Ranked pairs

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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 Condorcet method.

Procedure

The RP (Ranked Pair) 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 strength 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 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 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 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.[vs 1]

Lock

The next step is to examine each pair in turn to determine the pairs to "lock in". This can be visualized by drawing an arrow from the pair's winner to the pair's loser in a directed graph. 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, 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 (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

Tennessee and its four major cities: Memphis in the south-west; Nashville in the centre, Chattanooga in the south, and Knoxville in the east

Imagine that Tennessee is having an election 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 lives in these four cities and that everyone wants to live as near to the capital as possible.

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, near the center of the state
  • Knoxville, with 17% of the voters
  • Chattanooga, with 15% of the voters

The preferences of the voters would be divided like this:

42% of voters
(close to Memphis)
26% of voters
(close to Nashville)
15% of voters
(close to Chattanooga)
17% of voters
(close to Knoxville)
  1. Memphis
  2. Nashville
  3. Chattanooga
  4. Knoxville
  1. Nashville
  2. Chattanooga
  3. Knoxville
  4. Memphis
  1. Chattanooga
  2. Knoxville
  3. Nashville
  4. Memphis
  1. Knoxville
  2. Chattanooga
  3. Nashville
  4. Memphis

The results would be tabulated as follows:

Pairwise election results
A
Memphis Nashville Chattanooga Knoxville
B Memphis [A] 58%
[B] 42%
[A] 58%
[B] 42%
[A] 58%
[B] 42%
Nashville [A] 42%
[B] 58%
[A] 32%
[B] 68%
[A] 32%
[B] 68%
Chattanooga [A] 42%
[B] 58%
[A] 68%
[B] 32%
[A] 17%
[B] 83%
Knoxville [A] 42%
[B] 58%
[A] 68%
[B] 32%
[A] 83%
[B] 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%
  • [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

Tally

First, list every pair, and determine the winner:

Pair Winner
Memphis (42%) vs. Nashville (58%) Nashville 58%
Memphis (42%) vs. Chattanooga (58%) Chattanooga 58%
Memphis (42%) vs. Knoxville (58%) Knoxville 58%
Nashville (68%) vs. Chattanooga (32%) Nashville 68%
Nashville (68%) vs. Knoxville (32%) Nashville 68%
Chattanooga (83%) vs. Knoxville (17%) 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 (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:

Pair Winner
Chattanooga (83%) vs. Knoxville (17%) Chattanooga 83%
Nashville (68%) vs. Knoxville (32%) Nashville 68%
Nashville (68%) vs. Chattanooga (32%) Nashville 68%
Memphis (42%) vs. Nashville (58%) Nashville 58%
Memphis (42%) vs. Chattanooga (58%) Chattanooga 58%
Memphis (42%) vs. Knoxville (58%) Knoxville 58%

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).

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 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.

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 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 preferential single-winner election methods:

Comparison of Preferential Electoral Systems
Monotonic Condorcet Majority Condorcet loser Majority loser Mutual majority Smith ISDA LIIA Clone independence Reversal symmetry Participation, Consistency Later-no‑harm Later-no‑help Polynomial time Resolvability
Schulze Yes Yes Yes Yes Yes Yes Yes Yes No Yes Yes No No No Yes Yes
Ranked pairs Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No No No Yes Yes
Alternative smith No Yes Yes Yes Yes Yes Yes Yes No Yes No No No No Yes Yes
Alternative schwartz No Yes Yes Yes Yes Yes Yes Yes No Yes No No No No Yes Yes
Kemeny-Young Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes No No No No Yes
Copeland Yes Yes Yes Yes Yes Yes Yes Yes No No Yes No No No Yes No
Nanson No Yes Yes Yes Yes Yes Yes No No No Yes No No No Yes Yes
Instant-runoff voting No No Yes Yes Yes Yes No No No Yes No No Yes Yes Yes Yes
Borda Yes No No Yes Yes No No No No No Yes Yes No Yes Yes Yes
Baldwin No Yes Yes Yes Yes Yes Yes No No No No No No No Yes Yes
Bucklin Yes No Yes No Yes Yes No No No No No No No Yes Yes Yes
Plurality Yes No Yes No No No No No No No No Yes Yes Yes Yes Yes
Contingent voting No No Yes Yes Yes No No No No No No No Yes Yes Yes Yes
Coombs[1] No No Yes Yes Yes Yes No No No No No No No No Yes Yes
MiniMax Yes Yes Yes No No No No No No No No No No No Yes Yes
Anti-plurality[1] Yes No No No Yes No No No No No No Yes No No Yes Yes
Sri Lankan contingent voting No No Yes No No No No No No No No No Yes Yes Yes Yes
Supplementary voting No No Yes No No No No No No No No No Yes Yes Yes Yes
Dodgson[1] No Yes Yes No No No No No No No No No No No No Yes

References

  1. ^ 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.
  • Tideman, T.N. (1987) Independence of clones as a criterion for voting rules. Social Choice and Welfare 4: 185-206.

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
  1. ^ a b c Anti-plurality, Coombs and Dodgson are assumed to receive truncated preferences by apportioning possible rankings of unlisted alternatives equally; for example, ballot A > B = C is counted as  A > B > C and  A > C > B. If these methods are assumed not to receive truncated preferences, then Later-no-harm and Later-no-help are not applicable.
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