Ranked Pairs: Difference between revisions
Cleaned up Smith compliance proof, and elaborated on what tiebreaker is most commonly associated with RP.
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{{Wikipedia|Ranked pairs}}
The "'''Ranked Pairs'''" method (sometimes abbreviated as "RP")
== Procedure ==
The
# 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 margin of victory first to smallest last.
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See the [[Ranked pairs#Notes|Notes]] section for information on finding the Ranked Pairs winner without constructing a graph.
* 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 Ranked Pairs once, and then use the resulting [[Condorcet ranking]] (when ignoring the defeats that were ignored by the ranked-pairs procedure) as the ranked-pairs ranking.
=== Tally ===
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candidates are assumed to be equally worse than the stated candidates.
Once
If "Vxy" is the number of Votes that rank x over y, then
"x" wins if Vxy > Vyx, and "y" wins if Vyx > Vxy.
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Therefore, A is the winner.
=== A Limit case ===
We assume 48% of voters vote A>B>C, 3% votes B>C>A, 49% votes C>A>B. A>B in 97% (locked), C>A in 52% (locked) and B>C in 51% (not locked since there is a cycle). Thus, we have C>A>B and C is the winner. However, if we make a [[Borda count]] (2 points for first place 1 point for second place) A has 145 points and C has only 101 points and thus one could think that A deserves more to win.
== Advantages and disadvantages ==
Ranked Pairs is [[Smith-efficient]], because every Smith set member pairwise beats everybody outside the set. As a result, every defeat by a Smith set member over a non-Smith candidate is locked before any opposite-direction defeat, so a non-Smith candidate can never win.
Ranked Pairs passes the [[Independence of Smith-dominated Alternatives]] criterion, because the only cycles for RP to potentially resolve will always be between Smith set members. Because of this, all candidates not in the Smith set can be eliminated before starting the procedure, reducing the number of operations needed to be done to find the winner. In addition, Ranked Pairs, like [[Schulze]], is equivalent to [[Minimax]] when there are 3 or fewer candidates with no pairwise ties between them, so if the Smith set has 3 or fewer candidates in it with no pairwise ties between them, [[Smith//Minimax]] can be run instead to find/demonstrate the RP winner.
While Ranked Pairs behaves similarly to [[Schulze]], Ranked Pairs passes [[local independence of irrelevant alternatives]] whereas Schulze does not. Some authors argue that the Ranked Pairs method is more intuitive and easier to understand than Schulze as well.<ref name="Munger 2023 pp. 434–444">{{cite journal | last=Munger | first=Charles T. | title=The best Condorcet-compatible election method: Ranked Pairs | journal=Constitutional Political Economy | volume=34 | issue=3 | date=2023 | issn=1043-4062 | doi=10.1007/s10602-022-09382-w | pages=434–444}}</ref>
One disadvantage of Ranked Pairs is there's no easy way to detect ties for first place, as determining whether there exists a way to break ties between pairwise victories so that a given candidate wins is NP-complete.<ref name="Brill">{{cite journal | last=Brill | first=Markus | last2=Fischer | first2=Felix | title=The Price of Neutrality for the Ranked Pairs Method | journal=Proceedings of the AAAI Conference on Artificial Intelligence | publisher=Association for the Advancement of Artificial Intelligence (AAAI) | volume=26 | issue=1 | date=2012-07-26 | issn=2374-3468 | doi=10.1609/aaai.v26i1.8250 | pages=1299–1305}}</ref> However, ties can still be broken fairly and efficiently using some secondary method that doesn't compromise Ranked Pairs' properties. The most common such tiebreaker is [[random voter hierarchy]], a generalization of [[random ballot]]. Cardinal methods like [[Graduated Majority Judgment|highest medians]] can also be used, at the cost of slightly weakening properties like ranked [[clone independence]].
== Notes ==
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When ignoring struckthrough (non-locked in) pairwise victories, C is the only candidate with no pairwise defeats, and thus is the RP winner. The RP ranking is C>A>B, since C pairwise beats all others, A pairwise beats everyone except C, and B pairwise loses to everyone (when ignoring the defeats ignored by the RP procedure).
== References ==
<references />
▲Ranked Pairs passes the [[Independence of Smith-dominated Alternatives]] criterion, because the only cycles for RP to potentially resolve will always be between Smith set members. Because of this, all candidates not in the Smith set can be eliminated before starting the procedure, reducing the number of operations needed to be done to find the winner. In addition, Ranked Pairs, like [[Schulze]], is equivalent to [[Minimax]] when there are 3 or fewer candidates with no pairwise ties between them, so if the Smith set has 3 or fewer candidates in it with no pairwise ties between them, [[Smith//Minimax]] can be run instead to find/demonstrate the RP winner.
== External Resources ==
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[[Category:Defeat-dropping Condorcet methods]]
[[Category:Monotonic_electoral_systems]]
[[Category:Ranked voting methods]]
[[Category:Condorcet methods]]
[[Category:Clone-independent electoral systems]]
{{fromwikipedia}}
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