It has been suggested that Maximize Affirmed Majorities be merged into this article or section. (Discuss)

It has been suggested that Maximum majority voting be merged into this article or section. (Discuss)

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. Image File history File links Please see the file description page for further information. ... Maximize Affirmed Majorities (MAM) is a voting method developed by Stephen Eppley that selects a single winner using votes that express preferences. ... Image File history File links Please see the file description page for further information. ... Maximum majority voting (MMV) is a voting method that selects a single winner using votes that express preferences. ... T. Nicolaus Tideman (born August 11, 1943 in Chicago, Illinois) is a Professor of Economics at Virginia Polytechnic Institute and State University. ... Voters at the voting booths in the US in 1945 Voting systems are methods (algorithms) for groups of people to select one or more options from many, taking into account the individual preferences of the group members. ...

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. Any election method conforming to the Condorcet criterion is known as a Condorcet method. ... The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. ...

Ranked Pairs is currently used by the Ice Games design competition.

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.

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:

1. Vxy > Vzw. In other words, the majority having more support for its alternative is ranked first.
2. 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

Imagine that the population of Tennessee, a state in the United States, is voting on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate live in one of these four cities, and that they would like the capital to be established as close to their city as possible. Image File history File links Tennessee_map_for_voting_example. ... Official language(s) English Capital Nashville Largest city Memphis Area  - Total  - Width  - Length  - % water  - Latitude  - Longitude Ranked 36th 109,247 km² 195 km 710 km 2. ... In politics, a capital (also called capital city or political capital — although the latter phrase has an alternative meaning based on an alternative meaning of capital) is the principal city or town associated with its government. ... In politics, an electorate is the group of entities entitled to vote in an election. ...

The candidates for the capital are:

• Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
• Nashville, with 26% of the voters
• Knoxville, with 17% of the voters
• Chattanooga, with 15% of the voters

The preferences of the voters would be divided like this: Nickname: The River City, The Bluff City Official website: http://www. ... Nickname: Music City Official website: http://www. ... Nickname: The Marble City, K-Town, Big Orange Country, Knox Vegas Official website: www. ... Nickname: Scenic City (official), River City, Chatty, Chatt-Town, Chattavegas Official website: http://www. ...

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 > 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).

A graph of the voting system example. ...

In this example, Nashville is the winner using RP.

Ambiguity resolution example

Let's say there was an ambiguity. For a simple situation involving canidates 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.

Summary

In the example election, the winner is Nashville. This would be true for any Condorcet method. Using the first-past-the-post system 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. Any election method conforming to the Condorcet criterion is known as a Condorcet method. ... The first-past-the-post electoral system is a voting system for single-member districts, variously called first-past-the-post (FPTP or FPP), winner-take-all, plurality voting, or relative majority. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

Criteria

Of the formal voting system criteria, the Ranked Pairs method passes the majority criterion, the monotonicity criterion, 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 independant of irrelevant alternatives, it does satisfy local independence of irrelevant alternatives. Wikipedia does not yet have an article with this exact name. ... The majority criterion is a voting system criterion, used to objectively compare voting systems. ... A voting system is monotonic if it satisfies the monotonicity criterion, given below. ... The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate. ... Given a vote where voters rank options in order of preference, a Condorcet loser is an option that loses all of its pairwise comparisons. ... Strategic nomination is the manipulation of an election through its candidate set (compare this to tactical voting, where the manipulation comes from the voters). ... A voting system is consistent if, when the electorate is divided arbitrarily into two parts and separate elections in each part result in the same alternative being selected, an election of the entire electorate also selects that alternative. ... Statement of Criterion Adding one or more ballots that vote X over Y should never change the winner from X to Y. Complying Methods Plurality voting, Approval voting, Cardinal Ratings, Borda count, and Woodalls DAC method all pass the Participation Criterion. ... Independence of irrelevant alternatives (IIA) is an axiom often adopted by social scientists as a basic condition of rationality. ...

Independence of irrelevant alternatives

Ranked Pairs fails independence from irrelevant alternatives. However, the method adheres to a less strict property, sometimes called local independence from irrelevant alternatives ("local IIA"). 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. Local IIA implies the Condorcet criterion. Independence of irrelevant alternatives (IIA) is an axiom often adopted by social scientists as a basic condition of rationality. ... In voting systems, the Smith set is the smallest set of candidates in a particular election who, when paired off in pairwise elections, can beat all other candidates outside the set. ... The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate. ...

External resources

• Ranked Pairs by Blake Cretney
• Voting methods survey by James Green-Armytage
• Descriptions of ranked-ballot voting methods by Rob LeGrand