Ranked Pairs

Ranked Pairs (RP) or Tideman (named after Nicolaus Tideman) is a voting system that selects a single winner using votes that express preferences. RP can also be used to create a sorted list of winners. RP passes the Smith criterion and Condorcet Criterion, and is by definition a Condorcet method. RP has many variations such as the Maximize Affirmed Majorities (MAM) and Maximum Majority Voting (MMV) voting methods.

Procedure
The RP procedure is as follows:
 * 1) Tally the vote count comparing each pair of candidates, and determine the winner of each pair (provided there is not a tie)
 * 2) Sort (rank) each pair, by the largest margin of victory first to smallest last.
 * 3) "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.

See the Notes section for information on finding the Ranked Pairs winner without constructing a graph.

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). A simpler way to create the sorted list is simply to run RP once, and then use the resulting Condorcet ranking (when ignoring the defeats that were ignored by the RP procedure) as the RP ranking.

Tally
To tally the votes, consider each voters' preferences. For example, if a voter states "A &gt; B &gt; 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 &gt; Vyx, and "y" wins if Vyx &gt; 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:


 * 1) Vxy &gt; Vzw. In other words, the majority having more support for its alternative is ranked first.
 * 2) Vxy = Vzw and Vwz &gt; 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).

The Situation
Imagine an election for the capital of Tennessee, a state in the United States that is over 500 miles east-to-west, and only 110 miles north-to-south. Let's say the candidates for the capital are Memphis (on the far west end), Nashville (in the center), Chattanooga (129 miles southeast of Nashville), and Knoxville (on the far east side, 114 northeast of Chattanooga). Here's the population breakdown by metro area (surrounding county):


 * Memphis (Shelby County): 826,330
 * Nashville (Davidson County): 510,784
 * Chattanooga (Hamilton County): 285,536
 * Knoxville (Knox County): 335,749

Let's say that in the vote, the voters vote based on geographic proximity. Assuming that the population distribution of the rest of Tennessee follows from those population centers, one could easily envision an election where the percentages of votes would be as follows:

The results would be tabulated as follows:


 * [A] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
 * [B] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption
 * [NP] 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 &gt; Knoxville, and they're the losers, Nashville vs. Knoxville will be added first, followed by Nashville vs. Chattanooga.

Thus, the pairs from above would be sorted this way:

Lock
The pairs are then locked in order, skipping any pairs that would create a cycle:


 * Lock Chattanooga over Knoxville.
 * Lock Nashville over Knoxville.
 * Lock Nashville over Chattanooga.
 * Lock Nashville over Memphis.
 * Lock Chattanooga over Memphis.
 * Lock Knoxville over Memphis.

In this case, no cycles are created by any of the pairs, so every single one is locked in.

Every "lock in" would add another arrow to the graph showing the relationship between the candidates. Here is the final graph (where arrows point from the winner).



In this example, Nashville is the winner using RP.

Ambiguity Resolution Example
Let's say there was an ambiguity. For a simple situation involving candidates A, B, and C.


 * A > B 68%
 * B > C 72%
 * C > A 52%

In this situation we "lock in" the majorities starting with the greatest one first.


 * Lock B > C
 * Lock A > B
 * We don't lock in the final C > A as it creates an ambiguity or cycle.

Therefore, A is the winner.

External Resources

 * Ranked Pairs
 * The Maximize Affirmed Majorities voting procedure (MAM)
 * Maximum Majority Voting
 * Descriptions of ranked-ballot voting methods
 * Voting methods resource page