Talk:Ranked pairs

capital letters
Dear Michael Hardy, you wrote: "Would the authors of articles on voting methods please stop worshipping capital letters with such incredibly fanatical intensity. It makes it hard to get links right."

I consider your recent changes to be vandalism. If you are really so upset about capital letters, then it is sufficient to replace e.g. Ranked Pairs by ranked pairs. But the way you replaced capital letters destroyed the links completely. -- Markus Schulze 5 Jan 2005

Winning votes?
This article seems to define Ranked Pairs as using winning votes as the measure of defeat strength. Isn't that inaccurate, as far as what Tideman defined the method to be? KVenzke 17:42, May 22, 2005 (UTC)


 * Probably. The winning-votes variant should be mentioned in a section. RSpeer 05:25, May 23, 2005 (UTC)


 * It would be a tough choice since WV satisfies more criteria. The way the article is written now, anything that Schulze method satisfies is also satisfied by Ranked Pairs. But this will have to be qualified if this article defaults to margins. KVenzke 15:02, May 23, 2005 (UTC)

Tideman prefers margins at present, and has probably assumed that margins would be the defeat strength definition when publishing articles about ranked pairs, although he is open to the idea of WV. I think that it makes sense to present both ranked pairs/Tideman and beatpath/Schulze as base methods that can work with different definitions of defeat strength.Hermitage 23:23, 8 Jun 2005 (UTC)


 * I disagree with Hermitage. I think it's much more important for the general public to treat MAM, Tideman's Ranked Pairs, Markus Schulze's beatpath, etc. as specific voting methods being advocated, which satisfy or fail specific criteria, than as abstract families of voting methods each having variants. MAM satisfies more criteria than either Tideman's Ranked Pairs or Schulze's beatpath, so it doesn't make sense to exclude MAM from Wikipedia or other discussions, or treat it as some unimportant variant. SEppley (talk) 16:29, 8 March 2010 (UTC)

Most comments above are outdated. Some time after they were written, someone revised the Ranked Pairs article to mention only "margins of victory" (consistent with the way Tideman defined Ranked Pairs). It now makes no mention of the alternative (which MAM and several other methods use). On top of that, someone deleted the Wikipedia article about MAM. MAM satisfies more criteria than either Ranked Pairs or beatpath (Schulze) and there are some significant differences (and several subtle differences) between MAM and Ranked Pairs. Deleting the MAM article without merging its information into the Ranked Pairs article looks like vandalism, an attempt to suppress information about the best voting method. SEppley (talk) 16:29, 8 March 2010 (UTC)

MAM
I think that it is somewhat confusing for Steve to refer to his method as "MAM", rather than as a particular version of ranked pairs (which it is, essentially). I read the definition of MAM on his web site a year or so ago, and I had to ask on the EM list whether it was equivalent to RP. Thus, if MAM is discussed at all on wikipedia, I'd like for it to be made clear that it is a particular version of RP, or an alternate name for RP. Rather than bringing up MAM on wikipedia as a specific method, it might make more sense simply to discuss the component parts of MAM (i.e. the WV defeat strength definition and different tie breaker) as separate issues within the range of "ranked pairs" methods. However, if someone wants to briefly mention MAM in the article as an alternate name, I don't mind. Hermitage 23:35, 8 Jun 2005 (UTC)

F451 has deleted the MAM and MMV external links from this page. My personal opinion is that the links are appropriate, but I do not feel strongly enough to revert the edit. Hermitage 23:13, 13 Jun 2005 (UTC)

I'm conflicted. I definitely don't think MAM and MMV are significant enough to be mentioned in an article's text on Wikipedia; I think that the external links are perhaps the one place they should exist. But I don't feel strongly enough either. And I don't hold F451 at fault for anything here - if he makes an edit that's so non-controversial and keeps the articles clean, what's the problem? RSpeer 05:42, Jun 14, 2005 (UTC)

I am the Steve (Eppley) who invented and named the Maximize Affirmed Majorities voting method (MAM). Above, Hermitage has it backwards: it is more confusing to use the name Ranked Pairs as a family of voting methods that includes MAM as just a variant than to treat Ranked Pairs and MAM as specific, different voting methods. MAM is significantly different. Ranked Pairs is the voting method defined by Tideman in 1987, and refined by Zavist & Tideman in 1989. Tideman did not define Ranked Pairs to be a family of voting methods. The most significant ways that MAM and Ranked Pairs differ are (1) MAM allows each voter's ranking of the candidates to be a "weak" ordering (allowing indifference) whereas Ranked Pairs expects "strict" orderings, and (2) when sorting the pairs or majorities into a "largest to smallest" order of precedence, Ranked Pairs measures the size of each pair coalition by subtracting the size of its opposing coalition (what some people call "margins of victory"), whereas MAM measures the size of each majority *without subtracting* the size of its opposing minority (what some people call "winning votes"). MAM was invented independently of Tideman's work. MAM was initially defined as the voting method that selects, from all possible strict orders of finish, the order of finish that minimizes the largest "thwarted" majority (in the minlexmax sense). While searching for a quick algorithm to find that order of finish, I was unaware of the relationship between Ranked Pair's algorithm and what I was looking for. (I experimented for months with several algorithms that did not work quite right but came close, which I described at the time in an ongoing series of emails to Mike Ossipoff. Mike was like me at the time: aware of Ranked Pairs but unaware of its relationship to MAM until after I discovered MAM's quick algorithm.) Originally MAM was named Minimize Thwarted Majorities since that is what I sought to do, but after discussion I renamed it Maximize Affirmed Majorities to emphasize the positive rather than the negative.)

I wish someone had asked for my opinion regarding these proposed changes before they wrecked the Wikipedia information about MAM. I invested a lot of time thinking about naming issues, and it's unlikely anyone else is as familiar with all the ways MAM and Ranked Pairs differ. One can only believe MAM isn't significant enough if he doesn't understand the ways it differs from the other methods. MAM satisfies more criteria than Ranked Pairs and more criteria than Schulze's beatpath. Computer simulations show that majorities rank MAM winners over beatpath winners more often than vice versa, and more voters over the long run rank MAM winners over beatpath winners than vice versa. MAM differs from Tideman's Ranked Pairs in some significant ways and several subtle ways, but the Wikipedia articles on Ranked Pairs and MAM (which I did not write) said so little about the details of either method that most readers were unaware of all the differences. SEppley (talk) 17:20, 8 March 2010 (UTC)


 * I agree with Hermitage that MAM is only a version of RP. Anyway, SEppley admits that he was aware of RP when "MAM was invented independently" by him. And whether "MAM satisfies more criteria than Schulze's beatpath" is a matter of opinion. Markus Schulze 18:12, 8 March 2010 (UTC)


 * Why does Markus think it matters that I had read about RP (a couple years beforehand)? I was not conscious of RP at the time. I remembered it in a negative way, since it fails criteria we were discussing in the Election-Methods maillist at the time. (Strong Defensive Strategy Criterion, etc.) What I remembered positively was the "independence from clone alternatives" criterion Tideman proposed, which was satisfied by other methods we in the maillist preferred, which were not RP-like. Until after I worked out for myself the algorithm that quickly finds the order of finish that minimizes the largest thwarted majority, I was unaware the algorithm would be RP-like. After someone pointed out the relationship, I reread Tideman's papers more carefully, this time reading his proofs of theorems. (His proofs have several errors, by the way.) I don't see how my awareness is relevant to whether MAM should be included in Wikipedia. I presume Markus would admit MAM is better than RP because MAM satisfies criteria he deems important that RP does not satisfy. Perhaps Markus has an incentive to bury information about MAM, since people unaware of MAM are more likely to pick the Schulze method. SEppley (talk) 12:14, 11 March 2010 (UTC)


 * Dear SEppley, you wrote: "MAM was invented independently of Tideman's work. (...) Mike was like me at the time: aware of Ranked Pairs but unaware of its relationship to MAM." What you are doing is: You are trying to claim credit for not understanding Tideman's ranked pairs method.
 * Anyway, the first time that you used the term "Maximize Affirmed Majorities" (MAM) or "Minimize Thwarted Majorities" (MTM) was on 23 February 2000. Already in that mail, you were well aware of Tideman's work and of its relationship to MAM/MTM.
 * Another problem with your "independent invention" is that it hasn't been published anywhere. Therefore, it is non-notable. Markus Schulze 17:45, 11 March 2010 (UTC)

Voting criteria pass and fail
This article lacks an evaluation based on criteria. Since RP is a condorcet method, listing whether or not it satisfies the Smith criteria would be relevant. I wonder if there are any volunteers for this?--Fahrenheit451 00:29, 15 Jun 2005 (UTC)


 * RP and MAM etc satisfy the Smith criterion (a.k.a. the top cycle criterion). Assume x is in the Smith set (a.k.a. top cycle) and y is not. By definition, every candidate in the Smith set is ranked by a majority over every candidate outside the Smith set. Thus the "x over y" majority cannot be part of a majority cycle. (Proof: Suppose to the contrary that x over y is part of a cycle. This means there exists a sequence of three or more candidates a1, a2, ..., aN such that a1 is y, aN is x, a majority does not rank a2 over a1, a majority does not rank a3 over a2, etc., and a majority does not rank aN over aN-1. Since a1 is not in the Smith set, a2 can't be either, by the definition of the Smith set. Similarly, since a2 is not in the Smith set, a3 can't be either. By induction, aN can't be in the top cycle. But that is a contradiction since aN is x. Thus the lemma has been established.) Since x over y is not part of a cycle, it follows that RP (and MAM etc) will affirm that x finishes ahead of y. SEppley (talk) 18:28, 8 March 2010 (UTC)

Hmm, I guess we could just copy all the criteria over from Schulze method, since they'd be the same. But one note: If this article defaults to a margins interpretation (which I hope it doesn't, in fact), then the method won't satisfy Plurality criterion, Strong Defensive Strategy criterion, or Weak Defensive Strategy criterion. ...And I notice there is no article for Plurality criterion. Hmmmm. KVenzke 05:22, Jun 15, 2005 (UTC)
 * KVenzke, do you mean Majority criterion? - McCart42 (talk) 22:57, 16 July 2005 (UTC)


 * No, Plurality says that if A has more first preferences than B has votes of any rank, then B must not win with greater probability than A. Margins versions of RP and Schulze don't satisfy this. KVenzke 14:17, July 17, 2005 (UTC)

Kevin Venzke is mistaken above where he claims the Schulze method (a.k.a. beatpath) and the "winning votes" variation of Ranked Pairs satisfy the same criteria. Schulze's method fails Peyton Young's "local independence of irrelevant alternatives," which is satisfied by Ranked Pairs (regardless of whether it uses winning votes or margins) and by MAM. The winning votes variation fails Mike Ossipoff's Strong Defensive Strategy and Weak Defensive Strategy criteria (and Steve Eppley's Minimal Defense criterion, which is very similar to Ossipoff's Strong Defensive Strategy criterion) since according to Tideman each vote in Ranked Pairs is a strict ordering of the candidates; whereas MAM and Schulze allow weak orderings. Schulze's method fails Steve Eppley's "immunity from majority complaints," which is satisfied by the "winning votes" variation and by MAM. (I've distinguished MAM here because MAM differs in several ways from the winning votes variation of Ranked Pairs: 1. MAM allows each vote to be a weak ordering, allowing voters to express indifference and save time by leaving candidates unranked or ranked equally worst. 2. MAM includes only the majorities when constructing the "largest to smallest" order of precedence, whereas Ranked Pairs includes all N*N-1 ordered pairs, including ties and minorities; note that ties can be larger than majorities in the variation of RP most similar to MAM, which uses winning votes and allows votes to be weak orderings; note also that, by including ties in the order of precedence, RP always constructs a strict order of finish by the time it has considered the smallest pair, whereas MAM postpones tiebreaking involving pairwise ties until a final stage (if necessary). 3. MAM's tiebreaker differs subtly from Tideman-Zavist's 1989 tiebreaker, allowing MAM to completely satisfy the strong Pareto criterion, which is not completely satisfied by the winning votes variation of RP.) SEppley (talk) 18:28, 8 March 2010 (UTC)

Ranked Pairs variants
Do the Ranked Pairs variants, Maximize Affirmed Majorities and Maximum majority voting, deserve their own article (especially given the amount of overlap due to the Tennessee voting example)? My vote is no. -- Dissident (Talk) 23:21, 25 December 2005 (UTC)


 * I agree. Maximize Affirmed Majorities (MAM) and Maximum majority voting (MMV) should be merged into Ranked Pairs, because MAM and MMV are only concreted formulations of Ranked Pairs. Markus Schulze 09:10, 26 December 2005 (UTC)


 * The merger is a good idea – especially if a comparison table of the variants is provided, similar to the one on Voting system. clacke 08:20, 21 April 2006 (UTC)


 * I disagree. Ranked Pairs is a specific method first described in 1987 by Tideman, refined in 1989 by Tideman & Zavist. MAM and MMV are not "concreted formulations" of Ranked Pairs; they are significantly different from Tideman's method, which is what allows them to satisfy desirable criteria that Ranked Pairs fails. Several people in the Election-Methods maillist and in some websites have used the name Ranked Pairs to mean the same method as MAM, or to mean a family of methods that includes both MAM and Tideman's Ranked Pairs, but those usages have sown confusion. Not only was the MAM wikipedia article deleted, all information about MAM was deleted from the Ranked Pairs wikipedia article! THAT WAS NOT A MERGER; IT MAY EVEN HAVE BEEN VANDALISM, since MAM satisfies more desirable criteria than either Ranked Pairs or Schulze's method. (Ranked Pairs fails Mike Ossipoff's Weak and Strong Defensive Strategy criteria and fails my Immunity from Majority Complaints criterion. Schulze's method fails Peyton Young's Local Independence of Irrelevant Alternatives criterion and my Immunity from Majority Complaints criterion, and computer simulations show that majorities rank MAM winners over Schulze winners more often than vice versa.) For these reasons, MAM is the superior method and thus merits its own wikipedia article. (That article should mention the several ways in which it differs from Ranked Pairs). The Ranked Pairs article should say that Ranked Pairs is just Tideman's method and is not MAM, and that the name Ranked Pairs has been used in various websites and maillists, somewhat misleadingly, by some people when they mean the MAM method. SEppley (talk) 14:32, 8 March 2010 (UTC)


 * I don't agree that "MAM satisfies more desirable criteria than Schulze's method". The Schulze method satisfies Woodall's CDTT criterion and guarantees that the winner is always chosen from the union of all sets with minimum worst defeat. See section 4.9 of my paper. Computer simulations by Norman Petry, Jobst Heitzig, and Barry Wright show that RP/MAM needlessly generates winners with strong worst defeats, while the winner of the Schulze method is almost always identical to the winner of the MinMax method. Anyway, WP:NOTAFORUM. Markus Schulze 18:35, 8 March 2010 (UTC)


 * Another problem of RP/MAM is that it has an exponential runtime when we want to calculate all winners. This fact makes it difficult to combine RP/MAM with some additional tiebreaker. Markus Schulze 18:57, 8 March 2010 (UTC)


 * Isn't it a problem of the Schulze method too? Ranked Pairs has O(n4) complexity, Schulze method has O(n3) complexity. Wat 20 01:21, 8 July 2012 (UTC)

Weaknesses
Can anyone add a description of the weaknesses of this method? What kinds of strategic nomination / voting is it suceptible to? --Doradus 21:50, 6 January 2006 (UTC)


 * As with other Condorcet methods, strategic nomination and voting are possible when there's a Condorcet cycle, or when voters have enough information about the votes to create one. I don't know any strategic issues that are specific to Ranked Pairs.


 * A practical problem I've seen is that the method is not very deterministic. When ties between defeats occur, which is very likely in small elections, you have to break them randomly, and the entire outcome of the election can hinge on these ties. This is an observation from my use of the method, though, and I don't know of a source to cite for it.


 *  r speer  / ɹəəds ɹ  22:45, 6 January 2006 (UTC)


 * Isn't there a way to make a tie a tie? Why are tiebreakers necessary?  --Doradus 16:29, 7 January 2006 (UTC)


 * But when you get a tie deep in the vote-counting process, not in the final result, what do you do with it? If you propagate the ties into the final result, you can get results like "((A > B > C > D) tied with (B > C > D > A)) tied with C > D > B > A". You'd be better off declaring the whole Condorcet cycle to be a tie - which in fact gives you a very theoretically nice method.  r speer  / ɹəəds ɹ  21:52, 7 January 2006 (UTC)


 * There is a variation called CIVS Ranked Pairs which allows ties and is completely deterministic. What it does is when checking for cycles in the graph, it only checks a preference against stronger preferences. Then adds all preferences of the same strength in the graph at the same time. If any cycles result from these same strength preferences, they become ties.


 * This variation is useful even if you use recommended non-deterministic tie-breaking, because it allows you to check whether the final result was deterministic or not. Wat 20 01:45, 8 July 2012 (UTC)

Broken link
The link in the Criteria section to the voting criteria table appears to be broken. It links an anchor on the Voting system page, but there does not appear to be any voting criteria table on that page. Qutezuce 08:56, 16 January 2006 (UTC)
 * I cleaned it up into the start of a more useful, self-contained section. Thanks! Scott Ritchie 22:07, 16 January 2006 (UTC)

Why the captial P in the title?
I ask this as a layman. -- nyenyec &#9742; 16:11, 10 February 2007 (UTC)

Well, as you can see from the discussion above, voting theorists tend to capitalize everything, sometimes to the point of silliness. But the decapitalized title "Ranked pairs" doesn't seem right to me; it makes it sound like the article is about a kind of pairs, instead of about a voting system referred to by the name "Ranked Pairs". In comparison, the Borda count really can be described as a "count", as it's a way to count up votes.  r speer  / ɹəəds ɹ  11:03, 11 February 2007 (UTC)

Second place
I started a discussion recently on the election-methods mailing list about how to determine "second place" in a Ranked Pairs election. An interesting fact I found is that in ranked pairs the second place winner can be determined either by looking at whomever is second in the (complete) graph OR by removing the winner from all ballots and rerunning the algorithm -- both will produce the same result. Scott Ritchie (talk) 22:03, 14 October 2008 (UTC)


 * This criterion is called "decreasing sequential independence". Markus Schulze 22:25, 14 October 2008 (UTC)


 * Both Ranked Pairs and MAM satisfy Peyton Young's criterion "local independence of irrelevant alternatives" (LIIA). This means that deleting the winner from the ballots will not change the relative order of finish of the remaining alternatives. (It also means that deleting the alternative that finishes in last place will not change the relative order of finish of the remaining alternatives.) It follows by induction that deleting the top N finishers will not affect the relative order of finish of the remaining alternatives (for any N). The article should be revised regarding how the order of finish is constructed, since it is unnecessary to iteratively delete the "winner." SEppley (talk) 14:43, 8 March 2010 (UTC)

Note also that Ranked Pairs is equivalent to the voting method defined as the method that finds the "best" possible order of finish, assuming "best" is defined to mean the order of finish that minimizes the largest reversed pair, where the size of a pair "x ahead of y" is measured by the so-called "margin of victory": subtracting the number of votes that rank y over x from the number of votes that rank x over y. (By "minimizes the largest" I mean in the lexical sense. Any two orders of finish can be compared by considering only the pairs on which they disagree; that is, all pairs x,y such that one order of finish places x ahead of y and the other order of finish places y ahead of x.) If 51 voters ranked x over y and 49 ranked y over x, the size of the "x over y" pair is 2, and the size of the "y over x" pair is -2. (Margin of victory is a misleading term, since it can be positive, zero or negative, and it is unnecessary to use the term.) My point here is that Ranked Pairs already returns an order of finish, making it unnecessary to use Ranked Pairs iteratively to construct the rest of the order of finish. SEppley (talk) 16:01, 8 March 2010 (UTC)


 * The Maximize Affirmed Majorities method (MAM) is also equivalent to finding the "best" order of finish, but MAM defines "best" in a significantly different way: the size of the opposing minority is not subtracted when measuring the size of a majority. (Minorities are not considered at all by MAM except in the rare case where two or more majorities are the same size, in which case MAM uses a form of tiebreaking that considers the opposing minorities.) In the example above, for MAM the size of the "x over y" majority is 51, and the "y over x" minority is not included in the order of precedence. (Ranked Pairs ranks all pairs&mdash;majorities, minorities and tied pairs&mdash;but MAM ranks only the majorities, and could perhaps have been named Ranked Majorities. There are several other subtle differences between MAM's tiebreaking and Ranked Pairs' tiebreaking, but those are beyond the scope of this note.) MAM allows votes to be "weak" orderings (that is, the voter can rank two or more candidates the same to express indifference between them) so it matters how majorities' sizes are measured, and MAM measures them in the way that permits satisfaction of several criteria failed by Ranked Pairs. For example, suppose 35% rank x over y, 20% rank y over x and 45% are indifferent between x & y, and suppose 55% rank y over z and 45% rank z over y. For Ranked Pairs 35%-20% is greater than 55%-45% so it gives the x over y pair greater precedence than the y over z pair. For MAM 55% is greater than 35%, so MAM gives the y over z majority greater precedence than the x over y majority. (Ranked Pairs as defined by Tideman requires votes to be strict orderings, which is another part of the reason it fails some of those criteria.) SEppley (talk) 16:01, 8 March 2010 (UTC)

Move to Ranked pairs

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 

The result of the move request was: page moved per request. GTBacchus(talk) 04:01, 2 June 2010 (UTC)

Ranked Pairs → — This doesn't seem to be a proper noun. It's uncapitalized both in the source publication as well as this external link. Jafeluv (talk) 17:01, 25 May 2010 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Odd number of candidates
In the method of Ranked pairs candidates are divided into pairs & voters have to choose the winner of each pair. What if the number of candidates is odd? That must be a serious drawback of this voting method. Nikolay95 (talk) 20:28, 13 September 2012 (UTC)


 * Candidates are not divided into pairs (as in a tournament?); pairing means that each candidate is compared to each other candidate. For 3 candidates there are 3 pairings; for 5 candidates, 10 pairings; for 7 candidates, 21 pairings. —Tamfang (talk) 22:11, 13 September 2012 (UTC)

Subsection ordering in Example Section
In the Example section, the subsection Summary follows subsection Ambiguity resolution example, yet the Summary subsection summarizes all subsection preceding Ambiguity resolution example without mention of the Ambiguity one. The Ambiguity resolution example subsection is effectively a second example, with separate example items (A&gt;B&gt;C&gt;A as opposed to the Tennessee cities in the rest of the main section). It would seem appropriate to move the Summary subsection before the Ambiguity resolution example subsection. Anyone have a reason in opposition to this? — al-Shimoni  (talk) 16:17, 17 November 2012 (UTC)

Parallel computation of ranking
Is there any algorithm which calculates a Ranked Pairs ranking in parallel? The algorithm in Eppley's MAM procedure definition paper is sequential. Wat 20 18:05 16 February 2013 (UTC)


 * The algorithm is fundamentally sequential, but parts of it can be parallelized. Counting ballots and computing the pairwise results can be done in parallel, provided that all of the results are collated in the end. Sorting the majorities can be parallelized using various divide-and-conquer sorting methods (merge is likely the best). Locking in majorities pretty much needs to be done sequentially. Computing transitive closure (to detect cycles) can be parallelized. None but the first really matters unless you have a large number of candidates though.

72.68.72.137 (talk) 04:38, 8 October 2014 (UTC)

Locking in Ties
The following line is in the "Lock" section of the algorithm description: "One way to resolve this issue is to allow cycles if they are needed to resolve ties (i.e., if a single new edge would not create a cycle, but multiple tied edges would), and then define the winners as the resulting Schwartz set."

Where does this come from? There seems to be no citation given. Can anyone give further description? — Preceding unsigned comment added by TheShepherd7 (talk • contribs) 04:06, 18 November 2017 (UTC)


 * This proposal has been added by Vectro in 2016. I am not aware that this proposal has ever been published somewhere. So it seems to be original research. Markus Schulze 08:38, 19 November 2017 (UTC)
 * Markus, I am attempting to describe your method, so I am surprised to see you consider it OR! Perhaps I have misunderstood something. I added a citation to a secondary source. TheShepherd, for more information you may wish to look here. Vectro (talk) 00:12, 29 November 2017 (UTC)
 * The Schulze method is not a concretization of ranked pairs. The Schulze method and ranked pairs are two different election methods in so far as they can lead to different winners even when both methods have decisive winners. Markus Schulze 19:22, 1 December 2017 (UTC)

Is there a typo or error in the conditions in "Procedure > Sort"?
"Procedure > Sort" states that:

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.

However, I'm having trouble seeing how the second criterion could ever be possible if there are no indifference or unstated candidates, as in the example, because surely Vxy = Vzw would imply that Vwz = Vyx.

Further, under "An example > Sort", the following is stated:

"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."

but this appears to be in direct contradiction with the given procedure; abbreviating Chattanooga, Knoxville, and Nashville with c, k, and n respectively, and letting x = z = n, y = k, and w = c, we have that Vnk = Vnc = 68%, and Vkn = Vcn = 32%, so neither criterion is satisfied, and thus neither majority can precede the other.

It seems like to have the conditions agree with the reason given, we need to add a third condition similar to "Vxy = Vzw, Vyx = Vwz, and Vwy > Vyw". Or is the reason given incorrect?

I also can't seem to find any sources that back up this procedure, so I can't tell if there is indeed an error or if I'm just misunderstanding it. The article states in reference #1 that the original article uses a different approach to ranking the strength of a victory. Edderiofer (talk) 14:54, 10 September 2018 (UTC)


 * I fully agree – thanks for pointing this out. I’ve rewritten the procedure (and also deleted the tag complaining about paucity of references – Tideman’s paper is crystal clear and authoritative, so why would anyone need more?) Colin.champion (talk) 14:16, 10 January 2022 (UTC)

Typography dispute
Hi, User:Beland and User:Colin.champion; I've noticed that in the past couple of days, you two have been editing and reverting the article over the typography of the article. To prevent edit-warring (see WP:WAR), I've created this section where you two can hopefully come to a consensus on the matter instead. Edderiofer (talk) 17:28, 30 January 2022 (UTC)
 * There is no point in looking for consensus with Beland, who is purely a drive-through editor. I use thin spaces to ensure correct spacing of italics; he or she made changes which do not have the desired effect. If there is some Wikipedia standard which says that incorrect spacing has to be tolerated then I haven’t a leg to stand on; otherwise I don’t see that the changes are any more than vandalism. Colin.champion (talk) 17:46, 30 January 2022 (UTC)
 * Whoa, there's no need for name-calling; I'm happy to discuss! First, just a note that the straight quote marks and apostrophes are required by MOS:STRAIGHT. I restored these after they were reverted, but they got reverted again anyway. I had put them in a separate edit on the assumption that they were not in dispute, and in case the whitespace changes were still disputed (which apparently they are). No worries, this happens sometimes, but if you could try not to revert them again, that would save some work.
 * In general, editors prefer that we use alternatives to numeric HTML entities because the vast majority of people don't know what they mean, and that's why I'm converting them across the project. A full non-breaking space is definitely unattractive in this case; that was a mistaken substitution. It turns out the exact substitute nnbsp is available, and I would recommend using that or narrow no-break space if you think a thin, non-breaking space is the best option. (Both because it's more meaningful to editors and because then this article will stop showing up on my numeric HTML entity report.)
 * Whether or not a thin space is needed after an italicized character is a bit of a tricky question. There are two reasons why we might decide not to add any space there at all. One is that MOS:MARKUP says to keep markup simple, and maybe precise control over whitespace is not something that we need to try to do at the expense of more complicated markup. The other is that the visual results are different depending on the web browser, installed fonts, operating system, zoom level, etc. For example, on my screen, a thin non-breaking space actually looks too wide. Trying to make the spacing look perfect on one system may cause it to look horrible on other systems. We might consider that making the output look good is a job for the font system and the web browser, and so for example if we're seeing characters overlapping with each other or just too close because of italics, maybe the right thing to do is to file a bug with the appropriate software vendor. Keeping markup simple might give us the best chance of looking acceptable on most systems, and not making things look very bad if software vendors improve their rendering code. It is for these reasons that I removed the thin spaces entirely.
 * That said, having no space there does look somewhat asymmetrical on my system. I'm not sure what looks best for other people, but what looks best for me is actually a hair space, not a thin space. If we don't mind complicating the markup and other people have the same issue as me, then I would recommend using hair space or hairsp. (These templates are also non-breaking, which we need in this context.)
 * And finally, a note about the use of math: I don't know if that change was reverted because it was disputed or simply because it was made in the same edit as the other whitespace changes. math makes whitespace in the supplied text non-breaking, and I think that template is good to use for equations because it makes the body of the equation easier to read in the wikitext, since it obviates the need to include bulky markup for non-breaking spaces. It also lets spell-checkers and grammar-checkers know how to treat this span of text.
 * So, given all of that, what's your preferred solution? -- Beland (talk) 00:02, 1 February 2022 (UTC)


 * Thanks for replying constructively. Firstly, it is not true that “straight quote marks and apostrophes are required by MOS:STRAIGHT” – MOS is a set of guidelines, not of policies. The last time curly quotes were discussed there was barely any consensus, and the retention of the guideline was justified in part by the fact that “The MOS does not prohibit or require. It says what is preferred, so that editors will generally move in that direction”. See MOS talk page. It adds that Wikignomes convert curly quotes to straight, but the opposite movement is not excluded.
 * Secondly, I am not fond of numeric HTML entities and seldom use them; this is a case in which I find them convenient, in conformity with the statement in the MOS that it should be “treated with common sense, and occasional exceptions may apply”. The deletion of my thin spaces, or their replacement by full-width spaces, produced effects which I found obtrusively ugly. I dare say there are better solutions. I am happy to try to use in their place; other templates such as  may be better still.
 * I bulk-reverted your edits because I felt that overall they reduced the readability of the article, and I didn’t think it was fair to impose on me the burden of going through them one by one to reduce the damage. I wouldn’t have objected to some of them if they hadn’t been bundled with the others. Colin.champion (talk) 10:28, 2 February 2022 (UTC)
 * Though I also like the appearance of curly quotes, I see nothing in the MOS that allows editors to change straight quotes to curly quotes, though I understand doing that incidentally as part of a mass revert. The full quote from the discussion was "People use curly quotes all over, and wikignomes convert them to straight. It could go the other way if we decide." This is not an endorsement of rogue gnomes going in the opposite direction of what the MOS prefers; it's an explanation of what would happen if the MOS were changed. If you want to be able to change to curly quotes, you would need to get consensus and actually change the MOS to say that curly quotes are preferred or that a given article can use either style consistently. A few people commenting on the talk page did not result in changing the MOS. A change seems unlikely to happen without an actual RFC, since it would affect so many articles, for the technical reasons given in the MOS that seem more important than aesthetics, and because an RFC has previously affirmed this preference. Based on the MOS I have programmed the site-wide spell checker to treat straight quotes as demarcating quotations, and curly quotes as an error, and so far I don't remember anyone else objecting to following the MOS, whether or not it agrees with their personal preference. After cleaning up tens of thousands of articles, the only common-sense exceptions I have come across are when discussing the characters themselves, when a non-English language uses bottom-top quote marks, and for characters from non-English languages which are not actually quote marks (like ʻokina).
 * On the question of whitespace following italics, I have changed this article to use hairsp; hopefully that looks acceptable to everyone. -- Beland (talk) 19:43, 2 February 2022 (UTC)


 * – that all looks fine, thanks. Colin.champion (talk) 16:01, 3 February 2022 (UTC)

Repeated use of the algorithm?
The article now says "Alternatively, you can use the procedure above to pick the winner, make that candidate first place, drop them from the election (i.e. drop all comparisons/edges that include them), and then repeat the process." Please supply a citation to a textbook or similar to support this. Absent support, this is too easily confused with original research. — Q uantling (talk &#124; contribs) 18:59, 25 February 2023 (UTC)
 * I have removed the part about using the algorithm repeatedly. If you want to re-insert it, please include a citation.  — Q uantling (talk &#124; contribs) 14:37, 28 February 2023 (UTC)

Full sorted list of preferences
Until the these recent edits (by ), the article emphasized how to rank all candidates from first to last, not merely how to choose the first. Shouldn't we be keeping the description for full ranking? — Q uantling (talk &#124; contribs) 18:41, 23 February 2024 (UTC)


 * Thank you for this recent edit. Is it the case that the newly described procedure for generating the full ranking -- rerunning the election without the first-place winner to get a second-place winner, etc. -- is equivalent to the previously described approach?  The previously described approach had us going through the pairwise election results only once, from strongest to weakest, keeping the first and also keeping every subsequent pairwise victory that does not create a contradiction (e.g. directed cycle) with the already kept pairwise victories.  That is, this approach makes a single directed acyclic graph, which is also a linear ordering, and the full ranking can be read from the graph without re-running the approach from the start with some candidates removed.
 * If they are not equivalent, should we discuss both? If they are equivalent, should we indicate that, and is this newer description considered better by the literature?  Thank you — Q uantling (talk &#124; contribs) 19:16, 23 February 2024 (UTC)
 * They're equivalent thanks to RP satisfying LIIA. The reason I replaced it with the current method is that I think River doesn't create a consistent linear ordering with the one-step approach. River is both easier to compute--O(n^2) instead of O(n^3)--and gets to satisfy independence of strongly-dominated alternatives (i.e. candidates who are "way behind" the other members of the Smith set don't affect the result). However, the single linear ordering approach is faster for RP, so I think it would make a good footnote. -- Closed Limelike Curves (talk) 19:31, 23 February 2024 (UTC)
 * Thank you for the quick response. Yes, please do add the appropriate footnote and any other edits that would explain to the article reader what you have explained here.  Thank you — Q uantling (talk &#124; contribs) 20:20, 23 February 2024 (UTC)

The river method is not a variant of ranked pairs.
The river method is not a variant of ranked pairs. The river method and the ranked pairs method have different heuristics; the river method tries to find the "best" arborescence (according to its own definition for "better"), while the ranked pairs method tries to find the "best" complete digraph (according to its own definition for "better"). The river method and the ranked pairs method are not related; the ranked pairs method has been proposed by Nicolaus Tideman in 1987, while the river method has been proposed by Jobst Heitzig in 2004. The river method and the ranked pairs method find different winners (even when there are no pairwise defeats of equal strengths). The river method and the ranked pairs method have different properties; the river method satisfies independence of Pareto-dominated alternatives, while the ranked pairs method violates this criterion; the river method violates reversal symmetry and local independence of irrelevant alternatives, while the ranked pairs method satisfies these criteria. The current version of this article misleads the reader into believing that the river method satisfies all criteria that are satisfied by the ranked pairs method plus independence of Pareto-dominated alternatives. The fact that both methods might look very similar doesn't justify the claim that the one method is a variant of the other method. Markus Schulze 10:06, 16 May 2024 (UTC)


 * I would agree with this. Further, I don't think River really belongs in any wikipedia article yet, as it is original research, so I do not think @Closed Limelike Curves should reinstate it. Jannikp97 (talk) 05:53, 17 May 2024 (UTC)

Capitalization of article title
It looks like the change from "Ranked pairs" to "Ranked Pairs" would need to be implemented as an actual page move rather than via {{DISPLAYTITLE}} or similar. And in light of the discussions above on exactly this topic, if you want such a change then I recommend that you propose the move rather than boldly making it. In the meantime, I recommend that we leave the article text references to these two words as they were. — Q uantling (talk &#124; contribs) 18:38, 24 June 2024 (UTC)


 * Got it; that's unusual. Do you know why it doesn't work? Closed Limelike Curves (talk) 19:40, 24 June 2024 (UTC)
 * I don't know why. If I had to guess, I'd say it's a policy thing.  It forces people to debate the proper capitalization.  In contrast, {{DISPLAYTITLE}} is if (part of) the title should be italics or superscript or similar, which I guess is not as controversial somehow.  That's my guess.  — Q uantling (talk &#124; contribs) 20:31, 24 June 2024 (UTC)