cycle in the graph with the existing locked in pairs. The completed graph shows the final ranking. The procedure can be illustrated using a simple example. Suppose that there are 27 voters and 4 candidates w, x, y and z such that the votes are cast as shown in the table of ballots. The vote tally can be expressed as a table in which the (w, x) entry is the number of ballots in which w comes higher than x minus the number in which x comes higher than w. In the example w comes higher than x in the first two rows and the last two rows of the ballot table (total 18 ballots) while x comes higher than w in the middle two rows (total 9), so the entry in the (w, x) cell is 18–9 = 9. Notice the skew symmetry of the table. The positive majorities are then sorted in decreasing order of magnitude. The next stage is to examine the majorities in turn to determine which pairs to "lock in". This can be done by building up a matrix in which the (x, y) entry is initially 0, and is set to 1 if we decide that x is preferred to y and to –1 if we decide that y is preferred to x.