Games - Ranked Pairs

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	Games - Ranked Pairs

Ranked Pairs (RP) or Tideman (named after its developer Nicolaus Tideman) is a voting method that selects a single winner using votes that express preferences. RP 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, RP guarantees that that candidate will win. Because of this property, RP is (by definition) a Condorcet method
. It is closely related to another Condorcet method, the Schulze method.

Ranked Pairs is currently used by the .

Procedure

The RP procedure is 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 the largest margin of victory first to smallest last. # "Lock in" each pair, starting with the one with the largest number of winning votes, and add one in turn to a graph as long as they do not create a cycle (which would create an ambiguity). The completed graph shows the winner.

RP can also be used to create a sorted list of preferred candidates. To create a sorted list, repeatedly use RP to select a winner, remove that winner from the list of candidates, and repeat (to find the next runner up, and so forth).

Tally

To tally the votes, consider each voters' preferences. For example, if a voter states "A > B > C" (A is better than B, and B is better than C), the tally should add one for A in A vs. B, one for A in A vs. C, and one for B in B vs. C. Voters may also express indifference (e.g., A = B), and unstated candidates are assumed to be equally worse than the stated candidates.

Once, tallied the majorities can be determined. If "Vxy" is the number of Votes that rank x over y, then "x" wins if Vxy > Vyx, and "y" wins if Vyx > Vxy.

Sort

The pairs of winners, called the "majorities", are then sorted from the largest majority to the smallest majority. A majority for x over y precedes a majority for z over w if and only if at least one of the following conditions holds:

#Vxy > Vzw. In other words, the majority having more support for its alternative is ranked first. #Vxy = Vzw and Vwz > Vyx. Where the majorities are equal, the majority with the smaller minority opposition is ranked first.

Lock

The next step is to examine each pair in turn to determine which pairs to "lock in". Using the sorted list above, lock in each pair in turn unless the pair will create a circularity in a graph (e.g., where A is more than B, B is more than C, but C is more than A).

An example

The situation

The results would be tabulated as follows:

Pairwise Election Results
		A
		Memphis	Nashville	Chattanooga	Knoxville
B	Memphis		58% 42%	58% 42%	58% 42%
	Nashville	42% 58%		32% 68%	32% 68%
	Chattanooga	42% 58%	68% 32%		17% 83%
	Knoxville	42% 58%	68% 32%	83% 17%
Pairwise election results (won-lost-tied):		0-3-0	3-0-0	2-1-0	1-2-0
Votes against in worst pairwise defeat:		58%	N/A	68%	83%

indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption
indicates voters who expressed no preference between either candidate

Tally

First, list every pair, and determine the winner:

Note that absolute counts of votes can be used, or percentages of the total number of votes; it makes no difference.

Sort

The votes are then sorted. The largest majority is "Chattanooga over Knoxville"; 83% of the voters prefer Chattanooga. Nashville (68%) beats both Chattanooga and Knoxville by a score of 68% over 32% (an exact tie, which is unlikely in real life for this many voters). Since Chattanooga > Knoxville, and they're the losers, Nashville vs. Knoxville will be added first, followed by Nashville vs. Chattanooga.

[ Visit the complete Wikipedia entry for Ranked Pairs ]

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