Ranked pairs
Ranked pairs or the Tideman method is an electoral system developed in 1987 by Nicolaus Tideman 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 candidate will win. Because of this property, RP is, by definition, a Condorcet method.
Procedure
The RP procedure is as follows:- Tally the vote count comparing each pair of candidates, and determine the winner of each pair
- Sort each pair, by the largest strength 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. The completed graph shows the winner.
To create a sorted list, repeatedly use RP to select a winner,
remove that winner from the list of candidates,
and repeat.
Tally
To tally the votes, consider each voter's preferences.For example, if a voter states "A > B > 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, and unstated
candidates are assumed to be equal to 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 fromthe largest majority to the smallest majority.
A majority for x over y precedes a majority for z over w
if and only if 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 pairs to "lock in".
- Lock in the first sorted pair with the greatest majority.
- Evaluate the next pair on whether a Condorcet cycle occurs when this pair is added to the locked pairs.
- If a cycle is detected, the evaluated pair is skipped.
- If a cycle is not detected, the evaluated pair is locked in with the other locked pairs.
- Loop back to Step #2 until all pairs have been exhausted.
Using the sorted list above, lock in each pair in turn unless
the pair will create a circularity in the graph
.
Winner
In the resulting graph for the locked pairs, the source corresponds to the winner. A source is bound to exist because the graph is a directed acyclic graph by construction, and such graphs always have sources. In the absence of pairwise ties, the source is also unique.An example
The situation
The results would be tabulated as follows:- 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
Tally
| Pair | Winner |
| Memphis vs. Nashville | Nashville 58% |
| Memphis vs. Chattanooga | Chattanooga 58% |
| Memphis vs. Knoxville | Knoxville 58% |
| Nashville vs. Chattanooga | Nashville 68% |
| Nashville vs. Knoxville | Nashville 68% |
| Chattanooga vs. Knoxville | Chattanooga: 83% |
Note that absolute counts of votes can be used, or
percentages of the total number of votes; it makes no difference since it is the ratio of votes between two candidates that matters.
Sort
The votes are then sorted.The largest majority is "Chattanooga over Knoxville"; 83% of the
voters prefer Chattanooga.
Nashville beats both Chattanooga and Knoxville by a score
of 68% over 32%.
Since Chattanooga > Knoxville, and they are the losers,
Nashville vs. Knoxville will be added first, followed by
Nashville vs. Chattanooga.
Thus, the pairs from above would be sorted this way:
| Pair | Winner |
| Chattanooga vs. Knoxville | Chattanooga 83% |
| Nashville vs. Knoxville | Nashville 68% |
| Nashville vs. Chattanooga | Nashville 68% |
| Memphis vs. Nashville | Nashville 58% |
| Memphis vs. Chattanooga | Chattanooga 58% |
| Memphis vs. Knoxville | Knoxville 58% |
Lock
The pairs are then locked in order, skipping any pairsthat 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.
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.
In this example, Nashville is the winner using RP, followed by Chattanooga, Knoxville, and Memphis in second, third, and fourth places respectively.
Ambiguity resolution example
For a simple situation involving candidates A, B, and C.- A > B: 68%
- B > C: 72%
- C > A: 52%
- Lock B > C
- Lock A > B
- C > A is ignored as it creates an ambiguity or cycle.
Summary
In the example election, the winner is Nashville. This would be true for any Condorcet method.Using the First-past-the-post voting and some other systems, Memphis would have won the election by having the most people, even though Nashville won every simulated pairwise election outright. Using Instant-runoff voting in this example would result in Knoxville winning even though more people preferred Nashville over Knoxville.