Ranked pairs (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. 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

Descriptions of ranked-ballot voting methods

by Rob LeGrand

Example JS implementation

by Asaf Haddad

Pair Ranking Ruby Gem

by Bala Paranj

A margin-based PHP Implementation of Tideman's Ranked Pairs

{{voting systems Electoral systems Monotonic Condorcet methods Single-winner electoral systems

Condorcet method
A Condorcet method (; ) is an election method that elects the candidate who wins a majority rule, majority of the vote in every head-to-head election against each of the other candidates, that is, a candidate preferred by more voters than any oth ...

.
Procedure

The RP (Ranked Pair) 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 strength of victory first to smallest last.In fact, there are different ways how the ''strength of a victory'' is measured. The approach used in this article is called winning votes. Another common approach also used by Tideman defining the ranked pairs method in 1987 is the variant using margins of a victory. The margin of victory, also called "defeat strength", is the difference of the number of votes of the two compared candidates. # "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 voter's 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 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 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 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 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. Condorcet cycle evaluation can be visualized by drawing an arrow from the pair's winner to the pair's loser in adirected graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a Graph (discrete mathematics), graph that is made up of a set of Vertex (graph theory), vertices connected by directed Edge (graph theory), edges often called ...

.
Using the sorted list above, lock in each pair in turn ''unless''
the pair will create a circularity in the graph
(for example, where A is more than B, B is more than C, but C is more than A).
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 by construction, and such graphs always have sources. In the absence of pairwise ties, the source is also unique (because whenever two nodes appear as sources, there would be no valid reason not to connect them, leaving only one of them as a source).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 captionTally

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 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 (68%) beats both Chattanooga and Knoxville by a score of 68% over 32% (a tie, unlikely in real life for this many voters). 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: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 away from the winner). Image:Tennessee-vote.svg 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% In this situation we "lock in" the majorities starting with the greatest one first. * Lock B > C * Lock A > B * C > A is ignored as it creates an ambiguity or cycle. Therefore, A is the winner.Summary

In the example election, the winner is Nashville. This would be true for anyCondorcet method
A Condorcet method (; ) is an election method that elects the candidate who wins a majority rule, majority of the vote in every head-to-head election against each of the other candidates, that is, a candidate preferred by more voters than any oth ...

.
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
Instant-runoff voting (IRV), also sometimes referred to as the alternative vote (AV), preferential voting, or, in the United States, ranked-choice voting (RCV), though these names are also used for other systems, is a type of ranked preferential ...

in this example would result in Knoxville winning even though more people preferred Nashville over Knoxville.
Criteria

Of the formal voting criteria, the ranked pairs method passes the majority criterion, the monotonicity criterion, the Smith criterion (which implies the Condorcet criterion), the Condorcet loser criterion, and the independence of clones criterion. Ranked pairs fails the consistency criterion and the participation criterion. While ranked pairs is not fully independence of irrelevant alternatives, independent of irrelevant alternatives, it still satisfies local independence of irrelevant alternatives.Independence of irrelevant alternatives

Ranked pairs fails independence of irrelevant alternatives. However, the method adheres to a less strict property, sometimes called independence of Smith-dominated alternatives (ISDA). It says that if one candidate (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is not in the Smith set. ISDA implies the Condorcet criterion.Comparison table

The following table compares Ranked Pairs with other Ranked voting, preferential single-winner election methods:Notes

References

External links

Descriptions of ranked-ballot voting methods

by Rob LeGrand

Example JS implementation

by Asaf Haddad

Pair Ranking Ruby Gem

by Bala Paranj

A margin-based PHP Implementation of Tideman's Ranked Pairs

{{voting systems Electoral systems Monotonic Condorcet methods Single-winner electoral systems