Talk:Monotonicity criterion

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There's a "message board argument" tone to many sentences in the latter half of the article. "And not, as claimed by earlier versions of this wikipedia article, X" sounds childish. —Preceding unsigned comment added by (talk) 13:53, 5 January 2011 (UTC)

Rights on the text[edit]

From the page:

Some parts of this article are derived from text at

Some other pages have stated this, and said "used by permission". Can someone confirm that this use is by permission?

I did not seek explicit permission (though I may get around to sending a thank-you email). The linked page, at the very bottom, gives more than enough permission to re-distribute under the GFDL. Indeed, the way that he phrased it there was no need to explicitly acknowledge his work, but I thought it would be good to do so, both to thank him, and because his site is a useful resource to link to. DanKeshet 21:23 7 Jul 2003 (UTC)

Thanks! That's good news. -- Anon.

Don't confuse Condorcet criterion and monotonicity criterion[edit]

The comment that "several variants of the Condorcet method (including Cloneproof Schwartz Sequential Dropping and Maximize Affirmed Majorities) are monotonic" is misleading since there is no connection between the Condorcet criterion and the Monotonicity criterion. If you only want to say that the Condorcet criterion and the Monotonicity criterion are compatible, then it is sufficient to say that Cloneproof Schwartz Sequential Dropping and Maximize Affirmed Majorities are monotonic. -- Markus Schulze

Hmm, I don't think the para as it was asserts a connection between the Condorcet Criterion and MC. And though CSSD and MAM are different, they really are minor tweaks on the Condorcet Method and usually make the same decision, so it seems weird to write about them as if they were as different as Plurality and Borda. Is there any wording which mentions that these two are both Condorcet methods that would be acceptable to you? — ciphergoth 14:05, 2004 Nov 21 (UTC)
ciphergoth asked: "Is there any wording which mentions that these two are both Condorcet methods that would be acceptable to you?" No. The fact that CSSD and MAM satisfy the Monotonicity criterion must only be used to promote CSSD or MAM. But it must not be used to promote Condorcet methods in general or even a concrete Condorcet method that doesn't satisfy the Monotonicity criterion. -- Markus Schulze 21 Nov 2004
As ciphergoth, I fail to see how the sentence you talk about promotes Condorcet methods in general.--Chealer 03:36, 2004 Dec 10 (UTC)

Old edit war[edit]

I've reverted from a lot of edits by user:polytope. Polytope: you are wellcome to contribute to articles, but your material was really not appropriate, for at least the following reasons:

1) Wikipedia has a strict Neutral point of view policy which your discussion of the CVD definitely did not fit under. Please read that link for more information on how to construct NPOV pages.

2) It was very technical and frankly incomprehensible to me. You have to do a much better job of explaining your terms.


Your decision is unfair, wrong, and unreasonable, so I intend to upload the page again, after at least 1/2 of a day or so. The spirit of fairness would be more present if I was the person editing the definitions. I expected details comments. Here are problems with your decision. My reasoning is more carefully done than yours. Note that the edit that you did is here:

I assume you do not want to even read this, but do however wish to keep censoring with an indifference to how thoroughly an audience moves against you if publicity shows up. I find your decision to be unreasonable since it has been made without an obvious relationship to the facts or evidence. That is plain since except for the mention of the CVD, no other comment relates back to the text of the page. That refers to grounds #1 and #2. The decision to replace the page instead of editing it is found by me to be unreasonable in respect of ground #2 since it seems that you could have satisfied yourself by altering one or both of the words "[IRV is] promoted [in USA]" and "[the CVD] complains [about...]". Also you refused to provide reasoning by e-mail after I requested that. You had two grounds but the weights were missing so the significance of your failure to comprehend what I wrote is something that I now do not have. Any of my text could lucid and optimal though may misunderstand. You may refer to this for the theory of how to make decisions about me: <A HREF=""></A>.

Decisions should be accompanied with reasoning and reasoning is too inadequate to be present. I do very much suspect that you had an improper purpose for taking such a strong action over that article. I had not used a wikipedia before. I note that your initial friendliness fast became a lockout where I could not get fuller information about your reasoning. Your website is remarkable in carefully censoring out the idea of monontonicity (, yet it seems to aim to describe principles of preferential voting (and fails completely since none of those can be ported over to mathematics in my opinion). The fact censoring of censoring out that same anti-CVD topic is indicative of a consideration of an irrelevant factor (categorized under improper purpose). I requested (using private e-mail) a statement of the reasoning that was fuller and I never got any. Your argument was that I should write here at this page; that I now write to, and your grounds were "I don't read my e-mail very often." (you wrote on about 10-July-2003). That was a harassing decision. You intended to communicate only using private e-mail and you explained that since you receive and read your e-mail quite well (but a little slowly), then I must therefore avoid using e-mail (and use the wikipedia). That can not be followed and it is unreasonable. I was later? clear in stating that e-mail should be used and the wikipedia must not be. Why didn't you identify your error of telling me to use this wikipedia page ?. Ground #2 implied that you did not understand my mathematics page.

As I wrote at the Election Methods List, you made a mistake at your website indicating that you lack a knowledge of the principles of 2 candidate preferential methods. It was about "representation" in 2 candidate elections. I assume that you test is oppressive and harsh since setting an impossible hard standard for me: a requirement that you understand me when I write for the public with the plainest clarity I can achieve. I only have your word for it that you failed to understand my page. I reject the idea that you could not understand all of the page. Perhaps your 1st ground was oppressive since it told me to look at a policy document but it unreasonably failed to identify a problem or error in my text (or my behaviour) that would be guided to a better state by the policy document. I.e. you warned of how I could improve under policy without identifying what should get better. The CVD comment about non-neutrality with the CVD appeared to be untrue.

I accuse you of making an improperly discriminatory decision because you have worse reasoning and harsher actions for authors of monotonicity pages and me (so far). You maybe do not want your Green party to keep losing. Your decision was more adverse to the public interest than this: you made a lot of major changes. I have a diff program and it is convenient for me to undo only some of changes. Most of my complaint is about your lack of reasoning rather than the decision to erase the document and restore a worse one. Nowhere do you give an opinion on the difference between the documents. You decision was wrong to replace one document with another and have no consideration for the 2nd new document. The reasoning is required by me and there was consideration of you preferred version. I don't know why you prefer it. It is certainly less useful. The text does not reject the Alternative Vote (you wrongly opted for that), since mysteriously Mr B.C. confuses a preference with a candidate. That is unexplained. I have asked Mr B.C. to tell the Election Methods List and not enough time has passed. It seemed to be deliberate since the word "alternative" was used. So there is some improper purpose to the text you prefer (in addition to the possible improper purpose of not letting a monotonicity page harm Greens across USA over an unknown period of time). Greens should not be advanced by unthinkable definitions of others. That is a bit neutral on Greens and able to cause harm to members of the Election Methods List. Already Mr Eric Gorr has been confused by a definition from the website. My new version that I will upload is here, only in the box at the top: The topic is mathematics and they are detailed. If you aer not, then you should quite withdraw. There is also a private dispute where I intend to comment adversely on your personal website but only fairly and accurately.

Craig Carey, New Zealand (no advice that I be anonymous) 'research at'. Wed 16-Jul-2003


Responding to just a few of your points: I was perfectly reasonable in refusing to provide additional reasoning by e-mail; there is nothing that says I should spend my time responding to personal e-mails about wikipedia, especially not after you accused me of being "perverted" and "corrupt" and suggested I was taking bribes from user:RobLa. Also, judge my contributions on their merit, not on my supposed political views about voting systems. In any case, your assumptions about my views are inaccurate, sometimes wildly so.

On to your actual contribution. Here are my specific problems:

1) In your definition, you introduce the concept of a "weighted vote", without any explanation whatsoever of what that is. Also, your text is gramatically incorrect, ambiguous, and frankly incoherent.

2) The section titled "Centuries of fair voting ideals lost as use of IRV and STV spread" is chock full of hyperbole, and inappropriately placed. If you want to discuss the results of IRV, it would be better placed on the Instant-runoff voting page. But don't be surprised if your section is heavily edited there, too. In addition, you introduce the concept of "vote-negating" with barely any explanation.

3) The section titled "A human right to have a vote" is pure hyperbole, and belongs nowhere in an encyclopedia. If this were an actual tact that some people had taken, appealing for changes in voting systems to the UN under the "equal suffrage" clause, then we could as an encyclopedia, report that. We can not engage in pure rhetoric, though.

4) The section titled "Parameterized 3 candidate method has IRV/AV be 33% outside of the fair region" has more unexplained jargon, as does the section "Dual of shadowing stays-losing polytope is a normal vector constraining polytope".

5) The section titled "Monotonicity is not best for theoreticians" is inappropriate for an encyclopedia, because we are here to report on how people study the monotonicity criterion, not argue that they should study it another way.

These are not all the problems I have with your text, but they are a start.

DanKeshet 13:51 16 Jul 2003 (UTC)

STV is never monotonic[edit]

I've removed text which stated that Single Transferable Vote can be non-monotonic depending on the methods used. No counting method (not even Meek's method) eliminates the non-monotonicity of STV - all counting methods reduce to IRV in single winner elections, and IRV is non-monotonic. Some counting methods, such as Meek's method, reduce the potential for tactical voting with STV, but they do not completely eliminate the non-monotonicity.

I amnot sure I agree with the removal. It is not enough to say that Meek's method cannot be montonic because 'STV' is never monotonic. That argument is circular. I will post a fuller answer at the STV talk page when I have researched it. Alan 09:46, 17 May 2005 (UTC).
When I added that text ("STV may or may not be monotonic, depending on the method used") I was actually thinking of Condorcet-STV, not Meek. Meek can be non-monotonic, though in real-world elections it is rare; see [1]. Since Condorcet-STV doesn't actually tabulate vote transferral directly, it is not susceptible to the same analysis that indicts other STV systems; in some sense, Condorcet-STV is STV in effect, not in implementation. TreyHarris 15:25, 17 May 2005 (UTC)
Is there any proven statement that 'any' STV-method isn't monotonic? Like "any method that fulfills ... (which we call a 'STV-method') ... is not monotonic". Otherwise the most rigid statement should be "satisfaction of monotonicity criterion hasn't been proven for any known STV method". To write "all known STV methods are not monotonic" requires to show counter-examples for all of them. Please show me e.g. an example that Schulze STV is not monotonic. It reduces to the monotonic Schulze method in single winner elections, so any proposed counter-example must have at least two seats to be filled. Best, --Richard (talk) 11:52, 24 April 2014 (UTC)

If a voting system can be non-monotnic then it is fair to say that the system is a monotonic. Clearly any voting system is non-monotic sometimes. For example if there is unanamouse agreement about who the best candidate is. So when someone says that a system is non-monotonic they mean that there are circumstances where ranking someone higher can hurt them, given a certain pattern of votes. —Preceding unsigned comment added by (talk) 09:47, 1 April 2011 (UTC)

"Clearly any voting system is non-monotic sometimes.". This is simply wrong. You might have written "Any voting system may be redesigned in some way (resulting in a different system), such that it becomes monotonic, or loses this property". Best, --Richard (talk) 11:33, 24 April 2014 (UTC)

Problem in the example?[edit]

In the example of a violation of the Monotonicity Criterion, it says that candidate C would be the Condorcet winner. However, it seems that there is no Condorcet winner in this case. In pairwise matchups C would beat A, A would beat B and B would beat C, meaning all three would be in the Smith Set. The winner would then depend on what form of Condorcet voting would take place. I would suggest solving this problem by either

1) Changing the second preference of A's voters to C (this will create 4 categories in the second election, but still result in a violation of monotonicity). Then in pairwise voting we would have C beats A and C beats B, meaning C is not the Condorcet winner.

2) Remark that there is no Condorcet winner and choose an appropriate resolution scheme, such as Minimax Condorcet, to choose a winner.

--Anon. 18:25:07, December 3, 2005 (UTC)

I cut the sentence completely. Why discuss what Condorcet methods will do instead of, say, Bucklin voting? It seems irrelevant to the article. KVenzke 17:46, 4 December 2005 (UTC)
Because condorcet methods pass the monotonicity criterion, that's why. You're right that Bucklin voting should be mentioned as well. The article could use more examples, such as a sentence explaining why Bucklin passes too. Scott Ritchie 18:35, 4 December 2005 (UTC)
Condorcet methods don't necessarily pass the monotonicity criterion. Consider Raynaud, Borda-Elimination (Nanson), or Smith//IRV. That's why it doesn't make sense to point out who the Condorcet winner is. KVenzke 04:16, 6 December 2005 (UTC)
Can you point out an example demonstrating that? It would be relevant to the article to show how they can fail. Scott Ritchie 00:10, 7 December 2005 (UTC)

Suppose that the method is Raynaud (wv), so that we repeatedly eliminate the candidate who has the most votes against him in a single pairwise contest.

36 abc
34 bca
30 cab
C is eliminated and A wins.

Now raise A on 5 bca ballots:

41 abc
29 bca
30 cab
B is eliminated and C wins.

I hope that's what you were looking for. KVenzke 22:58, 7 December 2005 (UTC)

Monotonic STV methods?[edit]

I have deleted the following sentence: "Most versions of the Single Transferable Vote which simplify to Instant Runoff when there is only one winner are not monotonic, however variants of STV such as CPO-STV and the Quota Borda system are." To the best of my knowledge, there is no STV method that has been proven to satisfy monotonicity. And the Quota Borda system has been proven to violate monotonicity. See example 3. Markus Schulze 09:22, 31 December 2005 (UTC)

Why not CPO-STV, or other methods that simplify to monotonic systems when there is one winner? Scott Ritchie 04:07, 2 January 2006 (UTC)
The Quota Borda system does simplify to a monotonic system when there is only one winner. Markus Schulze 09:50, 2 January 2006 (UTC)

Does The Example Fail To Fail The Monotonicity Criterion?[edit]

It is possible that I am missing something, but I am confused. It seems to me that the example given is not an example of a violation of the Monotonicity Criterion.

The Definition:

Douglas Woodall, calling the criterion mono-raise, defines it as:

A candidate x should not be harmed [i.e., change from being a winner to a loser] if x is raised on some ballots without changing the orders of the other candidates.

The Example:

1st Election
39 A,B
35 B,C
26 C,A
C is eliminated, and A wins.

2nd Election
49 A,B
25 B,C
26 C,A
B is eliminated, and C wins.

The Example shows that A is the winner in the first election, but in the second election, A has a raised position yet loses, which seems to violate monotonicity. However, the Definition also states that candidate x should be harmed if x is raised without changing the orders of the other candidates. I guess I am confused because in the Example the orders of the candidates change. C is in 3rd place in the first election and changes to 2nd place in the second election. This, to me, is an example of a change in order of the candidates. If this is the case, then either another example is needed or the definition of the MC needs to be clarified. Otherwise, based on my current interpretation, I am not convinced that IRV, for instance, fails the MC.

You are misunderstanding. "without changing the orders of the other candidates" means that votes which ranked y > z still rank y > z. In the example we start with 39 A > B > C, 35 B > C > A, and 26 C > A > B for the first election. We then take 10 of the B > C > A and change them to A > B > C, i.e. some voters increase their preference for A without changing the orders of the other candidates, all those votes still rank B > C. After the change we have 49 A > B > C, 25 B > C > A, and 26 C > A > B, which causes A to lose the election and IRV to violatate the monotonicity criterion. I'll be correcting the main article shortly. (talk) 18:56, 20 June 2008 (UTC)

Now, it could be interpretation is wrong and that the "changing the orders of the other candidates" means something else. It could mean a change in the secondary votes (for example, if the primary voters of C have secondary votes of A in the first election and change their secondary votes to B in the second election).

My point is that this article is confusing to someone who is not on expert on voting methods or voting system criterion. I am suggesting that this article could be clarified a little, but I do not consider myself knowledgeable enough to do so myself. Can someone either clarify the article or at least clarify this to me? Bryanmode 16:20, 6 November 2006 (UTC)

The reference to order changing is about the order of candidates on the ballot, not in some final ranking. Note that not all voting methods even produce a full ordered ranking -- the algorithm can simply halt when a winner (or N winners) are selected. And no, basic IRV is definitely not monotonic.
Say you have an IRV election with three candidates ("A", "B", "C") and the votes are as follows:
2 C > B > A
6 A > B > C
5 B > A > C
4 C > B > A
The first group and fourth group look the same, but they're separated because we're going to explore what happens when the first group changes their vote.
First round: 6 votes for A, 5 for B, 6 for C. B is eliminated.
Second round: 11 votes for A, 6 votes for C. A wins.
Now, A does such a good job that the two voters in the first bloc decide they were wrong about A. So, in the next election, with all the same candidates, the votes are:
2 A > C > B
6 A > B > C
5 B > A > C
4 C > B > A
First round: 8 votes for A, 5 for B, 4 for C. C is eliminated.
Second round: 8 votes for A, 9 for B. B wins.
By ranking A higher, these two voters caused A to lose when A would've otherwise won.
Note that the relative order of the other candidates on the ballot in the first bloc (C > B) is unchanged. That's what was required by the condition you were discussing.
Ka-Ping Yee created an excellent graphical representation of IRV's monotonicity failure. Rmharman 06:40, 28 November 2006 (UTC)

Edit needed (by someone with more expertise than me)[edit]

The following sentence is incorrect:

Tactical voting in this way presents an obvious risk if a voter's information about other ballots is wrong, however, and there is no evidence that voters actually pursue such counter-intuitive strategies in non-monotonic voting systems in real-world elections.

I don't have the time or indeed the expertise to edit the section properly, but somebody should. The second clause ("there is no evidence" etc.) has been quite plainly wrong since the German federal election of 2005, in which conservative voters in Dresden deliberately voted against their party of choice (the CDU) in order to maximize that party's number of seats in the federal parliament. This was possible due to Germany's voting system (mixed member proportional with overhang seats) and the fact that the vote in Dresden took place a week after the rest of the country due to the death of a candidate, enabling voters in Dresden to vote tactically in full knowledge of the results already achieved elsewhere. As a result of this, the German Constitutional Court ruled on July 3 2008 that the German voting system must be reformed to improve its monotonicity.

For more, see: [2] (in English) or various German wikipedia articles such as [3] —Preceding unsigned comment added by (talk) 04:19, 26 July 2009 (UTC)

Instant-runoff voting / Runoff voting (not monotonic)[edit]

The stated IRV example is not very realistic, because it contains a Condorcet cycle among the three Candidates in election one, with a very strong support for any of the three majorities in the cycle. I also criticize, that the example assumes a huge swing of the voters preferences. I suggest another example, without a Condorcet cycle, and just a small swing: Three candidates are running, a left, a right, and a center candidate. Hundred voters casting their ballots:

Number of votes 1st Preference 2nd Preference
28 Right Center
5 Right Left
35 Left Center
32 Center

The IRV-winner in that election would be Left, because Center gets eliminated first, and Left wins against Right according to the remaining preferences. But if the five voters, who voted 1st LeftRight and 2nd RightLeft, switch their 1st and 2nd choice, then Right gets eliminated first, and Center wins against Left, so Left loses due to the new votes in favor of Left. -- Richard 13:45, 13 May 2011 (UTC)

Maybe I do not understand something, but nothing matches in the calculations in the article and in the talk page as well. "But if the five voters, who voted 1st Left and 2nd Right" - there are no such voters in the table given as the example. Also, in the article, given examples fail to prove the thesis they were brought for. Changing ranks from Right to Left entrenches the Left, not sets them down.

Robs777 (talk) 18:54, 11 October 2012 (UTC)

I corrected the error in the description of the example that's given here. In the article I used a different example. The two additional votes for Left in the article example (switching Right, Left, Center to Left, Right, Center) show a greater support for Left. That hurts Right in the first round, cause it no longer reaches the 2nd round. But Center, now entering 2nd round, beats Left. So the greater support for Left causes Left to lose, the result changes from Left winner, Right 2nd to Center winner, Left 2nd. Center is least prefered by the Left, Right, Center voters. Best, --Richard (talk) 10:53, 24 April 2014 (UTC)

Real-life monotonicity violations[edit]

Currently this section is misleading, a single election can't be used to show that a method violates monotonicity, therefore I propose to change this section as follows: When the ballots of a real election are released, it is fairly easy to proof, if it was possible

  • to defeat the winner by raising him on some of the ballots
  • to push a looser by lowering him on some of the ballots

This could be called a real-life monotonicity violation.

The ballots (or information allowing them to be reconstructed) are rarely released for instant runoff elections, which means there are few recorded monotonicity violations for real IRV elections. -- Richard 13:45, 13 May 2011‎ (UTC)

But any real-life election can be modelled in examples. I calculated figues given here as an examples and they do not confirm lack of monotonicity. I just want to understand how it really works. I seems that everybody talk about this but nobody can fetch any examples. Robs777 13:02, 15 October 2012 (UTC)

They do confirm, as they show that when X is winner, upranking X on some ballots causes X to lose. It's worthless for X to have a greater support in the first round, and it's harmfull if that is causing X to lose. If the actual ballot tallies are not available, a real-life election can't be modelled in examples. Guessing what the actual votes could have been is fiction and not real-life. Best, --Richard (talk) 11:08, 24 April 2014 (UTC)

2009 Burlington, Vermont mayoral election[edit]

The current version is hard to understand, but not very informative, I propose also to change this subsection as follows: A real-life monotonicity violation was detected in the 2009 Burlington, Vermont mayor election under instant runoff voting, where the necessary information is available. In this election, the winner Bob Kiss could have been defeated by raising him on some of the ballots. E.g. if all voters who ranked Kurt Wright over Bob Kiss over Andy Montroll, would have ranked Kiss over Wright over Montroll, and additionally some people who ranked only Wright, would have ranked Kiss over Wright, then these votes in favor of Kiss would have have made him lose! [1] The winner in this scenario would have been Andy Montroll, who was also the Condorcet winner according to the original ballots.

If there are no complaints or suggestions to these three changes, I would change the article accordingly -- Richard 13:45, 13 May 2011‎ (UTC)

Real-life monotonicity violations[edit]

I have a problem with the section "Real-life monotonicity violations" and the 2009 Burlington description. AFAIK it is not a violation of monotonicity if you could adjust the vote tallies to affect the result, even if such examples show the latent possibility of such an outcome. To observe real life monotonicity you need to show change in votes between two elections. IIRC from reading the lit some years ago, that's why it's hard to find a real life example, precisely because changing patterns of votes are not so easy to detect, and it's impossible to know whether voters were switching from one candidate to another, ceteris parebus.--Red Deathy (talk) 13:44, 7 June 2011 (UTC)

No, you can't argue on the base of two different elections, because you can't say that voter V changed it's preferences if ballot submission is secret (what it always should be!). And you won't most likely (almost for sure) find any two real elections that fit into the mono-raise criterion ("... without changing the orders of the other candidates.", only raise one candidate, no other changes in all ballots). Best, --Richard (talk) 12:12, 24 April 2014 (UTC)


I altered the section on probability, chiefly because it read like someone making a case, rather than an NPOV statement of the available resources, some bits sounded like OR (The section asked for a citation on still reads a little OR, but I think it deserves a chance to be substantiated).--Red Deathy (talk) 13:57, 7 June 2011 (UTC)

Runoffs are much more common than IRV, but no focus on them[edit]

Backers of single winner reforms that violate later-no-harm criteria like to focus their ire on instant runoff voting. But in fact traditional runoff elections are more likely to generate nonmonotonic outcomes than IRV because the field abruptly is reduced to just two candidates for the runoff. In typical fashion in articles developed primarily by IRV critics, the focus of this article is much more on IRV than on runoff elections. Anyone interested in providing balance by examining the huge number of runoffs around the world (most presidential elections are chosen by runoffs, for example) would be doing disinterested readers a favor. RRichie (talk) 22:13, 11 October 2012 (UTC))

  1. ^ Burlington Vermont 2009 IRV mayor election